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Nonlinear optics at the single-photon level

by

Amir Feizpour

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

c© Copyright 2015 by Amir Feizpour

Abstract

Nonlinear optics at the single-photon level

Amir Feizpour

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2015

In this thesis I report the first experimental observation of an interaction between a post-

selected single photon and an electromagnetic field containing a few thousand photons.

This is the first observation of the nonlinear interactions due to a single freely propagating

photon. To achieve this pioneering result, we have theoretically and experimentally

studied nonlinear optics at the few-photon level. To the best of our knowledge, the

pulse energy levels used for our experiments are the lowest energies used for cross-phase

modulation in free space.

Photon-photon interaction is vanishingly weak; the strength of this interaction can be

enhanced in the presence of a material medium as mediator. In our experiment, we use

an atomic coherence effect, namely Electromagnetically-induced Transparency (EIT), to

produce a ‘giant’ nonlinear interaction between two optical fields. We have observed a

nonlinear phase shift of 18 ± 4 µrad per freely propagating single photon. This is done

by sending very weak coherent state pulses into the medium and subsequently detecting

single photons, inferring the existence of a photon in the interaction region.

The optical setup, electronics and software that have been developed during the com-

pletion of this thesis provide the basis for several exciting future light-matter interaction

experiments. We have produced phase-stable laser beams to create EIT in our cold Ru-

bidium sample prepared in a Magneto-optical Trap (MOT). The laser system and the

improved MOT setup have been built to be stable so that we can do very long data

ii

acquisition runs. We also have developed tools and software with high precision and

stability such that we can measure phase shifts down to micro-radian level.

iii

To my parents,

for their unconditional support and sacrifices

iv

Acknowledgements

A major part of my life as a graduate student involved “living” in a dark, window-

less room, called ‘lab’. It could have been quite different had I stayed in theoretical

quantum optics, with an office on the 12th floor looking over beautiful University of

Toronto campus and occasional sun light. My experience as an experimental physicist

has completely transformed me and helped me grow up to the person that I am today.

(For example, I think, I am a far more patient person now.) Regardless of all difficult

days and nights I spent in the ‘lab’, all the frustrations, rebuilding parts of the setup

again and again, trying new ways and failing, I do not regret my choice to work on

experimental quantum optics. I could sit down for hours and stare at an EIT window

that I prepared the situations for it to happen; or any other physical phenomena that I

or someone else predicted and happens right in front of my eyes in the lab. These are

priceless experiences and make all the trouble to reach them worthwhile.

I should tip my hat to Aephraim Steinberg, my supervisor, for giving me the sup-

port, trust and to help me develop into an independent researcher. I always admired

Aephraim’s excellent physical intuition and learned so much from him. I started working

in the lab back when Luciano Cruz was our post-doc and had started to set up for this

experiment; I learned my baby-steps as an experimentalist from him. At the same time

and after Luciano left, I worked with Xingxing Xing who taught me a lot about elec-

tronics, physics and specially about the narrow-band single photon source he had built.

We shared many good moments and carried out several projects together. In early days

of my PhD Christopher Paul and Chao Zhuang were our ‘atom guys’. They taught me

the first things about cold atoms and MOT. Most of my time during my PhD, however,

has been shared with Greg Dmochowski and Matin Hallaji. Three of us have carried out

several projects and spent so many hours together. I have also learned a lot about elec-

tronics from Alan Stummer who has built several electronic devices for us and without

his help many tasks would not have been possible (or at least would have been a lot more

v

expensive). I was fortunate to work on a few projects with our recent post-doc, Alex

Hayat. He has introduced a completely new and different point of view to our group

while he was here and I believe that working with him was a very valuable experience.

I have also worked with several undergraduate and junior graduate students. Lee

Liu who worked on stabilizing one of our diode lasers, Kelsey Allen who did research on

setting up a Dark SPOT MOT and a QUIC Ioff trap, James Bateman on the optical

centroid measurement, Josiah Sinclair and Saeid Oghbaei on Rydberg atom simulations,

and finally Xing Song and Ginelle Johnston on the narrow-band single-photon source.

I should specially thank Ginelle for giving me very critical and detailed feedback on

the language and presentation of several chapters of my thesis. Same goes to Matin

for working very hard over the last several months of my work in the lab and sharing

so many overnight runs and 3am pizzas. I should also thank Greg, Matin and Josiah

for proofreading parts of my thesis. Finally, I always enjoyed discussing physics and

spending time with other members of the group, Lee Rozema, Dylan Mahler, Shreyas

Potnis, Ramon Ramos, Yasaman Soudagar, and Rockson Chang.

vi

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Disambiguation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 “Giant Kerr Nonlinearity” 9

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Theory of the N-scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Maxwell-Bloch Model . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Linear Time-Invariant Model . . . . . . . . . . . . . . . . . . . . 13

2.3 Properties of the N-scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Dependence on EIT medium properties . . . . . . . . . . . . . . . 17

2.3.2 Dependence on signal pulse . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Propagation in an optically thick medium . . . . . . . . . . . . . 25

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Kerr nonlinearity as a measurement 29

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Quantum Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Strong measurement . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.2 Weak measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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3.3 Weak-value amplification of photon number . . . . . . . . . . . . . . . . 32

3.3.1 Full calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.2 Enhancement of signal-to-noise ratio . . . . . . . . . . . . . . . . 37

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Controversy over weak-value amplification . . . . . . . . . . . . . . . . . 40

3.5.1 SNR improvement in WVA . . . . . . . . . . . . . . . . . . . . . 40

3.5.2 Classical anomalous values? . . . . . . . . . . . . . . . . . . . . . 41

4 Apparatus 44

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Probe, coupling and signal preparation . . . . . . . . . . . . . . . . . . . 46

4.2.1 Master laser stabilization . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.2 AOM double-pass . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.3 Injection lock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Phase measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.1 Probe interferometry . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.2 Detection and demodulation . . . . . . . . . . . . . . . . . . . . . 55

4.3.3 Measurement bandwidth . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.4 Data acquisition and analysis . . . . . . . . . . . . . . . . . . . . 58

4.3.5 Phase noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.6 Measurement and atom cycle . . . . . . . . . . . . . . . . . . . . 62

4.4 Interaction region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.1 EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.2 Signal collection and background photon counts . . . . . . . . . . 65

4.4.3 Time-gating and Tagging . . . . . . . . . . . . . . . . . . . . . . . 66

4.4.4 Probe and signal telescopes . . . . . . . . . . . . . . . . . . . . . 69

4.4.5 Focus size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4.6 Level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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5 Results 74

5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.1 Cross-phase shift versus signal photon number . . . . . . . . . . . 75

5.2.2 Cross-phase shift versus signal detuning . . . . . . . . . . . . . . 76

5.3 Inferred photon number in the interaction region . . . . . . . . . . . . . . 78

5.3.1 Classical intensity fluctuations . . . . . . . . . . . . . . . . . . . . 81

5.4 Cross-phase shift due to a post-selected single photon . . . . . . . . . . . 83

5.5 Technical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.6 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Summary and outlook 87

6.1 Possible improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.1.1 Location of the setup . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.1.2 Brighter single-photon source . . . . . . . . . . . . . . . . . . . . 88

6.1.3 Higher optical density . . . . . . . . . . . . . . . . . . . . . . . . 88

6.1.4 Lower probe power . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1.5 Measurement rate (slow SA processor) . . . . . . . . . . . . . . . 89

6.1.6 Background photon rate . . . . . . . . . . . . . . . . . . . . . . . 90

6.1.7 AOM drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.1.8 Copropagating geometry for probe and coupling . . . . . . . . . . 91

6.1.9 Telescope re-design . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.10 Optical density fluctuation . . . . . . . . . . . . . . . . . . . . . . 92

6.1.11 Use of both D1 and D2 lines . . . . . . . . . . . . . . . . . . . . . 92

6.1.12 Coupling light leakage . . . . . . . . . . . . . . . . . . . . . . . . 93

6.1.13 Polarization spectroscopy . . . . . . . . . . . . . . . . . . . . . . 93

6.1.14 Maximum possible XPS in N-scheme . . . . . . . . . . . . . . . . 93

6.1.15 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

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6.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A Alignment procedures 99

A.1 Polarization Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 99

A.2 Master laser Lock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

A.3 AOM double-pass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.4 Injection lock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.4.1 Polarization-maintaining fiber . . . . . . . . . . . . . . . . . . . . 102

A.5 Probe and signal telescopes . . . . . . . . . . . . . . . . . . . . . . . . . 103

B Interaction of Electromagnetic Fields with Multi-level Atom 104

B.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

B.4 Interaction of EM Fields with Multi-level Atom . . . . . . . . . . . . . . 109

B.4.1 Multi-level atom and a EM field with isotropic polarization . . . . 109

B.4.2 Two-level atom and a polarized EM field . . . . . . . . . . . . . . 111

B.4.3 Lambda system with two polarized EM fields . . . . . . . . . . . 113

C Data analysis MATLAB code 116

Bibliography 139

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List of Tables

1.1 Recent experimental advances in Cross-phase Modulation (XPM) at few-

photon level. The quantity nph is the minimum average photon number

used. † The nonlinear phase shift measured here is due to 1 post-selected

single photon. ∗ The value reported is the inferred number of photons per

atomic cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Major recent experimental advances in other areas of nonlinear optics at

few-photon level. The quantity nph is the minimum average photon num-

ber used. ∗ The value reported is the inferred number of photons per

atomic cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6.1 Comparison of the current experimental parameters and an optimal ex-

perimentally achievable set. . . . . . . . . . . . . . . . . . . . . . . . . . 94

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List of Figures

1.1 Lowest order Feynman diagram for photon-photon interaction in vacuum. 2

2.1 Level structure for the EIT-enhanced cross-Kerr effect, the N-scheme.

Here, Ωp and Ωc are the Rabi frequencies of the Continuous Wave (CW)

probe and coupling fields; Ωs is the peak Rabi frequency of the signal field,

which is a Gaussian pulse with rms width of τs; Γ is the excited state decay

rate and γ is the ground-state dephasing rate. . . . . . . . . . . . . . . . 11

2.2 Time dependence of the per-photon Cross-phase Shift (XPS) for a variety

of EIT window widths. The linear scaling of the peak XPS versus EIT

window width breaks down once the response time of the EIT medium

becomes comparable to or larger than the signal pulse duration. However,

narrower window widths produce longer tails. Simulation parameters: Γ =

2π×6MHz, τs = 1/2√

2π×2000 kHz−1, nph = 100, d0 = 1, ∆p = 0, ∆c = 0,

∆s = −10Γ, σat = 1.2 × 10−13 m2, Ω0,p = 0.003Γ, γ = 1 × 10−5Γ, beam

waist is 10 µm and the wavelength is 780.24 nm. The atomic cloud has a

Gaussian spatial distribution. . . . . . . . . . . . . . . . . . . . . . . . . 18

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2.3 Peak (top) and integrated (bottom) XPS per photon as extracted from

figure 2.2. The peak XPS scales inversely with EIT window only when the

response time of the EIT medium is shorter than the signal pulse duration

while the integrated XPS grows inversely with window width owing to the

longer tails that arise from narrower EIT windows. Squares correspond

to simulation results and dashed lines show the prediction of the Linear

Time-Invariant (LTI) model presented in section 2.2.2. For window widths

comparable to the natural linewidth of the transition the EIT medium

response includes oscillations that are not included in the LTI impulse

response, resulting in a small discrepancy between the two approaches.

Also, the linear scaling of the integrated XPS can be interrupted if the

pumping and dephasing rates become comparable (inset). . . . . . . . . 19

2.4 Peak (top) and integrated (bottom) XPS per photon for various ground-

state dephasing rates, γ. As the dephasing rate increases, both peak and

integrated XPS decrease due to the degradation of the EIT window. Peak

XPS falls to nearly half of its ideal value when the dephasing rate becomes

equal to the pumping rate, R. Squares correspond to simulation results

while the dashed lines show the prediction of the LTI system response.

For this simulation R = 0.01Γ, τs = (0.6Γ)−1 and the rest of parameters

are the same as in figure 2.2. Note that EIT window width is 2(R + γ). 20

2.5 Time response of XPS (per photon) for various signal pulse bandwidths.

The linear scaling of the peak XPS with signal pulse bandwidth breaks

down when this bandwidth becomes comparable to or larger than the

EIT window width. Once the bandwidth of the signal pulse becomes

comparable to its detuning, ∆s = −10Γ, the peak XPS stops growing

and starts to fall. Simulation parameters: ∆EIT = 0.2Γ and the rest of

parameters are the same as in figure 2.2. . . . . . . . . . . . . . . . . . . 21

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2.6 Peak (top) and integrated (bottom) XPS per photon as a function of signal

field bandwidth (normalized to central detuning) as extracted from figure

2.5. Initially, increasing the pulse bandwidth causes the peak XPS to grow

proportionately due to the higher pulse intensity. However, once the pulse

bandwidth becomes larger than the EIT window width, the peak XPS

stops growing, similar to the behavior seen in figure 2.3. The maximum

integrated XPS occurs when the pulse half-width at half-maximum of the

intensity is equal to the detuning. The insets show the Fourier transform

of the signal pulse intensity (red dashed) along with the frequency depen-

dence of the AC Stark Shift (ACS) (blue solid) as a function of detuning

from the excited state. For very broadband pulses, there is a discrep-

ancy between the result of the LTI model and the numerical solution as

explained in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Peak (top) and integrated (bottom) XPS per photon for various signal

detunings, ∆S, when the Half-width at Half Maximum (HWHM) of the

signal pulse bandwidth is set equal to the detuning. The squares show

simulation results while the dotted line is a guide for the eye. Both peak

and integrated XPS have maxima close to ∆s = Γ/2. The inset shows the

Fourier transform of the signal pulse intensity (red dashed) along with the

frequency dependence of the ACS (blue solid) as a function of detuning

from the excited state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8 Space-time diagram of the interaction between the probe and signal. At

high Optical Density (OD) the group velocity mismatch of the probe and

signal causes a large portion of the probe to be affected. . . . . . . . . . 25

xiv

2.9 XPS due to a step-function signal field (top) and time dependence of pulsed

XPS (per photon) for different OD’s (bottom). As the OD, d0, increases

the peak XPS begins to grow but eventually saturates due to the group

velocity mismatch between the signal and the probe. However, larger

values of OD result in longer-lasting phase shifts; the temporal extent of

the flat region of the transient is determined by the duration of the probe

that is compressed in the medium, τL, when the signal pulse passes through

the medium at group velocity, c. Simulation parameters: R = 0.1Γ, τs =

(0.6Γ)−1 and all others as in figure 2.2. . . . . . . . . . . . . . . . . . . 26

2.10 Peak (top) and integrated (bottom) XPS per photon versus OD, d0, as

extracted from figure 2.9. Squares correspond to simulation results while

the dashed lines are predictions of an LTI model. The response function

adopted in equation 2.5 only partially accounts for the propagation effects

(through the dependence of the EIT medium response time, τ , on OD);

however, this is not sufficient to model the behavior of the system at high

OD’s. It is important to note that the response of the system is still linear

at high OD’s and a proper impulse response can account completely for the

saturation effect. The integrated XPS increases linearly with OD and an

LTI model agrees very well with the simulation results. τ0 is the response

time of the EIT medium in the limit of vanishing OD. . . . . . . . . . . 27

3.1 Quantum measurement. System and probe couple through a measurement

interaction and depending on the strength of the coupling compared to the

position uncertainty of the probe, it can be a strong (top) or weak (bottom)

measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

xv

3.2 The single-photon “system” is prepared in an equal superposition of arms

a and b by the first Beam-splitter (BS)1. After a weak XPM interaction

with the “probe”, prepared in a coherent state |α〉p, the system is post-

selected on a nearly orthogonal state by detecting the single photon in the

nearly-dark port, D1. The success probability of post-selection depends on

the imbalance δ in the reflection and transmission coefficients of BS2, and

the back-action of the probe on the system. Using the lower interferometer

to read out the phase shift of the probe amounts to a measurement of the

system observable nb, the photon number in arm b. The phase shifter θ is

used to maximize the sensitivity of the measurement. . . . . . . . . . . 33

3.3 The enhancement factor versus |α|2 φ0. The parameters used are φ0 =

2π × 10−5 and δ = 0.01. The enhancement factor is calculated by using

the state of Eq. (3.4) without any approximations. The dashed line shows

the enhancement factor if the average phase written by the probe on the

system, |α|2 φ0, is compensated; otherwise enhancement occurs whenever

|α|2 φ0 is close to an integer multiple of 2π (solid curve). The inset shows

the enhancement factor as a function of post-selection parameter, δ, in

two different regimes: i) |α|2 = 105, in which case the imparted phase on

the system by the probe, ε, is 0 (solid blue); ii) |α|2 = 102, where ε is

a small non-zero phase (dashed green). For large values of δ the weak-

measurement prediction is valid; however as δ decreases the back-action

from the probe plays a more dominant role. The dashed line shows the

prediction of the weak-measurement formalism. . . . . . . . . . . . . . . 36

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3.4 The Signal-to-noise Ratio (SNR) as a function of the single photon rate Γ.

The technical noise is modelled by an exponential correlation function with

an amplitude, η, 10 times larger than the quantum noise. The dashed line

shows the non-post-selected SNR for the phase shift due to one photon in

mode b. The post-selected SNR for δ1 = 0.1 (weak-measurement regime-

dash-dotted red) and δ2 = 0.01 (the optimum value of measured phase

shift- solid green) are also shown; the dotted line shows the quantum-

limited SNR for comparison. The non-post-selected SNR approaches a

maximum value, S0, due to low-frequency noise. However, for the post-

selected SNR, there is an enhancement by a factor of δ/2P , compared to

the non-post-selected SNR, S0, for measurements with high enough rate.

For low rates the enhancement is given by δ/2√P and therefore the weak

measurement results in the best possible post-selected SNR. Relevant

parameters include T/τc = 103, φ = 2π × 10−5, |α|2 = 105 and therefore

P1 = 0.01 and P2 = 3× 10−4. . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 The general view of the apparatus. The probe, coupling and signal beams

are prepared in the photon side and are sent to the atom side to interact

with the cold atom cloud. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Polarization spectroscopy setup to produce error signals for stabilizing

the Master Laser (ML). A circularly polarized pump beam saturates the

atoms and causes the polarization of the probe beam to rotate. Doing a

polarization analysis on the probe results in dispersion-like features that

can be used for locking the laser. . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Typical set of polarization spectroscopy signals. The green dots show the

real transitions along with the values of F ′. The red arrow is our typical

locking point for the ML. . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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4.4 A typical Acousto-optic Modulator (AOM) double-pass setup for scanning

the laser frequency using AOM without losing pointing accuracy. . . . . 50

4.5 Injection-locking to the 3 GHz sideband of a phase modulation is used to

produce signal and coupling beams. . . . . . . . . . . . . . . . . . . . . 51

4.6 The 0 and -1 orders of the AOM are used to produce the reference and near-

resonance probe components for our frequency-domain interferometer. . 54

4.7 The electronics for production, detection and demodulation of the probe

beating signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.8 Simulated peak XPS (left) and SNR (right) versus Intermediate Filter

(IF)-Bandwidth (BW) and signal pulse bandwidth. . . . . . . . . . . . . 57

4.9 Typically measured XPS time trace and corresponding regions for calcu-

lating the average and peak phase shift. . . . . . . . . . . . . . . . . . . 59

4.10 Acquisition time per trace versus the measurement window. . . . . . . . 60

4.11 Atom and measurement cycle. . . . . . . . . . . . . . . . . . . . . . . . 63

4.12 Interaction region, and probe and signal collections. . . . . . . . . . . . 64

4.13 The tagging procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.14 The signal (right) and probe (left) telescopes. . . . . . . . . . . . . . . . 68

4.15 Implementation of N-scheme in 85Rb atoms. . . . . . . . . . . . . . . . . 71

5.1 XPS versus average photon number per pulse. The nonlinear phase shift

depends linearly on the photon number at lower intensities. A fit to the

low-photon-number data yields a slope of 13 ± 1 µrad per photon while

the deviation at higher photon numbers arises due to higher-order nonlin-

earities. The inset shows a typical linear phase profile (green) and optical

density (red) as seen by the probe with the arrow indicating where the

on-resonance component of the probe laser is locked. Other relevant pa-

rameters include signal center detuning = −10 MHz, OD = 2, EIT widow

width = 2 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

xviii

5.2 The level scheme used to observe cross-phase modulation using the D2

line of 85Rb atoms. The ac-Stark shift due to the signal pulses, pulls the

probe out of EIT conditions and this appears as a refractive index change

proportional to the signal intensity. . . . . . . . . . . . . . . . . . . . . 77

5.3 XPS vs signal detuning. The nonlinear phase shift is caused by ACS due

to the signal pulses. Therefore, it has the same dependence on signal

detuning as the ACS. This scaling also depends on probe power because

more probe power results in a larger population in F = 3 ground state

which means a larger signal absorption. The overall effect is broadening

and smearing of the dispersion-like scaling at higher probe powers. . . . 78

5.4 Inferred, ninf , versus average photon number in the interaction region.

The overall collection efficiency is assumed to be 20% and the background

click rate is taken to be 10% for solid and dotted lines. The circles show

the photon number values inferred for the data points in figure 5.5 for

no-click (red) and click (blue) events. The overall efficiency percentage for

each data point (numbers beside circles) is slightly different which accounts

for the discrepancies between the data points and the solid curves. The

average photon number in the interaction region for the data points is lower

than the incident photon number because of the finite signal absorption.

The dotted green lines show the photon number which would be inferred,

were a number-resolving detector used. The solid blue line could also be

obtained from a weighted average of the dotted lines with non-zero number

of clicks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

xix

5.5 Post-selected single-photon XPS. Most notably, for an average incident

photon number of 0.5 (green-shaded region), the XPS for no-click and click

events are 2±3 and −13±6 µrad, respectively, which definitively shows the

effect of a single post-selected photon. For the other data points, the av-

erage incident photon number and/or the signal center detuning is varied.

Taking all the data points together, the magnitude of the post-selected

single-photon XPS is −18±4µrad. The inset shows the post-selected XPS

versus ninf (2π× 18MHz)/|∆s|, inferred photon number corrected for the

sign of the signal detuning. The solid line has a slope of −14 ± 1 µrad

per photon. Other relevant parameters include EIT window = 2 MHz and

OD = 3. The data in the region shaded in blue are tests for systematics

as explained in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

B.1 The level structure for a) section B.4.1, b) and c) section B.4.2. Hyperfine

structure of 85Rb is used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

B.2 Response of the multi-level atom to the Electromagnetic (EM) field at

times 50τ , 500τ and 5000τ : (top) real and imaginary parts of refractive

index (in arbitrary units), (bottom) The populations of the two ground

states F = 2 (left) and F = 3 (right). The horizontal axis is the detuning

from F = 2 → F ′ = 2 transition in terms of natural linewidth, Γ. The

F = 2 → F ′ = 3 transition is at +10.5Γ and the F = 2 → F ′ = 1, that

is a cyclic transition, is at −4.8Γ. The intensity of the laser is taken to be

0.13mW/cm3. All the population is initially in F = 2 ground state. . . . 110

B.3 Two level structure for which conditions of electromagnetically induced

transparency is studied. In this example D2 line of 87Rb is considered. . . 113

xx

B.4 Change of the atomic response to the EM field tuned to F = 1→ F ′ = 2 as

magnetic field is increased. It can be seen that by increasing the magnetic

field the EIT is revived. The real (left) and imaginary (right) parts of the

refractive index (in arbitrary units) versus detuning from F = 1→ F ′ = 2

transition in units of Γ. The intensity of the EM fields are 1.3 and 13

mW/cm3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

B.5 The effect of a large magnetic field and Zeeman splitting in EIT window

structure. If the Zeeman splitting is large the condition of two photon

resonance is satisfied for different sub-levels at different frequencies. . . . 115

xxi

List of Publications

• Short pulse cross-phase modulation in an electromagnetically-induced-transparency

medium Amir Feizpour, Greg Dmochowski, Aephraim M. Steinberg, arxiv:1406.0245

(submitted)

• Experimental demonstration of a flexible time-domain quantum channel Xingxing

Xing, Amir Feizpour, Alex Hayat, Aephraim M. Steinberg, Optics Express, Vol.

22, Issue 21, pp. 25128-25136 (2014)

• Scalable Spatial Super-Resolution using Entangled Photons Lee A. Rozema, James

D. Bateman, Dylan H. Mahler, Ryo Okamoto, Amir Feizpour, Alex Hayat, Aephraim

M. Steinberg, Phys. Rev. Lett. 112, 223602 (2014)

• Enhanced probing of fermion interaction using weak-value amplification Alex Hayat*,

Amir Feizpour*, and Aephraim M. Steinberg, Phys. Rev. A 88, 062301 (2013)

* equal contributions

• Multidimensional Quantum Information Based on Single-Photon Temporal Wavepack-

ets Alex Hayat, Xingxing Xing, Amir Feizpour and Aephraim M. Steinberg, Opt.

Express 20, 29174-29184 (2012)

• Amplifying Single-Photon Nonlinearity Using Weak Measurements Amir Feizpour,

Xingxing Xing and Aephraim M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011)

xxii

List of Acronyms

ACS AC Stark Shift

AOM Acousto-optic Modulator

APD Avalanche Photo-diode

BOP Bipolar Power Supply

BS Beam-splitter

BW Bandwidth

CGC Clebsch-Gordan Coefficient

CSV Comma-separated Values

CW Continuous Wave

DC Direct Current

DDS Direct Digital Synthesizer

DP Double-pass

ECDL External Cavity Diode Laser

EIT Electromagnetically-induced Transparency

EM Electromagnetic

xxiii

EOM Electro-optic Modulator

GHZ Greenberger-Horne-Zeilinger

HWHM Half-width at Half Maximum

FWHM Full-width at Half Maximum

HWP Half-wave Plate

HCF Hollow-core Fiber

IF Intermediate Filter

I In-phase Quadrature

ILL Injection-locked Laser

LTI Linear Time-Invariant

MMF Multi-mode Fiber

MOT Magneto-optical Trap

ML Master Laser

OI Optical Isolator

OD Optical Density

PBGF Photonic Band-gap Fiber

PBS Polarizing Beam-splitter

PD Photo-diode

PID Proportional-Integral-Derivative

PMF Polarization Maintaining Fiber

xxiv

Q Out-of-phase Quadrature

QED Quantum Electrodynamics

QWP Quarter-wave Plate

RF Radio Frequency

RWA Rotating-wave Approximation

SA Signal Analyzer

SMF Single-Mode Fiber

SNR Signal-to-noise Ratio

SP Single-pass

SPCM Single-photon Counting Module

TTL Transistor-to-transistor Logic

VCO Voltage-controlled Oscillator

VI Labview Virtual Instrument

WVA Weak-value Amplification

XPS Cross-phase Shift

XPM Cross-phase Modulation

xxv

Chapter 1

Introduction

Back to the Table of Contents

1.1 Background

Modern optical physics has principally revolved around two poles. On one side, in nonlin-

ear optics rich effects occur through the interactions of photons with one another. How-

ever, these effects have only been observed for pulses containing billions of photons due

to the extreme weakness of these nonlinear interactions. On the other side, in quantum

optics phenomena such as entanglement have been commonly studied, although photon-

photon interactions are negligible. In fact, in the absence of nonlinearities, classical and

quantum electromagnetism make the same predictions about intensities (or detection

rates) [1, 2]. It has therefore long been a dream to move into the realm of “quantum

nonlinear optics,” which would provide a foundation for developing complex many-body

interacting quantum optical systems provided by strong photon-photon interactions. In

this new regime of optics, we will be able to produce and detect novel entangled states

including few-photon bound states [3, 4], along with cluster [5], Greenberger-Horne-

Zeilinger (GHZ) [6] and Bell [7] states.

In addition to being fundamentally interesting, such interactions have implications

1

Chapter 1. Introduction 2

1

2

3

4

e

e

e e

Figure 1.1: Lowest order Feynman diagram for photon-photon interaction in vacuum.

for optical quantum information processing. While photonic qubits are ideal candidates

for quantum information storage and transmission, an efficient and scalable method for

processing optical quantum information has yet to be demonstrated. The weakly interact-

ing nature of light, which makes photonic qubits robust against decoherence, also renders

photons poor candidates for information processing. This is problematic as (nonlinear)

interactions are at the heart of logic gate operations. Photon-photon interactions could

enable new architectures for quantum logic [8, 9, 10, 11, 12], as well as non-demolition

measurement of photon number [13], deterministic quantum teleportation [14] and low

light level switching [15].

Photons can only interact with charged particles and since they do not have charge

their interaction with themselves is zero to the first order. The only way they can interact

is if they break into charged particles (leptons or quarks) through which they can interact;

see figure 1.1 for the lowest order Feynman diagram of such interaction. The probability

of this happening is extremely small and practically vanishing. This probability can

however be significantly enhanced in the presence of matter. A material medium can

play role as the mediator of the interaction between photons. For a list of the nonlinear

optical coefficients in different material see [16].

Naturally occurring nonlinear optical coefficients are insufficient for any of the ap-

plications mentioned above. Decades ago, important steps in this direction were taken

towards enhancement of nonlinear effects in the realm of cavity Quantum Electrodynam-

ics (QED) [17, 18]. This approach has proven more beneficial in the context of supercon-

ducting qubits and microwave photons than optical settings. Strengthened interactions

for freely propagating photons have recently been observed, using electromagnetically in-


duced transparency and slow light [19], Photonic Band-gap Fiber (PBGF) [20], atoms in

Hollow-core Fiber (HCF) [21, 22], and single atoms strongly coupled to mictoresonator

cavities [23]. Most recently, the application of “Rydberg blockades” has enabled very

large interaction strengths, already leading the observation of strongly modified quan-

tum statistics [24, 4, 25, 26]. For a recent review of the major advances of the field see

for example [27].

In parallel to the advances, fundamental noise limits have led to a controversy about

the applicability of strong interactions for quantum logic. Shapiro showed that a non-

linear phase shift that is strong enough to implement a quantum logic gate is always

accompanied by a large amount of phase noise. In other words, to achieve a nonlinear

phase shift on the order of π the phase noise is large enough that it significantly degrades

the fidelity of the final state [28]. In another work [29], Shapiro and Razavi showed that

the same arguments also apply to the case of logic gates based on weak cross-Kerr nonlin-

earity [11]. However, they do emphasize that their treatment does not necessarily apply

to the case of atomic systems under EIT conditions and a more careful treatment of the

noise is called for. They also bring up an important point that an “optimum response

function” can be calculated for which there is a trade off between the amount of extra

noise and the detectability of the nonlinear phase shift. He et al do similar treatments

for transverse degrees of freedom and suggest using long-range interactions as a poten-

tial solution to the problem [30]. Gea-Banacloche studied the interaction between two

single-photon pulses through EIT and concludes that to avoid noise one has to always

use pulse bandwidth much narrower than the EIT window and therefore the achievable

interaction strengths are “useless” [31]. Marzlin et al point out that using two copropa-

gating pulses can have the drawback that a strong coupling between the two can result

in a large reduction of the fidelity of the gate operation. They propose using pulses with

mismatched group velocities so that one can pass through the other [32]. However, Dove

et al showed recently that even this proposal would suffer from phase noise and will not


help realize quantum logic [33]. Although they suggest that EIT or cavity-based Kerr

nonlinearity might be exceptions to their otherwise general treatment.

Schmidt and Imamoglu proposed a scheme [19] based on EIT [34, 35] which allowed

for “giant”, resonantly-enhanced cross-Kerr effect while simultaneously eliminating self-

Kerr effect and linear absorption [34]. In an EIT system the excitation pathways interfere

destructively and atomic population is trapped in a so-called “dark state” [36]. Fields

passing through the medium on resonance see it as transparent with a sharp dispersive

feature which can be used for enhancement of optical nonlinearities. Shortly after this

proposal, Harris and Hau pointed out that in the so-called N-scheme there is a funda-

mental limit on the largest possible XPS resulting from the group velocity mismatch of

the probe and signal fields [37]. Based on a simple single-mode treatment the largest

XPS in N-scheme, regardless of most of the parameters and assuming that the signal

is detuned by half a linewidth and focused to atomic cross-section, is on the order of

100 mrad [34]. In this scheme, the probe is inside an EIT window and therefore expe-

riences a large group index but the signal is detuned from its transition and propagates

at the velocity of light. In order to solve this problem several schemes were proposed

in which both fields were placed in EIT windows. These include M [38, 39, 40], tripod

[41, 42, 43], inverted-Y [44], double-Lambda [45, 46], and N-tripod [47] schemes, and also

use of mixtures of two atoms [48] or symmetric use of the Zeeman levels [49] in order to

produce slow light situations for both fields. There has also been some concern about the

bandwidth mismatch problem in EIT-based nonlinearity [31]. I address these questions

in detail in chapter 2.

Table 1.1 displays most of the recent experimental attempts to observe XPM at the

few-photon level. It is immediately seen that before our recent result, presented in chapter

5, the lowest energy used for XPM in free space was on the order of 400 photons. Our

results presented in this thesis use average photon numbers as low as one per pulse. In

addition, the nonlinear phase shift written by a single freely propagating photon on a


Light Source Medium, Scheme Nonlinear effect nph Reference

Post-selected 85Rb vapor, D2 line, XPM 1† Steinberg (2014)

single photons N-scheme 18 µrad /photon [50]n2 = 5× 10−5 cm2/W

Classical pulses 85Rb vapor in XPM 16 Gaeta (2013)

(tens of ns) HCF, Ladder 5 mrad (300 µrad /photon) [21]Classical pulses Cs MOT, D2 line, XPM 106 Chen (2011)

(µs) M-scheme 900 mrad (1 µrad /photon) [40]Classical pulses 87Rb MOT, D2 line, XPM 400 Chen (2011)

(20 µs) N-scheme 5 mrad (13 µrad /photon) [51]Classical pulses 87Rb MOT, D2 line, XPM 107 Chen (2010)

(20 µs) N-scheme 20 mrad for 3 µW/cm2 [52]1 GHz Ti:Saph Nonlinear PBGF XPM 0.1 Edamatsu (2009)

0.1 µrad per photon [20]Classical CW 87Rb vapor, D1 line, XPM — Wang (2008)

Double Lambda n2 = 2× 10−5 cm2/W [46]Classical CW 87Rb MOT, D1 line, XPM — Peng (2008)

Tripod n2 = 7× 10−5 cm2/W [41]Classical CW 87Rb vapor, D1 line XPM — Wang (2004)

inside a ring cavity, n2 = 1× 10−5 cm2/W [53]Lambda

Classical pulses 87Rb MOT XPM 1000∗ Zhu (2003)

(ms) D1 and D2 lines, 120 mrad [54]N-scheme

Classical CW 87Rb vapor, D1 line XPM — Xiao (2002)

inside a ring cavity, n2 = 6× 10−6 cm2/W [55]Lambda

Table 1.1: Recent experimental advances in XPM at few-photon level. The quantity nphis the minimum average photon number used. † The nonlinear phase shift measured hereis due to 1 post-selected single photon. ∗ The value reported is the inferred number ofphotons per atomic cross-section.


Light Source Medium, Scheme Nonlinear effect nph Reference

Classical pulsed cold 87Rb, optical switch 0.17 Rempe (2014)

(µs) Rydberg blockade [56]Classical CW cold 87Rb photon blockade — Lukin-Vuletic

in crossed dipole trap, (single-photon filter) (2012)

Ladder [24]Rydberg excitation

Classical pulsed cold 87Rb optical switch 500 Lukin-Vuletic

(µs) in HCF, N-scheme (2009)

[57]Classical CW single Cs atom coupled to photon blockade — Kimble (2008)

microtoroidal resonator, (single-photon filter) [58]Cavity-QED

Classical pulses 87Rb MOT, optical switch 23∗ Harris (2003)

(300 ns) D1 and D2 lines, N-scheme [59]

Table 1.2: Major recent experimental advances in other areas of nonlinear optics at few-photon level. The quantity nph is the minimum average photon number used. ∗ Thevalue reported is the inferred number of photons per atomic cross-section.

probe beam has never been reported. Our work is the first to detect the nonlinear effect

due to a post-selected single photon on a classical beam. Table 1.2 shows some of the

major recent advances in other areas of nonlinear optics at the few-photon level.

1.2 Outline

This thesis reports our theoretical and experimental study of XPM at the few-photon

level. In chapter 2, I study the N-scheme and explore the behavior of the scheme

in experimentally relevant situations of broadband signal pulses, narrow EIT windows

and high optical density. In order to be practically relevant, a multi-mode treatment of

the problem is required. There have been several multi-mode treatments of EIT, which

examine the transients due to switching on optical fields [60, 61] as well as of sudden

changes in two-photon (Raman) resonance [62, 63]. In addition, the transient properties

of the associated nonlinearities, both absorptive (photon switching) [64, 65] and dispersive

(cross-Kerr effect) [66, 67, 68, 69, 70] have since been investigated. However, none of the


studies present results directly applicable to experimental settings. This chapter presents

a thorough and clear analysis of the EIT-enhanced XPS behavior with respect to several

practical parameters. Most of the material in this chapter are taken from [71].

In chapter 3, I use ideas inspired by weak measurement to study enhancement of

photon-photon interactions. I show that using Weak-value Amplification (WVA) [72], a

single photon can be made to “act like” many photons. I demonstrate that it is possible

to amplify an XPS to a value much larger than the intrinsic magnitude of the single-

photon-level nonlinearity. In so doing, I also demonstrate quantitatively how WVA may

improve the SNR in appropriate regimes. Material in this chapter is reprinted with

permission from A. Feizpour, X. Xing and A. M. Steinberg, PRL. 107, 133603 (2011).

Copyright (2011) by the American Physical Society.

Chapter 4 describes the work carried out to create an experimental setup for our

light-matter interaction experiments and lead to the measurement of XPM due to a single

post-selected photon.

In chapter 5, I present our calibration measurements and the pioneering result of

observing XPS due to post-selected single photons.

Finally, in chapter chapter 6, I discuss possible apparatus improvements that could

lead to future experiments.

1.3 Disambiguation

The following terms might be used interchangeably in this thesis, however, there are

slight differences between them:

The Kerr effect is when the refractive index of a material is linearly dependent on

the intensity of light passing through it [16]. As a result of the Kerr effect, the phase

picked up by an EM field passing through a nonlinear optical medium can depend on the

intensity of the field. This can be caused by a field on itself (self-phase modulation)


or by a different field (cross-phase modulation, also known as cross-Kerr effect).

With potential application to quantum logic, I am only interested in XPM in this thesis.

In this setting, usually a signal modulates the phase of a probe; the phase shift picked

up by the probe as a result of the interaction is called cross-phase shift.

Chapter 2

“Giant Kerr Nonlinearity”


2.1 Motivation

In the presence of a material medium the interaction between photons can be greatly

enhanced. The strength of the nonlinear interaction increases when near a resonance in

a material medium; however, the linear absorption also increases. Using EIT has the

benefit of eliminating the linear absorption on resonance while enhancing the strength of

the nonlinear optical coefficient. This enhancement is accompanied by detrimental effects

such as group velocity [37] and bandwidth [71] mismatch. In this chapter, I theoretically

study the behavior of an EIT-based optical nonlinear effect and its response to several

parameters of practical interest including EIT window width, signal pulse properties and

optical density of the atomic medium.

The results shown in this chapter have important consequences about the detectability

of XPS and implementation of quantum logic gate based on cross-Kerr effect. Early

schemes for optical quantum information processing required very large (on the order of

π) XPS [8]. As this has proven to be experimentally out of reach in single-pass geometries

so far, more recent proposals have replaced the need for such large phase shifts with

9

Chapter 2. “Giant Kerr Nonlinearity” 10

the less demanding requirement of any XPS detectable on a single shot [11, 12]. In

this proposal, each qubit single photon interacts individually with the same classical

electromagnetic field (‘bus’). A subsequent parity measurement on this bus projects the

two qubits into an entangeld state up to a local correction. The crucial step here is to be

able to detect the nonlinear phase shift of the single photon on the classical beam on a

single shot. In order to improve the detectability of the phase shift, one usually integrates

the effect over its duration. In other words, it is not only the peak size of the nonlinear

phase shift but also its duration that plays an important role. In the rest of this chapter,

I show theoretically that the limitations of the scheme studied lead to the saturation of

the peak nonlinear phase shift but simultaneously can result in longer-lasting effect of

the signal on the probe.

2.2 Theory of the N-scheme

Consider the level scheme shown in figure 2.1, in which CW in-phase probe and coupling

fields form a three-level Lambda system. If the two-photon resonance condition is sat-

isfied, i.e. δ = ∆p − ∆c = 0, and the coupling field is strong enough, Ω2c Γγ, then

destructive interference of multiple excitation pathways causes the medium to become

transparent to the probe light. That is, the interaction of the probe and coupling fields

with the medium results in new atomic eigenstates, one of which (the so-called dark

state) is decoupled from the optical fields. Atomic population is pumped into this dark

state, where it remains, at a rate of R = Ω2cΓ/2(4∆2 + Γ2), where ∆ = (∆p + ∆c)/2.

The steady-state spectral Full-width at Half Maximum (FWHM) of the EIT window is

determined by this pumping rate along with the ground state dephasing rate according

to ∆EIT = 2(R + γ) [69]. The presence of the signal field inside the medium completes

the ‘N-scheme’, serving to perturb the ground-state coherence created by the Lambda

system in two ways: first, the scattering of the signal photons from the excited state |es〉


|gp〉

|gc〉

|ep〉

|es〉

∆p∆c

∆s

Ωp

Ωc

Ωs

Γ

& 1/τs

Γ

Figure 2.1: Level structure for the EIT-enhanced cross-Kerr effect, the N-scheme. Here,Ωp and Ωc are the Rabi frequencies of the CW probe and coupling fields; Ωs is the peakRabi frequency of the signal field, which is a Gaussian pulse with rms width of τs; Γ isthe excited state decay rate and γ is the ground-state dephasing rate.

dephases the ground-state coherence at the rate of Ω2sΓ/4∆2

s; second, the ACS caused

by the signal pulse, ∆ACS = Ω2s/4∆s, detunes the system out of two-photon resonance

and causes the probe field to experience a different refractive index, thereby acquiring a

XPS. The signal detuning can be made large enough compared to both the excited state

linewidth and the bandwidth of the signal pulse so that the first contribution is negligible

and only the ACS perturbs the system significantly. If this ACS, ∆ACS, is smaller than

the EIT window width, ∆EIT , then the phase shift that the probe experiences is linear

in ∆ACS and, in turn, linear in the intensity of the signal field, |Ωs|2. This is the regime

in which the nonlinear interaction between the signal and the probe can be considered a

cross-Kerr effect.


2.2.1 Maxwell-Bloch Model

The Hamiltonian describing the interactions of figure 2.1 (in a rotating frame and using

the Rotating-wave Approximation (RWA)) is

H =~2

0 0 Ωp 0

0 2δ Ωc Ωs

Ω∗p Ω∗c 2∆p 0

0 Ω∗s 0 2(∆s + δ)

(2.1)

where Ωi = −~µ · ~Ei/~ is the Rabi frequency and Ei is the electric field for i = p, c, s;

~µ is the matrix element of the transition. For a multi-level treatment of light-matter

interaction see appendix B. The dynamics of the system can be obtained from solving

the Maxwell-Bloch equations,

∂tΩp + c∂zΩp = igN(z)Sp(z, t)

∂tΩc + c∂zΩc = igN(z)Sc(z, t)

∂tΩs + c∂zΩs = igN(z)Ss(z, t)

∂tSp = (i∆p − Γ/2)Sp(z, t) + i1

2Ωp(z, t) + i

1

2Ωc(z, t)Sgg(z, t)

∂tSs = (i∆s − Γ/2)Ss(z, t)− i1

2Ωc(z, t)See(z, t)

∂tSc = (i∆c − Γ/2)Sc(z, t)− i1

2Ωs(z, t)S

∗ee(z, t) + i

1

2Ωp(z, t)S∗gg(z, t)

∂tSgg = (iδ − γ)Sgg(z, t) + i1

2Ω∗c(z, t)Sp(z, t)− i1

2Ωp(z, t)S∗c (z, t) + i

1

2Ω∗s(z, t)Sge(z, t)

∂tSee = (i(∆s −∆c)− Γ/2)See(z, t) + i1

2Ωs(z, t)S

∗c (z, t)− i1

2Ω∗p(z, t)Sge(z, t)− i

1

2Ω∗c(z, t)Ss(z, t)

∂tSge = (i(∆s + ∆p −∆c)− Γ/2)Sge(z, t)− i1

2Ωp(z, t)See(z, t) + i

1

2Ωs(z, t)Sgg(z, t) (2.2)

which encapsulate the dynamics of both the atomic system and the electromagnetic fields.

In equations 2.2, c is the speed of light; N(z) is the atom density; Sp = Tr(ρ|gp〉〈ep|),

Sc = Tr(ρ|gc〉〈ep|) and Ss = Tr(ρ|gc〉〈es|) are the probe, coupling and signal transition


coherences; Sgg = Tr(ρ|gp〉〈gc|), See = Tr(ρ|ep〉〈es|) and Sge = Tr(ρ|gp〉〈es|) are the

coherences between the two ground-states, between the two excited states, and between

the probe ground-state and the signal excited state, respectively; ρ is the atomic density

matrix; and g = ω0µ2/ε0~ is the light-matter coupling constant, where ω0 is the center

frequency of the electromagnetic field. For the purposes of this paper ω0 and µ are

taken to be constants and equal for all transitions. In deriving the above equations of

motion, it is assumed that all optical fields are weak enough that the population remains

completely in the probe ground-state, |gp〉. Therefore, to first order in electric fields, the

equations of motion for populations can be neglected. I assume a Gaussian distribution

for atom density and set both one- and two-photon detunings to zero, ∆p = 0 and

∆c − ∆p = 0, respectively. In addition, the probe and coupling fields are assumed to

be CW (pulses with durations much longer than the simulation time) while the signal

pulse is taken to be Gaussian with rms duration of τs. The probe and coupling fields

have to be long enough to encompass all of the dynamics of the system, especially any

potentially long-lasting transient behavior. Note that the OD of a transition is given by

d0 = (2g/cΓ)∫N(z)dz = σat

∫N(z)dz where σat is the interaction cross section.

The equations of motion, eq. 2.2, can be solved using approximate analytical methods

[69] or numerical techniques. I take the latter route, using a first-order difference method

to discretize the spatial coordinate and then the 4th-order Runge-Kutta method to take

the time integral, which yields the solution to the density matrix of the combined light-

matter system for different sets of parameter choice. First, however, I present an alternate

and simpler approach to modelling the dynamics as a LTI system. The results of section

2.3 compare and contrast these two approaches.

2.2.2 Linear Time-Invariant Model

In this section I present a model for the dynamics of the cross-Kerr interaction, which ab-

stracts the underlying nonlinearities and treats the probe phase as the “output” of a LTI


system whose behavior is affected by an independent, potentially time-varying, “driv-

ing” signal field intensity. The impulse response characterizing this linear system may

be obtained by direct differentiation of the system’s step-response. This step-response

is precisely what has been reported in previous transient studies of EIT-enhanced XPS

[69]. There it was shown that, when the ACS is smaller than the EIT window width, the

rise time of the XPS is τ = (1 + d/4)/(R + γ), where d = d0R/(R + γ) is the depth of

the transparency (the difference in the OD seen by the on-resonance probe without and

with a resonant coupling beam). I, therefore, take the step-response, S(t), to have an

exponential shape,

S(t) =φss

|Ω|2 Θ(t)(1− exp(−t/τ)) (2.3)

where φss is the steady-state XPS for a weak signal field of intensity |Ω|2, and Θ(t) is

the Heaviside step-function. It is important to note that the shape of the response in an

optically thick medium or very broad EIT windows deviates from the exponential form.

For simplicity, I first consider optically thin media, leaving the details of optically thick

samples to section 2.3.3. The steady-state phase shift, φss, as predicted by single-mode

and step-response treatments, is

φss = ∆ACSω0

2c

∫dz

∂χpr(z)

∂∆p

∣∣∣∣∆c=0, δ=0

= ∆ACSω0

2c

4d2

~ε0Ω2c

(2γΓ + Ω2c)

2

∫N(z)dz

= ∆ACSd0ΓΩ2c

(2γΓ + Ω2c)

2= ∆ACS

d

∆EIT

(2.4)

where χpr is the steady-state susceptibility of the probe transition [34], ∆ACS = −|Ω|2/4∆s

is the ground-state ACS for ∆s Γ, and d/∆EIT is proportional to the slope of the re-

fractive index with respect to the detuning seen by the probe field. This expression also


assumes that the bandwidth of the signal is negligible compared to its detuning. The

impulse response can be obtained by differentiating the above step-response,

I(t) =∂S(t)

∂t=

φss

|Ω|2τΘ(t) exp(−t/τ) (2.5)

Let us now investigate the behavior of this system in response to a Gaussian signal

pulse. I describe the pulse by its time-dependent Rabi frequency,

Ωs(t) = Ω0,s

√1

τsΓexp(−t2/4τ 2

s ). (2.6)

With applications of single-photon nonlinearities in mind, I consider a fixed number of

signal photons, nph, constraining the pulse energy,

E =

(√π

Ω20,s

Γ2

A

σat

)~ω0 = nph~ω0, (2.7)

where A is the transverse area of the signal pulse. Assuming linearity, the temporal

profile of the XPS is the convolution of the impulse response and the intensity profile of

the signal pulse,

φ(t) = |Ωs(t)|2 ∗ I(t)

=φ0nph

2τeτ

2s /2τ

2

× exp(−t/τ)(

1 + erf(t/√

2τs − τs/√

2τ))

(2.8)

where erf(x) = 2/√π∫ x

0dx′ exp(−x′2) is the error function, ∗ indicates convolution and

φ0 =Γ

−4∆s

σatA

d

∆EIT

==Γ

−2∆s

σatA

d

4 + dτ (2.9)

is the integrated XPS per signal photon. The temporal profile of the XPS predicted by


the LTI model, eq. 5.10, suggests that there are two different timescales involved: the

response time of the EIT medium, τ , and the signal pulse duration, τs. Initially, when

t τ , the error function term alone dictates the temporal shape, having a timescale

given by τs. The rise of the phase shift always mimics the envelope of the signal pulse,

irrespective of τ . For later times, however, the temporal shape of the phase shift is given

by a combination of the signal pulse duration and the response time of the EIT medium.

In the limiting case of τs τ (when the signal pulse is much longer than the response

time of the medium), the probe phase follows the signal pulse envelope. This corresponds

to a quasi-steady-state scenario where the atomic coherences are able to follow the change

in two-photon detuning arising from the signal field. In the other extreme, when τs τ ,

the phase of the probe field rises quickly due to the short signal pulse and then relaxes

to its original steady-state value on a timescale given by τ alone. This corresponds to a

short impulse perturbing the system momentarily, leaving the atomic coherences to build

back up once it passes. For intermediate cases, the phase decays on a timescale which is

a combination of τ and τs.

In addition, the integrated XPS per photon, φ0, as predicted by the LTI model, eq.

5.11, is seen to be independent of the signal pulse duration (recall that the energy of the

signal pulse is held fixed). Importantly, the integrated XPS scales inversely with the EIT

window width for pumping rates much larger than the dephasing rate, R γ; peaks

when R = γ; and falls off for R γ. The only other parameters that φ0 depends on

are the OD d0, the signal pulse detuning ∆s, and how tightly the signal beam is focused

compared to the atomic cross section, σat/A. I now turn to the dynamics of EIT-enhanced

XPS and show that this linear model accurately predicts the behavior obtained from a

numerical solution of the complete system density matrix.


2.3 Properties of the N-scheme

In what follows, I show how different parameters of interest modify the behavior of EIT-

enhanced XPS in the presence of a pulsed signal field. I consider both the numerical

solution of section 2.2.1 as well as the LTI model of section 2.2.2 and show that the

latter captures the salient features of this nonlinear interaction. I begin by discussing

the effect of the transparency window width, ∆EIT , on the XPS time response and the

role that dephashing plays in this regard. In section 2.3.2, I investigate the effects of the

signal pulse duration and detuning, and I conclude by discussing in section 2.3.3 how

an optically thick medium alters these dynamics. In order to carry out the numerical

simulations, most of the medium parameters are chosen to be close to practically available

values in a cold Rubidium atom sample. However, it is important to remember that the

qualitative results presented here are general properties of the N-scheme regardless of the

specific medium chosen to implement it.

2.3.1 Dependence on EIT medium properties

I first address how the width of the transparency window affects the dynamics of the EIT-

enhanced XPS. In the original single-mode treatment, the size of the nonlinear phase

shift increased indefinitely as the EIT window was narrowed. In the subsequent multi-

mode, step-response analysis, the steady-state phase shift behaved similarly but this

steady state took longer to be established for narrower transparency windows. Figure

2.2 shows the temporal profile of the XPS experienced by a probe field in response to

a Gaussian signal pulse for a variety of EIT window widths, as obtained by numerical

simulation of equation 2.2. It is immediately evident that the rise time of the nonlinear

phase shift is independent of the EIT window width, mimicking instead the rise of the

signal pulse; also, as the window width narrows, the effect of the signal pulse on the

probe field is prolonged. For narrower EIT windows, more time is required for the probe


0 0.3 1 3 10 30 1000

0.5

1

1.5

2

2.5

3

3.5

4

time (µs)

φ(t)

/nph

(µr

ad)

EIT Window = 2π×12 kHzEIT Window = 2π×61 kHzEIT Window = 2π×121 kHzEIT Window = 2π×364 kHzEIT Window = 2π×1213 kHzEIT Window = 2π×3640 kHzSignal Pulse (a.u.)

Figure 2.2: Time dependence of the per-photon XPS for a variety of EIT window widths.The linear scaling of the peak XPS versus EIT window width breaks down once theresponse time of the EIT medium becomes comparable to or larger than the signal pulseduration. However, narrower window widths produce longer tails. Simulation parame-ters: Γ = 2π × 6MHz, τs = 1/2

√2π × 2000 kHz−1, nph = 100, d0 = 1, ∆p = 0, ∆c = 0,

∆s = −10Γ, σat = 1.2 × 10−13 m2, Ω0,p = 0.003Γ, γ = 1 × 10−5Γ, beam waist is 10 µmand the wavelength is 780.24 nm. The atomic cloud has a Gaussian spatial distribution.

phase to return to its original steady-state value. In many practical applications of the

EIT-enhanced cross-Kerr effect, this elongated tail permits a longer integration time and,

hence, improved SNR.

Figure 2.3 shows the peak and integrated XPSs extracted from figure 2.2 (squares) as

well as those predicted from the LTI model of section 2.2.2 (dashed line). Immediately

evident is the good agreement between these two different approaches. In both cases, I

see that the peak XPS scales linearly with 1/∆EIT only when the EIT window is wide

enough that τ τs, i.e. when the response time is shorter than the signal pulse duration;

once the window becomes narrower this linear scaling is disrupted, eventually plateauing

for τ τs. In fact, the peak XPS changes by a mere factor of two for a window width

variation that spans two orders of magnitude. Although the steady-state phase continues

to grow with decreasing ∆EIT , the time needed to reach this steady state also grows while


3640 1213 364 121 61 120

1

2

3

4

Simulation

LTI Model

Simulation

LTI Model

1/EIT

inte

gra

ted X

PS

EIT=4

(R= )

Figure 2.3: Peak (top) and integrated (bottom) XPS per photon as extracted from figure2.2. The peak XPS scales inversely with EIT window only when the response time of theEIT medium is shorter than the signal pulse duration while the integrated XPS growsinversely with window width owing to the longer tails that arise from narrower EITwindows. Squares correspond to simulation results and dashed lines show the predictionof the LTI model presented in section 2.2.2. For window widths comparable to the naturallinewidth of the transition the EIT medium response includes oscillations that are notincluded in the LTI impulse response, resulting in a small discrepancy between the twoapproaches. Also, the linear scaling of the integrated XPS can be interrupted if thepumping and dephasing rates become comparable (inset).

the interaction time (signal pulse duration) is held constant here. Therefore, once ∆EIT

is sufficiently narrow, decreasing the window width further does not help with increasing

the peak XPS, which accounts for the plateau seen in figure 2.3. On the other hand,

figure 2.3 also shows that the integrated phase continues to scale inversely with the EIT

window width irrespective of the medium response time and the signal pulse duration. I

am, therefore, led to conclude that the slow dynamics, far from degrading the effect, can

still lead to an enhanced integrated XPS that could be exploited to obtain better SNR

when detecting an EIT-based XPS, even when the peak XPS saturates.

It can be seen that the integrated XPS scales as 1/∆EIT and this scaling is interrupted


0.003 0.01 0.1 0.3 1 3 10 30

0

1

2

3

4

/R

Simulation

LTI Model

/R

Simulation

LTI Model

Figure 2.4: Peak (top) and integrated (bottom) XPS per photon for various ground-statedephasing rates, γ. As the dephasing rate increases, both peak and integrated XPSdecrease due to the degradation of the EIT window. Peak XPS falls to nearly half ofits ideal value when the dephasing rate becomes equal to the pumping rate, R. Squarescorrespond to simulation results while the dashed lines show the prediction of the LTIsystem response. For this simulation R = 0.01Γ, τs = (0.6Γ)−1 and the rest of parametersare the same as in figure 2.2. Note that EIT window width is 2(R + γ).

only by the ground-state dephasing rate, γ, which has only technical but no fundamental

limit. This dephasing limits the maximum depth of transparency, d = d0R/(R + γ),

as well as the minimum attainable EIT window width, 2(R + γ). These two quantities

correspond to the rise and run, respectively, of the refractive index profile experienced

by the probe field. Figure 2.4 shows the peak and integrated XPS for various values of

γ and a fixed pumping rate, R. The peak XPS falls by a factor of two at γ = R while

the integrated XPS does so at a value of γ smaller than R since it is affected by both the

refractive index slope and the shortened tail.


0 0.5 1 1.5 2 2.5 3 0

0.5

1

1.5

2

2.5

3

3.5

4

t (µs)

φ(t)

/nph

(µr

ad)

Signal FWHM =0.1 ΓSignal FWHM =1 ΓSignal FWHM =10 ΓSignal FWHM =30 Γ

Figure 2.5: Time response of XPS (per photon) for various signal pulse bandwidths.The linear scaling of the peak XPS with signal pulse bandwidth breaks down when thisbandwidth becomes comparable to or larger than the EIT window width. Once thebandwidth of the signal pulse becomes comparable to its detuning, ∆s = −10Γ, the peakXPS stops growing and starts to fall. Simulation parameters: ∆EIT = 0.2Γ and the restof parameters are the same as in figure 2.2.

2.3.2 Dependence on signal pulse

So far the only assumption I have made about the frequency content of the signal pulse

was that its bandwidth was small compared to the signal pulse detuning. In this section I

study how changing this frequency content can result in the modification of the behavior

of the EIT-enhanced XPS. For simplicity I assume that the signal pulse is transform-

limited: that is, that its bandwidth is proportional to 1/τs. Increasing the bandwidth,

therefore, corresponds to a temporally shorter pulse. Since the Kerr effect depends

linearly on the signal field intensity, one would expect to be able to maximize the XPS,

for a given pulse energy, by making the pulse as short, and therefore as intense, as possible.

However, in the case that the spatial extent of the signal pulse is larger than the atomic

medium, a shorter pulse yields a shorter interaction time and this must be weighed


0.002 0.007 0.02 0.07 0.21 0.42 1 2.1

0.25

1

4

Signal FWHM / 2 | |s

peak X

PS

(µ

rad)

Simulation

LTI Model

0.002 0.007 0.02 0.07 0.21 0.42 1 2.11

1.1

1.2

1.3

1.4

1.5

1.6

Signal FWHM / 2| |s

inte

gra

ted X

PS

(µ

rad.µ

sec)

Simulation

LTI Model

detuning detuning

Stark Shift (a.u.)

Signal Pulse (a.u.)

detuning

Stark Shift (a.u.)

Signal Pulse (a.u.)

Figure 2.6: Peak (top) and integrated (bottom) XPS per photon as a function of signalfield bandwidth (normalized to central detuning) as extracted from figure 2.5. Initially,increasing the pulse bandwidth causes the peak XPS to grow proportionately due to thehigher pulse intensity. However, once the pulse bandwidth becomes larger than the EITwindow width, the peak XPS stops growing, similar to the behavior seen in figure 2.3.The maximum integrated XPS occurs when the pulse half-width at half-maximum of theintensity is equal to the detuning. The insets show the Fourier transform of the signalpulse intensity (red dashed) along with the frequency dependence of the ACS (blue solid)as a function of detuning from the excited state. For very broadband pulses, there is adiscrepancy between the result of the LTI model and the numerical solution as explainedin the text.

against the larger intensity due to broadening the signal bandwidth (i.e. decreasing τs).

Figure 2.5 shows the temporal profile of the XPS for different signal pulse bandwidths

for a constant pumping rate of R = 0.1Γ. I find that when τs τ , the XPS replicates the

temporal profile of the signal pulse but the peak XPS is relatively small due to the low

intensity signal pulse. As one broadens the bandwidth of the pulse, the peak intensity

and therefore the peak XPS increase. However, this increase in peak XPS with signal

intensity is seen to saturate and even reverse once τs becomes sufficiently small. Figure

2.6 plots the peak and integrated XPS against signal pulse bandwidth normalized to its


0.05 0.15 0.5 1.04 3.12 10 150

5

10

15

20

25

30

Signal detuning, s/

peak X

PS

(µ

rad)

signal FWHM = 2|s|

0.05 0.15 0.5 1.04 3.12 10 150

5

10

15

Signal detuning, s/

inte

gra

ted X

PS

(µ

rad.µ

sec) signal FWHM = 2|

s|

detuning

Stark Shift (a.u.)

Signal Pulse (a.u.)

Figure 2.7: Peak (top) and integrated (bottom) XPS per photon for various signal detun-ings, ∆S, when the HWHM of the signal pulse bandwidth is set equal to the detuning.The squares show simulation results while the dotted line is a guide for the eye. Bothpeak and integrated XPS have maxima close to ∆s = Γ/2. The inset shows the Fouriertransform of the signal pulse intensity (red dashed) along with the frequency dependenceof the ACS (blue solid) as a function of detuning from the excited state.

detuning, ∆s. For pulse bandwidths narrower than the EIT window the peak XPS scales

linearly with signal bandwidth (and therefore linearly with intensity) as expected from

single-mode or step-response treatments. However, once the signal bandwidth exceeds

the EIT window width, the scaling begins to flatten out. This saturation is a consequence

of the tradeoff between shorter interaction time and higher peak intensity of the signal

pulse. Once the signal pulse has a bandwidth wider than the EIT window then it exits

the medium before the XPS reaches its peak value. Increasing the bandwidth any more

does not lead to a larger peak XPS. The integrated XPS remains flat throughout all of

this due to the fact that the energy of pulse and the window width are held constant


constant.

Once the bandwidth of the signal pulse grows to be comparable to its detuning, the

variation of the signal pulse amplitude versus frequency becomes important. The response

function used in section 2.2.2 does not take that frequency content into account and

therefore fails to predict the behavior of the system properly. I can, however, qualitatively

understand the behavior of XPS due to broadband pulses by recalling that the frequency

dependence of the Stark effect resembles a refractive index profile. That is, it is an odd

function passing though zero on resonance, with extremes Γ/2 away on either side of

resonance and scaling inversely with detuning away from resonance. Therefore, for a

given signal pulse detuning, as its bandwidth is broadened, a point will be reached when

frequency components begin to encroach on the peak of the ACS profile, leading to a

larger XPS. However, as the bandwidth is broadened further, this increase is quickly

reversed as frequency components begin to cross over to the other side of the resonance

addressed by this signal field. These frequency components then contribute strongly to

the ACS but with opposite sign, yielding a smaller net phase shift. The optimum phase

shift is obtained when the signal HWHM,√

log 2/√

2τs, is equal to the signal detuning,

∆s.

It is interesting to see how the XPS behaves as a function of signal detuning when

∆sτs is held constant at the value of√

log 2/√

2. Figure 2.7 shows the peak and integrated

XPS for the case when the signal HWHM is set equal to the detuning and then the two

are varied simultaneously. It can be seen that the largest optimum phase shift is achieved

close to ∆s =√

log 2/√

2τs = Γ/2. For this choice of detuning and signal bandwidth the

center of the pulse (in frequency space) coincides with the peak of the ACS profile and

its width covers those parts with the largest positive shift without spilling over onto the

other side of the resonance (inset of figure 2.7).


z

t

Signal

Probe

EIT

Med

ium

Phase-shifted portion of the probe

Figure 2.8: Space-time diagram of the interaction between the probe and signal. At highOD the group velocity mismatch of the probe and signal causes a large portion of theprobe to be affected.

2.3.3 Propagation in an optically thick medium

Thus far, I have neglected the effects that an optically thick medium would have on

the dynamics of EIT-enhanced XPS. Steady state analysis predicts that the XPS scales

linearly with the OD and so it is of interest to see how the dynamics are affected by

exploiting higher OD’s. Particularly in the presence of EIT, which eliminates linear

absorption, higher OD increases the nonlinear interaction with no detrimental effects

arising from absorption. However, increasing the optical thickness of the medium also

increases the difference in the group velocities of the probe and the signal pulses; the

probe experiences a slow-light effect while the signal field does not. This group velocity

mismatch poses a limit on the maximum attainable peak XPS as one increases the OD

[37]. Given these tradeoffs, here I discuss whether EIT-based XPS can still benefit from

optically thick media.

For a sufficiently high OD, the transit time of the probe field through the sample

becomes longer than the temporal duration of the signal pulse. In this case, there will


0 1 2 4 8 160

1

2

3

4

5

6

7

8

9

10

t (µs)

(t)

(µra

d)

OD =0.3

OD =1.2

OD =3.6

OD =7.2

OD =14.4

OD =28.8

OD =57.6

Figure 2.9: XPS due to a step-function signal field (top) and time dependence of pulsedXPS (per photon) for different OD’s (bottom). As the OD, d0, increases the peak XPSbegins to grow but eventually saturates due to the group velocity mismatch between thesignal and the probe. However, larger values of OD result in longer-lasting phase shifts;the temporal extent of the flat region of the transient is determined by the duration ofthe probe that is compressed in the medium, τL, when the signal pulse passes throughthe medium at group velocity, c. Simulation parameters: R = 0.1Γ, τs = (0.6Γ)−1 andall others as in figure 2.2.

be portions of the probe field inside the medium which experience the entire signal pulse

as it passes through and, therefore, these portions acquire the maximum phase shift

possible; see figure 2.8. The temporal length of this portion of the probe is equal to its

group delay, τL = L/vg = d0(R− 2γ2/Γ)/2(γ +R)2 where L is the length of the medium

and vg is the group velocity of the probe. This is reflected in figure 2.9, where I plot the

temporal profiles of the XPS for a variety of OD’s. It can be seen that for high OD, the

peak height of the phase shift plateaus but the duration of this peak XPS continues to

grow as the OD is increased. The net effect, as shown in figure 2.10, is such that while

the peak XPS saturates, the integrated XPS scales linearly with OD.


0.3 1.2 4.8 14.4 57.61

3

10

optical densitypeak X

PS

(µ

rad)

Simulation

LTI Model

0.3 1.2 4.8 14.4 57.6

0.4

4

40

optical density

inte

gra

ted X

PS

(µ

rad.µ

sec)

Simulation

LTI Model

< 0 > 0

Figure 2.10: Peak (top) and integrated (bottom) XPS per photon versus OD, d0, asextracted from figure 2.9. Squares correspond to simulation results while the dashedlines are predictions of an LTI model. The response function adopted in equation 2.5only partially accounts for the propagation effects (through the dependence of the EITmedium response time, τ , on OD); however, this is not sufficient to model the behaviorof the system at high OD’s. It is important to note that the response of the systemis still linear at high OD’s and a proper impulse response can account completely forthe saturation effect. The integrated XPS increases linearly with OD and an LTI modelagrees very well with the simulation results. τ0 is the response time of the EIT mediumin the limit of vanishing OD.

To determine this saturation value of the peak XPS, it is instructive to consider the

response of the system to a step signal, see figure 2.9 (top), which includes a linear rise

with time-scale τL, followed by an exponential approach to the steady-state value. The

slope of the rise, shown by the red line in figure 2.9 (top), is equal to φss/τL. Since

the impulse response is the derivative of the step response, this slope determines the

maximum achievable XPS for pulsed signal in the presence of high OD,

φmax =φss

τL

∫dt|Ωs(t)|2 = − Γ

4∆s

σatA

(2.10)


This is similar to the limit found by Harris and Hau due to group velocity mismatch in

N-scheme [37]. Unlike the case of the response to a step signal, where the propagation

effects show up in the rise time of the nonlinear phase shift [70], the response to a pulsed

signal has a rise time determined by the signal pulse and the propagation effects only

result in the saturation of the peak XPS in the time response.

It can also be seen that while the integrated XPS is well modelled by our LTI approach,

the peak XPS is under-estimated for sufficiently high OD’s. This does not result from a

breakdown of the linearity but rather because the response function assumed in section

2.2.2 did not account for such propagation effects. In an optically thick medium the

effect from each thin slab of the medium takes some time, determined by the group

velocity of the probe and the length of the medium, to reach the observer. Therefore,

the exponential rise assumed in equation 2.3 does not capture the additional group delay

effects present in media with high OD’s.

2.4 Summary

I showed that in the regime of narrow transparency windows perturbed by short signal

pulses the peak XPS saturates and the duration of the effect grows as the window becomes

narrower. While the rise time of the EIT-enhanced XPS is determined by the signal pulse

duration, its fall is given by the inverse EIT window width, resulting in an integrated

XPS that continues to scale inversely with the window width even for ∆EIT 1/τS. It

was also shown that in the case of high optical thickness, the group velocity mismatch

between the probe and the signal pulses results in the saturation of the peak phase shift

and the effect lasts for a time that increases linearly as the optical density. Furthermore,

I showed that the dynamics of the XPS can be understood in terms of an LTI model.

The intensity of the signal field and the phase of the probe field can be thought of as the

“drive” and “response” of a linear system, respectively.

Chapter 3

Kerr nonlinearity as a measurement


3.1 Overview

In this chapter, I study how the ideas inspired by weak measurement [72, 73, 74, 75, 76]

can be used to amplify photon-photon interaction strength. Weak measurement is an ex-

citing new approach to understanding quantum systems from a time-symmetric perspec-

tive, obtaining information from both their preparation and subsequent post-selection

[77, 78]. In recent years it has been widely studied to both address foundational ques-

tions in quantum mechanics [79, 80, 81] and for its potential application to measurements

such as spin Hall effect of light [82], beam deflection [83, 84], frequency measurement [85],

wave-function characterization [86, 87, 88], velocity measurement [89], phase estimation

[90], and angular rotation [91]. There are also numerous theory proposals trying to use

weak values to measure effects such as longitudinal phase shifts [92], spin-spin interaction

[93], electric charge [94], fermionic interaction [95], and EIT lensing effect [96], or to carry

out tasks including quantum logic [97], protecting entanglement from decoherence [98],

and phase stabilization [99].

If a quantum system is coupled weakly to a probe, then very little information can

29

Chapter 3. Kerr nonlinearity as a measurement 30

be obtained from a single measurement This type of measurement disturbs the system

by a negligible amount. In such situations, if the system is prepared in some initial state

|i〉 and post-selected in some final state |f〉, the “weak value”, 〈A〉w = 〈f |A|i〉/〈f |i〉,

describes the mean size of the effect an ensemble of such systems would have on a device

designed to measure the observable A. It should be noted that weak values are not

guaranteed to exist within the eigenvalue spectrum of the observable A. If the overlap

between the initial and final states is small, the weak value may be anomalously large.

In Aharonov, Albert and Vaidman’s famous example, the spin of an electron may be

measured to be 100 [72]; in a mathematically equivalent sense, I show that the effective

photon number in one arm of an interferometer may be found to be 100 even in the

presence of only one photon.

3.2 Quantum Measurement

The purpose of a quantum measurement is to obtain information about the system under

investigation. In a measurement, the quantum system is coupled to a classical probe and

information about the state of the system is gained by looking at the probe. The system

experiences a disturbance as a result of its mutual interaction with the probe.

The ‘system’ can start in an initial superposition of its eigenstates, |i〉; see figure 3.1.

The ‘probe’ is normally a (close to) classical entity with a distribution rms width of σx in

position, x, space. The interaction causes the probe to move to a new position. Suppose

the system and the probe are interacting through exp(−ıθAP ), where A is the system

observable of interest, P is the momentum operator of the probe and θ is the coupling

strength between the two. Through the measurement interaction the probe moves to aθ

where a is any of the eigenvalues of the system. The amount of information gained from

the measurement depends on the ratio of the probe displacement, aθ, compared to its

position uncertainty σx.


a b

MeasurementInteraction

a b

wea

k>lim

itst

rong

>lim

it

a b

b

strongly>entangled>state

slightly>entangled>state

Observationsystem

probe

Post-selection

a b 100

superposition,>|i>

a>different>superposition,>|f>

x

Figure 3.1: Quantum measurement. System and probe couple through a measurementinteraction and depending on the strength of the coupling compared to the positionuncertainty of the probe, it can be a strong (top) or weak (bottom) measurement.

3.2.1 Strong measurement

If the probe displacement is large compared to its uncertainty, then the result of the

interaction is a strongly entangled state between the probe and the system; see figure 3.1

(top). When strongly entangled to the probe, the most information about the state of the

system can be found only by simultaneously disturbing its state maximally. Although it

started in a coherent superposition, if one traces over the state of the probe, the state

of the system will become completely mixed. Finally, observing the state of the probe

projects the state of the system into the ‘observed’ eigenstate.

3.2.2 Weak measurement

For a weak enough interaction the displacement of the probe is small compared to its

position uncertainty, and the state of the probe and the system is only slightly entangled;

see figure 3.1 (bottom). In this case, observing the probe does not provide much infor-

mation about the system due to the large overlap of possible probe states; in return, the

system is minimally disturbed. However, if the system is post-selected to be in some final

state, |f〉, due to interference in the projected probe state, it can show average values


outside the eigenvalue spectrum of the observable A.

Suppose that the initial state of the system and probe is given by |i〉|ψp〉 and after

the interaction it evolves into exp(−ıθAP )|i〉|ψp〉. If the system is post-selected to be in

final state |f〉 and the interaction is weak enough that the propagator can be expanded

to the first order in θ, then the state of the probe after the post-selection is

|ψp〉 → 〈f |i〉(

1 + ıθ〈f |A|i〉〈f |i〉 P

)+O(θ2) (3.1)

The final state of the probe can be written as ≈ exp(ıθAwP ) where Aw = 〈f |A|i〉/〈f |i〉

is the ‘weak value’ of the observable A. As a result of the weak measurement interaction

and the post-selection, the probe will move to a position determined by Aw instead of

the eigenvalues of A.

The weak value, Aw, can have any values regardless of the eigenvalue spectrum of the

observable. If the overlap of the initial and final states is small the weak value can be

quite large. The cost of this WVA is discarding most of the data since a small overlap

leads to a low probability of successful post-selection.

There have been numerous studies extending the original weak measurement idea

including use of a qubit probe [100], use of incoherent measuring device [101], limits

on weak-value amplification based on higher-order corrections [102, 103], use of orbital

angular momentum as the probe [104], and optimal probe wave function [105].

3.3 Weak-value amplification of photon number

The Kerr nonlinearity can be viewed as a measurement in which a single-photon “system”

is coupled through the cross-Kerr effect to a classical “probe” field; see Fig. 3.2. The single

photon is sent through a 50-50 BS, thus prepared in the superposition |i〉 ≡ (|b〉−|a〉)/√

2

of modes a and b. The single photon interacts with a probe through a Kerr medium,

leading to a XPS that is modelled as exp(iφ0nbnc), where φ0 1 is the XPS per photon


|0〉

a

b

c

dθ

|1〉

|0〉

|α〉pφ0

BS1BS2

BS3 BS4

Trig

ger

Phase

Post-Selection

System State

PreparationXPM

δ

Read-out∣∣iαeiθ

⟩

|ψ〉pProbe

Figure 3.2: The single-photon “system” is prepared in an equal superposition of arms aand b by the first BS1. After a weak XPM interaction with the “probe”, prepared in acoherent state |α〉p, the system is post-selected on a nearly orthogonal state by detectingthe single photon in the nearly-dark port, D1. The success probability of post-selectiondepends on the imbalance δ in the reflection and transmission coefficients of BS2, andthe back-action of the probe on the system. Using the lower interferometer to read outthe phase shift of the probe amounts to a measurement of the system observable nb, thephoton number in arm b. The phase shifter θ is used to maximize the sensitivity of themeasurement.

and nb (nc) is the number operator for mode b (c). After the interaction with the

probe the system is post-selected to be in a state nearly orthogonal to the initial one,

|f〉 = t |b〉 + r |a〉, by triggering on the detection of a photon at D1. This port exhibits

imperfect destructive interference when the reflectivity r and transmissivity t, which is

chosen to be real and positive, are slightly imbalanced. A small post-selection parameter

δ ≡ 〈f | i〉 = (t− r)/√

2 1 is defined as the overlap of the initial and finale states. The

weak value of the photon number in mode b is given by

〈nb〉w =〈f | nb |i〉〈f | i〉 =

t/√

2

(t− r)/√

2' (1 + δ)/2

δ' 1

2δ. (3.2)

This means that whenever the post-selection succeeds (which occurs with probability δ2

ignoring the measurement back-action) the weak value of the photon number in mode

b is 1/δ times the strong value, 1/2. The post-selection parameter δ can be very small,


leading to a large weak value for the photon number in the system. Therefore, within

the weak-measurement formalism, the probe will experience a XPS equivalent to that of

many photons, even though the system never has more than one photon. In the rest of

this chapter, I will show explicitly that such a scheme does in fact lead to a large phase

shift, and quantify the improvement in the SNR as a function of the characteristics of

the technical noise.

3.3.1 Full calculation

The state of the system and probe after coupling is

|Ψ〉 =1√2

(|b〉s∣∣αeiφ0

⟩p− |a〉s |α〉p). (3.3)

For φ0 1, the overlap between the two possible final probe states is⟨α∣∣ αeiφ0

⟩'

ei|α|2φ0−|α|2φ20/2. The amplitude of this overlap, e−|α|

2φ20/2, has to be close to 1 for the

interaction to be weak, which implies |α|φ0 1. The phase of the overlap, |α|2 φ0,

describes the average phase-shift imparted to the system by the probe. This phase does

not result in dephasing of the system state and therefore, in principle, can be compensated

by adding a phase-shifter to the upper interferometer. Without compensation, WVA will

occur only when |α|2 φ0 is close to an integer multiple of 2π, where the overlap between

the initial and final states of the system is small. I define ε to be the difference between

|α|2 φ0 and the closest multiple of 2π.

If the system is post-selected to be in state |f〉, the state of the probe, |ψ〉p = s 〈f | Ψ〉,

collapses to a superposition of two coherent states,

|ψ〉p =√P−1

1

2

((1 + δ)

∣∣αeiφ0⟩− (1− δ) |α〉

), (3.4)

where P ' |α|2 φ20/4 + δ2 + ε2/4 is the post-selection probability. The final state of the

probe can be most easily understood by displacing it to the origin in phase space, defining


|χ〉 = D†(α) |ψ〉p, where D(α) is the displacement operator. For φ0, |α|φ0 1, one can

write

|χ〉 '√P−1 ((δ + iε/2) |0〉+ (iαφ0/2) |1〉) , (3.5)

where |0〉 and |1〉 are vacuum and single photon number states respectively. The weak

measurement formalism applies if δ2 (ε2 + |α|2 φ20)/4; in particular, as ε → 0, one

recovers the weak-measurement prediction |ψ〉p ' |α exp(iφ0/δ)〉, a coherent state with a

largely enhanced phase. On the other hand, if δ2 ε2/4 + |α|2 φ20/4 the post-selection is

significantly modified by the back-action of the probe on the system. It is instructive to

look at both regimes and the transition between them and determine what the maximum

possible enhancement is, taking the back-action into account.

Most of the interesting phenomena can be understood by investigating properties of

|χ〉. If δ or ε is much larger than |α|φ0, then the state |χ〉 is approximately equal to a

weak coherent state, |χ〉 ' |0〉+ iαφ0 |1〉 /(2δ+ iε). It can be seen that δ contributes to a

shift in the imaginary quadrature (phase of |ψ〉p) and ε contributes to a shift in the real

quadrature (average photon number). On the other hand, if |α|φ0 is much larger than

the two other terms, the state |χ〉 is approximately a single-photon number state.

The average phase shift can be measured by using the lower interferometer in Fig.

3.2, e.g. as the ratio of the difference of the photon numbers at D2 and D3 to the sum,

φ =〈M−〉p〈M+〉p

' δ

2Pφ0, (3.6)

where M± = n3±n2. This should be compared to the value of the phase shift φ0 imparted

to the probe by a single photon in path b. The phase that one measures after successful

post-selection is enhanced by a factor of δ/2P . Fig. 3.3 shows this enhancement factor as

a function of post-selection parameter δ and the average number of probe photons, |α|2.

For sufficiently small back-action, the weak measurement prediction for the amplification,

1/2δ, is correct. However, as δ becomes smaller, the amplification grows but so does the


0

25

50

2π 4π 6π 8π

EnhancementFactor(δ/(2P))

|α|2 φ0

0

δ

δ(i)opt ≃ |α|φ0/2

δ(ii)opt ≃ |α|2 φ0/21

2δ

2δ|α|2φ2

0

Figure 3.3: The enhancement factor versus |α|2 φ0. The parameters used are φ0 =2π × 10−5 and δ = 0.01. The enhancement factor is calculated by using the state of Eq.(3.4) without any approximations. The dashed line shows the enhancement factor if theaverage phase written by the probe on the system, |α|2 φ0, is compensated; otherwiseenhancement occurs whenever |α|2 φ0 is close to an integer multiple of 2π (solid curve).The inset shows the enhancement factor as a function of post-selection parameter, δ, intwo different regimes: i) |α|2 = 105, in which case the imparted phase on the system bythe probe, ε, is 0 (solid blue); ii) |α|2 = 102, where ε is a small non-zero phase (dashedgreen). For large values of δ the weak-measurement prediction is valid; however as δdecreases the back-action from the probe plays a more dominant role. The dashed lineshows the prediction of the weak-measurement formalism.

back-action, until at δopt =√|α|2 φ2

0 + ε2/2 a maximum amplification value is achieved

of 1/4δopt, half of the weak-measurement value. For small ε, the maximum phase shift

is equal to 1/2|α|, which is one-half the quantum uncertainty of the probe phase. Thus,

the WVA works up to the point where the single-shot quantum-limited SNR would be

on the order of 1. Taking a closer look at the form of state |χ〉, one can see that the

large phase shift is caused by destructive interference due to post-selection; the vacuum

term largely cancels out, enhancing the importance of the single-photon term. Note that

the large overlap of the two possible probe states corresponding to the two states of the

system is essential for this to occur.

The weakness condition |α|φ0 1 is often met in experimental situations, either


Quantum-nois

e lim

it

Γ = 1τc

Γ = 1τcP1

Γ = 1τcP2

Single-Photon Rate, Γ (in units of 1/τc)

Signal-to-N

oiseRatio

δ22P2

S0

δ12P1

S0

S0

Figure 3.4: The SNR as a function of the single photon rate Γ. The technical noise ismodelled by an exponential correlation function with an amplitude, η, 10 times largerthan the quantum noise. The dashed line shows the non-post-selected SNR for thephase shift due to one photon in mode b. The post-selected SNR for δ1 = 0.1 (weak-measurement regime- dash-dotted red) and δ2 = 0.01 (the optimum value of measuredphase shift- solid green) are also shown; the dotted line shows the quantum-limited SNRfor comparison. The non-post-selected SNR approaches a maximum value, S0, due tolow-frequency noise. However, for the post-selected SNR, there is an enhancement bya factor of δ/2P , compared to the non-post-selected SNR, S0, for measurements withhigh enough rate. For low rates the enhancement is given by δ/2

√P and therefore the

weak measurement results in the best possible post-selected SNR. Relevant parametersinclude T/τc = 103, φ = 2π×10−5, |α|2 = 105 and therefore P1 = 0.01 and P2 = 3×10−4.

because of the difficulty of approaching quantum-limited performance at high intensities

or to avoid additional undesired nonlinear effects. In Ref. [20], for instance, a XPS of

φ0 = 10−7 rad per photon was reported and unwanted nonlinear effects were observed

once the average number of probe photons |α|2 reached about 106. In this situation both

conditions of |α|φ0 1 and |α|2 φ0 1 are met and WVA can be used to enhance the

SNR.

3.3.2 Enhancement of signal-to-noise ratio

Unfortunately, WVA always comes at the cost of reducing the sample size (via post-

selection) by just enough to nullify any potential improvement in SNR, at least in the


case of statistical noise. Several recent experiments [82, 106, 83] observed that many real-

world measurements are limited by technical noise, which is not reduced by averaging

over more samples, and attempted to show that in such cases weak measurement can

indeed be of practical advantage. It still remains unclear exactly when such “technical”

noise could be overcome by using WVA. In Refs. [82, 106, 83], a very specific noise

model was assumed, in which rejection of photons through post-selection did not reduce

the ultimate signal strength, an assumption I do not make. Here I find that the SNR can

be increased, roughly to but not beyond the quantum limit, when the noise correlation

times are sufficiently long.

In practice, phase measurement is subject to both quantum and technical noise. While

the average measured phase is enhanced by a factor of δ/2P , one would expect the

uncertainty due to statistical noise to be inversely proportional to the square root of the

sample size, thus scaling as 1/√P (recall that P is the probability of successful post-

selection). The overall SNR is hence multiplied by a factor δ/2√P , which has a maximum

value of 1/2 (the actual photon number in arm b); in the case of pure quantum noise, for

instance, there is no advantage with post-selection. In what follows, using a more general

noise model, I study under what type of “technical” noise WVA can be beneficial.

Consider a non-post-selected measurement performed over a total time T . Single

photons are sent to the upper interferometer at a rate Γ and phase measurement is

triggered by the detection of a single photon. I term the outcome of the ith measurement

φim = φ+ηi, where the zero-mean fluctuating term ηi includes the quantum and technical

noise. The average measured phase shift is φm = 1/(ΓT )∑ΓT

i=1 〈φim〉 = φ. The uncertainty

in this average value is given by (∆φm)2 = 1/(ΓT )2∑ΓT

i,j=1 〈ηiηj〉. There are two possible

extremes to be considered. In the white-noise limit (noise correlation time τc much shorter

than the mean time between successive measurements, 1/Γ), the correlation function can

be modelled as a delta function: 〈ηiηj〉 = η2δij. In particular, this holds for quantum

(shot) noise. In this limit the noise scales statistically with the number of measurements,


∆φm = η/√

ΓT . The opposite extreme is that of noise with long-time correlations,

τc 1/Γ, in which case 〈ηiηj〉 = η2, and averaging cannot help reduce the uncertainty.

In the post-selected case, the sample size drops from ΓT to PΓT , and ∆φm increases

to η/√PΓT in the delta-correlated case while it remains constant at η in the presence

of long-time correlations. Given the enhancement factor of δ/2P , the SNR thus scales

as δ/2√P (always < 1, as remarked earlier) in the former case but δ/2P (which may be

1) in the latter case.

Fig. 3.4 shows the calculated SNR as a function of single photon rate, Γ, where the

noise is modelled with a correlation function 〈ηiηj〉 = δij/2 |α|2 + η2 exp(− |i− j| /Γτc) to

account for delta-correlated quantum noise and a technical contribution with correlation

time τc. The non-post-selected SNR shows a knee around Γτc = 1, separating the regimes

where measurements are not correlated (Γτc 1) and highly correlated (Γτc 1). The

SNR has a statistical scaling,√

Γ, in the former regime and remains constant in the latter.

The graphs for the post-selected cases are qualitatively similar, but the knee occurs near

PΓτc = 1, that is, when the noise in the successive post-selected measurements starts

to become correlated. Thus whenever the noise exhibits correlations over timescales

greater than the mean time between incident photons, the SNR can be improved via

post-selection.

3.4 Discussion

In this chapter, I studied the cross-Kerr effect as a measurement interaction and showed

how this idea can be exploited to amplify the effect of one photon on many. In this pro-

posal two distinct optical beams may be coupled deterministically, by using accessible

interactions, in such a way that no classical explanation is possible for the predicted am-

plification. This is in contrast to previous weak-measurement demonstrations in which

instead of entangling a “system” with a distinct “probe,” merely two degrees of free-


dom of the same physical photon were used as the system and probe. This resulted in

experiments which could be equally well understood in the framework of classical elec-

tromagnetism, with no need of the full quantum formalism of weak measurement1. I also

have carried out a full quantum mechanical calculation to show that the surprising weak-

measurement prediction of a single photon “acting like” a collection of many photons is

rigorously correct. Finally I studied the behavior of SNR in the presence of post-selection

and concluded that it can be substantially improved by WVA, when the noise possesses

long correlation times (e.g. 1/f noise).

3.5 Controversy over weak-value amplification

3.5.1 SNR improvement in WVA

The introduction of weak values about two decades ago and the fact that they can have

anomalously large values [72] have brought up some debate about their potential use in

metrology. Following two of the early experiments in which very tiny quantities were

measured using weak-value amplification [82, 83], people started to believe that weak

values are advantageous when the measurement is dominated by some sort of technical

noise [106]. However, the question of SNR in the presence of WVA remains a very

controversial subject.

Kedem claimed that using imaginary weak values can always help improve the SNR

regardless of the time correlation of the noise [109]. Knee et al put a rather obvious

result into more rigorous terms: WVA cannot overcome decoherence [110]. It is a well-

known fact that the WVA is originated from an interference between potential probe

states and in the presence of decoherence the interference will not happen. Tanaka and

Yamamoto used a parameter estimation approach to study potential advantages of WVA

1Some implementations have been carried out with probabilistic coupling between the system andthe probe [107, 108].


[111]. They calculated the Fisher information associated with a typical measurement

scenario and showed that if one has unlimited time to make measurements, there is no

advantage in using WVA in terms of the measurement uncertainty. Ferrie and Combes

took a similar but more general approach and show that WVA cannot be useful even in

case of finite amount of data [112].

This question still remains controversial mostly because the answer depends signifi-

cantly on the context of the measurement. Many of the research that prove uselessness of

WVA are based on ideal assumptions like having unlimited time or computational power.

Once one factors such practical limitations in, the WVA might prove even essential. This

calls for further analysis of the merit of WVA, taking into account practically relevant

parameters and limitations [113, 114].

3.5.2 Classical anomalous values?

Very recently Ferrie and Combes (FC) claimed to have found a classical analogue to

anomalous weak values [115]. Based on their result, they claim that since the resolution

to quantum paradoxes using weak measurement is based on having anomalous values

and since there is a classical analogue, the validity of those resolutions is “called to

question”. In their paper, they show that in a classical system, the prerequisites to

see anomalous values are pre- and post-selection, very uncertain measurement, and an

outcome-dependant disturbance. In this classical approach, an anomalous value means

that an outcome appears with a probability higher than what is ‘normally’ expected.

Their result is in some sense quite obvious: since the probability of the disturbance

depends on the outcome, it can make the post-selection more likely to succeed for certain

outcomes. This biased disturbance is an essential ingredient of their scenario, but they

do not elaborate on it enough and more importantly do not draw direct and detailed

connections to how this disturbance appears in the quantum mechanical case and in a

typical weak measurement scenario.


The approach taken by FC has brought up some discussions about weak values.

Vaidman has pointed out that the results shown by FC do not reproduce most of the

main features of weak values and therefore their conclusions are invalid [116]. Vaidman

believes that their approach fails to “provides functional dependence on the pre- and

post-selected states of the system.” He also points out that “the concept of weak value

arises due to wave interference and has no analogue in classical statistics. Moreover,

if weak values are observed with external systems [as probe] (and not with a different

degree of freedom of the observed system as it has been done until now) then the weak

value appears due to interference of a quantum entangled wave”.

In another comment, Aharonov and Rohrlich believe that FC’s approach “might bet-

ter be characterized as a parody of a weak value” [117]. They have three main criticisms:

1) complete boundary conditions (pre- and post-selection) for a classical system is redun-

dant or inconsistent; 2) the inherent scatter in the measurement results is a result of the

uncertainty principle and has no classical analogue; and 3) the FC scenario lacks any sort

of ‘measuring device’ and therefore it is not possible to have probe values corresponding

to anomalous weak values.

Cohen also has written a comment on the FC result emphasizing the importance of

“non-invasiveness” in quantum weak measurement in having consistent predictions [118].

He, for example, points out that in the case of identical pre- and post-selected states,

weak measurement predicts the expected value of the observable while the FC treatment

results in a weak-value of zero. He also criticizes the fact that for orthogonal states

the post-selection success probability in FC’s scenario depends linearly on the weakness

parameter while the dependence should be quadratic in the quantum mechanical case.

This is taken to mean that the measurement in the case of FC’s analysis is more ‘invasive’

than in the quantum case. In addition, he believes that the lack of coherence in FC’s

example renders it irrelevant to weak measurements.

It seems that the measurement-induced disturbance is an essential part of the argu-


ments here. Therefore, a closer look at how measurement disturbs the system and how

closely the example put forward by FC represents what really happens in a weak mea-

surement scenario is quite essential. Ipsen looks at this problem in some detail [119] and

concludes that a measurement that minimizes the disturbance to the system can yield

weak values consistent with Aharonov-Albert-Vaidman formalism.

Dressel carries out a “careful review of the role of the weak value for conditioned ob-

servable estimation and concludes that any classical disturbance is insufficient to explain

the weak values unless if it could simulate quantum interference” [120]. He also points

out that it is a well-known fact that “classical conditioned averages of noisy signals can

show anomaly if the quantity being measured is also disturbed”, referring to his own

earlier works [121]. In some sense, Dressel calls both the interpretation and the novelty

of FC’s claims to question.

In short, arguments presented by FC has brought up some discussion about the “real

meaning” of the weak values or at least have heated up the on-going debates about them.

However, the general census so far seems to be that the scenario they are talking about

does not reproduce many of the main characteristics of weak values.

In the remaining chapters of this thesis, I report on the first steps towards the imple-

mentation of the proposal put forward in this chapter.

Chapter 4

Apparatus


4.1 Overview

This setup is built for the purpose of light-matter interaction, and involves several main

components that work together. This chapter provides both the overview and the details

of all these different components1.

Figure 4.1 shows a general view of the experimental setup. There are two main parts

of the setup: the ‘photon side’ and the ‘atom side’; laser beams are prepared in the former

and sent to the later to study their interaction with atoms. These two parts are located

in two separate labs; see section 6.1.1 for a note on the location of the setup. There

is also an atom-compatible single-photon source [122, 123, 124, 125] on the photon side

but at the moment its production rate is too low for use; see section 6.1.2 for alternative

approaches.

In the photon side, the laser beams with appropriate properties are prepared and sent

through long (around 20m) Single-Mode Fiber (SMF) to the atoms side. There is a ML,

commercial New Focus Vortex - External Cavity Diode Laser (ECDL), that provides the

1For instructions about the alignment procedures see chapter A.

44

Chapter 4. Apparatus 45

NewFocusVORTEX

Pol

ariz

atio

n:S

pect

rosc

opy

Injection:Lock

Probe::AOM:double1pass

Probe:AOMsingle1pass

Coupling:AOMsingle1pass

Signal::AOM:double1passSignal:AOMsingle1pass

MOT

Probe Signal

Coupling

EO

M

SMF

SM

F

Polarizing:beam1splitter

Fiber:port1:collimator

SMF Single1mode:fiber

55mW

52uW

42mW

D4mW 8mW

6mW cp::HmW

AT

OM

:SID

E

H52uW

PH

OT

ON

:SID

E

D8mW

D52uW 4mW

nW pW

822uW

Figure 4.1: The general view of the apparatus. The probe, coupling and signal beamsare prepared in the photon side and are sent to the atom side to interact with the coldatom cloud.

reference and seed light for production of the probe, coupling and signal beams. The

ML is usually locked close to F = 2 → F ′ = 3 transition in 85Rb through polarization

spectroscopy and Proportional-Integral-Derivative (PID) feedback. One portion of this

beam goes through an AOM Double-pass (DP) and an AOM Single-pass (SP) to produce

frequency components needed for probing the atoms. Another portion of this beam, goes

through an Electro-optic Modulator (EOM) and serves as the seed light to an Injection-

locked Laser (ILL). The ILL can be tuned to lock to the carrier or any of the microwave

sidebands of the EOM output. The output of the ILL goes through an AOM-DP and an

AOM-SP to produce the signal beam. A portion of the same beam is used with an AOM-

SP to produce the coupling beam. Two SMF’s are used to send light to the atom side;

coupling and probe beams are coupled into the same fiber with orthogonal polarizations,

and the signal beam is coupled into a separate SMF. The probe and coupling beams

are separated on a Polarizing Beam-splitter (PBS) near the atoms and sent to the cloud

from two directions.

In the atoms side, the 85Rb atoms [126] are trapped, cooled and prepared in a MOT.


Two beams, ‘trapping’ and ‘repumper’, each incident on the atoms in 6 directions (coming

from three angles and their retro-reflections) provide a dissipative force to slow the atoms.

The trapping beam (beam diameter: 1 inch, power: 50mW in each direction) is red-

detuned by around 20 MHz from the cyclic transition F = 3 → F ′ = 4. Atoms go

through multiple scatterings when interacting with these beams and lose kinetic energy.

In order to keep the population in F = 3 ground state we use the ‘repumper’ beam (beam

diameter: 1cm, power: 10mW in each direction) tuned close to F = 2→ F ′ = 3, pumps

the population from F = 2 ground state into F = 3. The repumper beam copropagates

with the trapping in all six directions. There is also a magnetic field gradient provided by

two coils in anti-Helmholtz configuration; the center of the trap occurs where the magnetic

field is zero. Along the axis of symmetry of the coils and the other two directions, the

magnetic field gradients are 20 and 10 G/cm, respectively. We typically get a millimeter

size cloud with the density of around 1010 atoms/cm3. The optical density of the cloud

on F = 2→ F ′ = 3 transition is typically 1 to 3 depending on the quality of the MOT;

see section 6.1.3 for ways to increase the OD. Most of the MOT parameters (trapping

power and frequency, repumper power, magnetic gradient strength, etc.) are controlled

through digital-to-analog cards with 100 µs timing resolution for cycles. The digital card

can also send a synchronization trigger Transistor-to-transistor Logic (TTL) signal to

other parts of the experiment.

4.2 Probe, coupling and signal preparation

4.2.1 Master laser stabilization

The stability of the ML is crucial to the experiment because it acts as the reference

for all the beams that interact with atoms. We use polarization spectroscopy [127, 128]

and PID feedback to stabilize the ML, as shown in figure 4.2. An Optical Isolator (OI)

is placed right after the VORTEX laser to prevent any back-reflection from the optics


NewFocusVORTEX

OI

TopthepProbepAOMpDP

Toptheinjectionplock

Rbpvaporpcell

Magneticpshield

WTopthepscopeandpPIDpfeedback

HalfWwaveplate

QuarterWwaveplate

PolarizingpbeamWsplitter

BeamWsplitter

OI Opticalpisolator

PhotoWdiode

150uW 40uW

100uW

pump

probe

H

RporpL

Figure 4.2: Polarization spectroscopy setup to produce error signals for stabilizing theML. A circularly polarized pump beam saturates the atoms and causes the polarizationof the probe beam to rotate. Doing a polarization analysis on the probe results indispersion-like features that can be used for locking the laser.

reach the laser and cause instability. Sets of Half-wave Plate (HWP) and PBS are used

in various places to control the amount of power going to different elements.

For stabilizing the laser we need to have an error signal that goes through zero right

on the atomic transition. This cannot be achieved by detecting the transmission, but

could be done if we measure the phase of the light passing through the atoms. In

polarization spectroscopy we can use the polarization rotation of a probe beam induced

by atoms to obtain dispersive features corresponding to the phase picked up. In order

to do that a strong circularly polarized ‘pump’ saturates the atoms; therefore a ‘probe’,

counter-propagating to the pump, experiences a birefringence close to atomic transitions:

right- and left circular components of the probe experience different absorption and phase

shifts. As a result, the incident linear polarization changes as the probe passes through

the atoms. The advantage for using a counter-propagating geometry is for the pump and

the probe to interact only with the zero-velocity class; therefore, we have a Doppler-free


− 0 0 0 100

−0.1

0

0.1

0.2

85Rb, F=2 F’

Detuning (MHz)

Err

or

sig

nal (V

)

100 50 0 50 100 150 2000.05

0.1

0.15

0.2

85Rb, F=3 F’

Err

or

sig

nal (V

)

100 0 100 200 300 400

0.05

0.1

0.15

0.2

87Rb, F=2 F’

Err

or

sig

nal (V

)

100 0 100 200 3000

0.05

0.1

87Rb, F=1 F’

Detuning (MHz)

Err

or

sig

nal (V

)

3

2

1

4

3

2

2

1

0

3

2 1

Figure 4.3: Typical set of polarization spectroscopy signals. The green dots show thereal transitions along with the values of F ′. The red arrow is our typical locking pointfor the ML.

spectroscopy signal. There are, however, non-zero-velocity-class atoms that could result

in features in the polarization spectroscopy signal. When the probe and pump address

different excited states they create additional features, so-called ‘cross-overs’.

A HWP and a PBS are used to set the power going into the spectroscopy probe and

pump arms, see figure 4.2. The power in the probe arm is set in order to achieve a

large error signal for locking, but limited to avoid power-broadening of the feature. The

incident probe polarization, that is horizontal (H), on passing through the vapor cell

rotates into an elliptical polarization. A Quarter-wave Plate (QWP) at 45 degrees sets

the pump polarization to circular. Finally, there is another HWP-PBS set to measure in

the diagonal-antidiagonal basis. If there is no rotation, there will be equal powers in the

two output ports of the analyzer PBS; most rotations will result in a power imbalance

between these two ports. Two inversed-biased Photo-diode (PD)’s are used to measure

the power in the output ports of the analyzer. The difference in their output currents

produces the error-signal for the PID feedback. Figure 4.3 shows typical error signals

measured for the 4 transition manifolds available in a Rb vapor cell.


A crucial aspect of polarization spectroscopy is its high sensitivity to background

magnetic fields, which induce polarization rotation. Therefore, it is important to use

magnetic shields around the vapor cell. We have wrapped our Rb vapor cell with three

turns of high permeability µ-shield (0.004 Magnetic Foil). Doing so significantly enhances

the quality of the error signal and prevents drifts in the shape of the error signal. See

section 6.1.13 for other possible improvements.

The error signal is fed into a home-made ramp-generator and PID feedback box,

‘Schlosser Lock’ [129]. The box provides laser scanning and locking circuits, producing

piezo- and current modulation voltage ramps and using a PID feedback system. The piezo

and current modulations provide slow (up to 1kHz) and fast (up to 1MHz) feedback to

stabilize the laser.

Probe

The ML is usually locked to be around -30 MHz from F = 2 → F ′ = 3. An AOM-

DP with the center frequency around +65MHz and a AOM-SP at -100MHz is used to

produce the on-resonance probe beam. We can ramp the AOM-DP frequency to scan

the probe in a range of at most 80MHz around the transitions. The probe power we

typically use is 18 nW on-resonance and around 180 nW off-resonance; see section 6.1.4

for a discussion on potential benefits of using lower probe powers. The theoretical value

for the on-resonance probe saturation power is 43 nW for the level structure and the

focus size we use.

4.2.2 AOM double-pass

Once the master and slave lasers are stabilized, further frequency changes are done using

AOM’s. In an AOM, light diffracts from a traveling acoustic wave caused by a Radio

Frequency (RF) drive (picks up or gives up one or more phonons) and shifts in frequency

by integer multiples of the RF. The diffraction angle depends on the RF and the order


Polarizing beam-splitter

AOM +10

Quarter-waveplate

+1

+2

+2

f1f2

Mirror

Converging lens

Iris

Figure 4.4: A typical AOM double-pass setup for scanning the laser frequency usingAOM without losing pointing accuracy.

of diffraction. Therefore, using AOM’s to scan the frequency of the beams can change

their direction and result in mis-alignment. In order to avoid this problem one usually

uses an AOM-DP; see figure 4.4.

Crucial elements of an AOM-DP is a lens that is one focal length after the AOM

and a mirror immediately following it. Any diffracted beam, for example the +1 order,

reaching that lens will be parallel to the axis after the lens. The beam, parallel to the

axis, will hit the mirror at normal incidence and takes the same path to go back into the

AOM where it diffracts again. The result of this diffraction is that the +2 order beam will

propagate into the original incoming mode. There is a QWP between the lens and the

mirror that ensures that the incoming H polarization turns into V (it is important that

the input polarizations into an AOM be H or V, otherwise any temperature change in the

crystal- eg. varying the amplitude of the drive RF- will cause uncontrolled polarization

rotations). Then, the +2 order, reflects off of the input PBS and goes to the rest of the

setup.

In order to achieve the best mode quality, diffraction efficiency and modulation speed,

the beam is usually focused into the AOM crystal. The clear aperture of the crystal is

typically around 1mm and normally a 100 micron focus waist is used (lens with 150-200

mm focal length for a 1mm input beam). A tighter focus causes the beam divergence and

diffraction angles to be similar and therefore there will be vertical fringes in the beam


NewFocusVORTEX

OI

TofthefspectroscopyfandthefprobefAOMfDP

Half-waveplate

Polarizingfbeam-splitter

OI Opticalfisolator Fiberfport-fcollimator

EOM

Free-runningdiodeflaser

half-OI

Magnet

Rotatingpolarizer

TofcouplingfandfsignalfAOMfpasses

PMF

PMF Polarizationfmaintainingffiber

HWP1

HWP2PBS1

See

dflig

ht

See

dflig

ht

Figure 4.5: Injection-locking to the 3 GHz sideband of a phase modulation is used toproduce signal and coupling beams.

because of the overlap of different orders. A looser focus, however, means that parts of

the beam could be clipped and the modulation speed will be slower (the response time

of the frequency- or amplitude-modulation of the AOM output goes inversely as the spot

size because it is the time that a phonon needs to go across the beam).

4.2.3 Injection lock

We require phase-locked probe and coupling beams in order to create EIT. One way to

do this is to use independent lasers, detect their beating signal and lock that to a reliable

microwave reference [130, 131]. An alternative solution is to modulate the beam from

one laser and injection-lock the second laser to the side-peak of the modulation. The

modulation frequency in this case is around 3 GHz and the injection lock current can

be used to set to lock to the modulation carrier or any of the side-peaks (the spacing

between the two ground states of 85Rb is 3035 MHz [126]). The injection lock current


determines which seed peak has the highest gain and the best chance to win the mode

competition.

In order to drive the EOM we use a home-made microwave source [132]. We are

interested in locking to the first side-peak and therefore the microwave drive amplitude

(modulation index) has to be set to have the most power in that peak. In addition, the

amount of optical power seeded to the ILL and the microwave power have to be chosen

properly; it must be large enough that the injection-lock is possible and stable, but low

that only one peak has enough seed light power to lase, thus ensuring the single-mode

behavior of the ILL. Usually around 50 µW of seed light is enough to have a stable and

clean lock.

The injection-lock setup is shown in figure 4.5. The seed light is shone back into the

slave laser and forces it to emit light at the seed frequency through stimulated emission.

The output of the ILL passes through several optics and there is some chance that a

portion of that beam gets reflected back into the laser. The laser would be more stable

if an OI is used to prevent back-reflections reaching the laser. However, in order to be

able to send the seed light back into the laser, the end polarizer of the OI is removed

and instead a PBS and a HWP is used. The direction dependence of the Faraday effect

guarantees that for the setting that the seed light has a maximum transmission through

the OI in the forward direction, the slave laser light has the maximum transmission in

the reverse direction.

The EOM we are using is a fiber-based low-π-voltage EOSpace 10 GHz phase modu-

lator. The EOM requires a certain polarization to work properly and therefore the input

and output fibers to the crystal are Polarization Maintaining Fiber (PMF)’s. Therefore,

it is important to align the input polarization into the EOM fiber properly to avoid hav-

ing large polarization fluctuations. One easy way to do this is that the notch on the fiber

connector normally shows the right linear input polarization. For a finer way to do this

see chapter A.


The microwave frequency to drive the EOM is typically set to be around 3100 MHz

and we lock to the -1 sideband so that the resultant light is close to F = 3→ F ′ manifold.

The injection-locked beam is split off into two portions to produce the coupling and the

signal beams.

Coupling

One portion of beam from the ILL goes through an AOM-SP set to +103MHz to produce

the coupling beam; see figure 4.1. The purpose of the AOM-SP here is to use it as a

switch for coupling light. Since this beam is addressing the ground state F = 3, which

is the trapping ground state, it can blow the atoms in the MOT away while they are

being loaded. Therefore, it is essential to be able to turn on the coupling beam only

after MOT loading is over. The coupling detuning can be set using the microwave (EOM

drive) frequency.

Signal

We use very short, on the order of tens of ns, signal pulses close to F = 3 → F ′ =

4 transition. This choice of pulse duration is to ensure that we have an appropriate

bandwidth for our signal pulses. For a given pulse energy, shorter pulses have larger

intensities and result in stronger nonlinear effect. However, in terms of the peak XPS it

is not useful to have them more broadband than the EIT windows we use. In order to

get the right frequency, the beam goes through an AOM-DP centred at +71 MHz and

an AOM-SP at +75 MHz; see figure 4.1. The AOM-DP provides the ability to set the

center detuning of the signal pulses. Using the amplitude-modulation of the AOM-SP

short pulses are created.


Half-waveplate

PolarizingDbeam-splitter

FiberDport-Dcollimator

SMF Single-modeDfiber

AOM

-1

0FromDtheDprobeAOMDDP

FromDtheDcouplingAOMDSP

SMFD(~20m)

Quarter-waveplateBeam-splitter

Figure 4.6: The 0 and -1 orders of the AOM are used to produce the reference andnear-resonance probe components for our frequency-domain interferometer.

4.3 Phase measurement

The nonlinear optical effect we are interested in is a probe refractive index change pro-

portional to the signal photon number. This change in the refractive index shows up as

a small phase shift on the probe. Therefore, we need to be able to measure the probe

phase very accurately. For a phase measurement one usually uses an interferometry

technique, the most common of which is the spatial (Mach-Zender) interferometer. How-

ever, spatial interferometers are usually difficult to stabilize. This is due to probe and

reference beams going through independent paths and interacting with different optics.

Using alternative degrees of freedom of the probe, polarization or frequency, instead of

the spatial- momentum- degree of freedom, has the advantage that most sources of fluc-

tuation are common-mode between the probe and the reference components. Here we

use a frequency-domain interferometer, also called beat-note interferometry, to measure

our probe phase accurately [133].


4.3.1 Probe interferometry

In order to do the frequency-domain interferometry, we need two co-propagating fre-

quency components2; one close to the atomic resonance and the other far off-resonance.

The near-resonance component can be affected by the atoms and the far-off component

can act as a reference. Any phase or amplitude change due to atoms will appear as phase

or amplitude change on the beating signal of the two frequency components.

The beam from the ML goes through an AOM-DP so that we can tune the probe

frequency. It is then sent to an AOM-SP driven at 100 MHz, the 0 and -1 orders are

combined on a beam-splitter; see figure 4.6. The -1 order produces the near-resonance

probe component and the 0th order acts as the reference component. The frequency

and amplitude of the AOM-DP affects both components and the AOM-SP amplitude is

used to set the near-resonance probe power independently. In this geometry, the absolute

detuning of the two-components with respect to the atomic transition can be varied while

the difference of the two is kept at 100 MHz. Therefore, we can detect and demodulate

the 100 MHz beating signal even when we scan the probe detuning.

4.3.2 Detection and demodulation

The probe beating signal is detected on a fast Avalanche Photo-diode (APD) - PerkinElmer,

250 MHz bandwidth- and is sent to the Agilent CXA Signal Analyzer (SA) for demod-

ulation to obtain In-phase Quadrature (I) and Out-of-phase Quadrature (Q). We use

the Basic I/Q Analyzer option of the SA to demodulate the beating signal at 100 MHz.

For the demodulation the internal 100 MHz reference of the SA has to be in-phase with

the RF signal that drives the probe AOM-SP. To achieve this, the internal clock of the

probe AOM-SP driver, Tektronix AFG3102, is locked to a 10 MHz RF reference from

2In fact, one can use phase- or amplitude modulation to create side-bands for beat-note interferometry.This would have the advantage that no spatial interferometer would be involved (unlike what we havehere) but also has the disadvantage that several side-bands can be produced and one needs to take theirdetunings from other transitions into account.


Fast

APD

Probe

Interferometer

Figure 4.7: The electronics for production, detection and demodulation of the probebeating signal.

the SA; see figure 4.7. The result of demodulation is an output from the SA of the two

quadratures, I and Q, of the beating signal. This data output is given as a function of

time with a sampling period of 2n/30 µs where n is an integer determined internally

based on the measurement bandwidth choice to satisfy the Nyquist limit.

The demodulation starts by multiplying the beating signal, B(t) = b sin(Ωt + φ(t)),

with the reference RF signal, R(t) = r sin(Ωt + φ0), where Ω/2π is the frequency of

the beat-note, 100 MHz, and φ(t) is the phase shift we are interested to extract. The

reference phase, φ0, can be set to 0 or 90 degrees to measure the in- or out of phase

quadratures, I and Q respectively. The resulting signal goes through a narrow band

filter, here usually set to 2 MHz, to give s = (rb/2) cos(φ(t) − φ0) where I = s(φ0 = 0)

and Q = s(φ0 = π/2). Here, the action of the narrow-band filter is approximated by

integration, and the dynamics of φ(t) is assumed to be slower than the response time of

the filter.

Assuming that the reference probe component remains unchanged, the two quadra-

tures can be used to obtain information about the phase shift and the amplitude change of


2 4 6 8 10 12 14

1

2

3

4

5

6

7

8

9

10

IF−BWefMHzN

Sig

nalep

ulse

eban

dwid

thef

MH

zN

Peakephaseeshift

2

4

6

8

10

12

14

16

2 4 6 8 10 12 14

1

2

3

4

5

6

7

8

9

10

IF−BWefMHzN

Sig

nalep

ulse

eban

dwid

thef

MH

zN

SNR

0.5

1

1.5

2

EITeHWHM

Figure 4.8: Simulated peak XPS (left) and SNR (right) versus IF-BW and signal pulsebandwidth.

the near-resonance probe component. The quadrature sum of I and Q gives the amplitude

information and taking the ArcTan of the ratio of the two gives the phase information.

Our phase measurement is equivalent to an 8-port homodyne detection [134, 135].

4.3.3 Measurement bandwidth

One of the steps in the demodulation is to send the RF signal, B(t)R(t), through a

narrow-band filter. The measurement bandwidth is set using the IF-BW setting of the

SA to allow for the largest possible ratio of signal-to-noise energy. This is achieved by

setting the bandwidth narrow enough to avoid excess noise but wide enough to transmit

all the frequency content of the XPS, φ(t). In most our measurements, the signal pulse

bandwidth is chosen to be broader than the EIT window to ensure we saturate the size of

the peak nonlinear phase shift; see chapter 2 for more details. Therefore, the narrowest

feature in the problem is the EIT window and that sets the optimum measurement

bandwidth.

Figure 4.8 shows a simulation of the expected variation of the peak XPS and the SNR

with respect to IF-BW and signal pulse bandwidth. The narrow-band filter is assumed

to be Gaussian with rms width of IF-BW. The signal pulse is also a Gaussian with the

rms widths shown on the vertical axis. The XPS has a time-response given in chapter 2;

the result of the filtering is modeled as the convolution of that transient with the filter


time- response. The noise is modelled as white and the integrated noise power allowed

through the filter is used to obtain the SNR. It can be clearly seen that the best SNR is

obtained when the measurement bandwidth is equal to the EIT window HWHM. This

is true regardless of the signal pulse bandwidth.

4.3.4 Data acquisition and analysis

We use National Instrument Labview to control the SA remotely and acquire data. The

SA receives a trigger from the atom cycle when the atoms are ready to be probed and are

expanding freely. Upon the arrival of the trigger the SA takes a trace with a predeter-

mined duration of 1.5 ms. For each sample point during the measurement time, values of

I and Q are obtained and sent to the Labview Virtual Instrument (VI) as an array. This

is repeated for a set number of repetitions and the resultant 2D array (rows: I and Q

values, and columns: iteration) is saved as a Comma-separated Values (CSV) file. This

file is then loaded to MATLAB for analysis.

We typically need to be handling very large datasets; therefore, to avoid memory

overflow we need to occasionally dump the collected data. This is done through (Labview

‘queue’ functionality) a separate loop that waits for the data to be available and then

saves it to the hard disk drive. Using parallel loops for data collection and dumping

ensures a high acquisition rate. Other measures that can improve the data collection and

transfer rate include:

• Refreshing the SA display requires resources but is not needed during the data

collection (turning off the display makes the measurements roughly 1.5 times faster).

• Transferring the data as Real32 is significantly faster than transferring ASCII.

• Using parallel loops for different tasks such as collection and dumping saves pro-

cessing time.


0 5 10 15 20 250

20

40

60

80

100

120

140

index

XP

S (

mra

d)

b1 b2s

Figure 4.9: Typically measured XPS time trace and corresponding regions for calculatingthe average and peak phase shift.

• Turning off the auto-alignment function of the SA, instead manually aligning the

SA daily.

• Data transfer over Ethernet is normally fast enough (1 Gb/s). A direct link between

the SA and computer will prevent speed reduction due to other devices using the

Ethernet router.

• Having the SA on Single (as opposed to continuous) Sweep increases the measure-

ment speed significantly.

The data files are loaded into MATLAB where I and Q are used to obtain probe phase

and amplitude information. The probe amplitude values in the presence and absence of

atoms is compared to obtain information about the optical density of the medium. We

use ArcTan(Q/I) to calculate the phase from the measured data. However, it is important

to note that ArcTan has a range of −π/2 to π/2 and proper phase “unwrapping” has to

find values of phase outside of this range; see appendix C for the details.

Another concern is a slowly varying phase shift resulting from the relative phase

between the two arms of the probe interferometer, see figure 4.6, that appears as a

background phase. This is normally a constant phase over the measurement time and


0 1 2 3 4 5 6 7 80

50

100

150

200

250

300

350

Measurement window (ms)

Acq

uisi

tion

time

per

trac

e (m

s)

trigger spacing = 6ms



linear fit (y=38x)

Figure 4.10: Acquisition time per trace versus the measurement window.

can be subtracted off. Based on precise timings of the signal pulses, we know the locations

of the XPS peaks inside each measurement trace; see figure 4.9. The value we report for

the nonlinear phase shift is obtained from s − (b1 + b2)/2 where s is the average value

of the probe phase shift for the points during the XPS, shown in green, and b1 and b2

are the same for points before and after the XPS, respectively, shown in red. Assuming

that the shape of the XPS is roughly Gaussian in time and the range for s covers the

full-width at half-maximum, there is a scaling factor of 2/√πerf(1) between the average

phase shift obtained here and the peak value of the nonlinear phase shift. This makes

our measurement insensitive to any noise or fluctuations which are slower than each

measurement shot (we define a shot as the period of time over which we send only one

signal pulse; each measurement trace contains several shots).

One of the most important limitations of the current apparatus is the refresh rate of

the SA for taking new traces. Most of this “dead time” is due to the slow SA processor

for the I/Q Analyzer. Figure 4.10 shows the amount of time it takes for the SA to

take one trace versus the duration of the trace. The characterization procedure is as

follows: we measured how long it takes, t1, to acquire 100 traces using a timer function


inside our acquisition VI; the contribution of the trigger spacing, ttrig to the overall

measurement time was subtracted off so that the values on the y-axis are: t1/100− ttrig.

In these measurements the sampling period was constant and a longer measurement

window means a larger amount of data to acquire and process. The acquisition time

scales linearly with the measurement window with a slope of 38 ms/ms which means that

our best possible measurement duty cycle cannot exceed 1/(38 + ttrig), on average. This

is a very significant bottle-neck in how fast we can collect data. Taking into account

the best atom duty cycles we have observed (roughly 30%), solving this problem can

improve our measurement rate by a factor of roughly 10. There are some discussions

about potential solutions in section 6.1.5.

4.3.5 Phase noise

The measurement phase noise depends on several factors. First, the quality of the EIT

window; a deep transparency means less probe absorption and therefore a larger beating

signal and smaller phase shot noise. Furthermore, a narrower EIT window means that

the nonlinear effect would last for longer time, and as a result, more noise can be averaged

out. Second, the frequency noise on the two-photon resonance becomes phase noise when

passing through the atoms; assuming roughly 10 kHz of frequency noise, an OD of 3,

and an EIT window of 2 MHz we will have around 15 mrad phase noise (these values are

chosen similar to our typical experimental ones).

The following are measured values of single-shot noise and estimates of where they

arise (EIT window: 2 MHz which results in an XPS that lasts for roughly 250 ns, 90-

degree geometry for probe and coupling, OD: 2, on-resonance probe power reaching the

detector (30% collection efficiency): 6 nW):

• Bypassing all the optical parts and demodulating the 100 MHz from the signal

generator directly: 2 mrad

This is the intrinsic demodulation noise. We cannot have a performance better


than this unless if we can elongate the effect.

• Using all optical parts while the atoms are off (by turning off magnetic field gradi-

ent): 25 mrad

This is the contribution of quantum noise and any phase noise contributed by

the probe detector and other electronics.

• Typical experimental situation (with atoms and all optics): 60 mrad

Based on this value, most of the noise in our measurement is contributed by

atom density fluctuations or laser frequency noise.

• The expected value of quantum noise is roughly 10 mrad.

Even if we can solve all technical noise, this is the minimum phase noise per

shot we can expect unless if the effect can last for longer.

Improvements can be made to decrease our phase noise and enhance performance.

Going to a narrower EIT window and higher optical density will make the effect last for

longer and therefore reduce the quantum noise along with other fast noise. This requires

improving our frequency noise performance (mostly limited by AOM drivers linewidth

and drift) and a copropagating geometry for EIT beams to obtain deeper transparencies.

A better EIT window also results in less probe absorption which in turn reduces quantum

noise.

4.3.6 Measurement and atom cycle

In order to probe the atoms we need to turn off all the MOT beams and the magnetic

field gradient. The MOT lasers would move the population around and disturb the

ground state coherence induced by EIT. Also, the magnetic field gradient would cause

a spatially-varying Zeeman shift which could act as a source of dephasing. Therefore,

we need a proper atomic and measurement cycle in which we prepare atoms, and let


Trap

Repumper

Magnetic5fieldgradient

Probe5and5coupling

Signal

Tagging

no5click click excluded

Cooling5and5trapping20ms

Freeexpansion1.5ms

Populationpreparation0.5ms

2.4us

Figure 4.11: Atom and measurement cycle.

them fall while they are being probed. The amplitude and frequencies of the laser fields

are controlled using AOM’s, and can be changed in tens of ns. Also, the magnetic filed

gradient can be switched in around 300 µs. We use a Bipolar Power Supply (BOP) to

drive the anti-Helmholtz coils. We needed to place a series resistor-capacitor set (67 Ω,

3 µF) in parallel to the output to reduce the switch-off time from 10 to 0.3 ms.

We use a cycle of re-cooling, population preparation and free expansion (probing). For

20 ms all the MOT beams and the magnetic field gradient are on and at the full strength.

Then we take 0.5 ms in which both the magnetic field gradient is turned off and the

population is prepared in the probe ground state. This is done by turning the repumper

off, decreasing the trapping optical power, and detuning it closer to F = 3 → F ′ = 3

excited state. At the end, for 1.5 ms, the trapping beam is also turned off and coupling

and probe are turned on as long pulses. During this measurement time, signal pulses are

sent into the medium every 2.4 µs, each pulse corresponding to one shot. If the there is a

corresponding single-photon detection, a tag is written in the subsequent shot which will

be excluded in the phase measurement because of the systematic phase dynamics caused


MOT

Signalcollection

Coupling

SM

F

Fiber:port-:collimator

SMF Single-mode:fiber

Probe:telescope

Tel

esco

pe

MMF

SPCM

Probecollection

Signal:telescope

Tel

esco

pe

MMF

FastAPD

OI

MMF Multi-mode:fiber

10:90:beam-splitter:(T:R)

OI Optical:isolator

Figure 4.12: Interaction region, and probe and signal collections.

by the tag spike.

4.4 Interaction region

After all the beams are prepared on the photon side, they are sent through long (around

20m) SMF’s to interact with the atoms . In the atoms side, the probe and signal beams

pass through telescopes that set their polarization and shape for focusing into the cloud;

see figure 4.12. Each of the two beams are then collected on a 10-90 BS in the opposite

telescope, re-collimated, and sent into respective detectors.

4.4.1 EIT

The coupling beam is taken out of the probe telescope before any beam shaping and

polarization adjustment. It is then collimated into a 500 µm beam. It takes a different


path and enters the cloud at almost 90 degrees relative to the probe and signal beams.

This geometry allows us to use π-polarization for the coupling beam. However, it has

the disadvantage that it is not a Doppler-free geometry [136]: there is a dephasing term

given by the motion of atoms as |(~kpr−~kcp) ·~v| where ~k’s are the probe and coupling wave

vectors and ~v is the average velocity of atoms. In a co-propagating geometry for probe

and coupling beams, this term is negligible but in a 90-degree or counter-propagating

geometry this is non-zero and gives rise to a limitation on the narrowest EIT windows

available; see section 6.1.8.

The narrowest EIT window we have measured is around 250 kHz in the co-propagating

geometry (the narrowest window in 90-degree geometry is roughly 1 MHz). This means

that the ground state dephasing rate is approximately 60 kHz in the co-propagating

geometry. The major source of this dephasing is the AOM drivers we use; see section 6.1.7

for potential solutions. After injection-locking we have to use two AOM’s to frequency-

tune the probe and the coupling beams; however, the RF output of each AOM driver

has a linewidth of around 10 kHz and drifts by several tens of kHz over seconds. The

poor quality of these drivers directly results in noise on the two-photon detuning. Other

potential sources of dephasing are the residual magnetic field gradient, the motion of

atoms and any noise added through injection-locking.

4.4.2 Signal collection and background photon counts

The signal beam is collected into a Multi-mode Fiber (MMF) and sent to a Single-photon

Counting Module (SPCM). This is used as a method to detect if the weak signal pulses

contain photons or not after they pass through the interaction region. We take several

precautions to ensure detecting signal photons rather than background ones from room

light, or scattering of probe and coupling beams, etc. The signal collection parts are,

therefore, mostly enclosed in a box made of dark acrylic sheets to block out the room

light as much as possible.


There is an OI near the probe collection port. A small portion of probe beam reflected

from the tip of the probe collection fiber makes it all the way back into the signal

collection. Therefore, we require the OI to block out any reflection from the probe

collection. Background counts due to the reflection of the probe off of all other optical

surfaces is negligible.

The major remaining source of background photons is the scattering of the probe and

coupling beams from atoms (these counts are only seen when all three - signal, coupling

and atoms - are present). In principle, a better spatial filtering- such as coupling the signal

into a SMF instead of MMF can reduce these counts significantly. This, unfortunately

cannot be done in the current telescope design because of a beam-clipping on the 10-90 BS

mounts that make the beam shape quite poor for coupling into single mode. Therefore,

we have to use time gating to minimize the number of detected scattered photons. For

potential alternative solutions see section 6.1.6.

Gating the SPCM

The SPCM can be gated using a TTL signal and has roughly 2µs of timing accuracy.

We are interested in single-photon detections that happen during the presence of the

signal pulse, normally tens of ns long. The built-in gating function of the SPCM lacks

the necessary timing accuracy and we have to use an additional gating arrangement to

filter out unwanted single photons.

4.4.3 Time-gating and Tagging

The goal of the tagging is to indicate shots that result in single-photon detections. This

is done by shining a bright flash of light on the probe detector which appears as a spike

in the amplitude of the probe. This spike has to be resolvable on a single-shot so that

we can deterministically decide whether a given shot corresponds to a single-photon

detection. It is, in principle, possible to do the tagging using electrical signals instead of


Signalcollection

SPCM

Probecollection

FastAPD

SignalOpulseswfromOtheOphotonOsidez

AND

Triggerinput

DGSG

AOMODriver

AOM0

W1

W2

TTL

HalfWwaveplatePolarizingObeamWsplitter

FiberOportWOcollimator

SG SignalOgenerator BeamWsplitter

DG DelayOgenerator

FastPD

Signaltelescope

QuarterWwaveplate

NDOfilter

100OMHz

SW

SW RFOswitch

ControlOinput

Figure 4.13: The tagging procedure.

the flash of light. However, most of the methods we tried led to significant excess noise

and systematics that adversely affect our phase measurement adversely. An advantage of

the optical tagging method we use here is that the components that make up the tagging

setup are electrically decoupled from the rest of the setup. It is also important to note

that this flash of light must have a components at 100 MHz, or the phase measurement

is insensitive to its presence.

Figure 4.13 shows the details of the tagging procedure. Short signal pulses produced

on the photon side get split on a PBS where most of the light hits a fast detector. This

150 MHz PD turns the optical pulses into electrical ones (gate pulses) which trigger a

time-gating circuit. A smaller portion of the light is coupled into single-mode fiber which

goes to the signal telescope. To set the incident signal power we use an APD which can

measure powers down to nano-watt level. Since we normally need to operate at the pico-

watt level so that each ≈ 100 ns signal pulse contains a few photons, we set the signal

power using our APD and then use an ND filter with a calibrated attenuation of 610±10.

Few-photon-level signal pulses pass through the interaction region and get collected into

a MMF and are eventually detected on a SPCM. The SPCM sends out a TTL pulse

(height: 3.6V, duration: 18ns) in response to each single-photon detection. A length of

cable on the output of the SPCM and a signal generator (used as a delay generator) on

the output of the fast PD are used to synchronize the TTL and gate pulses. Then, a


Figure 4.14: The signal (right) and probe (left) telescopes.

logical AND operation is done on the TTL and gate pulses to reject any single-photon

detection that does not happen within the signal pulse duration.

A single-photon detection during a signal pulse results in a short pulse at the output

of the AND gate. This pulse can be delayed using another delay generator to adjust

the location of the tag within a measurement trace. The output of the delay generator

gates an RF switch that turns on the output of an AOM driver driven at 100 MHz for a

duration of 200 ns. A SMF is aligned so that the -1 and -2 diffraction orders are coupled

into the fiber and then are shone directly on the probe detector. This causes a bright

enough flash of light with a 100 MHz component on the probe detector. The tagging

100 MHz does not have a phase relation with the probe 100 MHz therefore the size of

the tags vary randomly. However, the tags are adjusted to be large enough so that they

always remain positive. Also, although the tagging beam does not pass through atoms,

the light used for it is far from all transitions.


4.4.4 Probe and signal telescopes

The probe and signal beams enter the atom cloud through two telescopes, this allows

for control over polarization and focusing; see figure 4.14. Light enters both telescopes

from SMF’s and using a fiber-to-free-space adapter, they are collimated with 5cm doublet

achromatic lenses into roughly 1cm beams. Polarizers and wave-plates are used to set

polarizations, 10-90 BS’s for collection, 20cm achromatic doublet lenses to focus down

the beam into a 13 µm spot size, and mirrors to steer the beams. The two telescopes

face each other and are aligned together so that each beam is collected in the opposite

telescope. Therefore, the 20 cm lenses, other than focusing down the beam in their own

telescope, play a role in the collection of the other beam by re-collimating it. Two other

mirrors are used, outside the telescopes, to control enough degrees of freedom for aligning

the telescopes to each other and to the cloud.

The polarizer in the probe telescope is a PBS used to separate the coupling and probe

beams. The coupling beam is re-collimated and sent to the cloud at roughly 90 degrees

relative to the probe and signal beams. The alignment of the coupling to the cloud is

more straightforward because the beam-waist is wider and furthermore, it can also be

fine-tuned by maximizing the width of the EIT window.

There is an overall 40% loss in the collection parts from the interaction region to

right after the 10-90 BS: roughly 10 − 12% loss is due to the transmission of the BS;

there is roughly an overall loss of 7% due to all other optics (lens, wave-plates, etc.)

and the remaining loss is because of a beam-clipping on the 10-90 BS mount. The

overall collection efficiency, from the interaction region into fiber is normally around 30%

(including reflections from cuvet walls, and losses due to all collection optics).

See section 6.1.9 for possible improvements of the telescope design.


4.4.5 Focus size

The beam waist inside the cloud is 13±1 µm which corresponds to a (two-sided) Rayleigh

range of roughly 1.4 mm. The choice of this focus size is to ensure that the Rayleigh

range of the beams matches the size of the cloud. Focusing the beams tighter would

not be helpful because it reduces the interaction length and increases the probe phase

shot noise (the intensity of the probe is limited and fixed by the saturation intensity of

atoms). Also, focusing the beams any less tightly would produce smaller intensity for a

given signal pulse energy and makes the nonlinear effect smaller.

In an optically thin medium, the XPS is proportional to the atoms density, ρ, the

effective interaction length (the smaller of the size of the cloud or the beam Rayleigh

range), and the beam area (proportional to the beam waist squared, w20). If the Rayleigh

range of the beam is equal to the size of the cloud, the XPS is only proportional to

the atom density. Therefore, increasing the atom density can improve the size of the

nonlinear effect up to the point where the medium can not be considered optically thin,

OD more than roughly 2. For a high enough atom density the peak phase shift saturates

due to the group velocity mismatch; see chapter 2. In order to take advantage of higher

atom densities, we need to use a smaller cloud so that the shorter interaction length

allows for a higher atom density (ρ ∝ w−20 ) before saturating the limit posed by group

velocity mismatch. On the other hand, focusing tighter has the disadvantage of increased

shot noise. The shot-noise limited SNR scales as w−10 ; this means that it is beneficial

to use a cloud as small and dense as possible, and focus to match the Rayleigh range

of the beam to the size of the cloud. For a discussion of the benefits of focusing tighter

and using a more dense atom cloud to achieve the largest possible XPS in N-scheme, see

section 6.1.14.


F=2,C-0.47MHz/G

F=3,C0.47CMHz/G

F'=1,C1.4CMHz/G

F'=2,C0.16CMHz/G

F'=3,C0.54CMHz/G

F'=4,C0.7MHz/G

Probe

Signal

Signal

Coupling

29M

Hz

63M

Hz

120M

Hz

4MH

z/G

Figure 4.15: Implementation of N-scheme in 85Rb atoms.

4.4.6 Level scheme

Figure 4.15 shows our implementation of N-scheme in 85Rb. This choice of level structure

and polarizations is to ensure having a polarization-dependent nonlinear effect to eventu-

ally implement our weak measurement proposal [112]. Ideally one would optically pump

the population and use an external Direct Current (DC) magnetic field to implement the

scheme as shown in the figure. However, it turns out that even without optical pumping

and DC magnetic field, there is a significant polarization dependence; the main reason

for this behavior is that the polarizations of the probe and coupling break the symmetry

for left- or right-circularly polarized signal fields.

Taking equation 2.4 into account, the XPS is proportional to d2sd

2p/d

2c where di’s are


transition dipole moments for signal, probe and coupling. For a multi-level system like

the one shown in figure 4.15 the contribution of all sub-structures have to be averaged

weighted by the population of each ground state.

φ ∝∑

i

pid2s,id

2p,i/d

2c,i (4.1)

where pi is the probability of the population being in the probe ground state of the i-th

N-substructure. For example, if the population is optically pumped into F = 2, mF =

+2 ground state, then only one N-substructure is involved in the nonlinear interaction.

However, if the population is equally distributed then there are four substructures that

can contribute. It is straightforward to estimate the strength of the nonlinear effect in

different scenarios using the Clebsch-Gordan Coefficient (CGC)’s. For equally distributed

population among ground states, the ratio of the strength of the interaction for σ+, φ+,

and σ−, φ−, signal polarizations is equal to

φ+

φ−=

1/2 · 2/9 · 24/5 + 3/8 · 4/27 · 54/5 + 15/56 · 2/45 · 216/5 + 3/28 · 2/135 · 216/5

1/56 · 2/9 · 24/5 + 3/56 · 4/27 · 54/5 + 3/28 · 2/45 · 216/5 + 15/56 · 2/135 · 216/5

= 3.56 (4.2)

For a 100% pumping to F = 2, mF = +2 this ratio is 28. The values of CGC’s for this

calculation are taken from [126].

In this section we have assumed low optical density and low probe power. However,

in the limit of high OD, the peak XPS is only proportional to the dipole moment of the

signal transition, d2s; see equation 2.10. This is a result of the group velocity mismatch

that the peak phase shift is independent of the background optical depth ∝ d2p, and the

width of the EIT window ∝ d2c . For the above example the ratio is 2.8 in situations of

high OD. On the other hand, finite probe power along with the fact that the polarization

of the probe is σ+ means that equal populations in all ground states is not a very good


assumption. In fact the probe can optically pump the population to higher mF states and

this in turn results in a larger contrast between φ+ and φ−; we experimentally measure

a ratio of 6 for φ+/φ−.

For the purposes of this thesis, the signal is set to have the same circular polarization,

σ+, as the probe to achieve the strongest nonlinear optical interaction. As shown in the

following chapter, this interaction is used to demonstrate the effect of a single post-

selected photon on a pulse containing thousands of photons.

See chapter 2 for the limitations of the current level scheme and chapter 1 for alter-

native level structures.

Chapter 5

Results


5.1 Overview

In this chapter some calibration and the main results of this thesis are presented. As one

of our calibration steps, we observe the nonlinear phase shift due to pulses containing on

average as few as one photon. This is the lowest energy (around 250 zepto-joule) ever

used for cross-phase modulation in free space. More importantly, we have demonstrated

that by illuminating a sample of atoms with a weak coherent state, but post-selecting

on subsequent detection of a photon at the far side of the sample, we can observe the

nonlinear effect of that one additional photon on a probe beam. We report the observation

of a 18 microradian per photon nonlinear phase shift per individual post-selected photon.

This represents the first direct measurement of the cross-phase shift due to single freely

propagating photons.

74

Chapter 5. Results 75

100

101

102

103

104

105

106

100

101

102

103

104

105

106

Average photon number per pulse

XP

S (

µra

d)

Experimental data

Linear fit

30 20 10 0 10 20 302

0

2

Probe detuning (MHz)

Pro

be p

hase s

hift

(rad)

30 20 10 0 10 20 300

2

Optical density

Figure 5.1: XPS versus average photon number per pulse. The nonlinear phase shiftdepends linearly on the photon number at lower intensities. A fit to the low-photon-number data yields a slope of 13 ± 1 µrad per photon while the deviation at higherphoton numbers arises due to higher-order nonlinearities. The inset shows a typicallinear phase profile (green) and optical density (red) as seen by the probe with the arrowindicating where the on-resonance component of the probe laser is locked. Other relevantparameters include signal center detuning = −10 MHz, OD = 2, EIT widow width = 2MHz.

5.2 Calibration

5.2.1 Cross-phase shift versus signal photon number

As a first calibration, we measure the value of the nonlinear phase shift per signal photon

using classical pulses. In order to obtain that, we send signal pulses with given energies,

and therefore average photon numbers, into the interaction medium and measure the

corresponding phase shift of the probe. The temporal profile of signal pulses are measured

on an APD and by integrating the pulse power with respect to time the average energy,

and therefore the photon number, of each signal pulse is inferred.

Figure 5.1 shows the results of the measurement of the nonlinear phase shift of the

probe versus average signal photon number per pulse. The XPS is linear in low photon

numbers and the slope of a fit to the phase shift versus signal photon numbers gives


13 ± 1 µrad per photon. The lowest energy per pulse that we use here corresponds to

an average of one photon per pulse which is the lowest energy ever used for cross-phase

modulation in free space. The nonlinear phase shift saturates for large photon numbers

where the ACS causes a large detuning comparable to or larger than the half-width of

the EIT window.

5.2.2 Cross-phase shift versus signal detuning

In this section we study the dependence of the nonlinear effect on the signal detuning,

∆s. Since ACS caused by the signal is the working principle of the effect, for small

enough signal amplitudes the dependence of the ACS has to carry on to the nonlinear

phase shift. In the limit of weak signal pulses and probe field, the ACS versus detuning

has a dispersion-like scaling that goes to zero on resonance and reaches extrema at half

linewidth

∆ACS =−|Ωs|2∆s

∆2s + (Γ/2)2

(5.1)

where Ωs is the signal Rabi frequency, and ∆s is its detuning. The quantity |Ωs|2 is

proportional to the intensity and therefore the photon number of the signal.

There is another contribution to the detuning-dependence of the nonlinear phase shift

which is a result of the finite signal absorption due to population of the coupling ground

state, F = 3; see figure 5.2. The population of each ground state, in an EIT system, is

determined by the ratio of the coupling and probe intensities. Signal absorption due to

this non-zero population leads to an effective photon number in the interaction region of

Neff = N01− e−ds(∆s)

ds(∆s)(5.2)

where ds(∆s) = d0Γ2/(4∆2s + Γ2) is the optical density of the signal transition at the

detuning ∆s and N0 is the incident photon number. Here d0 is the on-resonance optical


F=2

F=3

F'=3

F'=4

+ +

Δ

Pr Cp

Sg

Figure 5.2: The level scheme used to observe cross-phase modulation using the D2 lineof 85Rb atoms. The ac-Stark shift due to the signal pulses, pulls the probe out ofEIT conditions and this appears as a refractive index change proportional to the signalintensity.

density. The nonlinear phase shift is, therefore, proportional to the product of these two

contribution,

φ(∆s) = −2φm∆sΓ/2

∆2s + (Γ/2)2

1− e−ds(∆s)

ds(∆s), (5.3)

where φm is a proportionality constant and Γ = 6 MHz is the excited state linewidth. The

detuning dependence is measured for two different probe intensities keeping the coupling

intensity fixed. It can be seen that in the case of higher probe power the nonlinear phase

shift is smaller because of the higher signal absorption.

Figure 5.3 shows the scaling of the measured nonlinear phase shift with signal detun-

ing. There are also fits based on all these contributions. The fit parameters are d0 = 4±2,

φm = 500± 100 µrad for low probe power, and d0 = 5± 2, φm = 300± 40 µrad for high

probe power.


−20 −15 −10 −5 0 5 10 15 20−300

−200

−100

0

100

200

300

400

Signal detuning (MHz)

XP

S (

µrad

)

High probe powerLow probe power

Figure 5.3: XPS vs signal detuning. The nonlinear phase shift is caused by ACS due tothe signal pulses. Therefore, it has the same dependence on signal detuning as the ACS.This scaling also depends on probe power because more probe power results in a largerpopulation in F = 3 ground state which means a larger signal absorption. The overalleffect is broadening and smearing of the dispersion-like scaling at higher probe powers.

5.3 Inferred photon number in the interaction region

Our main goal is to observe the XPS on the probe due to a single signal photon. Although

in data presented in section 5.2.1, we have demonstrated XPS due to signal pulses with

average photon number of 1, these pulses still have ∼ 40% chance of having no photon

and ∼ 25% chance of having more than one photon. To make sure that the observed XPS

is not due to more than one photon, the energy of signal pulses can be set to even lower

values which substantially reduce the chance of having more than one photon. When

we send these weak signal pulses to a SPCM, we rarely detect a single photon, a ‘click’.

However, the occasional single-photon detection means that there has been one photon in

the interaction region. Because of the finite detection efficiency, having no clicks means

that the number of photons in the interaction region could have been non-zero. But, due


0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

AverageAphotonAnumberAinAinteractionAregion

infe

rred

Apho

tonA

num

ber

nph,no

nph,no

+1

nph,yes

ContributionAofAhigherphotonAnumbersAtoAclicks

ContributionAofAbackgroundphotonsAtoAclicks

1Aclick

2Aclicks

3Aclicks

4Aclicks

noAclick

2527

13

7

Figure 5.4: Inferred, ninf , versus average photon number in the interaction region. Theoverall collection efficiency is assumed to be 20% and the background click rate is takento be 10% for solid and dotted lines. The circles show the photon number values inferredfor the data points in figure 5.5 for no-click (red) and click (blue) events. The overallefficiency percentage for each data point (numbers beside circles) is slightly differentwhich accounts for the discrepancies between the data points and the solid curves. Theaverage photon number in the interaction region for the data points is lower than theincident photon number because of the finite signal absorption. The dotted green linesshow the photon number which would be inferred, were a number-resolving detector used.The solid blue line could also be obtained from a weighted average of the dotted lineswith non-zero number of clicks.

to the Poisson statistics of the incident photon numbers, having a click always means

that at least one extra photon was present in the interaction. Therefore, the average

probe phase shift in the cases that the detector clicks is expected to be larger than that

of no-click cases by exactly a single-photon XPS.

In order to calculate the most probable number of signal photons interacting with the

probe, one can use Baysian inference to obtain the conditional probability of having nph

photons in case of a no-click- (no) or a click event (yes),


P (nph|no) =1

N0

P (no|nph)P (nph) = e−(1−η)|α|2 |α|2nph

nph!(1− η)nph

P (nph|yes) =1

N1

P (yes|nph)P (nph) =e−|α|

2

1− e−η|α|2|α|2nph

nph!(1− (1− η)nph) (5.4)

where N0 and N1 are normalization factors; η is the overall collection and detection

efficiency; P (nph) = exp(−|α|2)|α|2nph/nph! is the incident photon number distribution

with average of |α|2. Using the conditional probabilities given above, one can calculate

the average photon number in the interaction region in each case,

nph,no = |α|2(1− η),

nph,yes = |α|2(

1 + ηe−η|α|

2

1− e−η|α|2)

= nph,no +η|α|2

Psig(yes), (5.5)

where Psig(yes) = 1 − Psig(no) = 1 − e−η|α|2

is the probability of a detection event.

For low count rates, η|α|2 1, the expression above reduces to nph,yes ≈ nph,no + 1;

the difference in the inferred photon number between click and no-click events is unity,

independent of both the detection efficiency and the average incident photon number. It is

straightforward to include the effect of background photons: Psig(no)→ Psig(no)Pbkg(no)

and Psig(yes)→ 1−Psig(no)Pbkg(no) where Pbkg(no) is the probability of getting no clicks

from background photons. Figure 5.4 plots the inferred versus the average photon number

in the interaction region, |α|2, for click and no-click events. It also shows the inferred

photon number if our detector had the ability to resolve the number of photons, for

comparison.


5.3.1 Classical intensity fluctuations

The results of this section so far assumed that there is no number uncertainty on top

of the quantum fluctuations. Therefore the assumption of Poisson distribution for the

incident light has been appropriate. However, any intensity fluctuation modifies the

incident light statistics and in turn changes the inferred intensities (or photon numbers).

Assume that the intensity of the incident light, I, has a Gaussian distribution centred at

I0 with and rms width of σI . Then, the conditional probabilities for click and no-click

events are

P (I|no) =1

N ′0P (no|I)P (I) =

1√2πσI

1− ζI1− ζI0

exp(−(I − I0)2

2σ2I

)

P (I|yes) =1

N ′1P (yes|I)P (I) =

1√2πσI

I

I0

exp(−(I − I0)2

2σ2I

) (5.6)

where N ′0 and N ′1 are normalization factors, and P (yes|I) = 1−P (no|I) = ζI is Fermi’s

golden rule (assuming small ζI). Using these conditional probabilities, the average in-

tensities for the cases of click and no-click are,

Ino = I0 +O(σ2I

I20

),

Iyes = I0 +σII0

, (5.7)

Therefore, a click means a higher intensity (photon number) compared to a no-click case.

The amount by which the inferred intensity for click events is higher, to the first order

in σI/I0, depends on the rms width of the incident intensity distribution.

A similar approach can be used to study various incident intensity distributions and

similar results are expected. In a general sense, the higher intensity (photon number)

for click events is a consequence of the electromagnetic field fluctuations, the minimum


−80

−60

−40

−20

0

20

40

60

80

100

XP

S.,

µra

d:

clicksno−clicks

y18.MHz5.Photons

v18.MHz0.5.Photons

v18.MHz1.Photon

v18.MHz2.Photons

v18.MHz5.Photons

y300.MHz5.Photons

no.signalpulses

atoms.offbsignal.pulses.on

Signal.detuning:Average.incident.photon.number:

checks.for.systematics40.ns.signal.pulse

100.ns.signal.pulse

−6 −4 −2 0 2 4 6−100

−80

−60

−40

−20

0

20

40

60

80

100

XP

S.,

µra

d:

(2 × 18MHz)/Δ

Figure 5.5: Post-selected single-photon XPS. Most notably, for an average incidentphoton number of 0.5 (green-shaded region), the XPS for no-click and click events are2± 3 and −13± 6 µrad, respectively, which definitively shows the effect of a single post-selected photon. For the other data points, the average incident photon number and/orthe signal center detuning is varied. Taking all the data points together, the magnitudeof the post-selected single-photon XPS is −18±4µrad. The inset shows the post-selectedXPS versus ninf (2π×18MHz)/|∆s|, inferred photon number corrected for the sign of thesignal detuning. The solid line has a slope of −14 ± 1 µrad per photon. Other relevantparameters include EIT window = 2 MHz and OD = 3. The data in the region shadedin blue are tests for systematics as explained in the text.

of which is given by quantum fluctuations corresponding to one photon. The extra

fluctuations can be caused by the intensity variation of the signal light source or as a

result of the interaction. However, for the remainder of this thesis I assume that those

contributions are negligible.


5.4 Cross-phase shift due to a post-selected single

photon

In our measurement we use a flash of light to reliably tag the shots when the detection

of the signal pulses resulted in a single-photon detection. Therefore, we can measure

the XPS in cases of no-click and click separately, see figure 5.5. For 0.5 incident signal

photons per pulse the inferred photon number in the interaction region for no-click events

is around 0.3 and for click events is roughly 1; see the square shaded in green. The phase

shift measured for no-click cases is unresolvable from zero, 2±3 µrad, and the click events

result in a non-zero phase shift of −13 ± 6 µrad. For other data points, the average

incident signal photon number and/or centre detuning of signal pulses are varied. The

most significant feature is that the XPS for click events are larger in the absolute size

than in no-click cases. The magnitude of the phase difference between click and no-click

events averaged over all data points, the nonlinear phase shift due to a post-selected

single photon, is −18± 4 µrad.

In order to confirm that the effect observed is not due to systematics, we have also

taken data in the absence of signal pulses or atoms, and with large signal detuning.

The data in the region shaded in blue displays these checks for systematics. The most

important feature to highlight is that the click- and no-click probe phase shifts for all

systematic checks are equal within error-bars.

Finally, the inset of figure 5.5 plots all the post-selected data versus inferred photon

number corrected for the sign of the detuning. The solid line is a parameter-free fit and

has a slope of −14± 1 µrad per photon. This slope is inferred from the fit in figure 5.1,

the optical density, and the detuning dependence of the nonlinear effect. We see excellent

agreement between the data presented here and the value of XPS per photon extracted

from figure 5.1.


5.5 Technical details

Background photon detection rate. In order to reduce the background photon rate for

cases of incident average photon number of 0.5 and 1, the signal pulse duration is chosen

to be 40 ns while it is 100 ns for the rest of the data. One would expect this change

to make the signal pulses more intense and therefore make the nonlinear phase shift

more than twice as big. However, as we shown in chapter 2, because of the bandwidth

mismatch of the signal pulses and the EIT window (2 MHz here), the change in the size

of the nonlinear phase shift is expected to be close to 1.5. Therefore, given the size of

our phase measurement uncertainty, the two values are expected not to be statistically

different. The background photon detection percentage is 13% and 6% for 100ns and

40ns signal pulses, respectively.

For the data presented here, the inferred photon number difference for cases of click

versus no-click is slightly different from one. Contribution of background count rate,

higher photon numbers and detection efficiency makes the difference non-ideal. However,

we can precisely estimate the inferred photon number for the situations of each data

point. We use these values to evaluate the inferred photon number difference in click-

and no-click cases. In order to obtain the nonlinear phase shift due to a post-selected

single photon, the values of phase difference are normalized to the values of photon

number difference.

Data collection. For each data point, we took approximately 300 million shots over

14 hours, 90 million of which resulted in clicks at the SPCM. Because of our tagging

procedure 90 million shots were discarded, and out of the remaining shots we observed 60

million click events and 150 million no-click events. The overall collection and detection

efficiency is around 20%, and for higher incident photon numbers more attenuation is

added to keep the click percentage to be around 20%− 30%.

Averaging the collected data. In order to ensure fair sampling of noise, and also since

each data point required a very long collection run, the data was taken in smaller portions


in a random order and was eventually averaged. However, duration of each short run was

different, mostly limited by the stability of the apparatus, and therefore the errorbar on

the average XPS for each was different. One would expect that data runs with a smaller

error should have a larger weight in the overall average. Suppose that for a data point,

N short runs are done and the ith one yielded xi ± σi. Then the average of the whole

collected data was defined as,

x =N∑

i=1

xiσ2i

/

N∑

i=1

1

σ2i

(5.8)

and the error in the mean is,

∆x =

(N∑

i=1

1

σ2i

)−1/2

(5.9)

This weighted average ensures that the fit of a constant value, x, to the N data points

has a minimum χ2.

5.6 Discussion of results

Previously, we showed theoretically that the temporal profile of the XPS expected for a

single-photon Gaussian signal pulse with an rms duration of τs interacting with an EIT

medium with a response time of τ is given by [71]

φ(t) =φ0

2τeτ

2s /2τ

2

exp(−t/τ)(

1 + erf(t/√

2τs − τs/√

2τ))

(5.10)

where erf(x) = 2/√π∫ x

0dx′ exp(−x′2) is the error function, and

φ0 =Γ

−4∆s

σatπw2

0

d

∆EIT

(5.11)


is the integrated XPS per signal photon. Here, Γ is the excited state linewidth, ∆s is the

signal detuning, σat is the atomic cross section as for the signal, w0 is the beam waist, d

is change in the optical density for the on-resonance probe without and with a resonant

coupling beam, and ∆EIT is the full-width at half-maximum of the transparency. For our

experimental parameters (Γ = 2π × 6 MHz, |∆s| = 2π × 18 MHz, A/σat = 3000, d = 2,

∆EIT = 2π × 2 MHz, τs = 40 ns, and τ = 250 ns) the XPS has a peak equal to 13 µrad.

The maximum achievable cross-phase shift per photon in the N-scheme, due to group

velocity mismatch issues, is

|φmax| =Γ

4|∆s|σatA, (5.12)

as shown by Harris and Hau [37]. For the parameters of our experiment this value is

28 µrad. The atomic cross section is calculated assuming equal population in ground state

Zeeman sub-levels and taking the level scheme in section 4.4.6 into account. The values

of CGC’s and dipole moments are taken from [126]. The phase shift we measure is lower

than this value because our optical density is not high enough to saturate the limit posed

by group velocity mismatch. We are, however, almost at the point where the propagation

effects start to play role and the peak phase shift deviates from linear scaling with optical

density. Another reason for smaller phase shift is the imperfect spatial overlap of the

probe and signal beams. Also, the probe power in our experiment is high enough that

the population in the signal ground state cannot be ignored. Therefore, there is some

signal absorption that causes the effective photon number in the interaction region to be

lower. For potential ways to improve the size and the detectability of the nonlinear phase

shift see chapter 6 (section 6.1.14 reports the maximum expected XPS in this system).

Chapter 6

Summary and outlook


This chapter provides an overview of the current experimental limitations of the

apparatus and provides recommendations and references to relevant literature for im-

provements. These improvements help enhance the ease of operation, measurement rate

and phase noise performance for future light-matter interaction experiments. Finally, I

summarize the achievements of this thesis.

6.1 Possible improvements

6.1.1 Location of the setup

Long data runs are required ro achieve good signal-to-noise ratio. Over each run several

parameters must be monitored and controlled. Limitations arise due to the setup span-

ning both the photon and atom sides (two physically separate labs). An upgrade we did

to the MOT setup (removing unnecessary optics, making the setup more compact and

fiber-coupling the MOT beams), has created space on the MOT optical table. Lasers and

optics from the photon side could be moved to the atoms lab. Removing the need for

long fibers and cables, in addition to localizing the apparatus to one room would make

87

Chapter 6. Summary and outlook 88

adjustments and monitoring significantly simpler. We note that the narrow-band single

photon source would remain on the photon side, nonetheless, having the rest of the setup

in one place would be beneficial.

6.1.2 Brighter single-photon source

The best heralded single-photon rate we can get from our source [122] is currently around

100 per second. In our post-selected XPS data we had to take 70 million shots to reach a

phase noise of 6 µrad (single-shot phase noise of 50 mrad); see section 5.5 for more details.

Assuming that we make no improvements to the apparatus, observing a resolvable XPS

due to real single-photons from our source would take roughly 8 days of continuous mea-

surement. Although not completely impossible, this is practically extremely challenging.

Some of the improvements suggested in this section can help make this measurement

easier by increasing the SNR, however, a brighter source of atom-compatible single pho-

tons can increase the measurement rate. Alternative single-photon source geometries

[137, 138, 139, 140, 141] should be explored before deciding the best future approach.

6.1.3 Higher optical density

Because of the limitations caused by group velocity mismatch in N-scheme, we do not

use a very high optical density in our measurements. We can use a different level scheme

in which the group velocity mismatch is less pronounced; see chapter 1 for more details.

In such cases we would have the benefits of a higher OD. A normal MOT setup is limited

in density by the radiation pressure due to the scattering of light from the atoms in the

inner parts of the cloud. In order to solve this problem a spatial or temporal dark-SPOT

MOT [142, 143] could be used along with compression periods [144, 145] to reach high

optical densities for short time intervals [146, 147]. Another useful approach is to use a

2D (cigar-shaped) MOT, as apposed to the current spherical one, in order to increase

the interaction length. This could be done using rectangular anti-Helmholtz coils [148],


using QUIC Ioff coils configuration [149], or using an optical trap [150].

6.1.4 Lower probe power

Currently, in order to optimize our phase measurement SNR, we need to use probe powers

comparable to saturation (3-6 nW on- and 55-60 nW off-resonance reaching the detector);

c.f. [133] and see 4.2.1 for more details. Having a high on-resonance probe power means

that there is more population in the coupling ground state and therefore more signal

absorption. It also limits the narrowest achievable EIT window width (the pumping rate

is in fact given by the quadrature sum of the Rabi frequencies of the probe and coupling

if the probe intensity is not negligible). Another technical problem that high probe power

causes is the photon scattering from the atoms that makes it to the signal collection.

It is important to be able to lower the probe power while maintaining or even im-

proving the SNR. One obvious solution is to lower the on-resonance and increase the

reference power. However, with our current detector, we have noticed that the phase

noise increases with a larger off-resonance probe power. The reason for this is still un-

known and needs to be studied in more details. However, one potential solution could

be use of a more sensitive detector which allows for lower on- and off-resonance probe

powers while maintaining the beating signal size.

6.1.5 Measurement rate (slow SA processor)

Our measurement rate is currently limited by the SA refresh rate; see section 4.3.4 for

details. This limitation is due to the slow processor speed of the SA when using the

I/Q analyzer option. Although a convenient solution, this is not the most efficient way

to carry out the phase measurement. Essentially, we require an I/Q demodulator and a

fast digitizer to record the data; a fast enough digitizer would significantly enhance our

measurement rate. this would result in making more accurate measurements over shorter

time intervals. A stand-alone digitizer would also have the advantage of multi-channel


operation with a common clock. One of the channels could be used to record the TTL

pulses coming from the single-photon detection events on the SPCM. This would remove

the need for optical tagging and save roughly 30% of the data that is discarded due to

the current tagging procedure.

Once we solve this issue, our measurement (using classical pulses as the signal) will

be limited by the atom duty cycle. This cycle could be optimized to achieve the best

optical density and fastest measurement rate. The best atom duty cycle that we have

observed, with an OD of roughly 2, is approximately 30%.

6.1.6 Background photon rate

We have background photon counts due to scattering of probe and coupling light from

atoms; see section 4.4.2 for more details. These counts deteriorate our ability to infer

the presence of a single-photon in the interaction region. Our time gating method (using

AND gate) reduces the contamination due to these counts significantly. However, these

counts are still preventing us from going to lower average incident photon numbers. These

photons are not polarized and therefore a polarization-filtering would not be helpful. The

best solution to reduce them is spatial filtering, which could be done through collecting

the signal light into a single-mode fiber. Currently, the clipping of the signal beam on the

10-90 BS distorts the shape of the beam and prevents acceptable coupling efficiencies.

Once we replace the plate BS with a cube one, we should be able to couple more efficiently

into single-mode fiber and reduce the background counts significantly. Use of a better

probe detector so that we can go to lower probe powers can also help reduce the number

of these counts.

6.1.7 AOM drivers

We use injection-locking to produce phase-locked probe and coupling fields. In order to

control the frequency and amplitude of these beams we use AOM’s. The phase noise


added through these devices is a major source of ground-state dephasing in our system;

see section 4.4.1 for more details. The problem is mainly caused by the oscillator used

in the drivers. The RF output of these oscillators has a linewidth of roughly 10 kHz and

drifts by tens of kHz over seconds. A better Voltage-controlled Oscillator (VCO) would

help reduce the linewidth, although not the drift. Given the typical dynamic range that

we require for these drivers, ≈70 MHz, it is difficult to achieve stabilities better than tens

kHz.

One potential solution is to use a Direct Digital Synthesizer (DDS) as the oscillator.

DDS is a frequency synthesizer that has excellent stability, phase noise performance and

dynamic range. Using a DDS along with the RF amplifiers currently used in our drivers

would help reduce our two-photon noise and achieve narrower EIT windows.

6.1.8 Copropagating geometry for probe and coupling

In order to have a polarization-dependent effect, we used a 90-degree geometry for our

probe and coupling beams; see sections 4.4.1 and 4.4.6 for details. This is not a Doppler-

free geometry and therefore results in extra ground-state dephasing [136]. For any appli-

cation that does not require polarization-dependence a copropagating geometry should

be used to obtain narrower EIT windows. However, it is important to remember that

in a copropagating geometry, due to different polarization of the beams, different N-

substructures will be involved and the peak achievable value of XPS might differ; see

section 4.4.6.

6.1.9 Telescope re-design

The current telescope design uses plate beam-splitters, the requirement for placement at

45 degrees leads to clipping of both the transmitted and reflected beams; see section 4.4.4

for details. The clipping leads to extra collection loss and distorts the beam shape; this

reduces the efficiency of coupling into single-mode fiber. Replacing these beam-splitters


with cube ones would help solve this problem.

Another improvement to the telescopes design would be use of a more sophisticated

focusing arrangement. In order to be able to focus to a tighter spot size, we need a design

with better aberration performance. In our current design, rings are apparent in the far

field due to spherical aberration. One could design a system of lenses (rather than only

one lens) for focusing with less aberration and the ability to focus to tighter spot.

6.1.10 Optical density fluctuation

We are interested in a nonlinear effect which is proportional to the optical density; as

a result fluctuations in OD increase the noise of our measurement. Information on the

probe amplitude given by our demodulation technique, giving both quadratures, allows us

to monitor the optical density. We can normalize the measured phase shifts to the optical

density to divide out any added uncertainty due to its fluctuation. In our measurements

so far we did not monitor the incident probe power on the atoms while collecting data.

As a result, power variations may appear to be optical density fluctuations and we can

not reliably compensate for the OD fluctuation. In our telescope design, we use the probe

light reflected from the 10-90 BS in the probe telescope to monitor its power; however,

we would need to digitize the probe power values in sync with our measurement runs.

6.1.11 Use of both D1 and D2 lines

Currently all our signal and EIT beams are near the D2 (780 nm) line in rubidium. In

order to separate the beams we need to use polarization or momentum degrees of freedom.

Having the signal or the EIT beams on the D1 (795 nm) transition gives an extra degree

of freedom for separating the signal beam from the probe and coupling. Normal filters

(typically with a few nm linewidth) could be used for selecting the signal beam, in such

a setup.


6.1.12 Coupling light leakage

We use the same fiber to send probe and coupling light from the photon to the atom

side. The two beams are then separated on a PBS, on the atom side, provided they have

orthogonal polarizations. A HWP and a QWP are used to minimize the leakage of the

probe and coupling light at the PBS into the incorrect modes. Polarization drifts, for

example due to small motions of the long fiber, can cause power fluctuations. Due to

higher power, drift in the power of the coupling beam could result in both an effect on

the atoms or extra noise on the probe detector. One potential solution is to use separate

fibers to bring in probe and coupling light. Also, having the setup in one lab could be

beneficial by removing the need for very long fibers.

6.1.13 Polarization spectroscopy

Although currently adequate, some improvements could be made to our current polariza-

tion spectroscopy set up; see section 4.2.1 for the details of the current setup. We can use

a proper housing for heating, magnetic field shielding and applying a DC magnetic field.

We could also expand the beams passing through the vapor cell and then focus them

onto the detectors, in order to be able to increase the optical power. The result would

be an increase error signal while avoiding the detrimental effects of power-broadening.

6.1.14 Maximum possible XPS in N-scheme

In chapter 2 it was shown that changing any parameters that result in group velocity- or

bandwidth mismatch will not enhance the peak value of the XPS. Therefore, increasing

optical density (above roughly 2) or narrowing down the EIT window (beyond the band-

width of the signal pulses) would not help increase the size of the nonlinear effect. The

only parameter that can affect the size of the nonlinear phase shift without increasing

the mismatch problem is the spot size.


Parameter Current ImprovedBeam waist, w0 (µm) 13 2Beam area, A (µm2) 530 13Rayleigh Range, zR (µm) 680 16Effective interaction length, 2zR (µm) 1360 32Required atom density for OD > 3 (cm−3) > 2× 1010 > 1012

Signal detuning (MHz) 18 3Saturation probe power, Isat ≈1 mW.cm−2 (nW) 5 0.13Probe photon number for saturation power over 300ns 6000 150Probe phase quantum noise (mrad) 13 80Measured (and expected) peak XPS (µrad) 18 3700(Quantum-limited) SNR 1/700 1/25

Table 6.1: Comparison of the current experimental parameters and an optimal experi-mentally achievable set.

Table 6.1 compares our current parameter setting to one with a experimentally achiev-

able tighter focus. The assumptions made to obtain the new improved parameter setting

are as follow:

• The effective interaction length for the nonlinear effect is determined by the range

over which the intensity remains high, twice the Rayleigh range. The atom cloud

is assumed to have a radius at least as large at the Rayleigh range. However, to

avoid absorption it is better to match the cloud size to the Rayleigh range.

• In order to saturate the limit posed by group velocity mismatch, the optical density

has to be at least 2. Therefore, the atom density has to be increased to compensate

for the shorter effective interaction length in the case of a tighter focus. However,

increasing the atom density beyond 1012 cm−3 for the case presented does not

increase the peak phase shift by considerable amount.

• The probing time is assumed to be 300ns which is the expected duration of XPS

for an EIT window of roughly 2 MHz. In order to increase this time, and therefore

decrease any fast noise, one could increase the optical density or narrow down

the EIT window. For example, increasing the atom density by a factor of 100 or


reducing the width of the EIT window by a factor of 100 will make the fast noise

10 times smaller; however, the change in peak phase shift would be negligible. The

bandwidth of the signal pulse is assumed to be wider than the EIT window in all

cases to ensure that the peak nonlinear phase shift is saturated. In choosing the

EIT window, one has to be careful about the limits set by the non-vanishing probe

power on how narrow the EIT window can become.

• The maximum achievable XPS is calculated based on the expression Γ∆sσat/(4∆2s+

Γ2)A (see chapter 2 for details). This expression assumes that the optical density

( 2) is high enough that the limit posed by group velocity mismatch does not

saturate. It also ignores any detrimental effects (eg. two-photon absorption) asso-

ciated with tuning the signal closer to resonance.

• The quantum noise is calculated based on the number of photons reaching the probe

detector. Increasing the probe collection efficiency (currently 30%) can reduce the

quantum noise by letting more probe photons reach the phase measurement stage.

• It might prove beneficial to use a low-finesse cavity on resonance with the signal

field to increase the interaction time of the signal and the probe. In the current

experimental setup, the cavity mirrors have to be placed outside the MOT cuvet

and therefore a high-finesse cavity can result in too much loss. However, a cavity

with a finesse around 5-10 can help increase the size of the nonlinear phase shift

without adding too much loss.

Values presented in table 6.1 suggest that the SNR can be enhanced by at least an order

of magnitude given the proposed changes. However, this is still too small to be resolvable

on a single shot. Reaching that resolution requires significant improvements to the optical

density or the EIT window width to reduce the quantum (or any other fast) noise. It

is also important to remember that focusing to 2 micron or smaller is non-trivial and

requires a careful design of a lens system to reduce the aberrations.


One practically important point is that the atom duty cycle to achieve the density of

1012 cm−3 should normally involve a spatial or temporal dark-SPOT MOT and compres-

sion, or use of Bose-Einstein condensate. These extra steps can make the preparation

time of atoms significantly longer than the current cycle. One potential solution is the

atoms be kept in a deep far-off-resonance optical trap after preparation, allowing them

to be probed for a long period.

6.1.15 Future directions

In order to obtain larger cross-phase shifts, alternative solutions that call for major

changes to the optical setup should be investigated. Recently, strong coupling has been

achieved in free space using Rydberg atoms and this direction seems to be very promising

to pursue [24, 4, 25, 26]. In addition, a major problem of light-matter interaction in free

space is the trade-off between how tightly the beam can be focused and the distance over

which the tight focus is maintained. This can be overcome using structures that confine

light such as HCF [21], nano-photonics structures [151], micro-resonators [152].

6.2 Summary

In this thesis, I reported the first experimental observation of the interaction between

post-selected single-photons and classical pulses, resulting in a nonlinear phase shift of

18± 4 µrad per post-selected single photon [50]. This demonstrates the first observation

of the photon-photon interaction due to single freely propagating photons. In these

observations, we have used the lowest pulse energies used for cross-phase modulation in

free space. This thesis also presents the details of the apparatus built to carry out these

experiments and suggestions for future improvements.

We have theoretically and experimentally studied the behavior of EIT-enhanced XPS

for pulsed signals in the N-scheme, and showed how different parameters, such as EIT


window width, pulse bandwidth, and optical thickness affect the transient behavior of

the system [71]. The results obtained here have important implications for quantum

logic gates based on such EIT schemes; they also permitted us to determine the optimal

pulse duration and detuning for these purposes. It was also shown that a treatment

based on linear time-invariant system response, taking the intensity of the signal as

the “drive” and the phase shift on the probe as the “output”, adequately models the

transient behavior of the XPS. It was shown that the peak value and the duration of

XPS are determined by several parameters; the peak XPS scales as the inverse of the EIT

window width and is linear in pulse bandwidth as long as the EIT window is broader than

the pulse bandwidth. However, for EIT windows narrower than the pulse bandwidth,

even though there is no increase in the peak XPS, the effect lasts for a longer time,

providing more time for detecting the phase shift and potentially improving the SNR.

The peak XPS also scales linearly in optical density as long as propagation effects can

be neglected. For optical densities above ∼ 2 (assuming negligible dephasing), the group

velocity mismatch of the probe and the signal starts to play a significant role in the

dynamics of the response and this poses a limitation on the maximum achievable peak

XPS. On the other hand, this group velocity mismatch causes the XPS to last longer.

In short, narrow EIT windows and high optical densities can enhance the detectability

of XPS by elongating the duration of the effect. Unlike the peak XPS, which is limited

by the EIT response time and propagation effects, the integrated phase shift follows

the prediction of the steady-state treatment. This integrated phase shift, which grows

linearly with OD and inversely with EIT window width, is a more relevant figure of merit

for the detectability of the XPS [11, 12].

I have also theoretically shown that in a scheme inspired by weak measurement one

photon may act like many photons, writing a very large XPS on a coherent state, and

that this amplification may greatly improve the SNR for measuring single-photon-level

nonlinearities [153]. Considering presently observable optical nonlinearities, this opens


the door to unambiguous weak measurement experiments, in which two distinct physical

systems could be deterministically coupled, leaving no room for an alternative classical

explanation. Accounting for the effects of back-action when the weakness criterion is

relaxed, it was found that the largest achievable phase shift per post-selected photon is

always of the order of the quantum uncertainty of the probe phase. More generally, it

was found that although post-selection cannot enhance the SNR in the presence of noise

with short (or vanishing) correlation times, particularly shot noise, it can be of great

use in the presence of noise with long correlation times. Given the prevalence of low-

frequency noise (e.g. 1/f noise) in real-world systems, this suggests that WVA may find

broad application in precision measurement. An experiment to study these predictions

is currently in progress in our lab.

Appendix A

Alignment procedures


In this chapter, I review the alignment procedure for the most major parts of the

setup. For details of the setup see chapter 4.

A.1 Polarization Spectroscopy

This setup for this part is shown in Figure 4.2.

• Put all the optics (except the vapor cell) and the detectors in place.

• Block the beam and set the Schlosser input set-point to zero.

• Unblock the beam and set the HWP to get equal powers on both detectors (be

careful about detector saturations). In case of using battery-powered detectors,

always use new and similar batteries. Using old and mixed batteries can cause

instability and drifts.

• Place the vapor cell in, normal to earth magnetic field and wrap it with µ-shield to

reduce spectroscopy signal drift.

99

Appendix A. Alignment procedures 100

• Align the pump to overlap with the probe on both sides of the cell (in order to

reduce the back-reflection of the pump from cell faces, put the cell at an angle).

• At this point one should be able to see the error signal; changing the overlap of the

pump and probe at this point can give you a larger signal, but it might also result

in getting a background through the back-reflection of the pump.

• Rotate the QWP to get the right signal shape and fine-tune the HWP to get the

best signal (necessary because the vapor cell walls slightly rotate the polarization).

The goal of this step is to have the largest slope of the error-signal at the desired

locking point.

• Change the input set point to have a zero-crossing at the desired locking point.

• Flip to locking mode.

A.2 Master laser Lock

• Scan the laser to find the appropriate transition.

• Use the ‘input setpoint’ to determine the locking point (where the error signal

crosses zero).

• Flip the switch to go from the scan mode to the feedback mode.

The following steps need to be done only once:

• The lock polarity has to be set to determine which slope (negative or positive) is

going to be used for locking.

• Set the low-frequency bandwidth to maximum (or the desired value).

• Set both I and P gains to zero.


• Start increasing the I gain until you see oscillations and then reduce the I gain by

25 to 50%.

• Repeat the last step for P gain.

• If no oscillation can be seen, the size of the error signal or the feedback ‘pre-gain’

might be too small.

A.3 AOM double-pass

The setup for this part is shown in Figure 4.4.

• Put the PBS, the first iris and the mirror in place.

• Align the mirror so that beam passing through the iris makes it back through it.

• Put in the QWP and set it so that the back-reflected beam reflects off of the PBS.

• Put the two lenses in the roughly right positions and try to collimate the beam

reflecting off of the PBS using the input lens.

• Put in the AOM at the focus of the beam and align it to get the most power in the

+1 order (one usually needs to change the height, angle and the tilt of the AOM).

• Fine-tune the mirror to get most power in +2 order coming out the AOM-DP.

• Monitor the output spot a few meters away, frequency-modulate the AOM very

slowly, for example at 1Hz, and use the back lens to minimize any motion of the

spot.

A.4 Injection lock

The setup for this part is shown in Figure 4.5.


• Rotate the input polarizer of the OI to have maximum transmission of the slave

laser beam through the OI.

• Rotate the HWP1 to have the maximum transmission of the slave laser beam

through the PBS1.

• Use two mirrors to couple the 5% reflection from the PBS1 into the EOM fiber (the

HWP1 can be rotated a little to make this step easier but has to be undone once

this step is over). This step is to ensure the mode-matching of the seed and slave

laser beams.

• Use two mirrors to couple seed beam into EOM fiber.

• Use a HWP2 to ensure that all the seed beam is reflecting from the PBS1.

• A telescope can be used to improve the mode matching between the seed- and slave

laser beams.

• Injection lock current can be tuned to lock to desired modulation peak.

A.4.1 Polarization-maintaining fiber

The PM fiber is strongly birefringent. A beam with an incorrect input polarization expe-

riences a large polarization rotation: the components of polarization along the ordinary

and extra-ordinary axes of the fiber obtain different phases which depend on the fre-

quency of the light, length of the fiber and the refractive index of the fiber along each

axis. The different phases will cause the polarization to turn into an elliptical one. Using

these facts and taking the following steps one can set the polarization correctly:

• Send the beam coming out of the fiber through a PBS and measure it on a detector

(polarization rotation looks like an amplitude variation in this case).


• Scan the laser frequency by at least 1GHz (the phase difference and therefore the

polarization rotation depends on frequency and the scan has to be large enough to

see the amplitude variation due to it on the detector).

• Rotate the input polarization using a HWP to have the minimum amplitude vari-

ation as a result of the scan.

A.5 Probe and signal telescopes

The telescopes are shown in Figure 4.14.

• Align one telescope to the cloud: use intense near-resonance beam to ensure the

beam is hitting the center of the cloud.

• Keep reducing the optical power and fine tune the alignment of the telescope to

the cloud.

• Use the z-translation capability of the focusing lens to obtain the smallest spot size

in the cloud (the portion of the cloud blown away by the beam is proportional to

the spot size of the beam inside the cloud which is being imaged using fluorescence

imaging).

• Use the mirrors and the focusing lens of the other telescope to fiber-couple the beam

from one telescope to another (the typical coupling efficiency from the interaction

region into fiber is 20-30% taking into account the transmission of the 10-90 BS).

• Since there are polarizers in both telescopes, the polarization has to be set properly

to see maximum fiber-coupling.

• The alignment can be fine-tuned looking at the nonlinear phase shift at this point,

but is usually unnecessary.

Appendix B

Interaction of Electromagnetic

Fields with Multi-level Atom


B.1 Overview

In this appendix the interaction of a multi-level atom with EM fields is studied. Then the

interaction of an ideal two-level atom with an isotropic EM field is discussed. In order

to take the polarization of the EM field into account one needs to include all Zeeman

sub-levels. At the end, all the ideas discussed in this chapter are applied to the case of a

real atomic lambda system including Zeeman sub-levels in the presence of magnetic field.

It is concluded that unless the EM field polarizations and magnetic field are not chosen

wisely, incoherent optical pumping dominates and destroys the EIT.

B.2 Approach

To setup the equations of motion for the interaction, the semi-classical master equation

approach is taken. For non-unitary evolution of the density matrix, ρ, of a quantum

104

Appendix B. Interaction of Electromagnetic Fields with Multi-level Atom105

mechanical system due to relaxing and decoherence processes usually the Lindblad form

of master equation is used

∂

∂tρ = − i

~[H, ρ] +

N2−1∑

i=1

γi(CiρC†i −

1

2ρC†iCi −

1

2C†iCiρ), (B.1)

where H is the Hamiltonian of the system and thus the first term gives the unitary part of

the evolution of the system. N is the dimension of the Hilbert space of the problem and

Ci’s are a complete set of N ×N matrices. One usually chooses the Ci’s to be projectors

of the form σkj = |k〉〈j|. With that choice of Ci’s each term in the summation would be

of the form

ρjj(|k〉〈k| − |j〉〈j|)−1

2

∑

l 6=j

(ρlj |l〉〈j|+ ρjl |j〉〈l|). (B.2)

Thus γi is the population decay rate from |j〉 to |k〉 and twice the dephasing rate of |j〉

and all other states. Therefore, all we need is to find the Hamiltonian of the system and

the decay and dephasing rates to be able to setup the equations of motion.

We are both interested in the time evolution and in the steady state behavior of the

system. To solve for the steady state we set ∂ρ∂t

= 0 and solves a system of algebraic

equations. There are two ways to get the time evolution of the system. The first one is

to integrate the system over time and obtain ρ(t) for any desired time. However, this

method is not very economic in dealing with large times. The second method to obtain

the time dependence of the system is the following: the dynamics of the system is given

by a relation of the type

∂ρ

∂t= Lρ (B.3)

where L is the Liouvillian super-operator containing unitary and non-unitary dynamics

of the system. If the Hamiltonian and the relaxation processes are time independent then

the solution to the above relation is


ρ(t) = eLtρ(0) (B.4)

Now, we can solve the eigen-value problem of the super-operator L and calculate the eLt

term for any desired time without the need to integrate over all times.

To solve the equations of motion numerical methods are used. Some parts of the codes

are written using the Quantum Optics and Computation Toolbox for MATLAB [154].

B.3 Equations of Motion

As a first step let us consider the interaction of an EM field with a multi-level atom in

general. The Hamiltonian of the system is of the form

H = Hatom +Hinteraction (B.5)

The interaction Hamiltonian in the dipole approximation is

HI = −d · E(t) (B.6)

where d is the dipole operator and E(t) = 12E0e

iωLt + c.c. is the electric field1 interacting

with the atom. The matrix element of the interaction Hamiltonian in the basis of the

bare atomic states |i〉 , |j〉 , ... is

(HI)ij = −〈i|d |j〉 · 1

2E0e

iωLt + h.c. (B.7)

1Electric fields with any time dependence of interest can be Fourier expanded as,

E(t) =

∫ ∞

0

(E0(ω)eiωt + c.c.)/2.

Therefore, what is calculated here is in fact the frequency domain response of the atomic system andwith an inverse-Fourier transform one can have the time domain response.


Thus the overall Hamiltonian of the system is

H = ~ω0I + ~∑

i

ωi0σii + ~∑

i,j

(Ωij

2eiωLtσij + h.c

)(B.8)

where ~ωi0 is the energy difference of the state |i〉 from the ground state, Ωij = −dij ·E0

~ is

the Rabi frequency of the transition from |i〉 to |j〉 and σij’s are the projection operators

as defined before. One can ignore the first term by redefining the zero of energy.

In order to remove the time dependence of the Hamiltonian we go to a rotating frame.

In the rotating frame the Hamiltonian and density matrix can be redefined as

H = eiAtHe−iAt − A (B.9)

and

ρ = eiAtρe−iAt (B.10)

that satisfy the following relation

∂

∂tρ = − i

~[H, ρ] (B.11)

In most cases there is at least one choice of a diagonal matrix A that makes H time-

independent. However, the time-independence is obtained using the RWA, that is ig-

noring the terms that contain high frequencies, e.g. 2ωL. Thus, the Hamiltonian of the

system in the dipole approximation and using RWA in a proper rotating frame is given

by

H = ~∑

i

(ωi0 − Aii)σii + ~∑

i,j

(Ωij

2σij + h.c

)(B.12)

One can plug in the rotating frame operators in the master equation (B.1) (dropping

the tilde’s for convenience) and solve for the dynamics of the system. Generalization


of the current treatment to the case of more than one beam is straight forward, except

that one needs at least one atomic level more than the number of EM fields and the

assumption that each of the fields is tuned closer to one of the transitions to be able to

remove the time dependence completely (using RWA).

We are usually interested in the atomic polarization density defined as

P = Tr(dρ) (B.13)

This quantity is related to the electric susceptibility, χ, as

P = ε0χE0 + c.c. (B.14)

which is in turn related to the complex refractive index as

n =√

1 + χ =⇒ n− 1 ' 1

2χ (B.15)

This refractive index describes the response of the atomic medium to an EM field. The

real part of the refractive index gives the dispersion properties of the medium and the

imaginary part of it gives its dissipative properties2. Generalization to more than one

EM field is again straight forward, except that

P = ε0(χ1E01 + χ2E02 + · · ·+ c.c.) (B.16)

and each EM field sees the refractive index given by its own χi.

2As χ and n− 1 are proportional the behaviour of one determines the other. Thus, throughout thisreport whenever refractive index is mentioned, in fact n− 1 or χ is meant.


F=2

F=3

F’=1

F’=2

F’=3

F’=4

a) b)

-2 -1 0 +1 +2

-2 -1 0 +1 +2

-2 -1 0 +1 +2

-2 -1 0 +1 +2

c)

F=2

F’=2

F=2

F’=2

Figure B.1: The level structure for a) section B.4.1, b) and c) section B.4.2. Hyperfinestructure of 85Rb is used.

B.4 Interaction of EM Fields with Multi-level Atom

In this section the formalism developed so far is applied to various cases of atom-light

interaction and the responses of different atomic configurations to EM fields are studied.

B.4.1 Multi-level atom and a EM field with isotropic polariza-

tion

The polarization of the EM field is assumed to be isotropic. Thus, the Zeeman levels are

not needed to be considered. The level structure considered is shown in figure B.1. It

is the D2 line structure of 85Rb. The overall Hamiltonian is given by eq. (B.12). The

energy differences, effective dipole moments for isotropic polarization and other atomic

parameters are taken from [126]. It can be easily shown that the rotating frame matrix,

A, is given by

A = diag[0, 0, ωL, ωL, ωL, ωL] (B.17)


−10 −5 0 5 10 15 20−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

detuning from F=2 −> F’=2 transition (×Γ)

Re(

χ)

−10 −5 0 5 10 15 20−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Im(χ

)

−10 −5 0 5 10 15 200

0.2

0.4

0.6

0.8

1


pop.

of F

=2

−10 −5 0 5 10 15 200

0.2

0.4

0.6

0.8

1


pop.

of F

=3

50τ

5000τ

50τ500τ

5000τ

50τ

500τ

5000τ

50τ

500τ

5000τ

500τ

Figure B.2: Response of the multi-level atom to the EM field at times 50τ , 500τ and5000τ : (top) real and imaginary parts of refractive index (in arbitrary units), (bottom)The populations of the two ground states F = 2 (left) and F = 3 (right). The horizontalaxis is the detuning from F = 2 → F ′ = 2 transition in terms of natural linewidth, Γ.The F = 2 → F ′ = 3 transition is at +10.5Γ and the F = 2 → F ′ = 1, that is a cyclictransition, is at −4.8Γ. The intensity of the laser is taken to be 0.13mW/cm3. All thepopulation is initially in F = 2 ground state.

One can plug in the Hamiltonian to the master equation, eq. (B.1), and solve for time

dependence of the system. Figure (B.2) shows the response of the atom in this case. The

laser is tuned around F = 2 → F ′ = 2 transition and is scanned over all dipole-allowed

transitions. One can obviously see the effect of pumping from F = 2 to F = 3 ground

states at frequencies close to F = 2→ F ′ = 2 and F = 2→ F ′ = 3 transitions. However,

the transition F = 2 → F ′ = 1 is a cyclic transition. If atom is excited to F ′ = 1 it

cannot spontaneously decay to F = 3 as it is dipole-forbidden. That is why the refractive

index and the populations at that frequency do not change much with time. Population

of all excited levels are very close to zero except for F ′ = 1 that absorbs part of the

population at F = 2→ F ′ = 1 transition frequency.

One can also solve for the steady state solution of the system. It might be surprising


to see that steady state solution is that all the population is pumped to F = 3 and

the refractive index vanishes, even at the cyclic transition frequency. However, the fact

is that even at the cyclic transition frequency there is a non-zero chance of getting to

non-cyclic transitions and then decaying to F = 3. However, this chance is very small,

due to very large detuning, and it takes a long time for the population to leave the cyclic

transition. In figure (B.2) it can be seen that at t = 5000τ the frequencies close to the

cyclic transition started to lose population.

The same procedure can be repeated for F = 3→ F ′ transition.

B.4.2 Two-level atom and a polarized EM field

In this section an EM field with a plane polarization is taken into account. In this case

it is important to keep track of the Zeeman sub-levels as they mostly determine the

behaviour of the system. Figure (B.1b,c) show the level structure considered in this

section.

The difference between the two structures in figure (B.1b,c) is just a matter of the

definition of the quantum axis. If there is a magnetic field in the problem (or any preferred

direction), it determines the quantum axis. Then the two structures are different in the

fact that structure in part (b) has a dark state (|F = 2,Mf = 0〉) and the one in part (c)

does not.

However, if there is no preferred direction then the z-axis can be determined arbi-

trarily. If one defines the z-axis parallel to the polarization of the EM field, then the

corresponding structure is the one in part (b). This structure has obviously a dark

state. The population from all Zeeman sub-levels of the ground state can be excited and

spontaneously decay to any of the ground state sub-levels, except the ones that go to

|F = 2,Mf = 0〉 sub-level. This sub-level is decoupled from the EM field and the pop-

ulation that goes to that is trapped there. The steady state of the problem is when all

the atoms go to the dark state and there is no further interaction between the EM field


and the atomic population.

On the other hand, in the absence of any preferred direction, one can choose the

z-axis perpendicular to the polarization direction of the EM field, for example y. The

structure in part (c) shows this case where the plane polarization is written in terms of

two circular polarizations. In this case it is more subtle to understand what happens to

the dark state. However, it is easy to check what happens using the formalism developed

here.

The Hamiltonians for part (b) and (c) can be easily written as in eq. (B.12). The

rotating frame matrix is given by

A = diag[0, 0, 0, 0, 0, 0, ωL, ωL, ωL, ωL, ωL] (B.18)

Plugging into the master equation one can solve for steady state of the system. For

the case of part (b) the expected result is obtained: all the population ends up in

|F = 2,Mf = 0〉. The steady state density matrix for the case of part (c) is pure,

tr(ρ2) = 1, and corresponds to the state

1

2(

√3

2|F = 2,Mf = −2〉+ |F = 2,Mf = 0〉+

√3

2|F = 2,Mf = +2〉) (B.19)

that is an eigen-state of Jy for j = 2 with eigen-value zero. So it seems that the steady

state solution is again the zero-eigen-value state along the EM field polarization. A closer

look at the level structure in part (c) makes the result obtained here more clear. All the

polarizations are paired in lambda type structures except two of them. One can expect

that these two are pumping the population out of |F = 2,Mf = ±1〉 sub-levels. That

is why the amplitude of these sub-levels in the steady state solution is zero. The two

lambdas connecting sub-levels |F = 2,Mf = 0,±2〉 form two EIT type structure that

traps the population in a certain dark state that is given above.


-1 0 +1F=1

-2 -1 0 +1 +2F=2

-2 -1 0 +1 +2F’=2

-1 0 +1F=1

-2 -1 0 +1 +2F=2

-2 -1 0 +1 +2F’=2

(a) (b)

Figure B.3: Two level structure for which conditions of electromagnetically inducedtransparency is studied. In this example D2 line of 87Rb is considered.

B.4.3 Lambda system with two polarized EM fields

Lastly, we study the role of Zeeman sub-levels in EIT. Figure B.3 shows the level struc-

tures to be discussed here. Part (a) clearly shows that there is an optical pumping dark

state. The steady state of the system is when all the population ends up in this sub-

level. Therefore one can not observe EIT with the two EM fields polarized parallel to

the magnetic field (if any).

However, if there is a magnetic field and the polarization of the EM field tuned to the

levels with a optical pumping dark state is perpendicular to that, then EIT is revived.

Part (b) shows the couplings in this case. The part of the Hamiltonian due to the

magnetic field can be added as

HB = µBgFMfBz (B.20)

that adds corrections to the diagonal elements of the Hamiltonian.

Figure B.4 shows the real and imaginary parts of refractive index for different values of

the magnetic field. It is clearly seen that the increase in the magnetic field is destroying

the degeneracy in F = 2 ground state and the EIT is being formed. However, if the

magnetic field is too large the individual Zeeman sub-levels start to play role that is

not collective any more. This means that the EIT gets deformed due to the different

detunings of the Zeeman sub-levels (that is considerable now). The condition of two


−5 −4 −3 −2 −1 0 1 2 3 4 5−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

detuning from |F=2,Mf=0 ⟩ → |F’=2,M

f=0 ⟩ (×Γ)

Re(

χ)

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.005

0.01

0.015


f=0 ⟩ (×Γ)

Im(χ

)

B=0.01G

B=0.01G

B=0.1G

B=0.1G

B=1G

B=1G

Figure B.4: Change of the atomic response to the EM field tuned to F = 1→ F ′ = 2 asmagnetic field is increased. It can be seen that by increasing the magnetic field the EITis revived. The real (left) and imaginary (right) parts of the refractive index (in arbitraryunits) versus detuning from F = 1 → F ′ = 2 transition in units of Γ. The intensity ofthe EM fields are 1.3 and 13 mW/cm3

photon resonance is lost for certain sub-levels and they start to absorb and some other

satisfy the two-photon resonance at other frequencies. Figure B.5 shows the EIT window

in case a magnetic field of 10G.


−5 −4 −3 −2 −1 0 1 2 3 4 5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08


f=0 ⟩ (×Γ)

Re(

χ)

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08


f=0 ⟩ (×Γ)

Im(χ

)

Figure B.5: The effect of a large magnetic field and Zeeman splitting in EIT windowstructure. If the Zeeman splitting is large the condition of two photon resonance issatisfied for different sub-levels at different frequencies.

Appendix C

Data analysis MATLAB code


The I and Q data are saved as comma-separated-variable files and can be loaded into

MATLAB for further analysis. In this chapter we go through some details of our data

analysis code.

We do not necessarily need to read the raw data files every time we do analysis

and occasionally need to re-run the code on the data already loaded to the MATLAB

workspace. Therefore, the first line of the code asks if new raw data has to be read or

it has to use the data in the workspace. The next part of the code, initialized all the

known parameters for the analysis, like the length of each shot, sampling period, so on.

readData = input(’read data? [default: 0]: ’);

% 0: don’t read data from files and use the variables in workspace, 1: read data from files

if isempty(readData)

readData = 0;

end

if readData

clc; clear;

116

Appendix C. Data analysis MATLAB code 117

readData = 1;

end

%% initialization

parentFolder = ’D:\IQ_Data\140823_WVA_Part19’;

fileName = ’test13’; % doesn’t need the "_#" at the end

startFromBin = 0 ; % analysis starts from 0

NBins = 10; % number of bins to read a1fter the startBin

NTracesPerBin = 10; % number of traces per bin

samplingPeriod = 2/30; % us, Sampling rate=30 MHz

NSamplesPerShot = 1*36;

t_shot = NSamplesPerShot * samplingPeriod; % duration of each shot

totNSamples = 22501;

%%% range setting for the averaging

i_s = 7;

i_e = 13;

endOfRange = 17; % start point of the background after

i_0 = 3; % number of points to average before and after

iTagRange = 5:NSamplesPerShot - 5; % location of tag range

tagSize = 0.01; % abs tag size less than this is a ’no-click’

ODi_point = 95; % where to read the initial OD

ODf_pointStart = totNSamples - 501;

% search for peak from this point on to find the final OD

startingDataPoint = 4500 - 1-2-6; % ignoring the first 300 us due to probe phase dynamics

endDataPoint = totNSamples - 180;

%%% amplitude of the beating signal

amp0 = 0.03; % V - amplitude of the probe without atoms (read of the vi)


%%% work out the number of photons

detSens = 1/0.77; % mV / nW (calibrated, 50 ohm termination)

pulsePeak = input(’enter the pulse amplitude (mV)[default: 0]: ’);

if isempty(pulsePeak)

pulsePeak = 0;

end

atten = input(’enter the value of attenuation (ND filter) [default: 1]: ’);

if isempty(atten)

atten = 1;

end

peakOpticalPower = pulsePeak/atten / detSens; % nW

pulseFWHM = 40; % nsec

collectionEff = 13/42; % measured

NPhotons = peakOpticalPower * pulseFWHM /2.5e-1 / collectionEff * (sqrt(pi)*erf(1)/2);

% 2.5e-1 x 1e-18 J, sqrt(pi)*erf(1)/2: gaussian pulse correction

dNPhotons = sqrt((0.015)^2 + (0.02)^2 + (0.05)^2 + (0.05)^2)*NPhotons;

% error: ND filter, det sen, fwhm, peak power

fprintf(’the number of signal photons is: %0.1f +/- %0.1f \n’,NPhotons,dNPhotons);

% error is calculated using 10% error in 5 parts

%%% find the location of each shot

x = (0:endDataPoint - startingDataPoint + 1)*samplingPeriod;

idx = zeros(1,round((endDataPoint - startingDataPoint + 1)*samplingPeriod/t_shot));

N_shotsPerTrace = length(idx)-1;

for j=1:length(idx)

[a idx(j)] = min(abs(x-(j-1)*t_shot));

end

lengthOfaShot = mean(idx(2:end)-idx(1:end-1));


fprintf(’\nlength of a shot (has to be an integer): %0.2f\n’,lengthOfaShot);

If we ask the code to read the data from the raw files, this next part will be executed

and corresponding variables will be initialized.

%% read in data from file

if readData

ODi_bins = zeros(1,NBins);

ODf_bins = zeros(1,NBins);

ODf_point_bins = zeros(1,NBins);

avgOD_diff= zeros(1,NBins);

idxODf_bins = zeros(1,NBins);

XPS_ps = zeros(1,NBins);

XPS_nps = zeros(1,NBins);

XPS_nps_shotsAfterExcluded = zeros(1,NBins);

XPS_ps_shotsAfterExcluded = zeros(1,NBins);

XPS_shotsAfter = zeros(1,NBins);

xpsOverBins_avgUpToNow = zeros(1,NBins);

xpsOverBins_stdUpToNow = zeros(1,NBins);

xpsOverBins_ps_avgUpToNow = zeros(1,NBins);

xpsOverBins_ps_stdUpToNow = zeros(1,NBins);

xpsOverBins_nps_avgUpToNow = zeros(1,NBins);

xpsOverBins_nps_stdUpToNow = zeros(1,NBins);

XPS_nps_shotsAfterExcluded_avgUpToNow = zeros(1,NBins);

XPS_nps_shotsAfterExcluded_stdUpToNow = zeros(1,NBins);

XPS_ps_shotsAfterExcluded_avgUpToNow = zeros(1,NBins);

XPS_ps_shotsAfterExcluded_stdUpToNow = zeros(1,NBins);

XPS_shotsAfter_avgUpToNow = zeros(1,NBins);

XPS_shotsAfter_stdUpToNow = zeros(1,NBins);


N_ps_shots = zeros(1,NBins);

N_nps_shots = zeros(1,NBins);

N_nps_shotsExcluded = zeros(1,NBins);

N_Ps_shotsExcluded = zeros(1,NBins);

phaseOfOneShot = zeros(lengthOfaShot,NBins);

ampOfOneShot = zeros(lengthOfaShot,NBins);

phaseOfOneShot_ps = zeros(lengthOfaShot,NBins);

ampOfOneShot_ps = zeros(lengthOfaShot,NBins);

phaseOfOneShot_nps = zeros(lengthOfaShot,NBins);

ampOfOneShot_nps = zeros(lengthOfaShot,NBins);

phaseOfOneShot_ps_shotsAfterExcluded = zeros(lengthOfaShot,NBins);

ampOfOneShot_ps_shotsAfterExcluded = zeros(lengthOfaShot,NBins);

phaseOfOneShot_nps_shotsAfterExcluded = zeros(lengthOfaShot,NBins);

ampOfOneShot_nps_shotsAfterExcluded = zeros(lengthOfaShot,NBins);

xpsOverTraces = zeros(N_shotsPerTrace,NTracesPerBin);

ampOverTraces = zeros(N_shotsPerTrace,NTracesPerBin);

tagAmpOverTraces = zeros(N_shotsPerTrace,NTracesPerBin);

tagPhaseOverTraces = zeros(N_shotsPerTrace,NTracesPerBin);

avgPhaseOverTraces = zeros(totNSamples,NBins);

avgAmpOverTraces = zeros(totNSamples,NBins);

xpsOverBins = zeros(N_shotsPerTrace*NTracesPerBin,NBins);

ampOverBins = zeros(N_shotsPerTrace*NTracesPerBin,NBins);

tagAmpOverBins = zeros(N_shotsPerTrace*NTracesPerBin,NBins);

tagPhaseOverBins = zeros(N_shotsPerTrace*NTracesPerBin,NBins);

xpsOverBinsNormToOD = zeros(N_shotsPerTrace,NBins);

timeVec = zeros(NBins,6);

figure;


for k = 1:NBins

tic;

fileRead = strcat(parentFolder,’\’,fileName,’_’,num2str(startFromBin+k-1),’.csv’);

fprintf(’\nloading: %s \n’,fileRead);

In order to save time while we are doing long data runs we need to start analyzing

the data we already have. Therefore, the code has the ability to wait for a new file to

become available and then load and use it for analysis.

while exist(fileRead,’file’) ~= 2

fprintf(’->’);

pause(10);

end

fileInfo = dir(fileRead);

while fileInfo.bytes == 0

fileInfo = dir(fileRead);

fprintf(’->’);

pause(2);

end

fprintf(’ loaded!\n’);

rawIQData = csvread(fileRead);

After loading the files, I and Q data are separated, phase and amplitude are calculated

and phase unwrapping is done. We use ArcTan to calculate the phase from I and Q which


has a range of −π/2 to π/2. But the phase can go beyond these values and the ‘wrapping’

of the phase in that range has to be undone.

%% removing the traces that the vi coughs up and saves zeros and

%% NANs instead of actual values

fprintf(’number of unsaved traces: %i \n’, sum(isnan(sum(rawIQData))~=0));

rawIQData(:,isnan(sum(rawIQData))) = 0;

IData = rawIQData(1:2:end,:);

QData = rawIQData(2:2:end,:);

clear rawIQData;

rawAmpData = sqrt(IData.^2 + QData.^2);

rawPhaseData_1 = atan(QData./IData);

%% preventing NANs

rawPhaseData_1(IData==0) = pi/2;

%% unwrapping the phase

rawPhaseData = rawPhaseData_1;

for j = 2:size(rawPhaseData,1)

rawPhaseData(j,:) = rawPhaseData_1(j,:) - ...

pi * floor((rawPhaseData_1(j,:)-rawPhaseData(j-1,:))/pi+0.5);

end

%% size of the data

if NTracesPerBin ~= size(rawPhaseData,2)

error(’# traces does not match!’);

end

if totNSamples ~= size(rawPhaseData,1)

error(’# samples does not match!’);


end

fprintf(’number of NaN: %i \n’, sum(sum(isnan(rawPhaseData),1)~=0));

clear IData QData rawPhaseData_1;

timeVec(k,:) = clock;

disp(datestr(timeVec(k,:)));

%% average over N measurements runs first (i.e. across rows)

avgPhaseOverTraces(:,k) = mean(rawPhaseData,2);

avgAmpOverTraces(:,k) = mean(rawAmpData,2);

Using the amplitude information we can calculate the optical density (one caveat

here is that we assume that the probe power remains constant during the measurement

which is not necessarily true). The phase traces are divided into shots and the nonlinear

phase shift in each shot is calculated. Also, the tagged shots are determined, the shots

that are located after tags are discarded and the phase of ‘click’ and ‘no-click’ events are

calculated separately.

%%% monitor OD and phase shift

ODi_bins(k) = -log(avgAmpOverTraces(ODi_point,k)/amp0);

[ODf_bins(k) idxODf_bins(k)] = max(-log(avgAmpOverTraces(ODf_pointStart:end,k)/amp0));

ODf_point_bins(k) = ODf_pointStart + idxODf_bins(k) - 1;

odWithinBin = (ODi_bins(k) - ODf_bins(k))/(ODi_point - ODf_point_bins(k)) ...

* (startingDataPoint:endDataPoint) ...

+ (ODi_point * ODf_bins(k) - ODf_point_bins(k) ...

* ODi_bins(k)) / (ODi_point - ODf_point_bins(k));

phaseOfOneShotOverBins = zeros(lengthOfaShot, NTracesPerBin);


ampOfOneShotOverBins = zeros(lengthOfaShot, NTracesPerBin);

phaseOfOneShotOverBins_ps = zeros(lengthOfaShot, NTracesPerBin);

ampOfOneShotOverBins_ps = zeros(lengthOfaShot, NTracesPerBin);

phaseOfOneShotOverBins_nps = zeros(lengthOfaShot, NTracesPerBin);

ampOfOneShotOverBins_nps = zeros(lengthOfaShot, NTracesPerBin);

phaseOfOneShotOverBins_ps_shotsAfterExcluded = zeros(lengthOfaShot, NTracesPerBin);

ampOfOneShotOverBins_ps_shotsAfterExcluded = zeros(lengthOfaShot, NTracesPerBin);

phaseOfOneShotOverBins_nps_shotsAfterExcluded = zeros(lengthOfaShot, NTracesPerBin);

ampOfOneShotOverBins_nps_shotsAfterExcluded = zeros(lengthOfaShot, NTracesPerBin);

for l = 1:NTracesPerBin

%%% splicing up the traces into shots

truncatedPhaseData = rawPhaseData(startingDataPoint:endDataPoint,l);

truncatedAmpData = rawAmpData(startingDataPoint:endDataPoint,l);

phaseOfOneShotOverTraces = zeros(lengthOfaShot, N_shotsPerTrace);

ampOfOneShotOverTraces = zeros(lengthOfaShot, N_shotsPerTrace);

for j = 1:N_shotsPerTrace

ph_temp = truncatedPhaseData(idx(j):idx(j+1)-1);

phaseOfOneShotOverTraces(1:idx(j+1)-idx(j),j) = ph_temp - mean(ph_temp(1:i_0));

ampOfOneShotOverTraces(1:idx(j+1)-idx(j),j) = truncatedAmpData(idx(j):idx(j+1)-1);

xpsOverTraces(j,l) = 1/2/(i_e-i_s) * sum(phaseOfOneShotOverTraces(i_s:i_e-1,j)...

+ phaseOfOneShotOverTraces(i_s+1:i_e,j))...

- 1/4/(i_0 -1) * (sum(phaseOfOneShotOverTraces(1:i_0-1,j)...

+ phaseOfOneShotOverTraces(2:i_0,j))...

+ sum(phaseOfOneShotOverTraces(endOfRange:endOfRange+i_0-2,j)...

+ phaseOfOneShotOverTraces(endOfRange+1:endOfRange+i_0-1,j)));

ampOverTraces(j,l) = 1/2/(i_0-1) * sum(ampOfOneShotOverTraces(1:i_0-1,j) ...

+ ampOfOneShotOverTraces(2:i_0,j));


[tagAmp tagIdx] = max(abs(ampOfOneShotOverTraces(iTagRange,j)));

tagAmpOverTraces(j,l) = tagAmp - mean(abs(ampOfOneShotOverTraces(iTagRange(1) ...

- 1+tagIdx+[-4,-3,3,4],j)));

tagPhaseOverTraces(j,l) = mean(phaseOfOneShotOverTraces(tagIdx:tagIdx+3,j)) ...

- mean(phaseOfOneShotOverTraces(iTagRange,j));

end

phaseOfOneShotOverBins(:,l) = mean(phaseOfOneShotOverTraces,2);

ampOfOneShotOverBins(:,l) = mean(ampOfOneShotOverTraces,2);

successfulTagsIdx_traceL = abs(tagAmpOverTraces(:,l))>abs(tagSize);

successfulTagsIdx_traceL = [successfulTagsIdx_traceL(2:end); false];

% tag for each shot is in the shot after which is being excluded

idxForNpsCases_shotsAfterExcluded_traceL = ...

xor(not(successfulTagsIdx_traceL),[false; successfulTagsIdx_traceL(1:end-1)]);

idxForNpsCases_shotsAfterExcluded_traceL(not(successfulTagsIdx_traceL)==false) = false;

idxForNpsCases_shotsAfterExcluded_traceL(1) = false;

idxForPsCases_shotsAfterExcluded_traceL = ...

xor(successfulTagsIdx_traceL,[false; successfulTagsIdx_traceL(1:end-1)]);

idxForPsCases_shotsAfterExcluded_traceL(successfulTagsIdx_traceL==false) = false;

idxForPsCases_shotsAfterExcluded_traceL(1) = false;

phaseOfOneShotOverBins_ps(:,l) = ...

mean(phaseOfOneShotOverTraces(:,successfulTagsIdx_traceL),2);

ampOfOneShotOverBins_ps(:,l) = ...

mean(ampOfOneShotOverTraces(:,successfulTagsIdx_traceL),2);

phaseOfOneShotOverBins_nps(:,l) = ...

mean(phaseOfOneShotOverTraces(:,not(successfulTagsIdx_traceL)),2);

ampOfOneShotOverBins_nps(:,l) = ...


mean(ampOfOneShotOverTraces(:,not(successfulTagsIdx_traceL)),2);

phaseOfOneShotOverBins_ps_shotsAfterExcluded(:,l) = ...

mean(phaseOfOneShotOverTraces(:,idxForPsCases_shotsAfterExcluded_traceL),2);

ampOfOneShotOverBins_ps_shotsAfterExcluded(:,l) = ...

mean(ampOfOneShotOverTraces(:,idxForPsCases_shotsAfterExcluded_traceL),2);

phaseOfOneShotOverBins_nps_shotsAfterExcluded(:,l) = ...

mean(phaseOfOneShotOverTraces(:,idxForNpsCases_shotsAfterExcluded_traceL),2);

ampOfOneShotOverBins_nps_shotsAfterExcluded(:,l) = ...

mean(ampOfOneShotOverTraces(:,idxForNpsCases_shotsAfterExcluded_traceL),2);

end

if sum(sum(isnan(phaseOfOneShotOverBins_ps),1)~=0) ~= 0

jj = 1;

col2Remove = zeros(1,sum(sum(isnan(phaseOfOneShotOverBins_ps),1)~=0));

for j = 1:NTracesPerBin

if sum(isnan(phaseOfOneShotOverBins_ps(:,j))) ~= 0

col2Remove(jj) = j;

jj = jj+1;

end

end

phaseOfOneShotOverBins_ps(:,col2Remove) = [];

ampOfOneShotOverBins_ps(:,col2Remove) = [];

end

if sum(sum(isnan(phaseOfOneShotOverBins_ps_shotsAfterExcluded),1)~=0) ~= 0

jj = 1;

col2Remove = zeros(1,sum(sum(isnan(phaseOfOneShotOverBins_ps_shotsAfterExcluded),1)~=0));

for j = 1:NTracesPerBin

if sum(isnan(phaseOfOneShotOverBins_ps_shotsAfterExcluded(:,j))) ~= 0


col2Remove(jj) = j;

jj = jj+1;

end

end

phaseOfOneShotOverBins_ps_shotsAfterExcluded(:,col2Remove) = [];

ampOfOneShotOverBins_ps_shotsAfterExcluded(:,col2Remove) = [];

end

xpsOverBins(:,k) = reshape(xpsOverTraces,[],1);

ampOverBins(:,k) = reshape(ampOverTraces,[],1);

tagAmpOverBins(:,k) = reshape(tagAmpOverTraces,[],1);

tagPhaseOverBins(:,k) = reshape(tagPhaseOverTraces,[],1);

successfulTagsIdx = abs(tagAmpOverBins(:,k))>abs(tagSize);

successfulTagsIdx = [successfulTagsIdx(2:end); false];

idxForNpsCases_shotsAfterExcluded = ...

xor(not(successfulTagsIdx),[false; successfulTagsIdx(1:end-1)]);

idxForNpsCases_shotsAfterExcluded(not(successfulTagsIdx)==false) = false;

idxForNpsCases_shotsAfterExcluded(1:N_shotsPerTrace:end) = false;

idxForPsCases_shotsAfterExcluded = ...

xor(successfulTagsIdx,[false; successfulTagsIdx(1:end-1)]);

idxForPsCases_shotsAfterExcluded(successfulTagsIdx==false) = false;

idxForPsCases_shotsAfterExcluded(1:N_shotsPerTrace:end) = false;

XPS_ps(k) = mean(xpsOverBins(successfulTagsIdx,k));

XPS_ps_shotsAfterExcluded(k) = mean(xpsOverBins(idxForPsCases_shotsAfterExcluded,k));

XPS_nps(k) = mean(xpsOverBins(not(successfulTagsIdx),k));

XPS_nps_shotsAfterExcluded(k) = mean(xpsOverBins(idxForNpsCases_shotsAfterExcluded,k));

XPS_shotsAfter(k) = mean(xpsOverBins([false; successfulTagsIdx(1:end-1)],k));

%%% phase of one shot; time trace

phaseOfOneShot(:,k) = mean(phaseOfOneShotOverBins,2);


ampOfOneShot(:,k) = mean(ampOfOneShotOverBins,2);

phaseOfOneShot_ps(:,k) = mean(phaseOfOneShotOverBins_ps,2);

ampOfOneShot_ps(:,k) = mean(ampOfOneShotOverBins_ps,2);

phaseOfOneShot_nps(:,k) = mean(phaseOfOneShotOverBins_nps,2);

ampOfOneShot_nps(:,k) = mean(ampOfOneShotOverBins_nps,2);

phaseOfOneShot_ps_shotsAfterExcluded(:,k) = ...

mean(phaseOfOneShotOverBins_ps_shotsAfterExcluded,2);

ampOfOneShot_ps_shotsAfterExcluded(:,k) = ...

mean(ampOfOneShotOverBins_ps_shotsAfterExcluded,2);

phaseOfOneShot_nps_shotsAfterExcluded(:,k) = ...

mean(phaseOfOneShotOverBins_nps_shotsAfterExcluded,2);

ampOfOneShot_nps_shotsAfterExcluded(:,k) = ...

mean(ampOfOneShotOverBins_nps_shotsAfterExcluded,2);

%%% xps over bins up to now

xpsOverBins_avgUpToNow(k) = mean(mean(xpsOverBins(:,1:k),1));

xpsOverBins_stdUpToNow(k) = std(mean(xpsOverBins(:,1:k),1));

xpsOverBins_ps_avgUpToNow(k) = mean(XPS_ps(1:k));

xpsOverBins_ps_stdUpToNow(k) = std(XPS_ps(1:k));

xpsOverBins_nps_avgUpToNow(k) = mean(XPS_nps(1:k));

xpsOverBins_nps_stdUpToNow(k) = std(XPS_nps(1:k));

XPS_nps_shotsAfterExcluded_avgUpToNow(k) = mean(XPS_nps_shotsAfterExcluded(1:k));

XPS_nps_shotsAfterExcluded_stdUpToNow(k) = std(XPS_nps_shotsAfterExcluded(1:k));

XPS_ps_shotsAfterExcluded_avgUpToNow(k) = mean(XPS_ps_shotsAfterExcluded(1:k));

XPS_ps_shotsAfterExcluded_stdUpToNow(k) = std(XPS_ps_shotsAfterExcluded(1:k));


XPS_shotsAfter_avgUpToNow(k) = mean(XPS_shotsAfter(1:k));

XPS_shotsAfter_stdUpToNow(k) = std(XPS_shotsAfter(1:k));

Some values like the total number of shots, the number of ‘click’ and ‘no-click’ events,

optical density and values of nonlinear phase shift are printed and plotted as each file is

loaded and analyzed.

%%% number of un/successful shots

N_ps_shots(k) = sum(successfulTagsIdx);

N_nps_shots(k) = sum(not(successfulTagsIdx));

N_nps_shotsExcluded(k) = sum(idxForNpsCases_shotsAfterExcluded);

N_Ps_shotsExcluded(k) = sum(idxForPsCases_shotsAfterExcluded);

fprintf(’number of 1s: %i (%i%%)\n’, N_ps_shots(k),...

round(100*N_ps_shots(k)/(N_ps_shots(k)+N_nps_shots(k))));

fprintf(’number of 0s: %i (%i%%)\n’, N_nps_shots(k),...

round(100*N_nps_shots(k)/(N_ps_shots(k)+N_nps_shots(k))));

fprintf(’number of 1s included: %i (%i%%)\n’, N_Ps_shotsExcluded(k),...

round(100*N_Ps_shotsExcluded(k)/(N_Ps_shotsExcluded(k)+N_nps_shotsExcluded(k))));

fprintf(’number of 0s included: %i (%i%%)\n’, N_nps_shotsExcluded(k),...

round(100*N_nps_shotsExcluded(k)/(N_Ps_shotsExcluded(k)+N_nps_shotsExcluded(k))));

subplot(3,3,1); hold on;

plot(k-1,ODf_bins(k),’ro’,k-1,mean(-log(ampOverBins(:,k)/amp0),1),’go’);

title(’final OD (red), initial OD (blue), ODtrans (green)’);


errorbar(k-1,mean(xpsOverBins(:,k),1),...

std(xpsOverBins(:,k),1)/sqrt(length(xpsOverBins(:,k))),’.’);

errorbar(k-1,xpsOverBins_avgUpToNow(k),xpsOverBins_stdUpToNow(k)/sqrt(k),’cs’);


errorbar(k-1,XPS_shotsAfter_avgUpToNow(k),XPS_shotsAfter_stdUpToNow(k)/sqrt(k),’r.’);

title(’avg phase shift per bin (blue), avg phase over bins up to now (cyan)’);

plot(0:k-1,zeros(1,k),’:k’);


errorbar(k-1,XPS_ps_shotsAfterExcluded_avgUpToNow(k),...

XPS_ps_shotsAfterExcluded_stdUpToNow(k)/sqrt(k),’gs’);

errorbar(k-1,XPS_nps_shotsAfterExcluded_avgUpToNow(k),...

XPS_nps_shotsAfterExcluded_stdUpToNow(k)/sqrt(k),’rs’);

plot(0:k-1,zeros(1,k),’k’);

title(’1s, exc (gs), 0s, exc (rs)’);


errorbar(k-1,xpsOverBins_ps_avgUpToNow(k),xpsOverBins_ps_stdUpToNow(k)/sqrt(k),’c.’);

errorbar(k-1,xpsOverBins_nps_avgUpToNow(k),xpsOverBins_nps_stdUpToNow(k)/sqrt(k),’m.’);


title(’0s (m.), 1s (c.)’);


errorbar(k-1,N_ps_shots(k),sqrt(N_ps_shots(k)),’go’);

errorbar(k-1,N_nps_shots(k),sqrt(N_nps_shots(k)),’ro’);

title(’# 0s (r), # 1s (g)’);

subplot(3,3,7);

hist(reshape(tagAmpOverBins(:,1:k),[],1),20);

title(’hist of tags’);

subplot(3,3,8);

hist(reshape(tagPhaseOverBins(:,1:k),[],1),20);

title(’hist of ph of tags’);

subplot(3,3,3); hold off;

plot(1e6*mean(phaseOfOneShot_ps_shotsAfterExcluded(:,1:k),2),’g’,’LineWidth’,2); hold on;


plot(1e6*mean(phaseOfOneShot_nps_shotsAfterExcluded(:,1:k),2),’r’,’LineWidth’,2);

title(’phase of one shot’)

subplot(3,3,9); hold off;

plot(mean(ampOfOneShot_ps_shotsAfterExcluded(:,1:k),2),’g’,’LineWidth’,2); hold on;

plot(mean(ampOfOneShot_nps_shotsAfterExcluded(:,1:k),2),’r’,’LineWidth’,2);

title(’amp of one shot’)

hold off

drawnow

toc;

end

If we choose to read the data from the workspace rather than raw files, the block above

will be ignored and the following part will run. The two blocks are similar in terms of

the analysis done but the following one reads the data from the MATLAB workspace.

else

XPS_ps = zeros(1,NBins);

XPS_nps = zeros(1,NBins);

XPS_nps_shotsAfterExcluded = zeros(1,NBins);

XPS_ps_shotsAfterExcluded = zeros(1,NBins);

XPS_shotsAfter = zeros(1,NBins);

xpsOverBins_avgUpToNow = zeros(1,NBins);

xpsOverBins_stdUpToNow = zeros(1,NBins);

xpsOverBins_ps_avgUpToNow = zeros(1,NBins);

xpsOverBins_ps_stdUpToNow = zeros(1,NBins);

xpsOverBins_nps_avgUpToNow = zeros(1,NBins);

xpsOverBins_nps_stdUpToNow = zeros(1,NBins);

XPS_nps_shotsAfterExcluded_avgUpToNow = zeros(1,NBins);


XPS_nps_shotsAfterExcluded_stdUpToNow = zeros(1,NBins);

XPS_ps_shotsAfterExcluded_avgUpToNow = zeros(1,NBins);

XPS_ps_shotsAfterExcluded_stdUpToNow = zeros(1,NBins);

XPS_shotsAfter_avgUpToNow = zeros(1,NBins);

XPS_shotsAfter_stdUpToNow = zeros(1,NBins);

N_ps_shots = zeros(1,NBins);

N_nps_shots = zeros(1,NBins);

N_nps_shotsExcluded = zeros(1,NBins);

N_Ps_shotsExcluded = zeros(1,NBins);

figure;

for k = 1:NBins

successfulTagsIdx = abs(tagAmpOverBins(:,k))>abs(tagSize);

successfulTagsIdx = [successfulTagsIdx(2:end); false];

idxForNpsCases_shotsAfterExcluded = xor(not(successfulTagsIdx),...

[false; successfulTagsIdx(1:end-1)]);

idxForNpsCases_shotsAfterExcluded(not(successfulTagsIdx)==false) = false;

idxForNpsCases_shotsAfterExcluded(1:N_shotsPerTrace:end) = false;

idxForPsCases_shotsAfterExcluded = xor(successfulTagsIdx,[false; successfulTagsIdx(1:end-1)]);

idxForPsCases_shotsAfterExcluded(successfulTagsIdx==false) = false;

idxForPsCases_shotsAfterExcluded(1:N_shotsPerTrace:end) = false;

XPS_ps(k) = mean(xpsOverBins(successfulTagsIdx,k));

XPS_ps_shotsAfterExcluded(k) = mean(xpsOverBins(idxForPsCases_shotsAfterExcluded,k));

XPS_nps(k) = mean(xpsOverBins(not(successfulTagsIdx),k));

XPS_nps_shotsAfterExcluded(k) = mean(xpsOverBins(idxForNpsCases_shotsAfterExcluded,k));

XPS_shotsAfter(k) = mean(xpsOverBins([false; successfulTagsIdx(1:end-1)],k));

%%% xps over bins up to now

xpsOverBins_avgUpToNow(k) = mean(mean(xpsOverBins(:,1:k),1));

xpsOverBins_stdUpToNow(k) = std(mean(xpsOverBins(:,1:k),1));


xpsOverBins_ps_avgUpToNow(k) = mean(XPS_ps(1:k));

xpsOverBins_ps_stdUpToNow(k) = std(XPS_ps(1:k));

xpsOverBins_nps_avgUpToNow(k) = mean(XPS_nps(1:k));

xpsOverBins_nps_stdUpToNow(k) = std(XPS_nps(1:k));

XPS_nps_shotsAfterExcluded_avgUpToNow(k) = mean(XPS_nps_shotsAfterExcluded(1:k));

XPS_nps_shotsAfterExcluded_stdUpToNow(k) = std(XPS_nps_shotsAfterExcluded(1:k));

XPS_ps_shotsAfterExcluded_avgUpToNow(k) = mean(XPS_ps_shotsAfterExcluded(1:k));

XPS_ps_shotsAfterExcluded_stdUpToNow(k) = std(XPS_ps_shotsAfterExcluded(1:k));

XPS_shotsAfter_avgUpToNow(k) = mean(XPS_shotsAfter(1:k));

XPS_shotsAfter_stdUpToNow(k) = std(XPS_shotsAfter(1:k));

N_ps_shots(k) = sum(successfulTagsIdx);

N_nps_shots(k) = sum(not(successfulTagsIdx));

N_nps_shotsExcluded(k) = sum(idxForNpsCases_shotsAfterExcluded);

N_Ps_shotsExcluded(k) = sum(idxForPsCases_shotsAfterExcluded);

%%% plotting


plot(k-1,ODf_bins(k),’ro’,k-1,ODi_bins(k),’o’,k-1,mean(-log(ampOverBins(:,k)/amp0),1),’go’);

title(’final OD (red), initial OD (blue), ODtrans (green)’);


errorbar(k-1,mean(xpsOverBins(:,k),1),...

std(xpsOverBins(:,k),1)/sqrt(length(xpsOverBins(:,k))),’s’);

errorbar(k-1,xpsOverBins_avgUpToNow(k),xpsOverBins_stdUpToNow(k)/sqrt(k),’cs’);

errorbar(k-1,XPS_shotsAfter_avgUpToNow(k),XPS_shotsAfter_stdUpToNow(k)/sqrt(k),’r.’);

title(’avg phase shift per bin (blue), avg phase over bins up to now (cyan)’);

plot(0:k-1,zeros(1,k),’:k’);



errorbar(k-1,xpsOverBins_ps_avgUpToNow(k),...

xpsOverBins_ps_stdUpToNow(k)/sqrt(k),’c.’);

errorbar(k-1,xpsOverBins_nps_avgUpToNow(k),...

xpsOverBins_nps_stdUpToNow(k)/sqrt(k),’m.’);

errorbar(k-1,XPS_ps_shotsAfterExcluded_avgUpToNow(k),...

XPS_ps_shotsAfterExcluded_stdUpToNow(k)/sqrt(k),’gs’);

errorbar(k-1,XPS_nps_shotsAfterExcluded_avgUpToNow(k),...

XPS_nps_shotsAfterExcluded_stdUpToNow(k)/sqrt(k),’rs’);


title(’avg phase shift up to now: successful (c.),...

unsuccessful (m.), psShotsExc (gs), npsShotsExc (rs)’);


errorbar(k-1,N_ps_shots(k),sqrt(N_ps_shots(k)),’go’);

errorbar(k-1,N_nps_shots(k),sqrt(N_nps_shots(k)),’ro’);

end

end

Eventually, the final values read from all files are printed and plotted.

%%

figure;

plot(mean(xpsOverBins,2),’.’)

%% choose which data bins to use for averaging & shot range selection

fprintf(’\nnumber of bins per trace: %i \n’, size(xpsOverBins,2)’);

binsToUse = input(’enter the range of the bins to use [s1:e1, s2:e2, ...],...

[default is all]: ’);

if isempty(binsToUse)

binsToUse = 1:size(xpsOverBins,2);


end

fprintf(’number of bins used: %i \n’, length(binsToUse)’);

fprintf(’\nnumber of shots per trace: %i \n’, N_shotsPerTrace’);

shotsToKeep = input(’enter the range of the shots to keep [s1:e1, s2:e2, ...],...

[default is all]: ’);

if isempty(shotsToKeep)

shotsToKeep = 1:size(xpsOverBins,1);

end

XPS = mean(xpsOverBins(shotsToKeep,binsToUse),1);

Ntotal = numel(xpsOverBins);

N_shotsUsed = length(shotsToKeep);

fprintf(’number of shots used: %i \n’, N_shotsUsed’);

fprintf(’peak phase shift (urad): %0.1f +/- %0.1f (snr: %0.1f)\n\n’...

,2/sqrt(pi)/erf(1) * 1e6*mean(XPS),2/sqrt(pi)/erf(1) * 1e6*std(XPS)./sqrt(length(XPS))...

,mean(XPS)/(std(XPS)./sqrt(length(XPS))));

if NPhotons > 0

fprintf(’phase shift per photon: %0.1f +/- %0.1f \n\n’...

, 2/sqrt(pi)/erf(1) * 1e6*mean(XPS) / NPhotons...

, 2/sqrt(pi)/erf(1) * 1e6*abs(mean(XPS)) / NPhotons * sqrt((dNPhotons/NPhotons)^2 ...

+ (mean(XPS)/(std(XPS)./sqrt(length(XPS))))^(-2)));

end

fprintf(’peak phase shift, for shots after tags: %0.1f +/- %0.1f (snr: %0.1f)\n\n’...

,2/sqrt(pi)/erf(1) * 1e6*mean(XPS_shotsAfter(binsToUse)),...

2/sqrt(pi)/erf(1) * 1e6*std(XPS_shotsAfter(binsToUse))./...

sqrt(length(XPS_shotsAfter(binsToUse)))...

,mean(XPS_shotsAfter(binsToUse))...


/(std(XPS_shotsAfter(binsToUse))./sqrt(length(XPS_shotsAfter(binsToUse)))));

fprintf(’peak phase shift for 1s: %0.1f +/- %0.1f (snr: %0.1f)\n’...

,2/sqrt(pi)/erf(1) * 1e6*mean(XPS_ps(binsToUse))...

,2/sqrt(pi)/erf(1) * 1e6*std(XPS_ps(binsToUse))./sqrt(length(XPS_ps(binsToUse)))...

,mean(XPS_ps(binsToUse))/(std(XPS_ps(binsToUse))./sqrt(length(XPS_ps(binsToUse)))));

fprintf(’peak phase shift for 0s: %0.1f +/- %0.1f (snr: %0.1f)\n\n’...

,2/sqrt(pi)/erf(1) * 1e6*mean(XPS_nps(binsToUse))...

,2/sqrt(pi)/erf(1) * 1e6*std(XPS_nps(binsToUse))./sqrt(length(XPS_nps(binsToUse)))...

,mean(XPS_nps(binsToUse))/(std(XPS_nps(binsToUse))./sqrt(length(XPS_nps(binsToUse)))));

fprintf(’peak phase shift for 1s (excluding shots after tags):...

%0.1f +/- %0.1f (snr: %0.1f)\n’...

,2/sqrt(pi)/erf(1) * 1e6*mean(XPS_ps_shotsAfterExcluded(binsToUse)),...

2/sqrt(pi)/erf(1) * 1e6*std(XPS_ps_shotsAfterExcluded(binsToUse))./...

sqrt(length(XPS_ps_shotsAfterExcluded(binsToUse)))...

,mean(XPS_ps_shotsAfterExcluded(binsToUse))...

/(std(XPS_ps_shotsAfterExcluded(binsToUse))./...

sqrt(length(XPS_ps_shotsAfterExcluded(binsToUse)))));

fprintf(’peak phase shift for 0s (excluding shots after tags):...

%0.1f +/- %0.1f (snr: %0.1f)\n\n’...

,2/sqrt(pi)/erf(1) * 1e6*mean(XPS_nps_shotsAfterExcluded(binsToUse)),...

2/sqrt(pi)/erf(1) * 1e6*std(XPS_nps_shotsAfterExcluded(binsToUse))./...

sqrt(length(XPS_nps_shotsAfterExcluded(binsToUse)))...

,mean(XPS_nps_shotsAfterExcluded(binsToUse))...

/(std(XPS_nps_shotsAfterExcluded(binsToUse))./...

sqrt(length(XPS_nps_shotsAfterExcluded(binsToUse)))));

fprintf(’number of 1s: %0.2i +/- %0.2i (%i%%) \n’, sum(N_ps_shots(binsToUse)),...

std(N_ps_shots(binsToUse))*sqrt(length(N_ps_shots(binsToUse))),...

round(100*mean(N_ps_shots(binsToUse))/...

(mean(N_ps_shots(binsToUse))+mean(N_nps_shots(binsToUse)))));

fprintf(’number of 0s: %0.2i +/- %0.2i \n’, sum(N_nps_shots(binsToUse)),...


std(N_nps_shots(binsToUse))*sqrt(length(N_nps_shots(binsToUse))));

fprintf(’number of 1s included: %0.2i +/- %0.2i \n’,...

sum(N_Ps_shotsExcluded(binsToUse)),std(N_Ps_shotsExcluded(binsToUse))...

*sqrt(length(N_Ps_shotsExcluded(binsToUse))));

fprintf(’number of 0s included: %0.2i +/- %0.2i \n’,...

sum(N_nps_shotsExcluded(binsToUse)),std(N_nps_shotsExcluded(binsToUse))...

*sqrt(length(N_nps_shotsExcluded(binsToUse))));

fprintf(’average final OD: %0.1i +/- %0.1i \n’, mean(ODf_bins(binsToUse)),...

std(ODf_bins(binsToUse))/sqrt(length(N_nps_shotsExcluded(binsToUse))));

fprintf(’average transparency OD: %0.1i +/- %0.1i \n’,...

mean(mean(-log(ampOverBins(:,binsToUse)/amp0),1)),...

std(mean(-log(ampOverBins(:,binsToUse)/amp0),1))...

/sqrt(length(N_nps_shotsExcluded(binsToUse))));

%% visualization

%%

avgPhaseOverBins = mean(avgPhaseOverTraces(:,binsToUse),2);

avgAmpOverBins = mean(avgAmpOverTraces(:,binsToUse),2);

figure;


plot(avgPhaseOverBins); title(’phase’);

plot([startingDataPoint,endDataPoint],avgPhaseOverBins([startingDataPoint,endDataPoint])...

,’r^’,’LineWidth’,3);

plot(startingDataPoint+idx([shotsToKeep(1),shotsToKeep(end)]),...

avgPhaseOverTraces(startingDataPoint+idx([shotsToKeep(1),shotsToKeep(end)]),62)...

,’r^’,’LineWidth’,3);

subplot(1,2,2); plot(-log(avgAmpOverBins)); title(’amp’);

%%

figure;

subplot(2,2,1); hold on; % hist of XPS


hist(mean(xpsOverBins,2),20);

subplot(2,2,4); hold on; % hist of the tag values

hist(tagAmpOverBins,20);

subplot(2,2,3); hold on; % hist of the tag values

hist(reshape(tagAmpOverBins,[],1),20);


hist(reshape(tagPhaseOverBins,[],1),20);

%%

figure;

subplot(1,2,1); hold on

for j = 1:numel(binsToUse)

plot(1e6*phaseOfOneShot_ps(:,binsToUse(j)),’:c’);

plot(1e6*phaseOfOneShot_nps(:,binsToUse(j)),’:m’);

plot(1e6*phaseOfOneShot_ps_shotsAfterExcluded(:,binsToUse(j)),’:g’);

plot(1e6*phaseOfOneShot_nps_shotsAfterExcluded(:,binsToUse(j)),’:r’);

end

plot(1e6*mean(phaseOfOneShot_ps(:,binsToUse),2),’c’,’LineWidth’,2);

plot(1e6*mean(phaseOfOneShot_nps(:,binsToUse),2),’m’,’LineWidth’,2);

plot(1e6*mean(phaseOfOneShot_ps_shotsAfterExcluded(:,binsToUse),2),’g’,’LineWidth’,2);

plot(1e6*mean(phaseOfOneShot_nps_shotsAfterExcluded(:,binsToUse),2),’r’,’LineWidth’,2);

plot(i_s:i_e,1e6*mean(phaseOfOneShot_nps_shotsAfterExcluded(i_s:i_e,binsToUse),2)...

,’or’,’LineWidth’,2);

plot(1:i_0,1e6*mean(phaseOfOneShot_nps_shotsAfterExcluded(1:i_0,binsToUse),2)...

,’oc’,’LineWidth’,2);

plot(endOfRange:endOfRange+i_0-1,...

1e6*mean(phaseOfOneShot_nps_shotsAfterExcluded(endOfRange:endOfRange+i_0-1,binsToUse),2)...

,’oc’,’LineWidth’,2);


subplot(1,2,2); hold on

for j = 1:numel(binsToUse)

plot(ampOfOneShot_ps(:,binsToUse(j)),’:c’);

plot(ampOfOneShot_nps(:,binsToUse(j)),’:m’);

plot(ampOfOneShot_ps_shotsAfterExcluded(:,binsToUse(j)),’:g’);

plot(ampOfOneShot_nps_shotsAfterExcluded(:,binsToUse(j)),’:r’);

end

plot(mean(ampOfOneShot_ps(:,binsToUse),2),’c’,’LineWidth’,2);

plot(mean(ampOfOneShot_nps(:,binsToUse),2),’m’,’LineWidth’,2);

plot(mean(ampOfOneShot_ps_shotsAfterExcluded(:,binsToUse),2),’g’,’LineWidth’,2);

plot(mean(ampOfOneShot_nps_shotsAfterExcluded(:,binsToUse),2),’r’,’LineWidth’,2);

The workspace can be saved to make it easier to reanalyze the data.

%%

save_ans = input(’do you want to save? [default: n]: ’,’s’);

if save_ans == ’y’

save(strcat(parentFolder,’\mat\’,fileName));

end

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