danielle boddy durham university – atomic & molecular physics group observing the average...

Danielle Boddy

Durham University – Atomic & Molecular Physics group

Observing the Average Trajectories of Single

Photons in a Two-Slit Interferometer

Scientific paper

Journal Club Seminar 08-02-12

Sacha Kocsis, et al., Science 332, 1170 (2011)

Outline


Introduction: - Heisenberg uncertainty principle

- Double slit: Classical picture

- Double slit: Quantum picture

- The problem so far…

- Strong measurement

- Weak measurement

- Clever theorists

- Polarization pointer

- Making the measurement

- Pictorial view of the set-up

- What do you see on the screen?

- Mathematical view

- What do we actually see?

- Is the measurement weak?Experimental set-up: - Basic experimental set-up

Results: - Interference patterns

- Trajectories

- Intensity distributions

Conclusion: - Questions?

Introduction


In classical physics, dynamics of a particle’s evolution are governed by its position and velocity.

To simultaneous know the particle’s position and velocity is to know its past, present, and future.v(t’

)

x(0) x(t’) x(t)

Used with great success in the macroscopic world.

Heisenberg uncertainty principle


Experiment cannot simultaneously determine the exact value of a component of momentum, px, of a particle and also the exact value of its corresponding coordinate, x

ΔpxΔx ≥ ħ/2

This restriction is not on the accuracy to which px or x can be measured, but on the product of ΔpxΔx in a simultaneous measurement of both.

e.g. if Δpx = 0, then Δx = ∞

Double slit: Classical picture


B

A

screen

Double slit: Quantum picture


B

A xA

xB

screen

X

Double slit: Quantum picture


B

A

screen

X

The problem so far…


In a von Neumann measurement, an observable of a system is coupled to a measurement apparatus or ‘pointer’ via its momentum.

0 1

Measurement

0 1

Induces a measurement shift

Strong measurement


0 1

Measurement

0 1


Determining which slit (position) the photon passed through induced a large uncertainty in photon momentum.

pI PxtgH ˆˆ)(ˆ

Weak measurement


0 1

Measurement

0 1


Measurement of yields little information

Clever theorists


There is a limit in which you find out everything without disturbing the system

Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)

Follow the weak measurement with just the right strong measurement

Polarization pointer


Can’t use the momentum as a measurement pointer if we want to measure it.

Need a pointer that commutes with both the momentum and position.

Set the polarization as a pointer

Making the measurement


Use calcite to perform both the weak and strong measurement.

Calcite is birefringent.

Phase shift is induced between the

components of polarization.

Figure taken from WikipediaWeak measurement of the photon momentum

Strong measurement of the photon position

Pictorial view of the set-up


B

A

( + )2

1

Weak measurement

Set polarization

Strong measurement

( + )2

1

Post-select

x

Bx

( + )2

1

What do we see on the screen?


Mathematical view


After the double slits the polarizer sets the polarization to

Initial transverse two-slit wave function

VHD 2

1

pathpolD

Set polarization

Mathematical view


Interaction Hamiltonian

1̂ˆˆ SkgH xI

After the measurement the state evolves as

Dee tSkigtHi xI 1ˆˆˆ

'

Weak Measurement

We wish to weakly observe the transverse momentum

VVHHSPp 2

ˆˆ1

Mathematical view


But because interaction is weak →Taylor expand the Hamiltonian

Such that the state evolves as

VHA 2

1where

tSkige xtSkig x

1

ˆˆ ˆˆ11

Akigt

D x ˆ2

'

Weak Measurement

i.e. the state can be written in terms of the initial state and the weak measurement

Initial state

Mathematical view


We measure the rotation of the pointer by performing a strong measurement.

Project polarization into the circular basis to get

Strong Measurement & Post-selection

ViHR 2

1 ViHL 2

1

In order to measure the final position of the photon, we must measure the rotation of the polarization.

Mathematical view


Akxigt

Dxx xfff ˆ2

'

Strong Measurement & Post-selection

VeHe

xw

xw

x kigt

kigt

fˆ

2ˆ

2

2

At a specific position, xf , we can find the weak momentum value

Phase shift between polarization components tells us about w

xk̂ xk

LR

LR

wx II

IIk 1sin

kˆ

Where is the coupling strength of the calcite to the system

What do we actual see?


The bottom pattern is undeviated by the strong measurement, but the top pattern suffers a phase shift

xk

xk

Is the measurement ‘weak’?


How do you know if the measurement is ‘weak’?


In each square we can detect a photon

The width Δ of the square must be smaller than the fringe spacing

Can treat the weak value as constant over the width of the pixel

Δ

Δ

Outline







- Weak measurement

- Clever theorists





- Mathematical view


- Is the measurement weak?

Experimental set-up: - Basic experimental set-upResults: - Interference patterns

- Trajectories



Basic experimental set-up


Single photons

from quantum

dot

50:50 beam splitte

r

VHD 2

1

Polarizer Calcite

VH ,

Components pick up a relative phase shift

xk

depends on angle of the crystal’s optic axis, the length of crystal, incident angle of photons

xk

QWP Polarizing beam splitter

CCD

g2(0) = 0.17 ± 0.04



Single photons

from quantum

dot

50:50 beam splitte

r

Polarizer Calcite

Crystal parameters are chosen to induce a small momentum-dependent polarization rotation


CCD

VeHe

xx ki

ki

22

2

1



Single photons

from quantum

dot

50:50 beam splitte

r

Polarizer Calcite

To measure how much the pointer has rotated, project polarization into circular basis using the QWP


ViHR 2

1

ViHL 2

1

CCD



Single photons

from quantum

dot

50:50 beam splitte

r

Polarizer Calcite QWP Polarizing beam splitter

ViHR 2

1

ViHL 2

1

CCD

LR

LR

wx II

IIk 1sin

kˆ

Weak momentum value



Single photons

from quantum

dot

50:50 beam splitte

r


(PBS)

CCD

To measure trajectory, increase the separation between the calcite and polarizing beam splitter.

Calcite remains in a fixed position.



Single photons

from quantum

dot

50:50 beam splitte

r


(PBS)

CCD

Trajectories are reconstructed over the range (2.75 ± 0.05) to (8.2 ± 0.1) m

Measurement result is not affected since 0ˆˆˆˆˆ,ˆ IfreefreeIfreeI HHHHHH

Outline







- Weak measurement

- Clever theorists





- Mathematical view



Experimental set-up: - Basic experimental set-up


- Trajectories

- Intensity distributionsConclusion: - Questions?

Results: Interference patterns


Pixel on CCD where

each photon is detected

corresponds to the

photon’s x position.

26 μm pixel width

sets the precision

Results: Interference patterns


Can extract

each value of kx at

each pixel using

LR

LRx

II

II

k

k 1sin1

Results: Trajectories


41 imaging planes

80 trajectories

Repeat measurement for many imaging planes along z



Photons are not constrained to follow these precise trajectories

Represent the average behaviour



Trajectories originating from one slit do not cross the central line.

Trajectories cross over dark fringes at steep angles.

Separation of planes sets the scale over which features in the trajectories can be observed.

Results: Intensity distribution


Trajectories reproduce the global interference pattern well.

Overlay trajectories on top of the measured intensity distribution.

Outline







- Weak measurement

- Clever theorists





- Mathematical view



Experimental set-up: - Basic experimental set-up


- Trajectories



Conclusion


Observed trajectories provide an intuitive picture of the way in which a single particle interferes with itself.

Information has been gained about the average momentum of the particle at each position within the interferometer

Exact interpretation of these observed trajectories will require continued investigation

Using power of weak measurements, a new prospective on the double-slit experiment was provided.

Questions?


Thanks for listening, any questions?

Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60,

1351 (1988)

Sacha Kocsis, et al., Science 332, 1170 (2011)

danielle boddy durham university – atomic & molecular physics group observing the average...

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