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

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

B

A

screen

Double slit: Quantum picture

Journal Club Seminar 08-02-12

B

A xA

xB

screen

X

Double slit: Quantum picture

Journal Club Seminar 08-02-12

B

A

screen

X

The problem so far…

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

0 1

Measurement

0 1

Induces a measurement shift

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

pI PxtgH ˆˆ)(ˆ

Weak measurement

Journal Club Seminar 08-02-12

0 1

Measurement

0 1

Induces a measurement shift

Measurement of yields little information

Clever theorists

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

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?

Journal Club Seminar 08-02-12

Mathematical view

Journal Club Seminar 08-02-12

After the double slits the polarizer sets the polarization to

Initial transverse two-slit wave function

VHD 2

1

pathpolD

Set polarization

Mathematical view

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

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?

Journal Club Seminar 08-02-12

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

xk

xk

Is the measurement ‘weak’?

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

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-upResults: - Interference patterns

- Trajectories

- Intensity distributions

Conclusion: - Questions?

Basic experimental set-up

Journal Club Seminar 08-02-12

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

Basic experimental set-up

Journal Club Seminar 08-02-12

Single photons

from quantum

dot

50:50 beam splitte

r

Polarizer Calcite

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

QWP Polarizing beam splitter

CCD

VeHe

xx ki

ki

22

2

1

Basic experimental set-up

Journal Club Seminar 08-02-12

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

QWP Polarizing beam splitter

ViHR 2

1

ViHL 2

1

CCD

Basic experimental set-up

Journal Club Seminar 08-02-12

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

Basic experimental set-up

Journal Club Seminar 08-02-12

Single photons

from quantum

dot

50:50 beam splitte

r

Polarizer Calcite QWP Polarizing beam splitter

(PBS)

CCD

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

Calcite remains in a fixed position.

Basic experimental set-up

Journal Club Seminar 08-02-12

Single photons

from quantum

dot

50:50 beam splitte

r

Polarizer Calcite QWP Polarizing beam splitter

(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

Journal Club Seminar 08-02-12

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 distributionsConclusion: - Questions?

Results: Interference patterns

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

Can extract

each value of kx at

each pixel using

LR

LRx

II

II

k

k 1sin1

Results: Trajectories

Journal Club Seminar 08-02-12

41 imaging planes

80 trajectories

Repeat measurement for many imaging planes along z

Results: Trajectories

Journal Club Seminar 08-02-12

Photons are not constrained to follow these precise trajectories

Represent the average behaviour

Results: Trajectories

Journal Club Seminar 08-02-12

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

Journal Club Seminar 08-02-12

Trajectories reproduce the global interference pattern well.

Overlay trajectories on top of the measured intensity distribution.

Outline

Journal Club Seminar 08-02-12

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?

Conclusion

Journal Club Seminar 08-02-12

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?

Journal Club Seminar 08-02-12

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)