1. Durham University – Atomic & Molecular Physics group
Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer
Danielle Boddy
Durham University – Atomic & Molecular Physics group

2. Journal Club Seminar 08-02-12
Scientific paper
Sacha Kocsis, et al., Science 332, 1170 (2011)
University of Toronto.
June 2011.
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3. Journal Club Seminar 08-02-12
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?
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4. Journal Club Seminar 08-02-12
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)
However in the process of making observations the observer interacts with the system. Another astronomy example, when measuring the position of the moon by bouncing radar from it, yes the motion of the moon is disturbed by the measurement, but due to the very large mass of the moon, the disturbance can be ignored. It was naturally assumed by classical physicists that in the realm of microscopic systems the position and momentum of an object, such as an electron, could be determined precisely by observations in a similar way. Heisenberg and Bohr questioned this assumption.
Used with great success in the macroscopic world.
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5. 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 = ∞
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6. Double slit: Classical picture
screen
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7. Double slit: Quantum picturexA xB A X B
screen
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8. Double slit: Quantum pictureX B
screen
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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. 1
Measurement
Induces a measurement shift 1
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10. Journal Club Seminar 08-02-12
Strong measurement 1
Measurement
Induces a measurement shift 1
Determining which slit (position) the photon passed through induced a large uncertainty in photon momentum.
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11. Journal Club Seminar 08-02-12
Weak measurement 1
Measurement
Induces a measurement shift 1
Measurement of yields little information
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12. Follow the weak measurement with just the right strong measurement
Clever theorists
There is a limit in which you find out everything without disturbing the system
Follow the weak measurement with just the right strong measurement
Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)
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13. Journal Club Seminar 08-02-12
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
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14. 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.
Weak measurement of the photon momentum
Figure taken from Wikipedia
Strong measurement of the photon position
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15. Pictorial view of the set-upB A
( )
( )
( )
Set polarization
Weak measurement
Strong measurement
Post-select
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16. What do we see on the screen?
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Mathematical view
Set polarization
After the double slits the polarizer sets the polarization to
Initial transverse two-slit wave function
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Mathematical view
Weak Measurement
We wish to weakly observe the transverse momentum
Interaction Hamiltonian
After the measurement the state evolves as
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19. Journal Club Seminar 08-02-12
Mathematical view
Weak Measurement
But because interaction is weak →Taylor expand the Hamiltonian
Such that the state evolves as
where
Initial state
i.e. the state can be written in terms of the initial state and the weak measurement
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20. Strong Measurement & Post-selection Journal Club Seminar 08-02-12
Mathematical view
Strong Measurement & Post-selection
In order to measure the final position of the photon, we must measure the rotation of the polarization.
We measure the rotation of the pointer by performing a strong measurement.
Project polarization into the circular basis to get
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21. Strong Measurement & Post-selection Journal Club Seminar 08-02-12
Mathematical view
Strong Measurement & Post-selection
At a specific position, xf , we can find the weak momentum value
Phase shift between polarization components tells us about
Where is the coupling strength of the calcite to the system
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What do we actual see?
The bottom pattern is undeviated by the strong measurement, but the top pattern suffers a phase shift
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23. Is the measurement ‘weak’?
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24. 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 Δ Δ
Can treat the weak value as constant over the width of the pixel
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25. Journal Club Seminar 08-02-12
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?
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26. Basic experimental set-up
Polarizer
Polarizing beam splitter
Calcite
QWP
Single photons from quantum dot
50:50 beam splitter
CCD
g2(0) = 0.17 ± 0.04
Components pick up a relative phase shift
depends on angle of the crystal’s optic axis, the length of crystal, incident angle of photons
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27. Basic experimental set-up
Polarizer
Calcite
Polarizing beam splitter
QWP
Single photons from quantum dot
50:50 beam splitter
CCD
Crystal parameters are chosen to induce a small momentum-dependent polarization rotation
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28. Basic experimental set-up
Polarizer
Calcite
Polarizing beam splitter
QWP
Single photons from quantum dot
50:50 beam splitter
CCD
To measure how much the pointer has rotated, project polarization into circular basis using the QWP
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29. Basic experimental set-up
Polarizer
Calcite
Polarizing beam splitter
QWP
Single photons from quantum dot
50:50 beam splitter
CCD
Weak momentum value
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30. Basic experimental set-up
Polarizer
Calcite
Polarizing beam splitter (PBS)
QWP
Single photons from quantum dot
50:50 beam splitter
CCD
To measure trajectory, increase the separation between the calcite and polarizing beam splitter.
Calcite remains in a fixed position.
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31. Basic experimental set-up
Polarizer
Calcite
Polarizing beam splitter (PBS)
QWP
Single photons from quantum dot
50:50 beam splitter
CCD
Measurement result is not affected since
Trajectories are reconstructed over the range (2.75 ± 0.05) to (8.2 ± 0.1) m
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32. Journal Club Seminar 08-02-12
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?
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33. 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
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34. Results: Interference patterns
Can extract each value of kx at each pixel using
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35. Results: Trajectories
Repeat measurement for many imaging planes along z
41 imaging planes
80 trajectories
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36. Results: Trajectories
Photons are not constrained to follow these precise trajectories
Represent the average behaviour
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37. Results: Trajectories
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.
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38. Results: Intensity distribution
Overlay trajectories on top of the measured intensity distribution.
Trajectories reproduce the global interference pattern well.
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39. Journal Club Seminar 08-02-12
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?
Journal Club Seminar

40. Journal Club Seminar 08-02-12
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.
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41. Journal Club Seminar 08-02-12
Questions?
Thanks for listening, any questions?
Sacha Kocsis, et al., Science 332, 1170 (2011)
Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)
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