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**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

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**Journal Club Seminar 08-02-12**

Scientific paper Sacha Kocsis, et al., Science 332, 1170 (2011) University of Toronto. June 2011. Journal Club Seminar

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**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

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**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. Journal Club Seminar

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**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 = ∞ Journal Club Seminar

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**Double slit: Classical picture**

screen Journal Club Seminar

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**Double slit: Quantum picture**

xA xB A X B screen Journal Club Seminar

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**Double slit: Quantum picture**

X B screen Journal Club Seminar

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**Journal Club Seminar 08-02-12**

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 Journal Club Seminar

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**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. Journal Club Seminar

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**Journal Club Seminar 08-02-12**

Weak measurement 1 Measurement Induces a measurement shift 1 Measurement of yields little information Journal Club Seminar

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**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) Journal Club Seminar

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**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 Journal Club Seminar

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**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 Journal Club Seminar

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**Pictorial view of the set-up**

B A ( ) ( ) ( ) Set polarization Weak measurement Strong measurement Post-select Journal Club Seminar

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**What do we see on the screen?**

Journal Club Seminar

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**Journal Club Seminar 08-02-12**

Mathematical view Set polarization After the double slits the polarizer sets the polarization to Initial transverse two-slit wave function Journal Club Seminar

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**Journal Club Seminar 08-02-12**

Mathematical view Weak Measurement We wish to weakly observe the transverse momentum Interaction Hamiltonian After the measurement the state evolves as Journal Club Seminar

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**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 Journal Club Seminar

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**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 Journal Club Seminar

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**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 Journal Club Seminar

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**Journal Club Seminar 08-02-12**

What do we actual see? The bottom pattern is undeviated by the strong measurement, but the top pattern suffers a phase shift Journal Club Seminar

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**Is the measurement ‘weak’?**

Journal Club Seminar

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**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 Journal Club Seminar

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**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

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**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 Journal Club Seminar

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**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 Journal Club Seminar

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**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 Journal Club Seminar

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**Basic experimental set-up**

Polarizer Calcite Polarizing beam splitter QWP Single photons from quantum dot 50:50 beam splitter CCD Weak momentum value Journal Club Seminar

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**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. Journal Club Seminar

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**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 Journal Club Seminar

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**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

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**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 Journal Club Seminar

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**Results: Interference patterns**

Can extract each value of kx at each pixel using Journal Club Seminar

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**Results: Trajectories**

Repeat measurement for many imaging planes along z 41 imaging planes 80 trajectories Journal Club Seminar

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**Results: Trajectories**

Photons are not constrained to follow these precise trajectories Represent the average behaviour Journal Club Seminar

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**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. Journal Club Seminar

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**Results: Intensity distribution**

Overlay trajectories on top of the measured intensity distribution. Trajectories reproduce the global interference pattern well. Journal Club Seminar

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**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

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**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. Journal Club Seminar

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**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) Journal Club Seminar

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PHYSICS DEPARTMENT. If Δx is uncertainty in position measurement along x-direction and Δp x is uncertainty in momentum in same direction then the product.

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