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

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

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Outline Journal Club Seminar 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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Introduction Journal Club Seminar 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.

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Heisenberg uncertainty principle Journal Club Seminar Experiment cannot simultaneously determine the exact value of a component of momentum, p x, of a particle and also the exact value of its corresponding coordinate, x Δp x Δx ≥ ħ/2 This restriction is not on the accuracy to which p x or x can be measured, but on the product of Δp x Δx in a simultaneous measurement of both. e.g. if Δp x = 0, then Δx = ∞

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Double slit: Classical picture Journal Club Seminar B A screen

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Double slit: Quantum picture Journal Club Seminar B A xAxA xBxB screen X

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Double slit: Quantum picture Journal Club Seminar B A screen X

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The problem so far… Journal Club Seminar In a von Neumann measurement, an observable of a system is coupled to a measurement apparatus or ‘pointer’ via its momentum. 01 Measurement 01 Induces a measurement shift

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Strong measurement Journal Club Seminar Measurement 01 Induces a measurement shift Determining which slit (position) the photon passed through induced a large uncertainty in photon momentum.

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Weak measurement Journal Club Seminar Measurement 01 Induces a measurement shift Measurement of yields little information

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Clever theorists Journal Club Seminar 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

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Polarization pointer Journal Club Seminar 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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Making the measurement Journal Club Seminar 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 Wikipedia Weak measurement of the photon momentum Strong measurement of the photon position

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Pictorial view of the set-up Journal Club Seminar B A ( + ) Weak measurement Set polarization Strong measurement ( + ) Post-select ( + )

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What do we see on the screen? Journal Club Seminar

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

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

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Mathematical view Journal Club Seminar But because interaction is weak → Taylor expand the Hamiltonian Such that the state evolves as where Weak Measurement i.e. the state can be written in terms of the initial state and the weak measurement Initial state

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Mathematical view Journal Club Seminar We measure the rotation of the pointer by performing a strong measurement. Project polarization into the circular basis to get Strong Measurement & Post-selection In order to measure the final position of the photon, we must measure the rotation of the polarization.

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Mathematical view Journal Club Seminar Strong Measurement & Post-selection At a specific position, x f, 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? Journal Club Seminar The bottom pattern is undeviated by the strong measurement, but the top pattern suffers a phase shift

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Is the measurement ‘weak’? Journal Club Seminar

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

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Outline Journal Club Seminar 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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Basic experimental set-up Journal Club Seminar Single photons from quantum dot 50:50 beam splitter Polarizer Calcite Components pick up a relative phase shift depends on angle of the crystal’s optic axis, the length of crystal, incident angle of photons QWP Polarizing beam splitter CCD g 2 (0) = 0.17 ± 0.04

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Basic experimental set-up Journal Club Seminar Single photons from quantum dot 50:50 beam splitter Polarizer Calcite Crystal parameters are chosen to induce a small momentum-dependent polarization rotation QWP Polarizing beam splitter CCD

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Basic experimental set-up Journal Club Seminar Single photons from quantum dot 50:50 beam splitter Polarizer Calcite To measure how much the pointer has rotated, project polarization into circular basis using the QWP QWP Polarizing beam splitter CCD

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

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Basic experimental set-up Journal Club Seminar Single photons from quantum dot 50:50 beam splitter 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.

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Basic experimental set-up Journal Club Seminar Single photons from quantum dot 50:50 beam splitter 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

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Outline Journal Club Seminar 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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Results: Interference patterns Journal Club Seminar 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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Results: Interference patterns Journal Club Seminar Can extract each value of k x at each pixel using

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Results: Trajectories Journal Club Seminar imaging planes 80 trajectories Repeat measurement for many imaging planes along z

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

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

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Outline Journal Club Seminar 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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Conclusion Journal Club Seminar 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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Questions? Journal Club Seminar 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)

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