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Optical implementation of the Quantum Box Problem

AKA: Let's Make a Quantum Deal!. Optical implementation of the Quantum Box Problem. Kevin Resch Jeff Lundeen Aephraim Steinberg. Department of Physics, University of Toronto. Motivation: subensembles, postselection, and "Let's Make a Deal" Weak measurements

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Optical implementation of the Quantum Box Problem

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  1. AKA: Let's Make a Quantum Deal! Optical implementation of the Quantum Box Problem Kevin Resch Jeff Lundeen Aephraim Steinberg Department of Physics, University of Toronto

  2. Motivation: subensembles, postselection, and "Let's Make a Deal" Weak measurements The Quantum 3-Box Problem An optical implementation Can a particle be in 2 places at 1 time? Summary Outline Funding (sources not scared to admit association with this research):

  3. Motivation In QM, one can make predictions (probabilities) about future observables. Can one "retrodict" anything about past observables? What can one say about "subensembles" defined both by preparation & post-selection? N.B. Questions about postselected subensembles becoming more and more widespread in quantum optics, quantum info, etc. [Cf. Wiseman, PRA 65, 032111 ('02) and quant-ph/0303139]

  4. A+B+C A+B–C Pick a box, any box... What are the odds that the particle was in a given box?

  5. Weak Measurements

  6. Conditional measurements(Aharonov, Albert, and Vaidman) Measurement of A Reconciliation: measure A "weakly." Poor resolution, but little disturbance. …. can be quite odd … AAV, PRL 60, 1351 ('88) Prepare a particle in |i> …try to "measure" some observable A… postselect the particle to be in |f> Does <A> depend more on i or f, or equally on both? Clever answer: both, as Schrödinger time-reversible. Conventional answer: i, because of collapse.

  7. Initial State of Pointer Final Pointer Readout Hint=gApx System-pointer coupling x x Well-resolved states System and pointer become entangled Decoherence / "collapse" Large back-action A (von Neumann) Quantum Measurement of A

  8. A Weak Measurement of A Initial State of Pointer Final Pointer Readout Hint=gApx System-pointer coupling x x Poor resolution on each shot. Negligible back-action (system & pointer separable) Mean pointer shift is given by <A>wk.

  9. The Quantum 3-Box Problem

  10. The 3-box problem PA = < |A><A| >wk = (1/3) / (1/3) = 1 PB = < |B><B| >wk = (1/3) / (1/3) = 1 PC = < |C><C|>wk = (-1/3) / (1/3) = -1. Prepare a particle in a symmetric superposition of three boxes: A+B+C. Look to find it in this other superposition: A+B-C. Ask: between preparation and detection, what was the probability that it was in A? B? C? Questions: were these postselected particles really all in A and all in B? can this negative "weak probability" be observed? [Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)]

  11. Aharonov's N shutters PRA 67, 42107 ('03)

  12. An Optical Implementation

  13. The implementation – A 3-path interferometer Diode Laser Spatial Filter: 25um PH, a 5cm and a 1” lens GP A l/2 BS1, PBS l/2 MS, fA GP B BS2, PBS BS3, 50/50 CCD Camera BS4, 50/50 GP C MS, fC l/2 Screen PD

  14. The pointer... • Use transverse position of each photon as pointer • Weak measurements can be performed by tilting a glass optical flat, where effective q Mode A Flat gt cf. Ritchie et al., PRL 68, 1107 ('91).

  15. A+B–C (neg. shift!) Rail C (pos. shift) Rails A and B (no shift) A negative weak value Perform weak msmt on rail C. Post-select either A, B, C, or A+B–C. Compare "pointer states" (vertical profiles).

  16. Data for PA, PB, and PC... Rails A and B Rail C WEAK STRONG STRONG

  17. So can one really "detect" that a particle is in box A and that it is in box B ????

  18. Measuring joint probabilities • If PA and PB are both 1, what is PAB? • For AAV’s approach, one would need an interaction of the form OR: STUDY CORRELATIONS OF PA & PB... - if PA and PB always move together, then the uncertainty in their difference never changes. - if PA and PB both move, but never together, then D(PA - PB) must increase.

  19. Practical Measurement of PAB Use two pointers (the two transverse directions) and couple to both A and B; then use their correlations to draw conclusions about PAB. We have shown that the real part of PABW can be extracted from such correlation measurements:

  20. Extracting the joint probability... anticorrelated particle model exact calculation no correlations (PAB = 1)

  21. And a final note... The result should have been obvious... |A><A| |B><B| = |A><A|B><B| is identically zero because A and B are orthogonal. Even in a weak-measurement sense, a particle can never be found in two orthogonal states at the same time.

  22. You have won the fabulous vacation! We have implemented the quantum box problem and confirmed the predictions, including the strange "negative probability." New method for joint weak measurements Each photon appears to be definitely in each of two places but never both (cf.Aharonov et al., Phys. Lett. A 301, 130 (2002)on Hardy's Paradox) Much more to explore in the strange magical land of weak measurements! Summary

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