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

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

5. Weak Measurements

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

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

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

9. The Quantum 3-Box Problem

10. The 3-box problem 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? P A = wk = (1/3) / (1/3) = 1 P B = wk = (1/3) / (1/3) = 1 P C = wk = (-1/3) / (1/3) =  1. 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 CCD Camera MS,  A MS,  C Spatial Filter: 25um PH, a 5cm and a 1” lens BS1, PBS BS2, PBS BS3, 50/50 BS4, 50/50 Screen GP C GP B GP A /2 PD /2

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

15. Rails A and B (no shift) Rail C (pos. shift) A+B–C (neg. 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 P A, P B, and P C... Rail C Rails A and B WEAKSTRONG

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 P A and P B are both 1, what is P AB ? For AAV’s approach, one would need an interaction of the form OR: STUDY CORRELATIONS OF P A & P B... - if P A and P B always move together, then the uncertainty in their difference never changes. - if P A and P B both move, but never together, then  (P A - P B ) must increase.

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

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

21. And a final note... The result should have been obvious... |A> <B| = |A> <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. Summary You have won the fabulous vacation!You have won the fabulous vacation! We have implemented the quantum box problem and confirmed the predictions, including the strange "negative probability."We have implemented the quantum box problem and confirmed the predictions, including the strange "negative probability." New method for joint weak measurementsNew method for joint weak measurements Each photon appears to be definitely in each of two places but never both (cf. on Hardy's Paradox)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!Much more to explore in the strange magical land of weak measurements!