1. Let's Make a Quantum Deal! The 3-box problem Another case where airtight classical reasoning yields seemingly contradictory information Experimental consequences of this information Actual experiment! Weak measurements shed light on Hardy's Paradox as well "Weak probabilities" obey all the constraints we expected in that example. There is no contradiction, because if negative "probabilities" are accepted, all the joint probabilities can be constructed. Which-path experiments Recall the debate: is momentum disturbed or not? How could one determine whether or not momentum had changed? Weak measurement predictions share some of the properties claimed by Scully et al. and some of those claimed by Walls et al. More negative probabilities needed... Conclusion These "probabilities" are real, measurable things......but what in the world are they? 25 Nov 2003 (some material thanks to Kevin Resch, Reza Mir, Howard Wiseman,...) Weak measurements: from the 3-box problem to Hardy's Paradox to the which-path debate

2. Recall principle of weak measurements... H int =gAp x System-pointer coupling x Initial State of Pointer x Final Pointer Readout By using a pointer with a big uncertainty, one can prevent entanglement ("collapse").

3. By the same token, no single event provides much information... xx x Initial State of Pointer x Final Pointer Readout xx But after many trials, the centre can be determined to arbitrarily good precision...

4. Predicting the past... A+B What are the odds that the particle was in a given box (e.g., box B)? B+C A+B It had to be in B, with 100% certainty.

5. Consider some redefinitions... In QM, there's no difference between a box and any other state (e.g., a superposition of boxes). What if A is really X + Y and C is really X - Y? A + B = X+B+Y B + C = X+B-Y XY

6. A redefinition of the redefinition... X + B' = X+B+Y X + C' = X+B-Y XY So: the very same logic leads us to conclude the particle was definitely in box X.

7. What does this mean? Then we conclude that if you prepare in (X + Y) + B and postselect in (X - Y) + B, you know the particle was in B. But this is the same as preparing (B + Y) + X and postselecting (B - Y) + X, which means you also know the particle was in X. If P(B) = 1 and P(X) = 1, where was the particle really? But back up: is there any physical sense in which this is true? What if you try to observe where the particle is?

8. The 3-box problem: weak msmts 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? P A = wk = (1/3) / (1/3) = 1 P B = wk = (1/3) / (1/3) = 1 P C = wk = (-1/3) / (1/3) =  1. [Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)]

9. Remember that test charge... e-e- e-e- e-e- e-e-

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

11. The implementation – A 3-path interferometer (Resch et al., quant-ph/0310091) 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

12. 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). The position of each photon is uncertain to within the beam waist... a small shift does not provide any photon with distinguishing info. But after many photons arrive, the shift of the beam may be measured.

13. 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). [There exists a natural optical explanation for this classical effect – this is left as an exercise!]

14. Data for P A, P B, and P C... Rail C Rails A and B WEAKSTRONG

15. Is the particle "really" in 2 places at once? 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.

16. 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.

17. Non-repeatable data which happen to look the way we want them to... no correlations (P AB = 1) exact calculation anticorrelated particle model

18. 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.

19. The joint probabilities

20. OutcomeProb D + and C - 1/16 D - and C + 1/16 C + and C - 9/16 D + and D - 1/16 Explosion4/16 Hardy's Paradox D- e+ was in D+D- both were in? But … if they were both in, they should have annihilated! BS1 - e-e- BS2 - O-O- C-C- D-D- I-I- BS1 + BS2 + I+I+ e+e+ O+O+ D+D+ C+C+ W

21. Probabilitiese- ine- out e+ in1 e+ out0 10 01 1 11 But what can we say about where the particles were or weren't, once D+ & D– fire? Upcoming experiment: demonstrate that "weak measurements" (à la Aharonov + Vaidman) will bear out these predictions.

22. PROBLEM SOLVED!(?)

23. Quantum Eraser (Scully, Englert, Walther) Suppose we perform a which-path measurement using a microscopic pointer, z.B., a single photon deposited into a cavity. Is this really irreversible, as Bohr would have all measurements? Is it sufficient to destroy interference? Can the information be “erased,” restoring interference?

24. Which-path measurements destroy interference (modify p-distrib!)

25. How is complementarity enforced? The fringe pattern (momentum distribution) is clearly changed – yet every moment of the momentum distribution remains the same.

26. The debate since then...

27. Why the ambiguity?

28. Weak measurements to the rescue! To find the probability of a given momentum transfer, measure the weak probability of each possible initial momentum, conditioned on the final momentum observed at the screen...

29. Convoluted implementation... Glass plate in focal plane measures P(p i ) weakly (shifting photons along y) Half-half-waveplate in image plane measures path strongly CCD in Fourier plane measures for each position x; this determines wk for each final momentum p f.

30. Calibration of the weak measurement

31. A few distributions P(p i | p f ) Note: not delta-functions; i.e., momentum may have changed. Of course, these "probabilities" aren't always positive, etc etc... EXPERIMENT THEORY (finite width due to finite width of measuring plate)

32. The distribution of the integrated momentum-transfer EXPERIMENT THEORY Note: the distribution extends well beyond h/d. On the other hand, all its moments are (at least in theory, so far) 0.

33. CONCLUSIONS Weak-measurement theory can predict the output of meas-urements without specific reference to the measurement technique. They are consistent with the surprising but seemingly airtight conclusions classical logic yields for the 3-box problem and for Hardy's Paradox. They also shed light on tunneling times, on the debate over which- path measurements, and so forth. Of course, they are merely a new way of describing predictions already implicit in QM anyway. And the price to pay is accepting very strange (negative, complex, too big, too small) weak values for observables (inc. probabilities).

34. Some references Tunneling times et cetera: Hauge and Støvneng, Rev. Mod. Phys. 61, 917 (1989) Büttiker and Landauer, PRL 49, 1739 (1982) Büttiker, Phys. Rev. B 27, 6178 (1983) Steinberg, Kwiat, & Chiao, PRL 71, 708 (1993) Steinberg, PRL 74, 2405 (1995) Weak measurements: Aharonov & Vaidman, PRA 41, 11 (1991) Aharonov et al, PRL 60, 1351 (1988) Ritchie, Story, & Hulet, PRL 66, 1107 (1991) Wiseman, PRA 65, 032111 Brunner et al., quant-ph/0306108 Resch and Steinberg, quant-ph/0310113 The 3-box problem: Aharonov et al, J Phys A 24, 2315 ('91); PRA 67, 42107 ('03) Resch, Lundeen, & Steinberg, quant-ph/0310091 Hardy's Paradox: Hardy, PRL 68, 2981 (1992) Aharonov et al, PLA 301, 130 (2001). Which-path debate: Scully et al, Nature 351, 111(1991) Storey et al, Nature 367 (1994)etc Wiseman & Harrison, N 377,584 (1995) Wiseman, PLA 311, 285 (2003)