1. Some recent experiments on weak measurements and quantum state generation Aephraim Steinberg Univ. Toronto (presently @ Institut d'Optique, Orsay)

2. Let's Make a Quantum Deal! OUTLINE The 3-box problem Overture: an alternative introduction to retrodiction, the 3-box problem, and weak measurements Experimental results Nonlocality? Hardy's Paradox and retrodiction Retrodiction is claimed to lead to a paradox in QM "Weak probabilities" seem to "resolve" the "paradox"? Experiment now possible, thanks to 2-photon "switch" Which-path experiments (collab. w/ Howard Wiseman) Old debate (Scully vs. Walls, e.g.): When which-path measurements destroy interference, must momentum necessarily be disturbed? Weak values allow one to discuss this momentum shift, and reconcile some claims of both sides (Negative values essential, once more...) And now for something completely different Non-deterministic generation of |0,3> + |3,0> "maximally path-entangled states" Phase super-resolution (Heisenberg limit?)

3. U of T quantum optics & laser cooling group: PDFs: Morgan MitchellMarcelo Martinelli (back  Brazil) Optics: Kevin Resch(  Zeilinger) Jeff Lundeen Krister Shalm Masoud Mohseni (  Lidar) Reza Mir[  real world(?)]Rob Adamson Karen Saucke (back   ) Atom Traps: Jalani FoxStefan Myrskog (  Thywissen) Ana Jofre (  NIST?) Mirco Siercke Samansa ManeshiSalvatore Maone (  real world) Chris Ellenor Some of our theory collaborators: Daniel Lidar, János Bergou, Mark Hillery, John Sipe, Paul Brumer, Howard Wiseman

4. Recall principle of weak measurements... By using a pointer with a big uncertainty (relative to the strength of the measurement interaction), one can obtain information, without creating entanglement between system and apparatus (effective "collapse"). H int =gAp x System-pointer coupling

5. 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...

6. 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.

7. 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

8. 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.

9. 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?

10. A Gedankenexperiment... e-e- e-e- e-e- e-e-

11. 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)]

12. An "application": N shutters Aharonov et al., PRA 67, 42107 ('03)

13. The implementation – A 3-path interferometer (Resch et al., Phys Lett A 324, 125('04)) 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). 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.

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

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

17. 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.

18. 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. Resch &Steinberg, PRL 92,130402 ('04)

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

20. The joint probabilities

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. Problem: Consider a collection of bombs so sensitive that a collision with any single particle (photon, electron, etc.) is guarranteed to trigger it. Suppose that certain of the bombs are defective, but differ in their behaviour in no way other than that they will not blow up when triggered. Is there any way to identify the working bombs (or some of them) without blowing them up? " Quantum seeing in the dark " (AKA: The Elitzur-Vaidman bomb experiment) A. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993) P.G. Kwiat, H. Weinfurter, and A. Zeilinger, Sci. Am. (Nov., 1996) BS1 BS2 D C Bomb absent: Only detector C fires Bomb present: "boom!"1/2 C1/4 D1/4 The bomb must be there... yet my photon never interacted with it.

23. What do you mean, interaction-free? Measurement, by definition, makes some quantity certain. This may change the state, and (as we know so well), disturb conjugate variables. How can we measure where the bomb is without disturbing its momentum (for example)? But if we disturbed its momentum, where did the momentum go? What exactly did the bomb interact with, if not our particle? It destroyed the relative phase between two parts of the particle's wave function.

24. 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 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- ? But … if they were both in, they should have annihilated! D+ e- was in

25. What does this mean? Common conclusion: We've got to be careful about how we interpret these "interaction-free measurements." You're not always free to reason classically about what would have happened if you had measured something other than what you actually did. (Do we really have to buy this?)

26. How to make the experiment possible: The "Switch"  PUMP   Coinc. Counts 22  PUMP - 2 x  LO  LO   +  PUMP 2 x  LO = 2  LO -  PUMP =  K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Rev. Lett. 87, 123603 (2001).

27. GaN Diode Laser PBS Det. H (D-)Det. V (D+) DC BS 50-50 BS1 50-50 BS2 Switch H V CC PBS Experimental Setup (W)(W)

28. H Pol DC V Pol DC 407 nm Pump

29. What do we need to measure? Ideally If D + clicks  - photon is in I - IFM + If D - clicks  + photon is in I + IFM - There is never a photon in I - and I + Annihilation at W Sometimes both D + and D - click The paradox For a Real Apparatus Vis Hor Int Vis Vert Int Vis Switch Vis Switch + Vis Int 2  7/4

30. Vs + Vmz 2  7/4 Required Interferometer Visibilities To Show Hardy’s Paradox

33. 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. [Y. Aharanov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, quant-ph/0104062]

34. MeasurementPointer Position Uncertainty IdealDirac Delta RealWidth << Change in Position WeakWidth >> Change in Position Weak Measurements  X  P  h/2   small disturbance  little system – pointer entanglement No system disturbance  simultaneous measure of different weak values. Y. Aharonov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, e-print quant-ph/0104062 (2001). Pointer(X) = exp[-(X - gA W ) 2 /  X]

35. Weak Measurements in Hardy’s Paradox Y. Aharanov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, e-print quant-ph/0104062 (2001). Det. H (D-)Det. V (D+) /2 N(I - ) N(O  ) N(I + ) N(O + ) Pol. BS2 + BS2 -

36. An experimental implementation of Hardy’s Paradox is now possible. A single-photon level switch allows photons to interact with a high efficiency. A polarization based system is now running. Once some stability problems solved, we will look at the results of weak measurements in Hardy’s Paradox.

37. PROBLEM SOLVED!(?)

38. Which-path controversy (Scully, Englert, Walther vs the world?) Suppose we perform a which-path measurement using a microscopic pointer, e.g., 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?

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

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

41. The debate since then...

42. Why the ambiguity?

43. 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...

44. 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.

45. Calibration of the weak measurement

46. 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)

47. 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.

48. 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).

49. Highly number-entangled states ("low- noon " experiment). The single-photon superposition state |1,0> + |0,1>, which may be regarded as an entangled state of two fields, is the workhorse of classical interferometry. The output of a Hong-Ou-Mandel interferometer is |2,0> + |0,2>. States such as |n,0> + |0,n> ("high-noon" states, for n large) have been proposed for high-resolution interferometry – related to "spin-squeezed" states. A number of proposals for producing these states have been made, but so far none has been observed for n>2.... until now! (But cf. related work in Vienna) Morgan W. Mitchell et al., to appear

50. [See for example H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)] ˘ Practical schemes? Important factorisation: =+ A "noon" state A really odd beast: one 0 o photon, one 120 o photon, and one 240 o photon... but of course, you can't tell them apart, let alone combine them into one mode!

51. Trick #1 Okay, we don't even have single-photon sources. But we can produce pairs of photons in down-conversion, and very weak coherent states from a laser, such that if we detect three photons, we can be pretty sure we got only one from the laser and only two from the down-conversion... SPDC laser |0> +  |2> + O(  2 ) |0> +  |1> + O(  2 )  |3> + O(  2 ) + O(  2 ) + terms with <3 photons

52. Trick #2 How to combine three non-orthogonal photons into one spatial mode? Yes, it's that easy! If you see three photons out one port, then they all went out that port. "mode-mashing"

53. Trick #3 But how do you get the two down-converted photons to be at 120 o to each other? More post-selected (non-unitary) operations: if a 45 o photon gets through a polarizer, it's no longer at 45 o. If it gets through a partial polarizer, it could be anywhere... (or nothing) (or <2 photons) (or nothing)

54. The basic optical scheme

55. More detailed schematic of experiment

56. It works! Singles: Coincidences: Triple coincidences:

57. A 3  fringe in ++- coincidences

58. Too good to publish?

59. SUMMARY Three-box paradox implemented Some more work possible on nonlocal observables Hardy's paradox implemented Setting up to perform the joint weak measurements Wiseman's proposal re which-path measurements carried out Paper in preparation What to do next? (Suggestions welcome!) 3-photon entangled state produced. What next? (Probably new sources required.) Other things I didn't have time to tell you about: Process tomography working in both photonic and atomic systems. Next steps: adaptive error correction (bang-bang, DFS,...) Optimal (POVM) discrimination of non-orthogonal states Using decoherence-free-subspaces for optical implementations of q. algorithms BEC project.... plans to probe tunneling atoms in the forbidden region Coherent control of quantum chaos in optical lattices Tunneling-induced coherence " " "

60. 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)