1. Aephraim M. Steinberg Centre for Q. Info. & Q. Control Institute for Optical Sciences Dept. of Physics, U. of Toronto Thinking Inside The Box: some experimental measurements in quantum optics BEYOND workshop (tomorrow, 2007)

2. DRAMATIS PERSONÆ Toronto quantum optics & cold atoms group: Postdocs: An-Ning Zhang(  IQIS) Morgan Mitchell (  ICFO) (HIRING!)Matt Partlow(  Energetiq)Marcelo Martinelli (  USP) Optics: Rob AdamsonKevin Resch(  Wien  UQ  IQC) Lynden(Krister) ShalmJeff Lundeen (  Oxford) Xingxing XingReza Mir (  geophysics) Atoms: Jalani Fox (  Imperial)Stefan Myrskog (  BEC  ECE) (SEARCHING!)Mirco Siercke ( ...?)Ana Jofre(  NIST  UNC) Samansa ManeshiChris Ellenor Rockson Chang Chao Zhuang Xiaoxian Liu UG’s: Max Touzel, Ardavan Darabi, Nan Yang, Michael Sitwell, Eugen Friesen Some helpful theorists: Pete Turner, Michael Spanner, Howard Wiseman, János Bergou, Masoud Mohseni, John Sipe, Paul Brumer,...

3. The 3 quantum computer scientists: see nothing (must avoid "collapse"!) hear nothing (same story) say nothing (if any one admits this thing is never going to work, that's the end of our funding!) Quantum Computer Scientists

4. OUTLINE Measurement may mean many different things.... {Forget about projection / von Neumann} Bayes’s Thm approach to weak values Thoughts on tunnelling 3-box problem joint weak values Retrodiction paradoxes Welcher Weg controversies Using measurements for good rather than for evil (à la KLM) Tomography in the presence of inaccessible information The Wigner function of the triphoton

5. Can we talk about what goes on behind closed doors? (“Postselection” is the big new buzzword in QIP... but how should one describe post-selected states?)

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. AAV, PRL 60, 1351 ('88) …. can be quite odd …

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) H int =gAp x System-pointer coupling x Initial State of Pointer x Final Pointer Readout Strong: Weak:

9. Bayesian Approach to Weak Values Note: this is the same result you get from actually performing the QM calculation (see A&V).

10. Weak measurement & tunneling times

11. Conditional probability distributions

12. A problem... These expressions can be complex. Much like early tunneling-time expressions derived via Feynman path integrals, et cetera.

13. A solution...

14. Conditional P(x) for tunneling

15. What does this mean practically?

16. An early proposal for our tunneling experiment...

17. How many ways are there to be in two places at one time? We all know even a quantum particle may not affect particles at spacelike separations. But even a classical cause may have two effects which are spacelike from each other. On the other hand, a classical particle may not have such effects. Neither would a single photon split into two paths of an interferometer. If, from an ensemble of particles, each affects only one region of spacetime, then the difference between the two will grow noisier. Perhaps the nonlocality of a tunneling particle is something deeper? AMS, in Causality and Locality in Modern Physics (Kluwer: 1998); quant-ph/9710046

18. Of course, timing information must be erased AMS, Myrskog, Moon, Kim, Fox, & Kim, Ann. Phys. 7, 593 (1998); quant-ph/9810009

19. Quantum Let’s Make a Deal 2

20. 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.

21. 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

22. 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.

23. A Gedankenexperiment... e-e- e-e- e-e- e-e-

24. 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)]

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

26. 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

27. 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.

28. Rails A and B (no shift) Rail C (pos. shift) A+B–C (neg. shift!) A negative weak value for Prob(C) Perform weak msmt on rail C. Post-select either A, B, C, or A+B–C. Compare "pointer states" (vertical profiles). K.J. Resch, J.S. Lundeen, and A.M. Steinberg, Phys. Lett. A 324, 125 (2004).

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

30. 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.

31. 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)

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

33. The joint probabilities

34. 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. (So much for “serious” nonlocality of a tunneling particle as well...)

35. “Quantum Seeing in the Dark” 3

36. 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: “Interaction-free” measurement, aka “Vaidman’s bomb”) 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.

37. 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 (for Elitzur-Vaidman “interaction-free measurements”) D- –> e+ was in D+D- –> ? But … if they were both in, they should have annihilated! D+ –> e- was in

38. The two-photon switch... OR: Is SPDC really the time-reverse of SHG? The probability of 2 photons upconverting in a typical nonlinear crystal is roughly 10  (as is the probability of 1 photon spontaneously down-converting). (And if so, then why doesn't it exist in classical e&m?)

39. Quantum Interference

40. (57% visibility) Suppression/Enhancement of Spontaneous Down-Conversion

41. Photon-photon transmission switch On average, less than one photon per pulse. One photon present in a given pulse is sufficient to switch off transmission. The photons upconvert with near-unit eff. (Peak power approx. mW/cm 2 ). The blue pump serves as a catalyst, enhancing the interaction by 10 10.

42. 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)

43. H Pol DC V Pol DC 407 nm Pump Using a “photon switch” to implement Hardy’s Paradox

44. 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? In fact, this is precisely what Aharonov et al.’s weak measurement formalism predicts for any sufficiently gentle attempt to “observe” these probabilities...

45. Weak Measurements in Hardy’s Paradox Y. Aharonov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, PLA 301, 130 (2002); quant-ph/0104062 Det. H (D-)Det. V (D+) /2 N(I - ) N(O  ) N(I + ) N(O + ) Pol. BS2 + BS2 -

46. N(I - )N(O  ) N(I + ) 011 N(O + ) 1 11 0 10  0.039±0.023 0.925±0.024 0.087±0.021  0.758±0.083 0.721±0.074N(O + ) 0.882±0.0150.663±0.0830.243±0.068N(I + ) N(O  ) N(I - ) Experimental Weak Values (“Probabilities”?) Ideal Weak Values Weak Measurements in Hardy’s Paradox

47. The Bohr-Einstein (and Scully-Walls) Debates... 4

48. Which-path controversy (Scully, Englert, Walther vs the world?) Which-path measurements destroy interference. This is usually explained via measurement backaction & HUP. Suppose we use a microscopic pointer. Is this really irreversible, as Bohr would have all measurements? Need it disturb momentum? Which is «more fundamental» – uncertainty or complementarity? [Reza Mir et al., submitted to {everywhere} – work with H. Wiseman]

49. Which-path measurements destroy interference (modify p-distrib!) The momentum distribution clearly changes Scully et al prove there is no momentum kick Walls et al prove there must be some  p > h/a. Obviously, different measurements and/or definitions. Is it possible to directly measure the momentum transfer? (for two-slit wavefunctions, not for momentum eigenstates!)

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

51. The debate since then...

52. Why the ambiguity?

53. 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... Wiseman, PLA 311, 285 (2003)

54. 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.

55. Calibration of the weak measurement

56. 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)

57. 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. The former fact agrees with Walls et al; the latter with Scully et al. For weak distributions, they may be reconciled because the distri- butions may take negative values in weak measurement.

58. Measurement as a tool: Post-selective operations for the construction of novel (and possibly useful) entangled states... 5

59. Theory: H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002) ˘ Highly number-entangled states ("low- noon " experiment). 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... M.W. Mitchell et al., Nature 429, 161 (2004) States such as |n,0> + |0,n> ("noon" states) have been proposed for high-resolution interferometry – related to "spin-squeezed" states.

60. How does this get entangled? H H 2H Non-unitary

61. How does this get entangled? H V H & V Perfectly unitary

62. How does this get entangled? H 60˚ H & 60˚ Non-unitary (Initial states orthogonal due to spatial mode; final states non-orthogonal.)

63. Trick #1 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" Post-selective nonlinearity

64. But do you really need non-unitarity? V H PBS Has this unitary, linear-optics, operation entangled the photons? Is |HV> = |+ +> - |– –> an entangled state of two photons at all, or “merely” an entangled state of two field modes? Can the two indistinguishable photons be considered individual systems? To the extent that they can, does bosonic symmetrization mean that they were always entangled to begin with? Is there any qualitative difference in the case of N>2 photons?

65. Where does the weirdness come from? HHH+VVV (not entangled by some definition) A B C If a photon winds up in each of modes {A,B,C}, then the three resulting photons are in a GHZ state – 3 clearly entangled subsystems. You may claim that no entanglement was created by the BS’s and post- selection which created the 3003 state... but then must admit that some is created by the BS’s & postselection which split it apart.

66. Trick #2 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(  3 ) + O(  2 ) + terms with <3 photons * But we’re working on it (collab. with Rich Mirin’s quantum-dot group at NIST)

67. 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)

68. The basic optical scheme

69. It works! Singles: Coincidences: Triple coincidences: Triples (bg subtracted):

70. Complete characterisation when you have incomplete information 4b 6

71. Fundamentally Indistinguishable vs. Experimentally Indistinguishable But what if when we combine our photons, there is some residual distinguishing information: some (fs) time difference, some small spectral difference, some chirp,...? This will clearly degrade the state – but how do we characterize this if all we can measure is polarisation?

72. Quantum State Tomography Distinguishable Photon Hilbert Space Indistinguishable Photon Hilbert Space ? ? Yu. I. Bogdanov, et al Phys. Rev. Lett. 93, 230503 (2004) If we’re not sure whether or not the particles are distinguishable, do we work in 3-dimensional or 4-dimensional Hilbert space? If the latter, can we make all the necessary measurements, given that we don’t know how to tell the particles apart ?

73. The sections of the density matrix labelled “inaccessible” correspond to information about the ordering of photons with respect to inaccessible degrees of freedom. The Partial Density Matrix Inaccessible information Inaccessible information The answer: there are only 10 linearly independent parameters which are invariant under permutations of the particles. One example: For n photons, the # of parameters scales as n 3, rather than 4 n Note: for 3 photons, there are 4 extra parameters – one more than just the 3 pairwise HOM visibilities.

74. A polarization measurement scheme N.B.: You could imagine different schemes – e.g., a beam-splitter network followed by independent polarizers – but you get no new information.

75. Experimental Results When distinguishing information is introduced the HV-VH component increases without affecting the state in the symmetric space No Distinguishing InfoDistinguishing Info  H  H  +  V  V  Mixture of  45  –45  and  –45  45  R.B.A. Adamson, L.K. Shalm, M.W. Mitchell, AMS, PRL 98, 043601 (2007)

76. More Photons… So the total number of operators accessible to measurement is If you have a collection of spins, what are the permutation-blind observables that describe the system? They correspond to measurements ( * ) of angular momentum operators J and m j... for N photons, J runs to N/2 E.g., for 3 photons, you need 20 msmt’s: 16 for “spin-3/2” and 4 for “spin-1/2” Open question: what is the relationship between bounds you can place on distinguishability from these 4 numbers and from the 3 pairwise singlet proj’s? ( * ) - Group theoretic approach thanks to Pete Turner R.B.A. Adamson, P.S. Turner, M.W. Mitchell, AMS, quant-ph/0612081

77. Hot off the presses (well, actually, not on them yet): Density matrix of the triphoton (80% of population in symmetric subspace) Extension to n-particle systems: R.B.A. Adamson, P.S. Turner, M.W. Mitchell, AMS, quant-ph/0612081

78. A better description than density matrices? 4b 7

79. Wigner distributions on the Poincaré sphere ? Any pure state of a spin-1/2 (or a photon) can be represented as a point on the surface of the sphere – it is parametrized by a single amplitude and a single relative phase. (Consider a purely symmetric state: N photons act like a single spin-N/2) This is the same as the description of a classical spin, or the polarisation (Stokes parameters) of a classical light field. Of course, only one basis yields a definite result, so a better description would be some “uncertainty blob” about that classical point... for spin-1/2, this uncertainty covers a hemisphere, while for higher spin it shrinks.

80. Wigner distributions on the Poincaré sphere Can such quasi-probability distributions over the “classical” polarisation states provide more helpful descriptions of the “state of the triphoton” than density matrices? “Coherent state” = N identically polarized photons “Spin-squeezed state” trades off uncertainty in H/V projection for more precision in phase angle. [Following recipe of Dowling, Agarwal, & Schleich, PRA 49, 4101 (1993).] {and cf. R.L. Stratonovich, JETP 31, 1012 (1956), I’m told} Please to note: there is nothing inside the sphere! Pure & mixed states are simply different distributions on the surface, as with W(x,p).

81. Wigner distributions on the Poincaré sphere a.a slightly “number”-squeezed state b.a highly phase-squeezed state c.the “3-noon” state movie of the evolution from 3- noon state to phase- squeezed, coherent, and “number”- squeezed states... Some polarisation states of the fully symmetric triphoton (theory– for the moment), drawn on the J=3/2 Bloch sphere: [Following recipe of Dowling, Agarwal, & Schleich, PRA 49, 4101 (1993).] {and cf. R.L. Stratonovich, JETP 31, 1012 (1956), I’m told} 3H,0V 2H,1V 1H,2V 0H,1V

82. Beyond 1 or 2 photons... 15-noon squeezed state 2 photons: 4 param’s: Euler angles + squeezing (eccentricity) + orientation 3-noon 3 photons: 6 parameters: Euler angles + squeezing (eccentricity) + orientation + more complicated stuff A 1-photon pure state may be represented by a point on the surface of the Poincaré sphere, because there are only 2 real parameters.

83. Making more triphoton states... HV(H+V) = (R + iL) (L + iR) (R+iL+L+iR)  R 2 +L 2 (R+L) = R 3 + R 2 L + RL 2 + L 3 E.g., In HV basis, H 2 V + HV 2 looks “number-squeezed”; in RL basis, phase-squeezed.

84. The Triphoton on the rack (?)

85. Another perspective on the problem

86. 1.Post-selected systems often exhibit surprising behaviour which can be probed using weak measurements. 2.These weak measurements may “resolve” various paradoxes... admittedly while creating new ones (negative probability)! 3.Post-selection can also enable us to generate novel “interactions” (KLM proposal for quantum computing), and for instance to produce useful entangled states. 4.POVMs, or generalized quantum measurements, are in some ways more powerful than textbook-style projectors 5.A modified sort of tomography is possible on “practically indistinguishable” particles The moral of the story Calculated Wigner-Poincaré function for a variety of “triphoton states”: