Measuring & manipulating quantum states (Fun With Photons and Atoms)

1. Measuring & manipulating quantum states(Fun With Photons and Atoms) Aephraim M. Steinberg Centre for Q. Info. & Q. Control Institute for Optical Sciences Dept. of Physics, U. of Toronto GA Tech, March 2007

2. DRAMATIS PERSONÆ Toronto quantum optics & cold atoms group: Postdocs: Matt Partlow( ...) Morgan Mitchell ( ICFO) An-Ning Zhang( IQIS) Marcelo Martinelli ( USP) Optics: Rob Adamson Kevin Resch(Wien UQIQC) Lynden(Krister) Shalm Jeff Lundeen (Oxford) Xingxing Xing Atoms: Jalani Fox (Imperial) Stefan Myrskog (BEC ECE) Ana Jofre(NIST) Mirco Siercke ( ...) Samansa Maneshi Chris Ellenor Rockson Chang Chao Zhuang Xiaoxian Liu Recent ug’s: Shannon Wang, Ray Gao, Sabrina Liao, Max Touzel, Ardavan Darabi Some helpful theorists: Daniel James, János Bergou, Pete Turner, John Sipe, Paul Brumer, Howard Wiseman, Michael Spanner,...

3. Quantum Computer Scientists 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 theend of our funding!)

4. OUTLINE The grand unified theory of physics talks: “Never underestimate the pleasure people get from being told something they already know.”

5. OUTLINE Some things you (may) already know Quantum info, entanglement, nonunitary op’s Some things you probably haven’t seen before... Preparing entangled states of N photons Subtleties of measuring multi-photon states Motional-state tomography on trapped atoms Decoherence & progress on echoes Something slightly insane, in case you found the rest of the talk too boring... Ask me after the talk if you want to know about: Dr. Aharonlove OR “How I learned to stop worrying and love negative (& complex) probabilities...”

6. What makes a computer quantum? (One partial answer...) (NB: Product states only require 2n coeff’s, but non-entangled quantum computation could equally well be performed with classical waves) We need to understand the nature of quantum information itself. How to characterize and compare quantum states? How to most fully describe their evolution in a given system? How to manipulate them? across the Danube (...Another talk, or more!)

7. How to build a quantum computer Photons don't interact (good for transmission; bad for computation) Try: atoms, ions, molecules, ... or just be clever with photons! Nonlinear optics: photon-photon interactions Generally exceedingly weak. Potential solutions: Cavity QED Better materials (1010 times better?!) Measurement as nonlinearity (KLM) Novel effects (slow light, EIT, etc) Photon-exchange effects (à la Franson) Interferometrically-enhanced nonlinearity “Moderate” nonlinearity + homodyne measurement

8. Measurement as a tool: KLM... MAGIC MIRROR: Acts differently if there are 2 photons or only 1. In other words, can be a “transistor,” or “switch,” or “quantum logic gate”... INPUT STATE OUTPUT STATE a|0> + b|1> + c|2> a|0> + b|1> – c|2> TRIGGER (postselection) ANCILLA special |i > particular |f > Knill, Laflamme, Milburn Nature 409, 46, (2001); and others since. Experiments by Franson et al., White et al., Zeilinger et al...

9. 1 Building up entanglement photon by photon by using post-selective nonlinearity

10. Highly number-entangled states("low-noon" experiment). Theory: H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002) ˘ + = A "noon" state A really odd beast: one 0o photon, one 120o photon, and one 240o photon... but of course, you can't tell them apart, let alone combine them into one mode! 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. Important factorisation:

12. Trick #1 "mode-mashing" 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. Post-selective nonlinearity

13. Trick #2 SPDC laser 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... |0> + e |2> + O(e2)  |3> + O(3) + O(2) + terms with <3 photons |0> +  |1> + O(2) * But we’re working on it (collab. with Rich Mirin’s quantum-dot group at NIST)

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

15. Experimental Setup

16. It works! Singles: Coincidences: Triple coincidences: Triples (bg subtracted): M.W. Mitchell, J.S. Lundeen, and A.M. Steinberg, Nature 429, 161 (2004)

17. 1b 4b Complete characterisation when you have incomplete information

18. Fundamentally Indistinguishablevs.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?

19. 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 ?

20. 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: The sections of the density matrix labelled “inaccessible” correspond to information about the ordering of photons with respect to inaccessible degrees of freedom. For n photons, the # of parameters scales as n3, rather than 4n Note: for 3 photons, there are 4 extra parameters – one more than just the 3 pairwise HOM visibilities. R.B.A. Adamson, L.K. Shalm, M.W. Mitchell, and A.M. Steinberg, PRL 98, 043601 (07)

21. Experimental Results No Distinguishing Info Distinguishing Info When distinguishing information is introduced the HV-VH component increases without affecting the state in the symmetric space Mixture of 45–45 and –4545 HH + VV

22. A typical 3-photon density matrix

23. My favorite cartoon about Alice (almost) and Bob

24. 1c 4b A better description than density matrices?

25. Wigner distributions on the Poincaré sphere ? 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. (Consider a purely symmetric state: N photons act like a single spin-N/2) 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.

26. 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. Dowling, Agarwal, & Schleich, PRA 49, 4101 (1993).

27. Beyond 1 or 2 photons... squeezed state 3-noon 15-noon 3 photons: 6 parameters: Euler angles + squeezing (eccentricity) + orientation + more complicated stuff 2 photons: 4 param’s: Euler angles + squeezing (eccentricity) + orientation A 1-photon pure state may be represented by a point on the surface of the Poincaré sphere, because there are only2 real parameters.

28. Measured Wigner Function Measured density matrix Squeezing Preliminary Experimental Results NOTE: To study a broad range of entangled states, a more flexible “mode-masher” is needed – the tradeoff is lower efficiency, which decreases SNR THEORY EXPERIMENT N00N state + background Ideal N00N state

29. 2 Quantum CAT scans

30. Tomography in Optical Lattices [Myrkog et al., PRA 72, 013615 (05) Kanem et al., J. Opt. B7, S705 (05)] Rb atom trapped in one of the quantum levels of a periodic potential formed by standing light field (30GHz detuning, 10s of mK depth) Complete characterisation of process on arbitrary inputs?

31. First task: measuring state populations

32. Time-resolved quantum states

33. Quantum state reconstruction D x Wait… Shift… 1 1 p p Q(0,0) = P g n W(0,0) = (-1) P S Measure ground state population n (former for HO only; latter requires only symmetry) Cf. Poyatos,Walser,Cirac,Zoller,Blatt, PRA 53, 1966 ('96) & Liebfried,Meekhof,King,Monroe,Itano,Wineland, PRL77, 4281 ('96)

34. Husimi distribution of coherent state

35. Data:"W-like" [Pg-Pe](x,p) for a mostly-excited incoherent mixture

36. Recapturing atoms after setting them into oscillation...

37. ...or failing to recapture themif you're too impatient

38. Oscillations in lattice wells (Direct probe of centre-of-mass oscillations in 1mm wells; can be thought of as Ramsey fringes or Raman pump-probe exp’t.)

39. Extracting a superoperator:prepare a complete set of input states and measure each output Likely sources of decoherence/dephasing: Real photon scattering (100 ms; shouldn't be relevant in 150 s period) Inter-well tunneling (10s of ms; would love to see it) Beam inhomogeneities (expected several ms, but are probably wrong) Parametric heating (unlikely; no change in diagonals) Other

40. Towards bang-bang error-correction:pulse echo indicates T2 ≈ 1 ms... 0 500 ms 1000 ms 1500 ms 2000 ms coherence introduced by echo pulses themselves (since they are not perfect p-pulses) Free-induction-decay signal for comparison echo after “bang” at 800 ms echo after “bang” at 1200 ms echo after “bang” at 1600 ms (bang!)

41. Cf. Hannover experiment Far smaller echo, but far better signal-to-noise ("classical" measurement of <X>) Much shorter coherence time, but roughly same number of periods – dominated by anharmonicity, irrelevant in our case. Buchkremer, Dumke, Levsen, Birkl, and Ertmer, PRL 85, 3121 (2000).

42. Echo from compound pulse Instead of a single abrupt phase shift, try different shapes. Perhaps a square (two shifts) can provide destructive interference into lossy states? Perhaps a “soft” (gaussian) pulse has fewer dangerous frequency components? Ongoing: More parameters; find best pulse. E.g., combine amplitude & phase mod. Also: optimize # of pulses.

43. Optimizing inter-band couplings Optimized 1 -> 2 coupling versus size of shift, for three pulse shapes: Even with relatively simple pulse shapes, could we get 65% excitation by choosing the optimal lattice depth? (Not in a vertical geometry, as the states become unbound.)

44. Why does our echo decay? 3D lattice (first data) Except for one minor disturbing feature: We see similar plateaux with both 1D and 3D lattices; temporal (laser) noise in addition to spatial fluctuations? Ongoing work to eliminate what noise we can & understand the rest! Finite bath memory time: So far, our atoms are free to move in the directions transverse to our lattice. In 1 ms, they move far enough to see the oscillation frequency change by about 10%... which is about 1 kHz, and hence enough to dephase them.

45. 2 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?)

46. 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.

47. 3a “Quantum Seeing in the Dark”

48. " Quantum seeing in the dark " D C BS2 BS1 The bomb must be there... yet my photon never interacted with it. (AKA: “Interaction-free” measurement) A. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993) P.G. Kwiat, H. Weinfurter, and A. Zeilinger, Sci. Am. (Nov., 1996) 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? Bomb absent: Only detector C fires Bomb present: "boom!" 1/2 C 1/4 D 1/4

49. Hardy's Paradox(for Elitzur-Vaidman “interaction-free measurements”) D+ D- C+ C- BS2+ BS2- I+ I- O- O+ W BS1+ BS1- e- e+ D+ –> e- was in D- –> e+ was in D+D- –> ? But … if they were both in, they should have annihilated!

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