1. Aephraim Steinberg Centre for Quantum Info. & Quantum Control Institute for Optical Sciences Department of Physics University of Toronto Measuring & manipulating coherence in photonic & atomic systems PITP/CQIQC Workshop: “Decoherence at the Crossroads”

2. DRAMATIS PERSONAE Toronto quantum optics & cold atoms group: Postdocs: Morgan Mitchell (  ICFO) Matt PartlowAn-Ning Zhang Optics: Rob AdamsonKevin Resch(  Zeilinger   ) Lynden(Krister) ShalmMasoud Mohseni (  Lidar) Xingxing XingJeff Lundeen (  Walmsley) Atoms: Jalani Fox (...  Hinds)Stefan Myrskog (  Thywissen) Ana Jofre(  Helmerson)Mirco Siercke Samansa ManeshiChris Ellenor Rockson ChangChao Zhuang Some helpful theorists: Daniel Lidar, János Bergou, Pete Turner, John Sipe, Paul Brumer, Howard Wiseman, Michael Spanner,...

3. OUTLINE Some things you may already know Some things you probably haven’t heard... but on which we’d love (more) collaborators! A few words about Quantum Information, about photons, and about state & process tomography Two-photon process tomography How to avoid quantum state & process tomography? Tomography of trapped atoms, and attempts at control Complete characterization given incomplete experimental capabilities How to draw Wigner functions on the Bloch sphere? “Never underestimate the pleasure people get from hearing something they already know”

4. Quantum tomography: why? 0

5. Quantum Information What's so great about it?

6. Quantum Information What's so great about it?

7. 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

8. What makes a computer quantum? across the Danube (...Another talk, or more!) 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? The danger of errors & decoherence grows exponentially with system size. The only hope for QI is quantum error correction. We must learn how to measure what the system is doing, and then correct it.

9. The Serious Problem For QI The danger of errors grows exponentially with the size of the quantum system. Without error-correction techniques, quantum computation would be a pipe dream. A major goal is to learn to completely characterize the evolution (and decoherence) of physical quantum systems in order to design and adapt error-control systems. $The tools are "quantum state tomography" and "quantum process tomography": full characterisation of the density matrix or Wigner function, and of the " $ uperoperator" which describes its time-evolution.

10. Density matrices and superoperators Two photons: HH, HV, VH, VV, or any superpositions. State has four coefficients. Density matrix has 4x4 = 16 coefficients. Superoperator has 16x16 = 256 coefficients.

11. Quantum process tomography on photon pairs 1

12. Entangled photon pairs (spontaneous parametric down-conversion) The time-reverse of second-harmonic generation. A purely quantum process (cf. parametric amplification) Each energy is uncertain, yet their sum is precisely defined. Each emission time is uncertain, yet they are simultaneous.

13. HWP QWP PBS Argon Ion Laser Beamsplitter "Black Box" 50/50 Detector B Detector A Two waveplates per photon for state preparation Two waveplates per photon for state analysis SPDC source Two-photon Process Tomography [Mitchell et al., PRL 91, 120402 (2003)]

14. Hong-Ou-Mandel Interference How often will both detectors fire together? r r t t + r 2 +t 2 = 0; total destructive interf. (if photons indistinguishable). If the photons begin in a symmetric state, no coincidences. {Exchange effect; cf. behaviour of fermions in analogous setup!} The only antisymmetric state is the singlet state |HV> – |VH>, in which each photon is unpolarized but the two are orthogonal. This interferometer is a "Bell-state filter," needed for quantum teleportation and other applications. Our Goal: use process tomography to test this filter.

15. “Measuring” the superoperator } Output DM Input HH HV VV VH } } } etc. 16 analyzer settings 16 input states Coincidencences

16. “Measuring” the superoperator Input Output DM HH HV VV VH etc. Superoperator Input Output

18. Comparison to ideal filter Measured superoperator, in Bell-state basis: A singlet-state filter would have a single peak, indicating the one transmitted state. Superoperator after transformation to correct polarisation rotations: Dominated by a single peak; residuals allow us to estimate degree of decoherence and other errors.

19. Can we avoid doing tomography? 2

20. Often, only want to look at a single figure of merit of a state (i.e. tangle, purity, etc…) Would be nice to have a method to measure these properties without needing to carry out full QST. Todd Brun showed that m th degree polynomial functions of a density matrix f m (  ) can be determined by measuring a single joint observable involving m identical copies of the state. (T. A. Brun, e-print: quant-ph/0401067) Polynomial Functions of a Density Matrix

21. For a pure state, P=1 For a maximally mixed state, P=(1/n) Quadratic  2-particle msmt needed Measuring the purity of a qubit Need two identical copies of the state Make a joint measurement on the two copies. In Bell basis, projection onto the singlet state HOM as Singlet State Filter + Pure State on either side = 100% visibility H H H H H H H HOM Visibility = Purity Mixed State = 50% visibility + V H H H H VV V Linear Purity of a Quantum State P = 1 – 2   –    –   Singlet-state probability can be measured by a singlet-state filter (HOM)

22. Use Type 1 spontaneous parametric downconversion to prepare two identical copies of a quantum state Vary the purity of the state Use a HOM to project onto the singlet Compare results to QST Coincidence Circuit Single Photon Detector Single Photon Detector Type 1 SPDC Crystal Singlet Filter Quartz Slab Quartz Slab Experimentally Measuring the Purity of a Qubit

23. Prepared the state |+45> Measured Purity from Singlet State Measurement P=0.92±0.02 Measured Purity from QST P=0.99±0.01 Results For a Pure State

24. Can a birefringent delay decohere polarization (when we trace over timing info) ? [cf. J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, and P. G. Kwiat, Phys. Rev. Lett., 90, 193601 ] Case 1: Same birefringence in each arm V V H H 100% interference Visibility = (90±2) % Case 2: Opposite birefringence in each arm V V H H 25% interference Visibility = (21±2) % Preparing a Mixed State The HOM isn’t actually insensitive to timing information.

25. The HOM is not merely a polarisation singlet-state filter Problem: Used a degree of freedom of the photon as our bath instead of some external environment The HOM is sensitive to all degrees of freedom of the photons The HOM acts as an antisymmetry filter on the entire photon state Y Kim and W. P. Grice, Phys. Rev. A 68, 062305 (2003) S. P. Kulik, M. V. Chekhova, W. P. Grice and Y. Shih, Phys. Rev. A 67,01030(R) (2003) Not a singlet filter, but an “Antisymmetry Filter”

26. Randomly rotate the half-waveplates to produce |45> and |-45> |45> |45> or |-45> Preliminary results Currently setting up LCD waveplates which will allow us to introduce a random phase shift between orthogonal polarizations to produce a variable degree of coherence Could produce a “better” maximally mixed state by using four photons. Similar to Paul Kwiat’s work on Remote State Preparation. Coincidence Circuit Visibility = (45±2) % Preparing a Mixed State

27. Tomography in optical lattices, and steps towards control... 3

28. Rb atom trapped in one of the quantum levels of a periodic potential formed by standing light field (30GHz detuning, 10s of  K depth) Tomography in Optical Lattices [Myrskog et al., PRA 72, 103615 (’05) Kanem et al., J. Opt. B 7, S705 (’05)] Complete characterisation of process on arbitrary inputs?

29. Towards QPT: Some definitions / remarks "Qbit" = two vibrational states of atom in a well of a 1D lattice Control parameter = spatial shifts of lattice (coherently couple states), achieved by phase-shifting optical beams (via AO) Initialisation: prepare |0> by letting all higher states escape Ensemble: 1D lattice contains 1000 "pancakes", each with thousands of (essentially) non-interacting atoms. No coherence between wells; tunneling is a decoherence mech. Measurement in logical basis: direct, by preferential tunneling under gravity Measurement of coherence/oscillations: shift and then measure. Typical experiment: Initialise |0> Prepare some other superposition or mixture (use shifts, shakes, and delays) Allow atoms to oscillate in well Let something happen on its own, or try to do something Reconstruct state by probing oscillations (delay + shift +measure)

30. First task: measuring state populations

31. Time-resolved quantum states

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

33. ...or failing to recapture them if you're too impatient

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

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

36. Husimi distribution of coherent state

37. Data:"W-like" [P g -P e ](x,p) for a mostly-excited incoherent mixture

38. Atomic state measurement (for a 2-state lattice, with c 0 |0> + c 1 |1>) left in ground band tunnels out during adiabatic lowering (escaped during preparation) initial statedisplaceddelayed & displaced |c 0 | 2 |c 0 + c 1 | 2 |c 0 + i c 1 | 2 |c 1 | 2

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

41. Why does our echo decay? 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. Inter-well tunneling should occur on a few-ms timescale... should one think of this as homogeneous or inhomogeneous? “How conserved” is quasimomentum?

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

43. A better "bang" pulse for QEC? Under several (not quite valid) approximations, the double-shift is a momentum displacement. We expected a momentum shift to be at least as good as a position shift. In practice: we want to test the idea of letting learning algorithms search for the best pulse shape on their own, and this is a first step. A = –60 ° t t = 0 T = 900  s measurement time position shift (previous slides) initial state A = –60 ° t variable hold delay =  t = 0 T = 900  s pulse measurement double shift (similar to a momentum shift) initial state

44. time ( microseconds ) single-shift echo (≈10% of initial oscillations) double-shift echo (≈30% of initial oscillations) Echo from compound pulse Future: More parameters; find best pulse. Step 2 (optional): figure out why it works! Also: optimize # of pulses (given imper- fection of each) Pulse 900 us after state preparation, and track oscillations

45. 0.1090.003+0.007 i-0.006+0.028 i0.14+0.037 i 0.003-0.007 i0.2590.018+0.024 i0.011-0.034 i -0.006-0.028 i0.018-0.024 i0.414-0.019-0.037 i 0.14-0.037 i0.011+0.034 i-0.019+0.037 i 0.202 A pleasant surprise from tomography… To characterize processes such as our echo pulses, we extract the completely positive map or “superoperator,” shown here in the Choi-matrix representation: () Upper left-hand quadrant indicates output density matrix expected for a ground-state input Ironic fact: when performing tomography, none of our inputs was a very pure ground state, so in this extraction, we never saw P e > 55% or so, though this predicts 70% – upon observing this superoperator, we went back and confirmed that our echo can create 70% inversion!

46. What if we try “bang-bang”? (Repeat pulses before the bath gets amnesia; trade-off since each pulse is imperfect.)

47. Some coherence out to > 3 ms now...

48. How to tell how much of the coherence is from the initial state? The superoperator for a second-order echo:

49. Some future plans... Figure out what quantity to optimize! Optimize it... (what is the limit on echo amp. from such pulses?) Tailor phase & amplitude of successive pulses to cancel out spurious coherence Study optimal number of pulses for given total time. (Slow gaussian decay down to exponential?) Complete setup of 3D lattice. Measure T 2 and study effects of tunneling BEC apparatus: reconstruct single-particle wavefunctions completely by “SPIDER”-like technique? Generalize to reconstruct single-particle Wigner functions? Watch evolution from pure single-particle functions (BEC) to mixed single-particle functions due to inter-particle interactions (free expansion? approach to Mott? etc?)

50. Measurement as a tool: Post-selective operations for the construction of novel (and possibly useful) entangled states... 4a

51. 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, 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.

52. 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(  3 ) + O(  2 ) + terms with <3 photons * But we’re working on it (collab. with Rich Mirin’s quantum-dot group at NIST)

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

54. 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)

55. The basic optical scheme

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

57. Complete characterisation when you have incomplete information 4b

58. 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?  Left  Arnold  Right  Danny OR  –  Arnold&Danny ?

59. 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 ?

60. 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:

61. When distinguishing information is introduced the HV-VH component increases without affecting the state in the symmetric space Experimental Results (2 photons) No Distinguishing InfoDistinguishing Info  H  H  +  V  V  Mixture of  45  –45  and  –45  45 

62. 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

63. 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).] 3H,0V 2H,1V 1H,2V 0H,1V

64. Conclusions Plea For Help 1.Quantum process tomography can be useful for characterizing and "correcting" quantum systems (ensemble measurements). 2.It’s actually quite “expensive” – there is still much to learn about other approaches, such as “adaptive” tomography, and “direct” measurements of quantities of interest. 3.Much work remains to be done to optimize control of systems such as optical lattices, where a limited range of operations may be feasible, and multiple sources of decoherence coexist. 4.Can we do tomography on condensed atoms, e.g., in a lattice? In what regimes will this help observe interesting (entangling) dynamics? 5.The full characterisation of systems of several “indistinguishable” photons offers a number of interesting problems, both for density matrices and for Wigner distributions.