Measuring & manipulating coherence in photonic & atomic systems

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

2. DRAMATIS PERSONAE Toronto quantum optics & cold atoms group: Postdocs: Morgan Mitchell ( ICFO) Matt Partlow An-Ning Zhang Optics: Rob Adamson Kevin Resch(Zeilinger ) Lynden(Krister) Shalm Masoud Mohseni (Lidar) Xingxing Xing Jeff Lundeen (Walmsley) Atoms: Jalani Fox (...Hinds) Stefan Myrskog (Thywissen) Ana Jofre(Helmerson) Mirco Siercke Samansa Maneshi Chris Ellenor Rockson Chang Chao 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 A few words about Quantum Information, about photons, and about state & process tomography Some things you probably haven’t heard... but on which we’d love (more) collaborators! 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. 0 Quantum tomography: why?

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

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

7. 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!)

8. What makes a computer quantum? 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. across the Danube (...Another talk, or more!)

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

10. 1 Quantum process tomography on photon pairs

11. 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.

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

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

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

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

17. 2 Can we avoid doing tomography?

18. 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. Polynomial Functions of a Density Matrix (T. A. Brun, e-print: quant-ph/0401067) • Todd Brun showed that mth degree polynomial functions of a density matrix fm() can be determined by measuring a single joint observable involving m identical copies of the state.

19. HOM as Singlet State Filter Pure State on either side = 100% visibility + H H H H Mixed State = 50% visibility H H H V H H H + V V H V HOM Visibility = Purity Linear Purity of a Quantum State • 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 P = 1 – 2  –  –  Singlet-state probability can be measured by a singlet-state filter (HOM)

20. Experimentally Measuring the Purity of a Qubit • 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 Single Photon Detector Quartz Slab Type 1 SPDC Crystal Singlet Filter Coincidence Circuit Quartz Slab Single Photon Detector

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

22. Case 1: Same birefringence in each arm Visibility = (90±2) % H V 100% interference Case 2: Opposite birefringence in each arm V H V H H Visibility = (21±2) % V 25% interference Preparing a Mixed State 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 ] The HOM isn’t actually insensitive to timing information.

23. Not a singlet filter, but an “Antisymmetry Filter” • 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. A68, 062305 (2003) • S. P. Kulik, M. V. Chekhova, W. P. Grice and Y. Shih, Phys. Rev. A 67,01030(R) (2003)

24. |45> |45> or |-45> Could produce a “better” maximally mixed state by using four photons. Similar to Paul Kwiat’s work on Remote State Preparation. Coincidence Circuit Preparing a Mixed State Randomly rotate the half-waveplates to produce |45> and |-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 Visibility = (45±2) %

25. 3 Tomography in optical lattices, and steps towards control...

26. Tomography in Optical Lattices [Myrskog et al., PRA 72, 103615 (’05)Kanem et al., J. Opt. B 7, 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?

27. 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)

28. First task: measuring state populations

29. Time-resolved quantum states

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

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

32. 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.)

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. Atomic state measurement(for a 2-state lattice, with c0|0> + c1|1>) initial state displaced delayed & displaced left in ground band tunnels out during adiabatic lowering (escaped during preparation) |c0 + i c1 |2 |c0|2 |c0 + c1 |2 |c1|2

37. 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

38. Towards bang-bang error-correction:pulse echo indicates T2 ≈ 1 ms... 0 500 ms 1000 ms 1500 ms 2000 ms decay of 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!)

39. 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?

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

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

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

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

44. 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 T2 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?)

45. 4a Measurement as a tool: Post-selective operations for the construction of novel (and possibly useful) entangled states...

46. 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:

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

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

49. 4b Complete characterisation when you have incomplete information

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