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

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DRAMATIS PERSONÆ Toronto quantum optics & cold atoms group: Postdocs: Matt Partlow ( ...) Morgan Mitchell ( ICFO) An-Ning Zhang( IQIS) Marcelo Martinelli ( USP) Optics: Rob AdamsonKevin Resch( Wien UQ IQC) Lynden(Krister) ShalmJeff Lundeen ( Oxford) Xingxing Xing Atoms: Jalani Fox ( Imperial)Stefan Myrskog ( BEC ECE) Ana Jofre( NIST)Mirco Siercke ( ...) Samansa ManeshiChris 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,...

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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

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

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

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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? (One partial answer...) (NB: Product states only require 2n coeff’s, but non-entangled quantum computation could equally well be performed with classical waves)

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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 (10 10 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

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special | i > a|0> + b|1> + c|2>a|0> + b|1> – c|2> Measurement as a tool: KLM... INPUT STATE ANCILLA TRIGGER (postselection) OUTPUT STATE particular | f > Knill, Laflamme, Milburn Nature 409, 46, (2001); and others since. Experiments by Franson et al., White et al., Zeilinger et al... 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”...

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

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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! States such as |n,0> + |0,n> ("noon" states) have been proposed for high-resolution interferometry – related to "spin-squeezed" states. M.W. Mitchell et al., Nature 429, 161 (2004)

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

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

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

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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

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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?

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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.

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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)

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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)

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The basic optical scheme

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Experimental Setup

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

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Complete characterisation when you have incomplete information 4b 1b

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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?

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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 ?

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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. R.B.A. Adamson, L.K. Shalm, M.W. Mitchell, and A.M. Steinberg, PRL 98, 043601 (07)

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

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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

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A typical 3-photon density matrix

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My favorite cartoon about Alice (almost) and Bob

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A better description than density matrices? 4b 1c

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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.

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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).

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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

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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.

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

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Quantum CAT scans 2

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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 [Myrkog et al., PRA 72, 013615 (05) Kanem et al., J. Opt. B7, S705 (05)] Complete characterisation of process on arbitrary inputs?

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First task: measuring state populations

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Time-resolved quantum states

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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

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Husimi distribution of coherent state

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

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Recapturing atoms after setting them into oscillation...

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

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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.)

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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

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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

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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 coherence introduced by echo pulses themselves (since they are not perfect -pulses) (bang!)

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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.

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Echo from compound pulse Ongoing: More parameters; find best pulse. E.g., combine amplitude & phase mod. Also: optimize # of pulses. 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?

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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.)

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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. 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!

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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?)

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

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“Quantum Seeing in the Dark” 3a

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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) 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.

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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

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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?)

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Quantum Interference

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

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

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Probabilitiese- ine- out e+ in1 e+ out0 10 01 1 11 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...

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N(I - )N(O ) N(I + ) 011 N(O + ) 1 11 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

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The moral of the story 1.Post-selected systems often exhibit surprising behaviour which can be probed using weak measurements. 2.Post-selection can also enable us to generate novel “interactions” (KLM proposal for quantum computing), and for instance to produce useful entangled states. Multi-photon entangled states may be built “from the ground up” – no need for high-frequency parent photons 3.Multi-photon entangled states may be built “from the ground up” – no need for high-frequency parent photons 4.A modified sort of tomography is possible on “practically indistinguishable” particles; there remain interesting questions about the characterisation of the distinguishability of >2 particles. 5.There are many exciting open issues in learning how to optimize control and error correction of quantum systems

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An Experimental Implementation of Hardy’s Paradox Jeff Lundeen, Kevin Resch Aephraim Steinberg University of Toronto June 2003 Funding by: NSERC, PRO,

An Experimental Implementation of Hardy’s Paradox Jeff Lundeen, Kevin Resch Aephraim Steinberg University of Toronto June 2003 Funding by: NSERC, PRO,

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