Playing Games with Quantum Information: Experiments with Photons and Laser-Cooled Atoms

Transcription

1 Playing Games with Quantum Information: Experiments with Photons and Laser-Cooled Atoms Interns: Grad Students: Postdocs: Supervisor: Jeff Lundeen Univ. of Toronto Dept. of Physics CAP 2003 Rockson Chang, Kjanen, Karen Saucke Stefan Myrskog, Ana Jofre, Jalani Fox, Mirco Siercke, Samansa Maneshi, Rob Adamson, Reza Mir, Chris Ellenor Morgan Mitchell, Marcello Martine Aephraim Steinberg

2 Outline The Superoperator Introduction/Review of quantum information Process Tomography with atoms in lattices Process Tomography of a quantum logic device with photons Two-Photon Switch

3 Quantum Information What's so great about it? Information is physical (which is quantum) Factoring Searching Modelling quantum systems Cryptography

4 What is a computer quantum? 1. Superposition: Ψ>= c 0 0>+c 1 1> (the qubit) Single qubit gates ie. Hadamards, Pi pulses, Waveplates H 2. Entanglement: Ψ T > = (c 0 0> 0>+c 1 1> 1>) Ψ a > Ψ b > Two qubit gates ie. Cnot, Kerr Effect, Ion Repulsion 3. Parallelism: Ψ>= c o 000>+c 1 001>+c 2 010>+c 3 011>+ f(ψ)>= c o f(000)>+c 1 f(001)>+c 2 f(010)>+c 3 f(011)>+ 2 n coefficients f(x)

5 Systems For Quantum Information Laser-cooled neutral atoms in lattices Ψ>= c 0 E 1 >+c 1 E 2 > U p E Intensity Standing Wave E 2 E 1 Polarized photons Ψ>= c 0 H>+c 1 V> λ/2 Kerr V H H Problem: Kerr Effect is too small V

6 The Real Problem The danger of errors grows exponentially with the size of the quantum system. With error-correction there is a threshold for the error-rate above which quantum computation is possible. 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.

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10 State Tomography in an Optical Lattice Measuring State Populations Initial Lattice Ground State Thermal After adiabatic decrease Excited State Well Depth 0 t(ms) Preparing a ground state 2 bound states t 1 t bound state 0 7 ms t 1 t 1 +40

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14 QPT of Driving Oscillations Operation: Resonantly shake the lattice. x t Observed Bloch Sphere Modelled Bloch Sphere from theory (Harmonic oscillator plus decoherence from previous measurement)

15 What about two-qubit processes? The lattice experiment can do process tomography on any single-qubit process. Two-particle logic gates and processes: Polarized photon pairs from spontaneous parametric downconversion. Now each density operator has 4x4=16 elements and the superoperator has 16x16=256 elements. Measurement Procedure: Prepare a complete set of 16 r For each r measure: H-H, H-V, H-RHC, V-45, etc.

16 Spontaneous Parametric Downconversion Downconversion Momentum is conserved.. Pump s k s k i k PUMP A pump photon is spontaneously converted into two lower frequency photons in a material with a nonzero χ (2) i..as well as energy ω ω s PUMP ω i ϕ PUMP = ϕ s + ϕ i

17 General Two-Photon State Production Ψ> = c 1 H> H>+c 2 e iφ V> V> Ψ> =c 1 H> H>+c 2 e iφ2 H> V> +c 3 e iφ3 V> H>+c 4 e iφ4 V> V> λ/2 λ/4 λ/2 λ/4 λ/2 λ/4 ie. With just one λ/2 we can create the singlet state Ψ> = H> V> - V> H> - By adjusting 6 waveplates we can produce a complete set of input states to measure the superoperator

18 Two-photon Process Tomography Translatable Retro-reflector Two SPDC crystals to Create HH or VV. Two waveplates per photon for state analysis HWP QWP QWP HWP Detector A PBS BBO two-crystal downconversion source. Black Box Process HWP QWP HWP QWP QWP HWP PBS Argon Ion Laser Two waveplates per photon for state preparation Detector B

19 Our Black Box The (not-so) simple 50/50 beamsplitter Codename: Bell-state Filter Bell - State F + > = HH> + VV> F - > = HH> VV> Y + > = HV> + VH> Y - > = HV> VH> r r t + = 0 t Coincidence Counts No (symmetric) No (symmetric) No (symmetric) Yes! (anti-symmetric) Uses: Quantum Teleportation, Quantum Repeaters, CNOT Our Goal: use process tomography to test this filter.

20 Process Tomography Apparatus Black Box Process

21 Hong-Ou-Mandel Interference DecScan Position (microns) Coincidences (per 50 s) > 85% visibility for HH and VV polarizations HOM acts as a filter for the Bell state: Ψ = (HV-VH)/v2 Goal: Use Quantum Process Tomography to find the superoperator which takes ρ in ρ out

22 Measuring the superoperator 1. Input: Ψ> = H> V> 2. Measurement Coincidencences 16 analyzer settings } Real Imaginary 3. Output: ρ out =

23 Repeat for 16 Input States Coincidencences Output ρ Input 16 input states } } } HH HV 16 analyzer settings etc. VV VH

24 The Urban Representation Input Output DM Superoperator HH HV VV VH etc. Input Output

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26 The PEI Representation Input Output DM Superoperator HH Input HV VV VH etc. Output

27 Testing the superoperator LL = input state Predicted N photons = 297 ± 14 Detector A HWP QWP QWP HWP PBS BBO two-crystal downconversion source. "Black Box" 50/50 Beamsplitter HWP QWP QWP HWP PBS Argon Ion Laser Detector B Observed N photons = 314

28 So, How's Our Bell-State Filter? In: Bell singlet state: Ψ = (HV-VH)/v2 1/2-1/2 Ψ = ( )= -1/2 1/2 Out: Ψ, but is a different maximally entangled state: Ψ

29 Model of real-world beamsplitter multi-layer dielectric Singlet filter AR coating f 1 f 2 45 unpolarized 50/50 dielectric beamsplitter at 702 nm (CVI Laser) birefringent element + singlet-state filter + birefringent element Best Fit: φ 1 = 0.76 p φ 2 = 0.80 p

30 Comparison to measured Superop Observed Predicted χ2 = 477 N-d.f. = 253 +/- 1σ (statistical)

31 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.

32 ϕ LO ω ω ϕ LO ϕ PUMP The Switch K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Rev. Lett. 87, (2001). 2ω ω ω Coinc. Counts ϕ PUMP - 2 x ϕ LO ϕ PUMP 2 x ϕ LO + = 2ϕ LO - ϕ PUMP = π

33 The Switch Phase chosen so that coincidences are eliminated ω ω 2ω 2ω ω ω OFF ON PQE XXXI

34 Hardy s Paradox L. Hardy, Phys. Rev. Lett. 68, 2981 (1992) C + D + D - C - Outcome Prob BS2 + I + BS2 - D + and C - 1/16 D - and C + 1/16 O + BS1 + W I - BS1 - O - C + and C - D + and D - Explosion 9/16 1/16 4/16 e + e -

35 SUMMARY ATOMS PHOTONS State Tomography in a 2 bound-state lattice Quantum process tomography: Superoperator for natural decoherence and single qubit rotations Quantum process tomography for two polarized photons Superoperator for a not so perfect Bell-state filter THE SWITCH Quantum interference allows huge enhancements of optical nonlinearities. Useful for quantum computation? Two-photon switch useful for studies of quantum weirdness (Hardy's paradox, weak measurement, )