University of New Mexico

Transcription

1 Quantum State Reconstruction via Continuous Measurement Ivan H. Deutsch, Andrew Silberfarb University of New Mexico Poul Jessen, Greg Smith University of Arizona Information Physics Group

2 Quantum Information Processing Classical Input State Preparation ψ in QUANTUM WORLD Control ψ out Measurement Classical Output

3 Evaluating Performance State Preparation How well did we prepare? ρ 0 Control dynamics [ ] How well did we implement a map? ρ t = M t ρ 0 Sense external fields: Metrology Under what Hamiltonian H( Θ) did the system evolve?

4 Quantum State Estimation Fundamental problem for quantum mechanics: Measure state ρ Finite dimensional Hilbert space, dim d Any measurement, at most log 2 d bits/system e.g. Spin 1/2 particle, d=2: one bit. Density operator, d 2-1 real parameters Hermitian matrix with unit trace. Required fidelity: b(d 2-1) bits of information. b bits/matrix-element. Require many copies ρ N = ρ N

5 Spin-1/2: density matrix 3 indep. real numbers Tr[ ρ]= 1 Re[ ρ ρ 12 ] 11 ρ = ρ + Im[ ρ 12 ] Stern-Gerlach measurement: Q-Axis Apparatus Observable Information z z x y S N Ŝ z ρ 11, ρ 22 x y N S Ŝ x Re [ ρ 12 ] similarly Ŝ y Im[ ρ 12 ] Note: can equally well rotate system 3 ways & measure ˆ S z Heisenberg picture

6 Quantum State Reconstruction Basic Requirements: Informationally Complete Measurements { M (i) (i) } m j { (i) j = 1,2,..., j } max p j { (i)} ρ Assign probabilities based on sampling ensemble - Standard approach: Strong measurement Each ensemble p (i) j = n (i) j N with possible accuracy of N log 2 d bits

7 Challenges for Standard Reconstruction Many measurements Each measurement on ensemble, d-1 independent results. Require at least d+1 measurements of different observables. Strong backaction Projective measurements destroy state. Need to prepare identical ensembles for each M i Wasted quantum information Need only blog 2 d bits. Wasteful when N>>b. Much more quantum backaction than necessary. Quantum State erased after reconstruction. Alternative -- Continuous measurement

8 Continuous Weak Measurement Approach N ˆ ρ N = ˆ ρ 0 Ensemble identically coupled to probe (ancilla) Measurement record Probe noise: Gaussian variance M(t) = NO t Signal σ 2 = 1 κ t + { σ W Noise Ultimate resolution -- Long averaging δm 2 = 1 κt Projection noise N O 2 << δm 2 No correlations

9 Advantages of Continuous Measurement Weak probes -NMR -Electron transport -Off resonance atom-laser interaction Real-time feedback control System Probe Detector Controller

10 The Basic Protocol Continuously measure observable O. Apply time dependent control to map new information onto O. Use Bayesian filter to update state-estimate given measurement record. Optimize information gain.

11 Mathematical Formulation Work in Heisenberg Picture (control independent of ρ 0 ) O t = Tr( O(t) ρ 0 )= O(t) ρ 0 Coarse-grain over detector averaging time. O i = 1 t t i + t O( t )d t M i = N O i ρ 0 + σ W i t i Measurement series Need to generate complete set of operators O i to find ρ 0 Time dependent Hamiltonian, { }, generates SU(d) Include decoherence (beyond usual Heisenberg picture) H i Stochastic linear estimation: Given { M i } find ρ 0

12 Bayesian Filter Bayesian posterior probability P( ρ 0 { M i })= AP( { M i }ρ 0 )P(ρ 0 ) Conditional probability Single measurement Gaussian ( ) 2 PM ( i ρ 0 )= C i exp M i NO i ρ 0 2σ 2 Multidimensional Gaussian P( { M i }ρ 0 )= PM ( i ρ 0 )= C exp 1 2 i δρr δρ R = N 2 σ 2 O i O i i δρ = ρ{ M i } ρ 0 ρ{ M i } = N M σ 2 i R -1 O i i Least square fit

13 Information Gain M 1 M 1,M 2 { } { } { M 1,M 2,M 3 } { M 1,M 2,M 3,M 4 } M 1,M 2 Optimize Entropy in Gaussian Distribution: S = 1 log( detr )= log λ α 2 α Eigenvalues of : R λ α = SNR for measurement of observable along principle axis α. full rank R (d 2 1) Informationally complete

14 Physical System Ensemble of alkali atoms. Total spin F, dim= 2F+1 Measurement: Couple to off-resonant laser Polarization dependent index of refraction: Faraday Rotation e x = σ + + σ θ - e θ = σ + + e iθ σ - Magnetically polarized atomic cloud Measures average spin projection: O = F z

15 Basic Tool: Off Resonance Atom-Laser Interaction e g np 3/2 ns 1/2 = ω L ω eg F ω L Monochromatic Laser ( ) Re Ee iω L t Alkali Atom Hyperfine structure F = J + I Tensor Interaction Atomic Polarizability ˆ V = ˆ α ij E i * E j t ˆ α = F e d ˆ ˆ ge d eg h( eg iγ /2) Irreducible decomposition V ˆ = ˆ α (0) E 2 r ˆ α (1) ( E * E) r ˆ α (2) E * i E j + E * j E i 2

16 Irreducible Tensor Decomposition Irreducible Component Interaction Effect on Atoms Effect on Photons scalar ˆ α (0) ~ I ˆ V ˆ (0) = c 0 E 2 I ˆ Stateindependent level shift. Scalar index of refraction. vector ˆ α (1) ~ F ˆ V ˆ (1) = c ( 1 E * E) F ˆ Zeeman-like, shift linear in m-level. Faraday rotation about atomic spin. ˆ α (2) ~ ˆ tensor ( F ± ) 2, ˆ F ± ˆ F z (3 ˆ F z 2 F 2 ). V ˆ (2) = c 2 3E ˆ F 2 E 2 ˆ F 2 Nonlinear spin dependence. Birefringence depending on 2 nd order moments. Note: For detunings >> E HF : c 0,c 1 ~ 1/ c 2 ~ E HF / 2 Γ scat

17 Signal -- Continuous observation of Larmor Precession Experiment: P.S. Jessen, U of A G. Smith et al. PRL 96, Collapse and revival of Larmor precession H ˆ = g F µb F ˆ y + c ˆ ε L F ˆ ( ) 2 i hγ s 2 ˆ 1 Larmor Precession AC Stark Shift Photon Scattering

18 Enhanced Non-Linearity on Cs D1 Transition Non-linear (tensor) light shift V NL = β hγ s ˆ ε probe F 2 - maximize β HF by probing on D1 line - Collapse & revival in Larmor precession (D1 probe) Master Equation Simulation signal [arb. units] Experiment time [ms]

19 Controlability Our example system effective Hamiltonian: H = B(t) F + hγ sβf x 2 For magnetic fields alone uncontrollable (algebra closes) F x F y F z : [ F i,f j ]= iε ijk F k The nonlinearity allows full controlability: [ F 2 n n 1 x,f y ] F x F z F y But Light shift nonlinearity is tied to decoherence.

20 Measurement Strategy Time dependent B(t) to cover operator space, su(2f+1). In given trial, system decays due to decoherence. F z su(2f+1) Errors along primary axis given by the eigenvalues of R λ 1-1/2 λ 2-1/2 P(ρ 0 ) P(M ρ 0 ) Maximize information gain in time before decoherence, subject to costs

21 Quantum State Reconstruction linear polariz. In the Lab: y B(t) θ - use NL light shift + time varying B-field to implement required evolution - measure Faraday signal F z (t) z Probe (Measure axis) Fix magnitude and B z =0 π Choose 50 points along trajectory to specify angles. Interpolate up to full sample rate. θ -π time

22 6P 3/2 6P 1/2 6S 1/2 D1 Spins of 133 Cs D2 Cesium Doublet Scattering time: 1 ms Detuning: D1 = 6 GHz, D2 = 4 GHz Measurement duration: 4 ms Integration steps: 1000, τ D = 4µs F F F=4 F=3 Some atomic physics Off-resonance: >> Γ = 2π 5.2 MHz Scattering rate: γ s = Nonlinear light shift requires detuning not large compared to hyperfine splitting. 6S 1/2 6P 1/2 6P 3/2 E HF = 9.2GHz E HF = 1.2GHz I I sat Γ E HF = 0.25, 0.20, 0.15 GHz

23 Example: D1, F=3, Stretched state F = 3, m = 3 Fidelity SNR

24 Example: D2, F=4, Cat state, 1 2 ( F = 4,m = 4 + F = 4,m = 4 ) Fidelity SNR

25 Quantum State Reconstruction (Cs, F = 3) Preliminary Result Single Measurement Record signal (polarization rotation) theory experiment fidelity = 0.62 (random guess = 0.38) % error

26 Quantum State Reconstruction (Cs, F = 3) Preliminary Result 128 Average signal (polarization rotation) theory experiment fidelity = 0.82 (random guess = 0.38) % error systematic errors time [ms]

27 What if control parameters are not known exactly? Background or unknown B-field Assume a distribution P(B i (t)) then Inhomogeneities in field intensity - include in model. Optimization naturally seeks spin-echo solution.

28 Robustness to Field Fluctuations 1 Fidelity.7 0 1% 2.5% Percent error control B-field. Use known state to estimate field and adjust simulation.

29 Extensions Improved optimization (convex sets). Generalized measurements: Ellipticity spectroscopy. F (2) Limited reconstruction: e.g. second moments. Observation of entangling dynamics. Toward feedback control of quantum features.

30 Conclusions Continuous weak measurement allows quantum state reconstruction using a single ensemble. Maximizes information gain/disturbance tradeoff. Can be useful when probes are too noisy for single quantum system strong-measurement but sufficient signal-to-noise can be seen in ensemble. State-estimation beyond the Kalman filter. New possibilties of quantum feedback control. e-print: quant-ph/ , to appear in PRL

31 Trace Tessier René Stock Seth Merkel Collin Trail Iris Rappert Andrew Silberfarb