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 http://info.phys.unm.edu
Quantum Information Processing Classical Input State Preparation ψ in QUANTUM WORLD Control ψ out Measurement Classical Output
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?
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
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
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
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
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 1 2 3 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
Advantages of Continuous Measurement Weak probes -NMR -Electron transport -Off resonance atom-laser interaction Real-time feedback control System Probe Detector Controller
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.
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
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
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
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
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
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
Signal -- Continuous observation of Larmor Precession Experiment: P.S. Jessen, U of A G. Smith et al. PRL 96, 163602 2004. 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
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 0 0.5 1 1.5 2 2.5 3 3.5 4 time [ms]
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.
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
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
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 2 4 2 E HF = 0.25, 0.20, 0.15 GHz
Example: D1, F=3, Stretched state F = 3, m = 3 Fidelity 1.9.8.7.6.5 50 100 150 200 250 300 350 400 450 500 SNR
Example: D2, F=4, Cat state, 1 2 ( F = 4,m = 4 + F = 4,m = 4 ) Fidelity 1.9.8.7.6.5.4.3 SNR
Quantum State Reconstruction (Cs, F = 3) Preliminary Result Single Measurement Record signal (polarization rotation) theory experiment fidelity = 0.62 (random guess = 0.38) % error
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]
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.
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.
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.
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/0405153, to appear in PRL
http://info.phys.unm.edu/deutschgroup Trace Tessier René Stock Seth Merkel Collin Trail Iris Rappert Andrew Silberfarb