Quantum State Reconstruction via Continuous Measurement

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 http://info.phys.unm.edu

2. QUANTUM WORLD Quantum Information Processing State Preparation Classical Input Control Measurement Classical Output

3. • State Preparation How well did we prepare ? • Control dynamics How well did we implement a map ? • Sense external fields: Metrology Under what Hamiltonian did the system evolve? Evaluating Performance

4. Require many copies Quantum State Estimation • Fundamental problem for quantum mechanics: Measure stater Finite dimensional Hilbert space, dim d • Any measurement, at mostlog2dbits/system e.g. Spin 1/2 particle,d=2: one bit. • Density operator, d2-1 real parameters Hermitian matrix with unit trace. • Required fidelity: b(d2-1) bits of information. b bits/matrix-element.

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

6. • Assign probabilities based on sampling ensemble - Standard approach: Strong measurement Each ensemble with possible accuracy of bits Quantum State Reconstruction Basic Requirements: • “Informationally Complete” Measurements

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 Mi Wasted quantum information • Need onlyblog2dbits. Wasteful whenN>>b. • Much more “quantum backaction” than necessary. • Quantum State erased after reconstruction. Alternative -- Continuous measurement

8. • Ensemble identically coupled to probe (ancilla) Ultimate resolution -- Long averaging “Projection noise” No correlations Continuous Weak Measurement Approach Measurement record Probe noise: Gaussian variance

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

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. • Work in Heisenberg Picture (control independent of r0) • Coarse-grain over detector averaging time. Measurement series • Need to generate “complete set” of operators to find Time dependent Hamiltonian, , generatesSU(d) Include decoherence (beyond usual Heisenberg picture) Given find • Stochastic linear estimation: Mathematical Formulation

12. Conditional probability Single measurement Gaussian Multidimensional Gaussian Least square fit Bayesian Filter Bayesian posterior probability

13. Optimize Entropy in Gaussian Distribution: = SNR for measurement of observable along principle axisa. Eigenvalues of : full rank Informationally complete Information Gain

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 q ex = + + - eq = + + eiq- Magnetically polarized atomic cloud Measures average spin projection:

15. Basic Tool: Off Resonance Atom-Laser Interaction nP3/2 Monochromatic Laser Alkali Atom Hyperfine structure F nS1/2 Tensor Interaction Atomic Polarizability Irreducible decomposition

16. Irreducible Tensor Decomposition Note: For detunings :

17. Signal -- Continuous observation of Larmor Precession Larmor Precession AC Stark Shift Photon Scattering Experiment: P.S. Jessen, U of A G. Smith et al. PRL 96, 163602 2004. Collapse and revival of Larmor precession

18. Enhanced Non-Linearity on Cs D1 Transition Non-linear (tensor) light shift • maximize b µDHF by probing on D1 line - Collapse & revival in Larmor precession (D1 probe) Master Equation Simulation signal [arb. units] Experiment time [ms]

19. Controlability For magnetic fields alone uncontrollable (algebra closes) The nonlinearity allows full controlability: Our example system effective Hamiltonian: But… Light shift nonlinearity is tied to decoherence.

20. Measurement Strategy Errors along primary axis given by the eigenvalues ofR 1-1/2 P(0) 2-1/2 P(M|0) • Time dependent B(t) to cover operator space, su(2F+1). Fz • In given trial, system decays due to decoherence. su(2F+1) Maximize information gain in time before decoherence, subject to “costs”

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

22. Spins of 133Cs Some atomic physics 6P3/2 F’ • Off-resonance: 6P1/2 F’ • Scattering rate: D2 D1 • Nonlinear light shift requires detuning not large compared to hyperfine splitting. F=4 6S1/2 F=3 6S1/2 Cesium Doublet 6P1/2 6P3/2 • Scattering time: 1 ms • Detuning: D1 , D2 • Measurement duration: 4 ms • Integration steps: 1000,

23. Example: D1, F=3, Stretched state 1 .9 .8 .7 .6 .5 Fidelity 50 100 150 200 250 300 350 400 450 500 SNR

24. Example: D2, F=4, Cat state, 1 .9 .8 .7 .6 .5 .4 .3 Fidelity SNR

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

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

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

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

29. Generalized measurements: Ellipticity spectroscopy. F(2) Extensions • Improved optimization (convex sets). • 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/0405153, to appear in PRL

31. http://info.phys.unm.edu/DeutschGroup Trace Tessier Andrew Silberfarb Collin Trail René Stock Iris Rappert Seth Merkel