Efficient Quantum-State Estimation by Continuous Weak Measurement and Dynamical Control

We demonstrate a fast, robust, and nondestructive protocol for quantum-state estimation based on continuous weak measurement in the presence of a controlled dynamical evolution. Our experiment uses optically probed atomic spins as a test bed and successfully reconstructs a range of trial states with fidelities of $\sim 90\%$. The procedure holds promise as a practical diagnostic tool for the study of complex quantum dynamics, the testing of quantum hardware, and as a starting point for new types of quantum feedback control.

Figure 1

Continuous measurement and quantum-state estimation for atomic spins. (a) Schematic of our experiment. PBS: polarization beamsplitter; PD: photodetector; $\lambda /4$: quarter-wave plate. The control magnetic field has constant magnitude and is confined to the $x\mathrm{\text{−}}y$ plane; the time-dependent angle $\varphi$ is optimized for best estimation fidelity. (b) Simulated (dark line) and observed (light line) measurement signals for test states $|m_F=-3⟩$ (top), $|{\psi }_c⟩$(middle) and ${\rho }_{\mathrm{mix}}$ (bottom). (c) Input and estimated density matrices (absolute values) corresponding to the simulated and observed measurement signals.

Figure 2

Wigner function representations of an evolving spin state. (a) Snapshots of the quantum-state estimate as an initial $|m_F=-3⟩$ state undergoes Larmor precession in the presence of the probe. The nominal precession angle $\theta$ is indicated in each case. (b) Pronounced squeezing of the spin wave packet occurs at later times, and at $\theta =2868°$ the system has evolved into a coherent superposition $|{\psi }_c⟩$ of spin-up or -down wave packets. The viewpoint is fixed, except for the snapshots at $\theta =630°$ where it has been changed to show the wave packet on the opposite side of the sphere, and at $\theta =2868°$ where it has been adjusted to show both lobes of $|{\psi }_c⟩$.

Figure 3

Evolving fidelity, state estimate, and purity. Plots show the fidelity (light line) of the estimate ${\rho }_{\mathrm{ML}}$ and the largest eigenvalue (dark line) of the evolving state ${\rho }_0$ as a function of elapsed measurement time, for inputs (a) $|m_F=-3⟩$, (b) $|{\psi }_c⟩$ and (c) ${\rho }_{\mathrm{mix}}$. Inserts show the estimated density matrices (absolute values only) at a few representative times.