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arXiv:quant-ph/0402137 · Physical Review A

Deterministic Dicke-state preparation with continuous measurement and control

John K. Stockton, Ramon van Handel, Hideo Mabuchi

Published Aug 26, 2004• 160 citations
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Abstract

We characterize the long-time projective behavior of the stochastic master equation describing a continuous, collective spin measurement of an atomic ensemble both analytically and numerically. By adding state-based feedback, we show that it is possible to prepare highly entangled Dicke states deterministically.
DOI
10.1103/physreva.70.022106

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In quantum mechanics, standard or strong measurement approaches generally result in the collapse of an ensemble of wavefunctions into a stochastic mixture of eigenstates. On the other hand, continuous or weak measurements have the propensity to dynamically control the evolution of quantum states over time, guiding the trajectory of the state into non-trivial superpositions and maintaining state purity. This kind of measurement-induced state steering is of great theoretical and experimental interest for the harnessing of quantum bits or "qubits", which are the fundamental unit of the emerging quantum computer. We explore continuous measurement-based quantum state stabilization through linear feedback control for a single quantum bit. By applying a time-varying Rabi drive that includes a linear feedback term, we show that the fixed points of the continuous measurement may be relocated. Analytical derivations of Itô Stochastic Master Equations being ideal models, we employ them to derive the projected ensemble average while utilizing numerical simulations to characterize the stability of the set of possible fixed points, as well as their modified collapse time-scales. We include the effects of realistic experimental non-idealities, such as environmental energy relaxation, dephasing, time-delay, and inefficient measurement. Ultimately, we discuss potential experimental implementations from collaborating universities.
Abstract Achieving unit fidelity in quantum state preparation is often impossible in the presence of environmental decoherence. While continuous monitoring and feedback control can improve fidelity, perfect state preparation remains elusive in many scenarios. Inspired by quantum speed limits, we derive a fundamental bound on the steady-state average fidelity achievable via continuous monitoring and feedback control. This bound depends only on the unconditional Lindblad dynamics, the Hamiltonian variance, and the target state. We also adapt the bound to the case of Markovian feedback strategies. We then focus on preparing Dicke states in an atomic ensemble subject to collective damping and dispersive coupling. By imposing additional constraints on control Hamiltonians and monitoring strategies, we derive tighter fidelity bounds. Finally, we propose specific control strategies and validate them using reinforcement learning. Benchmarking their performance against our theoretical bounds highlights the relevance and usefulness of these bounds in characterizing quantum feedback control strategies.
We combine the concept of Bell measurements, in which two systems are projected into a maximally entangled state, with the concept of continuous measurements, which concerns the evolution of a continuously monitored quantum system. For such time-continuous Bell measurements we derive the corresponding stochastic Schrödinger equations, as well as the unconditional feedback master equations. Our results apply to a wide range of physical systems, and are easily adapted to describe an arbitrary number of systems and measurements. Time-continuous Bell measurements therefore provide a versatile tool for the control of complex quantum systems and networks. As examples we show that (i) two two-level systems can be deterministically entangled via homodyne detection, tolerating photon loss up to 50%, and (ii) a quantum state of light can be continuously teleported to a mechanical oscillator, which works under the same conditions as are required for optomechanical ground-state cooling.
We present a Dicke state preparation scheme which uses global control of NNN spin qubits: our scheme is based on the standard phase estimation algorithm, which estimates the eigenvalue of a unitary operator. The scheme prepares a Dicke state nondeterministically by collectively coupling the spins to an ancilla qubit via a ZZZZZZ interaction, using $\ensuremath{\lceil}{log}_{2}N\ensuremath{\rceil}+1$ ancilla qubit measurements. The preparation of such Dicke states can be useful if the spins in the ensemble are used for magnetic sensing: we discuss a possible realization using an ensemble of electronic spins located at diamond nitrogen-vacancy centers coupled to a single superconducting flux qubit. We also analyze the effect of noise and limitations in our scheme.
Dicke states ∣J=N/2,m⟩|J=N/2,m\rangle∣J=N/2,m⟩ of a collection of NNN atoms were central to Dicke's theory of superradiance. Except for the fully polarized end states, they are entangled many-body states; in particular, the single-excitation Dicke state is the W state well known in quantum information science. However, deterministic preparation of Dicke states with a prescribed spin projection remains challenging, especially without relying on postselection or heralding. Here we present a detuning-programmed Hamiltonian protocol using atoms or qubits in a dispersive cavity subject to a coherent transverse drive. In the absence of the drive, the off-resonant cavity produces an effective collective-spin interaction. By combining this cavity-mediated interaction with the coherent drive, tuning the atom--drive detuning enables the protocol, in principle, to target any allowed Dicke state along the symmetric Dicke ladder. Starting from the transverse-drive ground state, adiabatic ground-state interpolation prepares the selected Dicke state by ramping down the drive strength while ramping up the cavity-mediated interaction. After preparation, tuning the cavity into resonance with the atoms or qubits provides a direct way to probe the collective-emission response of the prepared Dicke state. We discuss implementation with superconducting circuit QED and show that the prepared states, especially the central Dicke state with m=0m=0m=0, provide resources for quantum sensing with Heisenberg-limited scaling.