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arXiv:2306.00868

Stochastic Mean-field Theory for Conditional Spin Squeezing by Homodyne Probing of Atom-Cavity Photon Dressed States

ZhiQing Zhang, HaiZhong Guo, Lingrui Wang, Gang Chen, Chongxin Shan, Klaus Mølmer, Yuan Zhang

Published Jun 1, 2023• 1 citation
arXivHTMLPDF

Abstract

Projective measurements of collective observables can be employed to herald the preparation of entangled states of quantum systems, and the resulting conditional dynamics is usually handled by stochastic master equation (SME) for small systems, and by an approximate Gaussian-state formalism for large systems. In this work, we present an alternative technique by developing a stochastic variant of cumulant mean-field theory, benchmark it against an exact stochastic collective density matrix approach by the simulations of hundreds of identical two-level atoms. More importantly, we demonstrate its full power by studying the conditional spin squeezing of thousands of three-level atoms coupled strongly with an optical cavity subject to individual decay and dephasing, and by simulating the experimental protocol to reveal formation and detection of the spin squeezed state. The proposed technique might be further extended to study more exotic quantum-measurement effects of large quantum systems, such as deterministic spin squeezing with quantum feedback, spin squeezing of optical clock transitions, and retrodictive spin squeezing by posterior measurements, and so on.

DOI
10.48550/arxiv.2306.00868

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Projective measurements of collective observables can be employed to herald the preparation of entangled states of quantum systems, and the resulting conditional dynamics is usually handled by stochastic master equation (SME) for small systems, and by an approximate Gaussian-state formalism for large systems. In this work, we present an alternative technique by developing a stochastic variant of cumulant mean-field theory, benchmark it against an exact stochastic collective density matrix approach by the simulations of hundreds of identical two-level atoms. More importantly, we demonstrate its full power by studying the conditional spin squeezing of thousands of three-level atoms coupled strongly with an optical cavity subject to individual decay and dephasing, and by simulating the experimental protocol to reveal formation and detection of the spin squeezed state. The proposed technique might be further extended to study more exotic quantum-measurement effects of large quantum systems, such as deterministic spin squeezing with quantum feedback, spin squeezing of optical clock transitions, and retrodictive spin squeezing by posterior measurements, and so on.
In this article, we present a stochastic collective density-matrix approach to numerically study the conditional dynamics of large systems of identical particles in the presence of individual and collective dissipation, as described by stochastic master equations. To illustrate the application potential of the approach, we use it to study the conditional spin squeezing of hundreds of identical atoms in a bad optical cavity subject to homodyne detection. We demonstrate that the conditional spin squeezing can be vividly illustrated by a Gaussian-like distribution on the quantum states of the atomic ensemble, and we examine the influence of various processes, such as the individual atomic dissipation, on this distribution. In the future, the present approach can be used to reveal measurement effects of multiple-level atoms system, to explore generation of exotic nonclassical states, such as cat states, and to gauge approximate approaches, such as the stochastic mean-field approach.
Stochastic master equations are often used to describe conditional spin squeezing of atomic ensemble, but are limited so far to the systems with few atoms due to the exponentially increased Hilbert space. In this article, we present an exact numerical solution of these equations for systems with identical atoms by mapping identical density matrix elements to a single quantity characterized by collective quantum numbers, and apply it to the system with hundred atoms in a bad cavity subject to a homodyne detection. We demonstrate that the spin squeezing can be vividly illustrated by the Gaussian-like distribution of the collective density matrix elements, and we examine the influence of the probe field strength and polarization, the detection efficiency, the spontaneous emission rate and the number of atoms. Our exact approach can play an important role in gauging the approximate approaches applied for systems with more atoms, such as Gaussian-state formalism and stochastic mean-field approach, and it permits also exploration of entanglement effects beyond these approaches.
Abstract We analyze the generation of spin-squeezed states via coupling of three-level atoms to an optical cavity and continuous quantum measurement of the transmitted cavity field in order to monitor the evolution of the atomic ensemble. Using analytical treatment and microscopic simulations of the dynamics, we show that one can achieve significant spin squeezing, favorably scaling with the number of atoms N. However, contrary to some previous literature, we clarify that it is not possible to obtain Heisenberg scaling without the continuous feedback that is proposed in optimal approaches. In fact, in the adiabatic cavity removal approximation and large N limit, we find the scaling behavior N − 2 / 3 for spin squeezing and N − 1 / 3 for the corresponding protocol duration. These results can be obtained only by considering the curvature of the Bloch sphere, since linearizing the collective spin operators tangentially to its equator yields inaccurate predictions. With full simulations, we characterize how spin-squeezing generation depends on the system parameters and departs from the bad cavity regime, by gradually mixing with cavity-filling dynamics until metrological advantage is lost. Finally, we discuss the relevance of this spin-squeezing protocol to state-of-the-art optical clocks.
We describe an efficient numerical method for simulating the dynamics of interacting spin ensembles in the presence of dephasing and decay. The method builds on the discrete truncated Wigner approximation for isolated systems, which combines the mean-field dynamics of a spin ensemble with a Monte Carlo sampling of discrete initial spin values to account for quantum correlations. Here we show how this approach can be generalized for dissipative spin systems by replacing the deterministic mean-field evolution by a stochastic process, which describes the decay of coherences and populations while preserving the length of each spin. We demonstrate the application of this technique for simulating nonclassical spin-squeezing effects or the dynamics and steady states of cavity QED models with 105{10}^{5}105 interacting two-level systems. This opens up the possibility to perform accurate real-scale simulations of a diverse range of experiments in quantum optics or with solid-state spin ensembles under realistic laboratory conditions.
Squeezing of collective atomic spins has been shown to improve the sensitivity of atomic clocks and magnetometers to levels significantly below the standard quantum limit. In most cases the requisite atom-atom entanglement has been generated by dispersive interaction with a quantized probe field or by state-dependent collisions in a quantum gas. Such experiments typically use complex multilevel atoms like Rb or Cs, with the relevant interactions designed so that atoms behave like pseudospin-12\frac{1}{2}21​ particles. We demonstrate the viability of spin squeezing for collective spins composed of the physical angular momenta of 106{10}^{6}106 Cs atoms, each in an internal spin-4 hyperfine state. A peak metrological squeezing of at least 5dB is generated by quantum backaction from a dispersive quantum nondemolition (QND) measurement, implemented using a two-color optical probe that minimizes tensor light shifts without sacrificing measurement strength. Other significant developments include the successful application of composite pulse techniques for accurate dynamical control of the collective spin, enabled by broadband suppression of background magnetic fields inside a state-of-the-art magnetic shield. The absence of classical noise allows us to compare the observed quantum projection noise and squeezing to a theoretical model that properly accounts for both the relevant atomic physics and the spatial mode of the collective spin, finding good quantitative agreement and thereby validating its use in other contexts. Our work sets the stage for experiments on quantum feedback, deterministic squeezing, and closed-loop magnetometry. The implementation of real-time feedback may also create an opportunity for new types of quantum simulation, wherein the evolution of a quantum system is conditioned on the outcome of a time-continuous QND measurement. Such a scheme has the potential to access new regimes near the quantum-classical boundary, with opportunities to study long-standing issues related to quantum-classical correspondence in chaotic systems.
Abstract The quantum spin states of an atomic ensemble are of special interest to both fundamental studies and precision measurement applications. Here, we propose a scheme to prepare collective quantum states of an atomic ensemble placed in an optical cavity via homodyne measurement of a probing light field. The effective interactions of atoms mediated by photons are enhanced by the optical cavity, and the output probe light could also be entangled with the collective spin states. By selectively measuring the quadrature of the output light, we can prepare various quantum states, including superposition states of Dicke states and Dicke squeezed states. It is also demonstrated that the fidelity of the prepared quantum state can be enhanced by repetitive homodyne detection and using longer probe laser pulses. With the experimental parameters, we discuss some imperfections, including background noise, fluctuation of the measurement outcomes, and free-space scattering.We believe that our scheme should be feasible for experimental realization with current technologies, which may be used in the future study of quantum mechanics and quantum metrology.