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Number-phase Wigner representation for scalable stochastic simulations of controlled quantum systems

Michael Hush, Stuart S. Szigeti, Andre R. R. Carvalho, J. J. Hope

Published Aug 1, 2011• 2 citations
Publisher

Abstract

Simulation of conditional master equations for many-body systems is important for state estimation and control of large quantum systems. Many-body conditional master equations have been simulated recently with stochastic methods, but the convergence time was restricted by the use of representations based on coherent states. We demonstrate that applying a stochastic method based on the number-phase Wigner (NPW) representation can converge over an order of magnitude longer than it's coherent equivalent. This technique can provide a drastic improvement in the convergence of simulations of controlled many-body quantum systems.

DOI
10.1109/iqec-cleo.2011.6193774

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Simulation of conditional master equations for many-body systems is important for state estimation and control of large quantum systems. Many-body conditional master equations have been simulated recently with stochastic methods, but the convergence time was restricted by the use of representations based on coherent states. We demonstrate that applying a stochastic method based on the number-phase Wigner (NPW) representation can converge over an order of magnitude longer than it's coherent equivalent. This technique can provide a drastic improvement in the convergence of simulations of controlled many-body quantum systems.
Simulation of conditional master equations for many-body systems is important for state estimation and control of large quantum systems. Many-body conditional master equations have been simulated recently with stochastic methods, but the convergence time was restricted by the use of representations based on coherent states. We demonstrate that applying a stochastic method based on the number-phase Wigner (NPW) representation can converge over an order of magnitude longer than it’s coherent equivalent. This technique can provide a drastic improvement in the convergence of simulations of controlled many-body quantum systems.
Simulation of conditional master equations is important to describe systems under continuous measurement and for the design of control strategies in quantum systems. For large bosonic systems, such as Bose-Einstein condensates and atom lasers, full quantum-field simulations must rely on scalable stochastic methods. Currently, these methods have a convergence time that is restricted by the use of representations based on coherent states. Here, we show that typical measurements on atom-optical systems have a common form that allows for an efficient simulation using the number-phase Wigner (NPW) phase-space representation. We demonstrate that a stochastic method based on the NPW can converge orders of magnitude longer and more precisely than its coherent equivalent. We then examine how these methods can be used in multimode simulations, demonstrated by a simulation of a two-mode Bose-Hubbard model. Finally, we combine these techniques to demonstrate a full-field simulation of a realistic multimode quantum system controlled by active feedback.
Engineers are looking to produce technologies that exploit the full quantum nature of a system, including quantum computing, quantum communication and quantum metrol- ogy. Quantum control allows engineers to modify the behaviour of the full quantum state of a system, which will be an important part of making these technologies commercially viable. If these technologies are to be competitive, they must be scaled up to large sizes. We produce a novel toolset which can be used to simulate large con- trolled quantum systems efficiently. We target this toolset at a large quantum system that can immediately benefit: Bose-Einstein Condensates (BECs). We first develop the Controlled Fokker-Planck Equation (CFPE), which allows scalable simulation of a controlled BEC. We find using coherent-state-based methods to generate a CFPE produces solutions with severely limited integration times, as monitoring of a BEC is intrinsically involves continuous 'number-like' measurements. To solve this problem, we develop a novel representation based on number state which acts in harmony with the measurement: the Number-Phase Wigner (NPW) representation. The NPW method efficiently and accurately simulates controlled BECs. It converges over 4 orders of mag- nitude longer than its closest coherent-state-based competitor. Using the NPW we are able to perform the first full-field analysis of a BEC under a cooling feedback control. We find that quantum noise produces heating in BECs that was not previously pre- dicted using approximate methods. We remove the effects of quantum noise with the creation of a novel control.
Phase-space representations based on coherent states (PPP,QQQ, Wigner) have been successful in the creation of stochastic differential equations (SDEs) for the efficient stochastic simulation of high-dimensional quantum systems. However, many problems using these techniques remain intractable over long integration times. We present a number-phase Wigner representation that can be unraveled into SDEs. We demonstrate convergence to the correct solution for an anharmonic oscillator with small dampening for significantly longer than other phase-space representations. This process requires an effective sampling of a nonclassical probability distribution. We describe and demonstrate a method of achieving this sampling using stochastic weights.
We reformulate the full quantum dynamics of spin systems using a phase-space representation based on SU(2) coherent states which generates an exact mapping of the dynamics of any spin system onto a set of stochastic differential equations. This representation is superior in practice to an earlier phase-space approach based on Schwinger bosons, with the numerical effort scaling only linearly with system size. By also implementing extrapolation techniques from quasiclassical equations to the full quantum limit, we are able to extend useful simulation times severalfold. This approach is applicable in any dimension including cases where frustration is present in the spin system. The method is demonstrated by simulating quenches in the transverse-field Ising model in one and two dimensions.
The general idea of a stochastic gauge representation is introduced and compared with more traditional phase-space expansions, like the Wigner expansion. Stochastic gauges can be used to obtain an infinite class of positive-definite stochastic time-evolution equations, equivalent to master equations, for many systems including quantum time evolution. The method is illustrated with a variety of simple examples ranging from astrophysical molecular hydrogen production, through to the topical problem of Bose–Einstein condensation in an optical trap and the resulting quantum dynamics.