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arXiv:0901.4391 · Physical Review A

Scalable quantum field simulations of conditioned systems

Michael Hush, Andre R. R. Carvalho, J. J. Hope

Published Jul 10, 2009• 11 citations
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Abstract

We demonstrate a technique for performing stochastic simulations of conditional master equations. The method is scalable for many quantum-field problems and therefore allows first-principles simulations of multimode bosonic fields undergoing continuous measurement, such as those controlled by measurement-based feedback. As examples, we demonstrate a 53-fold speed increase for the simulation of the feedback cooling of a single trapped particle, and the feedback cooling of a quantum field with 32 modes, which would be impractical using previous brute force methods.

DOI
10.1103/physreva.80.013606

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We demonstrate a technique for performing stochastic simulations of conditional master equations. The method is scalable for many quantum-field problems and therefore allows first-principles simulations of multimode bosonic fields undergoing continuous measurement, such as those controlled by measurement-based feedback. As examples, we demonstrate a 53-fold speed increase for the simulation of the feedback cooling of a single trapped particle, and the feedback cooling of a quantum field with 32 modes, which would be impractical using previous brute force methods.
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.
We develop an approach for efficient and scalable simulations of measurement and control of quantum systems built upon existing phase-space methods, namely, the truncated Wigner approximation.. We benchmark against existing particle-filter methods by simulating measurement-based feedback cooling in a two-mode system, whose low-dimensional nature permits a computation of an exact solution. The advantage of our method is multimode scalability, which we demonstrate through the successful simulation of measurement-based feedback cooling of an incoherent quasi-one-dimensional thermal ensemble to quantum degeneracy. As the underlying principle of our approach exploits a general correspondence between measurement and coherent feedback, we anticipate it is also applicable across a broad range of other quantum control scenarios.
We consider the problem of controlling the motion of an atom trapped in an optical cavity using continuous feedback. In order to realize such a scheme experimentally, one must be able to perform state estimation of the atomic motion in real time. While in theory this estimate may be provided by a stochastic master equation describing the full dynamics of the observed system, integrating this equation in real time is impractical. Here we derive an approximate estimation equation for this purpose, and use it as a drive in a feedback algorithm designed to cool the motion of the atom. We examine the effectiveness of such a procedure using full simulations of the cavity QED system, including the quantized motion of the atom in one dimension.
We give a detailed theoretical derivation of the master equation for the coherent Ising machine. This is a quantum computational network with feedback, that approximately solves NP-hard combinatoric problems, including the traveling salesman problem and various extensions and analogs. There are two possible types of master equation, either conditional on the feedback current or unconditional. We show that both types can be accurately simulated in a scalable way using stochastic equations in the positive-P phase-space representation. This depends on the nonlinearity present, and we use parameter values that are typical of current experiments. While the two approaches are in excellent agreement, they are not equivalent with regard to efficiency. We find that unconditional simulation has much greater efficiency, and is more scalable to large sizes. This is a case where too much knowledge is a dangerous thing. Conditioning the simulations on the feedback current is not essential to determining the success probability, but it greatly increases the computational complexity. To illustrate the speed improvements obtained with the unconditional approach, we carry out full quantum simulations of the master equation with up to 1000 nodes.
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.