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Efficient simulation of controlled large quantum systems

Michael Hush

Published Jan 1, 2012• 1 citation

Abstract

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.

DOI: 10.25911/5d514d3164727

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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 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.

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.

We introduce an approximate phase-space technique to simulate the quantum dynamics of interacting bosons. With the future goal of treating Bose-Einstein condensate systems, the method is designed for systems with a natural separation into highly occupied (condensed) modes and lightly occupied modes. The method self-consistently uses the Wigner representation to treat highly occupied modes and the positive-

P

representation for lightly occupied modes. In this method, truncation of higher-derivative terms from the Fokker-Planck equation is usually necessary. However, at least in the cases investigated here, the resulting systematic error, over a finite time, vanishes in the limit of large Wigner occupation numbers. We tested the method on a system of two interacting anharmonic oscillators, with high and low occupations, respectively. The hybrid method successfully predicted atomic quadratures to a useful simulation time 60 times longer than that of the positive-

P

method. The truncated Wigner method also performed well in this test. For the prediction of the correlation in a quantum nondemolition measurement scheme, for this same system, the hybrid method gave excellent agreement with the exact result, while the truncated Wigner method showed a large systematic error.

A method of simulating the full quantum field dynamics of multimode, multi-component Bose-Einstein condensates is developed. The truncated Wigner representation is used to obtain a probabilistic theory. The representation is extended to to use a functional calculus and can be applied to quantum fields in the arbitrary number of dimensions. Detailed proofs of the corresponding theorems are given. This approach produces c-number stochastic equations which may be solved using conventional stochastic integration methods, with the limitation that it is only valid for large mode occupation numbers. The equations describe the spatial evolution of spinor components and properly account for nonlinear interactions and losses. The method is further used to model several experiments where quantum effects are significant, and cannot be calculated with simpler approximations. It is shown to describe leading quantum corrections accurately, including effects such as phase noise, quantum squeezing, entanglement, and interactions with engineered nonlinear reservoirs. In addition to this approach, other positive phase-space methods are also analysed for sampling highly nonclassical states. In conclusion, the phase-space representations developed in this thesis provide a relatively fast and accurate method of simulating dynamics of large nonlinear quantum systems, such as trapped Bose-Einstein condensates. The required stochastic equations are easily and scalably integrated in parallel. Algorithms for efficient computation on modern graphical processing unit hardware are presented.

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.