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

Abstract

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.

From the Weyl quantization of a particle on the circle to number–phase Wigner functions

“…In fact, it has been motivated by a question raised by the referee of [1]. Namely, the referee pointed out that the Weyl quantization formalism developed in [1] should be closely related to the Wigner function and the Wigner representation of quantum phase investigated previously by the others [2][3][4]. So, here we follow this suggestion and we intend to display, how one can define the Wigner function which depends on the number and the phase.…”

Controlling spontaneous-emission noise in measurement-based feedback cooling of a Bose–Einstein condensate

“…These approximations were all developed and verified in [31,32]. After the application of these approximations, equation (A.12) simplifies to dN [n(·), ϕ(·)] = 1 h dx dyh (x, y, u) − i∂ n(y) (n(x) + δ(0)/2) (n(y) + δ(0)/2)…”

“…However, the quasi-probability distribution in equation (B.5) can have negative values, and thus cannot be efficiently sampled [104]. Instead, we sample from an approximate distribution that is valid when the number of particles N is much larger than the number of simulated modes M. The details of these approximations and a derivation of the resulting approximate distribution can be found in [31,32]; here we simply present the result N α 0 [n(·), ϕ(·)] ≈ x √ 2 n 0 (x) n(x) e −n 0 (x)+ −2((ϕ(x)−ϕ 0 (x)) ψ (1) (n(x)+1) n(x)! πψ (1)…”

“…In doing so, it also provides the first practical application of the NPW particle filter. Previous work has verified the accuracy of this simulation technique and compared its performance to other established methods [31]. To make this paper as accessible as possible, we provide an overview of the relevant numerical techniques, including the NPW particle filter, in section 2.…”

Gaussian Quantum Trajectories for the Variational Simulation of Open Quantum-Optical Systems

“…On the other hand, we see that the NΘ-methods are able to capture the phase diffusion on the level of the ensemble, as well as TWA. This relative success of the NΘ-methods with respect to the XP-methods is somewhat reminiscent of a similar observation for number-phase phase-space methods of monitored quantum systems [68,69]. What distinguishes the NΘ-Gaussian method from TWA, however, is that the NΘ-Gaussian method is able to show the composition of the ensemble: it maintains information of individual trajectories, which is lost in TWA (we note that in practice, under appropriate conditions, a single TWA sample may still be representative of experimental realizations [56,70]).…”