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Quasiprobability distributions emerging in the stochastic wave-function method of Carusotto et al. [Phys. Rev. A 63, 023606 (2001)] are investigated. We show that there are actually two types of quasiprobabilities. The first one, the "diagonal Hartree-Fock state projection" representation, is useful in representing the initial conditions for stochastic simulation in the most compact form. It defines antinormally ordered expansion of the density operator and normally ordered mapping of the observables to be averaged. We completely characterize the equivalence classes of this phase-space representation. The second quasiprobability distribution, the "nondiagonal Hartree-Fock state projection" representation, extends the first one in order to achieve stochastic representation of the quantum dynamics. We demonstrate how the differential identities of the stochastic ansatz generate the automorphisms of this phase-space representation. These automorphisms turn the stochastic representation into a gauge theory. The gauge transformations of the quasiprobability master equation are described. In particular, it is the analyticity of the stochastic ansatz that allows one to transform the master equation into the genuine Fokker-Planck equation. We demonstrate how the different variants of the stochastic wave-function method can be constructed, first by choosing a certain optimality criteria or constraints, and then by satisfying these criteria with a suitable choice of gauge. The problem of boundary terms is considered. It is demonstrated that the simple scheme with Fock states of Carusotto et al. is not subjected to this problem.

A new approach to theory and simulation of the non-Markovian dynamics of open quantum systems is presented. It is based on identification of a parameter which is uniformly bounded on wide time intervals: the occupation of the virtual cloud of quanta. By 'virtual' we denote those bath excitations which were emitted by the open system, but eventually will be reabsorbed before any measurement of the bath state. A useful property of the virtual cloud is that the number of its quanta is expected to saturate on long times, since physically this cloud is a (retarded) polarization of the bath around the system. Therefore, the joint state of open system and virtual cloud (we call it dressed state) can be accurately represented in a truncated basis of Fock states, on a wide time scale. At the same time, there can be an arbitrarily large number of the observable quanta (which survive up to measurement), especially if the open system is under driving. However, it turns out that the statistics of the bathmeasurement outcomes is classical (in a suitable measurement basis): one can employ a Monte Carlo sampling of these outcomes. Therefore, it is possible to efficiently simulate the dynamics of the observable quantum field. In this work we consider the bath measurement with respect to the coherent states, which yields the Husimi function as the positive (quasi)probability distribution of the outcomes. The joint evolution of the dressed state and the corresponding outcome is called the dressed quantum trajectory. The Monte Carlo sampling of these trajectories yields a stochastic simulation method with promising convergence properties on wide time scales.

2016

The phase-space description of bosonic quantum systems has numerous applications in such fields as quantum optics, trapped ultracold atoms, and transport phenomena. Extension of this description to the case of fermionic systems leads to formal Grassmann phase-space quasiprobability distributions and master equations. The latter are usually considered as not possessing probabillistic interpretation and as not directly computationally accessible. Here, we describe how to construct c-number interpretations of Grassmann phase-space representations and their master equations. As a specific example, the Grassmann B representation is considered. We disscuss how to introduce c-number probability distributions on Grassmann algebra and how to integrate them. A measure of size and proximity is defined for Grassmann numbers, and the Grassmann derivatives are introduced which are based on infinitesimal variations of function arguments. An example of c-number interpretation of formal Grassmann equations is presented.

Abstract. When conducting the numerical simulation of quantum transport, the main obstacle is a rapid growth of the dimension of entangled Hilbert subspace. The Quantum Monte Carlo simulation techniques, while being capable of treating the problems of high dimension, are hindered by the so-called "sign problem". In the quantum transport, we have fundamental asymmetry between the processes of emission and absorption of environment excitations: the emitted excitations are rapidly and irreversibly scattered away. Whereas only a small part of these excitations is absorbed back by the open subsystem, thus exercising the non-Markovian self-action of the subsystem onto itself. We were able to devise a method for the exact simulation of the dominant quantum emission processes, while taking into account the small backaction effects in an approximate self-consistent way. Such an approach allows us to efficiently conduct simulations of real-time dynamics of small quantum subsystems immersed in non-Markovian bath for large times, reaching the quasistationary regime. As an example we calculate the spatial quench dynamics of Kondo cloud for a bozonized Kodno impurity model.

In this work we investigate the exact classical stochastic representations of many-body quantum dynamics. We focus on the representations in which the quantum states and the observables are linearly mapped onto classical quasiprobability distributions and functions in a certain (abstract) phase space. We demonstrate that when such representations have regular mathematical properties, they are reduced to the expansions of the density operator over a certain overcomplete operator basis. Our conclusions are supported by the fact that all the stochastic representations currently known in the literature (quantum mechanics in generalized phase space and, as it recently has been shown by us, the stochastic wave-function methods) have the mathematical structure of the above-mentioned type. We illustrate our considerations by presenting the recently derived operator mappings for the stochastic wave-function method.

We are at the midst of second quantum revolution where the mesoscopic quantum devies are actively employed for technological purposes. Despite this fact, the description of their real-time dynamics beyond the Fermi's golden rule remains a formiddable theoretical problem. This is due to the rapid spread of entanglement within the degrees of freedom of the surrounding environment. This is accompanied with a quantum noise (QN) acting on the mesoscopic device. In this work we propose a possible way out: to exploit the fact that this QN is usually bandlimited. This is because its spectral density is often contained in peaks of localized modes and resonances, and may be constrained by bandgaps. Inspired by the Kotelnikov sampling theorem from the theory of classical bandlimited signals, we put forward and explore the idea that when the QN spectral density has effective bandwidth B, the quantum noise becomes a discrete-time process, with an elementary time step τ ∝ B −1 . After each time step τ , one new QN degree of freedom (DoF) gets coupled to the device for the first time, and one new QN DoF get irreversibly decoupled. Only a bounded number of QN DoFs are significantly coupled at any time moment. We call these DoFs the Kotelnikov modes. As a result, the real-time dissipative quantum motion has a natural structure of a discretetime matrix product state, with a bounded bond dimension. This yields a microscopically derived collision model. The temporal entanglement entropy appears to be bounded (area-law scaling) in the frame of Kotelnikov modes. The irreversibly decoupled modes can be traced out as soon as they occur during the real-time evolution. This leads to a novel bandlimited input-output formalism and to quantum jump Monte Carlo simulation techniques for real-time motion of open quantum systems. We illustrate this idea on a spin-boson model.

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