Open quantum systems (OQSs) cannot always be described with the Markov approximation, which requires a large separation of system and environment time scales. Here, we give an overview of some of the most important techniques available to tackle the dynamics of an OQS beyond the Markov approximation. Some of these techniques, such as master equations, Heisenberg equations and stochastic methods, are based on solving the reduced OQS dynamics, while others, such as path integral Monte Carlo or chain mapping approaches, are based on solving the dynamics of the full system. We emphasize the physical interpretation and derivation of the various approaches, explore how they are connected and examine how different methods may be suitable for solving different problems.
We introduce a simple setup corresponding to the matter-wave analogue of impurity atoms embedded in a photonic crystal and interacting with the radiation field. Atoms in a given internal level are trapped in an optical lattice, and play the role of the impurities. Atoms in an untrapped level play the role of the radiation field. The interaction is mediated by means of lasers that couple those levels. By tuning the lasers parameters, it is possible to drive the system through different regimes, and observe phenomena such as matter-wave superradiance, non-Markovian atom emission, and the appearance of bound atomic states.
Multiple time correlation functions are found in the dynamical description of different phenomena. They encode and describe the fluctuations of the dynamical variables of a system. In this paper we formulate a theory of non-Markovian multiple-time correlation functions (MTCF) for a wide class of systems. We derive the dynamical equation of the reduced propagator, an object that evolve state vectors of the system conditioned to the dynamics of its environment, which is not necessarily at the vacuum state at the initial time. Such reduced propagator is the essential piece to obtain multiple-time correlation functions. An average over the different environmental histories of the reduced propagator permits us to obtain the evolution equations of the multiple-time correlation functions. We also study the evolution of MTCF within the weak coupling limit and it is shown that the multiple-time correlation function of some observables satisfy the Quantum Regression Theorem (QRT), whereas other correlations do not. We set the conditions under which the correlations satisfy the QRT. We illustrate the theory in two different cases; first, solving an exact model for which the MTCF are explicitly given, and second, presenting the results of a numerical integration for a system coupled with a dissipative environment through a non-diagonal interaction.PACS numbers: 3.65 Yz, 42.50 Lc Introduction and motivation. Many research contexts are focused on the dynamics of a system (S) that is affected by an environment (B) from which it cannot be considered isolated. Examples of such situations are encountered in statistical physics, condensed matter and quantum optics. We found a concrete example in the description of the dynamics of an atom (S) immersed in an electromagnetic field (B) [1,2].In some circumstances, the analysis of the dynamics of the system is done using the expectation values of its observables over state vectors of the whole system, and then averaging over the environmental degrees of freedom. However, in some other situations, like when studying the response of a system to an external EM field, some additional information is needed. In particular, for the analysis of the spectroscopic properties of a system some multiple-time correlation function (MTCF) has to be computed, usually a two-time correlation function.The dynamics of the system S is usually described through its reduced density operator. Such operator verifies some master equation that in the Markovian case is of Lindblad type [1,3,4,5,6,7]. Complementary to the master equation approach, a series of Monte-Carlo type of approaches based on the so called stochastic Schrödinger equations [1,4,8,9,10] have been developed in the last decade. In such schemes, the dynamics of system state vectors is integrated, and after an average is made over many realizations of environment histories that eventually are understood as a noise and takes into account the environment influence on S. In the non-Markovian case, within the context of nuclear magnetic resonance, the ...
We propose a scheme involving cold atoms trapped in optical lattices to observe different phenomena traditionally linked to quantum-optical systems. The basic idea consists of connecting the trapped atomic state to a non-trapped state through a Raman scheme. The coupling between these two types of atoms (trapped and free) turns out to be similar to that describing light-matter interaction within the rotating-wave approximation, the role of matter and photons being played by the trapped and free atoms, respectively. We explain in particular how to observe phenomena arising from the collective spontaneous emission of atomic and harmonic oscillator samples such as superradiance and directional emission. We also show how the same setup can simulate Bose-Hubbard Hamiltonians with extended hopping as well as Ising models with long-range interactions. We believe that this system can be realized with state of the art technology.
We consider a thermofield approach to analyze the evolution of an open quantum system coupled to an environment at finite temperature. In this approach, the finite temperature environment is exactly mapped onto two virtual environments at zero temperature. These two environments are then unitarily transformed into two different chains of oscillators, leading to a one dimensional structure that can be numerically studied using tensor network techniques. arXiv:1504.07228v1 [quant-ph]
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