We consider an open quantum system in [Formula: see text] governed by quasiperiodic Hamiltonian with rationally independent frequencies and under the assumption of Lyapunov–Perron reducibility of the associated Schrödinger equation. We construct the Markovian Master Equation and the resulting CP-divisible evolution in the weak coupling limit regime, generalizing our previous results from the periodic case. The analysis is conducted with the application of projection operator techniques and concluded with some results regarding stability of solutions and existence of quasiperiodic global steady state.
In this chapter we study some properties of systems of coupled quadratic quantum Hamiltonians. The well known example is coupled harmonic oscillators appearing in the Quantum Brownian Motion. An open system is a physical (S) interacting with the environment (E) such that the total system
(S∪E) is isolated from the rest of the universe. So we have two independent Hamiltonians
H^S,
H^E and an interaction
V^ between (S) and (E). The main problem discussed for open systems is to understand the dynamical action of the interaction
V^ with the environment on the time dependent evolution of (S) which is now irreversible because a part of its energy escapes to the environment. An important physical effect of the interaction of the environment on a quantum system is decoherence which give an explanation for the mechanism driving the quantum-to-classical transition. We restrict ourself to few mathematical problems related with this very active field of research in physics [155, 200, 231, 270]. In a first part we study the purity of a state of (S) (or the linear entropy) in a short time approximation. In a second part we discuss the separability of bipartite systems. In a third part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for coupled quadratic Hamiltonians by a bilinear inter-action. We give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time t is written as a convolution integral for the reduced density at initial time
t0=0, with a Gaussian kernel, for
0≤t<tc where
tc∈]0,∞] is a critical time such that reversibility is lost for
t>tc if
tc<+∞. Some manifestations of decoherence, for
t>0 small enough, are proved on our simple model which is similar to a representation of the Schrödinger cat.
We study the Lindblad master equation in the space of operators and provide simple criteria for closeness of the hierarchy of equations for correlations. We separately consider the time evolution of closed and open systems and show that open systems satisfying the closeness conditions are not necessarily of Gaussian type. In addition, we show that dissipation can induce the closeness of the hierarchy of correlations in interacting quantum systems. As an example we study an interacting optomechanical model, the Fermi–Hubbard model, and the Rabi model, all coupled to a fine-tuned Markovian environment and obtain exact analytic expressions for the time evolution of two-point correlations.
We study open quantum systems whose evolution is governed by a master equation of Kossakowski-Gorini-Sudarshan-Lindblad type and give a characterization of the convex set of steady states of such systems based on the generalized Bloch representation. It is shown that an isolated steady state of the Bloch equation cannot be a center, i.e., that the existence of a unique steady state implies attractivity and global asymptotic stability. Necessary and sufficient conditions for the existence of a unique steady state are derived and applied to different physical models, including two- and four-level atoms, (truncated) harmonic oscillators, and composite and decomposable systems. It is shown how these criteria could be exploited in principle for quantum reservoir engineeing via coherent control and direct feedback to stabilize the system to a desired steady state. We also discuss the question of limit points of the dynamics. Despite the nonexistence of isolated centers, open quantum systems can have nontrivial invariant sets. These invariant sets are center manifolds that arise when the Bloch superoperator has purely imaginary eigenvalues and are closely related to decoherence-free subspaces.
A well-known situation in which a non-Markovian dynamics of an open quantum system
S arises is when this is coherently coupled to an auxiliary system
M in contact with a Markovian bath. In such cases, while the joint dynamics of $S\text{\ensuremath{-}}M$ is Markovian and obeys a standard (bipartite) Lindblad-type master equation (ME), this is in general not true for the reduced dynamics of
S. Furthermore, there are several instances (e.g., the dissipative Jaynes-Cummings model) in which a closed ME for the
S's state cannot even be worked out. Here, we find a class of bipartite Lindblad-type MEs such that the reduced ME of
S can be derived exactly and in a closed form for any initial product state of $S\text{\ensuremath{-}}M$. We provide a detailed microscopic derivation of our result in terms of a mapping between two collision models.
We develop a framework to analyze the dynamics of a finite-dimensional quantum system S in contact with a reservoir R . The full, interacting SR dynamics is unitary. The reservoir has a stationary state but otherwise dissipative dynamics. We identify a main part of the full dynamics, which approximates it for small values of the SR coupling constant, uniformly for all times t≥0 . The main part consists of explicit oscillating and decaying parts. We show that the reduced system evolution is Markovian for all times. The technical novelty is a detailed analysis of the link between the dynamics and the spectral properties of the generator of the SR dynamics, based on Mourre theory. We allow for SR interactions with little regularity, meaning that the decay of the reservoir correlation function only needs to be polynomial in time, improving on the previously required exponential decay.In this work we distill the structural and technical ingredients causing the characteristic features of oscillation and decay of the SR dynamics. In the companion paper [27] we apply the formalism to the concrete case of an N -level system linearly coupled to a spatially infinitely extended thermal bath of non-interacting Bosons.