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Register a new userSelf-consistent microscopic derivation of Markovian master equations for open quadratic quantum systems
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We provide a rigorous construction of Markovian master equations for a wide class of quantum systems that encompass quadratic models of finite size, linearly coupled to an environment modeled by a set of independent thermal baths. Our theory can be applied for both fermionic and bosonic models in any number of physical dimensions, and does not require any particular spatial symmetry of the global system. We show that, for non-degenerate systems under a full secular approximation, the effective Lindblad operators are the normal modes of the system, with coupling constants that explicitly depend on the transformation matrices that diagonalize the Hamiltonian. Both the dynamics and the steady-state (guaranteed to be unique) properties can be obtained with a polynomial amount of resources in the system size. We also address the particle and energy current flowing through the system in a minimal two-bath scheme and find that they hold the structure of Landauers formula, being thermodynamically consistent.
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Local master equations are a widespread tool to model open quantum systems, especially in the context of many-body systems. These equations, however, are believed to lead to thermodynamic anomalies and violation of the laws of thermodynamics. In contrast, here we rigorously prove that local master equations are consistent with thermodynamics and its laws without resorting to a microscopic model, as done in previous works. In particular, we consider a quantum system in contact with multiple baths and identify the relevant contributions to the total energy, heat currents and entropy production rate. We show that the second law of thermodynamics holds when one considers the proper expression we derive for the heat currents. We confirm the results for the quantum heat currents by using a heuristic argument that connects the quantum probability currents with the energy currents, using an analogous approach as in classical stochastic thermodynamics. We finally use our results to investigate the thermodynamic properties of a set of quantum rotors operating as thermal devices and show that a suitable design of three rotors can work as an absorption refrigerator or a thermal rectifier. For the machines considered here, we also perform an optimisation of the system parameters using an algorithm of reinforcement learning.
Open Quantum Walks (OQWs) are exclusively driven by dissipation and are formulated as completely positive trace preserving (CPTP) maps on underlying graphs. The microscopic derivation of discrete and continuous in time OQWs is presented. It is assumed that connected nodes are weakly interacting via a common bath. The resulting reduced master equation of the quantum walker on the lattice is in the generalised master equation form. The time discretisation of the generalised master equation leads to the OQWs formalism. The explicit form of the transition operators establishes a connection between dynamical properties of the OQWs and thermodynamical characteristics of the environment. The derivation is demonstrated for the examples of the OQW on a circle of nodes and on a finite chain of nodes. For both examples a transition between diffusive and ballistic quantum trajectories is observed and found to be related to the temperature of the bath.
The problem of a driven quantum system coupled to a bath and coherently driven is usually treated using either of two approaches: Employing the common secular approximation in the lab frame (as usually done in the context of atomic physics) or in the rotating frame (prevailing in, e.g., the treatment of solid-state qubits). These approaches are applicable in different parts of the parameter space and yield different results. We show how to bridge between these two approaches by working in the rotating frame without employing the secular approximation with respect to the driving amplitude. This allows us to uncover novel behaviors in regimes which were previously inaccessible or inaccurately treated. New features such as the qualitative different evolution of the coherence, population inversion at a lower driving amplitude, and novel structure in the resonance fluorescence spectrum of the system are found. We argue that this generalized approach is essential for analyzing hybrid systems, with components that come from distinctly different regimes which can now be treated simultaneously, giving specific examples from recent experiments on quantum dots coupled to optical cavities, and single-spin electron paramagnetic resonance.
We consider an open quantum system described by a Lindblad-type master equation with two times-scales. The fast time-scale is strongly dissipative and drives the system towards a low-dimensional decoherence-free space. To perform the adiabatic elimination of this fast relaxation, we propose a geometric asymptotic expansion based on the small positive parameter describing the time-scale separation. This expansion exploits geometric singular perturbation theory and center-manifold techniques. We conjecture that, at any order, it provides an effective slow Lindblad master equation and a completely positive parameterization of the slow invariant sub-manifold associated to the low-dimensional decoherence-free space. By preserving complete positivity and trace, two important structural properties attached to open quantum dynamics, we obtain a reduced-order model that directly conveys a physical interpretation since it relies on effective Lindbladian descriptions of the slow evolution. At the first order, we derive simple formulae for the effective Lindblad master equation. For a specific type of fast dissipation, we show how any Hamiltonian perturbation yields Lindbladian second-order corrections to the first-order slow evolution governed by the Zeno-Hamiltonian. These results are illustrated on a composite system made of a strongly dissipative harmonic oscillator, the ancilla, weakly coupled to another quantum system.
An open quantum system that is put in contact with an infinite bath is pushed towards equilibrium, while the state of the bath remains unchanged. If the bath is finite, the open system still relaxes to equilibrium, but it induces a dynamical evolution of the bath state. In this work, we extend the weak-coupling master equation approach of open quantum systems interacting with finite baths to include imprecise measurements of the bath energy. Those imprecise measurements are not only always the case in practice, but they also unify the theoretical description. We investigate the circumstances under which our equation reduces to the more standard Born-Markov-secular master equation. As a result, we obtain a hierarchy of master equations that improve their accuracy by including more dynamical information about the bath. We discuss this formalism in detail for a particular non-interacting environment where the Boltzmann temperature and the Kubo-Martin-Schwinger relation naturally arise. Finally, we apply our hierarchy of master equations to study the central spin model.
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