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Register a new userNon-Markovian Dynamics of Open Quantum Systems: Stochastic Equations and their Perturbative Solutions
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We treat several key stochastic equations for non-Markovian open quantum system dynamics and present a formalism for finding solutions to them via canonical perturbation theory, without making the Born-Markov or rotating wave approximations (RWA). This includes master equations of the (asymptotically) stationary, periodic, and time-nonlocal type. We provide proofs on the validity and meaningfulness of the late-time perturbative master equation and on the preservation of complete positivity despite a general lack of Lindblad form. More specifically, we show how the algebraic generators satisfy the theorem of Lindblad and Gorini, Kossakowski and Sudarshan, even though the dynamical generators do not. These proofs ensure the mathematical viability and physical soundness of solutions to non-Markovian processes. Within the same formalism we also expand upon known results for non-Markovian corrections to the quantum regression theorem. Several directions where these results can be usefully applied to are also described, including the analysis of near-resonant systems where the RWA is inapplicable and the calculation of the reduced equilibrium state of open systems.
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Perturbation theory (PT) is a powerful and commonly used tool in the investigation of closed quantum systems. In the context of open quantum systems, PT based on the Markovian quantum master equation is much less developed. The investigation of open systems mostly relies on exact diagonalization of the Liouville superoperator or quantum trajectories. In this approach, the system size is rather limited by current computational capabilities. Analogous to closed-system PT, we develop a PT suitable for open quantum systems. This proposed method is useful in the analytical understanding of open systems as well as in the numerical calculation of system properties, which would otherwise be impractical.
Non-Markovian reduced dynamics of an open system is investigated. In the case the initial state of the reservoir is the vacuum state, an approximation is introduced which makes possible to construct a reduced dynamics which is completely positive.
We introduce jumptime unraveling as a distinct method to analyze quantum jump trajectories and the associated open/continuously monitored quantum systems. In contrast to the standard unraveling of quantum master equations, where the stochastically evolving quantum trajectories are ensemble-averaged at specific times, we average quantum trajectories at specific jump counts. The resulting quantum state then follows a discrete, deterministic evolution equation, with time replaced by the jump count. We show that, for systems with finite-dimensional state space, this evolution equation represents a trace-preserving quantum dynamical map if and only if the underlying quantum master equation does not exhibit dark states. In the presence of dark states, on the other hand, the state may decay and/or the jumptime evolution eventually terminate entirely. We elaborate the operational protocol to observe jumptime-averaged quantum states, and we illustrate the jumptime evolution with the examples of a two-level system undergoing amplitude damping or dephasing, a damped harmonic oscillator, and a free particle exposed to collisional decoherence.
We construct a large class of completely positive and trace preserving non-Markovian dynamical maps for an open quantum system. These maps arise from a piecewise dynamics characterized by a continuous time evolution interrupted by jumps, randomly distributed in time and described by a quantum channel. The state of the open system is shown to obey a closed evolution equation, given by a master equation with a memory kernel and a inhomogeneous term. The non-Markovianity of the obtained dynamics is explicitly assessed studying the behavior of the distinguishability of two different initial systems states with elapsing time.
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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