A novel scheme for the steady state solution of the standard Redfield quantum master equation is developed which yields agreement with the exact result for the corresponding reduced density matrix up to second order in the system-bath coupling strength. We achieve this objective by use of an analytic continuation of the off-diagonal matrix elements of the Redfield solution towards its diagonal limit. Notably, our scheme does not require the provision of yet higher order relaxation tensors. Testing this modified method for a heat bath consisting of a collection of harmonic oscillators we assess that the system relaxes towards its correct coupling-dependent, generalized quantum Gibbs state in second order. We numerically compare our formulation for a damped quantum harmonic system with the nonequilibrium Green's function formalism: we find good agreement at low temperatures for coupling strengths that are even larger than expected from the very regime of validity of the second-order Redfield quantum master equation. Yet another advantage of our method is that it markedly reduces the numerical complexity of the problem; thus, allowing to study efficiently large-sized system Hilbert spaces.
A novel scheme for the steady state solution of the standard Redfield quantum master equation is developed which yields agreement with the exact result for the corresponding reduced density matrix up to second order in the system-bath coupling strength. We achieve this objective by use of an analytic continuation of the off-diagonal matrix elements of the Redfield solution towards its diagonal limit. Notably, our scheme does not require the provision of yet higher order relaxation tensors. Testing this modified method for a heat bath consisting of a collection of harmonic oscillators we assess that the system relaxes towards its correct coupling-dependent, generalized quantum Gibbs state in second order. We numerically compare our formulation for a damped quantum harmonic system with the nonequilibrium Green's function formalism: we find good agreement at low temperatures for coupling strengths that are even larger than expected from the very regime of validity of the second-order Redfield quantum master equation. Yet another advantage of our method is that it markedly reduces the numerical complexity of the problem; thus, allowing to study efficiently large-sized system Hilbert spaces.
We describe a method to obtain the reduced density matrix (RDM) correct up to second order in system-bath coupling in nonequilibrium steady-state situations. The RDM is obtained via a scheme based on analytic continuation, using the time-local Redfield-like quantum master equation, which was earlier used by the same authors [J. Chem. Phys. 136, 194110 (2012)] to obtain the correct thermal equilibrium description. This nonequilibrium modified Redfield solution is then corroborated with the exact RDM obtained via the nonequilibrium Green's function technique for the quantum harmonic oscillator. Lastly, the scheme is compared to different quantum master equations (QMEs), namely the time-local Redfield-like and the Lindblad-like QMEs, in order to illustrate the differences between each of these approaches.
We put forth a new class of quantum master equations that correctly reproduce the asymptotic state of an open quantum system beyond the infinitesimally weak system-bath coupling limit. Our method is based on incorporating the knowledge of the reduced steady state into its dynamics. The correction not only steers the reduced system toward a correct steady state but also improves the accuracy of the dynamics, thereby refining the archetypal Born-Markov weak-coupling second-order master equations. In case of equilibrium, we use the exact mean-force Gibbs state to correct the Redfield quantum master equation. By benchmarking it with the exact solution of the damped harmonic oscillator, we show that our method also helps correct the long-standing issue of positivity violation, albeit without complete positivity. Our method of a canonically consistent quantum master equation opens a new perspective in the theory of open quantum systems leading to a reduced density matrix accurate beyond the commonly used Redfield and Lindblad equations, while retaining the same conceptual and numerical complexity.
When the coupling of a quantum system to its environment is nonnegligible, its steady state is known to deviate from the textbook Gibbs state. The Bloch-Redfield quantum master equation, one of the most widely adopted equations for solving the open quantum dynamics, cannot predict all the deviations of the steady state of a quantum system from the Gibbs state. In this paper, for a generic spin-boson model, we use a higher-order quantum master equation (in system-environment coupling strength) to analytically calculate all the deviations of the steady state of the quantum system up to second order in coupling strength. We also show that this steady state is exactly identical to the corresponding generalized Gibbs state, the so-called quantum mean force Gibbs state, at arbitrary temperature. All these calculations are highly general, making them immediately applicable to a wide class of systems well modeled by the spin-Boson model, spanning a diverse range of topics, from nanomaterials to various condensed-phase processes, and quantum computing (e.g., environment-induced corrections to the steady state of a superconducting qubit). As an example, we use our results to study the dynamics and the steady state of a solid-state double-quantum-dot system under physically relevant choices of parameters.
This thesis describes the development of practical and efficient computational approaches to the quantum dynamics of complex systems. Most of the work presented here relies on the generalized quantum master equation (GQME) formalism, which provides a simple equation of motion of reduced dimensionality for a set of dynamical quantities, e.g., nonequilibrium averages and equilibrium time correlation functions. The reduced dimensionality of the GQME comes at a cost: the introduction of the memory kernel, which accounts for the influence of all ``excluded'' degrees of freedom. Focusing first on the second-order perturbative treatment of the memory kernel known as Redfield theory, I present a collaborative effort to extend its applicability into highly non-Markovian regions via a mode freezing approach. In this method, a portion of bath modes characterized by low frequencies are treated as sources of static disorder and used to calculate modified Redfield dynamics. Application of the method to the spin-boson and FMO complex models indicates that the Redfield+frozen modes scheme consistently produces dynamics that are as good or better than bare Redfield dynamics. Next, we explore GQME approach coupled to the self-consistent solution of the memory kernel, which requires the calculation of auxiliary kernels. Previous implementations of the method had shown impressive boosts in efficiency and, when approximate methods were used to calculate the auxiliary kernels, accuracy over direct calculation of nonequilibrium averages. We show that this method, when formulated from the Mori perspective, is equally applicable to nonequilibrium averages and equilibrium correlation functions. In addition, we examine the dependence of the improvements afforded by the GQME framework on the choice projection operator and kernel closure. In particular, we demonstrate that improvements in efficiency, which rely on short memory lifetimes, are sensitively dependent on the choice of projection operator, and that the choice of kernel closure directly dictates the improvements in accuracy. In addition, we present evidence that indicates that the success of the GQME formalism when the auxiliary kernels are calculated via semi- and quasi-classical methods is largely due to the exact sampling of bath operators at t=0 required by the calculation of specific kernel closures. Next, we provide analytical arguments that delineate when the GQME framework coupled to the self-consistent solution of the memory kernel is likely to provide improvements in efficiency and accuracy. Finally, we present a path integral framework that can efficiently render the partially Wigner-transformed canonical density operator for systems coupled linearly to harmonic baths. This approach permits the direct calculation of any thermodynamic quantity and can be integrated into dynamical schemes like the Ehrenfest, surface hopping, or linearized semi-classical initial value representation methods to calculate equilibrium correlation functions.
Fourth-order quantum master equations (FQMEs) are derived in both time nonlocal and local forms for a general system Hamiltonian, with new detailed expressions for the fourth-order kernel, where the bath correlation functions are explicitly decoupled from the system superoperators. Further simplifications can be made for the model of linearly coupled harmonic oscillator bath. Consideration of the high temperature Ohmic bath limit leads to a general Markovian FQME with compact forms of time independent superoperators. Two examples of this equation are then considered. For the system of a quantum particle in a continuous potential field, the equation reduces to a known form of the quantum Fokker–Planck equation, except for a fourth-order potential renormalization term that can be neglected only in the weak system-bath interaction regime. For a two-level system with off-diagonal coupling to the bath, fourth-order corrections do not alter the relaxation characteristics of the second-order equation and introduce additional coherence terms in the equations for the off-diagonal elements.
In the framework of the theory of open quantum systems, we derive quantum master equations for the ultrastrong system-bath coupling regime and, more generally, the strong-decoherence regime. In this regime, the strong decoherence is complemented by slow relaxation processes. We use a generalization of the F\"orster and modified Redfield perturbation theories known in the theory of excitation-energy transfer. Also, we show that the mean-force Gibbs state in the corresponding limits is stationary for the derived master equations.