Numerical evaluation and robustness of the quantum mean force Gibbs state

Abstract

We introduce a numerical method to determine the Hamiltonian of Mean Force (HMF) Gibbs state for a quantum system strongly coupled to a reservoir. The method adapts the Time Evolving Matrix Product Operator (TEMPO) algorithm to imaginary time propagation. By comparing the real-time and imaginary-time propagation for a generalized spin-boson model, we confirm that the HMF Gibbs state correctly predicts the steady state. We show that the numerical dynamics match the polaron master equation at strong coupling. We illustrate the potential of the imaginary-time TEMPO approach by exploring reservoir-induced entanglement between qubits.

Quantum–classical correspondence in spin–boson equilibrium states at arbitrary coupling

“…This generalised version of the spin-boson model [33,34] describes a vast range of physical contexts, including excitation energy transfer processes in molecular aggregates described by the Frenkel exciton Hamiltonian [35][36][37][38][39][40][41], the electronic occupation of a double quantum dot whose electronic dipole moment couples to the substrate phonons in a semi-conductor [42], an electronic, nuclear or effective spin exposed to a magnetic field and interacting with an (anisotropic) phononic, electronic or magnonic environment [43][44][45][46][47], and a plethora of other aspects of quantum dots, ultracold atomic impurities, and superconducting circuits [48][49][50][51]. In all these contexts, an effective 'spin' S interacts with an environment, where S is a vector of operators (with units of angular momentum) whose components fulfil the angular momentum commutation relations [S j , S k ] = ih l ϵ jkl S l with j, k, l ∈ {x, y, z}.…”

Hamiltonian of mean force in the weak-coupling and high-temperature approximations and refined quantum master equations

“…[8], see also Refs. [10,48,49]. In particular, it is diagonal in the so called "pointer basis" (the eigenbasis of A in the case of Eq.…”

Quantum-classical correspondence in spin-boson equilibrium states at arbitrary coupling

“…While this is widely thought to be the case, some open questions remain about formal proofs showing the convergence of the dynamics towards the steady state predicted by the MF state [32]. For example, for quantum systems, this convergence has only been proven in the weak [22] and ultrastrong limits [31], while for intermediate coupling strengths there is numerical evidence for the validity of the MF state [36]. Here we proceed to numerically verify the convergence of the dynamics towards the MF state for the case of the classical spin at arbitrary coupling strength.…”

“…Setting. This generalised version of the spin-boson model [35,36] describes a vast range of physical contexts, including excitation energy transfer processes in molecular aggregates described by the Frenkel exciton Hamiltonian [37][38][39][40][41][42][43], the electronic occupation of a double quantum dot whose electronic dipole moment couples to the substrate phonons in a semi-conductor [34], an electronic, nuclear or effective spin exposed to a magnetic field and interacting with an (anisotropic) phononic, electronic or magnonic environment [44], and a plethora of other aspects of quantum dots, ultracold atomic impurities, and superconducting circuits [45][46][47][48]. In all these contexts, an effective "spin" S interacts with an environment, where S is a vector of operators (with units of angular momentum) whose components fulfil the angular momentum commutation relations [S j , S k ] = i l jkl S l with j, k, l = x, y, z.…”