Driving spin-boson models from equilibrium using exact quantum dynamics

Abstract

We present an application of the Extended Stochastic Liouville Equation (ESLE) [Phys. Rev. B 95, 125124], which gives an exact solution for the reduced density matrix of an open system surrounded by a harmonic heat bath. This method considers the extended system (the open system and the bath) being thermally equilibrated prior to the action of a time dependent perturbation, as opposed to the usual assumption that system and bath are initially partitioned. This is an exact technique capable of accounting for arbitrary parameter regimes of the model. Here we present our first numerical implementation of the method in the simplest case of a Caldeira-Leggett representation of the bath Hamiltonian, and apply it to a spin-boson system driven from coupled equilibrium. We observe significant behaviours in both the transient dynamics and asymptotic states of the reduced density matrix not present in the usual approximation.arXiv:1712.06397v1 [quant-ph]

Exactly thermalized quantum dynamics of the spin-boson model coupled to a dissipative environment

“…where H 0 = H tot (t 0 ) is the initial Hamiltonian of the combined open system and its environment, Z 0 = Tr(e −βH 0 ) is the equilibrium partition function of the total system, and β = 1/k B T is the inverse temperature. Following the seminal work of Graber, Schramm, and Ingold [13], it was recently shown [10,15] that it is possible to thermalize the reduced density matrix of the open system via a novel application of the influence functional formalism in which the environment variables are integrated out for arbitrary real time t. The resulting pair of SDEs describing the thermalization in imaginary time and subsequent dynamics in real time of the stochastic reduced density matrix are known as the extended stochastic Liouville-von Neumann equations (ESLNs), with the evolution of the reduced density matrix being driven by complex correlated Gaussian noises in both cases. Expressing the equation of motion of the physical reduced density matrix as an ensemble average over stochastic paths via a Hubbard-Stratonovich transformation in this way is commonly referred to as stochastic unraveling [31][32][33].…”

“…It turns out that some allowed choices result in numerical instability during the early-time dynamics, even though the correlation functions are fully satisfied. In our previous work [15], a method for noise generation was proposed which we shall review and further develop here, introducing a modified noise generation scheme that diminishes the exponential growth of the trace of the density matrix which seems to characterize these methods. This is the latest in a series of proposals aimed at tackling this problem [16,17].…”

“…The model consists of a twolevel spin system surrounded by bosonic degrees of freedom that describe the environment, and can naturally be applied to qubits coupled to an environment [18][19][20][21][22], electronic energy transfer in biological systems [16], Josephson junctions [23][24][25], cold atoms [26,27], and solid-state artificial atoms [28]. The spin-boson model has already been considered previously by us in the context of the ESLN [15]; however, due to a recently discovered implementation error, the numerical results were inaccurate. Here we present further implementation development and update our numerical results.…”

Proposal of a Computational Approach for Simulating Thermal Bosonic Fields in Phase Space

“…For simplicity, the theory was only applied to the free field case [38], in order to demonstrate the technicalities associated with coupling a different NHC thermostat [44,45] to each field mode. However, this theory is intended for situations where the field is coupled to a spin system [47][48][49][50][51][52][53][54][55][56][57]. In this case, the use of NHC thermostats in the dynamics of the thermal state of the field would make it possible to simulate processes that, to our knowledge, have not been investigated so far.…”

“…In addition to the case when the field is coupled to another system [47][48][49][50][51][52][53][54][55][56][57] there are a number of cases when it is desirable to simulate the dynamics of the thermal field state [27-34] on a computer. To this end, we employ a Nosé-Hoover chain (NHC) thermostat [44,45], which may be theoretically defined in terms of a quasi-Lie bracket [63][64][65][66].…”