Exact non-Markovian master equation for the spin-boson and Jaynes-Cummings models

Abstract

We provide the exact non-Markovian master equation for a two-level system interacting with a thermal bosonic bath, and we write the solution of such a master equation in terms of the Bloch vector. We show that previous approximated results are particular limits of our exact master equation. We generalize these results to more complex systems involving an arbitrary number of two-level systems coupled to different thermal baths, providing the exact master equations also for these systems. As an example of this general case we derive the master equation for the Jaynes-Cummings model. In order to describe these systems one needs to consider a non-static bath, i.e. a bath that keeps track of the interaction with the TLS. Accordingly, some memory effects build up and the dynamics is non-Markovian. Several tentatives have been made to provide a non-Markovian master equation for the spin-boson model, exploiting e.g. the noninteracting-blip approximation [3], the time convolution-less technique (TCL) [1,6], or the stochastic approach [7]. However, only approximated results were obtained. The lack of an exact analytical description lead to investigate the problem by means of numerical techniques, among which we mention hierarchical equations of motion [8], * Electronic address: ferialdi@fmf.uni-lj.si arXiv:1609.00645v3 [quant-ph]

Qubit dynamics with classical noise

“…Clearly C1 deviates from this behaviour. A deviation of exponential time decay in non-markovian settings is known [2,12,4,9,10,25,8], but we have not yet analyzed the markovian properties of our current model. Both in quantum and classical stochastic (timedependent) noise models, it is how to define the noise correlation function (which in turn determines decay rates).…”

“…), they represent diverse physical systems, ranging from qubits, spins or atoms interacting with radiation in quantum information theory and quantum optics [10,15] to donor-acceptor systems in quantum chemical and quantum biological processes [28,19]. The analysis of open two-level systems is far from trivial [16] and new results are emerging regularly [8,13,14]. One possible realization of such a two-level system is given by a quantum particle in a double well potential, having minima (say at spatial locations x 1 and x 2 ).…”

Solvable class of non-Markovian quantum multipartite dynamics

“…For tackling this issue a broad class of phenomenological and theoretical approaches has been formulated [3], dealing with both time-convoluted and convolutionless master equations [11]. Examples include the dynamics induced by stochastic Hamiltonians defined by non-white noises [12], phenomenological single memory kernels [13][14][15][16], interaction with incoherent degrees of freedom [17][18][19][20][21][22] and arbitrary ancilla systems [23,24], related quantum collisional models [25][26][27][28][29][30][31][32], quantum generalizations of semi-Markov processes [33,34], and random (convex) superpositions of unitary and unital maps [35][36][37], together with some exact derivations from underlying (microscopic or effective) unitary dynamics [38][39][40][41][42][43][44][45][46].…”

A solvable class of non-Markovian quantum multipartite dynamics

“…For tackling this issue a broad class of phenomenological and theoretical approaches has been formulated [3], dealing with both time-convoluted and convolutionless master equations [11]. Examples include the dynamics induced by stochastic Hamiltonians defined by non-white noises [12], phenomenological single memory kernels [13][14][15][16], interaction with incoherent degrees of freedom [17][18][19][20][21][22] and arbitrary ancilla systems [23,24], related quantum collisional models [25][26][27][28][29][30][31][32], quantum generalizations of semi-Markov processes [33,34], and random unitary dynamics [35,36], together with some exact derivations from underlying (microscopic or effective) unitary dynamics [37][38][39][40][41][42][43][44][45].…”