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arXiv:0903.5181 · Physical Review A

Numerical and analytical approach to the quantum dynamics of two coupled spins in bosonic baths

Alessandro Sergi, Ilya Sinayskiy, Francesco Petruccione

Published Jul 15, 2009• 27 citations
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Abstract

The quantum dynamics of a spin chain interacting with multiple bosonic baths is described in a mixed Wigner-Heisenberg representation. The formalism is illustrated by simulating the time evolution of the reduced density matrix of two coupled spins, where each spin is also coupled to its own bath of harmonic oscillators. In order to prove the validity of the approach, an analytical solution in the Born-Markov approximation is found. The agreement between the two methods is shown.

DOI
10.1103/physreva.80.012108

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The quantum dynamics of a spin chain interacting with multiple bosonic baths is described in a mixed Wigner-Heisenberg representation. The formalism is illustrated by simulating the time evolution of the reduced density matrix of two coupled spins, where each spin is also coupled to its own bath of harmonic oscillators. In order to prove the validity of the approach, an analytical solution in the Born-Markov approximation is found. The agreement between the two methods is shown.
When modelling real physical systems, one should always consider the influence of the environment.The study of open quantum systems aims precisely at the assessment of the role of the environment in the evolution of these systems.The computational study of these systems is in general a formidable task.However, in many instances one can approximate the environment by means of classical dynamics.For such cases, when the environment is in a thermal state, one can generalize non-Hamiltonian equations of motion to quantum-classical dynamics in order to control the thermodynamic temperature.Such a generalization is achieved within the partial Wigner representation of quantum mechanics, and leads to a quantum-classical evolution of the system.We have studied how to simulate a thermal bath, by means of the least possible number of additional extended variables [1].This has been achieved upon reformulating the Nosé-Hoover Power (NHP) thermostat in quantum-classical theory.When applied to the dissipative evolution of a quantum spin the NHP thermostat, obtained numerical results in agreement with those obtained using Nosè-Hoover chains.However, since a fewer number of variables are needed to achieve the correct sampling of the canonical distribution at equilibrium, the NHP thermostat promises to be better suited for the simulation of low dimensional open quantum system on discrete grids.Following this the quantum dynamics of a XXZ two-spin chain interacting with multiple bosonic baths have been studied, within the mixed Wigner-Heisenberg representation of quantum mechanics.In particular, we have simulated the dynamics of the reduced density matrix ρ s (t), along with the time evolution of the quantum entanglement of the two spins via the concurrence C (ρ).
We calculate the reduced density matrix for a system of coupled harmonic oscillators in a bosonic heat bath using the Born-Markov approximation and show that the expectation values of position and momentum evolve like classical quantities. We consider the cases where every oscillator is coupled to its own reservoir and where all oscillators are coupled to the same heat bath, pointing out the differences between these two models. In particular, for systems coupled to a common reservoir, we present an uncommon dissipation mechanism, which disappears, when the dynamics of the system are gouverned by the low frequency modes of the environment (Caldeira-Leggett limit). We also address the interesting phenomenon of reservoir-induced interactions.
The dynamics of simple qubit systems in a chain configuration coupled at both ends to separate bosonic baths at different temperatures is studied. An exact analytical solution of the master equation in the Born-Markov approximation for the reduced density matrix of the qubit system is constructed. The unique non-equilibrium stationary state for the long time behavior of the reduced density matrix in obtained. Dynamical and steady state properties of the concurrence between the first and the last spin are studied.
We study a new model for treating quantum dissipative systems, in which the bath is modeled as a collection of spins coupled to the system of interest. We develop a quasiclassical method to study this model, approximating the quantum Heisenberg equations by the classical ones, supplemented with stochastic initial conditions carefully chosen so that the results obtained from the classical equations are as close as possible to the quantum results. Using this method we compare the dynamics of such a spin–spin system with that of a spin–boson system, in which the bath is modeled as a collection of harmonic oscillators. We verify numerically that when the system-bath coupling is spread over many bath spins (the Brownian motion limit), the spin–spin model can be mapped on the spin–boson model (although with a temperature dependent spectral density). We also demonstrate that the two dissipative models are qualitatively very different in a non-Brownian motion regime.