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Physical review. B, Condensed matter

Dissipative quantum dynamics in a boson bath

Y. C. Chen, Joel L. Lebowitz, Carlangelo Liverani

Published Sep 1, 1989• 37 citations
Publisher

Abstract

We investigate the dynamics of a quantum particle coupled to a boson heat bath. Using a real-time path-integral formalism, we obtain the Wigner distribution of the particle in the form of a power series in the strength of the anharmonicity ${V}_{0}$. This series is shown to converge for all ${V}_{0}$ when t is fixed and for small ${V}_{0}$ uniformly in t. The latter proves the convergence to equilibrium for small anharmonicities. The effects of initial conditions on the evolution are studied by explicitly considering two types of initial states: product states and mixed Gibbs states. We show that in certain cases the evolution of the mixed states which avoid many pathologies of the product states, arising from the removal of the cutoff on the frequency distribution of the bath, can be related to that of product states when the latter are started at t=-\ensuremath{\infty}. We also solve exactly a simple stationary nonequilibrium model, a harmonic system in contact with two thermal baths, and derive an alternative criterion for a practically useful quasiclassical approximation. Finally, some connections to the Josephson junction are discussed. These include (a) the Green-Kubo-Einstein relation between the mobility and the diffusion constant of the washboard potential and (b) the time evolution of a rf superconducting quantum interference device.

DOI
10.1103/physrevb.40.4664

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We investigate the dynamics of a quantum particle coupled to a boson heat bath. Using a real-time path-integral formalism, we obtain the Wigner distribution of the particle in the form of a power series in the strength of the anharmonicity V0{V}_{0}V0​. This series is shown to converge for all V0{V}_{0}V0​ when t is fixed and for small V0{V}_{0}V0​ uniformly in t. The latter proves the convergence to equilibrium for small anharmonicities. The effects of initial conditions on the evolution are studied by explicitly considering two types of initial states: product states and mixed Gibbs states. We show that in certain cases the evolution of the mixed states which avoid many pathologies of the product states, arising from the removal of the cutoff on the frequency distribution of the bath, can be related to that of product states when the latter are started at t=-\ensuremath{\infty}. We also solve exactly a simple stationary nonequilibrium model, a harmonic system in contact with two thermal baths, and derive an alternative criterion for a practically useful quasiclassical approximation. Finally, some connections to the Josephson junction are discussed. These include (a) the Green-Kubo-Einstein relation between the mobility and the diffusion constant of the washboard potential and (b) the time evolution of a rf superconducting quantum interference device.
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
Quantum information processing has greatly increased interest in the phenomenon of environmentally-induced decoherence. The spin boson model is widely used to study the interaction between a spin-modelling a quantum particle moving in a double well potential-and its environment-modelled by a heat bath of harmonic oscillators. This paper extends a previous analysis of the static spin boson study to the driven spin boson case, with the derivation of an exact integro-differential equation for the time evolution of the propagator of the reduced spin density matrix. This is the first main result. By specializing to weak damping we then obtain the next result, a set of Bloch-Redfield equations for the equilibrium fixed spin initial condition. Finally we show that these equations can be used to solve the classic dissipative Landau-Zener problem and illustrate these solutions for the weak damping case. The effect of dissipation is seen to be minimised as the speed of passage is increased, implying that qubits need to be switched as fast as possible.
We investigate the effect of dissipation in a Bose-Josephson junction (BJJ) coupled to baths of bosons at its two sites. Apart from the dynamical transition due to repulsive interactions, the BJJ undergoes a quantum phase transition by increasing the coupling strength with the bath modes. We analyze this system by mapping to an equivalent spin model coupled to the bosonic modes. The excitation energies and fluctuation of number imbalance are obtained within a Holstein-Primakoff approximation, which exhibits vanishing of the energy gap and enhanced quantum fluctuations at the critical point. We study the dynamics of BJJ using a time-dependent variational method and analyze stability of different types of steady states. As a special case we study in detail the phase space dynamics of BJJ coupled to a single mode, which reveals diffusive and incoherent behavior with increasing coupling to the bath mode. The dynamical steady states corresponding to the π oscillation and self-trapped state become unstable when their oscillation frequencies are in resonance with the bath modes. We study the Josephson dynamics in the presence of an Ohmic bath with Gaussian noise to incorporate the thermal fluctuations and obtain the Josephson oscillation frequency and damping analytically. We also observe the transition to the symmetry-broken state for strong coupling as well as decay of π oscillation and a self-trapped state to the ground state due to dissipation. Variation of the phase fluctuation with temperature of the bath shows similar behavior as observed in experiment. Finally we discuss the experimental setup to study the observable effects of dissipation in BJJ.
We explore how the dissipative geometric phase evolves within the spin-boson model, focusing specifically on coupling to an Ohmic bath in the weakly coherent regime. To determine the non-unitary time evolution of the system's reduced density matrix, we employ the non-interacting blip approximation (NIBA). For the localized pure initial state used throughout, we derive a compact Bloch-sphere expression showing that the mixed-state geometric phase is a weighted azimuthal winding of the dissipative trajectory. We then map geometric-phase accumulation across various system-bath coupling strengths, temperatures, static biases, and bath cutoff frequencies. Benchmarking representative results against the numerically exact time-evolving matrix product operator (TEMPO) technique shows that NIBA reproduces the population dynamics almost indistinguishably. It also captures the geometric phase quantitatively and qualitatively, although TEMPO reveals a clearer trend in the stationary coherence. Our results show that quantum dissipation suppresses the geometric phase through two complementary mechanisms: thermal noise reduces the state's purity, while static bias localizes the dynamics and reduces the accessible geometric area.
We investigate the effect of dissipation in a Bose Josephson junction (BJJ) coupled to bath of bosons at two sites as an application of the generalized spin-boson model. Apart from the dynamical transition due to repulsive interactions, the BJJ undergoes a quantum phase transition by increasing the coupling strength with the bath modes. We analyze this system by mapping to an equivalent spin model coupled to the bosonic modes. The excitation energies and fluctuation of number imbalance are obtained within Holstein-Primakoff approximation, which exhibits vanishing of energy gap and enhanced quantum fluctuations at the critical coupling. We study the dynamics of BJJ using time-dependent variational method and analyze stability of different types of steady states. As a special case, we study in details the phase space dynamics of BJJ coupled to a single-mode, which reveals diffusive and incoherent behaviour with increasing coupling to the bath mode. The dynamical steady states corresponding to the Pi-oscillation and self-trapped state become unstable in the region where their oscillation frequencies are in resonance with the bath modes. We investigate the time evolution of number imbalance and relative phase in presence of Ohmic bath with Gaussian noise to incorporate thermal fluctuations. Apart from the damping of Josephson oscillations and transition to symmetry broken state for strong coupling, we observe the decay of Pi-oscillation and self-trapped state to the ground state as a result of dissipation. Variation of phase fluctuation with temperature of the bath shows similar behaviour as observed in the experiments.
We consider systems described by compact, periodic variables, such as Josephson junctions and small normal metal rings. We examine how they can be coupled to a heat bath, via either momentum or position coupling, and how the two descriptions can be related by a unitary transformation. We show how it is critical to transform not only the Hamiltonian, but also the initial conditions. We then demonstrate that for certain types of initial conditions, paths of different winding number can interfere. Still other, ostensibly reasonable initial conditions, have no such interference. We conclude with a discussion of appropriate models to describe periodic systems.