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Physical Review A

Correlation functions for dissipative two-state systems: Effects of the initial preparation

Maura Sassetti, Ulrich Weiß

Published May 1, 1990• 69 citations
Publisher

Abstract

We study the influence of system-bath correlations in the initial state on the dynamics of biased two-state systems. The formal solution for the equilibrium correlation function in the form of a power series in the number of transitions between the system states is determined. For Ohmic damping and a special value of the coupling strength (K=1/2), the exact solution for all times and temperatures is presented. We find that at low temperatures the long-time tails of the decay into equilibrium are strongly affected by the initial preparation. Correspondingly, the low-frequency behavior of the neutron-scattering function is qualitatively changed by the presence or absence of correlations in the initial state.

DOI
10.1103/physreva.41.5383

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We study the influence of system-bath correlations in the initial state on the dynamics of biased two-state systems. The formal solution for the equilibrium correlation function in the form of a power series in the number of transitions between the system states is determined. For Ohmic damping and a special value of the coupling strength (K=1/2), the exact solution for all times and temperatures is presented. We find that at low temperatures the long-time tails of the decay into equilibrium are strongly affected by the initial preparation. Correspondingly, the low-frequency behavior of the neutron-scattering function is qualitatively changed by the presence or absence of correlations in the initial state.
We study the dynamical equilibrium correlation function of the polaron-dressed tunneling operator in the dissipative two-state system. Unlike the position operator, this coherence operator acts in the full system-plus-reservoir space. We calculate the relevant modified influence functional and present the exact formal expression for the coherence correlations in the form of a series in the number of tunneling events. For an Ohmic spectral density with the particular damping strength K=12,K=\frac{1}{2},K=21​, the series is summed in analytic form for all times and for arbitrary values of temperature and bias. Using a diagrammatic approach, we find the long-time dynamics in the regime K<1.K<1.K<1. In general, the coherence correlations decay algebraically as ${t}^{\ensuremath{-}2K}$ at T=0.T=0.T=0. This implies that the linear static susceptibility diverges for K< 12K<~\frac{1}{2}K< 21​ as $\stackrel{\ensuremath{\rightarrow}}{T}0,$ whereas it stays finite for K>12K>\frac{1}{2}K>21​ in this limit. The qualitative differences with respect to the asymptotic behavior of the position correlations are explained.
The time correlation functions for a Gaussian wave-packet preparation of the dissipative harmonic oscillator evolving from three initial conditions for the heat bath are calculated and compared with each other for Ohmic heat baths. The three initial distributions for the bath are the factorized, partially factorized, and unfactorized distributions. Explicit analytical formulas are derived and then used to study the effect of the three initial distributions on the subsequent dynamics. We find that the transient behavior does not depend sensitively on the initial condition as long as the initial Gaussian wave function of the system is centered at the equilibrium point. Differences become noticeable as the center of the wave packet is significantly shifted from the equilibrium point. These observations justify to some extent the prevalent use of factorized initial conditions for studying real time quantum dynamics in dissipative systems. The total energy in the system is also calculated for the three initial states and its relation to features in the decay is pointed out.
Abstract The emerging quantum technologies heavily rely on the understanding of dynamics in open quantum systems. In the Born approximation, the initial system-bath correlations are often neglected which can be violated in the strong coupling regimes and quantum state preparation. In order to understand the influence of initial system-bath correlations, we study the extent to which these initial correlations and the distance of separation between the qubits influence the dynamics of quantum entanglement and coherence. It is shown that at low temperatures, the initial correlations have no role to play while at high temperatures, these correlations strongly influence the dynamics. Furthermore, we have shown that the distance of separation between the qubits in presence of a collective bath helps to maintain entanglement and coherence at long times.
We investigate the dynamical correlation function of a quantum-mechanical two-state system that is coupled to a bosonic heat bath, utilizing the equivalence between the spin-boson Hamiltonian and the 1/r2{1/r}^{2}1/r2 Ising model. The imaginary-time correlation function is obtained from Monte Carlo simulations on the Ising system and then continued to real time by a Pad\'e approximation. In the unbiased system, the transition from oscillatory to strongly damped behavior is found to occur at a coupling strength close to $\ensuremath{\alpha}=1/2$. The biased system favors coherent relaxation and displays a significantly larger crossover value ${\ensuremath{\alpha}}_{c}.$ We introduce the quasiparticle picture to describe the relevant behavior at intermediate time scales. Within this approximation, we map out phase diagrams for the unbiased and biased systems.
The thermodynamics and dynamics of the spin-boson model are formulated in terms of the exact formal solutions. The low-temperature features of the free energy, the static susceptibility, and the equilibrium correlation function are presented for Ohmic and super-Ohmic dissipation, and various universal properties are deduced. In the Ohmic case the Wilson ratio and the spectral properties at low frequencies (Shiba's relation) are calculation in the range 01 of the coupling strength. The corresponding universal relations are determined also for super-Ohmic dissipation.
We study the equilibrium correlation function of the polaron-dressed tunnelling operator in the dissipative two-state system and compare the asymptoptic dynamics with that of the position correlations. For an Ohmic spectral density with damping strength K = ½, the correlation functions are obtained in analytic form for all times at any T and any bias. For K < 1, the asymptotic dynamics is found by using a diagrammatic approach within a Coulomb gas representation. At T = 0, the tunnelling or coherence correlations drop as t −2 K , whereas the position correlations show universal decay ∝ t −2 . The former decay law is a signature of unscreened attractive charge-charge interactions, while the latter is due to unscreened dipole-dipole interactions.