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International Journal of Quantum Information

ARBITRARILY SLOW RELAXATIONS IN A SPIN-BOSON MODEL UNDER THE NONINTERACTING-BLIP APPROXIMATION

Filippo Giraldi, Francesco Petruccione

Published Dec 1, 2012• 3 citations
Publisher

Abstract

A two-level system coupled to a thermal bath of bosons in the noninteracting-blip approximation, is studied in dependence on the low frequency behavior of the spectral density of the reservoir. The long time evolution of the polarization and the populations exhibits a discontinuity in correspondence of the power four of the super-ohmic regime. By approaching this boundary, the long time inverse power law relaxations become arbitrarily slow. The role of the temperature changes in the transition to powers larger than four, enhancing the amplitude of the long time polarization.

DOI
10.1142/s0219749912410018

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A two-level system coupled to a thermal bath of bosons in the noninteracting-blip approximation, is studied in dependence on the low frequency behavior of the spectral density of the reservoir. The long time evolution of the polarization and the populations exhibits a discontinuity in correspondence of the power four of the super-ohmic regime. By approaching this boundary, the long time inverse power law relaxations become arbitrarily slow. The role of the temperature changes in the transition to powers larger than four, enhancing the amplitude of the long time polarization.
We analyze the short- and long-time behavior of the bath correlation function for open quantum systems interacting with thermal baths. The spectral densities under study are ohmic-like at low frequencies, exhibit possible perturbations of the low-frequency power-law profiles and are arbitrarily shaped at higher frequencies. The low-frequency perturbations are provided by arbitrarily positive or negative powers of logarithmic forms as additional factors for the power laws of the ohmic-like spectral densities. If the spectral density decays sufficiently fast at high frequencies the short-time behavior of the bath correlation function is algebraic. The model provides long-time relaxations of the bath correlation function which are arbitrarily faster or slower than inverse power laws. In fact, the long-time relaxations are described by inverse power laws and by arbitrary powers of logarithmic forms as additional factors. Such relaxations are regularly related to the low-frequency structure of the spectral density, except for even or odd natural values of the ohmicity parameter. In these exceptional conditions the long-time relaxations of the bath correlation function are faster than those obtained via the regular relation.
A generalized approximation scheme is proposed to describe the dynamics of the spin-boson problem. Being nonperturbative in the coupling strength nor in the tunneling frequency, it gives reliable results over a wide regime of temperatures and coupling strength to the thermal environment for a large class of bath spectral densities. We use a path-integral approach and start from the exact solution for the two-level system population difference in the form of a generalized master equation (GME). Then, we approximate interblip and blip-sojourn interactions up to linear order, while retaining all intrablip correlations to find the kernels entering the GME in analytical form. Our approximation scheme, which we call weakly interacting blip approximation, fully agrees with conventional perturbative approximations in the tunneling matrix element (noninteracting-blip approximation) or in the system-bath coupling strength in the proper parameter regime.
The effective theories for many quantum phase transitions can be mapped onto those of classical transitions. Here we show that the naive mapping fails for the sub-Ohmic spin-boson model which describes a two-level system coupled to a bosonic bath with power-law spectral density, J(omega) proportional, variantomega(s). Using an epsilon expansion we prove that this model has a quantum transition controlled by an interacting fixed point at small s, and support this by numerical calculations. In contrast, the corresponding classical long-range Ising model is known to display mean-field transition behavior for 0 < s < 1/2, controlled by a noninteracting fixed point. The failure of the quantum-classical mapping is argued to arise from the long-ranged interaction in imaginary time in the quantum model.
The dynamical behavior of a two-level system coupled linearly to a dissipative bath of two-level systems whose tunneling splittings and coupling parameters are distributed according to an Ohmic spectrum is studied using numerical path integral methods as well as the noninteracting blip approximation for an equivalent harmonic environment characterized by a scaled spectral density. Comparisons with the behaviors of the conventional spin-boson model of similar coupling strength reveal a weaker quenching of coherence in the present case. Finally, the boundary in parameter space between coherent and incoherent dynamics is mapped.
The loss of coherence in a general open quantum system interacting with a bosonic environment is analyzed. The reservoir is initially in a thermal state. The reduced dynamics is described by a non-Markovian time-local master equation. We consider spectral densities that are sub- or super-Ohmic at low frequencies and arbitrarily shaped at high frequencies. In the super-Ohmic regime, for noninteger frequency powers larger than 2, long time survival of coherence appears. In the latter regime, at vanishing temperature, the asymptotic amount of surviving coherence is stabilized to its initial value, up to a phase factor, by properly increasing the bandwidth and decreasing the low-frequency profile of the spectral density. For noninteger positive frequency powers less than 2, stretched exponential-like decoherence is found over long times. The relaxations to the asymptotic configurations become arbitrarily slow by approaching the frequency power 2 of the super-Ohmic regime. The same dependence on temperature, spectral density, and scale frequency appears for purity and concurrence of two qubits and coherence of a qubit.
The transition from weakly damped coherent motion to localization in the context of the spin-boson model has been the subject of numerous studies with distinct behavior depending on the form of the phonon-bath spectral density $J\left(\ensuremath{\omega}\right)\ensuremath{\propto}{\ensuremath{\omega}}^{s}$. Sub-Ohmic (s<1s<1s<1) and Ohmic (s=1s=1s=1) spectral densities show a clear localization transition at zero temperature and zero bias, while for super-Ohmic (s>1s>1s>1) spectral densities this transition disappears. In this paper, we consider the influence of the phonon-bath spectral density on the nonequilibrium dynamics of a quantum dot with electron-phonon interactions described by the extended Holstein model. Using the reduced density matrix formalism combined with the multilayer multiconfiguration time-dependent Hartree approach, we investigate the dynamic response, the time scales for relaxation, as well as the existence of multiple long-lived solutions as the system-bath coupling changes from the sub- to the super-Ohmic cases. Bistability is shown to diminish for increasing powers of sss similar to the spin-boson case. However, the physical mechanism and the dependence on the model parameters such as the typical bath frequency ${\ensuremath{\omega}}_{c}$ and the polaron shift $\ensuremath{\lambda}$ are rather distinct.