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International Journal of Modern Physics B

DYNAMICS OF A LARGE-SPIN-BOSON SYSTEM IN THE STRONG COUPLING REGIME

Till Vorrath, Tobias Brandes, Bernhard Krämer

Published Nov 10, 2003• 2 citations
Publisher

Abstract

We investigate collective effects of an ensemble of biased two-level systems interacting with a bosonic bath in the strong coupling regime. The two level systems are described by a large pseudo-spin J. An equation for the expectation value M(t) of the z-component of the pseudo spin is derived and solved numerically for an ohmic bath at T=0. In case of a large cut-off frequency of the spectral function, a Markov approximation is justified and an analytical solution is presented. We find that M(t) relaxes towards a highly correlated state with maximum value ±J for large times. However, this relaxation is extremely slow for most parameter values so as if the system was "frozen in" by interaction with the bosonic bath.

DOI
10.1142/s0217979203020624

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We investigate collective effects of an ensemble of biased two-level systems interacting with a bosonic bath in the strong coupling regime. The two level systems are described by a large pseudo-spin J. An equation for the expectation value M(t) of the z-component of the pseudo spin is derived and solved numerically for an ohmic bath at T=0. In case of a large cut-off frequency of the spectral function, a Markov approximation is justified and an analytical solution is presented. We find that M(t) relaxes towards a highly correlated state with maximum value ±J for large times. However, this relaxation is extremely slow for most parameter values so as if the system was "frozen in" by interaction with the bosonic bath.
We investigate collective effects of an ensemble of biased two level systems interacting with a bosonic bath in the strong coupling regime. The two level systems are described by a large pseudo-spin J. An equation for the expectation value M(t) of the z-component of the pseudo spin is derived and solved numerically for an ohmic bath at T=0. In case of a large cut-off frequency of the spectral function, a Markov approximation is justified and an analytical solution is presented. We find that M(t) relaxes towards a highly correlated state with maximum value ±J\pm J±J for large times. However, this relaxation is extremely slow for most parameter values so as if the system was "frozen in" by interaction with the bosonic bath.
The spin-boson model, describing a two-level system coupled to a bosonic bath, is extensively studied as a paradigmatic dissipative quantum system, exhibiting rich dynamical behavior and even a localization transition in the strong coupling regime. Here, we additionally consider dephasing as a source of Markovian dissipation on top of the non-Markovian dynamics due to an Ohmic bath, and investigate the dynamics of the spin. We show that the characteristic frequency of the spin dynamics, while strongly renormalized by the bosonic bath, changes in a simple fashion (or does not change at all) with dephasing. To obtain these results, we develop an exact nonperturbative method known as the stochastic Schr\"odinger equation, mimicking the Ohmic bath via a stochastic magnetic field combined with the Lindblad quantum master equation due to dephasing, which allows us to numerically compute the dynamics. Furthermore, we derive weak-coupling analytic results utilizing the well-known noninteracting blip approximation. Our findings are relevant to quantum simulation of the spin-boson model in the regime of strong coupling in trapped ions and circuit QED architectures among others.
We develop a semianalytical approach beyond the Born-Markov approximation to study the quench dynamics of the spin-boson model in the strong-coupling regime ($\ensuremath{\alpha}\ensuremath{\le}1/2$) for the Ohmic bath. The basic idea in our approach is to write an effective time-dependent model for the dynamics of the system coupled to the bosonic bath after integrating out high-frequency bath modes. By applying this procedure to the Heisenberg equations of motion, we derive a set of flow equations for the system parameters as a function of time. The final flow equations look similar to those of the equilibrium renormalization-group theory; however, in our derivation the scaling parameter is set by the real time. We solve the equations of motion with time-dependent renormalized parameters and show that the resulting dynamics is in decent agreement with the exact NRG calculations as well as the noninteracting blip approximation that is a well-known good solution in this limit.
We consider multiple noninteracting quantum mechanical two-level systems coupled to a common bosonic bath and study its quantum phase transition with Monte Carlo simulations using a continuous imaginary time cluster algorithm. The common bath induces an effective ferromagnetic interaction between the otherwise independent two-level systems, which can be quantified by an effective interaction strength. For degenerate energy levels above a critical value of the bath coupling strength $\ensuremath{\alpha}$ all two-level systems freeze into the same state and the critical value ${\ensuremath{\alpha}}_{c}$ decreases asymptotically as 1/N1/N1/N with increasing NNN. For a finite number NNN of two-level systems the quantum phase transition (at zero temperature) is in the same universality class as the single spin-boson model; in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the system shows mean-field critical behavior independent of the power of the spectral function of the bosonic bath. We also study the influence of a spatial separation of the spins in a bath of bosonic modes with linear dispersion relation on the location and characteristics of the phase transition as well as on correlations between the two-level systems.
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.