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Physical Review A

Dynamics of the spin-boson model at zero temperature and strong dissipation

Jiushu Shao

Published May 2, 2022• 7 citations
Publisher

Abstract

Based on the stochastic decoupling framework and auxiliary function averaging technique, a functional integral equation for the zero-temperature population dynamics of the spin-boson model is established. This equation naturally displays the hierarchical structure of the dissipative dynamics and is amenable to nonperturbative approximations. It is shown that at strong dissipation the spin-boson model displays exponential decay with the rate decreasing from the maximum at the coherence-incoherence crossover to zero at the critical dissipation, a manifestation of quantum phase transition. Besides, the scaled time does not change in the whole regime of strong dissipation.

DOI
10.1103/physreva.105.052201

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Based on the stochastic decoupling framework and auxiliary function averaging technique, a functional integral equation for the zero-temperature population dynamics of the spin-boson model is established. This equation naturally displays the hierarchical structure of the dissipative dynamics and is amenable to nonperturbative approximations. It is shown that at strong dissipation the spin-boson model displays exponential decay with the rate decreasing from the maximum at the coherence-incoherence crossover to zero at the critical dissipation, a manifestation of quantum phase transition. Besides, the scaled time does not change in the whole regime of strong dissipation.
The zero-temperature dynamics of the spin-boson model with strong dissipation has been a challenging problem for more than 20 years. To solve this and quantum dynamics of dissipative systems at large, we recently proposed a mixed random-deterministic method. This scheme has been successfully used to simulate the time evolution of the spin-boson model at zero temperature for weak to moderate dissipation. For a better numerical performance, the approach is further modified so that it is flexible to convert a certain part of the random treatment to a deterministic one a la hierarchical equations. Applying the new method to the strong dissipated spin-boson model at zero temperature, we observe that the population in the localized state obeys a simple decay dynamics and the time scale is proportional to the reciprocal of the cutoff frequency.
Starting from the stochastic Liouville equation, we obtain an integral functional equation for describing the finite-temperature dynamics of the asymmetric spin-boson model. Like the zero-temperature case, this equation displays a hierarchical structure characterizing the multiple-timescale nature of quantum dissipative dynamics. We are thereby able to establish a perturbation series and illustrate the contributions with the conventional concepts of blips and sojourns. We clarify the speciality of the Toulouse limit by presenting the analytical properties of intrablip and blip-sojourn interactions. This speciality leads to a closed-form equation for the blip, from which the exact result of the dissipative dynamics can be obtained. We also demonstrate the exact solution by summing the perturbative series. Moreover, we calculate the exact equilibrium distribution at the Toulouse limit and compare it with the Boltzmann distribution valid for weak dissipation.
We report an anomalous decoherence phenomenon of a quantum dissipative system in the framework of a stochastic decoupling scheme along with a hierarchical equations-of-motion formalism without the usual Born-Markov or weak-coupling approximations. It is found that the decoherence of a two-qubit spin-boson model can be reduced by increasing the bath temperature in strong-coupling regimes. For the weak-coupling situation, we find that the bath temperature may enhance the decoherence. This result is contrary to the common recognition that a higher bath temperature always induces a more severe decoherence and suggests that a decoherence dynamical transition occurs in this two-qubit spin-boson model. We also demonstrate that the critical transition point can be characterized by the behavior of the frequency spectrum of the quantum coherence indicator.
The dissipative dynamics of a quantum bistable system coupled to a Ohmic heat bath is investigated beyond the spin-boson approximation. Within the path-integral approach to quantum dissipation, we propose an approximation scheme which exploits the separation of time scales between intra- and interwell (tunneling) dynamics. The resulting generalized master equation for the populations in a space localized basis enables us to investigate a wide range of temperatures and system-environment coupling strengths. A phase diagram in the coupling-temperature space is provided to give a comprehensive account of the different dynamical regimes.
We investigate the real-time dynamics of the sub-Ohmic spin-boson model across a broad range of coupling strengths, using the numerically exact inchworm quantum Monte Carlo algorithm. From short- and intermediate-time dynamics starting from an initially decoupled state, we extract signatures of the zero-temperature quantum phase transition between localized and delocalized states. We show that the dynamical phase diagram thus obtained differs from the equilibrium phase diagram in both the values of critical couplings and the associated values of the critical exponent. We also identify and quantitatively analyze two competing mechanisms for the crossover between coherent oscillations and incoherent decay. Deep in the sub-Ohmic regime, the crossover is driven by the damping of the oscillation amplitude, while closer to the Ohmic regime the oscillation frequency itself drops sharply to zero at large coupling.
Through a decomposition of the bath correlation function, the hierarchical equations of motion are extended to the Ohmic spin-boson model at zero temperature. For two typical cutoff functions of the bath spectral density, the rate kernel of spin dynamics is numerically extracted by a time-convolution equation of the average magnetic moment. A characteristic time is defined accordingly as the inverse of the zeroth-order moment of the rate kernel. For a given Kondo parameter in the incoherent regime, the time evolution of average magnetic moments gradually collapses onto a master curve after rescaling the time variable with the characteristic time. The rescaled spin dynamics is nearly independent of the cutoff frequency and the form of cutoff functions. For a given cutoff frequency, the characteristic time with the change of the Kondo parameter is fitted excellently as a function of the renormalized tunneling amplitude. Despite a significant difference in definition, our result is in good agreement with the characteristic time of the noninteracting blip approximation.