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arXiv:cond-mat/0612396 · Europhysics Letters (EPL)

Spin-boson dynamics: A unified approach from weak to strong coupling

Francesco Nesi, Elisabetta Paladino, Michael Thorwart, Milena Grifoni

Published Oct 18, 2007• 12 citations
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Abstract

We present a novel approximation scheme to describe the influence of a harmonic bath on the dynamics of a two-level particle over almost the whole regime of temperatures and coupling to the environment, for a wide class of bath spectral densities. Starting from the exact path integral solution for the two-level system density matrix, effective intra-blip correlations are fully included, while inter-blip and blip-sojourn interactions are considered up to first order. In the proper regimes, an excellent agreement with conventional perturbative approaches and ab initio path-integral results is found.

DOI
10.1209/0295-5075/80/40005

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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 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 dissipative quantum dynamics of a two-level system interacting with a structured reservoir consisting of a damped harmonic mode is investigated by means of the numerically exact path integral Monte Carlo method. This approach provides benchmark results in a broad range of parameter space, in particular, in those domains which are not accessible by approximate methods and alternative numerical schemes, that is, strong coupling between system and harmonic mode and from low to high temperatures. For low temperatures the numerical data are quantitatively in agreement with the noninteracting blip approximation only in the regimes of very weak and very strong coupling. It turns out that the entangled dynamics of the two-level system and the harmonic mode is relatively robust so that its signatures are observable up to relatively strong friction and high temperatures. Nonequilibrium initial preparations of the reservoir with respect to the initial state of the system give rise, for strong interactions, to a stepwise decay of the population, thus displaying coherent dynamics of the bath. The impact of an additional Ohmic bath coupled directly to the two-level system is studied as well, including the case where both reservoirs carry different temperatures.
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 introduce and apply a numerically exact method for investigating the real-time dissipative dynamics of quantum impurities embedded in a macroscopic environment beyond the weak-coupling limit. We focus on the spin-boson Hamiltonian that describes a two-level system interacting with a bosonic bath of harmonic oscillators. This model is archetypal for investigating dissipation in quantum systems, and tunable experimental realizations exist in mesoscopic and cold-atom systems. It finds abundant applications in physics ranging from the study of decoherence in quantum computing and quantum optics to extended dynamical mean-field theory. Starting from the real-time Feynman-Vernon path integral, we derive an exact stochastic Schr\"odinger equation that allows us to compute the full spin density matrix and spin-spin correlation functions beyond weak coupling. We greatly extend our earlier work [P. P. Orth, A. Imambekov, and K. Le Hur, Phys. Rev. A 82, 032118 (2010)] by fleshing out the core concepts of the method and by presenting a number of interesting applications. Methodologically, we present an analogy between the dissipative dynamics of a quantum spin and that of a classical spin in a random magnetic field. This analogy is used to recover the well-known noninteracting-blip approximation in the weak-coupling limit. We explain in detail how to compute spin-spin autocorrelation functions. As interesting applications of our method, we explore the non-Markovian effects of the initial spin-bath preparation on the dynamics of the coherence ${\ensuremath{\sigma}}^{x}(t)$ and of ${\ensuremath{\sigma}}^{z}(t)$ under a Landau-Zener sweep of the bias field. We also compute to a high precision the asymptotic long-time dynamics of ${\ensuremath{\sigma}}^{z}(t)$ without bias and demonstrate the wide applicability of our approach by calculating the spin dynamics at nonzero bias and different temperatures.
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.