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arXiv:2108.04915

Caldeira-Leggett oscillator bath simulation of the Prokof'ev-Stamp spin bath in strong coupling limits

Seyyed M. H. Halataei

Published Aug 10, 2021• 0 citations
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Abstract

Open quantum systems are subject to interaction with their surrounding environments. The environments are in general complex and intractable. At low temperatures, quantum environments are mapped onto two simpler universality classes of models, namely oscillator bath and spin bath models. The two models are commonly recognized as completely distinct at strong coupling limits. In particular, it is believed, that they cause two different dissipative relaxation rates when they act on qubits. It is also believed that at such a limit the relaxation rate caused by the spin bath model cannot be simulated by the oscillator bath model. In this paper, I show, in contrast, that the oscillator bath model can simulate the effect of the spin bath model in the strong coupling limit of the spin bath. I demonstrate that, by choosing the right parameters for the oscillator bath model such that it induces the same rate and amount of bias energy fluctuations as those of the spin bath, the oscillator bath model can produce a relaxation rate equivalent to that of the spin bath model. This result implies that, as far as the relaxation rate is concerned, the distinction between the oscillator bath model and the spin bath model is far less than has been previously recognized.

DOI
10.48550/arxiv.2108.04915

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Open quantum systems are subject to interaction with their surrounding environments. The environments are in general complex and intractable. At low temperatures, quantum environments are mapped onto two simpler universality classes of models, namely oscillator bath and spin bath models. The two models are commonly recognized as completely distinct at strong coupling limits. In particular, it is believed, that they cause two different dissipative relaxation rates when they act on qubits. It is also believed that at such a limit the relaxation rate caused by the spin bath model cannot be simulated by the oscillator bath model. In this paper, I show, in contrast, that the oscillator bath model can simulate the effect of the spin bath model in the strong coupling limit of the spin bath. I demonstrate that, by choosing the right parameters for the oscillator bath model such that it induces the same rate and amount of bias energy fluctuations as those of the spin bath, the oscillator bath model can produce a relaxation rate equivalent to that of the spin bath model. This result implies that, as far as the relaxation rate is concerned, the distinction between the oscillator bath model and the spin bath model is far less than has been previously recognized.
Open quantum systems are subject to interaction with their surrounding environment. In many applications, at low temperatures, quantum environments fall into two universality classes of models: Caldeira-Leggett oscillator bath models and Prokof'ev-Stamp spin bath models. The two classes are commonly recognized to be distinct and to have strikingly different effects on principal systems, except at weak coupling limits. However, I show here, in contrast, that oscillator bath models can simulate the effect of spin bath models in strong coupling limit of the spin bath. I choose parameters of the oscillator bath models such that they produce incoherent relaxation, just as in the spin bath models, with relaxation rates for a two-state system (a qubit) equivalent to those of the spin bath models in the strong coupling limit.
Open quantum systems often interact with complex, intractable quantum environments. Establishing a minimal solvable model that captures the essential effects of these environments on central quantum systems has long been of interest in physics, dating back to the origins of quantum mechanics. The Caldeira-Leggett oscillator bath model has become a prominent minimal model that effectively represents many quantum environments. However, it has been argued that this model cannot capture the main effects of spin environments on central two-level systems when the couplings between the spins and the central systems are strong. In contrast, I adopt an appropriate spectral density function for the Caldeira-Leggett model and demonstrate that it can effectively replicate the essential effects of spin environments in the regime of interest. This suggests that the class of the Caldeira-Leggett model is more universal than previously recognized.
Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.
The quantum dynamics of mesoscopic or macroscopic systems is always complicated by their coupling to many `environmental' modes. At low T these environmental effects are dominated by localized modes, such as nuclear and paramagnetic spins, and defects (which also dominate the entropy and specific heat). This environment, at low energies, maps onto a `spin bath' model. This contrasts with `oscillator bath' models (originated by Feynman and Vernon) which describe delocalized environmental modes such as electrons, phonons, photons, magnons, etc. The couplings to N spin bath modes are independent of N (rather than the ~O(1/( N ) 1/2 ) dependence typical of oscillator baths), and often strong. One cannot in general map a spin bath to an oscillator bath (or vice versa); they constitute distinct `universality classes' of quantum environment. We show how the mapping to spin bath models is made, and then discuss several examples in detail, including moving particles, magnetic solitons, nanomagnets, and SQUIDs, coupled to nuclear and paramagnetic spin environments. We then focus on the `central spin' model, which couples a central two-level system to a background spin bath. It is the spin bath analogue of the famous `spin-boson' oscillator model, and describes, e.g., the tunnelling dynamics of nanoscopic and mesoscopic magnets and superconductors. We show how to average over (or `integrate out') spin bath modes, using an operator instanton technique, to find the central spin dynamics. The formal manouevres involve four separate averages - each average corresponds physically to a different `decoherence' mechanism acting on the central spin dynamics. Each environmental spin has its own topological `spin phase', which by interacting with the phase of the central system, decoheres it - this can happen even without dissipation. We give analytic results for the central spin correlation functions, under various conditions. We then describe the application of this theory to magnetic and superconducting systems. Particular attention is given to recent work on tunnelling magnetic macromolecules, where the role of the nuclear spin bath in controlling the tunnelling is very clear; we also discuss other magnetic systems in the quantum regime, and the influence of nuclear and paramagnetic spins on flux dynamics in SQUIDs. Finally, we discuss decoherence mechanisms and coherence experiments in superconductors and magnets. We show that a spin bath environment causes decoherence even in the T 0 limit. Control of this decoherence will be essential in the effort to construct `qubits' for quantum computers.
We calculate the reduced density matrix for a system of coupled harmonic oscillators in a bosonic heat bath using the Born-Markov approximation and show that the expectation values of position and momentum evolve like classical quantities. We consider the cases where every oscillator is coupled to its own reservoir and where all oscillators are coupled to the same heat bath, pointing out the differences between these two models. In particular, for systems coupled to a common reservoir, we present an uncommon dissipation mechanism, which disappears, when the dynamics of the system are gouverned by the low frequency modes of the environment (Caldeira-Leggett limit). We also address the interesting phenomenon of reservoir-induced interactions.