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arXiv:2307.14330 · Physical review. B./Physical review. B

Effective Hamiltonian theory: An approximation to the equilibrium state of open quantum systems

Nicholas Anto-Sztrikacs, Brett Min, Marlon Brenes, Dvira Segal

Published Sep 29, 2023• 14 citations

Abstract

We extend and benchmark the recently developed Effective-Hamiltonian (EFFH) method [PRX Quantum 4, 020307 (2023)] as an approximation to the equilibrium state (``mean-force Gibbs state'') of a quantum system at strong coupling to a thermal bath. The EFFH method is an approximate framework. Through a combination of the reaction-coordinate mapping, a polaron transformation, and a controlled truncation, it imprints the system-bath coupling parameters into the system's Hamiltonian. First, we develop a variational EFFH technique. In this method, the system's parameters are renormalized by both the system-bath coupling parameters (as in the original EFFH approach) and bath temperature. Second, adopting the generalized spin-boson model, we benchmark the equilibrium state from the EFFH treatment against numerically exact simulations and demonstrate a good agreement for both polarization and coherences using the Brownian spectral function. Third, we contrast the (normal and variational) EFFH approach with the familiar (normal and variational) polaron treatment. We show that the two methods predict a similar structure for the equilibrium state, albeit the EFFH approach offers the advantage of simpler calculations and closed-form analytical results. Altogether, we argue that for temperatures comparable to the system's frequencies, the EFFH methodology provides a good approximation for the mean-force Gibbs state in the full range of system-bath coupling, from ultraweak to ultrastrong.

DOI: 10.1103/physrevb.108.115437

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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.

We present the reaction-coordinate polaron-transform (RCPT) framework for generating effective Hamiltonian models to treat nonequilibrium open quantum systems at strong coupling with their surroundings. Our approach, which is based on two exact transformations of the Hamiltonian followed by its controlled truncation, ends with a new Hamiltonian with a weakened coupling to the environment. This new effective Hamiltonian mirrors the initial one, except that its parameters are dressed by the system-bath couplings. The power and elegance of the RCPT approach lie in its generality and in its mathematical simplicity, allowing for analytic work and thus profound understanding of the impact of strong system-bath coupling effects on open quantum system phenomena. Examples interrogated in this work include canonical models for quantum thermalization, charge and energy transport at the nanoscale, performance bounds of quantum thermodynamical machines such as absorption refrigerators and thermoelectric generators, as well as the equilibrium and nonequilibrium behavior of many-body dissipative spin chains.

Calculation of the equilibrium state of an open quantum system interacting with a bath remains a challenge to this day, mostly due to a huge number of bath degrees of freedom. Here, we present an analytical expression for the reduced density operator in terms of an effective Hamiltonian for a high temperature case. Comparing with numerically exact results, we show that our theory is accurate for slow baths and up to intermediate system-bath coupling strengths. Our results demonstrate that the equilibrium state does not depend on the shape of spectral density in the slow bath regime. The key quantity in our theory is the effective coupling between the states, which depends exponentially on the ratio of the reorganization energy to temperature and, thus, has opposite temperature dependence than could be expected from the small polaron transformation.

In the study of open quantum systems, the polaron transformation has recently attracted a renewed interest as it offers the possibility to explore the strong system-bath coupling regime. Despite this interest, a clear and unambiguous analysis of the regimes of validity of the polaron transformation is still lacking. Here we provide such a benchmark, comparing second order perturbation theory results in the original untransformed frame, the polaron frame, and the variational extension with numerically exact path integral calculations of the equilibrium reduced density matrix. Equilibrium quantities allow a direct comparison of the three methods without invoking any further approximations as is usually required in deriving master equations. It is found that the second order results in the original frame are accurate for weak system-bath coupling; the results deteriorate when the bath cut-off frequency decreases. The full polaron results are accurate for the entire range of coupling for a fast bath but only in the strong coupling regime for a slow bath. The variational method is capable of interpolating between these two methods and is valid over a much broader range of parameters.

Pseudomode approaches allow for an exact and unapproximated description of a quantum system interacting arbitrarily strongly with a bath. In general, a system coupled to pseudomodes will not thermalize to the system's Gibbs state: This is to be expected when the system-bath coupling is non-perturbative, but conflicts with common thermodynamic intuition when the system-bath coupling is asymptotically weak. We explore under which circumstances pseudomode models satisfy detailed balance (and subsequently thermalize to the system Gibbs state) and how specific choices of parameters can force "weak" detailed balance that is restricted to a limited frequency range. A combination of Hermitian and non-Hermitian pseudomodes that yields a flat effective-temperature profile is also considered. The results and criteria established here are relevant for the construction of pseudomode models in contexts where thermodynamic consistency is required.