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.
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.
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.
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 reduced operator approximation: a simple, physically transparent and computationally efficient method of modelling open quantum systems. It employs the Heisenberg picture of the quantum dynamics, which allows us to focus on the system degrees of freedom in a natural and easy way. We describe different variants of the method, low- and high-order in the system–bath interaction operators, defining them for either general quantum harmonic oscillator baths or specialising them for independent baths with Lorentzian spectral densities. Its wide applicability is demonstrated on the examples of systems coupled to different baths (with varying system–bath interaction strength and bath memory length), and compared with the exact pseudomode and the popular quantum state diffusion approach. The method captures the decoherence of the system interacting with the bath, while conserving the total energy. Our results suggest that quantum coherence effects persist in open quantum systems for much longer times than previously thought.
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.
It is known that the origin of the deviations from standard thermodynamics proceed from the strong coupling to the bath. Here, it is shown that these deviations are related to the power spectrum of the bath. Specifically, it is shown that the system thermal-equilibrium-state cannot be characterized by the canonical Boltzmann's distribution in quantum mechanics. This is because the uncertainty principle imposed a lower bound of the dispersion of the total energy of the system that is dominated by the spectral density of the bath. However, in the classical case, for a wide class of systems that interact via central forces with pairwise-self-interacting environment, the system thermal equilibrium state is exactly characterized by the canonical Boltzmann distribution. As a consequence of this analysis and taking into account all energy scales in the system and reservoir interaction, an effective coupling to the environment is introduced. Sample computations in different regimes predicted by this effective coupling are shown. Specifically, in the strong coupling effective regime, the system exhibits deviations from standard thermodynamics and in the weak coupling effective regime, quantum features such as stationary entanglement are possible at high temperatures. Moreover, it is known that the spectrum of thermal baths depends on the non-Markovian character. Hence, non-Markovian interactions have a important role in the thermal equilibrium state of physical systems. For example, in quantum optomechanics is looked up to cool the mechanical system through an auxiliary system which generally is a cavity. This cooling process takes into account the non-Markovian interaction and as it is shown here, it is more effective than if we use only the Markovian approximation in the equation of motion for the different modes.
In this thesis we develop a systematic formalism that allows the calculation of transport properties for small quantum systems coupled to multiple baths with different temperatures and/or chemical potentials. Our approach is based on a generalization of the projector operator technique, previously used to study the evolution towards equilibrium of systems weakly coupled to a single bath. Applying this technique to a system weakly coupled to multiple baths, we find a dynamical equation for the reduced density matrix of the system (the reduced density matrix is obtained by tracing out from the total density matrix the baths’ degrees of freedom). Integration of this dynamical equation gives the time evolution of the biased system. After a time interval long enough compared to the characteristic time-scales of the problem, the system arrives to a non-equilibrium steady-state. The density matrix in this non-equilibrium steady state can also be calculated directly, avoiding the time-integration of the dynamical equation. In the second part of the thesis we study non-equilibrium steady states and heat transport through short spin chains coupled to two thermal baths with different temperature. Two new definitions for the local temperature in nonequilibrium are proposed, based on local spin and energy at different sites. The definition based on the local energy seems to be more reliable. We find that for small biases, the heat and spin currents increase proportionally to the bias, however non-linear behavior is observed for large biases. The temperature profile of the chain depends strongly on the parameters of the system.