We introduce a numerical method to determine the Hamiltonian of mean force (HMF) Gibbs state for a quantum system strongly coupled to a reservoir. The method adapts the time evolving matrix product operator (TEMPO) algorithm to imaginary-time propagation. By comparing the real-time and imaginary-time propagation for a generalized spin-boson model, we confirm that the HMF Gibbs state correctly predicts the steady state. We show that the numerical dynamics match the polaron master equation at strong coupling. We illustrate the potential of the imaginary-time TEMPO approach by exploring reservoir-induced entanglement between qubits.
We introduce a numerical method to determine the Hamiltonian of mean force (HMF) Gibbs state for a quantum system strongly coupled to a reservoir. The method adapts the time evolving matrix product operator (TEMPO) algorithm to imaginary-time propagation. By comparing the real-time and imaginary-time propagation for a generalized spin-boson model, we confirm that the HMF Gibbs state correctly predicts the steady state. We show that the numerical dynamics match the polaron master equation at strong coupling. We illustrate the potential of the imaginary-time TEMPO approach by exploring reservoir-induced entanglement between qubits.
When the coupling of a quantum system to its environment is nonnegligible, its steady state is known to deviate from the textbook Gibbs state. The Bloch-Redfield quantum master equation, one of the most widely adopted equations for solving the open quantum dynamics, cannot predict all the deviations of the steady state of a quantum system from the Gibbs state. In this paper, for a generic spin-boson model, we use a higher-order quantum master equation (in system-environment coupling strength) to analytically calculate all the deviations of the steady state of the quantum system up to second order in coupling strength. We also show that this steady state is exactly identical to the corresponding generalized Gibbs state, the so-called quantum mean force Gibbs state, at arbitrary temperature. All these calculations are highly general, making them immediately applicable to a wide class of systems well modeled by the spin-Boson model, spanning a diverse range of topics, from nanomaterials to various condensed-phase processes, and quantum computing (e.g., environment-induced corrections to the steady state of a superconducting qubit). As an example, we use our results to study the dynamics and the steady state of a solid-state double-quantum-dot system under physically relevant choices of parameters.
Two qubits strongly coupled to a common bosonic reservoir can become entangled with each other, despite having no direct interaction. In equilibrium, such coupling-induced coherences can be described by the mean-force Gibbs state. Here we derive approximate, analytic expressions for the two-qubit mean-force Gibbs state, and use these to characterize equilibrium qubit-qubit entanglement mediated by a thermal reservoir. Entanglement, which is highest at lowest temperatures, is a non-monotonic function of the system-reservoir coupling strength. Moreover, we find that broadening the reservoir spectral density beyond a single mode, as is realistic for typical baths, can enhance the qubit entanglement. Our results provide a comprehensive understanding of reservoir-mediated two-qubit entanglement in thermal equilibrium and provide a benchmark to compare with numerical methods, as well as demonstrating the utility of strong system-reservoir coupling as a resource.
The Gibbs state is widely taken to be the equilibrium state of a system in contact with an environment at temperature T. However, non-negligible interactions between system and environment can give rise to an altered state. Here, we derive general expressions for this mean force Gibbs state, valid for any system that interacts with a bosonic reservoir. First, we derive the state in the weak coupling limit and find that, in general, it maintains coherences with respect to the bare system Hamiltonian. Second, we develop a new expansion method suited to investigate the ultrastrong coupling regime. This allows us to derive the explicit form for the mean force Gibbs state, and we find that it becomes diagonal in the basis set by the system-reservoir interaction instead of the system Hamiltonian. Several examples are discussed including a single qubit, a three-level V-system, and two coupled qubits all interacting with bosonic reservoirs. The results shed light on the presence of coherences in the strong coupling regime, and provide key tools for nanoscale thermodynamics investigations.
The Gibbs state is widely taken to be the equilibrium state of a system in contact with an environment at temperature T. However, non-negligible interactions between system and environment can give rise to an altered state. Here we derive general expressions for this mean force Gibbs state, valid for any system that interacts with a bosonic reservoir. First, we derive the state in the weak coupling limit and find that, in general, it maintains coherences with respect to the bare system Hamiltonian. Second, we develop a new expansion method suited to investigate the ultrastrong coupling regime. This allows us to derive the explicit form for the mean force Gibbs state, and we find that it becomes diagonal in the basis set by the system-reservoir interaction instead of the system Hamiltonian. Several examples are discussed including a single qubit, a three-level V-system and two coupled qubits all interacting with bosonic reservoirs. The results shed light on the presence of coherences in the strong coupling regime, and provide key tools for nanoscale thermodynamics investigations.
The Hamiltonian of mean force (HMF) provides the standard starting point for strong-coupling thermodynamics, yet explicit operator forms are known only in restricted settings. We present a quenched density framework that uses the Hubbard-Stratonovich transformation to rewrite the reduced equilibrium state as an average over local propagators in imaginary time. This approach rigorously separates the statistical definition of the environment from the algebraic structure of the system response. We apply this framework to the minimal case of a harmonic environment with a coupling commuting with the system Hamiltonian. In this scenario the correction to the HMF has an exact, closed-form expression. We validate this result against finite-bath trace-out calculations and stochastic imaginary-time sampling in a five-level projector-coupled model.
The time-evolving matrix product operator (TEMPO) method has proven to be a powerful method to study the long-time dynamics of bosonic impurity problems where a small system is linearly coupled to a noninteracting bosonic bath. However, current developments of TEMPO have mostly focused on the case of diagonal system-bath coupling, i.e., $\sum_k \Aop(V_k \bdop_k + \hc)$, with $\Aop$ a Hermitian operator of the system. Based on the process tensor framework, we extend TEMPO to the more general case of off-diagonal system-bath coupling in the form $\sum_k (V_k\Aop\bdop_k + \hc)$, where $\Aop$ could be non-Hermitian. As applications, we study the real-time dynamics of a spin that is coupled to a sub-ohmic bath via the Jaynes-Cummings-type system-bath coupling and compare it against the standard spin-boson model, where we show that the commonly used rotating-wave approximation could be very poor for this bath. We also study the imaginary-time evolution of a bosonic impurity with nonzero on-site interaction that is coupled to a sub-ohmic bath, to illustrate the flexibility of our method. Our method provides a unified framework to understand different variants of TEMPO, and is a promising building block for an impurity solver in the bosonic dynamical mean field theory for the normal phase with a scalar hybridization function.