We introduce a response theory for open quantum systems within nonequilibrium steady states subject to a Hamiltonian perturbation.Working in the weak system-bath coupling regime, our results are derived within the Lindblad-Gorini-Kossakowski-Sudarshan formalism.We find that the response of the system to a small perturbation is not simply related to a correlation function within the system, unlike traditional linear response theory in closed systems or expectations from the fluctuation-dissipation theorem.In limiting cases, when the perturbation is small relative to the coupling to the surroundings or when it does not lead to a change of the eigenstructure of the system, a perturbative expansion exists where the response function is related to a sum of a system correlation functions and additional forces induced by the surroundings.Away from these limiting regimes, however, the secular approximation results in a singular response that cannot be captured within the traditional approach but can be described by reverting to a microscopic Hamiltonian description.These findings are illustrated by explicit calculations in coupled qubits and anharmonic oscillators in contact with bosonic baths at different temperatures.
We introduce a response theory for open quantum systems within nonequilibrium steady states subject to a Hamiltonian perturbation.Working in the weak system-bath coupling regime, our results are derived within the Lindblad-Gorini-Kossakowski-Sudarshan formalism.We find that the response of the system to a small perturbation is not simply related to a correlation function within the system, unlike traditional linear response theory in closed systems or expectations from the fluctuation-dissipation theorem.In limiting cases, when the perturbation is small relative to the coupling to the surroundings or when it does not lead to a change of the eigenstructure of the system, a perturbative expansion exists where the response function is related to a sum of a system correlation functions and additional forces induced by the surroundings.Away from these limiting regimes, however, the secular approximation results in a singular response that cannot be captured within the traditional approach but can be described by reverting to a microscopic Hamiltonian description.These findings are illustrated by explicit calculations in coupled qubits and anharmonic oscillators in contact with bosonic baths at different temperatures.
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.
Quantum response theory is a cornerstone of statistical physics.However, the standard Kubo formalism is restricted to isolated equilibrium systems.We here generalize Kubo's results to open quantum systems in nonequilibrium steady states.We derive three different, but equivalent, forms of the quantum response function.We discuss for each of them the role of the noncommutativity of quantum operators and introduce a steady-state extension of the Kubo transformation.We show in particular that the equilibrium response vanishes for perturbations that commute with the unperturbed Hamiltonian, while the steady-state response does not, highlighting the profound difference between the two linear response theories.
We introduce a functional perturbative method for treating weakly nonlinear systems coupled with a quantum field bath. We demonstrate using this method to obtain the covariance matrix elements and the correlation functions of a quantum anharmonic oscillator interacting with a heat bath. We identify a fluctuation-dissipation relation based on the nonequilibrium dynamics of this nonlinear open quantum system. To establish its connection with dynamical equilibration, we further examine the energy flows between the anharmonic oscillator and the bath field. The vanishing of the net flow is an indication of the existence of an equilibrium state for such an open-system configuration. The results presented here are useful for studying the nonequilibrium physical processes of nonlinear quantum systems such as heat transfer or electron transport.
The absence of a simple fluctuation-dissipation theorem is a major obstacle for studying systems that are not in thermodynamic equilibrium. We show that for a fluid in a nonequilibrium steady state characterized by a constant temperature gradient the commutator correlation functions are still related to response functions; however, the relation is to the bilinear response of products of two observables, rather than to a single linear response function as is the case in equilibrium. This modified fluctuation-response relation holds for both quantum and classical systems. It is both motivated and informed by the long-range correlations that exist in such a steady state and allows for probing them via response experiments. This is of particular interest in quantum fluids, where the direct observation of fluctuations by light scattering would be difficult. In classical fluids it is known that the coupling of the temperature gradient to the diffusive shear velocity leads to correlations of various observables, in particular temperature fluctuations, that do not decay as a function of distance, but rather extend over the entire system. We investigate the nature of these correlations in a fermionic quantum fluid and show that the crucial coupling between the temperature gradient and velocity fluctuations is the same as in the classical case. Accordingly, the nature of the long-ranged correlations in the hydrodynamic regime also is the same. However, as one enters the collisionless regime in the low-temperature limit the nature of the velocity fluctuations changes: they become ballistic rather than diffusive. As a result, correlations of the temperature and other observables are still singular in the long-wavelength limit, but the singularity is weaker than in the hydrodynamic regime.
Continuing our inquiry into the conditions when fluctuation-dissipation relations (FDR) may appear in the context of nonequilibrium dynamics of open quantum systems (over and beyond the conventional FDR from linear response theory) we turn to non-Gaussian systems and consider this issue for an anharmonic quantum oscillator interacting with a scalar quantum field bath. We present the general nonperturbative expressions for the rate of energy (power) exchange between the anharmonic oscillator and its thermal bath. For the cases that a stable final equilibrium state exists, and the nonstationary components of the two-point functions of the anharmonic oscillator have negligible contributions to the power balance, we can show nonperturbatively that equilibration implies an FDR for the anharmonic oscillator. We then use a weakly anharmonic oscillator as an example to illustrate the validity of those two assumptions and show that in the weak anhamonicity limit, they are satisfied according to our first-order perturbative results..
The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet been possible to overcome the perturbative approach, and the most developed theory is the linear one. Extensions of the exact response theory developed in the field of nonequilibrium molecular dynamics could prove useful in quantum mechanics, as perturbations of small systems or far-from-equilibrium states cannot always be taken as small perturbations. Here, we introduce a quantum analogue of the classical Dissipation Function. We then derive a quantum expression for the exact calculation of time-dependent expectation values of observables, in a form analogous to that of the classical theory. We restrict our analysis to finite-dimensional Hilbert spaces, for the sake of simplicity, and we apply our method to specific examples, like qubit systems, for which exact results can be obtained by standard techniques. This way, we prove the consistency of our approach with the existing methods, where they apply. Although not required for open systems, we propose a self-adjoint version of our Dissipation Operator, obtaining a second equivalent expression of response, where the contribution of an anti-self-adjoint operator appears. We conclude by using new formalism to solve the Lindblad equations, obtaining exact results for a specific case of qubit decoherence, and suggesting possible future developments of this work.