When a quantum system is coupled to several heat baths at different temperatures, it eventually reaches a non-equilibrium steady state (NESS) featuring stationary internal heat currents.From one side, these currents are responsible to cause decorehence and produce entropy in the system.However, their existence also induce correlations between different parts of the system.In this work, we explore this two-folded aspect of NESSs.Using phase-space techniques we calculate the Wigner entropy production on general linear networks of harmonic nodes.Working in the ubiquitous limit of weak internal coupling and weak dissipation, we obtain simple closedform expressions for the entropic contribution of each individual quasi-probability current.Our analysis also shows that, it is exclusively the (reversible) internal dynamics which maintain the stationary (irreversible) entropy production.From the informational point of view, we address how to quantify the amount of information that disconnected parts of a quantum chain share in a non-equilibrium steady-state.As we show, this is more precisely captured by the conditional mutual information (CMI), a more general quantifier of tripartite correlations than the usual mutual information.As an application, we apply our framework to the paradigmatic problem of energy transfer through a chain of oscillators subject to self-consistent internal baths that can be used to tune the transport from ballistic to diffusive.We find that the entropy production scales with different power law behaviors in the ballistic and diffusive regimes, hence allowing us to quantify what is the "entropic cost of diffusivity".We also compute the CMI for arbitrary sizes and thus find the scaling rules connecting information sharing and diffusivity.Finally, we discuss how this new perspective in the characterization of non-equilibrium systems may be applied to understand the issue of local equilibration in non-equilibrium states.
When a quantum system is coupled to several heat baths at different temperatures, it eventually reaches a non-equilibrium steady state (NESS) featuring stationary internal heat currents.From one side, these currents are responsible to cause decorehence and produce entropy in the system.However, their existence also induce correlations between different parts of the system.In this work, we explore this two-folded aspect of NESSs.Using phase-space techniques we calculate the Wigner entropy production on general linear networks of harmonic nodes.Working in the ubiquitous limit of weak internal coupling and weak dissipation, we obtain simple closedform expressions for the entropic contribution of each individual quasi-probability current.Our analysis also shows that, it is exclusively the (reversible) internal dynamics which maintain the stationary (irreversible) entropy production.From the informational point of view, we address how to quantify the amount of information that disconnected parts of a quantum chain share in a non-equilibrium steady-state.As we show, this is more precisely captured by the conditional mutual information (CMI), a more general quantifier of tripartite correlations than the usual mutual information.As an application, we apply our framework to the paradigmatic problem of energy transfer through a chain of oscillators subject to self-consistent internal baths that can be used to tune the transport from ballistic to diffusive.We find that the entropy production scales with different power law behaviors in the ballistic and diffusive regimes, hence allowing us to quantify what is the "entropic cost of diffusivity".We also compute the CMI for arbitrary sizes and thus find the scaling rules connecting information sharing and diffusivity.Finally, we discuss how this new perspective in the characterization of non-equilibrium systems may be applied to understand the issue of local equilibration in non-equilibrium states.
Non-equilibrium steady-states are characterized by the existence of macroscopic heat currents flowing between two or more reservoirs. However, the existence of these currents also implies that information is constantly being transmitted from one reservoir to the other. In this letter we address the transport of information in the context of quantum non-equilibrium steady states. We first show how the ability of the excitations to transfer information can be quantified by the conditional mutual information (CMI), a more general measure of tripartite correlations. We then apply this to an exactly soluble model allowing for both ballistic and diffusive behavior. Our approach allows us to compute the CMI for arbitrary sizes and thus find the scaling rules connecting information transport and diffusivity. Finally, we discuss how this new perspective in the characterization of non-equilibrium systems may be applied to understand the issue of local equilibration in non-equilibrium states.
Abstract The conditional mutual information (CMI) I ( A : C | B ) quantifies the amount of correlations shared between A and C given B . It therefore functions as a more general quantifier of bipartite correlations in multipartite scenarios, playing an important role in the theory of quantum Markov chains. In this paper we carry out a detailed study on the behavior of the CMI in non-equilibrium steady-states (NESS) of a quantum chain placed between two baths at different temperatures. These results are used to shed light on the mechanisms behind ballistic and diffusive transport regimes and how they affect correlations between different parts of a chain. We carry our study for the specific case of a 1D bosonic chain subject to local Lindblad dissipators at the boundaries. In addition, the chain is also subject to self-consistent reservoirs at each site, which are used to tune the transport between ballistic and diffusive. As a result, we find that the CMI is independent of the chain size L in the ballistic regime, but decays algebraically with L in the diffusive case. Finally, we also show how this scaling can be used to discuss the notion of local thermalization in non-equilibrium steady-states.
The conditional mutual information (CMI) I(A:C∣B) quantifies the amount of correlations shared between A and C \emph{given} B. It therefore functions as a more general quantifier of bipartite correlations in multipartite scenarios, playing an important role in the theory of quantum Markov chains. In this paper we carry out a detailed study on the behavior of the CMI in non-equilibrium steady-states (NESS) of a quantum chain placed between two baths at different temperatures. These results are used to shed light on the mechanisms behind ballistic and diffusive transport regimes and how they affect correlations between different parts of a chain. We carry our study for the specific case of a 1D bosonic chain subject to local Lindblad dissipators at the boundaries. In addition, the chain is also subject to self-consistent reservoirs at each site, which are used to tune the transport between ballistic and diffusive. As a result, we find that the CMI is independent of the chain size L in the ballistic regime, but decays algebraically with L in the diffusive case. Finally, we also show how this scaling can be used to discuss the notion of local thermalization in non-equilibrium steady-states.
We investigate the link between the irreversibility generated by a stationary dissipative process and the correlations established within a composite quantum system. We provide two equivalent expressions for the entropy generated in the non-equilibrium steady state of coupled quantum harmonic oscillators that allow for a simple interpretation of the onset of irreversibility. We then unveil a quantitative relation between the entropy production rate and correlations, both total and quantum, built between the two oscillators. In the small-coupling limit, the entropy production rate is shown to be proportional to both the mutual information and the quantum correlations. We apply our results to the analysis of the optomechanical interaction between a nano-mechanical resonator and a cavity field, and show that the behavior of the entropy production captures the peculiar features of the model, such as sideband cooling and amplification.
The physics of systems out of equilibrium is a topic of great interest, mainly due to the possibility of exploring phenomena that can not be observed in equilibrium systems. Driven-dissipative phase transitions open the opportunity of studying phases with no classical counterparts, and these can be experimentally realized in quantum optical platforms. Since these transitions occur in systems kept out of equilibrium, they are characterized by a finite entropy production rate. However, due to technical difficulties regarding the zero temperature limit and the non-gaussianity of such models, very little is known about how entropy production behaves around criticality. Using a quantum phase-space method, based on the Husimi Q-function, we put forth a framework that allows for the complete characterization of the entropy production in driven-dissipative transitions. This new theoretical framework is tailored specifically to describe photon loss dissipation, which is effectively a zero temperature process for which the standard theory of entropy production breaks down. It makes no assumptions about Gaussianity about the model or the state. It works for both, steady-states as well as the dynamics and as an application, we study both situations in the paradigmatic driven-dissipative Kerr model, which presents a discontinuous phase transition. For general driven-dissipative critical systems, where one can define a thermodynamic limit, we find that the entropy production rate and flux naturally split into two contributions: an extensive one and a contribution due to quantum fluctuations only. Moreover, we identify a contribution to the entropy production due to unitary dynamics, and we find that the behavior of this contribution at the non-equilibrium steady-state (NESS) matches the behavior of entropy production rate observed in classical systems. The quantum contributions are found to diverge at the critical point.
Thermodynamics is a phenomenological theory describing the energy conversion of work and heat. At its origins thermodynamics was developed in order to understand and improve heat engines. In conventional thermodynamics, however, only such processes are completely describable which are slow enough to keep the system of interest in an equilibrium state with its thermal environment at all times. On the contrary, all real physical processes are accompanied by non-equilibrium phenomena. These are mathematically described with the help of the irreversible entropy production, i.e. the dissipated work divided by the temperature of the bath. The second law of thermodynamics merely states that the irreversible entropy production is on average always positive. The modern trend of miniaturization, however, leads to smaller and smaller devices. On short length scales thermal noise as well as quantum fluctuations become important. Thus, usual thermodynamic quantities as work and heat acquire stochastic nature. Moreover, in the quantum regime a completely new theory had to be invented, since classical notions of work and heat are no longer valid. The present dissertation contributes to this prevailing field by the derivation of analytical expressions for the entropy production in open and closed quantum system far from thermal equilibrium. The theoretical treatment is motivated by an experimental point of view. The obtained results were derived from the reduced dynamics of the quantum mechanical system, which may be coupled to a thermal bath. To this end, it was dealt with methods and approaches of statistical physics, conventional thermodynamics, quantum information theory and the theory of open quantum systems. In this dissertation the most important physical systems are treated. First, an isolated system is considered before later on a heat bath is included. The limit of weak as well as of strong coupling are analyzed in detail. For isolated systems it turns out that a geometric approach is able to capture the thermodynamic properties, completely. To this end, the irreversible entropy production is sharply estimated from below by the angle between the real non-equilibrium state and the corresponding equilibrium one. Further, it is shown that the Heisenberg uncertainty relation for energy and time can be formulated more precisely in terms of that angle. As experimental system cold ion traps are introduced, in which the theoretical predictions can be verified. For weakly coupled systems, then, the irreversible entropy production is identified and a fluctuation theorem is derived by merely thermodynamic arguments. As experimental system we, again, propose the cold ion traps, where the heat bath can be simulated with the help of LASER light. In the limit of strong coupling a semiclassical description of the reduced quantum system is discussed. Again, an analytically exact expression for the entropy production and the corresponding fluctuation theorem is derived. A physical system, in which strong coupling to the environment can be analyzed, is given by Josephson junctions. This dissertation discusses the entropy production for isolated, weakly coupled, and overdamped quantum systems. For these physical regimes analytical relation are derived and experimental systems proposed, with which the predictions can be verified.