We develop a population and flux landscape theory for general non-equilibrium quantum systems. We illustrate our theory by modelling the quantum transport of donor-acceptor energy transfer. We find two driving forces for the non-equilibrium quantum dynamics. The symmetric part of the driving force corresponds to the population landscape contribution which mainly governs the equilibrium part of dynamics while the anti-symmetric part of the driving force generates the non-equilibrium curl quantum flux which leads to the detailed-balance-breaking and time-irreversibility. The multi-loop structure of the flux emerges forms the flux-landscape. We study the trend of changes in population and flux-landscape with respect to the voltage (temperature difference induced by environments) and electronic coupling. Improving the voltage and electronic coupling in general facilitates the quantum transport by reducing the population landscape barriers between major states and increasing the mean value of the flux. A limit-cycle mode emerges when the underlying flux-landscape becomes funnelled with a significant gap between the largest flux loop and the rest of them. On the kinetic level, we find that multiple kinetic paths between quantum states emerge and illustrate the interference effects. The degree of interference is determined by the landscape and flux. Furthermore, we quantify kinetic rate which strongly correlates with the population landscape and flux. For quantum transport, we demonstrate that as the coherence or the quantum entanglement is enhanced, the flux and energy transfer efficiency are increased. Finally it is surprising that the non-equilibriumness quantified by voltage has a non-trivial contribution on strengthening the entanglement, which is attributed to the non-local feature of the quantum curl flux.
We develop a population and flux landscape theory for general non-equilibrium quantum systems. We illustrate our theory by modelling the quantum transport of donor-acceptor energy transfer. We find two driving forces for the non-equilibrium quantum dynamics. The symmetric part of the driving force corresponds to the population landscape contribution which mainly governs the equilibrium part of dynamics while the anti-symmetric part of the driving force generates the non-equilibrium curl quantum flux which leads to the detailed-balance-breaking and time-irreversibility. The multi-loop structure of the flux emerges forms the flux-landscape. We study the trend of changes in population and flux-landscape with respect to the voltage (temperature difference induced by environments) and electronic coupling. Improving the voltage and electronic coupling in general facilitates the quantum transport by reducing the population landscape barriers between major states and increasing the mean value of the flux. A limit-cycle mode emerges when the underlying flux-landscape becomes funnelled with a significant gap between the largest flux loop and the rest of them. On the kinetic level, we find that multiple kinetic paths between quantum states emerge and illustrate the interference effects. The degree of interference is determined by the landscape and flux. Furthermore, we quantify kinetic rate which strongly correlates with the population landscape and flux. For quantum transport, we demonstrate that as the coherence or the quantum entanglement is enhanced, the flux and energy transfer efficiency are increased. Finally it is surprising that the non-equilibriumness quantified by voltage has a non-trivial contribution on strengthening the entanglement, which is attributed to the non-local feature of the quantum curl flux.
We present a fluctuation-dissipation theorem (FDT) for nonequilibrium quantum systems with detailed-balance breaking, which deviates from the conventional form of the FDT for equilibrium systems preserving detailed balance. Using the phase space formulation of quantum mechanics and the potential-flux landscape framework, we find that the response function of nonequilibrium quantum systems to external perturbations contains a nontrivial contribution from the quantum curl flux quantifying detailed-balance breaking in the steady state, in addition to the correlation function of observables in the steady state representing the contribution of spontaneous fluctuations which is also present in the equilibrium FDT. We illustrate our general formalism with a harmonic oscillator coupled to two heat baths, and show that the nonequilibrium FDT reduces to the conventional expression at the equilibrium condition when the flux contribution vanishes.
Projected ensembles---the collections of conditional pure states induced on a system by measuring an entangled environment---have become central objects in quantum science, underlying deep thermalization, quantum state designs, the emergence of classicality, and exhibiting interesting phase transitions. Here we develop a general and exact theory of their dynamics, yielding a systematic approach to their dynamics and equilibrium and non-equilibrium stationary states. We derive an exact continuity equation on quantum state space for the projected ensemble, together with microscopic kinetic equations and analytical expressions for the probability flux and source terms generated by the system-environment interaction. The resulting dynamics admits a classical representation: isolated systems obey Hamiltonian transport and Liouville's theorem, while open systems are described by a kinetic theory of probability transport. The stationary continuity equation provides a general characterization of equilibrium and non-equilibrium stationary projected ensembles, which can be analyzed through the method of characteristics. The theory provides an analytical framework for studying the emergence, structure, and timescales of equilibrium and non-equilibrium stationary projected ensembles, with applications ranging from deep thermalization and the emergence of classicality to random quantum-state generation and quantum device benchmarking.
Abstract We review one of the most versatile theoretical approaches to the study of time-dependent correlated quantum transport in nano-systems: the non-equilibrium Green’s function (NEGF) formalism. Within this formalism, one can treat, on the same footing, inter-particle interactions, external drives and/or perturbations, and coupling to baths with a (piece-wise) continuum set of degrees of freedom. After a historical overview on the theory of transport in quantum systems, we present a modern introduction of the NEGF approach to quantum transport. We discuss the inclusion of inter-particle interactions using diagrammatic techniques, and the use of the so-called embedding and inbedding techniques which take the bath couplings into account non-perturbatively. In various limits, such as the non-interacting limit and the steady-state limit, we then show how the NEGF formalism elegantly reduces to well-known formulae in quantum transport as special cases. We then discuss non-equilibrium transport in general, for both particle and energy currents. Under the presence of a time-dependent drive—encompassing pump–probe scenarios as well as driven quantum systems—we discuss the transient as well as asymptotic behavior, and also how to use NEGF to infer information on the out-of-equilibrium system. As illustrative examples, we consider model systems general enough to pave the way to realistic systems. These examples encompass one- and two-dimensional electronic systems, systems with electron–phonon couplings, topological superconductors, and optically responsive molecular junctions where electron–photon couplings are relevant.
This review presents a thermodynamic perspective on quantum coupled transport processes in nanoscale systems. Our analysis is formulated within the framework of entropy production rate, the central quantity governing non-equilibrium processes and expressed through conjugate force-flux pairs. Although thermodynamic laws are universal across classical and quantum domains, the discussion is developed within a microscopic open quantum system framework, focusing on quantum dots (QDs) coupled to electronic reservoirs. We first examine elementary single transport processes and highlight their strong thermodynamic constraints in the near-equilibrium regime. This motivates the study of coupled transport, where multiple force-flux pairs coexist and interact, leading to richer thermodynamic behaviour. Using entropy production as the guiding principle, we analyse coupled energy and particle transport in a minimal two-terminal single-QD setup and show how conventional thermoelectric phenomena, including Seebeck and Peltier effects as well as thermoelectric heat engines and refrigerators, naturally emerge as thermodynamic cross-effects. We then extend the framework to a three-terminal coupled quantum dot (CQD) geometry, which provides a versatile platform for studying coupled transport and reduces, under suitable constraints, to the well-known Sánchez-Büttiker configuration. Beyond standard cross-effects, we discuss the phenomenon of inverse currents in coupled transport (ICC), where a current flows against mutually parallel thermodynamic forces without violating the second law. We show that ICC requires breaking the symmetry between energy and particle transport and identify the conditions for its realization in coupled quantum-dot systems with attractive interdot interactions.
Abstract The entropy produced when a quantum system at thermal equilibrium is driven to a non-equilibrium state can be considered as having components from coherence generation and from a population mismatch. The entropy component which is related to coherence represents a truly quantum effect and can be calculated using two different decompositions, based on either the relative entropy of coherence or on the thermal state corresponding to the Hamiltonian dephased in the eigen basis of the actual state. However, the decomposition based on the relative entropy was unable to quantify highly coherent processes or processes occurring in the low temperature regime. We experimentally explored a general driving scheme (a non-equilibrium process) on a nuclear magnetic resonance quantum processor, and were able to isolate the contribution of coherence to irreversible entropy generation. Furthermore, we numerically explored an infinitesimal driving process to distinguish between high and low temperature limits. The thermal equilibrium temperature and the final energy gap were varied to establish the relationship among temperature, energy level spacing, entropy production and coherence generation. Our results indicate that the contribution to entropy from coherence is higher in the low temperature regime. We also verified a generalized Clausius inequality, which affirms that the irreversible entropy production is lower-bounded.
A toolbox of theoretical methods pertinent to the study of non-equilibrium many-body quantum mechanics is presented with an eye to specific applications in cold atoms systems and solids. We discuss the generalization from unitary quantum mechanics to the non-unitary framework of open quantum systems. Theoretical techniques include the Keldysh close-time-path integral and its associated correlation functions, the quantum kinetic equation, and numerical integration of equations of motion both unitary and non-unitary. We explore how the relaxation of the assumption of equilibrium yields a whole new array of sometimes counterintuitive effects. We treat such examples as the non-equilibrium enhancement of BCS superfluidity by driving, bistability and coherent population transfer in Feshbach coupled fermions, and the dynamic stimulation of quantum coherence in bosons confined to a lattice. These systems are considered with an eye to enhancing some useful quantum properties and making them available in wider parameter regimes.