We formulate a full-counting statistics description to study energy exchange in multi-terminal junctions from weak to strong system-bath coupling regime. Our approach can describe energy transport in quantum systems that are coupled in a non-additive manner to multiple reservoirs. We derive a Redfield-type equation for the counting-field dependent reduced density operator, valid under the Markovian approximation for the baths. Under the secular approximation, we confirm that the cumulant generating function satisfies the heat exchange fluctuation theorem. Our treatment thus respects the second law of thermodynamics---though for non-additive interaction models the entropy production rate within an individual thermal bath cannot be identified. We exemplify our method on a multi-terminal two-level quantum system. We write down explicit expressions for the energy current and its noise in the additive (weak coupling) and non-additive (strong coupling) cases and verify the so-called thermodynamic uncertainty relation. Lastly, we use our formalism to describe the smallest quantum absorption refrigerator, operating through engineered reservoirs, acting cooperatively.
Recently, a "unified" quantum master equation was derived and shown to be of the Gorini-Kossakowski-Lindblad-Sudarshan form. This equation describes the dynamics of open quantum systems in a manner that forgoes the full secular approximation and retains the impact of coherences between eigenstates close in energy. We implement full counting statistics with the unified quantum master equation to investigate the statistics of energy currents through open quantum systems with nearly degenerate levels. We show that, in general, this equation gives rise to dynamics that satisfy fluctuation symmetry, a sufficient condition for the Second Law of Thermodynamics at the level of average fluxes. For systems with nearly degenerate energy levels, such that coherences build up, the unified equation is simultaneously thermodynamically consistent and more accurate than the fully secular master equation. We exemplify our results for a "V" system facilitating energy transport between two thermal baths at different temperatures. We compare the statistics of steady-state heat currents through this system as predicted by the unified equation to those given by the Redfield equation, which is less approximate but, in general, not thermodynamically consistent. We also compare results to the secular equation, where coherences are entirely abandoned. We find that maintaining coherences between nearly degenerate levels is essential to properly capture the current and its cumulants. On the other hand, the relative fluctuations of the heat current, which embody the thermodynamic uncertainty relation, display inconsequential dependence on quantum coherences.
This tutorial presents an illustrative overview behind the idea of the two time mea- surement protocol scheme in connection with the method of Full Counting Statistics, a widely used tool in the context of Quantum Thermodynamics and quantum noises. This article presents the mathematical tools to derive the quantum master equation for the de- formed reduced density operator for the system in the most general way by tracing out the bath degrees of freedom for different types of interaction Hamiltonians generally used to model the open quantum system problems in the limit of weak system bath coupling. The derivation of the microscopic Q.M.E has also been described. Finally, the article provides an application of the Full counting statistics method for the well known Resonant Level Model of thermoelectric transport and its connection with the Fluctuation symmetry.
We study a microscopic Hamiltonian model describing an N-level quantum system S coupled to an infinitely extended thermal reservoir R. Initially, the system S is in an arbitrary state while the reservoir is in thermal equilibrium at temperature T. Assuming that the coupled system S+R is mixing with respect to the joint thermal equilibrium state, we study the Full Counting Statistics (FCS) of the energy transfers S->R and R->S in the process of return to equilibrium. The first FCS describes the increase of the energy of the system S. It is an atomic probability measure, denoted
PS,λ,t, concentrated on the set of energy differences
σ(HS)−σ(HS) (
σ(HS) is the spectrum of the Hamiltonian of S,
t is the length of the time interval during which the measurement of the energy transfer is performed, and
λ is the strength of the interaction between S and R). The second FCS,
PR,λ,t, describes the decrease of the energy of the reservoir R and is typically a continuous probability measure whose support is the whole real line. We study the large time limit
t→∞ of these two measures followed by the weak coupling limit
λ→0 and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and on a representation of
PR,λ,t by quantum transfer operators.
We develop a comprehensive framework for characterizing fluctuations in quantum transport and nonequilibrium thermodynamics using two complementary approaches: full counting statistics and first-passage times. Focusing on open quantum systems governed by Markovian Lindblad dynamics, we derive general ensemble relations that connect the two approaches at all times, and we clarify how the steady states reached at long times relate to those reached at large jump counts. In regimes of metastability, long-lived intermediate states cause violations of experimentally testable cumulant relations, as we discuss. We also formulate a fluctuation theorem governing the probability of rare fluctuations in the first-passage time distributions based on results from full counting statistics. Our results apply to general integer-valued trajectory observables that do not necessarily increase monotonically in time. Three illustrative applications, a two-state emitter, a driven qubit, and a variant of the Su-Schrieffer-Heeger model, highlight the physical implications of our results and provide guidelines for practical calculations. Our framework provides a complete picture of first-passage time statistics in Markovian quantum systems, encompassing multiple earlier results, and it has direct implications for current experiments in quantum optics, superconducting circuits, and nanoscale heat engines.
A general formalism for computing the full counting statistics of energy exchanged between ``
N'' squeezed thermal photon reservoirs weakly coupled to a cavity with ``
M'' photon modes is presented. The formalism is based on the two-point measurement scheme and is applied to two simple special cases: the relaxation dynamics of a single mode cavity in contact with a single squeezed thermal photon reservoir and the steady-state energy transport between two squeezed thermal photon reservoirs coupled to a single cavity mode. Using analytical results, it is found that the short time energy exchange statistics is significantly affected by noncommutativity of the initial energy measurements with the reservoirs' squeezed states, and may lead to negative probabilities if not accounted for properly. Furthermore, it is found that for the single reservoir setup, generically there is no transient or steady-state Jarzynski-W\'ojcik energy exchange fluctuation theorem. In contrast, for the two reservoir cases, although there is no generic transient energy exchange fluctuation theorem, a steady-state energy exchange fluctuation theorem with a nonuniversal affinity is found to be valid. Statistics of energy currents are further discussed.