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.
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.
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.
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.
We formulate a full-counting statistics description to study energy exchange in multi-terminal junctions. Our approach applies to quantum systems that are coupled either additively or non-additively (cooperatively) to multiple reservoirs. We derive a Markovian Redfield-type equation for the counting-field dependent reduced density operator. 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. We exemplify our formalism on a multi-terminal two-level quantum system, and apply it to realize the smallest quantum absorption refrigerator, operating through engineered reservoirs, and achievable only through a cooperative bath interaction model.
Continuously measured quantum systems are characterized by an output current, in the form of a stochastic and correlated time series, which conveys crucial information about the underlying quantum system. The many tools used to describe current fluctuations are scattered across different communities: quantum opticians often use stochastic master equations, while a prevalent approach in condensed-matter physics is provided by full counting statistics. These, however, are simply different sides of the same coin. Our goal with this tutorial is to provide a unified toolkit for describing current fluctuations. This not only provides novel insights, by bringing together different fields in physics, but also yields various analytical and numerical tools for computing quantities of interest. We illustrate our results with various pedagogical examples and connect them with topical fields of research, such as waiting-time statistics, quantum metrology, thermodynamic uncertainty relations, quantum point contacts, and Maxwell’s demons. Published by the American Physical Society 2024
We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad-Gorini-Kossakowski-Sudarshan generator. By combining the formalism with full counting statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be defined with respect to the reservoir. We exemplify our findings with simple but pedagogical examples.
Abstract Processes at the nanoscale happen far away from the thermodynamic limit, far from equilibrium and are dominated by fluctuations and, perhaps, even quantum effects. This book establishes a consistent thermodynamic framework for such processes by combining tools from non-equilibrium statistical mechanics and the theory of open quantum systems. The book is accessible for graduate students and of interest to all researchers striving for a deeper understanding of the laws of thermodynamics beyond their traditional realm of applicability. It puts most emphasis on the microscopic derivation and understanding of key principles and concepts as well as their interrelation. The topics covered in this book include (quantum) stochastic processes, (quantum) master equations, local detailed balance, classical stochastic thermodynamics, (quantum) fluctuation theorems, strong coupling and non non-Markovian effects, thermodynamic uncertainty relations, operational approaches, Maxwell's demon and time-reversal symmetry, among other topics. Furthermore, the book treats a few applications in detail to illustrate the general theory and its potential for practical applications. These are single-molecule pulling experiments, quantum transport and thermoelectric effects in quantum dots, the micromaser and related set-ups in quantum optics.