In recent years, the study of heat to work conversion has been re-invigorated by nanotechnology. Steady-state devices do this conversion without any macroscopic moving parts, through steady-state flows of microscopic particles such as electrons, photons, phonons, etc. This review aims to introduce some of the theories used to describe these steady-state flows in a variety of mesoscopic or nanoscale systems. These theories are introduced in the context of idealized machines which convert heat into electrical power (heat-engines) or convert electrical power into a heat flow (refrigerators). In this sense, the machines could be categorized as thermoelectrics, although this should be understood to include photovoltaics when the heat source is the sun. As quantum mechanics is important for most such machines, they fall into the field of quantum thermodynamics. In many cases, the machines we consider have few degrees of freedom, however the reservoirs of heat and work that they interact with are assumed to be macroscopic. This review discusses different theories which can take into account different aspects of mesoscopic and nanoscale physics, such as coherent quantum transport, magnetic-field induced effects (including topological ones such as the quantum Hall effect), and single electron charging effects. It discusses the efficiency of thermoelectric conversion, and the thermoelectric figure of merit. More specifically, the theories presented are (i) linear response theory with or without magnetic fields, (ii) Landauer scattering theory in the linear response regime and far from equilibrium, (iii) Green-Kubo formula for strongly interacting systems within the linear response regime, (iv) rate equation analysis for small quantum machines with or without interaction effects, (v) stochastic thermodynamic for fluctuating small systems. In all cases, we place particular emphasis on the fundamental questions about the bounds on ideal machines. Can magnetic-fields change the bounds on power or efficiency? What is the relationship between quantum theories of transport and the laws of thermodynamics? Does quantum mechanics place fundamental bounds on heat to work conversion which are absent in the thermodynamics of classical systems?

Machines are only Carnot efficient if they are reversible, but then their power output is vanishingly small. Here we ask, what is the maximum efficiency of an irreversible device with finite power output? We use a nonlinear scattering theory to answer this question for thermoelectric quantum systems; heat engines or refrigerators consisting of nanostructures or molecules that exhibit a Peltier effect. We find that quantum mechanics places an upper bound on both power output, and on the efficiency at any finite power. The upper bound on efficiency equals Carnot efficiency at zero power output, but decays with increasing power output. It is intrinsically quantum (wavelength dependent), unlike Carnot efficiency. This maximum efficiency occurs when the system lets through all particles in a certain energy window, but none at other energies. A physical implementation of this is discussed, as is the suppression of efficiency by a phonon heat flow. [6,7]. It places fundamental bounds on the efficiency and power output of heat engines and refrigerators made from such systems, such as Carnot's thermodynamic bound on efficiency or Pendry's quantum bound on entropy flow [8].The efficiencies of heat engines, η eng , and refrigerators, η fri , are particularly important (η fri is called the coefficient of performance, COP). These efficiencies are the ratio of power output to power input. For a heat engine, the output is the electrical power, P gen , and the input is the heat flow out of a reservoir (the left (L) reservoir in Fig. 1c), J L . For a refrigerator, it is the inverse. For left (L) and right (R) reservoirs at temperatures T L and T R , Carnot's bounds on these efficiencies are

We investigate the nonlinear scattering theory for quantum systems with strong Seebeck and Peltier effects, and consider their use as heat-engines and refrigerators with finite power outputs. This article gives detailed derivations of the results summarized in Phys. Rev. Lett. 112, 130601 (2014). It shows how to use the scattering theory to find (i) the quantum thermoelectric with maximum possible power output, and (ii) the quantum thermoelectric with maximum efficiency at given power output. The latter corresponds to a minimal entropy production at that power output. These quantities are of quantum origin since they depend on system size over electronic wavelength, and so have no analogue in classical thermodynamics. The maximal efficiency coincides with Carnot efficiency at zero power output, but decreases with increasing power output. This gives a fundamental lower bound on entropy production, which means that reversibility (in the thermodynamic sense) is impossible for finite power output. The suppression of efficiency by (nonlinear) phonon and photon effects is addressed in detail; when these effects are strong, maximum efficiency coincides with maximum power. Finally, we show in particular limits (typically without magnetic fields) that relaxation within the quantum system does not allow the system to exceed the bounds derived for relaxation-free systems, however, a general proof of this remains elusive.

Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, K(tau), of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution -2tau(2) to the form factor which agrees with random-matrix theory.

We investigate the geometric phase or Berry phase (BP) acquired by a spin-half which is both subject to a slowly varying magnetic field and weakly-coupled to a dissipative environment (either quantum or classical). We study how this phase is modified by the environment and find that the modification is of a geometric nature. While the original BP (for an isolated system) is the flux of a monopole-field through the loop traversed by the magnetic field, the environment-induced modification of the BP is the flux of a quadrupole-like field. We find that the environment-induced phase is complex, and its imaginary part is a geometric contribution to dephasing. Its sign depends on the direction of the loop. Unlike the BP, this geometric dephasing is gauge invariant for open paths of the magnetic field. Introduction. The Berry phase (BP) is a fundamental quantum-mechanical phenomenon related to the adiabatic theorem. Berry [1] showed that the phase acquired by an eigenstate of a slowly varying Hamiltonian H(t) is related to the geometric properties of the loop traversed by H(t). In the presence of dissipation the condition of adiabaticity and the existence of the Berry phase require careful analysis. The widespread criterion of adiabaticity, based on a comparison of the rate of change of the Hamiltonian with the gap in the spectrum should be modified to involve the matrix elements of the system-environment coupling, cf.[2]. Here we study the interplay of the varying field and the dissipation, analyzing the BP in the limiting case of weak system-environment coupling. This analysis is relevant to the recent and proposed experiments to manipulate quantum two-level systems (qubits). Our findings could be tested in solid-state qubits, such as superconducting nanocircuits [3,4,5,6].Berry [1] considers a two-level spin-half system in a magnetic field [7], which is varied slowly along a closed path: H spin = − 1 2 B(t)σ. The rate of the field's change is characterized by the time to complete the loop, t P . In the adiabatic limit, Bt P ≫ 1, the relative phase acquired by the eigenstates is a sum Φ = |B(t)|dt + Φ BP of the dynamical and Berry phases. The latter is geometric, it depends on the geometry of the loop but not on the details of its traversal (for an isolated spin-half it is given by the solid angle subtended by the loop B(t)). In the spin language, the evolution is a rotation of the spin by an angle Φ about B.If the spin is not isolated, the dynamics are more complicated. For a static field, B, dissipation induces energy and phase relaxation processes (with the time-scales T 1 , T 2 respectively) and a Lamb-like shift of the level splitting, δB Lamb , which modifies the dynamical phase.

Abstract. The perturbative master equation (Bloch-Redfield) is extensively used to study dissipative quantum mechanics -particularly for qubitsdespite the 25 year old criticism that it violates positivity (generating negative probabilities). We take an arbitrary system coupled to an environment containing many degrees-of-freedom, and cast its perturbative master equation (derived from a perturbative treatment of Nakajima-Zwanzig or Schoeller-Schön equations) in the form of a Lindblad master equation. We find that the equation's parameters are time-dependent. This time-dependence is rarely accounted for, and invalidates Lindblad's dynamical semigroup analysis. We analyze one such Bloch-Redfield master equation (for a two-level system coupled to an environment with a short but non-vanishing memory time), which apparently violates positivity. We show analytically that, once the time-dependence of the parameters is accounted for, positivity is preserved.

We investigate the effect of the environment on a Berry phase measurement involving a spin-half. We model the spin+environment using a biased spin-boson Hamiltonian with a time-dependent magnetic field. We find that, contrary to naive expectations, the Berry phase acquired by the spin can be observed, but only on timescales which are neither too short nor very long. However this Berry phase is not the same as for the isolated spin-half. It does not have a simple geometric interpretation in terms of the adiabatic evolution of either bare spin-states or the dressed spinresonances that remain once we have traced out the environment. This result is crucial for proposed Berry phase measurements in superconducting nanocircuits as dissipation there is known to be significant.PACS numbers: 03.65. Vf, 03.65.Yz, 85.25.Cp It was recently suggested [1] that it should be possible to observe the Berry phase (BP) [2] in a superconducting nanostructure, and possibly use it to control the evolution of the quantum state [3,4]. This intriguing sugestion however did not consider the coupling to the environment, which is never negligible in such structures [5]. To truly understand the feasibility of the proposed experiment, we must know the effect of the environment on the BP. Originally the BP was defined for systems whose states were separated by finite energy gaps. Here we ask whether a BP can be observed in a system whose spectrum is continuous because it is not completely isolated from its environment. All real systems are coupled, at least weakly, to their environment and as a result never have a truly discrete energy level spectrum. The usual requirement for adiabaticity is that the parameters of the Hamiltonian are varied slowly compared to the gap in the spectrum. Here there is no gap so naively one would say that adiabaticity is impossible and hence the BP could never be observed. However experiments have observed the BP, both directly and indirectly [6], so this argument must be too naive. We therefore take a simple model in which a quantum system, which when isolated exhibits a BP, is coupled to many other quantum degrees of freedom. We then ask two questions. Firstly, under what conditions can the BP be observed? Secondly, is the observed BP the same as that of the isolated system? While others have investigated systems with a BP coupled to other degrees of freedom [7,8,9], we believe we are the first to explicitly address these two questions.We distinguish between the system and the environment in the following way. We have complete experimental control over the system, but almost no control over the environment. The most that we can do to the environment is to ensure the "universe" (system + environment) is in thermal equilibrium, with a temperature T . We will assume we have enough control over T to take it to zero, and thus prepare the universe in its ground state. However any procedure to measure a BP in an isolated system must involve measuring a phase difference from a superposition of two states. When the sys...

We add simple tunnelling effects and ray-splitting into the recent trajectory-based semiclassical theory of quantum chaotic transport. We use this to derive the weak-localization correction to conductance and the shot-noise for a quantum chaotic cavity (billiard) coupled to n leads via tunnelbarriers. We derive results for arbitrary tunnelling rates and arbitrary (positive) Ehrenfest time, τE. For all Ehrenfest times, we show that the shot-noise is enhanced by the tunnelling, while the weaklocalization is suppressed. In the opaque barrier limit (small tunnelling rates with large lead widths, such that the Drude conductance remains finite), the weak-localization goes to zero linearly with the tunnelling rate, while the Fano factor of the shot-noise remains finite but becomes independent of the Ehrenfest time. The crossover from RMT behaviour (τE = 0) to classical behaviour (τE = ∞) goes exponentially with the ratio of the Ehrenfest time to the paired-paths survival time. The paired-paths survival time varies between the dwell time (in the transparent barrier limit) and half the dwell time (in the opaque barrier limit). Finally our method enables us to see the physical origin of the suppression of weak-localization; it is due to the fact that tunnel-barriers "smear" the coherent-backscattering peak over reflection and transmission modes.