Double quantum dot coupled to a quantum point contact: a stochastic thermodynamics approach

Abstract

We study the nonequilibrium properties of an electronic circuit composed of a double quantum dot (DQD) channel capacitively coupled to a quantum point contact (QPC) within the framework of stochastic thermodynamics. We show that the transition rates describing the dynamics satisfy a nontrivial local detailed balance and that the statistics of energy and particle currents across both channels obeys a fluctuation theorem. We analyze two regimes where the device operates as a thermodynamic machine and study its output power and efficiency fluctuations. We show that the electrons tunneling through the QPC without interacting with the DQD have a strong effect on the device efficiency.The microscopic processes underlying such transitions involve the tunneling of an electron between the QPC reservoirs which exchanges a fraction of its energy with the DQD. As a result, the transition rates cannot be written anymore as a sum where each term only involves one single reservoir. Despite this non additivity of the rates, the local detailed balance (LDB) [37, 39] is shown to hold, and is explicitly written in terms of the fluxes of entropy from the reservoirs involved in the transitions.Furthermore, since some electrons tunnel between the QPC reservoirs without exchanging energy with the DQD, they do not induce transitions in the DQD but need to be taken into account in the counting statistics. We show that their statistics is well described by the modified quantum master equation and corresponds to the Levitov-Lesovik formula [40-44] to second order perturbation theory in the coupling between the QPC reservoirs. Though such processes do not reveal themselves in the master equation for the DQD populations, they are shown to strongly impact the circuit performance.We proceed by analyzing two different regimes where the circuit operates as a thermoelectric and as a current converter, respectively. We identify the optimal working conditions to reach large average output power and high macroscopic efficiency and study the statistical properties of the output power as well as of the efficiency, as recently proposed in [45][46][47].The paper is organized as follows: the model is introduced in section 2. In section 3, we derive the modified quantum master equation to calculate the counting statistics of energy and matter currents, and analyze the microscopic processes contributing to the transition rates. In section 4, using the LDB property of the rates, we identify the entropy flows associated to each microscopic processes in the circuit, and the steady state FT is derived. The expressions for the average energy and matter currents as well as the average irreversible entropy production are also provided. The thermodynamic analysis of the circuit operating as a thermoelectric and as a current converter is done in section 5. Conclusions are drawn in section 6. +--+ From these expressions, one readily verifies the conservation laws J J J J J J J 0, and . 69

Conservation laws and symmetries in stochastic thermodynamics

“…Similarly, the fundamental affinities can be found by solving the linear equations Let us now discuss some examples of LDB, as can be found, e.g., in Refs. [4,[20][21][22][23]. When the system transitions are caused by exchanges of energy and particles y = ( 1 , .…”

Quantum Thermodynamics with Degenerate Eigenstate Coherences

“…The generating function of work and currents (31) can be evaluated numerically by solving the dressed quantum master Equation (27) for the specific model (40), namely Equation (C9). The corresponding joined probability distribution is then obtained by a Fourier transform.…”

“…We now write for our model the dressed Lindblad master Equation (27) describing the dressed system density matrix ρ(ξ, λ, t), where the counting fields ξ and λ account for, respectively, the currents of energy and particles out of the left reservoir [9,38]. We use the local energy eigenbasis |0 (empty), |t (top occupied), |b (bottom occupied) and |2 (doubly occupied) with system energy eigenvalues E 00 = 0, E t = , E b = , and E 2 = 2 + U, respectively.…”

“…For many processes that only depend on populations or for steady state dynamics where eigenstates coherences are always vanishing, a classical stochastic thermodynamics (ST) [19][20][21] can be easily built for the population dynamics. This provides a consistent framework for the study of the thermodynamics of open quantum systems at both the average and the single trajectory level [22][23][24][25][26][27][28]. However, various time-dependent processes do depend on eigenstate coherences.…”

Fluctuation-Dissipation Relations Far from Equilibrium

“…In particular, the framework of stochastic thermodynamics gives a thermodynamic description of small systems subject to fluctuations [6][7][8][9][10], with applications to interdisciplinary areas including nanoscopic electronics [11][12][13][14][15], complex biomolecules such as molecular motors [16][17][18][19][20][21], and chemical reaction networks [22][23][24][25]. In this framework, the stochastic observables of experimental interest are the time-averaged thermodynamic currents ð1=τÞΦ Above and in what follows, h·i denotes an average over stochastic trajectories sampled from a stationary ensemble.…”