Thermodynamics of a continuously monitored double quantum dot heat engine in the repeated interactions framework

TL;DR

This work analyzes a continuously monitored double quantum dot heat engine coupled to two fermionic reservoirs, showing that a thermodynamically consistent description can be obtained by deriving the GKSL master equation within the repeated interactions framework. The study demonstrates that measurement-induced dephasing from a quantum point contact can both boost the particle current and modify its fluctuations, while the entropy-production lower bound from the thermodynamic uncertainty relation remains intact. By constructing minimal RI representations for the reservoirs and the QPC, the authors bridge local GKSL descriptions with a collision-like thermodynamic framework, ensuring consistency with the first and second laws. The findings highlight dephasing as a potential resource for stabilizing currents at fixed entropic cost and provide a foundation for more realistic models of measurement backaction in quantum thermodynamics.

Abstract

Understanding the thermodynamic role of measurement in quantum mechanical systems is a burgeoning field of study. In this article, we study a double quantum dot (DQD) connected to two macroscopic fermionic thermal reservoirs. We assume that the DQD is continuously monitored by a quantum point contact (QPC), which serves as a charge detector. Starting from a minimalist microscopic model for the QPC and reservoirs, we show that the local master equation of the DQD can alternatively be derived in the framework of repeated interactions and that this framework guarantees a thermodynamically consistent description of the DQD and its environment (including the QPC). We analyze the effect of the measurement strength and identify a regime in which particle transport through the DQD is both assisted and stabilized by dephasing. We also find that in this regime the entropic cost of driving the particle current with fixed relative fluctuations through the DQD is reduced. We thus conclude that under continuous measurement a more constant particle current may be achieved at a fixed entropic cost.

Figures (7)

• Figure 1: The DQD consists of two quantum dots coupled via a tunneling interaction of amplitude $t$. Each quantum dot is locally exchanging energy and particles with an independent, macroscopic thermal fermionic reservoir, at rate $\gamma_\mathrm{H(C)}$. The reservoirs are fully characterised by their respective chemical potential $\mu_\mathrm{H(C)}$ and temperature $T_\mathrm{H(C)}$.
• Figure 2: In the framework of repeated interactions, the hot and cold reservoirs are replaced by periodically refreshed units consisting of single qubits with energy splitting $\epsilon_1-\mu_H$ and $\epsilon_2-\mu_C$, respectively. For the interaction time $\tau$, a single unit per reservoir interacts with the DQD. Subsequently the two units are replaced by new units with the same initial state. The inital population of the qubit levels are set by the chemical potentials and temperatures of the reservoirs we aim to model.
• Figure 3: The DQD in the two-terminal set-up is continuously monitored via an interaction between one quantum dot in the DQD and a QPC with measurement strength $\Gamma$. The particle current through the QPC depends on the occupation of the monitored quantum dot.
• Figure 4: In the framework of repeated interactions, we model the interaction of the DQD with the QPC with periodically refreshed units comprised of two qubits, i.e. one qubit per QPC lead. The energy splitting of the qubits is $(\Omega-\mu_\mathrm{R(L)})$, respectively. They are coupled via a tunneling interaction with amplitude $\mathcal{T}$. The qubits are initialized so that the qubit in the right lead is occupied while the qubit in the left lead is empty, resulting in a unidirectional current from right to left.
• Figure 5: Work, heat and particle currents as a function of the measurement strength. (a) In the NESS, the total work current out of the DQD $-\dot{W}_\mathrm{tot}$ (black solid line) exhibits an extremum at the measurement strength $\Gamma_\mathrm{ext}$ and is larger than its value in the absence of measurement up a measurement strength of $\Gamma_0$, as indicated by the dashed grey lines and the grey box. Here, also the work current between DQD and QPC $-\dot{W}^\mathrm{DQD}_\mathrm{QPC}$ (blue solid line) and the heat current between the hot reservoir and the DQD $\dot{Q}_\mathrm{H}$ (red solid line) are shown. (b) The average particle current attains a maximum at the measurement strength $\Gamma_\mathrm{ext}$ and exceeds its magnitude in the absence of measurement up a measurement strength of $\Gamma_0$, as indicated by the dashed grey lines and the grey box. Parameters: reservoir temperatures $T_\mathrm{H}$, $T_\mathrm{C}$ = 3, 1 $T=(T_\mathrm{H}+T_\mathrm{C})/2$; reservoir chemical potentials $\mu_H$, $\mu_C$ = 0.5$k_\mathrm{B}T$, 1.5$k_\mathrm{B}T$; quantum dot energies $\epsilon_1$, $\epsilon_2$ = 2$k_\mathrm{B}T$, 2.1$k_\mathrm{B}T$; interdot hopping amplitude $t$ = 0.025$k_\mathrm{B}T$; coupling to reservoirs $\gamma_\mathrm{C}, \gamma_\mathrm{H}$ = $\gamma$ = 0.025$k_\mathrm{B}T$.