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

Laetitia P. Bettmann<EMAIL_ADDRESS>School of Physics, Trinity College
Dublin, College Green, Dublin 2, Ireland  Michael J. Kewming<EMAIL_ADDRESS>School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
John Goold<EMAIL_ADDRESS>School of Physics, Trinity College Dublin, College
Green, Dublin 2, Ireland

(August 28, 2024)

###### 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.

##  I Introduction

In recent decades, technological developments have led to the creation of
artificial mesoscopic and nanoscopic steady-state heat engines Benenti  _et
al._ (2017); Myers  _et al._ (2022); Pekola and Karimi (2021); Pekola (2015);
Arrachea (2022). Without the need for moving parts in such engines, stochastic
energy obtained in the form of a heat transfer is converted into useful work
by currents of microscopic particles. Applications are diverse, ranging from
atomic and molecular junctions Dubi and Di Ventra (2011) to quantum dots
Josefsson  _et al._ (2018). At the scale at which these devices operate, not
only do average values of thermodynamic quantities matter, but thermal and
quantum fluctuations become just as relevant and must be taken into account
for a complete thermodynamic understanding Esposito  _et al._ (2009); Campisi
(2014). To study the devices experimentally, one often relies on continuous
quantum measurement to obtain information about the temporal evolution of the
internal state. At the same time, even with the most non-invasive detectors
such as quantum point contacts (QPCs) acting as charge sensors Levinson
(1997); Aleiner  _et al._ (1997); Stodolsky (1999); Büttiker and Martin
(2000), performing weak measurements inevitably leads to measurement
backaction. Moreover, by monitoring an observable incompatible with energy,
the energy transport in quantum devices can be altered too Bhandari and Jordan
(2022). This points to the fundamental role that measurement and the
subsequent quantum trajectory description arising from measurement play in the
interpretation of stochastic quantum thermodynamics quantities Horowitz
(2012); Manzano  _et al._ (2015); Campisi  _et al._ (2015); Elouard  _et al._
(2017a, b); Manzano  _et al._ (2018); Di Stefano  _et al._ (2018); Strasberg
(2019); Alonso  _et al._ (2016); Belenchia  _et al._ (2020); Rossi  _et al._
(2020); Menczel  _et al._ (2020); Miller  _et al._ (2020); Liu and Su (2020);
Kewming and Shrapnel (2022); Manzano and Zambrini (2022).

While dephasing occurs in coherent quantum systems in many scenarios, such as
electron-phonon interaction Fedichkin and Fedorov (2004, 2005) and background
charge noise Itakura and Tokura (2003), it also arises from capacitive
coupling to QPCs Fujisawa  _et al._ (2006); Ihn  _et al._ (2009). In the
latter scenario, dephasing is a direct consequence of the measurement
backaction of the QPC on the monitored quantum system Gurvitz (1997); Levinson
(1997); Aleiner  _et al._ (1997); Stodolsky (1999); Büttiker and Martin
(2000). Interestingly, there are situations where, counter-intuitively,
quantum transport is aided by dephasing, e.g. Contreras-Pulido  _et al._
(2014); Plenio and Huelga (2008); Rebentrost  _et al._ (2009a); Scholes and
Smyth (2014); Zerah-Harush and Dubi (2020); Kilgour and Segal (2015, 2016);
Sowa  _et al._ (2017); Lacerda  _et al._ (2021); Mendoza-Arenas  _et al._
(2013); Engel  _et al._ (2007); Collini  _et al._ (2010); Žnidarič  _et al._
(2017); Chiaracane  _et al._ (2022); Caruso  _et al._ (2009, 2016); Chin  _et
al._ (2012); Collini (2013); Biggerstaff  _et al._ (2016), by suppressing
coherent single-particle interference effects. Exploiting this mechanism to
improve transport efficiency is therefore interesting for applications in
quantum technology, including controlled quantum systems, but also because it
challenges the conception that disturbances due to couplings to the
environment, under all circumstances, hinder performance. Interestingly, the
presence of a dephasing noisy environment has been shown to be also beneficial
for performance in some biological systems, such as photosynthetic systems
Faisal  _et al._ (2008); Adolphs and Renger (2006); Gaab and Bardeen (2004);
Mohseni  _et al._ (2008); Rebentrost  _et al._ (2009b, a).

A widely used tool to study the temporal evolution as well as steady-states of
quantum systems are Gorini - Kossakowski - Sudarshan - Lindblad (GKSL) master
equations (MEs) Lindblad (1976); Gorini  _et al._ (1976). Often the effect of
an environment on the quantum system of interest can be approximated as local.
Then, one assumes that the dissipators, which account for the interaction of
the system with the individual components of the environment, can be derived
independently of each other. A discussion of local versus global MEs can be
found in Refs. Cattaneo  _et al._ (2019); González  _et al._ (2017); Hofer
_et al._ (2017). Importantly, as argued in Ref. Levy and Kosloff (2014), a
local GKSL ME — for which global detailed balance does not generally hold
Chiara  _et al._ (2018) — does not guarantee a consistent thermodynamic
description of non-equilibrium steady state (NESS), if the mechanism for its
emergence is not fully taken into account. This issue was addressed by Barra
Barra (2015) who—building off previous works Esposito  _et al._ (2010); Reeb
and Wolf (2014)—developed a consistent thermodynamic framework for calculating
the average flows of work, heat and entropy in the repeated interactions
framework (collisional models) Attal and Pautrat (2006); Karevski and Platini
(2009); Ciccarello  _et al._ (2022); Campbell and Vacchini (2021); Scarani
_et al._ (2002); Ziman  _et al._ (2002) for systems driven locally by
interactions with the environment. This framework is an intuitive and
convenient methodology which requires assumptions on the environment similar
to those of the local GKSL ME. However, it is important to point out that the
compatibility of local MEs with thermodynamics is an active area of research
and there are a variety of alternative approaches Trushechkin and Volovich
(2016); Strasberg (2019); Soret  _et al._ (2022); Hewgill  _et al._ (2021);
Potts  _et al._ (2021); Cattaneo  _et al._ (2019). In the present work, we
investigate the influence of continuous measurement via a QPC on the NESS
transport and thermodynamic properties of a double quantum dot (DQD) van der
Wiel  _et al._ (2002) coupled to two independent macroscopic thermal fermionic
reservoirs. Importantly, DQDs exhibit quantum coherence which makes them
susceptible to dephasing due to measurement-induced backaction. A similar set
up was studied also within the stochastic thermodynamics approach in Ref.
Cuetara and Esposito (2015). Here, we use a thermodynamically consistent
formalism Barra (2015) in the framework of repeated interactions, starting
from a minimalist microscopic model of the QPC and the reservoirs. We show
that the commonly used additive local GKSL ME for the DQD, accounting for both
dissipation to the reservoirs as well as dephasing due to the coupling to the
QPC, can be alternatively derived in the repeated interactions framework.
Thus, the characteristics of the environment relevant to the time evolution of
the DQD state and its NESS — within the assumptions made in deriving the local
GKSL ME — are well captured in our minimalist model, suggesting that it can be
used as a starting point for more sophisticated models in the future. We find
that the net output power of the DQD operated as a heat engine is, in certain
regimes, assisted by dephasing and can be tuned via the QPC’s measurement
strength. Further, we study the entropic cost of precision in the particle
current through the DQD within the context of a thermodynamic uncertainty
relation (TUR) Barato and Seifert (2015); Pietzonka and Seifert (2018);
Pietzonka  _et al._ (2017, 2016); Horowitz and Gingrich (2017). TURs set a
fundamental lower bound on the trade-off between the relative fluctuations of
a thermodynamic current in classical Markovian NESS heat engines and the
entropy production rate. Moreover, TURs valid for quantum systems were
developed in Ref. Timpanaro  _et al._ (2019) and subsequently studied for
continuous measurements, and open quantum systems Carollo  _et al._ (2019);
Hasegawa (2020, 2021); Van Vu and Saito (2022). Importantly, recent works on
TURs have been using the DQD as a working model Agarwalla and Segal (2018);
Liu and Segal (2019); Prech  _et al._ (2022), albeit without the presence of a
QPC. We find that in the parameter regime in which the output power is
increased due to the dephasing measurement-induced backaction, the precision
of the particle current through the DQD, quantified by its relative
fluctuations, is enhanced relative to the entropic cost of driving it. The
fundamental lower bound set by the TUR, however, remains intact.

The paper is organised as follows: First, we study the NESS of the DQD in the
absence of continuous measurement (II). We introduce the microscopic
Hamiltonian describing the DQD, the two reservoirs, and their local
interaction with the DQD, and then give the well-known GKSL ME for the DQD
(II.1). Next, we discuss the issue of the apparent thermodynamic inconsistency
of the local GKSL equation for this set-up (II.2). We then show that the GKSL
ME for the DQD can be alternatively derived in the framework of repeated
interactions (II.3). Finally, we compute the NESS flows of work, heat and
entropy production following Ref. Barra (2015) and show that our result is
consistent with the first and second law of thermodynamics (II.4). Next, we
introduce continuous measurement of the DQD via a QPC. We first give the
microscopic Hamiltonian (III.1) and then state the unconditional GKSL ME for
the reduced density matrix for the DQD, now also accounting for dephasing due
to measurement backaction. We then propose a minimalist composite unit for the
QPC in the framework of repeated interactions and show that the GKSL ME
derived in this framework coincides with the GKSL ME derived starting from the
microscopic Hamiltonian (III.2). Next, we compute the NESS flows of work, heat
and entropy production in the presence of the QPC and show that our result is
consistent with the first and second law of thermodynamics (5). We address the
parameter regimes allowing for enhanced power output via dephasing assisted
particle transport in Sec. (III.4). We show the consequences of the latter to
reduce relative fluctuations of the particle current at a fixed entropic cost
(III.5). Finally, we conclude by summarizing and giving the outlook (IV).

##  II DQD operated as heat engine

Before investigating the effect of measurement-induced backaction due to the
QPC on the boundary-driven DQD NESS, we first provide a pedagogical overview
on computing the particle and energy fluxes of the NESS using the framework of
repeated interactions, in the absence of a QPC. We show that although the time
evolution of the reduced DQD density matrix in this approach is governed by a
local GKSL ME that does not obey global detailed balance, the average NESS
energy fluxes between the DQD and the reservoirs are compatible with the first
and second laws of thermodynamics.

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
$T_{\mathrm{H(C)}}$.

###  II.1 DQD and the GKSL ME

The global microscopic Hamiltonian of the DQD in contact with the two
reservoirs, a resonant-level transport model, is of the form

$\hat{H}=\hat{H}_{\mathrm{DQD}}+\sum_{\alpha\in\\{\mathrm{H,C}\\}}\hat{H}_{\mathrm{\alpha}}+\hat{H}_{\mathrm{DQD,R}},$
(1)

where $\hat{H}_{\mathrm{DQD}}$ denotes the DQD Hamiltonian,

$\hat{H}_{\alpha}=\sum_{k}\epsilon_{\alpha,k}\hat{c}_{\alpha,k}^{\dagger}\hat{c}_{\alpha,k}$
(2)

is the reservoir $\alpha$ Hamiltonian expressed in terms of the creation
(annihilation) operators $\hat{c}_{\alpha,k}^{\dagger}$ ($\hat{c}_{\alpha,k}$)
of the reservoir single-particle states with energy $\epsilon_{\alpha,k}$ and
$\hat{H}_{\mathrm{DQD,R}}$ is the tunneling Hamiltonian between the DQD and
the reservoirs. The DQD Hamiltonian is given by

$\displaystyle\hat{H}_{\mathrm{DQD}}$
$\displaystyle=\epsilon_{1}\hat{c}_{1}^{\dagger}\hat{c}_{1}+\epsilon_{2}\hat{c}_{2}^{\dagger}\hat{c}_{2}+t\left(\hat{c}_{1}^{\dagger}\hat{c}_{2}+\hat{c}_{2}^{\dagger}\hat{c}_{1}\right).$
(3)

It is expressed in terms of the creation (annihilation) operators
$\hat{c}_{1(2)}^{\dagger}$ ($\hat{c}_{1(2)}$) of single quantum dot states
with energies $\epsilon_{1(2)}$, and $t$ denotes the interdot tunneling
amplitude. The tunneling Hamiltonian between the DQD and the reservoirs is
given by

$\displaystyle\begin{split}\hat{H}_{\mathrm{DQD,R}}=&\sum_{k}g_{\mathrm{H},k}\hat{c}_{1}^{\dagger}\hat{c}_{\mathrm{H},k}+g^{*}_{\mathrm{H},k}\hat{c}^{\dagger}_{\mathrm{H},k}\hat{c}_{1}\\\
&+\sum_{k}g_{\mathrm{C},k}\hat{c}_{2}^{\dagger}\hat{c}_{\mathrm{C},k}+g^{*}_{\mathrm{C},k}\hat{c}^{\dagger}_{\mathrm{C},k}\hat{c}_{2},\end{split}$
(4)

where $g_{\alpha,k}$ denotes the tunneling rate between mode $k$ in reservoir
$\alpha$ and the respective quantum dot. The set-up is such that the DQD is
driven into a NESS in the long-time limit by its coupling to the two
reservoirs with chemical potentials $\mu_{\mathrm{H(C)}}$ and temperatures
$T_{\mathrm{H(C)}}$ at its boundaries.

Usually, the NESS is derived from a GKSL master equation, which is only valid
under the following assumptions Breuer (2003): weak coupling of the reservoirs
to the DQD (Born approximation), the reservoirs are memory-less (Markov
approximation), and the bandwidths of the reservoirs are much larger than
those of the DQD. The last assumption yields frequency-independent
interactions between the reservoirs and the DQD (wide-band limit). Finally, we
suppose that the reservoirs are kept in local thermal equilibrium states and
are static and uncorrelated with the DQD and among each other at all times.
Tracing out the reservoir degrees of freedom one obtains the additive local
GKSL ME governing the time evolution of the DQD,
$\hat{\rho}_{\mathrm{DQD}}=\mathrm{Tr_{R}}\left[\hat{\rho}_{\mathrm{tot}}\right]$,
($\hbar=1$)

$\displaystyle\begin{split}\dot{\hat{\rho}}_{\mathrm{DQD}}=&-i[\hat{H}_{\mathrm{DQD}},\hat{\rho}_{\mathrm{DQD}}]\\\
&+\underbrace{\gamma_{\mathrm{H}}f_{\mathrm{H}}\left(\epsilon_{1}\right)\mathcal{D}[\hat{c}_{1}^{\dagger}]\hat{\rho}_{\mathrm{DQD}}+\gamma_{\mathrm{H}}\left(1-f_{\mathrm{H}}\left(\epsilon_{1}\right)\right)\mathcal{D}[\hat{c}_{1}]\hat{\rho}_{\mathrm{DQD}}}_{\mathcal{L}_{\mathrm{H}}\left(\hat{\rho}_{\mathrm{DQD}}\right)}\\\
&+\underbrace{\gamma_{\mathrm{C}}f_{\mathrm{C}}\left(\epsilon_{2}\right)\mathcal{D}[\hat{c}_{2}^{\dagger}]\hat{\rho}_{\mathrm{DQD}}+\gamma_{\mathrm{C}}\left(1-f_{\mathrm{C}}\left(\epsilon_{2}\right)\right)\mathcal{D}[\hat{c}_{2}]\hat{\rho}_{\mathrm{DQD}}}_{\mathcal{L}_{\mathrm{C}}\left(\hat{\rho}_{\mathrm{DQD}}\right)},\\\
&=\mathcal{L}(\hat{\rho}_{\mathrm{DQD}}),\end{split}$ (5)

where the reservoir Fermi functions
$f_{\mathrm{H(C)}}\left(\epsilon_{1(2)}\right)=\left[1+\exp\left(\left(\epsilon_{1(2)}-\mu_{\mathrm{H(C)}}\right)/k_{\mathrm{B}}T_{\mathrm{H(C)}}\right)\right]^{-1}$,
where $k_{\mathrm{B}}$ is the Boltzmann constant, are evaluated at the quantum
dot energies $\epsilon_{1(2)}$ (resonant tunneling), and the tunneling rates
between the DQD and the two reservoirs are denoted by
$\gamma_{\mathrm{H(C)}}$. The set-up as well as the interactions are
schematically depicted in Fig. (1). The dissipators are defined as
$\mathcal{D}[\hat{L}]\hat{\rho}_{\mathrm{DQD}}=\hat{L}\hat{\rho}_{\mathrm{DQD}}\hat{L}^{\dagger}-\frac{1}{2}\\{L^{\dagger}\hat{L},\hat{\rho}_{\mathrm{DQD}}\\}$.
Here, we use the common notations $\left[\cdot,\cdot\right]$ for the
commutator and $\\{\cdot,\cdot\\}$ for the anti-commutator. The jump operators
in Eq. (5) are local and account for the boundary-driving of the DQD by the
reservoirs since they drive transitions between single quantum dot eigenstates
at the boundary rather than between delocalized energy eigenstates of
$\hat{H}_{\mathrm{DQD}}$. In this approach it is assumed that the dissipators
for the two reservoirs can be derived independently of each other. While this
approximation is commonly made, it is important to point out that in general
the exact evolution cannot be expressed as a sum of dissipators describing the
action of each reservoir alone Mitchison and Plenio (2018). Moreover, the
interaction $\hat{H}_{\mathrm{DQD,R}}$ does not commute with the DQD
Hamiltonian $\hat{H}_{\mathrm{DQD}}$, i.e.
$\left[\hat{H}_{\mathrm{DQD,R}},\hat{H}_{\mathrm{DQD}}\right]\neq 0$. This is
because the jump operators $\hat{L}$ here are not eigenoperators of
$\hat{H}_{\mathrm{DQD}}$, due to the inter-system coupling Chiara  _et al._
(2018). Thus, the local GKSL ME (5) derived from the interaction
$\hat{H}_{\mathrm{DQD,R}}$ violates global detailed balance, although locally
detailed balance holds.

###  II.2 Thermodynamic inconsistency of the local GKSL ME

In this section, we follow the discussion in the work by Levy and Kosloff Levy
and Kosloff (2014) on the the apparent violation of the second law of
thermodynamics starting from the local GKSL ME (5) for the DQD coupled to two
reservoirs. Assume for this example the absence of a chemical potential
gradient. For simplicity we set $\mu_{\mathrm{H}},\mu_{\mathrm{C}}=0$.
Commonly, the heat current from the hot reservoir into the DQD is defined as

$\displaystyle\begin{split}\dot{Q}_{\mathrm{H}}=&\mathrm{Tr}\left[\hat{H}_{\mathrm{DQD}}\mathcal{L}_{\mathrm{H}}\left(\hat{\rho}_{\mathrm{DQD}}\right)\right].\end{split}$
(6)

Evaluating the trace, it may be rewritten as

$\dot{Q}_{\mathrm{H}}=\left(f_{\mathrm{H}}(\epsilon_{1})-f_{\mathrm{C}}(\epsilon_{2})\right)\mathcal{F},$
(7)

where $\mathcal{F}$ is a function of the DQD parameters and is always
positive. Clearly, for
$\beta_{\mathrm{C}}\epsilon_{2}<\beta_{\mathrm{H}}\epsilon_{1}$,
$\left(f_{\mathrm{H}}(\epsilon_{1})-f_{\mathrm{C}}(\epsilon_{2})\right)<0$,
since $\mu_{\mathrm{H}},\mu_{\mathrm{C}}=0$, so that the heat current from the
hot reservoir is negative. Since in this NESS
$\dot{Q}_{\mathrm{H}}=-\dot{Q}_{\mathrm{C}}$, the second law of thermodynamics
appears to be violated, as the entropy production rate, for
$\dot{Q}_{\mathrm{H}}<0$,

$\displaystyle\sigma=\left(\beta_{\mathrm{C}}-\beta_{\mathrm{H}}\right)\dot{Q}_{\mathrm{H}}<0.$
(8)

This apparent violation arises because the above definition of the heat
current from the hot reservoir is valid only for global MEs which satisfy the
condition of global detailed balance. The above example illustrates that the
local GKSL ME alone may not be suitable for describing the thermodynamics of
the DQD and other approaches are needed.

###  II.3 GKSL ME in the framework of repeated interactions

A description of boundary-driven systems such as the above reconciled with the
second law of thermodynamics, without having to compromise on using a local
GKSL ME, was put forward by Barra in Ref. Barra (2015) for the framework of
repeated interactions. We will briefly review it here. Note, however, that a
variety of alternative approaches exist Trushechkin and Volovich (2016);
Strasberg (2019); Soret  _et al._ (2022); Hewgill  _et al._ (2021); Potts  _et
al._ (2021); Cattaneo  _et al._ (2019).

The framework of repeated interactions provides a prescription to model a
system’s interaction with an environment in terms of replacing it by a
collection of identical units interacting with the system one after the other
for a fixed interaction time $\tau$. Initially the system is uncorrelated with
all interaction units and, importantly, the interaction units remain
uncorrelated among each other for all times. The system then evolves for an
interaction time $\tau$ in the presence of the interaction with a single unit,
subsequently the unit is replaced, and the system then interacts with the
fresh unit, again for an interval of length $\tau$, and so on. In Fig. (2) we
schematically show how we model the two thermal reservoirs by units, each
consisting of a thermal qubit with an energy splitting of
$\epsilon_{1}-\mu_{\mathrm{H}}$ and $\epsilon_{2}-\mu_{\mathrm{C}}$,
respectively, during each interaction interval. The populations of the qubit
levels are set by the chemical potentials and temperatures,
$\mu_{\mathrm{H(C)}}$ and $T_{\mathrm{H(C)}}$, of the original reservoirs.

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.

This microscopic set-up is described by the bare Hamiltonians for the DQD
$\hat{H}_{\mathrm{DQD}}$ defined in Eq. (3) and the qubit-units for the two
reservoirs

$\displaystyle\hat{H}_{\alpha}$
$\displaystyle=(\epsilon_{\alpha}-\mu_{\alpha})\hat{c}_{\alpha}^{\dagger}\hat{c}_{\alpha},$
(9)

where $\alpha\in\\{\mathrm{H,C}\\}$, and the (single) interactions between the
DQD and reservoir qubits, via

$\displaystyle\hat{v}_{\mathrm{H}}$
$\displaystyle=\sqrt{\gamma_{\mathrm{H}}}\left(\hat{c}_{\mathrm{H}}^{\dagger}\hat{c}_{1}+\hat{c}_{1}^{\dagger}\hat{c}_{\mathrm{H}}\right),$
(10) $\displaystyle\hat{v}_{\mathrm{C}}$
$\displaystyle=\sqrt{\gamma_{\mathrm{C}}}\left(\hat{c}_{\mathrm{C}}^{\dagger}\hat{c}_{2}+\hat{c}_{2}^{\dagger}\hat{c}_{\mathrm{C}}\right).$
(11)

The total Hamiltonian is given by

$\hat{H}=\hat{H}_{\mathrm{DQD}}+\sum_{\alpha}\hat{H}_{\alpha}+\sum_{\alpha}\hat{V}_{\alpha}$
(12)

where $\hat{V}_{\alpha}=\hat{v}_{\alpha}/\sqrt{\tau}$. Importantly, the
periodic refreshing of the units introduces time-dependence into the total
Hamiltonian. Barra shows that because of this, an inherent external work is
required to produce the dissipative evolution described by the local GKSL
equation Barra (2015). Since the units representing the two reservoirs consist
of a single thermal qubit each, their respective density matrix is initialised
to

$\displaystyle\hat{\omega}_{\beta_{\mathrm{H}}}$
$\displaystyle=\left(\begin{array}[]{cc}f_{\mathrm{H}}(\epsilon_{1})&0\\\
0&1-f_{\mathrm{H}}(\epsilon_{1})\end{array}\right),$ (15)
$\displaystyle\hat{\omega}_{\beta_{\mathrm{C}}}$
$\displaystyle=\left(\begin{array}[]{cc}f_{\mathrm{C}}(\epsilon_{2})&0\\\
0&1-f_{\mathrm{C}}(\epsilon_{2})\end{array}\right).$ (18)

The total density matrix $\hat{\rho}$ at the start of each subsequent
interaction is given by

$\hat{\rho}=\hat{\rho}_{\mathrm{DQD}}\otimes\underbrace{\hat{\omega}_{\beta_{\mathrm{H}}}\otimes\hat{\omega}_{\beta_{\mathrm{C}}}}_{\hat{\omega}_{\mathrm{R}}}.$
(19)

After initialisation $\hat{\rho}$ evolves unitarily for the duration of the
interaction time $\tau$ with

$\hat{U}=\exp\left(-i\tau\hat{H}\right).$ (20)

After the $n^{\mathrm{th}}$ interaction, the difference
$\Delta\hat{\rho}_{\mathrm{DQD}}\left(n\tau\right)=\hat{\rho}_{\mathrm{DQD}}(n\tau+\tau)-\hat{\rho}_{\mathrm{DQD}}(n\tau)$
is given by

$\Delta\hat{\rho}_{\mathrm{DQD}}\left(n\tau\right)=\mathrm{Tr}_{\mathrm{E}}\left[\hat{U}\hat{\rho}_{\mathrm{DQD}}\left(n\tau\right)\otimes\hat{\omega}_{\mathrm{R}}\hat{U}^{\dagger}-\hat{\rho}_{\mathrm{DQD}}\left(n\tau\right)\otimes\hat{\omega}_{\mathrm{R}}\right].$
(21)

The GKSL ME is recovered by expanding the unitary $\hat{U}$ to second order in
$\tau$ and taking the limit of infinitely many interactions
$n\rightarrow\infty$ and infinitesimal interaction time $\tau\rightarrow 0$.
One then finds

$\dot{\hat{\rho}}_{\mathrm{DQD}}=-i\left[\hat{H}_{\mathrm{DQD}},\hat{\rho}_{\mathrm{DQD}}\right]+\sum_{\alpha\in\\{\mathrm{H},\mathrm{C}\\}}\mathcal{D}_{\alpha}\left(\hat{\rho}_{\mathrm{DQD}}\right).$
(22)

Importantly, the action of the reservoirs on the system emerges as a sum of
dissipators due to the interaction with individual reservoirs defined by

$\displaystyle\begin{split}\mathcal{D}_{\alpha}\left(\hat{\rho}_{\mathrm{DQD}}\right)=&\mathrm{Tr}_{\alpha}\left[\hat{v}_{\alpha}\left(\hat{\rho}_{\mathrm{DQD}}\otimes\hat{\omega}_{\beta_{\alpha}}\right)\hat{v}_{\alpha}\right]\\\
&-\frac{1}{2}\mathrm{Tr}_{\alpha}\\{\hat{v}_{\alpha}^{2},\left(\hat{\rho}_{\mathrm{DQD}}\otimes\hat{\omega}_{\beta_{\alpha}}\right)\\}.\end{split}$
(23)

By plugging Eq. (10) and Eq. (11) into Eq. (23), we recover the local GKSL ME
stated in Eq. (5). Therefore, we know that at the level of the DQD, the
dynamics dictated by the local GKSL equation and in the framework of repeated
interactions are equivalent. However, the utility of the repeated interactions
framework allows us to recover a thermodynamically consistent picture, as we
now show.

###  II.4 Thermodynamic consistency

We are now in a position to compute the flows of heat and work between the DQD
and the reservoirs, as well as the entropy production rate in the NESS Barra
(2015).

The NESS of the DQD is uniquely defined by

$\dot{\hat{\rho}}_{\mathrm{DQD}}^{\mathrm{NESS}}=\mathcal{L}\left({\hat{\rho}}_{\mathrm{DQD}}^{\mathrm{NESS}}\right)=0,$
(24)

where the Liouvillian $\mathcal{L}$ is defined in Eq. (5). Firstly, we are
interested in showing agreement with the first law of thermodynamics. To this
aim, we begin by defining the work and heat currents from the reservoir
$\alpha$ as

$\displaystyle\dot{Q}_{\alpha}$
$\displaystyle=-D_{\alpha}\left(\hat{H}_{\alpha}\right),$ (25)
$\displaystyle\dot{W}_{\alpha}$
$\displaystyle=D_{\alpha}\left(\hat{H}_{\mathrm{DQD}}+\hat{H}_{\alpha}\right),$
(26)

where

$\displaystyle D_{\alpha}\left(\hat{A}\right)$
$\displaystyle=\mathrm{Tr}\left[\left(\hat{v}_{\alpha}\hat{A}\hat{v}_{\alpha}-\frac{1}{2}\\{\hat{v}_{\alpha}^{2},\hat{A}\\}\right)\hat{\rho}^{\mathrm{NESS}}_{\mathrm{DQD}}\otimes\hat{\omega}_{\beta_{\alpha}}\right].$
(27)

In the NESS, the average particle current from the hot reservoir into the DQD
is given by

$\displaystyle J^{N}_{\mathrm{H}}$
$\displaystyle=\gamma_{\mathrm{H}}\left(f_{\mathrm{H}}(\epsilon_{1})-\langle\hat{c}_{1}^{\dagger}\hat{c}_{1}\rangle\right)\,,$
(28) $\displaystyle=\frac{4|t|^{2}\Delta
f\gamma_{\mathrm{H}}\gamma_{\mathrm{C}}\tilde{\gamma}}{\gamma_{\mathrm{H}}\gamma_{\mathrm{C}}\left(4\Delta\epsilon^{2}+\tilde{\gamma}^{2}\right)+4|t|^{2}\tilde{\gamma}^{2}},$
(29)

where $\Delta\epsilon=\epsilon_{2}-\epsilon_{1}$ denotes the detuning between
the two quantum dots in the DQD, $\Delta
f=f_{\mathrm{H}}(\epsilon_{1})-f_{\mathrm{C}}(\epsilon_{2})$ is the difference
in the Fermi distributions of the reservoirs evaluated at the quantum dot
energies, and $\tilde{\gamma}=(\gamma_{\rm H}+\gamma_{\rm C})$. Using Eq.
(26), we find that the work currents into the DQD in the NESS are given by

$\displaystyle\dot{W}_{\mathrm{H}}$
$\displaystyle=\left(\frac{\gamma_{\mathrm{H}}\Delta\epsilon}{\tilde{\gamma}}+\mu_{\mathrm{H}}\right)J^{N}_{\mathrm{H}},$
(30) $\displaystyle\dot{W}_{\mathrm{C}}$
$\displaystyle=\left(\frac{\gamma_{\mathrm{C}}\Delta\epsilon}{\tilde{\gamma}}-\mu_{\mathrm{C}}\right)J^{N}_{\mathrm{H}},$
(31) $\displaystyle\dot{W}_{\mathrm{tot}}$
$\displaystyle=-\left[\left(\epsilon_{1}-\mu_{\mathrm{H}}\right)-\left(\epsilon_{2}-\mu_{\mathrm{C}}\right)\right]J^{N}_{\mathrm{H}}.$
(32)

Similarly, the heat currents into the DQD in the NESS, according to Eq. (25),
are found to be

$\displaystyle\dot{Q}_{\mathrm{H}}$
$\displaystyle=\left(\epsilon_{1}-\mu_{\mathrm{H}}\right)J^{N}_{\mathrm{H}},$
(33) $\displaystyle\dot{Q}_{\mathrm{C}}$
$\displaystyle=-\left(\epsilon_{2}-\mu_{\mathrm{C}}\right)J^{N}_{\mathrm{H}},$
(34) $\displaystyle\dot{Q}_{\mathrm{tot}}$
$\displaystyle=\left[\left(\epsilon_{1}-\mu_{\mathrm{H}}\right)-\left(\epsilon_{2}-\mu_{\mathrm{C}}\right)\right]J^{N}_{\mathrm{H}}.$
(35)

Note the linear dependence of the energy current on the particle current. This
so-called tight-coupling limit, where the determinant of the Onsager response
matrix vanishes, is a consequence of the weak-coupling approximation made in
deriving the GKSL equation and does not hold in general. Nevertheless, the
tight-coupling limit is of high interest in nanoelectronics since in this
limit the system may operate at Carnot efficiency Humphrey  _et al._ (2002).

The first law states that
$\dot{W}_{\mathrm{tot}}+\dot{Q}_{\mathrm{tot}}=\langle\dot{\hat{H}}_{\mathrm{DQD}}\rangle$.
Within this framework, the expectation value of the first derivative with
respect to time of the DQD Hamiltonian is
$\langle\dot{\hat{H}}_{\mathrm{DQD}}\rangle=\sum_{\alpha}D_{\alpha}\left(\hat{H}_{\mathrm{DQD}}\right)$.
In the NESS, this quantity vanishes. Clearly,
$\dot{W}_{\mathrm{tot}}=-\dot{Q}_{\mathrm{tot}}$, so that overall our
calculation of heat and work currents into the DQD in this framework is
consistent with the first law of thermodynamics.

Moreover, if this device is operated as a heat engine, then its efficiency
$\eta$ is defined as the ratio between the time derivative of total work
performed by the system DQD and the time derivative of the heat flowing into
the DQD from the hot reservoir,

$\displaystyle\begin{split}\eta&=\frac{-\dot{W}_{\mathrm{tot}}}{\dot{Q}_{\mathrm{H}}}\,,\\\
&=1-\frac{\epsilon_{2}-\mu_{\mathrm{C}}}{\epsilon_{1}-\mu_{\mathrm{H}}}.\\\
\end{split}$ (36)

Note that the DQD operates as a heat engine, if the work current leaving the
DQD $-\dot{W}_{\mathrm{tot}}>0$. This is ensured if
$\left(f_{\mathrm{H}}(\epsilon_{1})-f_{\mathrm{C}}(\epsilon_{2})\right)>0$, so
that $J^{N}_{\mathrm{H}}>0$, and
$\left(\epsilon_{1}-\mu_{H}\right)-\left(\epsilon_{2}-\mu_{C}\right)>0$. By
the first condition,

$\frac{\left(\epsilon_{2}-\mu_{\mathrm{C}}\right)/T_{\mathrm{C}}}{\left(\epsilon_{1}-\mu_{\mathrm{H}}\right)/T_{\mathrm{H}}}>1$
(37)

and we find that $\eta$ is bounded from above by the Carnot efficiency
$\eta_{C}$, since

$\displaystyle\eta<1-\frac{T_{\mathrm{C}}}{T_{\mathrm{H}}}=\eta_{C}.$ (38)

We can finally show that this model satisfies the second law of
thermodynamics. This can be readily achieved by recognising that in the NESS,
the entropy production rate can be expressed in standard thermodynamic form

$\displaystyle\sigma$
$\displaystyle=-\sum_{\alpha\in\\{\mathrm{H,C}\\}}\beta_{\alpha}\dot{Q}_{\alpha}$
(39)
$\displaystyle=\left(\frac{\epsilon_{2}-\mu_{\mathrm{C}}}{k_{\mathrm{B}}T_{\mathrm{C}}}-\frac{\epsilon_{1}-\mu_{\mathrm{H}}}{k_{\mathrm{B}}T_{\mathrm{H}}}\right)J^{N}_{\rm
H}$ (40) $\displaystyle\geq 0,$ (41)

where $\beta_{\alpha}$ is the inverse temperature of reservoir $\alpha$. One
can easily show that it is strictly positive, thus we find that this result is
indeed consistent with the second law of thermodynamics.

##  III DQD continuously monitored by QPC

We now illustrate a setup where the occupation of one of the quantum dots in
the DQD is continuously monitored by a QPC, see Fig. (3). The goal of this
section is to recover a thermodynamically consistent picture of such a system
using the framework of repeated interactions. In this way, we consider the QPC
as performing boundary-driving on the DQD.

###  III.1 Monitored DQD and the GKSL ME

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.

The global Hamiltonian of this configuration is given by

$\hat{H}=\hat{H}_{\mathrm{DQD}}+\sum_{\alpha\in\\{H,C\\}}\hat{H}_{\alpha}+\hat{H}_{\mathrm{QPC}}+\hat{H}_{\mathrm{DQD,R}}+\hat{H}_{\mathrm{DQD,QPC}},$
(42)

where the DQD Hamiltonian $\hat{H}_{\mathrm{DQD}}$ is defined in Eq. (3), the
Hamiltonian $\hat{H}_{\alpha}$ of the $\alpha$ reservoir is defined in Eq.
(2), and the tunneling Hamiltonians describing the coupling between the DQD
and the thermal reservoirs is defined in Eq. (4). Here, the QPC is modeled as
a tunnel junction between a left and right reservoir comprised of non-
interacting fermions

$\displaystyle\hat{H}_{\mathrm{QPC}}$ $\displaystyle=\sum_{\alpha\in\\{\rm
L,R\\},k}\hat{\omega}_{\alpha,k}\hat{a}_{k}^{\dagger}\hat{a}_{k}+\sum_{k,k^{\prime}}\mathcal{T}_{k,k^{\prime}}\left(\hat{a}_{\mathrm{R},k}^{\dagger}\hat{a}_{\mathrm{L},k^{\prime}}+\hat{a}_{\mathrm{L},k^{\prime}}^{\dagger}\hat{a}_{\mathrm{R},k}\right)\,.$
(43)

In the first term, $\hat{\omega}_{\alpha,k}$ corresponds to the energies for
the left and right reservoir at mode $k$ and $\hat{a}_{\alpha,k}^{\dagger}$
($\hat{a}_{\alpha,k}$) are the respective QPC fermionic creation
(annihilation) operators. In the second term $\mathcal{T}_{k,k^{\prime}}$ is
the tunneling matrix element between states $k$ and $k^{\prime}$ in left and
right reservoir. When the DQD is occupied, then the tunneling rates in the QPC
are modulated. The interaction Hamiltonian between the quantum dot in the DQD
and the QPC is given by

$\displaystyle\hat{H}_{\mathrm{DQD,QPC}}=$
$\displaystyle\sum_{k,k^{\prime}}\chi_{k,k^{\prime}}\hat{c}_{1}^{\dagger}\hat{c}_{1}\left(\hat{a}_{\mathrm{R},k}^{\dagger}\hat{a}_{\mathrm{L},k^{\prime}}+\hat{a}_{\mathrm{L},k^{\prime}}^{\dagger}\hat{a}_{\mathrm{R},k}\right).$
(44)

where $\chi_{k,k^{\prime}}$ is the change in the tunneling amplitude between
states $k$ and $k^{\prime}$ when the first quantum dot is occupied. Now we
follow Ref. Goan  _et al._ (2001) in assuming that the chemical potentials
$\mu_{\mathrm{L}}$ and $\mu_{\mathrm{R}}$ of the left and right QPC leads,
respectively, the resulting bias $eV=\mu_{\mathrm{R}}-\mu_{\mathrm{L}}$ and
the temperature of the QPC are chosen such that
$|eV|,k_{\mathrm{B}}T_{\mathrm{QPC}}\ll\mu_{\mathrm{L}},\mu_{\mathrm{R}}$.
Moreover, we assume that both reservoirs in the QPC are at the same, albeit
low temperature $T_{\mathrm{QPC}}$. Thus, the Fermi functions in the right and
left lead of the QPC, $f_{\mathrm{R}}$ and $f_{\mathrm{L}}$, respectively,
will behave similar to step functions, and the narrow transport window, in
which $f_{\mathrm{R}}\approx 1$ and $f_{\mathrm{L}}\approx 0$, becomes sharp,
resulting in unidirectional current in the QPC. Supposing the wide band limit
for both reservoirs and the leads of the QPC and taking the Born-Markov
secular approximation Breuer (2003), the GKSL ME is then given by Goan  _et
al._ (2001)

$\displaystyle\dot{\hat{\rho}}_{\mathrm{DQD,\Gamma}}=\mathcal{L}(\hat{\rho}_{\mathrm{DQD,\Gamma}})+\mathcal{D}\left[\mathcal{T}_{+}+\chi_{+}\hat{c}_{1}^{\dagger}\hat{c}_{1}\right]\hat{\rho}_{\mathrm{DQD,\Gamma}}$
$\displaystyle+\mathcal{D}\left[\mathcal{T}_{-}^{*}+\chi_{-}^{*}\hat{c}_{1}^{\dagger}\hat{c}_{1}\right]\hat{\rho}_{\mathrm{DQD,\Gamma}}\,,$
(45)

where $\mathcal{L}(\hat{\rho}_{\mathrm{DQD,\Gamma}})$ is defined in Eq. (5),
and $\mathcal{T}_{\pm}$ and $\chi_{\pm}$ are assumed to be energy independent
tunneling rates along/against the chemical potential gradient and real Goan
_et al._ (2001). Note that

$\mathcal{D}\left[\mathcal{T}_{\pm}+\chi_{\pm}\hat{c}_{1}^{\dagger}\hat{c}_{1}\right]\hat{\rho}_{\mathrm{DQD,\Gamma}}=|\chi_{\pm}|^{2}\mathcal{D}\left[\hat{c}_{1}^{\dagger}\hat{c}_{1}\right]\hat{\rho}_{\mathrm{DQD,\Gamma}},$
(46)

which can be intuitively understood as that the evolution of the DQD density
matrix does not depend on the baseline current through the QPC, and allows us
to rewrite the master equation as

$\displaystyle\begin{split}\dot{\hat{\rho}}_{\mathrm{DQD,\Gamma}}&=\mathcal{L}(\hat{\rho}_{\mathrm{DQD,\Gamma}})+\Gamma\mathcal{D}\left[\hat{c}_{1}^{\dagger}\hat{c}_{1}\right]\hat{\rho}_{\mathrm{DQD,\Gamma}}\,,\\\
&=\mathcal{L}_{\Gamma}\left({\hat{\rho}}_{\mathrm{DQD,\Gamma}}\right)\end{split}$
(47)

where $\Gamma=|\chi_{+}|^{2}+|\chi_{-}|^{2}$ is the dephasing rate due to the
measurement. We can therefore neglect the contribution due to the QPC dynamics
given by $\mathcal{T}$ as it does not change the dynamics of the DQD.
Following Goan  _et al._ (2001), we can define an explicit expression for
$\Gamma$ in terms of the physical QPC parameters given that

$\displaystyle|\mathcal{T}_{\pm}|^{2}$
$\displaystyle=D_{\pm}=2\pi|\mathcal{T}_{00}|^{2}g_{\mathrm{L}}g_{\mathrm{R}}eV_{\pm}$
(48) $\displaystyle|\mathcal{T}_{\pm}+\chi_{\pm}|^{2}$
$\displaystyle=D_{\pm}^{\prime}=2\pi|\mathcal{T}_{00}+\chi_{00}|^{2}g_{\mathrm{L}}g_{\mathrm{R}}eV_{\pm}\,,$
(49)

where $\mathcal{T}_{00}$ and $\chi_{00}$ are the tunneling amplitudes for
energies near the chemical potentials, $g_{\mathrm{L}}$ and $g_{\mathrm{R}}$
are the energy-independent density of states for the left and right
reservoirs, and effective finite temperature external potential bias is

$eV_{\pm}=\frac{\pm eV}{1-\exp\left(\mp eV/k_{B}T_{\mathrm{QPC}}\right)}\,.$
(50)

Here, the average electron currents through the QPC when the quantum dot is
unoccupied and occupied are $eD=e(D_{+}-D_{-})$ and
$eD^{\prime}=e(D_{+}^{\prime}-D_{-}^{\prime})$ respectively. Combining these,
and assuming the tunneling rates $\mathcal{T}_{00}$ and $\chi_{00}$ are real,
then one can show that the dephasing rate
$\Gamma=(\sqrt{D}-\sqrt{D^{\prime}})^{2}$ is given by the expression

$\Gamma=2\pi
g_{\mathrm{L}}g_{\mathrm{R}}\chi_{00}^{2}eV\coth\left(\frac{eV}{2k_{B}T_{\mathrm{QPC}}}\right)\,.$
(51)

There are thus many parameters that can be varied to tune the dephasing rate
i.e the temperature $T_{\mathrm{QPC}}$, $\chi_{00}$, the density of states
$g_{\mathrm{R(L)}}$ and the chemical potential bias $eV$. Note, that even in
the limit of zero bias, $\Gamma\neq 0$ due to the fact that the tunneling rate
varies depending on the occupation of the quantum dot.

The NESS of the DQD in the presence of the QPC is uniquely defined by

$\dot{\hat{\rho}}_{\mathrm{DQD,\Gamma}}^{\mathrm{NESS}}=\mathcal{L}_{\Gamma}\left({\hat{\rho}}_{\mathrm{DQD,\Gamma}}^{\mathrm{NESS}}\right)=0,$
(52)

where the Liouvillian $\mathcal{L}_{\Gamma}$ is defined in Eq.
(LABEL:eq:_me_DQD_QPC). As studied in section (II.2), the above local GKSL ME
is not guaranteed to be thermodynamically consistent. Therefore, we now
develop an approach for the QPC analogous to that in section (II.3) for the
thermal reservoirs, which were replaced by a string of identical auxiliary
units that interact with the DQD for a finite time and are then replaced.

###  III.2 GKSL ME in the framework of repeated interactions

In order to deal with the QPC in the framework of repeated interactions, we
recast it as a single unit comprised of two qubits with energy splitting
$\Omega-\mu_{\mathrm{R(L)}}$, respectively, coupled via tunneling with
strength $\mathcal{T}$. The bare Hamiltonians of the DQD and reservoir units,
Eqs. (3) and (9), respectively, remain unchanged, while the Hamiltonian of the
QPC in Eq. (43) turns into

$\displaystyle\hat{H}_{\mathrm{QPC}}$
$\displaystyle=\left(\Omega-\mu_{\mathrm{R}}\right)\hat{a}_{\mathrm{R}}^{\dagger}\hat{a}_{\mathrm{R}}+\left(\Omega-\mu_{\mathrm{L}}\right)\hat{a}_{\mathrm{L}}^{\dagger}\hat{a}_{\mathrm{L}}+\mathcal{T}\left(\hat{a}_{\mathrm{R}}^{\dagger}\hat{a}_{\mathrm{L}}+\hat{a}_{\mathrm{L}}^{\dagger}\hat{a}_{\mathrm{R}}\right),$
(53)

where we choose $\mu_{\mathrm{R}}\geq\mu_{\mathrm{L}}$. It is assumed that at
the beginning of each interaction the global density matrix is a product state
of the individual states of all four unities: DQD, hot and cold reservoir
units and the QPC unit. The latter is also assumed to be factorized in terms
of two thermal density matrices associated with each L,R qubit (see Fig. (4)),
$\hat{\omega}_{\beta_{L}}$ and $\hat{\omega}_{\beta_{R}}$, respectively. Thus,
at the start of each interaction

$\hat{\rho}=\hat{\rho}_{\mathrm{DQD,\Gamma}}\otimes\underbrace{\hat{\omega}_{\beta_{\mathrm{H}}}\otimes\hat{\omega}_{\beta_{\mathrm{C}}}\otimes\hat{\omega}_{\beta_{R}}\otimes\hat{\omega}_{\beta_{L}}}_{\hat{\rho}_{\mathrm{E}}}.$
(54)

where $\hat{\omega}_{\beta_{\mathrm{H}}}$ and
$\hat{\omega}_{\beta_{\mathrm{C}}}$ are defined in Eq. (15) and (18),
respectively. This configuration is representative of the QPC in the limit for
which the ME was derived originally, i.e.
$|eV|,k_{\mathrm{B}}T_{\mathrm{QPC}}\ll\mu_{\mathrm{L}},\mu_{\mathrm{R}}$,
such that the energy window for transmission is narrow and the current through
the QPC is unidirectional. In this limit, and assuming that the temperature of
the QPC is low, the Fermi functions of the right and left lead are
$f_{\mathrm{R}}(\Omega)=1$ and $f_{\mathrm{L}}(\Omega)=0$, respectively, so
that we define

$\displaystyle\hat{\omega}_{\beta_{\mathrm{R}}}$
$\displaystyle=\left(\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right),$ (57)
$\displaystyle\hat{\omega}_{\beta_{\mathrm{L}}}$
$\displaystyle=\left(\begin{array}[]{cc}0&0\\\ 0&1\end{array}\right).$ (60)

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.

Since we assume that the QPC unit couples only to the first quantum dot with
coupling strength $\sqrt{\Gamma}$, we define the (single) interaction between
the monitored dot and the QPC via

$\displaystyle\hat{v}_{\mathrm{QPC}}$
$\displaystyle=\sqrt{\Gamma}\hat{c}_{1}^{\dagger}\hat{c}_{1}\left(\hat{a}_{\mathrm{R}}^{\dagger}\hat{a}_{\mathrm{L}}+\hat{a}_{\mathrm{L}}^{\dagger}\hat{a}_{\mathrm{R}}\right),$
(61)

and the interactions with the hot and cold reservoir, $\hat{v}_{\mathrm{H}}$
and $\hat{v}_{\mathrm{C}}$, as defined in Eqs. (10) and (11), respectively.

We are now in a position to study the repeated interactions of the DQD by
following the recipe outlined in Sec. (II.3) but including the QPC. The
formalism developed in Ref. Barra (2015) assumes that the density matrices of
the units are thermal. While the density matrices describing the left and
right side of the QPC, $\hat{\omega}_{\beta_{\mathrm{R(L)}}}$, are thermal
with respect to their respective bare Hamiltonian
$\hat{H}_{\mathrm{QPC,R(L)}}=\left(\Omega-\mu_{\mathrm{R(L)}}\right)\hat{a}_{\mathrm{R(L)}}^{\dagger}\hat{a}_{\mathrm{R(L)}}$,
the total QPC density matrix
$\hat{\rho}_{\mathrm{QPC}}=\hat{\omega}_{\beta_{\mathrm{R}}}\otimes\hat{\omega}_{\beta_{\mathrm{L}}}$
is not thermal with respect to $\hat{H}_{\mathrm{QPC}}$. This can be easily
seen as $\left[\hat{H}_{\mathrm{QPC}},\hat{\rho}_{\mathrm{QPC}}\right]\neq 0$,
because of the hopping interaction of the two thermal qubits. Nevertheless,
Eqs. (22,25, 26) can be analogously derived also for the interaction between
the QPC unit proposed here and the DQD. One can thus easily obtain the GKSL ME
for the DQD via

$\dot{\hat{\rho}}_{\mathrm{DQD,\Gamma}}=-i\left[\hat{H}_{\mathrm{DQD}},\hat{\rho}_{\mathrm{DQD,\Gamma}}\right]+\sum_{\alpha\in\\{\mathrm{H},\mathrm{C},\mathrm{QPC}\\}}\mathcal{D}_{\alpha}\left(\hat{\rho}_{\mathrm{DQD,\Gamma}}\right),$
(63)

where $\mathcal{D}_{\alpha}\left(\hat{\rho}_{\mathrm{DQD,\Gamma}}\right)$ is
defined in Eq. (23). In fact, one can readily show that this recovers Eq.
(LABEL:eq:_me_DQD_QPC).

###  III.3 Thermodynamics of the boundary-driven monitored DQD in the
framework of repeated interactions

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$k_{\mathrm{B}}T$.

We now study the influence of continuous measurement via the QPC on the energy
currents into the DQD as well as the entropy production rate. The first and
second law of thermodynamics can be validated, and one can analyse the
efficiency of the continuously monitored DQD operated as a heat engine.

Firstly, we confirm that this model is thermodynamically consistent with the
first law. Using the formalism described in section (II.4) we find the heat
and work currents —Eqs. (64) and (65) respectively— are given by

$\displaystyle\begin{split}\dot{Q}_{\mathrm{QPC}}=&-eV\Gamma\langle\hat{c}_{1}^{\dagger}\hat{c}_{1}\rangle,\\\
\dot{Q}_{\mathrm{H}}=&\left(\epsilon_{1}-\mu_{H}\right)J^{N}_{\mathrm{H,\Gamma}},\\\
\dot{Q}_{\mathrm{C}}=&-\left(\epsilon_{2}-\mu_{C}\right)J^{N}_{\mathrm{H,\Gamma}},\\\
\end{split}$ (64)

and

$\displaystyle\begin{split}\dot{W}_{\mathrm{QPC}}=&\frac{\Gamma\Delta\epsilon}{\Gamma+\tilde{\gamma}}J^{N}_{\mathrm{H,\Gamma}}+eV\Gamma\langle\hat{c}_{1}^{\dagger}\hat{c}_{1}\rangle,\\\
\dot{W}_{\mathrm{H}}=&\left(\frac{\gamma_{\mathrm{H}}\Delta\epsilon}{\Gamma\tilde{\gamma}}+\mu_{H}\right)J^{N}_{\mathrm{H,\Gamma}},\\\
\dot{W}_{\mathrm{C}}=&\left(\frac{\gamma_{\mathrm{C}}\Delta\epsilon}{\Gamma+\gamma_{\mathrm{C}}+\gamma_{\mathrm{H}}}-\mu_{C}\right)J^{N}_{\mathrm{H,\Gamma}},\\\
\end{split}$ (65)

where $eV=\mu_{\mathrm{R}}-\mu_{\mathrm{L}}$ and the particle current is given
by

$\displaystyle J^{N}_{\mathrm{H,\Gamma}}$
$\displaystyle=\gamma_{\mathrm{H}}\left(f_{\mathrm{H}}(\epsilon_{1})-\langle\hat{c}_{1}^{\dagger}\hat{c}_{1}\rangle\right),$
(66) $\displaystyle=\frac{4\Delta
f\gamma_{\mathrm{C}}\gamma_{\mathrm{H}}\left(\Gamma+\tilde{\gamma}\right)|t|^{2}}{\tilde{\gamma}\left(4\Delta\epsilon^{2}+\left(\Gamma+\tilde{\gamma}\right)^{2}\right)+4\tilde{\gamma}\left(\Gamma+\tilde{\gamma}\right)|t|^{2}}.$
(67)

Here, we have used the $\Gamma$ subscript to indicate the presence of the QPC.
We can readily show that our results are consistent with the first law of
thermodynamics for the NESS since

$\displaystyle\begin{split}\langle\dot{\hat{H}}_{\mathrm{DQD}}\rangle_{\mathrm{tot}}&=\sum_{\alpha=\mathrm{QPC,H,C}}\dot{W}_{\mathrm{\alpha}}+\dot{Q}_{\mathrm{\alpha}}=0\,.\end{split}$
(68)

The QPC’s work and heat current can be divided into two contributions: one
associated with the particle current through the DQD,
$J^{N}_{\mathrm{H,\Gamma}}$, and the other associated with the change in
particle current through the QPC as a function of the monitored dot
occupation, $\Delta
J^{N}_{\mathrm{QPC}}=\Gamma\langle\hat{c}_{1}^{\dagger}\hat{c}_{1}\rangle$, so
that

$\displaystyle\dot{Q}_{\mathrm{QPC}}=$ $\displaystyle-eV\Delta
J^{N}_{\mathrm{QPC}},$ (69) $\displaystyle\dot{W}_{\mathrm{QPC}}=$
$\displaystyle\frac{\Gamma\Delta\epsilon}{\Gamma+\tilde{\gamma}}J^{N}_{\mathrm{H,\Gamma}}+eV\Delta
J^{N}_{\mathrm{QPC}}$ (70) $\displaystyle=$
$\displaystyle\dot{W}^{\mathrm{DQD}}_{\mathrm{QPC}}+\dot{W}^{\mathrm{QPC}}_{\mathrm{QPC}}.$
(71)

Note that the energy current associated with $\Delta J^{N}_{\mathrm{QPC}}$
follows Watt’s law.

Here we are interested in the influence of continuous measurement on the
thermodynamics of the DQD operated as a heat engine, in which the particle
current through the DQD, $J^{N}_{\mathrm{H,\Gamma}}$, is the carrier of
energy. Thus, we argue that the heat and work currents due to the change in
the QPC particle current, $\Delta J^{N}_{\mathrm{QPC}}$, that, importantly, is
contained within the QPC unit, must not be taken into account for the
efficiency of the DQD heat engine and the entropic cost of driving its
particle current.

We now study the heat and work current related to the particle current
throught the DQD as as a function of the measurement strength $\Gamma$, which
quantifies the strength of the dephasing backaction. To this aim, we define

$\dot{W}_{\mathrm{tot}}=\dot{W}_{\mathrm{H}}+\dot{W}_{\mathrm{C}}+\dot{W}^{\mathrm{DQD}}_{\mathrm{QPC}}.$
(72)

We find that all thermodynamic currents are either reduced or assisted by the
dephasing due to the measurement-induced backaction, as depicted in Fig. (5).
Moreover, we notice that the total output power $-\dot{W}_{\mathrm{tot}}$ is
extremal at $\Gamma_{\rm ext}$, which we will revisit shortly. It suffices to
say that because we are operating the DQD heat engine in the tight-coupling
limit, the non-trivial dependence on $\Gamma$ can be traced back to the the
particle current Eq. (66).

This raises the question whether there is any advantage in the efficiency of
this heat engine due to $\Gamma$. As argued above, one can define the
efficiency of the engine as in Eq. (36) taking into account energy currents
associated with the particle current through the DQD. In the limit of tight
coupling, the efficiency is independent of the dephasing strength—but this may
not hold if the weak-coupling assumption is relaxed.

Lastly, the entropy production rate associated with the thermodynamic cost of
driving the NESS particle current through the DQD is given by

$\displaystyle\begin{split}\sigma_{\mathrm{DQD}}=&-\left(\beta_{H}\dot{Q}_{\mathrm{H}}+\beta_{C}\dot{Q}_{\mathrm{C}}\right),\\\
=&\left(\frac{\epsilon_{1}-\mu_{\mathrm{H}}}{k_{\mathrm{B}}T_{\mathrm{H}}}-\frac{\epsilon_{2}-\mu_{\mathrm{C}}}{k_{\mathrm{B}}T_{\mathrm{C}}}\right)J^{N}_{\rm
H,\Gamma}\geq 0.\end{split}$ (73)

The total entropy production rate
$\sigma_{\mathrm{tot}}=\sigma_{\mathrm{DQD}}+\sigma_{\mathrm{QPC}}$ is in
agreement with the second law of thermodynamics since

$\displaystyle\begin{split}\sigma_{\mathrm{QPC}}=&-\beta_{\mathrm{QPC}}\dot{Q}_{\mathrm{QPC}},\\\
=&\beta_{\mathrm{QPC}}eVJ^{N}_{\mathrm{QPC}}\geq 0.\end{split}$ (74)

###  III.4 Dephasing enhanced particle transport

Figure 6: Quantum dot occupations and coherences as a function of the
measurement strength. (a) The single quantum dot occupations
$n_{1(2)}=\langle\hat{c}_{1(2)}^{\dagger}\hat{c}_{1(2)}\rangle$ exhibit an
extremum at the measurement strength $\Gamma_{\mathrm{ext}}$ and return to
their original value in the absence of measurement at measurement strength
$\Gamma_{0}$, as indicated by the dashed grey lines and the grey box. (b) At
$\Gamma_{\mathrm{ext}}$ the real and imaginary part of the coherence
$\langle\hat{c}_{1}^{\dagger}\hat{c}_{2}\rangle$ coincide and
$\mathrm{Im}\langle\hat{c}_{1}^{\dagger}\hat{c}_{2}\rangle$, which in the NESS
is proportional to the average particle current, exhibits an extremum.
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$k_{\mathrm{B}}T$.

In the previous section we observed a non-trivial behaviour of the particle
current $J^{N}_{\rm H,\Gamma}$ as a function of the measurement strength
$\Gamma$. The measurement introduces dephasing in the DQD which typically
suppresses the coherences in the DQD density matrix Levinson (1997). Thus, for
large measurement strength we expect the particle current to vanish due to the
quantum Zeno effect Itano  _et al._ (1990) leading to localization. For a
significant range in measurement strength that depends on the system’s
parameters as well as the reservoir couplings we find, however, dephasing
enhanced particle transport Contreras-Pulido  _et al._ (2014); Plenio and
Huelga (2008); Rebentrost  _et al._ (2009a); Lacerda  _et al._ (2021);
Mendoza-Arenas  _et al._ (2013); Engel  _et al._ (2007); Collini  _et al._
(2010). This can be understood by studying the particle current in the NESS.
We start from the Heisenberg equations of motion for the single quantum dot
occupations, $n_{1(2)}=\langle\hat{c}_{1(2)}^{\dagger}\hat{c}_{1(2)}\rangle$,
and coherence $\langle\hat{c}_{1}^{\dagger}\hat{c}_{2}\rangle$

$\displaystyle\begin{split}\frac{\mathrm{d}\langle\hat{c}_{1}^{\dagger}\hat{c}_{1}\rangle}{\mathrm{d}t}&=\gamma_{\mathrm{H}}\left(f_{\mathrm{H}}-\langle\hat{c}_{1}^{\dagger}\hat{c}_{1}\rangle\right)+i\left(t\langle\hat{c}_{1}^{\dagger}\hat{c}_{2}\rangle+t^{*}\langle\hat{c}_{2}^{\dagger}\hat{c}_{1}\rangle\right),\\\
&=J^{N}_{\mathrm{H,\Gamma}}+J^{N}_{\mathrm{12}}\,,\end{split}$ (75)
$\displaystyle\begin{split}\frac{\mathrm{d}\langle\hat{c}_{1}^{\dagger}\hat{c}_{2}\rangle}{\mathrm{d}t}&=\left[i\Delta\epsilon-\frac{\tilde{\gamma}+\Gamma}{2}\right]\langle\hat{c}_{1}^{\dagger}\hat{c}_{2}\rangle+it^{*}\left(\langle\hat{c}_{2}^{\dagger}\hat{c}_{2}\rangle+\langle\hat{c}_{1}^{\dagger}\hat{c}_{1}\rangle\right),\end{split}$
(76)
$\displaystyle\begin{split}\frac{\mathrm{d}\langle\hat{c}_{2}^{\dagger}\hat{c}_{1}\rangle}{\mathrm{d}t}&=\left[-i\Delta\epsilon-\frac{\tilde{\gamma}+\Gamma}{2}\right]\langle\hat{c}_{2}^{\dagger}\hat{c}_{1}\rangle-
it\left(\langle\hat{c}_{2}^{\dagger}\hat{c}_{2}\rangle+\langle\hat{c}_{1}^{\dagger}\hat{c}_{1}\rangle\right),\end{split}$
(77)
$\displaystyle\begin{split}\frac{\mathrm{d}\langle\hat{c}_{2}^{\dagger}\hat{c}_{2}\rangle}{\mathrm{d}t}&=\gamma_{\mathrm{C}}\left(f_{\mathrm{C}}-\langle\hat{c}_{2}^{\dagger}\hat{c}_{2}\rangle\right)-i\left(t\langle\hat{c}_{1}^{\dagger}\hat{c}_{2}\rangle+t^{*}\langle\hat{c}_{2}^{\dagger}\hat{c}_{1}\rangle\right)\,,\\\
&=J^{N}_{\mathrm{C,\Gamma}}-J^{N}_{\mathrm{12}},\end{split}$ (78)

where $J^{N}_{\mathrm{H/C,\Gamma}}$ denotes the NESS average particle current
flowing into the DQD from the hot/cold side and $J^{N}_{\mathrm{12}}$ denotes
the NESS average particle current from the first to the second quantum dot in
the DQD. In the NESS we set all the time derivatives to zero and clearly see
$J^{N}_{\mathrm{H,\Gamma}}=-J^{N}_{\mathrm{C,\Gamma}}=-J^{N}_{\mathrm{12}}$
and therefore in the following we restrict to the discussion of
$J^{N}_{\mathrm{H,\Gamma}}$. The effect of the measurement strength $\Gamma$
on the single quantum dot occupations and the coherence are shown in Fig. (6).
Note that while the single quantum dot occupations $n_{1(2)}$ depend on
$\Gamma$, the total DQD occupation $n=n_{1}+n_{2}$ is conserved. This can be
understood intuitively: the QPC performs work on the DQD as seen in Fig. (5).
As a result, the populations of eigenstates of the DQD Hamiltonian
$H_{\mathrm{DQD}}$ with total occupation $n=1$ are redistributed among the two
single quantum dot eigenstates. Since the QPC does not exchange particles with
the DQD, the interaction conserves the total occupation $n$. Solving the above
system of equations, we find that the single quantum dot occupations $n_{1}$
and $n_{2}$ are extremal at the measurement strength

$\Gamma_{\mathrm{ext}}=2|\Delta\epsilon|-\tilde{\gamma}.$ (79)

Thus, for any choice of $\Delta\epsilon$ and $\tilde{\gamma}$ for which
$\Gamma_{\mathrm{ext}}>0$ the particle current can be enhanced by increasing
the measurement strength. Further, we find that the measurement strength at
which the dot occupations match those in the absence of measurement, i.e.
$n_{1/2}\left(\Gamma_{0}\right)=n_{1/2}\left(0\right)$, is given by

$\Gamma_{0}=\frac{4\Delta\epsilon^{2}}{\tilde{\gamma}}-\tilde{\gamma},$ (80)

meaning that the particle current is assisted up to $\Gamma_{0}$.
Consequently, the particle current from the hot reservoir into the DQD
exhibits a maximum at $\Gamma_{\mathrm{ext}}$ and is enhanced up to a
measurement strength $\Gamma_{0}$, as shown in Fig. (5). Hence, a boost in the
particle current can obtained via continuous monitoring of the occupation of
the first quantum dot.

###  III.5 Thermodynamic uncertainty relation

Figure 7: Relative fluctuations and the thermodynamic uncertainty relation
ratio (TURR) of the particle current as a function of the measurement
strength. (a) Dephasing due to measurement-induced backaction can reduce the
relative fluctuations of the particle current up to a measurement stength of
$\Gamma^{\prime}_{0}$. The relative fluctuations are minimal at measurement
strength $\Gamma^{\prime}_{\mathrm{ext}}$. (b) Similarly, the TURR expressing
the trade-off between the entropic cost and the relative fluctuations of the
particle current is minimal at $\Gamma^{\prime\prime}_{\mathrm{ext}}$ and is
reduced up to $\Gamma^{\prime\prime}_{0}$, which is significantly larger than
$\Gamma_{0}$ and $\Gamma^{\prime}_{0}$. 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$k_{\mathrm{B}}T$.

As a final consideration, we study how TURs Barato and Seifert (2015);
Pietzonka and Seifert (2018); Pietzonka  _et al._ (2017, 2016); Horowitz and
Gingrich (2017) are impacted by the interplay between the average particle
current $J^{N}_{\rm H,\Gamma}$ and the measurement strength $\Gamma$. These
relations describe a trade-off between the relative fluctuations of a
fluctuating thermodynamic current in classical Markovian NESS heat engines and
the entropic cost associated with their driving, and are commonly expressed as

$\frac{D\sigma}{J^{2}}\geq 2,$ (81)

where $J$ is the average of the fluctuating current, $D$ is its diffusion
coefficient and $\sigma$ denotes the entropy production rate. In the following
we will refer to $D\sigma/J^{2}$ as the TUR ratio (TURR). Eq. (81) strictly
holds only for classical systems, and has been shown to be violated in some
quantum systems Kalaee  _et al._ (2021); Cangemi  _et al._ (2020); Agarwalla
and Segal (2018); Ptaszyński (2018); Agarwalla and Segal (2018); Liu and Segal
(2019); Prech  _et al._ (2022). It should be noted, however, that the role of
quantum coherence (as is conjectured to be responsible for such violations)
can be either constructive or destructive for the fluctuations of
thermodynamic currents, depending on the context Rignon-Bret  _et al._ (2021).

Here, we numerically study the TUR for the particle current
$J^{N}_{\mathrm{H,\Gamma}}$ where the diffusion coefficient $D$ can be
calculated using well known methods Kewming  _et al._ (2022); Esposito  _et
al._ (2009); Schaller (2014); Wiseman and Milburn (2009) using the power
spectrum $S(\omega)$, which often is more accessible experimentally because it
describes the amount of power in each frequency component of the measured
signal, evaluated at zero frequency $D=S(\omega=0)$. It is important to note
that we are not computing the power spectrum with respect to the measured
signal in the QPC, but rather relating to the counting statistics between the
system and the hot reservoir, i.e in the particle current
$J^{N}_{\mathrm{H,\Gamma}}$. Analytically, it can be expressed in a compact
vector notation Kewming  _et al._ (2022)

$D=S(0)=M-2\bra{1}\mathcal{L}_{1}\mathcal{L}^{+}\mathcal{L}_{1}\ket{\hat{\rho}},$
(82)

where $\bra{1}$ is the vectorised identity, $\ket{\hat{\rho}}$ is the
vectorised steady-state density matrix and

$\mathcal{L}_{1}\hat{\rho}=\sum_{k}\mu_{k}\hat{L}_{k}\hat{\rho}\hat{L}_{k}^{\dagger}.$
(83)

The operator $\hat{L}_{k}$ is the measurement operator with which the system’s
density matrix is updated conditioned on a click in the detector with
associated measurement outcome $\mu_{k}$. $\mathcal{L}^{+}$ is the Drazin
inverse of the Lindbladian $\mathcal{L}$ of the GKSL ME and is defined as

$\mathcal{L}^{+}=\sum_{j}\frac{1}{\lambda_{j}}\ket{x_{j}}\bra{y_{j}}.$ (84)

Here, $\lambda_{j}$, $\ket{x_{j}}$ and $\bra{y_{j}}$ are the non-zero
eigenvalues, the right eigenvectors and the left eigenvectors of $\mathcal{L}$
in matrix form, respectively. Finally, the dynamical activity $M$ is defined
as

$M=\sum_{k}\mu_{k}^{2}\mathrm{Tr}\left[\hat{L}_{k}\hat{\rho}\hat{L}_{k}^{\dagger}\right].$
(85)

Here, we numerically compute the diffusion coefficient of the fluctuating
particle current between the DQD and the hot reservoir. To this end we choose
the two measurement operators
$\hat{L}_{k}\in\left(\sqrt{\gamma_{\mathrm{H}}f_{\mathrm{H}}(\epsilon_{1})}\hat{c}^{\dagger}_{\mathrm{1}},\sqrt{\gamma_{\mathrm{H}}\left(1-f_{\mathrm{H}}(\epsilon_{1})\right)}\hat{c}_{\mathrm{1}}\right)$
with their associated measurement outcomes $\mu_{k}\in\left(+1,-1\right)$ to
represent the detection of single particle hopping into and out of the DQD
from the hot reservoir, respectively. Again, we note this does not correspond
to the statistics of the observed current measured by the QPC, but rather the
counting statistics between the system and the hot reservoir. We find that the
relative fluctuations of the particle current, given by
$D/{\left(J^{N}_{\mathrm{H,\Gamma}}\right)}^{2}$, can be reduced up to a
certain measurement strength $\Gamma^{\prime}_{0}$, translating to a
stabilization of the particle current, as shown in Fig. (7), given that the
parameters describing the system and its coupling to the environment are
chosen appropriately. Note, that since we compute $D$ only numerically, we
cannot provide analytical expressions for the measurement strength
$\Gamma^{\prime}_{\mathrm{ext}}$ at which the relative fluctuations are
minimal and the measurement strength $\Gamma^{\prime}_{0}$ at which they
return to their value in the absence of measurement.

Furthermore, we find that the TURR
$D\sigma_{\mathrm{DQD}}/{J^{N}_{\mathrm{H,\Gamma}}}^{2}$ of the particle
current can be pushed towards its lower bound for arbitrarily weak measurement
strength $\Gamma$, as depicted in Fig. (7), in certain parameter regimes. Our
results suggest that the TURR may be reduced by the dephasing measurement
backaction even beyond the regime of dephasing assisted transport, i.e. for
$\Gamma>\Gamma_{0}$. Further, note that here the TURR changes at constant
engine efficiency. This, however, certainly is a pathological effect arising
due to tight-coupling between the energy and particle current.

##  IV Discussion

When considering quantum heat engines, fluctuations in thermodynamic currents
can be as important as their respective mean values. Increasing precision by
reducing fluctuations of a current on small scales is of great interest, also
from a fundamental thermodynamics point of view, as the fluctuations of the
current are closely related to the entropic cost of driving it, as expressed
in TURs. Here, we investigated the influence of continuous measurement via a
QPC on the relative fluctuations of a thermodynamically driven NESS particle
current and its entropic cost in a minimalist model that exhibits quantum
coherence: the DQD in a two-terminal setup. We show that the local GSKL ME
describing the non-unitary evolution of the DQD in the presence of
interactions with two thermal reservoirs, and under continuous measurement via
a QPC, can be derived in the framework of repeated interactions. This holds in
the limit where the Born-Markov secular approximation is valid and particle
transport in the QPC is unidirectional. We showed that under this framework
the first and second law of thermodynamics are satisfied and that the
efficiency of the DQD operating as a heat engine is bounded by the Carnot
efficiency. We argued here that the entropy production rate associated with
driving the particle current through the DQD can be computed from the heat
currents from the thermal reservoirs and their respective temperatures alone.
This is noteworthy because this entropy production rate is also consistent
with the second law of thermodynamics, but most importantly is independent of
the QPC temperature, which is often assumed to be near zero. In this limit,
the entropy production rate in the QPC diverges. The question here is thus how
the entropy production in a more realistic model of the measurement device can
be adequately taken into account in the thermodynamic description of the
continuously monitored system — and the question remains open.

Finally, using the GKSL ME to compute the NESS particle current, we find
dephasing assisted particle transport accompanied by a reduction in relative
fluctuations up to a certain measurement strength. This may mean that the cost
of driving a particle current given its signal to noise ratio can be reduced
by measurement-induced dephasing. In this way, the entropic cost of the
precision of the particle flux, which is often more accessible experimentally
than any energy flux, may be reduced, while the lower bound set by the TUR
remains intact. Our model, although minimalist, reproduces the commonly used
local GKSL ME for this setup while allowing the calculation of energy fluxes
into the DQD in a thermodynamically consistent manner. Thus, it may serve as a
starting point for more sophisticated models that will have a higher level of
detail in the future, such as realistic spectral densities for the reservoirs,
as is possible, for example, within the mesoscopic leads approach Lacerda  _et
al._ (2022); Brenes  _et al._ (2020, 2022) or using more realistic collision
based models Purkayastha  _et al._ (2021, 2022); Wojtowicz  _et al._ (2022).

##  V Acknowledgements

The authors acknowledge Mark Mitchison and Krissia Zawadzki for fruitful
discussions and feedback. LPB and JG acknowledge SFI for support through the
Frontiers for the Future project. MJK acknowledges the financial support from
a Marie Skłodwoska-Curie Fellowship (Grant No. 101065974). JG is funded by a
Science Foundation Ireland-Royal Society University Research Fellowship. This
work was also supported by the European Research Council Starting Grant
ODYSSEY (Grant Agreement No. 758403).

## References

  * Benenti  _et al._ (2017) G. Benenti, G. Casati, K. Saito, and R. S. Whitney, Physics Reports Fundamental aspects of steady-state conversion of heat to work at the nanoscale, 694, 1 (2017).
  * Myers  _et al._ (2022) N. M. Myers, O. Abah, and S. Deffner, AVS Quantum Science 4, 027101 (2022).
  * Pekola and Karimi (2021) J. P. Pekola and B. Karimi, Reviews of Modern Physics 93, 041001 (2021).
  * Pekola (2015) J. P. Pekola, Nature Physics 11, 118 (2015).
  * Arrachea (2022) L. Arrachea, “Energy dynamics, heat production and heat-work conversion with qubits: towards the development of quantum machines,”  (2022).
  * Dubi and Di Ventra (2011) Y. Dubi and M. Di Ventra, Reviews of Modern Physics 83, 131 (2011).
  * Josefsson  _et al._ (2018) M. Josefsson, A. Svilans, A. M. Burke, E. A. Hoffmann, S. Fahlvik, C. Thelander, M. Leijnse, and H. Linke, Nature Nanotechnology 13, 920 (2018).
  * Esposito  _et al._ (2009) M. Esposito, U. Harbola, and S. Mukamel, Reviews of Modern Physics 81, 1665 (2009).
  * Campisi (2014) M. Campisi, Journal of Physics A: Mathematical and Theoretical 47, 245001 (2014).
  * Levinson (1997) Y. Levinson, Europhysics Letters (EPL) 39, 299 (1997).
  * Aleiner  _et al._ (1997) I. L. Aleiner, N. S. Wingreen, and Y. Meir, Physical Review Letters 79, 3740 (1997).
  * Stodolsky (1999) L. Stodolsky, Physics Letters B 459, 193 (1999).
  * Büttiker and Martin (2000) M. Büttiker and A. M. Martin, Physical Review B 61, 2737 (2000).
  * Bhandari and Jordan (2022) B. Bhandari and A. N. Jordan, Physical Review Research 4, 033103 (2022).
  * Horowitz (2012) J. M. Horowitz, Phys. Rev. E 85, 031110 (2012).
  * Manzano  _et al._ (2015) G. Manzano, J. M. Horowitz, and J. M. R. Parrondo, Phys. Rev. E 92, 032129 (2015).
  * Campisi  _et al._ (2015) M. Campisi, J. Pekola, and R. Fazio, New Journal of Physics 17, 035012 (2015).
  * Elouard  _et al._ (2017a) C. Elouard, D. A. Herrera-Martí, M. Clusel, and A. Auffèves, npj Quantum Information 3, 1 (2017a).
  * Elouard  _et al._ (2017b) C. Elouard, N. K. Bernardes, A. R. R. Carvalho, M. F. Santos, and A. Auffèves, New Journal of Physics 19, 103011 (2017b).
  * Manzano  _et al._ (2018) G. Manzano, J. M. Horowitz, and J. M. R. Parrondo, Phys. Rev. X 8, 031037 (2018).
  * Di Stefano  _et al._ (2018) P. G. Di Stefano, J. J. Alonso, E. Lutz, G. Falci, and M. Paternostro, Physical Review B 98, 144514 (2018).
  * Strasberg (2019) P. Strasberg, Physical Review E 100, 022127 (2019).
  * Alonso  _et al._ (2016) J. J. Alonso, E. Lutz, and A. Romito, Phys. Rev. Lett. 116, 080403 (2016).
  * Belenchia  _et al._ (2020) A. Belenchia, L. Mancino, G. T. Landi, and M. Paternostro, npj Quantum Information 6, 1 (2020).
  * Rossi  _et al._ (2020) M. Rossi, L. Mancino, G. T. Landi, M. Paternostro, A. Schliesser, and A. Belenchia, Phys. Rev. Lett. 125, 080601 (2020).
  * Menczel  _et al._ (2020) P. Menczel, C. Flindt, and K. Brandner, Phys. Rev. Research 2, 033449 (2020).
  * Miller  _et al._ (2020) H. J. Miller, G. Guarnieri, M. T. Mitchison, and J. Goold, Physical Review Letters 125, 160602 (2020).
  * Liu and Su (2020) F. Liu and S. Su, Phys. Rev. E 101, 062144 (2020).
  * Kewming and Shrapnel (2022) M. J. Kewming and S. Shrapnel, Quantum 6, 685 (2022).
  * Manzano and Zambrini (2022) G. Manzano and R. Zambrini, AVS Quantum Science 4, 025302 (2022).
  * Fedichkin and Fedorov (2004) L. Fedichkin and A. Fedorov, Physical Review A 69, 032311 (2004).
  * Fedichkin and Fedorov (2005) L. Fedichkin and A. Fedorov, IEEE Transactions on Nanotechnology 4, 65 (2005).
  * Itakura and Tokura (2003) T. Itakura and Y. Tokura, Physical Review B 67, 195320 (2003).
  * Fujisawa  _et al._ (2006) T. Fujisawa, T. Hayashi, R. Tomita, and Y. Hirayama, Science 312, 1634 (2006).
  * Ihn  _et al._ (2009) T. Ihn, S. Gustavsson, U. Gasser, B. Küng, T. Müller, R. Schleser, M. Sigrist, I. Shorubalko, R. Leturcq, and K. Ensslin, Solid State Communications Fundamental Phenomena and Applications of Quantum Dots, 149, 1419 (2009).
  * Gurvitz (1997) S. A. Gurvitz, Physical Review B 56, 15215 (1997).
  * Contreras-Pulido  _et al._ (2014) L. D. Contreras-Pulido, M. Bruderer, S. F. Huelga, and M. B. Plenio, New Journal of Physics 16, 113061 (2014).
  * Plenio and Huelga (2008) M. B. Plenio and S. F. Huelga, New Journal of Physics 10, 113019 (2008).
  * Rebentrost  _et al._ (2009a) P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, and A. Aspuru-Guzik, New Journal of Physics 11, 033003 (2009a).
  * Scholes and Smyth (2014) G. D. Scholes and C. Smyth, The Journal of Chemical Physics 140, 110901 (2014).
  * Zerah-Harush and Dubi (2020) E. Zerah-Harush and Y. Dubi, Physical Review Research 2, 023294 (2020).
  * Kilgour and Segal (2015) M. Kilgour and D. Segal, The Journal of Chemical Physics 143, 024111 (2015).
  * Kilgour and Segal (2016) M. Kilgour and D. Segal, The Journal of Chemical Physics 144, 124107 (2016).
  * Sowa  _et al._ (2017) J. K. Sowa, J. A. Mol, G. A. D. Briggs, and E. M. Gauger, Physical Chemistry Chemical Physics 19, 29534 (2017).
  * Lacerda  _et al._ (2021) A. M. Lacerda, J. Goold, and G. T. Landi, Physical Review B 104, 174203 (2021).
  * Mendoza-Arenas  _et al._ (2013) J. J. Mendoza-Arenas, T. Grujic, D. Jaksch, and S. R. Clark, Physical Review B 87, 235130 (2013).
  * Engel  _et al._ (2007) G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, Nature 446, 782 (2007).
  * Collini  _et al._ (2010) E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes, Nature 463, 644 (2010).
  * Žnidarič  _et al._ (2017) M. Žnidarič, J. J. Mendoza-Arenas, S. R. Clark, and J. Goold, Annalen der Physik 529, 1600298 (2017).
  * Chiaracane  _et al._ (2022) C. Chiaracane, A. Purkayastha, M. T. Mitchison, and J. Goold, Physical Review B 105, 134203 (2022).
  * Caruso  _et al._ (2009) F. Caruso, A. W. Chin, A. Datta, S. F. Huelga, and M. B. Plenio, The Journal of Chemical Physics 131, 105106 (2009).
  * Caruso  _et al._ (2016) F. Caruso, A. Crespi, A. G. Ciriolo, F. Sciarrino, and R. Osellame, Nature Communications 7, 11682 (2016).
  * Chin  _et al._ (2012) A. W. Chin, S. F. Huelga, and M. B. Plenio, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, 3638 (2012).
  * Collini (2013) E. Collini, Chemical Society Reviews 42, 4932 (2013).
  * Biggerstaff  _et al._ (2016) D. N. Biggerstaff, R. Heilmann, A. A. Zecevik, M. Gräfe, M. A. Broome, A. Fedrizzi, S. Nolte, A. Szameit, A. G. White, and I. Kassal, Nature Communications 7, 11282 (2016).
  * Faisal  _et al._ (2008) A. A. Faisal, L. P. J. Selen, and D. M. Wolpert, Nature Reviews Neuroscience 9, 292 (2008).
  * Adolphs and Renger (2006) J. Adolphs and T. Renger, Biophysical Journal 91, 2778 (2006).
  * Gaab and Bardeen (2004) K. M. Gaab and C. J. Bardeen, The Journal of Chemical Physics 121, 7813 (2004).
  * Mohseni  _et al._ (2008) M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, The Journal of Chemical Physics 129, 174106 (2008).
  * Rebentrost  _et al._ (2009b) P. Rebentrost, M. Mohseni, and A. Aspuru-Guzik, The Journal of Physical Chemistry B 113, 9942 (2009b).
  * Lindblad (1976) G. Lindblad, Communications in Mathematical Physics 48, 119 (1976).
  * Gorini  _et al._ (1976) V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Journal of Mathematical Physics 17, 821 (1976).
  * Cattaneo  _et al._ (2019) M. Cattaneo, G. L. Giorgi, S. Maniscalco, and R. Zambrini, New Journal of Physics 21, 113045 (2019).
  * González  _et al._ (2017) J. O. González, L. A. Correa, G. Nocerino, J. P. Palao, D. Alonso, and G. Adesso, Open Systems & Information Dynamics 24, 1740010 (2017).
  * Hofer  _et al._ (2017) P. P. Hofer, M. Perarnau-Llobet, L. D. M. Miranda, G. Haack, R. Silva, J. B. Brask, and N. Brunner, New Journal of Physics 19, 123037 (2017).
  * Levy and Kosloff (2014) A. Levy and R. Kosloff, EPL (Europhysics Letters) 107, 20004 (2014).
  * Chiara  _et al._ (2018) G. D. Chiara, G. Landi, A. Hewgill, B. Reid, A. Ferraro, A. J. Roncaglia, and M. Antezza, New Journal of Physics 20, 113024 (2018).
  * Barra (2015) F. Barra, Scientific Reports 5, 14873 (2015).
  * Esposito  _et al._ (2010) M. Esposito, K. Lindenberg, and C. V. d. Broeck, New Journal of Physics 12, 013013 (2010).
  * Reeb and Wolf (2014) D. Reeb and M. M. Wolf, New Journal of Physics 16, 103011 (2014).
  * Attal and Pautrat (2006) S. Attal and Y. Pautrat, Annales Henri Poincaré 7, 59 (2006).
  * Karevski and Platini (2009) D. Karevski and T. Platini, Physical Review Letters 102, 207207 (2009).
  * Ciccarello  _et al._ (2022) F. Ciccarello, S. Lorenzo, V. Giovannetti, and G. M. Palma, Physics Reports Quantum collision models: Open system dynamics from repeated interactions, 954, 1 (2022).
  * Campbell and Vacchini (2021) S. Campbell and B. Vacchini, Europhysics Letters 133, 60001 (2021).
  * Scarani  _et al._ (2002) V. Scarani, M. Ziman, P. Štelmachovič, N. Gisin, and V. Bužek, Physical Review Letters 88, 097905 (2002).
  * Ziman  _et al._ (2002) M. Ziman, P. Štelmachovič, V. Bužek, M. Hillery, V. Scarani, and N. Gisin, Physical Review A 65, 042105 (2002).
  * Trushechkin and Volovich (2016) A. S. Trushechkin and I. V. Volovich, Europhysics Letters 113, 30005 (2016).
  * Soret  _et al._ (2022) A. Soret, V. Cavina, and M. Esposito, “Thermodynamic consistency of quantum master equations,”  (2022).
  * Hewgill  _et al._ (2021) A. Hewgill, G. De Chiara, and A. Imparato, Physical Review Research 3, 013165 (2021).
  * Potts  _et al._ (2021) P. P. Potts, A. A. S. Kalaee, and A. Wacker, New Journal of Physics 23, 123013 (2021).
  * van der Wiel  _et al._ (2002) W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Reviews of Modern Physics 75, 1 (2002).
  * Cuetara and Esposito (2015) G. B. Cuetara and M. Esposito, New Journal of Physics 17, 095005 (2015).
  * Barato and Seifert (2015) A. C. Barato and U. Seifert, Physical Review Letters 114, 158101 (2015).
  * Pietzonka and Seifert (2018) P. Pietzonka and U. Seifert, Physical Review Letters 120, 190602 (2018).
  * Pietzonka  _et al._ (2017) P. Pietzonka, F. Ritort, and U. Seifert, Physical Review E 96, 012101 (2017).
  * Pietzonka  _et al._ (2016) P. Pietzonka, A. C. Barato, and U. Seifert, Physical Review E 93, 052145 (2016).
  * Horowitz and Gingrich (2017) J. M. Horowitz and T. R. Gingrich, Physical Review E 96, 020103 (2017).
  * Timpanaro  _et al._ (2019) A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Physical Review Letters 123, 090604 (2019).
  * Carollo  _et al._ (2019) F. Carollo, R. L. Jack, and J. P. Garrahan, Phys. Rev. Lett. 122, 130605 (2019).
  * Hasegawa (2020) Y. Hasegawa, Physical Review Letters 125, 050601 (2020).
  * Hasegawa (2021) Y. Hasegawa, Physical Review Letters 126, 010602 (2021).
  * Van Vu and Saito (2022) T. Van Vu and K. Saito, Physical Review Letters 128, 140602 (2022).
  * Agarwalla and Segal (2018) B. K. Agarwalla and D. Segal, Physical Review B 98, 155438 (2018).
  * Liu and Segal (2019) J. Liu and D. Segal, Physical Review E 99, 062141 (2019).
  * Prech  _et al._ (2022) K. Prech, P. Johansson, E. Nyholm, G. T. Landi, C. Verdozzi, P. Samuelsson, and P. P. Potts, (2022), arXiv:2212.03835v1 [cond-mat.mes-hall] .
  * Breuer (2003) H.-P. Breuer, Physical Review A 68, 032105 (2003).
  * Mitchison and Plenio (2018) M. T. Mitchison and M. B. Plenio, New Journal of Physics 20, 033005 (2018).
  * Humphrey  _et al._ (2002) T. E. Humphrey, R. Newbury, R. P. Taylor, and H. Linke, Physical Review Letters 89, 116801 (2002).
  * Goan  _et al._ (2001) H.-S. Goan, G. J. Milburn, H. M. Wiseman, and H. Bi Sun, Physical Review B 63, 125326 (2001).
  * Itano  _et al._ (1990) W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Physical Review A 41, 2295 (1990).
  * Kalaee  _et al._ (2021) A. A. S. Kalaee, A. Wacker, and P. P. Potts, Physical Review E 104, L012103 (2021).
  * Cangemi  _et al._ (2020) L. M. Cangemi, V. Cataudella, G. Benenti, M. Sassetti, and G. De Filippis, Physical Review B 102, 165418 (2020).
  * Ptaszyński (2018) K. Ptaszyński, Physical Review B 98, 085425 (2018).
  * Rignon-Bret  _et al._ (2021) A. Rignon-Bret, G. Guarnieri, J. Goold, and M. T. Mitchison, Physical Review E 103, 012133 (2021).
  * Kewming  _et al._ (2022) M. J. Kewming, M. T. Mitchison, and G. T. Landi, arXiv:2205.02622 [quant-ph] (2022).
  * Schaller (2014) G. Schaller,  _Open Quantum Systems Far from Equilibrium_, Lecture Notes in Physics, Vol. 881 (Springer International Publishing, Cham, 2014).
  * Wiseman and Milburn (2009) H. M. Wiseman and G. J. Milburn,  _Quantum Measurement and Control_ (Cambridge University Press, Cambridge, 2009).
  * Lacerda  _et al._ (2022) A. M. Lacerda, A. Purkayastha, M. Kewming, G. T. Landi, and J. Goold,  _Quantum thermodynamics with fast driving and strong coupling via the mesoscopic leads approach_, Tech. Rep. arXiv:2206.01090 (arXiv, 2022).
  * Brenes  _et al._ (2020) M. Brenes, J. J. Mendoza-Arenas, A. Purkayastha, M. T. Mitchison, S. R. Clark, and J. Goold, Physical Review X 10, 031040 (2020).
  * Brenes  _et al._ (2022) M. Brenes, G. Guarnieri, A. Purkayastha, J. Eisert, D. Segal, and G. Landi, (2022), arXiv:2211.13832v1 [quant-ph] .
  * Purkayastha  _et al._ (2021) A. Purkayastha, G. Guarnieri, S. Campbell, J. Prior, and J. Goold, Physical Review B 104, 045417 (2021).
  * Purkayastha  _et al._ (2022) A. Purkayastha, G. Guarnieri, S. Campbell, J. Prior, and J. Goold, Quantum 6, 801 (2022).
  * Wojtowicz  _et al._ (2022) G. Wojtowicz, A. Purkayastha, M. Zwolak, and M. M. Rams, “Accumulative reservoir construction: Bridging continuously relaxed and periodically refreshed extended reservoirs,”  (2022).