# Non-linear regime for enhanced performance of an Aharonov-Bohm heat engine

Géraldine Haack  Department of Applied Physics, University of Geneva, 1211
Genève, Switzerland  Francesco Giazotto<EMAIL_ADDRESS><EMAIL_ADDRESS>NEST, Instituto Nanoscienze-CNR and Scuola Normale
Superiore, I-56127 Pisa, Italy

###### Abstract

Thermal transport and quantum thermodynamics at the nanoscale is nowadays
garnering an increasing attention, in particular in the context of quantum
technologies. Experiments relevant for quantum technology are expected to be
performed in the non-linear regime. In this work, we build on previous results
derived in the linear response regime for the performance of an Aharonov-Bohm
(AB) interferometer operated as heat engine. In the non-linear regime, we
demonstrate the tunability, large efficiency and thermopower that this
mesoscopic quantum machine can achieve, confirming the exciting perspectives
that this AB ring offers for developing efficient thermal machines in the
fully quantum regime.

††preprint: AIP/123-QED

##  I Introduction

Thermoelectricity in quantum system is nowadays gaining an increasing interest
since the enormous advances achieved in quantum technology. In this context, a
crucial question is related to determine if the performances of thermoelectric
heat engines are affected by quantum mechanics, in particular, by influencing
their output power and thermodynamic conversion efficiency, especially in the
non-linear response regime Houten92 ; Giazotto06 ; Benenti17 ; Whitney18 ;
Whitney14 ; Whitney15 ; Lopez16 . Yet, it is well established that phase
coherence and quantum effects do play a stark role in governing the overall
behavior of mesoscopic heat engines Blanter97 ; Hofer15 ; Lambert16 ;
Samuelsson17 ; Hwang18 ; Haack19 . An emblematic example of phase-tunable
quantum device is represented by the celebrated Aharonov-Bohm (AB)
interferometer Aharonov59 ; Buttiker84 ; Yeyati95 , where the charge particle
response can be influenced by either electric or magnetic potentials. AB-type
interferometers represent suitable building blocks for the implementation of
efficient and versatile phase-coherent quantum heat engines. In particular, in
a previous work Ref. [Haack19, ], the authors investigated the performance of
AB-based quantum heat engines operated in the linear-response regime,
highlighting the excellent tunability and behavior provided by magnetic fields
and gate voltage control. However, experiments relevant for quantum technology
applications often require the operation of a device well beyond the linear-
response regime.

Figure 1: Panel a). Open-circuit configuration of the AB ring in the non-
linear response regime. Left and right contacts are biased in temperatures,
respectively at $T_{h}$ and $T_{c}$, and set to same chemical potential
$\mu_{0}$. The tranmission probability of the AB ring $T_{AB}(E)$ is energy-
dependent, and also depends on the length (time) imbalance between the two
arms, on $V_{g}$, an external gate voltage, on the transmission parameter
$\epsilon$ of the T-junctions and on $\Phi_{AB}$, the magnetic flux piercing
the interferometer. Due to thermoelectric properties of the AB ring, a
thermovoltage $V_{th}$ ($eV_{th}$ being the corresponding energy with $e$ the
electrical charge) is developped between the contacts, in response to the
temperature gradient. Panel b). Closed-circuit configuration for operating the
AB ring as a quantum heat engine, when the interferometer is connected to a
generic load resistor $R_{L}$ through dissipationless superconducting lines.

Here, we build upon previous results derived in the linear-response regime by
investigating the behavior of an AB interferometer operated in the full non-
linear regime as a quantum heat engine. We will demonstrate the good
tunability intrinsic to the device, the sizable thermodynamic conversion
efficiency, and the very large thermopower that this phase-controllable
thermoelectric quantum machine can achieve. All this confirms the AB
interferometer as a prototypical tool to develop efficient thermoelectric
machines operating in the fully quantum regime.

After briefly recalling the model in Sec. II, this contribution is organized
as follows. In Sec. III, we estimate the thermoelectric properties of the AB
ring in the non-linear response regime, through the thermovoltage and
differntial Seebeck coefficient. This is done in an open-circuit
configuration, as a function of the AB magnetic flux, external gate voltage
and temperature bias. In Sec. IV, we turn to the closed-circuit configuration
that allows us to operate the AB ring setup as a heat engine. Specifically, we
provide pedagogical steps towards the characterization of a quantum thermal
machine in the non-linear response regime, based on a scattering-matrix
approach. In Sec. V, we compute the heat current, power, efficiency and
determine the optimal load resistance for a given temperature bias in order to
maximize both power and efficiency of the AB heat engine. We conclude with
perspectives towards efficient phase-coherent quantum thermal machines.

##  II Model

We start by introducing our model for the AB ring shown in Fig. 2, and first
proposed as an efficient quantum heat engine in a previous work by the authors
Haack19 . The AB ring is connected to two metallic contacts that can be biased
in voltage or in temperature. Transmission probabilities between the contacts
and the ring can be tuned for both contacts independently by controlling the
transmission probability of the T-junctions Buttiker84 . Electrons traveling
through the ring acquire two fundamentally different phase factors. One is of
dynamical origin, it depends on the energy-dependent wave vector $k(E)$ of the
electrons and on the length asymmetry between the two arms, whereas the other
one is a geometrical phase arising from the presence of a perpendicular
magnetic field to the sample, giving rise to a magnetic flux threading the
ring $\Phi_{AB}$. The momentum $k(E)$ can be controlled by an external gate
voltage $V_{g}$, whereas the magnetic flux $\Phi_{AB}$ is controlled by an
external magnetic field $B$. The total transmission probability was derived in
full generality in Ref. [Haack19, ] and optimal working ranges for all
parameters were estimated. In this work, we focus on two key parameters, the
external gate voltage $V_{g}$ and the magnetic flux $\Phi_{AB}$, and fix fix
the other parameters based on Ref. [Haack19, ]. We recall below the energy-
dependent expression of the AB ring transmission probability $T_{AB}(E)$,
highlighting its dependence on $V_{g}$ and on the AB flux normalized by the
flux quantum $\Phi_{0}=h/(e)$, $\varphi\equiv 2\pi\Phi_{AB}/\Phi_{0}$:

$\displaystyle T_{AB}(E,V_{g},\varphi)$ $\displaystyle=$
$\displaystyle\frac{f_{1}(E,V_{g})+f_{2}(E,V_{g})\cos\varphi}{1+\big{[}f_{3}(E,V_{g})-f_{4}(E,V_{g})\cos\varphi\big{]}^{2}}\,.$

The exact expressions for the functions $f_{i},i=1,\ldots,4$ are provided in
App. A. In Ref. [Haack19, ], the functioning of this device was investigated
in the linear response regime to a temperature bias across the AB ring,
$\Delta T=T_{h}-T_{c}$. The linear response regime is valid when $\Delta
T\ll(T_{h}+T_{c})/2$ and amounts to neglect higher-order non-linear terms in
$\Delta T$. The authors showed that quantum transport through the device can
be fully tuned by controlling the AB flux $\varphi$. Optimal working regime
for thermoelectricity in the linear response regime could be achieved with low
values of $\epsilon$, corresponding to the T-junctions behaving as two
resonant tunnel barriers, similar to the Breit-Wigner regime in a Fabry-Pérot
interferometer. In this following work, we extend our investigation of this
fully tunable phase-coherent thermoelectric device to the non-linear response
regime, beyond small temperature bias.

Figure 2: $I-V$ characteristics in the non-linear regime for $T_{h}=2$ K;
$T_{c}=0.05$ K, $\varphi=2\pi\Phi_{AB}/\Phi_{0}=\pi$, $\epsilon=0.1$, and
$\delta\tau=0.3$. The curves correspond to $V_{g}=0.63$ mV (solid),
$V_{g}=1.1$ mV (dashed), and $V_{g}=1.5$ mV (dash-dotted). Thermovoltages for
different $V_{g}$ correspond to the crossing points with the $x$-axis, i.e.
for $I=0$.

##  III AB thermoelectricity in the non-linear regime

We start with the thermoelectric properties of the AB ring in an open-circuit
configuration. This setup allows us to characterize the thermoelectric
properties of the AB ring. Thermoelectricity refers to the ability of a device
to convert a voltage bias into a heat current (Peltier effect) or vice versa,
to convert a temperature bias into an electrical current (Seebeck effect).
Here, we focus on the latter, and will characterize the behavior of the
Seebeck coefficient $S$ as a function of the AB ring’s parameters. For
clarity, we recall its definition:

$\displaystyle S=\frac{V_{th}}{\Delta T}\Bigg{|}_{I=0}\,.$ (2)

This coefficient tells us how much voltage the device is able to develop from
a given temperature bias $\Delta T$ at zero charge current, $I=0$. Because
this voltage is determined at zero charge current and originates from a
temperature gradient, it was named thermovoltage, and will be labeled $V_{th}$
in the following. In general, this thermovoltage can also be built in presence
of a voltage bias $V$ between the two contacts.

Figure 3:  Thermoelectric characteristics of the AB ring in the open-circuit
configuration. Panel a). Color plot of the thermovoltage $V_{th}$ developed
across the AB interferometer vs magnetic flux $\varphi/2\pi$ and gate voltage
$V_{g}$ calculated for $T_{c}=0.05$ K, $T_{h}=2$K,
$\epsilon=0.1,\delta\tau=0.3$. Numerical result with steps of $2\pi/50$ for
both parameters. Panel b). Color plot of thermovoltage $V_{th}$ as a function
of the hot temperature $T_{h}$ and $V_{g}$, calculated at $\varphi=\pi$ and
for fixed $T_{c}=0.05$ K. Dashed line indicates a cut at $V_{g}=1.26$ mV.
Panel c). Differential thermopower (Seebeck coefficient in the non-linear
regime), shown in blue solid curve with scales on the left vertical axis. It
is calculated along the thermovoltage cut indicated as a dashed black curve in
Panels b and c, left vertical axis.

To illustrate the concept of thermovoltage, we show in Fig. 2, panel b, the
$I-V$ characteristics, namely the steady-state charge current in response to a
temperature bias as a function of the voltage $V$ for three different gate
voltages $V_{g}$ (solid, dashed and dashed-dotted curves). We have fixed the
magnetic flux $\varphi=\pi$, as well the temperature bias across the sample,
$T_{h}=2K,T_{c}=0.05K$. In all the work, we consider both contacts to be set
at the same chemical potential $\mu_{0}$. Within a scattering-matrix approach,
the average charge current is simply given by Lesovik11 ; Benenti17 :

$\displaystyle I(V,V_{g})$ $\displaystyle=$
$\displaystyle\frac{2e}{h}\int_{-\infty}^{\infty}dE\,T_{AB}(E,V_{g})\,\left[n(E,T_{h})-n(E+eV,T_{c})\right]\,,$

with $n(E,T_{i})=(1+e^{(E-\mu_{i})/(k_{B}T)})^{-1}$ the Fermi distribution of
hot or cold contact and $k_{B}$ the Boltzmann constant. In the steady state,
we have $\mu_{h}=\mu_{0}$ and $\mu_{c}=\mu_{0}+eV_{th}$ (both contacts are
initially set to the same chemical potential $\mu_{0}$ and we have to account
for $V_{th}$ in the steady state). It follows by definition that the
thermovoltage is the solution of

$\displaystyle I(V_{th},V_{g})=0\,,$ (4)

and corresponds to the crossing points of the $I-V$ curves with the x-axis in
Fig. 2b). It can already be seen that the external voltage $V_{g}$ provides an
important tunability for controlling $V_{th}$. To further explore the
thermoelectric response of the AB ring in the non-linear regime, we show in
Fig. 3 the thermovoltage as a function of the gate voltage and magnetic flux
(panel a), as a function of the temperature bias (panel b as a function of
$T_{h}$, $T_{c}$ being fixed), and finally the differential Seebeck
coefficient (panel c), which is the generalization of the Seebeck coefficient
in the non-linear regime. The main features to retain from these figures is
the high tunability of the thermovoltage with all parameters of the AB ring,
and the possibility to switch on and off the thermoelectric response of the AB
ring with the magnetic flux (see panel a). Remarkably, the differential
Seebeck coefficient can be two orders of magnitudes larger than the one in the
linear response regime. In panel c, the maximum of the blue curve reaches 15
mV/K, whereas the maximum attainable in the linear response regime was of the
order of 300 $\mu$V/K Haack19 . This confirms the excellent thermoelectric
response of this phase-coherent mesoscopic device, and motivates the rest of
this contribution where we investigate its behavior as a quantum heat engine
in a closed-circuit configuration.

Figure 4: Closed-circuit characteristics of the AB heat engine. Panel a).
Color plot of the power $P=(V_{th}^{(cl)})^{2}/R_{L}$ as a function of gate
voltage $V_{g}$ and magnetic flux $\varphi$. Panel b) Cuts of power as a
function of magnetic flux for different values of the gate voltage,
$V_{g}=0.79$ mV (red dot-dashed), $V_{g}=1.4$ mV (black dotted) and
$V_{g}=2.7$ mV (orange dashed). Cuts are also shown on panel a). Non-linear
behaviour of power Values for the other parameters for panels a) and b):
$T_{h}=2$ K, $T_{c}=0.05$ K, $\epsilon=0.1$, $\delta\tau=0.3,R_{L}=20\,R_{q}$.
Panels c). Parametric plot of the efficiency normalized by the Carnot
efficiency $\eta_{C}$ versus power $P$, as a function of the gate voltage
$V_{g}=[0,\pi(\mu_{0}/e)]$ in steps of $\Delta V_{g}=2\pi/50(\mu_{0}/e)$.
Different colors correspond to different load resistances: green (circles) for
$R_{L}=15R_{q}$, blue (diamonds) for $R_{L}=20R_{q}$ and red (squares) for
$R_{L}=40R_{q}$. Values of the other parameters are the same as for panels a)
and b) with fixed magnetic flux, $\varphi=\pi$. Panel d). Optimal load
resistance as a function of $V_{g}$, for maximizing efficiency and power, see
main text.

##  IV AB thermoelectricity in a closed-circuit configuration

The closed-circuit configuration requires additional careful analysis for
assessing correctly the thermovoltage and heat and charge currents. When the
circuit is closed with a load resistance as shown in Fig. 2, the Kirchhoff’s
laws have to be satisfied, ensuring charge and energy conservation: the sum of
the charge currents flowing through the two branches of the circuit, one with
the AB ring, the other one with the load resistance $R_{L}$, has to be equal
to 0. As a consequence, the thermovoltage in this new configuration is
obtained by solving a different equation as compared to Eqs. (III) and (4),
which reads:

$\displaystyle\frac{2e}{h}\int_{-\infty}^{\infty}dET_{AB}(E)\left[n(E,T_{h})-n(E+eV_{th}^{(cl)},T_{c})\right]=-\frac{V_{th}^{(cl)}}{R_{L}}\,,$

and its solution for the thermovoltage is denoted $V_{th}^{(cl)}$ for clarity.
We compute numerically values for the thermovoltages in the closed-circuit
configuration as a function of gate voltage $V_{g}$ and magnetic flux
$\varphi$, for fixed temperatures, fixed imbalance and fixed transmission
probabilities of the T-junctions. This allows us to compute the heat current
flowing through the AB ring in the non-linear response regime according to
Houten92 ; Lesovik11 ; Lopez16 . Let us precise that in the context of the AB
ring operated as heat engine, we have to consider the heat current flowing
into the hot contact, $J_{h}$, as it corresponds to the resource for the
engineBenenti17 . It corresponds to the energy flow that electrons can
dissipate into the hot reservoir, taking as reference the chemical potential
of that same reservoir, here $\mu_{0}$:

$\displaystyle J_{h}$ $\displaystyle=$
$\displaystyle\frac{2}{h}\int_{-\infty}^{\infty}dE(E-\mu_{0})\,T_{AB}(E)\left[n(E,T_{h})-n(E+eV_{th}^{(cl)},T_{c})\right]\,,$

For clarity, we recall that heat current is not conserved, $J_{h}\neq J_{c}$
with $J_{c}$ the heat current flowing into the cold contact. In the case of
the AB ring operated as heat engine with the building up of a thermovoltage on
the right (cold) lead, conservation law instead reads:

$\displaystyle J_{h}+J_{c}=V_{th}^{(cl)}I\,,$ (7)

with

$\displaystyle J_{c}$ $\displaystyle=$
$\displaystyle\frac{2}{h}\int_{-\infty}^{\infty}dE(E-(\mu_{0}+eV_{th}^{(cl)})\,$
(8) $\displaystyle
T_{AB}(E)\left[n(E+eV_{th}^{(cl)},T_{c})-n(E,T_{h})\right]\,.$

We also compute the power $P=(V_{th}^{(cl)})^{2}/R_{L}$ generated by the AB
heat engine, which allows us in the next section to discuss several figures of
merit for the AB heat engine in the non-linear response regime. The load
resistance is typically expressed as a multiple of the quantum of resistance
$R_{q}=h/(2e^{2})$. The numerical integration over energy for computing the
heat current was performed over ten times the Fermi energy $\mu_{0}$. Let us
remark that it takes values of the same order of magnitude as the one in the
linear response regime for $T=1K$, namely of the order of 0.1 -1 pW, see App.
B. Figure 4 panel a) shows the power for a fixed value of the load resistance,
$R_{L}=20\,R_{q}$ as a function of gate voltage and magnetic flux, and panel
b) highlights the power characteristics for different gate voltages as a
function of the magnetic flux. The cuts evidence the possibility to fully turn
on and off the heat engine, from 0 to maximal power.

##  V Power and efficiency of the AB heat engine in the non-linear response
regime

We now investigate the power versus efficiency trade-off in the non-linear
regime, as a function of the gate voltage $V_{g}$ and the load resistance
$R_{L}$. Given a temperature bias, we determine the optimal load resistance to
maximize both power and efficiency, the latter being simply defined as the
ratio of power (output of the engine in $[W]$) and heat current into the hot
reservoir (input of the engine, also in $[W]$):

$\displaystyle\eta=P/J_{h}\,.$ (9)

In Fig. 4 c), we show the Lasso-type parametric plot of efficiency versus
power, that evidences the trade-off between both quantities. A heat engine
with maximal efficiency, Carnot efficiency, does not produce any power as it
corresponds to the point of reversibility for the machine Benenti17 . Hence a
compromise must be found or determined depending on the primary goal of the
engine (to produce more power with less efficiency or be more efficient with
less output power). Here, we show that the AB heat engine can be easily tuned
through external gate voltage $V_{g}$ (panel c) to optimize either efficiency
or power. These two quantities also depend on the load resistance, as
evidenced by the different colored curves in the same panel. For the AB heat
engine, it seems that a load resistance of $R_{L}\sim 15-20\,R_{q}$ may be
optimal given the values we have fixed for the other parameters.
Interestingly, an efficiency up to $\sim$ 40$\%$ of Carnot efficiency can be
achieved by adjusting $V_{g}$ and $R_{L}$. This value has to be compared to
the efficiency in the linear response regime for similar temparture gardient,
where the authors showed a maximal efficiency of the order of $30\%$ the
Carnot efficiency Haack19 . The increase in efficiency can directly be
attributed to the non-linear response of the AB engine, as similar temperature
bias accross the setup were considered (while keeping the same parameters for
asymmetry, T-junctions parameter and magnetic flux). In panel d), we show the
optimal load resistance to maximize both efficiency and power, as a function
of the gate voltage. To achieve this, we maximize $P+\eta$,over $R_{L}$ for
fixed $V_{g}$ and fixed temperature gradients. This procedure does not reflect
the absolute optimal $R_{L}$ as a function of all parameters (of interest in
possible future experiments), but rather the high tunability of the AB ring as
heat engine to achieve sizeable power and efficiency in the quantum regime.

##  VI Conclusions and perspectives

In conclusion, we have analyzed a coherent mesoscopic heat engine consisting
of an Aharonov-Bohm quantum interferometer operated in the full non-linear
regime. The device thermoelectric response turns out to be sizable, and the
interferometer is a able to provide a thermopower which is about 50 times
larger than in the same structure operated in the linear regime. Moreover,
also the heat engine thermoelectric efficiency at maximum power is somewhat
large, obtaining values as high as $\sim 40\%$ of the Carnot efficiency in the
linear regime. The AB heat engine can provide magnetic flux- and
electrostatic-driven control of charge and heat current as well as of its
thermoelectric response under physical conditions which are accessible from
the experimental point of view. Suitable candidates for the implementation of
the Aharonov-Bohm quantum heat engine are represented by metallic or
GaAs/AlGaAs two dimensional electron gas heterostructures Webb85 ; Washburn86
; Yacobi96 ; Yamamoto12 , which are expected to yield devices with robust
performances over a wide range of system parameters. The above results show
that an Aharonov-Bohm loop operated as a quantum heat engine in the non-linear
regime represents a prototypical platform for the implementation of a unique
class of phase-tunable thermoelectric quantum machines operating at cryogenic
temperatures. Yet, in the context of quantum technologies Acin18 , this
coherent structure might be at the core of a number of innovative
thermolectric quantum devices, for instance, highly-sensitive photon sensors
where radiation-induced heating one of the electrodes forming the system is
detected via the resulting thermovoltage.

###### Acknowledgements.

GH acknowledges support from the Swiss National Science Foundation, through
the PRIMA grant PR00P2$\\_$179748 and the National Centers for Competences and
Research for Quantum Science and Technology (QSIT) and SwissMap. FG
acknowledges the European Research Council under Grant Agreement No.
899315-TERASEC, and the EU’s Horizon 2020 research and innovation program
under Grant Agreement No. 800923 (SUPERTED) and No. 964398 (SUPERGATE) for
partial financial support.

## Data Availability Statement

The techniques and values used to generate the data that support the findings
of this study are available within the article and from the corresponding
author upon reasonable request.

##  Appendix A Transmission probability of the AB ring

In this section, we provide the expressions of the functions $f_{i}$ in Eq.
(II) of the transmission probability of the AB ring. These coefficients are
determined and expressed according to Ref. [Haack19, ], where the authors
previously derived $T_{AB}$ in full generality:

$\displaystyle f_{1}(E,V_{g})$ $\displaystyle=$ $\displaystyle
1-\cos\chi\cos\delta\chi$ (10) $\displaystyle f_{2}(E,V_{g})$ $\displaystyle=$
$\displaystyle\cos\delta\chi-\cos\chi$ (11) $\displaystyle f_{3}(E,V_{g})$
$\displaystyle=$
$\displaystyle\frac{2(1-\epsilon)\cos\chi-(1-\epsilon-\sqrt{1-2\epsilon})\cos\delta\chi}{2\epsilon\sin\chi}$
$\displaystyle f_{4}(E,V_{g})$ $\displaystyle=$
$\displaystyle\frac{\big{(}1-\epsilon+\sqrt{1-2\epsilon}\big{)}}{2\epsilon\sin\chi}\,,$
(13)

with $\chi=\chi_{1}+\chi_{2}$ and $\delta\chi=\chi_{1}-\chi_{2}$ and
$\chi_{i}=k_{i}L_{i}$ are the dynamical phases that electrons acquire while
traveling in each arm $i=1,2$. Here $k_{i}$ is the energy-dependent wave
vector and $L_{i}$ the length of the corresponding arm

$\displaystyle k_{1}(E)$ $\displaystyle=$
$\displaystyle\tilde{k}_{\mu_{0}}+\frac{E-{\mu_{0}}}{\hbar
v_{d}}\,,\,k_{2}(E)=\tilde{k}_{\mu_{0}}+\frac{E-(\mu_{0}+eV_{g})}{\hbar
v_{d}}\,.$ (14)

For an asymmetric AB ring, the difference and sum of the dynamical phases then
read:

$\displaystyle\\!\\!\\!\\!\chi_{1}+\chi_{2}$ $\displaystyle=$
$\displaystyle(2L+\delta L)\left(\tilde{k}_{\mu_{0}}+\frac{E-\mu_{0}}{\hbar
v_{d}}\right)-\frac{eV_{g}L}{\hbar v_{d}}$ (15)
$\displaystyle\\!\\!\\!\\!\chi_{1}-\chi_{2}$ $\displaystyle=$
$\displaystyle\delta L\left(\tilde{k}_{\mu_{0}}+\frac{E-\mu_{0}}{\hbar
v_{d}}\right)+\frac{eV_{g}L}{\hbar v_{d}}\,.$ (16)

Here we have defined $L_{1}\equiv L+\delta L$, $L_{2}\equiv L$, $v_{d}$ is the
electronic drift velocity and the Fermi wave vector is tuned through energy
offsets applied onto each arm ($k_{\mu_{0}}\rightarrow\tilde{k}_{\mu_{0}}$).

##  Appendix B Heat current

For completeness, we provide here a color plot of the heat current as a
function of voltage gate and magnetic flux, similar to the power shown in Fig.
4. We also indicate the same cuts for values of $V_{g}$ given in the main
text.

Figure 5: Heat current through the AB heat engine in a closed-circuit
configuration. Panel a). Color plot of the heat current $J_{h}$ (see Eq.
(IV)), as a function of gate voltage $V_{g}$ and magnetic flux $\varphi$.
Panel b). Cuts of heat current as a function of magnetic flux for different
values of the gate voltage, $V_{g}=0.79$ mV (red dot-dashed), $V_{g}=1.4$ mV
(black dotted) and $V_{g}=2.7$ mV (orange dashed). Other parameters are fixed:
$T_{h}=2$ K, $T_{c}=0.05$ K, $\epsilon=0.1$, $\delta\tau=0.3$

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