# Quantum Rayleigh problem and thermocoherent Onsager relations

Onur Pusuluk  Department of Physics, Koç University, Sarıyer, İstanbul, 34450
Turkey  Özgür E. Müstecaplıoğlu  Department of Physics, Koç University,
Sarıyer, İstanbul, 34450 Turkey

###### Abstract

The role of quantum coherence and correlations in heat flow and equilibration
is investigated by exploring the Rayleigh’s dynamical problem to equilibration
in the quantum regime and following Onsager’s approach to thermoelectricity.
Specifically, we consider a qubit bombarded by two-qubit projectiles from a
side. For arbitrary collision times and initial states, we develop the master
equation for sequential and collective collisions. By deriving the Fokker-
Planck equation out of the master equation, we identify the quantum version of
the Rayleigh’s heat conduction equation. We find that quantum discord and
entanglement shared between the projectiles can contribute to genuine heat
flow only when they are associated with so-called heat-exchange coherences.
Analogous to Onsager’s use of Rayleigh’s principle of least dissipation of
energy, we use the entropy production rate to identify the coherence current.
Both coherence and heat flows can be written in the form of quantum Onsager
relations, from which we predict coherent Peltier and coherent Seebeck
effects. The effects can be optimized by the collision times and collectivity.
Finally, we discuss some of the possible experimental realizations and
technological applications of the thermocoherent phenomena in different
platforms.

##  I INTRODUCTION

In his 1891 paper [1], Lord Rayleigh focused, for the first time, on the whole
dynamical process by which a thermodynamic steady state is attained in a gas
of particles. He captured the essential physics by developing a simplified
model where tiny and light projectiles collide with heavy masses at rest (Fig.
1 (a)). A further limitation of his studies on the one-dimensional bombardment
of the masses by projectiles allowed him to reveal the role of the
environment’s initial conditions in the heat conduction. In particular, the
mass and velocity of the projectiles striking the system appeared explicitly
in the thermal diffusivity coefficient of Rayleigh’s heat equation. This
equation forms the basis for Onsager reciprocal relations between heat and
charge flows [2], which play the central role in thermoelectricity [3].

This paper builds on the pioneering works of Lord Rayleigh and Onsager. We
focus on the role of the initial quantum properties of the environment in heat
conduction (Fig. 1 (b)). Specifically, we simplify the complex problem of heat
conduction dynamics in a gas of coherent quantum particles to a one-
dimensional collision model [4, 5, 6, 7, 8, 9, 10]. Such a simplification
allows us to derive the heat conduction equation with the explicit distinction
of quantum coherent and purely thermal contributions, which turns out to be
additive. Accordingly, we are able to establish analogs of Onsager relations
between heat flow $\mathcal{J}_{h}$ and quantum coherence flow
$\mathcal{J}_{c}$, which read

$\displaystyle\mathcal{J}_{h}$ $\displaystyle=$ $\displaystyle
L_{hh}\,\Delta(1/T)+L_{hc}\,\Delta(-C)/T,$ (1) $\displaystyle\mathcal{J}_{c}$
$\displaystyle=$ $\displaystyle L_{ch}\,\Delta(1/T)+L_{cc}\,\Delta(-C)/T,$ (2)

where $L_{hh}$ and $L_{cc}$ are the heat and quantum coherence conductivities,
under temperature and quantum coherence differences $\Delta T$ and $\Delta C$,
respectively. The interference of heat and coherence flows is reflected by the
emergence of $L_{hc}$ and $L_{ch}$ coefficients that we dub as thermocoherent
coefficients. We rigorously show that $\mathcal{J}_{c}$ delivered by the
quantum projectiles is completely received as heat by the mass so that
$\mathcal{J}_{c}=\mathcal{J}_{h}$ and reciprocity $L_{hc}=L_{ch}$ holds.
Thermocoherent Onsager equations immediately reveal the possibility of mutual
induction or control of heat and coherence flows. For example, heat flow
between equal temperature reservoirs by a quantum coherence potential
gradient, as an extension of the Peltier effect, or coherent current
generation by thermal gradient as an extension of the Seebeck effect, are
possible thermocoherent phenomena that can be predicted. Remarkably, further
phenomenological particle current terms could be introduced in the
thermocoherent relations.

Figure 1: Three-dimensional illustration of (a) Rayleigh’s one-dimensional
collision model [1] in a box filled with a gas at some temperature and (b) its
extension into the quantum regime where particles also behave as waves. The
value $T$ shown in the thermometer depends on the classical degrees of freedom
(DOF) of individual projectiles, mass $m$ and velocity $v$ in the former.
Quantum projectiles consist of a pair of locally thermal two-level systems
whose classical DOF are the population of the energy levels. These projectiles
have an additional DOF, i.e., a quantum DOF represented by $\lambda$, which
can also affect the rate of heat exchange with the thermometer.

The systematic investigation of thermocoherent coefficients and quantum
coherence conductivity that we present here can be significant for the growing
body of research on the thermodynamic means of generating and protecting
quantum coherence and correlations (QCCs) [11, 12, 13, 14, 15, 16, 17], as
well as for the studies of QCCs to manipulate thermodynamic energy transfer
[18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35].

Another fundamental question that we ask here is which QCCs take a role in
thermocoherent effects and heat conduction in a quantum gas of particles. To
address this question, we consider all particles as two-level systems (qubits)
in our collision model, and take the projectiles as pairs of qubits in generic
quantum and classical correlated states. After introducing the quantum
Rayleigh problem, describing the collisions between the projectiles and the
target qubit in Sec. II, we will present the the projectile states explicitly
in Sec. III. Section IV develops the thermalization dynamics of the target
qubit in terms of the master equation (Sec. IV.1), which is used to determine
the quantum heat current (Sec. IV.2). The quantum coherence current is
identified in Sec. IV.3 and quantum thermocoherent Onsager relations are
established in Sec. IV.4. We discuss the experimental and application
significance of the results in Sec. V and conclude in Sec. VI.

##  II QUANTUM RAYLEIGH PROBLEM

Figure 2: Quantum version of Rayleigh’s collision model with initially
correlated bath qubits labeled $\\{1,2,3,...\\}$. Identical and non-
interacting bath qubits are locally at thermal equilibrium at temperature
$T_{B}$, while sharing either classical correlation, quantum discord, or
quantum entanglement in pairs (linked circles). Projectiles consisting of such
pairs strike the target qubit $S$ at random times $t_{j}$, which has an energy
level separation of $\omega_{S}$ and an initial temperature $T_{S}$. $S$
interacts with each bath qubit in the $j$th pair either (a) sequentially or
(b) collectively. The whole collision with a single projectile takes over a
time interval of duration $\tau\ll t_{j+1}-t_{j}$. Sequential collision is
described by the product of two-qubit operators
$U_{SB_{2j}}(\tau/2)U_{SB_{2j-1}}(\tau/2)$,while collective collision is
described by a single three-qubit operator $U_{SB_{2j-1}B_{2j}}(\tau)$.

When Lord Rayleigh examined the collisional route to equilibration, as
depicted in Fig 1 (a), in terms of a simplified one-dimensional bombardment of
a massive atom at rest by tiny projectiles, he stated some corollaries, which
are natural for a classical physicist. In particular he wrote that “If we
suppose that the projectiles are dispatched in pairs of closely following
components, we should expect that the effect would be the same as of a
doubling of the mass” [1]. Here, we explicitly generalize the Rayleigh problem
to the case of pairwise projectiles, which will not be a trivial case of
doubling of mass, if the projectiles have a wavelike nature, in particular,
carry a pairwise quantum coherence and correlations, as illustrated in Fig 1
(b). For brevity of terminology, we call the target qubit, representing the
stationary massive (thermometer) atom, only as qubit and the pair of qubits
bombarding it as projectiles.

The projectiles (marked with $B$) are identical copies each prepared in an
arbitrary, possibly nonthermal state, while the qubit (marked with $S$) is
initially in a thermal equilibrium state

$\rho_{S}(0)=q_{g}|g\rangle\langle g|+q_{e}|e\rangle\langle e|,$ (3)

where
$q_{g/e}=\exp(-\beta_{S}E_{g/e})/[\exp(-\beta_{S}E_{g})+\exp(-\beta_{S}E_{e})]$
and $\beta_{S}=1/k_{\mathrm{B}}T_{S}$ with $k_{\mathrm{B}}$ being the
Boltzmann constant. The qubit Hamiltonian reads

$H_{S}=E_{g}|g\rangle\langle g|+E_{e}|e\rangle\langle e|.$ (4)

Projectiles arrive at the qubit at times $t_{j}$ and the collision time is
fixed to $\tau$. We envision two different bombardment scenarios. In the first
case, the qubit interacts with each projectile qubit sequentially, as
illustrated in Fig. 2 (a). The collision at a time $t_{j}$ is then described
by

$U(\tau)=U_{SB_{2j-1}}(\tau/2)\,U_{SB_{2j}}(\tau/2),$ (5)

where each unitary $U_{SB_{j}}$ is generated by an interaction Hamiltonian of
the form

$\displaystyle
H_{SB_{j}}=J(\sigma_{x}^{(S)}\sigma_{x}^{(B_{j})}+\sigma_{y}^{(S)}\sigma_{y}^{(B_{j})}).$
(6)

Here, $J$ is the coupling constant, and $\sigma_{x},\sigma_{y}$ are the Pauli
matrices.

The second scenario (Fig. 2(b)) focuses on a collective collision described by
the following evolution

$U(\tau)=U_{SB_{2j-1}B_{2j}}(\tau)=e^{-\frac{i\tau}{\hbar}(H_{SB_{2j-1}}+H_{SB_{2j}})},$
(7)

where the qubit collides with the both qubits in the $j$th projectile
simultaneously. The process is described by the Hamiltonian in Eq. (6). The
total energy is preserved in both collision scenarios, yet the energy exchange
between the projectiles and the qubit varies. Accordingly, both the
equilibration and heat conduction are sensitive to the collective nature of
collisions.

##  III INITIAL STATES OF THE PROJECTILES

Figure 3: Amount of the quantum coherences and correlations found in quantum-
correlated local thermal bath states $\rho_{D}$ (left panels) and $\rho_{E}$
(right panels) with
${\footnotesize{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}=\lambda/\lambda_{max}=\mu/\mu_{max}$.
$E_{g}=1$, $E_{e}=2$, and $\beta_{B}=2$ in all four panels. In the upper
panels, the relative entropy of coherence [36] and $l_{1}$ norm of coherence
[36] are shown, respectively, by dashed black and gray curves. Dotted gray,
dash-dotted red, and blue curves in the lower panels correspond to classical
correlations [37], quantum discord [37, 38], and entanglement of formation
[39].

Projectiles are assumed to be prepared in a thermal equilibrium state at
temperature $T_{B}$ first; then they are either correlated classically or
injected with some quantum coherence. Depending on which states carry the
quantum coherence, the projectile can have profoundly distinct quantum
correlations. Specifically, we will examine three different states of the
projectiles before the collision with the qubit, which are expressed as

$\displaystyle\rho_{C}$ $\displaystyle=p_{g}|gg\rangle\langle
gg|+p_{e}|ee\rangle\langle ee|,$ (8a) $\displaystyle\rho_{D}$
$\displaystyle=(p_{g}|g\rangle\langle g|+p_{e}|e\rangle\langle e|)^{\otimes
2}+(\lambda|ge\rangle\langle eg|+\mathrm{h.c.}),$ (8b) $\displaystyle\rho_{E}$
$\displaystyle=p_{g}|gg\rangle\langle gg|+p_{e}|ee\rangle\langle
ee|+(\mu|gg\rangle\langle ee|+\mathrm{h.c.}),$ (8c)

where $|\lambda|\leq\lambda_{\text{max}}:=p_{g}p_{e}$ and
$|\mu|\leq\mu_{\text{max}}:=\sqrt{p_{g}p_{e}}$ because of the positivity of
$\rho_{D}$ and $\rho_{E}$, respectively, and
$p_{g/e}=\exp(-\beta_{B}E_{g/e})/[\exp(-\beta_{B}E_{g})+\exp(-\beta_{B}E_{e})]$
with $\beta_{B}=1/k_{\mathrm{B}}\,T_{B}$. Projectile qubits are identical with
the qubit so that their Hamiltonian $H_{B}$ is the same as in Eq. (4).
Injected coherence in neither of these states destroys the local thermal
equilibrium of each qubit of the projectile. Their reduced (local) states are
given by the Gibbs state
$\exp[-\beta_{B}H_{B}]/\mathrm{tr}[\exp[-\beta_{B}H_{B}]]$. Accordingly, we
can associate $T_{B}$ as a local temperature to each projectile qubit. This is
the first criterion behind the choice of the projectile states $\rho_{C}$,
$\rho_{D}$, and $\rho_{E}$ in Eq. (8). Second, each of these three state
families illustrates a different kind of correlation as discussed below.

Projectile pairs that share only classical correlations are described by
$\rho_{C}$ [37]. Pairs in $\rho_{D}$ and $\rho_{E}$ possess nonclassical
correlations that can be well captured by so-called quantum discord [37, 38],
though they have quite different natures. $\rho_{D}$ is a separable state like
$\rho_{C}$ but its preparation using incoherent operations requires the
consumption of a local quantum coherence initially found in the projectiles
[40]. It describes a sensitivity to local dynamics in such a way that local
measurements disturb the pre-measurement state of both projectiles. Unlike the
classical correlations contained in $\rho_{C}$, the total correlations carried
in this state cannot be extracted by any local quantum measurements [37, 38,
41]. On the other hand, $\rho_{E}$ is an entangled state, i.e. it cannot be
factorized into a product state. Its quantum correlations can be measured not
only by the quantum discord but also by the entanglement of formation [39],
which always gives zero for $\rho_{D}$. To emphasize this difference, whenever
we use the term “discordant projectile state” in what follows, we will refer
to $\rho_{D}$. On the other hand, “entangled projectile state” will stand for
$\rho_{E}$.

Quantum coherence and correlations in these states are compared in Fig. 3. The
nonzero values of $\lambda$ and $\mu$ generate quantum coherence and discord
respectively in $\rho_{D}$ and $\rho_{E}$, which differ only quantitatively
(cf. Fig. 3 (a)-(d)). Quantum correlation in $\rho_{D}$ becomes a maximum in
the case of $|\lambda|$ is equal to $\lambda_{\mathrm{max}}$ and decreases
when $\lambda$ approaches zero (cf. Fig. 3 (c)). In the limit of vanishing
$\lambda$, the pair becomes uncorrelated. Similarly, entangled state
$\rho_{E}$ reduces to classically correlated state $\rho_{C}$ at $\mu=0$.
Simple monotonic behavior of coherence and correlations (with either positive
or negative parameters) and lack of entanglement in the discordant state
allows us to investigate influence of QCCs in thermocoherent relations
explicitly. This is the last criterion that determines the three projectile
states given in Eq. (8).

Table 1: The coefficients in the time-dependent qubit state (9) are defined depending on the collision scenario and correlation type under consideration. Note that the value of $\alpha$ given for sequential collisions converges to unity in the weak coupling or short interaction limit. To avoid potential confusion, it should also be emphasized that $\alpha$ is a function of collectivity, strength, and duration of the collisions, but the dependence on these parameters is omitted in the text for the sake of readability. Scenario |  Correlations |  $\alpha$ |  $\Gamma_{g/e}$ |  $\gamma(t)$
---|---|---|---|---
Sequential Collision |  Quantum Discord |  $\frac{2\cos(J\tau)}{1+\cos^{2}(J\tau)}$ |  $p_{g/e}\frac{1}{1+2\,\alpha\lambda}+\frac{\alpha\lambda}{1+2\,\alpha\lambda}$ |  $e^{-\frac{pt}{\alpha}(1+2\,\alpha\lambda)\sin(J\tau)\sin(2J\tau)}$
|  Others |  $\frac{2\cos(J\tau)}{1+\cos^{2}(J\tau)}$ |  $p_{g/e}$ |  $e^{-\frac{pt}{\alpha}\sin(J\tau)\sin(2J\tau)}$
Collective Collision |  Quantum Discord |  1 |  $p_{g/e}\frac{1}{1+2\,\alpha\lambda}+\frac{\alpha\lambda}{1+2\,\alpha\lambda}$ |  $e^{-\frac{pt}{\alpha}\,(1+2\,\alpha\lambda)\sin^{2}(2\sqrt{2}J\tau)}$
|  Others |  1 |  $p_{g/e}$ |  $e^{-\frac{pt}{\alpha}\sin^{2}(2\sqrt{2}J\tau)}$

##  IV DYNAMICS TO STEADY STATE

###  IV.1 Master Equation and Anomalous Heat Current

We derive a master equation (31) for arbitrary collision times and initial
states in Sec. A.1 in the Appendix. It is straightforward to solve this
equation analytically for both sequential and collective collision scenarios.
In the Schrödinger picture, the solution takes the following general form

$\displaystyle\begin{aligned}
\rho_{S}(t)&=\big{(}\Gamma_{g}(1-q_{e}\gamma(t))+\Gamma_{e}q_{g}\gamma(t)\big{)}|g\rangle\langle
g|\\\
&\qquad+\big{(}\Gamma_{e}(1-q_{g}\gamma(t))+\Gamma_{g}q_{e}\gamma(t)\big{)}|e\rangle\langle
e|,\end{aligned}$ (9)

where $\Gamma_{g/e}$ and $\gamma(t)$ are defined in Table 1, depending on the
collision scenario and the correlation type under consideration. In
particular, classical correlations and quantum entanglement in the projectile
are not found to play any role in the target qubit’s open system dynamics.
Recall that the discordant pair becomes uncorrelated when $\lambda$ goes to
zero, and the coefficients in Table 1 turn out to be independent of the
correlation type in this limit.

The steady state of Eq. (31) is

$\displaystyle\begin{aligned} \rho_{S}^{\infty}=\Gamma_{g}|g\rangle\langle
g|+\Gamma_{e}|e\rangle\langle e|,\end{aligned}$ (10)

which is equal to the local (reduced) states of the qubits of the projectile
in $\rho_{C}$ or in $\rho_{E}$ states. Neither classical correlations nor
entanglement can make any difference in the thermalization of the qubit to the
local temperature $T_{B}$. In Sec. IV.3, we will revisit the case of entangled
projectiles. We will particularly show that these projectiles can affect the
dynamics of the qubit and its steady state if the collision block is extended
to include two projectile pairs instead of one.

Figure 4: Temperature of the qubit bombarded with projectiles in discordant
states $\rho_{D}$. Upper panels show $\beta_{S}^{-1}$ as a function of time
$t$ and
${\footnotesize{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}=\lambda/\lambda_{max}$
in the weak-coupling limit ($J=0.05/\tau$). Here, we work with dimensionless
time with a unit system where $p=1$. The value of $\beta_{S}^{-1}$ is explored
for different coupling strengths when $t\rightarrow\infty$ in the lower
panels. $E_{g}=1$, $E_{e}=2$, $\beta^{-1}_{B}=\beta^{-1}_{S}(t=0)=0.5$ in all
four panels. The sequential (collective) collision model is considered in the
left (right) column.

On the other hand, Eq. (10) corresponds to a thermal state for collisions with
discordant projectiles when $\lambda>-1/(2\alpha)$ (cf. Table 1). The qubit
reaches a final temperature

$T_{S}(\infty)=\frac{(E_{g}-E_{e})/k_{\mathrm{B}}}{\ln[(p_{e}+\alpha\lambda)/(p_{g}+\alpha\lambda)]},$
(11)

which shows that the discordant projectiles thermalize the qubit to a
temperature that is different than their local temperature $T_{B}$.
$T_{S}(\infty)$ can be greater or less than $T_{B}$ depending on the sign of
$\alpha\lambda$, as shown in Figs. 4 (c) and (d). In the high-temperature
limit, we get $T_{S}(\infty)\approx T_{B}(1+2\,\alpha\lambda)$. Note that
$-1/(2\alpha)<\lambda\leq p_{g}p_{e}$, $-1\leq\alpha\leq 1$, and
$p_{g}p_{e}\approx 1/4$ for high temperatures. The heating and cooling with
quantum discord depends on the sign of $\lambda$ for both collective and
sequential collision scenarios. In the case of sequential interactions,
however, the sign of $\alpha(J\tau)$ also becomes a factor.

$T_{S}(\infty)\neq T_{B}$ allows for enhancement or reduction in the heat flow
between the qubit and discordant bath. When $T_{S}(0)=T_{S}(\infty)$, the
qubit remains at its initial temperature so that the heat flow is inhibited,
despite the temperature difference of $~{}2\,\alpha\lambda\,T_{B}$.
Conversely, an anomalous heat emerges when $T_{S}(0)=T_{B}$. The absence of an
initial temperature gradient does not prevent the change in the temperature of
the qubit given by

$T_{S}(t)=\frac{(E_{g}-E_{e})/k_{\mathrm{B}}}{\ln\Big{[}\frac{p_{e}+\alpha\lambda(1+\gamma(t)(p_{e}-p_{g}))}{p_{g}+\alpha\lambda(1+\gamma(t)(p_{g}-p_{e}))}\Big{]}}.$
(12)

It goes to $T_{B}(1+2\,\alpha\lambda)/(1+2\,\alpha\lambda\gamma(t))$ in the
high-$T_{B}$ limit. $T_{S}(t)$ is further reduced to
$T_{B}(1+2\,\alpha\lambda(1-\exp[-p\,t\eta/\alpha]))$ for small values of
$\lambda$. Here and in the following, $\eta$ stands for
$\sin(J\tau)\sin(2J\tau)$ and $\sin^{2}(2\sqrt{2}J\tau)$, respectively, for
sequential and collective collisions. For the sake of readability, the
dependence of $\eta$ on the collectivity, strength, and duration of the
collisions is omitted in the text.

Thermalization of the target qubit to $T_{S}(\infty)\neq T_{B}$ does not mean
that there will be a persistent heat current between the projectile bath and
the qubit; as we will explicitly calculate, the heat flow vanishes at the
steady state. Nevertheless, the possibility of heating or cooling a qubit, in
a gas of qubits initially at thermal equilibrium, by injecting pairwise
discord to the rest of the qubits can be regarded as a transient
thermocoherent effect.

Figure 5: Net heat current between the qubit and discordant heat bath when
$\beta_{S}^{-1}(0)=0.6$ and $\beta_{B}^{-1}=0.5$. The sequential (collective)
collision model is considered in the left (right) column. Upper panels show
$\mathcal{J}$ in the weak-coupling limit ($J=0.05/\tau$), while its value for
different coupling strengths is explored at $t=0.1$ in the lower panels. The
remaining parameters are the same as in Fig. 4. Figure 6: Anomalous heat
current between the qubit and discordant heat bath when
$\beta_{S}^{-1}(0)=\beta_{B}^{-1}=0.5$. The sequential (collective) collision
model is considered in the left (right) column. The upper panels show
$\mathcal{J}$ in the weak-coupling limit ($J=0.05/\tau$), while its value for
different coupling strengths is explored at $t=0.1$ in the lower panels. The
remaining parameters are the same as in Fig. 4.

###  IV.2 Quantum Heat Conduction

To investigate the heat flow, we use the quantum thermodynamic definition
[42],

$\displaystyle\mathcal{J}(t)=\mathrm{tr}[H_{S}\,\frac{\mathrm{d}}{\mathrm{d}t}\rho_{S}(t)],$
(13)

where a positive current means that heat flows from the bath of projectiles to
the qubit and vice versa when $\mathcal{J}<0$. This is simply a consequence of
the fact that the collisions are energy-preserving, which implies that they
have zero work cost and that any energy exchange is pure heat. We further
justify it more rigorously in the Appendix by rewriting the master equation
(31) in the Lindblad form in Appendix A.2, which is used to determine the
quantum heat conduction equation via the Fokker-Planck equation in Appendix B.

Let us express the heat current explicitly, by substituting Eq. (32) into Eq.
(13),

$\mathcal{J}(t)=\mathcal{J}_{0}\,\gamma(t),$ (14)

where the time-independent amplitude is given by

$\mathcal{J}_{0}=\big{(}(q_{g}p_{e}-q_{e}p_{g})+\alpha\lambda(q_{g}-q_{e})\big{)}(E_{e}-E_{g})\,\eta/\alpha,$
(15)

and $\gamma(t)$ depends on $\alpha$, $\lambda$, and $J\tau$ (see Table 1).

The term preceding $\alpha\lambda$ in $\mathcal{J}_{0}$ drops when $T_{S}(0)$
is set to $T_{B}$, and a transient anomalous heat current can be identified to
be

$\mathcal{J}(t)=\lambda\,\eta\,(q_{g}-q_{e})(E_{e}-E_{g})\,\gamma(t).$ (16)

which can only emerge in the case of discordant projectiles. In the high
temperature limit it reduces to

$\mathcal{J}_{h}(t)\approx\gamma(t)\big{(}\tilde{L}_{hh}\Delta\beta-\tilde{L}_{hc}\,\beta_{S}(0)\,\Delta
C\big{)},$ (17)

where $\Delta\beta=(\beta_{S}(0)-\beta_{B})$, $\Delta C=\alpha(0-2\,\lambda)$,
and

$\tilde{L}_{hh}=\tilde{L}_{hc}=(E_{e}-E_{g})^{2}\,\eta/(4\alpha).$ (18)

We conclude that in addition to the initial local temperature difference
between the projectiles and the qubit, quantum coherence in a discordant
projectile acts as an additional source, analogous to chemical potential, to
the heat flow. While the thermal gradient yields the normal heat flow
direction between hot and cold bodies, the discordant source can give rise to
anomalous heat behavior. For example, it can lead to inhibition of heat flow
when these two terms cancel each other if $T_{S}(0)=T_{S}(\infty)\approx
T_{B}(1+2\,\alpha\lambda)$. On the other hand, only the second term survives
when $T_{S}(0)=T_{B}$, and a heat flow persists between two bodies at equal
temperatures. These examples are thermocoherent effects in the transient
regime, and heat flow vanishes in the steady state.

Quantitative investigations of Eqs. (14) and (16), are presented in Figs. 5
and 6, respectively. The net current disappears in spite of the presence of
temperature gradient for some nonzero values of $\alpha\lambda$ in Fig. 5; the
same values of $\alpha\lambda$ sustain a current in the absence of temperature
gradient in Fig. 6. On the other hand, a comparison of Figs. 4 and 6 reveals
that what is responsible for the qubit’s temperature change in the absence of
initial temperature gradient is the anomalous heat current defined in Eq.
(16).

###  IV.3 Quantum Coherence Current

Extending Lord Rayleigh’s principle of the least dissipation of energy,
Onsager showed in Ref. [2] that the rate of increase of the entropy plays the
role of potential in irreversible processes. Here, we consider the quantum
version of this extension [25, 27, 30, 32, 43] that relates the standard
thermodynamic definition of the entropy production to the quantum-information-
theoretical quantities

$\displaystyle\begin{aligned}
\Pi(t)&=-\frac{\mathrm{d}}{\mathrm{d}t}S[\rho_{S}(t)||\rho_{S}^{\infty}]\\\
&=-\mathrm{tr}[\mathcal{D}_{h}(\rho_{S})\ln(\rho_{S}(t))]+\mathrm{tr}[\mathcal{D}_{h}(\rho_{S})\ln(\rho_{S}^{\infty})]\;\\\
&\equiv\dot{S}_{\mathrm{vN}}[\rho_{S}(t)]+\Phi(t),\end{aligned}$ (19)

where $S_{\mathrm{vN}}[\rho]$ is the von Neumann entropy that is equal to
$-\mathrm{tr}[\rho\,\ln\rho]$, and $S[\rho||\sigma]$ is the quantum relative
entropy which measures the distinguishability of the two density matrices
$\rho$ and $\sigma$. The dynamical version of the second law of thermodynamics
[44] is always satisfied during the heat-exchange process under consideration
since the entropy production rate $\Pi$ obtained in this way never becomes
negative,

$\Pi(t)=\mathcal{J}(t)\big{(}\beta_{S}(t)-\beta_{S}(\infty)\big{)}\geq 0\,,$
(20)

even in the absence of initial temperature difference. This equation also says
that the generalized thermodynamic force that generates the net heat current
(14) is simply how far the system’s temperature is from the steady-state
temperature. On the other hand, both $\beta_{S}(t)$ and $\beta_{S}(\infty)$
depend on the value of $\Delta C=-2\,\alpha\lambda$. To split the entropy
production rate (20) into a term that only depends on the temperature
difference, and one that is induced by coherence as in Eq. (17), we can expand
it in the high-temperature limit,

$\Pi(t)\approx\gamma(t)\Big{(}\Delta\beta\,\mathcal{J}_{h}(t)-\beta_{S}(\infty)\,\Delta
C\,\mathcal{J}_{c}(t)\Big{)},$ (21)

where $\mathcal{J}_{h}(t)$ given in Eq. (17) is already properly identified as
heat flow by our Lindblad and Fokker-Planck analyses in the Appendix. After
dropping the multiplication of $\mathcal{J}(t)$ and its conjugate displacement
$\gamma(t)\Delta\beta$ in $\Pi(t)$, we can identify the coherence flow
$\mathcal{J}_{c}(t)$ driven by $-\gamma(t)\beta_{S}(\infty)\,\Delta C$ as

$-\mathcal{J}_{c}(t)\equiv\gamma(t)\big{(}\tilde{L}_{ch}\Delta\beta-\tilde{L}_{cc}\,\beta_{S}(0)\,\Delta
C\big{)},$ (22)

with $\tilde{L}_{cc}=\tilde{L}_{ch}=\tilde{L}_{hc}$. The minus sign preceding
$\mathcal{J}_{c}(t)$ means that the heat and coherence currents are in
opposite directions.

Figure 7: Macroscopic coherence current $\mathcal{J}_{c}(t=0)$ defined in Eq.
(22) (pink solid curve) and microcurrents generated during the first
collective collision of the qubit $S$ with a projectile pair. The change in
the coherence shared between the projectile qubits $B_{1}$ and $B_{2}$
(double-dash-dotted black curve) exactly matches up with
$\mathcal{J}_{c}(t=0)$. This coherence drops in $S$-$B_{1}$ and $S$-$B_{2}$
joint systems without a localization in any single-qubit subsystem and then
returns back to the $B_{1}$-$B_{2}$ joint system. While all the elements of
$\rho_{B_{1}B_{2}}$ are real, off-diagonal elements of $\rho_{SB_{j}}$ are
purely imaginary. The dashed blue (dash-dotted green) curve shows the change
in the imaginary part of the upper (lower) off-diagonal element of
$\rho_{SB_{j}}$. $\lambda=\lambda_{max}$, $E_{1}=1$, $E_{2}=2$, $J=0.05$ in
all panels. The $\\{\beta_{B},\beta_{S}\\}$ pair is (a) $\\{10,10\\}$, (b)
$\\{10,4\\}$, (c) $\\{4,4\\}$, and (d) $\\{4,2\\}$. Note that
$\mathcal{J}_{c}(t=0)$ is equal to twice the steady-state coherence current
$\mathcal{J}_{c}$ given in Eq. (29) when the second projectile bath shares the
same temperature and coherence as the qubit. Figure 8: (a) Temperature of the
qubit bombarded with projectiles in states $\rho_{E}\otimes\rho_{E}$ and (b)
resulting anomalous heat current when $\beta_{S}^{-1}(0)=\beta_{B}^{-1}=0.5$.
$J=0.2/\tau$, while the remaining parameters are the same as in Fig. 4.

As the qubit $S$ always remains in a thermal state, the presence of a
coherence flow described by $\mathcal{J}_{c}(t)$ seems a puzzling issue at
first glance. However, unlike the classical irreversible processes, quantum
coherence can flow in a many-body system without being localized in space. The
single collision of the qubit S with a projectile pair opens two such
delocalized channels through which a micro-current of shared coherence
circulates the global three-qubit system. The collision model framework allows
us to make a microscopical surgery of $\mathcal{J}_{c}(t)$ focusing on this
micro-current. As shown in Fig. 7, the initial coherence shared between
projectile qubits $B_{1}$ and $B_{2}$ drops in $S$-$B_{1}$ and $S$-$B_{2}$
joint systems without a localization in any single-qubit subsystem and then
returns back to $B_{1}$-$B_{2}$ joint system. Although this micro-current
disappears when we reset the projectile pair at the end of a single collision,
multiple collisions create an average delocalized coherence that flows being
shared between the qubit and projectile bath.

A natural question to ask at this point is that why the coherence of the
entangled projectiles does not affect the heat flow by inducing a similar
coherence flow?

Although the non-zero values of $\lambda$ and $\mu$ seem to generate quantum
coherences in projectile states $\rho_{D}$ and $\rho_{E}$ which differ only in
the amount in Fig. 3 (a) and (b), these coherences were found to pertain to
different physical processes in our previous studies [21, 26] of qubit baths.
The coherence of discordant projectiles was classified as heat-exchange
coherence (HEC), whereas the coherence of entangled projectiles was
categorized as squeezing coherence. HECs are also known as “internal” [31] or
“horizontal” [24, 33, 35] coherences in the literature. Please see Ref. [45]
for the details.

On the other hand, two entangled projectile pairs share a HEC of $2\mu^{2}$ in
the state $\rho_{E}\otimes\rho_{E}$. As a matter of fact, when we extend the
collision block to include two projectile pairs, target qubit’s temperature
acquires a $\mu^{2}$ dependence after collective collisions with entangled
projectiles (Fig. 8 (a)). Also, as shown in Fig. 8 (b), such energy-preserving
collisions can generate an anomalous heat current which depends on $\mu^{2}$
in the case of $\Delta\beta=0$.

Based on these observations, we surmise that the mutual induction of heat and
coherence currents originates from energy-preserving collisions with locally
thermal projectiles as long as their quantum correlations are associated with
HECs.

###  IV.4 Quantum Thermocoherent Onsager Relations

$-\Delta C/T$ appears in Eqs. (17) and (22) as if quantum coherence were a
thermodynamic potential, like the chemical potential $\mu_{e}$ in the
thermoelectric effect [3],

$\displaystyle\mathcal{J}_{Q}$ $\displaystyle=$ $\displaystyle
L_{11}\nabla(1/T)+L_{12}\nabla(-\mu_{e}/T),$ (23)
$\displaystyle\mathcal{J}_{N}$ $\displaystyle=$ $\displaystyle
L_{21}\nabla(1/T)+L_{22}\nabla(-\mu_{e}/T),$ (24)

where $\mathcal{J}_{Q}$ and $\mathcal{J}_{N}$ are, respectively, heat and
particle currents. The entropy production rate in thermoelectric phenomena can
be written in terms of these currents and the corresponding thermodynamic
forces,

$\Pi^{T\\!E}=\mathcal{J}_{Q}\cdot\nabla(1/T)+\nabla(-\mu_{e}/T)\cdot\mathcal{J}_{N},$
(25)

and the Onsager relation [2] requires that $L_{12}=L_{21}$.

However, our single-bath model is limited to the transient regime, and it is
necessary for us to explore the case of multiple baths for a more practical
and direct analogy to steady-state thermoelectricity. For that aim, we include
a second bath in the master equation (32). Assume that it consists of
projectiles in state (8b), with $\beta_{B^{\prime}}$ and $\lambda^{\prime}$.
When there is no difference either in the strength or in the duration of
collisions, the steady-state heat flow from the first bath to the qubit can be
written in the high-temperature limit as

$\mathcal{J}_{h}\approx L_{hh}\Delta\beta-
L_{hc}\,\beta_{S|BB^{\prime}}^{\infty}\,\Delta C,$ (26)

where $\Delta\beta=(\beta_{B^{\prime}}-\beta_{B})$, $\Delta
C=2\,\alpha(\lambda^{\prime}-\lambda)$,
$\beta_{S|BB^{\prime}}^{\infty}=(\beta_{B}+\beta_{B^{\prime}})/(2\,(1+\alpha\lambda+\alpha\lambda^{\prime}))$
is the steady-state temperature, and $L_{hh}=L_{hc}=\tilde{L}_{hc}/2$. The
first law of thermodynamics is satisfied in the form
$\mathcal{J}_{h}+\mathcal{J}_{h^{\prime}}=0$.

The second law in Eq. (19), for the case of double baths [27, 32], reads

$\displaystyle\begin{aligned}
\Pi(\infty)&=\dot{S}_{\mathrm{vN}}[\rho_{S|BB^{\prime}}^{\infty}]+\mathrm{tr}[\mathcal{D}_{h}(\rho_{S|BB^{\prime}}^{\infty})\ln(\rho_{S|B}^{\infty})]\\\
&\qquad+\mathrm{tr}[\mathcal{D}_{h^{\prime}}(\rho_{S|BB^{\prime}}^{\infty})\ln(\rho_{S|B^{\prime}}^{\infty})]\\\
&=-\mathcal{J}_{h}\beta_{S|B}^{\infty}-\mathcal{J}_{h^{\prime}}\beta_{S|B^{\prime}}^{\infty},\end{aligned}$
(27)

where the effective temperature $\beta_{S|B}^{\infty}$
($\beta_{S|B^{\prime}}^{\infty}$) is the thermalization temperature attained
only in the presence of the first (second) bath. This equation says that
entropy is generated only in the baths. When we write it as the bilinear
product of generalized forces and generalized fluxes in the high-temperature
limit, we get

$\Pi(\infty)\approx\Delta\beta\,\mathcal{J}_{h}-\big{(}\beta_{S|B}^{\infty}\,C^{\prime}-\beta_{S|B^{\prime}}^{\infty}\,C\big{)}\,\mathcal{J}_{c},$
(28)

where the coherence current $\mathcal{J}_{c}$ is identified as

$-\mathcal{J}_{c}\equiv L_{ch}\Delta\beta-
L_{cc}\,\beta_{S|BB^{\prime}}^{\infty}\,\Delta C,$ (29)

with the coefficients $L_{cc}=L_{ch}=L_{hc}$. The minus sign preceding
$\mathcal{J}_{c}$ means that the heat and coherence flow in opposite
directions.

Equations (26) and (29) bear a striking similarity to the thermoelectric
effect described in Eqs. (23) and (24). This leads us to introduce the term
“thermocoherent effect” to dub the present phenomena. But there is only a
formal analogy between the particle current $\mathcal{J}_{N}$ in Eq. (24) and
the coherent current $\mathcal{J}_{c}$ which has no fundamental analog in
classical physics.

The explicit system under consideration is not artificial, but is a faithful
representation of an actual natural thermalization scheme. Also, the mutual
induction of heat and coherence currents revealed here is not restricted to
the high-temperature regime. Please note that the time-independent amplitude
$\mathcal{J}_{0}$ that is given for more general temperatures in (15) is split
into two terms that depend on $\Delta\beta$ and $\Delta C$. The same is true
for steady-state heat and coherence flows from the first bath to the qubit at
any temperature. However, the linear dependence of the flows on the potentials
$\Delta\beta$ and $-\beta\Delta C$ only appears in the high-temperature limit.
Therefore, the quantum thermocoherent Onsager relations (26) and (29) are only
applicable in this regime.

##  V APPLICATIONS AND POTENTIAL EXPERIMENTS

Discordant states that we found to yield thermocoherent effects can be
produced directly by thermal resources [13]. It is experimentally produced in
nuclear magnetic resonance (NMR) experiments with CHCl3 molecules [29] and
187Yb+ ion trap simulations [34] to demonstrate heat flow reversals.
Thermocoherent Onsager relations provide a natural explanation for such heat
flow reversals, e.g., it is straightforward to find some $\tilde{L}_{jk}$ and
$\Delta C$ values for which $T_{S}(\infty)<T_{S}(0)<T_{B}$ or
$T_{B}<T_{S}<T_{S}(\infty)$. Also, a more comprehensive set of thermocoherent
effects that we explored systematically here could be examined in similar
setups.

The thermocoherent Peltier effect can be exploited to inject quantum coherence
into a region to make compact nonthermal cooling or heating environments,
where injecting heat would be inefficient. Heat transfer in a quantum or
nanoscale can be controlled and induced using well-focused coherent lasers.
Such applications may pave the way for a temperature manipulation over
different regions within a single molecule. A promising example can be
polymerase chain reaction (PCR) technology, where Peltier devices are commonly
used to control such temperature-sensitive biochemical reactions.

The thermocoherent Seebeck effect may have implications for coherent current
generation by a thermal gradient. According to Eqs. (22) and (29), the
temperature difference can produce a flow of coherence even when $\Delta C=0$.
Such a feat allows the thermal production of quantum coherence and
correlations either inside a single bath [13] or between different baths [32],
which has wide-ranging ramifications not only for quantum computers and heat
engines, but also for phononic devices [46]. Thermal diodes and transistors
that can rectify heat flows on the micro- and nanoscales [46] may be
particularly used in the thermocoherent preparation of qubits for information
processing and thermodynamic tasks. Hence, our findings may motivate further
studies to design quantum technologies based on phononics rather than
electronics.

Optimization of thermocoherent coefficients can be used to design “dicoherent
materials” to preserve coherence in a reservoir. Here we showed simple
optimization possibilities in terms of the collision time and collectivity.
Collective collisions in the short time limit result in thermocoherent
coefficients proportional to $8\,J^{2}\tau^{2}$, whereas $L_{ch}$ and $L_{hc}$
are found four times less for sequential collisions. Hence, the thermocoherent
resistivity of the target atom increases with a decrease in collision
collectivity, making sequential collisions a better strategy to protect the
quantumness of projectiles. On the other hand, experimental setups that allow
simultaneous collisions are more suitable to utilize the interconversion
between heat and quantum coherence for specific purposes.

Figure 7 shows that the optimization of thermocoherent coefficients to design
dicoherent materials is neither sensitive to the temperature of the
projectiles nor to the temperature of the qubit. In all four panels, the
magnitude of the coherence and heat currents become zero at the value of
$\tau$ that we pinpoint by the second vertical gray line. This corresponds to
the perfect preservation of the initial coherence and temperature of the
projectiles. The magnitude of the currents reaches its maximum when we halve
this value of $\tau$. This maximum value, and therefore the thermocoherent
effect, can be increased or decreased by adjusting the temperatures.

The thermocoherent phenomena promise improved control over chemical reactions
and nanofabrication techniques. Onsager, in his pioneering paper [2],
investigated a chemical monomolecular triangle reaction $A\rightleftarrows
B\rightleftarrows C\rightleftarrows A$ to exemplify his reciprocity relations.
Although this investigation was restricted to the currents for population
fractions, a molecule can exist in the quantum coherent superposition of two
configurations, e.g., configurations $A$ and $B$ can share quantum coherence
and correlations like the qubit pairs forming our projectiles. According to
our results, such coherences generate new flows from or to configuration $C$,
which can coherently change the chemical equilibrium. Searching for such
mechanisms in biological systems or utilizing them for nanofabrication
applications can be envisioned.

##  VI CONCLUSIONS

This paper follows Lord Rayleigh and Onsager’s footsteps to systematically
establish quantum Onsager relations and associated thermocoherent effects in a
simple system where a qubit is bombarded from a side by two-qubit projectiles.
We allow for arbitrary collision times and consider a set of quantum states of
the projectiles with classical and quantum correlations. In addition,
sequential and collective collision scenarios are examined. We find that
entangled and discordant projectiles contribute to heat exchange provided that
the quantum correlations shared between them are associated with so-called
heat exchange coherences. Proper identification of the discord contribution to
energy exchange as heat is given by deriving the heat conduction equation,
developed by the Fokker-Planck equation out of the master equation. By
calculating the heat current analytically and deducing the coherence current
from the entropy production rate, we showed that quantum Onsager relations can
be derived both in transient and in steady-state regimes; coherent Peltier and
coherent Seebeck effects can also be predicted. Thermocoherent effects can be
further optimized by utilizing collision times and collectivity.

We discussed a set of applications of these effects, such as coherence-based
cooling or heating, coherent heat management at the quantum scale, coherence
protection by dicoherent materials, and thermocoherent control of molecular
processes. Proof of principle experiments can be proposed in existing nuclear
magnetic resonance systems, trapped ions, and circuit or cavity QED setups.

Note added. Recently, we learned that similar reciprocal relations were also
developed between heat and squeezing fluxes in Ref. [47].

###### Acknowledgements.

O.P. acknowledges support by the Scientific and Technological Research Council
of Turkey (TÜBİTAK), Grant No. (120F089).

##  Appendix A THE MASTER EQUATION

The typical micromaser or collision model master equations that rely on short
interaction time approximation [48, 49, 50] are one of the key elements of the
theory of quantum thermalization [21, 26, 51, 52, 53]. The derivation of the
master equation here generalizes them to the arbitrary collision times.

###  A.1 Derivation

After a collision with the qubit, the projectile is discarded for a faithful
representation of the Rayleigh problem where another collision with the same
pair is a rare event (cf. Fig. 1). Each collision is therefore with a fresh
projectile so that we can write a factorized joint qubit-projectile state
$\rho(t_{j})=\rho_{S}(t_{j})\otimes\rho_{B}$ at the arrival time $t_{j}$,
where $\rho_{B}$ is the initial state of the projectile, described in Sec.
III.

Assuming random arrival times, typical for gas of particles, and introducing a
probability of $p\,\delta t$ for a collision to occur in a time interval
$(t,t+\delta t)$, where $p$ is the rate of the Poisson point process, the
evolution of the joint state $\tilde{\rho}$ in the interaction picture can be
written as

$\tilde{\rho}(t+\delta t)=p\,\delta
t\,U(\tau)\tilde{\rho}(t)U^{\dagger}(\tau)+(1-p\,\delta t)\tilde{\rho}(t).$
(30)

Taking the limit $\delta t\rightarrow 0$ and tracing out the projectile
degrees of freedom, we end up with

$\displaystyle\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\rho}_{S}(t)=\mathrm{tr}_{B}[p\,U(\tau)\tilde{\rho}(t)U^{\dagger}(\tau)-p\,\tilde{\rho}(t)].\end{aligned}$
(31)

Here, $\tilde{\rho}(t)=\tilde{\rho}_{S}(t)\otimes\tilde{\rho}_{B}$, where
$\tilde{\rho}_{S}(t)$ and $\tilde{\rho}_{B}$ are the interaction picture
states of the qubit and the projectile, respectively.

###  A.2 Lindblad Form

Now, we decompose the master equation above into the two dissipative terms for
the discordant projectiles,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\rho_{S}(t)=-\frac{\text{i}}{\hbar}[H_{S},\rho_{S}]+\mathcal{D}_{h}(\rho_{S})+\mathcal{D}_{d}(\rho_{S}),$
(32)

where the dissipators can be written in the following Lindblad forms [54]:

$\displaystyle\mathcal{D}_{h}(\rho_{S})$
$\displaystyle=\sum_{j=1}^{2}\kappa_{j}\big{(}A_{j}\rho_{S}A^{\dagger}_{j}-\frac{1}{2}\\{A^{\dagger}_{j}A_{j},\rho_{S}\\}\big{)},$
(33a) $\displaystyle\mathcal{D}_{d}(\rho_{S})$
$\displaystyle=\sum_{j,k=1}^{2}\bar{\kappa}_{j,k}\big{(}\bar{A}_{j}\rho_{S}\bar{A}^{\dagger}_{k}-\frac{1}{2}\\{\bar{A}^{\dagger}_{k}\bar{A}_{j},\rho_{S}\\}\big{)},$
(33b)

where $A_{1}=\sigma^{-}$, $A_{2}=\sigma^{+}$,
$\bar{A}_{1}=\sigma^{-}\sigma^{+}$, and $\bar{A}_{2}=\sigma^{+}\sigma^{-}$.

$\mathcal{D}_{d}(\rho_{S})$ can be cast into the simple form

$\mathcal{D}_{d}(\tilde{\rho}_{S})=\begin{pmatrix}0&-c{\tilde{\rho}_{S}}_{1,2}\\\
-c{\tilde{\rho}_{S}}_{2,1}&0\end{pmatrix},$ (34)

where $c:=\bar{\kappa}_{1,2}-(\bar{\kappa}_{1,1}+\bar{\kappa}_{2,2})/2$. We
find $c=\sin^{4}(J\tau)/2\geq 0$ and
$c=2(p_{g}^{2}+p_{e}^{2})\sin^{4}(\sqrt{2}J\tau)\geq 0$ for sequential and
collective collisions, respectively. Moreover,
$\mathrm{tr}[H_{S}\,\mathcal{D}_{d}(\rho_{S})]=0$. Accordingly,
$\mathcal{D}_{d}(\rho_{S})$ describes energy preserving dephasing collisions
of projectiles with the qubit, which does not contribute to the heat flow.

$\mathcal{D}_{h}(\rho_{S})$ describes two competing processes associated with
favorable and unfavorable collisions [1], following the terminology of Lord
Rayleigh. In contrast to dephasing collisions, these processes result in
energy exchange between the projectiles and the qubit in the form of heat, as
we will further justify below by calculating a heat conduction equation for
the system. We find the dissipation rates
$\kappa_{1}=\kappa_{2}=(p_{g/e}+\alpha\lambda)\,\eta/\alpha$. Favorable
(unfavorable) collisions result in a transition from
$|e\rangle\rightarrow|g\rangle$ ($|g\rangle\rightarrow|e\rangle$) at a rate
proportional to $p_{g}$ ($p_{e}$). Under the detailed balance of favorable and
unfavorable collisions, the competition ends with an equilibrium state that
can be described by an empirical temperature $T_{S}(\infty)$ (cf. Eq. (11)).
$T_{S}(\infty)$ can be different than the local temperature of projectile
qubits due to the contribution of $\alpha\lambda$ terms in $\kappa_{1/2}$.

##  Appendix B THE FOKKER-PLANCK EQUATION

Remarkably, identification of the energy flow described by Eq. (13) as heat is
not trivial in quantum thermodynamics, especially in the case of nonthermal
baths. A variation of the state may not always increase entropy, and worklike
contributions in Eq. (13) need to be removed to properly define the heat flow
[55].

To ensure the projectile state’s discordant contribution can be identified as
a genuine heat flow for proper thermocoherent description, we further examine
the generalized Fokker-Planck equation using Haken’s representation [56].
Assuming the initial thermal condition for the qubit, we can exactly drop
$\mathcal{D}_{d}$ out of the open system dynamics. The phase-space
distribution $P(\upsilon,\upsilon^{*},m,t)$ then satisfies

$\displaystyle\frac{\partial P}{\partial
t}=L\Big{(}\upsilon,\upsilon^{*},m,\frac{\partial}{\partial\upsilon},\frac{\partial}{\partial\upsilon^{*}},\frac{\partial}{\partial
m},\Big{)}P,$ (35)

which yields

$\displaystyle\begin{aligned}
P&=\Gamma_{2}\,\delta(m-1)\,\delta^{(2)}(v)+\Gamma_{1}\,\delta(m+1)\,\delta^{(2)}(v)\\\
&\qquad+\Gamma_{2}\,\delta(m+1)\frac{\partial^{2}}{\partial\upsilon\partial\upsilon^{*}}\delta^{(2)}(v).\end{aligned}$
(36)

Here $\Gamma_{1/2}=\langle g/e|\rho_{S}(t)|g/e\rangle$, $m$ and $\upsilon$
are, respectively, the inversion and polarization variables, and the
differential operator $L$ is equal to

$\displaystyle\begin{aligned}
L&=\mathrm{i}\,(E_{e}-E_{g})\Big{(}\frac{\partial}{\partial\upsilon}\upsilon-\frac{\partial}{\partial\upsilon^{*}}\upsilon^{*}\Big{)}\\\
&\;+(p_{g}+\alpha\lambda)\frac{\eta}{2\alpha}\Big{[}(e^{2\frac{\partial}{\partial
m}}-1)(1+m)+\frac{\partial}{\partial\upsilon}\upsilon+\frac{\partial}{\partial\upsilon^{*}}\upsilon^{*}\Big{]}\\\
&\;+(p_{e}+\alpha\lambda)\frac{\eta}{2\alpha}\Big{[}(e^{-2\frac{\partial}{\partial
m}}-1)(1-m)+2\frac{\partial^{2}}{\partial\upsilon\,\partial\upsilon^{*}}\\\
&\;+2\Big{(}e^{-2\frac{\partial}{\partial
m}}+\frac{\partial^{2}}{\partial\upsilon\,\partial\upsilon^{*}}-\frac{1}{2}\Big{)}\Big{(}\frac{\partial}{\partial\upsilon}\upsilon+\frac{\partial}{\partial\upsilon^{*}}\upsilon^{*}\Big{)}\\\
&\;+\frac{\partial^{4}}{\partial\upsilon^{2}\partial\upsilon^{*2}}e^{2\frac{\partial}{\partial
m}}(1+m)\Big{]}.\end{aligned}$ (37)

We can identify the heat conduction terms in Eq. (35) by neglecting the drift
terms in $L$, as well as the derivatives beyond the second order. The
generalized Fokker-Planck equation then turns out to be the heat equation

$\displaystyle\frac{\partial P}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{\eta}{\alpha}\Big{[}(p_{e}+\alpha\lambda)\Big{(}\frac{\partial^{2}}{\partial\upsilon\,\partial\upsilon^{*}}-2\frac{\partial^{2}}{\partial
m\,\partial\upsilon}\upsilon-2\frac{\partial^{2}}{\partial
m\,\partial\upsilon^{*}}\upsilon^{*}\Big{)}$ (38)
$\displaystyle\qquad+\big{(}1+m(p_{g}-p_{e})+2\,\alpha\lambda\big{)}\frac{\partial^{2}}{\partial
m^{2}}\Big{]}P,$

where each thermal conductivity coefficient can be explicitly split into a
purely thermal population term and a quantum coherent $\alpha\lambda$ term.
This quantum heat conduction equation justifies that initial quantum coherence
of projectile qubits yielding discordant correlations contributes to the
genuine heat flow.

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