Breakdown of detailed balance for thermal radiation by synthetic elds S.-A. Biehs Institut f ur Physik Carl von Ossietzky Universit at D-26111 Oldenburg Germany

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Breakdown of detailed balance for thermal radiation by synthetic ﬁelds

S.-A. Biehs∗

Institut f¨ur Physik, Carl von Ossietzky Universit¨at, D-26111 Oldenburg, Germany

G. S. Agarwal†

Institute for Quantum Science and Engineering and Department of Biological

and Agricultural Engineering Department of Physics and Astronomy,

Texas A & M University, College Station, Texas 77845, USA

In recent times the possibility of non-reciprocity in heat transfer between two bodies has been

extensively studied. In particular the role of strong magnetic ﬁelds has been investigated. A much

simpler approach with considerable ﬂexibility would be to consider heat transfer in synthetic electric

and magnetic ﬁelds which are easily applied. We demonstrate the breakdown of detailed balance for

the heat transfer function T(ω), i.e. the spectrum of heat transfer between two objects due to the

presence of synthetic electric and magnetic ﬁelds. The spectral measurements carry lot more physical

information and were the reason for the quantum theory of radiation. We demonstrate explicitly

the synthetic ﬁeld induced non-reciprocity in the heat transfer transmission function between two

graphene ﬂakes and for the Casimir coupling between two objects. Unlike many other cases of heat

transfer, the latter case has interesting features of the strong coupling. Further the presence of

synthetic ﬁelds aﬀects the mean occupation numbers of two membranes and propose this system for

the experimental veriﬁcation of the breakdown of detailed balance.

Reciprocity and detailed balance are at the heart of

Kirchhoﬀ’s law stating that the absorptivity equals emis-

sivity for any frequency and angle of incidence. In fact,

the second law of thermodynamics enforces the reci-

procity or better detailed balance of the radiative heat

transfer between two objects. Here it is unimportant if

far-ﬁeld heat transfer where Planck’s blackbody deter-

mines the upper limit is considered or near-ﬁeld heat

transfer where the blackbody limit is not a limit any-

more [1–3] as experimentally tested by a great number of

experiments [4–12] within the last decade. How the sec-

ond law enforces detailed balance can be understood [13]

by considering the heat ﬂux between two objects by ﬁrst

taking the transferred power from object ato object b

Pa→b=Z∞

dω

2π¯hωna(ω)Tab(ω) (1)

where ¯his the Planck constant, na(ω) =

(exp(¯hω/kBTa)−1)−1is the photonic occupation

number, kBis the Boltzmann constant, and Tais the

temperature of object a. The quantity Tab(ω) is a heat

transfer function (HTF) for the heat ﬂow from object

ato object b. Similarly, the heat ﬂow from object bto

object ais given by

Pb→a=Z∞

dω

2π¯hωnb(ω)Tba(ω) (2)

with nb(ω) = (exp(¯hω/kBTb)−1)−1and Tbthe temper-

atur of object b. In thermal equilibrium the objects have

the same temperature Ta=Tband therefore there is no

net heat ﬂow which means that Pa→b=Pb→aand hence

Z∞

dω

2π¯hωna(ω)Tba(ω)− Tab(ω)= 0.(3)

Since this expression holds for any value of temperature

Ta=Tband therefore for diﬀerent spectral weighting

by nait can be concluded that the validity of the sec-

ond law of thermodynamics is equivalent to the relation

Tab(ω) = Tba(ω) regardless of any symmetry [14]. That

means that even when time reversal symmetry is broken

by applying a magnetic ﬁeld or using topological Weyl

semi-metals, for instance, detailed balance of the energy

HTF must be fulﬁlled. However, in non-reciprocal sys-

tems the detailed balance of thermal radiation can be

nearly completely violated when considering three ob-

jects [15] which also oﬀers applications for optimized non-

reciprocal thermo-photovoltaic energy conversion [16].

Similarly, the HTFs do not need to fulﬁll Tab(ω) = Tba(ω)

when at least a third object cor a non-reciprocal environ-

ment are present. In such many-body systems therefore

several interesting eﬀects for thermal radiation in general

and radiative heat exchange in nanoparticle systems [3]

in particular could be highlighted like persistent heat cur-

rents and heat ﬂuxes [17–19], persistent spin and angu-

lar momenta [18–20], giant thermal resistance [13, 21],

a normal and anomalous Hall eﬀect for thermal radia-

tion [22–24], as well as a diode eﬀect with non-reciprocal

surface waves [25]. In all these studied systems, in order

to realize a non-reciprocal heat ﬂux or a violation of de-

tailed balance the presence of a third body seems to be

a necessary condition. However, within the framework

of ﬂuctuational electrodynamics and the scattering for-

malism [26, 27] a formal proof detailed in Ref. [28] shows

that Tab(ω) = Tba(ω) if the environment and the objects

fulﬁll both Lorentz reciprocity [29]. Therefore in princi-

ple for radiative heat transfer between two objects with

non-reciprocal properties in a reciprocal environment de-

tailed balance can be broken even though in practive this

arXiv:2210.13049v1 [cond-mat.mes-hall] 24 Oct 2022

has not been observed so far.

Interestingly, the presence of synthetic electric and

magnetic ﬁelds oﬀers the possibility to break the detailed

balance of energy transmission functions even for only

two coupled resonators which results in a non-reciprocal

energy transmission as shown theoretically and veriﬁed

experimentally [30]. The synthetic ﬁelds are generated

by external modulation of the resonance frequency of

the two resonators which ﬁrst of all generates side bands

which can be understood by the presence of a synthetic

electric ﬁeld in the synthetic frequency domain [31].

When the modulation of the two resonators is phase-

shifted a synthetic magnetic ﬁeld for the photons is gen-

erated [32] which enables the Aharonov-Bohm eﬀect for

photons [33], for instance. Now, dynamic modulations

of temperatures or material properties have also been

considered for modulation of radiative heat exchange be-

tween two or more objects [34–37] showing that the mod-

ulation of the temperature or chemical potential can re-

sult in a shuttling eﬀect [38] and the modulation of mate-

rial properties can be used to modulate the radiative heat

ﬂux between two or more objects [39, 40]. However, in

all those works the HTF for the radiative heat exchange

between two bodies are again strictly fulﬁlling detailed

balance, i.e. Tab(ω) = Tba(ω).

In this letter, by using a quantum Langevin equa-

tion approach to treat heat transfer we show that syn-

thetic ﬁelds can lead to a breakdown of detailed balance

for the HTF between two resonant objects, i.e. we ex-

plicitely show that Tab(ω)6=Tba(ω). We further show

that this broken detailed balance does not result in a

non-reciprocal heat ﬂux, i.e. we still have Pa→b=Pb→a

and the validity of Eq. (3). We will discuss these features

for the radiative heat ﬂux between two graphene ﬂakes in

which case the synthetic ﬁelds are realized by modulating

Fermi energies. Furthermore, we propose to measure the

broken detailed balance in the strong-coupling regime of

two Casimir-force coupled membranes as used in recent

experiments like in Ref. [41].

In the following we describe the near-ﬁeld radiative

heat ﬂux between two graphene ﬂakes as well as Casimir

force coupled membranes by two coupled oscillators [42–

44]. The oscillator frequencies ωa/b then correspond to

the frequencies of the main optical or vibrational modes

of the graphene ﬂakes or the membranes and their damp-

ing is described by the damping constants κa/b. The

coupling strength between the oscillators gquantiﬁes the

interaction strength of the graphene ﬂakes or membranes

due to the ﬂuctuational electromagnetic ﬁelds which are

at the origin of the radiative heat transfer and Casimir

force. Then the coupled oscillators can be described by

a set of two quantum Langevin equations [45, 46]

˙a=−iωaa−κaa−igb +Fa,(4)

b=−iωbb−κbb−iga +Fb(5)

−Ω −Ω

+Ω+Ω gκ

β/2 θ

β/2

κg

β/2 θ

β/2

bath a bath b

heat

FIG. 1. Sketch of the forward heat ﬂux Pa→bin the consid-

ered two couples oscillators with periodic modulation in the

synthetic dimension with the synthetic electric and magnetic

ﬁelds Eand B.

for the lowering operators aand bof the two coupled

oscillators. Furthermore, both oscillators are assumed to

be coupled to their own baths which enter here through

the bath operators Fa/b into the description.

Now, we introduce synthetic electric and magnetic

ﬁelds via the identical frequency modulation of both os-

cillators

ωa→ωa+βcos(Ωt) and ωb→ωb+βcos(Ωt+θ) (6)

with modulation frequency Ω, amplitude βand with

a phase shift θ. By Fourier transforming the coupled

Langevin equations into frequency space we obtain the

set of equations in the compact form

ψ=MF+β

2iMQ+ψ++β

2iMQ−ψ−(7)

by introducing the vectors ψ=a(ω), b(ω)t,ψ±=

a(ω±Ω), b(ω±Ω)t, and F=Fa(ω), Fb(ω)t, and the

matrices

M=A−1with A=Xaig

ig Xb(8)

so that

M=1

XaXb+g2Xb−ig

−ig Xa(9)

introducing Xa/b =−i(ω−ωa/b) + κa/b and the diagonal

matrix Q±= diag1,e±iθ. This compact form makes

obvious that we have an inﬁnite set of equations in fre-

quency space due to the coupling to the sidebands ±Ω,

±2Ω, etc. introduced by the modulation. These side-

bands can be understood as generated by an electric syn-

thetic ﬁeld along the synthetic frequency axis (see Fig. 1).

Furthermore, the phase shift itself can be interpreted by

a synthetic magnetic ﬁeld [30] which adds a phase Q+for

“upward” and Q−for “downward” transitions in the fre-

quency bands. Recently, it has been shown theoretically

and experimentally that this synthetic magnetic ﬁeld re-

sults in non-reciprocal energy transmission in coupled os-

cillator systems [30]. From the mathematical expression

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摘要：

BreakdownofdetailedbalanceforthermalradiationbysyntheticeldsS.-A.BiehsInstitutfurPhysik,CarlvonOssietzkyUniversitat,D-26111Oldenburg,GermanyG.S.AgarwalyInstituteforQuantumScienceandEngineeringandDepartmentofBiologicalandAgriculturalEngineeringDepartmentofPhysicsandAstronomy,TexasA&MUniversity,Co...

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Breakdown of detailed balance for thermal radiation by synthetic elds S.-A. Biehs Institut f ur Physik Carl von Ossietzky Universit at D-26111 Oldenburg Germany

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