Light emission in delta- T-driven mesoscopic conductors M. H ubler1and W. Belzig1 1Fachbereich Physik Universit at Konstanz D-78457 Konstanz Germany

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Light emission in delta-T-driven mesoscopic conductors

M. H¨ubler1and W. Belzig1

1Fachbereich Physik, Universit¨at Konstanz, D-78457 Konstanz, Germany

(Dated: April 14, 2023)

The scattering picture of electron transport in mesoscopic conductors shows that ﬂuctuations

of the current reveal additional information on the scattering mechanism not available through

the conductance alone. The electronic ﬂuctuations are coupled to the electromagnetic ﬁeld and

a junction at ﬁnite bias or temperature will emit radiation. The nonsymmetrized current-current

correlators characterize the emission and absorption spectrum. Recent interest is focused on the so-

called delta-Tnoise, which is the nonequilibrium noise caused by a temperature diﬀerence between

the terminals. Here, we generalize the notion of delta-Tnoise to the nonsymmetrized current-

current correlator at ﬁnite frequencies. We investigate the spectral density for energy-independent

scattering and for a resonant level as an example of energy-dependent scattering. We ﬁnd that a

temperature diﬀerence ∆Tleads to a partial reduction of the noise for certain frequency ranges. This

is a consequence of temperature broadening in combination with a frequency shift of the involved

Fermi distributions. In the case of energy-independent scattering, the lowest order is a quadratic

∝(∆T)2correction of the thermallike noise spectrum. For the resonance, an additional contribution

to the delta-Tnoise spectrum arises that is ∝∆Tto the lowest order.

I. INTRODUCTION

The electron transport in mesoscopic conductors is

investigated using the statistics of the electron current,

where the ﬁrst moment corresponds to the average

current and the second moment to the noise [1–3].

Unavoidable sources of noise are thermal noise at a

ﬁnite temperature - the so-called Nyquist-Johnson noise

[4, 5] and nonequilibrium shot noise [6]. The former

is caused by thermal ﬂuctuations in the occupation

number and the latter by the stochastic partitioning of

charge carriers. The noise at a tunnel junction can be

used for primary thermometry [7].

At ﬁnite frequencies, the noise involves current op-

erators taken at diﬀerent times. In general, these

operators do not commute, so the symmetrized cor-

relator is studied as an observable [1, 8]. A detector

that distinguishes between the transfer of an energy

quanta ~ωfrom or to the conductor can access the

nonsymmetrized correlator [8, 9]. Indeed, when the

ﬂuctuations interact with an electromagnetic ﬁeld, the

energy transfer rate is connected to the nonsymmetrized

noise spectrum. Negative frequencies account for the

radiated power when one photon is generated in the

radiation ﬁeld and, vice versa, positive frequencies for

the absorbed power when one photon is annihilated.

In a thermally occupied radiation ﬁeld, the measured

noise power spectrum is a sum of the nonsymmetrized

noise spectra at negative and positive frequencies.

The prefactors are determined by the Bose-Einstein

distribution and, consequently, by the temperature of

the electromagnetic ﬁeld [8, 9].

Shaping a possible ac-excitation can strongly inﬂu-

ence the noise properties [10] and can be interpreted

as electron-hole pair excitation on the Fermi sea [11].

A noise reduction due to driving was experimentally

observed [12, 13] and measurements at ﬁnite frequency

reveal a squeezed nonequilibrium state [14].

A fundamental nonequilibrium noise due to a tem-

perature diﬀerence ∆Twas recently demonstrated

by Lumbroso and coworkers [15–17] in atomic and

molecular junctions. This noise, dubbed delta-Tnoise, is

related to the voltage-driven shot noise and inherits the

properties of partition noise [18]. Using the scattering

approach, they obtain an approximation of the noise,

which is then decomposed into a thermal and a delta-T

component. The thermal component corresponds to

thermal noise at the average temperature, and the

lowest order delta-Tcomponent is similar to the quan-

tum shot noise except for diﬀerent numerical prefactor

and scales with (∆T)2instead of the voltage squared.

Another study [17] measured and calculated the noise

of a voltage-and temperature-biased metallic tunnel

junction. This setup operates at a very low temperature

and is not restricted to small relative temperature

diﬀerences. At the limit, when one terminal is at zero

temperature and no voltage is applied, the noise has the

form of thermal noise with an additional factor 2 ln 2 [18].

In a quantum Hall bar furnished with a quantum

point contact, the delta-Tnoise can serve as an instru-

ment to discriminate between electron and quasiparticle

tunneling [19, 20]. Tunneling of chiral fractional quan-

tum Hall edge states exhibits a negative delta-Tnoise, in

contrast to a positive contribution in the noninteracting

case. A sign inversion, from negative back to positive,

may also be forced by changing the transmission of

the quantum point contact or applying a voltage. The

negative signal is attributed to the scaling dimension of

the leading charge tunneling operator [21, 22]. Their

results suggest that the negative sign is a property due

to many-body interactions. In comparison, a quantum

dot in the SU(2) Kondo region has no negative delta-T

arXiv:2210.04984v2 [cond-mat.mes-hall] 12 Apr 2023

barrier

0.0

0.5hot emitter

0.0

0.2

−

∆T-noise

0 5

~ω/[kB(Th+Tc)]

0.0

0.2cold emitter

FIG. 1. The mesoscopic conductor consists of a hot reservoir

with a temperature Thand a cold reservoir with a tempera-

ture Tc, separated by a potential barrier. In the hot reservoir,

more electrons are excited at higher energies, as illustrated by

electrons farther away from the surface. The current-current

correlator depends on the scattering amplitudes (simpliﬁed

depicted by the arrows) of electrons (depicted by circles with

a minus sign) and holes (depicted by circles with a plus sign)

as well as the probability distribution of the terminal from

which they originated. Current-current ﬂuctuations are re-

lated to the energy transfer rate between the mesoscopic con-

ductor and an electromagnetic ﬁeld, i.e. to the emission and

absorption spectrum. The energy diﬀerence ~ωbetween the

electron and the hole corresponds to the energy of the anni-

hilated or excited photon. We separate the spectrum into an

equilibrium part (thermallike), a superposition of a hot and

cold emitter, and a nonequilibrium part, given by a delta-

Tnoise spectrum. The delta-Tnoise spectrum is negative

at some frequencies, thus diminishing the thermallike noise

spectrum. In the shown case, energy-independent scattering

is assumed.

noise [23], thus the eﬀect does not occur in this case

despite the presence of strong correlations. Further,

delta-Tnoise was employed to study experimentally the

heat transport of edge modes [24].

An investigation of the relative sizes shows that

delta-Tnoise never exceeds the thermallike noise under

the zero-average current condition [25]. In [25] they stud-

ied a resonant level as an example for energy-dependent

scattering. In the limit of a small resonance width, the

size of delta-Tnoise approaches the thermallike noise.

Furthermore, they investigated noise of heat transport,

which is not subject to a limit like charge noise. More

recently, bounds on the spin and heat current noise were

investigated [26].

In this work, we address the nonsymmetrized ﬁnite-

frequency noise spectrum of a temperature-biased

mesoscopic conductor. We are interested in separating

the light emission and absorption into a thermallike

and delta-Tspectrum. Figure 1 gives an illustrative

summary of the considered system and our ﬁndings

in the case of energy-independent scattering. The

mesoscopic conductor is described within the scattering

approach, where one terminal assumes a hot tempera-

ture and the other a cold temperature. We deﬁne the

thermallike noise spectrum as the average of the thermal

noise spectra at the hot and the cold temperature.

Consequently, the delta-Tnoise spectrum is deﬁned

similar to the excess noise spectrum [1, 27, 28]. Two

distinct contributions to the delta-Tnoise are obtained,

S∆T

1(ω) comes from the correlations of occupied and

free electronic states with diﬀerent Fermi statistics,

and S∆T

2(ω) from occupied and free states with the

same statistics but associated with a diﬀerent rate per

recombination event than assumed in the thermallike

noise. If the scattering is energy independent or the

same from both sides, the latter contribution vanishes.

Our main result is that the delta-Tnoise spectrum

can get negative at some frequencies, reducing the

thermallike noise spectrum. Below, we investigate the

spectra for energy-independent scattering and a single

resonant level model. In the resonant case S∆T

1(ω) has a

suppressed negative part for the chosen parameters and

additionally shows a contribution S∆T

2(ω).

The work is structured as follows. In the second

section II, we introduce the scattering approach and

summarize the connection to light emission and ab-

sorption. Afterwards, we deﬁne the thermallike noise

spectrum and discuss the delta-Tnoise spectrum for

general scattering matrices. As an application, we

consider energy-independent scattering in section III

and then a resonant level in section IV. Our results are

recapped in section V.

II. SCATTERING APPROACH TO

DELTA-T-DRIVEN CONDUCTORS

We consider a mesoscopic two-terminal conductor

modeled in the scattering approach as two macroscopic

electron reservoirs connected by waveguides to a scatter-

ing region [1, 29]. Uncorrelated electrons leave the reser-

voir and transverse through the scattering region, where

they are elastically scattered. Interactions between the

electrons and charging eﬀects are disregarded. We de-

note the hot terminal as 1 and the cold one as 2. The

Fermi functions fα(E) = {exp[βα(E−µα)] + 1}−1gov-

ern the energy distribution of emanating electrons from

terminals α∈ {1,2}. The parameters β1= 1/kBThand

β2= 1/kBTcare determined by the temperatures Thand

Tcof the hot and cold reservoir, respectively. We suppose

that there is no applied voltage and set µ1=µ2= 0 in

the following. The transport is described by the unitary

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Lightemissionindelta-T-drivenmesoscopicconductorsM.Hubler1andW.Belzig11FachbereichPhysik,UniversitatKonstanz,D-78457Konstanz,Germany(Dated:April14,2023)Thescatteringpictureofelectrontransportinmesoscopicconductorsshowsthatuctuationsofthecurrentrevealadditionalinformationonthescatteringmechanismnot...

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Light emission in delta- T-driven mesoscopic conductors M. H ubler1and W. Belzig1 1Fachbereich Physik Universit at Konstanz D-78457 Konstanz Germany

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