Generalizing Planck's Law: Nonequilibrium Emission of Excited Media

0.15 MB·English

by  Klaus Henneberger

Generalizing Planck’s law:
Nonequilibrium emission of excited media
∗
K. Henneberger
University of Rostock, Institute of Physics, 18051 Rostock, Germany
Using a quantum-kinetic many-body approach, exact results for the interacting system of field
and matter in a specified geometry are presented. It is shown that both the spectral function
of photons and the field fluctuations split up into vacuum- and medium-induced contributions,
for which explicit expressions are derived. Using Poynting’s theorem, the incoherent emission is
analyzed and related to the coherent absorption as may be measured in a linear transmission-
9
reflection experiment. Their ratio defines the medium-induced population of the modes of the
0
transverse electromagnetic field and so generalizes Planck’s law to an arbitrarily absorbing and
0
dispersingmediuminanonequilibriumsteadystate. Forquasi-equilibrium,thispopulationdevelops
2
intoaBose distributionwhosechemicalpotentialmarksthecrossoverfrom absorption togainand,

n also, characterizes the degree of excitation. Macroscopic quantum phenomena such as lasing and
a quantumcondensation are discussed on this footing.
J

9 PACSnumbers: 42.50.Ct,42.50.Nn,78.45.+h
1

] INTRODUCTION the medium are in equilibrium but in different steadily
l
e excited states arbitrarily far from equilibrium. (c) Ab-
-
sorption as measured by a linear transmission-reflection
r Since the establishment of Kirchhoff’s law till today,
t experiment is investigated on the same footing and its
s emissionandabsorptionarethefundamentalobservables
. relation to emission is shown.
t in spectroscopic investigations. Assuming both ideal
a
thermal radiation outside and thermodynamic equilib-
m
rium inside a medium and, also, between outside and
- GENERAL THEORY
d inside, Kirchhoff’s law e(ω) = n(ω,T)a(ω) fixes the ra-
n tio of emission e to absorption a as a universal func-
o tion n of frequency ω and temperature T. Thus, emis- We assume steady state conditions and start with the
c field operator of the vector potential in Coulomb gauge,
sion can be traced back to (i) absorption being observ-
[ which obeys the inhomogeneous wave equation
able in transmission-reflection experiments using classi-

3 cal (coherent) light, and (ii) the spectral distribution n 1 ∂2
v of thermal light, which later was found to be a Bose ∆− c2∂t2 Aˆ(r,t)=−µ0 [ˆj(r,t)+jext(r,,t) ] (1)
6 (cid:20) (cid:21)
function by Planck. However, if the very restricting as-
8 and the equal time commutation rules
6 sumptionofcompletethermodynamicalequilibriumisre-
5 moved,theemissionofamediumistobeconsideredasa Aˆ(r,t), ∂ Aˆ(r′,t) = i~c←→t (r−r′), (2)
0. purequantumeffect,andquantum-opticalandquantum- (cid:20) ∂t (cid:21) ε0
1 kineticmethods havetobe usedtodescribetheinteract- ←→
where t is the transverse delta function tensor. In
7 ing many-body system of field and matter. Using the
0 eq. (1), the total transverse current density operator is
Keldyshtechnique for the photonGreen’s function (GF)
: split into the current density operator ˆj of the medium
v [1] and neglecting effects of spatial dispersion (SD), a
i formula for the intensity emitted by a steadily excited and an externally controlled current density represented
X by given c-number functions j (r,t).
semiconductor into the surroundingvacuum was derived ext
r The Keldysh components of the photon GF
a in ref. [2].
In this article this result will be generalized in the D>(r,r′,t−t′)=D<(r′,r,t′−t)
ij ji
following fundamentally important aspects: (a) Rigor-
1
ous results for the emission will be given in terms of = hAˆ (r,t)Aˆ (r′,t′)i−hAˆ (r,t)ihAˆ (r′,t′)i , (3)
i~ i j i j
the exactly defined polarization function of an isotropic
h i
describe the field-field fluctuations. Formallysolving the
butotherwisearbitrarymediumasanenergeticallyopen
Dysonequationforthesecomponents,oneobtainstheso-
and spatially inhomogeneous finite system. Hence, this
called optical theorem after Fourier transformation with
approach applies independently of any further specific
′
respect to t−t →ω
material properties or models and, besides others, also
SD is exactly considered. (b) Instead of an (electromag-
netic) vacuum, an arbitrary nonequilibrium distribution Di≷j(r,r′,ω)= d3r1d3r2 ×
of photons is assumed to be present. In this way a sit- Z
Dret(r,r ,ω)P≷(r ,r ,ω)Dadv(r ,r′,ω), (4)
uation will be addressed where neither the radiation nor ik 1 kl 1 2 lj 2
2
where P≷ are the Keldysh components of the polariza- fromvacuumtomediumand,consequently,thesolutions
kl
tion function, which is givenby the functionalderivative of(5)evolvecontinuously,too,fromthe asymptoticones
(in compact notation: P =δj/δA) of the induced (aver- (comparealsothe discussionin[4]). Onlyif oneassumes
aged)currentdensityj=hˆjitothemeanfieldA=hAˆi. an abrupt switch of the polarization at the surface, one
Throughout this paper, P = δj/δA is to be taken at hastomakesurethe continuityofthe solutionsof(5)by
A = 0, i.e., in the linear approximation. This is evident imposing Maxwell’s boundary conditions.
since linear absorption is addressed, on the one hand,
and since the incoherent emission is defined as the one
TRANSMISSION AND REFLECTION
without any incident classical field, i.e., A(r,t) = 0, on
the other hand.
In the following it is assumed that up to a negligible
errorthe surface of the medium, i.e., the length L of the
SLAB GEOMETRY slab, can be fixed in a way that any polarization van-
ishes outside and, hence, any Keldysh component of the
A slab of thickness L will be considered, which is polarizationfunctionvanishesoutside,too. Thenthefor-
infinitely extended and homogeneously excited in the ward propagating solution of the homogeneous equation
transverse y-z-direction. For notational simplicity, TE- (5) has the structure
polarizedlightpropagatingfreelyinthetransversedirec- eiq0x+re−iq0x for x<−L
tion is considered. Due to cylindrical symmetry around A(x) = 2 (6)
the x-axis, the transversevectorpotential Aˆ(r,t) can be ( teiq0x for x> L,
2
chosen in the z-direction.
where r,t are reflection and transmission coefficients
for the field amplitudes. Due to the symmetry x ↔
−x, A(−x) is a solution, too (backward propagating,
CLASSICAL FIELD PROPAGATION
i.e., incidence from right). The solutions inside need
not to be specified. Neglecting spatial dispersion, i.e.,
Assuming steady state conditions, the propagation assuming Pret(x,x′) = Pret · δ(x − x′), yields inside
equation for the average field, after Fourier transform- A(x) = aeiqx +be−iqx, where the wave vector is given
ing with respect to (y,z)→q⊥ and t−t′ →ω, in this by q2 =q2−Pret. In this case, the results of ref. [2] can
geometry has the structure 0
be reproduced.
ManyattemptshavebeenmadetoconsiderSDbytrac-
dx′Dret,−1(x,x′)A(x′)=−µ0jext(x), (5) ingbackthis problemto the oneofthe bulk limit, where
Z duetofullspatialhomogeneitythepolarizationfunctions
P(q,ω)canbehandled. Mostcommonistheuseofbulk
Dret,−1(x,x′)= ∂2 +q2 δ(x−x′)−Pret(x,x′). (polariton)solutionsfor the waves(6)inside. This,how-
∂x2 0 ever,is notconsistentwitheq. (5)andintroducesahigh
(cid:18) (cid:19)
degreeofarbitrarinessintothe problem,since additional
Here and in what follows, the variables q⊥ and ω, boundary conditions (ABC’s) [3] have to be used. The
which enter all equations parametrically only, are omit- dielectric approximation [4] uses the bulk susceptibility
ted where possible. The wave vector in vacuum is q02 = inside. It is free of arbitrariness but thought to con-
[(ω+iδ)/c)]2−q⊥2, and Dret,−1 is the inverse of the re- flict with energy conservation [10]. The present author
tardedphotonGF.Theretardedpolarizationfunctionof suggested to construct solution (6) inside as induced by
the medium Pret(x,x′,q⊥,ω) is related to the linear sus- sources in a surface region [5]. This procedure is free of
ceptibility χ according to Pret(x,x′) = −ω2χ(x,x′)/c2. arbitrariness if the surface region is very small and the
It reflects the translational invariance in transverse di- sources can be handled as δ functions. This condition is
rections and enables to exactly include SD. obviously fulfilled in some specific materials, where this
For a traditional transmission-reflection experiment, approachworkedwell [6]. However,for excitons in semi-
the external source on the right-hand side of (5) is to be conductorsitwasshownbymicroscopicinvestigations[7]
put zero, and instead an external wave incoming from that all the mentioned proposals fail to reproduce spec-
left or right is assumed. Consequently, the homoge- troscopic details correctly.
neouspropagationequation(5)hastwolinearlyindepen-
dent solutions, which give the solution of the reflection-
transmission problem for incidence from left or right. POYNTING’S THEOREM FOR CLASSICAL
These solutions are fixed by their asymptotics and there FIELDS
is noneedtoimpose anyfurther boundaryconditions on
them (even not Maxwell’s), since the polarization of the Now Poynting’s theorem will be exactly addressed on
medium increases continuously in the transition region the grounds of eqs. (5,6) only, i.e., without referring to
3
anyoftheabovementionedapproachesthatconsiderSD case,Poynting’stheoremisstillgivenbyeq.(7),butdue
approximatively. Quite generally it reads to the non-commuting field operators (Eˆ and Bˆ as well
∂ asˆj=ˆj0−eAˆ and Eˆ) the symmetrized Poynting vector
U(r,t)dV + S(r,t)df =− W(r,t)dV . (7) Sˆ = 1 (Eˆ × Bˆ − Bˆ × Eˆ) and the energy dissipation
∂tZ Z Z 2µ0
Wˆ = 1(ˆjEˆ +Eˆˆj), respectively, have to be used. Their
Here,thechangeofthefieldenergyU willnotcontribute 2
quantum statistical averages are given by the Keldysh
to any of the quantities discussed below and, hence, can
be omitted. Moreover, Poynting vector S = 1 (E×B), components of the photon GF (3).
µ0 For slab geometry and TE polarization, the quantum
E = −∂∂At, B = curlA, and the absorption W = j·E statistical averages of both the Poynting vector Sˆ and
hviaavej=bee∂nPitnhtreopdoulcaerdiz,awtihoenrePt=heχcEur.rent density defines theenergydissipationWˆ aredefinedvia(ω,q⊥)-integrals
∂t (for details of the derivation see, e.g., ref. [2]). In the
Integrating over the slab in eq. (7) and Fourier trans-
following, these integrals will be omitted for notational
forming with respect to t→ω, this simplifies to
simplicity,andtheirspectrallyanddirectionallyresolved
L contributions (both scaled by 1/~ω) for slab geometry
2
L L denoted by s and w (in contrast to coherent absorption
S ,ω −S − ,ω =− dxW(x,ω) (8)
2 2 a), respectively, will be considered instead:
(cid:18) (cid:19) (cid:18) (cid:19) Z
−L
2
1 ∂
for the geometry investigated here. In eq. (8), S(x,ω) is s(x)=−µ ∂x′ D>(x,x′)+D<(x,x′) ,(11)
the x-component of the Poynting vector. 0 (cid:26) (cid:27)x′=x
(cid:2) (cid:3)
1
w(x)= dx′[P>(x,x′)D<(x′,x)
2
Z
COHERENT ABSORPTION −P<(x,x′)D>(x′,x)]. (12)
D≷(x,x′,q⊥,ω) and P≷(x,x′,q⊥,ω) are the Keldysh
At first, Poynting’s theorem (8) will be addressed for
components of the photon GF and of the polarization
average fields (classical light). Assuming a monochro-
function, respectively. Their parametric dependence on
dmωi0ar)etδciqct⊥iow,qna⊥,v,0ei.+eo.f,AfrA∗0e((qxxu,,eqqn⊥⊥c,y,0ω,ωω)00)=δin(cωid12+[eAnt0ω(0xin),δqqt⊥⊥h,,e0−,q(ωq⊥00,0),δ]q,(⊥ω,t0h−)e- ticnhoerrtehvlaeartieeaqdbu)laeestnioqenr⊥sg,.yωflNioso,wtaesstbhaeanftdorbedo,istnhsiopttahteeixopnilniccwiothlyaerrewenrttiitmt(eoenr-
static part ∝ δ(ω) of the energy balance (8) can be re-
independentinthesteadystate,incontrasttotheclassi-
gardedseparately. Usingthisansatzand(6)in(8)yields
calcase. The ω−andq⊥-integralsomitted hereindicate
after straightforward calculation for each ω → ω and
0 merelythatfieldfluctuationsofallfrequenciesanddirec-
q⊥,0 →q⊥ tions contribute to them.
1−|r|2−|t|2 =a=
i
dxdx′A∗(x)Pˆ(x,x′)A(x′). (9) MEDIUM- AND VACUUM-INDUCED
2q CONTRIBUTIONS
0 Z
On the RHS, due to well-known GF identities [1]
The polarization function
Pˆ =Pret−Padv =P>−P< , (10)
4ω
P≷(x,x′)=P≷(x,x′)−iδ n≷δ(x−x′) (13)
the coherentabsorption(a>0) or gain(a<0) balances m c2
generation iP>(x,x′) and recombination iP<(x,x′) of
comprises besides the medium part an infinitely weak
excitationsinthemedium,whereastheLHSbalancesthe
(δ → 0) part which describes the vacuum in absence of
incoming intensity (∝1) into absorption and the sum of
themediumandensuresthecorrectvacuumlimitforthe
the reflected and transmitted intensity. The latter may opticaltheorem (4)[2]. n(q⊥,ω)=n< =n>−1describes
be regardedasthe intensity re-emittedcoherently to the
a given nonequilibrium distribution of photons owing to
incominglightandistobecontrastedwiththeincoherent externalpreparationas,e.g.,byanappropriateenclosure
(or correlated) emission, which will be addressed now.
(heath bath) or incoherent radiation incident from out-
side. Therefore n in the following will be referred to as
the distribution of the externally controlled or simply of
POYNTING’S THEOREM FOR FIELD
external photons. As a consequence, the optical theorem
FLUCTUATIONS
(4) comprises a medium-induced and a vacuum-induced
contribution according to
The incoherent emission is defined as the one without
externalsources,i.e.,forvanishing averagefields. Inthis D≷(x,x′)=D≷(x,x′)+n≷Dˆ (x,x′), (14)
m 0
4
where NONEQUILIBRIUM PHOTON DISTRIBUTION
Dm≷(x,x′)= As usual [1], a distribution b(ω,q⊥) ≡ b< = b> −1
willbeattributedtotheglobalgeneration/recombination
dx¯dx¯′Dret(x,x¯)P≷(x¯,x¯′)Dadv(x¯′,x′) (15)
m P≷(ω,q⊥) in (22) by definition
Z
and P≷ =b≷Pˆ , (23)
4ω
Dˆ0(x,x′)=−iδ c2 dx¯Dret(x,x¯)Dadv(x¯,x′) . (16) where iPˆ = i(P> −P<) = 2q0a is directly related to
Z the coherent absorption a in eq. (9). In contrast to
Correspondingly, the spectral function Dˆ ≡ Dret − n, the occupation b characterizes globally the distribu-
Dadv = D> −D< decomposes into a medium-induced tion of the medium-induced (transverse) optical excita-
and a vacuum-induced contribution, too, tions (e.g., polaritons in semiconductors) over the ab-
sorption/gainspectrum.
Dˆ(x,x′)=Dˆm(x,x′)+Dˆ0(x,x′), (17) Using (23) in (14), e.g., for (x,x′) > L/2, the field
fluctuations take the explicit form
where
D≷(x,x′)=b≷Dˆ (x,x′)+n≷Dˆ (x,x′), (24)
Dˆ (x,x′)= m 0
m
where the medium-induced part of the spectral function
dx¯dx¯′Dret(x,x¯)Pˆm(x¯,x¯′)Dadv(x¯′,x′). (18) in contrast to Dˆ is related to the classical absorption a
0
Z as
Note that Dˆ is to be contrasted with the spectral
function of th0e pure vacuum Dˆvac(x,x′) = cos[q0(x − 2iq0Dˆm(x,x′)=aeiq0(x−x′). (25)
′
x)]/2iq . Also, it contains an improper integral diverg-
0
Now Poynting’s theorem (8) provides the incoherent
ingas1/δ,whichiscompensatedbytheprefactorδ →0.
Therefore,evaluatingthex¯-integralforDˆ ,itissufficient energy flow S(L/2) = −S(−L/2) from the energy dissi-
0
toregard|x¯|>L/2,where Dadv(x¯,x′)=Dret(x′,x¯)∗ is pation (21): 2s(L/2) = −w. Then from (8) and (21),
for any frequency ω > 0 and propagation direction
given in terms of solution (6) as
q = (q0,q⊥), we obtain for the spectrally and direc-
′ ′
Dret(x,x′)= Θ(x−x)A(x)A(−x ) +...x↔x′... (19) tionally resolved energy flow
2iq t
0
i
s=− (n<P>−n>P<)=(b−n)a. (26)
Inserting (19) in (16) yields 2q
0
Dˆ (x,x′)= 1 [A(x)A∗(x′)+A(−x)A∗(−x′)]. (20) Ineq.(26),−nadescribesabsorption(a>0)oremission
0
2iq0 (a < 0) as the response of the medium to (and stimu-
lated by) the given nonequilibrium distribution of exter-
Using (13) - (20) in (12) for ω > 0 yields for the energy
nal photons n. Putting n = 0, the emission ba is the
dissipation
response of the medium to vacuum fluctuations. It is
i to be regarded as spontaneous emission with respect to
w = dxw(x)= [n<P>−n>P<], (21)
q external photons, but it is stimulated by b. It is not di-
Z 0 rectly related to the coherent absorption. Instead, the
where the global generation/recombination P≷(ω,q⊥) recombination iP<, and so b, has to be calculated from
are defined as the particle GF’s of the interacting many-body system
of field and matter on just the same footing as has been
P≷ = dxdx′A∗(x)P≷(x,x′)A(x′) . (22) carried out here for the photon GF.
m
Z Assuming s = 0 , i.e., vanishing energy flow be-
Itisnoteworthythat,inserting(14)in(12),allthecontri- tween the medium and its surrounding, one arrives at
butions induced by the medium according to (15) cancel b = n, which generalizes Kirchhoff’s law to nonequilib-
exactly and, thus, contribute neither to the (incoherent) rium, and b develops towards a Bose function b(ω) =
energy dissipation nor to the emission. (exp[β~ω]−1)−1 for thermodynamic equilibrium.
Equation(21)balancesgloballyopticalexcitationiP> If the externally given incoherent radiation field inci-
of the medium accompanied by absorption of external dent from outside is strong, i.e., for n ≫ b, measuring
photons (∝ n) and recombination iP< accompanied by the energyflow s wouldprovidethe same informationas
emission of photons (∝ [1 + n], i.e., spontaneous and aclassical(coherent)reflection-transmissionexperiment,
externally stimulated emission). namely the absorption a=s/n.
5
In the following, the opposite case b ≫ n → 0, i.e., P≷ → P≷ +P δ δ(x −x′)δ(~ω −µ) will appear
m m cond q⊥,0
emission s = ba into the surrounding vacuum is con- in addition to the normal generation iP> and recombi-
m
sidered. Then the distribution b is accessible to direct nation iP< considered above. The Kroneckerδ indi-
m q⊥,0
observationinexperimentsmeasuringsimultaneouslythe cates that this term is to be excluded from the q⊥ inte-
(incoherent)emissionandthelinearcoherentabsorption, grals. ThestrengthP isdeterminedbythefractionof
cond
i.e., transmission and reflection. It generalizes Planck’s quasiparticlesin the condensate. Since it appearsidenti-
formula for the black body radiation to the nonequilib- cally in both the generation iP> and the recombination
m
riumradiationofanexcitedmediumin the steady state. iP<,itsinfluencecancelsintheclassicalabsorptionaac-
m
cording to (9) and (10). Consequently, those effects will
not appear directly in classical absorption experiments,
QUASIEQUILIBRIUM where at best they show up as smooth changes in the
spectral shape of the absorption a. The same applies
Of particular interest are those steadily excited states to the absorption (a > 0) or emission (a < 0) as the
which can be regarded as quasi-equilibrium states. As response of the medium to (and stimulated by) the ex-
such,e.g.,forsemiconductors,excitongasesgeneratedat ternally given nonequilibrium distribution of photons n
low up to moderate excitation and light-emitting diodes in (26).
working at high excitation are to be mentioned. For However,an additional sharp peak in the emission
quasi-equilibrium, due to the Kubo-Martin-Schwinger
1
condition [8], the distribution b(ω,q⊥) develops into a ba= iP< (27)
Bose distribution b(ω) = (exp[β(~ω−µ)]−1)−1, be- 2q0
ing independent of q⊥. The chemical potential µ starts at ~ω = µ, whose strength is ∝ P dxΘ(L/2 −
cond
at µ = 0 for complete thermal equilibrium and char- |x|)|A(x)|2, would give evidence for a condensate, since
acterizes the degree of excitation beyond the thermal the normal part of the emission just at tRhis frequency
one for µ > 0. Then the crossover from absorption tends towards zero.
(a > 0) to gain (a < 0) appears independently of
q⊥ at ~ω = µ, where the singularity in b is compen-
sated by the zero in a. Hence, expanding b−1 and a CONCLUSION
at ~ω = µ yields that at crossover the emission stays
finite and is given by the slope of the absorptionaccord- Using a quantum-kinetic many-body approach, exact
ing to s(µ,q⊥) = kBT {∂a(ω,q⊥)/∂ω}~ω=µ. Since both results have been presented for the interacting system
a=1−|r|2−|t|2 and b switch their signs, the emission of field and matter in a specified geometry. The spec-
s stays positive as it should be in the whole frequency tralfunction of photons splits up into a vacuum-induced
region. Measurings and a=1−|r|2−|t|2 wouldenable andamedium-inducedcontribution(seeeqs.(16-18),for
to check whether quasiequilibrium is realized. If so, the whichtheexplicitexpressions(20)and(25),respectively,
chemical potential µ is fixed throughthe crossoverpoint have been obtained. It is noteworthy that both kinds of
and, after that, the temperature and excitation density states are globally (i.e., inside and outside) defined and
can be obtained directly from experimental data. their split does not correspond to a spatial seperation of
the inside from the outside. There are, of course, many
differentoptions to split upthe spectralfunction of pho-
LOW TEMPERATURE tons, e.g., into the one of the pure vacuum Dˆ and
vac
the remaining. But only the decomposition according to
For T → 0, the Bose function degenerates to a eqs. (16-18) yields an exact cancellation of the medium-
step function and the emission s(ω,q⊥) → −Θ(µ − inducedcontributioninthebalance(12)andsothephys-
~ω)a(ω,q⊥)vanishescompletelyintheabsorptionregion ically clear and simple structure of the emission (26).
~ω >µandreflectsexactlythegain−ainthegainregion Also, according to eq. (24), the field fluctuations split
~ω <µ. upintoavacuum-inducedandamedium-inducedcontri-
Concerning low temperatures, however, the theory bution whose strengths are given by the distributions n
given above needs to be supplemented if effects of quan- and b, respectively.
tum condensation occur. As such, the crossover from The distribution n describes the population of the
Bose-Einstein condensation of excitons at moderate ex- vacuum-induced states and is externally given through
citation to the one of Cooper-like electron-hole pairs at preparation of the surrounding, e.g., either as a heath
high excitation has been addressed [9]. Its consequences bath or by incoherent radiation incident from outside.
for emission and absorption spectra will be discussed Measuring the net energy flow −na induced by an inco-
in detail in a forthcoming paper. However, some ba- herentradiationnincidentfromtheoutsideprovidesthe
sic features should be commented on here. If quan- absorption a just as a classical (coherent) transmission-
tum condensation occurs, an anomalous contribution to reflection experiment.
6
The distribution b describes the population of the symmetriesthegeometryprovides,themoredifficultthis
medium-induced states. It is fixed by the steady-state willbe. Takingthis into account,it appearsallthe more
excitation conditions of the medium and characterizes surprising that for the present case of steady-state slab
itsgloballydefinedtransverseopticalexcitationsas,e.g., geometry the evaluation can be taken such a long way.
excitonic polaritonsin semiconductors. For a medium in What we know today is at least that the splitting of the
thermalequilibrium, b tends towardsa Bose function, to spectral function in the presented form is a completely
which a chemical potential can be attributed in the case universalpropertyofthephotonGF[12]andwillthusal-
of quasi-equilibrium. Thus, in a sense, real photons, i.e. ways affect Poynting’stheorem. Analyzing the latter for
aftertheirinteractionwiththefermionsofthemediumis anon-steadystate,wesupposeonewillfindtheinterplay
exactly considered,behavestatistically like idealbosons. of Dˆ and Dˆ differ and worth examining closer.
m 0
E.g., for semiconductors, this applies likewise to optical
excitations below (excitonic polaritons) and above the The author wishes to thank the Deutsche Forschungs-
fundamental gap, whereas neither excitons nor ionized gemeinschaft for support through Sonderforschungsbe-
electron-holepairscanberegardedas(ideal)bosons. For reich 652. Stimulating discussions with the members of
n = 0, i.e., without any preparation of the surrounding the semiconductortheory groupatthe University of Ro-
of the medium, emission is governed by the distribution stock and with H. Stolz and W. Vogel are gratefully ac-
b. It generalizes the Planck distribution for ideal pho- knowledged.
tons in thermodynamic equilibrium to interacting ones
in nonequilibrium.
Our results prove that even lasing can be regarded as
quasi-thermal emission. This has already been demon- ∗
Electronic address: [email protected]
stratedinref.[2]neglectingspatialdispersionandisnow
[1] L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964);
exactly confirmed. All the novel features of this light [Sov.Phys.JETP 20,1018(1965)]; V.Korenman,Phys.
result exclusively from an extremely strong renormaliza- Rev. Lett. 14, 293 (1965); D.F. DuBois, in Nonequilib-
tion of the globally defined coherent absorption a in the riumQuantumStatistical MechanicsofPlasmasandRa-
gain region due to high compensation of output losses diation, Lectures in Theoretical Physics, edited by W.E.
there. Brittin(GordonandBreach,NewYork,1967),p.469;K.
HennebergerandH.Haug,Phys.Rev.B38,9759(1988)
Emission out of a quantum condensate appears as an
[2] K. Henneberger and S.W. Koch, Phys. Rev. Lett. 76,
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