Might be useful to think of this semiclassically; take for example an LRC circuit and couple it alla the fluctuation-dissipation relations to a heat bath.  The ensuing isentropic fluctuations in the system at equilibrium have power spectra peaked with the LRC's resonance. So, going further with this, one could as a first approximation, look at the how the isentropic fluctuation at a higher temperature for the LRC leads to a particular power spectrum of radiation at that temperature and compare it to the power spectrum it'd be getting back from the lower temperature environment (alla detailed balance that would be the same as the spectrum it would emit to the bath through dissipation in the resistor  of the isentropic fluctuations at that lower temperature). Hope that may help your thinking!

Physics ± Uspekhi 49 (4) 401 ± 405 (2006) #2006 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences

The equilibration time, or the particle lifetime in a level, is

usually denoted by T1 and is called the longitudinal relaxation

time [4]. The quantity 1=T1 is the rate with which the

population difference between two states related by condi-

tion (1) returns to its equilibrium value as a result of energy

transfer from the particle ensemble to the thermostat [4, 5].

According to Ref. [5], both spontaneous emission [3] and

nonradiative relaxation make a contribution to T1. The total

time an atomic ensemble takes to completely lose the

excitation energy is, as a rule, close to T1. This proposition

is partly at variance with the first Bohr postulate, which states

that a closed electron orbit must be stable and that the

electron residing in this orbit does not radiate energy and

would stay in the orbit (level) for an indefinitely long time.

It is noteworthy that spontaneous emission as well as

nonradiative relaxation can occur when the light field acting

on the particles of the medium has already been turned off. In

the presence of a resonance light field, of significance are the

processes of stimulated emission and (stimulated) absorption

of the light field quanta.

When the density of a medium is high and the field is

absent, the role played by radiative transitions is negligible in

comparison with the nonradiative ones and the medium can

stay in the thermodynamic equilibrium. This implies that the

atoms obey the Boltzmann distribution [6] over the levels:

n2

n1

. exp

ÿ DE

kT

; .2.

where k is the Boltzmann constant, T is the temperature, and

n1 and n2 are the respective numbers of electrons in the lower

and upper transition levels.

Exposure of a medium to the light of an external source

can disturb the thermodynamic equilibrium of the medium.

In this case, the relation between the spontaneous emission

probability and the probabilities of absorption W12 and

stimulated emission W21 is characterized by a factor that

involves the spectral intensity of the incident radiation or its

spectral density with the constraint that the probabilities of

stimulated emission and absorption are equal [3, 6].

The problem of the role of nonradiative relaxation in the

processes under discussion was investigated by Karlov [4] on

the basis of the analysis of population variation in a two-level

system exposed to a resonance electromagnetic field. Accord-

ing to Ref. [4], the variation in the particle number density n2

in the upper level is given by

dn2

dt

. ÿ

w21 . 1

t1

n2 . w12n1 ÿW21n2 .W12n1 ; .3.

where w12 is the probability of a spontaneous electron

transition from the ground level to the excited one, w21 is the

nonradiative relaxation probability, and 1=t1 is the rate of

particle escape from the second level due to spontaneous

emission.

It is significant, and this is the central point of the present

paper, that irrespective of whether the ensemble has com-

pletely lost the excitation energy, the medium permanently

retains the memory of the existence of a transition between

the levels E2 and E1 and of the value of the Bohr frequency n21

[see relation (1)]. In other words, the Bohr frequency is an

invariable characteristic of the medium under investigation,

irrespective of whether the electron occupation of the

transition levels is equilibrium or nonequilibrium. Unam-

biguously related to the Bohr frequencies are the oscillators

(as a rule, these are the optical electrons of atoms, molecules,

mesoatoms, and nanoparticles) distributed over the volume

of the medium. The eigenfrequencies of these oscillators are

determined by elastic forces and are independent of the

external field frequency [7].

It is pertinent to note that in the absence of exact

resonance between the excitation field frequency n and the

Bohr frequency n21, which characterizes the two-level system

under study, stimulated electron transitions are effected

involving multiphoton processes, specifically, three-photon

electron Raman scattering (RS) [8 ± 10] and six-photon

parametric scattering (SPS) [10, 11]. We recall that an

elementary event of the three-photon process involves the

absorption of two photons of the exciting radiation at the

frequency n and the emission of a photon at the combination

frequency n3 (in the literature, this process is sometimes called

hypercombination or Raman [12] scattering). In the case of a

two-level medium characterized by the Bohr frequency n21,

the frequency n3 of the three-photon electron RS is, in

accordance with the energy conservation law, expressed as

n3 . 2n ÿ n21 .4.

(the factor h is omitted).

Several research groups have reported the experimental

observation of this process in Refs [8 ± 10], which give

bibliographic references to three-photon scattering. For

information about its theoretical treatment, the reader is

referred to Ref. [13].

In the case of SPS in a two-level medium, according to the

energy conservation law, the three absorbed photons of

exciting radiation correspond to the production of three

photons at new frequencies. The process should not violate

the equilibrium conditions in the medium. In this case, of

significance is the fulfillment of the momentum conservation

law for the six photons participating in the elementary

interaction event. This process was considered in Refs [10, 11].

3. The correspondence principle as an endeavor

to make full use of the notions of classical

theories

We now turn our attention to another point made by Niels

Bohr [14]. The point at issue is the complementarity and

correspondence principles. According to Bohr, there exists

the same relation of complementarity between the unequi-

vocal application of stationary states and the mechanical

analysis of intraatomic motion as between a light quantum

and the electromagnetic radiation theory. The correspon-

dence principle expresses the endeavor to harness the con-

cepts of classical theories (mechanics, electrodynamics) to the

highest degree, despite the contradiction between a light

quantum and the electromagnetic theory of radiation [14].

The aim of this paper is to interpret, in accordance with

the Bohr correspondence principle, the features of the

spectral structure of the medium-scattered (secondary)

radiation excited by the beams of resonance or near-

resonance light quanta (photons) and to search for the

universal rules for processes of this kind with the use of

the notions of the classical theory that pertain to the

resonance medium. As shown below, using the processes

that correspond to relation (4) or the like (see Section 4)

promotes a clear classification and interpretation of the

spectra of in-medium-scattered radiation and luminescence

402 V E Ogluzdin Physics ± Uspekhi 49 (4)

spectra, as well as light amplification. The proposed model

enables a valid explanation of several features of the

resonance and near-resonance interaction of monochro-

matic radiation with different (atomic, molecular, nanodi-

mensional) media.

In an endeavor to make the fullest use of the notions of

classical theories, we mention that the mechanical Lorentz

model of a classical harmonic oscillator exists precisely for the

description of the properties of an atomic (molecular)

medium [15].

Having discussed the Bohr frequency, the relation of

complementarity, and the Bohr correspondence principle,

we now recall that the electrons responsible for the optical

properties of a medium may be represented as an ensemble of

similar or variously sized classical harmonic Lorentz oscilla-

tors, which in turn requires invoking the results of the

dispersion theory [15, 16] related to these oscillators. Our

prime concern is with how a photon beam of monochromatic

radiation of frequency n interacting with a nearly resonance

medium can propagate through a medium that obeys the

dispersion laws. In the material media under investigation, an

ensemble of classical harmonic oscillators corresponds to

every Bohr frequency n21.

Our interest is with three different situations:

A) n . n21, n.n. ' 1,

B) n > n21, n.n. < 1,

C) n < n21, n.n. > 1,

where n21 is the resonance transition frequency, which

naturally corresponds to the Bohr frequency [see relation

(1)], and n.n. is the refractive index of the medium.

In case A, according to the dispersion theory, the

refraction index n.n. of the medium with a volume oscillator

distribution is close to unity [15, 16]. According to Ref. [15],

the reflection coefficient of this medium peaks at the

resonance frequency. When a population inversion is pro-

duced in the medium at the transition with the frequency n21,

radiation amplification is bound to occur.

In case B, a special feature limiting the photon propaga-

tion is related to the fact that the refractive index n.n. < 1,

according to the dispersion theory. The propagation of

photons of monochromatic radiation turns out to be

impossible in this spectral region, because otherwise they

would propagate with a supraluminal speed v . c=n.n. (c is

the speed of light in empty space), which is at variance with

the existing notions. Only by effecting the saturation of the

medium, i.e., the level population equalization of the

transition under investigation, is it possible to transmit the

radiation through this medium. Otherwise, according to

Ref. [15], efficient reflection of the incident radiation occurs

at the boundary of a medium with a volume distribution of

classical Lorentz oscillators.

Finally, in case C, according to the Sellmeier relation [16],

an unbounded growth of the refractive index n.n. occurs in

the low-frequency spectral region relative to the resonance

frequency. This circumstance may be responsible for a very

strong moderation of the photons that penetrate the medium

and may therefore impede their propagation.

4. Several experimental facts and their

discussion on the basis of the proposed model

The problem of the resonance and near-resonance interaction

of radiation with different media has attracted the attention

of researchers for a long time (Wood, Rozhdenstvenskii).

The systematic study of resonance interaction became

possible with the advent of frequency-tunable cw or pulsed

lasers. Analyzing the experimental results in this area of

research might seem impossible due to the extensive body of

publications. My task is to extract those results from the

wealth of experimental data which correspond to the multi-

photon model considered in the present paper. For this, I use

the published materials obtained with my participation, as

well as the findings of other researchers available in the

literature.

We enlarge on the experimental data that serve to

illustrate case A. Here, it is pertinent to recall Wood's

classical experiments (see Ref. [17]) in which he observed the

opacity of amedium (sodium vapor) to the incident resonance

radiation. With an increase in the density of the atomic

sodium vapor, the region of vapor glow shrinks to the point

of beam entry into a vapor-containing cell and, in the view of

the observer, turns into the glow of a thin atomic surface layer

(or into the reflection of light by an atomic surface layer).

That is, the forward propagation of monochromatic radia-

tion at the resonance frequency (n . n21) is limited. According

to Ref. [18], the opacity (strong attenuation) of the medium to

resonance radiation can be attributed, from the standpoint of

classical electrodynamics, to the high magnitude of the

scattering cross section s. In the case where the frequency of

exciting radiation coincides with the Bohr frequency of the

medium (n . n21), the scattering by bound atomic electrons

limits the directional radiation propagation through the cell

with an increase in density of the alkali metal.

When a two-level inverted system is irradiated by photons

whose frequency n is equal to the Bohr frequency of the

medium (n . n21), the scheme of stimulated (induced) emis-

sion of photons with the same frequency n is realized, which

was proposed by Einstein in 1916 [19]. The production of two

photons in an elementary event in an inversely populated

resonance medium with the simultaneous absorption of one

photon corresponds to the expression

n . n 0

21

. 2n ; .5.

identical to equality (4), where n . n 0

21 and n 0

21

. .E2 ÿ E1.=h

is the Bohr frequency of the inverted transition of the two-

level medium under investigation. Relation (5) uncovers the

mechanism of light radiation amplification involving the

doubling of the number of photons in every elementary

event. This process has found wide application in many

problems of quantum radiophysics and electronics related to

the amplification and production of coherent monochro-

matic radiation [4, 12]. Interestingly, even at the dawn of the

age of laser physics, the model of the classical harmonic

Lorentz oscillator that we invoke was employed in the

discussion of lasing [20].

We now turn to the analysis of experimental data that

serve to illustrate case B: n > n21, n.n. < 1.

From the situation whereby the propagation of photons

at the frequency n through the medium is limited in the

specified spectral range, it is possible to find a way out by

assuming the possibility of the production of a precursor [15],

which testifies to a change in the properties of the medium for

the photons that follow behind the leading edge of the beam.

The experimental data on the interaction of nearly resonance,

monochromatic radiation with atomic potassium vapor

carried out with the participation of the author of the present

paper [8, 10, 11, 21, 22] allow concluding that in the case of a

resonance (two-level) medium, the role of the precursor

April, 2006 The role of Bohr frequencies in the scattering, luminescence, and generation of radiation in different media 403

should be assigned to the photons whose frequency n3

corresponds to the above relation (4): n3 . 2n ÿ n21, where

n21 is the Bohr frequency of the medium. Because n3 > n and

n > n21 in this case, we have n.n3. > n.n.. Furthermore,

according to the dispersion theory [15, 16], we expect that

n.n3. ! 1 as the frequency increases, which removes the

limitation on the radiation propagation.

The observation of such processes in nearly resonance

conditions in atomic alkali-metal vapors was reported by

other researchers in Ref. [9]. Reference [9] contains a

bibliographic list of mainly experimental works dedicated to

the observation of three-photon process (4) in atomic media.

Indeed, as a result of process (4), equilibration of the level

populations of the medium investigated occurs (electrons are

transferred from the ground level 1 to the excited level 2),

which testifies to the formation of conditions in the medium

for the unimpeded transmission of the radiation at the

frequency n3 and immediately after at the frequency n. The

radiation with the frequency n3, which is successfully recorded

on spectrograms, is indication that the process is appreciably

intense [8, 9].

The illusory contradiction that follows from the smallness

of the estimated cross section s3 of the three-photon process

given in Ref. [12, p. 173] and its accessibility for observations

in atomic alkali-metal vapors [8, 9], is attributable to the fact

that the lower-order processes in a medium with a refraction

index n.n. < 1 are insignificant: they do not mask the

radiation at the frequency n3.

The self-focusing of laser beams with a Gaussian intensity

distribution over the beam section, which is observed in

alkali-metal vapors [21 ± 23] in this spectral region, according

to the model under discussion, may be represented as the

tunneling regime [22] for the photons with the frequency n

through the channel prepared by process (4). In conformity

with this process, the first to appear on the beam axis as the

precursors are photons with the frequency n3.

We next analyze case C (n < n21, n.n. > 1) with recourse

to available experimental data. First, we consider a medium

consisting of two-level atoms (by the example of atomic

potassium vapor [10, 11, 22]) and, second, the luminescence

observed in nanodimensional media [with the proviso that

n < nm1 and n.n. > 1, where nm1 are the eigen (Bohr)

frequencies of the nanodimensional oscillators of the med-

ium; m . 2; . . .] [24, 25]. According to Refs [24, 25], the

conditions required in case C [n < nm1, n.n. > 1] in the

presence of luminescence are automatically satisfied for the

set of Bohr frequencies nm1, which are located, as a rule, in the

blue (anti-Stokes) spectral region relative to the frequency n of

the exciting radiation. The luminescence spectrum itself is

shifted to the Stokes spectral region relative to n [4, 26].

It is pertinent to note that in the case of potassium atoms,

we are dealing with calibrated oscillators and an invariable

Bohr frequency for a given transition. In the case of the

luminescence of a nanodimensional (or molecular) medium,

due to the different dimensions of the nanoparticles we obtain

a set of Bohr frequencies nm1 (m . 2; . . .).

1. Therefore, for a medium of two-level atoms, expression

(4) for the three-photon electron RS is of the form

n3 . 2n ÿ n21, where the quantity n21 corresponds to the

Bohr frequency of the transition under investigation, n is the

exciting radiation frequency, and n < n12. In this case,

according to Refs [15, 16], n.n.41, which should lower the

radiation propagation velocity in this spectral region. For

laser radiation beams with a Gaussian intensity distribution

over the beam section, it has been possible to observe the

Vavilov ± Cherenkov effect in alkali-metal vapors in this

spectral region. This effect is caused either by the supralum-

inal propagation of the nonlinear polarization induced in the

medium [10] or by the propagation of supraluminal photons

through the resonance medium [11], which actually travel

with a speed that does not exceed the speed of light c. Owing

to process (4), which takes the Bohr frequency into account, a

channel in which n.n. ! 1 forms on the axis of the light beam

due to population equilibration, which furnishes the condi-

tions for Cherenkov radiation [27] with the use of this

unconventional supraluminal source.

In considering cases A, B, and C, it has been possible to

show that it is precisely the Bohr frequencies of a medium with

a volume distribution of classical harmonic oscillators that

define the properties of interaction of monochromatic

radiation beams with such media. For similar calibrated

oscillators of the medium (atomic alkali-metal vapor), the

experimental results are evident and are easily amenable to

interpretation. The process of nonradiative relaxation ensures

dissipation of the energy stored by the oscillators of the

medium due to process (4). We note that the Bohr frequencies

themselves are, as a rule, not recorded on spectrograms except

in the case of exact resonance (5), whereby n . n 0

21.

Therefore, experimental investigations into the propaga-

tion of laser radiation beams in a two-level atomic medium

confirm that process (4) plays the decisive role in the

interaction of laser radiation with the nearly resonance two-

level medium.

2. The situation is more complicated for a medium with a

volume distribution of oscillators with different dimensions.

In this case, the spectrum of Bohr frequencies is broader,

entailing a complication of the radiation spectra scattered by

the medium.

We address ourselves to one more phenomenon, specifi-

cally to the effect of fluorescence (fast luminescence) [26], and

consider the consequences of applying the proposed model to

this process. The reason for this approach is as follows: until

recently [24, 25], the relation between the observed spectra

and the specific oscillators of the medium was not taken into

account in the interpretation of the above effect. We try to

establish this relation. Relation (4) is rewritten as

nm . 2n ÿ nm1 ; .6.

where m . 2; . . . denotes the number of one of the numerous

frequency spectral components of the luminescent emission,

which consists of the frequencies nm and is scattered into an

angle of 4p steradians in a medium with a volume oscillator

distribution, and nm1 . .Em ÿ E1.=h are the characteristic

Bohr optical frequencies for electrons (of a complex organic

molecule or a nanodimensional particle) embedded in one or

other environment. This environment is, as a rule, size-

noncalibrated aggregates of different dimensions consisting

of some combination of atoms, molecules, or fragments of a

crystalline structure with numerous defects. In accordance

with the problem formulated, we should show how the Bohr

frequencies nm1 of suchlike aggregates are related to the

luminescence spectrum. As in the case of a two-level atom,

we reason about the necessity of borrowing a part of the

medium-exciting radiation to effect the dynamic compensa-

tion of the dispersion of refractive indices of individual

oscillators in the medium, which turn out to have different

dimensions in this case [n.nm1. ! 1]. In our view, relation (6)

describes this process, as evidenced by the luminescent

404 V E Ogluzdin Physics ± Uspekhi 49 (4)

emission at the frequencies nm. In the recently published

papers [24, 25] of the present author, the luminescence of

silicon nanoparticles suspended in ethanol is described in

detail employing the present approach.

Upon discussing the problems related to resonance and

near-resonance media and the Bohr frequencies, we have to

revert to relation (3) considered in Section 2. The relevant case

pertains to the absorption W12 and stimulated emission W21

probabilities. Clearly, these quantities should take the three-

photon processes (4) ± (6) considered above into account [11].

5. Conclusion

We have considered several practical problems of scattering,

light amplification, and luminescence in resonance or nearly

resonance media with a volume distribution of classical

oscillators. The eigenfrequencies of these oscillators corre-

spond to the complete set of Bohr frequencies of the media

under investigation. The properties inherent in the interaction

of light quanta beams with such media are described.

Irrespective of whether the conditions for exact resonance

between the exciting radiation frequency and the Bohr

frequency of the medium are satisfied, the participation of

three photons in an elementary radiation ± medium interac-

tion event ensures the ground-to-excited (or vice versa)

oscillator state transition. The correspondence principle and

the Bohr principle of complementarity, which express the

endeavor to utilize the concepts of classical theories

(mechanics, electrodynamics) to the highest degree, in the

above-considered example are directly related, on the one

hand, to the quantum properties of light radiation and on the

other hand to the mechanical properties of the classical

harmonic Lorentz oscillator and the dispersion theory.

Using relations (4) ± (6) facilitates the identification of the

spectral structure of the radiation scattered by a resonance or

nearly resonance medium in different directions. On the one

hand, the question arises of how closely the exciting radiation

frequency n can be brought into coincidence with the

frequency n21 or nm1 of the transition under investigation.

On the other hand, in the case of a so-called nearly resonance

interaction (both for two-level atoms and for luminescence),

it is not necessary to be concerned about the exact coincidence

of the frequency n with the Bohr frequency n21. Recording the

output radiation at the frequency n3 (nm) in an experiment

with a resonance or nearly resonance medium supports the

proposed scheme. It is noteworthy, however, that the

experimental detection of radiation with the Bohr frequency

n21 or nm1 in the absence of exact resonance, as a rule, turns

out to be difficult or impossible, for instance, due to

nonradiative relaxation. Experimental data nevertheless

allow calculating this frequency.

Relations (4) ± (6), which are borne out in numerous

experiments on the amplification and scattering of laser

radiation, the interaction of laser radiation with nearly

resonance two-level media, and the fluorescence of nanodi-

mensional particles, have a universal nature in our view.

There is good reason to apply this law for the interpretation

of experiments in atomic, molecular, and nanodimensional

media. Special emphasis should be placed on the role played

by the Bohr frequencies n21 (nm1), which are integral

characteristics of the medium under investigation. In parti-

cular, using this rule in fluorimetric analysis provides an

independent expert evaluation of the characteristic frequen-

cies of the intrinsic oscillators of the medium investigated.

Reverting to the notion of elementary classical oscillators of

the medium, which are responsible for the Bohr frequencies,

seems to be expedient in the study of the above-listed media

and processes, because it enables gaining new information

about the characteristics of the medium.

The author expresses his deep gratitude to V I Kogan for

his support of the ideas in this paper concerning the practical

analysis of the data of a physical experiment.

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