On statistics of an ensemble of oscillators under excitation. III. Quantum non-linear oscillators with fast phase relaxation in a non-monochromatic external field

Chemical Physics 69 (1982) 459-471 North-HolIaud Pubkihiug Company

ON STATISTICS

OF AN ENSEMBLE OF OSCLELATORS UNDER EXCITATION.

III. QUANTUM NON-LINEAR N A NON-MONOCHROMATIC V.N. SAZONOV, Department

OSCILLATORS

WITH FAST PMASE RELAXATION

?3X”kERNAL FIELD

A.A. STUCHEBRUKHOV

and S.V. ZATSEPIN

of Theoretical Physics, PN. Lebedev Physical Institute. B-333 Moscow, 1I 7-924 USSR

Received 2 September 1981; in fd

form 15 hfarch 1982

The statistics of an ensemble of oscillators excited by an external field in a heat ba+h is considered with the help of the double averaging method and the quasi-probability distribution function method. The parametric resonance and the forceinduced resonance cases ye considered. The present models show that there are conditions when non-linearity of osdlators, or fast phase relaxation, or non-monockomaticity

of the external field promotes the excitation of oscilJ.ators. EX-

periments in which the increase of excitation ofpolyatomic molecules in an IR laser field was caused by a buffer gas have been discussed in the literature. This effect might accrue due to the i!‘tcieaX of the phase relaxation rate. A possible control experiment to verify this statement is given.

1. introduction We consider an ensemble of osciIlators excited by an external tIeId in 2 heat bath or thermostat. This problem is of interest particularly in connection with non-thermal action of laser radiation on resonant vibrational degrees of freedom of molecules, observed in some experiments. In our model the heat bath describoes non-resonant molecules and/or degrees of freedom which are not excited by laser radiation; the external field describes the action of laser radiation; the quantum non-linear oscillator describes the excited vibrational mode. Some properties of the ensemble of oscillators were investigated earlier in two previous papers of this series [1,2], where references on experimental and other theoretical papers are also given. In the present paper we consider a more realistic model than in refs. [I ,2]. For most real quantum systems, phase relaxation proceeds much faster than population relaxation. So we conaider a nonlinear quantum oscillator wi+& different phase population relaxation times. Any real laser radiation has non-zero bandwidth. So we consider a non-monochromatic external field. The external field can produce both a force-induced (ordinary) resonance and a parametric resonance. The force-induced resonance corresponds to the direct action of laser radiation at the excited mode in a molecule. In the case of parametric resonance, laser radiation excites directly the vibration in some other mode. Eigenfrequenties of other modes of the molecule will be changed periodically due to non-linear interaction between the modes. These changes may result in parametric excitation of that mode for which the frequency of periodic changes is a resonant one [3]. Both in refs. [1,2,4,5] and in this paper we are not interested in molecular multiphoton processes and dissociation caused by laser radiation high intensity. We are interested in excitation of 2 vibrational mode up to high, but not too high, levels by laser radiation of moderate intensity; we investigate factors which promote the excitation_ Though this problem has already been treated, +he method of double averaging ]1,6] which we utilize here makes it possible to obtain new results. In section 2 we describe our model and discuss the physical sense of various approximations. Our main results are given in section 3 and discussed iu section 4. The calculations and secondary results are listed in the appendix.

of

0301-0104/82/0000-0000/$02.7S

0 i982 North-Holland

460

V.N_ Smonov

2. lkscripiion

ef d.~Sfah-stiCS

of an

ensembk

of oSCii&fOrS

under

.%X&&On.

.%

of the model

We pick out the vibrational mode which is excited by laser radiation Fd regard it as a one-dimensional quanturn nonlinear oscillator excited by an external field. The hamIltonIan Ho of the oscillator in the absence of the field is given by Ei, = fiQ?rr

i f?n&~~

[ 1 t V&G)] )

(1)

where VI@) is a non-linear term. We introduce bound eigenstates [n> and an energy spectrumE(n): go In> = E(n) In>_By definition the transition frequency w,(n) is equal to w&r)

= h-‘[E(n

f 1) - E(n)] = zi-$iE/ti

+ $d2E/&z~_

In the ri&t-hand side of eq. (2) we have negIeected all derivatives harmonic defect o(n) is introduced by U(R) = -;;i-rd%/drr”.

(2) d”%/dn”

with m 2 3 (see beIow). The an-

(3)

Due to the non-linear term VI(x) the frequency w,,.(n) depends on n, and o(n) # 0. In the case of a linear oscillator c+,(n) = wve, and o(n) = 0. We consider only bound states of the hamiltonian (l), because it is assumed that there is no excitation in the continuum part of the spectrum_ Let g be the quantum number operator: 6 In> = n In). We can writei =E(ri) [7]. Indeed, any reasonable functi_on E(n) can be expanded in a power series of n and, because of this expansion, for any eigenstate in> we haveHnIn1 =E($ln). The resonance induced by an external force F(f) is described by the term @f in the total bamiltonian~ = kO + A??~,where Hf = -SF(t). We consider a non-monochromatic force with constant amplitude:

(4) where wp and at-(r) are the frequency and the frequency deviation of the external force. In eq. (4) we have introduced a function f, which is dose to the Rabi precession frequency of the transition 0 + 1; the value off defines the amplitude of the external force. Parametric resonance in the mode of interest will occur only if the following conditions are satisfied: (1) a vibration is excited in some other mode by direct action of laser radiation; (2) there is 2 non-linear coupling of this mode and our mode. Now, the frequency uve in eq. (1) is a function of time t, and the total hamiltoniank can be written in the formk

=ir,

+LT,,

where

We shall use the term “parametric signal” for fir,. Then wp and S2n in eqs. (5) are the frequency and the Frequency deviation of the parametric signal. The C-number function n(n), corresponding to the operator ,G= /.@i), defmes the amplitude of the parametric signal. We shall consider neither the directly excited mode nor non-linear c,oupling of this mode and our mode in detail. These two factors are described by the appearance of the termHn in the total hamiltonian~ of our mode. The laser radiation is non-monochromatic; so the parametric signal is also taken in a non-monochromatic form: fin(r) f 0. The non&near coupling of the modes should increase with the increase of the enera of the modes; so we assume that c((x) increases with n [2,5,8].

er ai.~Srafisticsof an ensemble

VN. Sazonov

The conditions for parametric molecules [39,10]. f and cc(n) have +he dimension of a week external force;

m the near-resonance

resonance

of osdkztorsunder

excitation.

seem to be artificial, but these conditions

of a frequency

and are in a sense analogous

III

461

are satisfied for many real

to each other. We consider the case

regime, when

lSwp(n) I, ibw*(rz) I 4 w&z).

(7)

Here and below So,(n)

= Up - 2w,(n),

S&(n)

= 0,

- WV(“)

are the frequency deiunings in the parametric and force-induced resonance cases respectively. According to conditions (6) the electric field of the laser radiation must be much smaller than the intramolecular field, i.e. laser intensi3 i< lOr3--iO14 W/cm2. The frequency of the parametric signal which is the result of non-linear coupling of the modes may be equal to c+, 2wp, 3we, etc. The near-resonance regime (7) is possible for many real molecules both in the parametric and force-induced resonance cases. Apart from (6) and (7) we assume that iQ(n)l Q ISw&z)I,

Is0&)i_

(8)

Taking into consideration eq. (8) and negtectiug derivatives of order m 2 3 in eq. (2), we consider the phenomena caused simultaneously by quantum and non-linear effects at the first non-vanishing order of 3i only. (Note that the anharmonic defect CX(IZ), and +the second term in eq. (2) are of first order in E.) Due to this restriction we cannot consider for example multiphoton transitions. We also assume the frequency deviations to be random Li-correlated gaussian processes:
+ 7)) = ES@),

where E has the dimension bandwidth to be narrow:

of a frequency

E < 16wpp(?r) I.

(9) and is close to the external force spectral bandwidth_

We assume this

(10)

Typical frequency

values are we = wv = fw, = 103 cm-l for a CO, laser, ~SGJ,,~~= l-20 cm-l, & Z= I-5 cm-l, E = 0.1 cm-l. The value ofp was not measured directly, we think it to be .K= 10-2-10 cm-l [8] _Ail the conditions mentioned above cm be satisfied at these parameter values. The evolution of the ensemble of oscillators cau be described by a density matrix 6, which obeys the equation

f = 10m3-1 cm-l,

dfi/dt - (ljift) [fi, j] = i,,

(11)

where ri = I+ u + iif or fi =&o f fir,; the collision integral operator i, describes the interaction of our oscillator with the heat bath. We solve eq. (1 I) utilizing the double averaging method [1,6] and the quasi-probability distribution function representation (see e.g. ref. [ll]). This representation is valid though our oscillator is not linear (see appendix). If we use the Glauber distribution function P(n, IJJ,?) for example, then after averaging once, eq. (11) in the case of force-induced resonance takes the form

VA’.Samnov et al.1Srotistics of an ensembleof oscillatorsunder excitation.III

462 (vl 2 andN os&ator.

are discussed below). The qua.+robabiIity distribution function contains & ido~xndioa about the For example, with the help of the P-Function the population pn of the eigenstate In> can be written as

p, = f J dq 7 dm(rnnjn!)e-mP(m, 5

q)_

(13)

0

When A + 0 (classical J.hn.it),the quasi-prcbability

distriiution fbncti~n coincides with the distribution function @ in classical -tietic theory, i.e.P(t, ~,I,LJ)= @(t, I,, cp), where I and cpare action-angle variables. The right-hand side of eq. (12) is called the “collision integral”. The influence of the heat bath can be described by a collision integral in differential (Landau) form (IZ), if the state of oscillator changes only slightly per interaction of the oscillator with the heat bath particles. The co!.lision integral deFnds on three parameters: Tl= vi1

, the population relaxation time; T2 = vF1, the

phase relaxation time, and N(7) = [exp(tiwJT)

- 11 -I;

(14)

the third parameter is the value of the quantum number n in the thermodynamical equiiibriurn state with temperature T_ Expression (14) forN(T) is valid oinly for a linear oscillator. However we can use (14) in the non-linear case because the energy spectrum of our oscillator is close to an equidistant spectrum in the sense of condition (8). The conventional form of the collision integral (see refs. [12,13]) implies that ZJ~= v2_We have generalized this form to take into account different population and phase relaxation times. We consider here the case of weak relaxation: max[vI, vz] Q ISr+Jlt)lj277.

(15)

Eq. (15) does not restrict the -magnitude of vz/vI, wbicb may be very large. Typical values are Tl,t ;= 1OJ-1O-8 s _ X 10P6-3 X 10m3 cm-l) and conditions (1.5) can be satisfied. (p1,z = = In the appendix we shall solve some equations of type (12) with the help of the double averaging method, utilizing (8) and (15). We shall be interested mainljr in the properties of an ensemble of oscillators in the steady state regime when vl,$ S 1. These properties do not depend on the initial state of the ensemble. Eq. (12) contains terms appearing due to quantum, thermostatic and spectral width effects, which are proportional to (Y, V, or v?, and E respectively.

3. Results 3.1. Irzfluence of fast phase reharion To make the influence of fast phase relaxation clear we discuss in this section the simplest case of a linear oscillator (w,(n) = wv, a(n) = 0, p(n) = p) excited by a monochromatic parametric signal ($ = 0, E = 0). Under the conditions discussed in section 2 and the additional conditions TS- AC+,

P < 16w,l,

(16)

the level popdations p, are given by the Gibbs distribution pn = [I - ex&h,/Te)] with the effective temperature

exp(-tiw,n/T,), given

Te = T[! + (v~v1)p~26wp],

(17)

by 08)

V.N. &zonov

&al./ Statistics of an ensemble of oscillators under excitation.

III

463

where T is the heat bath (thermostat) temperature. A steady-state distribution arises only below a certain threshold, i-e_ when J.I< p, = iS~+,i_ Notice that the threshold value of p does not depend on vl, u2 if SW, > vl, v2 according to (15). (See also the appendix.) Let the phase relaxation rate be larger than the population relaxation rate, i.e. IJ~/z#~> 1. Then it is possible that T, 3- T, and we have considerable excitation of oscillators without respect to the condition !-t 4 6w,. The relaxation rate v2 cannot be very large due to condition (IS), however this condition does not restrict the value OfV~/V~_ The more difficult and general cases, when the simple relations (17) and (la) are not valid, are considered in the appendix. in any case the increase of the phase relaxation rate promotes excitation of the oscillator both for parametric and force-induced resonance. 3.2. Influence case of an ensemble under conditions [1,14]. Let us discuss in this section the case of quantum non-linear oscillators in parametric resonance. The non-linearity is displayed in the dependence of the frequency detuning i%+,(n) and the frequency modulation amplitude ,@I) on the level number n. For polyatomic molecules the effective value Swp(n) diminishes due to the high density of the levels in the quasicontinuum part of the spectrum. The effective value of p(n) grows, being proportional to a mode coupling constant. (In the case of force-induced resonance analogous assumptions are discussed in refs. [I ,4,5,14] .) When v1 = ~2 and E = 0, the Glauber P-function is given by P(m,cp) =A exp

(19)

where A is 2 normalization constant_ Making use of eqs. (19) and (13) it is clear that the excess level population increases with the increase of the ratio ~1(n)/6wp(rz)_ 3.3. Injluence

of non-monochromatic&y

The consequences of frequency deviation are most evident in the case of a linear oscillator in force-induced constant up to resonance (6w&r) = 6~~). Let v1 = V* and E > IQ_ Then the level population pn is approximately n=nO,* where

ng = 2 [v2/Sw~)(

1 + 2&Q]

Ii2

(20)

and Pn decreases when n >n6. Note that nz may be much greater than f2/4&oz, i.e. the mean value ofn of the excited linear oscillator when the ftite bandwidth of the laser radiation is neglected_ In the case of parametric resonance the frequency deviation causes an increase of the level population for n > tzo, where no = (N + $)(4VlSW&.r2),

(21)

while the level population for n < rro does not change significantly. Some other results are considered in detail in the appendix.

4. Diiussion The external field in our model promotes either parametric or force resonance (the case of simultaneous parametric and force resonance was considered in ref. [2] for a linear quantum oscillator). Hereafter we discuss

V.N. &zono~ et al./ Statistic of an ensemble of oscihtors rmder excitntion. ZZZ

464

boih cases together because their qua&r&e properties are dose. The frequency deviation increases the excitation of the oscillators_ It is easy to understand this effect by general considerations. In practice, the basic frequency of the external field does not coincide with the resonance frequency of an excited mode (conditicn (10)). The fmite bandwidth leads to the appearance of harmonics with frequencies closer to the resonant one. so the increase of the spectral bandwidth promotes the excitation of oscillators. The unexpected circumstance is the appearance of considerable excess population in high levels while the population of low levels is close to thelmaI. The non-linearity of the oscillator promotes excitation when the self-vibratiokl frequency of the oscillator is close to the resonance frequenq in the region of high-energy levels, i.e. when the frequency detuning ~Swpp&)l diminishes with increasing n. An interesting situation arises when k+(n)I increases with increasing n. In this case the exte~rnal field may lead to a decrease of the vibrational energy in the excited resonant mode [Is]. It should be noticed that there is an excess population in the levels where ISw,&r) 1is small, irrespective of the concrete behaviour of Go+_,#z)_ Besides this, the non-tiea&’ promotes excitation if the vahre of p(n) increases with ?i. As noticed above, fast phase relaxation may also promote excitation of oscillators. Fast phase relaxation leads to an increase of the spectra! bandwidth of an oscillator, i.e. it leads to an enlargement of the frequency interval, where the external field results in a considerable excitation of oscillators. Therefore the widening of the oscillator bandwidth promotes excitation when the basic frequency of the extend field does not strictly coincide prith the self-vibrational frequency of the oscillator_ The same phenomenon takes also place for a two-level system. This is easily shown with the use of the Bloch equations (see ref. [ 161). The increase of y1 also promotes a widening of the bandwidth. But the excitation of a quantum system diminishes in that case because v1 leads to a direct relaxation of the level population. It was found experimentally that the addition of a buffer gas increases ‘the excitation of vibrational modes of some polyatomic [17-211 and diatomic [22] molecules in a gas mixture under laser radiation. In refs. [18-211 dissociation of molecules caused by the addition of a buffer gas was also observed. However, dissociation and multiphoton transitions are neglected here because the additional excitation, i.e. the excitation caused by a buffer gas, is a primary effect. Dissociation must be treated as a consequence of additional excitation. Indeed, in ref. [223 the additional excitation was observed but dissociation was not, because the CO-laser intensity in ref. [22] was I= lo3 W~cm2. This intensity is too small to produce dissociation by itself and even in the presence of a buffer gas. Let us discuss the possible reasons of additional excitation, One hypothesis [23] explains the excitation of molecules by the stimulation of transitions between states close in energy, i.e. by mixing of different states. In ref. 1231 the foEowing mixing agents were considered: (1) the microwave field (see also ref. [24]), (2) the electric field, (3) the magnetic field (see also refs. [25,26]), and (4) Langmuir pulsations in the plasma. In addition, collisions in a gas may also stimulate transitions between close states and Iead to en increase in excitation_ Experimentelly, an increase in excitation due to an electric fieId [%I and a magnetic field [27] has been found. If the addition of a buffer gas leads to an increase in excitation due to state mixing, then the relative value of this increase must fall with increasing laser radiation intensity (see the corresponding figures in refs. [27,‘_8]). At the same time, collisions with buffer gas molecules Cause the diffusion of excited molecules over the energy n.xis. As a result, part of the excited molecules appears in the region of high-energy levels. In our model these molecules will be excited quickly by laser radiation because cl(n) increases and SG+,(~) or So&) decreases with increasing ?I_ Another possible explanation is based upon one of the results of the present paper, formulated as follows: phase relaxation

may promote

excitation.

(22)

If the effect observed in refs. [17--231 is caused by (22) then the increase of excitation should be expected not orrry by the addition of a buffer gas, but also in the case of increasing laser bandwidth while the laser intensity (or fluence) remains constant. Experiments of this type are of considerable interest. Probably the conclusion (22) can be obtained also in the model developed in refs. [29,30], because the physical sense of this model is similar to ours.

465

VA. &zonov et al./ Statistics of an ensembleof oscilkztors underexcitation.III

The GlauberP-function

is given by

; = J d2flP@)IfiXflI.

(23)

dq = d Imp d Rep, where p is a complex c-number. states of the unperturbed hamiltonian Z?o _ Ip) = expf-+

I/3 13 C

The states

13) are obtained

by a superposition

[P”/(Lz!)~/~] In ).

of the eigen-

(24)

Expression (24) formally coincides with the expansion of the coherent state in the eigenstates of a linear oscfilator [11,31] _In our case the states I@ and In) correspond to a non-linear oscillator. However, eq. (23) remains valid. Indeed, we do not consider dissociation and the continuum part of the energy spectrum. We only deal with bound states, and so we can consider the set of in) as a complete set. So our coherent states have all the important properties of coherent states of a linear osciilator, e.g. (i):

and (ii): if@ ti Ifl) = 0 for any /3, then operatori = 0. These two properties make it possible to prove the validity of eq. (23) just as in the case of a linear oscillator (see ref. [31]). Note that we can formaUy introduce coherent states (24) for any well of attractive ener-9. But it is not convenient to use such states in cakulations if the well is far from a quadratic one. Indeed, in this case the ankarmonic defect lc@z)l (see (3)) is not smail (compared with (8)), and a state initiaiIy coherent very quickly becomes incoherent due to the influence of second derivatives in eq. (12) [32]_ When lOg~)l is large enough, the second derivatives in eq. (12) destruct initially coherent states, even if there is no heat bath, i.e. ZQ2 = 0. We introduce the operators St, i and quantum number operator fi as in the case of a linear oscillator bfln)

=(n+

I)‘&+

I),

din) =rGIn

It is convenient to rewrite the harrditcnian of

- I), 2

fi =&_

free oscillator

through its energy spectrum

E(n) utilising the ex-

pansion of E(n) in a power series

fipqj+

Tg

$. n=O

We use the fOlIowing approximation f = (J$?mw,)1/2(~~

.

of the G-operator

f c;),

(25)

valid only for a hnear osciUator. But when the anharmortic defect a(n) is small, the non-linearity is displayed first ofall in the dependence of ‘he frequency w,(n) and the parametric modulation amplitude g(n) on th:: quantum number n. SmalI corrections in eq. (2.5) arising from non-linearity of the oscillator can be neglected [32]. The collision integra! operatoric in e@ (11) has the weli-known [12,13] form which

is strictly

& = +y [(N -I- 1)(&j&

- c;‘@ - $&Q) f N(S/jii

- iiia^ti - /j&&Q].

W?

tie T2 = v2-I coincides with the population relaxation time Tl= Vi1 and is equal to v-l_ To generalize 1, for the case of different relaxation times it is necesw to use the P-representation and introduce new slow variables, n and y, instead of Refl and Imfl:

This form implies that the phase rejaxation

p = n’/2exp(-irp

- i$),

VX Sazonov et a1.f Scatisiics ofan ensemble of oscil:arorsunder exdarioion.III

466

where $ = Ge or *$ = $J2.

NOW

Ic=v$-[n(N~+P)]+v*p& This form of I, coincides wi’& (26) if VI= ~2 = V. A.I. l3e parametric resonance First we derive the kinetic equation in the P-representation iir the form

utilizing the methods

of refs. [11,29] -

We present the operatorkn fiP = ; n [(G’ f &(fi) /&)

= F

!!!q dJ+

i- ~($o(~

d ii_In=0 -

Eq. (1) with the hamiltonian

$=Im

+vRe

+ ;)*I cos GP, (28)

fi =F&, + ap has the following form in the P-representation

T[&F(lp12-$/3)

a

( > %fl

a2P

P+vNapaS1.

Expanding the right-hand side of eq. (29) in powers of fi and taking into consideration that the terms (l/fiw,)dfE/dnl and (Ji/Q$.ujdnl are proportional to fil-1 we obtain the quasi-classical equation

+ 0 (T&“‘+))

+ O(@p”‘(n)) f

3;

Re

(30)

where JZ=D*fl> a prime denotes differentiation with respect ton, w&z) is given by eq. (2). The symbol 6 on the right-hand side of eq. (30) denotes terms of second order in A and terms containing derivatives with Ia 3. Due to the resonance condition (7) and the small value of the external force (condition (6)) it is possible to average eq. (30) over the period 2rr/lwp_ After averaging over this period, small and fast deviations vanish and, utilizing slow variables, we obtain

(31)

467

VJV. Sazonov at aI./ Statistics of an ensemble of oscillators cmder excitation. III

where 1,(27) denotes the collision integral in the form of (27). Eq. (31) is the basic equation for the parametric resonance problem. Let us consider the parametric resonance of a linear oscillator when v1 # v2 and in the absence of frequency deviation (C+,(t) = 0). In this case,it is more convenient to deal with the Wigner function of the oscillator which is given by the relation W(p) = (2/7i) ~dtfl’P@‘)exp(-2 Eq. (31) in Wigner function

representation

ai,

aw awaf-7 awag _+_-__-_-_=-*--, at

ay an an

an ap

ii = fn(6wp -

Ifl- /3’I”).

p cos

(32)

reads

a& a9

(33)

2y),

(34)

- -In - -th =vp([(N+$w/&z] J*

+ W),

jp =jsh = [v?(N+

;)/4f2]aw/ap.

(35)

Eq. (33) does not contain the quantum non-linear additions and therefore it is exact. Notice that eq. (33) is the Fokker-Hanck equation with the left-hand side written in hamiltonian form with the hamiltonian given by eq. (34). The thermal currents in eq. (33) arise due to the collision integral. We solve this_equation with the help of the double averaging method. According to this method we introduce new variables H, c+?.This may be done if (36)

XH”>rFljXQP)~f. Making use of eq. (34) it is seen that condition

(36) is fulfilled if 16wpi #p. We may use the variable ~0instead of S (see ref. [l]) because all trajectories embrace the point u = u = 0 {the defmition of the variables u and u is given below eq. (54)). In the steady-state regime when aW/at = 0 we have the following equation for IV@, q) -(a

rv/apjfi = acjih, pya(i7,

(p) - a@,

Note that W(IZ,(p) is a periodic function over y and obtain

9/a@, 0.

(37)

of the variable cp_Taking this fact into consideration

we integrate

eq. (37)

(3% IIJ accordance with the double averaging method the distribution function W(fi, tp) depends only on the variable H in the case of weak relaxation (15). This helps us to obtain the following equation for W (N+ ~)[v+, The solution

+ (q - zy)(6w;

-$)1/2]

dW/ti+

2vlW= 0.

of eq. (39) is given by

(

x exp - (IV+ &8o,

“V&Jp

- Jl cos 2p)

+ (VI - PJ(SC$

- $)‘/2]

).

(40)

In the case of high temperature and weak excitation (16) the distribution (40) results in the simple equations (17) and (18). At this step we consider the parametric resonance of a no&near oscillator in the case v1 = v2 and Q2, = 0. The initial equation is given by (31). We shall soIve eq. (31) in the steady-state regime with the help of the double

YN. Saazcnovet al./ Statirtics of an ensemble of osciikttorsunder excitation. III

465

averaging method.

Jn the case of the nontieas

oscillator the new variable is given by

Condition (36) now leads to 6wp(n) + (p(n)n)‘_ In accordance tain eq. (38) for the P-function with new currentsj, andiP: j, =jJ’ +I:, where the quantum

with the procedure

desc!%ed

above one can ob-

.

j,- =jih +$, currents are given by

(43) and P(g) should be substituted for iV(g) and N for N + : in the currents iAh,jbh_ To obtain $e equation for the P-function eq. (42) should be integrated over qC It is necessary fust to obtain the function n(H, @I, which is impossible in general. However in the case of weak Nan-linearity, (3), n(g,p) is obtained as

where 110 is defined by

E =4

j” 6op(n)dn. cl

Now, the integration

of eq. (36) over cp results in a simple differential

$-vLJ,no(~)dP(fi)~~ where i@)

is &en

equation

i P(C) = 0,

by (45). The solution

(46) of eq. (46) has the form (471

Under the conditions [44), (45) and (33, formula (47) gives the distribution function (19) in the variables n and qx Finally we consider a parametric resonance in a non-monochromatic field, i.e. C+,(f) f 0 in the case of a linear oscillator. In the Wigner function representation the kinetic equation has the form (33)-(35) with the hamiitonian h7*: &* = fn [C$(f) + 6wp - p cos 2y]. If we assume first that Q, v2 = 0, then the equation tions ri = -ai*jap

= -w

Sin a,

4 = a&*/an

obtained

= 4 [s2Jt)

is equivalent

i hp

to the system of characteristic

- ~~3s *]

_

equa(4%

v_X Sazotzov et al.1 Statisticsof an ensemble of osdkztors under .gcitation. III

469

Eqs. (48) are the Langevii equations for the random force L+,(f>, the correlation function of which is given by (9). This enables us to write the Fokker-Pianck equation corresponding to the system (48). (The Langevin equations were considered in refs. 133,341 in connection with the excitation of vibrational modes of polyatomic molecules.) The fluctuations of G$,(t) result in diffusion of the distribution rc/(n, (p) over the variable cp.Taking into account the influence of the heat bath one can obtain eqs. (33) and (34), where the currentsj, andiF are equal t0 jn =j;y

iv =$h

+@qn,$qa~.

In accordance with the double averaging method we intrcduce new variables 1?(34) and p. The averaging prosdure of eq. (30) over the variable cpgives the differential equation

(ddrV/&)&

~4~2)

(491

f W = 0,

with A I= [(N f :)j2Y1] [V&.Jp + (V1 - rJi)@o$ - $)I/21 ,

A, = eu2/4vl(Sw;

- /.c2).

B is given by (34). The solution of eq. (49) has the form

where C is a normalization constant. With the help of relations (34) and (50) one can obtain the corresponding Wigner function. The distribution function W(n,p) for quantum numbers n no is given by

This formula shows that an increase of e, which is proportional to the spectral bandwidth of the parametric signal, leads to an excess population in the region of high levels. A.2 ITheforce-induced resonance The kinetic equation for the P-function in the ease of force-iiduced resonance with the hamiltonian

ti = I$-, f i& where f& = -4f (ii? + i)cos Gahas the form

We introduce slow variables II and ~3and average eq. (51) over the period of the external force. Keeping the terms proportional to IF with m < 1 we get so+&> + fit-0) +f (52) 2&2

VS. Sirzono~ et al./ Statisticsof an ermmble of osci&tors under excit~kbn. ZZZ

470

The frequency detuning 6o+(n), the anharmonic defect a(n> and the collision integral 1e.27) were given in (7) (3) and (27) respectively. The frequency deviation C&-(t) at the fmt derivative aP/& describes the influence of the non-monochromatic external fore. The quantum corrections zre given by the term ain) in the right-hand side of eq. (52). When E -+ 0 and y1,2 = 0, eq. (50) gives the following characteristic equations 2 = -fiz

I/‘*

rp 7

J = -[6&+(?2)

+ s2&)]

-

cf/znl~2)cos

p*

(53)

(53) is a system of Langevin equations for the random force of(t). This enables us to write the equivalent tb.e Fokker-Plan& equation for the distribution function P(rz, 9, t). We consider fast and small deviations of the laser frequency. _&suming that the collisional and quantum terms in eq. (50) are caused by some independent random forces we keep these terms in the Fokker-Planck equation in their previous form. The final equation for the P-function with time-independent coefficients at the derivatives is given by (12). It is convenient to rewrite eq. (12) in the form (54) where

The currents jr, andi” contain terms which are proportional to the small coefficients vl, ~2, o and E. In accordance with ref. [l] we again introduce new variables (2, S’), where S is the length of the arc of the trajectory determined by the condition f?(zi, u) = constant. In the case vi(n) = Q”, r&z) = v20n, where vlo and v2,, are constants, after averaging of eq. (54) over S we obtain the simple equation

The integrals on the left-h_rnd side of eq. (55) are taken over the region of the (u, u) plane, restricted by a trajectory for a given value ofH. These integrals can easily be evaluated in the case of the weak force approximation when 2 f’ < “6W&). In this case we can solve eq. (Si) in quadratures.

The steady-state

distribution

function

P(u,v)

is given by

(56) where

For a dis-xrssion of the result obtained

it is convenient

to introduce

the parameter

K:

The conditions (10) and (15) of course restrict the value of VI. v2, and E but not that of K. The main consequence may be obtained from eq. (56) directly in the case of large values of K. In the region of quantum numbers ?Z where K > ~Sw~(~)/N~~ the Glauber function p(,, V) is nearly constant. This restilts in the estimate (20) in the case ?I = g2.

VN. Snonov

et cl./ Statistics of an ensemble of osciikztors under &citation.

III

471

References [I] VN.

Sazonov and S.V. Zatsepm,Chem.

Phys. 52 (1980)

305.

[2] VH. Sazonov and AA. Stuchebrukhov,Chem. Phys. 56 (1981) 391. [3] N.G. Basov, AN. Oraevsky and A.V. Pankratov, Kvantovaya Elektron. 3 (1976) 814 [English tmnsl. Soviet J. Quantum Electron. 6 (1976) 44331. [4] VN. Sazonov, Zh. Eksp. Teor. Fiz. 77 (1979) 1751 [EngLisb transl. Soviet Phys. JETP 50 (1979) 8761. [S] V.N. Sazonov, Zh. Eksp. Teor. Fiz. 79 (1980) 39 [English transl. Soviet Phys. JETP 52 (1980) 19 J. i6] RR. Khokhlov, Lisp. Fiz. Nallk 87 (1965) 17. [7] VN. Sazonov, Teor. Mat. Fiz_ 35 (1978) 361. [8] SA.Misgkov and VN. Sazonov, Kvantovaya Elektron. 8 (1981) 1760. [9] AS.Akhmanov, AA.Vigasin and V_N. Seminogov, DOW. Akad. Nauk SSSR 237 (1977) 1066. [lo] S.V.AnreIkin and A.N. Oraevsky, Kvantovaya Elektron. 8 (1981) 2061. [ll] RJ. Giauber, Quantum optics and electronics (Gordon and Breach, New York, 1964). [12] M. Lax,Phys_ Rev. 145 (i96iJ) 110. 1131 B.Ya. Zeldovich. AM. Perelomov and VS. Popov, Zh. Eksp. Teor. Fiz. 55 (1968) 589 [English tmnsl. Soviet Phys. JETP 28 (1969) 308]_ 1141 VN. Sazonov and S-V. Zatsepin, Tear. Mat. Fiz. 41(1979) 111. [Xi] AA. Stuchebrukhov, Kvantovaya Elektron. 8 (1981) 1906. [16] L. Allen and JH. Eberly, Optical resonance and two-level atoms (Wiley, New York, 1975). [17] C.P.Quigley,Opt.Letters 3 (i978) 123. [18] CR. Quick and C. Witting J. Chem. Phys. 69 (1979) 4201. [I9 J ph. Avouris, M. Lay and LY. Chen, Chem. Phys. 63 (1979) 624. [20] J.C. Stephenson,PS. King, ME. Goodman and J. Stone, J. Chem. Phys. 70 (1979) 4496. 1211 T.G. Abzianidze, AS. Egiazarov, A.K. Petrov and YuM. Samsonov, Kvantovaya Elektron. 8 (1981) 565. 1221 J. Kosanetzky,H. Vormann,H. Diinnwald, W. Rohrbcck and W. Urban, Chem. Phys. Letters 70 (1980) 60. [231 V.N. Sazonov. Kvantovaya Elektron. 5 (1978) 563. [24] LX Goodman, 0. Paulsen and WJ. ChiIds, Opt. Commun. 28 (1979) 309. [25] VN. Sazonov, Pis’ma ZETF 35 (1982) 5. [26] VN. Sazonov, KratJc. Soobsh. po Fiz., Fizich, inst. im PN. Lebedeva AN SSSR (1982) No. 4 (English transL by Aherton Press, Inc.: Soviet Phys. Lebedev Inst. Reports). [27] R. Duperrex and H. van den Bergh, J. Chem. Phys. 73 (1980) 585. [ZS] P. GOzel and H. van den Bergh, J. Chem. Phys. 74 (1981) 1724. 1291 K.G. Kay, J. Chem. Phys. 75 (1981) 1690. [30] J. Stone,E. Thiele and ME. Goodman, J. Chem. Phys. 75 (1981) 1712. [31] JR. Klauder and E.C.G. Sudarshan,FundarnrentaIs of quantum optics (Benjamin, New York, 1968). 1321 V.N. Sazonov, Tear. Mat. Fiz. 31 (1977) 107. f33] S. Mukamel,I.Oppenheim and I. Ross,Phys. Rev. Al7 (1978) 1988. [34] JS J. Chou and ER. Grant, J. Chem. Phys. 74 (1981) 384.