Relation between quantum statistics of phonons and scattered light

North-Holland

Relation between quantum statistics of phonons and scattered light

Alexander S. Shumovskya,b and B. Tanatara

Department of Physics, Bilkent University, Bilkent, 06533Ankara, Turkey b BogolubovLaboratory ofTheoretical Physics, Joint Institutefor Nuclear Research,

P.O.Box 79, Moscow 101000, Russian Federation

Received 29 July 1993; revised manuscript received 27 September 1993; acceptedfor publication 28 September 1993 Communicated by B. Fncke

The resonance model ofthe Raman process with the generation ofa Stokes component is solved exactly. The asymptotic behav-ior ofthe solution is discussed. Mandel’s Q-factor is calculated as a function oftime both for the Stokes and the Rayleigh compo-nents fortheir dependence on thequantumstatistical properties of the vibration mode (phonons). A qualitative difference be-tween the cases of uncorrelated and correlated phonons is found whichmay have interesting experimental implications.

1. Introduction bration mode in Raman scattering. For simplicity,

we suppose the resonance steady state process with During the last years quantum statistical proper- the generation of an inelastic Stokes component only. ties of scattered light in the Raman process have at- The initial state of the Rayleigh mode is assumed to tracted considerable interest [1,2]. In particular, the be a number state, while the vibration mode can be anticorrelation between the Stokes and Rayleigh lines initially in a number state or in a squeezed vacuum in the resonance scattering have been examined state. The Stokes field is initially in the vacuum state. [3,4], and the generation of squeezed light has been The simplest model of three bounded oscillators is considered [5—7].At the same time, strong quan- used for the description of the process under con-turn fluctuations of energy have been observed ex- sideration [13]. Using the representation of the

perimentally [8,9]. Schrödinger equation in terms of new orthogonal

It is known that Raman scattering is an example polynomials [14,15], we examine the dynamics of of an optical parametric process in which one of the Mandel’s factor ofscattered light, and show the qual-interacting waves is a medium vibration mode ofbo- itative difference between the two choices of the

mi-son type (phonons) [10]. In the case of condensed tial states of the vibration mode.

matter such a mode is usuallyin thermal equilibrium The rest of this paper is organized as follows. We with a given temperature. The state of that mode is first introduce the model Hamiltonian for which we determined by different mechanisms of microscopic calculate the dynamical properties. The evaluation interactions in the medium and in some cases can of the eigenvalues and eigenfunctions, and a discus-lead to a strong number fluctuation [11]. An ex- sion of how to construct the time-dependent Mandel ample is provided by a polariton-type system in factor is given in the next section. We then present which the equilibrium state is a squeezed one [12]. our results and conclude with an emphasis on the ex-Undoubtedly, the statistical properties of the vibra- perimental implications.

tion mode must have influence on the statistics of the scattered light.

In the present paper we consider the quantum 2. Model and its eigenvatue spectrum properties of scattered light and its dependence on

time-dependent Mandel factor defined by the following defining some orthogonal polynomials [14]

expression, pnm(x) Here

Q1(g)~Vt(Cg Cg) <CgCg>t (1) q7’m=(n—j+l)j(m+j)

<Cg~Cg>t

These polynomials may also be defined through the Here c~and c5 are the Bose operators, V1(c~Cg) is differential equation

the time-dependent number variance, and

< >

de-notes a time-dependent expectation value. It is pos- 82F

itive in the case of super-Poisson statistics and neg- t3 — [1 +t2(n—m—2)]

ative for a sub-Poisson number distribution. The zero

+[x—tn(m+1)]F=0 value corresponds to the coherent state (Poisson

distribution), for the following generating function,

The time dependence of the functions on the right

ti hand side of eq. (1) is given by the dynamics ofour F(x, t)= ~ P7’m(x)-~.

model Hamiltonian, J=0 .i.

H=~ WgCg~Cg+y(c~~ Cr+ h.c.), (2) Let us represent the polynomials P7’tm (x) in the form

g

j where the index g=v, s or r denotes the vibration, PYm(x)= ~ ~m(J)Xl.

1=0 Stokes, or Rayleigh modes, respectively. We will

as-sume the resonance condition Then for the coefficients ~ in the above equation we

(Or=O)s+Wv (3) get

in the subsequent considerations. Because of the cr7.~P~s+i

(J)

=0,

conservation law [13] [N, H] =0, where i--i

V nm~nm

‘-~ q~’ ~,k~_2S+l(k—1). (7)

N=C7Cr+~(C~Cs+C~Cv) k=2s—1

an exact eigenstate of the Hamiltonian ofeq. (2) can Because of this relation it is not difficult to express

be chosen as these coefficients ~ in terms of the Bernoulli

poly-nomials [15].

n, m>= ~ 27’mln—j>rlj>slm+j>v, (4) Using the polynomials P7’m(x) we can represent

3=0 the equation for the eigenvalues in the form

where I >~is the number state of the gth mode. The _{P~’1(x)=0. (8)}

coefficients

A7’m

in eq. (4) are determined by the

re-cursion relation [15] Taking into account the relations given in eq. (7),

one can observethat (i) foranyfixed n and m eq. (8) has exactly n+ 1 real solutions; (ii) the set of =x’~’tmA7’m—

)~7~

[

(n

—1+

1 )j(m

+j)

]1/2, (5) solutions {x7’tm

}

is ordered symmetrically with re-together with the normalization condition spect to the zero value~ =0; (iii) the value~ =0

for whichEnmwrfl+Wvm is the root of eq. (8) for

),fl,nI 12=1 . evennonly; and (iv) between any two neighboring

J=o roots X?.m and x7~is situated only one root of eq.

(8). We note that the above expression (8) deter-HereXP~,tm=(E?~~m—(Orfl—w~m)/y, andE7z,mis an ei- mines the eigenvalues of some Jacobi matrix. For a

genvalue of Hamiltonian (2) corresponding to the

special case the problem of diagonalization of the eigenstate given in eq. (4). The above relation (5) corresponding matrix was solved numerically by can also be represented by the equivalent expression Walls and Barakat [16].

of the eigenstate given by eq. (4) are represented in n n n,m the form n[x~]2>~~ [x7’~}2~ ~ j=O j=O t7’m(x1) =~‘mP7’m(x1) / •~-~—1/2 =~n(n+l)[n2+n(3+2m)+4m+2] , (10)

x[~u,~nj

, (9)

j)’~

J

j whereXma,,is the maximum root of eq. (8) at given

n and m. Because of the above established symmetry where x1 mx?’tm is any solution of eq. (8). The coef- of the solutions of eq. (8) we obtain

ficient 2~~”is determined by the normalization

con-dition. For example, the eigenstate corresponding to ~ ~Wrfl+w~m the solution xP~,m=0for even n is described by the

following set of coefficients, — ~—~r,J(n+l)[n2+n(2m+3)+4m+2]

2nm.—A~.m{l 0 .m(2) ~flm(4) 0

— ‘ ‘ F~’m‘ ‘ F~’m‘“ ‘ F~’m~ which leads to the asymptotic behavior

where

E~<wrn+wvm~rs/n3+2n2m (11)

(m+2p—l)!!

~‘m(2p) = (2p— 1 )!!(2p)!!

(m— 1)!! for large n and m. A similar estimation of the lowest

and eigenvalue for the special case m=0 was given

pre-viously [11,141. It follows from the inequality of eq.

F~,m=

[

(2p)!j3(n)(m±

2p’~T’~’2

(10) that starting from some numbers n0 and m0, we

P

JJ

have at least one branch of collective excitations with

We stress that the same polynomials and polynomial negative energy, while the energy ofthe vacuum state equations ariseinthe more general case ofnonlinear E°’°=0. Moreover, E~~I7~~ —on when n—p on. Thus,

interaction of any order p ~ 2 of the type the spectrum of eigenvalues of the problem under consideration becomes unstable with increasing n.

H1~~

~(

~‘

)

The inequality of eq. (10) gives us an estimate as

= a0~

fl

a1~+

H

an upper bound to~ TTsjug the Frobenius

theo-~min’ ‘~

k=1 k=1

rem [17] one can also construct a lower bound, (decay ofone boson into p bosons). The onlychange

x~~max(

here is the parameter q~)in eq. (6). For instance,_{in the case when all kth modes are initially in the k} v/~+V’~~~)< -_~~~/n3+2n2m,

vacuum state, one gets qJ’°=j”(n—j+ 1).

which leads to 4y

3. Asymptotics of eigenvalues wr+wvm— ~ ~,Jn3+2n2m~<E?~

The representation of the eigenvalue problem in ~

~

+2n 2m (12)

terms of the polynomials P(x) allows us to estimate ~ Wr~+o)~m—

the asymptotic behavior of the eigenvalues~ for

large n and m. Apart from providing us with useful for large n and m. Because of these estimates from information on the numerical calculations, the in- above and below, we have the exact asymptotic be-vestigation of the asymptotic behavior leads also to havior for the minimum eigenvalue at fixed n and m.

some interesting properties of the model which we Due to the symmetry of the roots of eq. (8) similar

discuss below. asymptotics can also be derived for the maximum

From the results of the previous section it follows eigenvalue. We summarize the asymptotic behavior

E~‘~—yn312, n—~oo,infixed, f,n~5mm,

E~—’yn3’2, n—*on,in fixed, — 1 ,, \n/2U

— It1 m

12 V’n41

E~‘—‘w~m—yO(m/

),

m—~on,n fixed,

E~ -~w~m+yO(m1/2), m—~on,n fixed. for the number and squeezed vacuum initial states

of the vibration mode, respectively. Here~tand v are

From the above asymptotic behavior one readily ob- the squeezing parameters and H(x) is the Hermite serves that only n is a critical parameter for stability polynomial [18].

of the spectrum of elementary excitations in the Now all time dependent terms in the Mandel fac-present model. In this connection, we remark that tor given in eq. (1) must be calculated as the cor-increasing n means an increase in the number of responding expectation values with the time depen-photons in the Rayleigh mode. But at high intensity dent wave function (eqs. (13), (14)).

of the Rayleigh mode att =0, it is necessary to take The results of our calculations are presented in figs. into account the generation of anti-Stokes phonons. 1 and 2 for Mandel’ Q-factors of the Rayleigh and It is then reasonable to suppose that the inclusion of Stokes modes, respectively. In fig. 1 a we show the

theanti-Stokes process into the model Hamiltonian Rayleigh mode Mandel factor Qr(t) for the initial would stabilize the spectrum of collective excitations. number state of the vibration mode (m=2), and

different values ofthe initial number of photonsn=2 (solid line) andn=20 (dotted line) in the Rayleigh mode. One can see that the increase ofnin this case 4. Time dependent Mandel factor leads to a periodical change of statistics of elastic scattered photons from the sub-Poissonian to the Using the results of previous sections we can now super-Poissonian statistics.

express the exact time dependent wave function of A qualitatively different behavior is observed when

Hamiltonian (2) in the form the vibration mode is initially in a squeezed vacuum

state (fig. lb). In that case as n increases for a fixed ~ exp[ i(flWr+ mw )tJ mean number of quanta in the vibration mode

I vI~,

n,m0 the statistics of scattered light becomes completely

super-Poissonian without any change of sign of

x

~ C~tmexp(—iyx7’mt) Mandel’s Q-factor. Figure 2 shows Mandel’s

Q-fac-1=1 tor for the Stokes mode. Qualitatively the same

be-havior is observed also for the Stokes mode as

dis-X~A7~m(x?~rn)Inj>rIj>sIm+j>v. (13) cussedbefore. Here we have enumerated the rootsx7.m of eq. (8)

starting from the maximum value. The coefficients 5. Conclusion C7’m are determined by the initial conditions so that

+‘ . We have obtained a qualitative difference in the

~ C7’m27’m(x1) p,,fm, ifj=0, quantum statistical properties of scattered light

de-= pending on the statistics of the vibration mode. Our

=0, otherwise. (14) choice of the initial state of the vibration mode can

be considered as simulating the harmonic (uncor-According to the setting of the problem given in sec- related) and strongly correlated vibrations in a me-tion 1 we mustput here dium. Hence, the experimental investigation of the

= quantum statistical properties of scattered light in

“ “ the Raman correlation spectroscopy with a number

a 1_) I I I I I I I I a 1.0 III I I I I ~II I I I I

m=2

4’

~

~ ~

~

~! ~

~ ~I~IL ~ ~

—2 III III 1111111 III 11111 —2 IL IL III 1111 III

0 1 2 3 4 5 0 1 2 3 4 5

7t (xlO) 7t (xiO)

b 10 IIIII1IIIIIIIIIIIIIIIIII b 10 III1IIII1çIrrr

:

z

:

.:~—

~

jIj~i~

~

~:~~ii

~ 4:— i~Ii i1iI~~ ~

J

I

2 3

¶ t

~ 2 I

1v12=2

~ —20 1 I2I 3I 4I I I•5 —20 1 2I 3I 4 5 yt (xlO) 7t (xiO)

Fig. 1. (a) Time dependent Mandel factorQr(t)for the Rayleigh Fig.2. (a) Time dependentMandel factor Q~(t)for the Stokes mode when the vibration mode is initially in the number state mode when the vibration mode is initially in the number state

(in=2), and the initial number of photons n=2 (solid line) and (m= 2), and the initial number ofphotons n=2 (solid line) and n=20 (dotted line) for the Rayleigh mode. (b) Time dependent n=20 (dotted line) for the Rayleigh mode. (b) Timedependent Mandel factorQr(t) for the Rayleigh mode when the vibration Mandel factor Qg(t) for the Stokes mode when the vibration mode

mode is initially in the squeezed state vi 2=2.0, and the initial is initially in the squeezed state Iv 12=2.0, and the initial number number ofphotons n=2 (solid line) and n=20 (dotted line) for of photons n=2 (solid line) and n=20 (dotted line) for the

the Rayleigh mode. Rayleigh mode.

yield important information on the correlations in (10), it is not surprising to observe the collapse— the medium as well as in the molecules. rivival patterns as in the case of the Jaynes— It is not difficult to see from figs. I and 2 that the Cummings model [19,20]. It should be noted that collapse—revival phenomenon occurs in the system a similarbehavior was obtained in the numerical cal-for sufficiently largen.Since an increase innimplies culations ofDrobny and Jex [211, for the case of the an increase in the number of terms in the sum of eq. initial coherent state of the Rayleigh mode. In this

connection we emphasize that the collapse—revival [4] N. Bogolubov Jr., A. Shumovsky and Tran Quang, J. Phys. phenomenon is a general property of the model de- (Paris) 48 (1987) 1671.

[5] H.P. Yuen, Phys. Rev. A 13 (1976) 2226.

scribed by Hamiltonian (2) irrespective of the

mi-

[6]A. Shumovskyand Tran Quang, Phys.Lett.A 131(1988)

tial state of the Rayleigh mode. Some other cases are 471.

also considered in ref. [15]. [7] C.C. Gerryand J.H. Eberly, Phys. Rev. A 42 (1990) 6805. [8] l.A. Walmsley and M.G. Raymer, Phys. Rev. A 33 (1986)

382.

[9] M.D. Duncan, R. Mahon, L.L. Tankersley and J. Reintjes,

Acknowledgement J. Opt. Soc. Am. B 7(1990)1336.

[10] Y.R. Shen, The principles of nonlinear optics (Wiley, New A.S. would like to thank Professors C. Bowden, R. York, 1984).

Bullough, S. Carusotto, 0. Keller, F. Persico and~ [11] AS. Shumovsk~’,in: Modem nonlinear optics, ed. M. Evans (Wiley, New York, 1993).

Rupasov for fruitful discussions. [12] A. Chizhov, B. Govorkov Jr. and A. Shumovsky, Int. J. Mod. Phys. B, to bepublished.

[13] S. Carusotto, Phys. Rev. A 40 (1989)1848.

References [14] Yu. Orlov, I. Pavlotsky, A. Shumovsky, V. Suslin and V.

Vedenyapin, Int. J.Mod.Phys. B, to be published. [15] A. ShumovskyandB. Tanatar, Phys. Rev. A, to bepublished. [11 A.S. Shumovsky and Tran Quang, in: Interaction of [16] D.F. Walls and R. Barakat, Phys. Rev. A 1 (1970) 446.

electromagnetic field with condensed matter, eds. N.N. [17] M. Marcus and H. Minc, A survey of matrix theory and Bogolubov, A.S. Shumovsky and V.1. Yukalov (World matrix inequalities (Allyn and Bacon, Boston, 1964). Scientific, Singapore, 1990) p. 103. [18] R. Loudon and P.L. Knight, J. Mod. Opt. 34 (1987) 709. [2] J. Mostowski and M.G. Raymer, in: Contemporary [19] HI. Yoo and J.H. Eberly, Phys. Rep. 118 (1985) 239.

nonlinear optics, eds. G.P. Agarval and R.W. Boyd [20] Fain Le Kien and A.S. Shuxnovsky, tnt. J. Mod. Phys. B 5

(AcademicPress,NewYork, 1992) p. 187. (1991) 2287.