Vertex correction effects on nonlinear optical response

Solid State Communications, Vol.55,No.5, pp.397-400, Printed in Great Britain.

VERTEX

CORRECTION

EFFECTS

H. T s u n e t s u g u

1985.

ON N O N L I N E A R

0038-I098/85 $3.00 + .00 Pergamon Press Ltd.

OPTICAL

RESPONSE

and E. Hanamura

Department of A p p l i e d Physics, U n i v e r s i t y of Tokyo Hongo, Bunkyo-ku, Tokyo 113 Japan

(Received

ii April,

1985 by H. Kamimura)

The effects of vertex c o r r e c t i o n s due to frequency m o d u l a t i o n are exactly e v a l u a t e d on the e m i s s i o n spectra from the two-level e l e c t r o n i c system p u m p e d by the laser of a r b i t r a r y intensity and the absorption spectra of the weak probe field of this system. Here the frequency m o d u l a t i o n is a s s u m e d to obey the G a u s s i a n - M a r k o f f i a n process. The two-time c o r r e l a t i o n function of the p o l a r i z a t i o n s which describe the e m i s s i o n and a b s o r p t i o n spectra is r e p r e s e n t e d as the product of two p r o p a g a t o r s and a vertex correction. This c o r r e c t i o n is d e s c r i b e d in the form of the summation of the products of the m a t r i x c o n t i n u e d fractions. This correction is n u m e r i c a l l y d e m o n s t r a t e d to be quite large for the e m i s s i o n spectrum in the case of the slow or strong frequency m o d u l a t i o n and weak laser field, and for the absorption spectrum in the case of intermediate m o d u l a t i o n and laser intensity.

Nonlinear optical phenomena have beeen s u c c e s s f u l l y d e s c r i b e d in terms of optical Bloch equation with constant longitudinal (T I) and t r a n s v e r s e (T 2) relaxation times[l]. The radiatige decay which determines T has been studied carefully and t~e constant relaxation time was justified. On the other hand, the latter c o n s t a n t T^ is o b t a i n e d under the a s s u m p t i o n that z the relevant reservoir elementary excitations have an infinite spectrum width. Strong d e v i a t i o n from a c o n s t a n t T~ has been shown to become o b s e r v a b l e whe~ the d etun i n g A or the Rabi frequency X become larger than the s p e c t r u m width of the reservoir elementary excitations [2,3]. In fact, the free induction decay signal was o b s e r v e d to decay more slowly than the rate p r e d i c t e d by the c onst a n t T~[4]. For the purpose of d e s c r i b i n g -these phenomena beyond the optical Bloch equation, the frequency modulation model has been used [3]. This may be c o n s i d e r e d in many cases as a stochastic variable which obeys Gaussian process. In the first c o m m u n i c a tion[5], we have been able to obtain exactly the e x p e c t a t i o n value of the electronic density o p e r a t o r under arbitrary strength of the incident laser field and for a r b i t r a r y f r e q u e n c y modulation. The emission and the a b s o r p t i o n spectra, however, are represented in terms of the t w o - t i m e c o r r e l a t i o n function of the electronic polarization

operators. Therefore we had to use the quantum regression theorem to c a l c u l a t e the spectra from the s i n g l e - t i m e expectation value of the electronic density operator. This is not justified when the density operator has non-Markoffian p r o p e r t y due to the finite memory time T c of the f r e q u e n c y modulation. In this c~mmunication, we derive the exact e x p r e s s i o n of the two-time c o r r e l a t i o n function of the e l e c t r o n i c system taking into account the effects of the finite m e m o r y time of the frequency m o d u l a t i o n for a r b i t r a r y laser i n t e n s i t y and for a r b i t r a r y m o d u l a t i o n rate and magnitude. These effects are represented as the vertex c o r r e c t i o n s in the Green function formalism of the response functions. The importance of these vertex corrections is n u m e r i c a l l y d e m o n s t r a t e d in the emission spectra and the absorption spectra of the weak probe field. The emission spectrum from the two-level electronic system irradiated by a laser field E ( t ) = E 0 e x p ( - i ~ t ) + c . c . is given besides an u n i m p o r t a n t factor by -~ iut lem (u) =2 limRe )0dte <~ba (T) 0ab (T+t) >, T-~oo

(i) and the a b s o r p t i o n probe field is 397

spectrum

of

a

weak

398

VERTEXCORRECTIONEFFECTSON NONLINEAROPTICALRESPONSE I ab(")

Vol. 55, No. 5 (7)

~O(T,~)=[C+~W(T)D]O(T,T),

>. (2)

=2$~Rej;dteiVt<[Pab(r+t),Pba(')]

where

Therefore we need calculate the expectacorrelation value of two-time tion functions of the density operators. Let us define the vector with eight elements of two-time correlation functions:

c=

-y+iA -ix/2

ix/2

-ix/2

0

0

ix/2

ix/2 -y-iA

0

-ix/2

ix/2

2Y -ix/2 -2Y I

,

i

D=

0 0

O -i 0. 1

[

(8)

(3) First

is obtained by solving under the initial condition @(O,O) with the finite single element (0) in the thermal equilibrium with:eo 1aser pumping. Second we integrate eq.(4) to calculate @(t+T,T) under the initial condition t=o. @CT,T) at Finally we take the ensemble average over the stochastic variable &w(t). Here also we use the Gaussian model of the frequency modulation: eq.

These eight elements compose the equations of motion in t and T in the closed Here are the form. and p populations of pRb t e exciteaab and the ground a states, rzspectively, and 8 = tPR$ off-diagonal density matrix components the the polarization of describing electronic system. @(t+r,r) obeys the following equation of motion: ~~(t+r,T)=[A+6w(t+T)B]~(t+T, where A=

1

P=

<6w(t1)

6w(t2) >

=(dt0)2exp[-ycltl-t2I].

(9)

with the characteristic magnitude 6w and the inverse correlation time Y . We perform the Laplace transformatio%s both in t and r of @(~+T,T) as 0[s,al

,

PO 0

(4)

T),

'#'(T,T)

(7)

=l~dte-Stl~dTe-~TO(t+r, T).

P

0

ix/2

0

-ix/2

ix/2

-ix/2

0

-y+iA

-ix/2

Here we expand @(~+T,T) in the perturbation series of the frequency modulation 6w(t), and we take the expectation value of O[s,u] in the reservoir coordinate in terms of the Gaussian property eq.(9). Then we can sum up the perturbation series to the infinite order:

ix/2

-y-iA 0 I

(10)

,

<@[s,a]> 1

(5)

=Go[s]~[s,olKo[a]~(O,O).

(11)

Here y is the radiative decay constant, X =uE the Rabi frequency OJ: the transition dipole moment between the states a and ;lm,"n~o~;~~" the detuning. The equal ion relation leads to .-

I;;,,(T) 1

Gn[s+nYcl 1

= s+nyc-A-B

(6)

(n+l) (6~)~ El (n+2) (6~)' B s+(n+l)Yc-A-Bs+ n+2 )Y,-A... (12a)

where the 2nd, 3rd, 5th and 8th elements vanish and skipped, and this obeys the equation of motion

K [o+ny ] is obtained by replacing A and B" withC C and D, respectively, in eq.(l2a) and

Vol. 55, No. 5

399

VERTEX CORRECTION EFFECTS ON NONLINEAR OPTICAL RESPONSE

his,o] =I+ ~ n l ( 6 ~ ) 2 n B G l [ S + Y c ] - - - B G n [ S + n Y c ] n=l

×Kn[O+nYc]D.-.KI[O+Yc]D.

NV=5olO

(12b) •" - -

G0[s] and K0[o] describe the electronic p r o p a g a t i o n in the c o r r e l a t i o n function before and after the time T of the photon emission or absorption, respectively. H[s,o] denotes the vertex c o r r e c t i o n due to the process in which a finite number of reservoir e l e m e n t a r y excitations are e x c i t e d before T and a n n i h i l a t e d after T. C o m p a r i n g eqs.(1) and (ii), we can calculate the emission spectrum from the seventh elements of ~7[s,0] s=-i(v-~) Iem(V)=2Reo<~7 [s,o]> IO= 0 .

NV=I

v

I--t

j _-o 0

(,o-~)/~,,

(13) Fig.l. E m i s s i o n spectra for the case of 6~/Yc=10 and X / 7 =i. NV=I0, 5, 1 and 0 mean the o r d e r ~ to which the vertex c o r r e c t i o n s are taken into account, and B is the spectrum o b t a i n e d f~gm th~ optical Bloch equation with (T 2) ~E(~m)

We can also evaluate in the similar way the o r d e r - e x c h a n g e d two-time c o r r e l a t i o n functions ~(t+T,~) and its Laplace t r a n s f o r m a t i o n ~[s,o]. Then the absorption spectrum can be also o b t a i n e d from eq.(2) in the following form:

/Yc"

Iab(V) =_'

=2Reo{<~7 [s,a]>_<~7[s,o] > }so=0 l(V-~) (14) Both the emission and a b s o r p t i o n spectra are mostly c h a r a c t e r i z e d by two parameters of 6~/Y~th~ represents ~ J T ~ h a ~ t e ~ / s ~ c s frequency modulation, i.e., slow or strong one in the case of 6~/y_ > 1 and fast or weak one in the case of ~w/T^i and X/Yc > l works on and ~ in the almost sa~e way in the absorption process. As a result, the vertex c o r r e c t i o n is almost c a n c e l l e d out in the a b s o r p t i o n s p e c t r u m defined in terms of the difference between ~_/ • and ~7" As seen from Fig.2, the vertex c o r r e d t i o n is quite large rather in the

v

H

-5

(v-~)/x' Fig.2. A b s o r p t i o n spectra for the case of 6~/y_=l and X/Y_=1. NV=8, 5, 1 and 0 mean tee orders 4 o which the vertex c o r r e c t i o n s are taken into account, and B is the result _~_ the~ optical Bloch e q u a t i o n with (T 2) = ( ~ ) Z / Y c.

i c% so in the g X/Y c XI0~ I, we can justify the lowest-order t r u n c a t i o n in the continued fraction representation and the n e g l e c t i o n of the v e r t e x corrections [7]. We have been able to c a l c u l a t e exactly in this c o m m u n i c a t i o n the emission and a b s o r p t i o n spectra in a r b i t r a r y degree of the frequency m o d u l a t i o n and at arbitrary intensity of the pump

5

400

VERTEX CORRECTION EFFECTS ON NONLINEAR OPTICAL RESPONSE

laser. Here t h e e f f e c t of the v e r t e x c o r r e c t i o n s h a s b e e n for the f i r s t t i m e exactly taken into account. This formulation covers both the optical Bloch equation in the fast or weak modulation a n d the w e a k p u m p f i e l d a n d the d r e s s e d a t o m r e p r e s e n t a t i o n in the s t r o n g p u m p f i e l d s u c h as X > > y _ o r 6~ at the t w o l i m i t i n g c a s e s [3]. Th~s effect of t h e v e r t e x c o r r e c t i o n w i l l be also

discussed transient induction the time paper.

Vol. 55, No. 5

for the optical coherent phenomena such as the free d e c a y a n d the p h o t o n e c h o in region in the forth-coming

The authors acknowledge Prof. G.S. A g a r w a l for p o i n t i n g out the e f f e c t of the v e r t e x c o r r e c t i o n s to them.

References

i) Y.R. Shen: The Principles of N o n linear Optics, (John W i l e y & Sons, 1984). 2) S. M u k a m e l : Phys. R e p t . 9 3 , 1 (1982). 3) E. H a n a m u r a : J. Phys. Soc. Jpn 52, 2258, 3 6 7 8 ( 1 9 8 3 ) . 4) R.G. D e V o e and R.G. B r e w e r : Phys. Rev. Lett. 50, 1 2 6 9 ( 1 9 8 3 ) .

5) H. T s u n e t s u g u , T. T a n i g u c h i and E. H a n a m u r a : S o l i d S t a t e C o m m u n . 52, 663(1984). 6) B.W. M o l l o w : Phys. Rev. 188, 1969 (1969). 7) E. H a n a m u r a : J. Phys. Soc. J p n 52, 3265(1983).