Coherent and incoherent laser spectroscopy of spatial and temporal fluctuations

S~olid State Communications, Vol.51,No.9, pp.697-700, 1984 Printed in Great Britain.

0038-1098/84 $3.00 + .00 Pergamon Press Ltd.

COHERENT and INCOHERENT LASER SPECTROSCOPY of SPATIAL and TEMPORAL FLUCTUATIONS Eiichi Hanamura Department of Applied Physics, University of Tokyo Hongo, Bunkyo-ku, Tokyo 113 Japan (Received

29 May,

1984 by T. Tsuzuki)

Information on spatial and temporal fluctuations of the transition frequency is theoretically shown to be obtained from transient four-wave mixing (FWM) at k=2kl-k 2 or 2k2-k, under coherent excitation by two laser pulses with wav~ vectors k I and k . The same information is available from the FWM by two C ~ i n c o h e r e n t laser beams k ] and k 2 split off from a single source as a function of t~e relative delay time. In these cases, we can select a specified diagram for the FWM which depends on double sites. This makes it possible to observe the spatial fluctuations in addition to the temporal ones.

Four wave mixing (FWM) under stationary laser excitation, especially coherent anti-Stokes Raman scattering has been powerful to detect an elementary excitation in molecules and solids. The transient, degenerate and resonant FWM corresponds to photon-echo phenomena or free-induction decay in the lowintensity limit for the inhomogeneously broadened electronic system [ i] . From these measurements, we can get the various relaxation rates in the medium. We have developed time evolution formalism for resonant secondary radiation [2]. The competitive behavior could be discussed for Raman scattering and luminescence channels. The emission spectrum is obtained by evaluating time evolution of the density operator. In the leftward (rightward) propagating state, ~1-photon is absorbed (emitted) at time t~ (s I) and ~ - p h o t o n is emitted (absorbed)~at~time tf(s2). There are 4C2 (=6) contributions depending upon £he temporal orders of (t],t3) relative to (Sl,S2). This time evolution formalism is extended in this communication to the four wave mixing process. The probability to find out at time t that e l- and ~3-photons are absorbed and ~2 and ~4-photons are emitted is written as fdllows; p(t) = ~(t)Pa0#%(t),

-exp[-i(Ha+~2-~l)(t3-t2)](-iV(2)) •e x p [ - i ( ~ - ~ l ) ( t 2 - t l ) ] ( - i V ( 1 ) ) .

(2)

Here p0 is the initial density matrix of the t~tal system of the medium and the radiation field. Let us consider the nearly resonant case so that only the ground la> and excited Ib> electronic states are considered ,in the rotating wave approximation; V ~ J = - p . E (t)exp(ik .r) and V ~ J=-, •E~ (t) exp (-i~n. r ) . W~e can also r e p r e s e n ~ the effect of the reservoir elementary excitations on the electronic system as the frequency modulation 6~: (T) at the i-th localized electron. Th~n we may replace Hb(t-s) by

(~b-iy)(t-s)+ f~6~i(T)dT,

where

y is

the radiative decay constant and is related to the longitudinal relaxation time TI=I/(2Y). The probability of observing m4-photon with k4=kl-k2+k 3 ~ t is obtained as I(~4't)=~P(t)'v_

at We

have a single-site contribution represented by Fig.la and a double-site one by Fig.lb. Under stationary excitations, there are R C 4 = 7 0 diagrams for the single site contribution dependeng upon the time orders of t4>t3>t2>t I relative to s4>s3>s2>sl in Fig.la. In the case of a double site contributions represented by Fig.lb, there are 2x2x4C 2

(i)

(t (t4 [t3 It 2 ~(t) = I dt I dt I dt dt j_~ 4j_~ 3)_~ 2 _~ 1

x4C2=144 diagrams.

We have

also contri-

butions to FWM from the diagrams of Figs.lc and id. These are fourth (second) order perturbations in #(t) and second (fourth) order ones in ~%(t) for

•exp[-i (Ha+~4-~3+~2-~ I) (t-t 4 ) ] (-iV (4) ) •exp[-i(Hb-~3+~2-0~ I) (t4-t 3)](-iv (3))

697

698

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LASER SPECTROSCOPY OF SPATIAL AND TEMPORAL FLUCTUATIONS

intensity at ~ and k=2k2-k I, only single diagram of Fig.lb with tl=Sl=0 and tp=tR =s2=s3=T contributes and~de~cribes -th~ photon echo at t=2T. For observation at k=2kl-k 2, we have a contribution only from a diagram of Fig.lb with tl=tg=s.= s2=0 and t3=s3=T. This is t~e -fr~e induction decay just after the second pulse. As far as the effect of the reservoir is taken into account as relaxation constants, these measurements will fix only the transverse relaxation time T_. More generally, we should describ~ the environment effect in terms of the frequency modulation [3]. The first pulse at t=0 induces a single

b

8 t4

t4

' ~ ~6~2

t~~S~

S

d SB

h

t2=..c04..~.o ... I

transition:

V(1)(tl)=~(1)6(tl-0)

second

one

at

tions

V(2) (t2)=~(2)6(t2 -T)

t=T

does

(t3)=~(2)*6(t3--T). emitted

light

Let

intensity

~4

gives

transiV (3) and

us

observe

per

k=2k2-kl:I(t)~/d~4P(t). over

and the

double

second

the at

Integration

2w6(t-s4).

Then

we

can

perform all the time integrals and obtain the emission intensity under phase maching condition:

t~

s~

N

I(t) = w [ I~(i)121~(2)141v(4)I 2 i,j Fig.l Diagrams of the density matrix contributing to the four wave mixing. (a) Single site contribution of emitting ~d=~l-~p+~ 3 photon and (b) double site o~e o~ emitting ~4=~2+~3-~i . (c) and (d): Double site contributions due to sixth (second) order perturbations in the leftward (rightward) propagating states in the density matrix and its Hermite conjugate, respectively.

P(t) of eq.(1). As a result, the spectrum under stationary excitations is e x p r e s s e d in terms of many relaxation constants in a complicated way. When we observe the transient response due to the FWM by two coherent laser pulses, on the other hand, we can choose only a few contributions selectively. For example, we consider degenerate resonant FWM by two coherent laser pulses El(t)exp(ik 1 .r-i~t)

and

E2(t)exp(ik2.r-i~t).

The

latter is temporally delayed by T relative to the former. Here the electronic system may be considered effectively as a two-level la> and Ib> nearly resonant to ~ . The excitation energy ~ b a has the static inhomogeneous distributlon due to ~ d o p p l e r effect in the gas system and the crystalline defects in solids. It suffers also the dynamical frequency fluctuation 6~i(t) at the i-th site, e.g., due to the electron-acoustic phonon interaction or the electron-spin wave interaction. When we observe the light scattering

.exp[-(t-2T)2 2] T -exp (-2yt) +c.c. JT x Here tion

we of

(3)

assumed the Gaussian distributhe transition frequency A~baE

~ba-~ with the deviation o and the peak at A ~ a : g (Amba) = ~ o 1e x p [ (A~ba_A~a)2 /2a2]. from

The factor e x p [ - ( t - 2 T ) 2 O 2] comes the

multiplication

-ix(t-T)] g(x) complex

with

conjugate

of x=A~a

with

fdx e x p [ i x T . and

x=~a.

the

means taking the thermal average of 0 over the reservoir coordinates, e.g., the phonon fields or the spin waves. The correlation time T and the corrrelatlon length R of the frequency modulation 6~: (T) a r e U d e t e r m i n e d by the inverses of t~e effective spectrum width and of the most effective wave number of the elementary excitations involving in the frequency modulation, respectively. The linear dimension £ of the crossing region of two laser beams may be reduced to an order of the wavelength I. Then this nonlinear coherent phenomenon is effectivel~ induced only in a few thousands ~ around the product maximum of the two laser beam intensities at the trajecries. If Rc<<£c, we can take the

•

c

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thermal average over i- and j-th sites independently. Then the i n f o r m a t i o n at the single site is available as in the usual spectroscopy. In this case, only the temporal c o r r e l a t i o n of the frequency m o d u l a t i o n is observable. When many freedoms of e l e m e n t a r y e x c i t a t i o n s are p a r t i c i p a t i n g in frequency m o d u l a t i o n in solids, we may take the G a u s s i a n model: < ~ i ( t l ) 6 ~ i ( t 2 ) > = ~ 2 e x p ( - I t l - t 2 1 / T c ). Then I(t) of eq.(3) is e v a l u a t e d as I(t)

TEMPORAL

1.0 (¢O-

0.5 09 •

O _J

0.2 0.1

- (t-T -e

0.2 (4)

with A E ~ N 2 1 ~ ( 1 ) ( ~ ( 2 ) ) 2 V ( 4 ) I2. For TT . y'>>y in many cases of solids, so Cthat the signal shows double e x p o n e n t i a l decay as a function of T with a kink at T as shown in Fig.2a. This is in agreement with the result of the usual photon echo for the Gaussian-modulated two-level system [3]. On the other hand, for R >>£ , 6~. and 6~. in e ~ c ~ are cancel l ~ ed ~ u t ~o that 3the due to the frequency m o d u l a t i o n vanishes but only the radiative decay 2y remains finite. In the case of the electron-phono~ interaction, R is e s t i m a t e d to be 300A at the lattic~ temperature 1 ° K. The case of Rc~£ c is most interesting. The multi-photon processes due to such the i- and j-th e l e c t r o n pairs as R..>R ~£ 1 C " C suffers from both the frequency ~odulation and the radiative decay. On the other hand, those due to such the pairs as R..
0.05

)/TC} Icoh

01

the

_

b

- (t-T)/TC +2 (ATc)2 (l-e T/TC) {l-e }].

of

8

0.5

-T/T c

a fraction

699

FLUCTUATIONS

10

= 2Aexp[-(t-2T)2g2-2yt]

•exp[-2 A 2 Tct+2 ( AT c )2 {2-e

AND

Rij>R c and Ico h

/ (Iinc + I c o h ) is one of the pairs Rij
0

L

I

I

1.0

2.0

3.O

T/~c Fig.2 Photon echo signal intensity at k = 2 k 2 - k I under two laser pulses with and k 2 s e p a r a t e d by time interval T p l o t t e d as a function of T, (a) for the case in which the coherent range R c of the f r e q u e n c y m o d u l a t i o n is much smaller than the linear dimension £ of the c r o s s i n g region of two laser b~ams, and (b) for the case of Rc=Z c. Icoh/(Icoh + I. ) denotes a fraction of the region inc in which the frequency modulation is almost in phase in the beam c r o s s i n g region in which the nonlinear c o h e r e n t p o l a r i z a t i o n is induced. This fraction decays with the l o n g i t u d i n a l decay time T 1 denoted by d o t t e d line. y/(~)2T c =0.01 and 6~Tc=l were used.

frequency of the incoherent ^ l a s e r beam is d i s t r i b u t e d over A~>>y'HAZT c and y. Let us consider two CW incoherent laser beams k I and k 2 split off from a signal source. The latter laser beam k^ is delayed by T relative to the f o r m e ~ o n e kI. These laser beams have short c o r r e l a t i o n time TIE(A~) -I SO that these two laser beams are considered as continuous trains of two pulses with the pulse width T£ and the s e p a r a t i o n T. The degenerate FWM intensity I(t) contains <>.

Here

<<...>>

means

the

statisticl average over the stochastic variables of the laser phase and amplitude. We may assume that it c o n t r i b u t e s dominantly when these time arguments coincide with each other w i t h i n the time interval ~. Contributions obtained under the + G a u s s i a n approximation of dividing it into the product of three second-order moments show the s t a t i o n a r y background signals in addition to the

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LASER SPECTROSCOPY OF SPATIAL AND TEMPORAL FLUCTUATIONS

relevant signal [4]. The w a y of t a k i n g the statistical a v e r a g e o v e r the l a s e r field variables depends sensitively on the characteristics of the incoherent laser sources. In a n y c a s e s of the incoherent laser with a short correlat i o n t i m e T[, t h e c o r r e l a t i o n m e a s u r e ment presen£~d h e r e w i l l s h o w m o r e or less t h e r e l e v a n t s i g n a l as a f u n c t i o n o f T i n d i c a t i n g the r e l a x a t i o n p r o c e s s of m a t e r i a l s . W h e n the p r e s e n t sixorder correlation f u n c t i o n is a c c e p t e d , time integrals for I(t) are p e r f o r m e d easily. Then the stationary emitted intensity I(T) is o b t a i n e d only as a function of the delay time T of t w o b e a m s in the s a m e f o r m as eq. (3) w i t h

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t=2T. Therefore we w i l l be able to obtain the same information on the correlation range R a n d time of the . C frequency modulatlon as in t~e c a s e under two coherent laser pulses of eq. (3) . We conclude that the i n f o r m a t i o n not only on the temporal but also s p a t i a l f l u c t u a t i o n is o b t a i n e d f r o m the degenerate resonant FWM measurement by coherent laser pulse or i n c o h e r e n t CW lasers. The a u t h o r t h a n k s Prof. T. Y a j i m a and Dr. N.Morita for the fruitful d i s c u s s i o n s a n d c r i t i c a l r e a d i n g of the manuscript.

References [i] P . Y e a n d Y.R. Shen: Phys. Rev. 25, 2183(1982) and T. Y a j i m a a~d Y. T a i r a : J. Phys. Soc. Jpn 47, 1620(1979). [2] T. T a k a g a h a r a , E. H a n a m u r a and R. Kubo: J. Phys. Soc. Jpn 43, 802, 811, 1522(1977) and 44, 728, 742 (1978).

[3] E. H a n a m u r a : J. Phys. Soc. Jpn 52, 2258, 2267, 3265, 3 6 7 8 ( 1 9 8 3 ) . [4] N. M o r i t a a n d T. Y a j i m a : to a p p e a r in Phys. Rev. A.