IL NUOVO CIMENTO VoL. 7 D, N. 3 Matzo 1986

On Molecular Coherent Rotational States (').

L. FO~I)A, N. lViA~0~-BOR~T~ (*') and 1~. I~OSn~A ('*)

International School ]or Advanced Studies . Trieste, Italy International Center for Theoretical Physics - Trieste, Italy

(ricevuto il 30 Ottobre 1985)

Summary. - -Fol lowing the experimental detection of coherent rota- tional states for the case of CS 2 molecules, in this paper consideration is given to the theoretical evaluation of their formation and their time evolution. Use is made of the sudden approximation by choosing as Hamil- tonian the semi-classical dipole electric-field effective interaction.

PACt. 33.10. - Calculation of molecular spectra.

1. - T h e e x p e r i m e n t a l r e s u l t .

The format ion of coherent ro ta t ional states (CRS) has been revealed by

means of a very ingenious experiment by H:ERITAGE~ GUSTA~SON and LIN (1)

for the case of the CS2 gas, whose molecules exhibit an energy spectrum

very close to the ideal ro ta t ional band rule o~I(I ~- 1). The experiment followed two theoret ical papers (~,~) by the same authors

in which the conjecture has been pu t forward of the format ion of CRS in the

molecules of a (~ collisionless ~> gas when the gas is excited with a short optical

(*) Supported in part by the University of Trieste and by the National Institute for Nuclear Physics (INFN), Section of Trieste. {**) Permanent address: Faculty of :Natural Science and Technology and J. Stefan Institute, University E. Kardclj, Ljubljana, Yugoslavia. (1) J .P . HERITAGE, T. K. GUSTAFSO~ and C. H. L ~ : Phys. ]~ev. LeSt., 34, 1299 (1975). (~) C.H. LIN, J. P. HERITAGE and T. K. GUSTAFSON : Appl. Phys. Lett., 19, 397 (1971). (a) C. l:I. LI~, J .P . It~RITAG~, T.K. GUSTA]~SON, R.Y. CHIAO and J. P. McTAG~: Phys. t~ev. A, 13, 813 (1976).

2 9 - I~ N u o v o (Yimento D. 437

438 L. FONDA, N. MANKO~-BOR~TNIK and M. ROSINA

pulse. A collisionless gas is one for which most of the dynamics of the phenom- enon occurs on t ime intervals much shorter t han the collision t ime among the molecules, yielding the possibility of exper imental detection for the phenomenon itself.

I f the optical pulse is of short duration, as compared wi~h the rotat ional

period of the molecules, instantaneous transitions occur among the rotat ional quantum states resulting in final states with well-established phase relationship.

In reL (~), use has been made of the exper imental set-up shown in fig. 1 in which the CS~ vapour was excited with an intense ul t rashor t ( ~ 5 ps) infra-red laser pulse of wave-length 1.06 ~m generated by a mode-locked glass: Nd +3 laser. They used then a weak probe pulse of wave-length 0.53 Fin, obtained by harmonic generation from the fundamenta l laser pulse. This probe pulse was delayed in t ime by a variable optical pa th and polarized by the polarizer P1. The probe pulse, af ter crossing the CS~ chamber, was analysed by a polarizer

to oscilloscope

[[]Pd.1 L1 y 45~

1.06 I~m excitation path |

locked pu[se~ / r ~ U" kJ L~ ZrJ b "

0.53 ~m pt'obe ,~ | ,_L,

pulse path ~Pr3 --~-I r~Pd2 [ ~

oscEloscope TEK 7904 Fig. 1. - Schematic of the experimental arrangement used to detect the quantum beat. BS1 and BS2 are beam splitters, M 1 and M 2 are mirrors. The potassium dihydrogen phosphate (KDP) crystal was used to generate the second harmonic of the 1.06 ~m pulses. F1 and F2 filter out the residual 1.06 ~m radiation, gl and g~ represent glass plates and Pdl, Pd2 and Pd3 the photodetectors. Prisms Prl, Pr2 and Pr3 formed the adjust- able delay line. P1 and P~ are polarizers oriented as indicated and 2/2 represents a half-wave plate. L~ and L~ represent 1.5 m focal-length lenses. The CS~ vapour cell was 1 m in length, (From ref. (1)).

P, which lets through only photons with polarization perpendicular to tha t given by P1. The photodetector Pd3 measured then the probe energy trans- mi t ted through the analysing polarizer P , , as a function of the probe pulse delay, giving a measurement of the polarization change of the probe pulse which in tu rn depends on the coherence of the molecular rota t ional motion. Their exper imental results are shown in fig. 2 where birefringence bursts show up 38 and 76 ps a~ter the initial laser excitation. The birefringence bursts

O N MOLECULAR COHERENT ROTATIONAL STATES ~

displayed at 38 and 76 ps are a t t r i bu ted to the first and second coherent re- construct ion of the CI~S ini t ia ted by the 5 ps exci ta t ion laser pulse. The de- creased ampl i tude of the peak at 76 ps with respect to the first one at 38 ps indicated the incoming of dephasing processes, like molecular collisions, and the interference provoked by overlapping vibra t ional levels a t 396.7 and 656.5 cm -1 above the ground state, as indicated also in the insert to fig. 2. Note t ha t the t empera tu re of CS~ was 325 K. At this t empera tu re , about twen ty ro ta t ional states (0 < I < 40) are re levant in the format ion of the CI~S, while collision t imes u r e a few hundred picoseconds, so thu t the gas can be considered as (( eollisionless ,) with regard to the phenomenon under study.

10

lo-

E m

A.

~, 3 .b //////////////////////,///I l ia 26187 cm -1 a A 2

. . . . . 1 8 8 6 8 c m -1

. - - 94.34 cm -1

/

I = 6

= 4

= 2 = o

10o0 "~ vibrational

-~ J levels 01~0 flu 396.7 cm

exci tat ion

l oo~

O.Ocrn -1

o

D

10 -~. I I I I I I I I I t I I I 0 50 100

relative clelay(10--IZ s)

Fig. 2. - The measured t ime dependence of the fraction of energy transmit ted through the analysing polarizer. The t ime origin was taken to be the observed peak in the t ransmit ted energy. The appropriate portion of the term diagram for the CS~ molecule is also shown. One component of the overall Raman-mixing process is indicated (for KJ z = TJ~). The dashed lines indicate vir tual states lying below the excited electronic states. (From ref. (1)).

29* - IZ N u o v o C i m e n l o D .

~40 L. FONDA, :N. MA.2ffKO(~-BORSTNIK and. M. ]{OSINA

In ref. (2,3) LI~ et al. have given the theoret ical evaluat ion of this phenomenon

using per turba t ion theory to the first order in the dipole-electric-field inter-

action H~miltoni~n. However, even though the interact ion Hamil tonian lasts a short t ime, due to the intensi ty of the optical pulse a many-step process actually occurs with contributions to all orders of per turbat ion theory. In this paper we want to repeat the calculations of ref. (m) using the sudden approximat ion with contributions to all orders in the interaction Humiltonian. The use of this approximat ion is of course justified by the short durat ion in t ime of the laser exci tat ion pulse.

In sect. 2 we obtain the wave function of the molecular CRS. Section 3 covers the evaluation of the variat ion of the index of refraction. Units ~ ---- c ---- 1 will be used.

2. - M o l e c u l a r c o h e r e n t r o t a t i o n a l s t a t e s .

The quantum process which underlies the exci tat ion provided by the optical pulse on our gas is the so-called l~aman scattering which is the result of a vir tual electric-dipole photon absorption followed by a vi r tual electric-dipole photon emission. In this paper we shall consider an effective interact ion Hamil-

ionian describing semi-classically as a single step the above-considered photon absorption-emission.

For linear molecules, such us CS2, interact ing with a (classical) linearly polarized optical field, the to ta l t{amiltonian is then given by

O)

H(t) ---- Ho + H' ( t ) ,

/ L = ~oJ~,

H ' ( t ) = - - �89 < E ( t ) . E ( t ) } [A~ cos ~ 0 + ~•

< E . E ) means average over a few optical cycles of the modulus square of the (classical) electric field E characterizing the laser pulse, 0 is the angle between E and the molecular body-fixed symmet ry axis. Aa is the anisotropy given by ~ , , - ~• where ~, and a• are the polarizabilities parallel and perpendicular to the molecular symmet ry axis, respectively. E(t) has an impulsive shape of short durat ion in t ime. Mote tha t we have dropped from Ho the Hamil- tonian describing the system before the optical pulse (i.e. for t imes t < 0), all i rrelevant degrees of freedom, leaving only the rota t ional ones. Besides, only the rota t ional band K ---- 0 contributes.

:For the chosen gas CS2, whose molecules are linear and symmetrical , the statistics of the identical sulfur nuclei allows the existence of only even rotat ional states (I = even).

The interact ion Hamil tonian H'(t) gives transitions I - - > I and I - > 1 •

ON MOLECULAR C O H e r e N T ROTATIONAL STATES

The projected quan tum number M is instead left unchanged. mat r ix elements of the I t e rmi t i an operator

cos ~ 0 - ~ Y~o § (4~)~ I:oo

are given by

(2)

441

The re levant

a~, {[1 + 2P- M~][(! Jr 1)'-- M']p, <r z 0]r - - 21 + 3 (2I + 1)(21 -~ 5) /

<r162 = 0~., 2(I ~ -- M ~) + 2 I - 1

( 2 I - - 1 ) ( 2 I @ 3) '

(~MM' {(I2-- M~)[(I--1)'~-- M~]} ~, <r176 01r ) -- 21 -- 1 (-21-- 3)(2I + 1)

where ~)IM is the normalized angula r -momentum eigenstate. The the rma l equil ibrium at t ime t = O, just before the gas is exci ted with

thc optical pulse, is character ized by a stat ist ical mixture defined mathemat ica l ly

by the densi ty operator

(3) o(o) = z-1 exp [ - Ho/k~'],

where Z is the par t i t ion funct ion at t empera tu re T:

Z = Tr exp [-- tto/kT] = ~ (21 + 1) exp [-- I(I + 1)r 1

where the sum runs only on even I values. This fact will Mways be unders tood

in what follows. The unper tu rbed densi ty operator can be wri t ten as

(4) 5(0) = Y_. p, lr <r IM

where the weights Pl are given by

(5) p1 = z -1 exp [ - coz(I + 1) /kT] .

Previous to the optical dis turbance, the @-matrix has only diagonal non- vanishing elements on the angular -momentum quan tum states. After the optical dis turbance, the off-diagonal elements get popula ted as a result of the

4 4 2 L. FONDA, N. MAN1KOC-BORSTNIK and M. ROS~NA

where l~o~M(t)> is the state of the system as obtained at t ime t from initial boundary conditions

(7) I~'~(o)> = lr �9

In the sudden approximation ]~M(t)> is writ ten as

t~v'~(t)> = exp [-- iHot] U(t)lr ,

(s) U(t) = exp [-- i ~ ' ( t ' ) d t ' ] ,

0

which, using the completeness of the set {r can be writ ten as

(9) lw'~(t) > = ~ <r ] U(t)Ir exp [ - iT,, t] ]r I ' M "

with E~ = o~I(I Jr 1). Since the Hamil tonian leaves the projected angular momentum M unchanged, we have of course

(10) <r [U(t)Ir = ~,~' <r162 �9

The general structure of state (8) is immediately obtained using the explicit expression for H'(t):

(11) lyJ'~(t)} = exp [-- iHot] exp [iDo(t)] exp [iD2(t)Y~o]lr

where t

16zt Ddt) = ~ (~) f <E(t').E(t')}dt', (12) o

t

5 , i dt'. Do(t) = (~6~) D ~ ( t ) § 0

Using the expression obtained for a pure rotat ional band for the quadrupole moment operator ~0:

(13) 0o = �89

where the constant Qo is the intrinsic qnadrupole moment of the molecule, one gets

(14) I~zM(t)> = exp [-- iHot] exp [iDo(t)] exp [-- iqo(t)Oo]lr

where qo(t) = -- 2D2(t)/Qo.

ON ~OLEC~LA/%COHE~ENT ROTATIONAL STATES 443

If T is the duration in time of the optical pulse, for times t > ~ we see from (12) that

Do,~(t) = Do,~(v) = eonst, t > ~ ,

and the only t ime dependence in (14) is given by exp I--filet] applied to the state

(15) exp [iDo]" exp [-- iqoOo]ltm>.

&ccording to the definition by PE~ET,O~OV (4), state (15), apart from the trivial constant exp [/Do], is a coherent state belonging to the manifold generated by the action, on our initial state IteM>, of the generic element exp [--iqo(~o] of the Abe]Jan group R whose infinitesimal generator is given by the quadrupole moment operator ~o.

We hay% therefor% seen how it comes about that an optical pulse excites a gas, like CS~, into a coherent superposition of rotational states. The coef- ficients <r162 Of the CRS (9) are evaluated as follows:

(16) t

0

---- exp [iDo(t)] f dD ~ ,~(~) Ym(~) exp [iDa(t)Y~0(S2)]. 4 ~

The angular integral appearing in (16) is solved by expanding exp [iD, Y~o]:

exp [iD, Y~o(Q)] = ~ [4z(2I + 1)]t~,M(D2):Ym(Q), ZM

~m(D~) = [4z(2I q- 1)]-t.fd~ tr~*x(~2) exp [iD2 Y~o(~2)] �9 4 ~

One easily obtains

. , [2I + l \ t ,:~,-,,IV(t)lr = e~p [,,:Do(O],,.,,,.Z (2v + ~ ) t , ~ ' T i ) "

�9 e , . ( / ' 0; o0) o , , . ( / 'M; M~'),z.,.(D#)).

The second Clebsh-Gordan coefficient yields the Kroneeker ~.'o, so that one needs to evaluate only a~o(D2). Using the relation

T'~o(0,9) = ( ~ 1)~-Pz(cos 0),

(4) A . M . PEREL0•OV: Commas. Math. Phys., 26, 222 (1972). See also H. U~: Prog. Theor. Phys., 44, 153 (1970); G. R0SENST~.EL and D. J. ROWE: Am. Phys. (N. Y.), 126, 343 (1980); D.J . ROWE: INS International Symposium Mr..Fuji (1982), p. 331.

444 L, FONDA, ~q. ~r aI14 ~r I~OSIIWA

one gets the analytic expression

1

1 fdzP,(z) --1

exp 5 �89 ,1EI2F, ++ 1 )ox

where ~ is the confluent hypergeometric function. This ends up the eva- luation of the coefficients <r162 >. We obtain

(~7) . , [2I 4- 1\~ <r162 = exp [iDo(t)] ,.~" (2I" § • Cn.(I '0; 00)-

] 11"/2 3 3i 5 �89

3. - Measurement o f the index o f refraction.

The index of refraction is given classically by

n = ( l+4~Z) �89 , %~2Va[1 4 ~ a ] - 1 -

where N is the number of molecules of CS2 per cubic centimeter and a the polarizability. Since 4~rN~ is of the order of 10 -3 for the experiment con- sidered, we c~n use the usual simplified version

n = 1 + 2z3?a,

which quantum-mechanically, for our case, reads

(18) n(t) ---- l -}- 2~h r Tr [@(t)(Aa cos 20 + ~.)] = no + An(t).

no is the index of refraction obtained at t ime t if no interaction has occurred, in which case [?re( t ) )~ exp [--iE1t]lr >. One easily gets the time-indepcn- denr result

ON MOLECULAR COHERENT ROTATIONAL STATES ~

The var ia t ion An(t) is also easily evaluated:

(20) An(t) = 2~NA~ ~ P,,(I<r V(t)lr ~ - ~,,)<r 0ir + 1 M I t

~- 2~hrA~ Z p~,(exp [-- i~zt] (r U(t) ltz,}"

�9 (r ~(t) lr <r 01r + c.c.),

where ~9~ is the so-called Raman f requency:

(21) -QI-~ 6co ~- 4~I .

The general t ime s t ruc ture of An(t) is ga thered f rom (20)�9 For t > ~, where

is the t ime dura t ion of the opt ical pulse, we have seen tha~ bo th Do and D~ are constant in t ime. There follows t ha t also the coefficients (r162 are t ime independent for t > z. The only t ime dependence is, therefore, provided by the exponentials exp [ i i~ t ] .

Using (21) we see t hen t ha t An(t) can be wri t ten , in general, as

(22) [ o ]

An(t) = n~ + 27~NAa exp [i 6~ot] ~ exp [i 4o~It] G~ ~- e.c. ,

1 e v e n

where the constant background t e rm n B is actual ly zero to the first order of pe r tu rba t ion theo ry and the coefficients GI are cons tant in time.

The sum appearing on the r ight-hand side of (22) is periodic with period 7c/4w, on the o ther hand the curve An(t) duplicates itself exact ly only af te r a period ~/co because of the interference between the factor exp [i6wt] and the sum. We obtain t hen short bursts in the refract ive- index change at integral multiples of ~/dw, bu t only every four bursts the detailed pa t t e rn is completely reproduced. Since eo ~--2u.0.327-101~ s -1 for the CS2 molecule, one gets the value 38 ps for the coherence period ~/4w.

The exper imenta l results obta ined in ref. (1) exhibi t exact ly the above s t ructure with bursts separated by the coherence period 38 ps. The choice of the gas CS2 is par t icular ly favourable because of its small ro ta t ional con- s tunt w, which makes long~ be t t e r detectable, delays between bursts. Moreover, CS~ has a large anisot ropy (A~ ~- 75.10 _25 cm 8) which gives sizeable refractive- index changes. The ia te rpre ta t ion of the coherent s t ructure of An(t) goes as follows:

In a semi-classical picture, at equil ibrium, the CS2 gas consists of an en- semble of ro ta t ing molecnles. The applicat ion of an intense optical pulse induces an instantaneous molecular al ignment along the direction of polarization of the optical pulse. Following the applicat ion of the pulse, the molecules continue t o ro ta te gradual ly loosing this al ignment and periodically recovering it, wi th period 7~[4co, for a re la t ively long t ime, unt i l collisions randomize the rota t ional motion.

~6 L. FOI~DA, 1~. MANKO~-BOR~TNIK and ~. ROSINA

�9 R I A S S U N T 0

A seguito della rivelazione sperimentale dell'esistenza eli stati rotazionali ooerenti per le moleoole del gas CS 2, in questo artioolo si considerano dal punto di vista teorlco la loro formazione e la loro evoluzione temporale. Si usa la ~ suclden approximation ~> assumendo per hamiltoniana l ' interazione semi-classica effettiva eli dipolo elettrieo.

Pe3IOM0 He Ho~y~IeHO.

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