A theory of resonant light scattering

Volume 41,number 3

A THEORY OF RESONANT LIGHT SCATTERING”

Received 5 April 2976

By a some&hat novel choice for an interaction representation it has been demonstrated outside a Wigner-Weisskopf anmti that in two physically well defined situations the spectrum of light scatterkd by a moleculehas the same form as that c&Mated using a one-photon wavepacket. By characterizing the excit&g light using well known concepts from qnanturn optics we are able to describe the scattered radiation in terms of quantities which are measurable properties of seal light sources and molecules.

I. Intmduction

There has recently be3n considerable new interest in the nature of resonant and near resonant light scattering by molecules [l-4]. A main objective of such work is to provide an exact description of the measurable consequences of resonance Raman scatterin-g and to clearly delineate the limiting processes of resonance Raman and fluorescence. Calculations of photon counting rates usually begin with an initial state describing a molecule in its ground state and a light field state corresponding to a single non-stationary photon. The formal theory of scattering then yields a solution for the time dependent state of the whole system in certain physically meaningful situations. One parameter of the results is the energy width of the photon, For example used in ref. [l] was a lorentzian onephoton wavepacket having a well defimed energy width characterizing the exponential time decay of a light pulse. Admittedly this procedure was attractive since it yielded results which seemed readily interpretable and that did not depend on any Wigner-Weisskopf assumption. The interpretation of these results is however only qualitatively related to real experiments since the relationship between such a one-photon wavepacket and the measurable parameters of a real light source is not at all evident. In addition the assumption that the molecule need be considered to interact with only one photon at a time is unfounded since Glauber [5] has shown that at least in a chaotic light source the photon wavepackets tend to overlap in space. Rather #an trying to adjust the parameters of a model that contains a qu~itative representation of the Sight field it seems better to solve the problem in such manner that the real measurable characteristics of the light are included in the formalism for light scatter&g. fn what follows we present a method of solution for the near resonant scattering of light having well defined coherence properties.

2. The hamiltonian and time evolution operators A cruciai part of our method of solution is the manner in which the hamiltonian is partitioned_ The total hamiltonian for a system consisting of a molecule and a radiation field is, in the dipole approximation: * This research was supported by a PHS grant GM 12592 and by the NSFfifRL

program at the University of Pennsyknia.

407

Be= BI,

+Be,

p* E(r&

0)

where the electric field operator is defined as: E(Q) = EC+)(r,,) + E(-)(qJs

E(-) (Q)

=

[E’+)(rajl

f

)

(2) EC+)(q$ = i (z~~/~“’

c

k

~2;‘” & eik - r” ok J:

(33)

with ok being the annihilation operator for mode k. In a Raman experiment we usually have present light in two frequency regions - the incident and the scattered field. So we choose to write the hamiltonian in a form that *displays the interactions of the system with these two fields separately: 91 =Q,+Blf%Ji!~L

P*Ei(ruI - ICL*ES(~~) -su

- tc*Ei(rc>,

(41

where s and i signify scattered and incident f&Ids. The time evolution operator in the interaction representation, U[ (b), SatisfEs: iRdU;I(t)dt = VI(t)V; (t),

(5)

with Vi(tj=-eiMor~.~i(f~)2-iMof=-C11(~)

lEi(ro,t),

(61

where Ei(rb, tj is the Heisenberg operator for the electric field of the incident fight. We will need to calculate the

average number of photons in a mode k, and this is done by averaging aJ(*)nk(r) = eiBCr’a;i;fr~e-isrrr in the state 1qI(t)>

= U,(t)

(7)

I \kI. (@)) for k

not contained in i, and where SC, = %!, - RI’).

3. The spectral content of a scattered coherent pulse B&re proceeding to calculate the spectrum of radiation scattered from an arbitrary light beam it is useful to investigate the case closest to that described by a one-photon wavepacket, namely the interaction of a molecule with a coherent pulse of hght. Let us characterize the pulse quantum mechanically by a coherent state [5;6,p.1481

so that if the molecule was initiahy in its ground state IO), the w~v~f~~tion becomes in first order:

t If in (9) we now keep only the ITi part of the incident field, which is equivalent to making the rotating wave approximation, one fmds

f

where use has been made of the fact that a coherent state is an eigenstate of the annihilation operator. &e&u, t? is the eigenvalue of the operator Ai’-,*for the coherent state. Suppose that our molecule is characterized by three levels, IO),Ii)and I]?,and that the incident field contains only frequencies near oiO _ Furthermore we assume for simplicity (but not necessity) that state Ii) is undamped. Under

408

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CHEMiCAL PWSKS

LEX-i-ERS

1 August 1976

these circumstances the probability that a photon is scattered from the incident beam to a photon of frequency x Wij after a time long compared to all lifetimes can be calculated. Note first that the teti e-i BILf’10; VaCS; C~~)i),

0

0

represents no real time evolution of the system since there are no scattered photons in the initial state. It can only cause an energy shift of the ground state due to the interaction of the ground state molecule with virtual photons in S. If we call the corrected ground molecule state energy zero, then the amplitude for scattering one photon into a frequency near wjj in S, leaving the’exciting field in some other state, is given by ik-l ((fl&Jil C~~(f))i~~dl’(k;ile-i~~(r-“)li;vacs~~~~-L,(~~, t’), 0 where {Pk)i represents a final state of the incident light pulse. The matrix element of the propagator exp [-iV3$(t- t’)] may be calculated using resolvent operator techniques [7,8], by taking R(z)=(z-

02)

(13)

sI,r’,

so that ,-i

CZl(t-t’)

= (2ni)-’

sdz e-iZ(‘-‘f)R

(z),

tat’.

(14)

c The matrix element of R of interest here is

(kvjlR(z)li;vacs) ,

= - pij -
(z-w- I 3-i iF-) I ’

WI

where the width I”fis calculated by considering only modes in S. Using (14) and (15) one now finds that (12) becomes

If one now chooses a form for 6:, (ru, t’) &~(r~,t’)=:Eoe-i(wo-fi7)f’,

(17)

which represents a pulse with an initial sharp rise followed by an exponential decay at ‘0 one finds that the square modulus of the ampli~de (12) is given at long times by

In (f8) we are not interested in the final incident field state and should therefore integrate over all p giving a factor of unity for the coherent state part, since:

by the properties of coherent states f5]. The ~~ribution (18) is the same as that found previously using a onephoton lorentzian wavepacket however the physic is quite different. There is no assumption of only one photon in the pulse and the assumption that the exciting pulse is weak means that P=Ibrj*'&llQ~13

cw

since the perturbation treatment is a power series expansion in p when all terms are resonant. We point out further409

Volume 41, number 3

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PHYSICS LETTERS

more that (12) is the convolution of the exciting pulse with the internal propagator, a result much quoted in calculating transition amplitudes [4], If instead of (17) one uses a pulse with a gaussian time profile centered at a time z 3 0 SD that at t = 0 the pulse EELSan electric field of essentially zero, then the lower limit on the time integral may be taken as t = - m and the scattered light has a frequency Lstribution which is the product of a lorentzian centered on the molecular resonance and a gaussian centered on the exciting light frequency.

4. Scattering

from a weak stationary

light beam

Spectral lines are usually measured under steady state excitation and for this reason we will calculate the line shape for radiation scattered at = oii when a molecule is put in a stationary field. In order that time ordering is t&en properly into a account I; is convenient to calculate the pure state density operator for the molecule in the field. This density operator ill the interaction representation satisfies pl(f) = ~~(00) - in-‘jdt’

I?

(21)

p*(t’);

0

where Z(P) is the Liouville operator define6 by A?(8) ___= [VI (t’), . __1. Suppose now that we calculate the mean number of photons scattered into a particular mode k E S. We iterate (21) and keep only the first term yielding a non-zero result and fmd:

(22)

where pi is the density operator for the exciting field and where we have made the rotating wave approximation. The vector nature of Ei and pi has been surpressed for simplicity. Since the average value of the product of field operatdrs in (22) is the first order correlation function G (I) for the exciting field (22) becomes using (7)

t

t

+fT2 s

-- $1. (23) s dt, It1 0 0 The trace of a product of operators is invariant under cyclic permutation so one may perform the trace in (23) keeping only resonant terms and find bz$(t)ak(f))

dt,

= Re2~dtl~

0

dtzlpi0i2 Gtl)(ro, t2;ros tl)(k; jle’iwl(t-tl)ji;

dt, It1 w 0 The

vacsIei~Y(t-t2)li;k)=(~;kle-iQ

(24) becomes 410

f27.

0

matrix elements of the propagators 6;

vacsN; vacslei6[~(t-r2)~;~~

0

have already been evaluated using (14) and (I 5) and since 1(~42)li;vacs)*,

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CHEMICAL PhYSIC L?l%TTERS

Volume 41, nuinber 3

In order to proceed further we must now specify the nature of the light source through G(l). Two types of commonly used Iight sources both have the same first order correlation function. These area pressure broadened chaotic source ]6, pp. 102-1071 and a random-phase modulated coherent state model for a laser 191. For both of these sources G(l$,,

t2;ro,

tl)

=

Co

(27)

eiwO(r2-~lf-~lr2-~ll,

where 5 is on the order of the collision time in a pressure broadened source and for the laser it represents a phase diffusion constant related to the fluctuations of the laser oscillator. if one now substitutes (27) into (26) and evahrates the matrix element of E,‘+ one finds at times iong after the beam has been turned on

i.e. the average number of photons increases linearly in time when the time is long. The quantity that one measures with a monochromator and a photomultip~er is, however, d(=~(fdak(t))ldt=const.

[(w~+~~--~)?“+$~~]-~[(~~~~~-wo)~+~~]-~

uk/v,

(29)

and therefore apart from constants the distribution is the same as that obtained from a coherent Pul*.

5. IXscussion It has been demonstrated that the formal theory of scattering is not necessary to understand *he nat~e of light scattering by molecules in weak fields. In fact there is a danger innusingwavepackets, designed for non-relativistic particles, for an ultra-relativistic particle su?h as a photon [lo]. In our development we have circumvented any such subtle points, whieb may lead to asking improper questions, by using representations for the light field which are well known in quantum optics. In doing so we see that the one photon wavepacket concept actually restricts the physical situation to light pulses which are tirst order coherent [I 1J_ Nevertheless the form of the scattering equations is the same in both cases. It is just that measurable properties of the light field appear in the expressions developed here. The approach suggested here for solving problems of light scattering is expected to be useful also in the strong signal limit and for higher order interactions in which e.g. G” will appear in the development. In summary, we have shown that previous calculations of resonance light scattering that used a hypothetical one-photon packet having width are correct in form for any light source. The restriction on the validity of the result is the weli known weak-signal limit where the interaction between the molecule and the incident photons is less than the resonant linewidth geq_(29)], and not that there is only one photon. The expe~ent that most accurately mimics the one-photon packet concept uses a coherent pulse such as obtained from a mode-locked laser the appropriate parameter being the inverse time of the pulse. The first order correlation properties of such a pulse depend only on its spectral width. Such is the case for any light source and we find that the scattering depends only on the spectral width parameter g for stationary sources as well. Thus the physical interpretation for 411

Volume 41, aumbef 3

CHEMICAL PWYSKS LE2X%RS

1 August 1976

the paranrreter descfltring the JigI@ field in the denominator of aU *&e scattering formulas is simply the spectral width of light. This is because the scattering is first order in induced processes.

References [l] J.M. Friedman ?nd R.M. Hocbtrasser, Chem. Phys. 6 (1974) 155. [2] 3.0. Berg, CA. Lartgixoffand G.W. Robinson, Ckem. whys. Letters 29 (1974) 305. f3l S. Mukane and J. Sortner, 3. C&em. Phys. 62 (2975) 3609. f4] 3. Jortner and S. piiukamet.in: Proceedings of the First Intermtioual Congress on Quantum Chemisfry, edr R. Daudel and 3. Pullman (Reidel, Dordrecht, 1974) pp- 145-209. [S] R.3. Glauber, in: quantum optics, eds. SM. Kay and A. Maitiand (Academic Press, New York, 1970) pp. 53-125. [6] R. Laudon, The qur?ntumtheory of l&ht (Clare~don Press, Oxford, 1973) p. 148. 171 M-L. Goldberger znd R.M. Watson, Collision theory (Wiley, New York, 1964) ch. 8. 181 L. Mower. Phys Rev, 142 (19615) 799: 165 (1968) 145. 191 R.3_ Gfauber, in: Qu~tum optics ar,delectronics,eds C_De~~itt~A. Blandin 2nd C. Cohen Tyznoudji (Gordon and Breach, New York, 1964) pp. 165-X71. [lo] V.B. Berestetskii, EM. Lifshitz and L.P. Pitaeveskii, Relativistic quantum theory, part i (Per&unon Press, LandonfAddisan Wesley, Reading, 1971) pp. l-4. [ll] U.M. TituIr?erand RJ. Glauber, Phys Rev. 145 (1966) 1041.