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Enhancement of quantum interference effects
Chemical Physics 13 (1976) 271-284
North-HollandPublishingCompany
ENHANCEMENT OF QUANTUM INTERFERENCE EFFECTS Albert VILLAEYS* 2nd Karl F. FREED TheJanresFranck fnstit~rteand 73e Department of Cknistry,
The University of Chicago,Oricago, Illinois 60637, USA
Received31 July 1975 Revised manuscript received 1 December 1975 We present a method of excitation which enhances quantum beats of certain frequencies and diminishes those at other frequencies. This approach uneven produce an effect when the original quantum beats appear to be washed out in large molecules because of the average over many pairs of interfering levels. The excitation of the system involves the absorption of a single photon from a coherent pair of pulses, leading to an additional interference effect with a nonzero quantum yie!d. The derivations are presented for the atomic-like few !evel syslems, but the more general molecular cases are readily obtained by superposition of aU pairs of interfering levels. The analysis incorporates the position and nature of the photon detector.
1. Introduction Quantum beats represent a dramatic example of the superposition principle of quantum mechanics. Thus, each new type of observation of quantum beat phenomena is heralded as a “new effect” in reaffirmation of the laws of quantum mechanics. Quantum beats have heretofore been observed with certainty only in atomic systems [l] where the excited state level density is so sparse that it is readily possible to focus upon a pair of interfering levels. In large molecules, on the other hand, the excited state level density can become sufficiently large that it is virtually impossible to not coherently excite a superposition of excited levels, thereby producing quantum beat phenomena in the subsequent decay processes. It is, perhaps, the ubiquity of interference phenomena in excited states of large molecules that has prompted the search for quantum beats in the deczy of intermediate case molecules [2] Although quantum mechanics dictates the necessity formthe occurrence of quantum beats when levels are coherently excited, the difficulty of observing these phenomena in large molecules is readily understood. Let &denote the zeroth order excited state with oscillator strength and suppose the coherent excitation produces the initial probability amplitudes c, = (#r 1tin) corresponding to the states Grr at energies En with decay rates I’, . The.ernission probability at time I contains the quantum beat part (see section 5)
,c,ICnI*ICmI 2c0s2 [CEnoEm) [ifi1exp[-(rn + rm) t/21, exhibiting pair-wisebeating between each pair of coherently excited levels. For large molecules with many interfering levels the summation over levels n and m can be converted to an integration over beat frequencies w,, = (E, -E,)b with some continuous distribution, p(w,,), of beat frequencies. For large enough molecules it is reasonable to assume that Ic,, I2 Ic,, 12, p and (r, + I’mj are slowly enough varying with w,, that they can be taken outside of the integration. Consequently, the rapidly oscillating cos2(w,, 1) in the integcand just averages to onehalf, and no quantum beat phenomena are observed. With smaller, but still relatively large molecules, p(w,,) may exhibit some structure, In this case the integra. tion
Wmw s wmti
dw p(w) cos2 (wf),
* On leave from Centre de Recherche NuclGaaires, JAwatoire StrasbourgCronenbourg, France.
de Physique des Rayonnements et d’Electronique Nucliaire, 67
212
A Wkys, K.F. Fret-d/Enhcemenr of quanrum interference effects
m still exhibit some time dependence,i.e.,some quantum beat effects,but the resultsinay be decidedly not simple oscillafory. A simple example can be found for a gaussian distribution p(o)=@/n)“* exp[-cu(w-a:)]. Delory and Tric [3] have shown that the use of energy levels, E,,, from a random matrix still leads to the persistence of quantum beat phenomena. This observation is a direct consequence of the existence of an average distribution (the Wignerlaw) of nearest eigenvalue spacings (E,-E,,l) from random matrices [4] (and therefore for next nearest neighbor spacings, etc.). The persistence of quantum beats with random matrix energy levelsis, thus, a consequence of this average level spacing. However, in large molecules it is necessary to specify the rovibronic quantum numbers, so the excited levels fall into separate symmetry classes. Thus, even if random matrix energy level distributions are employed for each symmetry class individually, we are still left with a final summation over symmetry classes 0: [4],
w a
dw P&J) cos?-(cut),
which can then average out any beat phenomena. Noticing that the symmetry class designation 01involves, in part, the conserved quantum number J, the difficulty of observing quantum beats in the decay of large molecules is not at all surprising. Langhoff, Berg and Robinson [S] have, therefore, considered the decay of large molecules which correspond to the intermediate case and which therefore have sparser level distributions. They have shown how the properties of the molecular decay in intermediate case moleculescan depend sensitivelyupon the nature of the exciting light in conformity with the suggestion of Rhodes [6]. Robinson et al. have, thus, promoted the study of these decays as information (in addition to that obtained from spectra) aiding in untangling the complicated vibronic coupling patterns in large molecules. In the 92s phase, unfortunately, the presence of rotational quantum states requires an additional averaging which can comp!icate and mask the decay patterns determined in the absence of rotations. We are thus left with the problem, that in large enough molecules, the quantum beat phenomena are expected to be generaliy washed out, while when they are observable, they will give complicated-averaged-behavior that will be difficult, if not impossible, to unambiguously interpret as arising from qtiantum beats. Hence, it is necessary to devise a “quantum beat filter” to enhance beats st certain frequencies we of the distribution p,(w) and to suppress others. This would change the averaged beat pattern when one is observed, but it would not appear to affect the case of completely washedaut quantum beats. However, by basing the “filtering” process on yet another, but controllable, interference effect we cm still produce an effect when the original quantum beats appear to be averaged out. The method of quantum beat fdtering that we propose here involves the use of multiple pulse excitation. It leads to an enhancement of those beats “in phase”, i e., w = nwp where wp is the pulse frequency and n = 1,2..., with the pulse frequency and a diminution of those which are “out of phase”, etc. The case of two pulses is treated explicitly, others follow directly. The excitation of the system involves absorption of a single photon from the coherent pair of pulses,leading to an additional interference effect due to the uncertainty of where the absorbed ‘photon originates. As Fano [7] has explained, interference effects can be produced between separate light beams when time resolved experiments are considered. Tne two (or multiple) pulse experiments will naturally alter the beat pattern in atomic systems where ordinary quantum beats ate observable, thereby leading to a “new effect” which may be useful as a tool for probing other processes. The beat filter will also change the averaged interference pattern for small enough intermediate case molecules, but it will also produce an effect in the washedout quantum beat fit. In this latter case the multiple pulse experiment will lead to a change in quantum yield from that expected from a simple multiple of the one pulse results. (Recall that an angular averaging usually erases quantvm interference effects.) Although quantum yields are difficult to precisely measure, the experiments are .stiJ feasible. The derivations presented in subsequent sections consider the atomic-li;;e few level systems. The more general molecular case; are readily obtained by summing the interference terms over all pairs of interfering levels includ. ing a sin aveiall rotational quantum levels
A. Villueys, K.F. Freed/Enhancement
of quantum
273
interferetlceeffecfs
We employ the wave packet formalism that has been discussed by Jartner and Mukamel[8,9]. This enables us to properly describe the actual mode of preparation of the atomic or molecular excited states. In addition, we generalize previous theories to incorporate any statistical properties of the incident light, to deal with multiple pulse experiments, and to include the properties and location of the photon detector. In order to accomplish these, we employ a slightly different derivation of the wave packet formalism than that given by Jortner and Mukamel. Thus, section 2 presents this derivation 191, and it also serves to introduce the notation and important results that are subsequently required. Sections 3 and 4 describe the photon beams and atomic or molecular system, respectively, while the two-pulse quantum beats are derived in section 5. The last section presents illustrative calcula. tions and the discussion.
2. General time evolution of the total system The dynamical behavior of matter in the presence of radiation is dependent on the hamiltonian H of the total system. We decompose H in two parts, one corresponding to the zeroth order model hamiltonian and the other to the interaction H'=H-Ho. This partitioning should be used in a treatment of either nuclear, atomic, or molecular resonances and determines the time scale of the evolution. Ho must be chosen in order to provide a good starting description of the energy spectrum of the system and to yield an interactive H' which acts as a perturbation and which induces transition between the levels on a time scale that can be observed. The model hamiltonian Ho is taken to be
(1)
Ho =HhrfHF> where HFand Hbfare the hamiltonians corresponding to the free radiation field and to the material system, respectively. Then H’ represents the interaction part, Hint, between the matter and the field. The evolution of the system from time tn to a future time t is described by the unitary time operator [lo]
U+(t- ro) = e(f- to)
cqr-ro),
(2)
where 0 is the well known Heaviside function and U(t-to) the time evolution operator. When the total hamiltonian Hisnot time dependent, we can obtain an exact integral representation by introducing the Fourier transform m U+(t- fn) = (i/2n)s dE exp[-iE(r-CD
r,,)] C+(E),
(3)
wh.ere G+(E) = el;+ (E- H+ie)-l
=
G(P)
is, of course, related to the resolvent of the hamiltonian ff. The time evolution of the state tems, given the initial state I $(t,$ is I G(r)) = (i/2rr)s
-m
dE exp[-iE(t-
r,,)] G(E’)l$ (r,,)).
(4)
1$(r))of the total sys(5)
(here we employ units where A = c = 1). In a real system, we can assume that at the initial time, Co,the field and the material system are noninteracting as long as the photons and the matter are very far from one another. Then, the initial state of the system can be described as the product I 9(toN = lg) IQ) exp(-i
{KI r),
(6)‘
where Ig) represents the matter in its ground electronic state -but perhaps excited vibrationally and rotationally -with energy Eg S 0,and 11~5) describes the wave packet of the field with energy {K). Here we give description of
274 the
A. Viillaeys, K.F. Freed/~rllrancernent
of quantum interference effects
excitation in’terms of totally general wave packet states I’$)= $&] K
\{K)).
(7)
More specific models, involving explicit choices for the wave packet shape function A{&), can be described later. A direct way to introduce, into the time evolution, the interaction between the light beam and the matter is given by collision theory which considers an adiabatic switching on of the interaction [ 1 I]. In the case we consider here, which corresponds to the case of time ordered interactions, such formalism cannot be trivially applied due to the fact it simultaneously describes the establishment of the interaction for the two pulses. The time delay between the pulses is only taken into account by having a different phase for the two packets. There we lose the interference phenomena that occurs in the time interval between the arrival of the two pulses at the molecule. Therefore, we describe the evolution considering that, at time f qO, I G(O)) corresponds to a nonstationary state of the total hamiltonian where the photon wave packets are far removed from the molecules. From eq. (5),the integral representation for the I {IC’}, gi) component can be written as
The wetl known projection operators technique [8,9,12-141 is uSed here to obtain a decomposition of the reduced Green function PC(E) P into diagonal and non-diagonal parts relative to the field states in the P subspace. This decomposition is necessary because we can only obtain an explicit expression for the matrix elements for the diagonal part. For one light pulse, treated more often in the literature (see for example [S]), this decomposition does not exist because the P subspace corresponds to the vacuum state for the radiation field and of course PG(E) P is diagonal with respect to the field states. Let Q be the projection operator onto the {I {K), gj)‘) sub. space. Th.is implies that P=
c {K-l)
s
Q=F
ICK-~~,~~)((K--!),?~I~P’,
m
I
3 IIKI,gi)(IKI,giI>
(9)
where P corresponds to the excited states of the matter with one photon absorbed while P’ includes other remaining states which never enter into the problem considered below. The standard relations PtQ=l,
00)
(IK’bgiIG(~‘)ICK),g)=(IK’},giIQcQIrK},g),
(11)
and
are invoked. Following Mower [9?12] we have Qi;(E)Q
= Q(E-QHQ)-‘Q
+ Q(E-Q~Q)-‘QQ~titPPG(E)PPHti,QQ&QHQ)-’Q,
(12)
andweneglec?thefirsttermQ(E-QHQ)-lQinQG(E)Q Since . it does not involve interaction of the radiation with matter and is later shown to not contribute to the relevant cross section. In the P subspace the reduced Green function PC(E)P can be expressed as [E-PHoP-PH~tQ(E-Q~Q)~lQ~~~P]
PG(E)P=P.
(13)
The projected level shift operator is separated into two parts
PH,tQ(E-Q~Q)-lQH,,P=CRQ(E)~d+ fRQ@)),,d-
04)
The fast term in (14) correspdnds to the diagonal part relative to the I {K -11) photon states .’
!Re(E))d=
iKGII c [~H-l,gi)(IK-lj,giIH,tQ(E-Q~Q)-‘QH-,tlIK-l),g;)(IK-l~,giI, 6
(15)
A. Vilbeys, K.F. Freed/Enhancement
and the second one to the nondiagonal
part
c
c
{Rp (E&d =
27.5
of quannrm interference effects
I{K--i),gi)({K-l},giIHintQ(E-QHe)-lQ
IK- lI& (K’- 11, Sj (16)
x H,,IIK'-l),gj}({K'-l),gjI + terms frOnlP'.
Then using the identity (A-B)-‘=A-I+,4-1B(A-B)-1, PG(E)P
(17)
is found to be (after dropping the terms from P’ that do not enter into the subsequent calculation)
P(E-PM’-PiQ(E)P]-‘P,
PG(E)P=PGd(E)PtPGd(E)PP{RQ(E)),a
(18)
with
[E-PHP-P{Rg(E)),P]
PG&)P=
-‘I?
(19)
PG(E)P can be expanded in powers of P{Ra (E)jndP by iteration. Therefore, the evolution in the P subspace (atoms or molecules excited) is different from that which results when we consider the scattering with a single photon. One of the differences is caused by the term P{Ra(E))ndP. But it can be described by the same reduced resolvent considering the sum over all the possible virtual absorption a;ld emission processes that begin and end in the states ofP. With such a partitioning the probability amplitude (8) is given by the sum of terms from the two parts of the matrix elements of Q G(E) Q in (14), that is,
(~K”l.g;l~G(~)Ql{K~,g)=~+B,
(20)
where
“=,p
(~K"],gjl~~t({K'-l],Itl)
E-{K”]-E&
n,n
P ={,&
2
&$ 3
X-
({K'-13, ttllPG,#)PI{K'-I].
I?11 K2
F IcII ,r KS
‘CK”3~gi’H~t”K”‘-1”“’
(iK’-l),tlIHintiIKj,g) E--(K] ’
(21)
({K”‘-~),~IPG~(E)PI~K”‘-~},~)
E-(K”)+.,z
(b”-11, PI&t1 k$, gj)(@h gjl&,tl {K2--1),4) E-iK+Egi
tl)
({Kj--lhrlH~ll{Kj,g)
(~K~-~I.~I~G(~)PI~K~--I~,~)
E-b]
The contribution of the matrix element fl to the probability amplitude is much smaller than that from IYbecause p is proportional to the fourth order in the interaction of radiation and matter. Thus, at lower photon fluxes, we can neglect j relative to QIas Q is of second order in H-,, . Therefore, ({K’}, gil$ (f)} is expressed in terms of the diagonal part PG,(E)P alone
X
({K”)~giIH~tl
{K’-l},1?1)
({K’--l),II(Hintl(K),g)
I{K’-~),M~PG~(E~)P(~K’-~),~I) Ef-{~“)-Egi
Et-{K]
(22)
As the radiative coupling exhibits a weak energy dependence, the interaction matrix element can be taken outside the integration. This expression can be evaluated either in energy space or by utilizing the properties of the convoIution product in time space. The latter case yields
276
A. Vilbeys, KF.
Freed/Enhancement
of q!ru+Im intufehce
effeCB
where F(I--~)=(i/2rr)jd~!?exp[-iE(r-r)] _m
~~A{K}(E+-{K))-~, K
(24)
and (25)
We CXI now define the probability of photon emission at time t, or equivalently, the probability that the matter be in the ground state,
(26) The main physical observables ;Llededuced fromP&t). They are the photon counting rate, the number of photons emitted per unit time, defined by (27) and the quantum yield, the total number of emitted photons that have been emitted by time t = m, Y = P,(-).
(28)
For time resolved experiments with a detector at some point z, the photon number operator can be introduced, N(z) = VW P(z),
(29)
where q(z) is the detection operator [1.5],
p(z) = I$exp (ikz) uk,
(30)
ad where Q. represents the photon annihi!ation operator for photons with energy k. The operator N(z) corresponds to the case of broad band detection of photons with a point detector located at z. The fmite size of the detector can be introduced by an integration_fd!AJdz over all points in the true detector, while varying frequen cy response can be included by putt&g an efficiency amplitude cl(k) into the summand in (30). There is no difficulty in obtaining this general case by performing these operations on the final result for the point detector. When the detector is not in the forward dirdction, the ignored first term in (12) cannot contribute to (3!). In our case of a point detector the mean number of detected photons at time f ia given by ~(N(t))=(~(f)lrpf(z)~(z)i~(r)).
(31)
The formalism has the advantage, in the case of time resolved experiments, that it does not consider those pho‘. tons -.scattered and transmitted -which are otltslde of the region of system plus detector. These distant photons are always included in ({K”}, gil e(t), because the scattered photons are somewhere in space for all times. Introducing the hermitian conjugate relationship between $ and q we obtain
A. Villncys, RF. Freed/Enhancement
of quantum interference
277
effects
(N(t)) =
(32)
where the identity operator has been placed between (p”and ‘pin (3 1) and only those terms involving an absorption and remission are seen to contribute to lowest order in Htit. Using(22) it is found that (recall that the units are such that c = 1) ~({Kn-l),kg,glexp(ik~r)[~(r))= ko x
c fiK,_ll”cl KI
(ilZs)y dEexp(-iEf)
F
exp(i&-,z)ACK1 0
~Iw”-l~,k~,g;IH-,,ICn:‘-1~,nI~ ({K’--l),~(~G~(~)~({K’-1),tl)
E+-{K”-l)-kO-Egi
..B
(&l)J&&K).g) Ef-{I(}
-.
(33)
Usingthe same assumption of the weak energy dependence of the interaction matrix element and integrating over k givesthe result
X(IK’-l),nlH,,l{~),g~exp(-i[[a”-1]+Eg~j]~)~d~P(r’-i)~(r), (34) 0 with r’ = t-z. The detected intensity measured by the idealized detector can then be expressed in the general form
(35)
In order to obtain explicit expression for the observables,it is necessary to consider a specific model for the field as well as for the matter, and this is presented in the next section.
3. Description of the light beam
The general characteristic features of a light beam must be described statistically and, therefore, treated in the density matrix framework or by a technique which is mathematically identical. We consider the light beam to be well collimated and polarized. It can often be described by a density matrix that is diagonal in the wave packet representation PF =
(36)
I@)($1,
where 14) is the state of the photon wave packet. More general cases require summations of terms X,I@A) w~(Q~I and correspond to mixed states for the field. In such cases the notation can be simplified by displaying one tern as in (36); the summation over Xcan be effected at the end of the calculation. This diagonalform is not restrictive
in the sense that the photon wave packet (36) can vary in space and time. One reason why it is necessary to de. scribe the field in terms of wave packets is related to the existence of a finite lifetime for the radiating lev& in the source. The.light beam can be represented in terms of P superposition of monochromatic one photon states [17] I@) = F f(k) exp(-M)
ai,,lO),
(37)
278
A. Vilbeys, K.F. Freed/En!lancernent of quantum interference effects
where f(k),is the spectral distribution of the pulse, centered at frequency Ak with a linewidth I’k. k is the photon wave vector, e the polarization vectdr, and k its energy. The wave packet Iwidth r, is related to the coherence time 7, of the pulse by rc = 2nQ-‘ .
(38)
The operator ai denotes the one photon creation operator, 2 is its spatial coordinate and 10) represents the vacuum field state of no photons. The wave packet (37) can be normalized. Remembering that the one photon states “‘k,$ IO),
(3%
are normalized to unity (k, elk’, e’) = 6kk,6,,, ,
(40)
the normalization condition for the one wave packet (37) is given by (41) Here we consider excitation by two successive photon wave packets, separated by a time delay 7’. Such a beam can be described by the initial photon state l&j =
k:,,j(k')f(k")
exp(ik”T) a;,a&.lO).
(42)
The polarization of the two photons has not specifically been indicated because here we are only interested by measurement in time and energy; however, it is trivial to include this if desired. For simplicity, the notation a;,a;t-” 10) = (k’, k”),
(43)
is employed, but it must be remembered that (43) actually contains the two possibilities af;,a+,.,lO) = lk’, k”)=&,a;,[O)
= (k”, k’),
(4.4)
due to the indistinguishability of the photons. In fact we neglect here the possibility that we observe two photons in the same mode, but in the problem discussed below this only modifies the numerical factors. The wave packet (36) describes a two photon state which must be normalized. Its normalization factor Nil’* depends, of course, upon the time delay T and is givenby
N2 = (02I$$Operator algebra shows that it can be written as
IV*= 1 t F
If(
exp(ik’T) $
If(k”)l’exp(-ik”T).
(45)
For simplicity we use N, = 1, but more general analyses require explicit evaluation of (45). In this work the Fourier transform of the wave packet distribution f(k) is required. In order to obtain an explicit expression for these quantities, it is necessary to choose some particular distribution for f(k). We assume for simplicity, that this distribution can be characterized by a simple pole, Ek =
Ak -iI’k’k/2,
(46)
as the case, for example, for the lorentzian profile. Other choices may readily be used if desired [18]. ‘The requisite Fourier transform is given bi (47)
A. Villueys,K.F. Freed/Enhancement of quanhm interference effects
279
and is obtained from (24) by integration. For the particular wave packet (42) this gives F(t’-7))=
~f(k’)exp[-ik’(t’-~)I
(48)
;l(k”)exp[-ik”(i-r-T)],
where
F f(k')
exp(-ik’t’)
zf(k”)
exp[ik”(T-r’)]
= 0(t’) exp (-S&)32
= e(f’-T)
(f),
exp[-iEk(i-T)]%?
(f),
and 32 (f) means the residue of f(k) with respect to the pole Ek. The photon correlation function introduced by Jortner and Mukamel [8] or by Langhoff and.Robinson [S] is simply the Fourier transform of the spectral distribution in the case of one pulse excitation. Our correlation function gives their results in the one pulse excitation -A {Kj ++ f(k)> IK) ++ k - but can describe more general kinds of excitation as, for example, beams with many photons. In fact F(t’-T) expresses the Fourier transform of CI~~A{,J (Et-{K))-~. Thus, we can directly describe tie emission probability by a single convolution product instead of a double convolution product which
would be indispensablein their formalism.
4. Description of the material system The states of matter can generally be specified by describing its resonances. These resonances correspond to the excited states of the zero order hamiltonian for the matter which are coupled to many dissipative continua that act as exit channels. When the spacing between these resonances is less than their decay rates, interferences can, in principle, be observed between these discretvexcited states. Firstly, the molecular system is described in terms of resonances. Because of the particular structure of the reduced Green function Q G(E) Q, the radiant states, which carry oscillator strength, play a special role. To be explicit, a model hamiltonian for the matter is introduced. In the basis set of eigenvectors Is), let p be the projection operator onto all the radiant states P=p fp’.
(4%
Therefore, it is possible to introduce a new reduced resolvent p G(E)p which contains all the dynamics of the system. By standard algebra [8,9,11-131 it is found that PG~(E)P=[E-~H~-~CR~(E)}~P-PR~‘(E)PI~’,
(50)
with the usual level-shift operator I?&)=
Vt Vp’(E-p’Hp’)-$‘V.
(51)
The spectral decomposition of p Gd (E)p can be written in terms of the zeroth order radiant states lk, s),
(52) and is very useful in order to obtain the Fourier transformg (7) for the molecular correlation function in (36). The simplest system, which exhibits interference phenomena, is the two resonances model, that is, an atom or molecule characterized by two Jesonances. These resonances correspond to the poles E, of the spectral decompcsition of p Cd(E)p. Only the discrete eigenvectors or^the hamiltonian Hpp are treated exactly and correspond to the poles of the Green function. As discussed in the introduction, this model is quite general if we merely sum the interference contributions over all possible pairs of interfering levels. Thus, for simplicity, we focus only on a pair
A. Villueys, K.F. Freed/Enltor;ce9nent o/quantum interference effhts
28b
1. The spectrum of the zero order hamiltonian. Q[&+&] Q correspondsto the continuous part while [p(H& +H$p+p’(H&+!f~)p’] correspondsto rhe discrete Fig.
part.
Q(HO,+H,)Q
p~&W
p
p’#H,)p
The interferences me obwrved between the levels !k, r) and Ik, s). gi represents zdl the discrete matter levels which unnot be reached by the excitation.
of levels. Therefore, the two resonances model describes a material system with only two discrete levels treated exactly, all the others being considered as dissipative continua or quasi continua. This is, for example, the case of the v-ibronic levels for the statistical limit case in large molecules. The material model can be described by the scheme in fig. 1. These two discrete states can be atomic Zeeman levels interacting with the radiation fieId as is the case in the cadmium experiment [l] or two coupled vibronic levels that are coupled both to the radiation field - one 01 both of then -and to the vibrational levels of a lower electronic configuration. If these two levelsare coherently excited by the incident light, it is possible to observe quantum beats in a time resolved experiment. We give the description of the two photon beam experinent where one photon is absorbed and the other is transmitted. When only one photon is in the-beam, we have k ~0. Using (53) in (26) the Fourier transform of the molecular response can then be written as
= c (k,, kg,gil&lk, rn) (k, tlIfimtlk’, k”,g) %#,(k, ?I, m) exp(-iEaT). m. n.a With only two resonances interferences can, in principle, be observed whenever
(53)
In order for this interference to be a maximum, the two resonances must have the same width, E% G k + E. - AE- iI’/2.
I&, i k t I?,-,t AE-ilJ2,
(54)
This, in turn, implies that mq R,, ko,gilH-,,lk
I
m) (k,
sexp[-I’r/2-i(k+EO)]
G$-#, k”,@y(T) z
m, n
[CR.1(k,n,m)exp(-iAEr)+rR,2(k,n,m)exp(IlAET)].
(55)
The second condition for maximization of interference effects is the factorization of C
n.m
$,
Ac
kg,gilH-,tlk, m) (k, nlH-,tlk’,k”,g) qa, (k, n, m) (kl, k,,,
g,liv-,,l k, m) (k, !I If&I k’, k”, gj 32 Oz(k, n, m).
(56)
n, m
Such conditions are satisfied, for example, for a molecule in which one radiant state is strongly coupled with a quasidegenerate state of the intermediate structure or for two nondegenerate weakly coupled radiant states which can-j he same oscillator strength and have the same line width. The general case of many levels yields many poles in (53). Eq. (53) is valid in this general case, and the maximization criteria (54) and (56) then only apply to the interferences from a particular pair of levels. :
Villaeys,K.F. Freed/Enhancernenf
A.
of quarmm inferfererlcc
effects
281
(a)
m
n
Fig. 2. (a) The diagrams represent the two possibilities for the absorption, evolution in the matter subspace, and subsequent emission of a photon. In one of the cases the fust photon k’ is absorbed and the second transmitted. while in the Eecond case we have the opposite situation. (b) Diagrams b show the more complex real process which consists of absorption and remission of the lirst photon and
,
subsequent absorption and remissionof the second photon.
(b) 5. Quantum beats
As we have seen before, the scattered light is described by (35). w e nov, explicitly derive tie detected inter@] in the particular case of the two pulse excitation and energy-level system, described in previous sections, in order to evaluate the enhancement of the interference terms over that obtained from single pulse experiments. (N(o) is given by
X CR,(k, 171,n)jdiF(i.-r)
exp[-i(F& f k)~] ‘, 0 where the poles are separated into contributions from the field and the molecular part E, +k=E,.
(57)
(58)
Recalling the notation Jk’, k”) for the two photon states the interaction matrix elements are described as tk, nIH-,,Ik’, k”,g) =p*(g.,~)[~&
(k,,k,vgl&,,lkv
t c&l,
(59’)
(59”)
~~)=~(gir “)Sk,k;
The first matrix element (59’) gives two possibilities: in the tirst the photon k’ is absorbed and the photon k” is transmitted, whereas in the second one k” is absorbed and k' transmitted. The other matrix element (59”) contains only one possibility, when inserted into (7 l), due to the diagonality of p Gd(E)P with respect to I k). If the detector can only observe photons that are scattered by the target, this implies kl = k. Then the interfering terms are only prodded by the diagrams in fig.2a, and we neglect the real process corresponding to the diagram in fig. 2b. Using the Fourier transform for the excitation pulse shape, we obtain, after integration in (54),
W(f))= @ ~‘m~~l(k~)~(f)~(gi,m)p*(g,n)~o(k,m.~z)ex~(-~~~/2)(~~-E I
I
I
X {O(i)exp(iklT)[exp[-i(za-Ek)
t’]-l]+B(r’-T)exp(SkT)[exp[--i(&-Ek)(i-T)]-l]l
“, (6q)
which is the totally general result for many poles (i.e., many atomic and molecular levels). Again (60) is valid when many excited levels are present. With the assumption (56) to maximize the interference terms, the term C ~(gi,m)~*(g,n)‘#.(k,m,n) m,n
282
A. Viflaeys, KF. Freed/Enhancemerrf oJqoanfum interJerence effects
can be removed as an overall factor. Then this gives W(t’))-
$J $I:(k,)‘R(f)l*l I
~~(gi,m)p*(g,n)CA,(k.m,11)12exp(-r~kr’) m, n
X {e(f)exp(iklT)[exp[-i(E,-Ek)r’]
z a
(&-W1
-11 +O(r’-T)exp(iEJ)[exp[-i(Fa-Ek)(r’-T)]--l]]
2. (61)
in some instances the factor (E, -I$)-~ can be taken outside the Ea. This occurs especially when the width of the excitation spectra is larger than the energy gap between the poles. Then, if the expression for the poles (54) is introduced and integration is performed over k,, the scattered intensity is W(r’),GAjO(t’)[exp(-I’r’)cos2(AEt’)
t exp(-r,kt’)-2exp[-(I’+rk)
X cos(hsl’)cos[(Eo-Ak)r’]]+O(t)-T)[exp[-r(1’-T)] -2exp[-(r+rjc)(tl-T)/2]
cos[AE(t’-T)]
+ O(t’-T)[2cos(AEr’)cos[AE(t’-T)]
t’/2;
cos2 [AE(t’-T)]
+exp[-I’k
(t’-T)]
cos[(E,-A,)(&7-T)]]
cos[(E-Ak)
T] exp[-r~-(rk-r)T/2]]},
(62)
where A is given by (63) remembering that the spectral distribution is normalized. In the expression (62) we have neglected the interfering part decaying with a lifetime less than or eqeal to ril corresponding to the coherence time within the photon pulse. These terms are A{ O(t’- T) 2[exp (-r,t’) -cos{AE(~‘-T)]
-
cos
(AE t’)
cos
COS[(Q,-A~)(~‘-T)]
[(Ak -E,)
t’] exp [-(I’+ r,) r’/2]
exp[-r(t’-T)/2-rk(t’tT)/2]},
(64)
and they can be ignored as they are small and as our basic focus is on the basic molecular processes. Eq. (62) has specialized to two levels, the many level case is obtained by summation over all pairs of interfering levels. We now discuss the expression (62) for the scattered intensity. The first three terms describe the scattering of the first pulse. They correspond respectively to the absorption with a lifetime r,‘, the coherence time of the photon, and ihe emission with a lifetime r-’ , the lifetime of the unstable atomic or molecular state. The third term d,escribes the interferences in the rise time of the second pulse due to off.resonance excitation; these interferences disappear with a lifetime (I’+I’&’ and its recurrence time is (E,YAk)-‘, that is, the inverse of the enerw difference between !he mkan energy of the excitation pulse and the mean energy of the resonance. After time r = T, the second pulse is applied, and we observe the same evolution as before, except that we have an interference term due to the two successive excitations. This term, 2 cos(AE’~)cos[AE(t’-T)j
cos[&,-Ak)T]
exp[-l?‘-(rk-r)3/2].
(65)
gives an enhancement or diminution of the quantum beats in the scattered light and represents the reversible evolution in *he material system.
6. Numerical calculatians and discussion In order. to have ZQ?estimation of the enhancement of the quantum beats, the intensity of the scattered light has been.compuJed’as a function of time. The results are given in fig. 3. As mentioned before, in the,expressi?n (62),we have neglected all the interfering terms (64) hecaying with a
A. Villaeys, K.F. Freed/Enhancemcnr
283‘
of quonnrm inrerfercncc effects
Fig. 3. Plot of the detected intensities (solid curve) and the
contribution of the interference term (dashed curve) for three different cases. (a) The cscitint pulses are in phase. The contribution of the interference (b) The
term is always
positive.
exciting pulses ore n/2 out of phase. The contribu-
tion of the inkrference term is alternatively positive and negative. (c) The exciting pulses arc pi out of phase. The contribu-
0
1
2
3
r
k
t
tion of the interference term is always negative. In all the cases. we consider w 3 A,? and E. = Ak so
j
COS[(&-Ak)rl
wt/2rr
=
cos[(Eo-ak)(t-T)]=
1.
lifetime of the order or less than I’,‘. In the numerical calculations we must take these terms into account; if not,
a discontinuity would be introduced in the expression for the detected intensity at time r=T. But, after a short time of the order of I’i’, their contributions become negligible and the expression (62) gives the dominant variation of I(<). Because we focus our interest on the basis molecular processes, we consider the case of resonant excitation, that is to say cos[(Eo-Ak) Therefore,
t’] = cos[(EO-Ak)(r’-T)]
the oscillating terms corresponding
= 1. to interferences
between the molecular and fieId resonances
are not
retained. Given the smallness of the lifetime of these oscillating terms, they result kssentially in oscillation in the
rise time of the excitation. We observe in fig. 3 that the contribution of the interfering term gives a contribution which depends, of course, on the time delay between the two pulses. In fig. 3a the two successive pulses are in phase; the contribution of the interfering term is positive and gives an enhancement of the quantum beats. In the second case, corresponding to the fig. 3b, the pulses are n/2 out of phase, and the contribution is alternatively positive and negative. In the last case - fig. 3c - the pulses are R out of phase, and the contribution is always negative. The best enhancement is ther. obtained in the first case. If quantum beats are alleged to be observed in intermediate case molecule,s, the above analysis shows how they must be altered by the two pulse experiment. This two pulse approach also implies a definite dependence of the observation on the pulse separation T. However, in the limit where quantum beats are “washed out”, because of the large number of pairs of interfering levels, the interference term (65) of (62) gives a nonvanishing contribution when averaged over all pairs of interfering levels. (This result is clearly true ifi the general case of (60) where the approximations have not been invoked to maximize the interference terms.) For the model of a continuous distribution of beat frequencies up to some maximum AE,,(and E0 = A,) the average of (65) over this distribution of beat frequencies is nonvanishing if AL’,, T < 1. Hence, the “washed otit” limit still yields a quantum yield which differs from twice that arising from a single pulse by the beat frequency averaged interference term. This then provides the opportunity for verify@ the existence of the quantum interferences in this limit. In the opposite atomic limit, the additional interferences should readily be observable and should be a useful tool for studies of atomic relaxation processes.
.2a4
A. P~ikys, K.F. Freed,&nhancemer~~ofquanrum
irxerference
effecfs
It is,‘perhaps, relevant to note that a somewhat related phenomena has been considered by Comey and Series [ 191. They show thatinterference effects are produced in atoms which are excited by modulated light when the modulation frequency corresponds to an atomic transition frequency. Modulation is limited to frequencies of up to possibly a MHz. On the other hand, the use of a train of optical pulses can have spacings in the nanosecond or subnanosecond timescale to extend the phenomenon into this domain.
Acknoeledgement We are grateful to Abe Nitzan, Isaac Abella and Don Chemoff for helpful discussions and to Don Chemoff for performing the numerical calculations.This research is supported, in part, by NSF Grant ME’S7541549. K.F.F. acknowledgesthe support of a Camilleand Henry Dreyfus Foundation Teacher-Scholar Grant. AN. would like to thank G. Ourrisson,President of the University Louis Pasteur of Strasbourg, and ADRERUSfor a travel grant.
References [!] J.N. Dodd, RD. Kaul and D.M. Warrington, Proc. Thys. Sac. 84 (1964) 176; J-N. Dodd, W.J. Sandle and D. Ztiermann, Proc. Phys. Sac. 92 (1967) 497; H.J. Andra, Phys. Rev. Lett. 25 (1970) 325; S. Haroche, J.A. Paisner and A.L. Schawlow, Phys. Rev. Lett. 30 (1973) 948. [2] GE. Busch, P.M. Rentzepis and J. Jortner, J. Chem. Phys. 56 (1972) 361. [3] J.M. Delory and C. Tric, Chem. Phys. 3 (1974) 54. [4] C.E. Porter, Statistical Theories of Spectra: Fluctuations (Academic Press, New York, 1965). IS] C.A. Langhoff and G.W. Robinson, Mol. Phys. 26 (1973) 249; Chem. Phys. 5 (1973) 1; 6 (1974) 34; J.O. Berg, CA. Langholf and G.W. Robinson, Chem. Phys. Lett. 29 (1974) 305; J.M. Friedman and R.hI. Hochstrasser, Chen. Phys. 6 (1974) 155. [6] W. Rhodes, J. Chem. Phys. 50 (1969) 2885; Chem. Phys. Lett. 11 (1971) 179. 171 U. Fano, Am. J. Phys. 29 (1961) 539. [B] J. Jortner and S. Mukamel, in: The World of Quantum Chemistry, eds. R. Daudcl and B. Pullman (Reidel, Boston, 1974). [9] A. Villaeys, unpublished work; K.F. Freed, Topics Appl. Phys., in press. [lo] A. Messiah, Quan:um Mechanics (Wiley, New York, 1961). (111 S. Mukamel and J. Jortner, I. Chem. Phys. 61 (1974) 227. [ 121L. hIower, Phys. Rev. 142 (1966) 799; 165 (1968) 145. [ 131 M.L. Coldberger and K.M. Watson, Collision Theory (Wiley, New York, 1964). [14] K.F. Freed, J. Chem. Phys. 52 (1970) 1345. [15] L. Mandel, Phys. Rev. 144 (1966) 1071. [16] R-J. Glau’;xr, in: Quantum Optics, eds. S.M. Kay and A. hlaitland (Academic Press, 1970). [17] V.M. Titulaer and R.J. Glauber, Phys. Rev. 145 (1966) 1041. [18] L.R. Doddand I.E. h!cCarthy, Phys. Rev. A 134 (1964) 1136. [19] A. Comey and G.W. Series, Proc. Phys. Sac. (London) 83 (1964) 207,213.
North-HollandPublishingCompany
ENHANCEMENT OF QUANTUM INTERFERENCE EFFECTS Albert VILLAEYS* 2nd Karl F. FREED TheJanresFranck fnstit~rteand 73e Department of Cknistry,
The University of Chicago,Oricago, Illinois 60637, USA
Received31 July 1975 Revised manuscript received 1 December 1975 We present a method of excitation which enhances quantum beats of certain frequencies and diminishes those at other frequencies. This approach uneven produce an effect when the original quantum beats appear to be washed out in large molecules because of the average over many pairs of interfering levels. The excitation of the system involves the absorption of a single photon from a coherent pair of pulses, leading to an additional interference effect with a nonzero quantum yie!d. The derivations are presented for the atomic-like few !evel syslems, but the more general molecular cases are readily obtained by superposition of aU pairs of interfering levels. The analysis incorporates the position and nature of the photon detector.
1. Introduction Quantum beats represent a dramatic example of the superposition principle of quantum mechanics. Thus, each new type of observation of quantum beat phenomena is heralded as a “new effect” in reaffirmation of the laws of quantum mechanics. Quantum beats have heretofore been observed with certainty only in atomic systems [l] where the excited state level density is so sparse that it is readily possible to focus upon a pair of interfering levels. In large molecules, on the other hand, the excited state level density can become sufficiently large that it is virtually impossible to not coherently excite a superposition of excited levels, thereby producing quantum beat phenomena in the subsequent decay processes. It is, perhaps, the ubiquity of interference phenomena in excited states of large molecules that has prompted the search for quantum beats in the deczy of intermediate case molecules [2] Although quantum mechanics dictates the necessity formthe occurrence of quantum beats when levels are coherently excited, the difficulty of observing these phenomena in large molecules is readily understood. Let &denote the zeroth order excited state with oscillator strength and suppose the coherent excitation produces the initial probability amplitudes c, = (#r 1tin) corresponding to the states Grr at energies En with decay rates I’, . The.ernission probability at time I contains the quantum beat part (see section 5)
,c,ICnI*ICmI 2c0s2 [CEnoEm) [ifi1exp[-(rn + rm) t/21, exhibiting pair-wisebeating between each pair of coherently excited levels. For large molecules with many interfering levels the summation over levels n and m can be converted to an integration over beat frequencies w,, = (E, -E,)b with some continuous distribution, p(w,,), of beat frequencies. For large enough molecules it is reasonable to assume that Ic,, I2 Ic,, 12, p and (r, + I’mj are slowly enough varying with w,, that they can be taken outside of the integration. Consequently, the rapidly oscillating cos2(w,, 1) in the integcand just averages to onehalf, and no quantum beat phenomena are observed. With smaller, but still relatively large molecules, p(w,,) may exhibit some structure, In this case the integra. tion
Wmw s wmti
dw p(w) cos2 (wf),
* On leave from Centre de Recherche NuclGaaires, JAwatoire StrasbourgCronenbourg, France.
de Physique des Rayonnements et d’Electronique Nucliaire, 67
212
A Wkys, K.F. Fret-d/Enhcemenr of quanrum interference effects
m still exhibit some time dependence,i.e.,some quantum beat effects,but the resultsinay be decidedly not simple oscillafory. A simple example can be found for a gaussian distribution p(o)=@/n)“* exp[-cu(w-a:)]. Delory and Tric [3] have shown that the use of energy levels, E,,, from a random matrix still leads to the persistence of quantum beat phenomena. This observation is a direct consequence of the existence of an average distribution (the Wignerlaw) of nearest eigenvalue spacings (E,-E,,l) from random matrices [4] (and therefore for next nearest neighbor spacings, etc.). The persistence of quantum beats with random matrix energy levelsis, thus, a consequence of this average level spacing. However, in large molecules it is necessary to specify the rovibronic quantum numbers, so the excited levels fall into separate symmetry classes. Thus, even if random matrix energy level distributions are employed for each symmetry class individually, we are still left with a final summation over symmetry classes 0: [4],
w a
dw P&J) cos?-(cut),
which can then average out any beat phenomena. Noticing that the symmetry class designation 01involves, in part, the conserved quantum number J, the difficulty of observing quantum beats in the decay of large molecules is not at all surprising. Langhoff, Berg and Robinson [S] have, therefore, considered the decay of large molecules which correspond to the intermediate case and which therefore have sparser level distributions. They have shown how the properties of the molecular decay in intermediate case moleculescan depend sensitivelyupon the nature of the exciting light in conformity with the suggestion of Rhodes [6]. Robinson et al. have, thus, promoted the study of these decays as information (in addition to that obtained from spectra) aiding in untangling the complicated vibronic coupling patterns in large molecules. In the 92s phase, unfortunately, the presence of rotational quantum states requires an additional averaging which can comp!icate and mask the decay patterns determined in the absence of rotations. We are thus left with the problem, that in large enough molecules, the quantum beat phenomena are expected to be generaliy washed out, while when they are observable, they will give complicated-averaged-behavior that will be difficult, if not impossible, to unambiguously interpret as arising from qtiantum beats. Hence, it is necessary to devise a “quantum beat filter” to enhance beats st certain frequencies we of the distribution p,(w) and to suppress others. This would change the averaged beat pattern when one is observed, but it would not appear to affect the case of completely washedaut quantum beats. However, by basing the “filtering” process on yet another, but controllable, interference effect we cm still produce an effect when the original quantum beats appear to be averaged out. The method of quantum beat fdtering that we propose here involves the use of multiple pulse excitation. It leads to an enhancement of those beats “in phase”, i e., w = nwp where wp is the pulse frequency and n = 1,2..., with the pulse frequency and a diminution of those which are “out of phase”, etc. The case of two pulses is treated explicitly, others follow directly. The excitation of the system involves absorption of a single photon from the coherent pair of pulses,leading to an additional interference effect due to the uncertainty of where the absorbed ‘photon originates. As Fano [7] has explained, interference effects can be produced between separate light beams when time resolved experiments are considered. Tne two (or multiple) pulse experiments will naturally alter the beat pattern in atomic systems where ordinary quantum beats ate observable, thereby leading to a “new effect” which may be useful as a tool for probing other processes. The beat filter will also change the averaged interference pattern for small enough intermediate case molecules, but it will also produce an effect in the washedout quantum beat fit. In this latter case the multiple pulse experiment will lead to a change in quantum yield from that expected from a simple multiple of the one pulse results. (Recall that an angular averaging usually erases quantvm interference effects.) Although quantum yields are difficult to precisely measure, the experiments are .stiJ feasible. The derivations presented in subsequent sections consider the atomic-li;;e few level systems. The more general molecular case; are readily obtained by summing the interference terms over all pairs of interfering levels includ. ing a sin aveiall rotational quantum levels
A. Villueys, K.F. Freed/Enhancement
of quantum
273
interferetlceeffecfs
We employ the wave packet formalism that has been discussed by Jartner and Mukamel[8,9]. This enables us to properly describe the actual mode of preparation of the atomic or molecular excited states. In addition, we generalize previous theories to incorporate any statistical properties of the incident light, to deal with multiple pulse experiments, and to include the properties and location of the photon detector. In order to accomplish these, we employ a slightly different derivation of the wave packet formalism than that given by Jortner and Mukamel. Thus, section 2 presents this derivation 191, and it also serves to introduce the notation and important results that are subsequently required. Sections 3 and 4 describe the photon beams and atomic or molecular system, respectively, while the two-pulse quantum beats are derived in section 5. The last section presents illustrative calcula. tions and the discussion.
2. General time evolution of the total system The dynamical behavior of matter in the presence of radiation is dependent on the hamiltonian H of the total system. We decompose H in two parts, one corresponding to the zeroth order model hamiltonian and the other to the interaction H'=H-Ho. This partitioning should be used in a treatment of either nuclear, atomic, or molecular resonances and determines the time scale of the evolution. Ho must be chosen in order to provide a good starting description of the energy spectrum of the system and to yield an interactive H' which acts as a perturbation and which induces transition between the levels on a time scale that can be observed. The model hamiltonian Ho is taken to be
(1)
Ho =HhrfHF> where HFand Hbfare the hamiltonians corresponding to the free radiation field and to the material system, respectively. Then H’ represents the interaction part, Hint, between the matter and the field. The evolution of the system from time tn to a future time t is described by the unitary time operator [lo]
U+(t- ro) = e(f- to)
cqr-ro),
(2)
where 0 is the well known Heaviside function and U(t-to) the time evolution operator. When the total hamiltonian Hisnot time dependent, we can obtain an exact integral representation by introducing the Fourier transform m U+(t- fn) = (i/2n)s dE exp[-iE(r-CD
r,,)] C+(E),
(3)
wh.ere G+(E) = el;+ (E- H+ie)-l
=
G(P)
is, of course, related to the resolvent of the hamiltonian ff. The time evolution of the state tems, given the initial state I $(t,$ is I G(r)) = (i/2rr)s
-m
dE exp[-iE(t-
r,,)] G(E’)l$ (r,,)).
(4)
1$(r))of the total sys(5)
(here we employ units where A = c = 1). In a real system, we can assume that at the initial time, Co,the field and the material system are noninteracting as long as the photons and the matter are very far from one another. Then, the initial state of the system can be described as the product I 9(toN = lg) IQ) exp(-i
{KI r),
(6)‘
where Ig) represents the matter in its ground electronic state -but perhaps excited vibrationally and rotationally -with energy Eg S 0,and 11~5) describes the wave packet of the field with energy {K). Here we give description of
274 the
A. Viillaeys, K.F. Freed/~rllrancernent
of quantum interference effects
excitation in’terms of totally general wave packet states I’$)= $&] K
\{K)).
(7)
More specific models, involving explicit choices for the wave packet shape function A{&), can be described later. A direct way to introduce, into the time evolution, the interaction between the light beam and the matter is given by collision theory which considers an adiabatic switching on of the interaction [ 1 I]. In the case we consider here, which corresponds to the case of time ordered interactions, such formalism cannot be trivially applied due to the fact it simultaneously describes the establishment of the interaction for the two pulses. The time delay between the pulses is only taken into account by having a different phase for the two packets. There we lose the interference phenomena that occurs in the time interval between the arrival of the two pulses at the molecule. Therefore, we describe the evolution considering that, at time f qO, I G(O)) corresponds to a nonstationary state of the total hamiltonian where the photon wave packets are far removed from the molecules. From eq. (5),the integral representation for the I {IC’}, gi) component can be written as
The wetl known projection operators technique [8,9,12-141 is uSed here to obtain a decomposition of the reduced Green function PC(E) P into diagonal and non-diagonal parts relative to the field states in the P subspace. This decomposition is necessary because we can only obtain an explicit expression for the matrix elements for the diagonal part. For one light pulse, treated more often in the literature (see for example [S]), this decomposition does not exist because the P subspace corresponds to the vacuum state for the radiation field and of course PG(E) P is diagonal with respect to the field states. Let Q be the projection operator onto the {I {K), gj)‘) sub. space. Th.is implies that P=
c {K-l)
s
Q=F
ICK-~~,~~)((K--!),?~I~P’,
m
I
3 IIKI,gi)(IKI,giI>
(9)
where P corresponds to the excited states of the matter with one photon absorbed while P’ includes other remaining states which never enter into the problem considered below. The standard relations PtQ=l,
00)
(IK’bgiIG(~‘)ICK),g)=(IK’},giIQc
(11)
and
are invoked. Following Mower [9?12] we have Qi;(E)Q
= Q(E-QHQ)-‘Q
+ Q(E-Q~Q)-‘QQ~titPPG(E)PPHti,QQ&QHQ)-’Q,
(12)
andweneglec?thefirsttermQ(E-QHQ)-lQinQG(E)Q Since . it does not involve interaction of the radiation with matter and is later shown to not contribute to the relevant cross section. In the P subspace the reduced Green function PC(E)P can be expressed as [E-PHoP-PH~tQ(E-Q~Q)~lQ~~~P]
PG(E)P=P.
(13)
The projected level shift operator is separated into two parts
PH,tQ(E-Q~Q)-lQH,,P=CRQ(E)~d+ fRQ@)),,d-
04)
The fast term in (14) correspdnds to the diagonal part relative to the I {K -11) photon states .’
!Re(E))d=
iKGII c [~H-l,gi)(IK-lj,giIH,tQ(E-Q~Q)-‘QH-,tlIK-l),g;)(IK-l~,giI, 6
(15)
A. Vilbeys, K.F. Freed/Enhancement
and the second one to the nondiagonal
part
c
c
{Rp (E&d =
27.5
of quannrm interference effects
I{K--i),gi)({K-l},giIHintQ(E-QHe)-lQ
IK- lI& (K’- 11, Sj (16)
x H,,IIK'-l),gj}({K'-l),gjI + terms frOnlP'.
Then using the identity (A-B)-‘=A-I+,4-1B(A-B)-1, PG(E)P
(17)
is found to be (after dropping the terms from P’ that do not enter into the subsequent calculation)
P(E-PM’-PiQ(E)P]-‘P,
PG(E)P=PGd(E)PtPGd(E)PP{RQ(E)),a
(18)
with
[E-PHP-P{Rg(E)),P]
PG&)P=
-‘I?
(19)
PG(E)P can be expanded in powers of P{Ra (E)jndP by iteration. Therefore, the evolution in the P subspace (atoms or molecules excited) is different from that which results when we consider the scattering with a single photon. One of the differences is caused by the term P{Ra(E))ndP. But it can be described by the same reduced resolvent considering the sum over all the possible virtual absorption a;ld emission processes that begin and end in the states ofP. With such a partitioning the probability amplitude (8) is given by the sum of terms from the two parts of the matrix elements of Q G(E) Q in (14), that is,
(~K”l.g;l~G(~)Ql{K~,g)=~+B,
(20)
where
“=,p
(~K"],gjl~~t({K'-l],Itl)
E-{K”]-E&
n,n
P ={,&
2
&$ 3
X-
({K'-13, ttllPG,#)PI{K'-I].
I?11 K2
F IcII ,r KS
‘CK”3~gi’H~t”K”‘-1”“’
(iK’-l),tlIHintiIKj,g) E--(K] ’
(21)
({K”‘-~),~IPG~(E)PI~K”‘-~},~)
E-(K”)+.,z
(b”-11, PI&t1 k$, gj)(@h gjl&,tl {K2--1),4) E-iK+Egi
tl)
({Kj--lhrlH~ll{Kj,g)
(~K~-~I.~I~G(~)PI~K~--I~,~)
E-b]
The contribution of the matrix element fl to the probability amplitude is much smaller than that from IYbecause p is proportional to the fourth order in the interaction of radiation and matter. Thus, at lower photon fluxes, we can neglect j relative to QIas Q is of second order in H-,, . Therefore, ({K’}, gil$ (f)} is expressed in terms of the diagonal part PG,(E)P alone
X
({K”)~giIH~tl
{K’-l},1?1)
({K’--l),II(Hintl(K),g)
I{K’-~),M~PG~(E~)P(~K’-~),~I) Ef-{~“)-Egi
Et-{K]
(22)
As the radiative coupling exhibits a weak energy dependence, the interaction matrix element can be taken outside the integration. This expression can be evaluated either in energy space or by utilizing the properties of the convoIution product in time space. The latter case yields
276
A. Vilbeys, KF.
Freed/Enhancement
of q!ru+Im intufehce
effeCB
where F(I--~)=(i/2rr)jd~!?exp[-iE(r-r)] _m
~~A{K}(E+-{K))-~, K
(24)
and (25)
We CXI now define the probability of photon emission at time t, or equivalently, the probability that the matter be in the ground state,
(26) The main physical observables ;Llededuced fromP&t). They are the photon counting rate, the number of photons emitted per unit time, defined by (27) and the quantum yield, the total number of emitted photons that have been emitted by time t = m, Y = P,(-).
(28)
For time resolved experiments with a detector at some point z, the photon number operator can be introduced, N(z) = VW P(z),
(29)
where q(z) is the detection operator [1.5],
p(z) = I$exp (ikz) uk,
(30)
ad where Q. represents the photon annihi!ation operator for photons with energy k. The operator N(z) corresponds to the case of broad band detection of photons with a point detector located at z. The fmite size of the detector can be introduced by an integration_fd!AJdz over all points in the true detector, while varying frequen cy response can be included by putt&g an efficiency amplitude cl(k) into the summand in (30). There is no difficulty in obtaining this general case by performing these operations on the final result for the point detector. When the detector is not in the forward dirdction, the ignored first term in (12) cannot contribute to (3!). In our case of a point detector the mean number of detected photons at time f ia given by ~(N(t))=(~(f)lrpf(z)~(z)i~(r)).
(31)
The formalism has the advantage, in the case of time resolved experiments, that it does not consider those pho‘. tons -.scattered and transmitted -which are otltslde of the region of system plus detector. These distant photons are always included in ({K”}, gil e(t), because the scattered photons are somewhere in space for all times. Introducing the hermitian conjugate relationship between $ and q we obtain
A. Villncys, RF. Freed/Enhancement
of quantum interference
277
effects
(N(t)) =
(32)
where the identity operator has been placed between (p”and ‘pin (3 1) and only those terms involving an absorption and remission are seen to contribute to lowest order in Htit. Using(22) it is found that (recall that the units are such that c = 1) ~({Kn-l),kg,glexp(ik~r)[~(r))= ko x
c fiK,_ll”cl KI
(ilZs)y dEexp(-iEf)
F
exp(i&-,z)ACK1 0
~Iw”-l~,k~,g;IH-,,ICn:‘-1~,nI~ ({K’--l),~(~G~(~)~({K’-1),tl)
E+-{K”-l)-kO-Egi
..B
(&l)J&&K).g) Ef-{I(}
-.
(33)
Usingthe same assumption of the weak energy dependence of the interaction matrix element and integrating over k givesthe result
X(IK’-l),nlH,,l{~),g~exp(-i[[a”-1]+Eg~j]~)~d~P(r’-i)~(r), (34) 0 with r’ = t-z. The detected intensity measured by the idealized detector can then be expressed in the general form
(35)
In order to obtain explicit expression for the observables,it is necessary to consider a specific model for the field as well as for the matter, and this is presented in the next section.
3. Description of the light beam
The general characteristic features of a light beam must be described statistically and, therefore, treated in the density matrix framework or by a technique which is mathematically identical. We consider the light beam to be well collimated and polarized. It can often be described by a density matrix that is diagonal in the wave packet representation PF =
(36)
I@)($1,
where 14) is the state of the photon wave packet. More general cases require summations of terms X,I@A) w~(Q~I and correspond to mixed states for the field. In such cases the notation can be simplified by displaying one tern as in (36); the summation over Xcan be effected at the end of the calculation. This diagonalform is not restrictive
in the sense that the photon wave packet (36) can vary in space and time. One reason why it is necessary to de. scribe the field in terms of wave packets is related to the existence of a finite lifetime for the radiating lev& in the source. The.light beam can be represented in terms of P superposition of monochromatic one photon states [17] I@) = F f(k) exp(-M)
ai,,lO),
(37)
278
A. Vilbeys, K.F. Freed/En!lancernent of quantum interference effects
where f(k),is the spectral distribution of the pulse, centered at frequency Ak with a linewidth I’k. k is the photon wave vector, e the polarization vectdr, and k its energy. The wave packet Iwidth r, is related to the coherence time 7, of the pulse by rc = 2nQ-‘ .
(38)
The operator ai denotes the one photon creation operator, 2 is its spatial coordinate and 10) represents the vacuum field state of no photons. The wave packet (37) can be normalized. Remembering that the one photon states “‘k,$ IO),
(3%
are normalized to unity (k, elk’, e’) = 6kk,6,,, ,
(40)
the normalization condition for the one wave packet (37) is given by (41) Here we consider excitation by two successive photon wave packets, separated by a time delay 7’. Such a beam can be described by the initial photon state l&j =
k:,,j(k')f(k")
exp(ik”T) a;,a&.lO).
(42)
The polarization of the two photons has not specifically been indicated because here we are only interested by measurement in time and energy; however, it is trivial to include this if desired. For simplicity, the notation a;,a;t-” 10) = (k’, k”),
(43)
is employed, but it must be remembered that (43) actually contains the two possibilities af;,a+,.,lO) = lk’, k”)=&,a;,[O)
= (k”, k’),
(4.4)
due to the indistinguishability of the photons. In fact we neglect here the possibility that we observe two photons in the same mode, but in the problem discussed below this only modifies the numerical factors. The wave packet (36) describes a two photon state which must be normalized. Its normalization factor Nil’* depends, of course, upon the time delay T and is givenby
N2 = (02I$$Operator algebra shows that it can be written as
IV*= 1 t F
If(
exp(ik’T) $
If(k”)l’exp(-ik”T).
(45)
For simplicity we use N, = 1, but more general analyses require explicit evaluation of (45). In this work the Fourier transform of the wave packet distribution f(k) is required. In order to obtain an explicit expression for these quantities, it is necessary to choose some particular distribution for f(k). We assume for simplicity, that this distribution can be characterized by a simple pole, Ek =
Ak -iI’k’k/2,
(46)
as the case, for example, for the lorentzian profile. Other choices may readily be used if desired [18]. ‘The requisite Fourier transform is given bi (47)
A. Villueys,K.F. Freed/Enhancement of quanhm interference effects
279
and is obtained from (24) by integration. For the particular wave packet (42) this gives F(t’-7))=
~f(k’)exp[-ik’(t’-~)I
(48)
;l(k”)exp[-ik”(i-r-T)],
where
F f(k')
exp(-ik’t’)
zf(k”)
exp[ik”(T-r’)]
= 0(t’) exp (-S&)32
= e(f’-T)
(f),
exp[-iEk(i-T)]%?
(f),
and 32 (f) means the residue of f(k) with respect to the pole Ek. The photon correlation function introduced by Jortner and Mukamel [8] or by Langhoff and.Robinson [S] is simply the Fourier transform of the spectral distribution in the case of one pulse excitation. Our correlation function gives their results in the one pulse excitation -A {Kj ++ f(k)> IK) ++ k - but can describe more general kinds of excitation as, for example, beams with many photons. In fact F(t’-T) expresses the Fourier transform of CI~~A{,J (Et-{K))-~. Thus, we can directly describe tie emission probability by a single convolution product instead of a double convolution product which
would be indispensablein their formalism.
4. Description of the material system The states of matter can generally be specified by describing its resonances. These resonances correspond to the excited states of the zero order hamiltonian for the matter which are coupled to many dissipative continua that act as exit channels. When the spacing between these resonances is less than their decay rates, interferences can, in principle, be observed between these discretvexcited states. Firstly, the molecular system is described in terms of resonances. Because of the particular structure of the reduced Green function Q G(E) Q, the radiant states, which carry oscillator strength, play a special role. To be explicit, a model hamiltonian for the matter is introduced. In the basis set of eigenvectors Is), let p be the projection operator onto all the radiant states P=p fp’.
(4%
Therefore, it is possible to introduce a new reduced resolvent p G(E)p which contains all the dynamics of the system. By standard algebra [8,9,11-131 it is found that PG~(E)P=[E-~H~-~CR~(E)}~P-PR~‘(E)PI~’,
(50)
with the usual level-shift operator I?&)=
Vt Vp’(E-p’Hp’)-$‘V.
(51)
The spectral decomposition of p Gd (E)p can be written in terms of the zeroth order radiant states lk, s),
(52) and is very useful in order to obtain the Fourier transformg (7) for the molecular correlation function in (36). The simplest system, which exhibits interference phenomena, is the two resonances model, that is, an atom or molecule characterized by two Jesonances. These resonances correspond to the poles E, of the spectral decompcsition of p Cd(E)p. Only the discrete eigenvectors or^the hamiltonian Hpp are treated exactly and correspond to the poles of the Green function. As discussed in the introduction, this model is quite general if we merely sum the interference contributions over all possible pairs of interfering levels. Thus, for simplicity, we focus only on a pair
A. Villueys, K.F. Freed/Enltor;ce9nent o/quantum interference effhts
28b
1. The spectrum of the zero order hamiltonian. Q[&+&] Q correspondsto the continuous part while [p(H& +H$p+p’(H&+!f~)p’] correspondsto rhe discrete Fig.
part.
Q(HO,+H,)Q
p~&W
p
p’#H,)p
The interferences me obwrved between the levels !k, r) and Ik, s). gi represents zdl the discrete matter levels which unnot be reached by the excitation.
of levels. Therefore, the two resonances model describes a material system with only two discrete levels treated exactly, all the others being considered as dissipative continua or quasi continua. This is, for example, the case of the v-ibronic levels for the statistical limit case in large molecules. The material model can be described by the scheme in fig. 1. These two discrete states can be atomic Zeeman levels interacting with the radiation fieId as is the case in the cadmium experiment [l] or two coupled vibronic levels that are coupled both to the radiation field - one 01 both of then -and to the vibrational levels of a lower electronic configuration. If these two levelsare coherently excited by the incident light, it is possible to observe quantum beats in a time resolved experiment. We give the description of the two photon beam experinent where one photon is absorbed and the other is transmitted. When only one photon is in the-beam, we have k ~0. Using (53) in (26) the Fourier transform of the molecular response can then be written as
= c (k,, kg,gil&lk, rn) (k, tlIfimtlk’, k”,g) %#,(k, ?I, m) exp(-iEaT). m. n.a With only two resonances interferences can, in principle, be observed whenever
(53)
In order for this interference to be a maximum, the two resonances must have the same width, E% G k + E. - AE- iI’/2.
I&, i k t I?,-,t AE-ilJ2,
(54)
This, in turn, implies that mq R,, ko,gilH-,,lk
I
m) (k,
sexp[-I’r/2-i(k+EO)]
G$-#, k”,@y(T) z
m, n
[CR.1(k,n,m)exp(-iAEr)+rR,2(k,n,m)exp(IlAET)].
(55)
The second condition for maximization of interference effects is the factorization of C
n.m
$,
Ac
kg,gilH-,tlk, m) (k, nlH-,tlk’,k”,g) qa, (k, n, m) (kl, k,,,
g,liv-,,l k, m) (k, !I If&I k’, k”, gj 32 Oz(k, n, m).
(56)
n, m
Such conditions are satisfied, for example, for a molecule in which one radiant state is strongly coupled with a quasidegenerate state of the intermediate structure or for two nondegenerate weakly coupled radiant states which can-j he same oscillator strength and have the same line width. The general case of many levels yields many poles in (53). Eq. (53) is valid in this general case, and the maximization criteria (54) and (56) then only apply to the interferences from a particular pair of levels. :
Villaeys,K.F. Freed/Enhancernenf
A.
of quarmm inferfererlcc
effects
281
(a)
m
n
Fig. 2. (a) The diagrams represent the two possibilities for the absorption, evolution in the matter subspace, and subsequent emission of a photon. In one of the cases the fust photon k’ is absorbed and the second transmitted. while in the Eecond case we have the opposite situation. (b) Diagrams b show the more complex real process which consists of absorption and remission of the lirst photon and
,
subsequent absorption and remissionof the second photon.
(b) 5. Quantum beats
As we have seen before, the scattered light is described by (35). w e nov, explicitly derive tie detected inter@] in the particular case of the two pulse excitation and energy-level system, described in previous sections, in order to evaluate the enhancement of the interference terms over that obtained from single pulse experiments. (N(o) is given by
X CR,(k, 171,n)jdiF(i.-r)
exp[-i(F& f k)~] ‘, 0 where the poles are separated into contributions from the field and the molecular part E, +k=E,.
(57)
(58)
Recalling the notation Jk’, k”) for the two photon states the interaction matrix elements are described as tk, nIH-,,Ik’, k”,g) =p*(g.,~)[~&
(k,,k,vgl&,,lkv
t c&l,
(59’)
(59”)
~~)=~(gir “)Sk,k;
The first matrix element (59’) gives two possibilities: in the tirst the photon k’ is absorbed and the photon k” is transmitted, whereas in the second one k” is absorbed and k' transmitted. The other matrix element (59”) contains only one possibility, when inserted into (7 l), due to the diagonality of p Gd(E)P with respect to I k). If the detector can only observe photons that are scattered by the target, this implies kl = k. Then the interfering terms are only prodded by the diagrams in fig.2a, and we neglect the real process corresponding to the diagram in fig. 2b. Using the Fourier transform for the excitation pulse shape, we obtain, after integration in (54),
W(f))= @ ~‘m~~l(k~)~(f)~(gi,m)p*(g,n)~o(k,m.~z)ex~(-~~~/2)(~~-E I
I
I
X {O(i)exp(iklT)[exp[-i(za-Ek)
t’]-l]+B(r’-T)exp(SkT)[exp[--i(&-Ek)(i-T)]-l]l
“, (6q)
which is the totally general result for many poles (i.e., many atomic and molecular levels). Again (60) is valid when many excited levels are present. With the assumption (56) to maximize the interference terms, the term C ~(gi,m)~*(g,n)‘#.(k,m,n) m,n
282
A. Viflaeys, KF. Freed/Enhancemerrf oJqoanfum interJerence effects
can be removed as an overall factor. Then this gives W(t’))-
$J $I:(k,)‘R(f)l*l I
~~(gi,m)p*(g,n)CA,(k.m,11)12exp(-r~kr’) m, n
X {e(f)exp(iklT)[exp[-i(E,-Ek)r’]
z a
(&-W1
-11 +O(r’-T)exp(iEJ)[exp[-i(Fa-Ek)(r’-T)]--l]]
2. (61)
in some instances the factor (E, -I$)-~ can be taken outside the Ea. This occurs especially when the width of the excitation spectra is larger than the energy gap between the poles. Then, if the expression for the poles (54) is introduced and integration is performed over k,, the scattered intensity is W(r’),GAjO(t’)[exp(-I’r’)cos2(AEt’)
t exp(-r,kt’)-2exp[-(I’+rk)
X cos(hsl’)cos[(Eo-Ak)r’]]+O(t)-T)[exp[-r(1’-T)] -2exp[-(r+rjc)(tl-T)/2]
cos[AE(t’-T)]
+ O(t’-T)[2cos(AEr’)cos[AE(t’-T)]
t’/2;
cos2 [AE(t’-T)]
+exp[-I’k
(t’-T)]
cos[(E,-A,)(&7-T)]]
cos[(E-Ak)
T] exp[-r~-(rk-r)T/2]]},
(62)
where A is given by (63) remembering that the spectral distribution is normalized. In the expression (62) we have neglected the interfering part decaying with a lifetime less than or eqeal to ril corresponding to the coherence time within the photon pulse. These terms are A{ O(t’- T) 2[exp (-r,t’) -cos{AE(~‘-T)]
-
cos
(AE t’)
cos
COS[(Q,-A~)(~‘-T)]
[(Ak -E,)
t’] exp [-(I’+ r,) r’/2]
exp[-r(t’-T)/2-rk(t’tT)/2]},
(64)
and they can be ignored as they are small and as our basic focus is on the basic molecular processes. Eq. (62) has specialized to two levels, the many level case is obtained by summation over all pairs of interfering levels. We now discuss the expression (62) for the scattered intensity. The first three terms describe the scattering of the first pulse. They correspond respectively to the absorption with a lifetime r,‘, the coherence time of the photon, and ihe emission with a lifetime r-’ , the lifetime of the unstable atomic or molecular state. The third term d,escribes the interferences in the rise time of the second pulse due to off.resonance excitation; these interferences disappear with a lifetime (I’+I’&’ and its recurrence time is (E,YAk)-‘, that is, the inverse of the enerw difference between !he mkan energy of the excitation pulse and the mean energy of the resonance. After time r = T, the second pulse is applied, and we observe the same evolution as before, except that we have an interference term due to the two successive excitations. This term, 2 cos(AE’~)cos[AE(t’-T)j
cos[&,-Ak)T]
exp[-l?‘-(rk-r)3/2].
(65)
gives an enhancement or diminution of the quantum beats in the scattered light and represents the reversible evolution in *he material system.
6. Numerical calculatians and discussion In order. to have ZQ?estimation of the enhancement of the quantum beats, the intensity of the scattered light has been.compuJed’as a function of time. The results are given in fig. 3. As mentioned before, in the,expressi?n (62),we have neglected all the interfering terms (64) hecaying with a
A. Villaeys, K.F. Freed/Enhancemcnr
283‘
of quonnrm inrerfercncc effects
Fig. 3. Plot of the detected intensities (solid curve) and the
contribution of the interference term (dashed curve) for three different cases. (a) The cscitint pulses are in phase. The contribution of the interference (b) The
term is always
positive.
exciting pulses ore n/2 out of phase. The contribu-
tion of the inkrference term is alternatively positive and negative. (c) The exciting pulses arc pi out of phase. The contribu-
0
1
2
3
r
k
t
tion of the interference term is always negative. In all the cases. we consider w 3 A,? and E. = Ak so
j
COS[(&-Ak)rl
wt/2rr
=
cos[(Eo-ak)(t-T)]=
1.
lifetime of the order or less than I’,‘. In the numerical calculations we must take these terms into account; if not,
a discontinuity would be introduced in the expression for the detected intensity at time r=T. But, after a short time of the order of I’i’, their contributions become negligible and the expression (62) gives the dominant variation of I(<). Because we focus our interest on the basis molecular processes, we consider the case of resonant excitation, that is to say cos[(Eo-Ak) Therefore,
t’] = cos[(EO-Ak)(r’-T)]
the oscillating terms corresponding
= 1. to interferences
between the molecular and fieId resonances
are not
retained. Given the smallness of the lifetime of these oscillating terms, they result kssentially in oscillation in the
rise time of the excitation. We observe in fig. 3 that the contribution of the interfering term gives a contribution which depends, of course, on the time delay between the two pulses. In fig. 3a the two successive pulses are in phase; the contribution of the interfering term is positive and gives an enhancement of the quantum beats. In the second case, corresponding to the fig. 3b, the pulses are n/2 out of phase, and the contribution is alternatively positive and negative. In the last case - fig. 3c - the pulses are R out of phase, and the contribution is always negative. The best enhancement is ther. obtained in the first case. If quantum beats are alleged to be observed in intermediate case molecule,s, the above analysis shows how they must be altered by the two pulse experiment. This two pulse approach also implies a definite dependence of the observation on the pulse separation T. However, in the limit where quantum beats are “washed out”, because of the large number of pairs of interfering levels, the interference term (65) of (62) gives a nonvanishing contribution when averaged over all pairs of interfering levels. (This result is clearly true ifi the general case of (60) where the approximations have not been invoked to maximize the interference terms.) For the model of a continuous distribution of beat frequencies up to some maximum AE,,(and E0 = A,) the average of (65) over this distribution of beat frequencies is nonvanishing if AL’,, T < 1. Hence, the “washed otit” limit still yields a quantum yield which differs from twice that arising from a single pulse by the beat frequency averaged interference term. This then provides the opportunity for verify@ the existence of the quantum interferences in this limit. In the opposite atomic limit, the additional interferences should readily be observable and should be a useful tool for studies of atomic relaxation processes.
.2a4
A. P~ikys, K.F. Freed,&nhancemer~~ofquanrum
irxerference
effecfs
It is,‘perhaps, relevant to note that a somewhat related phenomena has been considered by Comey and Series [ 191. They show thatinterference effects are produced in atoms which are excited by modulated light when the modulation frequency corresponds to an atomic transition frequency. Modulation is limited to frequencies of up to possibly a MHz. On the other hand, the use of a train of optical pulses can have spacings in the nanosecond or subnanosecond timescale to extend the phenomenon into this domain.
Acknoeledgement We are grateful to Abe Nitzan, Isaac Abella and Don Chemoff for helpful discussions and to Don Chemoff for performing the numerical calculations.This research is supported, in part, by NSF Grant ME’S7541549. K.F.F. acknowledgesthe support of a Camilleand Henry Dreyfus Foundation Teacher-Scholar Grant. AN. would like to thank G. Ourrisson,President of the University Louis Pasteur of Strasbourg, and ADRERUSfor a travel grant.
References [!] J.N. Dodd, RD. Kaul and D.M. Warrington, Proc. Thys. Sac. 84 (1964) 176; J-N. Dodd, W.J. Sandle and D. Ztiermann, Proc. Phys. Sac. 92 (1967) 497; H.J. Andra, Phys. Rev. Lett. 25 (1970) 325; S. Haroche, J.A. Paisner and A.L. Schawlow, Phys. Rev. Lett. 30 (1973) 948. [2] GE. Busch, P.M. Rentzepis and J. Jortner, J. Chem. Phys. 56 (1972) 361. [3] J.M. Delory and C. Tric, Chem. Phys. 3 (1974) 54. [4] C.E. Porter, Statistical Theories of Spectra: Fluctuations (Academic Press, New York, 1965). IS] C.A. Langhoff and G.W. Robinson, Mol. Phys. 26 (1973) 249; Chem. Phys. 5 (1973) 1; 6 (1974) 34; J.O. Berg, CA. Langholf and G.W. Robinson, Chem. Phys. Lett. 29 (1974) 305; J.M. Friedman and R.hI. Hochstrasser, Chen. Phys. 6 (1974) 155. [6] W. Rhodes, J. Chem. Phys. 50 (1969) 2885; Chem. Phys. Lett. 11 (1971) 179. 171 U. Fano, Am. J. Phys. 29 (1961) 539. [B] J. Jortner and S. Mukamel, in: The World of Quantum Chemistry, eds. R. Daudcl and B. Pullman (Reidel, Boston, 1974). [9] A. Villaeys, unpublished work; K.F. Freed, Topics Appl. Phys., in press. [lo] A. Messiah, Quan:um Mechanics (Wiley, New York, 1961). (111 S. Mukamel and J. Jortner, I. Chem. Phys. 61 (1974) 227. [ 121L. hIower, Phys. Rev. 142 (1966) 799; 165 (1968) 145. [ 131 M.L. Coldberger and K.M. Watson, Collision Theory (Wiley, New York, 1964). [14] K.F. Freed, J. Chem. Phys. 52 (1970) 1345. [15] L. Mandel, Phys. Rev. 144 (1966) 1071. [16] R-J. Glau’;xr, in: Quantum Optics, eds. S.M. Kay and A. hlaitland (Academic Press, 1970). [17] V.M. Titulaer and R.J. Glauber, Phys. Rev. 145 (1966) 1041. [18] L.R. Doddand I.E. h!cCarthy, Phys. Rev. A 134 (1964) 1136. [19] A. Comey and G.W. Series, Proc. Phys. Sac. (London) 83 (1964) 207,213.