Molecular excitation processes by fluctuating light

Volume 65. number 3

1 September 1979

CHEMICAL PHYSICS LETTERS

MOLECULAR

EXCITATION

PROCESSES

BY FLUCTUATING

LIGHT

Maurizio BENFATTO and Paolo GRIGOLINI Gruppo A’aziomle di Stmttura della Materin de1 CXR.. 56-6100P&a. Itat)

Istituto di fisica.

Received 39 Xf~rch 1979

The “reduced” model theory is extended in sue11a way as to allow a quzmtitathe study of exitorion processes by pulses hating a width greater than their uncertainty-limited katue.

I_ introduction In the past few years significant attention has been focused on the dependence of molecular escitation processes on the charxteristics of the light sources [I] _For instance_ it has been stressed by Rhodes that the pu!se coherence duration strongIy affects the main characteristics of molecular decay [2] _ Some qualitative features of the theory of Rhodes, and equivalent approaches as well [3,4], have recently been reproduced through the expfoitation of the “reduced” mode1 theory (R&IT) [5] _ Such an approach consists in replacing a complex thenal bath. for instance of molecular kind, with a simpler one. Justification for this replacement is given by noticing that both cornples

and “reduced” thermal baths resuit in the same effects on the molecular state of interestA limitatiqn of the previous work [5] lies in the fact that only minimum uncertainty excitation pulses were taken into account. In the present paper, however, the RMT is extended in such a way as to allow a straightforward attainment of quantitative results also in the case of excitation by fluctuating pulses_ A detailed discussion of the general theory will be given in a later pubIication_ In this ietter we limit ourselves to comparing some preliminary results with significant findings of refs. [ 1,2,5] _

2. An outline of the general theory The hamittonian x=&+xR

for the molecular system under investigntron may be written in the following form +zCc, +;fcSR +x,,

where ZCs is the hamiltonian SC, = le)e,(el+

,

(1)

of the system of interest,

ig>eg@l

(Ee> Q-

(3

XR is the radiation hamiltonian.

XB is the hamiltonian of the intramolecular dissipatke manifold whose coupling with the state le) results in the radiationless decay under study. The interaction between the system of interest and its dissipative manifold is expressed as follows (3) The state ]nz)_ in turn, tnay interact with dissipative thermal baths. According

to ref. [.Sj such an interaction will be

VoIurnc65. number3

1 September1979

CHEMICAL PHYSICS LmR!3

dealt with by replacing the energies of the states lm), l,,r , with the complex values enl - ir, ) i-e_ sve shak perform the markovian assumption on the decay of the states ImX By performing also the rotating wave approximation, we shal1write XSR in the following way (Z - (8,) is proportional to the negative (positive) frequency part of the radiation field) XsR = le) CgE__ + [g> (elE,

_

(4)

The positive and negative frequency part of the electric field are defied tions G@!&)

= $+E_(~

+ +%&),

$,rr>=

&F+,(f

++S_(7)=

(5) (pE_!&)Y

where pp is the radiation field density matrk J’) - --,+ (l) = 2

through the foIlowing correIation func-

exp(-_7Lf) exp(-iwf)

,

(5’)

For the s&e of simplicity, we assume that

_

(‘5)

-Q_denotes_ therefore. the taser bandwidth and w is the dominant puke frequency_ We now foiIow the Zwanzig approach (61. We then need a suitabIe projection operatorP so as to replace the quantum-mechanical LiouviIIe equation

iapiar = xXp

(Wp = 30 - /.?x) _

(7)

with the generalized master equation concerning the part of interest alone, p I = Pp_ In the present case. the most suitable projection operator appears to be the one defined by (s = TrtP)p) Pp=pp~Ie~~elsle~~el~ig~
le)
_

G9

By applying the Zwanzig approach 17.5j to the quantum-mechanica J_iouviIIeequation_ eq_ (7),r$ated to the interaction picture and assuming (1 - P)p,(O) = 0, we obtain (7CI = -5csR + KSb, 7f0 = K - K,, A = exp(isc,f) X A exp(-*or)) ia&/&

= PST&

-

i j

Z(t, 5.)gI(5) ctr ,

(9)

0 where

r *t,

7) = P%f(r)

exp_

-i

[J

(1 - f’)%(r*)

dr’

0

1

(I - P$;(r)P

00)

_

As far as tfte intramokcuiar interaction is concerned_ the evaluation of %(t_ r) cur be traced back to that of the sCwn&or&r r’orrehtion function [S]

GEim(f) = z1) Iv,,, I’ exp[-i(e,,r m

- if,,)r]

.

(11)

which_ for the sake of simplicity, is assumed to bc (G = 2u/7) &’mu-a(r) = 12 exp(-y2)

exp(--ie&

= ;C’g’

esp(-7r)

exp(-ie,,lr)

_

(11’)

As far as tI,e interaction between the state le) and radiation tieid is concerned, we can follow the stochastic approach [Sj_ Then, the electric field & can be regarded as a cIassica1stochastic variable driven by the FokkerPlanck equation

Volume 65, number 3

1 September 1979

CHEMICklL PHYSICS LETTERS

The smplest diffusion operator satisfying the requirements of eqs_ (5) and (5’) seems to be the one endowed with an equilibrium state Ipo) (rPlpO) = 0) and two degenerate states lpl) and Ip,) (TPlpl) = yLIpI); rP lp$ = 7Llp$)By following the Rh4T [S] we can repIace eq. (7) with

iWW/at

= [Jcs, u0p)l

+ [Jcnl,u@p)l

+ Wi)Qpr

The density matrix a&) depends on the radiation The incoherent interaction is defined as follows:

0, uOp)l

stochastic

- irpu&)

03

-

variable and can be written

as a(Q)

= C,, u,~ IQ. (14)

~(i)(Xp,t)=(le)(gl&(i)tIg)(ei$!i)), where (@))*

= 81i_) = xv exp(-iwt){Ip$


(polj

_

(1%

The hamiltonian SC,, appearing in eq_ (13) is written as xAl=

(Qf -iy)11I1)WZl_

(16)

For the sake of simplicity we assume below that Enr = E,. The replacement of eq. (7) with eq- (13) can be justified by noticing that the Iatter equation when written in the interaction picture and subjected to the Zwanzig approach [5-71 through the use of the projection operator P' defined as (17)

,

P’u(Xp)=Ipo~Cle~~eIu~le~~eI+Ig~(glu0lg~~gl+le~~elu0lg~(qI~lg~(g~u~le~~el~

results in the same “memory kernel” as the one of eq_ (10) evaluated by using the stochastic approach [S] _ We can now define a twenty seventh-order vector Xwhose first nine elements are: (elGOle>, , (glal le). (gl~~lg), (eli?olInZ>, (MliTole), (nilif21g), , Cnfl~o~M~. With the symbol 5 we denote the “stochastic” density matrix u written in the rotating fnme of reference. Then, from eq. (I 3) we obtain idXjdt = AX,

(18)

where

0% and(Aw=e=-eEg-w) 0 -IV

A, =

--w

aw-irL

1v

0

0

\v - -LL’

u

0

0

0-

0

0

v

0

0

0

0

0

-v

1%’

0

-Aw--rq_

0

1%'

--IV

0

0

0

0

0

0

-iy

0

V

0

0

0

0

-iy

0

v

0

0

0

-IV

0

0

0

\L’

0

-0

0

0

v

-V

-v

0

We now assume that the “reduced”

-v

-V

0

0

0

0

0

1V

V

--1v Aw-i(ytyL3 0 0

0 0 -Aw--i(+-y=) 0

-v

0

0 --2iy_

system consisting of the states le>. lg) and lM> is undergoing a transverse 533

Volume 65, number3

CHELWCALPHYSICS LEITERS

I September1979

relazrtion through 3n energy fluctuation process affecting the state I&, whose rate is -y7_ In the presence of a usual density matrix pBy focusing our attention on the relative rotating frame of reference we can show that the vector X’ consisting of the nine elements (elCIe>, !elGlg>,
idX’/dt = A’X’ ~

(21)

where A’ = A when rz = rL_ Such a remark shows that when attention is focused, for example. on the excitedstate time evolution, the effect of light fluctuation cart easily be accounted for by regarding the state lg> as being affected by an energy ffuctuation process and the matrix eIements and Cgt~kr>, with a fg, as suffering a decay endowed with the rate rL_

3_ Applications to the problem of mokcuikr excitation We hare shown [S] that the non-linear regime can significantly be affected by a radiationless relaxation process of no~markovian type_ Then a straightforward interpretation of the osciilating behaviour in terms of the Rabi frequency alone is not allowed. Fig. 1 shows, moreover, that the oscillating behavior tends to disappear as the light bandwidth is increased. Such a result can be regarded as an extension of the ones of ref. [93_ Kimble and Mandle 191 show, in fact, that the Rabi oscillation tends to disappear as the laser bandwidth is increased_ The mat significant resuhs of this Ietter are ikztrated in figs. 2 and 3, Their discussion requires some remarks. fn ref- IZ] Rhodes considered the case of an exciting right whose band~v~dth7L Q-10-3 cxwL) is kept fried.. He studied the effect of exciting the molecule with pulses ranging in duration, r, from IWi2 s to 1CM s_ The theory ofref- fsj succeeded in providing a quantitative evahration of physical conditions such that the first case dealt with by Rhodes [2], z = 10-tz, has roughly to be regarded as being a ease of excitation by minimum-uncertainty puIses_According fo ref_ [2], in ref_ [5] minimum-uncertainty pukes of short duration were shown to result in an oscillating decay_ Ref- {5] shows, furthermore, that the osciiiating time behavior is e_xhibited to a Iesser extent as the time duration of the minimum-uncertainty pulse increases_ On the contrary, as the detuning parameter departs from the maximum absorption v&e, a more noticeable oscillating behaviour is recovered [S] _ In the GCWof a bng duration pulse, r * 10-G s, however, according to ref. [2] any oscillatory character shouId disappear. Since in this second case the Iight bandwidth is very much larger than its “coherence width” [2], a quantitative study of this physica condition requires that the theory of ref, [5] be replaced by that of present

Fig- 1. Time evofuti5nof the popuhtion of the excited state: I& JCdifferentvaluerof the tier bandwidth2~~ G ~4, w = 2~~The ker banduidthis expraed in units of 7_

Volume 65, number 3

CHEMICXL

PHYSICS LETTERS

1 September

1979

ooa

3.

4 rt

5

6

7.

6

Fig. 2- Time evolution of the popuktion of the excited state 1.~1in the presence of the radiation field at different values of the light bandwidth 7~. and detuning parameter Aw.The full lines indicate the case where 7~ = 0. The dashed lines indicate the case where TJ_ = 100 -y_The lines denoted I indicate the use where the detunlng parameter attains the maximum absorptionpoint, Aw=(G2 - 1)'Rr/2_Thelines denoted 2 indicate the case where Aw = 0. For the sake of clarity the population for -ye = 100 7 has been mukiptied by the factor 82.43. G = 4, w = 0.1 -r_ The pulse time duration is T = 5/-y_

P?C

:;* :I I

:

: 8 I

td

~

Fig. 3. Time evoIution of the population of the excited stale le) in the presence of an excitation pulse nhose time duration is Y = OS/-y. The full line indicates the case where TI_ = 0, nnd the dashed line that where -ye = 100 -y. The population in the latter case has been multiplied by the factor 2493. C = -L, w = 0.1 -y_

Ietter- Fig_ 2 shows that, whatever the detuning parameter, the excitation process does indeed result in the same non-oscillating decay. Perhaps a fully exponential decay could then be recovered by taking into account both the puke fall time and adiabatic interaction even with external- “thermal baths” of non-radiative nature CIO]_Such a result seems to be in qualitative agreement with ref. [2]. In the case of shortduration pulses, an increase in the laser bandwidth appears to have irrelevant effects on the moIecular decay (tig. 3)535

Volume 65, number 3

CHEMICAL PHYSICS LElTERS

I September 1979

Rcferencs [ I] [21 [3 j 14j [S)

[6] 17j [Sj i9j [IO]

536

W_ Rhpdes. inr Excited states. ed_ E-C_ Lim (Academic Press, New York), to be published. and references thereinW_ Rhodes, Chem- Phys- 4