Furaday Discuss. Chem. SOC., 1983, 75, 117-130 Pulsed Laser Preparation and Quantum Superposition State Evolution in Regular and Irregular Systems BY RONALD D. TAYLOR AND PAUL BRUMER * Department of Chemistry, University of Toronto, Toronto, Ontario M5S 1A1, Canada Received 20th December, 1982 The nature of the prepared state is crucial to an understanding of isolated-molecule intramolecular dynamics. A model is utilized to compare the created quantum superposition state in pulsed laser excitation from (1) a ground electronic state with regular nuclear wave- functions to an excited electronic state with regular nuclear wavefunctions with that from (2) the regular ground state to an excited state with irregular nuclear wavefunctions. All results are in the quantum-mechanical small-molecule limit.Visual inspection of the created state and its evolution shows distinct qualitative differences in these two cases, although various theoretically studied quantities do not reveal an obvious distinction. The differences are expected to be experimentally observable in tirne-resolved emission. 1. JNTRODUCTION The vibrational dynamics of isolated rnoIecuIes has historicaliy been formdated differently in two distinct energy regimes. At low energies motion is assumed separable and well approximated by a zeroth-order (normal, local, etc.) modes description. At higher vibrational energies an initial distribution is assumed to relax efficiently to a statistical result. In particular, the relaxation rate is expected to exceed the rates of competitive processes.This historical viewpoint has gained support from recent formal and computational developments in ergodic theory and in the clussical dynamics of N-degree-of-freedom non-linear Hamiltonian systems. The importance of these results to intramolecular dynamics is twofold. First, formal theory provides a concise definition of the ideal cases of regular quasiperiodic behaviour and of mixing systems.’ Secondly, COM- putational results on typical non-lincar systems show motion at low energies which is close to regular quasiperiodic ( N conserved isolating integrals of motion), whereas higher-energy dynamics shows relaxat ion dynamics characterized by fewer (but not necessarily conserved isolating integrals. Although formal techniques do not provide a useful route for identifying a system as ergodic or mixing, several com- putational diagnostics are available allowing one to identify dynamics which are not regular.Thus, although technical problems exist in classical intramolecular dynamics, the conceptual hasis for a transition in behaviour with increasing energy seems well established. The status of quantum intramolecular dynamics is quite different. In this case efforts to define formal, conceptually useful, ideal systems displaying finite time relaxation behaviour are in progress. Similarly, computationa1 diagnostics to identify a quantum analogue of regular and irregular motion are only in their earIy * I. W. Killarn Research Fellow,118 SUPERPOSITION STATE PREPARATION stages of development. A variety of different approaches, such as the examination of localized wavepacket dynamic^,^ eigenvalue distributions,s properties of Wigner functions,6 mapping dynamics etc., have been proposed to address these problems.Our approach, exemplified in this study, is to explore the exact quantum dynamics of model systems which are well characterized in the classical limit in an effort to ascertain similarities and differences between the quantum dynamics of systems which are (classically) regular as compared with those which are (classically) mixing. Further motivation for the study discussed in Section 4, on systems described in Section 3, is provided in Section 2 below. 2. MOTIVATION AND ORIENTATION Both classical quasiperiodic and statistical behaviour are observable in systems as small as N = 2.Thus it suffices to consider the quantum dynamics of isolated small molecules. An ideal experiment on the intramolecular dynamics of an isolated small molecule is readily envisaged. Here, a preparation device leaves the molecule, with bound molecular eigenstates {p,} and energy eigenvalues {Ej}, in a superposition state a v(0) = c CjPP i The subsequent time evolution, neglecting radiative emission, is given by i In this case intramolecular dynamics depends solely upon the molecular properties {p,}, {E,}, the interaction between the molecule and preparation device described by {c,}, and their cumulative effect in the sum [eqn (2)]. Observation of the evolving state entails projection onto states defined by the particular measurement. Such observations are necessarily dzpendent upon the coefficients {c,}.Previous efforts to explore regular versus irregular quantum behaviour have thus far focused on several aspects of eqn (2), such as the nature of the energy eigen- value spectrum,s the structural characteristics of p, in the semiclassical limit and the evolution of V(t) with the initial state taken as a localized wavepacket or mixed state.lo Little effort has, however, been directed towards the subject of this paper, the examination of similarities and differences in {cj} resulting from the response of " quantum-regular " and " quantum-irregular " systems to similar preparation devices, The importance of the preparation step to small-molecule dynamics is well recognized l1 and easily motivated. Consider, for example, an experiment which prepares, in two systems, one highly coupled and one uncoupled, a ~ ( 0 ) comprised of two exact molecular eigenstates.No matter what the extent of coupling between arbitrarily envisioned zeroth-order modes, an emission experiment, on either system, would only reveal qualitatively similar quantum beats or elementary properties of individual molecular eigenstates. Similarly,12 if experiments on uncoupled or highly coupled systems prepare a ~ ( 0 ) comprised of a large number of states, but with similar {c,} energy envelopes, then the ~ ( t ) evolve with qualitatively similar P(t) = 1(V(0)lV(t))12. Thus, if one performs similar experiments on two molecules and dctects characteristics of' regular behaviour in one system and irregular behaviour in the other, the origin of these differences can be attributed to the different nature of the created superposition states.These arguments motivate a comparison of state preparation and evolution in " quantum-mixing " and " quantum-regular " systems. In the absence of suitableR. D. TAYLOR AND P. BRUMER 119 definitions of these terms, and in the light of our expectation of an intimate relationship between systems which are classically mixing and " quantum mixing," we utilize below systems which are fully characterized in the classical limit. These systems (particles in boxes, circles and stadia) bear only a small resemblance to realistic N = 2 molecular vibrational systems, but they will display the distinct characteristics of regular and irregular quantum behaviour in preparation and evolution, should such differences exist.Indeed, we strongly advocate quantum studies on such sys- tems whose classical behaviour is fully characterized since regularity, mixing etc. is expected to be a property of the Hamiltonian. Below, we consider electronic excitation via pulsed laser irradiation, both as a model for these popular experiments and as a controlled means of preparing super- position states. Alternative means of preparation are under inve~tigation.'~ Our focus here is on ~ ( 0 ) and V ( t ) and not on realistic measurements of the evolution, to be reported e1~ewhere.l~ Hence, radiative emission, for example, is neglected. We emphasize that our interest is in quantum behaviour, with no effort being made to examine behaviour under semiclassical conditions.3. METHODOLOGY 3.1. MODEL SYSTEMS To explore differences in state preparation we model two cases: (I) a transition from a ground electronic state with regular vibrational eigenfunctions to an excited electronic state with regular vibrational eigenfunctions and (2) a transition from the regular ground state to an excited electronic state with irregular vibrational eigen- functions. Here the regular-irregular terminology is based on behaviour in the classical limit. Model N = 2 systems which are regular in the classical limit are readily constructed. This is not the case, however, for mixing systems, a condition proved for only a limited number of Hamiltonians. Of particular interest is the stadium billiard, i.e.a particle confined to apotential-free region by an infinite potential boundary comprised of two parallel lines and two circular caps (see fig. 1). Numerical eigensolutions for this system, regarded as a simple model for irregular N = 2 vibrational motion in an excited electronic state, are available. Regular vibrational motion in the excited state is modelled by a particle confined to a rectangle. Sides of the rectangle are chosen to be of irrationally related lengths to eliminate degeneracies. Finally, ground-electronic-state regular dynamics is chosen as that of the particle con- fined to a circle of radius rc, ensuring no symmetry relationships between ground- and excited-state eigenfunctions. In addition to the circle+rectangle (denoted the CR case) and circle-tstadium (denoted the CS case) systems an additional system, where the rectangle has a 2 : 1 ratio of sides, was studied.This system (denoted the CRD case), containing highly degenerate energy levels, is expected to be even further removed l5 from quantum ergodic behaviour. System potentials are shown in fig. 1. The Schrodinger equation in each of these cases is given by - h2 - (a2q/ax2 + a2q/ay2) = Ep; E = h2K2/2m 2m (3) with q = 0 on the appropriate boundary. Stadium eigenfunctions were determined numerically by an extension of the program of McDonald and Kaufman l6 to include states of all symmetry. Rectangle solutions are well known as is the ground-state solution of the particle-in-a-circle with eigenfunction qo(r,O) and eigenvalue h2K,2/2m given, with J,(Kr) the ith Bessel function, by qo(r,@ = J o ( ~ o ~ ) / ~ ~ ~ ~ c J l ~ ~ ~ ~ c ~ l for r < r,; qo(r,8) = 0 for r > rc (4)120 SUPERPOSITION STATE PREPARATION with Kor, = 2.404 825 5.This is the only state of the circle required below, since the system is assumed to initially reside in this ground state. In each of the circle, stadium and rectangle the numerical value of an energy eigenstate is inversely proportional to the (area)+ contained within the boundary, allowing for a scaling of energies into ranges appropriate to molecular problems. With rn chosen as a typical reduced mass (here m = 19 610 a.u. = 10.7 a.m.u.) the areas of the figures were set in the following manner. First, the area of the circle was G E S E E S c a s e - c s C R 0 Fig.1. Shape of vibrational potential for ground electronic state (GES) and excited electronic state (EES) for various cases. Dimensions utilized were a, = 1.1423 a.u. for the CS cases, 1, = 4.5692 a.u., 1, = 2.0394 a.u. for the CR cases and 1, = 21, = 4.3171 a.u. for the CRD cases. Dark circles within EES figures denote relative size and location of GES for 3 cases. adjusted so that Eo = k2Ki/2rn = 9.12 x a.u., i.e. the ground " vibrational " state of the circle is at a typical vibrational energy. The resultant Y, = 0.127 19 a.u. is typical of the size of probability density associated with a low-lying vibrational state of a vibrator. Secondly, the area of the stadium was determined by placing highly excited states of the stadium, known l6 to be characterized by highly convoluted nodal patterns, in an energy range such that they were accessible from Eo by visible radiation.In particular, the centre of the excited vibrational manifold (at K = 51.2989 for a stadium l6 whose area is z) was located 0.0735 a.u. above E,, yielding a readjusted stadium area of 9.3 185 a.u. The excited-state rectangle areas were set equal to that of the stadium. Once again the resultant dimensions are typical of the probability distribution associated with a highly excited vibrator. The final specification required to describe the model fully is the location of the centre of the ground-state circle with respect to the centre of the excited-state rectangles or stadium. For each of the CR, CRD and CS cases three relative locations were examined to allow a study of the dependence of the prepared state on circle location.This provides preliminary insight into whether statistical behaviour may be linked with the insensitivity of prepared state to aspects of the preparation. In particular, the circles were positioned, with respect to the centre of the excited-state potential, at C1R. D. TAYLOR AND P. BRUMER 121 = (1.2695,0.6984), C2 = (1.8407,0.4128) and C3 = (0.4128,0.4128). We distinguish these models as C1 R (ground-state circle Cl , excitation to the non-degenerate rectangle), C3S (ground-state circle C3, excitation to the stadium) etc. Relative positions of these circles are also shown in fig. 1 . The number of states q,, contributing to ~ ( 0 ) due to pulsed laser preparation is dependent upon the density of qn states and the laser pulse time.To observe quantum intramolecular relaxation, which for small molecules is dephasing on a timescale less than the density of states p, requires that hp exceeds the laser pulse time. An appropriate choice of laser characteristics allows, therefore, for a study with ~ ( 0 ) containing relatively few states. The relevant 16-23 states " picked up " by the laser pulse described later below consist of 16 states in the stadium between K,, = 29.6067 a.u. and K,, = 29.9456, 21 states of the non-degenerate rectangle in the range K,, = 29.5573-29.9966 and 23 states of the degenerate rectangle between K,, = 29.5509 and K,, = 30.0045. A wide variety of different (n,,n,) quantum numbers appear in the latter two sets. The resultant densities of states are 0.14-0.15 state cm-', a reason- able density in the small-molecule limit.This leads to a relatively short " recurrence time " of ca. 7.4 x 10-13 s, necessitating excitation pulses on the sub-picosecond time- scale. Less experimentally demanding pulses, i.e. in the nanosecond range, could have readily been used by altering the model to increase the state density. Such machinations are, however, unnecessary theoretically since the resultant excitation of 16-23 states would behave functionally the same but on a different timescale. 3.2. STATE PREPARATION AND TIME EVOLUTION The full Hamiltonian for the interaction of our " molecular " model with an external laser field is given by H = H,,, + V, where H, is the Hamiltonian for the model and V is its interaction with the laser field. Here " electronic " eigenstates of H,,, are circle + rectangle or circle + stadium with " vibrational '' eigenstates describcd above.The system is assumed initially to reside in the ground state of the circle, denoted y o ; {q,,,n # 0} denotes the set of " vibrational " eigenstates of the excited " electronic " state. A similar indexing convention applies to the energy eigenvalues. Matrix elements of the interaction potential V,, = (q,l Vlq,} are obtained within the following approximations: (1) all electronic transitions are assumed to be Franck- Condon in nature; (2) vibrational states within the excited electronic state are not radiatively coupled to one another; (3) radiative emission is neglected; (4) there is no interstate coupling between vibrational levels of the excited and ground electronic states.Under these circumstances : V," = 0 V,,= V,,,,-O; m#O,n#O ( 5 ) where pel is the electronic transition moment and ~ ( t ) defines the coherent, transform- limited, laser pulse. In particular, we choose ~ ( t ) as gaussian, to model a typical experimental environment; l8 i.e. where wL is the laser frequency, t, is the temporal location of the pulse maximum, 4 2122 SUPERPOSITION STATE PREPARATION is the field strength and the phase constant 6 is chosen, without loss of generality, as 6 = -w,t,. The full temporal width, at half maximum, of the pulse is 1.665~. Computations below utilized E = 109V m-l, t , = 22, wL = 0.0735 a.u., and pel = I .27 debye. Two pulse widths, at z = 8309 a.u.and 11 000 a.u., were studied. At any time t the wavefunction associated with H, is given by y(t) = co(t)poe-iEot/fi + 2 cn(t)p,e-iEntlfi. (7) n= 1 Substitution of eqn (7) into the time-dependent Schrodinger equation gives the stan- dard set of coupled equatioins for cJ(t) which, within the rotating wave approximation, is given by ikC,(t) = C c,(t) von(t) e-iwnot = pel e 2 c,,(t)f,,cos[m,,(t - t,)]e-(l- rm)2/72 e-%t (8) n=l n= 1 ikkn(t) = co(t) vn0(t) eiwnol = pel~~O(t)fnO cos[coL(t - t,)]e-(t- rJ2/r2 e i o n o r where mno = (En - Eo)/h. Below, eqn (8) is solved numerically to determine cJ(t), with co(0) = 1, hence by- passing approximations associated with perturbation treatments. Comparison with first-order perturbation-theory results, discussed below, indicates agreement between numerical results and first-order perturbation theory to 6% at large t .3.3 THE PREPARED STATE As described, preparation is confined to times when the laser field induces changes in the level populations. Hence the prepared state is comprised of cqn (2) with cl(t) asymptotically constant, The qualitative nature of this state is readily ascertained from perturbation-theory. Justification for using perturbation-theory results to gain this qualitative insight lies in its relative accuracy for the particular conditions dis- cussed above. After expanding cos[w,(t - t,)] in exponentials in eqn (8) and neg- lecting terms of the form exp[&i(o, + cono)(t - tm)] (rotating-wave approximation), a direct application of first-order perturbation theory yields co(t) = 1 cn(t) = (~~2/7~/4ih)p,~f,,~ e'wnotm e-(wt - 0n0)'~'14 equivalent to a form previously obtained by Rhodes.19 For ( t - t,) and t, $ z, eqn (9) approaches the asymptotic limit co(oo) = 1 c,(Q) = (qiel~dn/2ih)fno e - h --W,d2x2/4 e -i(Eo -En)tm/ft, (10) Practical studies show this approximation to be valid, for the systems studied, for t , z 22, t and is no wider in energy than permitted by the frequency distribution, and possibly narrower if the Franck-Condon factors at large IwL - wn,l are negligible. State populations Ic,(co)12 display the frequency distribution of the gaussian laser pulse modulated by Franck-Condon factors.Finally states are prepared with specific phases which may be interpreted as follows: the system is prepared as a coherent super- position state (Le.each state enters with the same phase) at t = t, which then evolves 42. Consistent with expectation, the prepared state is centred at mL,R. D. TAYLOR AND P. BRUMER I23 freely as e - W f - 'm)l'. Such a view is a useful mnemonic but does not, of course, provide a realistic description of the true preparation dynamics. 4. COMPUTATIONAL RESULTS Eqn (8) was solved numericaliy, with c,(O) = 1, for each of the nine cases described above. Analysis of the results was carried out by direct examination of P(t) = I< v/(O)I y(t)}12, c,,(oo), ly(t), etc. A technique to measure quantitatively quantum stocfiasticity in prepared pure states, proposed by Heller,20 was also applied. Space limitations prevent the presentation of more than a sampling of results.Further analysis is provided e1~ewhere.l~ 4.1. FRANCK-CONDON FACTORS As described above, the essential features of the prepared state, as viewed in the vrr representation, arise from the laser frequency distribution and the molecular Franck- Condon factorsf,,. Suggestions 2o have been made to the effect that the nature of the Franck-Condon factor distribution amongst pn states should provide a useful means of distinguishing excitation to regular as compared with irregular states. Similarly, substantially different analytic approximations to fo,, for excitation to regular and irregular states have been proposed,21 leading one to expect differences in fOn in excitation to regular or irregular states. Table 1 provides a summary of a11 Franck-Condon factors contributing to the nine cases under study.Several features are apparent. First, the fon do not display qualitative differences between excitation Table 1. Franck-Condon factors, fo, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 I5 16 17 I8 19 20 21 22 23 -0.012 0.032 -0.01 2 0.01 3 - 0.024 0.003 - 0.019 -0.005 0.01 8 0.006 -0.017 -0.005 0.003 0.014 0.01 8 -0.004 -0.001 -0.001 0.022 0.002 0,002 -0.023 -0.005 -0.019 -0.009 -0.006 -0.016 -0.003 -0.029 -0.019 0.002 -0.003 -0.020 -0.001 0.01 8 0.003 0.010 -0.023 -0.008 -0.008 -0.01 1 0.020 -0.01 3 0.028 -0.003 0.010 0.023 -0.005 0.001 0.032 -0.028 0.01 5 -0.01 6 0.0 -0.029 -0.001 0.002 0.02s 0.021 0.002 0.0 0.01 1 -0.01 6 0.017 0.018 0.024 0.0 -0.032 0.0 0.029 0.024 -0.001 -0.030 0.01 5 0.031 0.020 -0.025 -0.001 -0.017 0.002 -0.005 0.019 0.027 0.030 0.004 0.008 -0.001 -0.01 1 0.006 -0.025 0.030 0.028 0.004 0.014 - 0 .O26 --0.006 -0.01 2 -0.012 0.01 3 0.030 0.001 -0.005 -0.023 0.007 -0.030 0.01 3 -0.020 -0.006 0.024 -0.010 0.01 8 -0.034 0.008 -0.005 0.030 0.018 --0.007 0.001 0.008 -0.015 0.020 0.017 0.0 0.01 7 0.029 -0.006 0.021 -0.029 -0.006 - 0.023 - 0.01 5 -0.025 0.016 -0.008 -0.018 -0.009 0.003 --0.002 -0.032 -0.028 0.017 -0.012 -0.004 -0.012 0.026 -0.024 0.006 -0.010 0.004 -0.003 -0.015 0.016 -0.006 0.026 0.017 0.011 0.019 0.019 0.016 0.003 -0.002 0.026 0.032 -0.009 -0.008 -0.009 0.002 -0.030 0.013 0.002 -0.007 0.013 -0.002 0.006 -0.001 0.0 0.019 -0.030 0.006 0.017 States are indexed in order of increasing energy.124 SUPERPOSITION STATE PREPARATION to the rectangle and to the stadium states.For example, severalf,,, in both the rectangle and stadium cases are negligible compared to adjacent levels. Secondly, the fon values depend, for a given excited state, on the location of the ground-state circle. Thus an insensitivity to circle location in the CS cases is not observed in this detailed examination, via fon, of the excited-state wavefunctions. An immediate consequence of these results is that high-resolution spectroscopy measurements of the intensity distribution amongst neighbouring states, proportional to lfOnI2, will not display obvious qualitative features distinguishing excited states of the rectangle type from excited states of the stadium type. It is unnecessary, perhaps, to emphasize that such eigenstates are visually quite distinct, in displaying recognizable nodal patterns for the rectangles and erratic nodal patterns for the stadium.4.2. P(ly(v/) AND PSTo(v/lry) The survival probability P(t‘) = I(y/(O)ly/(t’))[’ has been used 430~12 as one measure of quantum dephasing. The behaviour of this function, quantitatively examined later below, is qualitatively similar, in all cases, to that seen elswhere.12 That is, P(t’) shows an initial falloff to values close to zero followed by recurrences of varying sizes. Studies of times up to t x 10hp show no recurrences prior to hp, major recurrences for the degenerate rectangle case at t x hp and prominent recurrences in other cases at t z 2hp. The quantity P(y/Iy/), the long-time average of P(t’), plays a major role in a proposed 20*22 means of quantifying quantum stochasticity .In particular, one com- pares P(y/ly/) with PgTO(y/yl y / ) , the latter corresponding to a statistical average which incorporates knowledge of system dynamics only up to time T. Specifically, = 2 (pnv)2 = 2 ( 1 ~ ~ 1 ~ ) ~ ( 1 la) T+w n n Application of eqn (1 lb) with Tchosen as the minimum of the initial P(t‘) falloff, which differs for each case, is discussed below. Adopting this choice ensures that knowledge about the energy envelope of y/ in each of the systems is restricted to the broadest envelope. Additional calculations, with T z hp 30 x lo3 a.u., i.e. a maximum time in Kay’s approach,1° were also carried out. For this T knowledge of the energy envelope is still confined to relatively broad features.In all cases t ‘ = 0 is defined at t = 22 + t,, i.e. where the sizable c,(t) are no longer changing, to within 0.1%. Two comparisons are possible, first of piTo*v with p r , fluctuations of the latter about the former being attributed 2o to non-stochasticity, and secondly of P(y/l v / ) with PtTo(v/Ily). Here P(y/Iy/) 3 PtTo(y/Iy/), deviations of P(y/Iy/) from P$To(y/Iy/) providing a suggested measure of non-stochasticity.20 Fig. 2 displays typical p , and pzTo for a stadium case (CIS) and a non-degenerate rectangle case ( C l R ) with 2 = 8309.5 a.u. and T set at the initial P(t’) minimum for each case. These figures, typical of those obtained, show no substantial qualitative difference between the two cases. Table 2 provides a comparison of P(yIy) and PSTo(y/Iy/) for three C R and CS cases for two different values of z.Examination of the table shows most states to be classified as highly non-stochastic using this criterion, the most nan-stochastic being one of the C2S cases. Although there is some indication that, on the average, CS cases show P(yIy) closer to PSTTo(y/1y/) than do the C R cases, this feature does not persist in comparisons with Tchosen as 30 000 a.u. (not shown) in which the opposite is the case. In light of these results no effort was made to extend the derivation of PsTo to cases involving degeneracies in order to examine CRD cases.R. D. TAYLOR AND P. BRUMER - - lbl - - - (10 O O h o - 0 (1 00 0 0 b O O 0 I 125 0.03 Pn 0.02 0.01 0 0.03 0.0 2 P n 0.01 0 E/ 1 0 - 2 a. u. Fig. 2. Ic,,12 (sticks) and p:To (circles) for (a) C1 S and (b) C2R.Note that the overall gaussian envelope is due to the laser profile. Circles clearly identify all energy eigenvalues. These results indicate that the direct cxarnination of Ic,12, or offon contained therc- in, do not identify strong qualitative differences in the states created in excitation to the regular rectangle wavefunctions or excitation to the irregular stadium wavefunctions. Explicit examination of Iv/(t)12 and of P(t') do, however, reveal differences, as described below. Table 2. P ( I , Y ~ ~ ) as compared with Psro(yilly) c1s c2s c3s ClR C2R C3R c1 s c2s c 3 s C1R C2R C3R z = 8309.5 a.u. 19.0 0.0636 15.5 0.049 1 20.0 0.091 1 13.5 0.0982 14.0 0.1597 23 .O 0.1756 z = 11 000a.u. 22.5 0.1685 21 .o 0.1238 26.5 0.1990 16.5 0.1949 22.0 0.5062 24.5 0.4030 0.1062 0.1062 0.1478 0.1967 0,3352 0.3530 0.2201 0.1975 0.2927 0.3510 0.8472 0.8153 66.9 116.3 62.2 100,3 109.8 101 .o 30.6 59.5 47.1 80.1 67.4 102.3 Dev = lOO(P-- PsTo)/PsTo, measuring thc percent deviation of P from PSTo.126 SUPERPOSITION STATE PREPARATION 4.3.THE SURVIVAL PRORABILITY P(t’) The quantity P(t’) was caIcuIated for each of the cases described above. Here, once again, t ’ = 0 corresponds to time t = 22 -1- t , when the laser pulse is no longer effective in changing state populations. We focus on the details of the initial de- phasing. Perturbation theory suggests, with A a cumdative constant, that We recailpi2 for comparison purposes? properties of P(t’) for the case where Ic,J2 are purely gaussian, i.e.where ]fnOl2 is constant in eqn (12). Under these circumstances P(t‘) = e--1‘21T2 for t’ < hp with a dephasing time t l j e = z [i.e. P(t‘ = tlJ = I/e]. This behaviour would result if the coefficients in the created state simply mimicked the laser frequency profile. Consider then the observed dephasing times, denoted ?l/e(calc,), shown in table 3. The results demonstrate that: (a) for all degenerate Table 3. Calculated dephasing times case tl,,(CdIC.) Dev‘ c1s c2s c3s ClRD C2RD C3RD CIR C2R C3R c1s c2s c3s Cf RD C2RD C3RD C1R C2R C3R T = 8309.5a.u. 8676 7860 8500 6000 5909 6992 6967 8783 8814 z = I 1 000a.u. 11 292 9784 10 546 7096 6985 8489 8865 b 11 384 4.4 -5.4 2.3 -27 - 29 -16 -16 16 6.1 2.6 -11.0 -4.1 -35 -36 - 23 -19 - 3.5 Dev = 100 [rll,(calc) - TIIT; all times in atomic units.’Initial dephasing faIloff not well defined due to interference. rectangle cases t,,,(caIc) deviates substantially from 2 ; (b) for all but one of the stadium cases t,,,(calc) = z to within 6%; ( c ) the nondegenerate rectangle case shows be- haviour characteristic of both the stadium and degenerate rectangle cases. There is, then, a clear propensity for the stadium P ( t r ) to behave, in its initial faIloff, as if If,,]” were constant; i.e. P(t ‘) adopts the falloff imposed by the laser frequency profile. This behaviour is shown more clearly in fig. 3 where the short-time falloff in several representative cases is compared with e--f”lT2, An analysis of the cosine transformR. D. TAYLOR AND P. BRUMER 127 0 5 10 15 t’/103 a.u.20 Fig. 3. Initial P(t’) falloff for C1S ( x ), C1R (0), ClRD ( A ) and C3R (0) with z = 11 000. Solid line is exp(-t’2/t2) with z = 11 000 a.u. P(co) of P(t‘) provides information complementary to that seen in the time represent- ation. That is, t,,,(calc) < z arises from contributions to P(m) at large frequencies, which exceed contributions expected if P(t ’) displayed a purely gaussian falloff. 4.4. THE PROBABILITY DENSITY I v/(t)12 The above techniques provide projected views of the created state and its evolution, best studied either by modelling realistic measurements or by examining 1 ty(t)12. Consideration of the latter does indeed display qualitative differences between states prepared in the rectangle case or the stadium case. Space limitations prevent but the briefest display of these differences, to be discussed in detail e1~ewhere.l~ Fig.4(a) and (b) provide a comparison, typical of those obtained, of the created state Ity(t = t, + 2z)I2 in coordinate space for CIS and ClR, z = 11 000 a.u. Several features are apparent. First, the density is spread, in both cases, throughout the allowed region, no enhanced density being observed in the region of the ground-state circle. This is hardly surprising since effective cancellation in the excited super- position state, to create a localized excited state, can only occur with a sufficiently large number of states. Secondly, the excited-state stadium wavefunction shows randomly distributed regions of slightly heavier probability whereas the excited-state rectangle wavefunction shows a heavy concentration of density in the upper half of the rectangle.Some recognizable aspects of the nodal character of the individually contributing eigenstates are also evident. Fig. 4(c) and (d), showing the same wavefunctions at a time approximately z later, display the principal observed qualitative distinction between excitation to regular as compared with irregular states. That is, the heavy probability in the rectangle moves together, in a rather regular way. In sharp con- trast, density in the stadium simply “ shimmers,” with local regions of higher proba- bility density moving randomly about. We further note that although the motion of128 SUPERPOSITION STATE PREPARATION 1 I I I I I 1 1 1 1 -3 c 0 .- I 0 * c - IR.D. TAYLOR AND P. BRUMER 129 Iv/(t)12 in the rectangle sometimes appears to display a pseudo-period, the P(t’) behaviour in such cases still indicates that v/(t) differs extensively from ~ ( 0 ) . This distinctive difference between excitation to regular and irregular eigenstates should be manifest in, for example, time-resolved emission to relatively localized states. Calculations to examine this feature are in progress.14 5. DISCUSSION It is well known that different dynamical behaviour can be manifest in the same system. For example, distributions display 23 finite time relaxation in classical regular systems and isolated quasiperiodic trajectories exist in classical mixing systems. Thus, in addition to knowledge of the formal system properties, knowledge of the nature of the prepared state is vital. This study has been designed to examine the relationship between system properties and the created state in a specific model experiment.The results make clear that qualitatively different types of states are created in the simple models examined, but that this difference is not readily reflected in several of the quantities examined. We have noted a propensity in the stadium system for P(t’) to assume dephasing characteristics of the imposed laser field, in contrast to the degener- ate rectangle system. This property can be shared, however, by the CR cases. Furthermore, we anticipate that time-dependent emission studies to localized states should be able to distinguish CR type from CS type cases. There are several issues central to current studies of quantum intramolecular dynamics : (a) establishing the quantum behaviour of molecular systems, (b) estab- lishing a formal quantum ergodic theory with ideal systems of some molecular utility and (c) understanding the classical-quantum correspondence in the irregular regime.The aim of this paper has been to contribute to the first two of these topics through a model with ideal classical properties and reasonably well characterized quantum states. The import of our results to realistic molecule systems [(a) above] depends on the extent to which the observed features carry over to real molecules. Barring the selected potentials, some concern might be expressed about the small number of eigenstates contributing to ~ ( 0 ) . In this regard we believe that this regime is, in fact, a useful one for experimental studies of small molecules.Here the number of states is sufficient to observe dephasing and experimental requirements are modest in demanding longer pulses as the density of states increases (e.g. z z 5 ns for p z lo4/ cm-I). Dephasing occurs, as is evident from the calculation, on timescales similar to z, or longer if they,,, distribution is narrow. The relevance of these results for a useful formal quantum ergodic theory is also clear. Any such theory, if it is to follow the spirit of classical ergodic theories, must identify a system as regular, ergodic, etc. Features, such as those reported in this paper, of the prepared states in model systems provide necessary input into such studies.Financial support, provided by N.S.E.R.C. and the Dreyfus Foundation, is gratefully acknowledged. We acknowledge extensive interaction with Prof. M. Shapiro and helpful conversations with Prof. K. Kay and thank Profs. Kaufman and McDonald for providing the stadium eigenfunction program. L. S. Kassel, Homogeneous Gas Reactions (Chem. Cat. Co, New York, 1932). For reviews see M. V. Berry, in Topics in Nonlinear Dynamics, ed. S. Jorna (American Institute130 SUPERPOSITION STATE PREPARATION of Physics, New York, 1978); P. Brumer, A h . Chem. Phys., 1981, 47, 201; M. Tabor, A h . Chem. Phys., 1981,00,000. G. Contopoulos, L. Galgani and A. Giorgilli, Phys. Reu. A, 1978, 18, 1183. E. J. Heller, in PotentiaZ Energy Surfaces and Dynamics Calculafions, ed. D. G . Truhlar (Plenum Press, New Yurk, 1981); J. S. Hutchinsan and R. E. Wyatt, Phys. Rev., 1981, A S , 1567. M. V. Berry and M. Tabor, Puoc. Roy. Soc. London A , 1477, 356, 375; D. W. Nuid, M. L. Koszykowsky and R. A. Marcus, Chem. Phys. Lett., 1980, 73, 269. M. V. Berry, Philos. Trans. R. Soc. London, Ser. A , 1977,287,237; J. S . Hutchinson arid R. E. Wyatt, G e m . Phys. Letr., 1980,72, 384. Utilizing the more general density matrix approach would allow for mixed states which include more averaging than desirabIe for studying pure quantum intramolecular relaxation. ’ The possibility that statistical behaviour implies less sensitivity to details of the creation step is not excluded. lo K. G. Kay, J. Chem. Phys., 1980,72, 5955. l1 See, e.g., J. Jortner and R. D. Levine, Adv. Chem. Phys., 1981, 47, 1 ; E. J. Heller and W. M. l2 f. Brumer and M. Shapiro, Chem. Phys. Left., 1980,72, 528; 1982,90,481. l3 P. Brumer, work in progress. l4 R. D. Taylor and P. Brumer, work in progress ; D. Gruner and P. 3rumer, work in progress. l5 S. Nordholm and S. A. Rice, J . Chem. P h y ~ . , 1975, 61, 203. S. W. McDonald and A. N. Kaufman, Phys. Rev. Lett., 1979,42, 1189. l7 K. F. Freed, C h m . Phys. Lert., 1976,42,600. l8 The coherent transform limited pulse is, of course, an experimental ideal. l9 W. Rhodes, in Radiationless Transiriom, ed. S. G. Lin (Academic Press, New York, 1980). 2* E. J. HelIer, J. Chem. Phys,, 1980, 72, 1337. 21 W-K. Liu and D. M. Noid, Chem. Phys. Lett., 1980,74, 152. 22 A criterion similar in appearance to this equation has been proposed by Kay.” However, as See K. G. Kay, Towards a ’ H. J. Korsch and M. V. Berry, Physica, 1981, 30, 627. Gelbart, J. Chern. Phys., 1980, 73, 626. formulated, it is not expected to be applicable to pure states. Cumprehensiue Semiclassical Ergodic Theory (pr e pr in t ) , 23 C. Jaffe and P. Brumer, to be published.