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Weisskopf-Wigner model for wave packet excitation

We consider a laser induced molecular excitation process as a decay of a single energy state into a continuum. The analytic results based on Weisskopf-Wigner approach and perturbation calculations are compared with numerical wave packet results. We find that the decay model describes the excitation process well within the expected parameter region.


Introduction
It has been known for a long time that when an isolated bound state is interacting with a continuum of quantum states, the occupation of the bound state experiences exponential decay; the first proper treatment was given by Weisskopf and Wigner in 1930 [1]. A simple example of such a situation is a two-state atom coupled to the electromagnetic modes of the vacuum, which leads to spontaneous emission; see e.g. refs. [2,3]. The general model has been described in detail within the formalism of scattering theory in ref. [4], but this approach provides only a steady state treatment. In a recent paper [5] we considered this problem in a genuine time dependent setting and discussed the decay process as a method to prepare a moving wave packet on a molecular electronic state in accordance with our earlier treatment of coupled channel molecular wave packet dynamics; for a review see ref. [6]. Now we proceed to study the validity of this exponential decay description of laser induced wave packet excitation in molecules.
The prototype model for the Weisskopf-Wigner decay is shown in fig. 1(a); the discrete state embedded in a continuum acquires a Lorentzian line shape, which corresponds to exponential decay of its occupation. The theoretical description leading to exponential decay is based on an approximation which represents the continuum spectral density by a single dominating pole [7,8]. In the general case this is only an approximation, which is expected to hold in the perturbative limit of weak coupling between the discrete state and the continuum, in which case the result of time dependent perturbation theory agrees with the Weisskopf-Wigner result. When we go beyond this limit, the result is nonexponential decay [4] or a continuous transition of the decay into a periodic oscillational behaviour of the Rabi type as discussed in ref. [9]. Thus we need to establish for molecular excitation the parameter regions where a) the perturbation approach is applicable, and b) where the excitation process can be described as exponential decay into the continuum. Figure 1: (a) An isolated energy level at E 0 is coupled to a system with a continuous energy spectrum E. (b) In the molecular excitation model the isolated state is the ground state of the molecule, U 1 , and the continuum is provided by the linear excited state potential U 2 .
In this paper we develop further the decay model introduced in ref. [5] and featured in fig. 1(b). The lowest vibrational level of the molecular electronic ground state corresponds to the discrete state; it can be described adequately by the harmonic potential U 1 . The molecular counterpart of the continuum is an excited electronic state which can be adequately described by the linear potential U 2 near the molecular equilibrium position. Usually state 2 is a dissociating one, but it could also represent a bound state in a energy region where the vibrational states have a very small separation (quasicontinuum). These potentials are a good basis for the molecular processes within the Born-Oppenheimer approximation. If we also apply the dipole and rotating wave approximations to the laser-molecule interaction, we can shift the excited state potential down by one laser pho-ton, which then leads to the elimination of the rapidly oscillating field term from the coupling between the states 1 and 2; for more details see ref. [6].
If the parameters of our model are chosen such that the vibrational excited states of the harmonic potential remain mostly unoccupied during the interaction, the decay of the oscillator ground state can be taken to be a model of the level scheme in fig. 1. The couplings between the oscillator state and the eigenstates of the slope are not constant; the Condon factors describing the overlap between the ground and excited state wave functions have an energy dependence which complicates the model. Thus for simplicity (one parameter less to consider) we arrange the sloping potential U 2 so that a Franck-Condon transition at the origin of the coordinate system gives the maximum overlap between the wave functions. Then we expect the Condon factor to depend only slowly on energy around E 0 and the model will simulate the behaviour of the Weisskopf-Wigner situation.
This paper is organised as follows. Section 2 presents the basic model. In sec. 3 we give the results of our numerical computations and discuss the occurrence of exponential decay. It is found that this can be observed in a parameter range where the decay rate is less than a definite value. This indicates that the exponential behaviour can indeed be achieved only in a perturbative regime as expected. We also find that the occupation of the state 2 emerges in the form of a localized wave packet even beyond this perturbative parameter regime. In sec. 4 we discuss the analytic result for the decay and compare this with our results obtained from numerical integration of our wave packet model. Finally we summarize and discuss our results in sec. 5.

The model
We consider a one dimensional system, spatial coordinate x, with two internal states {|1 , |2 }. The Hamiltonian of the system is written as where the coupling V is chosen real. We define (in suitably chosen dimensionless units, cf. refs. [5,6]) The Hamiltonian (1) describes our chosen model as shown in Fig. 1(b). Since in our scaling the energy of the lowest level of the harmonic potential is the energy of the linear potential resonates with the ground state at x = 0, see fig. 1. The eigenfunction of the ground state of the harmonic oscillator is with the energy eigenvalue (3). In the domain we are investigating, the higher eigenfunctions are not supposed to play any role, and they are consequently not needed, as already mentioned in sec. 1.
The eigenvalues of the linear state U 2 form a continuum {−∞, ∞} with the corresponding eigenfunctions easily obtained by a Fourier transform The functional form is that of the Airy function, and a suitable normalization is given by where the energy parameter is It is easy to prove the energy normalization and the completeness With these relations we see that the density of states is unity. We assume that the system is prepared in the initial state which is then coupled to the continuum by the parameter V . Using the result of the time dependent perturbation theory (Fermi Golden rule) the leakage into state |2 , i.e. into our final state |f , is expected to occur at the rate where the continuum function is evaluated at the energy (3) and the parameter dependence is almost entirely in the Condon factor ϕ 0 |ψ E 0 . From the normalization of eq. (6) we expect the decay rate to scale as where we have defined an effective decay parameter Γ 0 . When Γ 0 is below some limiting value, we expect that eq. (11) gives a reasonable approximation to the decay rate of the initial state (10). The result is modified by the Condon factor which will be discussed below.

Numerical calculations
We have used the numerical approach described in sec. 2.3.3 of ref. [6] to propagate the initial state wave packet (10) on the coupled energy surfaces of eq. (1). However, instead of switching the coupling on suddenly we have used the coupling term where t 0 indicates in practice the beginning of the excitation process, and we have set T = 0.05 in our scaled units. We have especially looked into the possible occurrence of exponential decay and the emergence of an outgoing wave packet on state |2 in our model. Some typical results of our calculations are shown in figs. 2 and 3. They present the logarithm of the occupation P 1 of the state |1 as a function of time. Both figures show descending straight lines which indicate exponential decay. For very large values of Γ 0 the initial state occupation P 1 becomes clearly oscillatory as demonstrated in fig. 4.
There is no absolute criterion when exponential decay is observed; the occurrence of linear behaviour in the semilogarithmic plots is subject to an arbitrary criterion. Running various parameter combinations, however, indicates that monotonic exponential decay is observed for values of V 2 0 smaller than a constant times α, thus indicating that the parameter (12) plays a decisive role in the phenomenon. In fig. 5 we show the data points that correspond to exponential and nonexponential decay in the (α,V 2 0 ) plane. The exponential decay points fall into a region for which we have roughly which is thus taken to be the limit of validity of the simple pole approximation for decay. Typically the nonexponential behaviour appears in P 1 and P 2 as oscillations, and the exponential behaviour as a steady change; this is demonstrated in fig. 6.
We have also studied how the emerging probability on level |2 forms a wave packet. For the case V 0 = 0.7, α = 2 (Γ 0 ≃ 1.5) we observe a clear wave crest emerging in fig. 7. This is followed by a long, slowly diminishing tail representing the final leakage out of state |1 . We have, however, observed that the formation of a wave packet is not conditioned on the occurrence of exponential decay. In fig. 8 we still see a well developed wave packet for the parameters V 0 = 1.5 and α = 3 (Γ 0 ∼ 4.7) far exceeding that for which perturbation theory would be expected to work.
We have not carried out a systematic investigation of the occurrence of a well formed wave packet. This is as subjective a phenomenon as the exponential decay. Because we initially find no probability on state |2 and ultimately there is no longer any probability leaking out of state |1 , there must be a localized wave packet emerging out of the system. However, a few words connecting the results of this paper to our previous work [10,11,12,13] are appropriate here.
In both figures 7 and 8 we see a clear, peaked wave crest wave packet. On the other hand, figures 2 and 3 show that for all parameter values the ground state occupation behaviour is not exponential when t ≃ t 0 . This is because initially the excitation process follows the well known area theorem, which for two resonant levels and a coupling V 0 states that for t ≥ t 0 . This is the origin of Rabi oscillations (if the levels are off resonance, then the oscillation frequency and amplitude are modified). For more details see e.g. [6]. These oscillations simply indicate that both states are "equal", i.e., transitions can take place also back from state |2 to state |1 . In the Weisskopf-Wigner regime the second state acts like a reservoir and population can move only from state |1 to state |2 . This is the point where the dynamics of the wave packet on state |2 comes to the rescue. In our earlier studies we have given special attention to cases where the dynamics inhibits the Rabi oscillations, because the excited state population is accelerated away from the resonance region faster than the Rabi oscillations take place [10,11,12,13]. Although our earlier studies were made for short pulses, the basic principle is the same. Thus the reservoir model works best for the excitation when α is large and wave packet motion on state |2 takes place swiftly after excitation.
We have now established a loose criterion for obtaining exponential decay. Next we need to consider the validity of the perturbation result (11). The exponential behaviour is only a necessary but not a sufficient condition for eq. (11) to hold. For suitably large values of V 0 the perturbation result may fail even if the criterion (14) is fulfilled and the decay is exponential. For this purpose we calculate the full perturbation result for our

Analytic considerations
The perturbative rate (11) contains the Condon factor which is the overlap of the initial and final states; see e.g. refs. [14,15,16,17]. This scalar product is most efficiently calculated in momentum space, where we need the Fourier transform of the state (4). Calculating the continuum wave function ψ E 0 for the energy (3) we set E = 0, which gives where we have used the substitution k = iα/ √ 2 − u. Using the Airy function we can write (after setting u = α 1/3 s in eq. (16)) The result (18) seems to contradict the simple dependence on α from (16), but the asymptotic relation Ai(x) ≃ x −1/4 exp(−2x 3/2 /3)/ √ 4π, valid for large x, restores the correct scaling with α in the perturbative limit. In fact, the asymptotic result for the Franck-Condon factor becomes lim This is exactly what the Condon reflection principle would predict [14,16,18]. According to it where x 0 is the point where U 2 (x 0 ) = U 1 (x 0 ); in our case x 0 approaches zero as α → ∞, and our ground state wave function contribution becomes |ϕ 0 (x = 0)| 2 = (2π 2 ) −1/4 . The reflection principle simply means that the excited state wave functions for steep potentials are so localised to x = x 0 that they behave like δ functions inside the overlap integrals. Thus we get that The interesting conclusion from eq. (16) is that the decay rate is not the simple function of α suggested by eq. (12) except in the asymptotic limit. The behaviour of Γ(α), however, is found to be simply monotonic with increasing α as shown in the fig. 9.  Figure 7: The excited state wave packet P 2 (x, t) for V = 0.7 and α = 2. This is in the exponential decay regime since Γ 0 = 1.5. Here t 0 = 1.0.

Discussion
We have shown that in the perturbation limit and for steep excited state potentials the Weisskopf-Wigner model can describe adequately laser induced molecular excitation processes. Our numerical results are in good agreement with analytical perturbation theory results in the parameter region where we expect them to be applicable.
Although we have considered mainly the molecular excitation process, there are some interesting atomic systems that resemble the molecular model shown in fig. 1(b). One can trap single ions electromagnetically, and cool them down to the lowest motional state of the trap potential, which in this energy range corresponds to a harmonic potential [19,20,21,22]. Furthermore, one can also trap neutral atoms with magnetic fields, and cool  Figure 8: The excited state wave packet P 2 (x, t) for V 0 = 1.5 and α = 3. This is in the nonexponential decay regime since Γ 0 = 4.7.
them evaporatively into the density-temperature region where Bose-Einstein condensation takes place [23,24]. The behaviour of the condensed atoms can be described as a single wave function [25,26]. Our model, therefore, resembles the recent studies for output couplers for the condensates, in which rf fields couple the trapping state to a nontrapping (continuum) state-such a system can be regarded as a first generation atom laser [27,28]. Thus our model may provide some insights when one wishes to move from pulsed atom lasers towards cw atom lasers.  (21)]. All selected data points correspond to exponential decay.