1. Introduction

A standard model for exponential decay in quantum mechanics is based on the assumption of an isolated discrete energy level embedded in a continuous spectrum. This was taken as a prototype model by Fano¹, and its result agrees with the perturbative calculation of decay due to Weisskopf and Wigner². The model utilized by Fano is an atomic physics version of field theory models used to describe the thresholds of particle production^3,4,5. In the collision theory context this approach is well treated in Ref. 6.

The Weisskopf-Wigner calculation is the standard approach to spontaneous radiation emission in atomic and molecular physics. In addition, the theory of exponential decay has been used in a multitude of physical contexts including the theory of nonra-diative decay of molecular systems⁷. Presently, the possibility to selectively excite and continuously monitor selected molecular levels^8,9 renews the interest in intramolecular energy transfer. It is thus of interest to reconsider some models of molecular potential configurations and establish the validity of exponential decay.

In our earlier paper¹⁰ we considered a model of an initially bound wave packet state coupled resonantly to a sloping potential simulating the continuum provided by a dissociating molecular state. We found that, starting from the ground state of the bound potential, the probability leaks out in an exponential fashion for weak coupling. The rate was found to agree with the perturbative result obtained from the Weisskopf-Wigner theory. This was taken to be a generic model for a system decaying slowly.

In modern laser experiment, it is, however, possible that the initial state resides on an excited electronic energy level. The vibrational degrees of freedom on this level are not necessarily in their ground state but may possess an initial energy of excitation. For sufficiently long monochromatic pulses, we may expect to excite mainly a single energy eigenstate, which is determined by the Franck-Condon factor. For short enough excitation pulses¹¹, the wave packet is lifted from its ground state essentially without distortion and ends up in an oscillating state. As the excited level usually does not have the same frequency as the ground state, the packet does not form a coherent state propagating without distortion, but is found to pulsate; it is a squeezed state of the oscillator.

In this paper we consider the case where an initial wave function is prepared on a bound excited state, and this is then coupled to a continuum by a second pulse. We limit our considerations here to the case of an initially excited eigenstate of the oscillator. This does not change with time, and hence the only evolution we need to follow is its slow leakage out into the continuum. We enquire whether such a state also can be found to decay exponentially, and does the rate agree with the Weisskopf-Wigner result in the perturbative limit?

In Sec. 2.1 we present the model, which consists of two electronic energy levels, one harmonic and one dissociating. Section 2.2 discusses the application of the Weisskopf-Wigner perturbative result to the decay. This is expected to be valid whenever exponential decay is observed. In Sec. 3.1 the time evolution of the model is solved numerically by wave packet propagation, and the regions of exponential decay are identified. Section 3.2 presents a comparison between the methods used in obtaining the decay, and in Sec. 4 we summarize our results.

2. The model

2.1 Energy levels of the model

As a model we choose a coupled pair of energy levels with one harmonic potential and one potential slope providing a continuum. The Schrödinger equation in scaled units is

where the excited state is a slope

and the potential of the initial state is harmonic:

This corresponds to a scaled molecular oscillation with the parameters

The energy eigenvalue of the nth eigenfunction is given by

If we can assume the oscilllator eigenstates to leak independently into the continuum, we may consider each initially excited oscillator eigenstate

to be embedded in the continuous spectrum of the slope; the coupling is given by V. In Fig. 1 we show the harmonic potential U ₁(x), our initial wave packets |φ_n (x)|² (normalised suitably) and the corresponding excited state potentials U ₂.

We need also the states of the sloping potential. The eigenvalues of the linear state 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

It is easy to prove the energy normalization

and the completeness

With these relations we see that the density of states is unity.

 

Fig. 1. The potentials (U ₁ and U ₂) and initial wave packets |φ_n (x)|² for our model. We show the harmonic ground state potential U ₁(x) (blue solid line), the corresponding energies of the eigenstates ε_n (dashed lines) and the squared eigenfunctions (solid lines) for n = 1 to n = 4. We show also the corresponding excited state potentials U ₂(x) (dotted lines) for each n when α = 1.

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2.2 Perturbative decay

When φ_n (x) is an energy eigenstate of the harmonic oscillator potential we apply a treatment similar to that discussed in our earlier paper¹⁰. In the perturbative regime

we expect the Weisskopf-Wigner treatment to hold. Then the probability of occurrence of the initial state

decays exponentially.

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 Fermi golden rule, we expect the leakage into state |2〉, i.e., into our final state |f〉, to occur at the rate

where the continuum function is evaluated at the resonance energy (5)

and the parameter dependence is almost entirely in the Franck-Condon factor

For a preselected initial energy ε_n , we choose the parameter β in (2) such that the initial and final states have the same energy at x = 0, i.e., β = ε_n ; see Fig. 1.

From the normalization of Eq. (8) we expect the decay rate to scale as

where we have defined an effective decay parameter Γ₀. When Γ₀ is below some limiting value, we expect that Eq. (14) gives a reasonable approximation to the decay rate of the initial state (13). The result (17) is modified with the Franck-Condon factor, which is the overlap of the initial and final states. This scalar product is most efficiently calculated in momentum space, where we need the Fourier transform of the harmonic oscillator eigenstates, which in the k-representation are given by

The normalisation factor is

The overlap integral now becomes

An important part of the comparison between the analytical calculations and the wave packet results is the evaluation of the Franck-Condon factors (20). Their values are shown as a function of the steepness of the slope α in Fig. 2. For large values of α, the factors tend to zero, but, in contrast to the ground state, they may display oscillatory behaviour and, in fact, go to zero for finite values of α. Also note that |S ₀|² is clearly larger than |S_n |² for all nonzero values of n.

When α goes to infinity, the eigenfunctions of the slope (8) approach a delta-function, which samples the oscillator eigenfunction at the position x = 0; this is the Condon reflection principle¹². Applied here, it implies that the factor would go to zero for all odd order eigenfunctions, but the limit is reached only slowly as Fig. 2 demonstrates.

 

Fig. 2. The perturbation theory result for the Franck-Condon factors, as defined in Eq. (23), and calculated numerically for the 6 lowest harmonic oscillator states as a function of the slope parameter α.

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In order to obtain a more accurate version of the reflection principle we adopt the following procedure. According to the Condon reflection principle ^12,13,14 we have

where x ₀ is the point where U ₂(x ₀) = U ₁(x ₀). We calculate this expression at the crossing point of the two potential curves, for the chosen value of β, and compare this with the results obtained from the wave packet calculations.

3. Wave packet considerations

3.1 Numerical calculations

We have used the numerical approach described in Sec. 2.3.2 of Ref. 9 to propagate the initial state wave packet (13) 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. We have set T = 0.05 and t ₀ = 1 in our scaled units. Thus for very small times around t ₀ we can expect behaviour related to the finite switch-time, but the exponential decay should quickly take the dominant role.

We have especially looked into the possible occurrence of exponential decay and the emergence of probability on state |2〉 in our model. Some typical results of our calculations are shown in Figs. 3 and 4. They present the occupation P ₁ of the state |1〉 as a function of time. Both figures show descending straight lines which indicate exponential decay. In Fig. 3 we look at the decay of the oscillator ground state n = 0. As we can see, exponential decay is observed for all values of α used, and the decay is faster for smaller values in accordance with the perturbation parameter (17). As V ₀ = 0.1, we find that perturbation theory ought to hold for all α ≪ 0.01, according to Eq. (11). For the same V ₀, we find in Fig. 4, that the state n = 3 requires α ≥ 3 to ensure exponential decay. For lower values the leakage displays oscillatory features reminiscent of the Rabi oscillations between the levels. As the decay is weak, one is tempted to suspect an effect from finite switch-time, but e.g. for α = 1 in Fig. 4 variations in T did not create any significant changes in P ₁(t).

 

Fig. 3. The decay of the vibrational ground state (n = 0) population P ₁ for various values of α. For small times exp[-Γ(t - t ₀)] ≃ 1 - Γ(t - t ₀) when t ≥ t ₀, and the exponential decay appears as a linear change in P ₁(t). The results are obtained using the wave packet approach. Here we have set V ₀ = 0.1.

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Fig. 4. As Fig. 3, but now the decaying state is the excited vibrational state n = 3.

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There is no absolute criterion when exponential decay is observed. Running various parameter combinations, however, indicates that monotonic exponential decay is observed for values of $V_0^2$ smaller than a constant times α, thus indicating that the parameter (17) plays a decisive role in the phenomenon. In the perturbative limit, the decay rate becomes second order in the coupling, and hence it is illuminating to plot the quantity (Γ/$V_0^2$), which is expected to become equal to the square of the Franck-Condon factor in the limit of large α,

Figures 5–7 show the emergence of the probability on the final continuum level by displaying the probability

as a function of time and interaction strength V ₀. In the limit of exponential decay, this is supposed to increase smoothly towards its final value of unity. In Fig. 5 we show the results of the oscillator ground state as discussed already in Ref. 10. Here α = 2, and for coupling strengths V ₀ < 1, we see a smooth growth, which goes over into Rabi oscillations for stronger coupling. For the excited state, n = 4 (in Fig. 6) we observe a very similar behavior, but the oscillations extend to smaller values of V ₀. The decay of the same state, n = 4, can clearly vary with α as Fig. 7 demonstrates. Here α = 3, which corresponds to S_n ≃ 0.

 

Fig. 5. The excited electronic state population P ₂ as a function of time t and coupling strength V ₀. Here n = 0 and α = 2. In the decay model for wave packet excitation this describes the occupation of the continuum modes.

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Fig. 6. The excited electronic state population P ₂ as a function of time t and coupling strength V ₀. Here n = 4 and α = 2.

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Fig. 7. The excited electronic state population P ₂ as a function of time t and coupling strength V ₀. Here n = 4 and α = 3.

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3.2 Comparison of results

In Figs. 8–12 we compare the results of our various calculations for the initial oscillator states with n = 0, 1, 2, 3 and 4. For the even states we show the results of the simple reflection principle evaluated at x = 0; for the odd states these results vanish exactly. In all cases we show the results from the modified reflection principle (see Sec. 2.2), the numerical results from the evaluation of the Franck-Condon overlap integral and the results of the wave packet integrations extracted from the linear slopes of the decay curves. For the cases where the overlap integral is small, it is difficult to extract this rate numerically with sufficient accuracy.

 

Fig. 8. The excited state decay rate Γ(α) for case n = 0, obtained a) using wave packet calculations, and the perturbation theory result with b) numerically evaluated overlap integral, c) modified Condon reflection principle and d) simple Condon reflection principle.

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Fig. 9. The excited state decay rate Γ(α) for case n = 1, obtained a) using wave packet calculations, and the perturbation theory result with b) numerically evaluated overlap integral and c) modified Condon reflection principle.

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In all cases, we find that the wave packet calculations tend to agree with the numerically evaluated overlap results for large enough values of α. Except for regions where the overlap integral is small and the decay extremely slow (see Fig. 12 near α = 3), the agreement appears already for α > 1.

When α is large, the simple and modified reflection principles give the asymptotic behaviour correctly for the even order states. In contrast, the modified reflection principle fails in the asymptotic limit for odd states (revealed if Figs. 9 and 11 are given a logarithmic y-axis scale). Thus we conclude that the reflection principle offers very little advantage in the present case in general, and none for eigenstates with odd symmetry.

 

Fig. 10. The excited state decay rate Γ(α) for case n = 2, obtained a) using wave packet calculations, and the perturbation theory result with b) numerically evaluated overlap integral, c) modified Condon reflection principle and d) simple Condon reflection principle.

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Fig. 11. The excited state decay rate Γ(α) for case n = 3, obtained a) using wave packet calculations, and the perturbation theory result with b) numerically evaluated overlap integral and c) modified Condon reflection principle.

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Fig. 12. The excited state decay rate Γ(α) for case n = 4, obtained a) using wave packet calculations, and the perturbation theory result with b) numerically evaluated overlap integral, c) modified Condon reflection principle and d) simple Condon reflection principle.

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4. Conclusions

We have demonstrated, using a simple pedagogical model, that in the weak interaction limit the excitation of vibrationally excited molecular states into a dissociating state (continuum) can be considered as exponential decay (Weisskopf-Wigner) of a single quantum state embedded in a continuum. However, for these excited states the decay is very sensitive to the respective values of the overlap integrals S_n , i.e., the Franck-Condon factors. If we look at Fig. 1, it is obvious that, especially in the limit of large α, the spatial forms of the excited state eigenfunctions play a strong role. In our oscillator model, the main contributions to the higher oscillator state eigenfunctions are located near the classical turning points. Our particular choice of energy resonance, on the other hand, samples the eigenstates near x = 0, where they have an increasingly oscillatory behavior and small amplitudes as the vibrational quantum number n increases. This explains the oscillatory behavior of S_n (α), and why in general the magnitude of the overlap in our particular model is much smaller for n ≠ 0, compared to the n = 0 situation (Fig. 2).

The situation described above explains also why the Condon reflection principle is not such a good approximation for n ≠ 0 in our model, especially when n is odd. The assumption that the continuum eigenstate, i.e., the Airy function becomes a δ-function at the spatial resonance point is not valid. The small residual oscillations in the Airy function tail sample the oscillator states better than the main peak of the Airy function. However, the validity of the Condon reflection principle concerns the actual calculation of the S_n , which is not really the main consideration of our study.

As Figs. 6–8 demonstrate, the region for exponential decay is limited to very small values of the coupling strength V ₀. Figures 9–13 show that for suitably small α the perturbative overlap integral approach fails. As discussed and demonstrated in Ref. 10, this is not yet a sign of the disappearance of the exponential decay. It merely implies that the Fermi Golden rule approach fails, which is not equal to the failure of the Weisskopf-Wigner description. For the case n = 0, the monotone behavior of the overlap integral S ₀ with α (and the validity of the Condon reflection principle) makes it simple to define a precise condition for the validity of the exponential decay description and Fermi golden rule approach. Unfortunately our study shows that the validity criteria given in Ref. 10 for the n = 0 can not be easily extended to all n. Instead, each n must be considered separately, and unfortunately the Condon reflection principle can not be relied on in the process.

It should be noted that, as discussed above, the general aspects in the behavior of the overlap integrals S_n in our model are partly induced by our choice of the energy resonance condition given in Eq. (15). If we add E ₀ as a changing parameter to our model, we could especially explore the behavior of the overlap integrals S_n in more detail. However, such investigations should be performed with more realistic potentials and in relation to the actual molecular dissociation studies. Our interest does not lie in exploring the dissociation process in detail, but in demonstrating the Weisskopf-Wigner model of decay in a special situation. We have shown that the picture of a quantum state embedded into a continuum is clearly complicated by the additional aspects such as the spatial structure of the eigenfunctions related to the discrete and continuum states. Thus by enlarging the range of values for E ₀ would move our study and discussion further away from our main aims. From this point of view, it is natural to choose such values for E ₀ that the continuum eigenfunction corresponds to a resonance with the discrete state. Then all the α and n dependence in the decay process are modifications imposed by the spatial aspect of the problem.

In this paper we have given a pedagogical demonstration how the standard Weisskopf-Wigner description is expected to be modified by the role of the spatial eigenfunctions in determining the coupling between the bound state and the continuum. The best realistic counterpart so far for our model is the laser-induced dissociation process for diatomic molecules, and we have performed our discussion in that framework. However, one can expect similar situations to arise in the output coupler schemes for magnetically trapped Bose-Einstein condensates or other trapping systems for ions and atoms, in addition to molecular processes. The harmonic and linear potentials used in our study are not quite realistic for molecular processes, but they may be much closer to reality for condensates, atoms and ions (harmonic trapping potentials, and linear potentials due to gravity).