1 Introduction

Peculiar quantum effects of wave packet motion in anharmonic potentials have been predicted in several model systems [1, 2, 3]. We are going to investigate here the role of anharmonicity in the realistic case of a vibrating diatomic molecule having a finite number of bound eigenstates together with a dissociation continuum. Revivals and the formation of Schrödinger-cat states demonstrating the genuinely quantum nature of the problem will be visualized here with showing both the wave packet motion and the time evolution of the corresponding Wigner function.

Our description of molecular vibrations is based on the Morse Hamiltonian [4], that can be written in the following dimensionless form

where the shape parameter, s, is related to the dissociation energy D, the reduced mass of the molecule m, and the range parameter of the potential α via $s=\frac{\sqrt{2\mathit{mD}}}{\mathit{ħ\alpha }}-\frac{1}{2}.$ The dimensionless operator X in Eq. (1) corresponds to the displacement of the center of mass of the diatomic system from the equilibrium position.

The Hamiltonian (1) has [s] + 1 normalizable eigenstates (bound states), plus the continuous energy spectrum with positive energies. The wave functions of the bound eigenstates of H are ${\psi }_n\left(y\right)=\sqrt{\frac{\left[n!\left(2s-2n\right)\right]}{[\left(2s-n\right)!]}}{y}^{s-n}{e}^{-\frac{y}{2}}{L}_n^{2s-2n}\left(y\right),$ where y = (2s +1)e ^-x is the rescaled position variable, and $L_n^{2s-1}$ (y) is a generalized Laguerre polynomial. The corresponding eigenvalues are E_m (s) = -(s - m)², m = 0,1,… [s], where [s] denotes the largest integer that is smaller than s.

In the following we solve the Schrödinger equation

where time is measured in units of T = 2π/ω ₀, with ${\omega }_0=\alpha \sqrt{\frac{2D}{m}}$ being the circular frequency of the small oscillations in the potential. We note that spontaneous emission is negligibly weak at the frequencies related to energy eigenvalues E_m .

The initial states of our analysis will be Morse coherent states [5] associated with the wave functions

We expand these states in terms of a suitable finite basis:

where ₂F₁ is the hypergeometric function of the variable 1 - β. The first [s]+1 elements of the basis {|Φ_n 〉${\}}_{n=0}^N$ are the bound states, and the continuous part of the spectrum is represented by a set of orthonormal states which give zero overlap with the bound states. The energies of the states |Φ_n 〉, n > [s] follow densely each other, approximating satisfactorily the continuous energy spectrum. The details of the construction of our basis can be found in [6].

We note that the states in Eq. (4) are “single mode” coherent states in contrast to those of [7], where the dynamics of two-mode coherent states were investigated for various symmetry groups, including SU(1,1), which is in a close relation to the relevant symmetry group of the Morse potential [8].

The label β in Eq. (4) is in one to one correspondence with the expectation values

therefore we can use the notation |x ₀,p ₀〉 for the state |β〉 that gives 〈X〉 = x ₀ and 〈P〉 = p ₀. The localized wave packet corresponding to |x ₀,p ₀〉 is centered at x ₀ (p ₀) in the coordinate (momentum) representation.

 

Fig. 1. [2.1 MB] The absolute square of the wave functions corresponding to the Morse coherent states |x ₀,0〉, with x ₀ = 0.0 (ground state) and x ₀ = 0.5. The time evolution of the latter initial wave function is shown in the attached movie file. These plots correspond to the case of the NO molecule, where s = 54.54.

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In our calculation we have chosen the NO molecule as our model, where m = 7.46 a.u., D = 6.497 eV and α = 27.68 nm^-1 [4], yielding s = 54.54. That is, this molecule has 55 bound states, and we found that a basis of dimension N + 1 = 150 is sufficiently large to handle the problem. The absolute square of the wave functions |〈x|0, 0〉|² and |〈x|0.5, 0〉|² is depicted in Fig. 1, where V(x) is also shown. Fig. 1 indicates that initial displacements, x ₀, having the order of magnitude of unity will not lead to “small oscillations”.

The Morse coherent states [5] can be prepared by an appropriate electromagnetic pulse that drives the vibrational state of the molecule starting from the ground state into an approximate coherent state. An example can be found in [9], where the effect of an external sinusoidal field is considered.

 

Fig. 2. The expectation value of the dimensionless position operator as a function of time. The initial states were |ϕ(t = 0)〉 = |x ₀,0〉, with x ₀ = 1.0, x ₀ = 0.5 and x ₀ = 0.06.

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2 Behavior of expectation values as a function of time

Starting from |ϕ(t = 0)〉 = |x ₀,0〉 as initial states, first we consider the dependence of the 〈X〉(t) curve on x ₀. It is not surprising that for small values of x ₀ (≤ 0.06) these curves show similar oscillatory behavior as in the case of the harmonic oscillator, see Fig. 2. However, when anharmonic effects become important, a different phenomenon can be observed: the amplitude of the oscillations decreases almost to zero, then faster oscillations with small amplitude appear but later we re-obtain almost exactly 〈X〉(0), and the whole process starts again. Fig. 2 is similar to the collapse and revival in the Jaynes-Cumings (JC) model [10, 11], but in our case the non-equidistant spectrum of the Morse Hamiltonian is responsible for the effect. There are important situations when revivals and fractional revivals [1, 12, 13, 14] of the wave packet can be described analytically [2], but in a realistic model for a diatomic molecule the difficulties introduced by the presence of the continuous spectrum implies choosing an appropriate numerical solution. An experimental method that allows for the observation of molecular wave packet motion was described in [15].

The expansion of the initial state in our finite basis |x ₀, 0〉 = ∑ _n c_n (x ₀)|Φ_n 〉 shows that for values of x ₀ shown in Fig. 2 the maximal |c_n (x ₀)| belongs to n < [s]. That is, the expectation value

is dominated by the bound part of the spectrum. Damping of the amplitude of the oscillations is due to the destructive interference between the various Bohr frequencies and we observe revival when the exponential terms rephase again.

Quantitatively, we have determined the dominant frequencies in Eq. (6) for x ₀ = 0.5 and found that they fall into two families. The first family is related to the matrix elements 〈Φ_n |X|Φ_n+1 〉 and a has a sharp distribution around ω ₁ = 0.9ω ₀. The contribution of the second family to the sum in Eq. (2) is much weaker, these frequencies around ω ₁ = 1.81ω ₀ correspond to the matrix elements 〈Φ_n |X|_Φn+2〉. The width the first distribution Δω ₁ = 0.1ω ₀ allows us to estimate the revival time t_r ≈ 2π/Δω ₁ = 62.8T, while Δω ₂ = 0.17ω ₀ is responsible for the partial revival at t/T ≈ 30, see Fig 2. Following Refs. [1, 12], we denote by T_rev the time when the anharmonic terms in the spectrum induce no phase shifts, that is, the initial wave packet is reconstructed. At t_r = T_rev /2 all these phase factors are -1, while t/T = 30 corresponds to a quarter-revival, i.e., to time T_rev /4.

3 Time evolution of wave function and Wigner function of the system

In order to gain more insight concerning the physical process leading to the collapse and revival seen in Fig. 2, one can look at the coordinate representation of the wave function ϕ(x,t) = 〈x|ϕ(t)〉. The movie file associated with Fig. 1 visualizes the time evolution of the initial state |ϕ(t = 0)〉 = |x ₀,0〉 with x ₀ = 0.5. In this representative case ϕ(x, t) is an initially well localized wave packet that gradually falls apart into several packets and (not shown in the related multimedia file) then conglomerates again. The last frame of this movie represents a a superposition of two spatially separated wave packets, i.e. a Schrödinger-cat state.

Starting from the same initial state it is instructive to compare the time evolution of ϕ(x,t) to that of the Wigner function W(x,p,t) that reflects the state of the system in the phase space [16] and defined as

 

Fig. 3. A) [1.3 MB] and B) [2.2 MB]. Frames of two movie files, showing Wigner functions of the Morse system at the initial stage of the time evolution and the formation of a Schrödinger-cat state. The plots correspond to t/T = 0 and t/T = 30, respectively. The initial state was |ϕ(t = 0)〉 = |x ₀,0), with x ₀ = 0.5.

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Fig. 3 A) shows the initial stage of the time evolution, while Fig. 3 B) corresponds to t/T = 30. This second Wigner function is typical for Schrödinger-cat states [17]. W(x,p) in Fig. 3 B) corresponds to a superposition of two states that are well-localized in both momentum and coordinate, and represented by the two positive hills centered at x ₁ = -0.1, p ₁ = -18.0 and x ₂ = 0.3, p ₂ = 12.0. The strong oscillations between them shows the quantum interference of these states.

According to the movie file, there are a few periods around t/T = 30, while the state of the system can be considered to be a phase space Schrödinger-cat state. During this time the Wigner function is similar to the one shown in Fig. 3 B), and it rotates around the equilibrium position. Similar behavior of the Wigner function was found in [18] for the JC model. This effect is responsible for the partial revival around t/T = 30 shown in Fig. 2, where the frequency of the oscillations is twice that of the oscillations around t = 0: in the neighborhood of t/T = 30 there are two wave packets moving approximately the same way as the coherent state soon after t = 0.

4 Measuring nonclassicality

An appropriate measure of the nonclassicality 0 ≤ M_nc < 1 of a state |ϕ〉 can be defined [19] by the aid of the corresponding Wigner function

where I ₊(ϕ) and I _-(ϕ) are the moduli of the integrals of W(x,p) over those domains of the phase space where it is positive and negative, respectively. Fig. 4 shows M_nc as a function of time for the same initial states as in Fig. 2. For the small initial displacement of x ₀ = 0.06 we see that the Wigner function is positive almost everywhere, the state can be considered as a classical one during the whole time evolution.

 

Fig. 4. Nonclassicality as a function of time. The initial state was |ϕ(t = 0)〉 = |x ₀, 0〉, with x ₀ = 1.0, x ₀ = 0.5 and x ₀ = 0.06.

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For larger initial displacements we can easily identify two time scales. The shorter one is the period of the wave packet in the potential, while the second time scale can be identified with the revival time. Looking at the initial part of the curve M_nc (t), we observe that the state of the system is the most classical at those turning points where 〈X〉 > 0, see Fig. 1. On the other time scale, the collapse of the oscillations in 〈X〉 presents itself as the increase of M_nc and the revival turns the state into a more classical one. When the state of the system can be considered as a Schrödinger-cat state, M_nc (t) has a small local minimum, but it still has significant values indicating nonclassicality.

5 Conclusion

We have found that in the potential of the NO molecule, when anharmonic effects are important, the time evolution naturally leads to the formation of Schrödinger-cat states at certain stages of the time evolution. These highly nonclassical states correspond to the superposition of two molecular states which are well localized in the phase space.

This work was supported by the Hungarian Scientific Research Fund (OTKA) under contracts Nos. T32920, D38267, and by the Hungarian Ministry of Education under contract No. FKFP 099/2001.