In this paper we discuss the splitting instability in a periodically driven double well. The physical motivation of this study comes from the relevance of the concept of molecular structure in chemistry, but the model could be tested directly by means of heterostructures and microwaves. Let us recall the old problem of the explanation of the molecular localization ~ML! hypothesis, successfully used in chemistry as the concept of molecular structure, in the rigorous quantummechanics ~QM! framework @1#. QM requires that the probability distribution of stationary states have the same symmetry of the Hamiltonian, in marked contrast with the ML requirement. The qualitative explanation of this apparent contradiction is simple: since the molecule is not an isolated system, its states cannot be stationary @2#. The main problem is the understanding of the quantitative aspect of the phenomenon, as it results from the following question by Woolley @1#: ‘‘Why should the general quantum theory describing energy eigenstates turn out to be of such little use in chemistry, or put in another way, why should transitions out of the time-dependent molecular quantum states which empirically appear to be an essential ingredient of any useful quantum chemistry, be so slow?’’ Although it is generally accepted that the phenomenon should be explained by means of decoherence arguments @3#, it is also clear that explicit models are needed. Thus, by means of the study of an explicit model, we want to point out the role of instability in the localization phenomenon. Indeed, we expect the existence of metastable states in perturbed systems and we want to study the smallness of the interaction between a pair of such states for large instability. Let us consider the case of the ammonia molecule NH3, where the model for the motion of the nitrogen atom N is a double well with a large internal barrier @1#. In this model we have the pyramidal shape of the molecule ~molecular structure! if the state is localized in one of the wells. The inversion line of the molecular microwave emission gives the energy splitting of the stationary states. Experiments on ammonia gas show that the localization and the inversion line are dependent on pressure. In particular, the localization probability increases and the inversion line broadens and decreases as the pressure increases, giving the so-called redshift ~RS! effect @2#. Some previous explicit models @4# which are able to explain ML, are autonomous, i.e., they make use of timeindependent potentials. In particular in a recent paper @5# ,b y using an unstable autonomous model, both ML and RS are obtained. In the present paper we use a nonautonomous model ~time-dependent potential! so that the instability caused by the molecular, collisions is represented in a more realistic way. In particular, our model consists of a double-well potential with a time-dependent perturbation simulating the dynamical influence of the environment on the ammonia molecule, i.e., the collisions with the other molecules of the gas, where the collision frequency is related to the pressure. Let us notice that the present model is more physical than the previous ones for the reasons stated above, but it is still simplified. One simplification is the choice of a perturbation periodic in time. This choice is technical and is due to the recent improvement of methods for handling periodic problems. We point out that the classical resonance effect between different frequencies are not relevant for the results. In any case the results give an a posteriori justification of the model. Since the first~but not the second! order perturbation term vanishes, we set the perturbation of the same order of the square root of the splitting. We consider the large internal barrier regime, so that both the splitting and the perturbation are exponentially small. This choice of parameters is similar to previous ones, and allows us to apply the same comparison with experiments given by Claverie and Jona Lasinio @4#, although in that case the frequency parameter was absent. If the periodic perturbation is strong enough, and the Fourier coefficients are slowly decreasing for increasing index, we have localization for a frequency larger than a critical value but smaller than a large value. This upper bound should be related to the simplification of the model given by the periodicity of the time behavior. Since the model is linear, we have no spontaneous symmetry breaking, so that the perturbation is asymmetric, but not too much, in order to have the RS. We control the asymmetry of the perturbation by varying the coefficient of the time independent perturbation.
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Open Access
Jun 19, 2002
T Padmanabhan 
