content_image/382111.jpg
Another type of the initial state 

<img src='content_image/382111.jpg'>

allows for an additional spread over the transverse modes. The transverse wavefunction vanishes at the waveguide axis. The mode occupation has a minimum at n = 0, and the occupation of the ground transverse mode tends to zero for κ$_{1}$ < κ$_{2}$ ≪ 1. The results for a sequence similar to the third sequence described above are shown at Fig. 1 as well. In spite of the significant initial spread over both transverse and longitudinal modes, the deviations from the equilibrium are close to the ones for the third sequence, while the energy-to-energy variations for the former are indeed less than for the latter. 

Note that we have also successfully tested the conver- gence of the instantaneous quantum expectation values of the observables to their infinite-time averages. 

Summary and interpretation of results .– In this arti- cle, we solve analytically the problem of two short-range- interacting atoms in a circular, transversely harmonic multimode waveguide. We assess the ability of the sys- tem to thermalize from an initially excited state; a broad class of initial states has been analyzed. We find a sub- stantial suppression of thermalization, even for the in- finitely strong interactions. We associate this effect with the previously demonstrated unattainability of complete quantum chaos in the ˇ Seba-type billiards [11–15]. 

We conjecture that the effect of suppression of thermal- ization is generic for the integrable systems perturbed, no matter how strong, by a well localized perturbation. The following reasoning applies. In a quantum chaotic system, a given eigenstate | α 〉 consists of a large superpo- sition of the eigenstates | ⃗ n 〉 of the underlying integrable system, which are drawn indiscriminately from the micro- canonical shell. Number of principal components in such a superposition is a sensitive measure of the approach to a complete chaos and the subsequent thermalizability (see, for example [29]). The zero-point-localized perturbation, being a particular case of a separable perturbation, gen- erates the eigenstates of a form 〈 ⃗ n | α 〉 ∝ 1 / ( E$_{α}$ − E$_{⃗}$ $_{n}$) (see [22]). There is always one and only one perturbed energy E$_{α}$ in between any two unperturbed energies E$_{⃗}$ n (see [13]). In addition, for the case of the infinitely strong perturbation (where the approach to chaos is expected to be the closest) E$_{α}$ tends to the middle position between the two E$_{⃗}$ $_{n}$’s. In this case, the only energy scale that remains is the distance between the unperturbed levels, and, as a result, the number of principal components a given eigenstate | α 〉 consists of is always of the order of unity. This property extends to the case of finite-range interactions, whenever the interaction range is much less then the de Broglie wavelength of the colliding atoms. 

Our study is the first attempt to address thermaliz- ability of a quantum system with separable interactions. 

We are grateful to Felix Werner for the enlightening discussions on the subject. This work was supported by a grant from the Office of Naval Research ( N00014-09-1- 0502 ). 

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