Effects of a nonlinear perturbation on dynamical tunneling in cold atoms

Abstract

We perform a numerical analysis of the effects of a nonlinear perturbation on the quantum dynamics of two models describing non-interacting cold atoms in a standing wave of light with a periodical modulated amplitude A(t). One model is the driven pendulum, considered in ref. [1], and the other is a variant of the well-known Kicked Rotator Model. In absence of the nonlinear perturbation, the system is invariant under some discrete symmetries and quantum dynamical tunnelling between symmetric classical islands is found. The presence of nonlinearity destroys tunnelling, breaking the symmetries of the system. Finally, further consequences of nonlinearity in the kicked rotator case are considered.PACS numbers: 05.45.-a, 03.65.-w Tunneling is one of the most typical features of quantum mechanics, concerning oscillations between states that cannot be connected in the classical Hamiltonian dynamics by real trajectories. The original formulation of the problem involved states separated by potential barriers: the relevant features are semiclassically explained in terms of complex solutions of Hamilton equations for one-dimensional systems [2], while a proper treatment in higher dimensions is considerably subtler even when integrability is preserved [3]. Recently a novel kind of tunneling, involving transitions between classically separated regions in the presence of a non trivial structure of the phase space, has attracted considerable attention, both theoretically [4][5][6][7], and experimentally [8,1,9]. In particular a large fraction of these papers focus on physical settings realized with cold atoms in optical potentials. The recent widespread interest and experimental activity in Bose-Einstein condensation [10] suggest to check whether the presence of Gross-Pitaevskii nonlinearities [11] deeply influences the characteristic features of dynamical tunneling. Such a question was already raised and analyzed in the framework of the kicked oscillator [12], where it was observed how the nonlinear terms typically destroys quantum effects induced by symmetry [13,14]. The paper is organized as follows: we firstly give a few details and fix notations for the class of models we are going to consider, then analyze two cases: a driven pendulum and the kicked rotator; in the last section we briefly reconsider pioneering work made on the nonlinear kicked rotator [15,16] and supplement it with novel results for the quantum resonant case. I. GENERAL SETTING.We will consider models described by the following hamiltonianwhere p and ϑ are canonical momentum and position coordinates respectively, expressed in scaled dimensionless units, and A(t) is a periodic function of time. Under appropriate choices of the function A(t), the quantum version of this model describes an ensemble of noninteracting cold atoms in presence of a standing wave with a periodically modulated amplitude A(t). The connection between the scaled variables ϑ, p and the physical ones We consider two possible choices for the periodic function A(t):• The first model i...

Delocalized and resonant quantum transport in nonlinear generalizations of the kicked rotor model

“…The nonlinear coupling constant u varies from 1 to 400. As already noticed [20], the resonant behavior survives even in the presence of nonlinearity, although generically the spreading is slowed w.r.t. the linear case.…”

“…The correspondent system in the u → 0 limit is characterized by the phenomenon of dynamical localization [16] caused by quantum interference effects. Previous theoretical predictions and numerical simulations [13,20] indicate that, above a critical border u c ∼ 2π for the nonlinear coupling constant, dynamical localization is destroyed. The delocalization takes place in the form of anomalous subdiffusion with an exponent of 2/5: in [13] an asymptotic law p 2 (t) ∼ c(u) · t 2/5 , (where c(u) ∼ u 4/5 ) is predicted for both models; this is confirmed for both models by the data reported in Fig.…”

Routes towards the experimental observation of the large fluctuations due to chaos-assisted tunneling effects with cold atoms

“…Applications of chaos assisted tunneling to cold atom systems were discussed in [19], and applications of chaotic tunneling to cold atom systems were studied in [20]. In [21] the effects of nonlinearities due to mean-field interactions (Gross-Pitaevski) on dynamical tunneling were studied. In the mean time there were two pioneering experiments on chaos assisted tunneling with cold atoms [22,23].…”

Macroscopic Quantum Self-Trapping in Dynamical Tunneling

“…It allows us to predict the critical nonlinearity U crit from knowledge of the noninteracting system, and to understand the increase of the tunnelling period with U that we find numerically. While previous work reported detrimental effects of nonlinearities on dynamical tunnelling [29], a connection with MQST was not made. Surprisingly, at higher nonlinearities with U > U crit we find some parameter ranges where dynamical tunnelling reappears.…”