Autoresonant excitation of Bose-Einstein condensates

Abstract

Controlling the state of a Bose-Einstein condensate driven by a chirped frequency perturbation in a one-dimensional anharmonic trapping potential is discussed. By identifying four characteristic time scales in this chirped-driven problem, three dimensionless parameters P_{1,2,3} are defined describing the driving strength, the anharmonicity of the trapping potential, and the strength of the particles interaction, respectively. As the driving frequency passes the linear resonance in the problem, and depending on the location in the P_{1,2,3} parameter space, the system may exhibit two very different evolutions, i.e., the quantum energy ladder climbing (LC) and the classical autoresonance (AR). These regimes are analyzed both in theory and simulations with the emphasis on the effect of the interaction parameter P_{3}. In particular, the transition thresholds on the driving parameter P_{1} and their width in P_{1} in both the AR and LC regimes are discussed. Different driving protocols are also illustrated, showing efficient control of excitation and deexcitation of the condensate.

Bifurcations of autoresonant modes in oscillating systems with combined excitation

“…The suggested approach relies on the stability analysis of the particular solutions with power-law asymptotics at infinity. It has been shown that outside the bifurcation points there are several autoresonant modes with different phase shifts associated with simple roots to Equation (4). Depending on the sign of the value  ′′ ( ; , ), where = 0 √ , some of these modes are stable (see the shaded areas in Figure 4).…”

“…Theorem 7. Let be a simple root to Equation (4). Then, in Case I, system (1) has two-parameter family of autoresonant solutions ( ; 0 , ), ( ; 0 , ) with the asymptotics…”

“…Theorem 9. Let be a root of multiplicity 3 to Equation (4). Then, in Case III, there exists 0 > 0 such that a two-parameter family of autoresonant solutions ( ; 0 , ), ( ; 0 , ) to system (1), starting at = 0 from -neighborhood of the solution * ( ), * ( ), has the following asymptotics:…”

“…1,2 The autoresonance was first suggested in the problems associated with the acceleration of relativistic particles in the middle of the twentieth century, and nowadays it is considered as a universal phenomenon that plays the important role in a wide range of physical systems. [3][4][5][6][7] The study of relevant mathematical models leads to new and interesting problems in the field of nonlinear dynamics. [8][9][10][11] Mathematical models associated with the autoresonance have been actively studied recently.…”

Quantum versus classical effects in the chirped-drive discrete nonlinear Schrödinger equation

“…It was shown that the growth of the population of mode 2 is in fact linear in time (with superimposed oscillations), as illustrated in Fig. 1, and a nearly full population transfer takes place if P 1 exceeds a sharp threshold [27,30,36]:…”