Theory of nonlinear Landau-Zener tunneling

Abstract

A nonlinear Landau-Zener model was proposed recently to describe, among a number of applications, the nonadiabatic transition of a Bose-Einstein condensate between Bloch bands. Numerical analysis revealed a striking phenomenon that tunneling occurs even in the adiabatic limit as the nonlinear parameter $C$ is above a critical value equal to the gap $V$ of avoided crossing of the two levels. In this paper, we present analytical results that give quantitative account of the breakdown of adiabaticity by mapping this quantum nonlinear model into a classical Josephson Hamiltonian. In the critical region, we find a power-law scaling of the nonadiabatic transition probability as a function of $C/V-1$ and $\alpha $, the crossing rate of the energy levels. In the subcritical regime, the transition probability still follows an exponential law but with the exponent changed by the nonlinear effect. For $C/V>>1$, we find a near unit probability for the transition between the adiabatic levels for all values of the crossing rate.Comment: 9 figure

Role of particle interactions in a many-body model of Feshbach-molecule formation in bosonic systems

“…Compared to the well known nonlinear Hamiltonian that describes the tunneling dynamics of BEC atoms in a double-well potential or between two internal states [14], the nonlinearity (i.e., the Hamiltonian depends on the instantaneous wavefunction as well as its conjugate) of the present Hamiltonian (6) is much more complex. The nonlinearity stems both from the diagonal and off-diagonal terms and is preserved even without taking into account the particle interactions.…”

Towards a generalized Landau-Zener formula for an interacting Bose-Einstein condensate in a two-level system

“…Note also that the emergence of looped levels was previously studied for the quantum dimer [8]. Several approaches were made to derive a nonlinear Landau-Zener formula for this problem using methods from classical Hamiltonian mechanics [9,10]. For subcritical values of the nonlinearity |g| < g c , standard methods of classical nonadiabatic corrections yield good results for the near-adiabatic case (α/v 2 ≪ 1).…”

Phase diagram of Landau-Zener phenomena in coupled one-dimensional Bose quantum fluids

“…For example, the non-linear two-level system shows that the mean-field interactions among particles tend to increase the tunnelling probability and that there exists a critical value of the interaction strength beyond which the transition probability becomes nonzero even in the adiabatic limit [5]. On the other hand, a stationary phase approximation leads to a characteristic scaling or power law for the critical behavior that occurs as the nonlinear parameter equals the gap of avoided crossing energy levels [8]. Regarding the asymmetric LZ tunnelling in a periodic potential, JonaLasinio et.…”

“…We make reference to the two-mode mean-field and Bose-Hubbard schemes inherited from the Gross-Pitaevskii and full quantum approaches [4][5][6][7][8][9]. As a result of incorporating the linear variation in time between the two levels, all of those treatments suggested a breakdown of the adiabatic limit, that is, that the LZ transition probability does not vanish even in the adiabatic limit.…”