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
We present a general theory for adiabatic evolution of quantum states as governed by the nonlinear Schrödinger equation, and provide examples of applications with a nonlinear tunneling model for Bose-Einstein condensates. Our theory not only spells out conditions for adiabatic evolution of eigenstates, but also characterizes the motion of non-eigenstates which cannot be obtained from the former in the absence of the superposition principle. We find that in the adiabatic evolution of non-eigenstates, the Aharonov-Anandan phases play the role of classical canonical actions.PACS numbers: 03.65.Vf, 03.75.Fi, 71.35.Lk Adiabatic evolution has been an important method of preparation and control of quantum states [1,2,3]. The main guidance comes from the adiabatic theorem of quantum mechanics [4], which dictates that an initial nondegenerate eigenstate remains to be an instantaneous eigenstate when the Hamiltonian changes slowly compared to the level spacings. More precisely, the quantum eigenstate evolves only in its phase, given by the time integral of the eigenenergy (known as the dynamical phase) and a quantity independent of the time duration (known as the geometric phase). The linearity of quantum mechanics then immediately allows a precise statement about the adiabatic evolution of non-eigenstates through the superposition principle.Our concern here is how the adiabatic theorem gets modified in nonlinear evolution of quantum states. Nonlinearity has been introduced in various forms as possible modifications of quantum mechanics on the fundamental level [5]. Our motivation, however, derives from practical applications in current pursuits of adiabatic control of Bose Einstein condensates (BECs) [6], which can often be accurately described by the nonlinear Schrödinger equation. Here the nonlinearity stems from a mean field treatment of the interactions between atoms. Difficulties in theoretical study of adiabatic control of the condensate arise not only from the lack of unitarity in the evolution of the states but also from the absence of the superposition principle. This problem was recently addressed for BECs in some specific cases [7], and a similar problem was discussed in the past for soliton dynamics [8].In this Letter, we present a general adiabatic theory for the nonlinear evolution of quantum states (eigenstates or noneigenstates). Our study is conducted by transforming the nonlinear Schrödinger equation into a mathematically equivalent classical Hamiltonian problem, where nonlinearity is no longer a peculiar issue but rather a common character. We can thus apply the adiabatic theory for classical systems [9,10,11] to the study of adiabatic evolution of quantum states. The eigenstates become fixed points in the classical problem, whose adiabatic evolution can be understood from a stability analysis of such points. Aharonov-Anandan phases [12] of cyclic or quasi-cyclic quantum states play the role of canonical action in the classical problem, and are therefore conserved during adiabatic changes of the co...
We study the adiabatic limit and the semiclassical limit with a second-quantized two-mode model of a many-boson interacting system. When its mean-field interaction is small, these two limits are commutable. However, when the interaction is strong and over a critical value, the two limits become incommutable. This change of commutability is associated with a topological change in the structure of the energy bands. These results reveal that nonlinear mean-field theories, such as Gross-Pitaevskii equations for Bose-Einstein condensates, can be invalid in the adiabatic limit.
We study bright solitons in a Bose-Einstein condensate with a spin-orbit coupling that has been realized experimentally. Both stationary bright solitons and moving bright solitons are found. The stationary bright solitons are the ground states and possess well-defined spin-parity, a symmetry involving both spatial and spin degrees of freedom; these solitons are real valued but not positive definite, and the number of their nodes depends on the strength of spin-orbit coupling. For the moving bright solitons, their shapes are found to change with velocity due to the lack of Galilean invariance in the system.
We study the superfluidity of a spin-orbit coupled Bose-Einstein condensate (BEC) by computing its Bogoliubov excitations, which are found to consist of two branches: one is gapless and phonon-like at long wavelength; the other is typically gapped. These excitations imply a superfluidity that has two new features: (i) due to the absence of the Galilean invariance, one can no longer define the critical velocity of superfluidity independent of the reference frame; (ii) the superfluidity depends not only on whether the speed of the BEC exceeds a critical value, but also on cross helicity that is defined as the direction of the cross product of the spin and the kinetic momentum of the BEC.
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