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arXiv:cond-mat/0210504 · Physical Review A

Second-quantized Landau-Zener theory for dynamical instabilities

J. R. Anglin

Published May 19, 2003• 23 citations
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Abstract

State engineering in nonlinear quantum dynamics sometimes may demand driving the system through a sequence of dynamically unstable intermediate states. This very general scenario is especially relevant to the dilute Bose-Einstein condensates, for which ambitious control schemes have been based on the powerful Gross-Pitaevskii mean-field theory. Since this theory breaks down on logarithmically short time scales in the presence of dynamical instabilities, an interval of instabilities introduces quantum corrections, which may possibly derail a control scheme. To provide a widely applicable theory for such quantum corrections, this paper solves a general problem of time-dependent quantum-mechanical dynamical instability, by modeling it as a second-quantized analog of a Landau-Zener avoided crossing: a ``twisted crossing.''

DOI
10.1103/physreva.67.051601

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State engineering in nonlinear quantum dynamics sometimes may demand driving the system through a sequence of dynamically unstable intermediate states. This very general scenario is especially relevant to the dilute Bose-Einstein condensates, for which ambitious control schemes have been based on the powerful Gross-Pitaevskii mean-field theory. Since this theory breaks down on logarithmically short time scales in the presence of dynamical instabilities, an interval of instabilities introduces quantum corrections, which may possibly derail a control scheme. To provide a widely applicable theory for such quantum corrections, this paper solves a general problem of time-dependent quantum-mechanical dynamical instability, by modeling it as a second-quantized analog of a Landau-Zener avoided crossing: a ``twisted crossing.''
We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-level system, for the full many-particle system as well as in the mean-field approximation. Novel nonlinear eigenstates emerge in the mean-field description, which leads to a breakdown of adiabaticity: The Landau-Zener transition probability does not vanish even in the adiabatic limit. It is shown that the emergence of nonlinear eigenstates and thus the breakdown of adiabaticity corresponds to quasi-degenerate avoided crossings of the many-particle levels. The many-particle problem can be solved approximately within an independent crossings approximation, which yields an explicit generalized Landau-Zener formula. A comparison to numerical results for the many-particle system and the mean-field approximation shows an excellent agreement.
We present a comprehensive analysis of the nonlinear Landau-Zener tunneling. We find characteristic scaling or power laws for the critical behavior that occurs as the nonlinear parameter equals to the gap of avoided crossing energy levels. For the nonlinear parameter larger than the energy gap, a closed-form solution is derived for the nonlinear tunneling probability, which is shown to be a good approximation to the exact solution for a wide range of the parameters. Finally, we discuss the experimental realization of the nonlinear model and possible observation of the scaling or power laws using a Bose-Einstein condensate in an accelerating optical lattice.
Bose-Einstein condensation is a quantum statistical phase transition which was theoretically predicted almost a hundred years ago. After years of seminal research, physicists realized the first almost ideal Bose-Einstein condensates in ultracold dilute atomic gases in 1995. Since then, the theoretical and experimental methods concerning such systems have been developing rapidly, and many fascinating phenomena have been found in these novel quantum systems.\n\nBose-Einstein condensation occurs in a system consisting of massive bosons when a single quantum state becomes macroscopically occupied as the temperature is lowered below the transition temperature. In general, condensates consisting of repulsively interacting bosons exhibit superfluidity: Particle currents can flow in the system without dissipation and viscosity. Moreover, the velocity fields of condensates have to be irrotational, which severely restricts the rotational characteristics of these systems. Apart from the center of mass motion, the system may carry angular momentum in the form of elementary excitations or so-called quantized vortices.\n\nThis Thesis is a theoretical study of subjects related to stability and dynamics of quantized vortices in dilute atomic Bose-Einstein condensates. The precession and instability of off-centered vortices in trapped condensates is investigated both in the zero-temperature limit and at finite temperatures. Dynamical stability of multiply quantized vortices and vortex clusters is studied in axisymmetric trap geometries. Splitting of energetically and dynamically unstable multiply quantized vortices into singly quantized vortices is also studied. Finally, as a separate subject, tunneling of a condensate through a potential barrier is investigated. Majority of this work relies on numerical methods for solving the Gross-Pitaevskii and Bogoliubov equations, which are of central importance in the study of dilute atomic Bose-Einstein condensates.
Nonlinear Landau-Zener tunneling is an important nonlinear phenomenon. We propose to stimulate the nonlinear tunneling in a spin-orbit-coupled spinor Bose-Einstein condensate. The system provides an experimentally tunable nonlinearity as well as multiple avoided crossings with tunable gap size in nonlinear dispersion relations. The nonlinearity generates tilted cusp and loop structures around the avoided crossings, and the physical consequence of these nonlinear structures is the nonlinear Landau-Zener tunneling. The spin-momentum locking induced by the spin-orbit coupling leads to a time-resolved observation of the nonlinear tunneling by measuring atom populations.
Abstract Three analytic solutions to the Schrödinger equation for the time-dependent Landau–Zener (LZ) Hamiltonian are presented. They correspond to specific finite-time driving paths in a bounded parameter space of a two-level system. Two of these paths go through the avoided crossing of levels, either with a constant speed or with variable speed that decreases in the region of reduced energy gap, the third path bypasses the crossing such that the energy gap remains constant. The solutions yield exact time dependencies of the excitation probability for the system evolving from the ground state of the initial Hamiltonian. The LZ formula emerges as an approximation valid within a certain interval of driving times for the constant-speed driving through the avoided crossing. For long driving times, all solutions converge to the prediction of the adiabatic perturbation theory. The excitation probability vanishes at some discrete time instants.
We present a rigorous analysis of the generalized Landau-Zener problem for the two-level interacting Bose-Einstein condensates. We show that the dynamics of the system is accurately, in detail, described by a two-term variational ansatz that is valid for the whole time domain and is applicable for any set of involved parameters. Applying an exact third-order nonlinear differential equation we construct an advanced fifth-order polynomial equation for the final transition probability serving as a highly accurate generalized Landau-Zener formula.