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Physical Review A

Separatrix eigenfunctions

John R. Cary, Petre Rusu

Published Jun 1, 1992• 10 citations
Publisher

Abstract

Near-separatrix eigenfunctions for the double-well potential are analyzed. These functions are needed for the study of systems with perturbed or slowly varying double-well potentials. The probability density of separatrix eigenfunctions collapses to the classical result (a \ensuremath{\delta} function at the unstable fixed point) logarithmically with the number of quantum states. Matrix elements with respect to this basis are also studied. Unlike the wave functions, the unnormalized matrix elements are nonsingular in the limit of large quantum number.

DOI
10.1103/physreva.45.8501

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Near-separatrix eigenfunctions for the double-well potential are analyzed. These functions are needed for the study of systems with perturbed or slowly varying double-well potentials. The probability density of separatrix eigenfunctions collapses to the classical result (a \ensuremath{\delta} function at the unstable fixed point) logarithmically with the number of quantum states. Matrix elements with respect to this basis are also studied. Unlike the wave functions, the unnormalized matrix elements are nonsingular in the limit of large quantum number.
Quantum-mechanical evolution in a slowly varying double-well potential is analyzed in the semiclassical limit to determine the transition probability due to loss of adiabatic invariance of states having energy close to that of a separatrix in the equivalent classical system. Transformation to the rotating-axis basis reduces the problem to the integration of a set of coupled differential equations. In the adiabatic limit the integration is carried out by the method of successive approximations. For adiabatic parameter \ensuremath{\epsilon} of the order of the inverse, 1/N, of the number of quantum states with energy below the separatrix, the spread of the quantum adiabatic invariant is exponentially small in the inverse of the adiabaticity parameter. Surprisingly, the transition to this quantum adiabatic behavior occurs far below the minimum transition frequency of the system.
The results of analytical and numerical investigation of quantum dissipative dynamics a particle moving in a potential well with two equilibrium states is presented. A classical analogue of this problem is the capture into the resonance a particle in a double-well potential, when two states of equilibrium in the phase space are separated by a separatrix. The probability of finding the system in one of states of equilibrium for the quantum particle, depending on initial conditions and noise, is discussed.
We present a quantum phase-space model of the Bose-Einstein condensate (BEC) in a double-well potential. In a quantum two-mode approximation we examine the eigenvectors and eigenvalues and find that the energy correlation diagram indicates a transition from a delocalized to a fragmented regime. Phase-space information is extracted from the stationary quantum states using the Husimi distribution function. We show that the mean-field phase-space characteristics of a nonrigid physical pendulum arises from the exact quantum states, and that only 4--8 particles per well are needed to reach the semiclassical limit. For a driven double-well BEC, we show that the classical chaotic dynamics is manifest in the dynamics of the quantum states. Phase-space analogy also suggests that a $\ensuremath{\pi}$ phase-displaced wave packet put on the unstable fixed point on a separatrix bifurcates to create a superposition of two pendulum rotor states---a macroscopic superposition state of BEC. We show that the choice of initial barrier height and ramping, following a $\ensuremath{\pi}$ phase imprinting on the condensate, can be used to generate controlled entangled number states with tunable extremity and sharpness.
A new type of basis set for quantum mechanical problems is introduced. These basis states are adapted to describing the dynamics of a Hamiltonian Ĥ which is dependent upon a parameter X . A function f ( E ) is defined which is an analytic function of E , and which is negligibly small when | E |>>δ E , where δ E is large compared to the typical level separation. The energy-shell basis set consists of states |ξ n ( X )⟩ which are derived by applying the operator f (Ĥ( X )-Ē n ( X )) to elements of a fixed basis set, where Ē n ( X ) is an analytic approximation to an eigenvalue E n ( X ). The energy-shell basis states are combinations of states close to energy E n , but vary more slowly as a function of X than the eigenfunctions |ϕ n ( X )⟩ of Ĥ( X ). This feature gives the energy-shell basis states some advantages in analysing solutions of the time-dependent Schrödinger equation.