Shortcuts to adiabaticity for an interacting Bose–Einstein condensate via exact solutions of the generalized Ermakov equation

Abstract

Shortcuts to adiabatic expansion of the effectively one-dimensional Bose–Einstein condensate (BEC) loaded in the harmonic-oscillator (HO) trap are investigated by combining techniques of variational approximation and inverse engineering. Piecewise-constant (discontinuous) intermediate trap frequencies, similar to the known bang–bang forms in the optimal-control theory, are derived from an exact solution of a generalized Ermakov equation. Control schemes considered in the paper include imaginary trap frequencies at short time scales, i.e., the HO potential replaced by the quadratic repulsive one. Taking into regard the BEC’s intrinsic nonlinearity, results are reported for the minimal transfer time, excitation energy (which measures deviation from the effective adiabaticity), and stability for the shortcut-to-adiabaticity protocols. These results are not only useful for the realization of fast frictionless cooling, but also help us to address fundamental problems of the quantum speed limit and thermodynamics.

Shortcuts to Adiabaticity for Optical Beam Propagation in Nonlinear Gradient Refractive-Index Media

“…Moreover, from Equations ( 6 ) and ( 7 ), we obtain the auxiliary differential equation which connects the dynamic evolution of the beam width with the guiding coefficient . This resembles the generalized Ermakov equation obtained for effectively one-dimensional Bose-Einstein condensates (BECs) governed by the Gross–Pitaevskii equation including the nonlinear atom-atom interaction and the time-varying harmonic trap [ 58 ]. As a result, the width of the optical beam a can be manipulated by modulating the parabolic profile of refraction index through Equation ( 9 ), in the presence of Kerr-type nonlinearity.…”

“…Now, sharing the concept of STA in Refs. [ 16 , 47 , 58 ], our idea presented here is to first choose the trajectory of a by fixing the initial and final boundary conditions, and later the profile of refractive index is designed inversely, such that one can always implement the same task but within a shorter propagation distance.…”

“…Now it is ready for us to design STA based on the inverse engineering [ 47 , 58 ] to compress the optical beam within finite short propagation distance . To this end, we design by assuming the following simple polynomial ansatz [ 16 , 47 , 58 ] with , satisfying the boundary conditions Here the boundary conditions ( 16 ) should coincide with the initial and final values of beam width ( 14 ), given by the adiabatic reference, such that and , and the other boundary conditions are postulated for the smooth changes of refractive index at initial and final edges. Interestingly, these boundary conditions ( 17 ) also imply the minimum potential energy and zero kinetic energy at initial and final points, since the first and second derivatives of a are null.…”

“…However, they are not optimal at all. One can further optimize it, for instance, by minimizing the propagation distance [ 58 ].…”

“…However, when the Kerr-type nonlinearity is weak, the Gaussian assumption is still reasonable of soliton, see the similar analysis for Bose-Einstein condensates in Refs. [ 45 , 47 , 58 ]…”