Enhanced quantum reflection of matter-wave solitons

– Matter-wave bright solitons are predicted to reflect from a purely attractive potential well although they are macroscopic objects with classical particle-like properties. The non-classical reflection occurs at small velocities and a pronounced switching to almost perfect transmission above a critical velocity is found, caused by nonlinear mean-field interactions. Full numerical results from the nonlinear Schrödinger equation are complimented by a two-mode variational calculation to explain the predicted effect, which can be used for velocity filtering of solitons. The experimental realization with laser-induced potentials or two-component BoseEinstein condensates is suggested. In the framework of classical mechanics, a moving object will never turn back until it reaches a turning point, at which the radial velocity vanishes (the kinetic energy vanishes for one dimensional systems). At the microscopic scale, where the wave character of particles becomes important, quantum mechanics allows the reflection of a particle in a classically allowed region even when there is no classical turning point. Quantum reflection can occur above a repulsive potential barrier or an attractive potential well, and may take place in an attractive potential tail [1] or at a potential step [2]. The quantum reflection of cold atoms by a solid surface has triggered great interest both for the fundamental understanding of the implications of quantum mechanics and for potential applications of mirrors in atom optics [3]. Recently, Pasquini et al. reported the experimental observation of the quantum reflection of atoms from a dilute Bose-Einstein condensate (BEC) with up to 20 and 50% efficiency [4]. Matter-wave bright solitons are macroscopic quantum objects that may act as classical particle-like objects maintaining their integrity during collisions or while subjected to external forces. They have been prepared as self-bound droplets of atomic BECs with negative swave scattering lengths in quasi-one-dimensional waveguides [5, 6]. Previously we considered conditions for the creation of soliton trains [7] and their propagation in harmonic traps [8]. In this letter we show that a matter-wave soliton approaching an attractive potential well c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10408-4 322 EUROPHYSICS LETTERS Fig. 1 – Quantum reflection of a soliton incident on an attractive potential well centred at x = 0. The initial conditions are A = 1, x0 = −4 and v = 0.1. The upper panel shows the density and the lower one the phase evolution. may experience non-classical reflection. In contrast to the finite probabilities of quantum reflection of single atoms, the whole soliton reflects with very little radiative loss as seen in fig. 1, leading to a significant enhancement of reflection due to nonlinear interactions and macroscopic coherence. Above a critical velocity we observe a sharp transition to almost complete transmission while trapping is also possible in different parameter regimes. Solitons behave as classical particles according to perturbation theory when they move in weak external potentials [9] or by virtue of Ehrenfest’s theorem in semiclassical conditions where the potential varies slowly on the size scale of the soliton. Non-classical soliton scattering is expected when the kinetic energy dominates over the soliton binding energy or in specific resonant scattering scenarios and was previously discussed for step-like [10] and square potentials [11] and impurities [12–14]. The effect of pronounced switching from quantum reflection to transmission reported in this letter fits into neither of these categories as adiabatic conditions are broken by a strongly localised potential well but nonlinearity dominates over kinetic energy. Thus the soliton may remain intact. In the following we will discuss the various parameter regimes in detail. We report results of numerical simulations and study a two-mode variational model, which gives valuable insights into the underlying mechanism. We furthermore discuss possible experimental realizations with optical dipole potentials and incoherent solitons in two-component BECs and applications in velocity filtering. Consider a matter-wave soliton in a one-dimensional waveguide trap approaching the centre of a localised attractive potential well V (x). The dynamics is described well by the GrossPitaevskii equation (GPE), which we consider in the one-dimensional approximation [15] and write in dimensionless units as