We consider an effective scaling approach for the free expansion of a one-dimensional quantum wave packet, consisting in a self-similar evolution to be satisfied on average, i.e., by integrating over the coordinates. A direct comparison with the solution of the Gross-Pitaevskii equation shows that the effective scaling reproduces with great accuracy the exact evolution---the actual wave function is reproduced with a fidelity close to one---for arbitrary values of the interactions. This result represents a proof of concept of the effectiveness of the scaling ansatz, which has been used in different forms in the literature but never compared against the exact evolution.
We consider an effective scaling approach for the free expansion of a one-dimensional quantum wave packet, consisting in a self-similar evolution to be satisfied on average, i.e., by integrating over the coordinates. A direct comparison with the solution of the Gross-Pitaevskii equation shows that the effective scaling reproduces with great accuracy the exact evolution---the actual wave function is reproduced with a fidelity close to one---for arbitrary values of the interactions. This result represents a proof of concept of the effectiveness of the scaling ansatz, which has been used in different forms in the literature but never compared against the exact evolution.
We compare the exact evolution of an expanding three-dimensional Bose-Einstein condensate with that obtained from the effective scaling approach introduced in D. Guéry-Odelin [Phys. Rev. A 66, 033613 (2002)].This approach, which consists in looking for self-similar solutions to be satisfied on average, is tested here in different geometries and configurations.We find that, in case of almost isotropic traps, the effective scaling reproduces with high accuracy the exact evolution dictated by the Gross-Pitaevskii equation for arbitrary values of the interactions, in agreement with the proof-of-concept of M. Modugno, G. Pagnini, and M. A. Valle-Basagoiti [Phys. Rev. A 97, 043604 (2018)].Conversely, it is shown that the hypothesis of universal self-similarity breaks down in case of strong anisotropies and trapped geometries.
The large-scale expansion dynamics of quantum gases is a central tool for ultracold gas experiments and poses a significant challenge for theory. In this work we provide an exact reformulation of the Gross-Pitaevskii equation for the ultracold Bose gas in a coordinate frame that adaptively scales with the system size during evolution, enabling simulations of long evolution times during expansion or similar large-scale manipulation. Our approach makes no hydrodynamic approximations, is not restricted to a scaling ansatz, harmonic potentials, or energy eigenstates, and can be generalized readily to noncontact interactions via the appropriate stress tensor of the quantum fluid. As applications, we simulate the expansion of the ideal gas, a cigar-shaped condensate in the Thomas-Fermi regime, and a linear superposition of counterpropagating Gaussian wave packets. We recover known scaling for the ideal gas and Thomas-Fermi regimes, and identify a linear regime of aspect-ratio preserving free expansion; analysis of the scaling dynamics equations shows that an exact, aspect-ratio invariant, free expansion does not exist for nonlinear evolution. Our treatment enables exploration of nonlinear effects in matter-wave dynamics over large scale-changing evolution.
A method is proposed for a self-consistent evaluation of the coupling constant in the Gross-Pitaevskii equation without involving a pseudopotential replacement. A renormalization of the coupling constant occurs due to medium effects and the trapping potential, e.g., in quasi-1D or quasi-2D systems. It is shown that a simplified version of the Hartree-Fock-Bogoliubov approximation leads to a variational problem for both the condensate and a two-body wave function describing the behavior of a pair of bosons in the Bose-Einstein condensate. The resulting coupled equations are free of unphysical divergences. Particular cases of this scheme that admit analytical estimations are considered and compared to the literature. In addition to the well-known cases of low-dimensional trapping, crossover regimes can be studied. The values of the kinetic, interaction, external, and release energies in low dimensions are also evaluated and contributions due to short-range correlations are found to be substantial.
We show analytically that the profile of the Bose–Einstein condensate can evolve exact self-similarly when the time-dependent interatomic interaction and harmonic potential satisfy a certain condition for one-, two- and three-dimensional geometries, respectively. Based on the exact scaling laws for the amplitude and width of the condensate profile, we propose experimentally feasible ways to improve the accuracy in measuring the stationary-state profile of the condensate when its spatial size is so small that the direct measurement is not available, and to squeeze the condensate to high local particle density to check the validity of the Gross–Pitaevskii equation via the controllable exact self-similar evolution of the condensate profile.
We consider systems governed by the Gross Pitaevskii equation (GPE) with the Morse potential V(x)=D(e−2ax−2e−ax) as the trapping potential. For positive values of the coupling constant g of the cubic term in GPE, we find that the critical value gc beyond which there are no bound states scales as D3/4 (for large D). Studying the quantum evolution of wave packets, we observe that for ggc, on the otherhand, all initial wave packets escape from the potential and the dynamics is like that of a quantum free particle. For g<0, we find that there can be initial conditions for which the escaping wave packet can propagate with very little change in width i,e., it remains almost shape invariant.
The propagation of an initially Gaussian wave packet of width [Formula: see text] in Gross–Pitaevskii equation is extensively studied both for attractive and repulsive interactions. It is predicted analytically and verified numerically that for a free particle with attractive interaction, the dynamics of the width is governed by an effective potential which is sensitive to initial conditions. If [Formula: see text] is equal to a corresponding critical width [Formula: see text], then the packet will propagate in time with very little change in shape. These are in essence like coherent states. Whereas, if [Formula: see text], depending on the nature of the effective potential for chosen [Formula: see text] and the interaction strength [Formula: see text], the width of the packet in course of time, either oscillates with bounded width or will spread like a free particle. For a simple harmonic oscillator (SHO) also, we find that for [Formula: see text] smaller than a critical value, there always exists a coupling strength for which the packet simply oscillates about the mean position without changing its shape, once again providing a resemblance to a coherent state. We also consider the Morse potential, which interpolates between the free particle and the oscillator. For large attractive interactions, the two limiting dynamics (free and simple harmonic) are indeed observed but in the intermediate form of the potential where the nonlinear terms dominate in the dynamics, an initial Gaussian wave packet does not retain its shape. For repulsive interaction, the Gaussian packet always changes shape no matter what the system parameters are.