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arXiv:1501.02669

Gross Pitaevskii Equation with a Morse potential: bound states and evolution of wave packet

Sukla Pal, Jayanta K. Bhattacharjee

Published Jan 12, 2015• 0 citations
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Abstract

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 $g_c$ beyond which there are no bound states scales as $D^{3/4}$ (for large $D$). Studying the quantum evolution of wave packets, we observe that for $gg_c$, 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.

DOI
10.48550/arxiv.1501.02669

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We consider systems governed by the Gross Pitaevskii equation (GPE) with the Morse potential V(x)=D(e−2ax−2e−ax)V(x)=D(e^{-2ax}-2e^{-ax})V(x)=D(e−2ax−2e−ax) as the trapping potential. For positive values of the coupling constant ggg of the cubic term in GPE, we find that the critical value gcg_cgc​ beyond which there are no bound states scales as D3/4D^{3/4}D3/4 (for large DDD). Studying the quantum evolution of wave packets, we observe that for ggcgg_cggc​, on the otherhand, all initial wave packets escape from the potential and the dynamics is like that of a quantum free particle. For g<0g<0g<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.
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We show how a dynamical systems approach can, somewhat unexpectedly, be relevant in the quantum dynamics featuring oscillations and escape in the Morse potential. We compare the dynamics resulting from the approach with the results obtained from a direct numerical integration of the relevant Schrödinger equation to support our claim. An interesting finding of the numerical investigation is the marked increase in the probability of obtaining a significant fraction (more than 50%) of the wave packet in the classically forbidden range beyond a critical energy of the packet. The fact that the dynamical systems approach shows an instability near that critical energy is a definite indication of the relevance of dynamical systems to the quantum dynamics. At lower energies, the calculated mean position 〈x〉 and variance V from the dynamical system allow us to clearly establish the phenomenon of tunneling since the sum 〈x〉+sqrt[V] clearly exceeds, at various times, the classical bound on displacement for the corresponding energy.
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Consider a system of NNN bosons in three dimensions interacting via a repulsive short range pair potential N2V(N(xi−xj))N^2V(N(x_i-x_j))N2V(N(xi​−xj​)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let HNH_NHN​ denote the Hamiltonian of the system and let ψN,tψ_{N,t}ψN,t​ be the solution to the Schrödinger equation. Suppose that the initial data ψN,0ψ_{N,0}ψN,0​ satisfies the energy condition <ψN,0,HNkψN,0>≤CkNk < ψ_{N,0}, H_N^k ψ_{N,0} > \leq C^k N^k <ψN,0​,HNk​ψN,0​>≤CkNk for $k=1,2,... $. We also assume that the kkk-particle density matrices of the initial state are asymptotically factorized as N→∞N\to\inftyN→∞. We prove that the kkk-particle density matrices of ψN,tψ_{N,t}ψN,t​ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential VVV. We also prove the same conclusion if the energy condition holds only for k=1k=1k=1 but the factorization of ψN,0ψ_{N,0}ψN,0​ is assumed in a stronger sense.
This article concerns the singularly of perturbed Gross-Pitaevskii equations in trapped dipolar quantum gases, $$\displaylines{ - \varepsilon^2 \Delta u + V(x)u +{\lambda _1}{| u |^2}u + {\lambda _2}(K * {| u |^2})u= 0 \quad \text{in } \mathbb{R}^3, \cr u > 0,\quad u \in {H^1}({\mathbb{R}^3}), }$$ where ε\varepsilon ε is a small positive parameter, λ1,λ2∈R\lambda_1, \lambda_2\in \mathbb{R}λ1​,λ2​∈R, ∗*∗ denotes the convolution, K(x)=1−3cos⁡2θ∣x∣3 K(x) = \frac{{1 - 3{{\cos }^2}\theta }}{{{{| x |}^3}}}K(x)=∣x∣31−3cos2θ​ and θ=θ(x)\theta = \theta(x)θ=θ(x) is the angle between the dipole axis determined by (0,0,1)(0,0,1)(0,0,1) and the vector xxx. Moreover, the potential VVV satisfies lim inf⁡∣x∣→∞V(x)>inf⁡R3V(x)=0\liminf_{|x| \to \infty } V(x) > \inf_{\mathbb{R}^3} V(x) = 0liminf∣x∣→∞​V(x)>infR3​V(x)=0. Under certain assumptions on (λ1,λ2)∈R2(\lambda_1, \lambda_2)\in \mathbb{R}^2(λ1​,λ2​)∈R2, we construct a family of positive solutions uε∈H1(R3){u_\varepsilon } \in {H^1}({\mathbb{R}^3})uε​∈H1(R3) whose L∞{L^\infty }L∞ norm approaches 000 as ε→0\varepsilon \to 0ε→0. Our main results extend the results in Byeon and Wang [6] which dealt with singularly perturbed Schrodinger equations with a local nonlinearity, to the nonlocal Gross-Pitaevskii type equation. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/48/abstr.html