Bistability and Exact Reflectionless States in Nonlinear Scattering of a Bose--Einstein Condensate

TL;DR

The paper investigates nonlinear scattering of a quasi-1D Bose–Einstein condensate from a Rosen–Morse potential, deriving analytically exact reflectionless states that form two-fold and three-fold degenerate manifolds. It uses a two-term ansatz to construct these states and applies Bogoliubov–de Gennes stability analysis to show that only one branch per degenerate manifold is dynamically stable, revealing multistable transmission possible in this setting. It further demonstrates that a configuration with spatially localized nonlinearity supports an exact reflectionless state and exhibits a clear mono-to-bistability transition in the transmission as incident intensity increases. The results provide an analytic framework for multistable matter-wave and photonic transport in engineered RM-like landscapes, with direct relevance to ultracold atomic gases and photonic lattices.

Abstract

We investigate the mean-field scattering dynamics of a quasi-one-dimensional Bose--Einstein condensate interacting with a Rosen--Morse potential. For specific potential and nonlinearity parameters, we derive analytically exact, degenerate scattering states (doubly or triply degenerate) exhibiting perfect transmission. Using the Bogoliubov--de Gennes approach, we analyze the stability of these reflectionless degenerate states, demonstrating that only one solution within each degenerate manifold is dynamically stable. Furthermore, we study a configuration with spatially localized nonlinearity, identifying an exact reflectionless state under specific conditions. Numerical analysis shows that this state marks the system's transition from monostability to bistability as the incident wave amplitude increases. Our work establishes an analytic framework for these multistable transmission phenomena, directly relevant to coherent matter-wave transport in ultracold atomic systems and optical propagation in engineered photonic lattices.

Figures (6)

• Figure 1: Three-fold degenerate reflectionless stationary states for the parameter values $V_0=3/8, \mu=7/4, g=5/4$, and $k=1$ (dimensionless units). (a) Densities $|\phi_{\mathrm{I}}(x)|^2$ and $|\phi_{\mathrm{II},\pm}(x)|^2$. (b) Phase $\tilde{\theta}(x)=\arg \left[\phi(x) e^{-\mathrm{i} k x}\right]$. The horizontal dotted lines show the analytically predicted asymptotic phase shifts ($\theta_{\mathrm{I}},\theta_{\mathrm{II},\pm}$).
• Figure 2: Stability of the isolated reflectionless state $\phi_{\mathrm{I}}$. $\varepsilon_{\max}$ is plotted versus the interaction strength $g$ at fixed chemical potential $\mu=7/4$. Here $a$, $b$, $c$ denote the three regimes ($g<0$), ($0<g<g_c$), and ($g>g_c$), respectively, separated by two dashed vertical lines. The critical point $g_c=1.5$ is defined by $V_0(g_c)=0$.
• Figure 3: $\varepsilon_{\max}(g)$ versus the interaction strength $g$ for the reflectionless states $\phi_{\mathrm{II},+}$, $\phi_{\mathrm{II},-}$, and $\phi_{\mathrm{I}}$. The reflectionless states $\phi_{\mathrm{II},\pm}$ exist for $g>1/4$, whereas the reflectionless state $\phi_{\mathrm{I}}$ with fixed $V_0=3/8$ holds only for $g>5/8$ due to $k^2 = 8g /5 - 1$. The vertical dashed line at $g=5/4$ marks the three-fold degenerate point with $V_0=3/8, \mu=7/4, g=5/4$, and $k=1$.
• Figure 4: Transmission $T$ versus incident intensity $|\tilde{r}_0|^2$ for the RM potential with a localized nonlinearity $g(x)=g_0 \operatorname{sech}(x)$, calculated from Eq. (\ref{['eq:embedded-gpe']}). The red circle marks the analytically derived perfect-transmission state, corresponding to the upper turning point. Here the potential depth is $V_0=5/8$, and other physical parameters are determined by setting $\lambda=3/4$ in Eq. \ref{['eq:embedded-param']}, which yields $\mu=5/32, g_0=3/8$, and $k=\sqrt{5}/4$.
• Figure 5: (a) Spatial profiles of the densities $|\phi_{\mathrm{III},+}(x)|^2$ and $|\phi_{\mathrm{III},-}(x)|^2$ of the degenerate reflectionless states $\phi_{\mathrm{III},\pm}(x)$ for the RM well ($V_0 = 3/2$). Parameters: $g = 3/8$ (giving $k = \sqrt{g - 1/4}$ and $\mu = 3g/2 - 1/8$). (b) phase $\tilde{\theta}(x)=\arg \left[\phi(x) e^{-\mathrm{i} k x}\right]$; the horizontal line indicates the transmission phase $\arg (\tilde{t}_{\mathrm{III}})$.