Collective Feshbach scattering of a superfluid droplet from a mesoscopic two-component Bose-Einstein condensate

We examine the collective scattering of a superfluid droplet impinging on a mesoscopic Bose-Einstein condensate (BEC) as a target. The BEC consists of an atomic gas with two internal electronic states, each of which is trapped by a finite-depth external potential. An off-resonant optical laser field provides a localized coupling between the BEC components in the trapping region. This mesoscopic scenario matches the microscopic setup for Feshbach scattering of two particles, when a bound state of one submanifold is embedded in the scattering continuum of the other submanifold. Within the mean-field picture, we obtain resonant scattering phase shifts from a linear response theory in agreement with an exact numerical solution of the real-time scattering process and simple analytical approximations thereof. We find an energy-dependent transmission coefficient that is controllable via the optical field between 0 and 100%.

Figure 1

(Color online) Schematic setup of a trapped two-component BEC with square-well potentials $V_a\left(x\right)$, $V_b\left(x\right)$ (solid line) and a coupling laser beam $\Omega \left(x\right)$ (long-dashed line) versus position $x$ [15]. Scattering occurs in the two open (left and right) $b$ channels, while the $a$ channels are energetically closed with a threshold energy $\Delta$. The chemical potential is $\mu$ (heavy solid line), the dash-dotted lines show the bound excitations, and the numbered quasibound energy levels (dashed line) are responsible for the collective Feshbach resonances.

Figure 2

Collective Feshbach resonances in the scattering phases ${\delta }_e$ (solid line) and ${\delta }_o$ (dashed line) of a weak perturbation of a two-component BEC versus energy $E$. Note that the $\pi$ jumps of the collective Feshbach resonances occur at excitation energy $E_i=\mu +{ϵ}_i$ of the even and odd quasibound Bogoliubov modes of Fig. 1. The inset shows the Lorentzian behavior of the phase derivative ${\delta }_o^{\prime }\left(E\right)$ close to the resonance as in Eq. (4). For parameters see [15].

Figure 3

Transmission coefficient of a weak coherent perturbation in the $b$ component of a BEC in (A) square-well and (B) Pöschl-Teller traps versus energy $E$ from the linear response Bogoliubov calculation (solid line). The dashed line is the transmission coefficient obtained from propagating a Gaussian wave packet in the nonlinear GP equation (1), in real time. The dotted line in (A) is the Thomas-Fermi approximation. Feshbach resonances are seen clearly in all curves in (A) and (B) superimposed on a background caused by potential scattering. For parameters see [15, 35].

Figure 4

Collective Feshbach resonance: complete transmission (A) or total refection (B) of a small coherent atomic wave packet ${\psi }_b(x,t)$ with incident momentum $k_0=2.05$ (A) or $k_0=2.35$ (B), when scattering off a stationary, two-component BEC, trapped around $-1<x<1$, with a maximal density $n_b\left(x\right)\approx 500$, which is off scale. The dimensionless density $n_b(x,t)$ is depicted versus position $x$ (in natural units of the trap [15]) for three instants $t$: initially ($t_i$, dotted line), on impact ($t_0$, solid line), and finally ($t_f$, dashed line).