Numerical study of localized impurity in a Bose-Einstein condensate

Motivated by recent experiments, we investigate a single ${}^{133}\text{Cs}$ impurity in the center of a trapped ${}^{87}\text{Rb}$ Bose-Einstein condensate (BEC). Within a zero-temperature mean-field description we provide a one-dimensional physical intuitive model which involves two coupled differential equations for the condensate and the impurity wave function, which we solve numerically. With this we determine within the equilibrium phase diagram spanned by the intra- and interspecies coupling strength whether the impurity is localized at the trap center or expelled to the condensate border. In the former case we find that the impurity induces a bump or dip on the condensate for an attractive or a repulsive Rb-Cs interaction strength, respectively. Conversely, the condensate environment leads to an effective mass of the impurity which increases quadratically for small interspecies interaction strength. Afterwards, we investigate how the impurity imprint upon the condensate wave function evolves for two quench scenarios. At first we consider the case that the harmonic confinement is released. During the resulting time-of-flight expansion it turns out that the impurity-induced bump in the condensate wave function starts decaying marginally, whereas the dip decays with a characteristic time scale which decreases with increasing repulsive impurity-BEC interaction strength. Second, once the attractive or repulsive interspecies coupling constant is switched off, we find that white-shock waves or bisolitons emerge which both oscillate within the harmonic confinement with a characteristic frequency.

Figure 1

Geometric function $f\left({\omega }_{\text{Ir}}/{\omega }_{\text{r}}\right)$ reaches its maximum value at $1+m_{\text{B}}/m_{\text{I}}$.

Figure 2

Equilibrium phase diagram spanned by $g_{\text{B}}$ and $g_{\text{IB}}$ for (a) $N_{\text{B}}=20$ (b) $N_{\text{B}}=200$, and (c) $N_{\text{B}}=800$. Impurity is localized at the trap center (blue) or expelled to the condensate border (red) together with the unstable region (black) in dimensionless units.

Figure 3

Numerical density profile of BEC (a, b) and impurity (c, d) for the two-particle Rb-Rb coupling constant value $G_{\text{B}}=16000$ and for interspecies coupling constants $g_{\text{IB}}$ which increase from top to bottom according to the inlets. For increasing negative values of $g_{\text{IB}}$ the impurity-induced bump (a) in the condensate wave function decreases, whereas for positive values the corresponding dip (b) increases in dimensionless units.

Figure 4

(a) Height/depth and (b) width of the impurity bump/dip according to Eqs. (8)–(10) versus the impurity-BEC coupling constant $g_{\text{IB}}$ for the BEC coupling constant $G_{\text{B}}=16000$ calculated numerically by solving 1DDEs (6) and (7) in dimensionless units.

Figure 5

Effective mass of ${}^{133}\text{Cs}$ impurity versus the impurity-BEC coupling strength $g_{\text{IB}}$. The inset shows that the effective mass increases quadratically for small impurity-BEC coupling strength $g_{\text{IB}}$ in dimensionless units.

Figure 6

Impurity imprint height/depth after having released the trap versus time for (a) increasing negative and (b) decreasing positive values of the impurity-BEC coupling constant $g_{\text{IB}}$ from top to bottom. Inset: Relaxation time $t_{\text{rel}}$ decreases with increasing $g_{\text{IB}}$ in dimensionless units.

Figure 7

Dynamics of the impurity density represented in color scale after having switched off the harmonic trap for the initial $g_{\text{IB}}=-8$ for the BEC coupling strength $G_{\text{B}}=16000$ in dimensionless units.

Figure 8

Impurity imprint width after having released the trap versus time for decreasing positive values of the impurity-BEC coupling constant $g_{\text{IB}}$ from top to bottom in dimensionless units.

Figure 9

Coherent matter-wave time-of-flight evolution of the depleted density ${‖{\psi }_{\text{B}}\left(z,t\right)‖}_{\text{DD}}^2={‖{\psi }_{\text{B}}\left(z,t\right)‖}_{{\mathrm{g}}_{\text{IB}}}^2-{‖{\psi }_{\text{B}}\left(z,t\right)‖}_{{\mathrm{g}}_{\text{IB}}=0}^2$ represented in color scale after having switched off the trap for the BEC coupling constant $G_{\text{B}}=16000$: (a) $g_{\text{IB}}=20$ and (b) $g_{\text{IB}}=80$ in dimensionless units.

Figure 10

Density profile of the BEC represented in color scale after having switched off the impurity-BEC coupling constant for initial $g_{\text{IB}}=120$ for the BEC coupling constant $G_{\text{B}}=16000$ in dimensionless units.

Figure 11

Center-of-mass positions of excitations ${\overline{z}}_{\mathrm{L}}$ (solid circles) and ${\overline{z}}_{\mathrm{R}}$ (empty circles) according to Eq. (11) versus time after having switched off (a) negative decreasing and (b) positive increasing values of $g_{\text{IB}}$ from top to bottom. Black solid circles represent the region of colliding excitations, where mean positions are not perfectly detectable in dimensionless units.