Matter-wave dark solitons in a double-well potential

We study stability of the first excited state of quasi-one-dimensional Bose-Einstein condensates in a double-well potential, which is called “$\pi$ state.” The density notch in the $\pi$ state can be regarded as a standing dark soliton. From the excitation spectrum, we determine the critical barrier height, above which the $\pi$ state is dynamically unstable. We find that the critical barrier height decreases monotonically as the number of condensate atoms increases. We also simulate the dynamics of the $\pi$ state by solving the time-dependent Gross-Pitaevskii equation. We show that due to the dynamical instability the dark soliton starts to move away from the trap center and exhibits a large-amplitude oscillation.

Figure 1

(Color online) The condensate wave functions for the ground state (solid line) and the $\pi$ state (dashed line) of BECs in a double-well potential. We set the potential barrier $V_0∕\left(\hslash {\omega }_z\right)=10.0$. We consider condensates of ${}^{23}\mathrm{Na}$ atoms, the mass is $23\mathrm{amu}$ (atomic mass units) and the atom number $N=2000$. The values of parameters are as follows: The harmonic-oscillator frequency is ${\omega }_z∕\left(2\pi \right)=10\mathrm{Hz}$, the frequency of the radial confinement is ${\omega }_{\perp }∕\left(2\pi \right)=250\mathrm{Hz}$ and the $s$-wave scattering length is $a_s=3\mathrm{nm}$. The harmonic oscillator length of the axial confinement is $a_{\mathrm{z}}=\sqrt{\left[\hslash /\left(m{\omega }_z\right)\right]}$ and $g$ is the coupling constant in 1D.

Figure 2

Stability phase diagram calculated from the Bogoliubov equations (7) (solid line) and the two-mode approximation (dashed line).

Figure 3

The lowest or imaginary eigenvalue of the Bogoliubov equations (7) as a function of $V_0∕\left(\hslash {\omega }_z\right)$ with the fixed atom number $N=2000$.

Figure 4

The closed circles represent the excitation energy for $N=2000$ as a function of $V_0∕\left(\hslash {\omega }_z\right)$. ${\epsilon }_{\mathrm{tma}}$ is also plotted by the open circles.

Figure 5

The closed circles represent the excitation energy for $N=10$ as a function of $V_0∕\left(\hslash {\omega }_z\right)$. ${\epsilon }_{\mathrm{tma}}$ is also plotted by the opened circles.

Figure 6

(Color online) Time evolution of (a) the density and (b) the phase of the condensate wave function obtained by solving the time-dependent GP equation starting with the $\pi$ state. The barrier height is $V_0∕\left(\hslash {\omega }_z\right)=0$, where the system is dynamically stable. The dimensionless time is defined by $\tilde{t}\equiv t{\omega }_z$. The harmonic oscillator length of the axial confinement is $a_{\mathrm{z}}=\sqrt{\left[\hslash /\left(m{\omega }_z\right)\right]}$.

Figure 7

(Color online) Time evolution of (a) the density and (b) the phase of the condensate wave function in the $\pi$ state obtained by solving the time-dependent GP equation in the dynamically unstable region $V_0∕\left(\hslash {\omega }_z\right)=6.0$. The dimensionless time is defined by $\tilde{t}\equiv t{\omega }_z$. The harmonic oscillator length of the axial confinement is $a_{\mathrm{z}}=\sqrt{\left[\hslash /\left(m{\omega }_z\right)\right]}$.

Figure 8

(Color online) The kink position $z$ of the gray soliton as a function of the dimensionless time $\tilde{t}\equiv t{\omega }_z$. The cases of $V_0∕\left(\hslash {\omega }_z\right)=0$, 0.1 and 0.2 (in the dynamically stable region) correspond to the solid line. The dashed, dotted, and dashed-dotted lines represent the kink positions for $V_0∕\left(\hslash {\omega }_z\right)=2.0$, $V_0∕\left(\hslash {\omega }_z\right)=4.0$, and $V_0∕\left(\hslash {\omega }_z\right)=6.0$ (in the dynamically unstable region), respectively.