Stability of Bose-Einstein condensates in a Kronig-Penney potential

We study the stability of Bose-Einstein condensates with superfluid currents in a one-dimensional periodic potential. By using the Kronig-Penney model, the condensate and Bogoliubov bands are analytically calculated and the stability of condensates in a periodic potential is discussed. The Landau and dynamical instabilities occur in a Kronig-Penney potential when the quasimomentum of the condensate exceeds certain critical values as in a sinusoidal potential. It is found that the onsets of the Landau and dynamical instabilities coincide with the point where the perfect transmission of low energy excitations through each potential barrier is forbidden. The Landau instability is caused by the excitations with small $q$ and the dynamical instability is caused by the excitations with $q=\pi ∕a$ at their onsets, where $q$ is the quasimomentum of excitation and $a$ is the lattice constant. A swallow-tail energy loop appears at the edge of the first condensate band when the mean-field energy is sufficiently larger than the strength of the periodic potential. We find that the upper portion of the swallow-tail is always dynamically unstable, but the second Bogoliubov band has a phonon spectrum reflecting the positive effective mass.

Figure 1

Schematic picture of a current-carrying condensate in a KP potential. The solid lines, gray bars, and dashed arrow express the condensate density ${\mid {\Psi }_0\left(x\right)\mid }^2$, potential barriers, and supercurrent, respectively.

Figure 2

Amplitude $A\left(x\right)$ of the condensate wave function when $V_0=5g{n}_0{\xi }_0$, $a=15{\xi }_0$, and $Q=0.09{\xi }_0^{-1}$. The dashed and solid lines represent the exact solution Eq. (16) and the approximate solution Eq. (20), respectively.

Figure 3

Phase $S\left(x\right)$ of the condensate wave function when $V_0=5g{n}_0{\xi }_0$, $a=15{\xi }_0$, and $Q=0.09{\xi }_0^{-1}$. The dashed line, thick solid line and thin solid line, represent the exact solution Eq. (18), approximate solution Eq. (22), and asymptotic solution $Qx+\mathrm{sgn}\left(x\right)\phi ∕2$, respectively.

Figure 4

Superfluid momentum $Q$ as a function of the phase difference $\phi$. The thick black line, the gray line, and the thin black line represent the superfluid momenta with $(a,V_0)=(15{\xi }_{\mathrm{av}},5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, $(15{\xi }_{\mathrm{av}},7.5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, and $(15{\xi }_{\mathrm{av}},15g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, respectively.

Figure 5

Chemical potential $\mu$ as a function of the quasimomentum $K$. The thick black line, the gray line, and the thin black line represent the chemical potentials with $(a,V_0)=(15{\xi }_{\mathrm{av}},5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, $(15{\xi }_{\mathrm{av}},7.5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, and $(15{\xi }_{\mathrm{av}},15g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, respectively.

Figure 6

Energy $E$ of the condensate per site as a function of the quasimomentum $K$, namely the first condensate band. The thick black line, the gray line, and the thin black line represent the first condensate bands with $(a,V_0)=(15{\xi }_{\mathrm{av}},5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, $(15{\xi }_{\mathrm{av}},7.5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, and $(15{\xi }_{\mathrm{av}},15g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, respectively.

Figure 7

Group velocity $v_{\mathrm{g}}$ as a function of the quasimomentum $K$. The thick black line, the gray line, and the thin black line represent the group velocities with $(a,V_0)=(15{\xi }_{\mathrm{av}},5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, $(15{\xi }_{\mathrm{av}},7.5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, and $(15{\xi }_{\mathrm{av}},15g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, respectively.

Figure 8

Group velocity $v_{\mu }$ as a function of the quasimomentum $K$. The thick black line, the gray line, and the thin black line represent the group velocities with $(a,V_0)=(15{\xi }_{\mathrm{av}},5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, $(15{\xi }_{\mathrm{av}},7.5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, and $(15{\xi }_{\mathrm{av}},15g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, respectively.

Figure 9

Effective mass $m^{*}$. The thick black line, the gray line, and the thin black line represent the effective masses with $(a,V_0)=(15{\xi }_{\mathrm{av}},5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, $(15{\xi }_{\mathrm{av}},7.5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, and $(15{\xi }_{\mathrm{av}},15g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$, respectively.

Figure 10

Condensate quasimomentum $K_{\mathrm{M}}$ which gives the maximum group velocity.

Figure 11

Condensate quasimomentum $K_{\mathrm{E}}$ which gives the edge of the swallow-tail.

Figure 12

Schematic picture of the solutions of the single-barrier problem. Scattering processes in which (a) the excitation comes from the left-hand side, and (b) the excitation comes from the right-hand side. The solid curves, solid arrows, dashed curves, and dashed arrows describe the condensate wave functions, scattering components, localized components, and condensate supercurrent, respectively.

Figure 13

Transmission coefficient $T^{\mathrm{l}}$ as a function of $\epsilon$ and $\phi$ for $V_0=5g{n}_0{\xi }_0$. $T^{\mathrm{l}}$’s with $\phi =0$, $\phi =\pi ∕3$, $\phi ={\phi }_{\mathrm{A}}\simeq \pi ∕2$, $\phi =2\pi ∕3$, and $\phi =\pi$ are shown. This figure is almost the same as Fig. 2(a) of Ref. 23 where $V_0=10g{n}_0{\xi }_0$.

Figure 14

Phase shift ${\delta }^{\mathrm{l}}$ of $t^{\mathrm{l}}$ as a function of $\epsilon$ and $\phi$ for $V_0=5g{n}_0{\xi }_0$. ${\delta }^{\mathrm{l}}$’s with $\phi =0$, $\phi =\pi ∕3$, $\phi ={\phi }_{\mathrm{A}}\simeq \pi ∕2$, $\phi =2\pi ∕3$, and $\phi =\pi$ are shown.

Figure 15

Bogoliubov band of a current-carrying condensate in a KP potential for $a=15{\xi }_{\mathrm{av}}$, $V_0=5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}}$, and $K=\frac{K_{\mathrm{A}}}{2}$. The band structure is shown in (a) and the left-hand side of Eq. (79) as a function of energy is shown in (b).

Figure 16

Magnification of the swallow-tail energy loop in the first condensate band for $(a,V_0)=(15{\xi }_{\mathrm{av}},5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$ shown in Fig. 6.

Figure 17

First Bogoliubov band with $K⩽K_{\mathrm{A}}$ for $(a,V_0)=(15{\xi }_{\mathrm{av}},5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$.

Figure 18

First Bogoliubov band with $K_{\mathrm{A}}⩽K⩽K_{\mathrm{M}}$ for $(a,V_0)=(15{\xi }_{\mathrm{av}},5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$. (a) The real part of the first Bogoliubov band. (b) The imaginary part of the first Bogoliubov band.

Figure 19

First Bogoliubov band around the edge of the swallow-tail for $(a,V_0)=(15,5)$. (a) The real part of the first Bogoliubov band. (b) The imaginary part of the first Bogoliubov band.

Figure 20

Second Bogoliubov band around the edge of the swallow-tail for $(a,V_0)=(15,5)$.

Figure 21

Stability phase diagram of the condensate in a KP potential when $(a,V_0)=(15{\xi }_{\mathrm{av}},5g{n}_{\mathrm{av}}{\xi }_{\mathrm{av}})$. The light shaded area and the dark shaded area correspond to the regions of the Landau instability and the dynamical instability, respectively. This figure is focused to the swallow-tail energy loop.