Transcritical flow of a Bose-Einstein condensate through a penetrable barrier

The problem of the transcritical flow of a Bose-Einstein condensate through a wide repulsive penetrable barrier is studied analytically using the combination of the locally steady “hydraulic” solution of the one-dimensional Gross-Pitaevskii equation and the solutions of the Whitham modulation equations describing the resolution of the upstream and downstream discontinuities through dispersive shocks. It is shown that within the physically reasonable range of parameters, the downstream dispersive shock is attached to the barrier and effectively represents the train of very slow dark solitons, which can be observed in experiments. The rate of the soliton emission, the amplitudes of the solitons in the train, and the drag force are determined in terms of the Bose-Einstein condensate oncoming flow velocity and the strength of the potential barrier. Good agreement with direct numerical solutions is demonstrated. Connection with recent experiments is discussed.

Figure 1

Sketch of experimental setup in Ref. [21]. The potential barrier created by the laser beam is moved through the BEC along the $-x$ direction. The trap tightly confines the BEC in the radial $y$ and $z$ directions.

Figure 2

Plot of the function $\mu \left(v\right)$; for $V_m=0.5$ the critical values are equal to $v_{-}\approx 0.2$ and $v_{+}\approx 1.9$.

Figure 3

Plots of the fluid velocity $u\left(x\right)$ in hydraulic solution (11) for potential barrier (13) for which $v_{-}\approx 0.2$ and $v_{+}\approx 1.9$. (a) corresponds to subcritical velocity $v=0.1$ and (b) to supercritical velocity $v=2.0$. The physical branches are shown by solid lines and the nonphysical ones by dashed lines.

Figure 4

(a) Subcritical hydraulic solutions for $u$ at different values of $v<v_{-}$ (thin solid lines) and a separatrix solution at $v=v_{-}$ (thick solid line). (b) Supercritical hydraulic solution at different values of $v>v_{-}$ (thin solid lines) and separatrix solution at $v=v_{+}$ (thick solid line). Short-dashed lines: nonphysical branches. Long-dashed line: nonphysical separatrix.

Figure 5

Riemann invariants ${\lambda }_{\pm }$ as functions of $x$ in hydraulic solutions (29, 30) for potential (13) with $V_m=0.5$. Note that ${\lambda }_{+}^u\approx {\lambda }_{+}^d$.

Figure 6

Qualitative behavior of the Riemann invariants ${\lambda }_j$ as functions of $x$ in the full (hydraulic plus modulation) solution.

Figure 7

Qualitative behavior of the Riemann invariants in the partial (attached) downstream dispersive shock.

Figure 8

(a) Cutoff modulus $m^{\ast }$ in the downstream dispersive shock (dark soliton train). (b) Downstream dark soliton emission rate $f^{\ast }$ at the potential location vs oncoming BEC velocity $v$. Both panels correspond to $V_m=0.5$. At $v\approx 1.4$ the downstream dispersive shock wave detaches from the potential. Crosses in (b) correspond to the numerical simulation data. The dashed line in (b) corresponds to the frequency at the trailing edge of the detached downstream dispersive shock.

Figure 9

Dependence of the downstream dispersive shock trailing edge speed $s_{-}^d$ on transcritical BEC velocity $v$ calculated with formula (63). The detachment point $v=v^{\ast }\approx 1.4$ is found from the condition $s_{-}^d=0$.

Figure 10

Leading soliton parameters: (a) amplitude $A^d$ and (b) speed $s_{+}^d$—in the downstream dispersive shock (dark soliton train) vs oncoming BEC velocity $v$ for $V_m=0.5$. Solid line: modulation solution; crosses: direct numerical solution.

Figure 11

(a) Cutoff modulus $m^{\ast }$ in the downstream dispersive shock. (b) Downstream dark soliton emission rate $f^{\ast }$ vs potential strength $V_m$. Both panels correspond to $v=1.0$. Crosses in (b) correspond to the numerical simulation data.

Figure 12

Leading soliton parameters: (a) amplitude $A^d$ and (b) speed $s_{+}^d$—in the downstream dispersive shock (dark soliton train) vs potential strength $V_m$. Both panels correspond to $v=1.0$. Solid line: modulation solution; crosses: direct numerical solution.

Figure 13

Plot of the combined analytical ($\text{hydraulic}+\text{modulated}$ oscillatory) solution for the condensate density distribution in the wave pattern generated by a BEC flow with $v=1$ through the potential barrier with $V_m=0.5$ at $t=30$.

Figure 14

Numerical simulation of the condensate density distribution as a function of the space coordinate $x$ in the wave pattern generated by a BEC flow with $v=1$ through potential barrier (13) with $V_m=0.5$ and $\sigma =2$. The evolution time is equal to $t=30$.

Figure 15

Dependences of the velocities $v_{-}$ and $v_{+}$ at the boundaries of the transcritical region (solid lines) and of the velocities of $v^{\ast }$ and $v^{\ast \ast }$ (dashed lines) on the maximum value $V_m$ of potential.

Figure 16

Numerical simulation of the condensate density distributions as functions of $x$ in the wave pattern generated by potential barrier (13) with $V_m=0.5$ and (a) $v=1.4$ and (b) $v=1.8$. Evolution time $t=45$.

Figure 17

The drag force $F_{drag}$ vs (a) BEC oncoming flow $v$ in the transcritical region $\left(v_{-},v_{+}\right)$ and (b) potential strength $V_m$ for fixed $v=1$. Solid lines are drawn according to the approximate Eq. (97), and dots correspond to the full numerical solution of Eq. (93).