Formation and Dynamics of Dark Solitons and Vortices in Quantum Electron Plasmas

We present simulation studies of the formation and dynamics of dark solitons and vortices in quantum electron plasmas. The electron dynamics in the latter is governed by a pair of equations comprising the nonlinear Schrödinger and Poisson system of equations, which conserves the number of electrons as well as their momentum and energy. The present governing equations in one spatial dimension admit stationary solutions in the form a dark envelope soliton. The dynamics of the latter reveals its robustness. Furthermore, we numerically demonstrate the existence of cylindrically symmetric two-dimensional quantum electron vortices, which survive during collisions. The nonlinear structures presented here may serve the purpose of transporting information at quantum scales in ultracold micromechanical systems and dense plasmas, such as those created during intense laser-matter interactions.

Figure 1

The electron density $|\Psi {|}^2$ (upper panel) and electrostatic potential $\phi$ (lower panel) associated with a dark soliton supported by the system of Eqs. (3, 4), for $A=5$ (solid lines), $A=1$ (dashed lines), and $A=0.2$ (dash-dotted lines).

Figure 2

The time development of the electron density $|\Psi {|}^2$ (left panel) and electrostatic potential $\phi$ (right panel), obtained from a simulation of the system of Eqs. (1, 2). The initial condition is $\Psi =0.18+\tanh [20\sin (x/10)]\exp (iKx)$, with $K=v_0/2A$, $A=5$, and $v_0=5$.

Figure 3

The electron density $|\Psi {|}^2$ (upper panel) and electrostatic potential $\phi$ (lower panel) associated with a two-dimensional vortex supported by the system (6, 7), for the charge states $n=1$ (solid lines), $n=2$ (dashed lines), and $n=3$ (dash-dotted lines). We used $A=5$ in all cases.

Figure 4

The electron density $|\Psi {|}^2$ (left panels) and an arrow plot of the electron current $i(\Psi \mathbf{\nabla }{\Psi }^{*}-{\Psi }^{*}\mathbf{\nabla }\Psi )$ (right panels) associated with singly charged ($n=1$) two-dimensional vortices, obtained from a simulation of the time-dependent system of Eqs. (1, 2), at times $t=0$, $t=3.3$, $t=6.6$, and $t=9.9$ (from top to bottom). We used $A=5$. The singly charged vortices form pairs and keep their identities.

Figure 5

The electron density $|\Psi {|}^2$ (left panels) and an arrow plot of the electron current $i(\Psi \mathbf{\nabla }{\Psi }^{*}-{\Psi }^{*}\mathbf{\nabla }\Psi )$ (right panels) associated with double charged ($n=2$) two-dimensional vortices, obtained from a simulation of the time-dependent system of Eqs. (1, 2), at times $t=0$, $t=3.3$, $t=6.6$, and $t=9.9$ (from top to bottom). We used $A=5$. The doubly charged vortices dissolve into nonlinear structures and wave turbulence.