Colloquium: Nonlinear collective interactions in quantum plasmas with degenerate electron fluids

The current understanding of some important nonlinear collective processes in quantum plasmas with degenerate electrons is presented. After reviewing the basic properties of quantum plasmas, model equations (e.g., the quantum hydrodynamic and effective nonlinear Schrödinger-Poisson equations) are presented that describe collective nonlinear phenomena at nanoscales. The effects of the electron degeneracy arise due to Heisenberg’s uncertainty principle and Pauli’s exclusion principle for overlapping electron wave functions that result in tunneling of electrons and the electron degeneracy pressure. Since electrons are Fermions (spin-$1/2$ quantum particles), there also appears an electron spin current and a spin force acting on electrons due to the Bohr magnetization. The quantum effects produce new aspects of electrostatic (ES) and electromagnetic (EM) waves in a quantum plasma that are summarized in here. Furthermore, nonlinear features of ES ion waves and electron plasma oscillations are discussed, as well as the trapping of intense EM waves in quantum electron-density cavities. Specifically, simulation studies of the coupled nonlinear Schrödinger and Poisson equations reveal the formation and dynamics of localized ES structures at nanoscales in a quantum plasma. The effect of an external magnetic field on the plasma wave spectra and develop quantum magnetohydrodynamic equations are also discussed. The results are useful for understanding numerous collective phenomena in quantum plasmas, such as those in compact astrophysical objects (e.g., the cores of white dwarf stars and giant planets), as well as in plasma-assisted nanotechnology (e.g., quantum diodes, quantum free-electron lasers, nanophotonics and nanoplasmonics, metallic nanostructures, thin metal films, semiconductor quantum wells, and quantum dots, etc.), and in the next generation of intense laser-solid density plasma interaction experiments relevant for fast ignition in inertial confinement fusion schemes.

Figure 1

The plasma diagram in the $\log T\mathrm{\text{−}}\log n_e$ plane, separating the classical and quantum regimes. From 146.

Figure 2

The electron density $|W{|}^2$ (upper panel) and ES potential $\phi$ (lower panel) associated with a dark soliton supported by the system of Eqs. (64, 65), for $\mathcal{A}=5$ (solid lines), $\mathcal{A}=1$ (dashed lines), and $\mathcal{A}=0.2$ (dash-dotted line). From 171.

Figure 3

The time development of the electron density $|\Psi {|}^2$ (left-hand panel) and ES potential $\phi$ (right-hand panel), obtained from a simulation of the system of Eqs. (57, 58). The initial condition is $\Psi =0.18+\tanh [20\sin (x/10)]\exp (i{K}_{s}x)$, with $K_s=v_0/2\mathcal{A}$, $\mathcal{A}=5$, and $v_0=5$. From 171.

Figure 4

The electron density $|\Psi {|}^2$ (upper panel) and ES potential $\phi$ (lower panel) associated with 2D electron vortices supported by the system (67, 68), for the charge states $s=1$ (solid lines), $s=2$ (dashed lines), and $s=3$ (dash-dotted lines), with $\mathcal{A}=5$ in all cases. From 171.

Figure 5

The electron density $|\Psi {|}^2$ (left panel) and an arrow plot of the electron current $i(\Psi \nabla {\Psi }^{*}-{\Psi }^{*}\nabla \Psi )$ (right panel) associated with singly charged ($|s|=1$) 2D electron vortices, obtained from a simulation of the time-dependent system of Eqs. (57, 58), at times $t=0$, $t=3.3$, $t=6.6$, and $t=9.9$ (upper to lower panels), with $\mathcal{A}=5$. The singly charged vortices form pairs and keep their identities. From 171.

Figure 6

The electron density $|\Psi {|}^2$ (left panel) and an arrow plot of the electron current $i(\Psi \nabla {\Psi }^{*}-{\Psi }^{*}\nabla \Psi )$ (right panel) associated with double charged ($|s|=2$) 2D electron vortices, obtained from a simulation of the time-dependent system of Eqs. (57, 58), at times $t=0$, $t=3.3$, $t=6.6$, and $t=9.9$ (upper to lower panels), with $\mathcal{A}=5$. The doubly charged vortices dissolve into nonlinear structures and wave turbulence. From 171.

Figure 7

Small-scale fluctuations in the electron density resulting from steady turbulence simulations, for $H=0.4$ (a, b) and $H=0.01$ (c, d). Forward cascades are responsible for the generation of small-scale fluctuations seen in (a) and (c). Large-scale structures are present in the ES potential, seen in (b) and (d), essentially resulting from an inverse cascade. From 163.

Figure 8

Energy ${\mathcal{E}}_k$ per vector wave number of 3D EPOs in the forward cascade regime. A Kolmogorov-like spectrum ${\mathcal{E}}_k\sim k^{-11/3}$ is observed for $H=0.4$. The spectral index changes as a function of $H$. From 163.

Figure 9

Time evolution of the effective electron diffusion coefficient associated with the large-scale ES potential and the small-scale electron density, for $H=0.4$, $H=0.1$, and $H=0.01$. Smaller values of $H$ correspond to a small effective diffusion coefficient, which characterizes the presence of small-scale turbulent eddies that suppress the electron transport. From 163.

Figure 10

The profiles of the CPEM vector potential $W$ (top row), the electron number density $P^2$ (middle row), and the scalar potential $\Phi$ (bottom row) for $\lambda =-0.3$ (left column), $\lambda =-0.34$ (middle column), and $\lambda =-0.4$ (right column), with $H_e=0.002$. From 172.

Figure 11

The profiles of the CPEM vector potential $W$ (top row), the electron number density $P^2$ (middle row), and the scalar potential $\Phi$ (bottom row) for $H_e=0.007$ (left column) and $H_e=0.002$ (right column), with $\lambda =-0.34$. From 172.

Figure 12

The dynamics of the CPEM vector potential $A_{\perp }$ and the electron number density $|\psi {|}^2$ (upper panels) and of the electrostatic potential $\Phi$ (lower panel) for $H_e=0.002$. From 172.

Figure 13

The dynamics of the CPEM vector potential $A_{\perp }$ and the electron number density $|\psi {|}^2$ (upper panels) and the electrostatic potential $\varphi$ (lower panel) for $H_e=0.007$. From 172.

Figure 14

Dispersion curves for EB waves in a Fermi-Dirac distributed plasma, showing several EB modes and the UH branch. From 49.