Nonlinear instability and dynamics of polaritons in quantum systems

We present analytical and simulation studies of the nonlinear instability and dynamics of an electron–hole/anti-electron (hereafter referred to as polaritons) system, which are common in ultra-small devices (semiconductors and micromechanical systems) as well as in dense astrophysical environments and the next generation intense laser–matter interaction experiments. Starting with three coupled nonlinear equations (two Schrödinger equations for interacting polaritons at quantum scales and the Poisson equation determining the electrostatic interactions and the associated charge separation effect), we demonstrate novel modulational instabilities and nonlinear polaritonic structures. It is suggested that the latter can transport information at quantum scales in high-density, ultracold quantum systems.


DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
the polaritons obey Fermi-Dirac statistics and their dynamics is, in turn, governed by the nonlinear Schrödinger and Poisson equations. The latter are naturally deduced from the GQH equations within the framework of an eikonal representation [10].
Since the polaritons are building blocks of many physical systems as described above, it is timely to present some novel collective interactions involving nonlinear interactions among electrons and holes/anti-electrons in quantum mechanical systems. Specifically, in this paper we present analytical and simulation studies of nonlinearly interacting polaritons and demonstrate the possibility of a new class of modulational instabilities and localized nonlinear structures (bright and dark envelope excitations and quantum vortex pairs). The latter may be exploited to transport information at quantum scales in semiconductors and micromechanical systems.
Let us first present the mathematical model which governs the dynamics of a polariton system. The collective motion of the particles is in this model described by effective Schrödinger equations [10] for the electrons and holes/anti-electrons (denoted by the subscript 'e' and 'h', respectively), coupled with the Poisson equation, ih where W e = m e v 2 Fe |ψ e | 4/D /2n The total energy has been obtained from equations (1) and (2) by using the identity , which is equivalent to the Poisson equation (3).
For the numerical analysis, it is convenient to introduce normalized variables so that a set of key parameters can be identified. Hence, normalizing the wavefunctions ψ e and ψ h by n

DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
We next consider the stability of the system (1)-(3). Using the Fourier decomposition in (1)- (3), where ψ e0 and ψ h0 are the envelopes of the equilibrium electron and hole wavefunctions and ψ e± , ψ h± (|ψ e± |, |ψ h± | |ψ e0 |, |ψ h0 |) and φ are the envelopes of the small-amplitude perturbations of the wavefunctions and potential, respectively, and sorting the equations by different Fourier components, we have the dispersion relations for the electron and hole zeroth order We note from equation (3) that |ψ e0 | = |ψ h0 | (= n 1/2 0 ). The nonlinear dispersion relation for the small-amplitude density modulations is where we have Here the nonlinear wave modes are characterized by We have solved the dispersion relation (7) numerically and have presented the growth rate (the imaginary part γ of ) in figure 1. We have taken the coupling constant A e = 5 and have assumed a two-dimensional (D = 2) geometry in the x-y-plane. The used mass ratio m h /m e = 1 is typical for light holes while m h /m e = 5 is typical for heavy holes [11]. We have taken the wavevectors K e0 and K h0 with opposite signs and directed along the x-axis, K e0 = xk e0 and K h0 = xk h0 , where x is the unit vector in the x-direction. Hence, the electrons and holes are counter-streaming, and this gives rise to a streaming instability, as can be seen in figure 1. For the smaller wavenumber |k e0 | = |k h0 | = 0.5 λ −1 F , the growth rate is smaller than for the larger wavenumber |k e0 | = |k h0 | = 1.0 λ −1 F . Comparing the panels (a) and (b) with panels (c) and (d) of figure 1, we also see that the wave modes for the larger |k e0 | and |k h0 | have a wider spectrum of growing waves in oblique directions to the x-axis, while the wave modes for the smaller wavenumbers have growing wave modes primarily in the x-direction. When k e0 and k h0 are taken to be equal to each other, then the system is stable, i.e. the dispersion relation (7) has only realvalued roots in this case. In order to study the nonlinear saturation of the streaming instability, we have solved the time-dependent system of equations (1) For a localized structure, we have d/dr = 0, φ = 0, | e | = | h0 | = n the system decouples completely into systems similar to those used to model BECs. In order to assess the dynamics and interaction between vortices, we have solved the time-dependent system (1)-(3) numerically, and as initial conditions we have used electron density perturbations in the form of vortex-like structures. The results are presented in figures 4 and 5. In figure 4, we used the initial condition ψ e = n 1/2 Here  seen at y ≈ ±15 λ F and x ≈ 0 λ F at time t = 24 t F in figure 4. These hole vortex pairs have moved further to y ≈ ±20 λ F at time t = 30 t F and are associated with a negative potential due to the sharp depletion of the hole density associated with the vortex pairs. In figure 5, we used the initial condition ψ e = n  figure 4) electron vortex pair which moves towards positive y at x = 0. This vortex pair, which is located at y = 20 λ F at t = 81 t F , is associated with a localized positive potential. We could also see the formation of a long-lived and slowly moving vortex pair in the hole fluid, seen at y ≈ −10 λ F (and x = 0 λ F ) at t = 8.1 t F , which is associated with a negative potential. In conclusion, we have reported a new class of modulational instability and the formation of nonlinear structures in a polaritonic system. Our numerical analysis revealed the formation of electron and hole density humps (bright solitary waves) as possible nonlinear saturation of the modulational/streaming instability, while long-lived vortex pairs (dark solitons) can be excited in the electron-hole system via nonlinear interactions. Quantum vortices could be exploited to transport information at quantum scales in semiconductors and micromechanical devices, as well as in metal clusters [12].