Hydrodynamic description of trapped ultracold paraexcitons in ${\mathrm{Cu}}_2\mathrm{O}$

In this paper we present a theoretical model to describe the dynamics of paraexcitons in cuprous oxide at ultralow temperatures inside potential traps. A possible condensate is described by a generalized Gross-Pitaevskii equation. The noncondensed excitons evolve under a set of hydrodynamic equations, which were derived from a quantum Boltzmann equation. The model takes into account the finite lifetime of the excitons, the pump laser, the Auger-like two-body decay, as well as exciton-exciton and exciton-phonon scattering. The numerical results show the strong influence of the Auger effect on exciton temperatures and densities not only at high pump powers but also at ultralow temperatures. Furthermore, the excitons do not cool down to very low bath temperatures $(T_{\mathrm{B}}\ll 0.5\mathrm{K})$ under continuous wave excitation. We also compare the results of the theoretical model with experimental data.

Figure 1

Density $\tilde{n}\left(r\right)$, velocity $v_n\left(r\right)$, temperature $T\left(r\right)$, and effective chemical potential ${\tilde{\mu }}_{\mathrm{eff}}\left(r\right)$ for the stationary state (blue solid line) at a phonon temperature of $T_{\mathrm{Ph}}=0.5\mathrm{K}$ and a laser power of $P_{\mathrm{L}}=26.4\mu \mathrm{W}$. The dashed red curves represent the corresponding values of a distribution function in equilibrium with the same mean temperature $\langle T\rangle =0.62\mathrm{K}$ and the same exciton number $N=2.13\times {10}^7$. The dashed black curve shows the phonon temperature $T_{\mathrm{Ph}}$. The Auger rate is set to $A=2.0\times {10}^{-18}\mathrm{c}{\mathrm{m}}^3/\mathrm{n}\mathrm{s}$.

Figure 2

Density $\tilde{n}\left(r\right)$ and temperature $T\left(r\right)$ for the stationary state at a laser power of $P_{\mathrm{L}}=2.8\mu \mathrm{W}$, a phonon temperature of $T_{\mathrm{Ph}}=1.0\mathrm{K}$ (solid red curve), $T_{\mathrm{Ph}}=0.5\mathrm{K}$ (solid blue curve), and $T_{\mathrm{Ph}}=0.037\mathrm{K}$ (solid black curve), and an Auger rate of $A=2.0\times {10}^{-18}\mathrm{c}{\mathrm{m}}^3/\mathrm{n}\mathrm{s}$. The dashed lines of the same color show the respective phonon temperatures. The dash-dotted black line in the right panel shows the temperature for $T_{\mathrm{B}}=0.037\mathrm{K}$ and $A=0$.

Figure 3

Density $\tilde{n}\left(r\right)$, temperature $T\left(r\right)$, and effective chemical potential ${\tilde{\mu }}_{\mathrm{eff}}\left(r\right)$ for the stationary state at a phonon temperature of $T_{\mathrm{Ph}}=0.037\mathrm{K}$ and a laser power of $P_{\mathrm{L}}=262\mu \mathrm{W}$ as well as the evolution of the total exciton number $N\left(t\right)$. The Auger rates are $A=2.0\times {10}^{-18}\mathrm{c}{\mathrm{m}}^3/\mathrm{n}\mathrm{s}$ [18] (blue curves), $A=7.0\times {10}^{-17}\mathrm{c}{\mathrm{m}}^3/\mathrm{n}\mathrm{s}$ [33] (red curves), and $A=4.0\times {10}^{-16}\mathrm{c}{\mathrm{m}}^3/\mathrm{n}\mathrm{s}$ [34] (black curves). The blue crosses mark the development of the exciton number without the Auger effect. The black dashed line represents the phonon temperature.

Figure 4

Left panel: Exciton temperature $T\left(r\right)$ (solid blue curve) and the mean exciton temperature $\langle T\rangle =0.40\mathrm{K}$ (dashed blue curve) under cw excitation with a pump power of $P_{\mathrm{L}}=3\mu \mathrm{W}$ and a phonon temperature of $T_{\mathrm{Ph}}=0.037\mathrm{K}$ (dashed black curve) in the stationary state. The dashed red curve shows the experimentally determined spectral temperature $T_{\mathrm{S}}=0.41\mathrm{K}$ (compare Fig. 3 in Ref. [18]). Right panel: Experimentally determined exciton numbers and spectral temperatures $T_{\mathrm{S}}$ (blue circles) for different pump powers (compare Fig. 11 in Ref. [18]) in comparison with the exciton numbers and the mean temperatures $\langle T\rangle$ (red crosses) as predicted by our theory.

Figure 5

Calculated exciton number $N\left(t\right)$ (dashed blue line) and mean temperature $\langle T\rangle$ (dashed red line) under pulsed excitation for a phonon temperature of $T_{\mathrm{Ph}}=0.82\mathrm{K}$. The laser pulse has been modeled according to the data provided in Ref. [16]. The blue and red circles represent the theoretical values averaged over a gate width of $200\mathrm{n}\mathrm{s}$. The blue and red crosses represent the experimentally determined values with the same gate width [16]. The Auger rate is set to $A=2.0\times {10}^{-18}\mathrm{c}{\mathrm{m}}^3/\mathrm{n}\mathrm{s}$.