Quantum Langevin model for nonequilibrium condensation

We develop a quantum model for nonequilibrium Bose-Einstein condensation of photons and polaritons in planar microcavity devices. The model builds on laser theory and includes the spatial dynamics of the cavity field, a saturation mechanism, and some frequency dependence of the gain: quantum Langevin equations are written for a cavity field coupled to a continuous distribution of externally pumped two-level emitters with a well-defined frequency. As an example of application, the method is used to study the linearized quantum fluctuations around a steady-state condensed state. In the good-cavity regime, an effective equation for the cavity field only is proposed in terms of a stochastic Gross-Pitaevskii equation. Perspectives in view of a full quantum simulation of the nonequilibrium condensation process are finally sketched.

Figure 1

A pictorial representation of the model. Emitters lose energy at a rate $\gamma$ while energy is pumped in at a rate $d$. Photons can leave the cavity after a time ${\kappa }^{-1}$.

Figure 2

Intensity of the field (upper panels) and oscillation frequency of the condensate (lower panels) as a function of the pumping parameter $x=d/\gamma$. Both quantities are shown for different values of self-interaction $\lambda$ and natural detuning $\nu -{\omega }_0$. In all panels, $\gamma =10\kappa$ and $g\sqrt{n_A}=7\kappa$.

Figure 3

Dispersion ${\lambda }_k^{\mathrm{Bog}}$ of the collective modes as predicted by the eigenvalues of the Bogoliubov matrix ${\mathbb{A}}_{\mathbf{k}}$ in the interacting case with $\lambda n_A=0.1\kappa$ and $\nu -{\omega }_0=-10\kappa$. Left panels show magnified views of the low-$\mathbf{k}$ region of right panels. System parameters: $\gamma =100\kappa$, $g\sqrt{n_A}=25$, and $x=5$.

Figure 4

Imaginary (a) and real (b) part of the dispersion of the collective modes and steady-state momentum distribution (c)–(d) in the vicinity of the Mollow instability onset. (c) shows a magnified view of the low-$\mathbf{k}$ region of (d). System parameters: $\gamma =10\kappa$, $g\sqrt{n_A}=42\kappa$, $x=5$, $\lambda n_A=0.1\kappa$, $\nu -{\omega }_0=0$.

Figure 5

Steady-state momentum distribution. Left panels show magnified views of the low-$\mathbf{k}$ region of right panels. (a), (b) Noninteracting case $\lambda n_A=\nu -{\omega }_0=0$. (c), (d) $\lambda n_A=0$, $\nu -{\omega }_0=-10\kappa$. (e), (f) $\lambda n_A=0.1\kappa$, $\nu -{\omega }_0=0$. System parameters: $\gamma =100\kappa$, $g\sqrt{n_A}=25$, and $x=5$.

Figure 6

Normalized momentum- and frequency-resolved spectrum of the photoluminescence from the cavity. Left panel: Detuned $\nu -{\omega }_0=-35\kappa$ case with $\lambda n_A=0$. Right panel: Resonant cavity $\nu -{\omega }_0=0$ with photon-photon interactions $\lambda n_A=0.1\kappa$. Other system parameters: $\gamma =10\kappa$, $g\sqrt{n_A}=7\kappa$, $x=7$.

Figure 7

Diffusion coefficients $D_{\varphi {\varphi }^{*}}$ (solid lines), $\text{Re}\left[D_{\varphi \varphi }\right]$ (dashed lines), and $\text{Im}\left[D_{\varphi \varphi }\right]$ (dotted lines) appearing in the SGPE for a field $\psi$ equal to the mean-field steady state. The quantities are plotted as a function of the pumping parameter $x=d/\gamma$ for different regimes of photon-photon interactions (left to right). The top (bottom) row refers to the SGPE in the normal (Wigner) ordering case. In all panels, we have taken $\gamma =100\kappa$ and $g\sqrt{n_A}=25\kappa$.

Figure 8

Comparison between SGPE and the full model. First row and second row: Eigenvalues of the Bogoliubov matrix in functions of the momentum; solid lines refers to SGPE quantities, dashed ones to the full model. Last row: Momentum distributions. In all panels, $\lambda n_A=\nu -{\omega }_0=0$, $x=2$.