Noncritical Slowing Down of Photonic Condensation

We investigate the response of a photonic gas interacting with a reservoir of pumped dye molecules to quenches in the pump power. In addition to the expected dramatic critical slowing down of the equilibration time around phase transitions, we find extremely slow equilibration even far away from phase transitions. This noncritical slowing down can be accounted for quantitatively by fierce competition among cavity modes for access to the molecular environment, and we provide a quantitative explanation for this noncritical slowing down.

Figure 1

Top: time taken for the system to equilibrate after a quench in pump power by 1%, as a function of the pump power after the quench. Bottom: steady state populations $n_i$ of cavity modes $i$ ranging from [0, 0] to [1, 2] and [0, 3]. The mode populations feature sharp increases and drops under increase of pump power, and the equilibration times show clear peaks around those phase transitions. In addition to this, the equilibration is also strongly slowed down in the intervals labeled $D$ and $E$ far away from any known phase transition.

Figure 2

Moduli squares $|{\mathrm{\Phi }}_i(x){|}^2$ for $i=0$, 1, 2 of the three lowest eigenfunctions of a quantum harmonic oscillator are depicted in red. The corresponding harmonic potential and lines indicating the eigenenergies are depicted in grey. The excitation profile functions ${\mathbf{e}}_i(x)$ corresponding to each of these cavity modes are depicted in blue. Green bars (C0 and C2) approximate one-dimensional cuts through the areas in which modes [0, 0] and [0, 2] compete for excitations and clamp molecules when condensed. Yellow bars (G1) denote the approximate locations of molecules absorbing photons from or emitting photons into mode [0, 1]. Mode [0, 1] experiences competition through the whole of G1 with either [0, 0] or [0, 2].

Figure 3

Mode populations as a function of time after a quench in pump power from $3.16\times {10}^{-4}\kappa$ to $2.5\times {10}^{-1}\kappa$. Modes [0, 0] and [0, 2] reach their new steady state on a timescale of $10/\kappa$, but equilibration of mode [0, 1] is about 3 orders of magnitude slower. The dotted line depicts the analytic prediction of Eq. (3), that matches the simulated data very well over a range of 7 orders of magnitude. In this case, the equilibration time is determined by the dynamics of mode [0, 1].

Figure 4

Time taken to reach steady state after the pump power is changed from the value indicated at the left to the value depicted at the bottom of the figure. The axes also depict the stationary state populations for the lowest three cavity modes as also shown in Fig. 1 and the division into intervals $A$ to $E$. One can see a clear enhancement of equilibration time close to the phase transitions, but also in the interval $D$ and $E$ of postquench pump power.