Kardar-Parisi-Zhang universality in the coherence time of nonequilibrium one-dimensional quasicondensates

We investigate the finite-size origin of the coherence time (or equivalently of its inverse, the emission linewidth) of a spatially extended, one-dimensional nonequilibrium condensate. We show that the well-known Schawlow-Townes scaling of laser theory, possibly including the Henry broadening factor, only holds for small system sizes, while in larger systems the linewidth displays a novel scaling determined by Kardar-Parisi-Zhang physics. This is shown to lead to an opposite dependence of the coherence time on the optical nonlinearity in the two cases. We then study how subuniversal properties of the phase dynamics such as the higher moments of the phase-phase correlator are affected by the finite size and discuss the relation between the field coherence and the exponential of the phase-phase correlator. We finally identify a configuration with enhanced open boundary conditions, which supports a spatially uniform steady state and facilitates experimental studies of the coherence time scaling.

Figure 1

Coherence time ${\tau }_c\left(L\right)$ of a one-dimensional condensate plotted as a function of the system length $L/l^{*}$ measured in natural units. The coherence time is here normalized to ${\tau }_c\left(l^{*}\right)$ as computed for $L=l^{*}$. The three numerical data series correspond to the full CGLE (green squares) and the KSE Eq. (4) (blue circles) and KPZ Eq. (5) (red triangles) approximations. The interaction constant is set to zero, $g=0$, and the other CGLE parameters are $\gamma =0.1$, $m=10$, $P=2\gamma$, $D=\gamma$, $n_S=1000$. In the present $g=0$ case, the KSE has $\tilde{\mu },\alpha =0$. The parameters of the KPZ are chosen as $D/2\nu =1.92,\lambda =2$, so to match the same universal long-wavelength physics as the CGLE and KSE. For small sizes Bogoliubov theory holds and the coherence time scales proportionally to the system length as predicted by the Schawlow-Townes theory (black dashed line). For longer systems, instead, the nonlinearity of the phase dynamics determines a stronger broadening of the linewidth and a scaling ${\tau }_c\propto L^{1/2}$ [red line, see Eq. (19)].

Figure 2

Coherence time in natural units ${\tilde{\tau }}_c={\left({\varphi }^{*}\right)}^2{\tau }_c/t^{*}$ as a function of the Henry parameter $\alpha =2g{n}_0/\mathrm{\Gamma }$ for three different values of the system size. The numerical points are obtained from simulations of the CGLE (1) using the same parameters as in Fig. 1 except for $n_S=400$. The dotted lines correspond to the Henry-Schawlow-Townes formula Eq. (12), while the dashed lines represent the KPZ scaling Eq. (20). These theoretical predictions hold, respectively, for large $\alpha$ and small but finite $\alpha$. At zero $\alpha$, instead, one has to consider the KSE (4) with $\tilde{\mu }=0$, which renormalizes to a KPZ with $\mathcal{D}/2\nu \simeq 1.92,\lambda =2$, whose coherence time is indicated by the horizontal solid lines.

Figure 3

Temporal evolution of the skewness $\text{skew}\left(t\right)$ of the phase for different system sizes, from the CGLE with the same parameters as in Fig. 1. The dotted line is the value expected from the Baik-Rains distribution.

Figure 4

(a) Sketch of different boundary conditions for a discrete lattice geometry. The color code indicates the spatial intensity profile of the steady-state condensate: the enhanced open boundary conditions (eOBC) introduced in Eq. (24) allow us to obtain a uniform density. (b) Coherence time as a function of the system size in the two cases of PBC (green) and eOBC (orange), plotted in the same units as in Fig. 1. The system with eOBC supports Bogoliubov modes with longer lifetime and, thus, displays an earlier departure from the Schawlow-Townes scaling at smaller system sizes.

Figure 5

Real (left) and imaginary (right) part of the Bogoliubov dispersion of collective excitations Eq. (A1) as a function of momentum $k$. Blue (red) points refer to vanishing (finite) values of the optical nonlinearity $g$.