Temporal coherence of one-dimensional nonequilibrium quantum fluids

We theoretically investigate the time dependence of the first-order coherence function for a one-dimensional driven dissipative nonequilibrium condensate. Simulations on the generalized Gross-Pitaevskii equation show that the characteristic time scale of exponential decay agrees with the linearized Bogoliubov theory in the regime of large interaction energy. For very weak interactions, the temporal correlation deviates from the linear theory, and instead respects the dynamic scaling of the Kardar-Parisi-Zhang universality class. This nonlinear dynamics is found to be quantitatively captured by a noisy Kuramoto-Sivashinsky equation for the phase dynamics.

Figure 1

Characteristic momentum dependence of correlation time ${\tau }_c$ in (a) the linear regime of $\mu =2$ and (b) the nonlinear regime of $\mu =0$. Three typical power-law dependences of $k$ are plotted by red dashed, black dashed-dotted, and blue solid lines as guide to the eyes. The green squares and red circles are simulations on GGPE and KSE, respectively, and the blue open circles denote Bogoliubov theory. $k_c$ is the bifurcation momentum, and $\Delta x=0.5$ the discretization length in real space.

Figure 2

Extraction of correlation time ${\tau }_c$ by fitting the temporal correlations in terms of Eq. (19). The upper panels [(a)–(c)] show the correlations for three different values of $k$ when $\mu =2$. The lower ones [(d)–(f)] are for $\mu =0$. The black squares and green circles are from GGPE and KSE simulations, respectively. Numerical fittings to exponential decays are illustrated by the red dashed and blue solid lines. The orange dashed-dotted lines represent results from Bogoliubov theory as a reference.

Figure 3

Time evolution of representative correlation functions in (a) the linear regime with $\mu =2$ and (b) the nonlinear regime with $\mu =0$. The blue solid lines depict the correlations at short distances, $x/\Delta x=16$, while the black dashed-dotted lines present those at long distances, $x/\Delta x=48$, both of which are obtained via GGPE. The corresponding KSE simulations are plotted by the red squares and green circles, respectively.

Figure 4

Dynamic scaling behaviors of GGPE (upper panels) and KSE (lower panels) for different interaction energies $\mu$. Here $f(t/x^z)$ is the dynamic scaling function, and $z$ is the scaling exponent extracted from numerical simulations (see text). The three different curves in each panel denote short, intermediate, and long distances.

Figure 5

Boundary effects on the dynamic scaling function $f(t/x^z)$ at the point $x/\Delta x=32$ in the simulation of the GGPE (upper panels) and KSE (lower panels), where $z$'s are the same scaling exponents as in Fig. 4. The boundary-induced modification is prominent for the chain of $N=64$ (blue dots), while it is negligible for $N=128$ (red dashed curves) and 256 (green solid curves).