Nonlinear excitations, stability inversions, and dissipative dynamics in quasi-one-dimensional polariton condensates

We study the existence, stability, and dynamics of the ground state and nonlinear excitations, in the form of dark solitons, for a quasi-one-dimensional polariton condensate in the presence of nonresonant pumping and nonlinear damping. We find a series of remarkable features that can be directly contrasted to the case of the typically energy-conserving ultracold alkali-atom Bose-Einstein condensates. For some sizable parameter ranges, the nodeless (“ground”) state becomes unstable toward the formation of stable nonlinear single- or multi-dark-soliton excitations. It is also observed that for suitable parametric choices, the instability of single dark solitons can nucleate multi-dark-soliton states. Also, for other parametric regions, stable asymmetric sawtooth-like solutions exist. These are shown to emerge through a symmetry-breaking bifurcation from bubble-like solutions that we also explore. We also consider the dragging of a defect through the condensate and the interference of two initially separated condensates, both of which are capable of nucleating dark multisoliton dynamical states.

Figure 1

Spatial profiles of the densities $\left|\psi \right|{}^2$ of nodeless states and single dark solitons (insets). In the left (right) panel, the parameter $\alpha$ ($x_m$) is kept fixed, taking the value $\alpha =2$ ($x_m=3$), while $\sigma =3.5$ in both cases. As seen in the left panel, the profiles do not change appreciably above a critical value of $x_m\approx 3$, see, e.g., the profiles corresponding to $x_m=3$ [dotted (black) line] and $x_m=5$ [dotted-dashed (green) line], which are almost indistinguishable from each other. All quantities in this figure (and in all subsequent ones) are dimensionless.

Figure 2

Stability domains of nodeless clouds (NC) and dark solitons (DSs) for $\sigma =0.35$ and $\alpha ⩽3.5$. Dark solitons are stable only in the area indicated by the arrow. The (green) circles indicate the parameter locations for the excitation spectra depicted in Fig. 3.

Figure 3

Spectral plane for nodeless clouds (first and third rows of panels) and dark solitons (second and fourth rows of panels) with $\sigma =0.35$. The top two rows of panels correspond to fixed $\alpha =1$ and increasing values of $x_m$, as labeled, whereas the bottom two rows of panels correspond to fixed $x_m=3$ and increasing values of $\alpha$, as indicated. The different cases correspond to the parameter locations depicted by the (green) circles in Fig. 2; cf. the stability or instability inferred by the absence or presence of eigenfrequencies with Im$(\omega )>0$ in these spectra with the stability conclusions of Fig. 2.

Figure 4

Chemical potential as a function of $x_m$ for a nodeless cloud with $\alpha =1.0$ and $\sigma =0.35$. The insets show the time evolution of the cloud for various values of $x_m$.

Figure 5

Chemical potential as a function of $x_m$ for a dark soliton with $\alpha =1.0$ and $\sigma =0.35$. The insets show the time evolution of the cloud for various values of $x_m$.

Figure 6

Left panel: Profiles for the nodeless clouds and dark solitons (inset) for different values of $\sigma$ and fixed $\alpha =2$ and $x_m=2.5$. Right panel: Stability domain for nodeless clouds (NC) and dark solitons (DS) for $\alpha =2$.

Figure 7

Left panel: Profiles of highly asymmetric “sawtooth” structures for different values of $\sigma$ and fixed $\alpha =3$ and $x_m=5$. Right panel: Stability domain for sawtooth structures with $\alpha =3$.

Figure 8

Density plots showing the evolution of unstable nodeless clouds to a breathing sawtooth structure for $\sigma =0.5$ (left) and to a nonbreathing (stable) sawtooth structure for $\sigma =1$ (right) with $x_m=5$. The inset in the left panel shows the linearization spectrum of the unstable sawtooth steady state about which the system oscillates for long times. The inset in the right panel corresponds to the linearization spectrum for the stable sawtooth steady state that the system asymptotes to.

Figure 9

The left panel represents the symmetry parameter of Eq. (7) vs $\sigma$ for the symmetric bubble predecessor states and the asymmetric sawtooth offspring states resulting from the symmetry-breaking bifurcation. Dashed lines represent (oscillatory) unstable solutions whereas solid lines correspond to stable solutions. Shaded regions indicate the range where a Hopf bifurcation cascade takes place (notice that the symmetric solutions do not experience such a bifurcation). The (green) circles indicate the parameter locations for the solution profiles depicted in Fig. 10. The right panels show the evolution of the most relevant instability eigenvalues for both the bubble (right subpanel) and the sawtooth (left subpanel) solutions. The parameters $\alpha =3$ and $x_m=5$ are fixed.

Figure 10

Profiles of sawtooth (top panels) and bubble solutions (bottom panels) for different values of $\sigma$ when $\alpha =3$ and $x_m=5$ are fixed. It is observed that the asymmetry of sawtooth solutions decreases as $\sigma$ decreases and, as expected, their profiles approach the symmetric bubble ones as the bifurcation point is approached. The different cases correspond to the parameter locations depicted by the (green) circles in the left panel of Fig. 9.

Figure 11

Nucleation of dark solitons by a dragging defect traversing a nodeless cloud. Depicted are the density plots of the nodeless cloud as it is traversed by the dragging defect. The dashed line represents the trajectory of the defect. The case depicted in the left panel corresponds to a tighter confinement with trap frequency $\Omega =1$, and parameter values $\alpha =1.0$, $\sigma =0.35$, and $x_m=2.8,$ whereas $V_0=8$, $v=0.9$, and ${\epsilon }^2=4$. The right panel corresponds to a weaker harmonic trapping with trap frequency $\Omega =0.04$ and $\alpha =0.1$, $\sigma =0.08$, and $x_m=44.5$, while $V_0=1$, $v=0.5$, and ${\epsilon }^2=0.16$. The insets show the profile [dashed (green) line] and its corresponding phase [solid (red) line] at the indicated times.

Figure 12

Nucleation of dark-soliton pairs by nonlinear interference of colliding condensate fragments. The plots depict the density of the evolution of a cloud created in a double well and allowed to interfere by removing the central barrier. Parameter values are (left) $\alpha =1.0$, $\sigma =0.35,$ and $x_m=2.8,$ whereas $V_0=17$, ${\epsilon }^2=1$, and $\Omega =1$; (right) $\alpha =0.1$, $\sigma =0.08$, and $x_m=44.5,$ whereas $V_0=4$ and ${\epsilon }^2=16$, while the trap frequency is $\Omega =0.04$. The insets show the profile [dashed (green) line] and its corresponding phase [solid (red) line] at the indicated times.