Continuous and discontinuous dark solitons in polariton condensates

Bose-Einstein condensates of exciton-polaritons are described by a Schrödinger system of two equations. Nonlinearity due to exciton interactions gives rise to a frequency band of dark soliton solutions, which are found analytically for the lossless zero-velocity case. The soliton's far-field value varies from zero to infinity as the operating frequency varies across the band. For positive detuning (photon frequency higher than exciton frequency), the exciton wave function becomes discontinuous when the operating frequency exceeds the exciton frequency. This phenomenon lies outside the parameter regime of validity of the Gross-Pitaevskii (GP) model. Within its regime of validity, we give a derivation of a single-mode GP model from the initial Schrödinger system and compare the continuous polariton solitons and GP solitons using the healing length notion.

Figure 1

Dark polariton soliton envelopes $({\varphi }_X\left(x\right),{\varphi }_C\left(x\right))$ for exciton frequency ${\omega }_X=0$ and photon frequency ${\omega }_C=1$, which gives a threshold frequency ${\omega }_{\mathrm{LP}}\approx -0.618$ for the onset of the soliton, a transition frequency ${\omega }_X=0$ at which ${\varphi }_X$ becomes discontinuous, and a blowup frequency ${\omega }_C=1$ at which the far-field values of ${\varphi }_X$ and ${\varphi }_C$ become unbounded as shown in Fig. 3 (right). These graphs demonstrate the increasing soliton amplitude as $\omega$ increases through four values. When $\omega <{\omega }_X, {\varphi }_X$ is continuous, and when $\omega >{\omega }_X, {\varphi }_X$ is discontinuous. The values of ${\varphi }_X$ and ${\varphi }_C$ are related by Eq. (4b).

Figure 2

The cubic relation (4b) giving the photon field envelope value ${\varphi }_C$ vs the exciton field envelope value ${\varphi }_X$. Left: The pair $({\varphi }_X\left(x\right),{\varphi }_C\left(x\right))$ travels continuously along the graph of the monotonic cubic between its far-field values as $x$ increases from $-\infty$ to $\infty$. Right: The pair jumps discontinuously between the points $(-{\varphi }_0,0)$ and $(+{\varphi }_0,0)$, with ${\varphi }_0=\sqrt{{\varpi }_X/g}$. The transition from continuous to discontinuous ${\varphi }_X$ occurs when $\omega ={\omega }_X$. Graphs of the fields ${\varphi }_C\left(x\right)$ and ${\varphi }_X\left(x\right)$ are shown in Fig. 1. The singularity of the ODE (6) occurs at the critical points $\pm {\varphi }_1$.

Figure 3

The threshold frequency ${\omega }_{\mathrm{LP}}$ marks the onset of a dark polariton soliton, and the photon frequency ${\omega }_C$ is the blowup frequency, at which the far-field amplitude of the soliton becomes unbounded. Left: (${\omega }_C<{\omega }_X$). As the operating frequency $\omega$ traverses the soliton band $({\omega }_{\mathrm{LP}},{\omega }_C)$, the far-field amplitude of the exciton field ${\varphi }_X$ goes from 0 to $\infty$ according to (7). The nadir (low point) is zero. Right: (${\omega }_X<{\omega }_C$). The free exciton frequency ${\omega }_X$ is the transition frequency from continuous to discontinuous solitons. The nadir of the discontinuous soliton is pushed upwards to the value ${\varpi }_X$.

Figure 4

The effective potential well $V\left(x\right)$ that confines the photon field of an exciton-polariton soliton. Left: When the exciton field is continuous ($\omega <{\omega }_X$), $V\left(x\right)$ has a finite minimal value. Right: When the exciton field is discontinuous ($\omega >{\omega }_X$), $V\left(x\right)$ is unbounded at the point of discontinuity $x=0$. (${\omega }_X=0, {\omega }_C=1$, and ${\omega }_{\mathrm{LP}}\approx -0.618$, as in Fig. 1.)