Exciton-Polariton Gap Solitons in Two-Dimensional Lattices

We report on the two-dimensional gap-soliton nature of exciton-polariton macroscopic coherent phases (PMCP) in a square lattice with a tunable amplitude. The resonantly excited PMCP forms close to the negative mass $M$ point of the lattice band structure with energy within the lattice band gap and its wave function localized within a few lattice periods. The PMCPs are well described as gap solitons resulting from the interplay between repulsive polariton-polariton interactions and effective attractive forces due to the negative mass. The solitonic nature accounts for the reduction of the PMCP coherence length and optical excitation threshold with increasing lattice amplitude.

Figure 1

(a) Square lattice for polaritons created by the interference of two surface acoustic waves (SAWs) propagating along the $\widehat{x}=[010]$ and $\widehat{y}=[001]$ surface directions of a (100)-(Al,Ga)As MC. The lattice moves along the $(\widehat{x}+\widehat{y})=[110]$ direction with a velocity of $v_{\mathrm{lat}}=\sqrt{2}{v}_{\mathrm{SAW}}$, where $v_{\mathrm{SAW}}=3\mu \mathrm{m}/\mathrm{ns}$ is the SAW phase velocity. (b) One-particle band structure for polaritons in a shallow lattice (${\Phi }_{\mathrm{SAW}}=50\mu \mathrm{eV}$). $E_{2d}$ denotes the soliton energy and wave vector range. The energy is relative to the bottom of the unmodulated dispersion. The inset is a diagram of the first four mini-Brillouin zones of the square lattice. (c) Schematic representation of the OPO process in the modulated polariton dispersion along $\Gamma \to M$. The pump state is generated by tuning the laser energy and angle of incidence. Above the threshold of formation of the PMCP polaritons scatter into the signal and idler states, as indicated by the arrows. (d) Calculated squared moduli of the wave function $|\psi {|}^2$ of the $1s$ and $2p_{\mathrm{x}}2p_{\mathrm{y}}$ states at the $M$ point. The crosses mark the minima of the potential lattice. The whole pattern moves along the vertical direction with velocity $v_{\mathrm{lat}}$.

Figure 2

(a) $k$-space PL image of an incoherent polariton gas. The red point at ${\mathbf{k}}_{\mathbf{p}}=(k_{{\mathrm{p}}_{\mathrm{x}}},k_{{\mathrm{p}}_{\mathrm{y}}})=(0,1.7)\mu {\mathrm{m}}^{-1}$ marks the pump state, which was blocked during the experiment. The PL peaks at $\mathbf{k}=(\pm k_{\mathrm{SAW}},k_{{\mathrm{p}}_{\mathrm{y}}})=(0.78,1.7)\mu {\mathrm{m}}^{-1}$ and at $\mathbf{k}=(\pm k_{\mathrm{SAW}},k_{{\mathrm{p}}_{\mathrm{y}}}-k_{\mathrm{SAW}})=(0.78,0.9)\mu {\mathrm{m}}^{-1}$ are the diffracted pump beams. The white lines delineate the first four MBZs. (b) Real-space PL image of an incoherent polariton gas at a pump power $P_{\ell }<P_{\mathrm{th}}$. (c) Spatially resolved spectrum along the slit in (b). The dotted lines mark the positions of the lattice $s$ and $p$ states. (d)–(f) and (g)–(i) are the corresponding images at $P_{\ell }=P_{\mathrm{th}}$ and at $P_{\ell }=1.4P_{\mathrm{th}}$. The image intensities were amplified within the boxes in (i). In all cases, $P_{\mathrm{rf}}=22\mathrm{mW}$.

Figure 3

(a) Intensity profiles along a $M\to \Gamma \to M$ direction [cf. dashed lines 1 and 2 in Fig. 2f]. The topmost curve for the diffracted pump beam (line 1) defines the limits of the first MBZ. The additional curves are profiles for the signal PMCP (line 2) in lattices created with $P_{\mathrm{rf}}=22$, 35, and 56 mW (from the second to the bottom curves). (b) Experimental dependence of $P_{\mathrm{th}}$ on $P_{\mathrm{rf}}$ (squares). The empty circles show the calculated dependence of the minimum number of polaritons $N_{2d,\min }$ to form the soliton with ${\Phi }_{\mathrm{SAW}}$. The line is a guide to the eye. The parameters used are listed in the caption of Fig. 4. (c) Dependence of the PMCP coherence length ($L_{\mathrm{coh}}$) at threshold on $P_{\mathrm{rf}}$.

Figure 4

(a) Energy map of the state defined by Eq. (1) (color scale) as a function of parameters $r_0$ and $k_0$ calculated for $N_{2\mathrm{d}}=150$ polaritons, ${\Phi }_{\mathrm{SAW}}=200\mu \mathrm{eV}$, $m_p=6\times {10}^{-5}{m}_e$, ${\lambda }_{\mathrm{SAW}}=8\mu \mathrm{m}$, and particle interaction constant $g_{2d}=10\mu \mathrm{eV}\mu {\mathrm{m}}^2$. $M_1$ and $M_2$ mark local energy minima corresponding to GS solutions. The white line is a guide to the eye. (b) Experimental (squares) and calculated (circles) dependence of the $k$-coordinates of the maxima of the emission of peaks $k_0$ on $L_{\mathrm{coh}}$. The red squares are the experimental points obtained for $P_{\ell }=1.2P_{\mathrm{th}}$ and the open circles are the calculated points.