Exciton-polariton gap soliton dynamics in moving acoustic square lattices

The modulation by a surface acoustic wave (SAW) provides a powerful tool for the formation of tunable lattices of exciton-polariton macroscopic quantum states (MQSs) in semiconductor microcavities. The MQSs were resonantly excited in an optical parametric oscillator configuration. We investigate the temporal dynamics of these lattices using time and spatially resolved photoluminescence (PL). Photoluminescence images of the MQSs clearly show the motion of the lattice at the acoustic velocity. Interestingly, the PL intensity emitted by the MQSs as well as their coherence length oscillate with the position of the lattice sites relative to the exciting laser beam. The coherence length and the PL intensity are correlated. The PL oscillation amplitude depends on both the intensity and the size of the exciting laser spot and increases considerably for excitation intensities close to the optical threshold power for the formation of the MQS. The oscillations are explained by a model that takes into account the combined effects of SAW reflections, which dynamically distort the amplitude of the potential, and the spatial phase of the acoustic lattice within the exciting laser spot. This paper could pave the way to tailor polariton-based light-emitting sources with intensity variations controlled by the SAWs.

Figure 1

(a) Structure of the MC sample showing the acoustic square lattice generated by two orthogonal SAWs. Here, $v_{\mathrm{lat}}$ denotes the velocity of the moving lattice. The PL from GS states consists of 4 beams forming a 4° angle with the sample normal (yellow arrows). (b) Cross-section of the polariton dispersion along the Γ-$M$ direction of the two-dimensional dispersion of the square lattice. The inset shows the lines of the boundaries of the first four MBZs. The GS states form at the $M$ points. The arrows in the main plot depict the optical parametric process used for the resonant excitation of GS. (c) The time-integrated real-space image of the GS PL at the condensation threshold ($P_{\mathrm{L}}=P_{\mathrm{th}}$) shows blurred spots separated by ${\lambda }_{\mathrm{SAW}}$/√2, where ${\lambda }_{\mathrm{SAW}}$ is the SAW wavelength. The inset shows the corresponding time-integrated $k$-space image. (d) Impulse response of the RF power reflection coefficient (scattering parameter $s_{11}$) of the IDT recorded in the configuration displayed in the inset. The broad peak for delays 0 < $t<1\mu \mathrm{s}$ is generated while the SAW propagates through the 2.8 mm long excited IDT. The echo centered at $t=\tau$ is caused by SAW reflection at the opposing IDT or at the sample border (cf. inset). The distance $v_{\mathrm{SAW}}\tau$ corresponds to twice the distance between the center of the IDT and the reflection point.

Figure 2

Time-resolved (a)–(d) real-space and (e)–(h) corresponding $k$-space images of $|{\mathrm{\Psi }}_{\mathrm{GS}}{|}^2$ recorded for an excitation power $P_{\mathrm{L}}=P_{\mathrm{th}}$ at the delays $t_{1...4}$ as marked in (i). To help the visualization of lattice motion with velocity $v_{\mathrm{lat}}$, the dashed white circles show the position of the emission maxima at delay $t_2$. The intensity scale is the same as in Fig. 1. (i) Integrated intensity $I_{\mathrm{PL},\mathrm{T}}$ of PL images (blue curve) as well as the GS coherence length $L_{\mathrm{coh}}$ (red squares) as a function of time. One acoustic period $T_{\mathrm{SAW}}$ equals 2.7 ns.

Figure 3

Time-resolved total PL intensity $I_{\mathrm{PL},\mathrm{T}}$ for different excitation laser powers ($P_{\mathrm{L}}$ stated in terms of $P_{\mathrm{th}}$) recorded for the laser spot diameter ${\varphi }_{\mathrm{L}}=60\mu \mathrm{m}$ (solid curves) and ${\varphi }_{\mathrm{L}}=100\mu \mathrm{m}$ (dashed curve). The solid and dashed arrows mark the first and second maximum of the PL intensity, respectively. The inset shows the dependence of the PL intensity variation $\mathrm{\Delta }{I}_{\mathrm{PL},\mathrm{T}}$ on the laser power $P_{\mathrm{L}}$ for ${\varphi }_{\mathrm{L}}=60\mu \mathrm{m}$ (solid curve) and ${\varphi }_{\mathrm{L}}=100\mu \mathrm{m}$ (dashed curve). The lines are fits to the expression discussed in the text.

Figure 4

Dependence of the time-integrated PL intensity $I_{\mathrm{PL}}$ of the GS on the laser power $P_{\mathrm{L}}$ and the SAW amplitude ${\mathrm{\Phi }}_{\mathrm{SAW}}$ for ${\varphi }_{\mathrm{L}}=60\mu \mathrm{m}$. Here, 1.0 ${\mathrm{\Phi }}_{\mathrm{SAW}}$ corresponds to the used RF power of 30 mW in the experiments. The inset shows a close-up of the plot for the laser powers close to the GS condensation threshold $P_{\mathrm{th}}$. It shows that an increase of ${\mathrm{\Phi }}_{\mathrm{SAW}}$ by 10% enhances $I_{\mathrm{PL}}$ by 120% at the same $P_{\mathrm{L}}$. Furthermore, a decrease of ${\mathrm{\Phi }}_{\mathrm{SAW}}$ by 10% leads to a decrease of $I_{\mathrm{PL}}$ by 70%.

Figure 5

(a) and (b) schematically show the SAW potential minima at $t=0$ and 0.5 $T_{\mathrm{SAW}}$ after the acoustic square lattice has moved $0.5T_{\mathrm{SAW}}{v}_{\mathrm{lat}}$. The blue and red dashed circles have a diameter $D_{\mathrm{L}}=24\mu m(=3{\lambda }_{\mathrm{SAW}})$ and $D_{\mathrm{L}}=48\mu \mathrm{m}(=6{\lambda }_{\mathrm{SAW}})$, respectively, and represent the effective area of the laser spot where $P_{\mathrm{L}}\ge P_{\mathrm{th}}$. The center of the laser spots coincides with the center of the square lattice at $t=0$ and ${}^1/{}_2{T}_{\text{SAW}}$. While the square lattice moves, as indicated by the white arrow, the spatial phase of the square lattice is changing with respect to the laser spot and, thus, causing the total PL intensity $I_{\mathrm{PL},\mathrm{T}}$ to oscillate as (c) calculated for a laser spot with ${\varphi }_{\mathrm{L}}=60\mu \mathrm{m}$ and $D_{\mathrm{L}}=24\mu \mathrm{m}$ (blue curve) and $D_{\mathrm{L}}=48\mu \mathrm{m}$ (red curve). The $\mathit{xN}$ label indicates the intensity magnification factor $N$ with respect to the result for $D_{\mathrm{L}}=24\mu \mathrm{m}$. For both cases, $I_{\mathrm{PL},\mathrm{T}}$ varies with $f_{\mathrm{SAW}}$ around its time-averaged value. However, the variation $\mathrm{\Delta }{I}_{\mathrm{PL},\mathrm{T}}$ is bigger for the blue curve ($D_{\mathrm{L}}=24\mu \mathrm{m}$). As well, the phase of the blue curve is shifted by a half of the SAW period with respect to the red curve ($D_{\mathrm{L}}=48\mu \mathrm{m}$). The origin of the phase shift is discussed in the text.

Figure 6

(a) Calculated total PL intensity $I_{\mathrm{PL},\mathrm{T}}$ considering SAW reflections and the changing spatial phase of the acoustic square lattice with respect to the laser spot. The parameter $D_{\mathrm{L}}$ was set to 24 μm (blue curve) and 48 μm (green curve) with ${\varphi }_{\mathrm{L}}=60\mu \mathrm{m}$ and $D_{\mathrm{L}}=32\mu \mathrm{m}$ with ${\varphi }_{\mathrm{L}}=100\mu \mathrm{m}$ (red curve). The center of the laser spots was placed at the center of the square lattice at $t=0$. The $\mathit{xN}$ labels indicate the intensity magnification factor $N$ with respect to the result for $D_{\mathrm{L}}=24\mu \mathrm{m}$. Here, $\mathrm{\Delta }{I}_{\mathrm{PL},\mathrm{T}}$ reduces with increasing $D_{\mathrm{L}}$ and/or ${\varphi }_{\mathrm{L}}$. (b) Calculated dependence between $\mathrm{\Delta }{I}_{\mathrm{PL},\mathrm{T}}$ and $D_{\mathrm{L}}$ according to the model. (c) and (d) show the comparison of the model with the experimental results for the excitation powers $P_{\mathrm{L}}=P_{\mathrm{th}}$ and $6P_{\mathrm{th}}$, respectively. The experimental results have been scaled by constant factors to match the calculated ones.