Spontaneous parametric scattering of microcavity polaritons in momentum space

We measure the spontaneous parametric emission from a semiconductor microcavity after resonant pulsed coherent excitation. Resolving the emission in time and two-dimensional momentum space, the underlying scattering processes, which were theoretically treated by [Ciuti et al. Phys. Rev. B 63, 041303 (2001)], are validated. The predicted figure-8 shaped distribution of the final states of the parametric scattering is observed. We find a narrowing of the momentum distribution with time, which is a feature of memory effects in the scattering due to parametric correlation. Additionally, the influence of higher-order scattering and pump depletion are observed in the experiment.

Figure 1

Calculated correlation $\mid ⟨p_s^{†}{p}_i^{†}⟩\mid$ (left) and signal density $N_{\mathrm{s}}$ (right), on an linear grey scale from zero, as function of normalized time $t\gamma$ and of normalized frequency mismatch ${\omega }_{\delta }∕\gamma$. The data are results of Eqs. (3, 4) with the parameters ${\gamma }_p=0$, ${\gamma }_s={\gamma }_i=\gamma$. Additionally, dotted (solid) curves for fixed values of ${\omega }_{\delta }∕\gamma =0,5,10$ (of $t\gamma =0.25,0.5,\infty$) are shown, representing horizontal (vertical) cuts through the contour plot, on a linear scale with the zero at the lower (right) axis, respectively.

Figure 2

Top: Calculated energy of the LP relative to the heavy-hole exciton energy as function of in-plane wave vector. Labels on contour lines are in $\mathrm{meV}$. Bottom: Possible final states of the scattering of two pump polaritons at ${\mathbf{k}}_p=(k_p,0)$ into a signal and an idler polariton conserving energy and momentum. Results for various $k_p$ in $\mu {\mathrm{m}}^{-1}$, as labeled.

Figure 3

Energy mismatch $\mid \hslash {\omega }_{\delta }\mid$ for the parametric process for two different pump wave vectors $k_p$ as labeled on a linear grey scale from 0 (white) to $4\mathrm{meV}$ (black).

Figure 4

Calculated polariton-polariton correlation due to spontaneous parametric scattering for ${\mathbf{k}}_{\mathrm{p}}=(1.73,0)\mu {\mathrm{m}}^{-1}$ and fixed times after excitation as labeled. Shown are $\mathbf{k}$-space images on a linear grey scale from zero (white) to a fixed positive value (black) with relative scaling factors as given.

Figure 5

Calculated polariton state population due to spontaneous parametric scattering for ${\mathbf{k}}_p=(1.73,0)\mu {\mathrm{m}}^{-1}$ and fixed times after excitation as labelled. Shown are $\mathbf{k}$-space images on a linear grey scale from zero (white) a fixed positive value (black) with relative scaling factors as given.

Figure 6

Calculated photon emission intensity due to spontaneous parametric scattering for ${\mathbf{k}}_p=(1.73,0)\mu {\mathrm{m}}^{-1}$ and fixed times after excitation as labeled. Shown are $\mathbf{k}$-space images on a linear grey scale from zero (white) to a fixed positive value (black) with relative scaling factors as given.

Figure 7

Calculated polariton state population due to parametric scattering including the final state stimulation and polariton renormalization for ${\mathbf{k}}_p=(1.73,0)\mu {\mathrm{m}}^{-1}$ and $D_0=0.03$, and fixed times after excitation as labeled. Shown are $\mathbf{k}$-space images on a linear grey scale from zero (white) to the indicated $N_{\max }$ (black).

Figure 8

Calculated polariton state population at $t=10\mathrm{ps}$ due to parametric scattering including final state stimulation and polariton renormalization for ${\mathbf{k}}_p=(1.73,0)\mu {\mathrm{m}}^{-1}$. Shown are $\mathbf{k}$-space images on a linear grey scale from zero (white) to the indicated $N_{\max }$ (black).

Figure 9

Measured time-integrated and wave-vector-resolved emission spectra for $k_y=0$ and resonant pulsed excitation at ${\mathbf{k}}_p=(1.73,0)\mu {\mathrm{m}}^{-1}$ at different excitation intensities as labeled. Logarithmic grey scale covering 3 decades. The dotted line is the LP dispersion calculated with the three-coupled-oscillator model, and the solid line is the cavity mode dispersion.

Figure 10

Measured wave-vector resolved, time and spectrally integrated MC emission for resonant pulsed excitation at (a) ${\mathbf{k}}_p=(1.5,0)\mu {\mathrm{m}}^{-1}$ and (b) ${\mathbf{k}}_p=(1.73,0)\mu {\mathrm{m}}^{-1}$. The excitation intensities are $1–130I_0$ as labeled, and the grey scale has a linear mapping from zero (white) to $1–500$ $S_0$ (black), as labeled. $S_0$ is a fixed but uncalibrated unit of emission intensity per solid angle. The region of the specular reflection of the excitation pulse $(\otimes )$ was blocked, indicated by the white rectangle. The data in (b) has been multiplied by a factor of 5 for $k_x>1.8\mu {\mathrm{m}}^{-1}$.

Figure 11

Measured emission intensity for ${\mathbf{k}}_p=(0,1.73)\mu {\mathrm{m}}^{-1}$ as in Fig. 10b, at $\mathbf{k}=(0,k_y)$ and normalized to the excitation intensity. The various excitation intensities in units of $I_0$ are labeled. Inset: Emission integrated in the ranges of $k_y=0.8–1\mu {\mathrm{m}}^{-1}$ (solid squares) and of $k_y=1.5–2\mu {\mathrm{m}}^{-1}$ (open circles). A linear and quadratic scaling are shown as dotted lines for comparison.

Figure 12

Measured wave vector and time-resolved polariton density for resonant pulsed excitation at ${\mathbf{k}}_p=(0,1.73)\mu {\mathrm{m}}^{-1}$. Data for different times after excitation as given are displayed for the two excitation intensities $40I_0$ and $100I_0$. The grey scale has a linear mapping from zero (white) to $1–250$ $N_0$ (black), as labelled.

Figure 13

Measured wave vector and time-resolved polariton density in/out scattering rate for the conditions of Fig. 12. The grey scale has a linear mapping from $-0.1$ (white) to 1.0 of $1–70$ $R_0$ as given in the graphs. The rate unit is $R_0=N_0{\mathrm{ps}}^{-1}$.

Figure 14

Polariton scattering rate ${\partial }_{t}N{\mid }_{\mathrm{scat}}$ and population $N$ at $\mathbf{k}=(0,0.9)\phantom{\mid }\mu {\mathrm{m}}^{-1}$ (signal) and $\mathbf{k}=(3.46,-0.9)\phantom{\mid }\mu {\mathrm{m}}^{-1}$ (idler) as function of time after excitation at different intensities as labeled. Inset: measured signal scattering rate (squares) at $I_{\mathrm{exc}}=10I_0$ compared with calculations (lines, see text).

Figure 15

Calculated polariton scattering rate ${\partial }_{t}N{\mid }_{\mathrm{scat}}\left(t\right)$ and population $N\left(t\right)$ at $\mathbf{k}=(0,0.9)\phantom{\mid }\mu {\mathrm{m}}^{-1}$ (signal) and $\mathbf{k}=(3.46,-0.9)\phantom{\mid }\mu {\mathrm{m}}^{-1}$ (idler) as function of time after excitation at different pump densities $D_0$ as labeled.

Figure 16

Polariton scattering rate ${\partial }_{t}N{\mid }_{\mathrm{scat}}\left(t\right)$ and density $N\left(t\right)$ at $\mathbf{k}=(1.0,0.5)\phantom{\mid }\mu {\mathrm{m}}^{-1}$ (signal) and $\mathbf{k}=(2.46,-0.5)\phantom{\mid }\mu {\mathrm{m}}^{-1}$ (idler) as function of time after excitation at different excitation intensities as labeled.

Figure 17

Properties of the polariton density created by parametric scattering as function of excitation density, given in units of $I_0$ for the experimental data, and in units of ${\lambda }_X^{-2}$ for the calculations. Full symbols are for signal, empty symbols are for idler polaritons. Squares: ${\mathbf{k}}_s=(0,0.9)\phantom{\mid }\mu {\mathrm{m}}^{-1}$. Stars: ${\mathbf{k}}_s=(1,0.5)\phantom{\mid }\mu {\mathrm{m}}^{-1}$. Thick solid (dotted) line: calculations of signal (idler) state populations $N_{s,i}$ for ${\mathbf{k}}_s=(0,0.9)\phantom{\mid }\mu {\mathrm{m}}^{-1}$ including renormalization and stimulation. (a) Time delay of the density maximum. (b) Density maximum. The low-intensity scaling $\left(\propto I_{\mathrm{exc}}^2\right)$ and the estimated pump depletion density ${10}^{-3}{N}_{\mathrm{exc}}$ are given for comparison.