Absorption, photoluminescence, and resonant Rayleigh scattering probes of condensed microcavity polaritons

We investigate and compare different optical probes of a condensed state of microcavity polaritons in expected experimental conditions of nonresonant pumping. We show that the energy- and momentum-resolved resonant Rayleigh signal provides a distinctive probe of condensation as compared to, e.g., photoluminescence emission. In particular, the presence of a collective sound mode both above and below the chemical potential can be observed, as well as features directly related to the density of states of particle-hole-like excitations. Both resonant Rayleigh response and the absorption and photoluminescence are affected by the presence of quantum well disorder, which introduces a distribution of oscillator strengths between quantum well excitons at a given energy and cavity photons at a given momentum. As we show, this distribution makes it important that in the condensed regime, scattering by disorder is taken into account to all orders. We show that, in the low-density linear limit, this approach correctly describes inhomogeneous broadening of polaritons. In addition, in this limit, we extract a linear blueshift of the lower polariton versus density, with a coefficient determined by temperature and by a characteristic disorder length.

Figure 1

(Color online) Contour plot of the disorder averaged RRS intensity $⟨I_{\mathbf{p}\mathbf{q}}\left(\omega \right)⟩$ for $\mid \mathbf{p}\mid =\mid \mathbf{q}\mid$ as a function of the dimensionless momentum $\mid \mathbf{p}\mid a_x$ and rescaled energy $2(\omega -\mu )∕{\Omega }_R$, for zero detuning, Rabi splitting ${\Omega }_R=26\mathrm{meV}$, temperature $k_{B}T=20\mathrm{K}$, and a disorder strength characterize by an inverse scattering time $1∕\tau =1.16\mathrm{meV}$: (a) noncondensed regime (dimensionless density $\rho \simeq 0$), (b) condensed regime $(\rho \simeq 7.8\times {10}^{-3})$, and (c) condensed regime $(\rho \simeq 6.7\times {10}^{-2})$. (The parameters chosen for these plots are the same as those used later for spectral weight and photoluminescence.) The value of the chemical potential is explicitly marked [horizontal green (gray) line]. While in the noncondensed regime, RRS emission is always above the chemical potential, in the condensed phase, emission from the collective sound mode is seen both above and below the chemical potential.

Figure 2

(Color online) Plot showing the energy dependence of the excitonic squared coupling strength ${\mid g_{\alpha ,\mathbf{p}}\mid }^2$ on photons of momentum (a) $\mid \mathbf{p}\mid =0$ (from Ref. 25, for comparison) and (b) $\mid \mathbf{p}\mid =6.3\times {10}^5{\mathrm{cm}}^{-1}$, where all exciton states are found numerically. Results are taken from 160 different realizations of disorder potential, and for (b), coupling strengths from eight different photon momenta $(p_x,p_y)$ with the same value of $\mid \mathbf{p}\mid$ are combined. The mean squared averaged oscillator strength $g^2(\epsilon ,\mid \mathbf{p}\mid )$ for the same value of momentum [lower red (gray) points] and the density of states $\mathrm{DOS}\left(\epsilon \right)$ [upper green (gray) points] are also explicitly plotted. Inset: Fit of $g^2(\epsilon ,\mid \mathbf{p}\mid )$ to expression (7) with a renormalized energy ${\epsilon }_{\mathbf{p}}$.

Figure 3

(Color online) Contour plot of the mean (over 160 realizations of the disorder potential) squared oscillator strength $g^2(\epsilon ,\mid \mathbf{p}\mid )$ versus energy and momentum [or, equivalently, angle $\theta ={\tan }^{-1}(c\mid \mathbf{p}\mid ∕{\omega }_0)$]. Note that the scale in angle is not linear. The free particle dispersion $E_x+{\mid \mathbf{p}\mid }^2∕2m_x$ [solid green (gray) line] and the trace of the mean squared oscillator strength for two representative values of momenta, $\mid \mathbf{p}\mid =0$ $(\theta =0°)$ and $\mid \mathbf{p}\mid =6.3\times {10}^5{\mathrm{cm}}^{-1}$ $(\theta =82°)$ [red (gray) plus symbols] are explicitly plotted (cf. Fig. 2). The figure is adapted from Ref. 25.

Figure 4

Optical density Eq. (6) versus energy. The maximum value is around ${\epsilon }^{*}-E_x\simeq -0.94\mathrm{meV}$ and the FWHM ${\sigma }^{*}\simeq 0.94\mathrm{meV}$.

Figure 5

(Color online) Phase diagram for the dimensionless critical temperature $2k_B{T}_c∕{\Omega }_R$ versus the dimensionless density $\rho$ for effective zero detuning $\delta ={\omega }_0-{\epsilon }^{*}=0\mathrm{meV}$ (${\epsilon }^{*}$ is the energy at which the excitonic optical density has its maximum, and so as seen in Fig. 4, $\delta =0$ implies ${\omega }_0-E_x=-0.94\mathrm{meV}$) (solid black) and for positive detuning $\delta =+6\mathrm{meV}$ $({\omega }_0-E_x=+5.06\mathrm{meV})$ (dotted-dashed black) and ${\Omega }_R=26\mathrm{meV}$. The mean-field boundaries for the two different detunings are cut off by the expected linear dependence of the critical temperature, as indicated at extremely low densities [orange (gray) solid and dotted-dashed lines]. The horizontal dashed line marks the temperature of $k_{B}T=20\mathrm{K}$, which will be used in later figures.

Figure 6

(Color online) Left column: Contour plot of the spectral weight $W(\omega ,\mathbf{p})$ as a function of the dimensionless momentum $\mid \mathbf{p}\mid a_x$ and rescaled energy $2(\omega -\mu )∕{\Omega }_R$, for zero detuning $\delta ={\omega }_0-{\epsilon }^{*}=0({\omega }_0-E_x=-0.94\mathrm{meV})$, ${\Omega }_R=26\mathrm{meV}$, $k_{B}T=20\mathrm{K}$, and photon mass $m_{\mathrm{ph}}{\Omega }_R{a}_x^2∕2=0.01$. Right column: Plot of the quasiparticle DOS for the same choice of parameters as the left column, respectively: (a) noncondensed ($\rho \simeq 0$, $E_x-\mu \simeq 27\mathrm{meV}$) (the bare exciton and photon dispersions [blue (dark gray) dotted line] and the upper and lower polariton dispersions [red (gray) solid line] obtained from the effective coupled oscillator model are shown for comparison), (b) condensed ($\rho \simeq 7.8\times {10}^{-3}$, $E_x-\mu \simeq 11\mathrm{meV}$), and (c) condensed ($\rho \simeq 6.7\times {10}^{-2}$, $E_x-\mu \simeq 7\mathrm{meV}$).

Figure 7

(Color online) Contour plot of the spectral weight $W(\omega ,0)$ for $\mathbf{p}=0$ as a function of the detuning $\delta ={\omega }_0-{\epsilon }^{*}$ and energy $\omega -E_x$, for a fixed and very low value of density ($\rho \simeq 0$, $E_x-\mu \simeq 27\mathrm{meV}$) and $k_{B}T=20\mathrm{K}$ $(2k_{B}T∕{\Omega }_R=0.13)$. The bare exciton and photon dispersions [blue (dark gray) dotted line] and the upper and lower polariton dispersions [red (gray) solid line] obtained from the effective coupled oscillator model are explicitly shown.

Figure 8

(Color online) Contour plot of the incoherent PL $P(\omega ,\mathbf{p})$ as a function of the dimensionless momentum $\mid \mathbf{p}\mid a_x$ and rescaled energy $2(\omega -\mu )∕{\Omega }_R$, for detuning ${\omega }_0-E_x=-0.94\mathrm{meV}$, ${\Omega }_R=26\mathrm{meV}$, $k_{B}T=20\mathrm{K}$, and $m_{\mathrm{ph}}{\Omega }_R{a}_x^2∕2=0.01$: (a) noncondensed ($\rho \simeq 0$, $E_x-\mu \simeq 27\mathrm{meV}$), (b) condensed ($\rho \simeq 7.8\times {10}^{-3}$, $E_x-\mu \simeq 11\mathrm{meV}$), and (c) condensed ($\rho \simeq 6.7\times {10}^{-2}$, $E_x-\mu \simeq 7\mathrm{meV}$).

Figure 9

(Color online) Normalized density of states $\mathrm{DOS}\left(\epsilon \right)∕\nu$ and thermal occupation factor [red (gray) line starting at top left] for $E_x-\mu =5\mathrm{meV}$ and $k_{B}T=20\mathrm{K}$.

Figure 10

(Color online) Contour plot of the spectral weight $W(\omega ,0)$ for $\mathbf{p}=0$ as a function of the dimensionless density $\rho$ and the rescaled energy $\omega -{\omega }_0$, for zero detuning $({\omega }_0-E_x=-0.94\mathrm{meV})$ and $k_{B}T=20\mathrm{K}$ $(2k_{B}T∕{\Omega }_R=0.13)$. The rescaled chemical potential $(\mu -{\omega }_0)$ [green (light gray) dashed], the noncondensed lower and upper polariton modes from the two coupled oscillator model [blue (dark gray) dotted], and the critical density for condensation ${\rho }_c\simeq 3.7\times {10}^{-5}$ [vertical solid red (dark gray) line] are shown for comparison.

Figure 11

(Color online) Excitonic speckle pattern ${\phantom{\mid }{\mid K_{\mathbf{p},\mathbf{q}}^{\left(1o\right)}\left({\omega }_h\right)\mid }^2\mid }_{i{\omega }_h=-\omega -i\eta }$ for a single disorder realization (gray dotted line) and its disorder average (black cross symbols) versus energy for $\mid \mathbf{p}\mid =\mid \mathbf{q}\mid =36°$ and a 90° azimuthal angle, $\mathbf{p}=(0,6.3\times {10}^4){\mathrm{cm}}^{-1}$ and $\mathbf{q}=(6.3\times {10}^4,0){\mathrm{cm}}^{-1}$. For comparison, we also plot the excitonic optical density [green (light gray) plus symbols].