Spectroscopic Signatures of Quantum Many-Body Correlations in Polariton Microcavities

We theoretically investigate the many-body states of exciton polaritons that can be observed by pump-probe spectroscopy in high-$Q$ inorganic microcavities. Here, a weak-probe “spin-down” polariton is introduced into a coherent state of “spin-up” polaritons created by a strong pump. We show that the $\downarrow$ impurities become dressed by excitations of the $\uparrow$ medium, and that they form new polaronic quasiparticles that feature two-point and three-point many-body quantum correlations that, in the low density regime, arise from coupling to the vacuum biexciton and triexciton states, respectively. In particular, we find that these correlations generate additional branches and avoided crossings in the $\downarrow$ optical transmission spectrum that have a characteristic dependence on the $\uparrow$-polariton density. Our results thus demonstrate a way to directly observe correlated many-body states in an exciton-polariton system that go beyond classical mean-field theories.

Figure 1

Spectroscopic signature of a two-point many-body correlated state in the probe photon transmission $\mathcal{T}(\mathbf{k},\omega )$ (see text and [29]) as a function of momentum and energy. (a) In the absence of pumping. The upper polariton (UP) and lower polariton (LP) are shown as solid lines, whereas the dotted lines correspond to the bare photon ($C$) and exciton ($X$) dispersions. (b) With a ${\sigma }_{+}$ pump resonant with the LP at zero momentum. Resonant coupling to a biexciton ($X_2$) at $\omega +{\omega }_{\mathrm{LP}0}\simeq -E_B$ (dashed line) causes a splitting of the bare lower polariton into attractive and repulsive branches, as well as a blueshift of the upper polariton. For this illustration, we take the ${\sigma }_{+}$ polariton density $n=m_X{\mathrm{\Omega }}_R/8\pi$, detuning $\delta =-{\mathrm{\Omega }}_R/3$, and $E_B={\mathrm{\Omega }}_R$.

Figure 2

Normal incidence pump-probe transmission $\mathcal{T}(\mathbf{0},\omega )$ as a function of the photon-exciton detuning and the rescaled probe energy (relative to the LP energy) for increasing pump densities: (a) $n=0$, (b) $n=m_X{\mathrm{\Omega }}_R/16\pi$, and (c) $n=m_X{\mathrm{\Omega }}_R/4\pi$. In the experimentally realistic case of ${\mathrm{\Omega }}_R=3\mathrm{meV}$, this corresponds to densities of (b) $n=3\times {10}^{10}{\mathrm{cm}}^{-2}$ and (c) $n=1.25\times {10}^{11}{\mathrm{cm}}^{-2}$. In both cases, we take $E_B={\mathrm{\Omega }}_R$ and a broadening of $\mathrm{\Gamma }={\mathrm{\Omega }}_R/10$. Lower and upper polariton energies in the absence of the medium are shown as black solid lines. Dashed white lines indicate locations of the vacuum biexciton ($X_2$) and triexciton ($X_3$) resonances at $\omega +{\omega }_{\mathrm{LP}\mathbf{0}}=-E_B$ and $\omega +2{\omega }_{\mathrm{LP}\mathbf{0}}={ϵ}_T$, respectively, with triexciton energy of ${ϵ}_T\simeq -2.4E_B$ [29].

Figure 3

(a) Minimal splitting between LP quasiparticle branches in the transmission spectrum, and (b) photon-exciton detuning at the minimal splitting. The splitting $\mathrm{\Delta }{\omega }_3$ and the detuning ${\delta }_3$ for the lowest two branches—originating from the triexciton resonance—are shown as solid blue lines, where ${\delta }_3\to -0.48{\mathrm{\Omega }}_R$ (dotted line) is the limit $n\to 0$ due to the triexciton state. Splitting $\mathrm{\Delta }{\omega }_2$ and detuning ${\delta }_2$ for the biexciton resonance are shown as solid black lines. Dashed black lines depict corresponding results calculated when including only two-point correlations (i.e., the Hilbert space with, at most, one excitation of the medium).