Emission of time-ordered photon pairs from a single polaritonic Bogoliubov mode

Abstract

In many-body quantum systems, interactions drive the emergence of correlations that are at the heart of the most intriguing states of matter. A remarkable example is the case of weakly interacting bosonic systems, whose ground state is a squeezed vacuum state, and whose elementary excitations have a collective nature. In this work, we report on the direct observation of the peculiar microscopic quantum structure of these elementary excitations. We perform time- and frequency-resolved two-photon correlation measurements on the fluctuations of weakly interacting polaritons in a resonantly-driven microcavity, and observe that upon decreasing the average number of fluctuation quanta below unity, large pair correlations build up together with strong time-ordering of the emitted photons. This behavior is a direct signature of the particle-hole quantum superposition which is at the heart of Bogoliubov excitations.

Figures (3)

• Figure 1: Mechanism of time-ordering in the correlated photon pairs emitted by a single Bogoliubov mode -- Qualitative sketch of the time-evolution of the number of excitation quanta $n_b$ in a single mode Bogoliubov field (purple thick line), in the regime $\langle \hat{n}_b\rangle=\langle \hat{\beta}^\dagger\hat{\beta}\rangle <1$. The Bogoliubov field leaks outside the system by emitting photons across the imperfect mirrors (horizontal black thick line). Every time a normal [ghost] photon is detected ("photon clicks"), the Bogoliubov field $\hat{\beta}$ is projected ("Jumps") into a state with one less [one more] excitation. Normal (ghost) photons are shown in blue (red) wavy arrows, and the corresponding operator $\hat{\alpha}_N\propto \hat{a}_N$ ($\hat{\alpha}_G\propto \hat{a}_G$) is mentioned for each photon click event.
• Figure 2: Measurement of time-ordered correlations between normal and ghost photons in a $7\,\mu$m micropillar microcavity Spatially-resolved photoluminescence spectroscopy $I_{PL}(x,\omega)$ in the low-intensity regime (a), and spatially-resolved emission spectrum of the fluctuation $I_{b}(x,\omega)$ under a strong resonant CW drive on the $1s$ LP mode (b). The photon count rate is plotted in logarithmic color scale. The micropillar edges are shown as vertical dashed lines. In b), an additional spatial filter rejects the emission from the micropillars edges (cross hatched region). The calculated micropillar lower (LP) and upper (UP) polariton levels in the linear regime are shown in (a) with their mode labels as white and pink dashes respectively. The calculated Bogoliubov normal-UP (NUBP), normal-LP (NLBP) and ghost-LP (GLBP) modes are shown in (b) with their mode labels as pink, white and green dashes respectively. The single photon collection areas used for the detection of the normal (ghost) photon $C_{N,1p}$ ($C_{G,1p}$) in the correlation measurement are indicated in b). The vertical (horizontal) size of $C_{N(G),1p}$, the collected regions, indicates the collected spectral bandwidth $\hbar\Gamma_{bw}$ (the horizontal extension $\Delta x$ of the collected mode area). (c) Measured (round symbols) normal-ghost photon correlation function $g_{NG,1p}^{(2)}(\tau)$. $\tau_0(\omega_{N,G}) \in[-15,15]\,$ps is a small unknown delay offset due to optical path length dependence on $\omega_{N,G}$. The thick red, and dashed white lines are fits to the data obtained using the Bogoliubov model Eq.(\ref{['eq:g2_of_tau_single_mode']}), and the relaxed lineshape model respectively. The latter is also obtained from Eq.(\ref{['eq:g2_of_tau_single_mode']}), but $A$ and $\xi$ are left independent. $\tau_0$ is a fixed offset determined by the fit with the Bogoliubov model. The blue dashed line is a fit of the right side ($\tau\geq0$), using a the relaxed lineshape model and taking $\xi=0$ (symmetric peak).
• Figure 3: Correlation amplitude $A$ and asymmetry $\xi$ of the $1p$ Bogoliubov mode emission as a function of temperature T -- (a) Measurement of $g_{1p}^{(2)}(\tau-\tau_0)$ as a function of the micropillar temperature (blue to red symbols for temperatures increasing from $4.4\,$K to $9\,$K). $\tau_0(\omega_{N,G}) \in[-15,15]\,$ps is a small unknown delay offset due to optical path length dependence on $\omega_{N,G}(T)$. The solid lines are fits to the data using the relaxed lineshape model. b) Measured emission intensity $I_{1p,N}$ of the normal $1p$ Bogoliubov mode (left axis), and proportion $\rho_G$ of background emission at the $1p$ Ghost frequency (right axis), as a function of temperature. (c) Measured (symbols) correlation asymmetry $\xi_M$ as a function of the measured correlation characteristic amplitude $A_{Mc}$, corrected from the uncorrelated background photons. The theoretical $\xi(A)$ is plotted as a solid line. (d) Measured (symbols) $1p$ number $n_{b,M}$ of Bogoliubov excitation as derived from (b) as a function of $A_{Mc}$. The theoretical $n_b(A)$ is plotted as a solid line. In (b,c,d), the three color shades and symbols (circles, diamonds, triangles) corresponds to three separate experimental runs, the data in (a) come from the first run.