Spectrum of collective excitations of a quantum fluid of polaritons

We use a recently developed high-resolution coherent probe spectroscopy method to investigate the dispersion of collective excitations of a polaritonic quantum fluid. We measure the dispersion relation with high energy and wave-number resolution, which allows to determine the speed of sound in the fluid and to evidence the contribution of an excitonic reservoir. We report on the generation of collective excitations at negative energies, on the ghost branch of the dispersion curve. Precursors of dynamical instabilities are also identified. Our methods open the way to the precise study of quantum hydrodynamics of quantum fluids of light.

Figure 1

Properties of microcavity polaritons. (a) Energy dispersion of the upper (UP) and lower (LP) polaritons. (b) Bistability curve at a fixed pump frequency, indicating the working points A, B, C. The red dot highlights the turning point (TP) of bistability.

Figure 2

Experimental setup. The microcavity is placed in a cryostat at 4 K. It is illuminated by a pump laser orthogonal to the cavity. The probe laser, modulated at 5 MHz, with an adjustable frequency and in-plane wave vector $+k$, is detected either in transmission or in reflection (not shown here). When the probe resonates with the pump polariton fluid, it is transmitted and generates an additional 4WM signal at $-k$. The polariton fluid is imaged in the reciprocal space on a DMD displaying a pinhole of tunable position, switched between $+k$ and $-k$. By selecting the $+k(-k)$ angle, a direct (4WM) detection of the probe transmission is operated. The pinhole output is detected by a photodiode, connected to a spectrum analyzer demodulating the signal at 5 MHz.

Figure 3

Bogoliubov dispersion curves (a) Energy scan of the cavity reflectivity at fixed $k=0.40\mu {\mathrm{m}}^{-1}$ (corresponding to the vertical red dashed line in (b). (b) Dispersion in the linear regime (no pump laser). The red points show the minima of the reflectivity dip for each energy scan. The red arrow indicates the energy $\hslash {\omega }_0$ of the pump used in (b) and (c). (c) Dispersion in the bistable nonlinear regime (point B in the bistability curve of Fig. 1). (d) Dispersion measured close to the bistability turning point A. The shape of the curve is changed as compared to (c) and the gap is decreased.

Figure 4

Analysis of the shape of the spectrum linewidth. (a) Vertical slices taken along the blue dashed lines at $k=0.16\mu {\mathrm{m}}^{-1}$ and orange dashed lines at $k=0.64\mu {\mathrm{m}}^{-1}$ of the spectrum shown in (b). (b) Reflection map of the probe with respect to its wave vector $k$ and energy $\hslash \omega$, at a detuning $\hslash \delta =0.2\mathrm{meV}$. The horizontal red dotted line indicates the energy of the pump $\hslash {\omega }_0$. The red points highlight the minima of reflectivity $R_{\min }$; the green points the correspond to the reflectivity dips at $R=0.11\times (R_{\max }-R_{\min })$. The red and green curves are the corresponding theoretical fits, obtained for $\hslash gn=122$ and $114\mu \mathrm{eV},$ respectively. (c) Analytical dispersion curves corresponding to concentric rings of equal width in the Gaussian pump, of reflectivity calculated assuming a Gaussian probe. The red curve corresponds to $\hslash gn=122\mu \mathrm{eV}$; the linear green curve to $\hslash gn=114\mu \mathrm{eV}$.

Figure 5

Experimental measurements (dots) and fitting (solid line) of (a) the speed of sound $c_s$, (b) the reservoir factor $\alpha$, and (c) the long-lived excitonic reservoir energy $\hslash g_r{n}_r$ as a function of the pump detuning $\delta$.

Figure 6

Dispersion relation of the exciton reservoir. (a) Full reflection map of the probe. (b) (Right) Zoom in the region of the red frame in (a). (Left) Vertical slice along the red dashed line of the right panel. The normal (NB) $R<1$ and ghost (GB) $R>1$ branches are observed. The additional narrow flat branch, locally amplified at the pump energy and indicated by the red arrow, corresponds to the dispersion relation of the exciton reservoir, detected at the same energy as the pump. (c) Numerical calculation of the polariton dispersion relation for the same cavity and pump parameters as the experiment.

Figure 7

Spectra in direct and FWM detections, at a pump detuning $\hslash \delta =0.05\mathrm{meV}$. (a) (Right) Reflection map with the detection set at the wave vector of the probe (direct detection scheme). The pump energy $\hslash {\omega }_0$ is indicated by the horizontal red dotted line. (Left) Slice along the vertical red dashed line of the spectrum. The reflectivity dip $R<1$ corresponds to the normal Bogoliubov branch; the amplification peak $R>1$ to the ghost branch. (b) (Left) Transmission map with the detection set at the opposite wave vector of the probe (indirect detection scheme), under the same experimental conditions as (a). $k$ and $\hslash \omega$ correspond respectively to the wave number and energy emission of the FWM process. (Right) Vertical slice along the vertical red dashed line of the spectrum. Two FWM peaks are detected: the more intense one corresponds to the ghost branch originating from the parametric scattering of the probe at resonance with the normal branch at frequencies above the pump ($\omega >{\omega }_0$); the other, less intense one to the normal branch, also detected via parametric scattering but originating from the resonance of the probe with the ghost branch at frequencies below the pump ($\omega <{\omega }_0$). The $\left|k\right|<0.15\mu {\mathrm{m}}^{-1}$ part is masked because the direct transmission angle of the probe is close to the emission angle of the FWM, preventing the filtering of one signal from the other with the DMD.

Figure 8

Detection of the precursors of dynamical instabilities for an intensity near the point C of the bistability curve and a detuning $\hslash \delta =0.16\mathrm{meV}$. (a) (Right) Transmission map of the probe in direct detection. The zoom in the red dashed inset highlights a narrow plateau in the spectrum, signature of the onset of precursor of modulation instabilities at the pump energy $\hslash {\omega }_0$ indicated by the red arrow. (Left) Slice along the vertical red dashed line of the spectrum. At the pump energy, a narrowing and amplification of the probe is visible, related to the change in the imaginary part of the energy of the collective excitations. (b) Left: transmission map of the ghost branch obtained in indirect detection, under the same experimental conditions as (a). $k$ and $\hslash \omega$ are respectively the energy and wave vector of the FWM emission. The ghost branch is detected. The plateau is clearly visible at the crossing point with the normal branch, with a better resolution than in (a). The $\left|k\right|<0.25\mu {\mathrm{m}}^{-1}$ part is masked for the same reason as in Fig. 7. Right: slice along the vertical red dashed line of the spectrum. The sub-$\gamma$ linewidth of the resonance peak gives the imaginary part of the energy of the modulation instabilitiy precursor.