Optomechanical parametric oscillation of a quantum light-fluid lattice

Two-photon coherent states are one of the main building pillars of nonlinear and quantum optics. They are the basis for the generation of minimum-uncertainty quantum states and entangled photon pairs, applications not obtainable from standard coherent states or one-photon lasers. Here, we describe a fully resonant optomechanical parametric amplifier involving a polariton condensate in a trap lattice quadratically coupled to mechanical modes. The quadratic coupling derives from nonresonant virtual transitions to extended discrete excited states induced by the optomechanical coupling. Nonresonant continuous-wave laser excitation leads to striking experimental consequences, including the emergence of optomechanically induced intersite parametric oscillations and intersite tunneling of polaritons at discrete intertrap detunings corresponding to sums of energies of the two involved mechanical oscillations (20- and 60-GHz confined vibrations). We show that the coherent mechanical oscillations correspond to parametric resonances with a threshold condition different from that of standard linear optomechanical self-oscillation. The associated Arnold tongues display a complex scenario of states within the instability region. The observed phenomena can have applications for the generation of entangled phonon pairs and squeezed mechanical states relevant in sensing and quantum computation and for the bidirectional frequency conversion of signals in a technologically relevant range.

Figure 1

Polariton effective potentials and optomechanical sidebands. (a) is a scheme of the lattice consisting of a pumped central polariton trap and neighbor traps in a square array. DBR, distributed Bragg reflector; QW, quantum well. (b) presents the involved polariton levels corresponding to the strongly confined ground states of the pumped trap ($|1\rangle$) and one of its neighbor traps ($|2\rangle$), as well as a delocalized excited state ($|3\rangle$). $g_0$ is the linear optomechanical interaction coupling the $s$-symmetric ground states with the $p$-symmetric excited state with the concomitant emission of a $p$-like confined phonon. (c) [(d)] displays the spectral and spatial image corresponding to the low-pump-power (high-pump-power) condition. The effective trap potentials (shaded light red), lateral distribution of the exciton energy (top dashed Gaussian curve), and corresponding confined polariton wave functions (thin colored curves) derived from a Gross-Pitaevskii modeling of the quantum light fluid are also shown. The detuning between the pumped and neighbor trap ground states ($\delta E$) can be tuned through the excitonic-related repulsive interaction with the reservoir by varying the nonresonant incoherent pump power as shown in (e) (note that this detuning is given in units of the phonon energy $\hslash {\mathrm{\Omega }}_0$ in the top scale). The emission from ground states $|1\rangle$ and $|2\rangle$ and the excited state $|3\rangle$ can be identified. Note that clear and symmetrical spectral sidebands appear for both the ground and excited state precisely when one of the neighbor traps is red detuned with respect to the pumped trap by integer multiples 2 and 4 of the fundamental breathing mechanical cavity mode ${\nu }_{\mathrm{m}}^0\sim 19$ GHz.

Figure 2

Mechanical mode softening, intertrap frequency locking, and enhanced polariton transfer. Examples of PL emission precisely at the resonant detuning conditions $\delta E_4\sim 75$ GHz and $\delta E_2\sim 37$ GHz are shown in (a) and (b), respectively. Asterisks indicate PL from neighbor traps. The symmetric evenly spaced mechanical sidebands can be clearly observed at detunings corresponding to integer numbers $n=2$ and $n=4$ of the fundamental mechanical frequency $\hslash {\mathrm{\Omega }}_0$. Fits to the spectra [indicated with continuous orange curves in (a) and (b)] allow for the derivation of the applied incoherent nonresonant cw pump power dependence of the involved mechanical frequencies, the trap energy detuning, and their occupation [shown in (c)–(e), respectively]. The vertical yellow bands indicate the regions where the intertrap resonances are attained, leading to the observation of the mechanical sidebands.

Figure 3

Optomechanical instability and Arnold tongues of the effective parametric resonator. (a) shows the calculated PL spectra for the Hamiltonian $H_0+H_{\mathrm{OM}}$, as a function of the bare detuning $\delta E$ between traps 1 and 2. Here we set ${\omega }_1=0$, ${\omega }_2=\delta E/\hslash$, ${\omega }_3=24{\mathrm{\Omega }}_0$, and the threshold power as indicated in the main text. Optomechanical instabilities are observed at $\delta E=-4\hslash {\mathrm{\Omega }}_0$ and $\delta E=-2\hslash {\mathrm{\Omega }}_0$, corresponding to resonances involving the fundamental and first overtone of the mechanical vibrations (${\mathrm{\Omega }}_1\pm {\mathrm{\Omega }}_0$), or two fundamental mechanical vibrations ($2{\mathrm{\Omega }}_0$), respectively. The resolved additional peaks appear around both the ground and excited states and are separated by the frequency of the fundamental mode (${\mathrm{\Omega }}_0$). As a function of trap detuning, and depending on the occupancy of the involved levels $n_1$ and $n_2$, instability (stability) regions can be defined characterized by the presence (absence) of coherent mechanical oscillations. The instability regions correspond to so-called Arnold tongues of a parametric resonator, with characteristic stationary values of $\sqrt{{\langle x_0^2\rangle }^{\phantom{0}}}$, which is proportional to the square root of the phonon number (b), the dressed cavity detuning $\mathrm{\Delta }\omega$ (c), and the traps' level occupation, as displayed in (d)–(f). Note in (d) and (e) the strong optomechanically enhanced transfer of polaritons from trap 1 to trap 2 and in (c) the locking of their relative detuning within the instability region to the resonant value given by the sum of the two involved vibrations. Dotted lines in the color maps indicate the boundary of the Arnold tongue as estimated from Eq. (9).