Single-Polariton Optomechanics

This Letter investigates a hybrid quantum system combining cavity quantum electrodynamics and optomechanics. The Hamiltonian problem of a photon mode coupled to a two-level atom via a Jaynes-Cummings coupling and to a mechanical mode via radiation pressure coupling is solved analytically. The atom-cavity polariton number operator commutes with the total Hamiltonian leading to an exact description in terms of tripartite atom-cavity-mechanics polarons. We demonstrate the possibility to obtain cooling of mechanical motion at the single-polariton level and describe the peculiar quantum statistics of phonons in such an unconventional regime.

Figure 1

Scheme of the considered hybrid system. A photon confined mode of frequency ${\omega }_c$ couples both to a two-level system (${\omega }_a$ is the transition frequency) and to a mechanical resonator of frequency ${\omega }_m$. $g_{ac}$ ($g_{cm}$) is the coupling strength of the Jaynes-Cummings (radiation pressure) atom-cavity (cavity-mechanics) coupling.

Figure 2

Structure of the polaron eigenstates for the atom-cavity-mechanics tripartite system. (a) Each $n$-polariton subspace (left) couples independently to states of the mechanical resonator with different displacements (right). (b) Energy spectrum for $n\le 2$. (c) Zoom on the energy levels in the ${\mathcal{H}}_1$ subspace (one-polariton states).

Figure 3

Optical joint spectral density of polaronic states describing transitions between the states with 0 and 1 polaritons. Main panel: $g_{cm}/{\omega }_m={10}^{-1}$, $Q_{ac}={10}^4$. Inset: $g_{cm}/{\omega }_m={10}^{-3}$, $Q_{ac}={10}^6$. For clarity, we present only transitions between states with $m^{(n)}\le 5$ polarons. Blue, green, and red peaks represent transitions reducing, conserving, and increasing the number of phonons, respectively.

Figure 4

Atom-assisted optomechanical cooling. (a) Time evolution of the number of photons (red) and phonons (blue) for an initial mechanical state $|l=2⟩$. (b) Phonon second-order correlation function $G_2$ of the stationary state. (c) Stationary number of phonons as a function of ${\omega }_p$ for $Q_m={10}^6$, $Q_{ac}={10}^6$, $g_{cm}/{\omega }_m={10}^{-3}$, and $F_p/{\gamma }_{ac}=100$. In (b) and (c), the dashed blue lines represent the hybrid QED-optomechanics case with an atom ($g_{ac}\ne 0$), while the black solid lines correspond to the usual atomless scenario ($g_{ac}=0$). The inset in (c) depicts schematically the doubly resonant polariton cooling of motion.

Figure 5

Steady-state mechanical resonator statistics (bath at zero temperature). (a) Real part of the mechanical reduced density matrix and (b) the corresponding Wigner function for an optical coherent pump tuned to $({\omega }_p-{\omega }_{+}^{(1)})/{\omega }_m={10}^{-1}$. (c) Phonon occupation number and (d) second-order correlation function $G_2$ under incoherent pumping as a function of the polariton quality factor $Q_{ac}$. The incoherent pump rate is set to $F_{\mathrm{inc}}={\gamma }_{ac}$. Black solid, blue dashed, and red dotted lines correspond to numerical results for $Q_m={10}^1$, ${10}^2$, and ${10}^3$, respectively. The analytical solutions of the master equation are represented by circles.