Optomechanical generation of a mechanical catlike state by phonon subtraction

We propose a scheme to prepare a macroscopic mechanical oscillator in a catlike state, close to a coherent state superposition. The mechanical oscillator, coupled by radiation-pressure interaction to a field in an optical cavity, is first prepared close to a squeezed vacuum state using a reservoir engineering technique. The system is then probed using a short optical pulse tuned to the lower motional sideband of the cavity resonance, realizing a photon-phonon swap interaction. A photon number measurement of the photons emerging from the cavity then conditions a phonon-subtracted catlike state with a negative Wigner distribution exhibiting separated peaks and multiple interference fringes. We show that this scheme is feasible using state-of-the-art photonic crystal optomechanical system.

Figure 1

Optomechanical scheme for generation of a mechanical catlike state. (a) Illustration of a cavity-optomechanical system. A mechanical oscillator (frequency ${\mathrm{\Omega }}_m$, energy dissipation rate ${\mathrm{\Gamma }}_m$, displacement $\widehat{x}$) forms part of an optical cavity (frequency ${\omega }_c$, energy dissipation rate $\kappa$). Cavity photons couple to the oscillator through radiation-pressure interaction, and the output light from the cavity is analyzed. (b) Time-domain picture of the scheme. The oscillator is first prepared in a squeezed state by driving the cavity on the upper and lower motional sidebands. Then, a short pulse on the lower motional sideband drives an anti-Stokes photon-phonon scattering (beam splitter interaction), subtracting phonons from the mechanical state, which can be analyzed after a variable wait time. [(c), (d)] Frequency-domain pictures of the squeezing (c) and subtraction (d) stages. The Wigner distribution of the mechanical state is also shown.

Figure 2

State purity in optomechanical dissipative squeezing. (a) State purity $n_{\mathrm{eff}}$ vs squeezing parameter $r$ for different cooperativities $\mathcal{C}$. (b) The cooperativity required to achieve a given purity vs $r$. The steady state in optomechanical dissipative squeezing, in particular the variance of the squeezed quadrature but also the thermal component $n_{\mathrm{eff}}$, results from a tradeoff between optical damping and ratio of the drives. The two working points with $n_{\mathrm{eff}}=0.02$ used in this work are indicated in both panels.

Figure 3

Effect of initial state purity on the nonclassicality of phonon-subtracted squeezed thermal mechanical states. [(a), (b)] The output state macroscopicity and Wigner negativity vs squeezing parameter $r$ for state purities $n_{\mathrm{eff}}=0$ (blue), 0.02 (orange), and 0.1 (green) and for $m=1$ (solid), 2 (dashed), and 3 (dash-dotted) subtracted phonons. Squeezed vacuum $n_{\mathrm{eff}}=0$ is shown for reference. [(c)–(e)] Wigner distributions for $r=1$, $n_{\mathrm{eff}}=0.02$, and $m=1,2,3$. The macroscopicity $\mathcal{I}$ is indicated. In all panels, we set $\theta =0.1$.

Figure 4

Effect of optical losses on generated catlike mechanical state. (a) Heralding probability (successful detection of $m$ photons) vs total optical detection efficiency $\eta$, shown for initial squeezing $r=0.5$ (blue) and $r=1$ (orange) and for $m=1$ (solid), 2 (dashed), and 3 (dash-dotted) subtracted phonons. This probability incorporates the weak optomechanical beam-splitter interaction $\theta$, chosen as $\theta =0.05$ for $m=1$ cases and $\theta =0.1$ for $m=2$ cases. (b) The macroscopicity $\mathcal{I}$ corresponding to the same curves in panel (a). [(c)–(f)] Wigner distributions and macroscopicities of the heralded mechanical state, assuming $\eta =0.2$, for the points indicated in panels (a) and (b).