Proposal for observing nonclassicality in highly excited mechanical oscillators by single photon detection

The preparation of pure quantum states with high degrees of macroscopicity is a central goal of ongoing experimental efforts to control quantum systems. We present a state preparation protocol which renders a mechanical oscillator with an arbitrarily large coherent amplitude in a manifestly nonclassical state. The protocol relies on coherent-state preparation followed by a projective measurement of a single Raman scattered photon, making it particularly suitable for cavity optomechanics. The nonclassicality of the state is reflected by sub-Poissonian phonon statistics, which can be accessed by measuring the statistics of subsequently emitted Raman sideband photons. The proposed protocol would facilitate the observation of nonclassicality of a mechanical oscillator that moves macroscopically relative to motion at the single-phonon level.

Figure 1

Diagram of the Raman scattering processes in the system, where ${\mathrm{\Delta }}_c=0$ and frequencies are relative to the cavity resonance frequency. Anti-Stokes photons from the red (${\mathrm{\Omega }}_R$) drive and Stokes photons from the blue (${\mathrm{\Omega }}_B$) drive pass through a filter with bandwidth $W\ll {\omega }_m$. The backaction from detecting a sideband photon is ambiguous, i.e., it produces a superposition of a phonon-added and a phonon-subtracted mechanical state.

Figure 2

Mandel $Q$ parameter (11) for the conditional state (9) when the initial state ${\rho }_m$ is a displaced, thermal state, in the high displacement limit $\left|\beta \right|\gg 1$, as a function of $\left|r\right|$ and the initial average phonon number $n_m$. We have assumed the ideal phase relation $cos\left(2\varphi \right)=1$. The dotted line separates the regions of positive and negative $Q$.

Figure 3

Conditioned Mandel $Q$ parameter (A1) evaluated for $\left|\beta \right|=\{5,10,20\}$ (from the top) along the optimizing axis $cos\left(2\varphi \right)=1$, where negative values of $r$ represent the $cos\left(\varphi \right)=-1$ part of the solution. The dotted lines delineate the negative (nonclassical) region boundaries. Note that for lower $\left|\beta \right|$ the nonclassical region stretches to higher $n_m$ when $r<0$ and goes as low as $Q_{b,c}\sim -0.3$ when $\left|\beta \right|=5$ and $n_m=0$.

Figure 4

Schematic of steady-state operation with multitone driving. Rather than varying the drive strengths sequentially (i.e., pulsed operation), sufficiently narrow filters allow different Stokes and anti-Stokes sideband compositions to be measured simultaneously in the steady state. Panel (a) shows a five-drive configuration with optomechanical cooling and displacement. Isolating the overlapping projective measurement sidebands from the cooling and readout sideband applies in the limit $W\ll |{\mathrm{\Delta }}_{\mathrm{proj}}|$. Panel (b) shows an alternative four-drive configuration where a strong cooling sideband may be used as an anti-Stokes photon source for both read and write measurements simultaneously by passing through both filters.