Heralded Single-Phonon Preparation, Storage, and Readout in Cavity Optomechanics

We show how to use the radiation pressure optomechanical coupling between a mechanical oscillator and an optical cavity field to generate in a heralded way a single quantum of mechanical motion (a Fock state). Starting with the oscillator close to its ground state, a laser pumping the upper motional sideband produces correlated photon-phonon pairs via optomechanical parametric down-conversion. Subsequent detection of a single scattered Stokes photon projects the macroscopic oscillator into a single-phonon Fock state. The nonclassical nature of this mechanical state can be demonstrated by applying a readout laser on the lower sideband to map the phononic state to a photonic mode and performing an autocorrelation measurement. Our approach proves the relevance of cavity optomechanics as an enabling quantum technology.

Figure 1

(a) Schematics of the optomechanical system pumped at the upper and lower motional sidebands and of the correlation measurement on the filtered cavity photons. (b) Representation of the relevant mechanical and optical frequencies and linewidths. (c) Resonant transitions during the write and readout pulses. For a given cavity photon number $n_a$, the Fock states of the mechanical oscillator form a harmonic ladder. Emission at ${\omega }_c$ is enhanced by the cavity, which allows us to selectively address the Stokes (S) [anti-Stokes (AS)] transitions between phonon states when driving the upper (lower) mechanical sideband. (d) Pulse sequence (cooling step not shown). Detection of a Stokes photon within the write pulse duration $T_w$ is the heralding event. After a storage time $T_{\mathrm{off}}$, coincidences between anti-Stokes photons emitted at time $t_r$ and $t_r+\tau$ are measured. We define $t_r=0$ at the beginning of the readout pulse. (e) Conditional two-photon coincidence as a function of the initial mechanical occupancy ${\overline{n}}_0$ (log scale) for different values of the product ${\tilde{g}}_w{T}_w$, under the assumption of negligible mechanical damping [$T_w\ll (\gamma {\overline{n}}_{\mathrm{th}}{)}^{-1}$ and $T_{\mathrm{off}}=0$]. Open squares: only single-photon emission events are postselected; see Eq. (5). Solid lines: including the contribution from multiple photon emission.

Figure 2

(a) Conditional second-order correlation function $g_{\text{cond}}^{(2)}(\tau |t_r)$ at fixed $t_r=1\mathrm{ns}$ for increasing waiting time $T_{\mathrm{off}}$ between the write and readout pulses. The bath temperature is set to 1.6 K (${\overline{n}}_{\mathrm{th}}=6.4$). The transition from antibunching to bunching is a signature of the relaxation from a single-phonon Fock state to a thermal state. (b) Same-time correlation $g_{\text{cond}}^{(2)}(0)$ (i.e., two-photon emission probability) as a function of $T_{\mathrm{off}}$ for decreasing phonon bath thermal occupancy ${\overline{n}}_{\mathrm{t}\mathrm{h}}$, yielding coherence times up to $(\gamma {\overline{n}}_{\mathrm{th}}{)}^{-1}\sim 100\mu \mathrm{s}$.

Figure 3

Shaping single photons. (a) Field and (b) intensity two-time correlation as a function of the time delay $\tau$ between photons, for increasing readout pulse intracavity photon number ${\overline{n}}_r$ (from black to red). Bath temperature is 1.6 K and $T_{\mathrm{off}}=5\mathrm{ns}$. The coherence time can be tuned over 3 orders of magnitude from $10\mu \mathrm{s}$ down to 10 ns, corresponding to linewidths from 100 kHz to 100 MHz. (c) Color plot of $g_{\text{cond}}^{(2)}(\tau )$ versus ${\overline{n}}_r$ and $\tau$. Rabi oscillations set in when the laser-enhanced optomechanical coupling becomes larger than the cavity decay rate: $g_{-}>\kappa$. Dashed lines show the expected maxima for the Rabi period $(1/2)\sqrt{g_0^2{\overline{n}}_r-(1/16){\kappa }^2}$.