Quantum entanglement and teleportation in pulsed cavity optomechanics

Entangling a mechanical oscillator with an optical mode is an enticing and yet a very challenging goal in cavity optomechanics. Here we consider a pulsed scheme to create Einstein-Podolsky-Rosen-type entanglement between a traveling-wave light pulse and a mechanical oscillator. The entanglement can be verified unambiguously by a pump-probe sequence of pulses. In contrast to schemes that work in a steady-state regime under a continuous-wave drive, this protocol is not subject to stability requirements that normally limit the strength of achievable entanglement. We investigate the protocol's performance under realistic conditions, including mechanical decoherence, in full detail. We discuss the relevance of a high mechanical $Qf$ product for entanglement creation and provide a quantitative statement on which magnitude of the $Qf$ product is necessary for a successful realization of the scheme. We determine the optimal parameter regime for its operation and show it to work in current state-of-the-art systems.

Figure 1

Schematic of the system and the teleportation protocol: (a) A blue detuned light pulse ($A$) is entangled with the mirror ($B$). (b) A second light pulse ($V$) is prepared in the input state and interfering with $A$ on a beam splitter. Two homodyne detectors measure $P_{\mathrm{l}}^{\mathrm{out}}+X_{\mathrm{v}}$ and $X_{\mathrm{l}}^{\mathrm{out}}+P_{\mathrm{v}}$, yielding outcomes $m_X$ and $m_P$ respectively. Feedback is applied by displacing the mirror state in phase space by a unitary transformation $D_{X_{\mathrm{m}}}\left(m_X\right)D_{P_{\mathrm{m}}}\left(m_P\right)$. (c) To verify the success of the protocol, the mirror state is coherently transferred to a red detuned laser pulse and a generalized quadrature $X_{\mathrm{l}}^{\prime }\left(\theta \right)={X_{\mathrm{l}}^{\prime }}^{\mathrm{out}}\cos \theta +{P_{\mathrm{l}}^{\prime }}^{\mathrm{out}}\sin \theta$ is measured. Repeating steps (a)–(c) for the same input state but for different phases $\theta$ yields a reconstruction of the mirror's quantum state.

Figure 2

Optimized parameters for $Q_{\mathrm{m}}={10}^7$ and $n_0=50$ [red (dark) line] and $Q_{\mathrm{m}}={10}^5$ and $n_0=0$ [blue (light) line], where $n_0$ is the initial mechanical occupation and $\overline{n}$ is the mean bath occupation. This corresponds to the two cases of large $Q_{\mathrm{m}}$ with moderate pre-laser-cooling and lower $Q_{\mathrm{m}}$ with precooling into the ground state. Clearly, entanglement creation is possible in both cases. (a) Minimal ${\Delta }_{\mathrm{EPR}}$ as a function of $\overline{n}$. The upper axis gives the corresponding bath temperature for an oscillator with a resonance frequency of 3.8 MHz. To each point corresponds a triple ${\epsilon }_{\mathrm{opt}},{\eta }_{\mathrm{opt}},{\xi }_{\mathrm{opt}}$ [(b)–(d)], for which the minimal value is realized. The dotted, black line shows the upper bound up to which entanglement is present. The black dot marks the values for $\overline{n}=1100$ ($T=200\mathrm{mK}$), which coincide with the values given in the first row of Table . The dashed lines show the respective thermal noise contribution $(2\overline{n}+1){\epsilon }_{\mathrm{opt}}$ to ${\Delta }_{\mathrm{EPR}}$ [see Eq. (14)]. (b)–(d) Values of $\epsilon ,\eta ,\xi$ which optimize ${\Delta }_{\mathrm{EPR}}$ for a given $\overline{n}$. (e) Relative amount of noise induced by the coupling to the mechanical environment (${\epsilon }_{\mathrm{opt}}^{\prime }={\epsilon }_{\mathrm{opt}}/{\Delta }_{\mathrm{EPR}}$).