Transfer of non-Gaussian quantum states of mechanical oscillator to light

Non-Gaussian quantum states are key resources for quantum optics with continuous-variable oscillators. The non-Gaussian states can be deterministically prepared by a continuous evolution of the mechanical oscillator isolated in a nonlinear potential. We propose feasible and deterministic transfer of non-Gaussian quantum states of mechanical oscillators to a traveling light beam, using purely all-optical methods. The method relies on only basic feasible and high-quality elements of quantum optics: squeezed states of light, linear optics, homodyne detection, and electro-optical feedforward control of light. By this method, a wide range of novel non-Gaussian states of light can be produced in the future from the mechanical states of levitating particles in optical tweezers, including states necessary for the implementation of an important cubic phase gate.

Figure 1

Universal conversion of non-Gaussian quantum states of the mechanical oscillator generated by a nonlinear potential to traveling light: OPO, optical parametric oscillator generating squeezed light; C, optical circulator; SBS, balanced 50:50 beam splitter; BS, beam splitter with transmittance $T_c$; HD, homodyne detector; $g_x,g_p$, electronic units with variable gains; and $D_x,D_p$, displacement operations.

Figure 2

The improved transmittancy $T^{\prime }>T$ of the squeezed-light powered converter for good converter $T=0.9$ (top) and bad converter $T=0.1$ (bottom). ${\eta }_i,{\eta }_o$ are incoupling and outcoupling efficiencies.

Figure 3

The relative improvement between the transmittancy $T^{\prime }$ and the transmittancy ${\eta }_{o}T$ of the direct converter. ${\eta }_i$, incoupling efficiency, and ${\eta }_o$, outcoupling efficiency.

Figure 4

The minimal transmittancy $T$ of beam-splitter coupling to reach $T^{\prime }>0.5$ for the squeezed-state powered converter visualized by the lower light plane. ${\eta }_i$, incoupling efficiency, and ${\eta }_o$, outcoupling efficiency. The dark upper plane corresponds to the condition $T>1/\left(2{\eta }_o\right)$ for the direct converter without the proposed method.

Figure 5

Wigner function $W(0,p)$ of state from cubic mechanical nonlinearity with ${\kappa }_3\tau =1$ transferred by universal converter (solid black line) $T=0.5$, ${\eta }_i={\eta }_o=0.9$, variance $V_S=0.125$ and for ground initial state : (solid gray line) $T=0.5$, ${\eta }_i={\eta }_o=0.9$ and asymptotic variance $V_S=0$. For comparison, (dashed line) without universal converter $T=0.5$, ${\eta }_o=0.9$, (dotted line) ideal state from cubic nonlinearity with $T=1$ and ${\eta }_o=1$.

Figure 6

Contours of regions of negativity of the Wigner function $W(0,p)$ of state from cubic mechanical nonlinearity with ${\kappa }_3\tau =1$ transferred by universal converter with $T$, $\eta ={\eta }_i={\eta }_o$, and variance $V_S=0.125:\eta =1$ (solid black line), $\eta =0.9$ (dark gray line), $\eta =0.8$ (light gray line), $\eta =0.7$ (dashed light gray line), and $\eta =0.6$ (dotted light gray line).

Figure 7

Transfer of the mechanical bath variance $T_m^{\text{in}}$ as a function of the signal transfer $T_{B_{\text{in}}}$ calculated with parameters of the experiment reported in Ref. [39]. Solid and dashed lines represent respectively the full analytical solution (A5) and the approximate solution after adiabatic elimination of the cavity mode. The gray dashed line denotes the limit of possible transmittivity set by ${\eta }_o={\kappa }_e/\kappa \approx 0.83$.