Macroscopic Tunneling of a Membrane in an Optomechanical Double-Well Potential

The macroscopic tunneling of an optomechanical membrane is considered. A cavity mode which couples quadratically to the membranes position can create highly tunable adiabatic double-well potentials, which together with the high $Q$ factors of such membranes render the observation of macroscopic tunneling possible. A suitable, pulsed measurement scheme using a linearly coupled mode of the cavity for the verification of the effect is studied.

Figure 1

Membrane-in-the-middle optomechanical system. The dashed lines illustrate the adiabatic double-well potential induced by the field mode. The membrane is shown in a superposition of the left-well state and right-well state. The lower figure sketches a scheme for the detection of membrane tunneling.

Figure 2

Ratio of barrier height to ground state energy (solid line, left axis) and half minimum separation $x_{\min }$ in units of the zero-point width $x_{\mathrm{zpt}}$ (dotted line, right axis) as functions of the pumping rate $\eta$. The $x$ axis starts at the point where we have $D=0$ (see the main text). The coupling strength is $g_2/{\omega }_M=-2\times {10}^{-4}$, and the detuning is zero.

Figure 3

Double well and resulting splitting of energy levels. Also shown are the squared amplitudes of the two localized wave functions. Parameters used: $g_2/{\omega }_M=-2\times {10}^{-4}$ and $\eta /{\omega }_M=176.785$.

Figure 4

Tunneling rate $J$ of the two lowest lying states, normalized to ${\omega }_M$, as a function of the scaled pumping rate $\eta /{\omega }_M$ for $g_2/{\omega }_M=-2\times {10}^{-4}$, $\kappa /{\omega }_M=10$, and ${\delta }_c=0$. Note the logarithmic $y$ axis.

Figure 5

Typical outcome of a sequence of 20 position measurements of a membrane initially prepared on the right potential well. The solid line gives the coherent evolution (i.e., without measurements) of $⟨\widehat{x}⟩$ for that particular trajectory. Inset: Distribution of 200 measurement outcomes at ${\omega }_{M}t=20$, where only trajectories that started in the right well were postselected. Parameters are as in Fig. 3; measurement strength $1/\sigma =0.02$.