Limitations of a measurement-assisted optomechanical route to quantum macroscopicity of superposition states

Optomechanics is currently believed to provide a promising route towards the achievement of genuine quantum effects at the large, massive-system scale. By using a recently proposed figure of merit that is well suited to address continuous-variable systems, in this paper we analyze the requirements needed for the state of a mechanical mode (embodied by an end-cavity cantilever or a membrane placed within an optical cavity) to be qualified as macroscopic. We show that according to the phase-space-based criterion that we have chosen for our quantitative analysis, the state achieved through strong single-photon radiation-pressure coupling to a quantized field of light and conditioned by measurements operated on the latter might be interpreted as macroscopically quantum. In general, though, genuine macroscopic quantum superpositions appear to be possible only under quite demanding experimental conditions.

Figure 1

(a) Wigner function of the conditional state of the mechanical mirror achieved by projecting the cavity field into the origin of the phase space. (b) Measure of quantum macroscopicity $\mathcal{I}$ and mean number of phonons $\mathcal{M}$ in the conditional mechanical state, plotted against the (dimensionless) coupling rate $k$. The inset is for $k\in [0,5]$. In both panels, we have taken $T=0,\alpha =0.8,{\beta }_0=2$, and $t=\pi$. The Wigner function in (a) has been evaluated using $k=1$. All of the quantities being plotted are dimensionless.

Figure 2

(a) [(b)] Measure of quantum macroscopicity [Average number of phonons] for the conditional state of a mechanical mode obtained taking $\alpha =0.8,{\beta }_0=0$ and projecting the cavity field onto the origin of its phase space. From bottom to top [top to bottom] curve, we have $\overline{n}=0.1,{10}^{-2}$, and ${10}^{-4}$, respectively. (c) Maximum degree of macroscopicity in the conditional state of an end-cavity mechanical oscillator, plotted against the mean occupation number $\overline{n}$ of the initial mechanical state. We have taken $k=0.7$ and $\alpha =0.8$. All the quantities being plotted are dimensionless.

Figure 3

(a) Behavior of the degree of squeezing $\zeta \left(n\right)$ for $n=0$ and $n=1$ against the dimensionless coupling strength $k=g/{\omega }_m$ at $\tau =\pi$. (b) Distribution of the probability amplitudes in the state of the membrane achieved when taking $\alpha =0.7,x=1,k=1,t=\pi$. (c) Wigner function associated with the state $\left|{\psi }_{\pi }^{\prime }\right.〉\simeq \mathcal{N}(\left|0\right.〉+\left|\zeta \left(1\right)\right.〉)$ of the mechanical system for a value of $k$ such that $\left|\zeta \right(1\left)\right|=2$. (d) Measure of macroscopicity and mean phonon number plotted against the degree of squeezing $\zeta =\left|\zeta \right(1\left)\right|$. Only at large degrees of squeezing is the state of the mechanical system macroscopically coherent. All the quantities being plotted are dimensionless.

Figure 4

We show the measure of macroscopicity $\mathcal{I}$ and the mean phonon number $\mathcal{M}$ against the cavity-pump detuning ${\mathrm{\Delta }}_0$ for ${\gamma }_m/2\pi =100$ Hz, ${\omega }_m/2\pi =10$ MHz, $\kappa =88$ MHz, $\epsilon =6\times {10}^{12}$ Hz, and a mechanical system of $5\mathrm{ng}$ mass in a cavity of 1 mm length. The external pump has frequency ${\omega }_L/2\pi =3.7\times {10}^{14}$ Hz and the operating temperature is taken to be the rather optimistic value of $0.4$ K. Both $\mathcal{I}$ and $\mathcal{M}$ are dimensionless.