Amplification effects in optomechanics via weak measurements

We revisit the scheme of single-photon weak-coupling optomechanics using postselection, proposed by Pepper, Ghobadi, Jeffrey, Simon, and Bouwmeester [Phys. Rev. Lett. 109, 023601 (2012)], by analyzing the exact solution of the dynamical evolution. Positive and negative amplification effects of the displacement of the mirror's position can be generated when the Kerr phase is considered. This effect occurs when the postselected state of the photon is orthogonal to the initial state, which cannot be explained by the usual weak measurement results. The amplification effect can be further modulated by a phase shifter, and the maximal displacement state can appear within a short evolution time.

Figure 1

Photon enters the first beam splitter of Mach-Zehnder interferometer, followed by an optomechanical cavity $A$ and a conventional cavity $B$. The photon weakly excites the small mirror. After the second beam splitter, dark port is detected, i.e., postselection acts on the case where the mirror has been excited by a photon, and fails otherwise.

Figure 2

Average displacement $\langle q\left(t\right)\rangle /\sigma$ of the mirror as a function of ${\omega }_{m}t$ with $k=0.005$, $\gamma =0$ (solid line), and $\gamma =0.005$ (dashed line).

Figure 3

Wigner function of the state achieving the maximal negative amplification. In this figure, $x$ stands for $c+c^{†}$ and $y$ for $-i(c-c^{†})$. They are therefore dimensionless quantities.

Figure 4

Average displacement $\langle p\left(t\right)\rangle /\frac{\hslash }{2\sigma }$ of the mirror as a function of ${\omega }_{m}t$ with $k=0.005$.

Figure 5

Average displacement $\langle q\left(t\right)\rangle /\sigma$ for $\theta =0.001$ (a) and $\theta =-0.001$ (b); other parameters are the same as before, i.e., $k=0.005$, $\gamma =0$ (solid line), and $\gamma =0.005$ (dashed line).