Figure 1

(a) Standard optomechanical setup driven by a control and a probe laser. (b) Classical expectation of the OMIT signal as a function of the probe and control detuning, cf. Eq. (6). If the beat frequency of the two lasers $|{\omega }_c-{\omega }_p|\sim \Omega$, a signal is expected (dotted black lines). We study the OMIT signal at the circles. In particular, we find an OMIT signal at the second mechanical sideband (thick red circle) even at moderate single-photon coupling, where a classical analysis fails to show the signal. This is a clear signature of the optomechanical quantum nonlinearity. (c) Field amplitude in a frame rotating at the control laser frequency ${\omega }_c$. The control generates a constant transmission amplitude $\tilde{\alpha }$ (dashed red line). The probe induces oscillations at the beat frequency ${\omega }_p-{\omega }_c$. The amplitude of these oscillations, Eq. (4), is our signal. [Parameters: $g=g_0|\alpha |=0.08\Omega$, $\kappa =\Omega /8$, ${\Gamma }_M=0.01\Omega$.]Reuse & Permissions

Figure 2

Quantum nonlinearities and OMIT. (a) Energy level scheme of the optomechanical system with levels $|n_a,n_b⟩$, cf. main text. $n_a$ ($n_b$) denotes the number of photons (phonons). Here, for example, the probe couples $|0,0⟩\leftrightarrow |1,0⟩$ since the detuning ${\Delta }_p={\omega }_p-{\omega }_{\mathrm{cav}}^{\mathrm{eff}}=0$. The control couples $|0,1⟩\leftrightarrow |1,0⟩$ since ${\Delta }_c={\omega }_c-{\omega }_{\mathrm{cav}}^{\mathrm{eff}}=-\Omega$. (b) The OMIT signal. Orange circles: Numerical results for $g_0/\kappa =4$; black line: expectation for the standard, classical regime, Eq. (6); blue line: expectation of Eq. (6), but including Franck-Condon factors and $|\alpha {|}^2\to ⟨{\widehat{a}}^{†}\widehat{a}⟩$, see main text. [Parameters: $\kappa =\Omega /40$, ${\Gamma }_M={10}^{-3}\Omega$, ${\epsilon }_c={10}^{-2}\Omega$, ${\Delta }_c=-\Omega$, $n_{\mathrm{th}}=0$.]Reuse & Permissions

Figure 3

Optomechanical transistor for large $g_0/\kappa$. (a) The probe drives the cavity continuously. At time $t=0$, the control laser power is ramped up linearly until $\Omega t_{\mathrm{switch}}\gg 1$. This decreases the normalized probe beam transmission on a scale $\sim {\Gamma }_M^{-1}$, cf. (b). When switching off the control linearly, $|\delta \tilde{a}{|}^2$ increases on a scale $\sim {\kappa }^{-1}\ll {\Gamma }_M^{-1}$. (c) Occupation transfer between individual quantum states induced by the control. Oscillations at $\Omega$ are clearly visible in $p_{10}$ (i.e., for the state with 1 photon, 0 phonons). [Parameters: Same as in Fig. 2, ${\Delta }_p=0$, $\Omega t_{\mathrm{switch}}=100$.]Reuse & Permissions

Figure 4

Quantum to classical crossover. The OMIT signal at the second sideband (b) vanishes in the classical limit $g_0/\kappa \to 0$, hence being a clear signature of the quantum nonlinearity. It is visible even for moderate coupling strengths $g_0<\kappa$. The yellow regions indicate the OMIT features’s width $\sim {\Gamma }_M$. (a) Since ${\Delta }_p=\Omega$ and ${\Delta }_c=0$, probe and control couple the levels $|0,0⟩\leftrightarrow |1,1⟩$ and $|0,1⟩\leftrightarrow |1,1⟩$, respectively. (b) Since ${\Delta }_p=2\Omega$ and ${\Delta }_c=0$, $|0,0⟩\leftrightarrow |1,2⟩$ and $|0,2⟩\leftrightarrow |1,2⟩$ are coupled, respectively. Symbols: OMIT signal for different $g_0/\kappa$ where $g_0|\alpha |$ and $\kappa$ are kept fixed. Gray line: Classical expectation, cf. (6). Inset of (b): OMIT signal for $g_0/\kappa =1/10$. The axes are the same as in (b). The OMIT signal varies in a range $\sim {10}^{-7}$. Note that the signal strength can be increased by increasing $g_0|\alpha |$. (c) Amplitude of the Fano resonance at the second sideband (i.e., the difference between the maximum and minimum) versus the thermal phonon number $n_{\mathrm{th}}$ (normalized to the amplitude at $n_{t\mathrm{h}}=0$). [Parameters: $\kappa =\Omega /8$, ${\Gamma }_M={10}^{-3}\Omega$, ${\Delta }_c=0$, $n_{\mathrm{th}}=0$, $g_0{\epsilon }_c=1.25\times {10}^{-3}{\Omega }^2=\mathrm{const}$ (this value has been chosen to keep the Hilbert space manageable as $g_0/\kappa \to 0$). (c) Same as in (b), $g_0/\kappa =1/2$.]Reuse & Permissions