In an optomechanical resonator, an optical mode couples to mechanical vibrations via radiation pressure induced by circulating optical fields [1,2]. The optomechanical interactions can feature red and blue sideband couplings. Blue sideband coupling, for which the driving laser is one mechanical frequency above the optical resonance, generates photon–phonon pairs. Red sideband coupling, for which the driving laser is one mechanical frequency below the optical resonance, leads to interconversion between phonons and photons. These interactions enable the exploration of a variety of nonlinear and quantum optical processes in optomechanical systems. Earlier experimental studies have used blue sideband coupling to demonstrate self-induced mechanical oscillations in silica toroids [3,4]. More recent efforts have exploited red sideband coupling to realize nonlinear optical processes, such as optomechanically induced transparency (OMIT) [5–8], light storage [9–11], and coherent optical wavelength conversion [12–16]. In addition, squeezed light arising from radiation-pressure-induced ponderomotive squeezing has also been demonstrated [17,18].

One of the most celebrated phenomena in both quantum optics and nonlinear optics is parametric downconversion. Optical parametric downconversion can lead to the generation of photon pairs as well as the entanglement or squeezing of two optical modes [19]. An analogous process can in principle be realized for mechanical oscillators in a three-mode optomechanical system, in which two mechanical modes couple to a common optical mode. As illustrated in Fig. 1, the optomechanical coupling of one mechanical mode takes place via blue sideband coupling, generating phonon–photon pairs (Stokes scattering), whereas the coupling of the other mechanical mode takes place via red sideband coupling, converting the Stokes photon in the photon–phonon pair to a phonon. Together, these two processes can lead to the generation of phonon pairs in the two respective mechanical modes. The red and blue sideband couplings combined can also result in the formation of Bogoliubov mechanical modes that have the form of two-mode squeezed states [20,21].

 

Fig. 1. (a) A schematic of two mechanical modes coupling to a common optical mode via radiation pressure. (b) Two optical driving fields, $E_1$ and $E_2$, at the red and blue sidebands, respectively, couple to the respective mechanical modes. (c) A diagram of the red (left) and blue (right) sideband couplings, illustrating the processes of phonon–photon interconversion and phonon–photon pair generation.

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Note that normal modes, instead of Bogoliubov modes, are formed if the optical mode couples to both mechanical modes via red sideband coupling [22,23]. In addition, for a three-mode optomechanical system, in which two optical modes couple to a common mechanical mode via respective red and blue sideband couplings, the formation of Bogoliubov optical modes can lead to the entanglement or squeezing of the two optical modes [21,24–26].

In this Letter, we report experimental studies of Bogoliubov mechanical modes by coupling an optical whispering gallery mode (WGM) to two mechanical breathing modes in a silica microsphere. We have used OMIT to characterize the effective optomechanical coupling of the Bogoliubov mode. Self-induced mechanical oscillations due to the parametric instability of the Bogoliubov mode further demonstrate the key cascade process for the generation of phonon pairs, namely, Stokes scattering and the conversion of Stokes photons to phonons in another mechanical mode.

We consider an optomechanical system in which two mechanical modes (mode 1 and mode 2) with frequencies ${\omega }_{m1}$ and ${\omega }_{m2}$ and decay rates ${\gamma }_1$ and ${\gamma }_2$, respectively, couple to an optical mode with frequency ${\omega }_c$ and decay rate $\kappa$ [see Fig. 1(a)]. The optomechanical couplings for mode 1 and mode 2 are driven by strong laser fields $E_1$ and $E_2$ at the corresponding red and blue sidebands, respectively, [see Fig. 1(b)]. The linearized optomechanical interaction Hamiltonian in the resolved sideband limit and with the rotating wave approximation is given by

We have used silica microspheres as model optomechanical resonators [27]. Experimental studies were carried out at room temperature with a sphere diameter near 33 μm. We used the (1, 2) breathing mode with a frequency of 95.86 MHz and a linewidth of 66 kHz as mode 1 and the (1, 0) breathing mode with a frequency of 140.34 MHz and a linewidth of 110 kHz as mode 2. These two mechanical modes were coupled to an optical WGM near $\lambda =800\mathrm{nm}$ and with a linewidth of 24 MHz. A schematic of the experimental setup is shown in Supplement 1. To avoid thermal bistability, we also used the acousto-optic-modulators to generate laser pulses with duration of 7 μs and with a duty cycle less than 10%. An electro-optic modulator was used to generate a tunable signal field for the resonant excitation of the optical mode. For the OMIT experiment, we measured the emission from the optical mode as a function of the signal frequency using a heterodyne detection technique with a spectrum analyzer operating in a gated-detection mode, as discussed in detail in earlier studies [8,12]. The detection gate with duration of 1 μs was centered at 6 μs from the leading edge of the driving pulse (see the inset in Fig. 2).

 

Fig. 2. Optical emission as a function of the detuning between the signal and $E_1$. The black and red squares are obtained with and without $E_2$ (detuned from the blue sideband by 0.54 MHz), respectively. The emission power is normalized to that obtained at the cavity resonance without the driving fields. The input powers for $E_1$ and $E_2$ are $P_1=5.6\mathrm{mW}$ and $P_2=8.2\mathrm{mW}$, corresponding to $G_1/2\pi =0.5\mathrm{MHz}$ and $G_2/2\pi =0.4\mathrm{MHz}$, respectively. The solid lines are the theoretical calculation discussed in the text. The inset shows schematically the timing of the laser pulses and the detection gate.

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We first discuss OMIT studies of the individual mechanical modes. Figure 2 shows the emissions from the optical mode as a function of $\delta$, the detuning between the signal field and $E_1$. In the absence of $E_2$, the emission spectrum features a sharp dip when $\delta ={\omega }_{m1}$. The dip corresponds to the OMIT response arising from the optomechanical coupling of mode 1. In the next experiment, we turned on $E_2$, but detuned it away from the blue sideband resonance by 0.54 MHz. In addition to the OMIT dip associated with mode 1, the emission spectrum now also features a sharp peak at a frequency that is 0.54 MHz away from the OMIT dip. The peak induced corresponds to the optomechanically induced amplification associated with mode 2. In this case, a constructive interference between the intracavity signal field and the lower motional sideband of the driving field leads to the peak in the spectral response, providing an effective gain for the signal field [6]. For the above experiment, the detuning of $E_2$ from the blue sideband resonance decouples the red and blue sideband couplings.

We now combine the red and blue sideband couplings by bringing both driving fields to their respective sideband resonances. Figure 3(a) shows the emission from the optical mode as a function of $\delta$, obtained at various $P_1$ and at a fixed $P_2$. Figure 3(b) shows the results of a similar experiment where $P_1$ was fixed and $P_2$ was varied. As shown in Fig. 3, when $G_1>G_2$, the emission spectrum features a sharp dip. This dip is the OMIT response associated with the Bogoliubov mechanical mode, with the effective coupling rate determined by $G=\sqrt{G_1^2-G_2^2}$. With ${\gamma }_1={\gamma }_2$, the OMIT response is characterized by the cooperativity for the Bogoliubov mode given by $C_B=4G^2/\kappa {\gamma }_m$. When $G_2>G_1$, $G$ becomes imaginary, leading to a net gain for the signal field. The effective coupling rate is now given by $\tilde{G}=\sqrt{G_2^2-G_1^2}$. Note that with ${\gamma }_1\ne {\gamma }_2$, the OMIT response is characterized by an effective cooperativity $C_1-C_2$, with $C_1=4G_1^2/\kappa {\gamma }_1$ and $C_2=4G_2^2/\kappa {\gamma }_2$. The optomechanically induced change in the intracavity optical field is determined by the competition between the red and blue sideband couplings.

 

Fig. 3. Optical emission as a function of the detuning between the signal and $E_1$. Both $E_1$ and $E_2$ are at their respective sideband resonances. (a) $G_2/2\pi =0.4\mathrm{MHz}$. From top to bottom: $G_1/2\pi =0.23$, 0.355, 0.43, 0.5, 0.61, 0.72 MHz. (b) $G_1/2\pi =0.5\mathrm{MHz}$. From bottom to top: $G_2/2\pi =0.40$, 0.49, 0.58, 0.65 MHz. The solid lines are the theoretical calculation discussed in the text. The optomechanical coupling rates were derived from studies similar to those in Fig. 2.

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For a quantitative analysis of the OMIT experiments, we used the equations of motion for the expectation values of the relevant optical and mechanical annihilation and creation operators (see Supplement 1). The solid lines in Figs. 2 and 3 show the results of the theoretical calculation. The calculations in Fig. 2 allow us to determine the optomechanical coupling rates $G_1$ and $G_2$ with an error bar of $\pm 5\%$. These results were then used in the calculation for Fig. 3 with no other adjustable parameters, indicating good agreement between the theory and the experiment. Note that the periodic modulations in OMIT responses in Fig. 3 are due to the relatively short durations of the driving fields. The optomechanical system has not reached steady state at the time of the measurement. The modulation is the most pronounced and is extremely sensitive to $G$, when $G_1$ and $G_2$ are nearly equal, since the time scale for establishing the steady state is in part determined by $1/G$ or $1/\tilde{G}$ [8].

The OMIT experiments discussed above characterize the effective coupling between the Bogoliubov mechanical mode and the optical mode. These experiments, however, provide little information on the correlations between the two individual mechanical modes in the Bogoliubov mode. To probe these correlations in a classical regime, we have investigated the behaviors of the mechanical modes when $G_2>G_1$. In this limit, the Bogoliubov mechanical mode experiences a radiation-pressure-induced gain. The resulting parametric instability can lead to self-induced mechanical oscillations. With ${\gamma }_1={\gamma }_2={\gamma }_m$, the threshold for the parametric instability is ${\tilde{G}}_{\text{th}}=\sqrt{\kappa {\gamma }_m}/2$. The threshold for a more general condition is discussed in earlier studies [21,25]. Note that the strong coherent mechanical oscillations can be far above the thermal mechanical noise even at room temperature.

Figure 4(a) shows the intensities of the mechanical oscillations of the two individual mechanical modes as a function of $P_1$. For these experiments, we have used a fixed $G_2$ that significantly exceeds the threshold coupling rate ($G_2^{\text{th}}=\sqrt{\kappa {\gamma }_2}/2=0.81\mathrm{MHz}$) for observing self-induced oscillations of mode 2 (with $G_1=0$). To measure the intensity of the mechanical oscillations, we used a separate optical pulse, which is at the same frequency of the relevant driving field, but arrives 1 μs after the driving pulse. Similarly to an earlier study [12], displacement power density spectra were obtained with time-gated detection, with a gate length of 1 μs and with the detection gate at the center of the 3 μs long measurement pulse [see the inset in Fig. 4(a)]. The spectrally integrated area of the power density spectrum corresponds to the intensity of the mechanical oscillation. The delay between the measurement pulse and the driving pulse was chosen such that the optical field in the WGM vanishes, whereas the mechanical oscillation still persists during the measurement.

 

Fig. 4. (a) The intensities of self-induced mechanical oscillations as a function of $P_1$, with $G_2/2\pi =1\mathrm{MHz}$. Blue squares: mode 2. Red dots: mode 1. The inset shows the timing of the measurement pulse and the detection gate. (b) Heterodyne-detected optical emission, with $E_1$ serving as the local oscillator and with $G_1/2\pi =0.3\mathrm{MHz}$ and $G_2/2\pi =1\mathrm{MHz}$. The inset shows an expanded scan. The data shown were extracted with a frequency filter to filter out the beatings with $E_2$.

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The mechanical intensities as a function of $P_1$ shown in Fig. 4(a) reveal a special correlation between the two individual mechanical modes. With $G_2>G_1\ne 0$, coherent oscillations can be observed for both mechanical modes. In particular, the intensity for mode 1 can increase with $G_1$. This is unusual since $E_1$ leads to radiation-pressure-induced damping for mode 1. For the Bogoliubov mode, $E_2$ couples to mode 2 via Stokes scattering, generating photon–phonon pairs with phonons in mode 2 and photons at the optical resonance [see Fig. 1(c)]. $E_1$ at the red sideband, however, can convert the photons to phonons in mode 1. The self-induced oscillations observed for mode 1 thus demonstrate the conversion of Stokes photons to phonons in mode 1. This cascade process of Stokes scattering followed by an optical-to-motional state conversion is the key mechanism for the generation of phonon pairs. With the mechanical system cooled to its motional ground state, phonon-pair generation and the closely related heralded single-phonon generation, for which a Stokes photon heralds a corresponding phonon in mode 2, can then be realized [28]. Note that because of the radiation-pressure damping induced by $E_1$, the intensity for mode 1 eventually decreases with increasing $G_1$ at sufficiently large $G_1$, as shown in Fig. 4(a). A more detailed theoretical analysis of Fig. 4(a) using the experimental parameters is presented in the Supplement.

To confirm the coherent nature of the induced mechanical oscillations discussed above, we show in Fig. 4(b) the heterodyne-detected optical emission as a function of time, obtained under the experimental conditions in Fig. 4(a) and with $E_1$ as the local oscillator. As expected, the heterodyne-detected emission features oscillations with well-defined phases and with a frequency of 95.86 MHz. The gradual increase in emission with a rise time of several microseconds reflects the time scale for the system to reach steady state.

In summary, the experimental studies of OMIT and self-induced mechanical oscillations in a three-mode optomechanical system combining both red and blue sideband couplings have demonstrated key properties of Bogoliubov mechanical modes in a classical regime. These studies open the door to exploring the quantum properties of Bogoliubov modes, such as mechanical entanglement and phonon-pair generation when the mechanical system is cooled to its ground state.