1. Introduction

The cooling of nanomechanical resonators is important for exploring the quantum behavior of mesoscale systems, high precision metrology [1–3], and quantum computing [4, 5]. Developments in laser cooling of mechanical modes by active feedback [6], bolometric effects [7], and ponderomotive forces [8–12] have enabled access to the mechanical ground state [13, 14] and the exploration of quantum mechanical effects [15]. The cooling rate achievable by such optomechanical mechanisms depends on the optical density of states (DoS) available for light-scattering and also increases with the intensity of the cooling laser. Resonant enhancement of the pump, for instance through two-mode optomechanics, can thus increase the cooling rate by enhancing the number of intracavity photons. ‘Membrane in the middle devices’ [16–18] are a key example of two-mode systems where this enhancement of cooling has been demonstrated. Another major example are Brillouin optomechanical devices in which two optical modes are coupled through a traveling acoustic wave mode. Experimental efforts on Brillouin systems have shown phonon lasing [19, 20], cooling [21], and induced transparency [22], but the role of the dual DoS has not been explored. In this study, we experimentally validate the role of the dual optical DoS in Brillouin cooling, and reveal the unique technical constraints that can arise from the enforced phase-matching relationship and the thermal locking effect in high-finesse resonators [23]. Using this, we demonstrate a room-temperature system showcasing cooling near the saturation regime using only a few hundred microwatts of input power.

Let us consider a general optomechanical system having two optical modes; the lower mode ω₁ hosts the resonantly enhanced pump laser, while the higher mode ω₂ hosts anti-Stokes scattered light. These two optical modes are coupled by a phonon mode of frequency Ω_B with the relation Ω_B ≈ ω₂ − ω₁. There are now two possibilities for the momentum associated with this phonon mode. In conventional optomechanical cooling the resonator supports standing-wave phonons, such as in a breathing mode, having zero momentum q_B = 0. Alternatively, a whispering gallery resonator may support a traveling wave phonon mode with non-zero momentum q_B, 0 as found in Brillouin scattering interactions. For phase matching either process, momentum conservation k₁ + q_B = k₂ is necessary, which we illustrate in general in Fig. 1. When the phase matching condition is satisfied, resonant pump light can scatter into the anti-Stokes optical mode (ω₂, k₂) while annihilating phonons from the system. Stokes scattering is simultaneously suppressed since no suitable optical mode is available. This leads to the insight that the resolved sideband regime could also be achieved in momentum space, even if the optical modes are not resolved in frequency space, enabling cooling with high fidelity selection of only anti-Stokes scattering.

 

Fig. 1 (a) Generalized modal arrangement required for dynamical back-action cooling of a (Ω_B, q_B) phonon mode in a resonator supporting two optical modes (ω₁, k₁) and (ω₂, k₂). Here ω_L and ω_AS (ω_S) denote the frequencies of pump laser and the anti-Stokes (Stokes) scattered light. (b) Our experiments are performed in a whispering gallery microresonator that is optically coupled using a tapered fiber waveguide. Optomechanical cooling of an acoustic whispering gallery mode (WGM) having nonzero momentum q_B ≠ 0 is achieved by means of forward Brillouin scattering.

Download Full Size | PDF

The existence of two-mode systems with q_B ≠ 0 phonon modes was experimentally verified recently [20]. Optomechanical cooling in these systems has been experimentally demonstrated [21] and also theoretically analyzed [24, 25]. However, the expected role of optical DoS in the cooling process has not been experimentally investigated. In this work, we study the dependence of cooling rate on both pump and anti-Stokes optical DoS using the Brillouin opto-acoustic cooling system [21] in a silica whispering gallery resonator. We analyze the role of thermal locking [23] and also discuss through experiment the potential of reaching the strong coupling regime.

2. Experimental methods

Our experiments are performed in a whispering gallery resonator of optical Q > 10⁸ with a diameter of about 150 µm as shown in schematics Fig. 1(b) and Fig. 2(a). This device hosts two optical modes (ω₁, k₁) and (ω₂, k₂) that are phase-matched by a traveling phonon mode (Ω_B, q_B). The configuration of the optical modes is shown in ω−k space in Fig. 1(a). The resonator modes in consideration have loss rates κ₁/2π = 1.6 MHz and κ₂/2π = 1.5 MHz, while the phonon mode is at Ω_B/2π = 260.9 MHz with loss rate Γ_B/2π = 31.3 kHz, placing the system in the resolved sideband regime (Ω_B ≫ κ₁, κ₂). A tunable external cavity diode laser (ECDL) pumps the system near 1555 nm with 1.1 mW launched in the waveguide. Light is coupled to the resonator via tapered optical fiber with external coupling rate κ_ex to the resonator optical modes. The pump is tuned to the (ω₁, k₁) mode while spontaneous anti-Stokes scattering occurs to the (ω₂, k₂) mode. As shown in Fig. 1(a), Stokes scattering is suppressed since no suitable optical mode is available. Forward scattered optical power from the resonator couples out via the tapered fiber and can be measured on a forward photodetector; low-frequency transmission measurement shows the optical modes, while high-frequency measurement provides the frequency offset between pump and spontaneously scattered light. An electro-optic modulator (EOM) is used to generate a probe sideband, using which a network analyzer can monitor the detuning Δ₂ = (ω_L + Ω_B) − ω₂ between the scattered light and the (ω₂, k₂) mode by means of Brillouin scattering induced transparency (BSIT) [22]. This induced transparency is illustrated in the network analyzer inset of Fig. 2(a).

 

Fig. 2 (a) Experimental setup used for investigating role of optical DoS in Brillouin cooling. Top-right inset shows fiber-transmission measurement of the two optical modes (ω₁, k₁) and (ω₂, k₂) during a laser sweep. (b) The normalized phonon spectrum $\overline{S_n}\left(\omega \right)$ exhibits a cooling trend with increasing the pump laser. The small shift in mechanical resonance frequency occurs due to the optical spring effect [10, 25]. Solid lines are Lorentzian fits to the data. (c) The mode temperature T_eff is presented as a function of Δ₂. It shows that the cooled mode temperature does not follow a symmetric Lorentzian-type curve, which is indicative that the pump optical density of states also affects the cooling.

Download Full Size | PDF

The spectrum of the photocurrent S_II (ω) generated by pump and scattered light that appears on the forward detector is related to the symmetrized spectral density of the phonon mode $\overline{S_{bb}}\left(\omega \right)$ [13, 26] via the relation

Since the pump and anti-Stokes optical fields have a well defined frequency offset Ω_B, the detuning of the anti-Stokes field from the cavity resonance Δ₂ can be simply modified by tuning the frequency of the pump. Figure 2(c) shows the measured effective temperature of the phonon mode T_eff as a function of Δ₂. As we reduce Δ₂ from 2 MHz to 0 MHz, the mode reaches a final effective temperature of T_eff ~ 138.3 K starting from 290 K. As anticipated, the effective temperature rises again to 255 K as we continue detuning to -2 MHz since the anti-Stokes optical density of states (DoS) modifies the scattering efficiency. The defining cooling curve in Fig. 2(c) does not follow a symmetric Lorentzian shape, since the pump detuning Δ₁ is simultaneously modified in this experiment. This result shows a clear evidence of the dependence of the Brillouin cooling effect on the optical DoS of both the pump and the anti-Stokes optical modes. The measurable mechanical frequency Ω_B/2π simultaneously tunes from 260.87 MHz to 260.96 MHz through the associated optical spring effect, as a function of detuning [27].

3. Optomechanical damping with dual optical DoS

Let us now evaluate the characteristics of the phonon mode spectrum $\overline{S_{bb}}\left(\omega \right)$ under the influence of the cooling laser. The Hamiltonian formulation for such optomechanical systems with two optical modes has been presented previously [24, 25]. The resulting quantum Langevin equations enable derivation of the optomechanical damping rate Γ_opt as follows:

 

Fig. 3 Role of two optical DoS on optomechanical damping rate Γ_opt. (a) Ideally, the pump (ω_L) and anti-Stokes (ω_AS) detuning track together due to the fixed acoustic mode frequency. (b) Practically, thermal locking of the pump to its host optical mode causes the true anti-Stokes detuning to vary faster than the true pump detuning. ${\mathrm{\Delta }}_1^{\mathrm{o}}$ is the intended detuning while Δ₁ is the true detuning. (c) Equation 2 predicts that Γ_opt is a Lorentzian function with respect to either of the true detunings Δ₁ and Δ₂. (d) Thermal locking of the pump to the ω₁ mode causes Δ₁ to no longer track the intended detuning ${\mathrm{\Delta }}_1^{\mathrm{o}}$, resulting in a skewed cooling rate.

Download Full Size | PDF

4. Effect of thermal locking

In reality, however, ultra-high-Q resonators do exhibit significant thermal tuning in response to pumping even in the sub-milliwatt regime. This primarily takes the form of locking of the resonator mode to the pump laser [23], and is illustrated in Fig. 3(b). The result is that the intended pump detuning ${\mathrm{\Delta }}_1^{\mathrm{o}}={\omega }_L-{\omega }_1$, i.e. the detuning from the original pump resonance (ω₁, k₁), does not generate a Lorentzian shaped cooling rate but instead traces out a skewed response Fig. 3(d). Experiments where measurable Brillouin cooling is to be observed require pump power in the > 10 microwatt regime, and fall under this latter category. Figure 3(c) and 3(d) does oversimplify the situation to some degree, since the anti-Stokes mode also experiences thermal tuning due to heating of the resonator. However, this thermal red shift is generally smaller than the red shift experienced by the pump mode due to the distinct spatial mode profiles. As a result, the true pump detuning ${\mathrm{\Delta }}_1={\omega }_L-{\omega }_1^{\prime }$ for the locked mode at ${\omega }_1^{\prime }$ is always lower than the intended detuning ${\mathrm{\Delta }}_1^{\mathrm{o}}={\omega }_L-{\omega }_1$. This technical challenge to optomechanical cooling has not been reported previously since it is not important in resolved sideband one-mode systems, but uniquely impacts two-mode systems having high finesse.

5. Verification of DoS dependence

Figure 4(a) shows the experimentally measured dependence of the optomechanical damping rate Γ_opt on the true pump detuning Δ₁ and the anti-Stokes detuning Δ₂. To relate the measurement to Eq. (2), several experimental parameters are required. The intracavity photon number ${\overline{n}}_{\text{cav}}$ and the true pump detuning Δ₁ can be inferred by measuring the power absorbed from the waveguide by the resonator. Specifically, ${\mathrm{\Delta }}_1=\sqrt{\frac{{\kappa }_1{\kappa }_{\text{ex}}}{4}(\frac{P_{\text{wg}}}{P_{\text{in}}}-\frac{{\kappa }_1}{{\kappa }_{\text{ex}}})}$ where P_wg is the optical power in the waveguide and P_in is the optical power absorbed by the resonator. As mentioned above, the detuning Δ₂ can be directly measured using BSIT [22]. We can then predict the cooling rate Γ_opt based on Eq. (2) with all measurable parameters being known, except g₀/2π = 2.6 Hz which is extracted by fitting to the BSIT model. This prediction leads to the 3d surface shown in Fig. 4(a) and is in very good agreement with the experiment. Since both pump and anti-Stokes detuning change together [see Figs. 3(a) and 3(b)], the results are not dependent only on one optical DoS, but are affected by both simultaneously. As predicted, the true pump detuning varies far less than the anti-Stokes detuning due to thermal locking of the pump optical mode. Importantly, the highest optomechanical damping rate is much more strongly dependent on the detuning of the anti-Stokes mode than on the detuning of the pump mode. Attainment of the global optimum point where both pump and anti-Stokes detuning are zero is thus non-trivial and requires compensatory action to adjust the temperature of both modes in a synchronized manner. An example technique may be the use of a second thermal-tuning laser that adjusts the anti-Stokes mode family.

We can now attempt to independently characterize the role of the optical DoS of either optical mode. This can be achieved by describing a normalized cooling rate ${\zeta }_1={\mathrm{\Gamma }}_{\text{opt}}/{\overline{n}}_{\text{cav}}$ which is independent of the intracavity photon number ${\overline{n}}_{\text{cav}}$ and Δ₁, and a normalized cooling rate ${\zeta }_2={\mathrm{\Gamma }}_{\text{opt}}\left({\kappa }_2^2+4{\mathrm{\Delta }}_2^2\right)$ which is independent of the anti-Stokes detuning Δ₂. In addition, we define the dimensionless numbers η₁ ≜ ζ₁/ζ_1.max and η₂ ≜ ζ₂/ζ_2.max where ζ_1.max and ζ_2.max are the maximum value of ζ₁ and ζ₂ at resonances, i.e. at Δ₂ = 0 and Δ₁ = 0 respectively. In this particular experiment, the measurable values of ζ_1.max and ζ_2.max are 1.66×10⁻⁵ Hz and 1.55 × 10¹⁷ Hz³, respectively. These values, along with the 2d data plots in Figs. 4(b) and 4(c) and previously mentioned parameters of the system, enable full reconstruction of the 3d data in Fig. 4(a). In Fig. 4(b) we see that the normalized measurement η₁ exhibits clear Lorentzian dependence on the anti-Stokes detuning Δ₂. This result is similar to what is expected from resolved-sideband optomechanical cooling [12]. Additionally, since we know the dependence of the intracavity photon number ${\overline{n}}_{\text{cav}}$ on pump detuning Δ₁, we are able to plot [in Fig. 4(c)] the normalized measurement η₂ against the pump detuning Δ₁ directly, without being affected by the anti-Stokes mode detuning. There is thus a clear contribution of the DoS in both optical modes on the resulting phonon dissipation rate.

 

Fig. 4 (a) Data points (diamonds) are the measured optomechanical damping rate Γ_opt a function of pump and anti-Stokes detuning Δ₁ and Δ₂. The 3d surface is the predicted result based on Eq. (2), using the optomechanical single-photon coupling strength g₀/2π = 2.6 Hz. The inset shows the intracavity photon number ${\overline{n}}_{\text{cav}}$ with respect to Δ₁, as deduced from Eq. (2). (b) Dependence of η₁ and (c) η₂ (see text) on anti-Stokes detuning Δ₂ and pump detuning Δ₁, respectively. Red dashed lines are derived from Eq. (2).

Download Full Size | PDF

6. Saturation and strong coupling regimes

A special feature of two-mode optomechanical cooling systems is that the pump laser is resonant in the optical mode (ω₁, k₁). These systems can thus support a very large amount of intracavity photons $\left({\overline{n}}_{\text{cav}}>{10}^9\right)$ compared to what is achievable in single-mode optomechanics [11–13]. As a result, the ratio of the light-enhanced optomechanical coupling strength $G=g_0\sqrt{{\overline{n}}_{\text{cav}}}$ to the optical loss rate κ₂ can be increased significantly, implying that two-mode systems could potentially reach the strong coupling regime where the loaded optomechanical coupling exceeds the optical linewidth (G ≥ {κ₂, Γ_B}), even at room temperature. Additionally, the saturation limit where optomechanical damping rate is constrained by the optical loss rate (i.e. Γ_opt,max = κ₂) can be reached when cooperativity exceeds κ₂/Γ_B [28–30]. To explore this possibility, we arrange a system that has optical loss rates κ₁/2π ≈ κ₂/2π = 1.1 MHz and acoustic loss rate Γ_B/2π = 33.9 kHz with mechanical frequency Ω_B/2π = 164.8 MHz. Due to the optomechanical coupling, we anticipate the appearance of induced transparency in the anti-Stokes mode (ω₂, k₂) [22], which evolves into a mode splitting effect under strong coupling. Figure 5(a) shows an experimental observation of this induced mode splitting that is generated due to the large circulating pump power $\left({\overline{n}}_{\text{cav}}=5\times {10}^9\right)$ in the system. By fitting this observation to a mathematical model (Supplement of [22]) we can obtain G/κ₂ ≈ 0.362, which is very close to the strong coupling regime. The observed linewidth of the phonon mode is shown in Fig. 5(b), both before and after cooling. We see that the effective linewidth broadens to Γ_e/2π = 592.3 kHz which is comparable to the optical loss rate of the anti-Stokes mode κ₂/2π = 1.1 MHz. This result corresponds to a final mode temperature of 16.6 K. The cooling rate is expected to saturate at the optical loss rate κ₂ when the system reaches the strong coupling regime. Here, the measured effective mode temperature is very close to the estimated saturation temperature of 8.9 K.

 

Fig. 5 (a) Transmission of a probe laser tuned the anti-Stokes optical mode (ω₂, k₂) is measured through the optical waveguide, while a fixed pump laser is tuned on the lower (ω₁, k₁) optical mode. The red line shows optical absorption without optomechanical coupling (G ≈ 0). For G ≠ 0, induced transparency appears in the (ω₂, k₂) mode due to destructive interference mediated by phonons [22]. Blue markers show experimental data and blue-bold line is the fit to theory. (b) The mechanical noise spectrum is measured for Ω_B/2π = 164.8 MHz, for G/κ₂ ≈ 0 and G/κ₂ ≈ 0.362.

Download Full Size | PDF

7. Conclusions

We have provided direct experimental verification that Brillouin cooling is dependent on two optical DoS, of both pump and anti-Stokes optical resonances. Additionally, we demonstrated that thermal locking of the pump laser causes the optomechanical damping rate Γ_opt to be asymmetric with respect to the laser detuning. This is a unique technical challenge that can arise in Brillouin optomechanical cooling systems having high optical finesse. In spite of thermal locking, we show the effects of the pump and anti-Stokes optical DoS can be individually de-convolved. Finally, we experimentally confirm that resonant enhancement of the pump in a Brillouin optomechanical cooling system enhances the cooling rate significantly. Our room temperature experimental results show that the strong coupling regime is well within reach, and may also pave the way to exploration of the saturation regime and beyond-saturation cooling techniques [31].