Coupling a harmonic oscillator to an optical cavity provides an elegant and powerful means of tailoring its mechanical response [1]. Of particular interest is the regime of “optomechanical cooling,” which exploits this coupling to transfer energy from the mechanical motion to the light field, in the process cooling the center-of-mass motion of the mechanical oscillator. In most experimental configurations, this optomechanical coupling is dispersive in nature, i.e., the motion of the harmonic oscillator alters the cavity resonant frequency. Under these conditions, optomechanical cooling can be efficient in the sideband-resolved regime, when the mechanical frequency Ω is much higher than the linewidth of the optical resonance γ, i.e., γ≪Ω. For the best optical cavities, γ normally lies in the range of a few hundreds of kHz [2]. Increasing the mass or dimensions of a mechanical system implies, however, a decrease in the resonant frequency and consequently inefficient cooling rates [1]. Nevertheless, numerous optomechanical systems operate at low resonant frequencies, e.g., the mirrors used in LIGO [3], ultracold atomic gases [4], and suspended micro-mirrors [5]. In this regime, several alternative cooling schemes have been proposed and demonstrated over the last decade [6–9].

To date, optomechanical cooling remains one of the very few ways to manipulate the noise spectrum of mechanical oscillators, whose large dimensions make it difficult to use cryostats or whose thermal coupling with the surrounding environment is weak (i.e., optically levitated particles [10] or long suspended waveguides [11,12]). In the second category, tapered optical nanofibers play a pivotal role, offering an effective means of interfacing with fiber-based networks and coupling light into photonic devices such as whispering-gallery-mode (WGM) resonators [13], photonic crystal cavities, and photonic circuits [14]. Furthermore, recent investigations suggest that the understanding and controlling of the mechanical resonances of tapered nanofibers may be beneficial for experiments in atomic physics [11,15], sensing [16,17], and optomechanics [18]. The low thermal conductivity of glass means that cryogenic cooling of tapered fiber at high vacuum is inefficient, so that more complex active cooling procedures must be used [19].

In this paper, we report strong passive optomechanical cooling of a tapered glass-fiber “nanospike” coupled to a WGM bottle resonator. Although several papers have discussed optomechanical interactions between a suspended waveguide and a WGM resonator [18,20,21], passive cooling in such systems has thus far evaded experimental observation, mostly due to the difficulty of obtaining high-Q mechanical modes for the waveguide without compromising its optical properties. We report that appropriately tapered glass-fiber nanospikes offer an elegant solution, providing both adiabatic guidance of light and flexural resonances with quality factors Q>105 [22,23]. This represents an increase of two to three orders of magnitude compared with the values reported in traditional nanofibers [18]. When the nanospike is placed close to a bottle resonator and the pump laser is blue-detuned from the optical cavity resonance, clear optomechanical cooling is observed. The result is strong suppression of Brownian nanospike motion, indicating self-stabilized coupling to the WGM resonator.

The experimental setup is sketched in Fig. 1(a). The nanospike was fabricated by scanning an oxybutane flame along a length of single-mode step-index fiber (SM980) while gently pulling it. The profile of the nanospike was engineered to yield single-mode adiabatic guidance of light at 1150 nm (pump laser) and 1064 nm (probe 1), while preserving high mechanical stiffness and low mechanical loss. The procedure for reproducibly fabricating nanospikes with high-Q flexural resonances was previously reported in detail [22–24]. The experiment was conducted in vacuum (10−5mbar) to eliminate viscous damping by air. The nanospike was mounted on a stainless-steel holder using Kapton tape. Carefully cleaning the optical fiber before tapering ensured that the subwavelength waist withstood several tens of mW of optical power in high vacuum without any damage. Five stepper motors permitted fine-tuning of both the relative position and the orientation between the nanospike and the WGM bottle-resonator. An optical micrograph of nanospike+bottle-resonator is shown in Fig. 1(b). The resonator [25,26] was fabricated in a two-step process. First, a piece of single mode fiber (SMF) was thermally tapered to a diameter of ∼20μm; then, the taper was placed in an arc-splicer, where an electric discharge locally heated the taper waist, while the two ends of the taper were pushed toward each other. Surface tension caused the formation of the prolate shape shown in Fig. 1(b). Tuning the arc power and its duration allowed precise control of the resonator diameter and profile. In the vicinity of a cavity resonance, the pump light transmission is low, making it difficult to image the motion of the nanospike tip. To side-step this problem, light from a second weak and nonresonant probe laser [probe 1 in Fig. 1(a)] was launched into the fiber so as to allow the nanospike motion to be monitored using a quadrant photodiode (QPD). This permitted the tip motion to be reconstructed in two dimensions with nm-scale spatial resolution. To calibrate the displacement measured by the QPD, the nanospike (in the vacuum chamber) was precisely displaced using stepper-motors and the response from the top CCD monitored (see Fig. 1).

The first set of experiments was performed using a nanospike with a tip diameter of ∼500nm and a fundamental flexural resonance at 2.5 kHz (Q-factor of 1.2×105). The bottle resonator had a diameter of 46 μm and an intrinsic optical Q-factor of 8.2×107. Figure 1(c) shows the near-field intensity distribution of the optical mode, measured when the pump beam was locked on resonance and captured by a sensitive CCD camera. The measurement matches well with the results of finite element simulations [Fig. 1(d)].

Strong optomechanical cooling was observed when the nanospike was placed in the over-coupled regime with the pump wavelength set close to, but blue-detuned from, the cavity resonance. The laser detuning was stabilized using the thermo-optical nonlinearity of glass, i.e., by thermal self-locking [27]. The resulting mechanical spectra in the vicinity of the fundamental nanospike resonance are depicted in Fig. 2(a) for increasing values of pump power. A significant drop in the amplitude of the mechanical resonance, accompanied by linewidth broadening, is apparent at higher pump powers. The effective temperatures (Teff) of the nanospike “degree of freedom” were estimated by integrating the area underneath the power spectra [1]. As shown in Fig. 2(b), an increase of several orders of magnitude in the mechanical linewidth could be measured at only 250 μW pump power, with a minimum Teff value of 1.8 K. Note that these results refer to nanospike motion orthogonal to the WGM resonator surface; further, weaker optomechanical coupling is expected in the direction parallel to the WGM resonator surface. Nonetheless, this degree of freedom could still be cooled, as shown in the insets of Figs. 2(a) and 2(b), with a minimum achievable effective temperature of 68 K. We attribute the observed saturation of Teff at higher pump power [Fig. 2(b)] to residual optical absorption in the bottle resonator, although a more precise understanding of its origin will require further investigation.

Since the nanospike mechanical frequencies are much smaller than the cavity linewidth, the cooling should not differ substantially for higher-order mechanical modes [1]. Figure 2(c)shows the measured power spectra of the first higher-order flexural mode, using the pump power as a parameter [same data set as in Fig. 2(a)]. At zero pump power, this mode had a resonance frequency of 6.45 kHz and a Q-factor of 1840. The clear trend observed when increasing the power of the pump laser confirms simultaneous cooling of the higher-order mode. Because of the lower mechanical Q-factor, the minimum achievable Teff was 118 K [inset of Fig. 2(c)]. It is worth mentioning that multimode cooling is not easily achievable in the sideband-resolved regime because laser detuning must be precisely matched to the mechanical resonant frequency.

Laser cooling of the first few flexural mechanical modes of the nanospike resulted in strong stabilization of its coupling to the WGM resonator. At room temperature in the absence of stabilization, Brownian motion of the nanospike causes fluctuations as high as tens of nm in its position. This causes random fluctuations in the frequency and linewidth of the cavity resonance as well as the optical transmission through the nanospike. Figure 3(a) plots the displacement of the nanospike (after calibrating the response of the QPD) recorded over 100 ms for pump powers of 0, 10, and 250 μW. The panel on the right-hand side compares histograms for data collected over 100 s. The reduction in the thermal noise can be clearly observed. At low power, the Brownian motion of the nanospike has a mean-square displacement (MSD) of 530nm2, which agrees with estimates from the equipartition theorem. The effective mass (estimated by solving the Euler–Bernoulli equation) of the nanospike tip is meff,FM≈50pg for the fundamental mechanical mode and meff,HOM≈70pg for the first higher-order mode. When the pump power was increased to 250 μW, the value of the MSD dropped significantly to 37nm2 [Fig. 3(b)], a suppression factor of 11.6 dB. At this level, the MSD is dominated by higher-order mechanicalmodes.

The effect of stabilization was further explored by introducing a second frequency-tunable 1550 nm probe laser [probe 2 in Fig. 1(a)] and scanning its wavelength across the cavity resonances with and without laser cooling. Figures 3(d) and 3(e) compare 50 consecutive measurements of the resonance for a Teff of 300 and 6.7 K, revealing a clear overall increase in the system stability. In particular, nanospike cooling significantly reduced the measured standard deviation of the minimum transmission from σ300K=0.0117 to σ6.7K=0.0025 [see Figs. 3(f) and 3(g)].

Nanospike motion modulates both the resonant frequency and the decay rate of the optical cavity, introducing simultaneously dispersive and dissipative optomechanical coupling [6]. In addition, since the experiment was performed in vacuum, photothermal effects due to residual absorption of the pump power can also contribute to the observed cooling effect [9].

Measuring the frequency and linewidth of the optical mode as a function of the distance between the nanospike and the bottle-resonator allows the dispersive and dissipative optomechanical coupling parameters to be estimated (see Appendix B for the data). Comparing the results with the theory of generalized optomechanical coupling [6], however, we found that the measured cooling efficiency exceeded the predicted values by about two orders of magnitude, suggesting that photothermal interactions were strongly affecting the cooling process. In particular, since the bottle resonator takes a finite time (time constant τ=280μs, see Appendix D) to reach thermal equilibrium, thermal nonlinearities delay the buildup of optical energy in the cavity, producing nonconservative optical forces and strongly perturbing the optomechanical state.

To further clarify the cooling mechanism, we performed measurements using a bottle resonator with a larger diameter of 350 μm, the aim being to suppress photothermal effects. The increase in volume results in a greater heat capacity and, since the fabrication procedure does not require pre-tapering, a reduction in water diffusion into the glass and consequently less residual laser absorption. In addition, a greater thermal response time of τ=8.3s (about 40 times longer than in the smaller bottle resonator) was measured, which should further suppress photothermal coupling. An increase in the intrinsic optical Q-factor to a value of 2.6×108 was also observed.

The second set of experiments was performed using a nanospike with resonant frequency 1.9 kHz (Q-factor of 1.4×105) and a tip diameter of ∼700nm (meff≈780pg). Under these circumstances, as shown in Fig. 4(a), the measured optical spring effect for a launched power of 70 μW (blue points) agrees well with the predictions of the model of generalized optomechanical interactions (solid-blue line; see Appendix C for the parameter list). The model also correctly estimates the pump power (∼300μW) required to access the regime of mechanical self-oscillation for the fundamental mechanical mode orthogonal to the WGM surface [inset of Fig. 4(a), with cavity detuning of 1.6 MHz].

The good agreement between experiment and theory suggests that photothermal effects have little relevance in this parameter range and that the generalized optomechanical model correctly describes the system dynamics. This also suggests that the results in Figs. 2 and 3 are dominated by photothermal effects.

Using the same parameters for the measurement in Fig. 4(a) (launched pump with power 1 mW, blue-detuned by 0.2 MHz from the cavity resonance), dissipative cooling of the fundamental mechanical mode to an effective temperature of 46 K is predicted. Unfortunately, the large mechanical frequency shift due to the optical spring effect at this power level (∼65Hz, orders of magnitude larger than the mechanical linewidth), together with a significant drop in the efficiency of the thermal self-locking mechanism in this bottle resonator, prevented collection of clean mechanical spectra and verification of the theoretical predictions.

Vibration of the nanospike in one of its mechanical eigenmodes modulates the optical field in the bottle resonator and drives the other mechanical modes via the optical force. Since all the mechanical modes of the nanospike interact with the same cavity mode, they become optomechanically coupled [28]. Though this coupling can be neglected in systems where the eigenmodes have widely different resonant frequencies, this is not true in our case because the orthogonal nanospike modes are nearly degenerate in frequency. Using the optical spring effect, we experimentally characterized the strength of the optomechanical coupling. The results are presented in Fig. 4(b), where the measured mechanical frequencies for the two fundamental flexural modes orthogonal and parallel to the bottle-resonator surface are shown as a function of the cavity detuning at a power level of 40 μW. A clear anti-crossing can be observed at 4 MHz detuning, with a measured coupling rate of 1.8 Hz, i.e., two orders of magnitude greater than the mechanical decay rate. The solid lines in Fig. 4(b) are fits of the data to coupled mode theory (see Appendix E), showing good agreement with the measurements.

In summary, glass-fiber nanospikes permit observation for the first time of passive cooling of a freestanding optical waveguide evanescently coupled to a WGM resonator. Accurate knowledge of the position and velocity of the mechanical resonator is not required, in sharp contrast to active cooling schemes [18], which suffer from noise at very low motion temperature, when the amplitude of the mechanical oscillations is greatly reduced. We demonstrate that cooling of the nanospike motion is beneficial for stabilizing the coupling to an optical cavity well beyond the limits set by thermodynamics, a result that is relevant in numerous applications of WGM resonators [15–17,29]. Moreover, in this configuration, optical cooling happens with blue-detuned laser light, for which the cavity is intrinsically stable [27]. Finally, this approach is general and relatively simple and may be applied to any type of optomechanical system that has a high enough mechanical Q-factor, allowing efficient laser cooling of low-frequency mechanical oscillators.

Acknowledgment. We thank G. Epple, F. Marquardt, F. Sedlmeir, J. Harris, A. Kashkanova, and A. Shkarin for useful discussions.