Stabilization of A Nanomechanical Oscillator to Its Thermodynamic Limit

Tobias Kippenberg

Authors: Emanuel Gavartin, Pierre Verlot, Tobias Kippenberg

Affiliation: Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

Micro- and nanomechanical oscillators are used as very sensitive detectors of forces [1], masses [2,3] and charges [4]. In these applications the measured quantity induces a change in the oscillator, which is readout, thus allowing for detection. Routinely, changes of the resonance frequency of one mechanical mode of the oscillator are used as the transduced quantity [5]. Consequently, for this type of application, the detection capabilities depend crucially on the frequency stability as frequency noise limits the resolution of the system.

A fundamental limit to frequency noise arises from the thermal motion of the mechanical oscillator [6]. While the effect of thermal noise can be decreased by coherently driving the mechanical oscillator, a natural limit is set by the onset of mechanical nonlinearity. In practice, even when operated at their nonlinearity, nanomechanical oscillators operate far away from the thermal limit [7-9]. This is due to external perturbations acting on the oscillator [10], such as temperature fluctuations or molecular processes on the surface of the device.

In our recent work published in Nature Communications, we introduce a stabilization technique, which allows to efficiently remove external frequency noise and to stabilize a nanomechanical beam to its thermal limit [11]. The method is based on the intuition that external noise usually affects the geometry of the entire nanomechanical system. Under such conditions, the induced frequency noise is expected to be highly correlated for different mechanical modes. One of the modes, a ‘noise detector’ mode, can then be used to detect the frequency fluctuations and generate an error signal, which is used to stabilize a second mode used for the actual application. This principle is illustrated in Figure 1 (a) for a doubly clamped nanobeam, whose in-plane mode (mode 2) is used to detect the frequency noise and stabilize the out-of-plane mode (mode 1).

Figure 1 (a) Scheme of the stabilization. Mode 2 (in-plane mode) serves as the ‘noise detector’ and detects external phase fluctuations leading to frequency noise. The derived error signal serves to stabilize an independent mode 1 (out-of-plane mode), which is to be used for applications. (b) Optical micrograph of the hybrid system used in our work (false colors) consisting of an optical microcavity made of silica (green) and a doubly clamped nanomechanical beam (purple) made of high-stress silicon nitride.

We have implemented this scheme with an integrated hybrid nano-optomechanical system shown in Figure 1 (b) and described in detail in our earlier work [12]. It consists of a nanomechanical beam made of high-stress silicon nitride that is placed in the near-field of a whispering-gallery-optical mode confined in a disk-shaped microcavity. This allows an efficient readout and control of mechanical motion. The resonance frequencies of the nanobeam are in the MHz range and can be tuned by changing the optical power in the microcavity.

To stabilize the mechanical mode, we drive the out-of-plane and in-plane modes of the nanobeam close to their respective nonlinearities. We determine the resonance frequency of both modes, and by monitoring the frequency drift of the in-plane mode, serving as the ‘noise detector’, we obtain the correction signal. Using this signal, we could stabilize both modes including the out-of-plane mode, which is the mode of interest for applications. The results obtained in the time domain are illustrated in Figure 2 (a) and show a strong decrease of frequency noise in presence of stabilization. For a more quantitative analysis, we performed a Fourier transform of the time domain signal and compared it to the frequency noise model for the thermal limit. The results shown in Figure 2 (b) illustrate that with our technique the frequency noise of the out-of-plane mode is reduced to its thermal limit in the frequency range of 5-50 Hz. The apparent decrease of the frequency noise of mode 2 below the thermal limit is an artifact known as ‘squashing’ due to the in-loop measurement linked to the ‘noise detector’ mode [13].

Figure 2 (a) Time evolution for the frequency drift for mode 1 (upper graph) and mode 2 (lower graph). Red denotes the non-stabilized and blue the stabilized case. (b) Frequency noise excess as compared to the thermal limit (green dashed curve) for mode 1 (darker color) and mode 2 (lighter color). The non-stabilized case is shown in red and the stabilized in blue. Mode 1 is stabilized to the thermal limit in a frequency bandwidth of 5-50 Hz.

In our work we have further analyzed the frequency stability of our oscillator both in the non-stabilized and stabilized case. We determined the long-term stability by recording the Allan deviation, which showed that our scheme is equally efficient to stabilize against long-term drifts. Moreover, we implemented a recently proposed technique based on the quadratures of motion [14], which allows to directly determine the presence of external noise. All of the methods employed to determine the amount of excessive frequency noise agree very well quantitatively and show the efficiency of our scheme.

By choosing a second mechanical mode as a ‘noise detector’, we are able to stabilize one mode of interest in a non-perturbative manner. Importantly, by choosing two modes whose motion is perpendicular, we fully preserve the detection capabilities of our system. Moreover, our technique can also be used for other applications requiring an ultrastable micro- or nanomechanical system, such as in applications for time and frequency control [15]. Our method is very general and can be applied to other nanomechanical systems. The only prerequisites are the ability to detect multiple mechanical modes and a frequency restoring force, which is fulfilled for a large majority of micro- and nanoscale mechanical systems.

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