# Measuring and imaging nanomechanical motion with laser light

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## Abstract

We discuss several techniques based on laser-driven interferometers and cavities to measure nanomechanical motion. With increasing complexity, they achieve sensitivities reaching from thermal displacement amplitudes, typically at the picometer scale, all the way to the quantum regime, in which radiation pressure induces motion correlated with the quantum fluctuations of the probing light. We show that an imaging modality is readily provided by scanning laser interferometry, reaching a sensitivity on the order of \(10\, {\mathrm {fm/Hz^{1/2}}}\), and a transverse resolution down to \(2\,\upmu {\hbox {m}}\). We compare this approach with a less versatile, but faster (single-shot) dark-field imaging technique.

## Keywords

Quantum Correlation Force Fluctuation Mechanical Resonator Laser Interferometry Vacuum Fluctuation## 1 Introduction

Lasers are indispensable tools in science and technology today. They heal eyes, power the Internet, and print objects in 3D. They have also revolutionized atomic physics: Techniques such as laser cooling and optical frequency metrology have enabled the creation of new states of matter, precision tests of fundamental physical laws, and the construction of clocks more accurate than ever before. The lasers’ key feature—high spatial and temporal coherence of the emitted light—is a unique asset, too, for the measurement of distance and motion. The laser interferometric gravitational wave observatory (LIGO) has provided the most recent, spectacular demonstration of this fact, with the direct detection of gravity waves [1].

While LIGO is concerned with the apparent displacement of kg-scale test masses, laser-based techniques are also an excellent choice to track the motion of micro- and nanoscale objects. Indeed, lasers have been used to measure a microcantilever’s motion induced by the magnetic force of a single electron spin [2], providing only one example of the force and mass sensing capabilities of laser-transduced mechanical devices. The interaction of laser light and nanomechanical motion, which lies at the heart of any such measurement scheme, has, itself, moved to the center of attention recently. Research in the field of cavity optomechanics [3] explores the fundamental mechanisms—governed by the laws of quantum mechanics, of course—and the limitations and opportunities for mechanical measurements that they imply. Without even making an attempt at a comprehensive review of the vast activity in this field, we illustrate recent progress through a selection of our own results below.

## 2 Laser interferometry and spectroscopy

*P*are the wavelength, angular frequency, and power of the employed laser light, respectively. \(\eta _{\mathrm {d}}\) is the detection efficiency, which also absorbs penalties in the sensitivity due to optical losses, insufficient interference contrast, etc. Equation (1) implies that within a bandwidth \({\mathrm {BW}}\), the smallest displacements that can be recovered with unity signal-to-noise ratio are given by \(\delta x_{\mathrm {min}}/\sqrt{{\mathrm {BW}} }=\sqrt{S_{xx}}\).

Our instrument (detailed below ) employs a near-infrared laser and mW-scale probing powers and typically achieves a \(S_{xx}^{1/2}\sim 10\,{\mathrm {fm/\sqrt{Hz}}}\) displacement sensitivity, consistent with Eq. (1). This compares favorably with the picometer-scale thermal root-mean-square (RMS) displacement \(\delta x_{\mathrm {th}}=\sqrt{{k_{\mathrm {B}} T}/{m_{\mathrm {eff}} \varOmega _{\mathrm {m}}^2}}\) of the mechanical resonators we employ [4, 6], with nanogram mass \(m_{\mathrm {eff}}\) and MHz frequency \(\varOmega _{\mathrm {m}}/2\pi\) at room temperature *T*. In the Fourier domain, the spectral density of the thermal motion is spread over the mechanical linewidth \(\varGamma _{\mathrm {m}}=\varOmega _{\mathrm {m}}/Q\). Correspondingly, a nearly four-order-of-magnitude signal-to-noise ratio \(S_{xx}^{\mathrm {th}}(\varOmega _m)/S_{xx}\) between the peak thermal displacement spectral density \(S_{xx}^{\mathrm {th}}(\varOmega _m)=\frac{\delta x_{\mathrm {th}}^2}{{\varGamma _{\mathrm {m}}}/2}\) and the noise background \(S_{xx}\) can be reached already with quality factors in the millions. An example for such a measurement is shown in Fig. 1a.

Figure 1b shows an example of such a measurement, in this case performed on the radial-breathing mode of a silica whispering-gallery-mode resonator [7], with the help of a polarization spectroscopy technique [8]. It resolves not only thermal motion with a large signal-to-noise ratio (here, about \(58\,{\mathrm {dB}}\)), but also achieves an imprecision noise below that at the resonant standard quantum limit (SQL), \(S_{xx}^{\mathrm {SQL}}(\varOmega _{\mathrm {m}})=\frac{\delta x_{\mathrm {zpf}}^2}{{\varGamma _{\mathrm {m}}}/2}\). Note that this coincides with the peak spectral density of ground-state fluctuations [9], for this device with \(\varOmega _{\mathrm {m}}/2\pi =40{.}6\,{\mathrm {MHz}}\), \(\varGamma _{\mathrm {m}}=1{.}3\,{\mathrm {kHz}}\) and \(m_{\mathrm {eff}}=10\,{\mathrm {ng}}\) at the level of \(S_{xx}^{\mathrm {SQL}}(\varOmega _{\mathrm {m}})=(2{.}2\,{\mathrm {am}})^2/{\mathrm {Hz}}\) [10].

Cavity-enhanced laser interferometry has also been applied to nanomechanical resonators all the way down to the molecular scale. For example, it was shown that a fiber-based optical microcavity can resolve the thermal motion of carbon nanotubes [11]. Another successful sensing scheme consists in introducing nanomechanical resonators in the near field of optical whispering-gallery-mode resonators. It achieves imprecision well below that at the SQL of stressed silicon nitride nanostrings with picogram masses and \(Q\sim 10^6\) [12, 13, 14]. It is also expected that optical cavities suppress diffraction losses through preferential scattering into the cavity mode.

To track or steer coherent dynamics of mechanical resonators at the level of their vacuum fluctuations, yet higher sensitivities are required [14]. In particular, it is necessary to resolve the ground state—which entails averaging for a time \(4S_{xx}/x_{\mathrm {zpf}}^2\)—before it decoheres, e.g., by heating. The latter happens at a rate \(n_{\mathrm {th}} \varGamma _{\mathrm {m}}\), where \(n_{\mathrm {th}}=k_{\mathrm {B}} T/\hbar \varOmega _{\mathrm {m}}\gg 1\) is the mean occupation of the dominant thermal bath at temperature *T*. It follows from Eq. (2) that a resolution at the level of the zero-point-fluctuations is acquired at the measurement rate [9] \(\varGamma _{\mathrm {opt}}=4g^2/\kappa\), where \(g=x_{\mathrm {zpf}} (\partial \omega _{\mathrm {c}}/\partial x) a\), and \(|a|^2\) the number of photons in the cavity (assuming \(\eta _{\mathrm {c}}\eta _{\mathrm {d}}=1\), \(\varOmega \ll \kappa\)). The above-mentioned requirement can then be written as \(\varGamma _{\mathrm {opt}}\gtrsim n_{\mathrm {th}}\varGamma _{\mathrm {m}}\).

Interestingly, a completely new effect becomes relevant in this regime as well: the quantum fluctuations of radiation pressure linked to the quantum amplitude fluctuations of the laser light, representing the quantum backaction of this measurement [15]. And indeed the ratio of radiation pressure to thermal Langevin force fluctuations is given by \(\frac{S_{FF}^{\mathrm {qba}}(\varOmega _{\mathrm {m}})}{S_{FF}^{\mathrm {th}}(\varOmega _{\mathrm {m}})}=\frac{\varGamma _{\mathrm {opt}}}{n_{\mathrm {th}}\varGamma _{\mathrm {m}}}\). While these force fluctuations induce random mechanical motion that can mask a signal to be measured, it is important to realize that motion and light become correlated, at the quantum level, via this mechanism. As a consequence, the mere interaction of cavity light with a nanomechanical device can induce optical phase–amplitude quantum correlations, which squeeze the optical quantum fluctuations, in a particular quadrature, below the level of the vacuum noise. This effect is referred to as ponderomotive squeezing [16, 17, 18, 19].

The above examples show that laser-based measurements resolve the motion of nanomechanical oscillators all the way to the level of their vacuum fluctuations. In a simple classification (Fig. 1d), basic interferometers can readily resolve thermal motion, as required in many sensing and characterization experiments. Cavity-enhanced approaches achieve imprecision below the resonant SQL. To measure and control motion at the quantum level, displacements at the scale of the vacuum fluctuations must be resolved within the coherence time of the mechanical resonator. Then the imprecision (of an ideal setup) is more than \(n_{\mathrm {th}}\) times below the resonant SQL, and quantum backaction exceeds thermal force fluctuations and induces quantum correlations [3, 9].

*g*. It is thus possible to derive

*g*, for example, from probe transmission measurements [31, 32].

## 3 Laser-based imaging

As already indicated, it can be of great interest to also spatially resolve mechanical displacement patterns. With laser light, this can be accomplished in an extremely sensitive and virtually non-perturbing manner [33, 34, 35, 36, 37]. In the following, we present two methods that we have implemented for characterizing nano- and micromechanical resonators with micrometer transverse resolution, sufficient for resolving the spatial patterns of MHz mechanical modes.

### 3.1 Scanning laser interferometry

In this case, small measured voltages \(\delta V(t) \ll V_{\text {ff}}\) convert to displacement via \(\delta x(t) \approx \pm \delta V(t) \lambda / 4 \pi V_{\text {ff}}\). Modulating PZT2 continuously with known frequency and amplitude generates a reference displacement and provides an independent calibration tone (CT) in the spectra.

*u*and

*v*, respectively, and \(|\beta |<1\) quantifies the degree of hybridization between degenerate mode pairs. We find that the measured maximum RMS displacements, as calibrated by the CT, are in good agreement with the expected thermal motion (Fig. 3). Here, we have assumed a mass \(m_{\text {eff}} = \rho l^2 h/4 \sim 34\,{\mathrm {ng}}\), given the thickness \(h = 50\) nm and density \(\rho = 2.7\) g/cm\(^3\) of the membrane. Note that the modes \((n,m)=(1,2)\) and (2, 1) show hybridization with \(|\beta | \sim 0.2\).

Scanning laser interferometry is particularly useful to characterize complex mode structures, such as SiN membranes patterned with phononic crystal structures [4] (Fig. 4). A scan measured on a grid of \(100\times 100\) points with a \(5\,{\upmu {\hbox {m}}}\) spacing resolves also the \(9{.}3\,{\upmu {\hbox {m}}}\)-wide tethers in between two holes, as Fig. 4b shows. At the expense of measurement time, the grid spacing could be further reduced; however, the spatial resolution of the obtained image is eventually limited to the \(\sim 2\,{\upmu {\hbox {m}}}\) diameter of the laser spot. Figure 4c shows another mode of the same device imaged over a larger area. At a distance of \(500\,{\upmu {\hbox {m}}}\) from the center, the mode’s amplitude has decayed to the measurement noise level, illustrating the localization of the mode to the defect.

An advantage of measuring thermally excited modes is that information on all modes within the detector bandwidth is acquired simultaneously. This large set of data can be processed and represented in different ways. As an example, Fig. 4c shows an average spectrum of 400 measurement points on the defect. It clearly reveals a phononic bandgap between about 1.41 and \(1{.}68\,{\mathrm {MHz}}\), containing five defect mode peaks, as well as the calibration peak at \(1{.}52\,{\mathrm {MHz}}\). The left panel shows a displacement map corresponding to a specific frequency bin of this spectrum. We can also create an animation that composes the displacement maps for each of the frequency bins in the spectrum. It is provided as electronic supplementary material to this article (see supplementary material). It delivers an instructive illustration of the effect of the phononic crystal structure, contrasting the small number of localized modes inside the bandgap with a “forest” of distributed modes at frequencies outside the bandgap.

A disadvantage of the scanning laser interferometer is its long measurement time. For instance, a high-resolution scan, such as the one shown in Fig. 4b, takes more than 8 hours. This is because for each pixel of the image we probe thermal motion during several seconds, averaging over timescales longer than \(\varGamma _{\mathrm {m}}^{-1}\). Some acceleration is possible by either artificially increasing \(\varGamma _{\mathrm {m}}\), e.g., by controlled gas damping, or by driving the modes coherently using PZT1. The latter can furthermore provide information about the mechanical phase at each position, if mechanical frequency drifts are properly accounted for.

### 3.2 Dark-field imaging

*u*,

*v*) is subject to a phase shift proportional to the membrane displacement

*w*(

*u*,

*v*,

*t*). We assume that the incident electric field \(E_{0} e^{i\omega t}\) is constant across the membrane, since the incident beam diameter is 2.4 times larger than the membrane. Assuming furthermore \(w(u,v,t) \ll \lambda\), the reflected electric field reads \(E_{\text {r}}(u,v,t) \approx r E_{0} e^{i\omega t} \left[ 1 + i k w(u,v,t) \right]\), where

*r*is the absolute value of the reflection coefficient, \(k = 2 \pi / \lambda\) and \(\omega = c k\). A lens (focal length \(f_1 = 75\,{\mathrm {mm}}\)) performs an optical Fourier transform \(\mathcal {F}\) with respect to the coordinates (

*u*,

*v*), yielding

*w*. A second, subsequent lens (focal length \(f_2 = 50\) mm) performs another Fourier transform on the filtered light. The time-averaged intensity pattern

A piezoelectric actuator (PZT1) successively excites the eigenmodes of the SiN membrane inside a vacuum chamber, by slowly sweeping a strong drive tone across the frequency window of interest (here \(0{.}4\ldots 2\,{\mathrm {MHz}}\)). Figure 6 shows images of several modes recorded with an incident optical power of \(\sim 100\,\mathrm {\mu W}\) and a typical integration time of \(10\,{\mathrm {ms}}\). Comparison with mode patterns calculated from Eq. (5) allows inferring the mode numbers (*n*, *m*), and the degree of hybridization, as seen, for example, on the 1.683-MHz mode.

While it enables much shorter measurement times than the scanning laser interferometer, the dark-field imaging setup has a relatively low displacement sensitivity. For this reason, PZT1 has to be driven with a stroke of \(\gtrsim 300\) pm, significantly increasing the membrane oscillation amplitude, up to a regime where mechanical nonlinearities (e.g., Duffing-type frequency shifts) can play a role. In principle, the sensitivity can be enhanced by increasing the laser intensity \(I_0\), yet in practice it is often limited by background noise due to scattered light from optical components increasing equally with \(I_0\). Another important limitation is that diffraction from the sample’s geometry cannot be discriminated from modal displacements. In this simple implementation, the approach is thus unsuitable for devices with fine structures in their geometry, such as the patterned membranes.

## 4 Conclusion

In summary, we have described several laser-based techniques to measure and image nanomechanical motion. As we show, exquisite displacement sensitivity can be reached, well into the regime in which quantum backaction and the ensuing light-motion quantum correlations dominate over thermomechanical noise. This sensitivity is rivaled only by techniques based on superconducting microwave electromechanical systems, which operate at ultra-low (\(T\ll 1\,{\mathrm {K}}\)) cryogenic temperatures [39, 40]. Interest in this quantum domain has originally been motivated by observatories such as LIGO and can now, for the first time, be explored with optical and microwave experiments [3, 9, 15, 41, 42]. In addition, laser-based techniques can provide spatial imaging of mechanical displacement patterns. They constitute not only highly useful tools to develop and characterize novel micro- and nanomechanical devices [4, 6, 35, 36, 37]. Similar techniques could also be used to address individual elements in multimode devices [20] or (opto-)mechanical arrays [4, 43]—if need be, also in combination with cavity-enhanced readout [33, 44].

## Notes

### Acknowledgements

We would like to acknowledge our (former and present) colleagues Georg Anetsberger, Olivier Arcizet, Tobias Kippenberg, Jörg H. Müller, Eugene S. Polzik, Andreas Næsby Rasmussen, Remi Rivière, Anders Simonsen, Koji Usami, Stefan Weis, and Dalziel J. Wilson for their contributions to the work discussed here. Financial support came from the ERC starting grant Q-CEOM, a starting grant from the Danish Council for Independent Research, the EU FP7 grant iQUOEMS, and the Carlsberg Foundation.

## Supplementary material

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