Deviation from the Normal Mode Expansion in a Coupled Graphene-Nanomechanical System

Schwarz, Cornelia; Pigeau, Benjamin; Lépinay, Laure Mercier de; Kuhn, A.; Kalita, Dipankar; Bendiab, Nedjma; Marty, Laëtitia; Bouchiat, Vincent; Arcizet, Olivier

doi:10.1103/physrevapplied.6.064021

Cited by 26 publications

(37 citation statements)

References 46 publications

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“…The use of the normal mode expansion is justified by the low ( 10%) damping asymmetry observed on the uncoupled eigenmodes [26]. F(r 0 + δr) represents the action of the force field under investigation on the two fundamental eigenmodes, evaluated at the vibrating extremity position.…”

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A universal and ultrasensitive vectorial nanomechanical sensor for imaging 2D force fields

et al. 2016

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Miniaturization of force probes into nanomechanical oscillators enables ultrasensitive investigations of forces on dimensions smaller than their characteristic length scale. Meanwhile it also unravels the force field vectorial character and how its topology impacts the measurement. Here we expose an ultrasensitive method to image 2D vectorial force fields by optomechanically following the bidimensional Brownian motion of a singly clamped nanowire. This novel approach relies on angular and spectral tomography of its quasi frequency-degenerated transverse mechanical polarizations: immersing the nanoresonator in a vectorial force field does not only shift its eigenfrequencies but also rotate eigenmodes orientation as a nano-compass. This universal method is employed to map a tunable electrostatic force field whose spatial gradients can even take precedence over the intrinsic nanowire properties. Enabling vectorial force fields imaging with demonstrated sensitivities of attonewton variations over the nanoprobe Brownian trajectory will have strong impact on scientific exploration at the nanoscale.Introduction-Nanosciences were revolutionized by the invention of atomic force microscopy [1] which enabled first perceptions of the nanoworld, by measuring proximity forces exerted by a sample on a micromechanical oscillator. The subsequent emergence of nanomechanical oscillators and evolutions in readout techniques [2][3][4] lead to impressive improvements in force sensitivity [5], enabling detection of collective spin dynamics [6][7][8], single electron spin [9], mass sensing of atoms [10,11] or inertial sensing [12]. Attractive perspectives arise too when nanoresonators are hybridized to single quantum systems, such as molecular magnets [13], spin or solid states qubits [14][15][16][17]. This reduction of the probe size naturally motivates the exploration of force fields on dimensions smaller than their characteristic length scale where a great physical richness is expected, in particular for nanoscale imaging or investigations of fundamental interactions such as proximity forces or near field couplings. In this situation, measurements are performed in presence of strong force field gradients whose vectorial structure becomes crucially relevant and should be fully accounted to describe the measurement process itself [18].

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A universal and ultrasensitive vectorial nanomechanical sensor for imaging 2D force fields

et al. 2016

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“…1c,d at ω/2π = 4.17 MHz. Around the degeneracy, the modes are coupled and hybridized 5,29 . Indeed, we observe a sharp splitting in the Brownian mode of graphene ( Fig.…”

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“…An approximate, on-resonant expression for the power gain of G th c ∼ α mgγgω0 2 (42 dB) scales directly with coupling constant, inversely with effective mode mass m g and damping γ g of graphene and is in agreement with the measured value. It can be noted that G th c = Q 2 g (∼ 46 dB) sets the maximum limit of the power gain 24,29 .…”

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Motion Transduction with Thermo-mechanically Squeezed Graphene Resonator Modes

et al. 2018

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There is a recent surge of interest in amplification and detection of tiny motion in the growing field of opto and electro mechanics. Here, we demonstrate widely tunable, broad bandwidth and high gain all-mechanical motion amplifiers based on graphene/Silicon Nitride (SiNx) hybrids. In these devices, a tiny motion of a large-area SiNx membrane is transduced to a much larger motion in a graphene drum resonator coupled to SiNx. Furthermore, the thermal noise of graphene is reduced (squeezed) through parametric tension modulation. The parameters of the amplifier are measured by photothermally actuating SiNx and interferometrically detecting graphene displacement. We obtain displacement power gain of 38 dB and demonstrate 4.7 dB of squeezing resulting in a detection sensitivity of 3.8 fm/ √ Hz, close to the thermal noise limit of SiNx.

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“…Such signatures are absent in conservative force fields 17 and illustrate a large deviation from the normal mode expansion: thermal noise spectra cannot be described anymore as the sum of two spectra featuring Lorentzian mechanical susceptibilities driven by independent Langevin forces. Such a deviation may in some systems originate from the conservative coupling of two resonators with heterogeneous damping 49 , 50 , but it is not the case here since no damping modification is measured at any optical power. Instead, each couple of thermal noise spectra can be very well simultaneously fitted with the model (solid lines in Fig.…”

Section: Resultsmentioning

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Eigenmode orthogonality breaking and anomalous dynamics in multimode nano-optomechanical systems under non-reciprocal coupling

Lépinay¹,

Pigeau²,

Besga³

et al. 2018

Nat Commun

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Thermal motion of nanomechanical probes directly impacts their sensitivities to external forces. Its proper understanding is therefore critical for ultimate force sensing. Here, we investigate a vectorial force field sensor: a singly-clamped nanowire oscillating along two quasi-frequency-degenerate transverse directions. Its insertion in a rotational optical force field couples its eigenmodes non-symmetrically, causing dramatic modifications of its mechanical properties. In particular, the eigenmodes lose their intrinsic orthogonality. We show that this circumstance is at the origin of an anomalous excess of noise and of a violation of the fluctuation dissipation relation. Our model, which quantitatively accounts for all observations, provides a novel modified version of the fluctuation dissipation theorem that remains valid in non-conservative rotational force fields, and that reveals the prominent role of non-axial mechanical susceptibilities. These findings help understand the intriguing properties of thermal fluctuations in non-reciprocally-coupled multimode systems.

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Deviation from the Normal Mode Expansion in a Coupled Graphene-Nanomechanical System

Cited by 26 publications

References 46 publications

A universal and ultrasensitive vectorial nanomechanical sensor for imaging 2D force fields

A universal and ultrasensitive vectorial nanomechanical sensor for imaging 2D force fields

Motion Transduction with Thermo-mechanically Squeezed Graphene Resonator Modes

Eigenmode orthogonality breaking and anomalous dynamics in multimode nano-optomechanical systems under non-reciprocal coupling

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