our detection scheme relies 
 on coupling nanomechanical motion to a focused beam of electrons .
electron beams can be focused to spot sizes approaching the diameter 
 of nanotube resonators , ensuring a much higher interaction overlap 
 compared to usual capacitive or optical techniques used to detect 
 nanomechanical motion .
the principle of the detection works as follows : the collisions between the electron beam and the nanotube 
 yield to the emission of so - called secondary electrons ( ses ) , which 
 result from inelastic scattering mechanisms .
the displacements of 
 the nanotube within the electron beam create a strong modulation of 
 the secondary electrons current , whose fluctuations are detected by 
 means of a high bandwidth scintillator .
note that previous measurements 
 of nanotubes using electron beams did not resolve the power spectrum 
 nor the real - time evolution of their mechanical fluctuations .
the samples 
 are mounted onto a three - dimensional positioning stage hosted in a 
 commercial sem delivering a highly focused , ultralow noise electron 
 beam .
importantly , the sem chamber is 
 thoroughly pumped in order to secure a high vacuum level , preventing 
 any significant electron - beam assisted spurious deposition mechanism 
 on the nanotube .
the mechanical resonators 
 discussed below consist of single - clamped nanotubes that are anchored 
 at the edge of silicon wafers ( figure 1b , c ) .
this detection method can also be employed with 
 doubly clamped nanotubes but the results are not shown here . experimental 
 setup and systems .
the carbon 
 nanotube resonators are mounted inside an sem , where their motion 
 is detected via the ses emission , whose 
 fluctuations are collected at the video output of the sem and further 
 sent to a spectrum analyzer .
( b , c ) two sem micrographs showing typical 
 singly clamped carbon nanotube obtained with our chemical vapor deposition 
 ( cvd ) growing method .
the catalyst ( white flakes ) is spread on the 
 silicon substrate ( darker parts ) , which is further cvd processed , 
 resulting in the growth of ultralow diameter carbon nanotubes , some 
 of which are found to be singly clamped ( highlighted in red , false 
 colors ) .
we first operate the sem in the 
 conventional  scanning mode  with the electron beam 
 being scanned over the surface of the sample and the ses response 
 simultaneously acquired .
figure 2a shows a typical ses image obtained by scanning a 
 suspended carbon nanoresonator representative of those investigated 
 in this work ( d1 ) .
the image seems increasingly blurred toward the 
 upper end of the nanoresonator , which is interpreted as a consequence 
 of position noise .
the carbon nanotube 
 ( cnt ) resonator can be described as a singly clamped cantilever whose 
 deformations are dominated by its fundamental flexural mode according 
 to euler 
following the energy equipartition 
 theorem , this mode stores motional energy at thermal equilibrium , 
 resulting in random vibrations .
when the latter are large compared 
 to the spatial extension of the electron beam , the integrated current   becomes simply proportional to the probability p(rp , t ) to find the object at the electron beam position rp within the integration time t ( here i denotes the ses emission rate ) .
provided 
 that the image integration time is long with respect to the motion 
 coherence time , the signal becomes proportional to the asymptotic 
 probability , that is , the spatial probability density function ( pdf ) 
 associated with the position noise , . for a singly clamped , unidimensional euler 
bernoulli 
 beam vibrating in the scanning plane and at thermal equilibrium , this 
 probability is given by2with ecnt and e1 respectively denoting 
 the axis and vibrational direction of the carbon nanotube resonator , u is its fundamental mode shape and th2 is the thermal motion variance ( the origin of the referential being 
 taken at the anchor point of the resonator ) .
figure 2b shows two cross sections of figure 2a ( black , dashed arrow ) obtained 
 with two distinct scanning rates , corresponding to short and long 
 electron beam exposure , respectively .
the cross sections confirm the 
 gaussian scaling of the ses emission rate and enable one to extract 
 the same value of the thermal motion variance ( th )  ( 14 nm ) , independent from the 
 exposure duration , showing that the nanomechanical dynamics is negligibly 
 affected by the electron beam as further discussed below .
measurements 
 on other nanotube cantilevers show that th remains 
 constant upon rotating the nanotube along its axis .
this is because 
 nanotube cantilevers feature two fundamental modes polarized perpendicularly 
 with similar effective masses and similar resonant frequencies , so 
 that the variance of the projected thermal motion is independent of 
 the rotation angle .
( a ) magnified 
 sem micrograph showing a suspended carbon nanotube representative 
 of those used in the present work ( device d1 ) .
the image is clearly 
 blurred toward the upper end of the nanotube , characteristic of the 
 thermal excitation of its fundamental vibrational mode .
( b ) intensity 
 profile taken across the section denoted by the dashed arrow on panel 
 a , obtained with fast and slow scanning rate ( left and right , respectively ) . 

the straight line corresponds to a gaussian fit , enabling to extract 
 the motion variance th2  ( 14 nm ) .
the sem is operated in spot 
 mode and the electron beam is being set at the edge of the carbon 
 nanotube .
the resulting ses fluctuations are collected at 
 the sem video output and further sent to a spectrum analyzer .
two 
 peak are observed around the fundamental resonance frequency x/2 = 5.58 mhz and y/2 = 6.33 mhz , corresponding to the two perpendicular 
 directions of vibration of the nanotube resonator . to further establish the vibrational origin of 
 this motion imprecision
, we turn the sem into  spot mode  , 
 where the electron beam is fixed at a given position .
we set the electron 
 beam at the tip of the resonator and acquire the ses current fluctuations 
 using a spectrum analyzer .
figure 2c shows two peaks centered at 0/2  5.58 and 6.33 mhz , consistent with the 
 expected resonant behavior .
rotating the positioning stage with respect 
 to the electron beam direction enables one to selectively extinguish 
 either of the peaks , which can be used in order to address the corresponding 
 direction of vibration .
the signal - to - noise ratio of the low - frequency 
 resonance is 14 db , limited by broadband background scattering ( linear 
 spectral decay , arising from the detector cutoff frequency of the 
 sem ) .
we verified that these peaks vanish when decoupling the electron 
 beam from the nanoresonator , confirming their motional origin .
the measurements in figure 2b , c enable one to determine the basic mechanical properties 
 of the carbon nanotube resonator . the motion variance can be written 
 as a function of the lateral spring constant k , th2 = kbt / k , yielding k  2.1  10 nm with the temperature t = 300 k. on the other hand , 
 the effective mass meff is expressed in 
 terms of the spring constant and mechanical resonance frequency , meff = k/02  17 ag . 
 from the values of the lateral spring constant and nanotube length , 
 the radius can be evaluated on the order of r  
 2 nm , assuming that the nanotube contains one wall ( supporting information ) .
using the length and radius of the 
 nanotube and the mass density of pristine graphene , we obtain that meff  2.8 ag . the difference between 
 this value and the mass measured above
is attributed to a thin layer 
 of contamination adsorbed on the nanotube . as mentioned earlier , 

the above measurements are weakly sensitive to e - beam - induced dynamical 
 effects , which would result in strong distortions of the motion pdf . 
 because of the much reduced surface of the electron beam compared 
 to the motion variance , the average electrical power received by the 
 nanotube resonator strongly depends on the e - beam position ( see supporting information ) . to verify the impact 
 of the electron beam on the nanomechanical dynamics , we subsequently 
 image the nanotube resonator by setting the e - beam dwell time td to short and extended values ( td  2/0 and td  2/0 , figure 2b left and right , respectively ) . 

short dwell times enable to decrease the effective electrical power 
 received by the nanotube at the expense of decreased signal - to - noise 
 ratio , whereas extended dwell times correspond to higher e - beam effective 
 exposure .
the obtained motion pdf remains gaussian and with very similar 
 variances in both cases , excluding any significant dynamical contribution 
 arising from the electron beam fluctuations .
additionally , the electron 
 beam may be responsible for asymmetric electrothermal dynamical backaction 
 effects , which either cool or amplify the nanomechanical vibrations 
 depending on the side from which the nanotube is exposed .
however , figure 2b shows that the motion pdf remains symmetric at lower 
 dwell time , which indicates that dynamical backaction effects remain 
 negligible . in total , this study is compatible with thermally induced 
 mechanical fluctuations , corroborating previous studies having addressed 
 the temperature dependence of cnt random vibrations in similar conditions .
the potential of nanomechanical devices relies on their ultrasensitive 
 dynamical behavior , which requires the ability to operate them close 
 to their fundamental limits and in real - time . to do so
, we connect the scintillator output of the sem to an ultrafast 
 lock - in amplifier , which we use for demodulating the quadratures of 
 the electromechanical signal around the mechanical resonance frequency .
figure 3a shows the fluctuations 
 spectrum of the out - of - phase quadrature obtained with device d2 with 
 the demodulation frequency being set to 356 khz .
two peaks are observed 
 with comparable widths and heights . using the measured resonant frequency 
 and the spatial pdf measurement , we obtain th2  ( 31 nm ) and k  4.8  10 nm .
figure 3b , c further show the spectrum of the electromechanical 
 signal as demodulated around each resonance frequency .
the data adjust 
 very well to lorentzian models ( plain lines ) , suggesting that the 
 nanotube resonator behaves as a linearly damped , two - dimensional harmonic 
 oscillator .
( a ) motion quadrature spectrum of a free - running carbon nanotube resonator . 

the quadratures of the electromechanical signal are demodulated at 
 d/2 = 356 khz using an ultrafast lock - in 
 amplifier while the electron beam being set at the upper edge of the 
 device .
the spectrum is obtain as the fourier transform of the 1 s - averaged 
 autocorrelation of the out - of - phase electromechanical quadrature . 

two peaks are observed , associated with the motion imprecision in 
 each vibrational direction of the resonator .
straight lines correspond 
 to lorentzian adjustments ( individual in purple and orange , dual incoherent 
 sum in blue ) , enabling the extraction of both mechanical resonance 
 frequencies x/2 = 356.577 
 khz and y/2 = 370.243 khz 
 and the values of the apparent quality factors qx = x/x = 541 
 and qy = y/y = 591 .
the data are obtained by demodulating 
 the electromechanical signal around frequency x/2 and further computing the fourier transform 
 of its 1 s - averaged autocorrelation .
the straight line corresponds 
 to a single lorentzian fit with additional , incoherent background . 

( d ) electromechanical 
 energy autocovariance calculated as   with i as the ses current , t is the time ,  is the 
 measurement delay time , and  ...  is statistical average . 

the straight line stands for the theoretical adjustment set by eq 3 , yielding to the values 
 of the intrinsic quality factors qx = 582 and qy = 
 559 . to address the origin of the observed 
 mechanical line widths , we compute the autocovariance of the energy 
 of the electromechanical signal , with  
 the measurement delay time , and  ...  statistical average . 

the energy autocovariance has indeed the property to be insensitive 
 toward frequency noise ( see supporting information ) , enabling the pure extraction of the mechanical damping rates with 
 the additional benefit of minimal driving amplitude , therefore avoiding 
 possible nonlinear artifacts . for a linear , 
 stationary driven nondegenerate two - dimensional mechanical oscillator , 

this energy autocovariance is independent of t and 
 can be shown to read as ( see supporting information)3with /2 as the frequency splitting 
 between the two modes and with p , p2 = p , and   being the mechanical 
 damping rate , the motion variance , and the electromechanical coupling 
 rate associated with each vibrational direction ( p  { x , y } ) , respectively . 

the terms on the first line of eq 3 identify to the individual energy components associated 
 with each mode , whereas the second line simply corresponds to the 
 acoustic beat between the two motional polarizations .
figure 3d shows the electromechanical 
 energy autocovariance corresponding to the spectrum shown on figure 3a .
the experimental 
 data ( dots ) are found to adjust very well to the theoretical model 
 set by eq 3 ( plain line ) .
quality 
 factors qp = p/p obtained from the fits of the quadrature 
 spectrum ( p denoting the mechanical line 
 width associated with each vibrational direction , p  { x , y } ) , to the  intrinsic  
 quality factors qp = p/p , measured via the autocovariance of the energy . 
 the measurements presented on figure 3a , d are consecutively repeated a number of times and 
 used for extracting the corresponding damping parameters , yielding 
 to qx = 412  
 89 , qy = 570  
 123 , qx = 583  
 70 , and qy = 583  
 50 .
these values show no significant difference between the apparent 
 and intrinsic quality factors , which establishes that the measured 
 decoherence is dominated by dissipation mechanisms in the carbon nanotube 
 resonator .
mode 
 coupling forces , which arise from inertial nonlinear effects in singly 
 clamped beams , are weak enough so that 
 motional fluctuations do not induce sizable dephasing .
we 
 insist that this aspect is indispensable for resolving the nature 
 and origin of the vibrational state .
indeed , fundamental differences 
 in vibrations , such as those reported in refs ( 21 , 29 , 44 , and 45 ) , can be 
 resolved only by measuring their motion quadrature distribution.figure 4a - i ( respectively 4b - i ) shows the time 
 evolution of the motion quadratures ( x1(t ) , x2(t ) ) of x ( respectively ( y1(t ) , y2(t ) ) of y ) , defined as the cross - phase , slowly varying 
 components of mechanical motion , x(t ) = x1(t)cos xt + x2(t)sin xt and y(t ) = y1(t)cos yt + y2(t)sin yt .
the corresponding real - time displacements x(t ) and y(t ) are 
 shown on figure 4a - ii , b - ii . figure 4a - iii , b - iii show 
 the quadratures cross - correlation functions cx( ) = x1(t)x2(t + ) and cy( ) = y1(t)y2(t + ) associated with each trajectory .
these correlations 
 are found to vanish below the 10% level and can therefore be safely 
 neglected .
figure 4a - iv , v , b - iv , v show the histogram of the normalized motion quadratures , 
 which are all found to be gaussian distributed with unit variance 
 ( plain lines ) . in total , these measurements show that the quadratures 
 of the nanomechanical fluctuations in each vibrational direction describe 
 a brownian motion , consistent with a 
 two - dimensional mechanical resonator at thermal equilibrium .
these 
 results establish that nonlinear mechanical effects in singly clamped 
 nanotube resonators at room temperature remain weak .
figure 4c , d further shows the corresponding 
 motion trajectory and associated histogram in real - space , confirming 
 a bivariate , symmetric normal distribution of the position noise .
( a - i ) time 
 evolution of the motion quadratures for mode 1 .
( a - ii ) corresponding 
 real - time evolution of the position x(t ) .
( a - iii ) quadratures cross - correlation for mode 1 . ( a - iv ,
( c ) nanomechanical trajectory ( x(t ) , y(t ) ) in real - space . ( d ) histogram of the nanomechanical trajectory 
 ( x(t ) , y(t ) ) in real - space . ( e )
the upper and lower curves stand for the imaginary and real 
 parts of the spatial correlation function , respectively ( see text ) . to complete our study
, we evaluate 
 the spatial correlations defined as 2xycxy( ) = {x1(t ) + ix2(t)}{y1(t + 
  )  iy2(t + )}. the result is reported on figure 4e , where the real and imaginary 
 parts are shown separately .
the very low level of correlations indicates 
 that potential landscape nonlinearities have negligible effects , to 
 first order .
finally , we note that the 
 two - dimensional , nondegenerate nature of suspended nanocantilevers 
 provides them with the peculiar property to develop short - term spatial 
 correlations under random external driving , such as the one resulting 
 from measurement backaction .
these correlations manifest through strong 
 distortions in the electromechanical spectrum , which are not observed in our measurements ( see figure 3a and supporting information ) .
this indicates the absence 
 of any random external driving source and in particular confirms the 
 innocuity of the electron beam toward the vibrational state .
the above presented results have been 
 obtained under optimized experimental conditions with e - beam induced 
 dynamical effects being kept negligible ( see figure 2 ) . in particular , it is essential to maintain 
 excellent vacuum conditions in order to avoid unwanted contamination 
 processes , which are enhanced under e - beam exposure ( ref ( 37 ) ; see also supporting information ] . the deposited material 
 ( e.g. , amorphous carbon ) may indeed act as an efficient energy absorber , 
 yielding noninstantaneous heating of the cnt resonator .
an important 
 aspect of our work relies in the fact that we are able to straightforwardly 
 address the dynamical consequences of such effects .
it has been 
 previously shown that electro / opto - mechanical coupling can change 
 the effective temperature of the resonator in a cavity - free scheme .
the carbon nanotube motion evolves in a delayed force gradient leading to electromechanically induced dynamical 
 effects , which may alter the mechanical behavior .
figure 5a shows two mechanical spectra 
 acquired with the e - beam being positioned at two distinct locations 
 of an amorphous carbon  contamination island  grown 
 at the edge of the resonator ( device d3 ) .
efficient cooling ( broad 
 curve ) and heating ( narrow curve ) of a carbon nanotube resonator are 
 consequently observed .
e - beam induced dynamical backaction ( device d3 ) . ( a ) mechanical 
 spectra for a damped thermal state ( broad ) and a self - oscillating 
 state ( narrow ) .
( b ) 
 a hole in the phase - space of the associated quadratures is observed , 
 indicating a self - oscillating mechanical state .
a mechanical resonator undergoing ponderomotive 
 heating is susceptible to enter the instable regime of self - oscillation . in this case
, the mere mechanical spectrum may 
 not allow distinguishing between a stable and an instable regime due 
 notably to the unavoidable presence of frequency noise . here , we show 
 that by extracting in real - time the motion quadratures , we are able 
 to unveil signatures in the phase - space trajectory indicating a self - oscillation .
figure 5b shows 
 a hole in the pdf suggesting oscillations of the associated motion 
 quadratures . in an instable regime
, the nanotube is undergoing self - sustained 
 oscillations rather than the ordinary thermal random walk .
the phase - space 
 trajectory is then confined in a well - defined region with a nonzero 
 mean amplitude value .
the demodulation frequency detuning and the 
 residual frequency noise lead to the exploration of all four phase - space 
 quadrants .
similarly , figure 5c - i , ii depict the histograms of the motion quadratures , both 
 presenting a non - gaussian distribution as it would be expected for 
 a nonthermal state .
the present work demonstrates 
 that our novel measurement method enables the detection of the vibrations 
 of nanotube - based resonators with masses as low as 17 ag .
the measurement 
 of such ultralow mass resonators raises the question of the limits 
 of our approach . besides the strong , subnanometer confinement of the 
 electron probe ,
the other key element of our scheme lies in the layout 
 of the device .
the absence of any electron scatterer within the immediate 
 vicinity of the free - standing nano - object enables a very high ses 
 contrast , which is at the origin of the high motion sensitivity ( see supporting information ) . on a more fundamental 
 side ,
the measurement is responsible for a random backaction that 
 may affect the vibrational fluctuations of the measured objects .
while such effects are not observed in the present 
 work ( where the investigated devices are driven by thermal forces 
 ranging between ( 2 anhz ) and ( 10 
 anhz ) ) , they may become significant 
 for nanotube resonators with higher quality factors .
indeed , e - beam 
 quantum backaction acting on thick semiconducting scatterers has recently 
 been evaluated to be on the order of ( 1 anhz ) under standard operating conditions , which should be in reach , for example , at low temperature 
 where the mechanical quality factors are found to be enhanced by several 
 orders of magnitude .
though certainly 
 representing a limit from the sensing point of view , this points out 
 that singly clamped nanotube resonators are devices of choice for 
 probing and controlling quantum properties of electronic beams .
lastly , we would like to once more attract the attention on a very 
 important and useful characteristic of our singly clamped suspended 
 nanotube resonators , that is , their two - dimensional vibrational nature . 

this property makes these resonators sensitive to spatially induced 
 motion correlations , resulting in strong distortions in their electromechanical 
 spectrum .
these signatures ( such as the 
 non - lorentzian resonance lineshapes in response to an external piezo 
 drive , see supporting information ) enable 
 one to address the presence and nature of external driving forces 
 with no further calibration being required . in particular , it is interesting 
 to note that these nanomechanical objects are expected to surpass 
 the limits set by quantum backaction in principle , for example , by 
 parametric driving or implementing multimode 
 mechanical detection scheme , which has 
 so far never been observed and would represent an important step from 
 the perspective of quantum measurement .
this two - dimensional behavior 
 has also been highlighted as a strong asset in the context of ultrasensitive 
 nanomechanical detection , related to the corresponding ability to 
 self - discriminate the external noise mode in phase - coherent measurements , which will be highly beneficial to our systems .
we have shown that the focused electron - beam of a sem operated 
 in spot mode allows the detection of the noise dynamics of attogram - scale 
 singly clamped suspended carbon nanotubes resonators in real - time . 

we have demonstrated that an sem operated in spot mode behaves as 
 a stereoscope with our devices , enabling the tridimensional reconstruction 
 of their motion fluctuations in real - time .
we have presented a detailed 
 analysis of the two - dimensional noise trajectories both in space and 
 time and have shown that such small objects behave as brownian particles 
 evolving in a two - dimensional harmonic potential .
our work paves the 
 way toward the exploration of novel thermodynamic regimes at scales 
 which have been so far inaccessible experimentally .
the nanotubes used in this work are grown via chemical vapor deposition 
 on silicon substrates .
nanotubes are attached to the surface of the 
 substrate by van der waals forces .
some of the nanotubes extend beyond 
 the substrate edges , thus forming singly clamped resonators ( figure 1b , c ) with lengths 
 in the 100 nm to 10 m range .
 mechanical 
 resonators based on low - dimensional materials provide a unique platform 
 for exploring a broad range of physical phenomena . the mechanical 
 vibrational states are indeed extremely sensitive to charges , spins , 
 photons , and adsorbed masses . however , the roadblock is often the 
 readout of the resonator , because the detection of the vibrational 
 states becomes increasingly difficult for smaller resonators . here , 
 we report an unprecedentedly sensitive method to detect nanotube resonators 
 with effective masses in the 1020 kg range . 
 we 
 use the beam of an electron microscope to resolve the mechanical fluctuations 
 of a nanotube in real - time for the first time . 
 we obtain full access 
 to the thermally driven brownian motion of the resonator , both in 
 space and time domains . 
 our results establish the viability of carbon 
 nanotube resonator technology at room temperature and pave the way 
 toward the observation of novel thermodynamics regimes and quantum 
 effects in nanomechanics . 
Motion Detection
with a Focused Electron Beam
Damping Rate of Nanotube Resonators
Motion Statistics of Nanotube Resonators
Self-Oscillation
of Nanotube Resonators
Discussion
Conclusion
Methods