the complete disclosure of priority u . s . provisional application ser . no . 61 / 752 , 242 filed jan . 14 , 2013 is incorporated by reference herein for all purposes . embodiments described herein are also provided in the publication , kenig , et al ., “ passive phase noise cancellation scheme ,” phys . rev . lett ., prl 108 , 264102 ( 2012 ) which is hereby incorporated herein by reference in its entirety for all purposes . the presently claimed inventions are not limited to the embodiments illustrated in the figures . in one embodiment , a diagram of the noise reduction scheme is illustrated in fig1 . the output signal of a noisy oscillator is used to parametrically - drive a pair of coupled beams , by modulating their natural frequency in time . this injects energy that induces oscillations , similarly to the way one could oscillate on a swing by changing the position of their center of mass [ see r . lifshitz and m . c . cross , “ nonlinear dynamics of nanomechanical and micromechanical resonators ,” in review of nonlinear dynamics and complexity ( h . g . schuster , ed . ), vol . 1 ( meinheim ), pp . 1 - 52 , wiley , 2008 ]. a possible electro - mechanical implementation of parametric excitation is achieved by feeding the oscillator signal into a pair of electrodes , and incorporating piezoelectric material in the beams themselves , so that the time - varying voltage produced by the electrodes is converted into mechanical stress , which in turn changes the natural frequency of the resonators in time . in the current setup parametric actuation is applied to each beam individually , with a relative phase of 180 ° between the driving signals . the use of the 180 ° between the driving signals achieves maximal non - degenerate response . smaller relative phases , greater than zero , can also be employed . in order to apply parametric excitation , in these embodiments , the device needs to be engineered such that the frequency of the driving oscillator is close to the sum of the two frequencies associated with the two linear modes of the motion , ω 1 + ω 2 . furthermore , in order for the parametric response to be non - degenerate , the difference between these frequencies , which is determined by the coupling between the beams , needs to be notable relative to their bandwidth . if this is satisfied , and certain threshold values of the controllable parameters are exceeded , the resonators are set in motion that is composed of the two modes , each oscillating at a frequency close to its linear frequency . a key result is that non - degenerate parametric oscillations are not locked to the driving source , and that the dynamics contains a phase - translation invariance that corresponds to the existence of a limit - cycle [ see r . graham , “ theory of cross - correlation of signal and idler in parametric oscillators ,” phys . lett . a , vol . 32 , no . 6 , pp . 373 - 374 , 1970 ; m . d . reid et al ., phys . rev . a . vol . 40 , pp . 4493 - 4506 , 1989 ; b . c . sanders et al ., phys . rev . a , vol . 42 , pp 6767 - 6773 , 1990 ; p . d . drummond et al ., phys . rev . a , vol . 41 , pp . 3930 - 3949 , 1990 ; and j . j . slosser et al ., phys . rev . a ., vol . 50 , pp . 793 - 800 , 1994 ]. as shown in the following quantitative description , as the drive frequency is swept , the limit - cycle frequency reaches a maximal value . the vanishing derivative at this maximum can be shown to correspond with making the output signal at frequency around ω 1 − ω 2 completely resistant to noise in the drive frequency . this provides a way to clean the phase noise of the driving oscillator . we initiate the quantitative description of the device by analyzing the mathematical model for a pair of coupled resonators , like the ones noted in fig1 . the dimensionless equations for the deviations x n from the equilibrium positions of the beams are ( r . lifshitz and m . c . cross , “ nonlinear dynamics of nanomechanical and micromechanical resonators ,” in review of nonlinear dynamics and complexity ( h . g . schuster , ed . ), vol . 1 . ( meinheim ), pp . 1 - 52 , wiley , 2008 ; r . b . karabalin , m . c . cross , and m . l . roukes , “ nonlinear dynamics and chaos in two coupled nanomechanical resonators ,” phys . rev . b , vol . 79 , no . 16 , p . 165309 , 2009 ). { umlaut over ( x )} n + x n + x n 3 +∈{ dot over ( x )} n + ηx n 3 { dot over ( x )} n +∈ h n x n cos ( ω p t )+ δ ( x n − x k )= 0 , ( 1 ) for n , k = 1 , 2 . the resonators are taken to be identical , having the resonant frequency ω 0 that was scaled out of these equations . note that each resonator is parametrically excited with a drive amplitude h n and that the nonlinearity of the resonator originates from both the elastic restoring force and damping mechanism . we consider the linear damping to be weak , ∈& lt ;& lt ; 1 ( 1 /∈ is the quality factor of the resonator ), so correspondingly a small drive amplitude is sufficient excite the resonators . in this regime , we can focus on the slow - time modulation of the basic oscillatory motion described by a pair of complex equations for the amplitudes of the modes of motion [ r . lifshitz and m . c . cross , “ nonlinear dynamics of nanomechanical and micromechanical resonators ,” in review of nonlinear dynamics and complexity ( h . g . schuster , ed . ), vol . 1 . ( meinheim ), pp . 1 - 52 , wiley , 2008 ] where ω 1 2 = 1 and ω 2 2 = 1 + 2δ are the linear frequencies of the in - phase and out - of - phase modes , respectively , ∈ ω p = ω p − ω 1 − ω 2 is the small difference between the drive frequency and the sum of the linear mode frequencies , t =∈ t is the slow time scale , and g =( h 1 − h 2 )/ 2 [ note that the 180 □ phase shift performed on the oscillator output makes h 1 =− h 2 .]. upon transforming to magnitude - phase coordinates , a n = a n e iφ n , it is possible to find the amplitudes and frequency of the periodic solutions of eqs . ( 2 ), which are shown in fig2 as a function of the drive frequency for a particular choice of the other parameters . for example the coupling δ = 7 , corresponds physically to a linear mode splitting equal to √{ square root over ( 1 + 2δ )}− 1 ≅ 2 . 87 times the resonant frequency . we now demonstrate how the dynamics can be utilized to eliminate the phase noise of the driving oscillator . we model this noise as white noise of intensity f ω p in the frequency of the drive ( ω p → ω p + p ( t ), p ( t ) p ( t ′) = f ω p δ ( t − t ′)). this noise causes the phase of the driving source to diffuse as [ φ p ( t + τ )− φ p ( t )] 2 = f ω p | τ | the noise p induces perturbations to the phase difference ψ = φ 1 − φ 2 which we note as ψ ; these can be calculated by solving the linearized polar amplitude equations spectrally , as was done by yurke et al ., “ theory of amplifier - noise evasion in an oscillator employing a nonlinear resonator ,” phys . rev . a , vol . 51 , pp . 4211 - 4229 , 1995 , or using the method of demir et al , circuits and systems i : fundamental theory and applications , ieee transactions on , vol . 47 , no . 5 , pp . 655 - 674 , 2000 . this gives diffusion of the phase of the output signal , with the variance of the phase difference growing linearly in time [( ψ ( t + τ )− ψ ( t )] 2 = d ω p f ω p | τ | ( 3 ) with the constant d ω p quantifying the intrinsic phase sensitivity to drive frequency noise . this expression applies in the limit that the time τ is much longer than the decay time onto the limit cycle for the no - noise oscillator . this approximation corresponds to frequency offsets close to the oscillator frequency . this is the relevant regime to explore since the width of the oscillator spectral peak is typically narrower than for the driven resonator , where the width is determined by the quality factor , which in turn sets the decay time . the phase diffusion induced by white drive frequency noise corresponds to a lorenzian spectral peak with width d ω p f ω p times the resonator line width ∈ ω o . equation ( 3 ) is the expected result because after transients have decayed and the system has settled onto the limit cycle , the net effect of the stochastic dynamics is a brownian motion of the free phase . since we are concerned with the long time phase diffusion , the coefficient d ω p can also be calculated by examining the effect an infinitesimal stationary perturbation in ω p has on the oscillation frequency ω o , by linearizing the algebraic polar amplitude equations that determine the fixed points and oscillation frequency ω o . comparing with the expression for the phase diffusion we find d ω p =( dω o / ω p ) 2 . therefore , to reduce the frequency stability degradation due to noise in the drive frequency we are interested in operating at the maximum shown in fig2 ( b ). by doing this , the effect of drive frequency noise on the phase diffusion is eliminated , and the corresponding frequency spectrum is narrowed ( to a sharp peak in the absence of other noise terms ). it is also noteworthy that for the parameters used to plot the diffusion coefficient d ω p in fig3 , the frequency stability is also improved along most of the curve , since d ω p & lt ; 1 ( output frequency noise less than input frequency noise ), indicating that this passive device is broadly applicable to enhance the performance of existing oscillator designs . a noise reduction device was made of a pair of beams that are driven parametrically at a frequency near the sum of their linear mode frequencies , with a 180 ° phase shift between them . see fig4 . the motion is detected and analyzed spectrally . in fig4 , at the left is schematically shown an oscillator with a frequency around ω 1 + ω 2 . this is passed into a power splitter that provides out - of - phase excitation to the electrodes of two coupled anharmonic mechanical resonators through the piezoelectric effect . the coupled beam motion is detected piezoresistively at the other end . a voltage source provides a bias to transduce the time - varying changes in resistance arising from beam motion . the response is composed of the two modes of motion and exhibits non - degenerate response . the fig5 a , 5 b , 5 c , and 5 d show the amplitude of the two modes as a function of the drive frequency ( top , fig5 a ), and the difference between each mode frequency and the frequency at onset , as a function of half the drive frequency ( bottom , fig5 c ). left side ( fig5 a and 5c ) is the experimental measurements and the right side ( fig5 b and 5d ) is the theory . noise is added to the input source , and the output signal which is composed of the two frequencies is mixed and produces a signal at the frequency sum and the frequency difference . see fig6 . more particularly , a noise source is added at the device input , complementing the signal provided by the oscillator . a nonlinear element at the output achieves the squaring operation and a subsequent filter eliminates all but the desired response at ω 1 − ω 2 . a phase noise analyzer is used to characterize the spectral purity improvement at the output . a variety of embodiments of the squaring element are possible such as , for example , nonlinear electrical circuit elements and nonlinear mechanical elements . fig7 illustrates the phase noise reduction achieved for the embodiment schematically depicted in fig1 , 4 , and 6 . output noise is shown as a function of the added noise at the input . the arrow at − 40 dbc / hz shows an improvement of 20 db from “ noise cleaning ” is attained .