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MATCHING OF LOCAL EXPANSIONS IN THE THEORY OF NON-LINEAR VIBRATIONS

Yu. V. Mikhlin

Dnepropetrovsk State University, 320050 Dnepropetrovsk, Ukraine

Normal vibrations in non-linear systems are a generalization of normal (principal) vibrations of linear systems [1][2][START_REF] Manevich | The Method of Normal Oscillation for Essentially Nonlinear Systems[END_REF]. In this case all position co-ordinates can be defined well from any one of them. R. M. Rosenberg is credited with being the first to introduce broad classes of conservative systems allowing normal vibrations with rectilinear trajectories in a configurational space. In systems of a more general type, trajectories of normal vibrations are curvilinear. Assume that in a conservative system the potential energy is a positively definite polynomial in the co-ordinates. At small amplitudes a linear system is to be selected as the initial one, while at large amplitudes a homogeneous non-linear system allows normal vibrations with rectilinear trajectories. In the vicinity of a linear system, trajectories of normal vibrations can be determined as power series in the amplitude; while in the vicinity of a homogeneous non-linear system, they can be determined as power series in the inverse amplitude. In order to join together local expansions and to investigate the behavior of normal vibration trajectories at arbitrary amplitude values, fractional rational diagonal Pade´approximants are used. Necessary conditions for the convergence of a succession of Pade´approximants have been obtained, and that allows one to establish relations between quasi-linear and essentially non-linear expansions: that is, to decide which of them correspond to the same solution and which to different ones. Additional modes of vibrations exist only in a non-linear systems; as the amplitude decreases, they vanish at a certain limiting point.

INTRODUCTION: NORMAL VIBRATIONS OF NON-LINEAR SYSTEMS

Normal vibrations in the non-linear case are a generalization of normal (principal) vibrations of linear systems. In a normal mode a finite-dimensional system behaves like a conservative one, having a single degree of freedom. In this case all position co-ordinates can be defined well from any one of them by

x i = p i (x), x 0 x 1 , i = 2, 3, . . . , n, (1)
p i (x) being analytical functions.

Rosenberg [1,2] is credited as being the first to introduce broad classes of essentially non-linear (not quasi-linear) conservative systems allowing normal vibrations with rectilinear trajectories in a configurational space:

x i = k i x 1 , i = 2, 3, . . . , n. (2)
For instance, homogeneous systems the potential of which is an even homogeneous function of the variables belong to such a class. It is interesting to note that the number of modes of normal vibrations in the non-linear case can exceed the number of degrees of freedom. This remarkable property has no analogy in the linear (non-degenerate) case.

In systems of a more general type, trajectories of normal vibrations are curvilinear. Liapunov [START_REF] Liapunov | The General Problem of the Stability of Motion[END_REF] showed that solutions of this kind exist in non-linear finite-dimensional systems with an analytical first integral which are close to generating linear systems. New results concerning normal vibrations with curvilinear trajectories in essentially non-linear cases have been reported in references [START_REF] Manevich | The Method of Normal Oscillation for Essentially Nonlinear Systems[END_REF][START_REF] Yu | [END_REF]. Consider a conservative system

m i x¨i + P x i = 0, x˙i = dx i /dt, P z = 1P/1z, i = 1, 2, . . . , n, (3)
P = P(x) being the potential energy, assumed to be a positive definite function, and x = (x 1 , x 2 , . . . , x n ) T . The power series expansion for P(x) begins with terms having a power of at least two. Without reducing the degree of generalization, assume that m i = 1, since this can be always ensured by dilatation of co-ordinates. The energy integral for system (3) is

1 2 s n k = 1 x˙2 k + P(x 1 , x 2 , . . . , x n ) = h, (4)
h being the system energy. Assume that within the configuration space, bounded by a closed maximum equipotential surface P = h, the only equilibrium position is

x i = 0 (i = 1, 2, . . . , n).
Introduce a new independent variable x instead of t here. Since

d dt = d dx x˙and d 2 dt 2 = d 2 dx 2 x˙2 + d dx x¨,
and it follows from the integral of energy (4) that

x˙2 = 2(h -P)/ 6 1 + s n (x' 2 k ) 7 ,
the sought-for equations defining the trajectories in the configurational space are obtained to replace equations (3):

2x0 i h -P 1 + s n k = 2 (x' k ) 2 + x' i (-P x ) = -P x i , i = 2, 3, . . . , n, x = x i . (5)
These could be obtained, as well as Euler equations for the variational principle in the Jacobi form. Equations ( 5) are non-linear and non-autonomous, and have movable singular points: they are no simpler in the linear case. Nevertheless, these relatioships are suitable for the determination of nearly rectilinear trajectories of normal vibrations.

An analytical extension of the trajectories on a maximum isoenergy surface P = h is possible if the boundary conditions (i.e., conditions of orthogonality of a trajectory to the surface) are satisfied [2,[START_REF] Manevich | The Method of Normal Oscillation for Essentially Nonlinear Systems[END_REF]:

x' i [-P x (X, x 2 (X), . . . , x n (X))] = -P x i (X, x 2 (X), . . . , x n (X)). (6)
Here (X, x 2 (X), . . . , x n (X)) are the trajectory return points lying on the P = h surface where all velocities are equal to zero. If a trajectory x i (x) is defined, the law of motion with respect to time can be found by using

x¨+ P x (x, x 2 (x), . . . , x n (x)) = 0,
for which a periodic solution x(t) is obtained by inversion of the integral.

Mikhlin et al. [START_REF] Manevich | The Method of Normal Oscillation for Essentially Nonlinear Systems[END_REF][START_REF] Yu | [END_REF] developed nearly rectilinear trajectories of normal vibrations in the form of power series.

Assume, for example, that P = P 0 + eP 1 , where e is a small parameter, and that the undisturbed system permits solution (2). The co-ordinate axes can be rotated so that a generating solution will be defined by x i = 0 (i = 2, 3, . . . , n).

The solution can be found in the form of a convergent series:

x i = s a k = 1 s a j = 0 e k a (k) ij x j . ( 7
)
Specifically, if the generating system is homogeneous, i.e., P 0 is an homogeneous even function of the power r + 1 in all variables (r = 1, 3, 5, . . .), the coefficients a (k) ij will be well defined from equation ( 5) and the boundary conditions [START_REF] Baker | Pade´Approximants[END_REF] when the following determinants are non zero: i.e.,

k p = =q ij = $ 0,
where

q ij = d j i [p(p -1)2P 0 (1, 0, . . . , 0) + pP 0x (1, 0, . . . , 0) -P 0x i x i (1, 0, . . . , 0)], (8)
and d j i are the Kronecker symbols, p = 0, 1, 2, 3, . . . . At r = 1 conditions (8) become v i $ v j , which for Liapunov systems mean that the case of inner resonance in the generating linear system is not considered.

TWO-POINT PADE APPROXIMANTS FOR NORMAL VIBRATIONS

Now consider a problem of normal vibrational behavior in certain non-linear systems when the amplitude (or energy) of the vibrations is varied from zero to an extremely large value.

Assume that in a system z¨i + P z i (z 1 , z 2 , . . . , z n ) = 0 (9) the potential energy P(z 1 , z 2 , . . . , z n ) is a positive definite polynomial of z 1 , . . . , z n having a minimum power of two and a maximum power of 2m. On choosing a co-ordinate, say z 1 , substitute z i = cx i , where c = z 1 (0). Obviously, x 1 (0) = 1. Furthermore, without loss of generality, assume that x˙1(0) = 0. Equation (9) can be rewritten as

x¨i + V x i (c, x 1 , x 2 , . . . , x n ) = 0, (10)
where

V = s 2m -2 k = 0 c k V (k + z) (x 1 , x 2 , . . . , x n ),
and V (r + 1) contains terms of power (r + 1) of the variables in the potential

V(c, x 1 , x 2 , . . . , x n ) 0 P(z 1 (x 1 ), z 2 (x 2 ), . . . , z n (x n )).
The energy integral here is

1 2 c 2 0 x˙2 1 + s n i = z x˙2 i 1 + V(c, x 1 , x 2 , . . . , x n ) = h, (11)
h being the system energy. It is assumed below that the amplitude of vibration c = z 1 (0) is the independent parameter, while the energy is defined by equation (11). Therefore it is expedient to represent the energy h as a sum of summands corresponding to homogeneous components of the potential energy V:

h = s 2m k = 2 c k h k . ( 12
)
In order to determine trajectories of the normal vibrations of the initial system, equations ( 5) are used together with boundary conditions [START_REF] Baker | Pade´Approximants[END_REF].

At small amplitudes an homogeneous linear system with a potential energy V (2) is selected as the initial one while, at large amplitude, an homogeneous non-linear system with a potential energy V (2m) is selected. Both linear and non-linear homogeneous systems allow normal vibrations of the x i = k i x 1 type, where the constants k i are determined from the algebraic equations

k i V (r) x 1 (1, k 2 , . . . , k n ) = V (r)
x i (1, k 2 , . . . , k n ). (In the non-linear case the number of vibrations of this type can be greater than the number of degrees of freedom.)

In the vicinity of a linear system at small values of c trajectories of normal vibrations can be determined as power series of x and c (upon assuming that x 1 0 x),

x (1) i = s a j = 0

a (i) j (x)c j 0 s a j = 0 s a l = 0 a (i) jl x l c j , i = 2, 3, . . . , n, (13)
while in the vicinity of a homogeneous non-linear system (at large values of c), as power series of x and c -1 .

x (2) i = s a j = 0 b (i) j (x)c -j 0 s a j = 0 s a l = 0 b (i) jl x l c -j , i = 2, 3, . . . , n. (14)
The construction of series (13) and ( 14) is described above. Note that functions a j (x) and b j (x) can be obtained in quadrature formulas as well, since the variational equations for normal vibrations of homogeneous systems are reduced to hypergeometric ones.

The amplitude values x = 1, x (1) i (1) and x (2) i (1) (at x˙= x˙i = 0) define the normal vibrations mode completely. Therefore, for the sake of simplicity, only expansions of r (1) i = x (1) i (1) and r (2) i = x i (1) in terms of powers of c will be discussed below, which are obtained from equations ( 13) and ( 14) at x = 1 (denoted as a (i) j = a (i) j (1) and b (i) j = b (i) j (1)):

r (1) i = s a j = 0 a (i) j c j , r (2
) i = s a j = 0 b (i) j c -j . ( 15
)
In order to join together local expansions (15) and to investigate the behavior of normal vibration trajectories at arbitrary values of c, fractional rational diagonal Padeá pproximants are used [START_REF] Yu | [END_REF][START_REF] Baker | Pade´Approximants[END_REF][START_REF] Baker | Essentials of Pade´Approximants[END_REF]:

P (i) s = s s j = 0 a (i) j c j > s s j = 0 b (i) j c j , s = 1, 2, 3, . . . , i = 2, 3, . . . , n. ( 16
)
Along with the Pade´approximants in terms of positive powers of c one can write those in terms of powers of c -1 :

P (i) s = s s j = 0 a (i) j c j -s > s s j = 0 b (i) j c j -s , s = 1, 2, 3, . . . , i = 2, 3, . . . , n. (17)
Compare the expressions ( 16) and ( 17) with expansions (15):

s a j = 0 a (i) j c j 2 s s j = 0 a (i) j c j > s s j = 0 b (i) j c j , s s j = 0 a (i) j c j -s > s s j = 0 b (i) j c j -s 2 s a j = 0 b (i) j c -j , or 0 s a j = 0 a (i) j c j 10 s s j = 0 b (i) j c j 1 2 s s j = 0 a (i) j c j , 0 s a j = 0 b (i) j c -j 10 s s j = 0 b (i) j c j -s 1 2 s s j = 0 a (i) j c j -s . ( 18
)
By preserving in equations (18) only the terms with an order of c r (-s E r E s) and equating the coefficients of equal powers of c, n -1 systems of 2(s + 1) linear algebraic equations will be obtained for determination of a (i) j , b (i) j , j = 0, 1, 2, . . . . The determinants of these systems take the form of

s + 1 s + 1 ZXXXCXXXV ZXXCXXV -1 0 ••• 0 0 a 0 0 • • • 0 0 0 -1 ••• 0 0 a 1 a 0 • • • 0 0 s + 1g F f • • • • • • • • • • • • • • • • • • 0 0 • • • -1 0 a s -1 a s -2 • • • a 0 0 D (i) s = 0 0 • • • 0 -1 a s a s -1 • • • a 1 a 0 -1 0 ••• 0 0 b 0 b 1 • • • b s -1 b s 0 -1 ••• 0 0 0 b 0 • • • b s -2 b s -1 s + 1 g G G F f • • • • • • • • • • • • • • • • • • 0 0 • • • -1 0 0 0 • • • b 0 b 1 0 0 • • • 0 -1 0 0 • • • 0 b 0 = b I (-)
s + 1

I (-) s + 1 D (a) s + 1 D T (b) s + 1 b , (19)
where

-1 0 • • • 0 g 0 0 • • • 0 I (-1) s + 1 = 0 -1 • • • 0 , D (g) s + 1 = g 1 g 0 • • • 0 , • • • • • • • • • • • • • • • • • • 0 0 • • • -1 g s g s -1 • • • g 0
and s + 1 is the order of the determinant. Since the determinants are generally not equal to zero, the systems of algebraic equations have a single exact solution, a (i) j = b (i) j = 0. Select a Pade´approximant corresponding to the preserved terms in equation (15) having non-zero coefficients a (1) j and b (1) j . Assume that b (i) 0 $ 0, for otherwise at c:0, x (1) i :a. Without loss of generality, it can also be assumed that b (i) 0 = 1. Now the systems of algebraic equations for determination of a (i) j and b (i) j become overdetermined. All unknown coefficients a (i) 0 , . . . , a (i) n , b (i) 1 , . . . , b (i) n , i = 2, 3, . . . , n, are determined from 2s + 1 equations, while the ''error'' of this approximate solution can be obtained by substitution of all coefficients in the remaining equation. Obviously, the ''error'' is determined by the value of D (i) s , since at D (i) s = 0 non-zero solutions and, consequently, exact Pade´approximants will be obtained for equation (18) in the given approximation in terms of c.

Hence the following is a necessary condition for convergence of a succession of Padeá pproximants (16), at s:a, to fractional rational functions,

P (i) = s a j = 0 a (i) j c j > s a j = 0 b (i) j c j , b (i) 0 = 1: (20) namely, lim s:a
D (i) s = 0, i = 2, 3, . . . , n. (21)
Indeed, if the conditions (21) are not satisfied, non-zero values of the coefficients a (i) i and b (i) i in equation (20) will obviously not be obtained.

In addition, the limiting Pade´approximants P (i) will be suited for the description of a solution at any value of amplitude c if the functions P (i) have no poles.

Conditions (21) are necessary but not sufficient for the convergence of approximants (16) to functions (20); nevertheless, the role of conditions ( 21) is determined by the following considerations.

Since, in the general case, there are more than one quasi-linear local expansion (13) and essentially non-linear local expansions (14) alike, the numbers of expansions of the respective type being not necessarily equal, it is the convergence conditions (21) that allow one to establish a relation between the quasi-linear and essentially non-linear expansions: that is, to decide which of them corresponds to the same solution and which to different ones.

EXAMPLES

example 1

As a first example of an analysis based on the above technique, consider a conservative system with two degrees of freedom, the potential energy of which contains terms of the second and fourth powers of the variables z 1 and z 2 . Writing z 1 = cx and z 2 = cy, where c = z 1 (0) (x(0) = 1), one obtains 4) .

V = c 2 0 d 1 x 2 2 + d 2 y 2 2 + d 3 xy 1 + c 4 0 g 1 x 4 4 + g 2 x 3 y + g 3 x 2 y 2 2 + g 4 xy 3 + g 5 y 4 4 1 0 c 2 V (2) + c 4 V (
The equation for determining the trajectory y(x) is of the form 2y0(h -V) + (1 + y' 2 )(-y'V x + V y ) = 0, (

while the boundary conditions (13) can be written as

(-y'V x + V y )= h -v = 0.
On the surface h -V = 0 (at x˙= y˙= 0) one of the values of the variable x is x(0) = 1. The corresponding boundary condition is

(-y'V x + V y )= x = 1 = 0. ( 23
)
In view of the system symmetry with respect to the origin of co-ordinates, the second boundary condition (at x = -1) coincides with condition (23).

example 2

Johnson and Rand [START_REF] Johnson | [END_REF] gave results of numerical integration for a conservative system with potential energy V(x, y) = 1 2 (aj 2 + bh 2 ) + (3x 4 + y 4 ), (30)

Figure 2. A load on resilient supports under a force having a constant direction (two-degree-of-freedom system).

Figure 1 .

 1 Figure 1. The evolution of normal vibration modes for system (25) in parameters c and 8 = arctg r (c and r are amplitudes of position co-ordinates). Local normal mode expansions are designated by I, II (quasi-linear case) and III, IV, V and VI (essentially non-linear case).

Inasmuch as the trajectory will be represented in terms of c 2 or c -2 power series, introduce a parameter n (n = c 2 in the quasi-linear case and n = c -2 in the essentially non-linear case). Now the solutions of equation ( 22) can be found in the form of a series in terms of the small parameter n: y = s a j = 0 y j (x)n j .

In equation ( 22) and in the boundary conditions (23), V = V 0 + nV 1 and h = h 0 + nh 1 ; also, in the quasi-linear case, n = c 2 , V 0 = V (2) , V 1 = V (4) , h 0 = V (2) = x = 1 and h 1 = V (4) = x = 1 ; and in the essentially non-linear case, n = c -2 , V 0 = V (4) , V 1 = V (2) , h 0 = V (4) = x = 1 and h 1 = V (2) = x = 1 .

In the zeroth approximation in n, both the linear and the non-linear homogeneous systems admit normal vibrations in the form y = k 0 x. The constants k 0 are determined from an algebraic second (linear system) or fourth (non-linear system) degree equation which is obtained from equation ( 22):

For definiteness, let

Write the equations of motion for such a system:

In the linear limiting case (c = 0) two rectilinear normal modes of vibrations, y = k 0 x, k

) 0 = 1 and k (2) 0 = -1, are obtained by using equations (25), while a non-linear system (equations of motion contain only third power terms with respect to x and y) admits four such modes; k (3) 0 = 1•496, k (4) 0 = 0, k (5) 0 = -1•279 and k (6) 0 = -5. In order to determine nearly rectilinear curvilinear trajectories of normal vibrations, equation ( 22) is used along with the boundary conditions (23). In particular, the first approximation in n gives the following expression for the trajectory:

Here, y = kx is substituted in all functions V 0 and V 1 and their derivatives. Equations in the subsequent approximations in terms of n are obtained similarly. Such splitting by a small parameter should be performed in the boundary conditions (23) as well.

As mentioned above, the solution is sought in the form of a power series in terms of n and x (n = c 2 ) in the quasi-linear case and n = c -2 in the essentially non-linear case). One finds (the clumsy intermediate calculations are omitted) the following relationships for r = r(n) for various values of g and all types of vibrations (terms of the order n 2 and x 3 were retained):

Matching with the use of Pade´approximants of the type (20) shows that only for the pairs of expansions (I), (IV) and (II), (V) is an increase of the number s accompanied by a decrease in the determinant D s ; i.e., condition (21) is satisfied. Therefore, each of the couples corresponds to a periodic solution.

By matching of the local expansions the following Pade´approximants (d is an error of coefficient computation) are obtained:

The two additional modes of vibration, expansions (III) and (VI), exist only in a non-linear system; as n increases (amplitude c decreases), they vanish at a certain limiting point.

For the analysis of these vibration modes, assume a new variable s = (r -1•496)/ (r + 5). Represent n in terms of a series of powers of s (expansion (III)) or s -1 (expansion (VI)). Restricting oneself to terms with a power less than or equal to three, one obtains the following:

(28)

By using the variable s two expansions in terms of positive and negative powers have been obtained; therefore, fractional rational representations can be introduced as above. By comparing these expansions, the following Pade´approximants are obtained:

Now produced to the determination of the limiting point. Obviously, it can be found from 1n/1s = 0. From equation (29), at g = 2 the limiting point is n 2 1•21, c 2 0•91, at g = 0•5 it is n 2 11•10, c 2 0•30, and at g = 0•2 it is n 2 23•93, c 2 0•20. Hence, as g:0 the limiting point is characterized by the amplitude c:0. Therefore, the two additional vibration modes in a non-linear system can exist at rather small amplitudes of vibration. Note that quasi-linear analysis does not allow one to find these solutions, even at small amplitudes.

In the limit, when g = 0, a linear system decomposes into two independent oscillators having identical frequencies and admits any rectilinear modes of normal vibrations. Obviously, a full system (25) at g = 0 admits four modes of vibrations (in a non-linear case) y 2 = ky 1 , k = (1•496, 0, -1•279, -5).

Thus fractional rational Pade´approximants allow judgement of non-local behavior of normal vibrations in non-linear finite-dimensional systems. For systems (25) the evolution of modes of normal vibrations is shown in Figure 1 by using parameters c and 8 = arctg r (the figure shows periodicity in 8, the period being 2p). The analytical solutions (( 27) and (29) were employed) and numerical check computations (carried out by A. L. Zhupiev at g = 2) show good agreement. For solutions II-V, for instance, relationship (29) and the numerical calculation gave, in the scale selected, the same curve (see Figure 1).

Note that the systems considered above (systems with cubic non-linearity) can be obtained in calculations of non-linear vibrations of shells (when using the Bubnov-Galerkin technique) as well as in other problems. For instance, consider a problem of the vibrations of a load on resilient supports under a force having a constant direction (see Figure 2). Assume that the restoring forces on the left and right supports respectively are F 1 = d 1 q 1 + r 1 q 3 1 and F 2 = d 2 q 2 + r 2 q 3 2 , where q 1 and q 2 are vertical displacements of the supports and Q = d 3 x 1 + r 3 x 3 1 is the constantly directed force. Let m be the load weight, and let mr 2 be the moment of inertia with respect to an axis going through the center of mass.

The equations of motion for the system are (3) a = 32, b = 2, u = p/4, h = 10. Plots representing trajectories on a configurational plane (x, y) show that, as h increases, two normal vibration modes (two trajectories) are superseded by four: i.e., bifurcation takes place. Calculations (with E. V. Ladygina) carried out using Pade´approximants allow one to follow changes in the normal vibration modes as the energy increases from 0 to infinity (at a fixed value of u) and, importantly, to gain an insight into the evolution of the normal vibrations with the variation in parameter u.

The limiting linear system has two normal vibration modes y = k (1) x, k (1) varying with u. The limiting non-linear homogeneous system, which is determined by the part of the potential energy containing fourth power terms, also admits solutions in the form y = k (2) x, where in all cases k (2) = -z3, +z3, 0, a.

The results obtained are given below for a situation with a = 8 and b = 4. The evolution of normal vibration modes with vibration in energy (amplitude) from 0 to infinity and in u is shown in Figure 3, where c = x(0), r = y(0) and 8 = arctg r. Digits refer to the following cases: (1) u = p/4; (2) u = 2p/9; (3) u = p/6; (4) u = p/9; (5) u = p/18; (6)u = 0. The fractional rotational representations of the in-phase and out-of-phase vibration modes which exist in the system at any amplitude c, and are shown as (I) and (II) in Figure 3, are as follows: (III) in Figure 3 refers to additional vibration modes (at u = 0) which exist only at great amplitudes and that vanish at a certain point as the amplitude decreases.

The calculated results show that at u q 0 vibration modes are changed and the limiting point (a in the figure) is replaced by a bifurcation point b.


Olivier Maillet

Frantisek Vavrek

Andrew D Fefferman

Olivier Bourgeois

Eddy Collin

Classical decoherence in a nanomechanical resonator

Keywords: numbers: 05.40.-a, 62.25.Fg, 85.85.+j

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Decoherence can be viewed either in its quantum picture, where it stands for the loss of phase coherence of a superposition state [START_REF] Armour | Entanglement and Decoherence of a Micromechanical Resonator via Coupling to a Cooper-Pair Box[END_REF][START_REF] Remus | Damping and decoherence of a nanomechanical resonator due to a few two-level systems[END_REF], or as its classical equivalent, where the phase of an oscillating signal is smeared due to frequency fluctuations [START_REF] Cleland | Noise processes in nanomechanical resonators[END_REF]. Until recently, dissipation, which accounts for an energy loss over time [START_REF] Unterreithmeier | Damping of Nanomechanical Resonators[END_REF], was not distinguished from decoherence in nanomechanical resonators. The first reason for this oversight lies in the acquisition method: spectral acquisition, in which the driving frequency is swept through the mechanical resonance, does not separate the two processes, since both lead to broadened resonance lines. The response linewidth then leads to the definition of a decoherence time T 2 , in analogy with nuclear magnetic resonance (NMR) [START_REF] Abragam | Principles of Nuclear Magnetism[END_REF] or quantum information experiments [START_REF] Schneider | Observation of decoherence in a carbon nanotube mechanical resonator[END_REF][START_REF] Ithier | Decoherence in a superconducting quantum bit circuit[END_REF]. One way to unravel a dephasing mechanism is to perform a complementary time-domain ringdown measurement. Here, the free decay of the response amplitude yields an energy relaxation time, since the decay rate only depends on dissipation, not frequency fluctuations. This procedure is analogous to a T 1 measurement for spins or qubits. Having T 1 = T 2 then leads to the definition of a pure dephasing rate Γ φ [START_REF] Schneider | Observation of decoherence in a carbon nanotube mechanical resonator[END_REF].

While the reported T 1 and T 2 for a particular qubit are often significantly different [START_REF] Ithier | Decoherence in a superconducting quantum bit circuit[END_REF], such a difference in a nanomechanical resonator is still rare in the literature: usually mechanical systems seem to experience frequency fluctuations small compared to dissipation mechanisms, and do not exhibit visible spectral broadening due to dephasing [START_REF] O'connell | Quantum ground state and singlephonon control of a mechanical resonator[END_REF][START_REF] Faust | Nonadiabatic Dynamics of Two Strongly Coupled Nanomechanical Resonator Modes[END_REF][START_REF] Faust | Coherent control of a classical nanomechanical two-level system[END_REF][START_REF] Van Leeuwen | Time-domain response of atomically thin MoS2 nanomechanical resonators[END_REF]. Nonetheless, pure dephasing has been observed in Refs. [START_REF] Fong | Frequency and phase noise of ultrahigh Q silicon nitride nanomechanical resonators[END_REF][START_REF] Miao | Graphene Nanoelectromechanical Systems as Stochastic-Frequency Oscillators[END_REF][START_REF] Zhang | Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems[END_REF][START_REF] Moser | Nanotube mechanical resonators with quality factors up to 5 million[END_REF]. But direct comparisons between time-domain (T 1 ) and frequency-domain (T 2 ) are still rare: thus, the second and main reason for the scarcity of experimental studies on mechanical decoherence is a lack of data combining these techniques. To our knowledge, only one recent work reports, for a suspended carbon nanotube, a signature of a pure (yet nonlinear) dephasing mechanism demonstrating T 1 = T 2 [START_REF] Schneider | Observation of decoherence in a carbon nanotube mechanical resonator[END_REF], while another study which did not contain T 1 measurements presents similar features interpreted as nonlinear damping [START_REF] Eichler | Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene[END_REF]. Indeed, the effects of frequency noise (and its origin) in nanomechanical devices is a subject intensely investigated today, both for its fundamental aspects and the technical limitations it poses to actual devices [START_REF] Sansa | Frequency fluctuations in silicon nanoresonators[END_REF][START_REF] Gray | Low-frequency noise in gallium nitride nanowire resonators[END_REF][START_REF] Maizelis | Detecting and characterizing frequency fluctuations of vibrational modes[END_REF][START_REF] Fong | Frequency and phase noise of ultrahigh Q silicon nitride nanomechanical resonators[END_REF][START_REF] Miao | Graphene Nanoelectromechanical Systems as Stochastic-Frequency Oscillators[END_REF][START_REF] Zhang | Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems[END_REF][START_REF] Eichler | Symmetry breaking in a mechanical resonator made from a carbon nanotube[END_REF][START_REF] Moser | Nanotube mechanical resonators with quality factors up to 5 million[END_REF][START_REF] Gavartin | Stabilization of a linear nanomechanical oscillator to its thermodynamic limit[END_REF][START_REF] Gieseler | Thermal nonlinearities in a nanomechanical oscillator[END_REF][START_REF] Sun | Telegraph frequency noise in electromechanical resonators[END_REF]. Besides, with the advances of quantum nanomechanics, the decoherence of quantum mechanical states also becomes a challenging issue [START_REF] O'connell | Quantum ground state and singlephonon control of a mechanical resonator[END_REF][START_REF] Kleckner | Creating and verifying a quantum superposition in a micro-optomechanical system[END_REF][START_REF] Armour | Probing the quantum coherence of a nanomechanical resonator using a superconducting qubit: I. Echo scheme[END_REF][START_REF] Szorkovszky | Mechanical entanglement via detuned parametric amplification[END_REF][START_REF] Verhagen | Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode[END_REF].

In this article we report on a model experiment using a high quality nanoelectromechanical system (NEMS), top-down fabricated from high-stress silicon-nitride (SiN) [START_REF] Unterreithmeier | Damping of Nanomechanical Resonators[END_REF][START_REF] Fong | Frequency and phase noise of ultrahigh Q silicon nitride nanomechanical resonators[END_REF]. Using a gate electrode [START_REF] Kozinsky | Tuning nonlinearity, dynamic range, and frequency of nanomechanical resonators[END_REF], we capacitively control the oscillator's frequency fluctuations by applying voltage noise, leading to a completely quantitative description of mechanical decoherence. First we perform time and frequency domain measurements without noise applied on the gate, in order to ensure that no pure dephasing is observed over our whole dynamic range, from linear to highly nonlinear regimes: we thus establish our device as ideal for our study. In a second part, a low-frequency noise gate voltage is injected, leading to resonance frequency fluctuations increasing with the noise level: indeed, a significant spectral broadening is observed, while ringdown measurements leave T 1 unchanged, demonstrating pure dephasing with T 1 = T 2 . We present the complete formalism applying to lowfrequency fluctuations of the resonance frequency. Building on the new methods presented, we furthermore dis-cuss the possibility to extract information about the fluctuations statistics from the shape of the spectral response in actual devices suffering from dephasing.

II. THE NANOELECTROMECHANICAL SYSTEM

Fig. 1b) shows a scanning electron microscope picture of the measured NEMS. This device consists in a 295 µm long x 100 nm thick x 300 nm wide doubly-clamped high-stress silicon nitride (Si 3 N 4 ) suspended beam, with a 90 nm thick conducting aluminium layer on top. A gate electrode is made in the vicinity of the beam, separated by a gap g ≈ 3 µm, over a 250 µm length. The total mass of the device is M = 4.7 × 10 -14 kg. The experiment is performed at 4.2 K in a cryogenic vacuum (pressure < 10 -6 mbar).

We first perform a careful calibration of the whole setup following Ref. [START_REF] Collin | Insitu comprehensive calibration of a tri-port nano-electromechanical device[END_REF], giving access to displacements and injected driving force in real units. Fig. 1a) shows the schematic of the experimental setup. A drive voltage is applied through a 1 kΩ bias resistor, injecting a current in the aluminium layer of the NEMS. Actuation and detection follow the magnetomotive scheme [START_REF] Collin | Insitu comprehensive calibration of a tri-port nano-electromechanical device[END_REF][START_REF] Cleland | Fabrication of high frequency nanometer scale mechanical resonators from bulk Si crystals[END_REF], with a magnetic field B < 1 T. The voltage induced by the motion is measured by standard lock-in detection with its in-phase (X) and quadrature (Y) components.

We work on the fundamental flexural mode n = 0, having effective spring constant k 0 , mass m 0 and applied sinusoidal force amplitude F 0 .

The typical response to a sinusoidal driving force in the linear regime is displayed in Fig. 1c). The resonance frequency is measured at ω 0 = 2π × 0.66 MHz, which is in good agreement with calculated properties. In the following, in order to minimize electrical losses [START_REF] Cleland | External control of dissipation in a nanometer-scale radiofrequency mechanical resonator[END_REF], we keep a low magnetic field B = 0.1 T. The damping rate is then ∆ω 0 = 2π × 5.6 Hz, hence a quality factor Q ≈ 118 000. Here Q 1 so we can write the susceptibility in the standard Lorentzian approximation:

χ 0 (ω) = [2m 0 ω 0 (ω 0 -ω -i∆ω 0 /2)]
-1 . In our notations, we define in-phase X(ω) = Im[χ 0 (ω)]F 0 and quadrature Y(ω) = Re[χ 0 (ω)]F 0 components of the motion.

A DC voltage V g is then delivered with an arbitrary waveform generator (AWG) on the gate electrode, which forms a geometric capacitance C(x) with the beam itself. The energy stored by the capacitor can be written as

E C = C(x)V 2 g /2.
Thus the additional electrostatic force acting on the beam along the x axis is

F C = (V 2
g /2)∂C(x)/∂x. Expanding F C in a Taylor series [START_REF] Kozinsky | Tuning nonlinearity, dynamic range, and frequency of nanomechanical resonators[END_REF][START_REF] Collin | Insitu comprehensive calibration of a tri-port nano-electromechanical device[END_REF], we obtain the effect of the DC voltage on the resonance through a frequency pulling term:

ω 0 (V g ) -ω 0 (0) ≈ - 1 4m 0 ω 0 ∂ 2 C(0) ∂x 2 V 2 g = αV 2 g (1)
Here ∂ 2 C(0)/∂x 2 is defined as the coupling strength, and α > 0 is a constant only determined by the oscillator's intrinsic quantities and the coupling strength. We measure

∂C(0) ∂x V 2
g . This drive is non-resonant with the mode under study, and has thus no impact on the dynamics discussed in the present article. Higher-order terms contribute to the nonlinear (Duffing) coefficient [START_REF] Collin | Insitu comprehensive calibration of a tri-port nano-electromechanical device[END_REF]. They can be safely neglected here since in Sec. IV, we keep the drive low enough to remain in the linear (Lorentzian-shaped) regime [Fig. 1 c]. More details on fabrication, the magnetomotive scheme, setup and capacitive coupling can be found in Supplementary Material [32].

III. DAMPING AND DEPHASING IN THE NONLINEAR REGIME

When driven with high input forces, our NEMS' oscillation amplitude at resonance becomes high enough so its dynamics enters the so-called "Duffing" regime, arising from a geometric non-linearity: the tensioning effect. The frequency domain response bends toward higher frequency and the resonance is shifted quadratically with the motion, eventually leading to a bistable regime [START_REF] Venstra | Stochastic switching of cantilever motion[END_REF][START_REF] Mahboob | Dispersive and Dissipative Coupling in a Micromechanical Resonator Embedded with a Nanomechanical Resonator[END_REF]. In doubly-clamped SiN nano-beams this non-linearity is known to be important [START_REF] Nayfeh | Nonlinear Oscillations[END_REF], which means that already modest drive excitations can bring the device into the Duffing regime. This fact naturally brings in two questions: are there as well any nonlinear relaxation/decoherence processes to discover in SiN structures, like for nanotubes [START_REF] Schneider | Observation of decoherence in a carbon nanotube mechanical resonator[END_REF][START_REF] Eichler | Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene[END_REF] ? And how do we perform T 1 and T 2 measurements within a highly nonlinear/bistable dynamic system ?

In the following we describe how to measure T 1 and T 2 response times that we defined in analogy with NMR over the whole dynamic range, in order to compare them as a function of motion amplitude. The relaxation time T 1 obtained from ringdown measurements is only sensitive to dissipation mechanisms, by analogy with longitudinal relaxation time in the Bloch sphere for two-level systems. The ringdown is triggered by exciting the device at large motion amplitude at frequency ω r close to ω 0 , and suddenly detuning the driving force to a frequency ω r + ∆ where the device is off resonance. The beating of the mechanics with the local oscillator is then recorded [START_REF] Collin | Addressing geometric nonlinearities with cantilever microelectromechanical systems: Beyond the Duffing model[END_REF] (see Fig. 2 right panel). To obtain the plots, about N 100 to 1000 decays have been averaged. The T 2 time is obtained from frequency sweeps across the resonance, with each point averaged over a time long enough to make sure that we are sensitive to potential low-frequency fluctuations (see Fig. 2 left panel). T 2 is affected both by dissipation and phase fluctuations, in a similar manner to the transverse relaxation time in NMR or qubit measurements. But one fundamental difference is that here we are dealing with an almost harmonic classical oscillator, not a quantum two-level system, so the analogy is restricted to the phenomenology of the decoherence phenomenon. Moreover, while for qubits and spins the pure dephasing Γ φ is defined through the comparison between T 2 and 2T 1 [START_REF] Ithier | Decoherence in a superconducting quantum bit circuit[END_REF], in our system T 2 has to be compared to T 1 , similarly to the case of a classical spin [START_REF] Faust | Coherent control of a classical nanomechanical two-level system[END_REF]. Fig. 2 shows both frequency and time-domain measurements in a highly nonlinear regime, with a peak-to-peak maximum amplitude of about 600 nm, that is, more than three times the NEMS thickness.

Here the bistable resonance line can be fit according to the Duffing model for nonlinear oscillators [START_REF] Collin | Addressing geometric nonlinearities with cantilever microelectromechanical systems: Beyond the Duffing model[END_REF]. In the linear regime, the FWHM W of the X quadrature directly yields the Lorentzian parameter ∆ω 0 /(2π) and is used to define T 2 = (πW ) -1 . In the strongly nonlinear case, from the measured maximum amplitude x max one can calculate [START_REF] Collin | Addressing geometric nonlinearities with cantilever microelectromechanical systems: Beyond the Duffing model[END_REF] the intrinsic linewidth parameter ∆ω 0 used for the full nonlinear fit: ∆ω 0 = F 0 /(m 0 ω 0 x max ). This parameter can again be converted into a T 2 . The latter is displayed as a function of motion amplitude in Fig. 3 (blue dots). The time-decay oscillation can be fit using the Lindstedt-Poincaré method [START_REF] Collin | Addressing geometric nonlinearities with cantilever microelectromechanical systems: Beyond the Duffing model[END_REF]. The relaxation is found to be exponential over the entire range studied, leading to a T 1 showed in Fig. 3 (red dots).

The ringdown measurement yields a relaxation time T 1 58 ms ± 3 ms in the whole attainable amplitude range, which matches the T 2 time of spectral data defined as 2/∆ω 0 57 ms ± 3 ms: we thus verify T 1 = T 2 within ± 8 %. In other words, we do not identify any sources of nonlinear damping or dephasing in our SiN system, as opposed to carbon-based devices [START_REF] Schneider | Observation of decoherence in a carbon nanotube mechanical resonator[END_REF][START_REF] Eichler | Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene[END_REF]. This result is achieved over an unprecendently broad dynamic range, with a measurement technique that can be adapted to other devices, in particular graphene or nanotube ones. As far as the present work is concerned, this establishes our device as an ideal NEMS resonator.

IV. CONTROLLING PURE DEPHASING

In order to introduce a dephasing process, we deliver a Gaussian noise with the AWG connected to the gate electrode, instead of the DC voltage used previously for calibration purposes. Our aim is to model "slow" fluctuations of the resonance frequency, that is with correlation times longer than the relaxation time T 1 [START_REF] Zhang | Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems[END_REF]. In the aforementioned work, the interplay of the driving force with the frequency noise spectrum is characterized, in order to separate fast and slow noise contributions. Complementarily, we focus here on how the driven response evolves with respect to the level of slow fluctuations. Furthermore, in Ref. [START_REF] Zhang | Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems[END_REF], the frequency noise is experimentally taken to be Gaussian; we on the other hand extend the study to highly asymmetric statistics. The modelling is fairly generic (see Appendix B), and does apply as well to the case of slow Telegraph frequency noise [START_REF] Sun | Telegraph frequency noise in electromechanical resonators[END_REF].

To keep only low-frequency fluctuations while avoiding spurious 1/f noise coming from the setup, the delivered voltage noise is thus filtered [see Fig. 1a)] below 40 mHz and above 0.8 Hz (-3 dB cut-off frequencies). The resulting voltage fluctuations on the gate electrode, represented by the Gaussian random variable δV g , of mean value 0 and standard deviation value σ V = δV 2 g 1/2 , translate through the capacitive coupling into fluctuations of the resonance frequency. Following Eq. ( 1) the new random variable associated with these fluctuations is δΩ(t) = αδV 2 g (t), with a probability distribution [see

ρ σ (δΩ) = θ(δΩ) 1 √ 2πσδΩ 1/2 exp - δΩ √ 2σ (2)
where θ(δΩ) is the Heaviside step function, and

σ = √ 2ασ 2 V = [ δΩ 2 -δΩ 2 ] 1/2
is the standard deviation of the distribution, taken as a characteristic frequency noise level (in rad.s -1 ). This is the effective tunable parameter with which we characterize decoherence in our system. Mathematically, we show that the averaging of fluctuations can be treated similarly to the case of "inhomogeneous broadening" in NMR [START_REF] Abragam | Principles of Nuclear Magnetism[END_REF], spatial-disorder being replaced, as for qubits, by time-dependent disorder (see Appendix B). In this context, the T 2 time is re-named T * 2 . However for simplicity we keep here the notation without the star, since our aim is to model the impact of any decoherence mechanism, especially intrinsic ones. Even though our noise source is extrinsic. Knowing the distribution ρ σ , the altered susceptibility can thus be written as the convolution of the mechanical response χ 0 in the high Q Lorentzian limit by the distribution ρ σ :

χ σ (ω) = +∞ -∞ χ 0 (ω -δΩ)ρ σ (δΩ)dδΩ (3)
Note that σ is also defined through S Ω (ω), the spectrum of the frequency fluctuations, with δΩ 2 = (2π) -1 S Ω (ω)dω = 3σ 2 /2. Thus Eq. ( 3) is valid only if the noise power is integrable (which justifies mathematically the introduction of a higher cut-off frequency). In particular for 1/f noise, dephasing would depend on the measurement time since the lower bound of the spectrum would be fixed by the experiment duration itself (which justifies our lower cut-off filtering choice). Fig. 4 shows frequency domain measurements for different noise levels, with theoretical calculations without free parameters from Eq. ( 3), clearly showing spectral broadening. The convolution picture provides a rather intuitive interpretation: while the noise level σ responsible for pure dephasing is kept small enough compared to the mechanical damping rate ∆ω 0 , one can approximate the distribution ρ σ as a Dirac delta function, and then the convolution leaves the Lorentzian susceptibility unchanged. When σ becomes comparable to the damping rate, the distribution leaves its imprint in the mechanical response, as an evidence for mechanical dephasing: here, the resonance line is asymmetrically broadened, which is a signature for the non-Gaussian nature of the fluctuations (see discussion below). For large fluctuations (σ ∆ω 0 ) a significant deviation from the Lorentzian lineshape is observed (see discussion below) and the resonance peak looks like the distribution ρ σ itself, the initial susceptibility being almost a Delta function.

The time domain measurements (see Fig. 5) performed for different noise levels yield unambiguously the same decay time T 1 57 ms 1/(π × 5.6 Hz) within ± 5% for all data sets [see Fig. 5 and6a)]. Interestingly, T 2 can also be measured via time-domain measurements, by averaging first X and Y quadratures over N relaxations, and then computing R

2 N = X 2 N + Y 2 N .
Indeed the quadratures taken independently are sensitive to frequency fluctuations, while the direct measurement of R 2 N = X 2 + Y 2 N is not. The obtained curve (see Fig. 5) whose exact mathematical description is given in Appendix D, can be well fit experimentally with an exponential law yielding a decay time in very good agreement with the T 2 defined from frequency domain measurements [see Fig. 6a)]. For simplicity we call this time domain measurement of decoherence T 2 . For a quantitative discussion we plot in Fig. 6a) the T 2 = (πW ) -1 obtained from a simple estimate of the FWHM in the frequency domain [see Fig. 4a)], together with T 2 and T 1 . Starting at the initial value of 2/∆ω 0 ≈ 57 ms at σ = 0, T 2 decreases to less than half its noise-free value for σ = 2π × 33 Hz. Our data are in good agreement (within ± 10%) with the theoretical curve calculated from the resonance lines derived from Eq. ( 3), with no free parameters.

Clearly, as soon as a significant frequency noise is applied, we verify T 1 = T 2 , thus demonstrating the impact of artificially engineered dephasing. This can also be seen as a consequence of the dispersive coupling of the mechanics to fluctuations described by the convolution formalism: the area under the X quadrature resonance curve, which corresponds to the energy stored in the mechanical mode, is preserved (more mathematical details in Appendix B).

One can then extract the pure dephasing rate

Γ φ = T -1 2 -T -1
1 , as displayed on Fig. 6b). The theoretical curve fits within ± 15% the data, demonstrating good agreement. This final graph thus presents the relation between Γ φ and σ 2 ∝ S Ω (ω)dω, which is the classical mechanical analogue to the superconducting qubit expression which can be found in Ref. [START_REF] Ithier | Decoherence in a superconducting quantum bit circuit[END_REF]. Note that the shape of the curve in Fig. 6b) is also characteristic of the frequency noise probability distribution. The mathematical tools are described thoroughly in Appendix B, and can be easily applied as a versatile method describing decoherence even in complicated non-analytic cases like the one presented here.

While being a practical quantitative estimate of decoherence, the definition of T 2 = (πW ) -1 or the exponential fit of T 2 essentially amount to approximating the resonance lineshapes by Lorentzians, which is rather crude. However, defining a shape factor S = 2πW x max / Im[χ 0 (ω)]F 0 dω, one can verify how accurate this simple approximation is. For the initial Lorentzian susceptibility we obtain S = 2/π as we should. For our strongest level of noise, S only deviates by 20% from this value, which proves that the simple approach is reasonably accurate (see Appendix D).

0 5 0 1 0 0 1 5 0 0 1 < R > 2 < R 2 > R 2 / R
But obviously, there is information in the measured resonance lineshapes that deserves to be exploited [START_REF] Dykman | Classical theory of nonlinear oscillators interacting with a medium[END_REF]. Highlighting this point is precisely the reason why we created a frequency noise that is non-Gaussian and asymmetric. The complete fits of the lineshapes prove that the theory can handle non-conventional noises. Furthermore, we note that the asymmetry in the resonance is a direct image of the noise properties, as frequency fluctuations here take only positive values. While demanding, the deconvolution of noise from the measured lineshape should be feasible, reconstructing the actual fluctuations distribution. This could prove to be extremely useful in actual experimental systems where dephasing is caused by intrinsic unknown mechanisms. One mathematical technique which has been proposed in the literature is to compute the moments of the frequency response [START_REF] Maizelis | Detecting and characterizing frequency fluctuations of vibrational modes[END_REF], an approach also found in NMR [START_REF] Abragam | Principles of Nuclear Magnetism[END_REF].

In summary, we establish that without external influence, pure dephasing is negligible in high quality SiN nanomechanical devices. We measure T 1 and T 2 over an unprecedently large dynamic range, ruling out any possible nonlinear dephasing/damping mechanisms, as opposed to carbon-based systems. We then engineer slow fluctuations of the beam's resonance frequency with an external tunable source, unravelling the signatures of pure dephasing. By analogy with quantum bits, we quantitatively link the dephasing rate to the characteristic level of noise. We develop the methods that can be applied to a wide range of devices and experiments studying pure dephasing originating from slow frequency fluctuations. Finally, we demonstrate that the lineshapes of the mechanical resonances contain valuable information about the distribution of the frequency fluctuations, and discuss the possibility to extract this information for actual nanomechanical systems suffering from intrinsic sources of noise.

-2 , 0 -1 , 5 -1 , 0 -0 , 5 0 , 0 0 , 5 1 , 0 1 , 5 2 , 0 0 2 0 0 0 0 4 0 0 0 0 6 0 0 0 0 8 0 0 0 0 C o u n t s

δV g ( V ) a σ V 0 5 1 0 1 5 2 0 2 5 3 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 C o u n t s δΩ/ ( 2 π) ( H z ) b σ/ ( 2 π)

APPENDIX A: PROBABILITY DISTRIBUTION FOR FREQUENCY FLUCTUATIONS

The voltage noise injected on the gate electrode is represented by a random Gaussian variable δV g (t) valued in R, of mean value 0 and standard deviation value σ V = δV 2 g 1/2 . Then, the probability to get a voltage valued in the interval [δV g , δV g + dδV g ] is:

dp(δV g ∈ R) = 1 σ V √ 2π exp - δV 2 g 2σ 2 V dδV g . (A1)
By capacitive coupling, these voltage fluctuations are converted to frequency fluctuations on the NEMS' resonance frequency, modelled by a random variable δΩ(t) = αδV 2 g (t) which is by construction only valued in R + . By simple change of variable, the probability to get a frequency shift valued in the interval [δΩ, δΩ + dδΩ] is:

dp(δΩ ∈ R + ) = 1 √ 2πσδΩ exp - δΩ √ 2σ dδΩ. (A2)
One can easily see that this probability distribution is normalized. It can be proved using the normalization for the probability distribution in δV g , which is symmetric around 0 so we can write:

+∞ -∞ dp(δV g ) = 2 +∞ 0 1 σ V √ 2π exp - δV 2 g 2σ 2 V dδV g = +∞ -∞ ρ σ (δΩ)dδΩ = 1,
hence the expression obtained for the probability distribution for frequency fluctuations ρ σ in the main article, Eq. ( 2). The probability distributions are presented in Fig. 7. Here σ = √ 2ασ 2 V is by construction the standard deviation of the frequency noise. Indeed, one can analytically obtain M n [ρ σ ], the n th moment of the distribution ρ σ :

M n [ρ σ ] = +∞ -∞ δΩ n ρ σ (δΩ)dδΩ = (2n)! n! σ 2 √ 2 n . (A3)
The present work aims at presenting a framework as generic as possible. This is why the distribution ρ σ is chosen non-Gaussian, and non-centered (its mean is not zero). We therefore deliberately keep this aspect in our discussion, and do not re-center the distribution in the following. Then, using the canonical definition of the squared standard deviation value of δΩ:

δΩ 2 -δΩ 2 = M 2 [ρ σ ] -M 1 [ρ σ ] 2 = 3σ 2 2 - σ √ 2 2 = σ 2 . (A4)
Besides, the 2 nd and 4 th moments of the δV g Gaussian distribution are respectively σ 2

V and 3σ 4 V . Thus, for the sake of consistency with the experimental definition, the standard deviation can also be calculated as follows:

σ 2 = δΩ 2 -δΩ 2 = 2α 2 σ 4 V .
Note that all the moments of the distribution ρ σ are finite, making it suited to our study. The frequency fluctuations power spectral density S Ω (ω) reads, following Wiener-Khinchin relation:

S Ω (ω) = +∞ -∞ δΩ(t)δΩ(t + τ ) e -iωτ dτ. (A5)
One gets by swapping the integrals:

1 2π +∞ -∞ S Ω (ω)dω = δΩ 2 (t) = 3σ 2 2 . (A6)
Thus the integrated value of the spectrum is related quadratically to the quantity σ.

APPENDIX B: CONVOLUTION PICTURE

The susceptibility χ 0 is randomized by the fluctuating gate voltage δV g . It can be recasted from Newton's law as:

χ 0 (ω, δV g ) = 1 m 0 ω 2 0 -1 2m0 ∂ 2 C(0) ∂x 2 δV 2 g -ω 2 + iω∆ω 0 = 1 m 0 (ω 2 0 + 2ω 0 δΩ -ω 2 + iω∆ω 0 )
.

Assuming the fluctuations are small compared to the characteristic frequency ω 0 , one can then write in the Lorentzian limit (ω 2 0 -ω 2 ≈ 2ω 0 (ω 0 -ω), ω∆ω 0 ≈ ω 0 ∆ω 0 ):

χ 0 (ω, δΩ) ≈ 1 2m 0 ω 0 (ω 0 + δΩ -ω + i ∆ω0
2 ) = χ 0 (ω -δΩ).

(B1)

The susceptibility depends directly on the random variable δΩ. At a given frequency ω and for a given noise level σ, a statistically averaged measurement ... σ over frequency fluctuations can be formally written:

χ σ (ω) = χ 0 (ω -δΩ) σ = +∞ -∞ χ 0 (ω -δΩ)ρ σ (δΩ)dδΩ, ( B2
)
hence the convolution picture for the susceptibility. Eq. (B2) only assumes that fluctuations are slow enough to validate this adiabatic picture, and fast enough to be fully integrated in the measurement. The formalism can be applied to the Telegraph noise of Ref. [START_REF] Sun | Telegraph frequency noise in electromechanical resonators[END_REF], when the switching rate is slow, with a distribution ρ σ consisting of two Delta functions separated by the switching frequency span.

In the convolution picture, it appears that the frequency fluctuations do not constitute an additional dissipative channel. Indeed, the distribution ρ σ is normalized to 1 independently of the noise level σ. Then, using Eq. (B2):

+∞ -∞ Im [χ σ (ω)] dω = +∞ -∞ Im [χ 0 (u)] du +∞ -∞ ρ σ (δΩ)dδΩ = +∞ -∞ Im [χ 0 (ω)] dω. Likewise: +∞ -∞ Re [χ σ (ω)] dω = +∞ -∞ Re [χ 0 (ω)] dω = 0.
In other words, the area under X and Y quadratures curves, i.e. the energy of the detected signal is preserved, independently of the frequency fluctuations.

APPENDIX C: FREQUENCY FLUCTUATIONS SPECTRAL DENSITY

Our formalism does not require to know the frequency noise spectral density: only its integrated value is needed. Yet we can infer this spectrum, since our (Gaussian) noise source and our filter are calibrated.

All parts of the filtering system are obviously linear and causal. Then, while theoretical calculations are made over both negative and positive frequencies, experimental values and curves are given only for calculations over positive frequencies. The causality assumption makes the latter differ only by a factor of 2 from theory.

The random variables δV g (t) and δV g (t + τ ) are both Gaussian, centered, with the same standard deviation value σ V . Using Wick formula (a brief demonstration is given in the supplementary material), one derives the autocorrelation function for the frequency fluctuations, as a function of the autocorrelation for gate voltage fluctuations:

δΩ(t)δΩ(t + τ ) = α 2 δV g (t)δV g (t)δV g (t + τ )δV g (t + τ ) = α 2 σ 4 V + 2α 2 δV g (t)δV g (t + τ ) 2 = σ 2 2 + 2α 2 δV g (t)δV g (t + τ ) 2 .
Remarkably, the frequency fluctuations autocorrelation function does not go to zero at long times. This is the result of the non-zero mean value of the frequency noise, which is as a matter of fact the constant value in the final expression of the autocorrelation. Using Eq. (A5) one writes the frequency fluctuations spectrum as a function of the gate voltage fluctuations spectrum S V :

S Ω (ω) = σ 2 2 δ(ω)+ α 2 π +∞ -∞ S V (ω )S V (ω-ω )dω , (C1)
One can recognize in the last term the auto-convolution of the voltage fluctuations spectrum and then reconstruct the frequency fluctuations spectrum: the gate electrode is fed with a white noise standard deviation value σ A over an equivalent bandwith (over positive frequencies) BW A 2π × 9 MHz. For the reasons explained in the main article, this noise is filtered at the output of the AWG. The power gain curve of the filter is fit with an algebraic function F (ω) given in Supplemental Material [32]. The integral calculated from the algebraic fit (over positive frequencies) yields the equivalent bandwith of this filter BW F 2π × 1.2 Hz. The filtered noise is then amplified over its whole frequency range with a power gain G = 2.5 × 10 7 . Taking into account again causality and linearity of the injection chain, one can recalculate the standard deviation value of the gate voltage noise σ V :

σ 2 V = BW F BW A Gσ 2 A . (C2)
For an injected noise of standard deviation value σ A = 0.25 V at the AWG output port, one gets σ V = 460 mV, which matches within 10% the measured value of 490 mV. The spectrum at the filter output, including the amplification factor G is then:

S V (ω) = πσ 2 A G BW A F (ω) = πσ 2 V BW F F (ω). (C3)
Then the frequency fluctuations spectrum reads: S Ω (ω) = σ 2 2 δ(ω)

+ πα 2 σ 4 V BW 2 F +∞ -∞ F (ω )F (ω -ω )dω S C (ω)
.

From experimental parameters we can thus quantitatively infer the frequency fluctuations spectrum. The convolution part S C (ω) of this spectrum is given in Fig. 8.

The convolution weighs the spectrum at low frequencies, with maximum at zero. Through this definition we can verify that the integrated value of the spectrum gives again 3σ 2 /2:

1 2π +∞ -∞ S Ω (ω)dω = σ 2 2 + α 2 σ 4 V 2BW 2 F × 4BW 2 F = 3σ 2
2 .

APPENDIX D: SHAPE ANALYSIS

The resonance lines obtained under frequency fluctuations are clearly not Lorentzian. Yet we can approximate them as such to quantify decoherence in our system, and thus extract their full width at half maximum (FWHM), for not too large noise levels. Let us define a shape factor S as: with W in Hz units, hence the 2π factor. Here we take advantage of the fact that the area of a true Lorentzian of FWHM W = W 0 and height x max = x 0 is π 2 W 0 x 0 . Then, for no noise applied, S is simply 2/π. Fig. 9 displays the evolution of the shape factor as a function of σ, with experimental data matching Eq. (D1) within ± 10%. In our experimental range, S deviates at most by 20% from its noise-free value which corresponds to the Lorentzian shape. This implies that a simple Lorentzian analysis in frequency domain, and a single exponential fit in time domain, are rather accurate. To illustrate this we plot in Fig. 10 where ∆ is the frequency of the beating between the mechanics and the local oscillator, and ... N ≈ ... σ by construction. For this particular case, σ/(2π) is of the order of the fluctuation-free linewidth W 0 . As can be seen on the graphs, while the shapes are different the key parameters W and T 2 are essentially the same for the two types of curves. We furthermore verify T 2 /2 ≈ (2πW ) -1 when the applied noise is not too large. An equivalent alternative way of studying the shape of the resonance is to define the integrated width I = Im [χ 0 (ω)] F 0 dω/x max , following the procedure of Ref. [START_REF] Moser | Nanotube mechanical resonators with quality factors up to 5 million[END_REF]. Indeed, the shape factor is nothing but the ratio of the FWHM to the integrated width: while the resonance peak remains close to a Lorentzian, I ≈ π 2 2πW .

Corresponding Author: E. C., eddy.collin@neel.cnrs.fr

  l i t u d e ( n m ) D r i v e f r e q u e n c y ( H z )

FIG. 2 :

 2 FIG. 2: (Color online) Frequency and time domain response of the mechanical oscillator, in X and Y quadratures. The applied sinusoidal driving force of amplitude F0 = 1.1 pN enables the resonator to reach the bistable regime. The two branches of the hysteretic response are displayed in the frequency domain on the left panel (see arrows for the frequency sweep direction). The free decay of the steady state response (right panel) is measured after detuning the drive from the resonance ωr by ∆ = -2π × 250 Hz. Solid lines are theoretical fits from the Duffing oscillator model (left) and the Lindstedt-Poincaré method (right). The magenta curve on the right panel corresponds to the amplitude R = √ X 2 + Y 2 , fit to an exponential decay of time constant T1 (full curve).

  FIG. 3: (Color online) Relaxation (T1) and decoherence (T2) times as a function of motion amplitude. The dotted line is set at 58 ms.

Fig

  Fig.4b) inset]:

σ = 2 πFIG. 4 :

 24 FIG. 4: (Color online) Frequency sweep measurements in (a) phase (X) and (b) quadrature (Y) components of the mechanical response xσ(ω) = χσ(ω)F0, with a sinusoidal driving force of amplitude F0 = 83 fN. Solid lines are theoretical curves calculated from Eq. (3), and resonance line parameters W, f0 are shown in a). Inset in (b): frequency noise histogram at σ = 2π ×33 Hz calculated from δV 2 g measurements, for 4×10 6 counts. The solid line is the application of Eq. (2).

FIG. 5 : 2 N

 52 FIG. 5: (Color online) Time domain measurements of R 2 N (magenta) and R 2 N (black, empty dots), normalized to their steady state resonant amplitudes of motion R 2 0 N and R0 2 N . Solid lines are exponential fits. Data acquired for σ = 2π × 33 Hz, F0 = 83 fN.

FIG. 6 :

 6 FIG. 6: (Color online) (a) Measured decay times as a function of injected frequency noise level σ/(2π). The blue solid line is a theoretical calculation from Eq. (3). Blue dots are obtained from frequency domain measurements, while green dots are obtained from exponential fits of R 2 N in time domain measurements. Red dots are time domain measurements of energy relaxation, with the dotted line used as a guide for the eyes. (b) Pure dephasing rate as a function of frequency noise level. Blue dots account for frequency domain measurements, green dots represent data from time domain (T 2) measurements. The blue solid line is the theoretical curve calculated from Eq. (3).

FIG. 7 :

 7 FIG. 7: (Color online) (a) Gaussian distribution of a gate voltage noise injected with standard deviation value σV = 490 mV, for 10 6 samples. The solid line is the application of Eq. (A1). (b) Equivalent distribution for induced frequency fluctuations, of characteristic level σ = 2π × 5.1 Hz (note the log scale). The solid line is the application of Eq. (A2).

FIG. 8 :

 8 FIG. 8: (Color online) Convolution term SC (ω) of the frequency fluctuations spectrum, with the area under the curve filled in orange.

FIG. 9 :

 9 FIG.9:(Color online) Shape factor as a function of the level of noise applied. Black dots are obtained from measurements, while the red curve is the application of Eq. (D1). The dotted line is the Lorentzian case.

2 FIG. 10 :

 210 FIG. 10: (Color online) (a) Frequency-domain response in the X quadrature at σ = 2π × 5.1 Hz (red curve), with a Lorentzian fit of FWHM W = 8.3 Hz (black curve). (b) Time-domain response of R 2 normalized to the steady state resonant amplitude (red curve), with an exponential fit of time constant T 2/2 = 0.02 s (black curve).

  the actual calculated frequency and time domain curves (red) for σ = 2π × 5.1 Hz together with the simple fits (black). The time domain curve is easily obtained by writing:R(t) 2 N = X(t) 2 N + Y(t) 2 N ,(D2)withX(t) = exp(-t/T 1 ) X(ω 0 -δΩ) cos (∆ + δΩ)t -Y(ω 0 -δΩ) sin (∆ + δΩ)t , Y(t) = exp(-t/T 1 ) Y(ω 0 -δΩ) cos (∆ + δΩ)t+ X(ω 0 -δΩ) sin (∆ + δΩ)t ,

1 0 1 0 0 1 0 0 0 1 1 0

ACKNOWLEDGEMENTS

We thank J. Minet and C. Guttin for help in setting up the experiment, as well as T. Fournier, J.-F. Motte and T. Crozes for help in the device fabrication. We acknowledge support from the ERC CoG grant ULT-NEMS No. 647917, from MICROKELVIN, the EU FRP7 grant No. 228464, and from the French ANR grant QNM No. 0404 01.