Thermal nonlinearities in a nanomechanical oscillator · 2013-12-02 · The gradual phase shift. η...

Supplementary Information

Thermal nonlinearities in a nanomechanical oscillator

Jan Gieseler1, Lukas Novotny2 and Romain Quidant1,3

1. ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona),

Spain

2. Photonics Laboratory, ETH Zurich, 8093 Zurich, Switzerland

3. ICREA-Institucio Catalana de Recerca i Estudis Avancats, 08010 Barcelona, Spain

1 Gaussian model of the optical force

To understand the forces exerted on the particle by the electromagnetic field of a tightly focused laser

beam, we derive a simple analytical model based on the Rayleigh (dipole) approximation and a Gaus-

sian description of the trapping laser. The analytical model allows us to understand the origin of the

nonlinearities discussed in the main text.

1.1 Optical field distribution

To gain some insight into the nature of the nonlinearities of the optical trap, we we assume a Gaussian

field distribution [1]:

E(ρ, z) = E0

[1 + (z/z0)

2]−1/2

e−(

x2

w2x(z)

+ y2

w2y(z)

)+iϕ(z,ρ)

nx, (1)

where the separate beam waists wx and wy account for the asymmetry of the focus along x and y,

respectively. For clarity, we have defined the following quantities

wi(z) = wi

√1 + z2/z20 beam radius (2)

ϕ(z, ρ) = kz − η(z) + kρ2 /2R(z) phase (3)

R(z) = z(1 + z20/z

2)

wavefront radius (4)

η(z) = arctan (z/z0) phase correction. (5)

S1

Thermal nonlinearities in a nanomechanical oscillator

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The gradual phase shift η(z) as the beam propagates through the focus is known as the Gouy phase

shift.Eq. 1 provides us with an analytical expression of the focal field from which we can calculate the

nonlinear coefficients. Note that in this model, the asymmetry of the focus is only accounted for in the

amplitude but not in the phase. We fit the intensity distribution |E|2 obtained from (1) to the intensity

distribution obtained from the exact numerical calculation of the Debye integral [1]:

E(ρ, φ, z) =ikfe−ikf

2π

∫ θmax

0

∫ 2π

0E∞(θ, ϕ)eikz cos θeikρ sin θ cos(ϕ−φ) sin θdϕsθ, (6)

where E∞(θ, ϕ) is the field distribution on the reference sphere, f the focal length and k = λ/2π the

wave vector. Eq. (6) can be understood as an interference of plane waves at the focus [2] (For details

on Eq. 6 see also Ref. [1]). The maximum angle θmax depends on the NA of the lens NA = nm sin θmax,

where nm is the refractive index of the surrounding medium. Note that the parameters wx,wy and z0

are free fit parameters.

Figure S1 shows the field distribution in the focal planes. The first row shows the intensity distribu-

tion calculated with the Debye integral (6) in the x-z and x-y plane, respectively. For comparison, the

bottom row shows the same distributions obtained with the Gaussian model (1). The Gaussian model

does not include diffraction and therefore doesn’t have side-lobes. However, up to λ/2 away from the

center, the Gaussian model fits the exact solution well.

1.2 Derivation of optical forces

It is instructive to write the optical force as a sum of two terms, the gradient force Fgrad(r) and the

scattering force Fscatt(r) [1, 3]:

F(r) = Fgrad(r) + Fscatt(r). (7)

Under the assumption that we can represent the complex amplitude of the electric field in terms of a

real amplitude E0 and phase ϕ, the forces are given by

Fgrad(r) = α′/4 ∇I0(r) (8)

and

Fscatt(r) = α′′/2 I0(r)∇ϕ(r), (9)

where I0(r) = E20(r) is the field intensity and α′ and α′′ are the real and imaginary part of the polar-

isability, respectively. This approximation is valid if the phase varies spatially much stronger than the

S2

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1.0 0.5 0.0 0.5 1.01.0

0.5

0.0

0.5

1.0

x ΜmyΜm

Intensity xy plane

1.0 0.5 0.0 0.5 1.01.0

0.5

0.0

0.5

1.0

x Μm

zΜm

Intensity xz plane

1.00.5 0.0 0.5 1.01.0

0.5

0.0

0.5

1.0

x Μm

yΜm

Intensity xy plane Gauss

1.00.5 0.0 0.5 1.01.0

0.5

0.0

0.5

1.0

x ΜmzΜm

Intensity xz plane Gauss

1.0 0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

Position Μm

E2V2 Μ

m2

Intensity distribution

b

zxy

a

129 [V2/µm2]

0

E2

c

d e

Figure S1: Intensity distribution of tightly focused optical fields. (a) The intensity distribution com-

puted with the Debye integral (6) is shown as black lines for the three mayor axes. The coloured lines

are fits to the Gaussian model. The exact solution exhibits side lobes as a consequence of diffrac-

tion. Both models take the asymmetry of the focal spot into account. (b,c) The intensity distribution

computed with the Debye integral in the x-y plane and y-z plane, respectively. (d,e) The same fields

computed with the Gaussian model. In the calculation we assumed focusing in air with a NA= 0.8

objective, a filling factor (ratio between beam waist of incident beam and lens aperture) of 2 and wave-

length λ = 1064nm. For a detailed description of how to apply equation (6) see reference [1].

amplitude. This is the case for weakly focused fields and for the fields given by Eq. (1).

For a spherical particle with volume V = 4/3πa3 and dielectric constant ϵp embedded in a medium

with dielectric constant ϵm, the polarisability is given by Clausius-Mossotti relation (also Lorentz-Lorenz

formula) [3, 1]:

α = 3V ϵ0(ϵp − ϵm)/(ϵp + 2ϵm). (10)

Generally, α is a tensor of rank two. However, a sub-wavelength particle can be treated as a dipole and

it is legitimate to use a scalar representation.

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The gradient force is proportional to the dispersive (real) part α′, whereas the scattering force is

proportional to the dissipative (imaginary) part α′′ of the complex polarisability α. The phase ϕ(r)

can be written in terms of the local k vector ϕ(r) = k · r. Hence, the scattering force results from

momentum transfer from the radiation field to the particle. Momentum can be transferred either by

absorption or scattering of a photon. Photon scattering by the particle changes the electric field. This,

in turn, modifies the optical force acting on the particle. This backaction effect also known as radiation

reaction is accounted for by an effective polarisability [1]

αeff = α

(1− i

k3

6πϵ0α

)−1

. (11)

Consequently, even for a lossless particle (Im(ϵp) = 0), the scattering force does not vanish completely!

From (8), (9) and (1) we calculate the optical forces in the Gaussian approximation

Fgrad(r) = −α′effI0(r)

×

x z20/w2x(z

2 + z20)

y z20/w2y(z

2 + z20)

z[(z/z0)

2 +(1− 2x/w2

x − 2y/w2y

)] [z20

/2(z2 + z20)

]

(12)

and

Fscatt(r) =α′′eff

2I0(r)k

×

x /R(z)

y /R(z)

1 +(x2 + y2

)z20

/z2R(z)2 −

[x2 + y2 + 2z z0

]/2zR(z)

, (13)

where I0(r) = E20

[1 + (z/z0)

2]−1

exp(−2

[x2/w2

x(z) + y2/w2y(z)

]).

The field intensity at the focus, E20 , is related to the total power of the Gaussian beam by

P =

∫ ∞

−∞

∫ ∞

−∞⟨S⟩nzdxdy = cϵ0πwxwyE

20 /4 , (14)

where c is the speed of light, nz is the direction of beam propagation and ⟨S⟩ = ⟨H×E⟩ is the the

Poynting vector.

S4





For small displacements |r| ≪ λ, we expand equations (12) and (13) to get the first order nonlinear

terms

Fgrad(r) ≈ −

k(x)trap

[1− 2x2/w2

x − 2y2/w2y − 2z2/z20

]x

k(y)trap

[1− 2x2/w2

x − 2y2/w2y − 2z2/z20

]y

k(z)trap

[1− 4x2/w2

x − 4y2/w2y − 2z2/z20

]z

(15)

and

Fscatt(r) ≈α′′eff

α′eff

k(z)trap

k xz

k yz

γ0 + γzz2 + γxx

2 + γyy2,

(16)

where

γ0 = z0(z0k − 1), (17a)

γz = (2− z0k)/z0, (17b)

γx =[k/2− 2(z0 − k z20)

/w2x

]and (17c)

γy =[k/2− 2(z0 − k z20)

/w2y

]. (17d)

are constants which depend only on the optical field but not on the properties of the particle.The longitudinal and transversal trap stiffness are given by

k(x)trap = α′

effE20/w

2x, (18a)

k(y)trap = α′

effE20/w

2y and (18b)

k(z)trap = α′

effE20/2z

20 , (18c)

respectively. As expected, the trap stiffness increases with polarisability, laser power and field confine-

ment.Note that, for the particles considered in the main text, the scattering force is still negligible. Addi-

tionally, since Fscatt ∝ x2, the frequency shift due to the scattering force averaged over one oscillation

period ∆Ω ∝∫x ∝ x0

∫cosΩ0t = 0 vanishes. Wether a signature of the scattering force can be

observed in a levitated nanoparticle requires further investigation.

2 Dynamics of parametric feedback cooling and parametric driving

Figure S2 shows a schematic of the experimental configuration. A single nanoparticle is trapped at the

focus of the laser beam by means of the optical gradient force and cooled parametrically by a feedback

loop [4]. The gradient force is proportional to both the particle displacement q and the power of the

S5





2ΩΔφΣ

x

y

zzzzz

+feedback

20

parametric drivem,

Figure S2: Experimental configuration to measure the Duffing nonlinearity. Feedback cooling reduces

thermal motion. An additional parametric drive excites one mode while the other two modes remain at

low oscillation amplitudes.

trapping laser Popt. The feedback loop modulates the trapping laser intensity proportional to qq, where

q is the particle velocity. Thus, the feedback force is given by

Ffb = −ηΩ0q2q, (19)

where Ω0 and η are the oscillation frequency and the feedback gain, respectively. Thus, feedback

cooling adds a nonlinear damping Γfb ∝ q2 to the natural damping Γ0.In a time-domain picture, the feedback loop hinders the particle’s motion by increasing the trap

stiffness whenever the particle moves away from the trap center and reducing it when the particle falls

back toward the trap. In the frequency domain, this corresponds to a modulation at twice the trap

frequency with an appropriate phase shift.Note that depending on the latency of the feedback loop we can achieve damping or amplification

of the particle’s oscillation. In the absence of active feedback, the particle’s oscillation naturally locks

to the modulation phase in such a way as to achieve amplification [5]. Cooling therefore requires active

feedback to adjust the modulation phase constantly.

In our cooling scheme, frequency doubling and phase shifting is done independently for each of

the photodetector signals x, y and z. Since the three directions are spectrally separated, there is no

cross-coupling between the three signals, that is, modulating one of the signals does not affect the

other signals. Therefore, it is possible to sum up all three feedback signals and use the result to drive a

S6





single electro optic modulator (EOM) that modulates the power Popt of the trapping laser. Thus, using

a single beam we are able to effectively cool all spatial degrees of freedom.

2.1 Equation of motion

The particle motion consists of three modes, each corresponding to a spatial oscillation along one of

the three symmetry axes of the optical intensity distribution. For large oscillation amplitudes, the modes

couple through cubic nonlinearities in the optical force (c.f. section 1). However, under the action of

feedback cooling, the effective thermal amplitude qeff =(2kBTeff

/mΩ2

0

)1/2 of the particle oscillation is

much smaller than the size of the trap. As a consequence, coupling between the modes is negligible

and the particle dynamics is well described by a one-dimensional equation of motion:

q + Γ0q +Ω20

1 + ϵ cos (Ωmt)

parametric drive

+Ω−10 ηqq

feedback

+ ξq2Duffing term

q =

Ffluct

m≈ 0. (20)

The term in brackets is proportional to the optical power Popt and allows to control the particle

motion. It consists of four terms:

• The constant term is proportional to the mean optical power and defines the oscillation frequency

Ω0 of the trapped particle.

• Parametric driving with modulation depth ϵ and modulation frequency Ωm allows to drive the

particle motion. The most interesting phenomena occur when Ωm is close to twice the natural

frequency Ω0 of the particle.

• Nonlinear damping is due to parametric feedback cooling. Without parametric drive (ϵ = 0), the

nonlinear damping reduces the effective thermal energy from T0 to Teff . It is important to note

that feedback cooling reduces the effective thermal motion in all three spatial directions (because

the feedback signal contains frequency components at at Ωx, Ωy and Ωz). In contrast, parametric

driving only excites the mode which fulfils the condition Ωm ≈ 2Ω0.

• The Duffing nonlinearity is due to the shape of the optical potential (c.f. section 1). It becomes

significant when the particle’s oscillation amplitude is comparable to the beam waist w0.

The right hand side of equation (20) is a stochastic force due to random collisions with residual air

molecules and therefore depends on pressure. The fluctuation-dissipation relation links the strength

of the thermal force to the damping ⟨Ffluct(t)Ffluct(t′)⟩ = 2mΩ0Q

−1 kBTeff δ(t − t′). In the following

S7





analysis we consider that the parametric driving term and the nonlinear terms dominate the dynamics.

This is a valid assumption since under typical experimental conditions Q = Ω0/Γ0 ≫ 1 and, therefore,

the stochastic force which is ∝ Q−1/2 is much smaller than the deterministic terms in (20).

The response of the particle to the external modulation depends on the parameters of the external

driving force ϵ and Ωm, respectively. One can distinguish between resonant and non-resonant paramet-

ric driving. For the former the condition Ωm ≈ 2Ω0 holds, whereas for the latter this condition is violated.

For sufficiently strong driving, the system makes a transition reminiscent of a phase transition. The par-

ticle motion changes from thermal (incoherent) motion to sustained (coherent) oscillations with a fixed

frequency with respect to the external modulation. The full theory of a mechanical oscillator which is

described by Eq. 20 can be found in [6]. In the following we focus on the case when the particle is

parametrically driven into the nonlinear regime.

2.2 Secular perturbation theory for the parametrically driven Duffing oscillator

We are interested in solutions q(t) that are slow modulations of the linear resonance oscillations. There-

fore, we introduce a dimensionless slow time scale T = κΩ0t and displacement amplitude A(T ). With

the ansatz

q =q02AeiΩ0t + c.c. (21a)

and using A = dAdt = Ω0κ

dAdT we get:

q = Ω0q02

[κdA

dT+ iA

]eiΩ0t + c.c. (21b)

q = Ω20

q02

[κ2

d2A

dT 2+ i2κ

dA

dT−A

]eiΩ0t + c.c. (21c)

q2q =q308Ω0

[2κ|A|2dA

dT+ i|A|2A+ κA2dA

∗

dT

]eiΩ0t + c.c. (21d)

q3 =3q308

|A|2AeiΩ0t + c.c. (21e)

ϵ cos(Ωmt)q =ϵq04

A∗e(Ωm−2Ω0)t eiΩ0t, (21f)

where c.c. stands for complex conjugate and we dropped small corrections from fast oscillating terms.

For a consistent expansion [7, 6], we apply the rescaling

γ0 =Γ0

Ω0κ; η =

η

ξ; ϵ =

ϵ

κ(22)

S8





with scale factors κ = Γ0/Ω0 = Q−1 and q20 = κ/ξ. Plugging (21) into (20) yields

Ω20

q02

[κ2

d2A

dT 2+ i2κ

dA

dT−A

]eiΩ0t (23)

+ γ0κΩ20

q02

[κdA

dT+ iA

]eiΩ0t

+Ω20

q02AeiΩ0t

+q08Ω20ηκ

[2κ|A|2dA

dT+ i|A|2A+ κA2dA

∗

dT

]eiΩ0t + c.c.

+Ω20κ

3q08

|A|2AeiΩ0t + c.c.

+ϵq04

κA∗e−i(δΩm−2Ω0)t eiΩ0t = 0

Dropping terms higher order terms O(κ2) and fast oscillating terms we arrive at

dA

dT= − γ0

2A− 1

8η|A|2A+ i

3

8|A|2A+ i

ϵ

4A∗e−iδmT , (24)

where δm = δm/κ is the rescaled normalised detuning δm = (2− Ωm/Ω0).

2.2.1 Equations of motion for amplitude and phase

Introducing

A = q expi(ϕ− T δm/2) (25)

into (24) yieldsdq

dT+ iq

(dϕ

dT− δm

2

)= − γ0

2q − 1

8ηq3 + i

3

8q3 + i

ϵ

4qe−2iϕ(T ). (26)

For convenience, we split (26) into an equation

dq

dT= − γ0

2q − 1

8ηq3 +

ϵq

4sin(2ϕ) (27a)

anddϕ

dT=

3

8q2 +

δm2

+ϵ

4cos(2ϕ) (27b)

for the real and imaginary part, respectively. Applying the inverse scaling (22), we find the following

equations for the amplitude q = q0q and phase ϕ

dq

dt= −Γ0

2q − 1

8Ω0ηq

3 +Ω0ϵ

4q sin(2ϕ) (28a)

dϕ

dt= Ω0

3

8ξq2 +Ω0

δm2

+ Ω0ϵ

4cos(2ϕ) (28b)

S9





In the linear case (ξ = 0), Equation (28b) is also known as the Adler equation. It is this equation

that explains the steady state, and in fact also the transient, injection-locking behaviour of a harmonic

oscillator with an external signal [8].

2.2.2 Nonlinear frequency shift

The nonlinear frequency shift follows immediately from Eq. 28b. According to (21a) and (25), the

particle oscillates at Ω = Ωm/2. Assuming weak driving ϵ ≈ 0 and a steady state oscillation dϕ/dt = 0,

the nonlinear frequency shift is given by

∆ΩNL = Ω− Ω0 = Ωm/2− Ω0 = −Ω0δm/2 = Ω03

8ξq2. (29)

Since the nonlinear frequency shift (29) does not depend on the origin of the driving term (i.e

parametric, direct or thermal), it also applies to the thermally driven oscillator. However, while the

free running (thermally driven) oscillator also suffers from frequency shifts originating from amplitude

fluctuations of the orthogonal modes (c.f. Eq. 15), this is not the case when the oscillator is excited

by an external modulation with a fixed frequency because the amplitudes of the orthogonal modes are

maintained low by the feedback loop. As a consequence, the response of the directly driven oscillator

(shown as black dots in Fig.3d of the main text) is sharp, while the response of the thermally driven

oscillator (scatter plot in Fig.3d) is shifted to lower frequencies (because the non-zero mean square

amplitude of the orthogonal modes leads to a non-zero frequency shift) and broadened (because of

fluctuations of the amplitudes of the orthogonal modes).We estimate the frequency shift due to excitation of the orthogonal modes:

∆Ωy,z

2π=

3

4

kBT

m

(ξyΩy

+ξzΩz

)(30)

where we used that r2th = 2kBT/(mΩ20). For the experimental values m = 3 × 10−18kg, T = 300K,

Ωy/2π = 135kHz, Ωz/2π = 37kHz, ξy = −10.41µm2 and ξz = −0.98µm2 we find ∆Ωy,z/2π = 2.7kHz.

2.2.3 Linear threshold condition

Now, we consider the condition for which the system makes the transition from thermal motion to

sustained oscillations. For weak driving, the oscillation amplitude q is small and the nonlinear terms in

(28) can be neglected. Therefore, the steady state phase is given by

ϕss =1

2cos−1

(− δmϵ/2

)(31)

and the total gain (negative damping) is given by

gL = Ω0ϵ

2sin(2ϕss)− Γ0 = Ω0

√ϵ2/4− δ2m − Γ0. (32)

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If the gain becomes positive, the amplitude grows exponentially until entering into saturation due to

nonlinearities. Thus, the system makes the transition from thermal motion to continuous oscillation at

half the parametric modulation frequency, if the condition

ϵ >2

Q

√1 +Q2δ2m ≈ 2|δm| (33)

is fulfilled. Again, the approximation holds for Q ≫ 1.

2.2.4 Steady state solution above threshold

Ignoring initial transients, and assuming that the nonlinear terms in the equation are sufficient to satu-

rate the growth of the instability, the system enters into a steady-state with dq/dt = dϕ/dt = 0. From

(28) we find two algebraic equations for the steady state amplitude and phase above threshold:[δm +

3

4ξq2

]2+

[Q−1 +

1

4ηq2

]2=

1

4ϵ2 (34a)

and

tan(2ϕ) =Q−1 + 1

4ηq2

δm + 34ξq

2. (34b)

Solving (34a) for q2 we find

q2 =−1

ηδ2th

[3ξ

ηδm +Q−1 −

√ϵ2δ2th − δ2m + 3

ξ

ηQ−2

(2Qδm − 3

ξ

η

)]

≈ −1

ηδ2th

[3ξ

ηδm −

√ϵ2δ2th − δ2m

], (35)

where

δth =√9ξ2 + η2 /2η . (36)

The approximation holds for Q ≫ 1. This is the case under typical experimental conditions.

From (35) follows that a solution only exists if

ϵ2δ2th − δ2m > 0. (37)

This condition plays a similar role as the linear stability condition (33). However, whereas (33) gives the

parameter range for which the linear system becomes unstable and makes a transition from thermal

motion to sustained oscillations, (37) is the condition for which the sustained oscillations remain stable

once they have been excited.

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2.3 Modulation frequency sweeps

In the following we consider the particle response for fixed modulation depth ϵ as the modulation fre-

quency Ωm is swept over the resonance of the x motion at 2Ω0 ∼ 250kHz. Note that the data presented

here is for a different particle than the one in the main text. However, the parameters change only very

little from particle to particle.

Figure S3a shows a map of the particle energy as Ωm is increased from 240 kHz to 255 kHz, for

values of ϵ ranging from 1 × 10−3 to 24 × 10−3. For off resonant modulation, the particle energy

remains at the effective temperature Teff =mΩ2

02kB

x2 ≈ 17K. For modulation frequencies within the

lock-in range, the particle energy increases significantly. The lock-in region, as predicted by the linear

stability condition (33), is shown as black dashed lines.Figure S3b shows the frequency sweep performed in the opposite direction. The high frequency

threshold is still given by (33). However, the low frequency threshold is pushed towards lower frequen-

cies (white dashed line) in agreement with the nonlinear stability condition (37).Figure S3c shows a horizontal cut through subfigures a and b indicated by black dotted lines

(ϵ = 22× 10−3). When the instability region is approached from below, the energy stays at Teff until the

threshold is reached. Above threshold, the motion locks to the external modulation and the energy is

given by (35). As the modulation frequency is further increased, the particle energy makes a smooth

transition to the off-resonant energy. Conversely, when the instability region is crossed from above,

the energy smoothly increases as predicted by (35). Because of the (negative) Duffing nonlinearity,

the increasing oscillation amplitude, pushes the effective resonance frequency Ωeff. = Ω0

(1 + 3

8ξq2)

to

lower frequencies. Since the lock-in region depends on the detuning from resonance, it is also dragged

along. While the nonlinear stability condition (37) is fulfilled the system follows the upper branch (35).

If the modulation is further decreased this solution becomes unstable and the particle oscillates again

with a low (thermal) amplitude determined by Teff .

2.3.1 Measurement of the nonlinear coefficients

From (37), we find the lower threshold frequency

Ωth = Ω0 [2− ϵδth] . (38)

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where the particle energy falls back to the off-resonant value. At this modulation frequency the particle’s

oscillation amplitudes reaches it’s maximum value

x2max =−3ξ

η2δthϵ. (39)

Figure S3d,e show the threshold frequency Ωth and energy Emax = 12mΩ2

0x2max together with their

respective fits. From the fits and equations (38) and (39) we extract the nonlinear coefficients. Alter-

natively we can obtain the nonlinear coefficients from a fit of (35) to the modulation downward sweep

(blue dashed curve Figure S3c).For the particle of the main text, we performed frequency downward sweeps across the three res-

onances such as shown in Fig. S3b and determined the nonlinear coefficients from linear fits to the

threshold frequency and maximum value of the oscillation amplitude (c.f. Figs. S3d,e)

3 Minimum detectable frequency shift of a nonlinear oscillator

The frequency noise spectral density of an harmonic oscillator with damping constant Γ0, frequency Ω0

and Q-factor Q = Ω0/Γ0 is given by

Sf =Ω0

Q= Γ0. (40)

Thus, the higher the effective Q-factor, the higher the sensitivity. However, since cooling to Teff =

T Qeff/Q is required to reduce nonlinear frequency fluctuations, the effective Q-factor is reduced, too.

The best compromise is achieved when the linear and the nonlinear frequency fluctuations contribute

equally, that is

∆ΩL =∆ΩNL (41)

Ω0Q−1eff =

3

8ξΩ0r

2eff =

3ξΩ0kBTeff

4k(42)

⇒ Q(opt)eff =

√4

3

mΩ20

ξkBTQ = R−1/2Q. (43)

Assuming that feedback cooling keeps the oscillator in the linear regime, the frequency power spec-

tral density with optimal feedback gain is

S(opt)f =

Ω0

Q(opt)eff

=Ω0

QR1/2. (44)

We estimate the improvement due to feedback cooling, assuming that the frequency noise spectral

density is white (however, this is not the case for frequencies larger than Ωc as we have shown in Fig. 4b

S13





1886 K/ 120 nm

4 K/ 4 nm

a b

up down

x2

up sweep

down sweep

T0

Te↵

c d

e

lock-in region down lock-in region up

fit

Emax

240 255244 251.9249.3

17

303

1851

764

11

48

119

54

modulation frequency m2Π kHz

EnergyK

xnm

5 10 15 200

500

1000

1500

2000

modulation depth 103

E maxK

5 10 15 20

244245246247248249250

modulation depth 103

th2Πk

Hz

Figure S3: Frequency Sweep. (a) For fixed modulation depth and increasing modulation frequency,

the particle energy maps out a triangular region in the ϵ-Ωm plane. The black dashed line marks the

instability threshold (33). (b) For decreasing modulation frequency, the lower instability threshold is

pushed to lower frequencies. (c) Up and down sweep at ϵ = 22× 10−3 (black dotted line in subfigures

a and b). The blue dashed line is a fit to (35). (d) Threshold modulation depth (white line in subfigure

b) and (e) threshold particle energy as a function of modulation depth. The black arrows mark the data

points which correspond to subfigure c.

of the main text). In that case, we can define the a Q factor for the nonlinear frequency fluctuations

QNL = Ω0/∆ΩNL and findS(opt)f

S(NL)f

=

[∆ΩL

∆ΩNL

]1/2= R−1/2. (45)

S14





[1] Novotny, L. & Hecht, B. Principles of Nano-Optics (Cambridge University Press, 2006), 1st edn.

[2] Wolf, E. Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image

Field. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences

253, 349–357 (1959).

[3] Rohrbach, A. & Stelzer, E. H. K. Optical trapping of dielectric particles in arbitrary fields. JOSA A

18, 839–853 (2001).

[4] Gieseler, J., Deutsch, B., Quidant, R. & Novotny, L. Subkelvin Parametric Feedback Cooling of a

Laser-Trapped Nanoparticle. Physical review letters 109, 103603 (2012).

[5] Yariv, A. Quantum Electronics (Wiley, New York, 1989), 3rd edn.

[6] Lifshitz, R. & Cross, M. C. Nonlinear Dynamics of Nanomechanical and Micromechanical Res-

onators 1–52 (2008).

[7] Villanueva, L. G. et al. Surpassing Fundamental Limits of Oscillators Using Nonlinear Resonators.

Physical review letters 110, 177208 (2013).

[8] Siegman. Lasers (1986).

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Thermal nonlinearities in a nanomechanical oscillator · 2013-12-02 · The gradual phase shift. η...

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