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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)
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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
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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.
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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
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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
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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
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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)
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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)
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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
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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
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[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
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[3] Rohrbach, A. & Stelzer, E. H. K. Optical trapping of dielectric particles in arbitrary fields. JOSA A
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[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.
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[8] Siegman. Lasers (1986).
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