of 5
Nonlinear radiation pressure dynamics in an optomechanical crystal
Alex G. Krause,
1, 2
Jeff T. Hill,
1, 2, 3
Max Ludwig,
4
Amir H. Safavi-Naeini,
1, 2, 3
Jasper Chan,
1, 2
Florian Marquardt,
4, 5
and Oskar Painter
1, 2,
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr.,
Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125
3
Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305
4
Institute for Theoretical Physics, Universit
̈
at Erlangen-N
̈
urnberg, 91058 Erlangen
5
Max Planck Institute for the Science of Light, G
̈
unther-Scharowsky-Straße 1/Bau 24, D-91058 Erlangen, Germany
(Dated: April 28, 2015)
Utilizing a silicon nanobeam optomechanical crystal, we investigate the attractor diagram arising from the
radiation pressure interaction between a localized optical cavity at
λ
c
=
1552 nm and a mechanical resonance
at
ω
m
/
2
π
=
3
.
72 GHz. At a temperature of
T
b
10 K, highly nonlinear driving of mechanical motion is
observed via continuous wave optical pumping. Introduction of a time-dependent (modulated) optical pump is
used to steer the system towards an otherwise inaccessible dynamically stable attractor in which mechanical self-
oscillation occurs for an optical pump red-detuned from the cavity resonance. An analytical model incorporating
thermo-optic effects due to optical absorption heating is developed, and found to accurately predict the measured
device behavior.
The field of optomechanics, concerned with the interaction
of an optical cavity and a mechanical resonator [1], has been
of recent interest for its promise for use in sensors [2, 3], non-
linear optics [4, 5], and demonstrations of macroscopic quan-
tum mechanics [6, 7]. To lowest order, the mechanical dis-
placement linearly modulates the frequency of the optical res-
onance in a cavity-optomechanical system. This, however,
gives rise to an inherently nonlinear phase modulation, and
through radiation-pressure backaction on the mechanical el-
ement, yields nonlinear system dynamics [8]. Much of the
previous work has focused on the linearized regime where the
interaction with the optical field still gives rise to a host of in-
teresting phenomena such as a modified spring constant [9],
damping or amplification of the mechanics [10], and EIT-like
slow-light effects [11, 12]. Recently, several experiments have
pushed into the quantum regime using back-action cooling in
the linearized regime to damp nanomechanical resonators to
near their quantum ground state of motion [13, 14].
In this work, we instead demonstrate new features and
tools in the nonlinear regime of large mechanical oscilla-
tion amplitude.
Previous experimental works have shown
that a blue-detuned laser drive can lead to stable mechanical
self-oscillations [15–18], or even chaotic motion [19]. The-
oretical predictions of an intricate multistable attractor dia-
gram [8] have so far eluded experimental observation, except
for the elementary demonstration of dynamical bistability in a
photothermally driven system [20]. In the present work, we
are able to verify the predicted attractor diagram and further
utilize a modulated laser drive to steer the system into an iso-
lated high-amplitude attractor. This introduces pulsed control
of nonlinear dynamics in optomechanical systems dominated
by radiation pressure backaction, in analogy to what has been
Electronic address:
opainter@caltech.edu;
URL:
http://copilot.
caltech.edu
shown recently for a system with an intrinsic mechanical bista-
bility [21].
We employ a one-dimensional (1D) optomechanical crys-
tal (OMC) designed to have strongly interacting optical and
mechanical resonances [22]. The OMC structure is created
from a free-standing silicon beam by etching into it a peri-
odic array of holes which act as Bragg mirrors for both acous-
tic and optical waves [23]. A scanning electron micrograph
(SEM) of an OMC cavity is shown in Fig. 1a along with finite-
element-method (FEM) simulations of the co-localized opti-
cal (Fig. 1b) and mechanical (Fig. 1c) resonances. To reduce
radiation of the mechanical energy into the bulk, the OMC is
surrounded by a periodic ‘cross’ structure which has a full
acoustic bandgap around the mechanical frequency (Fig. 1a,
green overlay) [24].
The experimental setup is shown schematically in Fig. 1d.
The silicon chip containing the device is placed into a helium
flow cryostat where it rests on a cold finger at
T
4 K (the
device temperature is measured to be
T
b
10 K). Input laser
light is sent into the device via a tapered optical fiber, which,
when placed in the near-field of the device, evanescently cou-
ples to the optical resonance of the OMC [25]. The trans-
mitted light is detected on a high-bandwidth photodiode (D1)
connected to a real-time spectrum analyzer (RSA). We also
employ an electro-optic modulator (EOM) in the laser’s path
to resonantly drive the mechanical resonator. Finally, we can
send in a low-power, counter-propagating probe laser whose
detected spectrum (D2) is used to measure the mechanical am-
plitude and the pump-cavity detuning. Using this set-up the
optical resonance of the device studied in this work is mea-
sured to be at at
λ
c
=
1542 nm, with intrinsic (taper loaded)
energy decay rate of
κ
i
/
2
π
=
580 MHz (
κ
/
2
π
=
1
.
7 GHz).
The breathing mechanical mode, shown in Fig. 1c, is found
to be at
ω
m
/
2
π
=
3
.
72 GHz with bare energy damped rate of
γ
i
/
2
π
=
24 kHz.
The coupling of the optical resonance frequency to the
arXiv:1504.05909v2 [physics.optics] 27 Apr 2015
2
a
d
RSA
D1
WM
Fiber Taper
Cyrostat
PM
D2
pump laser
probe spectrum
probe laser
VO
A
VO
A
λ=1,542 nm
λ=1,542 nm
0
1
1 m
b
c
1 m
FIG. 1: (a) SEM of the optomechanical nanobeam surrounded by
phononic shield (green). (b) FEM-simulated electromagnetic energy
density of first-order optical mode; white outline denotes edges of
photonic crystal. (c) FEM-simulated mechanical mode profile (dis-
placement exaggerated). In (b) the colorscale bar indicates large (red)
and small (blue) energy density, whereas in (c) the scale bar indicates
large (red) and small (blue) displacement amplitude. (d) Simplified
schematic of experimental setup. WM: wavemeter,
∆φ
: electro-optic
phase modulator, OMC: optomechanical crystal, D1: pump light de-
tector, D2: probe detector, VOA: variable optical attenuator, PM:
power meter
mechanical displacement yields the interaction Hamiltonian,
H
int
=
~
g
0
ˆ
a
ˆ
a
ˆ
x
where ˆ
a
( ˆ
x
) is the optical (mechanical) field
amplitude, and
g
0
is the vacuum optomechanical coupling rate.
The physical mechanical displacement expectation is given by
x
=
x
zpf
ˆ
x
, where the zero-point amplitude of the resonator is
x
zpf
= (
~
/
2m
eff
ω
m
)
1
/
2
=
2
.
7 fm (estimated using a motional
mass m
eff
=
311 fg calculated from FEM simulation). Uti-
lizing a calibration of the per-photon cooling power [13] we
find that
g
0
/
2
π
=
941 kHz. These device parameters put our
system well into the sideband resolved regime
κ
/
ω
m

1.
The classical nonlinear equations of motion for the mechan-
ical displacement (
x
) and the optical cavity amplitude (
a
=
ˆ
a
)
are,
̈
x
(
t
) =
γ
i
̇
x
(
t
)
ω
2
m
x
(
t
)+
2
ω
m
g
0
x
zpf
|
a
(
t
)
|
2
,
(1)
̇
a
(
t
) =
[
κ
2
+
i
(
L
+
g
0
x
zpf
x
(
t
)
)]
a
(
t
)+
κ
e
2
a
in
,
(2)
where
a
in
=
P
in
/
~
ω
L
is the effective drive amplitude of the
pump laser (input power
P
in
and frequency
ω
L
),
κ
e
/
2 is the
fiber taper input coupling rate,
ω
c
is the optical cavity reso-
nance frequency, and
L
ω
L
ω
c
. Since we are interested in
the regime of self-sustained oscillations, where the motion of
the oscillator is coherent on time scales much longer than the
cavity lifetime, we can take the mechanical motion to be sinu-
soidal with amplitude
A
,
x
(
t
) =
A
sin
ω
m
t
. The optical cavity
field is then given by,
a
(
t
) =
κ
e
2
a
in
e
i
Φ
(
t
)
n
i
n
α
n
e
in
ω
m
t
,
(3)
where
Φ
(
t
) =
β
m
cos
ω
m
t
and
α
n
=
J
n
(
β
m
)
/
(
κ
/
2
+
i
(
n
ω
m
L
))
. Here
J
n
is the Bessel function
of the first kind,
n
-th order, and its argument is the unitless
modulation strength
β
m
= (
Ag
0
)
/
(
x
zpf
ω
m
)
. For
β
m

1 only
the terms oscillating at the mechanical frequency,
ω
m
, are
appreciable, so the interaction can be linearized, and only the
first-order radiation pressure terms are present. However, for
β
1 the higher harmonic terms at each
n
ω
m
have significant
amplitude and backaction force.
The thermal amplitude is too small to enter the nonlinear
regime in our devices
(
β
th
0
.
01
)
, however, backaction from
the pump laser can provide amplification to drive the mechan-
ical resonator into the high-
β
, nonlinear regime. The result-
ing mechanical gain spectrum in the amplitude-detuning plane
(the attractor diagram) can be solved for by calculating the en-
ergy lost in one mechanical cycle (
P
fric
=
m
eff
γ
i
̇
x
2
) and com-
paring it to that gained (or lost) from the optical radiation force
(
P
rad
=
(
~
g
0
/
x
zpf
)〈
|
ˆ
a
|
2
̇
x
) [8] . Figure 2a shows a plot of the
gain spectrum for the parameters of the device studied here
with a laser pump power of
P
in
=
151
μ
W. Imposing energy
conservation,
P
rad
/
P
fric
= +
1, yields the steady-state solution
contour lines. Although the entire contour is a physical solu-
tion, the equilibrium is only stable when the power ratio de-
creases upon increasing the mechanical amplitude,
∂β
P
rad
P
fric
<
0
(i.e. stability is found at the ’tops’ of the contours) [8].
In the device studied here there is a thermo-optic frequency
shift of the optical cavity caused by heating due to intra-
cavity optical absorption. The thermal time constant of the
device structure is slow relative to the optical cavity coupling
rate, but fast compared to the laser scan speed. Absorption
heating can thus be modeled as a shift of the laser detuning
proportional to the average intra-cavity photon number ( ̄
n
a
),
L
=
L
,
0
+
c
to
̄
n
a
, where
L
,
0
is the bare laser-cavity detuning
in absence of thermo-optic effects. The per photon thermo-
optic frequency shift of the optical cavity is measured to be
c
to
/
2
π
=
216 kHz. Including this effect, the shifted con-
tours are shown in Fig. 2b as a function of the bare detun-
ing
L
,
0
. The solid lines with arrows indicate the expected
path traversed by the mechanical resonator during a slow laser
scan from lower to higher laser frequency (left to right) at each
power. The dashed lines are contours that are either unstable,
or unreachable by this adiabatic laser sweep.
We first explore the lowest-lying contour of the attractor
diagram by measuring the optical transmission as the pump
laser is tuned from red to blue across the optical cavity res-
3
−1
0
1
2
3
4
5
6
0.8
0.9
1.0
Normalized Transmission
1.75
2
2.25
m
(MHz)
70
55
40
P
m
(dBm)
P
in
(
W)
0.1
1
10
100
−1
0
1
2
3
4
5
6
0
1
0.5
0
0.5
1.0
0
0.5
1.0
-145
-80
e
g
0
2
4
6
8
10
-20
20
0
-10
10
−1
0
1
2
3
4
5
6
β
m
a
c
0
2
4
6
8
10
β
m
f
b
d
experiment
theory
0.1
1
10
100
P
in
(
W)
FIG. 2: (a) Calculated gain spectrum for the OMC in the amplitude-detuning plane. Color scale indicates the ratio of power input to that lost
from friction
(
P
rad
/
P
fric
1
)
at
P
in
=
151
μ
W. Positive values are regions of mechanical self-oscillation. Solid line curves indicate power-
conserving solution contours at selected input powers: 0
.
65
μ
W (white), 6
.
5
μ
W (grey), 151
μ
W (black). (b) Same as (a) with contours
now shifted by estimated thermo-optic effects. Solid line curves indicate the path taken by the mechanical oscillator during the laser sweep.
Dashed lines are contours which are either unstable or unreachable by a slow sweep of laser detuning. (c) Image plot of the measured optical
transmission spectrum versus laser detuning and power. (d) Image plot of the theoretically calculated transmission spectra including thermo-
optic shifts and a slow drift in the optical resonance frequency over the course of the measurement from low to high power. Spectra in (c) and
(d) are scaled at each power level to span the range 0
1. (e) Plot of the normalized optical transmission from scans in (c) at
P
in
=
0
.
12
μ
W
(top), 0
.
65
μ
W (center), 151
μ
W (bottom). Blue points are measured data and red curve is the theoretical model. (f) Power spectral density of
detected signal near the mechanical frequency for
P
in
=
151
μ
W, showing frequency shifts of the mechanical mode,
m
ω
m
ω
m
,
0
, from its
bare frequency
ω
m
,
0
/
2
π
=
3
.
72 GHz. Color scale is detected power density in dBm/Hz. (g) Total integrated power of spectra in (f). Measured
data are plotted as green circles, with the theoretical model (up to a scale factor) shown as a solid red curve. The red arrow in each plot
indicates the laser scan direction.
onance with a fixed optical input power. A dip in transmis-
sion indicates that light is entering the cavity and being lost
through absorption or scattering. At low optical input powers
(
P
in
<
0
.
3
μ
W), only a single resonance dip associated with
the bare optical cavity is observed. Upon increasing the laser
power, radiation pressure backaction amplifies the thermal mo-
tion of the mechanical resonator beyond threshold and into a
large amplitude state. When this occurs a large fraction of the
intra-cavity photons are scattered, resulting in additional trans-
mission dips near each detuning
L
,
0
=
n
ω
m
where threshold
is reached. Physically, mechanical oscillations at the
n
-th side-
band detuning are generated by a multi-photon gain process in-
volving
n
photon-phonon scattering events. This stair-step be-
havior is seen in the measured transmission spectrum of both
Fig. 2c and Fig. 2e. The theoretically calculated spectra in-
cluding thermo-optic effects are shown in Fig. 2d, in excellent
agreement with the measured spectra after taking into account
a slow drift in the cavity resonance frequency as the measure-
ments were taken from low to high power.
Figure 2f shows the radio-frequency noise power spectrum
near the mechanical frequency of the optical transmission pho-
tocurrent at the highest measured input power (
P
in
=
151
μ
W).
We note that backaction effects blue-shift the mechanical res-
onance frequency by an appreciable amount (
2 MHz) from
its intrinsic value of
ω
m
/
2
π
=
3
.
72 GHz. This frequency shift,
m
, depends on the laser detuning in a more intricate fashion
than a linear calculation of the optical spring effect would sug-
gest. A measure of oscillation amplitude can be extracted from
the total power in this mechanical sideband, and its measured
dependence on detuning is shown in Fig. 2g. The measured
total transduced power oscillates due to the nonlinearity of the
detection process, which is nicely captured by the theoretical
model, obtained without fit parameters except for an overall
scale factor (Fig. 2g, red curve).
It is readily apparent from Fig. 2a that at large optical pow-
ers (black contour) there are a number of isolated attractor con-
tours at higher oscillation amplitudes. Here we utilize exter-
nal time-dependent driving of the mechanical mode to explore
the lowest-lying isolated attractor on the red side of the opti-
cal cavity (
L
<
0), where the linearized theory predicts only
damping of the mechanical mode. An electro-optic modulator
(EOM) is utilized to phase-modulate the incoming light field,
resulting in an oscillating force inside the cavity which drives
the mechanical resonator towards higher amplitudes. The ex-
perimental sequence is displayed in Fig. 3a. We start with the
pump laser switched on at a power of
P
in
=
43
μ
W, the laser
detuned to the red side of the cavity resonance, and the phase
modulation off (
β
EOM
=
0), which initializes the mechanical
resonator into a cooled thermal state with
β
m
0. The EOM
phase modulation is then turned on which rings up the me-