Observation of Quantum Motion of a Nanomechanical Resonator
Amir H. Safavi-Naeini, Jasper Chan, Jeff T. Hill, Thiago P. Mayer Alegre, Alex Krause, and Oskar Painter
*
Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
(Received 14 September 2011; published 17 January 2012)
In this Letter we use resolved sideband laser cooling to cool a mesoscopic mechanical resonator to near
its quantum ground state (phonon occupancy
2
:
6
0
:
2
), and observe the motional sidebands generated on
a second probe laser. Asymmetry in the sideband amplitudes provides a direct measure of the displace-
ment noise power associated with quantum zero-point fluctuations of the nanomechanical resonator, and
allows for an intrinsic calibration of the phonon occupation number.
DOI:
10.1103/PhysRevLett.108.033602
PACS numbers: 42.50.Wk, 42.65.
k, 62.25.
g
Experiments with trapped ions and neutral atoms [
1
–
3
],
dating back several decades, utilized techniques such as
resolved sideband laser cooling and motional sideband
absorption and fluorescence spectroscopy to cool and mea-
sure a single trapped particle in its vibrational quantum
ground state. These experiments generated significant in-
terest in the coherent control of motion and the quantum
optics of trapped atoms and ions [
4
], and were important
stepping stones towards the development of ion-trap based
quantum computing [
5
,
6
]. Larger scale mechanical ob-
jects, such as fabricated nanomechanical resonators,
have only recently been cooled close to their quantum
mechanical ground state of motion [
7
–
14
]. In a pioneering
experiment by O’Connell,
et al.
[
11
], a piezoelectric nano-
mechanical resonator has been cryogenically cooled (
T
b
25 mK
) to its vibrational ground state and strongly coupled
to a superconducting circuit qubit allowing for quantum
state preparation and readout of the mechanics. An
alternate line of research has been pursued in circuit and
cavity optomechanics [
15
], where the position of a me-
chanical oscillator is coupled to the frequency of a high-
Q
electromagnetic resonance allowing for backaction
cooling [
16
,
17
] and continuous position readout of the
oscillator. Such optomechanical resonators have long
been pursued as quantum-limited sensors of weak classical
forces [
9
,
15
,
18
–
20
], with more recent studies exploring
optomechanical systems as quantum optical memories
and amplifiers [
21
–
24
], quantum nonlinear dynamical ele-
ments [
25
], and quantum interfaces in hybrid quantum
systems [
26
–
29
].
Despite the major advances in circuit and cavity opto-
mechanical systems made in the last few years, all experi-
ments to date involving the cooling of mesoscopic
mechanical oscillators have relied on careful measurement
and calibration of the motion-induced scattering of light to
obtain the average phonon occupancy of the oscillator,
h
n
i
.
Approach towards the quantum ground state in such ex-
periments is manifest only as a weaker measured signal,
with no evident demarcation between the classical and
quantum regimes of the oscillator. A crucial aspect of
zero-point fluctuations (zpfs) of the quantum ground state
is that they cannot supply energy, but can only contribute to
processes where energy is absorbed by the mechanics. This
is different from classical noise, and techniques that at-
tempt to measure zero-point motion without being sensi-
tive to this aspect (i.e., standard continuous linear position
detection) can always be interpreted classically and de-
scribed by some effective temperature.
A more direct method of thermometry and characteriza-
tion of the quantized nature of a mechanical oscillator, one
particularly suited to small
h
n
i
and utilized in the above-
mentioned trapped atom experiments [
1
–
3
], is referred to
as motional sideband spectroscopy. This method relies on
the fundamental asymmetry in the quantum processes of
phonon absorption from (proportional to
h
n
i
) and emission
into (proportional to
h
n
iþ
1
) the mechanical oscillator. In
the case of atomic systems, this asymmetry can be mea-
sured in the motionally generated Stokes and anti-Stokes
sidebands in either the fluorescence or absorption spectrum
of the atom. The ratio of the Stokes to anti-Stokes sideband
amplitudes [
ðh
n
iþ
1
Þ
=
h
n
i
] deviates significantly from
unity as the quantum ground state is reached (
h
n
i!
0
),
and provides a self-calibrated reference for the phonon
occupancy. In the present experiment, we cool a nano-
mechanical resonator to near its quantum ground state,
and measure the asymmetry in the motional sidebands
utilizing a form of resolved sideband spectroscopy based
upon the filtering properties of a high-
Q
optical cavity with
linewidth narrower than the mechanical frequency.
The cavity optomechanical system studied in this Letter
consists of a patterned silicon nanobeam which forms an
optomechanical crystal (OMC) [
30
] capable of localizing
both optical and acoustic waves (see Fig.
1
). The cavity is
designed to have two optical resonances, one for cooling
and one for readout of mechanical motion. The cooling
mode is chosen as the fundamental mode of the patterned
nanobeam cavity, with a frequency
!
c
=
2
¼
205
:
3 THz
and a corresponding free-space wavelength of
c
¼
1460 nm
. The readout mode is the second-order mode of
the cavity with
!
r
=
2
¼
194
:
1 THz
(
r
¼
1545 nm
). An
in-plane mechanical breathing mode at
!
m
=
2
¼
3
:
99 GHz
, confined at the center of the nanobeam due to
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acoustic Bragg reflection, couples via radiation pressure to
both optical resonances.
An illustration of the experimental apparatus used to cool
and measure the OMC nanomechanical oscillator is shown
in Fig.
2
. In order to precool the oscillator, the silicon sample
is mounted inside a Helium flow cryostat. For a sample
mount temperature of 6.3 K, the thermal bath temperature of
the mechanical mode is measured to be 18 K (thermal
phonon occupation of
n
b
¼
94
phonons) through optical
measurements described below. At this temperature the
breathing mode damping rate to the thermal bath is found
to be
i
=
2
¼
43 kHz
. The optical resonances of the OMC
cavity are measured to have total damping rates of
c
=
2
¼
390 MHz
and
r
=
2
¼
1
:
0 GHz
for the cooling and read-
out modes, respectively. An optical fiber taper is used to
evanescently couple light to and from the OMC cavity.
Utilizing piezoelectric stages, the taper is positioned to
the side of the nanobeam cavity and placed in contact
with the surface of the silicon microchip surrounding the
suspended nanobeam. In this scheme, the fiber taper runs
approximately parallel to the nanobeam, and can be rigidly
mounted at a prescribed nanoscale gap from the nanobeam.
For the taper-to-nanobeam gap used here (
&
200 nm
), the
coupling rate to the fiber taper waveguide is approximately
e;c
=
2
¼
46 MHz
for the cooling mode and
e;r
=
2
¼
300 MHz
for the readout mode.
A Hamiltonian describing the coupled OMC cavity sys-
tem is given by
^
H
¼
@
ð
!
r
þ
g
r
^
x=x
zpf
Þ
^
a
y
^
a
þ
@
ð
!
c
þ
g
c
^
x=x
zpf
Þ
^
c
y
^
c
þ
@
!
m
^
b
y
^
b
, where
^
c
(
^
c
y
) and
^
a
(
^
a
y
) are the
annihilation (creation) operators for photons in the cooling
and readout modes, respectively, and
^
x
x
zpf
ð
^
b
y
þ
^
b
Þ
is
the displacement operator of the breathing mode with
^
b
y
(
^
b
) the phonon creation (annihilation) operator.
x
zpf
, the
mode’s zero-point fluctuation amplitude, is estimated to be
2.7 fm from FEM simulations. The zero-point optome-
chanical coupling rates are determined from measurements
of the optically-induced damping of the mechanical mode
[
13
]tobe
g
c
=
2
¼
960 kHz
and
g
r
=
2
¼
430 kHz
for
the cooling and readout modes, respectively.
As alluded to above, resolved sideband cooling in opto-
mechanical cavities follows physics which is formally
similar to the Raman processes used to cool ions to their
motional ground state [
1
]. A cooling laser, with frequency
!
l
¼
!
c
!
m
, is tuned a mechanical frequency below
that of the cooling cavity resonance of the OMC, giving
rise to an intracavity photon population
n
c
. Motion of the
mechanical oscillator causes scattering of the intracavity
cooling beam laser light into Stokes and anti-Stokes side-
bands at
!
c
2
!
m
and
!
c
, respectively. Since the anti-
Stokes sideband is resonant with the cavity at
!
c
, and
c
<!
m
, the anti-Stokes optical up-conversion
process is greatly enhanced relative to the Stokes
down-conversion process, leading to cooling of the me-
chanical mode. Assuming a deeply resolved sideband
system (
c
=!
m
1
), the backaction cooled mechanical
mode occupancy is approximately given by
h
n
i
c
¼
i
n
b
=
ð
i
þ
c
Þ
[
16
,
17
].
Optical scattering of the intracavity light field can also
be used to read out the motion of the coupled mechanical
FIG. 2 (color online). Schematic of the experimental set-up.
Two narrowband lasers (linewidth
300 kHz
) are used to inde-
pendently cool and readout the motion of the breathing mechani-
cal mode of the OMC cavity. The 1500 nm (readout) and
1400 nm (cooling) laser beams are passed through variable
optical attenuators (VOAs) to set the laser power, and combined
at a wavelength multiplexer (
-MUX) before being sent into the
cryostat through an optical fiber. Transmission of the 1500 nm
readout beam through the OMC cavity, collected at the output
end of the optical fiber, is filtered from the 1400 nm cooling
beam light via a bandpass filter, preamplified by an Erbium-
doped fiber amplifier (EDFA), and detected on a high-speed
photodetector (PD2) connected to a real-time spectrum analyzer
(RSA). An optical wave meter (
-meter) is used to monitor both
the cooling and readout laser frequencies. The optical reflection
from the cavity is used to perform EIT-like spectroscopy [
22
]on
both the readout and cooling cavity modes. Other components
are: amplitude-modulation (a-m) and phase-modulation (
-m)
electro-optic modulators, fiber polarization controller (FPC),
swept frequency radio-frequency signal generator (rf-sg), lock-
in amplifier (lock-in), and optical switches (SW).
FIG. 1 (color online). (a) A scanning electron micrograph of
the silicon nanobeam optomechanical cavity. Finite-element
method (FEM) numerical simulations of the electric field am-
plitude of the (b) first- and (c) second-order optical modes of the
cavity which are used for cooling and probing the mechanical
motion, respectively. (d) FEM numerical simulation showing the
displacement amplitude of the coupled breathing mechanical
mode.
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oscillator. For a quantum harmonic oscillator, the noise
power spectral density (PSD) of the oscillator’s position is
equal to [
20
],
S
xx
ð
!
Þ
=x
2
zpf
¼
h
n
i
ð
!
m
þ
!
Þ
2
þð
=
2
Þ
2
þ
ðh
n
iþ
1
Þ
ð
!
m
!
Þ
2
þð
=
2
Þ
2
;
(1)
where
is the total mechanical damping rate. The asym-
metric zero-point motion contribution to
S
xx
ð
!
Þ
[illus-
trated in Fig.
3(a)
] arises from the noncommutivity of
position and momentum operators in quantum mechanics.
This absorption-emission asymmetry has no classical ana-
logue; of course, at high phonon occupation numbers
where
h
n
ih
n
iþ
1
, the classically symmetric spectral
density is recovered. Since the optical cavity frequency is
linearly coupled to the position of the mechanical oscilla-
tor, the displacement noise spectrum is imprinted on the
photons leaving the cavity and can be measured optically.
Specifically, consider a readout laser with frequency
!
lr
and detuning
!
r
!
lr
from the readout cavity mode.
The optical power spectrum of the motional sidebands
of the transmitted readout beam leaving the cavity is given
by [
16
],
S
ð
!
þ
!
lr
Þ¼
e;r
2
r
A
ð
r
Þ
h
n
i
ð
!
m
!
Þ
2
þð
=
2
Þ
2
þ
e;r
2
r
A
ð
r
Þ
þ
ðh
n
iþ
1
Þ
ð
!
m
þ
!
Þ
2
þð
=
2
Þ
2
:
(2)
Here
A
ð
r
Þ
þ
and
A
ð
r
Þ
are the detuning-dependent anti-Stokes
and Stokes motional scattering rates, respectively, of the
readout laser, given by
A
ð
r
Þ
¼
g
2
r
r
n
r
=
½ð
!
m
Þ
2
þ
ð
r
=
2
Þ
2
.
As illustrated in Fig.
3(b)
and
3(c)
, the optical readout
cavity can be used to selectively filter the positive or
negative frequency components of
S
ð
!
Þ
. For a detuning
¼
!
m
for the readout laser,
A
ð
r
Þ
þ
A
ð
r
Þ
, resulting in a
Lorentzian signal with area
I
proportional to
h
n
iþ
1
.
Conversely, a detuning of
¼
!
m
results in
A
ð
r
Þ
A
ð
r
Þ
þ
,
producing a signal of area
I
þ
proportional to
h
n
i
.
Comparison of the area under the Lorentzian part of the
measured photocurrent PSD of the transmitted readout
laser for detunings
¼
!
m
, can then be used to infer
the mechanical mode occupancy,
I
=I
þ
1
¼
1
h
n
i
:
(3)
This simple argument neglects the backaction of the
readout beam on the mechanical oscillator. In particular,
the mechanical damping rate becomes detuning dependent,
with
ð
i
þ
c
Þð
1
C
r
Þ
for
¼
!
m
. Here
C
r
j
A
ð
r
Þ
þ
A
ð
r
Þ
j
=
ð
i
þ
c
Þ
is the effective cooperativity of the
readout beam in the presence of the strong cooling beam,
and can be found from the measured spectra by the rela-
tion,
C
r
¼ð
þ
Þ
=
ð
þ
þ
Þ
. The backaction of the
readout beam also results in a corresponding change in the
phonon occupancy, given by
h
n
i
¼h
n
i
c
=
ð
1
C
r
Þ
for
¼
!
m
. Here
h
n
i
c
is the mechanical mode occupancy
due to backaction from the cooling beam only. Adding in a
correction for the readout laser backaction, one finds the
following relation between the measured motional side-
bands and the phonon occupancy of the cooled mechanical
oscillator,
0
I
=I
þ
1
þ
C
r
1
1
C
r
¼
1
h
n
i
c
;
(4)
where for
C
r
1
, we recover the standard relation given
in Eq. (
3
).
Figure
4
summarizes the measurement results of the
calibrated mechanical mode thermometry and motional
sideband asymmetry for the silicon OMC cavity. These
measurements are performed with the cooling laser locked
a mechanical frequency to the red of the fundamental mode
of the OMC cavity, and the cooling laser power swept from
n
c
1
to 330 (maximum input power of
250
W
). A
much weaker readout laser (
C
r
1
) is used to both esti-
mate the mechanical mode phonon occupancy and to com-
pare the motional sideband amplitudes. Locking of the
FIG. 3 (color online). (a) Displacement noise PSD,
S
xx
,ofa
quantum simple harmonic oscillator, plotted against
!
for
clarity. (b) Scheme for measurement of the down-converted
(Stokes) motional sideband. Here the readout laser (vertical
arrow; frequency
!
lr
) is detuned a mechanical frequency above
that of the readout cavity resonance (broad solid curve).
(c) Corresponding scheme for measurement of the up-converted
(anti-Stokes) motional sideband. The linewidth of the readout
cavity (
r
) and the mechanical resonance (
) are indicated.
Insets to (b) and (c) show a zoomedout spectra indicating the
relative frequency of the cooling cavity mode and cooling laser.
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cooling and readout lasers utilizes a high resolution wave
meter (10 MHz resolution) to set the absolute laser fre-
quency and a weak probe beam to determine the laser-
cavity detuning. Here the weak probe beam is generated
from the cooling or readout laser via electro-optic modu-
lation, similar to the electromagnetically induced trans-
parency (EIT) spectroscopy described in Ref. [
22
]. The
laser-cavity lock for both cooling and readout lasers is
performed every few minutes, multiplexed in time between
measurements of the phonon occupancy. With the readout
laser set to a detuning
¼
!
m
from the readout cavity, a
Lorentzian spectrum with linewidth
þ
and integrated area
I
þ
is measured in the readout laser photocurrent PSD, from
which a mode occupancy of
h
n
i
þ
is inferred from a careful
calibration of the optomechanical cavity and photodetec-
tion system parameters [
13
]. Similarly, by placing the
readout laser at
¼
!
m
we obtain spectra with line-
width
and integrated area
I
/h
n
i
þ
1
, from which
we estimate
h
n
i
.
Figure
4(a)
plots the readout cooperativity
C
r
, calculated
from the measured
, versus the mechanical damping
rate
¼ð
þ
þ
Þ
=
2
. The ratios
=
þ
and
h
n
i
þ
=
h
n
i
are plotted in Fig.
4(b)
. From
h
n
i
, the laser
cooled phonon occupation number,
h
n
i
c
, is calculated and
plotted in Fig.
4(c)
versus
. As expected,
h
n
i
c
drops
approximately linearly with
, reaching a minimum value
of approximately
2
:
6
0
:
2
phonons. Further cooling be-
low a single phonon has been achieved in similar devices
[
13
]; however, in this case cooling is limited by the avail-
able power of the 1400 nm cooling laser. Also evident in
Fig.
4(c)
, at the higher cooling powers, is an increased
scatter and deviation of
h
n
i
c
from the ideal cooling curve
(dashed curve). This can be attributed to optical absorption
in the silicon nanobeam [
13
], which in this case produces a
power-dependent variation in
n
b
and
i
due to both the
readout and cooling laser beams.
In Fig.
4(d)
, the measured values of the expression
0
are
plotted versus the calibrated value of
h
n
i
c
. Also plotted are
the classical and quantum values of this expression, 0 and
1
=
h
n
i
c
, respectively. A clear divergence from the classical
result of
0
¼
0
is apparent, agreeing with the deviation
due to zero-point fluctuations of the mechanical oscillator.
This deviation is directly apparent in the measured
spectra, shown for
h
n
i
c
¼
85
, 6.3, and 3.2 phonons in
Fig.
4(e)
–
4(g)
, with the shaded region corresponding to
the noise power contribution due to zero-point motion. At
FIG. 4 (color). (a) Plot of the cooperativity of the readout beam as a function of damped mechanical linewidth. (b) Plot of the
measured ratios
=
þ
(blue
) and
h
n
i
þ
=
h
n
i
(pink
). (c) Plot of the mechanical mode phonon occupancy,
h
n
i
c
, as a function of
the optically damped mechanical linewidth,
. The dashed line is the predicted phonon number
i
n
b
=
from an ideal backaction
cooling model. Vertical error bars in (b) and (c) indicate uncertainty in the calibrated phonon occupancy due to uncertainty in the
system parameters and a 95% confidence interval on the Lorentzian fits to spectra. (d) Plot of the asymmetry (
0
) in the measured
Stokes and anti-Stokes sidebands of the readout laser for each calibrated measurement of
h
n
i
c
. The horizontal error bars arise from a
2% uncertainty in the transmitted readout laser beam power between detunings
¼
!
m
, and a 95% confidence interval in the
Lorentzian fits to the measured spectra. The vertical error bars in
h
n
i
c
are the same as in (c). The classical (blue curve) and quantum
mechanical (pink curve) relations for the sideband asymmetry are also plotted. (e)–(g) Plot of the measured Stokes (red curve) and
anti-Stokes (blue curve) readout beam spectra for (from top to bottom)
h
n
i
c
¼
85
, 6.3, and 3.2 phonons. For clarity, we have divided
out the readout backaction from each spectra by multiplying the measured spectra at detunings
¼
!
m
by
. Additionally, we
have plotted the horizontal axis in units of
, and rescaled the vertical axis for different
h
n
i
c
to keep the areas directly comparable. The
difference in the Stokes and anti-Stokes spectra, which arises due to the quantum zero-point fluctuation of the mechanical system, is
shown as a shaded region.
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the largest powers, we measure asymmetry in the motional
sideband amplitudes of
40%
in agreement with the in-
ferred
h
n
i
c
¼
2
:
6
phonons from calibrated thermometry.
While the quantum nature of a mechanical resonator will
come as little surprise to most physicists, its observation
through the zero-point motion is a significant step towards
observing and controlling the quantum dynamics of meso-
scopic mechanical systems. By demonstrating the funda-
mentally quantum behavior of an engineered mechanical
nanostructure, we have shown that realizable optomechan-
ical systems have the sensitivity and environmental isola-
tion required for such quantum mechanical investigations.
The authors would like to thank Aash Clerk, Markus
Aspelmeyer, and Simon Gro
̈
blacher for their valuable input
at various stages in this experiment. This work was sup-
ported by the DARPA/MTO ORCHID program through a
grant from AFOSR, and the Kavli Nanoscience Institute at
Caltech. J C and A S N gratefully acknowledge support
from NSERC .
*
opainter@caltech.edu;
http://copilot.caltech.edu
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