of 8
Pulsed Excitation Dynamics of an Optomechanical Crystal Resonator near
Its Quantum Ground State of Motion
Seán M. Meenehan,
1,2
Justin D. Cohen,
1,2
Gregory S. MacCabe,
1,2
Francesco Marsili,
3
Matthew D. Shaw,
3
and Oskar Painter
1,2
,*
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology,
Pasadena, California 91125, USA
3
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA
(Received 6 April 2015; published 6 October 2015)
Using pulsed optical excitation and read-out along with single-phonon-counting techniques, we measure
the transient backaction, heating, and damping dynamics of a nanoscale silicon optomechanical crystal
cavity mounted in a dilution refrigerator at a base temperature of
T
f
11
mK. In addition to observing a
slow (approximately 740-ns) turn-on time for the optical-absorption-induced hot-phonon bath, we measure
for the 5.6-GHz
breathing
acoustic mode of the cavity an initial phonon occupancy as low as
h
n
0
.
021

0
.
007
(mode temperature
T
min
70
mK) and an intrinsic mechanical decay rate of
γ
0
¼
328

14
Hz (
Q
m
1
.
7
×
10
7
). These measurements demonstrate the feasibility of using short pulsed
measurements for a variety of quantum optomechanical applications despite the presence of steady-state
optical heating.
DOI:
10.1103/PhysRevX.5.041002
Subject Areas: Optics, Quantum Physics
The recent cooling of nanomechanical resonators to their
motional quantum ground state
[1
3]
opens the possibility
of utilizing engineered mechanical systems strongly coupled
to optical or microwave fields for a variety of quantum
metrology and information-processing applications
[4]
,
among them the preparation of highly nonclassical mechani-
cal states
[5
7]
and coherent frequency conversion between
microwave and optical signals
[8
12]
. A particularly inter-
esting device architecture for realizing large radiation
pressure coupling between light and mechanics is the
thin-film optomechanical crystal (OMC)
[13,14]
,inwhich
optical and acoustic waves can be guided and colocalized via
patterning of the surface layer of a microchip. Based largely
upon the OMC concept, new ideas for phononic quantum
networks
[15]
and optomechanical metamaterials
[16]
have
been proposed, in which arrays of cavity-optomechanical
resonatorsarecoupled together via optical oracoustic degrees
of freedom, and in which laser light is used to parametrically
control the emergent network or material properties.
For many of the applications mentioned above, it is
essential to obtain both a low mechanical occupation and a
large ratio between the optomechanical interaction rate and
the thermal decoherence rate of the mechanical oscillator.
Operation at cryogenic temperatures is desirable as it offers
a simple route to obtaining low thermal occupations and
long mechanical coherence times. Recent measurements at
millikelvin (mK) bath temperatures of an OMC resonator
formed from single-crystal silicon
[13,17]
, however, have
shown substantial mechanical mode heating and damping
due to weak sub-band-gap optical absorption
[18]
.
Although optical
Q
factors in excess of
10
6
are realized
in these highly optimized structures
[17]
, the large impact
of even very weak optical absorption can be attributed to a
combination of the relatively large energy per photon
and the sharp drop in thermal (phonon) conductance with
temperature in the low-temperature limit. Further compli-
cations arise from the seemingly contradictory require-
ments of isolating the mechanical resonator from its
environment to obtain a high mechanical
Q
factor and
that of providing large thermal anchoring to a low-temper-
ature bath for cooling of the mechanical resonator.
In this work, we utilize pulsed optical excitation and
single-phonon counting
[19]
to study the transient dynam-
ics of optical backaction, heating, and damping of the
5.6-GHz mechanical mode of a silicon optomechanical
crystal resonator at mK bath temperatures. Phonon count-
ing, realized by photon counting of the optically filtered
motional sidebands of the reflected optical excitation pulse,
yields simultaneously a high time resolution (approxi-
mately 25 ns) and mechanical mode occupancy sensitivity
(
<
10
2
). Measurement of both Stokes and anti-Stokes
sidebands also yields an absolute calibration of the
*
opainter@caltech.edu
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 3.0 License
. Further distri-
bution of this work must maintain attribution to the author(s) and
the published article
s title, journal citation, and DOI.
PHYSICAL REVIEW X
5,
041002 (2015)
2160-3308
=
15
=
5(4)
=
041002(8)
041002-1
Published by the American Physical Society
occupancy of the resonator mode in terms of mechanical
vacuum noise
[20
22]
, allowing accurate measurement
of ultralow initial phonon-mode occupancies of
h
n
0
.
021

0
.
007
and mechanical decay times as long as
τ
¼
475

21
μ
s. Additionally, we observe a slow (approx-
imately 750-ns) turn-on time for the optical-absorption-
induced phonon bath that both heats and damps the
mechanical resonator mode. Taken together, these mea-
surements demonstrate the feasibility of using short pulsed
measurements for quantum optical state engineering of the
mechanics in silicon optomechanical crystals, despite the
presence of large steady-state optical heating.
The device studied in this work consists of a patterned
nanobeam, formed from the top silicon device layer of a
silicon-on-insulator wafer. The etched hole pattern in the
silicon nanobeam forms an optomechanical crystal, in
which photonic and phononic band gaps at the ends of
the beam colocalize optical and acoustic resonances with
frequencies of
ω
c
=
2
π
196
THz (free-space wavelength
λ
c
1530
nm) and
ω
m
=
2
π
5
.
6
GHz, respectively. A
scanning electron micrograph (SEM) of the device is
shown in Fig.
1(a)
, and a finite-element-method simulation
of the acoustic resonance is displayed in Fig.
1(b)
.
Coupling to the optical resonance is accomplished via
an end-fire coupling scheme, using a lensed optical fiber
inside of a dilution refrigerator to couple to the central
waveguide shown in the SEM, as described in Ref.
[18]
.
Surrounding the cavity is a 2D cross pattern
[14]
that
possesses a complete acoustic band gap in the 5
6-GHz
range, providing an additional acoustic shield for the
mechanical resonator mode while allowing phonons above
and below the phononic band gap to carry heat from the
nanobeam structure.
The experimental setup is shown schematically in
Fig.
1(a)
. Modulation of a laser-probe beam using an
electro-optic modulator (EOM) generates optical pulses
with frequency and duty cycle controlled by a variable-
delay electrical pulse generator. The red arrows indicate a
coherent pump at frequency
ω
l
, which is red detuned from
the optical resonance frequency
ω
c
by
Δ
ω
c
ω
l
¼
ω
m
.
In this case, absorption of a single phonon from the
nanomechanical resonator results in up-conversion of a
pump photon to the anti-Stokes sideband at
ω
c
, represented
by the black arrows. The cavity reflection is filtered to reject
thepumpfrequencyandsubsequentlydirectedtoa
high-efficiency single-photon detector (SPD) and a time-
correlated single-photon-counting system synchronized to
the pulse generator. This measurement repeats each pulse
period, building up a histogram with respect to photon arrival
time relative to the synchronization pulse during a certain
integration time. As the generation of an anti-Stokes photon
is one-to-one correlated with the annihilation of a phonon in
the mechanical resonator, the counting of sideband photons
is equivalent to the detection of phonon-annihilation events,
such that photon counting in this case is equivalent to
phonon counting. In particular, the anti-Stokes photon-count
(a)
(d)
(b)
(c)
FIG. 1. (a) Pulsed pump light at frequency
ω
l
(red arrows) is directed to an OMC cavity inside a dilution fridge via an optical
circulator. The cavity reflection is then filtered at the cavity frequency
ω
c
(black arrows) and directed to a SPD. (b) SEM image of the
silicon OMC cavity studied in this work. (c) Finite-element-method simulation of the localized acoustic resonance at 5.6 GHz of the
OMC cavity. Deformation of the beam structure is exaggerated to highlight the mechanical motion, with color indicating the regions of
high (red) and low (blue) displacement magnitude. The inset shows the three processes affecting the mode occupancy: (i) detuned pump
light at
ω
l
(red arrow) exchanges energy with the acoustic mode at rate
γ
OM
, generating scattered photons at
ω
c
(black arrow) in the
process; (ii) pump light drives excitation of electronic defect states in the silicon device layer, subsequently exciting a hot-phonon bath
that heats the localized acoustic resonance at rate
γ
p
; and (iii) phonons escape the cavity volume via the acoustic shield at intrinsic decay
rate
γ
0
, coupling the localized resonance to the fridge bath. (d) Noise-equivalent phonon number
n
NEP
(gray squares) and phonon
occupancy
h
n
i
(red circles) for red-detuned (
Δ
¼
ω
m
) CW pumping versus intracavity photon number
n
c
. The dashed red line indicates
the expected
n
NEP
contribution from SPD dark counts. The vertical dashed line at
n
c
45
ð
4
.
5
×
10
5
Þ
indicates the photon number
during the on state (off state) of the pulse. Solid green and purple lines show fits to the CW heating model for fridge bath temperatures of
T
f
¼
70
and 10 mK, respectively.
SEÁN M. MEENEHAN
et al.
PHYS. REV. X
5,
041002 (2015)
041002-2
rate is equal to
γ
OM
h
n
i
, where
γ
OM
4
g
2
0
n
c
=
κ
is the
sideband-scattering rate. (
g
0
is the optomechanical coupling
rate,
n
c
is the intracavity photon number, and
κ
is the optical
decay rate.) The photon-count rate in each time bin then
portrays the time evolution of
h
n
t
Þ
during the pulse on
state. Similarly, a coherent pump blue detuned from the
optical cavity (
Δ
¼
ω
m
) results in a one-to-one correspon-
dence between the generation of Stokes sideband photons
and phonon creation, with a total sideband-count rate equal
to
γ
OM
ðh
n
1
Þ
. All measurements presented herein
are performed at a fridge-mixing-chamber base temperature
of
T
f
¼
11
mK. Further details about device fabrication
and the experimental setup, and further explanation of
the phonon-counting technique, can be found in
Refs.
[17,19,23]
.
The signal-to-noise ratio of this phonon-counting
method is determined by the sideband-scattering rate
γ
OM
, the total system detection efficiency (
η
), the dark-
count rate of the SPD (
Γ
dark
), and the residual transmission
of the filters at the pump frequency relative to the peak
transmission (
A
). A useful parametrization of the sensitivity
to low
h
n
i
is the noise-equivalent phonon occupancy
n
NEP
,
defined as
[19]
n
NEP
¼
Γ
dark
ηγ
OM
þ
A

κω
m
2
κ
e
g
0

2
;
ð
1
Þ
where
κ
e
is the optical decay rate into the detection channel.
For the device under test, we have
κ
=
2
π
¼
443
MHz,
κ
e
=
2
π
¼
221
MHz, and
g
0
=
2
π
¼
710
kHz. A typical mea-
sured
n
NEP
for our setup, taken using a comparable device
at room temperature, is shown in Fig.
1(d)
as gray squares.
Here, to obtain the lowest
n
NEP
, optical prefiltering is used
to remove phase noise around the laser line and broadband
spontaneous emission from the optical probe beam, details
of which are given in Ref.
[23]
. For sufficiently high probe
power (
n
c
>
40
),
n
NEP
falls below
10
2
, enabling sensitive
detection of the mechanical resonator deep in its quantum
ground state. However, at subkelvin temperatures, optical
absorption heating produces a steady state
h
n
i
>
1
for
n
c
>
0
.
01
[red circles in Fig.
1(d)
] during continuous-wave
(CW) optical excitation. In order to maintain the OMC in
the mechanical ground state, the duty cycle of the pulse
train must be kept sufficiently low, and the modulation
depth sufficiently high, such that the mechanical mode
thermalizes to the dilution-refrigerator ambient bath
between successive pulses.
The CW behavior fits well to a thermal model consisting
of the three processes illustrated in Fig.
1(c)
: (i) the
radiation pressure coupling at rate
γ
OM
between the
mechanical mode and the effective zero-temperature probe
laser, (ii) coupling to an optical-absorption-induced hot-
phonon bath above the phononic shield band gap, and
(iii) coupling to the ambient fridge bath at an intrinsic
rate
γ
0
through the acoustic radiation shield. At the low
intracavity photon numbers of these measurements, we
believe the optical absorption heating is a result of
excitation of electronic defect states at the silicon surfaces
[24,25]
and subsequent phonon-assisted relaxation of these
states. As detailed in Ref.
[18]
, the resulting local hot-
phonon bath occupancy (
n
p
) is found to scale as
n
1
=
4
c
in
steady state, consistent with linear optical absorption and a
cubic drop in the thermal conductance with temperature
[26]
. The corresponding coupling rate of the mechanical
resonance to the high-frequency hot-phonon bath (
γ
p
)is
measured to scale as
T
p
exp
ð
ω
c
=k
B
T
p
Þ
for low bath
temperature (
T
p
<
4
K), corresponding to inelastic phonon
scattering with a quasiequilibrium hot-phonon bath above a
cutoff phonon frequency of
ω
c
=
2
π
35
GHz. As shown in
Fig.
1(d)
, extrapolation of this steady-state model for a
fridge base temperature of
T
f
¼
10
mK reveals a relevant
pulse-off-state regime of
n
c
<
10
4
in which absorption
heating effects should be negligible.
Figure
2(a)
shows the measured sideband photon-count
rate versus time with pulsed optical excitation, for both red-
(
Δ
¼
ω
m
) and blue-detuned (
Δ
¼
ω
m
) pumping. The on-
state pulse amplitude in these measurements is
n
c;
on
¼
45
,
corresponding to an optomechanical coupling rate of
γ
OM
=
2
π
¼
205
kHz. For the achievable pulse-extinction
ratio of 60
70 dB, this results in a residual off-state photon
number of
n
c;
off
<
4
.
5
×
10
5
. Vertical dashed lines indi-
cate the time bins corresponding to the start and stop of the
pulse, determined from observing the rising and falling
edges of the pulse when bypassing the cavity. The start and
stop times are taken to be the time bins for which the pulse
reaches 90% of its maximum value. The variable
t
refers to
the time relative to the start of the optical pulse, which
occurs around approximately
1
μ
s after the synchronization
signal generated by the pulse generator due to the time
delay of propagation through the fridge. The pulse period in
these measurements is fixed at
T
per
¼
5
ms. The count
rates immediately after switching off the pulse are elevated
relative to the dark counts preceding the pulse due to the
finite turn-off time of the electro-optic switches used to
generate the high-extinction optical pulses, as discussed
further in Ref.
[23]
. This turn-off time is much smaller than
the pulse periods used in this work and provides a
negligible correction to the heating and cooling dynamics
of the mechanical resonator.
Throughout the pulse, a pronounced asymmetry is
observed between count rates for red-detuned versus
blue-detuned pumping, which can be quantified by the
asymmetry parameter
ξ
¼
Γ
=
Γ
þ
1
, where
Γ

is the
sideband photon-count rate for a pump detuning
Δ
¼
ω
m
.
This asymmetry, shown versus
t
in Fig.
2(b)
, initially
decreases with time before leveling off and beginning to
increase for sufficiently long pulse times. The increase at
later times can be ascribed to the effect of optomechanical
backaction, which results in cooling or heating of the
mechanical resonator for red- or blue-detuned pumps,
PULSED EXCITATION DYNAMICS OF AN
...
PHYS. REV. X
5,
041002 (2015)
041002-3
respectively
[27]
. However, for
t
γ
1
OM
750
ns, the
effects of backaction are negligible and the phonon occu-
pancy
h
n
i
may be assumed equal for both pump detunings.
In this case, the asymmetry is simply related to the
occupancy by
ξ
¼h
n
i
1
and arises from the fundamental
asymmetry between phonon-absorption (
Γ
þ
h
n
i
)and
emission (
Γ
h
n
1
) processes
[20,28]
.Whilethis
fundamental motional sideband asymmetry has previously
been measured in a variety of opto- and electromechanical
systems using linear (e.g., heterodyne) detection schemes
[20
22,29]
, detailed theoretical treatments have demon-
strated that the observed asymmetry is naturally traced back
to the zero-point fluctuations of either the optical field or the
mechanical resonator, depending on the particular detection
scheme utilized
[22,30]
. In contrast to linear detection, the
use of phonon-counting techniques in this work allows the
observed asymmetry to be directly and unambiguously
attributed to quantum fluctuations of the mechanical oscil-
lator. A third potential source of sideband asymmetry is
technical laser noise, such as phase noise, which can result
in substantial sideband asymmetry even for large phonon
occupations. A quantitative analysis of such systematic
errors in our system is provided in Ref.
[23]
,wherewe
conclude that technical laser noise has a negligible impact on
the sideband asymmetry observed here.
To illustrate the two regimes of asymmetry, theoretical
plots of the two contributions to
ξ
ð
t
Þ
are shown in Fig.
2(b)
.
The dashed green line shows the expected
ξ
ð
t
Þ
when
backaction is neglected. In this case, the mechanical occu-
pancy will be equal for both detunings and the decrease in
ξ
ð
t
Þ¼h
n
i
1
ð
t
Þ
arises solely from the time-dependent heat-
ing of the mechanical resonator due to optical absorption,
calculated using the heating model described below. On the
other hand, the dash-dotted orange line shows
ξ
ð
t
Þ
when the
effects of backaction are included, assuming no optical
heating and a high bath temperature such that the total
mechanical bath occupancy is a constant
n
b
1
. In this
case,
ξ
ð
0
Þ¼
n
1
b
0
, and asymmetry arises at later times
due solely to backaction effects. From the asymmetry
measured in the first 25-ns time bin, we extract a minimum
phonon occupancy of
h
n
i
min
¼
0
.
021

0
.
007
(minimum
mode temperature of
T
min
70
mK). Extrapolation to zero
time yields a mode temperature closer to 40 mK but still
above the 11 mK of the mixing plate of the fridge, likely
indicating a higher local chip temperature. This measured
occupancy is lower than previous results in both cavity-
optomechanical
[3]
and electromechanical
[2]
systems by
more than an order of magnitude.
While the sideband asymmetry does not accurately reflect
the phonon occupancy for long times due to the effects of
backaction, the initial sideband asymmetry provides a
convenient method of calibrating the occupancy at all times
within the duration of the pulse. Once the asymmetry is used
to determine
h
n
i
in the initial time bin, we may immediately
determine the conversion factor between phonon occupancy
and sideband photon-count rate for either detuning. As this
factor is constant during the on state of the pulse, this allows
us to convert measured count rates to phonon occupancies
throughout the pulse without a direct calibration of either the
overall detection efficiency or the sideband-scattering rate.
The resulting calibrated occupancy versus
t
is shown in
Fig.
2(c)
. A simple model of the heating dynamics would
suggest an exponential increase toward the steady-state
(a)
(b)
(c)
FIG. 2. (a) Total photon-count rate versus time
t
for a red-
detuned (red circles,
Δ
¼
ω
m
) and blue-detuned (blue squares,
Δ
¼
ω
m
) probe. For both data sets,
n
c;
on
45
and
T
per
¼
5
ms.
Vertical dashed lines indicate the start and stop times of the
pulses. (b) Asymmetry
ξ
versus time within the pulse
t
. The
dashed green line shows the theoretically expected
ξ
ð
t
Þ
when
backaction effects are neglected and a single time-dependent
phonon occupancy is assumed for both detunings. The dash-
dotted orange line shows
ξ
ð
t
Þ
, including the effects of back-
action, neglecting optical heating and assuming
ξ
ð
0
Þ¼
0
. Error
bars show one standard deviation (s.d.) determined from the
measured count rates, assuming Poissonian counting statistics.
(c) Calibrated phonon occupancy
h
n
i
versus
t
for
Δ
¼
ω
m
(red
circles) and
Δ
¼
ω
m
(blue squares). The solid black lines show
fits to a model including a slow exponential turn-on of the hot-
phonon bath. The inset shows detail for
0
<t<
750
ns on a
linear scale for
Δ
¼
ω
m
. Error bars show one s.d. determined
from the measured count rates, assuming Poissonian counting
statistics. For the blue-detuned data, the error bars are smaller
than the data markers.
SEÁN M. MEENEHAN
et al.
PHYS. REV. X
5,
041002 (2015)
041002-4
phonon occupancy with total rate
γ
¼
γ
0
þ
γ
p
þ
γ
OM
.
However, as can be seen in the inset to Fig.
2(c)
,the
curvature of the occupancy curve is positive for short times.
This fact is inconsistent with the simple model described
above and is likely due to a finite equilibration time of the
hot-phonon bath. The data can be fit well for both detunings
by using a simple model that assumes that a small fraction of
the hot-phonon bath turns on effectively instantaneously
(i.e., very fast relative to the length of a single time bin) while
the remainder has a slow exponential increase to its steady-
state value. Thus, the effective rate equation for the phonon
occupancy during the pulse is given by
_
h
n
γ
h
n
γ
p
n
p
ð
1
δ
b
e
γ
S
t
Þ
;
ð
2
Þ
where
δ
b
and
γ
S
are the slow-growing fraction of the hot-
phonon bath and the corresponding turn-on rate, respec-
tively, and the steady-state hot-phonon bath parameters
γ
p
and
n
p
can be determined by fitting the CW
h
n
i
versus
n
c
datashowninFig.
1(c)
. Fitting the pulsed occupancy curve
yields
δ
b
¼
0
.
79

0
.
08
and
γ
S
=
2
π
¼
215

29
kHz, indi-
cating that the heating occurs slowly enough as to be
manageable during coherent quantum operations, as deter-
mined in the remaining discussion.
During the off state of the pulse, optical heating of the
mechanical resonator should be negligible and the phonon
occupancy should cool at the intrinsic damping rate
γ
0
.
Thus, the initial and final occupancies during the pulse
(
h
n
i
i
and
h
n
i
f
, respectively) should obey the relation
h
n
i
i
¼
e
γ
0
T
per
h
n
i
tf
, assuming the pulse period
T
per
is
much larger than the pulse width
T
pulse
. The ratio
h
n
i
i
=
h
n
i
f
, shown in Fig.
3(a)
versus
T
per
, displays the
expected exponential decay with
γ
0
=
2
π
¼
328

14
Hz
and a corresponding intrinsic mechanical quality factor
of
Q
m
1
.
7
×
10
7
. This decay rate corresponds well with
the value inferred from previous CW measurements of
occupancy at mK temperatures
[18]
. From our measure-
ment of
γ
0
and
h
n
i
min
, we estimate a thermal decoherence
time of
τ
th
¼½
γ
0
ð
1
þh
n
i
min
Þ
1
¼
475

21
μ
s.
While the low thermal occupancy and long coherence
time measured here are promising, the utility of cavity-
optomechanical systems for performing coherent quantum
operations between the optical and mechanical degrees
of freedom is ultimately predicated upon the ability to
simultaneously achieve
h
n
i
1
and large cooperativity
C
γ
OM
=
γ
b
, where
γ
b
¼
γ
0
þ
γ
p
is the total coupling rate
between the mechanical resonator and its thermal bath.
In the specific example of optomechanically mediated
coherent transfer of photons between optical and super-
conducting microwave resonators
[8,9]
, the relevant figure
of merit is the effective cooperativity
C
eff
C=n
b
, where
n
b
is the effective bath occupancy defined such that
γ
b
n
b
¼
γ
0
n
0
þ
γ
p
n
p
, which must be much larger than unity in
order to achieve low-noise photon conversion at the single
quantum level
[10,12]
. Using the measured values of
γ
0
and
γ
S
, we can calculate the maximum phonon occupancy
achieved during the pulse
h
n
i
max
for a given
T
pulse
and
T
per
and
Δ
¼
ω
m
[Fig.
4(a)
], as well as
C
eff
as a function of
t
[Fig.
4(b)
]. Although optical heating of our devices
prevents us from reaching the
C
eff
>
1
regime using a
CW pump, due to the slow turn-on of the hot-phonon bath
observed in this work, we find that
h
n
i
<
1
and
C
eff
>
1
can be maintained throughout the entire pulse period for
T
pulse
300
ns at pulse rates approaching 1 MHz.
Furthermore, we find
C
eff
1
during the initial 100 ns
of the pulse and reaches values as large as
C
eff
40
.
However, at these high repetition rates, such a pulsed
wavelength-conversion scheme is still unsuitable for
FIG. 3. Ratio of initial to final phonon number
h
n
i
i
=
h
n
i
f
versus
pulse period
T
per
. The solid red curve shows a fit to exponential
decay. Error bars show one s.d. determined from the measured
count rates, assuming Poissonian counting statistics. The inset
shows
h
n
i
versus
t
for the different values of
T
per
.
(a)
(b)
FIG. 4. (a) Maximum phonon occupancy during a pulse
h
n
i
max
versus pulse period
T
per
and pulse width
T
pulse
. The white contour
delineates the region where the effective cooperativity
C
eff
1
throughout the pulse. (b)
C
eff
versus time
t
within the pulse for
T
per
¼
5
ms and
T
pulse
¼
3
μ
s.
PULSED EXCITATION DYNAMICS OF AN
...
PHYS. REV. X
5,
041002 (2015)
041002-5
high-fidelity transfer of quantum information due to the
necessity of using sufficiently short pulses with bandwidths
larger than that of the wavelength-conversion process. The
requirement of minimal distortion of the signal necessitates
the use of longer pulses (
T
pulse
γ
1
OM
), with correspond-
ingly lower repetition rates of 1 kHz or less.
Another relevant application for optomechanical systems
in the quantum regime is the generation of arbitrary
non-Gaussian mechanical states via phonon addition and
subtraction utilizing appropriate sequences of red- and
blue-detuned optical pulses
[5]
. In particular, recent pro-
posals have explicitly shown how to herald single-phonon
Fock states
[7]
and entangled mechanical states
[6]
via
application of a short blue-detuned pulse to an optome-
chanical system in its motional quantum ground state and
subsequent detection of an emitted Stokes sideband photon.
While the pulsed excitation, single-sideband photon detec-
tion, and low initial phonon occupancy demonstrated in this
work provide the basic tools for realizing such proposals,
the fidelity of the generated states can be significantly
degraded by the effects of optical heating during the pulse.
A detailed analysis of heralded phonon addition and
subtraction is presented in Ref.
[23]
, wherein we compute
the conditional density matrix for the mechanical system
including the effects of time-dependent heating to lowest
order. We may then compute the fidelity of generation of a
single-phonon Fock state according to the proposal of
Ref.
[7]
, assuming an initial thermal state of the mechanics
with average phonon occupancy
h
n
i
i
. Considering a range
of pulse widths
0
<T
pulse
100
ns, sufficient to guarantee
the necessary condition
γ
OM
T
pulse
1
, we find that for
T
per
1
ms, there is no appreciable loss in fidelity due to
any increase in
h
n
i
i
, and fidelity is degraded primarily due to
transient heating during the pulse. Thus, we set
T
per
¼
1
ms
and calculate the fidelity of Fock-state generation as a
function of pulse width, displayed in Fig.
5(a)
.Forshort
(
T
pulse
<
10
ns) pulse widths, the fidelity approaches 98.5%
and remains above 80% for pulse widths approaching
100 ns. However, it is equally important to quantify the
expected time to herald such a state (i.e., the expected time
before detection of a sideband photon), which in this case is
given by
T
Fock
¼
T
per
=
ð
ηγ
OM
T
pulse
h
n
i
i
Þ
,where
η
is the total
detection efficiency of the phonon-counting measurement.
This is shown versus
T
pulse
in Fig.
5(b)
for two cases. The
solid line is calculated using the actual measured detection
efficiency of
η
0
.
3%
, while the dashed line is calculated
using a realistic estimate of the ideal efficiency of our current
measurement setup
η
5
.
5%
. The latter figure is calculated
using the measured efficiency of the SPD (68%), the fiber-to-
waveguide coupling efficiency (68%), and the waveguide-
cavity coupling efficiency (50%), adding an additional 2-dB
insertion loss per filter and 0.5 dB for the insertion loss
of the optical circulator, which correspond to the highest
efficiencies measured for these components in our lab. As
showninFig.
5
, even in this idealized case, the expected
time for the generation of a Fock state is
100
ms, which is
much longer than the measured lifetime of the mechanical
state. Thus, while it is feasible to use our current systems
for the heralded, high-fidelity generation of nonclassical
mechanical states, it is still necessary to reduce heating
effects (and thus allow for shorter pulse periods) in order to
utilize this procedure for useful quantum information-
processing tasks (e.g., scalable entanglement distribution
via the DLCZ protocol
[31,32]
).
The deep ground-state cooling and long mechanical
coherence times demonstrated in this work highlight the
suitability of Si nanobeam OMC cavities for a variety of
quantum engineering tasks. While the prospects for high-rate
(a)(b)
FIG. 5. (a) Fidelity
F
of the generation of a single-phonon Fock state versus pulse width
T
pulse
(pulse period
T
per
¼
1
ms). (b) Average
time required to herald a Fock state
T
Fock
versus pulse width
T
pulse
. The solid line is calculated using the total detection efficiency
measured in this work, while the dashed line uses the estimated ideal detection efficiency.
SEÁN M. MEENEHAN
et al.
PHYS. REV. X
5,
041002 (2015)
041002-6
quantum state transfer and heralded state preparation will be
vastly improved by increasing the thermal conductance of
our structures; for example, by utilizing quasi-2D OMC
devices
[33]
, the pulsed heating measurements performed in
this work demonstrate that such optomechanical applications
in the quantum regime are feasible with our current structure
for reasonable pulse parameters without any further miti-
gation of optical heating effects.
The authors would like to thank V. B. Verma, R. P.
Miriam, and S. W. Nam for their help with the single-photon
detectors used in this work. This work was supported by the
DARPA ORCHID and MESO programs, the Institute for
Quantum Information and Matter, an NSF Physics Frontiers
Center with support of the Gordon and Betty Moore
Foundation, the AFOSR through the
Wiring Quantum
Networks with Mechanical Transducers
MURI program,
and the Kavli Nanoscience Institute at Caltech. Part of the
research was carried out at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the
National Aeronautics and Space Administration.
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