of 9
Supplementary Information for: ”Pulsed excitation dynamics of a nanoscale
optomechanical resonator near its quantum ground-state of motion”
Se ́an 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, CA 91125, USA
2
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA
3
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
(Dated: July 6, 2015)
FABRICATION
The devices are fabricated from a silicon-on-insulator (SOI) wafer (SOITEC, 220 nm device layer, 3
μ
m buried oxide)
using electron beam lithography followed by reactive ion etching (RIE/ICP). The Si device layer is then masked using
ProTEK PSB photoresist to define a mesa region of the chip to which a tapered lensed fiber can access. Outside of the
protected mesa region, the buried oxide is removed with a plasma etch and a trench is formed in the underlying silicon
substrate using tetramethylammonium hydroxide (TMAH). The devices are then released in hydrofluoric acid (49 %
aqueous HF solution) and cleaned in a piranha solution (3-to-1 H
2
SO
4
:H
2
O
2
) before a final hydrogen termination in
diluted HF. In fabrication, arrays of the nominal design are scaled by
±
2 % to account for frequency shifts due to
fabrication imperfections and disorder.
EXPERIMENTAL SETUP
The full measurement setup is shown in Fig. S-1. A fiber-coupled, wavelength tunable external cavity diode laser is
used as the light source, with a small portion (
1%) of the laser output sent to a wavemeter (
λ
-meter) for frequency
stabilization. The remaining laser power is sent through both a C-band optical demultiplexer (DeMux) to reject
broadband spontaneous emission from the laser, and a high-finesse tunable filter to remove laser phase noise at the
mechanical frequency, which can both contribute to excess pump transmission noise on the single photon detector [1].
After pre-filtering, the laser pump is sent through an electro-optic phase modulator (
φ
-m) which may be driven by
either an RF signal generator, to generate optical sidebands for locking the filter cavities, or a vector network analyzer
(VNA) for obtaining the amplitude and phase response of the optical cavity.
To generate pulses, an optical switch (SW1) allows the pump tone to be sent through a series of electro-optic
amplitude modulators for the generation of optical pulses. The first is a fast high-extinction modulator (a-m) with a
rise and fall time of
25 ns. Though in principle the high-extinction modulator can provide 50
60 dB extinction on
its own, its transmission level is much less stable and difficult to lock at the maximum extinction point. Thus, we lock
the modulator at
30 dB of extinction and use two electro-optic switches, each providing
18
20 dB extinction,
to fully extinguish the pump. While these switches are much slower (
200 ns rise time,
30
μ
s fall time), the extra
switching time should have a negligible impact as it is much smaller than the pulse periods used here (
1 ms or
more). The modulators are collectively driven by a variable delay electrical pulse generator which also provides a
synchronization pulse to the single photon counting electronics.
A variable optical attenuator (VOA) controls the power input to the cavity, after which an optical circulator routes
the laser light to a lensed fiber tip inside the dilution refrigerator for end-fire coupling to the device. Subsequently,
the cavity reflection can be switched (SW3) into one of two detection setups. The first setup sends the signal
through an erbium-doped fiber amplifier (EDFA) followed by a high-speed photodetector (PD). The resulting amplified
photocurrent may be directed to a real-time spectrum analyzer (RSA) in order to measure the optical noise power
spectral density (NPSD) for mechanical characterization or to the VNA which is used in conjunction with the phase
modulator to measure the full complex response of the optical cavity for purposes of optical characterization as
described below. The second detection path sends the reflected signal through three additional tunable filters in order
to reject the pump frequency, and then back into the dilution refrigerator where it is detected by a superconducting
single photon detector (SPD). The output of this detector is sent to a time-correlated single photon counting (TCSPC)
module to build up a histogram of photon count events as a function of time relative to the synchronization pulse
received from the pulse generator.
2
a-m
switch
switch
Pulse
Generator
pulsing
dilution fridge
lensed
SPD
SW4
DeMux
RSA
EDF
A
PD
het
er odyne
det
ec
tion
laser
λ
-meter
SW2
SW3
φ
-m
SW1
VOA
5.6 GHz
~
TCSPC
Module
sync
VNA
amplitude
phase
FPC
feedback c
on
tr ol
beamsplitt
er
phot
odet
ec
tor
cir
cula
tor
optical isola
tor
pump pre-ltering
FIG. S-1:
Experimental setup.
Full setup for performing pulsed phonon counting.
λ
-meter: wavemeter, DeMux: C-band
optical demultiplexer,
φ
-m: electro-optic phase modulator, SW: optical switch, a-m: high-extinction electro-optic amplitude
modulator, switch: high-speed electro-optic switch, VOA: variable optical attenuator, EDFA: erbium-doped fiber amplifier,
PD: high-speed photodetector, RSA: real-time spectrum analyzer, VNA: vector network analyzer, SPD: single photon detector,
TCSPC: time-correlated single photon counting, FPC: fiber polarization controller..
The SPDs used in this work are amorphous WSi-based superconducting nanowire single-photon detectors developed
in collaboration between the Jet Propulsion Laboratory and NIST. The SPDs are design for high-efficiency detection
of individual photons in the wavelength range
λ
= 1520
1610 nm with maximum count rates as large as 10
7
counts
per second (c.p.s) [2]. The SPDs are mounted on the still stage of the dilution refrigerator at
700 mK. Single-mode
optical fibers (Corning SMF-28) are passed into the refrigerator through vacuum feedthroughs and coupled to the
SPDs via a fiber sleeve attached to each SPD mount. Proper alignment of the incoming fiber with the 15
μ
m
×
15
μ
m
square area of the SPD nanowire is ensured by a self-aligned mounting system incorporated into the design of the
SPD [2]. The radio-frequency output of each SPD is amplified by a cold-amplifier mounted on the 50 K stage of
the refrigerator as well as a room-temperature amplifier, then read out by a triggered PicoQuant PicoHarp 300 time-
correlated single photon counting module. By systematically isolating the input optical fiber from environmental light
sources and filtering out long wavelength blackbody radiation inside the fridge we have achieved dark count rates of
4 (c.p.s.). At just below the switching current of the detectors, we have measured a peak detection efficiency of
η
SPD
= 68%, with
<
20% variability depending on photon polarization.
The tunable filters used for both pre-filtering the pump and filtering the cavity reflection are commercially available,
piezo-tunable Fabry-Perot filters (Micron Optics, FFP-TF2), all with a
50 MHz bandwidth, a free-spectral range
of
20 GHz, and a tuning voltage of
<
18 V per free-spectral range. The filters each offer roughly 40 dB of pump
suppression (relative to the peak transmission) measured at a filter-pump varies by 1
2 dB from filter to filter. When
locking the post-cavity filters a switch is used to bypass the cavity (SW2), as a relatively large amount of CW power
is used during the locking procedure and we would like to avoid sending large amounts of power into the cavity unless
necessary. More importantly, the SPD is also bypassed (SW4), as allowing too much power to reach the SPD will
saturate it resulting in significantly elevated dark counts (Γ
dark
500
1000 c.p.s.) for 1
2 minutes after the signal
is turned off. Once the switches are set an RF signal generator is used to drive the phase modulator at
ω
m
, producing
an optical sideband which is aligned with the cavity (and motional sideband) frequency. To lock the filter chain, a
sinusoidal voltage (with an initial range of
±
10 V) is used to dither each filter while monitoring its transmission.
The offsets of the sinusoids are then adjusted, and their amplitudes reduced, to maximize transmission of the desired
sideband, fixing the voltages once all three filters are well aligned. Over time the filters will drift and the locking
3
procedure will need to be repeated, though in subsequent re-locking attempts a much smaller dithering amplitude
of
1 V is sufficient. As the piezo elements controlling each filter have a finite settling time, the filters will drift
much more rapidly following the initial locking attempt (during which large voltage shifts are applied) than following
subsequent re-locks. After several re-locks, the filters will typically become stable enough that the total transmission
at the sideband frequency changes by
<
5% over several minutes. At this point the phase modulator is turned off, the
pulse generator, cavity and SPD are switched back into the optical train and we begin accumulating pulsed data for
2 minutes before re-locking the filters. The total filter transmission is recorded at the end of a locking procedure and
again prior to re-locking, and if the transmission has shifted by more than a few percent the previous pulsed dataset
is discarded. All the time-resolved data shown in this chapter are taken in this manner, with all 2 minutes datasets
averaged together to produce the final pulsed histogram.
DEVICE CHARACTERIZATION
The measurements presented in this work rely on an accurate calibration of the optomechanical damping rate
γ
OM
,
which depends on the vacuum optomechanical coupling rate
g
0
, the total optical decay rate
κ
, and the intracavity
photon number
n
c
. The photon number for a given power and detuning in turn depends on the single pass fiber-
to-waveguide coupling efficiency
η
cpl
and waveguide-cavity coupling efficiency
η
κ
=
κ
e
(
κ
e
is the waveguide-cavity
coupling rate). The fiber collection efficiency is determined by measuring the calibrated reflection level far-off resonance
with the optical cavity on the optical power meter, and is found to be
η
cpl
= 0
.
68 for the device measured here (total
device reflection efficiency of
0
.
46%). To measure
κ
and
η
κ
, the laser is placed off-resonance from the cavity and
the VNA is used to drive the phase modulator and sweep an optical sideband across the cavity. By detecting the
reflection on a high-speed photodiode connected to the VNA input we obtain the amplitude and phase response of
the cavity from which can extract
κ/
2
π
= 443 MHz (corresponding to an optical quality factor of
Q
o
4
.
4
×
10
5
),
and
η
κ
= 0
.
5. With these three parameters measured, it is possible to determine
n
c
for an arbitrary input power to
the cavity.
To characterize the acoustic resonance, the EDFA is used to amplify the cavity reflection so that the optical
noise floor overcomes the photodetector’s electronic noise and the optical NPSD is measured on the RSA, where a
Lorentzian response due to transduction of the acoustic thermal Brownian motion can be observed at the acoustic
resonance frequency
ω
m
/
2
π
= 5
.
6 GHz. For a red- or blue detuned pump laser (∆
ω
c
ω
l
=
±
ω
m
) the linewidth
of this transduced noise peak is
γ
=
γ
b
±
γ
OM
, where
γ
b
is the bare coupling rate of the mechanical resonator to
its bath and
γ
OM
4
g
2
0
n
c
. Thus, by averaging the observed linewidth for red- and blue-detuned pumps we can
determine
γ
b
and thus extract the excess optomechanically induced damping rate
γ
OM
as a function of
n
c
. With
n
c
and
κ
determined, a linear fit of
γ
OM
reveals a vacuum optomechanical coupling rate of
g
0
/
2
π
= 710 kHz.
PHONON COUNTING
In the linearized sideband resolved regime with a red-detuned pump (∆ =
ω
m
) the total reflected cavity amplitude
will be sum of the coherent pump reflection
α
out
and a fluctuation term given, in the Fourier domain and in a frame
rotating at the pump frequency, by [3]
ˆ
a
out
(
ω
)
|
∆=
ω
m
r
(
ω
a
in
(
ω
) +
n
(
ω
a
i
(
ω
) +
s
(
ω
)
ˆ
b
in
(
ω
)
,
(S-1)
where ˆ
a
in
and ˆ
a
i
are the pump noise and intrinsic cavity noise, respectively (assumed for the moment to be vacuum
noise),
ˆ
b
in
is the noise operator for the mechanical bath, which has the correlation function
ˆ
b
in
(
ω
)
ˆ
b
(
ω
)
=
n
b
δ
(
ω
+
ω
) (
n
b
is the bath occupancy), and the effective scattering matrix elements are given by
r
(
ω
) = 1
2
κ
e
κ
±
γ
OM
κ
e
κ
1
±
i
(
ω
m
ω
) +
γ/
2
,
(S-2)
n
(
ω
;
±
) =
±
κ
i
κ
e
κ
(
γ
OM
±
i
(
ω
m
ω
) +
γ/
2
2
)
,
(S-3)
s
(
ω
) =
κ
e
κ
i
γ
i
γ
OM
±
i
(
ω
m
ω
) +
γ/
2
.
(S-4)
4
As the full reflected signal from the cavity includes the reflected pump amplitude, which is many orders of magnitude
larger than the sideband signal, we must first filter the cavity output to reject the pump. Provided the filter is
sufficiently high-finesse, it can be modeled in the frequency domain by a single Lorentzian function
F
(
ω
;
ω
f
) =
κ
f
/
2
i
(
ω
f
ω
) +
κ
f
/
2
,
(S-5)
where
κ
f
and
ω
f
are the resonant frequency of the filter, respectively. Now, explicitly considering the case of red-
detuned driving in the sideband-resolved regime, the filtered cavity output will be the product of
F
(
ω
;
ω
f
) with the
frequency domain output of the cavity. As the resonantly enhanced anti-Stokes sideband photons will be detuned by
ω
m
from the pump, we choose
ω
f
=
ω
m
. Using Eq. S-1, we obtain
ˆ
a
filt
(
ω
) =
F
(
ω
;
ω
m
)
(
α
out
δ
(
ω
) +
r
(
ω
; +)ˆ
a
in
(
ω
) +
n
(
ω
; +)ˆ
a
i
(
ω
) +
s
(
ω
; +)
ˆ
b
in
(
ω
)
)
.
(S-6)
Performing photon counting on the filtered output then results in an average count rate of
Γ(
t
) =
ˆ
a
filt
(
t
a
filt
(
t
)
=
1
2
π

−∞

−∞
e
i
(
ω
+
ω
)
t
ˆ
a
filt
(
ω
a
filt
(
ω
)
=
1
2
π
(
|
F
(0;
ω
m
)
|
2
|
α
out
|
2
+
κ
e
κ
γ
OM

−∞
|
F
(
ω
;
ω
m
)
|
2
S
bb
(
ω
;
n
)
)
A
|
α
2
out
|
+
κ
e
κ
γ
OM
n
,
(S-7)
where
A
=
1
2
π
|
F
(0;
ω
m
)
|
2
is the pump attenuation factor, and
S
bb
(
ω
;
n
) is the phonon spectral density [3]
S
bb
(
ω
) =
γ
n
(
ω
m
+
ω
)
2
+ (
γ/
2)
2
.
(S-8)
Note that we have assumed a filter bandwidth
κ
f

γ
, allowing us to approximate
|
F
(
ω
;
ω
m
)
|
2
≈ |
F
(
ω
m
;
ω
m
)
|
2
= 1
inside the integral over
S
bb
. A similar analysis for blue detuning (where
ω
f
=
ω
m
to filter the Stokes sideband)
yields a comparable result, with
n
〉→〈
n
+ 1. Note that this formula assumes that ˆ
a
in
and ˆ
a
i
are vacuum noise, and
thus do not contribute to counting of real photons.
The total count rate, including noise of the photon counter and reduction of the pump and signal due to measurement
inefficiency, is given for red-detuning by
Γ
tot
= Γ
dark
+ Γ
pump
+
η
κ
e
κ
γ
OM
n
,
(S-9)
where Γ
dark
is the dark count rate of the photon detector, Γ
pump
=
ηA
|
α
out
|
2
and
η
is the total measurement
efficiency. These expressions can be used to perform thermometry in a similar fashion to linear detection, either by
calibrating the cavity parameters and total measurement efficiency or by measuring the asymmetry between the red-
and blue-detuned count rates.
To assess the sensitivity of this counting scheme, it is convenient to express the measurement noise floor Γ
dark
pump
in terms of an equivalent number of mechanical quanta (that is the mechanical occupancy
n
that would be needed
to yield a signal-to-noise of one). This noise-equivalent phonon number is obtained by dividing the total noise floor
by the per-phonon count rate Γ
SB,0
=
η
(
κ
e
)
γ
OM
, yielding
n
NEP
=
Γ
dark
+ Γ
pump
Γ
SB,0
.
(S-10)
For a highly sideband-resolved system, the reflected pump in the case of ∆ =
±
ω
m
will be approximately given by
α
out
α
in
. This in turn can be expressed in terms of the intracavity photon number as
|
α
out
|
2
ω
2
m
n
c
e
. Thus,
n
NEP
as a function of
n
c
is given by
n
NEP
(
n
c
) =
κ
2
Γ
dark
4
ηκ
e
g
2
0
n
c
+
A
(
κω
m
2
κ
e
g
0
)
2
.
(S-11)
5
EFFECTS OF TECHNICAL LASER NOISE
The mechanical frequency of the nanobeam used in this experiment (
ω
m
/
2
π
= 5
.
6 GHz) raises concerns about the
effects of laser phase noise on the measurements, as the laser used in this experiment has previously been observed
to possess a prominent phase noise peak at 5 GHz [3]. In addition to phase noise, most diode lasers typically have
a small amount of broadband spontaneous emission. While this additional noise is orders of magnitude weaker than
the laser tone itself, it exists outside the wavelength region (
λ
1520
1570 nm) where the filters are guaranteed to
be high finesse, and thus can be transmitted with high efficiency to the SPDs.
Phase noise in particular is worrisome as it can not only lead to an excess noise floor but also to real heating of
the mechanics and systematic errors in thermometry due to noise squashing/anti-squashing [4, 5]. Phase noise can
be accounted for by assuming a total pump noise operator given by [3]
ˆ
a
in,tot
(
t
) = ˆ
a
in
(
t
) + ˆ
a
φ
(
t
)
,
(S-12)
where ˆ
a
in
(
t
) still represents vacuum noise and
ˆ
a
φ
(
t
) =
i
|
α
in
|
φ
(
t
)
,
(S-13)
where
|
α
in
|
is the amplitude of the pump and
φ
(
t
) is the stochastic phase of the pump, assumed to be a real, stationary
Gaussian process with zero mean. The phase noise is assumed to be delta-correlated in the frequency domain, such
that
φ
(
ω
)
φ
(
ω
)
=
S
φφ
(
ω
)
δ
(
ω
+
ω
)
,
(S-14)
where the expectation value here corresponds to an ensemble average. The phase noise input to the cavity then has
the correlation function
ˆ
a
φ
(
ω
a
φ
(
ω
)
=
S
αα
(
ω
)
δ
(
ω
+
ω
)
,
(S-15)
where
S
αα
(
ω
) =
|
α
|
2
S
φφ
(
ω
). In a sideband-resolved system, for either red or blue detuning ∆ =
±
ω
m
, we find that
the presence of phase noise heats the mechanical resonator, with an additional added phonon occupancy given by [3]
n
φ
|
∆=
±
ω
m
=
κ
e
κ
γ
OM
γ
n
φ
,
(S-16)
where we have defined
n
φ
=
S
αα
(
ω
m
), and where we have assumed that
S
αα
(
ω
) is sufficiently slow-varying in the
vicinity of
ω
=
ω
m
(specifically for
|
ω
ω
m
|
<
γ
) that we may approximate
S
αα
(
ω
) =
S
αα
(
ω
m
).
Including this additional noise term in the analysis of the previous section yields a phase noise contribution to the
total photon count rate of
Γ
φ
|
∆=
±
ω
m
=
η

−∞
2
π
S
αα
(
ω
)
|
F
(
ω,
±
ω
m
)
|
2
|
r
(
ω
;
±
)
|
2
.
(S-17)
If we assume that
S
αα
(
ω
) is slowly-varying in frequency for
|
ω
ω
m
|
<
γ
and that
κ
f

γ
, this simplifies to
Γ
φ
|
∆=
±
ω
m
=
ηn
φ
(
κ
f
4
(
1
2
κ
e
κ
)
2
+
κ
e
γ
OM
κ
(
γ
OM
κ
e
γκ
±
(
1
2
κ
e
κ
))
)
.
(S-18)
Using the fact that
n
φ
=
|
α
|
2
S
φφ
(
ω
m
)
ω
2
m
n
c
S
φφ
(
ω
m
)
e
, we obtain the contribution of phase noise to the noise-
equivalent phonon number
n
NEP
|
∆=
±
ω
m
=
(
ω
m
κ
2
κ
e
g
0
)
2
S
φφ
(
ω
m
)
(
κ
f
4
(
1
2
κ
e
κ
)
2
+
κ
e
γ
OM
κ
(
γ
OM
κ
e
γκ
±
(
1
2
κ
e
κ
))
)
.
(S-19)
Like the pump-bleed through, phase noise leads to a constant contribution to
n
NEP
, and leads to squashing or anti-
squashing of the noise depending on detuning and cooperativity, similar to heterodyne detection. Note, however, that
in the case
κ
e
= 0
.
5 the contribution of phase noise will not depend upon detuning. Thus, even in the presence of
large phase noise it is possible to avoid detuning dependent noise squashing/anti-squashing, though one will still have
a large overall phase noise floor.
6
It is useful for characterization purposes to calculate the phase-noise contribution to the observed count rates when
the laser is far-detuned from the cavity resonance (
|
|
ω
m
). Assuming that the laser-filter detuning is kept constant
at
±
ω
m
, the phase-noise count rate in this case is just
Γ
φ
|
|
|
ω
m
=
η

−∞
2
π
S
αα
(
ω
)
|
F
(
ω,
±
ω
m
)
|
2
=
η
κ
f
4
n
φ
,
(S-20)
with a corresponding noise-equivalent phonon number
n
NEP
|
|
|
ω
m
=
(
ω
m
κ
4
κ
e
g
0
)
2
κ
f
S
φφ
(
ω
m
)
.
(S-21)
To get rid of the excess noise, we insert both the bandpass filter (for filtering spontaneous emission) and a tunable
high-finesse filter (for filtering phase noise) immediately after the pump laser output as shown in Fig. S-1, enabling us
to reach
n
NEP

1 using a three-filter phonon counting setup as shown in Fig. 1d of the main text. A conservative
estimate of the residual phase noise can be made by assuming that the limiting value of
n
NEP
4
×
10
3
is entirely
due to phase noise (i.e. perfect filtering of the pump tone). Using Eq. S-21 we find
S
φφ
(
ω
m
)
<
8
×
10
19
Hz
1
. For
the pump power during the on-state of the pulse (
n
c,on
45), the corresponding effective phase noise occupancy is
n
φ
3
.
2
×
10
5
, which has a negligible effect on the measurements in this work.
HEATING MODEL
The simplest thermal model assumes that the optically induced bath turns on instantaneously when the pulse is
in the on-state. The corresponding rate equation for the phonon occupancy
n
, for red- (∆ =
ω
m
) and blue-detuned
(∆ =
ω
m
) pulses during the on-state is thus
̇
n
=
γ
n
+
γ
p
n
p
+
γ
0
n
0
+
1
2
(1
1)
γ
OM
,
(S-22)
where
γ
=
γ
0
+
γ
p
±
γ
OM
,
γ
p
and
n
p
are the coupling rate and occupancy of the hot phonon bath,
γ
0
and
n
0
are the
coupling rate and occupancy of the ambient fridge bath, and the extra factor of
γ
OM
for a blue-detuned pump accounts
for the possibility of spontaneous emission into the mechanical resonator due to the optomechanical interaction. This
rate equation has the simple solution
n
(
t
) =
n
(0)
e
γt
+
n
H
(
1
e
γt
)
,
(S-23)
n
H
=
γ
1
(
γ
p
n
p
+
γ
0
n
0
+
1
2
(1
1)
γ
OM
)
.
(S-24)
where
t
T
pulse
.
In principle,
γ
OM
can be determined independently as described above,
γ
0
can be determined by fitting the occu-
pancy decay during the pulse off-state (Fig. 3 in the main text), while
γ
p
and
n
p
can be subsequently determined
by fitting the steady-state occupancy curve shown in Fig. 1c of the main text. However, using these independently
determined values in a fit to the red- and blue-detuned data shown in Fig. 2a of the main text results in a poor fit
and inconsistent results. In particular, the apparent heating rate
γ
is much smaller than expected for a red-detuned
pulse and larger than expected for a blue-detuned pulse. This, along with the “kink” in the heating curve shown in
the inset of Fig. 2c of the main text, suggests additional complexity in the heating dynamics.
The simplest addition to the heating model is to assume a finite time for the hot phonon bath to come into
equilibrium, which is approximated by allowing a fraction of the hot phonon bath occupancy to turn on exponentially
over time. Thus, the phenomenological rate equation becomes
̇
n
=
γ
n
+
γ
p
n
p
(
1
δ
b
e
γ
S
t
)
+
γ
0
n
0
+
1
2
(1
1)
γ
OM
,
(S-25)
where
δ
b
is the slow growing fraction of
n
p
and
γ
S
the turn-on rate. Strictly speaking
γ
p
should depend on the
phonon distribution of the hot phonon bath, and thus would be expected to be time-dependent in this model as well.
7
However, the resulting rate equation becomes intractable in such a case and the effects should be negligible in the
regime
γ
OM

γ
p
, so we approximate
γ
p
equal to its steady-state value. This modified rate equation has the solution
n
(
t
) =
n
(0)
e
γt
+
n
H
(
1
e
γt
)
+
n
δ
(
e
γ
S
t
e
γt
)
, n
δ
=
γ
p
n
p
δ
b
γ
S
γ
,
(S-26)
which is used to obtain the fit shown in Fig. 2c of the main text with
n
(0),
γ
S
and
δ
b
as free parameters.
During the off-state of the pulse (
T
pulse
t
T
per
), the resonator will simply cool towards the ambient fridge
occupancy
n
0
at the intrinsic damping rate
γ
0
. Using the initial condition that
n
(0) =
n
0
for the first pulse,
and iterating many pulses we find that in the steady-state the initial phonon occupancy during a pulse (assuming
T
per

T
pulse
) is
n
(0) =
n
0
(
1
e
γ
0
T
per
)
+
n
H
(
1
e
γT
pulse
)
e
γ
0
T
per
+
n
δ
(
e
γ
S
T
pulse
e
γ
S
T
per
)
e
γ
0
T
per
1
e
γT
pulse
γ
0
T
per
.
(S-27)
Thus, once
n
0
,
γ
S
and
δ
b
are determined by fitting the occupancy curves, we may use Eqs. S-26 and S-27 to determine
the occupancy throughout the pulse for arbitrary
T
per
and
T
pulse
. This allows us to determine the maximum attained
phonon occupancy as a function of pulse parameters, as shown in Fig. 4a of the main text.
PHONON ADDITION/SUBTRACTION FIDELITY
In this section we present an analysis of heralded phonon addition and subtraction [6], including the effects of time-
dependent, optically-induced heating to lowest order, with a particular eye towards the generation of single-phonon
Fock states [7].
In a frame rotating at the pump frequency, the full Heisenberg-Langevin equations for the optomechanical system
are [3]
̇
ˆ
a
=
(
i
∆ +
κ
2
)
ˆ
a
+
ig
0
(
ˆ
b
+
ˆ
b
)
ˆ
a
+
κ
ˆ
a
in
,
(S-28)
̇
ˆ
b
=
(
m
+
γ
b
2
)
ˆ
b
+
ig
0
ˆ
a
ˆ
a
+
γ
b
ˆ
b
in
,
(S-29)
where ˆ
a
,
ˆ
b
are the photon and phonon annihilation operators, respectively, and ˆ
a
in
,
ˆ
b
in
are quantum noise operators
corresponding to the optical and mechanical baths.
We linearize about a classical steady-state by displacing ˆ
a
α
+ ˆ
a
, where
|
α
|
2
=
n
c
. For concreteness we will
consider the case ∆
ω
c
ω
l
=
ω
m
(blue-detuned pump). Moving into a frame rotating at the mechanical frequency
(i.e. ˆ
a
ˆ
ae
m
t
, and so on for all operators), and making the rotating wave approximation, valid in the weak coupling
(
g
0
n
c

κ
) and sideband-resolved (
κ/ω
m

1) limit, we obtain
̇
ˆ
a
=
κ
2
ˆ
a
+
iG
ˆ
b
+
κ
ˆ
a
in
,
(S-30)
̇
ˆ
b
=
γ
b
2
ˆ
b
+
iG
ˆ
a
in
+
γ
b
ˆ
b
in
,
(S-31)
where
G
=
g
0
n
c
and ˆ
a
in
,
ˆ
b
in
are the usual noise operators multiplied by
e
m
t
. The noise operators obey the
following commutation and correlation relations
[
ˆ
a
in
(
t
)
,
ˆ
a
in
(
t
)
]
=
[
ˆ
b
in
(
t
)
,
ˆ
b
in
(
t
)
]
=
δ
(
t
t
)
(S-32)
ˆ
a
in
(
t
a
in
(
t
)
=
δ
(
t
t
)
(S-33)
ˆ
b
in
(
t
)
ˆ
b
in
(
t
)
=
n
b
(
t
)
δ
(
t
t
)
,
ˆ
b
in
(
t
)
ˆ
b
in
(
t
)
= (
n
b
(
t
) + 1)
δ
(
t
t
)
.
(S-34)
Since we are working in the weak-coupling limit (
G

κ
) we may use the adiabatic solution for ˆ
a
(i.e.
̇
ˆ
a
0).
Moreover, we wish to include the effects of mechanical noise to lowest order. Considering the effect of a short optical
pulse of duration
τ
, we consider the case
γ
b
τ,γ
b

τ
0
ds n
b
(
s
)

1, as well as
γ
OM
τ

1 and
γ
OM

γ
b
, where here
γ
OM
4
G
2
refers to the
magnitude
of the optomechanical damping rate (the sign will be explicitly incorporated
for simplicity, since we’re only considering blue-detuning). Under these assumptions, as in Ref. [8], the mechanical