Pulsed excitation dynamics of an optomechanical crystal 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: March 18, 2015)
Using pulsed optical excitation and read-out along with single phonon counting techniques, we
measure the transient back-action, heating, and damping dynamics of a nanoscale silicon optome-
chanical crystal cavity mounted in a dilution refrigerator at a base temperature of
T
f
≈
11 mK.
In addition to observing a slow (
∼
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
〈
n
〉
= 0
.
021
±
0
.
007 (
T
b
≈
70 mK) and an intrinsic mechanical decay rate of
γ
0
= 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.
PACS numbers: 42.50.Wk, 42.65.—k, 62.25.—g
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], amongst them the preparation of highly non-classical mechan-
ical states [5–7] and coherent frequency conversion between microwave and optical signals [8–12]. A particularly
interesting device architecture for realizing large radiation pressure coupling between light and mechanics is the thin-
film optomechanical crystal (OMC) [13, 14], in which optical and acoustic waves can be guided and co-localized 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
resonators are coupled together via optical or acoustic degrees of freedom, and in which laser light is used to para-
metrically control the emergent network or material properties.
For many of the above mentioned applications, operation at cryogenic temperatures is desired as it offers a route
to obtaining the requisite low thermal occupancy and long mechanical coherence time. 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 mechanical damping due to weak sub-bandgap 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 complications
arise from the seemingly contradictory requirements of isolating the mechanical resonator from its environment to
obtain high mechanical
Q
-factor, and that of providing large thermal anchoring to a low temperature bath for cooling
of the mechanical resonator.
In this work we utilize pulsed optical excitation and single phonon counting [19] to study the transient dynamics
of optical back-action, heating, and damping of the 5
.
6 GHz mechanical mode of a silicon optomechanical crystal
resonator at mK bath temperatures. Phonon counting, realized by photon counting of the optically filtered motional
sidebands of the reflected optical excitation pulse, yields simultaneously a high time resolution (
∼
25 ns) and me-
chanical mode occupancy sensitivity (
<
10
−
2
). Measurement of both Stokes and anti-Stokes sidebands also yields an
absolute calibration of the occupancy of the resonator mode in terms of mechanical vacuum noise [20–22]. In addition
to measuring initial phonon mode occupancies as low as
〈
n
〉
= 0
.
021
±
0
.
007 and mechanical decay times as long as
τ
= 475
±
21
μ
s, we observe a slow (
∼
740 ns) turn-on time for the optical-absorption-induced phonon bath that both
heats and damps the mechanical resonator mode. Taken together, these measurements 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.
∗
opainter@caltech.edu
arXiv:1503.05135v1 [quant-ph] 17 Mar 2015
2
10
0
10
-2
10
2
10
0
10
-2
10
2
10
-4
10
-6
phonon number
n
c
a
d
n
c, on
T
f
= 10 mK
b
n
c, o
T
f
= 70 mK
optical waveguide
back mirror
optomechanical cavity
acoustic radiation shield
2
μ
m
c
laser
EOM
circulator
SPD
sideband lter
!
l
!
c
γ
0
(iii)
e
-
γ
p
(ii)
γ
ΟΜ
(i)
!
l
!
c
FIG. 1.
a
, Pulsed pump light at frequency
ω
l
(red arrows) is directed to an optomechanical crystal cavity (OMC) 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 single photon detector (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) exchanges
energy with the acoustic mode at rate
γ
OM
, generating scattered photons at
ω
c
(black) in the process, (ii) pump light drives
excitation of electronic defect states in the silicon device layer, subsequently exciting a hot phonon bath which 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
〈
n
〉
(red circles) for red-detuned (∆ =
ω
m
) continuous-wave (CW) pumping versus intracavity photon number
n
c
.
The red dashed line indicates the expected
n
NEP
contribution from SPD dark counts. The vertical dashed line at
n
c
≈
45
(4
.
5
×
10
−
5
) 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
= 70 mK and 10 mK, respectively.
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 bandgaps at the ends of the beam co-localize 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. 1b, and a finite-element method simulation of the acoustic resonance
is displayed in Fig. 1c. 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] which possesses a complete acoustic bandgap in the
5
−
6 GHz range, providing an additional acoustic shield for the mechanical resonator mode while allowing phonons
above and below the phononic bandgap to carry heat from the nanobeam structure.
The experimental set-up is shown schematically Fig. 1a. Electro-optic modulation of a laser probe beam 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 ∆
≡
ω
c
−
ω
l
=
ω
m
. In this case absorption of a single phonon from the nanomechanical resonator results in upconversion
of a pump photon to the anti-Stokes sideband at
ω
c
, represented by the black arrows. The cavity reflection is filtered
to reject the pump frequency and subsequently directed to a high-efficiency single-photon detector (SPD) and a time-
correlated single photon counting system synced to the pulse generator. This measurement repeats each pulse period,
building up a histogram with respect to photon arrival time relative to the sync pulse during a certain integration time.
As the number of anti-Stokes photons is directly proportional to the average phonon occupancy of the mechanical
resonator,
〈
n
〉
, the photon count rate in each time bin then portrays the time-evolution of
〈
n
〉
(
t
) during the pulse
on-state. All measurements presented herein were performed at a fridge base temperature of
T
f
= 11 mK. Further
details about device fabrication and the experimental setup can be found in Refs. [17, 19] and in the appendices.
The signal-to-noise ratio (SNR) of this phonon counting method is determined by the sideband scattering rate
γ
OM
≡
4
g
2
0
n
c
/κ
(
g
0
is the optomechanical coupling rate,
n
c
is the intracavity photon number, and
κ
is the optical
decay rate), 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 parameterization of
3
the sensitivity to low
〈
n
〉
is the noise-equivalent phonon occupancy
n
NEP
, defined as [19]
n
NEP
=
Γ
dark
ηγ
OM
+
A
(
κω
m
2
κ
e
g
0
)
2
,
(1)
where
κ
e
is the optical decay rate into the detection channel. For the device under test we have
κ/
2
π
= 443 MHz,
κ
e
/
2
π
= 221 MHz, and
g
0
/
2
π
= 710 kHz. A typical measured
n
NEP
for our setup, taken using a comparable device at
room temperature, is shown in Fig. 1d as gray squares. Here, to obtain the lowest
n
NEP
optical pre-filtering 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 App. E. 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 sub-kelvin temperatures optical
absorption heating produces a steady-state
〈
n
〉
>
1 for
n
c
>
0
.
01 (red circles in Fig. 1c) 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 1c: (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 bandgap, and (iii)
coupling to the ambient fridge bath at an intrinsic rate
γ
0
through the acoustic radiation shield. At the low intra-
cavity photon numbers of these measurements, we believe the optical absorption heating is a result of excitation of
electronic defect states at the silicon surfaces [23, 24], 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 [25]. 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(
−
̄
hω
c
/k
B
T
p
) for low bath temperature (
T
p
<
4 K), corresponding to inelastic phonon scattering with
a quasi-equilibrium hot phonon bath above a cut-off phonon frequency of
ω
c
/
2
π
≈
35 GHz. As shown in Fig 1d,
extrapolation of this steady-state model for a fridge base temperature of
T
f
= 10 mK reveals a relevant pulse-off-state
regime of
n
c
<
10
−
4
in which absorption heating effects are negligible.
Figure 2a shows the measured sideband photon count rate versus time with pulsed optical excitation, for both red-
(∆ =
ω
m
) and blue-detuned (∆ =
−
ω
m
) pumping. The on-state pulse amplitude in these measurements is
n
c,on
= 45,
corresponding to an optomechanical coupling rate of
γ
OM
/
2
π
= 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 indicate
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. Henceforth, the variable
t
refers to time relative to the synchronization signal
generated by the pulse generator, while
t
pulse
refers to time relative to the start of the optical pulse occurring around
t
≈
1
μ
s. The pulse period in these measurements is fixed at
T
per
= 5 ms.
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
= Γ
−
/
Γ
+
−
1, where Γ
±
is the sideband photon
count rate for a pump detuning ∆ =
±
ω
m
. While motional sideband asymmetry has previously been measured in a
variety of opto- and electromechanical systems using linear detection schemes [20–22, 26] , the use of phonon counting
techniques allows the observed asymmetry to be directly and unambiguously attributed to quantum fluctuations of
the mechanical oscillator [22, 27]. This asymmetry, shown versus
t
pulse
in Fig. 2b, 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, respectively [28]. However, for
t
pulse
γ
−
1
OM
≈
750 ns the effects of backaction can be neglected and
the phonon occupancy
〈
n
〉
may be assumed equal for both pump detunings. In this case, the asymmetry is simply
related to the occupancy by
ξ
=
〈
n
〉
−
1
, and arises from the fundamental asymmetry between phonon absorption
(Γ
+
∝ 〈
n
〉
) and emission (Γ
−
∝ 〈
n
〉
+ 1) processes [20, 29]. Theoretical plots of the two contributions to
ξ
(
t
) are
shown in Fig. 2b. The green dashed line shows the expected
ξ
(
t
) assuming that
〈
n
〉
(∆ =
−
ω
m
) =
〈
n
〉
(∆ =
ω
m
) (no
backaction), while the orange dash-dotted line shows
ξ
(
t
) in the absence of optical heating and in the case when
n
b
1 such that
ξ
(0) = 0 and asymmetry arises solely from backaction effects. From the asymmetry measured
in the first 25 ns time bin, we extract a minimum phonon occupancy of
〈
n
〉
min
= 0
.
021
±
0
.
007 (
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.
Once
〈
n
〉
is determined in the initial time bin it may be used to convert measured count rates to phonon occupancies
throughout the pulse for either pump detuning. The resulting calibrated occupancy versus
t
pulse
is shown in Fig. 2c.
A simple model of the heating dynamics would suggest an exponential increase towards the steady-state phonon