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arXiv:1606.05259v1 [physics.optics] 16 Jun 2016
Stokes Solitons in Optical Microcavities
Qi-Fan Yang
, Xu Yi
, Ki Youl Yang and Kerry Vahala
T. J. Watson Laboratory of Applied Physics, California Institute of
Technology, Pasadena, California 91125, USA.
These authors contributed equally to this work.
Corresponding author: vahala@caltech.edu
Solitons are wavepackets that resist dispersion
through a self-induced potential well. They are
studied in many fields, but are especially well
known in optics on account of the relative ease
of their formation and control in optical fiber
waveguides
1,2
.
Besides their many interesting
properties, solitons are important to optical con-
tinuum generation
3
, in mode-locked lasers
4,5
and
have been considered as a natural way to con-
vey data over great distances
6
. Recently, soli-
tons have been realized in microcavities
7
thereby
bringing the power of microfabrication methods
to future applications. This work reports a soli-
ton not previously observed in optical systems,
the Stokes soliton. The Stokes soliton forms and
regenerates by optimizing its Raman interaction
in space and time within an optical-potential well
shared with another soliton.
The Stokes and
the initial soliton belong to distinct transverse
mode families and benefit from a form of soliton
trapping that is new to microcavities and soliton
lasers in general. The discovery of a new optical
soliton can impact work in other areas of photon-
ics including nonlinear optics and spectroscopy.
Introduction.
Solitons result from a balance of wave
dispersion with a non-linearity. In optics, temporal soli-
tons are readily formed in optical fiber waveguides
1–3,6
and laser resonators
4,5
and have recently been observed
in dielectric microcavties
7
. In each of these cases non-
linear compensation of group velocity dispersion is pro-
vided by the Kerr effect (nonlinear refractive index). Be-
sides the Kerr nonlinearity, a secondary effect associ-
ated with soliton propagation is the so-called soliton self-
frequency shift caused by Raman interactions, which in-
duce a continuously increasing red-shift with propagation
in a waveguide
3
or a fixed shift of the soliton spectrum
in cavities
8
. More generally, the Raman interaction can
produce optical amplfication and laser action of waves
red-shifted relative to a strong pumping wave
9
. This
work reports a new Raman-related effect, soliton gen-
eration through time and space varying Raman amplifi-
cation created by the presence of a first temporal soliton.
Because the new soliton is spectrally red-shifted relative
to the initial soliton we call it a Stokes soliton. It is
observed in a silica microcavity and obeys a threshold
condition resulting from an optimal balancing of Raman
gain with cavity loss when the soliton pulses overlap in
space and time. Also, the repetition frequency of both
the initial and the Stokes soliton are locked by the Kerr
nonlinearity.
In this work the cavity will be a circular-shaped whis-
pering gallery microcavity and the first temporal soliton
will be referred to as the primary soliton. The primary
soliton here is a dissipative Kerr cavity soliton
7
, however,
other solitons would also suffice. Consistent with its for-
mation, the primary soliton creates a spatially varying
refractive index via the Kerr nonlinearity that serves as
an effective potential well, traveling with the soliton and
and counteracting optical dispersion. Moreover, on ac-
count of the Raman interaction, the primary soliton cre-
ates local Raman amplification that also propagates with
the primary soliton. The primary soliton is composed of
many longitudinal modes belonging to one of the trans-
verse mode families of the cavity. ∆
ν
P
will be the longi-
tudinal mode separation or free spectral range for longi-
tundinal modes near the spectral center of the primary
soliton. ∆
ν
P
also gives the approximate round trip rate
of the primary soliton around the cavity (
T
RT
= ∆
ν
1
P
is the round trip time).
Consider another transverse mode family besides the
one that forms the primary soliton. Suppose that some
group of longitudinal modes in this family satisfies two
conditions: (1) they lie within the Raman gain spectrum
created by the primary soliton; (2) they feature a free
spectral range (FSR) that is close in value to that of
the primary soliton (∆
ν
P
). Any noise in these longitu-
dinal modes will be amplified by Raman gain provided
by the primary soliton. If the round trip amplification of
a resulting waveform formed by a superposition of these
modes is sufficient to overcome round trip optical loss,
then oscillation threshold is possible. The threshold will
be lowest (Raman gain maximal) if the modes of the sec-
ond family phase lock to form a pulse overlapping in both
space and time with the primary soliton. This overlap is
possible since the round trip time of the primary soliton
and the new pulse are closely matched, i.e., condition (2)
above is satisfied. Also, the potential well created by the
primary soliton can be shared with the new optical pulse.
This latter nonlinear coupling of the primary soliton with
the new, Stokes soliton pulse results from Kerr-mediated
cross-phase modulation and further locks the round trip
rates of the two solitons (i.e., their soliton pulse repetition
frequencies are locked). As an aside, a third condition on
the new mode family that forms the Stokes soliton is that
it feature spatial overlap with the spatial intensity distri-
bution of the primary soliton transverse mode family. A
conceptual schematic of the process is presented in figure
1a.
2
Color gradient: Raman gain
1520
1560
1600
1640
1680
Relative power (10 dB / div)
Wavelength (nm)
Device 1
Device 2
Device 3
1520
1560
1600
1640
1680
Wavelength (nm)
Measurement
Simulation
Primary soliton FSR
FSR - offset (5MHz / div)
Device
2
Device
3
Device
1
Primary
Soliton
Primary Soliton
Stokes
Soliton
Stokes Soliton
Raman
Gain
Kerr Phase
Modulation
“Pump”
“Interaction”
“Laser”
CW
LASER
EDFA
AOM
ESA
OSA
A-CORR
OSCI
PD
PD
(a)
(c)
(b)
FIG. 1:
Experimental setup and description of Stokes soliton gener
ation, primary and Stokes soli-
ton spectra in three microcavity devices, and Stokes solito
n mode family FSR dispersion measure-
ment. a)
The Stokes soliton (red) maximizes Raman gain by overlapping in time an
d space with the primary soli-
ton (blue). It is also trapped by the Kerr-induced effective optical
well created by the primary soliton. The micro-
cavity (shown as a ring) is pumped with a tunable, continous-wave (C
W) fiber laser amplified by an erbium-doped
fiber amplifier (EDFA). An acousto-optic modulator (AOM) is used to
control the pump power. The output soliton
power is detected with a photo diode (PD) and monitored on an oscillos
cope (OSCI). Wavelength division multi-
plexers (not shown) split the 1550 nm band primary soliton and 1600 n
m band Stokes soliton so that their powers
can be monitored separately on the oscilloscope. An optical spectr
um analyzer (OSA), auto-correlator (A-CORR)
and electrical spectrum analyzer (ESA) also monitor the output.
b)
Primary and Stokes soliton spectra observed
in three devices.
c)
Free spectral range (FSR) versus wavelength measured for mod
e families associated with the
Stokes soliton in three microcavity devices. Extrapolation of the me
asured results is done using mode simulation
(dashed lines). The FSR near the spectral center of the primary s
oliton in each device is shown as a dashed hor-
izontal line. The spectral location of the Stokes soliton in (b) closely
matches the predicted FSR matching wave-
length. The background coloration gives the approximate waveleng
th range of the Raman gain spectrum.
The generation of a fundamental soliton by another
fundamental soliton in this way is new and also repre-
sents a form of mode locking of a soliton laser. It dif-
fers from mechanisms like soliton fission which also re-
sult in the creation of one of more fundamental solitons
3
.
Specifically, soliton fission involves a higher order soliton
breaking into mutliple fundamental solitons, nor is it a re-
generative process. Also, whereas Raman self-frequency
shifting in solitons is well known in optical fibers
10,11
and
has been recently observed in optical microcavities
8,12
,
the Raman mediated formation and regeneration of a
new soliton by an existing soliton is new. Finally, con-
cerning the trapping phenomena that accompanies the
Stokes soliton formation, the trapping of temporal soli-
tons belonging to distinct polarization states was pro-
posed in the late 1980s
13
and was observed in optical
fiber
14,15
and later in fiber lasers
16
. However, trapping of
temporal solitons belonging to distinct transverse mode
families, as oberved here, was proposed even earlier
17,18
,
but has only recently been observed in graded-index fiber
waveguides
19
and not so far in a laser or a cavity. The ob-
servation, measurement and modeling of Stokes solitons
is now described.
Observation of Stokes Solitons.
The experimen-
tal setup is shown in figure 1a. The microcavities are
about 3mm in diameter, fabricated from silica on sil-
3
Wavelength (nm)
Power (dBm)
1510
1550
1590
1630
(a)
1670
Pump
-80
-60
-40
-20
-80
-60
-40
-20
Soliton power (mW)
0
-80
-60
-40
-20
0.2
0.3
0.4
0.1
Total soliton power (mW)
Primary mode
Stokes mode
Primary soliton peak power (mW)
0.2
0.3
0.4
0.5
0
10
20
30
40
Equation 1
Measured
Primary (measured)
Stokes (measured)
Stokes (simulated)
Primary (simulated)
Simulated intracavity power (a.u.)
Polar angle (rad) / 2
π
0
0.1
Stokes
X 10
Primary
(b)
(c)
Stokes soliton (simulated)
Primary soliton (simulated)
0
1
0
1
0
1
FIG. 2:
Stokes soliton spectra, power and threshold measurements.
a
Soliton spectra are plotted at three
primary soliton powers (one below the Stokes soliton threshold). Th
e insets show the spatial mode families asso-
ciated with the primary and Stokes solitons. The blue and red curves
are simulations using the coupled Lugiato
Lefever equations.
b
Simulated primary and Stokes soliton pulses corresponding to the sp
ectral simulations in (a)
plotted versus their angular location in the cavity.
c
Measurement and theory of Stokes (red) and primary (blue)
soliton power versus total soliton power. The primary soliton peak p
ower (orange) versus total power is also plot-
ted to show threshold clamping at the onset of Stokes soliton oscillat
ion. The theoretical threshold peak power from
eqn. (1) is also shown for comparison.
icon and have an unloaded optical Q factor of 400
million
20
. The cavity is pumped by a continuous-wave
fiber laser to induce a primary soliton which is a dissi-
pative Kerr cavity (DKC) soliton with a repetition fre-
quency of 22GHz and pulse width that could be con-
trolled to lie within the range of 100-200 fs. The require-
ments for DKC soliton generation include a mode fam-
ily that features anomolous dispersion. Other require-
ments as well as control of DKC soliton properties are
described elsewhere
7,12
. DKC soliton spectra observed
in three different microcavity devices are shown in figure
1b (red, green and blue spectra centered near 1550nm).
As the primary soliton power is increased, the Stokes soli-
ton threshold is achieved and its spectrum is observed at
longer wavelengths relative to the primary soliton (fig-
ure 1b). Note that condition 1 (Raman gain) is satisified
as the new spectra lie within the Raman gain spectrum
created by the primary soliton (see shaded area in figure
2c).
The dispersion in the FSR of the Stokes soliton mode
family is measured in figure 1c to confirm condition 2
(matching FSR of primary and Stokes solitons). To mea-
sure the mode families a tunable external cavity diode
laser (ECDL) scans the spectral locations of optical res-
onances from 1520 nm to 1580 nm. The resonances ap-
pear as minima in the optical power transmitted past the
microcavity, and the location of these resonances is cali-
brated using a fiber-based Mach Zehnder interferometer.
The resulting data provide the dispersion in the FSR of
cavity modes versus the wavelength and readily enable
the identification transverse mode families. The results
of measurements on the three devices are summarized in
figure 1c. The FSR at the spectral center of the primary
soliton is plotted as a horizontal dashed line (red, green,
blue according to device) while the FSR of a neighboring
mode family is plotted versus wavelength. Even though
the range of the measurement is not sufficient to extend
into the wavelength band of the observed Stokes soliton,
an extrapolation of the data is performed using a modal
dispersion simulation
12
. As can be seen, the wavelength
where primary and Stokes solitons FSR are equal is close
to the spectral maximum of the corresponding Stokes
solitons in figure 1b. As a final comment, primary and
Stokes solitons are observed to have the same polariza-
tion.
The thresholding nature of Stokes soliton formation
is observed in figure 2a. The spectra show the primary
soliton spectra measured at three power levels and the
corresponding Stokes soliton spectra. One spectral trace
is measured for a power level below the Stokes soliton
4
RBW
10 kHz
RBW
10 kHz
Autocorrelation intensity (a.u.)
0
1
Delay (ps)
0
50
Scan time (
μ
s)
0
60
Soliton power (a.u.)
0
1
Primary soliton
Stokes soliton
(a)
Stokes soliton
off
Stokes soliton
off
Frequency offset (MHz)
-50
0
50
0
1
Electrical beatnote power
(20 dB / division)
Stokes soliton
on
(b)
(c)
Stokes soliton
on
FIG. 3:
Scanned soliton power, autocorrelation
measurements and detection of primary and
Stokes soliton microwave repetition rate. a,
Pri-
mary and Stokes soliton power scanned versus time.
Regions to the far left in the scan correspond to pump-
ing conditions that do not form a primary soliton. Re-
gions to the far right in the scan are beyond the exis-
tence limit of the primary soliton.
b,
Autocorrelation
measurements of the primary soliton both below and
above Stokes soliton oscillation threshold.
c,
Electrical
spectrum of the detected primary soliton pulse stream
(upper) and the combined primary and Stokes pulse
streams (lower). The zero on the frequency scale corre-
sponds to 22 GHz.
threshold. The spectral width and the power in all spec-
tral lines of the Stokes solitons grow with pumping above
threshold. This behavior is distinct from that of DKC
solitons, where increased soliton power is accompanied
only by increasing soliton spectral width (i.e., maximum
power per frequency line is fixed). Simulations of the
spectra and the corresponding pulses are also provided in
figure 2a and figure 2b. These use the coupled Lugiato-
Lefever equations (Methods Section). In figure 2c power
data are provided showing the primary soliton peak and
average power as well as the Stokes soliton power plotted
versus the total soliton power. Simulations are provided
for comparison with the power data. As an aside, the
Stokes soliton power can exceed the power of the pri-
mary soliton. This was observed in one device and is a
consequence of the threshold condition.
The threshold-induced gain-clamping condition im-
posed on the primary soliton (which functions as a pump
for the Stokes soliton) is readily observable in figure 2c.
As shown in the Methods Section, this threshold condi-
tion clamps the peak power of the primary soliton and is
given by the following expression,
P
th
=
κ
ext
p
κ
s
2
R
(1 +
1
2
γ
)
(1)
where
P
th
is the threshold peak output power of the pri-
mary soliton and other parameters are defined in the
Methods Section. Eqn. (1) is derived assuming a weak
Stokes soliton power relative to the primary soliton which
is an excellent assumption near threshold. Eqn. (1) is
plotted for comparison to the peak power data using the
same parameters used in the figure 2c simulation plots.
Another way to measure the relationship between
power in the primary and Stokes solitons is to monitor
the average power in their respective pulse trains while
the power in primary soliton is being scanned. Because
the primary soliton is a DKC soliton its power is varied
by scanning the detuning of the pump laser relative to
the pumping mode of the primary soliton
12
. The onset
of the DKC soliton oscillation occurs when the scanned
pump is red-detuned relative to the resonance, and its
power increases as the pump continues to detune further
to the red. Utlimately, the existence detuning limit is
reached and the primary soliton shuts off
12
. In figure 3a
a temporal scan shows the primary soliton increasing in
power and the appearance of the Stokes soliton towards
the end of the scan. Because it is the peak power of the
primary soliton that is clamped above the Stokes soliton
threshold, the
total
primary soliton power actually de-
creases once the Stokes turns on. This happens because
the primary soliton pulse width changes during the scan.
Autcorrelation measurements of the primary soliton
both with and without the Stokes soliton are presented in
figure 3b. Similar measurements were not possible for the
Stokes soliton on account of the wavelength limitations
of the autocorrelation system. However, broad optical
band detection of the soliton repetition rate was possible
for both the primary and Stokes solitons pulse streams.
Figure 3c shows the corresponding electrical spectrum
analyzer signal for detection of the primary soliton sig-
nal and for simultaneous detection of the both the pri-
mary and Stokes soliton signals. The frequency of the
beat note is identical in both cases confirming that the
corresponding soliton round-trip rates are locked.
Summary.
The Stokes soliton is only the second type
of soliton to be observed in microcavities (beyond dissi-
pative Kerr solitons
7
) and also represents the first time
soliton trapping has been observed in any microcavity. It
also represents the first observation of trapping by soli-
tons in different transverse modes in a laser. From a
practical viewpoint, the pilot and primary solitons over-
lap in space and time, and have a frequency separation
that can be engineered to fall within the mid IR range.
As a result, this soliton system is potentially interesting
for mid IR generation by way of difference frequency gen-
5
eration. Not all devices are observed to produce Stokes
solitons. However, dispersion engineering techniques are
being advanced
21
and should enable control of both ob-
servation of the Stokes soliton as well as its placement in
the optical spectrum. Indeed, the spectral placement of
Stokes solitons in figure 1b is largely the result of micro-
cavity diameter control to shift the FSR crossing point
(condition 2). The specific implementation described
here uses a compact microresonator on a silicon wafer
which also suggests that monolithic integration will be
possible. In an appropriately phase-matched multimode
waveguide (optical fiber or monolithic) it should also be
possible to observe non-cavity-based Stokes solitons.
Methods
Coupled Lugiato Lefever equations.
A pair of coupled equa-
tions describing the intracavity slowly-varying field ampl
itudes for
the primary and Stokes soliton system can be found from the
Lugiato-Lefever equation (LL equation)
7,22–24
augmented by Ra-
man terms
25,26
. By including cross-phase modulation and Raman
interaction terms the following system of equations result
s:
∂E
p
∂t
=
i
D
2
p
2
2
E
p
∂φ
2
+
ig
p
|
E
p
|
2
E
p
+
iG
p
|
E
s
|
2
E
p
ig
p
D
1
p
τ
R
E
p
(
|
E
p
|
2
+
A
pp
|
E
s
|
2
/A
ps
)
∂φ
(
κ
p
2
+
i
ω
p
)
E
p
ω
p
ω
s
R
|
E
s
|
2
E
p
+
κ
ext
p
P
in
,
(2)
∂E
s
∂t
=
δ
∂E
s
∂φ
+
i
D
2
s
2
2
E
s
∂φ
2
+
ig
s
|
E
s
|
2
E
s
+
iG
s
|
E
p
|
2
E
s
ig
s
D
1
p
τ
R
E
s
(
|
E
s
|
2
+
A
ss
|
E
p
|
2
/A
ps
)
∂φ
(
κ
s
2
+
i
ω
s
)
E
s
+
R
|
E
p
|
2
E
s
,
(3)
The slowly varying fields
E
j
(subscript
j
= (
p, s
) for primary or
Stokes soliton) are normalized to optical energy. To second
order
in the mode number the frequency of mode number
μ
in soliton
j
= (
p, s
) is given by the Taylor expansion
ω
μj
=
ω
0
j
+
D
1
j
μ
+
1
2
D
2
j
μ
2
where
ω
0
j
is the frequency of mode
μ
= 0, while
D
1
j
and
D
2
j
are the FSR and the second-order dispersion at
μ
= 0. Also,
δ
=
D
1
s
D
1
p
is the
F SR
difference between primary and Stokes
solitons at mode
μ
= 0.
κ
j
is the cavity loss rate and ∆
ω
j
is the
detuning of mode zero of the soliton spectrum relative to the
cold
cavity resonance. For the primary soliton, which is a DKC sol
iton,
the pump field is locked to one of the soliton spectral lines an
d this
“pump” line is taken as mode zero.
τ
R
is the Raman shock time,
κ
ext
j
is the external coupling coefficient and
P
in
is the pump power.
g
j
and
G
j
are self and cross phase modulation coefficients, defined
as,
g
j
=
n
2
ω
j
D
1
j
2
nπA
jj
, G
j
=
(2
f
R
)
n
2
ω
j
D
1
j
2
nπA
ps
.
(4)
where the nonlinear mode area
A
jk
is defined as
25
A
jk
=
∫∫
−∞
|
u
j
(
x, y
)
|
2
dxdy
∫∫
−∞
|
u
k
(
x, y
)
|
2
dxdy
∫∫
−∞
|
u
j
(
x, y
)
|
2
|
u
k
(
x, y
)
|
2
dxdy
,
(5)
where
u
j
is the transverse distribution of the mode.
f
R
=
0
.
18 is the Raman contribution parameter in silica.
R
=
cD
1
p
g
R
(
ω
s
, ω
p
)
/
4
nπA
ps
where
g
R
(
ω
s
, ω
p
) is the Raman gain in
silica. For solitons with a few THz bandwidth, other nonlin-
ear effects are negligible (e.g., higher order dispersion, t
he self-
steepening effect and Raman induced refractive index change
25
).
Phase-sensitive, four-wave-mixing terms have been omitte
d in Eqn.
(2) and (3). In principle, these terms could introduce locki
ng of
the Stokes and primary soliton fields (in addition to their re
peti-
tion rates). However, for this to occur the underlying spati
al mode
families would need to feature mode frequencies that align r
eason-
ably well (both in FSR and offset frequency) within the same ba
nd.
Such conditions do not seem likely even though they might occ
ur
accidentally or through dispersion engineering. In this wo
rk, one
device featured primary and Stokes solitons with overlappi
ng spec-
tra (see device 1 in figure 1b). As expected the mode frequence
s
associated with each soliton did not overlap.
The coupled equations are numerically studied using the spl
it-
step Fourier method. Over 600 modes are included in the simul
a-
tion for each soliton. Parameters are given below. Note that
the
detuning of the Stokes soliton determines the rotation fram
e, which
can be set to zero during the simulation.
Calculation of threshold.
The behavior of the Stokes soliton
system can be studied analytically beginning with the coupl
ed LL
equations. Near threshold the Stokes soliton is much weaker
in
power than the primary soliton. In this limit, the cross-pha
se
modulation and cross Raman interaction terms within the pri
mary
soliton equation can be neglected, while the self-phase mod
ulation
and self-Raman terms are neglected in the Stokes equation. T
he
primary soliton is then governed by the standard uncoupled L
L
equation, which features an analytical solution of the field
ampli-
tude (hyperbolic secant form)
7
. This solution is substituted into
the Stokes soliton equation which then takes a Shr ̈odinger-
like form
containing a sech
2
potential in addition to sech
2
Raman gain. The
specific solution to these two equations in the Stokes low pow
er
limit take the following form:
E
p
sech
Bφ, E
s
sech
γ
Bφ,
(6)
where
B
=
2∆
ω
p
/D
2
p
. The Stokes soliton solution is the general
solution for a soliton trapped in the index well created by th
e sech
2
primary soliton intensity where the power
γ
satisfies the equation,
γ
(
γ
+ 1) = 2
G
s
D
2
p
/g
p
D
2
s
.
(7)
Once the peak power of the primary soliton reaches a point tha
t
provides sufficient Raman gain to overcome roundtrip loss, th
e
Stokes soliton will begin to oscillate. The threshold condi
tion
emerges as the steady state Stokes soliton power balance. Th
is
is readily derived from the Stokes soliton equation and take
s the
form,
2
π
0
t
|
E
s
|
2
=
2
π
0
(
κ
s
2
R
|
E
p
|
2
)
|
E
s
|
2
= 0
(8)
By substituting the solutions (eqns. (6)) into eqn. (8), the
resulting
threshold in primary soliton peak output power is found to be
given
by eqn. (1).
Parameters.
The measured parameters are:
κ
p
/
2
π
= 736 kHz,
κ
ext
p
/
2
π
= 302 kHz,
λ
p
= 1550 nm,
D
1
p
/
2
π
= 22 GHz and
D
2
p
/
2
π
= 16
.
1 kHz. Neither the intrinsic or coupling loss of the
Stokes mode could be measured in the 1600 nm band. However, th
e
optical loss for the Stokes family could be measured in the 15
50 nm
band. Accordingly, intrinsic loss of
κ
o
s
/
2
π
= 1
.
11 MHz was used
while the external loss of
κ
ext
s
= 1
.
25 MHz was slightly tuned to ob-
tain the best fitting. The latter is expected to shift somewha
t from
1550 nm to 1600 nm bands due to small changes in phase matching
of the microcavity to waveguide coupling.
P
in
= 150 mW is used
throughout the simulation. Calculated parameters (based o
n mode
simulations) are:
D
2
s
/
2
π
= 21
.
7 kHz,
A
ss
= 69
.
8
μ
m
2
,
A
pp
= 39
.
7
μ
m
2
,
A
ps
= 120
μ
m
2
, and
δ
= 0 when
λ
s
= 1627 nm. Other con-
stants are:
n
= 1
.
45,
n
2
= 2
.
2
×
10
20
m
2
/
W,
g
R
= 3
.
94
×
10
14
m
/
W,
τ
R
= 3
.
2 fs. The calculated
γ
= 0
.
55.
Acknowledgments
The authors thank Steven Cundiff
at the University of Michigan for helpful comments on
this manuscript. The authors gratefully acknowledge the
Defense Advanced Research Projects Agency under the
QuASAR program and PULSE programs, NASA, the
Kavli Nanoscience Institute and the Institute for Quan-
tum Information and Matter, an NSF Physics Frontiers
6
Center with support of the Gordon and Betty Moore Foundation.
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