ARTICLE
Petermann-factor sensitivity limit near an
exceptional point in a Brillouin ring laser gyroscope
Heming Wang
1,4
, Yu-Hung Lai
1,2,4
, Zhiquan Yuan
1,4
, Myoung-Gyun Suh
1,3
& Kerry Vahala
1
✉
Exceptional points are singularities of open systems, and among their many remarkable
properties, they provide a way to enhance the responsivity of sensors. Here we show that the
improved responsivity of a laser gyroscope caused by operation near an exceptional point is
precisely compensated by increasing laser noise. The noise, of fundamental origin, is
enhanced because the laser mode spectrum loses the oft-assumed property of orthogonality.
This occurs as system eigenvectors coalesce near the exceptional point and a bi-orthogonal
analysis con
fi
rms experimental observations. While the results do not preclude other pos-
sible advantages of the exceptional-point-enhanced responsivity, they do show that the
fundamental sensitivity limit of the gyroscope is not improved through this form of operation.
Besides being important to the physics of microcavities and non-Hermitian photonics, these
results help clarify fundamental sensitivity limits in a speci
fi
c class of exceptional-point
sensor.
https://doi.org/10.1038/s41467-020-15341-6
OPEN
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA.
2
Present address: OEwaves Inc., 465 North Halstead
Street, Suite 140, Pasadena, CA 91107, USA.
3
Present address: Physics and Informatics Laboratories, NTT Research, Inc., East Palo Alto, CA 94303, USA.
4
These authors contributed equally: Heming Wang, Yu-Hung Lai, Zhiquan Yuan.
✉
email:
vahala@caltech.edu
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1
1234567890():,;
N
on-Hermitian Hamiltonians
1
,
2
describing open systems
can feature singularities called exceptional points
(EPs)
3
–
5
. EPs have been experimentally realized in several
systems
6
–
8
and applied to demonstrate non-reciprocal transmis-
sion
9
–
11
and lasing dynamics control
12
–
15
. Moreover, resonant
frequencies become strongly dependent on externally applied
perturbations near an EP, which has given rise to the concept of
EP-enhanced sensing in photonics
16
–
19
and electronics
20
,
21
.
While increased sensor responsivity has been demonstrated in
several systems
22
–
25
, signal-to-noise performance (sensitivity) has
been considered only theoretically
26
–
30
.
Recently, strong responsivity improvement near an EP was
reported in a Brillouin ring laser gyroscope by monitoring an
increase in the gyroscope scale factor (i.e., transduction factor of
rotation-rate into a signal)
24
. At the same time, however, mea-
surement of the gyroscope Allan deviation versus averaging time
showed that short-term laser frequency noise also increased near
the EP. This noise was random-walk in nature, suggesting a
fundamental origin. Moreover, it depended upon system bias
relative to the EP in such a way so as to precisely compensate for
the observed EP-enhanced transduction. As a result, the gyro-
scope
’
s angular random walk, the metric used to quantify short-
term rotation sensitivity, was observed to maintain a constant
value (i.e., independent of gyroscope bias relative to the EP). In
effect, the measurements showed that gyroscope sensitivity (i.e.,
weakest rotation signal measurable at a given detection band-
width) is not improved by operation near the EP even while the
gyroscope responsiveness through improved transduction (scale
factor) increases.
As with all laser gyroscopes, the Brillouin ring laser gyroscope
measures rotations through the Sagnac effect
31
. Clockwise (cw)
and counter-clockwise (ccw) lasing waves experience opposing
frequency shifts when the plane of the gyroscope rotates. By
mixing the two laser
fi
elds on a detector, their difference fre-
quency therefore reveals the rotation-induced frequency shift
added onto a constant bias frequency (which is at audio rates in
this case
24
). Frequency noise in the beat frequency therefore
determines the measurement sensitivity. This noise has both a
technical component (observable on longer time scales in the
Allan deviation
24
) as well as a random walk component that,
absent the EP, is known to result from fundamental linewidth
broadening of the Brillouin laser waves
32
,
33
. Signi
fi
cantly, sub-
sequent measurement of the random walk component showed
that none of the parameters which normally impact its magnitude
(e.g., laser power, cavity Q factor) varied near the EP, therefore
suggesting that frequency noise (and linewidth) is increased by
way of another mechanism.
Laser linewidth can also be broadened by the Petermann fac-
tor
34
–
39
. This mechanism is associated with non-orthogonality of
a mode spectrum, and its connection to EPs has been considered
in theoretical studies of microresonators
40
,
41
. However, despite
continued theoretical interest
42
,
43
, including the development of
new techniques for determination of linewidth in general laser
systems
44
, the observation of Petermann linewidth broadening
near exceptional points was reported only recently by the Yang
group in a phonon laser system
45
, and the link between
Petermann-factor-induced noise and EP sensor performance is
unexplored.
Here, it is shown that mode non-orthogonality induced by the
EP limits the gyroscope sensitivity via Petermann-factor line-
width broadening. Indeed, analysis and measurement con
fi
rm
near-perfect cancellation of the signal transduction improvement
by increasing Petermann-factor noise, so that the gyroscope
’
s
fundamental signal-to-noise ratio (SNR) and hence sensitivity is
not improved by operation near the EP. These results are further
con
fi
rmed using an Adler phase locking equation approach
46
,
47
,
which is also applied to analyze the combined effect of dissipative
and conservative coupling on the system.
Results
Biorthogonal noise enhancement theory
. The gyroscope uses a
high-Q silica whispering gallery resonator
48
in a ring-laser con-
fi
guration
32
. As illustrated in Fig.
1
a, optical pumping of cw and
ccw directions on the same whispering-gallery mode index
induces laser action through the Brillouin process. On account of
the Brillouin phase matching condition, these stimulated Bril-
louin laser (SBL) waves propagate in a direction opposite to their
corresponding pump waves
33
. Dissipative backscattering
49
cou-
ples the SBLs and the following Hamiltonian governs the above-
laser-threshold motion
24
:
H
¼
ω
cw
i
Δ
ω
EP
=
2
i
Δ
ω
EP
=
2
ω
ccw
ð
1
Þ
where
H
describes the dynamics via
id
Ψ
=
dt
¼
H
Ψ
and
Ψ
¼
a
cw
;
a
ccw
ðÞ
T
is the column vector of SBL mode amplitudes
(square of norm is photon number). Also,
Δ
ω
EP
is a non-
Hermitian term related to the coupling rate between the two SBL
modes and
ω
cw
(
ω
ccw
) is the active-cavity resonance angular
frequency of the cw (ccw) SBL mode above laser threshold. The
dependence of
ω
cw
,
ω
ccw
, and
Δ
ω
EP
on other system parameters,
most notably the angular rotation rate and the optical pumping
frequencies, has been suppressed for clarity.
A class of EP sensors operates by measuring the frequency
difference of the two system eigenmodes. This difference is
readily calculated from Eq. (
1
)as
Δ
ω
S
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Δ
ω
2
D
Δ
ω
2
EP
p
where
Δ
ω
D
ω
ccw
ω
cw
is the resonance frequency difference and
Δ
ω
EP
is the critical value of
Δ
ω
D
at which the system is biased at
the EP. As illustrated in Fig.
1
b, c the vector composition of
the SBL modes strongly depends upon the system proximity
to the EP. For
j
Δ
ω
D
j
Δ
ω
EP
the SBL modes (unit vectors) are
orthogonal cw and ccw waves. However, closer to the EP the
waves become admixtures of these states that are no longer
orthogonal. At the EP, the two waves coalesce to a single state
vector (a standing wave in the whispering gallery). Rotation of the
gyroscope in state II in Fig.
1
Δ
ω
D
jj
>
Δ
ω
EP
ðÞ
introduces a
perturbation to
Δ
ω
D
whose transduction into
Δ
ω
S
is enhanced
relative to the conventional Sagnac factor
31
. This EP-induced
signal-enhancement-factor (SEF) is given by
24
,
SEF
¼
∂
Δ
ω
S
∂
Δ
ω
D
2
¼
Δ
ω
2
D
Δ
ω
2
D
Δ
ω
2
EP
ð
2
Þ
where SEF refers to the signal power (not amplitude) enhance-
ment. This factor has recently been veri
fi
ed in the Brillouin ring
laser gyroscope
24
. The control of
Δ
ω
D
(and in turn
Δ
ω
S
) in that
work and here is possible by tuning of the optical pumping
frequencies and is introduced later.
Δ
ω
S
is measured as the beat frequency of the SBL laser signals
upon photodetection and the SNR is set by the laser linewidth. To
understand the linewidth behavior a bi-orthogonal basis is used
as described in Supplementary Notes 1 and 3. As shown there and
illustrated in Fig.
1
d, the peculiar properties of non-orthogonal
systems near the EP cause the unit vectors (optical modes) to be
lengthened. This lengthening results in an effectively shorter laser
fi
eld amplitude. Also, noise into the mode is increased as
illustrated in Fig.
1
e. Because the laser linewidth can be
understood to result from diffusion of the phasor in Fig.
1
e,
linewidth increases upon operation close to the EP. And the
linewidth enhancement is given by the Petermann factor (see
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Supplementary Note 2),
PF
¼
1
2
1
þ
Tr
ð
H
y
0
H
0
Þ
j
Tr
ð
H
2
0
Þj
!
¼
Δ
ω
2
D
Δ
ω
2
D
Δ
ω
2
EP
ð
3
Þ
where Tr is the matrix trace operation and
H
0
¼
H
Tr
ð
H
Þ
=
2is
the traceless part of
H
. As derived in Supplementary Note 2, the
fi
rst part of this equation is a basis independent form and is valid
for a general two-dimensional system. The second part is speci
fi
c
to the current SBL system. Inspection of Eqs. (
2
) and (
3
) shows
that SEF
=
PF. As a result the SNR is not expected to improve
through operation near the EP when the system is fundamental-
noise limited.
Petermann noise measurement
. To verify the above predictions,
the output of a single pump laser (~1553.3 nm) is divided into
two branches that are coupled into cw and ccw directions of the
resonator using a tapered
fi
ber
50
,
51
. Both pump powers are
actively stabilized. The resonator is mounted in a sealed box and a
thermo-electric cooler (TEC) controls the chip temperature
which is monitored using a thermistor (
fl
uctuations are held
within 5 mK). Each pumping branch has its frequency controlled
using acousto-optic modulators (AOMs). SBL power is also
monitored and controlled so that
fl
uctuations are within 0.6%.
Even with the control of temperature and power, the Allan
deviation at longer gate times re
fl
ects technical-noise drifting that
is observed to be more pronounced for operation near the EP. As
described in ref.
24
, the ccw pump laser frequency is Pound-
Drever-Hall (PDH) locked to one resonator mode and the cw
pump laser can then be independently tuned by the AOM. This
pump detuning frequency (
Δ
ω
P
) is therefore controlled to radio-
frequency precision. It is used to precisely adjust
Δ
ω
D
and in turn
Δ
ω
S
as shown in three sets of measurements in Fig.
2
a. Here, the
photodetected SBL beat frequency
Δ
ω
S
is measured using a fre-
quency counter. The data sets are taken for three distinct SBL
output amplitude ratios as discussed further below. A solid curve
fi
tting is also presented using
Δ
ω
S
¼
±
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Δ
ω
2
D
Δ
ω
2
EP
p
, where
Δ
ω
D
¼
γ
=
Γ
1
þ
γ
=
Γ
Δ
ω
P
þ
1
1
þ
γ
=
Γ
Δ
ω
Kerr
(see Supplementary Note 4).
Also,
γ
is the photon decay rate,
Γ
is the Brillouin gain band-
width
33
, and
Δ
ω
Kerr
is a Kerr effect correction that is explained
below. As an aside, the data plot and theory show a frequency
locking zone, the boundaries of which occur at the EP.
The frequency counter data are also analyzed as an Allan
deviation (Adev) measurement (Fig.
2
a inset). The initial roll-off
of the Adev features a slope of
1/2 corresponding to white
frequency noise
52
. This was also veri
fi
ed in separate measure-
ments of the beat frequency using both an electrical spectrum
analyzer and a fast Fourier transform. The slope of this region is
fi
tto
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S
ν
=
ð
2
τ
Þ
p
where
S
ν
is the one-sided spectral density of the
CW
CCW
Im(
a
)
Re(
a
)
SBL2
SBL1
|
e
CCW
〉
|
e
CW
〉
|
e
2
〉
|
e
1
〉
I
II
III
0
1
CW
CCW
I
I
II
II
III
I
II
I
I
II
II
III
I
I
II
II
III
ac
b
d
e
|Δ
D
|
>>
Δ
EP
|Δ
D
|
>
Δ
EP
|Δ
D
|
=
Δ
EP
CW Pump
CCW Pump
CW
SBL
CCW
SBL
Fig. 1 Brillouin laser linewidth enhancement near an exceptional point. a
Diagram of whispering-gallery mode resonator with the energy distribution of an
eigenmode superimposed. The eigenmode energy distribution corresponds to state III in
b
. Optical pumps on the coupling waveguide and whispering-
gallery SBL modes are indicated by arrows.
b
Mode energy distributions for three different states: far from EP (state I) the eigenmodes are traveling cw or
ccw waves; near EP (state II) the eigenmodes are hybrids of cw and ccw waves; at EP (state III) eigenmodes coalesce to a standing wave.
c
Bloch sphere
showing the eigenstates for cases I, II and III with corresponding cw and ccw composition.
d
Illustration of the cw-ccw and SBL1-SBL2 coordinate systems.
Unit vectors for states I and II are shown on each axis. As the system is steered towards the EP, the SBL axes move toward each other so that unit vectors
along the SBL axes lengthen as described by the two hyperbolas. This is illustrated by decomposing a unit vector of the non-orthogonal SBL coordinate
system using the orthogonal cw-ccw coordinates [e.g., (5/4, 3/4)
T
and (3/4, 5/4)
T
for state II]. Consequently, the
fi
eld amplitude is effectively shortened
in the SBL basis.
e
Phasor representation of the complex amplitude of a lasing mode for states I and II provides an interpretation of linewidth enhancement.
Phasor length is shortened and noise is enhanced as the system is steered to the EP, leading to an increased phasor angle diffusion and laser linewidth
enhancement (see Supplementary Note 3).
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3
white frequency noise. Adev measurement at each of the detuning
points in Fig.
2
a is used to infer the
S
ν
values that are plotted in
Fig.
2
b. There, a frequency noise enhancement is observed as the
system is biased toward an EP. Also plotted is the Petermann
factor noise enhancement (Eq. (
3
)). Aside from a slight
discrepancy at intermediate detuning frequencies (analyzed
further below), there is an overall excellent agreement between
theory and measurement. The frequency noise levels measured in
Fig.
2
b are consistent with fundamental SBL frequency noise (see
Methods). Signi
fi
cantly, the fundamental nature of the noise, the
good agreement between the PF prediction (Eq. (
3
)) and
measurement in Fig.
2
b, and separate experimental work
24
that
has veri
fi
ed the theoretical form of the SEF (Eq. (
2
)) con
fi
rm that
SEF
=
PF so that the fundamental sensitivity limit of the
gyroscope is not improved by operation near the EP.
Adler noise analysis
. While the Petermann factor analysis pro-
vides very good agreement with the measured results, we also
derived an Adler-like coupled mode equation analysis for the
Brillouin laser system. This approach is distinct from the bi-
orthogonal framework and, while more complicated, provides
additional insights into the system behavior. Adapting analysis
applied in the noise analysis of ring laser gyroscopes
47
, a noise
enhancement factor NEF results (see Supplementary Note 4),
NEF
¼
Δ
ω
2
D
þ
Δ
ω
2
EP
=
2
Δ
ω
2
D
Δ
ω
2
EP
ð
4
Þ
It is interesting that this result, despite the different physical
context of the Brillouin laser system, has a similar form to one
derived for polarization-mode-coupled laser systems
53
. The PF
and NEF predictions are shown in Fig.
2
b and the Adler-derived
NEF correction provides slightly better agreement with the data at
the intermediate detuning values.
Adler locking bandwidth analysis
. The Adler approach is also
useful to explain a locking zone dependence upon SBL amplitudes
observed in Fig.
2
a. As shown in Supplementary Note 4, this
variation can be explained through the combined action of the
Kerr effect and intermodal coupling coef
fi
cients of both dis-
sipative and conservative nature. Speci
fi
cally, the locking band-
width is found to exhibit the following dependence upon the
amplitude ratio
q
¼j
a
ccw
=
a
cw
j
of the SBL lasers,
Δ
ω
2
EP
¼
Γ
Γ
þ
γ
2
q
þ
1
q
2
j
κ
j
2
þ
q
1
q
2
j
χ
j
2
"#
ð
5
Þ
where
κ
is the dissipative coupling and
χ
is the conservative cou-
pling between cw and ccw SBL modes. The locking zone bound-
aries in terms of pump detuning frequency have been measured
(Fig.
3
inset) for a series of different SBL powers. Using this data,
the locking bandwidth is expressed in pump frequency detuning
(
Δ
ω
P
) units using
Δ
ω
C
ð
1
þ
Γ
=
γ
Þ
Δ
ω
EP
and plotted versus
q
in
the main panel of Fig.
3
. The plot agrees well with Eq. (
5
)(
fi
tting
shown in black) and gives
j
κ
j
=
0.93 kHz,
j
χ
j
=
8.21 kHz.
Finally, the center of the locking band is shifted by the Kerr
effect and (in pump frequency detuning
Δ
ω
P
units) can be
expressed as
ð
Γ
=
γ
Þ
Δ
ω
Kerr
, where
Δ
ω
Kerr
¼
η
a
ccw
jj
2
a
cw
jj
2
¼
ð
η
Δ
P
SBL
Þ
=
ð
γ
ex
_
ω
Þ
is the Kerr induced SBL resonance frequency
difference,
Δ
P
SBL
¼
P
ccw
P
cw
is the output power difference of
the SBLs, and
γ
ex
is the photon decay rate due to the output
coupling. Also,
η
¼
n
2
_
ω
2
c
=
ð
Vn
2
0
Þ
is the single-photon Kerr-
effect angular frequency shift with
ω
the SBL angular frequency,
n
2
the Kerr-nonlinear refractive index of silica,
V
the mode
volume,
n
0
the linear refractive index, and
c
the speed of light in
vacuum. If the white frequency noise
fl
oors in Fig.
2
are used to
infer the resonator quality factor, then a Kerr nonlinearity value
of 558 Hz
μ
W
−
1
is predicted (see Methods). This value gives the
line plot in the Fig.
3
inset (with no free parameters), which
agrees with experiment.
Discussion
Prior work has shown that Brillouin laser gyroscopes when
operated near an EP have an improved responsivity (equivalently,
an increase in the gyroscope
’
s scale factor for transduction of
rotation rate into the Sagnac frequency shift)
24
. At the same time,
these measurements have shown that the gyroscope
’
s sensitivity
did not improve near the EP. We have veri
fi
ed through mea-
surement and theory that mode non-orthogonality induced by
the EP explains this latter result. Speci
fi
cally, increasing mode
non-orthogonality occurs when the two system eigenvectors
(optical modes) begin to coalesce near the EP. This, in turn,
increases laser frequency noise from an increasing Petermann
factor and thereby reduces sensitivity. Curiously, these two
mechanisms, the enhanced transduction and enhanced noise,
feature an almost identical dependence upon the system
’
s
10
–2
10
–1
10
0
10
1
(s)
10
–1
10
2
10
1
10
0
σ
v
(Hz)
Allan deviation
Fitting
–20
–10
0
10
20
SBL beating frequency
Δ
S
/2
π
(kHz)
Experiment data
Fitting curve
Slope = –1/2
Exp.
PF
NEF
–300
–200
–100
0
100
200
300
Pump detuning frequency
Δ
P
/2
π
(kHz)
10
White frequency noise
S
(Hz
2
Hz
–1
)
2
5
1
0.5
20
a
b
Fig. 2 Measured linewidth enhancement of SBLs near the exceptional
point. a
Measured SBL beating frequency is plotted versus pump detuning
for three distinct locking zones, corresponding SBL amplitude ratios
q
: 1.15
(blue), 1 (orange), 0.85 (red). Solid curves are theoretical
fi
ttings. Inset is
a typical Allan deviation measurement of frequency (
σ
ν
ð
τ
Þ
) versus gate
time
τ
. Error bars give the standard error of the mean. The short-term part
is
fi
tted with
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S
ν
=
ð
2
τ
Þ
p
where
S
ν
is the one-sided power spectral density of
the white frequency noise plotted in
b
.
b
Measured white frequency noise
of the beating signal is determined using the Allan deviation measurement.
Data point color corresponds to the amplitude ratios used in
a
. The
Petermann factor PF (solid lines) and NEF (dashed lines) theoretical
predictions use parameters obtained by
fi
tting from
a
.
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proximity to the EP. In effect, the increased signal response in the
gyroscope arising from the EP does not lead to an improvement
in the minimum detectable signal (sensitivity).
It is interesting to note that a recent theoretical study of noise
limitations in a class of non-lasing EP sensors showed no fun-
damental sensitivity advantage for operation near the EP
28
.
Nonetheless it is still possible that other sensing modalities could
bene
fi
t from operation near an EP. Moreover, open systems offer
other potentially useful ways for transduction of rotation
54
. Also,
the proposal of conservative nonlinear mode coupling provides a
potential way to enhance the Sagnac effect
55
–
57
. The apparent
divergence of the linewidth near the EP is an interesting feature of
the current model and also one that agrees well with the data (at
least in the range of the measurement). Nonetheless, constraints
to this divergence set by the linewidth of the passive cavity loss
are a subject of further study. More generally, the excellent
control of the state space that is possible in the Brillouin system
can provide a new platform for studies of the remarkable physics
associated with exceptional points.
Methods
Linewidth and Allan deviation measurement
. In experiments, frequency is
measured in the time domain using a frequency counter and its Allan deviation is
calculated for different averaging times (Fig.
2
a). The Allan deviation
σ
ν
ð
τ
Þ
for a
signal frequency is de
fi
ned by
σ
ν
τ
ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2
M
1
ðÞ
X
M
1
k
¼
1
ν
k
þ
1
ν
k
2
v
u
u
t
ð
6
Þ
where
τ
is the averaging time,
M
is the number of frequency measurements, and
ν
k
is the average frequency of the signal (measured in Hz) in the time interval between
k
τ
and
ð
k
þ
1
Þ
τ
. The Allan deviation follows a
τ
1
=
2
dependence when the
underlying frequency noise spectral density is white
52
as occurs for laser frequency
noise limited by spontaneous emission. White noise causes the lineshape of the
laser to be a Lorentzian. White noise is also typically dominant in the Allan
deviation plot at shorter averaging times where
fl
icker noise and frequency drift are
not yet important. This portion of the Allan deviation plot can be
fi
t using
σ
ν
ð
τ
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S
ν
=
ð
2
τ
Þ
p
where
S
ν
is the white frequency noise one-sided spectral density func-
tion. This result can be further converted to the Lorentzian full-width at half
maximum (FWHM) linewidth
Δ
ν
FWHM
(measured in Hz) using the conversion,
S
ν
¼
2
σ
2
ν
ð
τ
Þ
τ
¼
1
π
Δ
ν
FWHM
ð
7
Þ
Experimental parameters and data
fi
tting
. The resonator is pumped at the
optical wavelength
λ
¼
1553
:
3 nm, which, subject to the Brillouin phase matching
condition, corresponds to a phonon frequency (Stokes frequency shift) of
approximately
Ω
phonon
=
ð
2
π
Þ¼
10
:
8 GHz. Quality factors of the SBL modes are
measured using a Mach-Zehnder interferometer, and a loaded Q factor
Q
T
¼
88
́
10
6
and coupling Q factor
Q
ex
¼
507
́
10
6
are obtained.
The theoretical formula for the white frequency noise of the beat frequency far
away from the EP reads,
S
ν
¼
Γ
γ
þ
Γ
2
_
ω
3
4
π
2
Q
T
Q
ex
1
P
cw
þ
1
P
ccw
n
th
þ
N
th
þ
1
ðÞ
ð
8
Þ
which results from summing the Schawlow-Townes-like linewidths of the SBL laser
waves
33
. In the expression,
N
th
and
n
th
are the thermal occupation numbers of the
SBL state and phonon state, respectively. At room temperature,
n
th
577 and
N
th
0. For the power balanced case (orange data set in Fig.
2
),
P
cw
=
P
ccw
=
215
μ
W and the predicted white frequency noise (Eq. (
8
)) is
S
ν
=
0.50 Hz
2
Hz
−
1
.
For the blue (red) data set,
P
cw
(
P
ccw
) is decreased by 1.22 dB (1.46 dB) so that
S
ν
¼
0.58 (0.60) Hz
2
Hz
−
1
is calculated. On the other hand, the measured values for the
blue, orange and red data sets in Fig.
2
b (i.e., white frequency noise
fl
oors far from
EP) give
S
ν
¼
0.44, 0.39, 0.46 Hz
2
Hz
−
1
, respectively. The difference here is
attributed to errors in Q measurement. For example, the experimental values of
noise can be used to infer a corrected coupling Q factor
Q
ex
658
́
10
6
. Using this
value below yields an excellent prediction of the Kerr nonlinear coef
fi
cient which
supports this belief.
The beating frequency in Fig.
2
ais
fi
t using the following relations:
Δ
ω
S
¼
sgn
ð
Δ
ω
D
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Δ
ω
2
D
Δ
ω
2
EP
q
Δ
ω
D
¼
γ
=
Γ
1
þ
γ
=
Γ
Δ
ω
P
þ
1
1
þ
γ
=
Γ
Δ
ω
Kerr
ð
9
Þ
where sgn is the sign function and
γ
=
Γ
,
Δ
ω
Kerr
, and
Δ
ω
EP
are
fi
tting parameters.
The
fi
tting gives
γ
=
Γ
¼
0
:
076 consistently, while
Δ
ω
Kerr
and
Δ
ω
EP
are separately
adjusted in each data set. These parameters feature a power dependence that is fully
explored in Fig.
3
and the related main text discussion.
The theoretical Kerr coef
fi
cient used in Fig.
3
can be calculated as follows.
Assuming
n
2
2
:
7
́
10
20
m
2
=
W,
n
0
¼
1
:
45 for the silica material, and
V
=
10
7
μ
m
3
(obtained through
fi
nite-element simulations for the 36mm-diameter disk
used here), gives
η
=
2
π
10
5
Hz. Using the
Q
ex
corrected by the white
frequency noise data (see discussion above),
γ
ex
=
2
π
¼
299 kHz so that
Δ
ω
Kerr
=
ð
2
π
Δ
P
SBL
Þ
42 Hz
μ
W
−
1
.When
γ
=
Γ
¼
0
:
076, the center shift of pump
locking band is
ð
Γ
=
γ
Þ
Δ
ω
Kerr
=
558 Hz
μ
W
−
1
. This value agrees very well with
experiment (Fig.
3
inset).
Data availability
The data that support the plots within this paper and other
fi
ndings of this study are
available from the corresponding author upon reasonable request.
Received: 6 November 2019; Accepted: 28 February 2020;
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Acknowledgements
This work was supported by the Defense Advanced Research Projects Agency (DARPA)
under PRIGM:AIMS program through SPAWAR (grant no. N66001-16-1-4046) and the
Kavli Nanoscience Institute.
Author contributions
H.W., Y-H.L., and K.V. conceived the idea. H.W. derived the theory with feedback from
Y-H.L., Z.Y., and K.V. Y-H.L. designed and perform the experiments with Z.Y. and H.W.
Z.Y. analysed the data with Y-H.L. and H.W. M-G.S. fabricated the devices. All authors
participated in writing the paper. K.V. supervised the research.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary information
is available for this paper at
https://doi.org/10.1038/s41467-
020-15341-6
.
Correspondence
and requests for materials should be addressed to K.V.
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