of 15
Petermann-factor limited sensing near an exceptional point
Heming Wang
1
,
, Yu-Hung Lai
1
,
, Zhiquan Yuan
1
,
, Myoung-Gyun Suh
1
and Kerry Vahala
1
,
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA.
These authors contributed equally.
Corresponding author: vahala@caltech.edu
Non-Hermitian Hamiltonians [1, 2] describing
open systems can feature singularities called ex-
ceptional points (EPs) [3–5]. Resonant frequen-
cies become strongly dependent on externally ap-
plied perturbations near an EP which has given
rise to the concept of EP-enhanced sensing in
photonics [6–9]. However, while increased sen-
sor responsivity has been demonstrated [10–12],
it is not known if this class of sensor results
in improved signal-to-noise performance [13–17].
Here, enhanced responsivity of a laser gyroscope
caused by operation near an EP is shown to be
exactly compensated by increasing sensor noise in
the form of linewidth broadening. The noise, of
fundamental origin, increases according to the Pe-
termann factor [18, 19], because the mode spec-
trum loses the oft-assumed property of orthog-
onality. This occurs as system eigenvectors coa-
lesce near the EP and a biorthogonal analysis con-
firms experimental observations. Besides its im-
portance to the physics of microcavities and non-
Hermitian photonics, this is the first time that
fundamental sensitivity limits have been quanti-
fied in an EP sensor.
EPs have been experimentally realized in several sys-
tems [20–22] and applied to demonstrate non-reciprocal
transmission [23–25], lasing dynamics control [26–29] and
improved optical-based sensors [6–9]. The connection of
the Petermann factor [18, 19, 30–32] to EPs was con-
sidered in theoretical studies of microresonators [33, 34].
However, despite continued theoretical interest [35, 36]
including the development of new techniques for deter-
mination of linewidth in general laser systems [37], the
observation of Petermann linewidth broadening near ex-
ceptional points was reported only recently by the Yang
group in a phonon laser system [38].
And the link
between Petermann-factor-induced noise and EP sensor
performance has not been considered.
Recently, strong sensing improvement near an EP was
reported in gyroscopes operating near an EP [12]. Here it
is shown that mode non-orthogonality induced by the EP
severely limits the benefit of this improvement. Indeed,
analysis and measurement confirm near perfect cancel-
lation of signal improvement by increasing noise so that
gyroscope signal-to-noise ratio (SNR) and hence sensi-
tivity is not improved by operation near the EP. The
impact of mode non-orthogonality on laser noise is an-
alyzed using the Petermann factor and compared with
measurements. These results are further confirmed using
an Adler phase locking equation approach [39] which is
also applied to analyze the combined effect of dissipative
and conservative coupling on the system.
The gyroscope uses a high-Q silica whispering gallery
resonator [40] in a ring-laser configuration [41].
As
illustrated in Fig.
1a, optical pumping of clockwise
(cw) and counter-clockwise (ccw) directions on the
same whispering-gallery mode index induces laser action
through the Brillouin process. On account of the Bril-
louin phase matching condition, these stimulated Bril-
louin laser (SBL) waves propagate in a direction oppo-
site to their corresponding pump waves [42]. Dissipative
backscattering couples the SBLs and above threshold the
following Hamiltonian governs their motion [12]:
H
=
(
ω
cw
i
ω
EP
/
2
i
ω
EP
/
2
ω
ccw
)
(1)
where
H
describes the dynamics via
id
Ψ
/dt
=
H
Ψ and
Ψ = (
a
cw
,a
ccw
)
T
is the column vector of SBL mode am-
plitudes (square of norm is photon number). Also, ∆
ω
EP
is a non-Hermitian term related to the coupling rate be-
tween 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 ∆
ω
EP
on other system parameters, most no-
tably the angular rotation rate and the optical pumping
frequencies, has been suppressed for clarity.
A class of EP sensors operate by measuring the fre-
quency difference of the two system eigenmodes. This
difference is readily calculated from Eq. (1) as ∆
ω
S
=
ω
2
D
ω
2
EP
where ∆
ω
D
ω
ccw
ω
cw
is the reso-
nance frequency difference and ∆
ω
EP
is the critical value
of ∆
ω
D
at which the system is biased at the EP. As il-
lustrated in Fig. 1b,c the vector composition of the SBL
modes strongly depends upon the system proximity to
the EP. For
|
ω
D
| 
ω
EP
the SBL modes (unit vec-
tors) 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 (
|
ω
D
|
>
ω
EP
) introduces a perturbation to
ω
D
whose transduction into ∆
ω
S
is enhanced relative
to the conventional Sagnac factor [43]. This EP-induced
arXiv:1911.05191v1 [physics.ins-det] 8 Nov 2019
2
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
a
c
b
d
e
|∆ω
D
|
>
∆ω
EP
|∆ω
D
|
=
∆ω
EP
|∆ω
D
|
>>
∆ω
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. A portion of the resonator is outlined corresponding to
state
III
in panel
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 eigenmode is a traveling cw or ccw wave; near EP
(state
II
) the eigenmode is a hybrid of cw and ccw waves; at EP (state
III
) the eigenmode is 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 field 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 Information).
signal-enhancement-factor (SEF) is given by [12],
SEF =
ω
S
ω
D
2
=
ω
2
D
ω
2
D
ω
2
EP
(2)
where SEF refers to the signal power (not amplitude)
enhancement. This factor has recently been verified in
the Brillouin ring laser gyroscope [12]. The control of
ω
D
(and in turn ∆
ω
S
) in that work and here is possi-
ble by tuning of the optical pumping frequencies and is
introduced later.
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 its linewidth behavior a
bi-orthogonal basis is used as described in the Supple-
mentary Information. As shown there and illustrated in
Fig. 1d, the peculiar properties of non-orthogonal sys-
tems near the EP cause the unit vectors (optical modes)
to be lengthened. This lengthening results in an effec-
tively shorter laser field amplitude. Also, noise into the
mode is increased as illustrated in Fig. 1e. Because the
laser linewidth can be understood to result from diffusion
of the phasor in Fig. 1e, linewidth increases upon oper-
ation close to the EP. And the linewidth enhancement is
given by the Petermann factor (see Supplementary Infor-
mation),
PF =
1
2
(
1 +
Tr(
H
0
H
0
)
|
Tr(
H
2
0
)
|
)
=
ω
2
D
ω
2
D
ω
2
EP
(3)
where Tr is the matrix trace operation and
H
0
=
H
Tr(
H
)
/
2 is the traceless part of
H
. As derived in the
Supplementary Information, the first part of this equa-
tion is a basis independent form and is valid for a general
two-dimensional system. The second part is specific to
the current SBL system. Inspection of Eq. (2) and Eq.
3
10
-2
10
-1
10
10¹
τ
(s)
10
-1
10
10¹
10
2
σ
v
(Hz)
Allan Deviation
Fitting
-20
-10
0
10
20
SBL beating Frequency
∆ω
S
/2
π
(kHz)
Experiment Data
Fitting Curve
slope = -1/2
Exp.
PF
NEF
-300
-200
-100
0
100
200
300
Pump Detuning Frequency
∆ω
P
/2
π
(kHz)
0.5
10
White Frequency Noise S
ν
(Hz²/Hz)
2
5
1
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 (green), 0.85 (red). Solid curves are theoretical fit-
tings. Inset is a typical Allan deviation of frequency
σ
ν
(
τ
) ver-
sus gate time
τ
. The short-term part is fitted with
S
ν
/
(2
τ
)
where
S
ν
is the one-sided power spectral density of the white
frequency noise plotted in panel
b
.
b
, Measured white fre-
quency noise of the beating signal determined using the Allan
deviation measurement. Data point color corresponds to same
amplitude ratios used in panel
a
. Petermann factor PF (solid
lines) and the NEF (dashed lines) theoretical predictions use
parameters obtained by fitting from panel
a
.
(3) shows that SEF = PF. As a result the SNR is not ex-
pected to improve through operation near the EP when
the system is fundamental-noise limited.
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 res-
onator using a tapered fiber [44, 45]. 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 ther-
mistor (fluctuations are held within 5 mK). Each pump-
ing branch has its frequency controlled using acousto-
optic modulators (AOMs). The ccw pump laser fre-
quency is Pound-Drever-Hall (PDH) locked to one res-
onator mode and the cw pump laser can then be inde-
pendently tuned by the AOM. This pump detuning fre-
quency (∆
ω
P
) is therefore controlled to radio-frequency
precision. It is used to precisely adjust ∆
ω
D
and in
turn ∆
ω
S
as shown in three sets of measurements in
Fig. 2a. Here, the photodetected SBL beat frequency
ω
S
is measured using a frequency counter. The data
sets are taken for three distinct SBL output amplitude
ratios as discussed further below. A solid curve fitting
is also presented using ∆
ω
S
=
±
ω
2
D
ω
2
EP
, where
ω
D
=
γ/
Γ
1+
γ/
Γ
ω
P
+
1
1+
γ/
Γ
ω
Kerr
(see Supplementary
information). Also,
γ
is the photon decay rate, Γ is the
Brillouin gain bandwidth [42], and ∆
ω
Kerr
is a Kerr ef-
fect 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. 2a inset).
The initial roll-off of the Adev features a slope of
1
/
2
corresponding to white frequency noise [46]. This was
also verified in separate measurements of the beat fre-
quency using both an electrical spectrum analyzer and a
fast Fourier transform. The slope of this region is fit to
S
ν
/
(2
τ
) where
S
ν
is the one-sided spectral density of
the white frequency noise. Adev measurement at each
of the detuning points in Fig. 2a is used to infer the
S
ν
values that are plotted in Fig. 2b. There, a frequency
noise enhancement is observed as the system is biased to-
wards 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 overall excellent agreement between the-
ory and measurement. The frequency noise levels mea-
sured in Fig. 2b are consistent with fundamental SBL
frequency noise (see Methods). Significantly, the funda-
mental nature of the noise, the good agreement between
the PF prediction (Eq. (3)) and measurement in Fig. 2b,
and separate experimental work [12] that has verified the
theoretical form of the SEF (Eq. (2)) confirm that SEF
= PF so that the fundamental SNR of the gyroscope does
not improve near the EP.
While the Petermann factor analysis provides very
good agreement with the measured results, we also de-
rived an Adler-like coupled mode equation analysis for
the Brillouin laser system. This approach is distinct
from the bi-orthogonal framework and, while more com-
plicated, provides additional insights into the system be-
havior. Adapting analysis applied in the noise analysis
of ring laser gyroscopes [47], a noise enhancement factor
NEF results (see Supplementary information),
NEF =
ω
2
D
+ ∆
ω
2
EP
/
2
ω
2
D
ω
2
EP
(4)
It is interesting that this result, despite the different
4
-2
-1
0
1
2
SBL Amplitute Ratio q (dB)
20
30
40
50
60
70
80
90
100
Locking Bandwidth
∆ω
C
(kHz)
P
cw
= 249
μ
W
P
cw
= 198
μ
W
P
cw
= 157
μ
W
P
cw
= 125
μ
W
Fitting curve
-150
-100
-50
0
50
100
150
P
SBL
(
μ
W)
-200
-100
0
100
200
Locking Zone (kHz)
Locking zone center
Theoretical Kerr shift
FIG. 3.
Locking zone bandwidth versus SBL ampli-
tude ratio.
Measured locking zone bandwidth is plotted ver-
sus amplitude ratio
q
of the SBL lasers. The cw power is held
constant at four values (see legend) to create the data com-
posite. The solid black curve is Eq, (5). Inset: the measured
locking zone boundaries are plotted versus the SBL power
differences (∆
P
SBL
=
P
ccw
P
cw
). Colors and symbols corre-
spond to the main panel. The center of the locking zone is also
indicated and is shifted by the Kerr nonlinearity which varies
as the SBL power difference. Black line gives the theoretical
prediction (no free parameters).
physical context of the Brillouin laser system, has a sim-
ilar form to one derived for polarization-mode-coupled
laser systems [48]. The PF and NEF predictions are
shown on Fig. 2b and the Adler-derived NEF correc-
tion provides slightly better agreement with the data at
the intermediate detuning values.
The Adler approach is also useful to explain a locking
zone dependence upon SBL amplitudes observed in Fig.
2a. As shown in the Supplementary Information, this
variation can be explained through the combined action
of the Kerr effect and intermodal coupling coefficients
of both dissipative and conservative nature. Specifically,
the locking bandwidth is found to exhibit the following
dependence upon the amplitude ratio
q
=
|
a
ccw
/a
cw
|
of
the SBL lasers,
ω
2
EP
=
(
Γ
Γ +
γ
)
2
[
(
q
+
1
q
)
2
|
κ
|
2
+
(
q
1
q
)
2
|
χ
|
2
]
(5)
where
κ
is the dissipative coupling and
χ
is the conser-
vative coupling between cw and ccw SBL modes. The
locking zone boundaries in terms of pump detuning fre-
quency have been measured (Fig. 3 inset) for a series of
different SBL powers. Using this data, the locking band-
width is expressed in pump frequency detuning (∆
ω
P
)
units using ∆
ω
C
(1 + Γ
)∆
ω
EP
and plotted versus
q
in the main panel of Fig. 3. The plot agrees well with
Eq. (5) (fitting shown in black) and gives
|
κ
|
= 0.93 kHz,
|
χ
|
= 8.21 kHz.
Finally, the center of the locking band is shifted
by the Kerr effect and (in pump frequency detuning
ω
P
units) can be expressed as
)∆
ω
Kerr
, where
ω
Kerr
=
η
(
|
α
ccw
|
2
−|
α
cw
|
2
)
= (
η
P
SBL
)
/
(
γ
ex
~
ω
) is
the Kerr induced SBL resonance frequency difference,
P
SBL
=
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
=
n
2
~
ω
2
c/
(
V n
2
0
) is the single-photon
Kerr-effect angular frequency shift with
ω
the SBL an-
gular frequency,
n
2
the Kerr-nonlinear refractive index
of silica,
V
the mode volume,
n
0
the linear refractive in-
dex, and
c
the speed of light in vacuum. If the white
frequency noise floors in Fig. 2 are used to infer the res-
onator quality factor, then a Kerr nonlinearity value of
558 Hz/
μ
W is predicted (see Methods). This value gives
the line plot in the Fig. 3 inset (with no free parameters)
which agrees with experiment.
We have verified through measurement and theory that
mode non-orthogonality sets a fundamental limitation to
a class of sensors operating near an EP. Remarkably, a re-
sulting noise enhancement precisely compensates the sen-
sor’s EP-enhanced response. It is nonetheless important
to note that when SNR is limited by technical noise con-
siderations, it still could be advantageous to operate near
the EP. It is also possible that other sensing modalities
could benefit from operation near an EP. More generally,
the excellent control of the state space that is possible
in the Brillouin system can provide a new platform for
study of the remarkable physics associated with excep-
tional points.
Methods
Linewidth and Allan deviation measurement.
In experi-
ments, frequency is measured in the time domain using
a frequency counter and its Allan deviation is calculated
for different averaging times (Fig. 2a). The Allan devia-
tion
σ
ν
(
τ
) for a signal frequency is defined by
σ
ν
(
τ
)
1
2 (
M
1)
M
1
k
=1
(
ν
k
+1
ν
k
)
2
(6)
where
τ
is the averaging time,
M
is the number of fre-
quency measurements, and
ν
k
is the average frequency of
the signal (measured in Hz) in the time interval between
and (
k
+ 1)
τ
. The Allan deviation follows a
τ
1
/
2
de-
pendence when the underlying frequency noise spectral
density is white [46] 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 flicker noise and fre-
quency drift are not yet important. This portion of the
5
Allan deviation plot can be fit using
σ
ν
(
τ
) =
S
ν
/
(2
τ
)
where
S
ν
is the white frequency noise one-sided spectral
density function. This result can be further converted
to the Lorentzian full-width at half maximum (FWHM)
linewidth ∆
ν
FWHM
(measured in Hz) using the conver-
sion,
S
ν
= 2
σ
2
ν
(
τ
)
τ
=
1
π
ν
FWHM
(7)
Experimental parameters and data fitting.
The resonator
is pumped at the optical wavelength
λ
= 1553
.
3 nm,
which, subject to the Brillouin phase matching condi-
tion, corresponds to a phonon frequency (Stokes fre-
quency shift) of approximately Ω
phonon
/
(2
π
) = 10
.
8
GHz. Quality factors of the SBL modes are measured us-
ing a Mach-Zehnder interferometer, and a loaded Q fac-
tor
Q
T
= 88
×
10
6
and coupling Q factor
Q
ex
= 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
=
(
Γ
γ
+ Γ
)
2
~
ω
3
4
π
2
Q
T
Q
ex
(
1
P
cw
+
1
P
ccw
)(
n
th
+
N
th
+ 1)
(8)
which results from summing the Schawlow-Townes-like
linewidths of the SBL laser waves [42]. In the expres-
sion,
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 bal-
anced case (green 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. For the blue (red) data set,
P
cw
(
P
ccw
) is decreased by 1.22 dB (1.46 dB) so that
S
ν
=
0.58 (0.60) Hz
2
/Hz is calculated. On the other hand, the
measured values for the blue, green and red data sets in
Fig. 2b (i.e., white frequency noise floors far from EP)
give
S
ν
= 0.44, 0.39, 0.46 Hz
2
/Hz, 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 coefficient which supports this belief.
The beating frequency in Fig. 2a is fit using the fol-
lowing relations:
ω
S
= sgn(∆
ω
D
)
ω
2
D
ω
2
EP
ω
D
=
γ/
Γ
1 +
γ/
Γ
ω
P
+
1
1 +
γ/
Γ
ω
Kerr
(9)
where sgn is the sign function and
γ/
Γ, ∆
ω
Kerr
and ∆
ω
EP
are fitting parameters. The fitting gives
γ/
Γ = 0
.
076 con-
sistently, while ∆
ω
Kerr
and ∆
ω
EP
are separately adjusted
in each data set. These parameters feature a power de-
pendence that is fully explored in Fig. 3 and the related
main text discussion.
The theoretical Kerr coefficient used in Fig. 3 can
be calculated as follows. Assuming
n
2
2
.
7
×
10
20
m
2
/
W,
n
0
= 1
.
45 for the silica material, and
V
= 10
7
μ
m
3
(obtained through finite-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
= 299 kHz so that
ω
Kerr
/
(2
π
P
SBL
)
42 Hz
W. When
γ/
Γ = 0
.
076,
the center shift of pump locking band is
)∆
ω
Kerr
=
558 Hz/
μ
W. This value agrees very well with experiment
(Fig. 3 inset).
Data availability.
The data that support the plots within
this paper and other findings of this study are available
from the corresponding author upon reasonable request.
Acknowledgements.
This work was supported by the De-
fense Advanced Research Projects Agency (DARPA) un-
der PRIGM:AIMS program through SPAWAR (grant no.
N66001-16-1-4046) and the Kavli Nanoscience Institute.
Author contributions.
HW, Y-HL and KV conceived the
idea. HW derived the theory with the feedback from
Y-HL, ZY and KV. Y-HL designed and perform the ex-
periments with ZY and HW. ZY analysed the data with
Y-HL and HW. M-GS fabricated the devices. All authors
participated in writing the manuscript. KV supervised
the research.
Competing interests.
The authors declare no competing
interests.
Author Information.
Current addresses of two co-
authors: Y-HL, OEwaves Inc., 465 North Halstead
Street, Suite 140, Pasadena, California 91107, USA; M-
GS, NTT Physics and Information Laboratory, 1950 Uni-
versity Ave., East Palo Alto, California 94303, USA.
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7
Supplementary information
The supplementary information is structured as follows. In Section I we briefly review the framework for working
with general non-Hermitian matrices and introduce the bi-orthogonal relations. In section II we derive the Petermann
factor for a 2
×
2 Hamiltonian. In Section III we show that the effective amplitude of the non-orthogonal modes
is reduced compared to conventional modes and also justify the physical picture of the increased noise. Finally in
section IV we present the full coupled mode-equations in a Langevin formalism. It is used to derive an Adler-like
equation that will lead to the corrected noise enhancement factor as well as a relation giving the locking bandwidth
as a function of the amplitude ratio of the two modes.
Non-Hermitian Hamiltonian and bi-orthogonal relations
Here we briefly review the framework for working with general non-Hermitian matrices. An
n
-dimensional matrix
M
has
n
eigenvalues
μ
1
, μ
2
, ... μ
n
. For simplicity we will assume that all of the eigenvalues are distinct, i.e.
μ
j
6
=
μ
k
if
j
6
=
k
. In this case
M
will have
n
right eigenvectors and
n
left eigenvectors associated with each
μ
j
:
M
|
v
R
j
=
μ
j
|
v
R
j
,
v
L
j
|
M
=
v
L
j
|
μ
j
(S1)
To make sense of the left eigenvectors, note that
M
|
v
L
j
=
μ
j
|
v
L
j
, thus the left eigenvector is the eigenstate as if loss
is changed to gain and vice versa. Since
M
is in general non-Hermitian, there is no guarantee that
|
v
L
j
=
|
v
R
j
, and
many of the decomposition results that hold in the Hermitian case will fail. However we note that,
μ
j
v
L
j
|
v
R
k
=
v
L
j
|
M
|
v
R
k
=
μ
k
v
L
j
|
v
R
k
〉 ⇒ 〈
v
L
j
|
v
R
k
= 0
,
j
6
=
k
(S2)
Thus left and right eigenvectors associated with different eigenvalues are bi-orthogonal. We also note that the right
eigenvectors are complete and form a set of basis (as
M
is non-degenerate and finite-dimensional), and we can
decompose the identity matrix and
M
as follows:
1 =
j
|
v
R
j
〉〈
v
L
j
|
v
L
j
|
v
R
j
M
=
j
|
v
R
j
〉〈
v
L
j
|
v
L
j
|
v
R
j
μ
j
(S3)
where each term is a “projector” onto the eigenvectors. Again we note that
v
L
j
|
v
R
j
may be negative and even complex,
which results in special normalizations of the vectors. For simplicity we will choose
v
L
j
|
v
R
j
= 1 by rescaling the vectors
and adjusting the relative phase (such vectors are sometimes said to be bi-orthonormal). With this normalization in
place the above decompositions simplify further as follows:
1 =
j
|
v
R
j
〉〈
v
L
j
|
M
=
j
|
v
R
j
〉〈
v
L
j
|
μ
j
(S4)
We note that, as a result of using bi-orthonormal left and right vectors, the vectors, themselves, are not normalized,
i.e.
v
L
j
|
v
L
j
and
v
R
j
|
v
R
j
need not be 1 for each
j
. There is one extra degree of freedom per mode for fixing the lengths,
but the length normalization factors do not affect the physical observables if such factors are kept consistently through
the calculations. In section III a “natural” normalization will be chosen when we give a physical meaning to these
factors.
Petermann factor of a two-dimensional Hamiltonian
In this section we derive the Petermann factor of a two-dimensional Hamiltonian
H
. Denote the two normalized
right (left) eigenvectors of
H
as
|
ψ
R
1
and
|
ψ
R
2
(
|
ψ
L
1
and
|
ψ
L
2
). The Petermann factors of these two eigenmodes can
8
then be expressed as [19]
PF
1
=
ψ
L
1
|
ψ
L
1
〉〈
ψ
R
1
|
ψ
R
1
PF
2
=
ψ
L
2
|
ψ
L
2
〉〈
ψ
R
2
|
ψ
R
2
(S5)
We will first prove that PF
1
= PF
2
, which can then be identified as the Petermann factor for the entire system. Note
that
|
ψ
L
1
and
|
ψ
R
2
are orthogonal and span the two-dimensional space. As a result, the identity can be expressed
using this set of basis vectors as follows:
1 =
|
ψ
L
1
〉〈
ψ
L
1
|
ψ
L
1
|
ψ
L
1
+
|
ψ
R
2
〉〈
ψ
R
2
|
ψ
R
2
|
ψ
R
2
(S6)
Now apply this expansion to
|
ψ
R
1
and obtain
|
ψ
R
1
=
1
ψ
L
1
|
ψ
L
1
|
ψ
L
1
+
ψ
R
2
|
ψ
R
1
ψ
R
2
|
ψ
R
2
|
ψ
R
2
(S7)
where
ψ
L
1
|
ψ
R
1
= 1 has been used. Left multiplication by
ψ
R
1
|
results in
ψ
R
1
|
ψ
R
1
=
1
ψ
L
1
|
ψ
L
1
+
ψ
R
1
|
ψ
R
2
〉〈
ψ
R
2
|
ψ
R
1
ψ
R
2
|
ψ
R
2
(S8)
Thus we obtain,
1
PF
1
= 1
ψ
R
1
|
ψ
R
2
〉〈
ψ
R
2
|
ψ
R
1
ψ
R
1
|
ψ
R
1
〉〈
ψ
R
2
|
ψ
R
2
(S9)
which is symmetric with respect to the indexes 1 and 2 and thereby completes the proof that PF
1
= PF
2
PF.
Next, PF is expressed using the Hamiltonian instead of its eigenvectors. We begin by noting that the identity
operator added to the Hamiltonian will not modify the eigenvectors. As a result, the trace can be removed from
H
without changing the value of PF:
H
0
H
1
2
Tr(
H
)
(S10)
where Tr is the matrix trace and
H
0
is the traceless part of
H
. Using the bi-orthogonal expansion,
H
0
has the form,
H
0
=
μ
(
|
ψ
R
1
〉〈
ψ
L
1
|−|
ψ
R
2
〉〈
ψ
L
2
|
)
(S11)
where
μ
is the first eigenvalue. Consider next the quantity Tr(
H
0
H
0
):
Tr(
H
0
H
0
) =
|
μ
|
2
(
ψ
L
1
|
ψ
L
1
〉〈
ψ
R
1
|
ψ
R
1
+
ψ
L
2
|
ψ
L
2
〉〈
ψ
R
2
|
ψ
R
2
〉−〈
ψ
L
2
|
ψ
L
1
〉〈
ψ
R
1
|
ψ
R
2
〉−〈
ψ
L
1
|
ψ
L
2
〉〈
ψ
R
2
|
ψ
R
1
)
(S12)
where we used the fact that Tr(
|
α
〉〈
β
|
) =
β
|
α
. To simplify the expression, note that each of the first two terms
equals PF. Moreover, the third term can be evaluated by expressing
|
ψ
L
1
as a combination of right eigenvectors using
Eq. (S7):
−〈
ψ
L
2
|
ψ
L
1
〉〈
ψ
R
1
|
ψ
R
2
=
ψ
R
1
|
ψ
R
2
〉〈
ψ
R
2
|
ψ
R
1
ψ
R
2
|
ψ
R
2
ψ
L
1
|
ψ
L
1
= PF
1
(S13)
Similarly, the fourth term also equals PF
1. Thus
Tr(
H
0
H
0
) =
|
μ
|
2
(4
PF
2)
(S14)
Finally, to eliminate the eigenvalue
μ
we calculate,
Tr(
H
2
0
) =
μ
2
(
ψ
L
1
|
ψ
R
1
2
+
ψ
L
2
|
ψ
R
2
2
) = 2
μ
2
(S15)
and the PF can be solved as
PF =
1
2
(
1 +
Tr(
H
0
H
0
)
|
Tr(
H
2
0
)
|
)
(S16)
which completes the proof.
We note that while a Hermitian Hamiltonian with
H
0
=
H
0
results in PF = 1, the converse is not always true.
Consider the example of
H
0
=
z
where
σ
z
is the Pauli matrix. This would effectively describe two orthogonal modes
with different gain, and direct calculation shows that PF = 1.
9
Field amplitude and noise in a non-orthogonal system
In this section we consider the physical interpretation of increased linewidth whereby the effective field amplitude
decreases while the effective noise input increases as a result of non-orthogonality. This analysis considers a hypothet-
ical laser mode that is part of the bi-orthogonal system. It skips key steps normally taken in a more rigorous laser
noise analysis in order to make clearer the essential EP physics. A more complete study of the Brillouin laser system
is provided in Section IV.
The two-dimensional system is described by the column vector
|
Ψ
= (
a
cw
,a
ccw
)
T
whose components are the
orthogonal field amplitudes
a
cw
and
a
ccw
. The equation of motion reads
id
|
Ψ
/dt
=
H
|
Ψ
, where
H
is the two-
dimensional Hamiltonian. Now assume that
|
Ψ
=
c
1
|
ψ
R
1
, i.e. only the first eigenmode of the system is excited. We
interpret
c
1
as the phasor for the eigenmode. We see that
|
c
1
|
2
=
Ψ
|
Ψ
/
ψ
R
1
|
ψ
R
1
is reduced from the true square
amplitude
Ψ
|
Ψ
by a factor of the length squared of the right eigenvector
ψ
R
1
|
ψ
R
1
. The equation of motion for
c
1
reads
i
dc
1
dt
=
i
d
ψ
L
1
|
Ψ
dt
=
ψ
L
1
|
H
0
|
ψ
R
1
c
1
=
μ
1
c
1
(S17)
Here, we are assuming that the mode experiences both loss and saturable gain that are absorbed into the definition
of the eigenvalue
μ
1
. To simplify the following calculations we set the real part of
μ
1
to 0, since any frequency shift
can be removed with an appropriate transformation to slowly varying amplitudes.
To introduce noise into the system resulting from the amplification process the equation of motion is modified as
follows:
id
|
Ψ
/dt
=
H
0
|
Ψ
+
|
F
. Here,
|
F
= (
F
cw
(
t
)
,
F
ccw
(
t
))
T
is a column vector with fluctuating components.
The noise correlation of these components is assumed to be given by,
F
cw
(
t
)
F
cw
(
t
)
=
F
ccw
(
t
)
F
ccw
(
t
)
=
θδ
(
t
t
)
F
cw
(
t
)
F
ccw
(
t
)
=
F
ccw
(
t
)
F
cw
(
t
)
= 0
(S18)
where
θ
is a quantity with frequency dimensions. We note that the assumption of vanishing correlations between
the fluctuations on different modes is not trivial. Even if the basis is orthogonal, the non-Hermitian nature of the
Hamiltonian means that dissipative mode coupling will generally be present in the system. This will be associated
with fluctuations that can induce off-diagonal elements in the correlation matrix. In the system studied here, we will
show in Section IV that the main source of noise comes from the phonons and fluctuations due to the non-Hermitian
Hamiltonian are negligible, thereby justifying the assumption made here. Taking account of the fluctuations, the
equation of motion for
c
1
can be modified as follows,
dc
1
dt
=
−|
μ
1
|
c
1
+
ψ
L
1
|
F
=
−|
μ
1
|
c
1
+
F
1
(S19)
where the fluctuation term for the first eigenmode is defined as
F
1
=
ψ
L
1
|
F
. Its correlation reads
F
1
(
t
)
F
1
(
t
)
=
θ
ψ
L
1
|
ψ
L
1
δ
(
t
t
)
(S20)
which, upon comparison to Eq. (S18), shows that the noise input to the right eigenvector field amplitude (
c
1
) is
enhanced (relative to the noise input to either the cw or ccw fields alone) by a factor of the length squared of the left
eigenvector
ψ
L
1
|
ψ
L
1
.
We are interested in the phase fluctuations of
c
1
. Here, it is assumed that the mode is pumped to above threshold
and is lasing. Under these conditions it is possible separate amplitude and phase fluctuations of the field. We rewrite
c
1
=
|
c
1
|
exp(
c
) and obtain the rate of change of the phase variable as follows:
c
dt
=
i
2
|
c
1
|
(
F
1
e
c
F
1
e
c
)
(S21)
which describes white frequency noise of the laser field (equivalently phase noise diffusion). The correlation can be
calculated as
̇
φ
c
(
t
)
̇
φ
c
(
t
)
=
θ
2
|
c
1
|
2
ψ
L
1
|
ψ
L
1
δ
(
t
t
) =
θ
2
Ψ
|
Ψ
ψ
R
1
|
ψ
R
1
〉〈
ψ
L
1
|
ψ
L
1
δ
(
t
t
) = PF
×
θ
2
Ψ
|
Ψ
δ
(
t
t
)
(S22)
where the non-enhanced linewidth is ∆
ω
0
=
θ/
(2
Ψ
|
Ψ
) [49] and the enhanced linewidth is given by ∆
ω
= PF
×
ω
0
.
From the above derivation, the PF enhancement is the result of two effects, the reduction of effective square amplitude
(
|
c
1
|
2
=
Ψ
|
Ψ
/
ψ
R
1
|
ψ
R
1
) and the enhancement of noise by
ψ
L
1
|
ψ
L
1
.
10
Up to now we have not chosen individual normalizations for
ψ
L
1
|
ψ
L
1
and
ψ
R
1
|
ψ
R
1
as they appear together in the
Petermann factor. Motivated by the fact that left and right eigenvectors can be mapped onto the same Hilbert space,
we select the symmetric normalization:
ψ
L
1
|
ψ
L
1
=
ψ
R
1
|
ψ
R
1
=
PF
(S23)
With this normalization the squared field amplitude is reduced and the noise input is increased by a factor of
PF,
resulting in the linewidth enhancement by a factor of PF. We note that other interpretations are possible through
different normalizations. For example, in Siegman’s analysis
ψ
L
1
|
ψ
L
1
= PF and
ψ
R
1
|
ψ
R
1
= 1 is chosen, and the
enhancement is fully attributed to noise increase by a factor of PF [19] .
Langevin formalism
Here we analyze the system with a Langevin formalism. An Adler-like equation will be derived that provides an
improved laser linewidth and and an expression for the locking bandwidth dependence on the field amplitude ratio.
The analysis will also include the Kerr effect.
First we summarize symbols and give their definitions. For readability, all cw subscript will be replaced by
1
and all ccw subscript will be replaced by
2. The modes are pumped at angular frequencies
ω
P
,
1
and
ω
P
,
2
. These
frequencies will generally be different from the unpumped resonator mode frequency. The cw and ccw Brillouin lasers
oscillate on the same longitudinal mode with frequency
ω
. This frequency is shifted for both cw and ccw waves by the
same amount as a result of the pump-induced Kerr shift. On the other hand, the Kerr effect causes cross-phase and
self-phase modulation of the cw and ccw waves that induces different frequency shifts in these waves. This shift and
the rotation-induced Sagnac shift are accounted for using offset frequencies
δω
1
=
η
(
a
1
a
1
+ 2
a
2
a
2
)
ωD/
(2
n
g
c
)
and
δω
2
=
η
(
a
2
a
2
+ 2
a
1
a
1
)
+ Ω
ωD/
(2
n
g
c
) relative to
ω
, where
η
=
n
2
~
ω
2
c/
(
V n
2
0
) is the single-photon nonlinear
angular frequency shift,
n
2
is the nonlinear refractive index,
V
is the mode volume,
n
0
is the linear refractive index,
c
is the speed of light in vacuum, Ω is the rotation rate,
D
is the resonator diameter and
n
g
is the group index.
Phonon modes have angular frequencies Ω
phonon
= 2
ωn
0
v
s
/c
where
v
s
is the velocity of the phonons. The loss rate of
phonon modes is denoted as Γ (also known as the gain bandwidth) and the loss rate of the SBL modes are assumed
equal and denoted as
γ
. In addition, coupling between the two SBL modes is separated as a dissipative part and
conservative part, denoted as
κ
and
χ
, respectively. These rates will be assumed to satisfy Γ

γ
|
κ
|
,
|
χ
|
to simplify
the calculations, which is
a posteriori
verified in our system. In the following analysis, we will treat the SBL modes
and phonon modes quantum mechanically and define
a
1
(
a
2
) and
b
1
(
b
2
) as the lowering operators of the cw (ccw)
components of the SBL and phonon modes, respectively. Meanwhile, pump modes are treated as a noise-free classical
fields
A
1
and
A
2
(photon-number-normalized amplitudes).
Using these definitions, the full equations of motion for the SBL and phonon modes read
̇
a
1
=
(
γ
2
+
+
iδω
1
)
a
1
+ (
κ
+
)
a
2
ig
ab
A
2
b
2
exp(
P
,
2
t
) +
F
1
(
t
)
̇
a
2
=
(
γ
2
+
+
iδω
2
)
a
2
+ (
κ
+
)
a
1
ig
ab
A
1
b
1
exp(
P
,
1
t
) +
F
2
(
t
)
̇
b
1
=
(
Γ
2
i
phonon
)
b
1
+
ig
ab
A
1
a
2
exp(
P
,
1
t
) +
f
1
(
t
)
̇
b
2
=
(
Γ
2
i
phonon
)
b
2
+
ig
ab
A
2
a
1
exp(
P
,
2
t
) +
f
2
(
t
)
(S24)
where
g
ab
is the single-particle Brillioun coupling coefficient. The fluctuation operators
F
(
t
) and
f
(
t
) associated with
the field operators have the following correlations:
F
1
(
t
)
F
1
(
t
)
=
F
2
(
t
)
F
2
(
t
)
=
γN
th
δ
(
t
t
)
F
1
(
t
)
F
1
(
t
)
=
F
2
(
t
)
F
2
(
t
)
=
γ
(
N
th
+ 1)
δ
(
t
t
)
f
1
(
t
)
f
1
(
t
)
=
f
2
(
t
)
f
2
(
t
)
= Γ
n
th
δ
(
t
t
)
f
1
(
t
)
f
1
(
t
)
=
f
2
(
t
)
f
2
(
t
)
= Γ(
n
th
+ 1)
δ
(
t
t
)
(S25)