of 11
Supplementary Information for:
Petermann-factor limited sensing near an exceptional point
in a Brillouin ring laser gyroscope
Wang, Lai, Yuan et al.
2
Supplementary Note 1. 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
(1)
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
(2)
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
(3)
M
=
j
|
v
R
j
〉〈
v
L
j
|
v
L
j
|
v
R
j
μ
j
(4)
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
|
(5)
M
=
j
|
v
R
j
〉〈
v
L
j
|
μ
j
(6)
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 Supplementary Note 3 a “natural” normalization will be chosen when we give a physical meaning
to these factors.
Supplementary Note 2. PETERMANN FACTOR OF A TWO-DIMENSIONAL HAMILTONIAN
Here 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 then be
expressed as [1]
PF
1
=
ψ
L
1
|
ψ
L
1
〉〈
ψ
R
1
|
ψ
R
1
(7)
PF
2
=
ψ
L
2
|
ψ
L
2
〉〈
ψ
R
2
|
ψ
R
2
(8)
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
(9)
3
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
(10)
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
(11)
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
(12)
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
)
(13)
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
|
)
(14)
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
)
(15)
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
Supplementary Eq. (10):
−〈
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
(16)
Similarly, the fourth term also equals PF
1. Thus
Tr(
H
0
H
0
) =
|
μ
|
2
(4PF
2)
(17)
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
(18)
and the PF can be solved as
PF =
1
2
(
1 +
Tr(
H
0
H
0
)
|
Tr(
H
2
0
)
|
)
(19)
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.
4
Supplementary Note 3. FIELD AMPLITUDE AND NOISE IN A NON-ORTHOGONAL SYSTEM
Here 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 hypothetical 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
Supplementary Note 4.
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
(20)
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
)
(21)
F
cw
(
t
)
F
ccw
(
t
)
=
F
ccw
(
t
)
F
cw
(
t
)
= 0
(22)
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 Supplementary Note 4 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
(23)
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
)
(24)
which, upon comparison to Supplementary Eq. (21), 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
)
(25)
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
)
(26)
where the non-enhanced linewidth is ∆
ω
0
=
θ/
(2
Ψ
|
Ψ
) [2] 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
.
5
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
(27)
With this normalization the squared field amplitude is reduced and the noise input is increased both 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 [1] .
Supplementary Note 4. LANGEVIN FORMALISM
Here we analyze the system with a Langevin formalism, which includes Brillioun gain, the Sagnac effect and the Kerr
effect. An Adler-like equation will be derived that provides an improved laser linewidth formula and an expression
for the locking bandwidth dependence on the field amplitude ratio.
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 stimulated Brillouin lasers
(SBLs) 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
)
(28)
̇
a
2
=
(
γ
2
+
+
iδω
2
)
a
2
+ (
κ
+
)
a
1
ig
ab
A
1
b
1
exp(
P
,
1
t
) +
F
2
(
t
)
(29)
̇
b
1
=
(
Γ
2
i
phonon
)
b
1
+
ig
ab
A
1
a
2
exp(
P
,
1
t
) +
f
1
(
t
)
(30)
̇
b
2
=
(
Γ
2
i
phonon
)
b
2
+
ig
ab
A
2
a
1
exp(
P
,
2
t
) +
f
2
(
t
)
(31)
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
)
(32)
F
1
(
t
)
F
1
(
t
)
=
F
2
(
t
)
F
2
(
t
)
=
γ
(
N
th
+ 1)
δ
(
t
t
)
(33)
f
1
(
t
)
f
1
(
t
)
=
f
2
(
t
)
f
2
(
t
)
= Γ
n
th
δ
(
t
t
)
(34)
f
1
(
t
)
f
1
(
t
)
=
f
2
(
t
)
f
2
(
t
)
= Γ(
n
th
+ 1)
δ
(
t
t
)
(35)
where
N
th
and
n
th
are the thermal occupation numbers of the SBL state and phonon state. In addition, there are
6
non-zero cross-correlations of the photon fluctuation operators due to the dissipative coupling:
F
2
(
t
)
F
1
(
t
)
=
2
κN
th
δ
(
t
t
)
(36)
F
1
(
t
)
F
2
(
t
)
=
2
κ
N
th
δ
(
t
t
)
(37)
F
2
(
t
)
F
1
(
t
)
=
2
κ
(
N
th
+ 1)
δ
(
t
t
)
(38)
F
1
(
t
)
F
2
(
t
)
=
2
κ
(
N
th
+ 1)
δ
(
t
t
)
(39)
All other cross correlations not explicitly written are 0.
4.1. Single SBL
We first study a single laser mode and its corresponding phonon field (
a
1
and
b
2
) by neglecting
κ
and
χ
. By
introducing the slow varying envelope with
a
1
=
α
1
e
iωt
and
b
2
=
β
2
e
i
(
ω
P
,
2
ω
)
t
, the following equations result:
̇
α
1
=
(
γ
2
+
iδω
1
)
α
1
ig
ab
A
2
β
2
+
F
1
(
t
)e
iωt
(40)
̇
β
2
=
(
Γ
2
+
i
∆Ω
2
)
β
2
+
ig
ab
A
2
α
1
+
f
2
(
t
)e
i
(
ω
P
,
2
ω
)
t
(41)
where we have defined the frequency mismatch ∆Ω
2
=
ω
P
,
2
ω
phonon
. Neglecing the weak Kerr effect term in
δω
1
this is a set of linear equations in
a
1
and
b
2
. The eigenvalues of the coefficient matrix
(
γ/
2
iδω
1
ig
ab
A
2
ig
ab
A
2
Γ
/
2
i
∆Ω
2
)
(42)
can be solved as
μ
1
,
2
=
1
4
(
Γ
γ
2
iδω
1
2
i
∆Ω
2
±
16
g
2
ab
|
A
2
|
2
+ (Γ
γ
2
iδω
1
+ 2
i
∆Ω
2
)
2
)
(43)
At the lasing threshold, the first eigenvalue
μ
1
has a real part of 0. This can be rewritten as
16
g
2
ab
|
A
2
|
2
+ (Γ
γ
2
iδω
1
+ 2
i
∆Ω
2
)
2
= (Γ +
γ
+ 2
iδω
1
+ 2
i
∆Ω
2
+ 4
i
Im(
μ
1
))
2
(44)
Solving this complex equation gives the SBL eigenfrequency as well as the lasing threshold,
μ
1
=
i
γ
∆Ω
2
+ Γ
δω
1
Γ +
γ
(45)
g
2
ab
|
A
2
|
2
=
Γ
γ
4
(
1 +
4(∆Ω
2
δω
1
)
2
(Γ +
γ
)
2
)
(46)
The threshold at perfect phase matching (∆Ω
2
=
δω
1
) is usually written in a more familiar form
g
0
|
A
2
|
2
=
γ/
2, where
g
0
is the Brillouin gain factor [3]. Comparison gives
g
ab
=
g
0
Γ
2
(47)
We also introduce the modal Brillioun gain function for a single direction:
g
1
g
0
1 + 4(
δω
1
∆Ω
2
)
2
/
(Γ +
γ
)
2
(48)
so that the threshold can be written as
g
1
|
A
2
|
2
=
γ
2
(49)
7
With the threshold condition solved, the matrix can be decomposed using the bi-orthogonal approach outlined in
Supplementary Note 1. The linear combination that describes the composite SBL mode can be found as
α
1
=
Γ
γ
+ Γ
[
α
1
i
g
ab
Γ
2
1 + 2
i
(∆Ω
2
δω
1
)
/
(Γ +
γ
)
A
2
β
2
]
(50)
where the factor Γ
/
(
γ
+ Γ) properly normalizes
α
1
so that
α
1
=
α
1
when only the SBL mode is present in the system,
and we have dropped its dependence on the phase mismatch ∆Ω
2
δω
1
for simplicity. The associated equation of
motion is
d
dt
α
1
=
i
γ
∆Ω
2
+ Γ
δω
1
Γ +
γ
α
1
+
F
1
(
t
)
(51)
where the frequency term now includes a mode-pulling contribution so that the SBL laser frequency is given by,
ω
S
,
1
=
ω
+
γ
∆Ω
2
+ Γ
δω
1
Γ +
γ
(52)
Also, we have defined a combined fluctuation operator for
α
1
,
F
1
(
t
) =
Γ
γ
+ Γ
[
F
1
(
t
)e
iωt
i
1
2
i
(∆Ω
2
δω
1
)
/
(Γ +
γ
)
1 + 2
i
(∆Ω
2
δω
1
)
/
(Γ +
γ
)
γ
Γ
f
2
(
t
)e
i
(
ω
P
,
2
ω
)
t
]
(53)
with the following correlations,
F
1
(
t
)
F
1
(
t
)
=
(
Γ
γ
+ Γ
)
2
(
F
1
(
t
)
F
1
(
t
)
+
γ
Γ
f
1
(
t
)
f
1
(
t
)
)
=
(
Γ
γ
+ Γ
)
2
γ
(
n
th
+
N
th
)
δ
(
t
t
)
(54)
F
1
(
t
)
F
1
(
t
)
=
(
Γ
γ
+ Γ
)
2
(
F
1
(
t
)
F
1
(
t
)
+
γ
Γ
f
1
(
t
)
f
1
(
t
)
)
=
(
Γ
γ
+ Γ
)
2
γ
(
n
th
+
N
th
+ 2)
δ
(
t
t
)
(55)
We can now write
α
1
(
t
) =
N
1
exp(
1
), where
N
1
is the photon number,
φ
1
is the phase for the SBL, and where
amplitude fluctuations have been ignored on account of quenching of these fluctuations above laser threshold. We
note that amplitude fluctuations may result in linewidth corrections similar to the Henry
α
factor, but we will ignore
these effects here. The full equation of motion for
φ
1
is
̇
φ
1
=
ω
S
,
1
ω
+ Φ
1
(
t
)
,
Φ
1
(
t
) =
i
2
N
1
(
F
1
(
t
)e
1
F
1
(
t
)e
1
)
(56)
The correlation of the noise operator is given by,
Φ
1
(
t
1
(
t
)
=
1
4
N
1
(
F
1
(
t
)
F
1
(
t
)
+
F
1
(
t
)
F
1
(
t
)
=
(
Γ
γ
+ Γ
)
2
γ
2
N
1
(
n
th
+
N
th
+ 1)
δ
(
t
t
)
(57)
and we identify the coefficient before the delta function,
ω
FWHM
,
1
=
(
Γ
γ
+ Γ
)
2
γ
2
N
1
(
n
th
+
N
th
+ 1)
(58)
as the full-width half-maximum (FWHM) linewidth of the SBL.
In the experiment, the frequency noise of the SBL beating signal is measured. To compare against the experiment,
we calculate the FWHM linewidth for the beating signal by adding together the linewidths in two directions:
ω
FWHM
= ∆
ω
FWHM
,
1
+ ∆
ω
FWHM
,
2
=
(
Γ
γ
+ Γ
)
2
(
1
2
N
1
+
1
2
N
2
)
γ
(
n
th
+
N
th
+ 1)
(59)
and then convert to the one-sided power spectral density
S
ν
:
S
ν
=
1
π
ω
FWHM
2
π
=
(
Γ
γ
+ Γ
)
2
~
ω
3
4
π
2
Q
T
Q
ex
(
1
P
cw
+
1
P
ccw
)(
n
th
+
N
th
+ 1)
(60)
where
Q
T
and
Q
ex
are the loaded and coupling
Q
factors, and
P
cw
and
P
ccw
are the SBL powers in each direction.
8
4.2. Two SBLs
Now we can apply a similar procedure to the two pairs of photon and phonon modes with coupling on the optical
modes. We write the equations of motion for the SBL modes:
d
dt
α
1
=
i
(
ω
S
,
1
ω
)
α
1
+
Γ
γ
+ Γ
(
κ
+
)
α
2
+
F
1
(
t
)
(61)
d
dt
α
2
=
i
(
ω
S
,
2
ω
)
α
2
+
Γ
γ
+ Γ
(
κ
+
)
α
1
+
F
2
(
t
)
(62)
where quantities with the opposite subscript are defined similarly. We note that the coupling term involves the optical
modes
α
1
and
α
2
only. However, no additional coupling occurs between the other components of the SBL eigenstates
α
1
and
α
2
, and these states do not change up to first order of
κ/γ
and
χ/γ
. Thus we can approximate the optical
mode
α
1
with the composite SBL mode
α
1
. Within these approximations the lasing thresholds are also the same as
the independent case [4]. The equations now become
d
dt
α
1
=
i
(
ω
S
,
1
ω
)
α
1
+ (
κ
+
i
χ
)
α
2
+
F
1
(
t
)
(63)
d
dt
α
2
=
i
(
ω
S
,
2
ω
)
α
2
+ (
κ
+
i
χ
)
α
1
+
F
2
(
t
)
(64)
where we have defined mode-pulled coupling rates
κ
=
κ
Γ
/
(
γ
+ Γ) and
χ
=
χ
Γ
/
(
γ
+ Γ).
We can write
α
j
(
t
) =
N
j
exp(
j
) with
j
=
1
,
2, and once again ignore amplitude fluctuations. The equations
of motion for the phases are,
d
dt
φ
1
= (
ω
S
,
1
ω
)
q
Im[(
κ
+
i
χ
)e
(
1
2
)
] +
i
2
N
1
(
F
1
(
t
)e
1
F
1
(
t
)e
1
)
(65)
d
dt
φ
2
= (
ω
S
,
2
ω
)
q
1
Im[(
κ
+
i
χ
)e
(
2
1
)
] +
i
2
N
2
(
F
2
(
t
)e
2
F
2
(
t
)e
2
)
(66)
where we have defined the amplitude ratio
q
=
N
2
/N
1
for simplicity. As we measure the beatnote frequency, it is
convenient to define
φ
φ
2
φ
1
from which we obtain,
dt
= (
ω
S
,
2
ω
S
,
1
) + Im
{[
q
(
κ
+
i
χ
) +
q
1
(
κ
i
χ
)
]
e
}
+ Φ(
t
)
(67)
where the combined noise term and its correlation are given by
Φ =
i
2
N
1
(
F
1
(
t
)e
1
F
1
(
t
)e
1
) +
i
2
N
2
(
F
2
(
t
)e
2
F
2
(
t
)e
2
)
(68)
Φ(
t
)Φ(
t
)
=
(
Γ
γ
+ Γ
)
2
[
(
1
2
N
1
+
1
2
N
2
)
γ
(
N
th
+
n
th
+ 1) +
2
N
1
N
2
(
N
th
+
1
2
)
Re(
κ
e
(
t
)
)
]
δ
(
t
t
)
(69)
Since both
N
th
and
κ/γ
is small, we will discard the last time-varying term and write
Φ(
t
)Φ(
t
)
〉≈
ω
FWHM
δ
(
t
t
)
(70)
where ∆
ω
FWHM
= Γ
2
/
(
γ
+ Γ)
2
[(2
N
1
)
1
+ (2
N
2
)
1
]
γ
(
N
th
+
n
th
+ 1) is the linewidth of the beating signal far from
the EP (see also the single SBL discussion).
This equation can be further simplified by introducing an overall phase shift with
φ
=
φ
φ
0
, where
φ
0
=
Arg
[
q
(
κ
+
i
χ
) +
q
1
(
κ
i
χ
)
]
and Arg(
z
) is the phase of
z
:
d
φ
dt
= ∆
ω
D
ω
EP
sin
φ
+ Φ(
t
)
(71)
with
ω
D
=
ω
S
,
2
ω
S
,
1
=
γ
Γ +
γ
(
ω
P
,
1
ω
P
,
2
) +
Γ
Γ +
γ
[
η
(
N
2
N
1
) +
ωD
n
g
c
]
(72)
9
ω
2
EP
=
q
(
κ
+
i
χ
) +
q
1
(
κ
i
χ
)
2
=
(
Γ
γ
+ Γ
)
2
[
(
q
+
1
q
)
2
|
κ
|
2
+
(
q
1
q
)
2
|
χ
|
2
+ 2
(
q
2
1
q
2
)
Im(
κχ
)
]
(73)
This is an Adler equation with a noisy input. It shows the dependence of locking bandwidth on the amplitude ratio
and coupling coefficients. Moreover, it is clear that in the absence of ∆
ω
EP
, the beating linewidth would be given by
ω
FWHM
. The locking term ∆
ω
EP
sin
φ
makes the rate of phase change nonuniform and increases the linewidth.
The following part of analysis is dedicated to obtain the linewidth from this stochastic Adler equation. We define
z
φ
= exp(
i
φ
) and rewrite
d
dt
z
φ
=
iz
φ
(∆
ω
D
+ ∆
ω
EP
z
φ
z
1
φ
2
i
+ Φ)
(74)
The solution to the Adler equation is periodic when no noise is present. To see this explicitly we use a linear fractional
transform:
z
t
=
(∆
ω
D
ω
S
)
z
φ
+
i
ω
EP
ω
EP
z
φ
+
i
(∆
ω
D
ω
S
)
, z
φ
=
i
(∆
ω
D
ω
S
)
z
t
ω
EP
ω
EP
z
t
(∆
ω
D
ω
S
)
,
|
z
φ
|
=
|
z
t
|
= 1
(75)
1
z
t
d
dt
z
t
=
i
ω
S
i
ω
EP
(
z
t
+
z
1
t
)
/
2
ω
D
ω
S
Φ
(76)
where we introduced ∆
ω
S
=
ω
2
D
ω
2
EP
(which has the same meaning in the main text). The noiseless solution
of
z
t
would be
z
t
= exp(
i
ω
S
t
), and
z
φ
can be expanded in
z
t
as
z
φ
=
i
ω
D
ω
S
ω
EP
+ 2
i
ω
S
ω
EP
p
=1
(
ω
D
ω
S
ω
EP
z
t
)
p
(77)
where we have assumed ∆
ω
D
>
ω
EP
for convenience so that convergence can be guaranteed (For the case ∆
ω
D
<
ω
EP
we can expand near
z
t
= 0 instead of
z
t
=
). Thus the signal consists of harmonics oscillating at frequency
p
ω
S
with exponentially decreasing amplitudes. The noise added only changes the phase of
z
t
(as the coefficient is
purely imaginary) and to the lowest order the only effect of noise is to broaden each harmonic.
The linewidth can be found from the spectral density, which is given by the Fourier transform of the correlation
function:
W
E
(
ω
)
∝F
τ
{〈
z
φ
(
t
)
z
φ
(
t
+
τ
)
〉}
(
ω
)
(78)
and the correlation is given by
z
φ
(
t
)
z
φ
(
t
+
τ
)
=
(
ω
D
ω
S
ω
EP
)
2
+ 4
ω
2
S
ω
2
EP
p
=1
(
ω
D
ω
S
ω
EP
)
2
p
z
t
(
t
)
p
z
t
(
t
+
τ
)
p
(79)
where we have discarded the
z
t
(
t
)
p
z
t
(
t
+
τ
)
q
(
p
6
=
q
) terms since they vanish at the lowest order of ∆
ω
FWHM
.
To further calculate each
z
t
(
t
)
p
z
t
(
t
+
τ
)
p
〉≡
C
p
(
τ
) we require the integral form of the Fokker-Planck equation: if
dX
(
t
) =
μ
(
X,t
)
dt
+
σ
(
X,t
)
dW
is a stochastic differential equation (in the Stratonovich interpretation), where
W
is
a Wiener process, then for
f
(
X
) as a function of
X
, the differential equation for its average reads,
d
dt
f
(
X
)
=
(
μ
+
σ
2
∂σ
∂X
)
f
(
X
)
+
1
2
σ
2
f
′′
(
X
)
(80)
Applying the Fokker-Planck equation to
C
p
(
τ
), with the stochastic equation for
z
t
, gives
dC
p
(
τ
)
=
p
i
ω
S
z
t
(
t
)
p
z
t
(
t
+
τ
)
p
+
p
[
ω
FWHM
2∆
ω
2
S
(
ω
EP
z
t
(
t
+
τ
) +
z
1
t
(
t
+
τ
)
2
ω
D
)
(∆
ω
EP
z
t
(
t
+
τ
)
ω
D
)
]
z
t
(
t
)
p
z
t
(
t
+
τ
)
p
p
(
p
+ 1)
ω
FWHM
2∆
ω
2
S
(
ω
EP
z
t
(
t
+
τ
) +
z
1
t
(
t
+
τ
)
2
ω
D
)
2
z
t
(
t
)
p
z
t
(
t
+
τ
)
p
(81)
(
ip
ω
S
p
2
ω
2
D
+ ∆
ω
2
EP
/
2
2∆
ω
2
S
ω
FWHM
)
C
p
(
τ
)
(82)
10
and
C
p
(0) = 1, where again
z
t
(
t
)
p
z
t
(
t
+
τ
)
q
(
p
6
=
q
) terms are discarded. Thus completing the Fourier transform for
each term gives the linewidth of the respective harmonics. In particular, the linewidth of the fundamental frequency
can be found through
W
E,
1
(
ω
)
ω
FWHM
(
ω
ω
S
)
2
+ ∆
ω
2
FWHM
/
4
(83)
with
ω
FWHM
=
ω
2
D
+ ∆
ω
2
EP
/
2
ω
2
S
ω
FWHM
=
ω
2
D
+ ∆
ω
2
EP
/
2
ω
2
D
ω
2
EP
ω
FWHM
(84)
We see that this result is different from the Petermann factor result, which is a theory linear in photon numbers and
does not correctly take account of the saturation of the lasers and the Adler mode-locking effect.
From the expressions of ∆
ω
D
[Supplementary Eq. (72)] and ∆
ω
EP
[Supplementary Eq. (73)], the beating frequency
can be expressed using the following hierarchy of equations:
ω
S
= sgn(∆
ω
D
)
ω
2
D
ω
2
EP
(85)
ω
D
=
γ
Γ +
γ
ω
P
+
Γ
Γ +
γ
ω
Kerr
(86)
ω
P
=
ω
P
,
1
ω
P
,
2
(87)
ω
Kerr
=
η
(
N
2
N
1
) =
η
P
SBL
γ
ex
~
ω
(88)
where sgn is the sign function and we take Ω = 0 (no rotation). For the Kerr shift, ∆
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. The center of the locking
band can be found by setting ∆
ω
D
= 0, which leads to ∆
ω
P
=
)∆
ω
Kerr
.
We would like to remark that the equation for locking bandwidth ∆
ω
EP
in the main text does not contain the
phase-sensitive term Im(
κχ
). This terms leads to asymmetry of the locking band with respect to
q
and 1
/q
and has
not been observed in the experimental data. We believe its contribution can be neglected. In other special cases,
Im(
κχ
) disappears if there is a dominant, symmetric scatterer that determines both
κ
and
χ
(e.g. the taper coupling
point), or becomes negligible if there are many small scatterers that add up incoherently (e.g. surface roughness). This
term can also be absorbed into the first two terms so the locking bandwidth is rewritten using effective
κ
,
χ
and a net
amplitude imbalance
q
0
. Thus power calibration errors in the experiment may be confused with the phase-sensitive
term in the locking bandwidth.
Supplementary Note 5. TECHNICAL NOISE CONSIDERATIONS
Here we briefly consider the impact of technical noise to the readout signal. Two important noise sources are
temperature drifts and imprecisely-defined pump frequencies, both of which change the phase mismatch ∆Ω. For a
single SBL, the phase mismatch is transduced into the laser frequency through the mode-pulling effect [Supplementary
Eq. (52) and (72)], which gives a noise transduction factor of
γ
2
/
(Γ +
γ
)
2
. With
γ/
Γ = 0
.
076 fitted from experimental
data the mode-pulling effect reduces pump noise by
23 dB. The Pound-Drever-Hall locking loop in the system also
suppresses noise at low offset frequencies. For counter-pumping of SBLs, the pumping source are derived from the same
laser, and their frequency difference is determined by radio-frequency signals, thus the system has a strong common-
mode noise rejection. Within the model described by Supplementary Eq. (72), the SBL frequency is dependent on
the pump frequency difference only, and features a very high common-mode noise rejection. Other effects that are not
considered in the model (i.e. drift of frequency difference between pump and SBL modes) are believed to be minor
for offset frequencies above 10Hz, where the Allan deviation shows a slope of
1
/
2 corresponding to white frequency
noise.
Supplementary References
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, 1253
(1989).
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Laser Physics
(Addison-Wesley Pub. Co., 1974).
11
[3] Li, J., Lee, H., Chen, T. & Vahala, K. J. Characterization of a high coherence, Brillouin microcavity laser on silicon.
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Express
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, 20170–20180 (2012).
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576
, 65–69 (2019).