Supplementary information

GYROSCOPE

10.3 Supplementary information

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 𝜇₁, 𝜇₂, ... 𝜇_𝑛. For simplicity we will assume that all of the eigenvalues are distinct, i.e. 𝜇_𝑗 ≠ 𝜇_𝑘 if 𝑗 ≠ 𝑘. In this case, 𝑀 will have 𝑛 right eigenvectors and 𝑛 left eigenvectors associated with each𝜇_𝑗:

𝑀|𝑣^R

𝑗i =𝜇_𝑗|𝑣^R

𝑗i, h𝑣^L

𝑗|𝑀 =h𝑣^L

𝑗|𝜇_𝑗, (10.10)

To make sense of the left eigenvectors, note that 𝑀^†|𝑣^L

𝑗i = 𝜇^∗

𝑗|𝑣^L

𝑗i, thus the left eigenvector is the eigenstate as if loss is changed to gain and vice versa. Since𝑀 is in general non-Hermitian, there is no guarantee that |𝑣^L

𝑗i = |𝑣^R

𝑗i, and many of the decomposition results that hold in the Hermitian case will fail. However we note that

𝜇_𝑗h𝑣^L

𝑗|𝑣^R

𝑘i=h𝑣^L

𝑗|𝑀|𝑣^R

𝑘i= 𝜇_𝑘h𝑣^L

𝑗|𝑣^R

𝑘i ⇒ h𝑣^L

𝑗|𝑣^R

𝑘i=0, ∀𝑗 ≠ 𝑘 . (10.11) Thus the 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 and𝑀 as follows:

1=Õ

𝑗

|𝑣^R

𝑗ih𝑣^L

𝑗| h𝑣^L

𝑗|𝑣^R

𝑗i

, (10.12)

𝑀 =Õ

𝑗

|𝑣^R

𝑗ih𝑣^L

𝑗| h𝑣^L

𝑗|𝑣^R

𝑗i

𝜇_𝑗, (10.13)

where each term is a “projector” onto the eigenvectors. Again we note thath𝑣^L

𝑗|𝑣^R

𝑗i may be negative and even complex, which results in special normalizations of the vectors. For simplicity we will choose h𝑣^L

𝑗|𝑣^R

𝑗i = 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=Õ

𝑗

|𝑣^R

𝑗ih𝑣^L

𝑗|, (10.14)

𝑀 =Õ

𝑗

|𝑣^R

𝑗ih𝑣^L

𝑗|𝜇_𝑗. (10.15)

We note that, as a result of using bi-orthonormal left and right vectors, the vectors themselves are not normalized, i.e. h𝑣^L

𝑗|𝑣^L

𝑗i and h𝑣^R

𝑗|𝑣^R

𝑗i 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 information 3, a “natural”

normalization will be chosen when we give a physical meaning to these factors.

Petermann factor of a two-dimensional Hamiltonian

Here we derive the Petermann factor of a two-dimensional Hamiltonian𝐻. Denote the two normalized right (left) eigenvectors of𝐻as|𝜓^R

1iand|𝜓^R

2i(|𝜓^L

1iand|𝜓^L

2i).

The Petermann factors of these two eigenmodes can then be expressed as [35]

PF₁=h𝜓^L

1|𝜓^L

1ih𝜓^R

1|𝜓^R

1i, (10.16)

PF₂=h𝜓^L

2|𝜓^L

2ih𝜓^R

2|𝜓^R

2i. (10.17)

We will first prove that PF₁ = PF₂, which can then be identified as the Petermann factor for the entire system. Note that |𝜓^L

1i and |𝜓^R

2i 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

1ih𝜓^L

1| h𝜓^L

1|𝜓^L

1i + |𝜓^R

2ih𝜓^R

2| h𝜓^R

2|𝜓^R

2i

. (10.18)

Now apply this expansion to|𝜓^R

1iand obtain

|𝜓^R

1i = 1 h𝜓^L

1|𝜓^L

1i|𝜓^L

1i + h𝜓^R

2|𝜓^R

1i h𝜓^R

2|𝜓^R

2i|𝜓^R

2i, (10.19)

whereh𝜓^L

1|𝜓^R

1i=1 has been used. Left multiplication byh𝜓^R

1|results in h𝜓^R

1|𝜓^R

1i= 1 h𝜓^L

1|𝜓^L

1i + h𝜓^R

1|𝜓^R

2ih𝜓^R

2|𝜓^R

1i h𝜓^R

2|𝜓^R

2i

. (10.20)

Thus we obtain,

1

PF₁ =1− h𝜓^R

1|𝜓^R

2ih𝜓^R

2|𝜓^R

1i h𝜓^R

1|𝜓^R

1ih𝜓^R

2|𝜓^R

2i

, (10.21)

which is symmetric with respect to the indexes 1 and 2 and thereby completes the proof that PF₁=PF₂ ≡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:

𝐻₀ ≡ 𝐻− 1

2Tr(𝐻), (10.22)

where Tr is the matrix trace and𝐻₀is the traceless part of𝐻. Using the bi-orthogonal expansion,𝐻₀has the form

𝐻₀ =𝜇(|𝜓^R

1ih𝜓^L

1| − |𝜓^R

2ih𝜓^L

2|), (10.23)

where𝜇is the first eigenvalue. Consider next the quantity Tr(𝐻^†

0𝐻₀):

Tr(𝐻^†

0𝐻₀)= |𝜇|²(h𝜓^L

1|𝜓^L

1ih𝜓^R

1|𝜓^R

1i + h𝜓^L

2|𝜓^L

2ih𝜓^R

2|𝜓^R

2i

−h𝜓^L

2|𝜓^L

1ih𝜓^R

1|𝜓^R

2i − h𝜓^L

1|𝜓^L

2ih𝜓^R

2|𝜓^R

1i), (10.24) where we used the fact that Tr(|𝛼ih𝛽|) = h𝛽|𝛼i. 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

1ias a combination of right eigenvectors using Eq. (10.19):

−h𝜓^L

2|𝜓^L

1ih𝜓^R

1|𝜓^R

2i= h𝜓^R

1|𝜓^R

2ih𝜓^R

2|𝜓^R

1i h𝜓^R

2|𝜓^R

2i h𝜓^L

1|𝜓^L

1i=PF−1. (10.25) Similarly, the fourth term also equals PF−1. Thus

Tr(𝐻^†

0𝐻₀) =|𝜇|²(4PF−2). (10.26)

Finally, to eliminate the eigenvalue 𝜇, we calculate Tr(𝐻²

0) = 𝜇²(h𝜓^L

1|𝜓^R

1i²+ h𝜓^L

2|𝜓^R

2i²) =2𝜇², (10.27) and the PF can be solved as

PF= 1

2 1+ Tr(𝐻^†

0𝐻₀)

|Tr(𝐻²

0) |

!

, (10.28)

which completes the proof.

We note that while a Hermitian Hamiltonian with 𝐻^†

0 = 𝐻₀results in PF = 1, the converse is not always true. Consider the example of𝐻₀ =𝑖 𝜎_𝑧where𝜎_𝑧is the Pauli matrix. This would effectively describe two orthogonal modes with different gain, and direct calculation shows that PF=1.

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 information 4.

The two-dimensional system is described by the column vector|Ψi ↔ (𝑎_cw, 𝑎_ccw)^𝑇 whose components are the orthogonal field amplitudes𝑎_cw and𝑎_ccw. The equation of motion reads𝑖 𝑑|Ψi/𝑑 𝑡 = 𝐻|Ψi, where 𝐻 is the two-dimensional Hamiltonian.

Now assume that|Ψi=𝑐₁|𝜓^R

1i, i.e. only the first eigenmode of the system is excited.

We interpret𝑐₁as the phasor for the eigenmode. We see that|𝑐₁|² =hΨ|Ψi/h𝜓^R

1|𝜓^R

1i is reduced from the true square amplitude hΨ|Ψiby a factor of the length squared of the right eigenvectorh𝜓^R

1|𝜓^R

1i. The equation of motion for𝑐₁reads 𝑖

𝑑𝑐₁ 𝑑 𝑡

=𝑖 𝑑h𝜓^L

1|Ψi 𝑑 𝑡

=h𝜓^L

1|𝐻₀|𝜓^R

1i𝑐₁= 𝜇₁𝑐₁. (10.29) Here, we are assuming that the mode experiences both loss and saturable gain that are absorbed into the definition of the eigenvalue 𝜇₁. To simplify the following calculations, we set the real part of𝜇₁to 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: 𝑖 𝑑|Ψi/𝑑 𝑡 =𝐻₀|Ψi + |𝐹i. Here, |𝐹i ↔

(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

h𝐹

∗

cw(𝑡)𝐹_cw(𝑡⁰)i =h𝐹

∗

ccw(𝑡)𝐹_ccw(𝑡⁰)i =𝜃 𝛿(𝑡−𝑡⁰), (10.30) h𝐹

∗

cw(𝑡)𝐹_ccw(𝑡⁰)i =h𝐹

∗

ccw(𝑡)𝐹_cw(𝑡⁰)i =0, (10.31) where𝜃 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 information 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 for𝑐₁can be modified as follows,

𝑑𝑐₁ 𝑑 𝑡

=−|𝜇₁|𝑐₁+ h𝜓^L

1|𝐹i=−|𝜇₁|𝑐₁+𝐹₁, (10.32) where the fluctuation term for the first eigenmode is defined as 𝐹₁ = h𝜓^L

1|𝐹i. Its correlation reads

h𝐹

∗

1(𝑡)𝐹₁(𝑡⁰)i =𝜃h𝜓^L

1|𝜓^L

1i𝛿(𝑡−𝑡⁰), (10.33) which, upon comparison to Eq. (10.30), shows that the noise input to the right eigenvector field amplitude (𝑐₁) 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 h𝜓^L

1|𝜓^L

1i.

We are interested in the phase fluctuations of 𝑐₁. Here, it is assumed that the mode is pumped to above threshold and is lasing. Under these conditions, it is possible to separate amplitude and phase fluctuations of the field. We rewrite 𝑐₁=|𝑐₁|exp(−𝑖 𝜙_𝑐) and obtain the rate of change of the phase variable as follows:

𝑑 𝜙_𝑐 𝑑 𝑡

= 𝑖 2|𝑐₁|

𝐹₁e^{𝑖 𝜙𝑐} −𝐹

∗ 1e^{−𝑖 𝜙𝑐}

, (10.34)

which describes white frequency noise of the laser field (equivalently phase noise diffusion). The correlation can be calculated as

h ¤𝜙_𝑐(𝑡) ¤𝜙_𝑐(𝑡⁰)i = 𝜃 2|𝑐₁|²h𝜓^L

1|𝜓^L

1i𝛿(𝑡−𝑡⁰) = 𝜃

2hΨ|Ψih𝜓^R

1|𝜓^R

1ih𝜓^L

1|𝜓^L

1i𝛿(𝑡−𝑡⁰)

=PF× 𝜃

2hΨ|Ψi𝛿(𝑡−𝑡⁰), (10.35)

where the non-enhanced linewidth is Δ𝜔₀ = 𝜃/(2hΨ|Ψi) [58] and the enhanced linewidth is given by Δ𝜔 = PF ×Δ𝜔₀. From the above derivation, the PF en- hancement is the result of two effects, the reduction of effective square amplitude (|𝑐₁|²= hΨ|Ψi/h𝜓^R

1|𝜓^R

1i) and the enhancement of noise by h𝜓^L

1|𝜓^L

1i.

Up to now we have not chosen individual normalizations for h𝜓^L

1|𝜓^L

1iandh𝜓^R

1|𝜓^R

1i 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:

h𝜓^L

1|𝜓^L

1i=h𝜓^R

1|𝜓^R

1i=

√

PF, (10.36)

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 analysish𝜓^L

1|𝜓^L

1i=PF andh𝜓^R

1|𝜓^R

1i=1 is chosen, and the enhancement is fully attributed to noise increase by a factor of PF [35] .

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 frequencies𝜔

P,1and𝜔

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 frequency𝜔. 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 frequencies𝛿𝜔

1=−𝜂

𝑎^†

1

𝑎1+2𝑎^†

2

𝑎2

−Ω𝜔 𝐷/(2𝑛_g𝑐)and 𝛿𝜔2 =−𝜂

𝑎^†

2

𝑎2+2𝑎^†

1

𝑎1

+Ω𝜔 𝐷/(2𝑛_g𝑐) relative to𝜔, where𝜂 = 𝑛₂ℏ𝜔²𝑐/(𝑉 𝑛²

0) is the single-photon nonlinear angular frequency shift,𝑛₂is the nonlinear refractive index,𝑉 is the mode volume,𝑛₀is the linear refractive index,𝑐is 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 frequenciesΩ_phonon =2𝜔𝑛₀𝑣_s/𝑐, where𝑣_sis the velocity of the phonons. The loss rate of phonon modes is denoted asΓ(also known as the gain bandwidth) and the loss rate of the SBL modes are assumed equal and denoted as𝛾. In addition, coupling between the two SBL modes is separated as a dissipative part and conservative part, denoted as𝜅and 𝜒, respectively. These rates will be assumed to satisfy Γ 𝛾 |𝜅|,|𝜒|to simplify the calculations, which is a posterioriverified in our system. In the following analysis, we will treat the SBL modes and phonon modes quantum mechanically and define𝑎

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

¤

𝑎1 =−𝛾

2 +𝑖𝜔+𝑖 𝛿𝜔

1

𝑎1+ (𝜅+𝑖 𝜒)𝑎

2−𝑖𝑔_{𝑎 𝑏}𝐴

2𝑏^†

2exp(−𝑖𝜔

P,2𝑡) +𝐹

1(𝑡), (10.37)

¤

𝑎2 =−𝛾

2 +𝑖𝜔+𝑖 𝛿𝜔

2

𝑎2+ (𝜅^∗+𝑖 𝜒^∗)𝑎

1−𝑖𝑔_{𝑎 𝑏}𝐴

1𝑏^†

1exp(−𝑖𝜔

P,1𝑡) +𝐹

2(𝑡), (10.38) 𝑏¤^†

1 =− Γ

2 −𝑖Ω_phonon

𝑏^†

1+𝑖𝑔_{𝑎 𝑏}𝐴^∗

1𝑎

2exp(𝑖𝜔

P,1𝑡) + 𝑓^†

1(𝑡), (10.39) 𝑏¤^†

2 =− Γ

2 −𝑖Ω_phonon

𝑏^†

2+𝑖𝑔_{𝑎 𝑏}𝐴^∗

2𝑎

1exp(𝑖𝜔

P,2𝑡) + 𝑓^†

2(𝑡), (10.40) where 𝑔_{𝑎 𝑏} is the single-particle Brillioun coupling coefficient. The fluctuation operators 𝐹(𝑡) and 𝑓(𝑡) associated with the field operators have the following correlations:

h𝐹^†

1(𝑡)𝐹

1(𝑡⁰)i = h𝐹^†

2(𝑡)𝐹

2(𝑡⁰)i =𝛾 𝑁_th𝛿(𝑡−𝑡⁰), (10.41) h𝐹

1(𝑡)𝐹^†

1(𝑡⁰)i = h𝐹

2(𝑡)𝐹^†

2(𝑡⁰)i =𝛾(𝑁_th+1)𝛿(𝑡−𝑡⁰), (10.42) h𝑓^†

1(𝑡)𝑓

1(𝑡⁰)i = h𝑓^†

2(𝑡)𝑓

2(𝑡⁰)i = Γ𝑛_th𝛿(𝑡−𝑡⁰), (10.43) h𝑓

1(𝑡)𝑓^†

1(𝑡⁰)i = h𝑓

2(𝑡)𝑓^†

2(𝑡⁰)i = Γ(𝑛_th+1)𝛿(𝑡−𝑡⁰), (10.44) where𝑁_thand𝑛_thare the thermal occupation numbers of the SBL state and phonon state. In addition, there are non-zero cross-correlations of the photon fluctuation

operators due to the dissipative coupling:

h𝐹^†

2(𝑡)𝐹

1(𝑡⁰)i =−2𝜅 𝑁_th𝛿(𝑡−𝑡⁰), (10.45) h𝐹^†

1(𝑡)𝐹

2(𝑡⁰)i =−2𝜅^∗𝑁_th𝛿(𝑡−𝑡⁰), (10.46) h𝐹

2(𝑡)𝐹^†

1(𝑡⁰)i =−2𝜅^∗(𝑁_th+1)𝛿(𝑡−𝑡⁰), (10.47) h𝐹

1(𝑡)𝐹^†

2(𝑡⁰)i =−2𝜅(𝑁_th+1)𝛿(𝑡−𝑡⁰). (10.48) All other cross correlations not explicitly written are 0.

Single SBL

We first study a single laser mode and its corresponding phonon field (𝑎

1and 𝑏

2) by neglecting𝜅and 𝜒. By introducing the slow varying envelope with𝑎

1 =𝛼

1e^{−𝑖𝜔𝑡} and𝑏

2 =𝛽

2e^{−𝑖(𝜔P,2−𝜔)𝑡}, the following equations result:

¤

𝛼1=−𝛾 2 +𝑖 𝛿𝜔

1

𝛼1−𝑖𝑔_{𝑎 𝑏}𝐴

2𝛽^†

2+𝐹

1(𝑡)e^𝑖𝜔𝑡, (10.49) 𝛽¤^†

2=− Γ

2 +𝑖ΔΩ

2

𝛽^†

2+𝑖𝑔_{𝑎 𝑏}𝐴^∗

2𝛼

1+ 𝑓^†

2(𝑡)e^{−𝑖(𝜔P,2−𝜔)𝑡}, (10.50) where we have defined the frequency mismatchΔΩ

2=𝜔

P,2−𝜔−Ω_phonon. Neglecting the weak Kerr effect term in𝛿𝜔

1, this is a set of linear equations in𝑎

1and𝑏

2. The eigenvalues of the coefficient matrix,

−𝛾/2−𝑖 𝛿𝜔

1 −𝑖𝑔_{𝑎 𝑏}𝐴

2

𝑖𝑔_{𝑎 𝑏}𝐴^∗

2 −Γ/2−𝑖ΔΩ₂

!

, (10.51)

can be solved as 𝜇_1,2 = 1

4

−Γ−𝛾−2𝑖 𝛿𝜔

1−2𝑖ΔΩ₂± q

16𝑔²

𝑎 𝑏|𝐴

2|²+ (Γ−𝛾−2𝑖 𝛿𝜔

1+2𝑖ΔΩ₂)²

. (10.52) At the lasing threshold, the first eigenvalue 𝜇₁ has a real part of 0. This can be rewritten as

16𝑔²

𝑎 𝑏|𝐴

2|²+(Γ−𝛾−2𝑖 𝛿𝜔

1+2𝑖ΔΩ₂)² =(Γ+𝛾+2𝑖 𝛿𝜔

1+2𝑖ΔΩ₂+4𝑖Im(𝜇₁))². (10.53) Solving this complex equation gives the SBL eigenfrequency as well as the lasing threshold,

𝜇₁=−𝑖

𝛾ΔΩ₂+Γ𝛿𝜔

1

Γ+𝛾

, (10.54)

𝑔²

𝑎 𝑏|𝐴

2|²= Γ𝛾

4 1+ 4(ΔΩ₂−𝛿𝜔

1)² (Γ+𝛾)²

!

. (10.55)

The threshold at perfect phase matching (ΔΩ₂ =𝛿𝜔

1) is usually written in a more familiar form𝑔₀|𝐴

2|²=𝛾/2, where𝑔₀is the Brillouin gain factor [33]. Comparison gives

𝑔_{𝑎 𝑏} = r

𝑔₀Γ

2 . (10.56)

We also introduce the modal Brillioun gain function for a single direction:

𝑔1 ≡ 𝑔₀

1+4(𝛿𝜔

1−ΔΩ₂)²/(Γ+𝛾)², (10.57) so that the threshold can be written as

𝑔1|𝐴

2|²= 𝛾

2. (10.58)

With the threshold condition solved, the matrix can be decomposed using the bi- orthogonal approach outlined in Supplementary information 1. The linear combi- nation that describes the composite SBL mode can be found as

𝛼1 = Γ 𝛾 +Γ

𝛼1−𝑖

𝑔_{𝑎 𝑏} Γ

2 1+2𝑖(ΔΩ

2−𝛿𝜔

1)/(Γ+𝛾)𝐴

2𝛽^†

2

, (10.59)

where the factor Γ/(𝛾 +Γ) properly normalizes𝛼

1 so that𝛼

1 =𝛼

1when only the SBL mode is present in the system, and we have dropped its dependence on the phase mismatchΔΩ₂−𝛿𝜔

1for simplicity. The associated equation of motion is 𝑑

𝑑 𝑡

𝛼1 =−𝑖

𝛾ΔΩ₂+Γ𝛿𝜔

1

Γ+𝛾

𝛼1+𝐹

1(𝑡), (10.60)

where the frequency term now includes a mode-pulling contribution so that the SBL laser frequency is given by

𝜔S,1=𝜔+

𝛾ΔΩ₂+Γ𝛿𝜔

1

Γ+𝛾

. (10.61)

Also, we have defined a combined fluctuation operator for𝛼

1, 𝐹1(𝑡) = Γ

𝛾+Γ

"

𝐹1(𝑡)e^𝑖𝜔𝑡 −𝑖

s1−2𝑖(ΔΩ

2−𝛿𝜔

1)/(Γ+𝛾) 1+2𝑖(ΔΩ

2−𝛿𝜔

1)/(Γ+𝛾) r

𝛾 Γ𝑓^†

2(𝑡)e^{−𝑖(𝜔P,2−𝜔)𝑡}

# , (10.62)

with the following correlations, h𝐹

† 1(𝑡)𝐹

1(𝑡⁰)i = Γ

𝛾+Γ 2

h𝐹^†

1(𝑡)𝐹

1(𝑡⁰)i + 𝛾 Γh𝑓^†

1(𝑡)𝑓

1(𝑡⁰)i

= Γ

𝛾+Γ 2

𝛾(𝑛_th+𝑁_th)𝛿(𝑡−𝑡⁰), (10.63) h𝐹

1(𝑡)𝐹

† 1(𝑡⁰)i =

Γ 𝛾+Γ

2

h𝐹

1(𝑡)𝐹^†

1(𝑡⁰)i + 𝛾 Γh𝑓

1(𝑡)𝑓^†

1(𝑡⁰)i

= Γ

𝛾+Γ 2

𝛾(𝑛_th+𝑁_th+2)𝛿(𝑡−𝑡⁰), (10.64) We can now write𝛼

1(𝑡) =p

𝑁1exp(−𝑖 𝜙

1), where𝑁

1is the photon number,𝜙

1is 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 for𝜙

1is 𝜙¤

1=𝜔

S,1−𝜔+Φ

1(𝑡), Φ

1(𝑡) = 𝑖 2p

𝑁1

(𝐹

1(𝑡)e^{𝑖 𝜙1} −𝐹

†

1(𝑡)e^{−𝑖 𝜙1}). (10.65) The correlation of the noise operator is given by,

hΦ1(𝑡)Φ

1(𝑡⁰)i = 1 4𝑁

1

(h𝐹

† 1(𝑡)𝐹

1(𝑡⁰)i + h𝐹

1(𝑡)𝐹

† 1(𝑡⁰)i

= Γ

𝛾+Γ 2

𝛾 2𝑁

1

(𝑛_th+𝑁_th+1)𝛿(𝑡−𝑡⁰), (10.66) and we identify the coefficient before the delta function,

Δ𝜔

FWHM,1= Γ

𝛾+Γ 2

𝛾 2𝑁

1

(𝑛_th+𝑁_th+1), (10.67) 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:

Δ𝜔_FWHM= Δ𝜔

FWHM,1+Δ𝜔

FWHM,2= Γ

𝛾+Γ 2

1 2𝑁

1

+ 1 2𝑁

2

𝛾(𝑛_th+𝑁_th+1), (10.68) and then convert to the one-sided power spectral density𝑆_𝜈:

𝑆_𝜈 = 1 𝜋

Δ𝜔_FWHM 2𝜋

= Γ

𝛾+Γ 2

ℏ𝜔³ 4𝜋²𝑄_T𝑄_ex

( 1 𝑃_cw

+ 1

𝑃_ccw

) (𝑛_th+𝑁_th+1), (10.69)

where𝑄_Tand𝑄_exare the loaded and coupling𝑄factors, and 𝑃_cwand𝑃_ccw are the SBL powers in each direction.

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 𝑑 𝑡

𝛼1=−𝑖(𝜔

S,1−𝜔)𝛼

1+ Γ

𝛾+Γ(𝜅+𝑖 𝜒)𝛼

2+𝐹

1(𝑡), (10.70) 𝑑

𝑑 𝑡

𝛼2=−𝑖(𝜔

S,2−𝜔)𝛼

2+ Γ

𝛾+Γ(𝜅^∗+𝑖 𝜒^∗)𝛼

1+𝐹

2(𝑡), (10.71) where quantities with the opposite subscript are defined similarly. We note that the coupling term involves the optical modes𝛼

1 and𝛼

2only. However, no additional coupling occurs between the other components of the SBL eigenstates 𝛼

1 and𝛼

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 [24].

The equations now become 𝑑

𝑑 𝑡

𝛼1=−𝑖(𝜔

S,1−𝜔)𝛼

1+ (𝜅+𝑖 𝜒)𝛼

2+𝐹

1(𝑡), (10.72) 𝑑

𝑑 𝑡

𝛼2=−𝑖(𝜔

S,2−𝜔)𝛼

2+ (𝜅^∗+𝑖 𝜒^∗)𝛼

1+𝐹

2(𝑡), (10.73) where we have defined mode-pulled coupling rates𝜅 =𝜅Γ/(𝛾+Γ)and𝜒 = 𝜒Γ/(𝛾+ Γ).

We can write𝛼_𝑗(𝑡) =p

𝑁_𝑗exp(−𝑖 𝜙_𝑗)with 𝑗 =1,2, and once again ignore amplitude fluctuations. The equations of motion for the phases are

𝑑 𝑑 𝑡

𝜙1=(𝜔

S,1−𝜔) −𝑞Im[(𝜅+𝑖 𝜒)e^{(𝑖 𝜙1−𝑖 𝜙2)}] + 𝑖 2p

𝑁1

(𝐹

1(𝑡)e^{𝑖 𝜙1}−𝐹

†

1(𝑡)e^{−𝑖 𝜙1}), (10.74) 𝑑

𝑑 𝑡

𝜙2=(𝜔

S,2−𝜔) −𝑞⁻¹Im[(𝜅^∗+𝑖 𝜒^∗)e^{(𝑖 𝜙2−𝑖 𝜙1)}] + 𝑖 2p

𝑁2

(𝐹

2(𝑡)e^{𝑖 𝜙2} −𝐹

†

2(𝑡)e^{−𝑖 𝜙2}), (10.75) where we have defined the amplitude ratio 𝑞 = p

𝑁2/𝑁

1 for simplicity. As we measure the beatnote frequency, it is convenient to define𝜙 ≡ 𝜙

2−𝜙

1from which we obtain

𝑑 𝜙 𝑑 𝑡

=(𝜔

S,2−𝜔

S,1) +Im 𝑞(𝜅+𝑖 𝜒) +𝑞⁻¹(𝜅−𝑖 𝜒)

e^{−𝑖 𝜙} +Φ(𝑡), (10.76)

where the combined noise term and its correlation are given by Φ = − 𝑖

2p 𝑁1

(𝐹

1(𝑡)e^{𝑖 𝜙1}−𝐹

†

1(𝑡)e^{−𝑖 𝜙1}) + 𝑖 2p

𝑁2

(𝐹

2(𝑡)e^{𝑖 𝜙2}−𝐹

†

2(𝑡)e^{−𝑖 𝜙2}), (10.77)

hΦ(𝑡)Φ(𝑡⁰)i = Γ

𝛾+Γ 2

𝛿(𝑡−𝑡⁰)

×

"

1 2𝑁

1

+ 1 2𝑁

2

𝛾(𝑁_th+𝑛_th+1) + 2 p𝑁

1𝑁

2

𝑁_th+ 1 2

Re(𝜅e^{−𝑖 𝜙(𝑡)})

# , (10.78) Since both𝑁_thand𝜅/𝛾is small, we will discard the last time-varying term and write hΦ(𝑡)Φ(𝑡⁰)i ≈Δ𝜔_FWHM𝛿(𝑡−𝑡⁰), (10.79) whereΔ𝜔_FWHM = Γ²/(𝛾+Γ)²[(2𝑁

1)⁻¹+ (2𝑁

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−𝜙₀, where 𝜙₀ = Arg

𝑞(𝜅+𝑖 𝜒) +𝑞⁻¹(𝜅−𝑖 𝜒)

and Arg(𝑧)is the phase of 𝑧:

𝑑 𝜙 𝑑 𝑡

= Δ𝜔_D−Δ𝜔_EPsin𝜙+Φ(𝑡), (10.80) with

Δ𝜔_D =𝜔

S,2−𝜔

S,1= 𝛾 Γ+𝛾

(𝜔

P,1−𝜔

P,2) + Γ Γ+𝛾

𝜂(𝑁

2−𝑁

1) + 𝜔 𝐷 𝑛_g𝑐

Ω

, (10.81) Δ𝜔²

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

𝑞²− 1 𝑞²

Im(𝜅 𝜒^∗)

# , (10.82) 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 ofΔ𝜔_EP, the beating linewidth would be given byΔ𝜔_FWHM. The locking termΔ𝜔_EPsin𝜙makes the rate of phase change nonuniform and increases the linewidth.

The following part of analysis is dedicated to obtaining the linewidth from this stochastic Adler equation. We define𝑧_𝜙 =exp(−𝑖 𝜙) and rewrite

𝑑 𝑑 𝑡

𝑧_𝜙 =−𝑖 𝑧_𝜙(Δ𝜔_D +Δ𝜔_EP

𝑧_𝜙−𝑧⁻¹

𝜙

2𝑖

+Φ). (10.83)

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 = (Δ𝜔_D−Δ𝜔_S)𝑧_𝜙+𝑖Δ𝜔_EP

Δ𝜔_EP𝑧_𝜙+𝑖(Δ𝜔_D−Δ𝜔_S), 𝑧_𝜙 =−𝑖

(Δ𝜔_D−Δ𝜔_S)𝑧_𝑡−Δ𝜔_EP

Δ𝜔_EP𝑧_𝑡− (Δ𝜔_D−Δ𝜔_S), |𝑧_𝜙|= |𝑧_𝑡|=1, (10.84) 1

𝑧_𝑡 𝑑 𝑑 𝑡

𝑧_𝑡 =𝑖Δ𝜔_S−𝑖

Δ𝜔_EP(𝑧_𝑡+𝑧⁻¹

𝑡 )/2−Δ𝜔_D Δ𝜔_S

Φ, (10.85)

where we introduced Δ𝜔_S = q

Δ𝜔²

D−Δ𝜔²

EP (which has the same meaning in the main text). The noiseless solution of𝑧_𝑡 would be 𝑧_𝑡 = exp(𝑖Δ𝜔_S𝑡), and𝑧_𝜙 can be expanded in𝑧_𝑡 as

𝑧_𝜙 =−𝑖

Δ𝜔_D −Δ𝜔_S Δ𝜔_EP

+2𝑖 Δ𝜔_S Δ𝜔_EP

∞

Õ

𝑝=1

Δ𝜔_D−Δ𝜔_S Δ𝜔_EP𝑧_𝑡

^𝑝

, (10.86)

where we have assumed Δ𝜔_D > Δ𝜔_EP for convenience so that convergence can be guaranteed (for the case Δ𝜔_D < −Δ𝜔_EP we can expand near 𝑧_𝑡 = 0 instead of 𝑧_𝑡 =∞). Thus the signal consists of harmonics oscillating at frequency 𝑝Δ𝜔_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) ∝ F𝜏{h𝑧^∗

𝜙(𝑡)𝑧_𝜙(𝑡+𝜏)i}(𝜔), (10.87) and the correlation is given by

h𝑧^∗

𝜙(𝑡)𝑧_𝜙(𝑡+𝜏)i =

Δ𝜔_D−Δ𝜔_S Δ𝜔_EP

2

+4 Δ𝜔²

S

Δ𝜔²

EP

∞

Õ

𝑝=1

Δ𝜔_D−Δ𝜔_S Δ𝜔_EP

2𝑝

h𝑧_𝑡(𝑡)^𝑝𝑧_𝑡(𝑡+𝜏)^−𝑝i, (10.88) where we have discarded the h𝑧_𝑡(𝑡)^𝑝𝑧_𝑡(𝑡+𝜏)^−𝑞i (𝑝 ≠𝑞)terms since they vanish at the lowest order ofΔ𝜔_FWHM.

To further calculate eachh𝑧_𝑡(𝑡)^𝑝𝑧_𝑡(𝑡+𝜏)^−𝑝i ≡𝐶_𝑝(𝜏), we require the integral form of the Fokker-Planck equation: if𝑑 𝑋(𝑡) =𝜇(𝑋 , 𝑡)𝑑 𝑡+𝜎(𝑋 , 𝑡)𝑑𝑊is a stochastic dif- ferential equation (in the Stratonovich interpretation), where𝑊 is a Wiener process, then for 𝑓(𝑋) as a function of𝑋, the differential equation for its average reads

𝑑 𝑑 𝑡

h𝑓(𝑋)i = h(𝜇+ 𝜎 2

𝜕 𝜎

𝜕 𝑋

)𝑓⁰(𝑋)i + h1

2𝜎²𝑓⁰⁰(𝑋)i. (10.89)

Applying the Fokker-Planck equation to𝐶_𝑝(𝜏), with the stochastic equation for 𝑧_𝑡, gives

𝑑𝐶_𝑝(𝜏) 𝑑 𝜏

=−𝑝h𝑖Δ𝜔_S𝑧_𝑡(𝑡)^𝑝𝑧_𝑡(𝑡+𝜏)^−𝑝i +𝑝h

"

Δ𝜔_FWHM 2Δ𝜔²

S

Δ𝜔_EP

𝑧_𝑡(𝑡+𝜏) +𝑧⁻¹

𝑡 (𝑡+𝜏)

2 −Δ𝜔_D

(Δ𝜔_EP𝑧_𝑡(𝑡+𝜏) −Δ𝜔_D)

#

×𝑧_𝑡(𝑡)^𝑝𝑧_𝑡(𝑡+𝜏)^−𝑝i

−𝑝(𝑝+1) hΔ𝜔_FWHM 2Δ𝜔²

S

Δ𝜔_EP

𝑧_𝑡(𝑡+𝜏) +𝑧⁻¹

𝑡 (𝑡+𝜏)

2 −Δ𝜔_D

²

𝑧_𝑡(𝑡)^𝑝𝑧_𝑡(𝑡+𝜏)^−𝑝i (10.90)

≈ −𝑖 𝑝Δ𝜔_S− 𝑝² Δ𝜔²

D+Δ𝜔²

EP/2 2Δ𝜔²

S

Δ𝜔_FWHM

!

𝐶_𝑝(𝜏), (10.91)

and𝐶_𝑝(0) =1, where againh𝑧_𝑡(𝑡)^𝑝𝑧_𝑡(𝑡+𝜏)^−𝑞i (𝑝 ≠ 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) ∝ Δ𝜔_FWHM (𝜔−Δ𝜔_S)²+Δ𝜔²

FWHM/4

, (10.92)

with

Δ𝜔_FWHM = Δ𝜔²

D+Δ𝜔²

EP/2 Δ𝜔²

S

Δ𝜔_FWHM= Δ𝜔²

D+Δ𝜔²

EP/2 Δ𝜔²

D−Δ𝜔²

EP

Δ𝜔_FWHM. (10.93) 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 Δ𝜔_D [Eq. (10.81)] and Δ𝜔_EP [Eq. (10.82)], the beating frequency can be expressed using the following hierarchy of equations:

Δ𝜔_S =sgn(Δ𝜔_D) q

Δ𝜔²

D−Δ𝜔²

EP, (10.94)

Δ𝜔_D = 𝛾 Γ+𝛾

Δ𝜔_P+ Γ Γ+𝛾

Δ𝜔_Kerr, (10.95)

Δ𝜔_P =𝜔

P,1−𝜔

P,2, (10.96)

Δ𝜔_Kerr =𝜂(𝑁

2− 𝑁

1) = 𝜂Δ𝑃_SBL 𝛾_exℏ𝜔

, (10.97)

where sgn is the sign function and we takeΩ = 0 (no rotation). For the Kerr shift, Δ𝑃_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 settingΔ𝜔_D =0, which leads toΔ𝜔_P =−(Γ/𝛾)Δ𝜔_Kerr.

We would like to remark that the equation for locking bandwidth Δ𝜔_EP in the main text does not contain the phase-sensitive term Im(𝜅 𝜒^∗). This terms leads to asymmetry of the locking band with respect to𝑞and 1/𝑞and has not been observed in the experimental data. We believe its contribution can be neglected. In other special cases, Im(𝜅 𝜒^∗) 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 𝑞₀. Thus power calibration errors in the experiment may be confused with the phase-sensitive term in the locking bandwidth.

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 fre- quencies, both of which change the phase mismatchΔΩ. For a single SBL, the phase mismatch is transduced into the laser frequency through the mode-pulling effect [Eq.

(10.61) and (10.81)], which gives a noise transduction factor of𝛾²/(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 sources 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 rejec- tion. Within the model described by Eq. (10.81), 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.