PreprintPDF Available

Nonlinear Multi-Resonant Cavity Quantum Photonics Gyroscopes Quantum Light Navigation

July 2023

July 2023

Authors:

Mengdi Sun

Virginia Tech (Virginia Polytechnic Institute and State University)

Vassilios Kovanis

Virginia Tech (Virginia Polytechnic Institute and State University)

Zin Lin

Virginia Tech

Preprints and early-stage research may not have been peer reviewed yet.

We propose an on-chip all-optical gyroscope based on nonlinear multi-resonant cavity quantum photonics in thin film $\chi^{(2)}$ resonators -- Quantum-Optic Nonlinear Gyro or QONG in short. The key feature of our gyroscope is co-arisal and co-accumulation of quantum correlations, nonlinear wave mixing and non-inertial signals, all inside the same sensor-resonator. We theoretically analyze the Fisher Information of our QONGs under fundamental quantum noise conditions. Using Bayesian optimization, we maximize the Fisher Information and show that $\sim 900\times$ improvement is possible over the shot-noise limited linear gyroscope with the same footprint, intrinsic quality factors and power budget.

Content uploaded by Mengdi Sun

Content may be subject to copyright.

Nonlinear Multi-Resonant Cavity Quantum Photonics Gyroscopes

for Quantum Light Navigation

Mengdi Sun∗and Vassilios Kovanis

Bradley Department of Electrical and Computer Engineering, Virginia Tech, Arlington, VA, USA

Marko Lonˇcar

John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

Zin Lin

Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, USA

(Dated: July 22, 2023)

We propose an on-chip all-optical gyroscope based on nonlinear multi-resonant cavity quantum

photonics in thin ﬁlm χ(2) resonators—Quantum-Optic Nonlinear Gyro or QONG in short. The key

feature of our gyroscope is co-arisal and co-accumulation of quantum correlations, nonlinear wave

mixing and non-inertial signals, all inside the same sensor-resonator. We theoretically analyze the

Fisher Information of our QONGs under fundamental quantum noise conditions. Using Bayesian

optimization, we maximize the Fisher Information and show that ∼900×improvement is possible

over the shot-noise limited linear gyroscope with the same footprint, intrinsic quality factors and

power budget.

I. INTRODUCTION

Gyroscopes are critical components of an inertial nav-

igation system for augmenting the GPS guidance or sal-

vaging GPS-denied operational environments [1]. In an

optical gyroscope, the rotation rate is measured through

the phase shift between two counter-propagating beams

in an optical loop. This approach was ﬁrst proposed by

Sagnac in 1913 [2, 3] and soon diﬀerent types of op-

tical gyroscopes were developed [4, 5]. Careful studies

have been performed on the sensitivity and the quantum

noise of these gyroscopes [6, 7], and remarkable levels

of sensitivity (<0.001◦/h) have been achieved in state-

of-the-art discrete component optical gyroscopes, includ-

ing ﬁber optic gyroscopes (FOG) [8–10], ring laser gy-

roscopes (RLG) [11], atom-laser gyroscope [12] and op-

tical cavity gyroscopes [13–15]. However, bulky compo-

nents and relatively high power consumption remain ma-

jor roadblocks to further exploiting discrete component

optical gyroscopes. On the other hand, on-chip optical

gyroscopes [16–20] exhibit great potential for fully inte-

grated inertial navigation platforms (free of fragile mov-

ing parts) and can outperform their discrete component

counterparts in size, weight, power consumption, ma-

neuverability, manufacturing scalability, robustness and

the ability to operate in harsh environments. However,

on-chip gyros are yet to reach sensitivity levels smaller

than 1◦/h. This is due to fundamentally limited optical

path lengths even in ultra-high quality factor resonan-

tors [16], leaving dubious prospects for further improve-

ments via increasing resonator size or quality factors. To

address the challenge of this seemingly intrinsic trade-

oﬀ between sensitivity and compactness, novel physics

∗mengdis@vt.edu

and designs have been investigated, including exceptional

point sensing [21], slow light [22, 23], dispersive enhance-

ments [24, 25], dynamic thermal drift cancellation [15, 26]

and nuclear magnetic resonance [27].

Meanwhile, driven by the emerging trend of quantum

technologies [28–30], quantum light sensors have been

identiﬁed as a promising option that can extend the fun-

damental sensitivity limits beyond the shot noise regime

[31–33]. These ideas were reinforced by decades of de-

velopment and analysis that lead to the construction of

very large laser interferometers with extreme sensitivity

that is capable of detecting gravitational waves from re-

mote cosmological events. Recently, squeezed light was

used in the LIGO in the US and the VIRGO in Italy

to substantially improve the sensitivity of the observing

runs that happened late in April 2019 [34, 35]. On the

other hand, recent advances in nanofabrication, integra-

tion and packaging of ultra-coherent laser sources [36],

low-loss photonic circuits [37, 38] and highly eﬃcient

photo-detectors [39] have opened up exciting opportu-

nities for realizing fully on-chip quantum devices. Along

this trend, we identify on-chip quantum light gyroscopes,

which combine high sensitivities, low power consumption,

and small form factors, as promising candidates for next-

generation rotation sensing.

In this paper, we theoretically introduce a new type of

on-chip quantum light gyroscope that exploits nonlinear

multi-resonant cavity quantum photonics in integrated

thin ﬁlm resonators with strong quadratic χ(2) nonlinear-

ities. We call our gyroscope Quantum-Optic Nonlinear

Gyro or QONG in short. Instead of externally inject-

ing quantum states of light into the gyroscope [32], one

of the distinguishing features of our gyroscope is that it

fuses quantum-coherent nonlinear interactions, quantum

light generation and non-inertial signal accumulation in-

side the same sensor-resonator, enabling ≳900×im-

arXiv:submit/5020283 [quant-ph] 22 Jul 2023

provements in gyroscopic sensitivity over the linear shot

noise limit. In our scheme, classical laser light (coherent

state) is injected into a doubly-resonant χ(2) cavity, and

output light is measured at the fundamental (ω1) and

second-harmonic frequencies (ω2= 2ω1). The sensitiv-

ity of the gyroscope is evaluated by Fisher information

(FI) [40, 41], and the latter is maximized by Bayesian

optimization [42]. Various parameter regimes associated

with both fundamental and second harmonic injection

schemes were investigated, which reveal correlated noise

suppression and sensitivity enhancements via paramet-

ric oscillations and critically sensitive three wave mixing

dynamics. We predict that, under quantum noise con-

ditions, a minimum detectable rotation rate (MDR) of

<0.01 ◦/h can be achieved using a thin ﬁlm lithium nio-

bate (TFLN) ring resonator with a diameter of 20 mm,

intrinsic quality factors Qi2= 106at the second har-

monic wavelength (795 nm), Qi1= 107at the fundamen-

tal wavelength (1590 nm). We discuss the scope, validity

and implications of our approach and results, while the

key sensitivity enhancement factors are summarized in

Table I of Section III D.

II. GYROSCOPIC MODEL

A. Linear resonant gyroscope as a baseline

We ﬁrst review a basic interferometric scheme probing

the gyroscopic shift of a linear resonant cavity, as outlined

in Fig. 1. We perform a quantum noise analysis similar to

Ref. [43] or Section 4 of Ref. [7]. Two identical counter-

propagating probes (seeded from the same on-chip laser)

are injected into the clockwise (CW) and the counter-

clockwise (CCW) modes of a ring resonator; at the exit,

the two probe ﬁelds are set to interfere via balanced ho-

modyne detection [44]. In the absence of rotation, the

CW and the CCW modes are degenerate and the exiting

ﬁelds register a vanishing diﬀerential photocurrent signal

at the detection setup [7]. Rotational motion induces a

frequency splitting proportional to the rotation rate Ω,

which in turn induces a phase diﬀerence between the out-

going CW and CCW probes. Subsequently, interference

of the two probe ﬁelds gives rise to a non-zero diﬀeren-

tial signal and the underlying Ω can be measured. For

conceptual simplicity, we assume that the frequency of

the probe laser is always locked to the degenerate fre-

quency of the unperturbed gyro [7]. In principle, this

can be achieved by self-injection locking the laser to an

independent rotation-insensitive cavity (such as a high-Q

spiral resonator [45]) having the exact same frequency as

the unperturbed gyro ring. It has been demonstrated

[46] that self-injection locking to a high-Q cavity can

produce an ultra-coherent integrated laser with a sub-

Hertz linewidth; therefore, we can readily approximate

the laser state as a quantum-mechanical coherent state.

In the Heisenberg picture, the ring resonator gyro obeys

the Heisenberg-Langevin equations [47]:

dˆacw

dt =−κ

2−γ

2+iδˆacw +iβˆaccw +√κˆ

bin

cw +√γˆcin

(1)

dˆaccw

dt =−κ

2−γ

2−iδˆaccw +iβˆacw +√κˆ

bin

ccw +√γˆcin

ccw

(2)

where ˆacw and ˆaccw are the annihilation operators for

the cavity CW and CCW modes excited by the injec-

tions ˆ

bin

cw and ˆ

bin

ccw respectively. ˆcin

cw and ˆcin

ccw represent

intrinsic loss channels (such as radiative losses). κand

γare the decay rates for the coupling and the intrinsic

losses. We approximate Rayleigh-type back-scattering as

a linear (conservative) coupling βbetween CW and CCW

modes inside the cavity [16]. δis the rotation-induced

resonant frequency shift due to the Sagnac eﬀect. Here

we have assumed single-photon normalization for each

eigenmode so that ˆa†ˆa, for example, represents the pho-

ton number operator inside the cavity.

We denote ⟨ˆ

A⟩=⟨ψ|ˆ

A|ψ⟩as the usual notation for

computing the expectation value of a physical observable

Awith respect to the quantum state |ψ⟩. In the linear

problem, we will consider coherent states of the same am-

plitude bin in the input waveguides and vacuum states in

the intrinsic loss channels for both CW and CCW light

[7]. The input quantum state of the gyro is then given

by |ψ⟩=|bin⟩cw |bin⟩ccw |0⟩cw |0⟩ccw . The classical coun-

terpart of the input operator ˆ

bin is the input amplitude

of a coherent state in the feeder waveguide, and can be

related to the input power Pby the formula:

|bin|2=⟨ˆ

bin†ˆ

bin⟩=P

ℏω(3)

The output operators in the waveguides are given by

[7]:

bout

cw =ˆ

bin

cw −√κ1ˆacw (4)

bout

ccw =ˆ

bin

ccw −√κ1ˆaccw (5)

The clockwise and counterclockwise signals are set

to interfere through a directional coupler/beam splitter

with a controllable phase shift ϕ, followed by photode-

tection. The signal incident on the photodetectors and

the photocurrent operators are then given by:

b+=ˆ

bout

cw eiϕ/2+iˆ

bout

ccwe−iϕ/2/√2 (6)

b−=iˆ

bout

cw eiϕ/2+ˆ

bout

ccwe−iϕ/2/√2 (7)

i+=ˆ

b†

+ˆ

b+(8)

i−=ˆ

b†

−ˆ

b−(9)

We measure the diﬀerential current signal:

i=ˆ

i+−ˆ

i−(10)

As a ﬁgure of merit, we will investigate the minimum

detectable frequency shift by calculating the ratio be-

tween the standard variation of the measured diﬀeren-

tial current and the derivative of the mean value of the

current over the rotation-induced frequency shift, as re-

ported by Dowling in 1998 [12]:

δmin =q⟨ˆ

i2⟩−⟨ˆ

i⟩2



∂⟨ˆ

i⟩

∂δ δ=0

(11)

Since the resonant frequency shift due to the Sagnac

eﬀect is given by δ=2πrΩ

λn0[7], the minimum detectable

rotation rate (MDR) is given by:

Ωmin =λn0

2πR q⟨ˆ

i2⟩−⟨ˆ

i⟩2



∂⟨ˆ

i⟩

∂δ δ=0

(12)

where R and n0are the radius and the refractive index

of the micro-ring. λis the wavelength of the input light.

We emphasize that Ωmin is a holistic measure that con-

siders the deterministic sensitivity of the noise-averaged

photocurrent with respect to Ω as well as the variance

of the measured current signals due to noise (Both are

critical to correctly characterizing the overall sensitivity

of the gyro; it has been pointed out [48] that an analysis

only of the deterministic sensitivity could often lead to

misleading conclusions).

If we ignore Rayleigh back-scattering β= 0, we can

derive a simple closed-form expression for MDR in the

linear gyroscope (LG):

ΩLG

min =√2λn0(κ+γ)2

32πRκ√N(13)

where N=P

ℏωis the incident number of photons

per unit time. ΩLG

min is minimized at κ=γ, yielding

MDRLG

min =√2cn0

4R√NQi, where the intrinsic quality factor is

deﬁned by Qi=ω

γ. Note that this is only an example to

illustrate the sensitivity dependence of the simplest linear

gyroscope without considering Rayleigh back-scattering,

which will be taken into account in the following dis-

cussions. We note that, in our analysis, we only con-

sider fundamental quantum noise: without loss of gen-

erality, we have assumed perfect beam splitters and de-

tectors external to the resonator, while we do consider

realistic losses inside the resonator. This linear quan-

tum result will serve as a baseline comparison for our

later analysis of a new mode of gyroscope that relies on

nonlinear quantum optical eﬀects. Note that the scaling

MDRLG

min ∼1

√NQirecovers the familiar shot noise limit

or the standard quantum limit [12]. In addition, the 1

dependence in Equation (13) indicates that a larger ring

radius Roﬀers better sensitivity, which is one of the com-

mon control knobs of classical linear optical gyroscope.

FIG. 1. The schematic of the linear micro-ring gyroscope.

The input and output light of the micro-ring cavity is injected

at two waveguide ports bin

1(CW) / bin

1(CCW) and bout

1(CW)

/bout

1(CCW). The radiation losses are expressed by the ﬁc-

ticious radiation channel cin

1(CW) / cin

1(CCW). The output

light is measured by homodyne detection.

B. Nonlinear Multi-Resonant Cavity Quantum

Photonics Gyro

We now consider the gyroscopic operation of a doubly

resonant ring resonator with quadratic χ(2) nonlinearities

(Fig. 2). Quadratic nonlinearities are well-known gener-

ators of quantum-coherent correlations such as squeez-

ing and entanglement [49, 50]. Our Nonlinear Multi-

resonant Cavity Quantum Photonics Gyro, or Quantum-

Optic Nonlinear Gyro (QONG) in short, fuses nonlinear

dynamics, quantum correlations, and non-inertial Sagnac

eﬀects in the same sensor-resonator, to maximally lever-

age any possible nonlinear quantum-optical eﬀects for gy-

roscopic sensitivity. Speciﬁcally, we investigate the fol-

lowing Heisenberg-Langevin equations:

dˆa1,cw

dt =−κ1

2+γ1

2−iδ1ˆa1,cw +iβ1ˆa1,ccw

+χˆa†

1,cwˆa2,cw +√κ1ˆ

bin

1,cw +√γ1ˆcin

1,cw

(14)

dˆa1,ccw

dt =−κ1

2+γ1

2+iδ1ˆa1,ccw +iβ1ˆa1,cw

+χˆa†

1,ccwˆa2,ccw +√κ1ˆ

bin

1,ccw +√γ1ˆcin

1,ccw

(15)

dˆa2,cw

dt =−κ2

2+γ2

2−iδ2ˆa2,cw +iβ2ˆa2,ccw

−1

2χˆa2

1,cw +√κ2ˆ

bin

2,cw +√γ2ˆcin

2,cw

(16)

dˆa2,ccw

dt =−κ2

2+γ2

2+iδ2ˆa2,ccw +iβ2ˆa2,cw

−1

2χˆa2

1,ccw +√κ2ˆ

bin

2,ccw +√γ2ˆcin

2,ccw

(17)

In these equations, the index j= 1,2 in the ﬁeld op-

erators (ˆaj,ˆ

bj, ˆcj) stands for the fundamental ω1and

second harmonic ω2= 2ω1resonances. κjand γjare

the decay rates of the coupling and the intrinsic loss

channels. Rayleigh scattering rate between CW and

CCW modes at each resonance jis again character-

ized by βj. The nonlinear coupling terms χˆa†

1ˆa2indi-

cate a multi-photon process in which one incident pho-

ton with ω2breaks down into two photons of half the

frequency ω1=ω2/2 (parametric down conversion [51]),

or its reverse, 1

2χˆa2

1, indicating that two photons with

ω1combine into one photon with the double frequency

ω2= 2ω1(second harmonic generation [52]). These are

quantum-coherent, energy-conserving, three-wave mix-

ing processes, which preserve the fundamental commuta-

tion relations [53]. The rotation-induced frequency shifts

(δ1,δ2) are diﬀerent for each resonance, have opposite

polarity between CW and CCW modes, and can be ap-

proximated by δ2= 2δ1(since δ=2πrΩ

λn0[7]). Note that

here the material dispersion of the lithium niobate is ne-

glected due to relatively small index diﬀerence (n=2.21

at 1590 nm and n=2.25 at 795 nm). In order to improve

the accuracy of the model, however, the dispersion ef-

fect should be taken into account in future exploration.

The key parameter in this model is the nonlinear modal

coupling strength [47, 54]:

χ=ϵ0

ℏZZZ 3χ(2)(r)

4√2u∗

1(z, r, θ)2u2(z, r, θ)rdrdθdz (18)

where u∗

1(z, r, θ) and u2(z, r, θ) are the electric ﬁeld

proﬁles (in polar coordinates) of the fundamental and the

second harmonic eigenmodes of the gyroscopic resonator.

Given that our sensor is a ring resonator of radius R, it

is instructive to decompose χinto cross-sectional modal

overlap ζand the remaining contributions. Following

[55], we approximate:

χ≈sℏω2

1ω2

ϵ02πR

ϵ1√ϵ2

3χ(2)

4√2(19)

ζ=RR u∗

1(z, r)2u2(z , r)drdz

RR |u∗

1(z, r)|2drdzqRR|u2(z, r)|2drdz

(20)

It is important to realize that the nonlinear Langevin

equations [47, 56] encode the time evolution of four cou-

pled inﬁnite-dimensional quantum operators; as such, it

is very challenging to obtain an exact solution either an-

alytically or numerically (we note that straightforward

numerical methods using a truncated Fock basis [57] are

not feasible because our system typically involves milli-

watts of optical power amounting to ∼1016 photons).

However, at milli-watt injection powers, quantum ﬂuc-

tuations can be considered “small signals” compared to

much stronger average ﬁeld intensities at steady state,

so that each operator can be decomposed into a classi-

cal scalar-valued amplitude and a quantum ﬂuctuation

operator, e.g. ˆa=α+ˆ

δa. The details of calculat-

ing steady state solutions are included in Appendix A.

The classical amplitude represents a steady-state solu-

tion to the mean-ﬁeld averaged Langevin equations at the

classical (large photon number) limit while the “small-

signal” ﬂuctuation operator approximately obeys the lin-

earized Langevin equations in the vicinity of the steady-

state mean-ﬁeld solution. Linearizing a nonlinear steady

state to study the ﬂuctuations in its vicinity is com-

monly known as small-signal modeling in electronics en-

gineering [58]. In a similar spirit, the small-signal treat-

ment of quantum ﬂuctuation operators in the Heisenberg-

Langevin picture is a simple but eﬀective approach widely

accepted for steady-state noise analysis in laser and non-

linear quantum optics literature with experimental sup-

port [7, 59–62]. Theoretically, it is important to note that

such an approach is justiﬁed as long as the steady state

we consider is a hyperbolic ﬁxed point whose neighbor-

hood is a topologically stable manifold that ensures small

ﬂuctuations (Hartman-Grobman theorem [63]). On the

other hand, a more sophisticated phase-space formalism,

which employs quasi-probability distributions, Fokker-

Planck equations and stochastic calculus, can be used to

study more complicated dynamics such as large ﬂuctu-

ations at non-hyperbolic critical points and self-pulsing

(limit-cycle) solutions [64]. Using the small-signal ap-

proximation, we can compute the diﬀerential photocur-

rent signals at both the fundamental and the second har-

monic resonances (see also Fig. 2):

i1=iA1ˆ

bout†

1,cwˆ

bout

1,ccwe−iϕ1−ˆ

bout†

1,ccwˆ

bout

1,cweiϕ1(21)

i2=iA2ˆ

bout†

2,cwˆ

bout

2,ccwe−iϕ2−ˆ

bout†

2,ccwˆ

bout

2,cweiϕ2(22)

Here A1and A2are constant factors determined by

the frequencies of the light and the responsivity of the

photodetectors. eiϕ1and eiϕ2are the propagation phase

shifts that each output light experiences, which can be set

to zero here. Here we measure both ˆ

i1and ˆ

i2to extract

maximal information out of the nonlinear wave-mixing

gyro.

The output of our quantum-optic nonlinear gyro

(QONG) is now characterized by a mean vector ⟨i⟩and

a covariance matrix ⟨∆i2⟩:

⟨i⟩=⟨ˆ

i1⟩

⟨ˆ

i2⟩(23)

⟨∆i2⟩= ⟨ˆ

1⟩−⟨ˆ

i1⟩2⟨ˆ

i1ˆ

i2⟩+⟨ˆ

i2ˆ

i1⟩

2− ⟨ˆ

i1⟩⟨ˆ

i2⟩

⟨ˆ

i1ˆ

i2⟩+⟨ˆ

i2ˆ

i1⟩

2− ⟨ˆ

i2⟩⟨ˆ

i1⟩ ⟨ˆ

2⟩−⟨ˆ

i2⟩2!

(24)

Assuming that the joint probability distribution of the

measured photocurrents follow a bi-variate Gaussian, we

can express the Fisher information [41] of our QONG:

I(δ)=(d⟨i⟩

dδ )T⟨∆i2⟩−1(d⟨i⟩

dδ ) (25)

The details of the statistial analysis of the diﬀerential

currents are included in Appendix B. Then the sensitivity

is determined by Cramer-Rao bound [65]:

δmin =1

pI(δ)(26)

Similar to the linear gyroscope, here MDR is given by:

Ωmin =λn0

2πR δmin (27)

Before we provide further estimation for particular

QONG implementation, we want to oﬀer a few remarks

of our modeling approach:

•In this paper, we have stuck to a Langevin descrip-

tion of our quantum gyro, which takes into account

quantum noise through the Langevin ﬂuctuation

operator ˆ

bor ˆcin each coupling or dissipation chan-

nel (with the rates determined by the ﬂuctuation-

dissipation theorem), preserving the fundamental

commutation relations [47]. An equivalent formu-

lation can also be considered in terms of a density

operator ˆρ[66], leading to a Lindblad Master equa-

tion of the form (under vacuum noise conditions):

dˆρ

dt =−i

ℏ[H, ˆρ] + Xγi[[ˆai,j ,ˆρ],ˆa†

i,j ] (28)

H=Xℏωi,j ˆa†

i,j ˆai,j +Xiℏκi

2(ˆ

b†

i,j ˆai,j + ˆa†

i,jˆ

bi,j )

+Xiℏχ

2(ˆa†

1,j 2ˆa2,j −ˆa2

1,j ˆa†

2,j )

+Xℏβiˆa†

i,cwˆai,ccw + ˆa†

i,ccwˆai,cw(29)

We note that while the Langevin form is widely

utilized in many experimental situations [55], more

sophisticated theoretical analysis, delineating the

open-system quantum dynamics [67], can be per-

formed using the density operator formalism and

the Master equation, which will be the subject

of future investigations. In particular, our sim-

ple perturbative approach restricts our solution to

examine the quantum ﬂuctuations around a sta-

ble hyperbolic ﬁxed point. On the other hand,

non-hyperbolic ﬁxed points and non-steady state

attractors (such as limit cycles) require more so-

phisticated non-perturbative treatment (while their

implications for quantum correlations and sensing

remain unexplored). One such treatment involves

expanding the density operator in a non-diagonal

coherent state basis (so-called positive P represen-

tation), deriving a Fokker-Planck equivalent of the

Lindblad Master equation and simulating the asso-

ciated stochastic dynamics [47]. However, to the

best of our knowledge, Fokker-Planck equations

corresponding to more than two bosonic operators

[64, 66] have not been well studied; our nonlinear

multi-resonant cavity quantum photonic gyro is de-

scribed by 4 coupled Langevin equations and will

lead to an 8+1 dimensional Fokker Planck equa-

tion, which requires substantial computational re-

sources and will be the subject of future investiga-

tions.

•In our approach, we have assumed idealized sources

and detectors in order to simplify our gyroscopic

model to physically most crucial components, and

thereby to unveil the fundamental information-

theoretic limits (in the same spirit as the analysis

presented in Ref. [43] or the Section 4 of Ref. [7]).

Future works will develop more detailed models

that can compute commonly accepted experimen-

tal metrics such as the integration-time depen-

dent Allan deviation curve [68], for example, by

incorporating the quantum theory of photodetec-

tion [69, 70], which can explicitly take into account

photo-electron generation rates and detector inte-

gration times.

•Last but not least, we note that our present model

focuses on χ(2) processes to delineate their eﬀects

on the gyroscopic sensitivity. A more thorough

gyro model may also consider χ(3) (Kerr-type self

modulation) nonlinearities, which may come into

eﬀect at ultra-high quality factors and are found to

limit the sensitivity of the (otherwise) linear gyro-

scope [7, 16]. While we shall take into account χ(3)

processes in detailed comprehensive models in the

future (see Section IV), we note that Eqs. 14-17 are

fully applicable to material platforms, such as thin

ﬁlm lithium niobate [54, 71], which possess promi-

nent χ(2). Furthermore, unlike their linear coun-

terparts, resonators with χ(2) can be engineered to

exhibit negative Kerr shifts via cascaded second-

order eﬀects [72], which can mitigate the intrinsic

positive Kerr shift; we shall investigate such cancel-

lation schemes in our future works. On the other

hand, we would like to emphasize that χ(3) pro-

cesses, including even the Kerr shift, need not be

treated as a nuisance, but as extra complexities and

additional degrees of freedom that can be optimized

to our advantage (see Section IV). For example, it

has been recently reported that the bistability ef-

fects associated with the Kerr-shift self-modulation

can even enhance sensitivities under appropriate

sensing schemes [73, 74].

FIG. 2. The schematic of the nonlinear micro-ring gyroscope.

The input light is injected at four waveguide ports bin

1(CW)

/bin

1(CCW) and bin

2(CW) / bin

2(CCW). The input/output

and loss channels for the second harmonic light are expressed

by blue arrows. The output light is measured by a joint mea-

surement using the diﬀerential currents at both output ports

(bout

1and bout

2). The accuracy of the measurement is deter-

mined by evaluating the quantum Fisher information of the

output light.

C. Thin ﬁlm lithium niobate as an implementation

platform

Our nonlinear multi-resonant cavity quantum photonic

gyro (or quantum-optic nonlinear gyro QONG in short)

can be implemented in any thin ﬁlm material platform,

including LiNbO3 [71], AlN [75], SiC [76], GaAs [77], etc,

which has prominent χ(2). In this work we consider thin

ﬁlm lithium niobate (TFLN) as a particularly promising

platform, as it has gained widespread popularity for re-

alizing quantum-grade ultra-low loss photonic integrated

circuits [78, 79]. Indeed, lithium niobate has been tra-

ditionally employed in quantum optics applications as

a nonlinear medium for generating squeezed light and

entangled photon states [80, 81]. However, traditional

LN crystals are bulky and suﬀer from relatively lim-

ited strength of light-matter interactions (leading to very

weak nonlinear coupling χ∼103Hz in Eqs. 19). Only

recently, high quality wafer-scale TFLN becomes widely

available for realizing integrated photonic circuits with

nonlinear and electro-optic functionalities [71]. Associ-

ated with large χ(2), low optical loss and strong nanopho-

tonic conﬁnement [82], TFLN devices oﬀer orders of mag-

nitude enhancements in nonlinear coupling χ∼106Hz

[54].

Fig. 3 shows the design of a TFLN ring resonator which

can used as an QONG. Note that feeder waveguides

of diﬀerent dimensions, frequency cutoﬀs, and disper-

sion characteristics, can be designed to selectively cou-

ple to the fundamental (1590nm) and the second har-

monic (795nm) modes [83], and their coupling rates can

be further tuned by TFLN electro-optics [84]. To realize

strong nonlinear coupling χbetween the two resonances,

two zeroth-order transverse electric eigenmodes (TE00)

can be (quasi-)phase-matched [54] via periodic poling [85]

that achieves crystal domain inversion, leading to peri-

odically varying nonlinear susceptibility χ(2) which com-

pensates wave vector mismatch between the fundamental

and the second harmonic modes kχ=k(ω2)−2k(ω1) [54].

Apart from the phase-matched resonator itself, a fully in-

tegrated QONG can be implemented in TFLN, incorpo-

rating ﬂip-chip bonded semiconductor lasers [86] and het-

erogeneously integrated uni-travelling carrier photode-

tectors [87]. Furthermore, we note that TFLN comes

with unique electro-optic control capabilities [71] which

can be used for tuning resonator parameters such as the

coupling rates to the waveguides [88], managing long

term temperature stability, cancelling thermal drifts and

electronic noise [15, 89], and performing signal processing

[90].

FIG. 3. The 3D schematic (not drawn to scale) of quasi-phase-

matching (QPM) achieved by periodic poling. Poled rings are

used to form quasi-phase matched structures. Semiconductor

lasers [86] and photodetectors [87] are also integrated on the

chip. The numerically simulated ﬁeld proﬁles of both the

fundamental frequency (1590 nm) and the second harmonic

(795 nm) cavity modes are shown in the inset.

In our gyroscopic model, intrinsic losses γand back-

scattering rates βshould be treated as “ﬁxed” parame-

ters which depend on the experimentally feasible char-

acteristics of a particular implementation platform, such

as residual material losses and surface roughness due to

fabrication imperfections. In thin ﬁlm lithium niobate,

intrinsic quality factors reaching ∼108have been demon-

strated [91], and we expect proportionate back-scattering

rates with the same order of magnitude. It is impor-

tant to realize that, apart from γand β, almost all other

parameters can be designed, engineered and optimized,

including injection powers, coupling rates, resonator ra-

dius, resonator waveguide cross section and dispersion as

well as quasi-phase matching processes. We will utilize

these parameters as degrees of freedom (DoF) in optimiz-

ing the Fisher Information and hence the minimum de-

tectable rotation (MDR) of our QONG (as compared to

the linear gyro). While any optimization algorithm can

be employed, gradient-free global optimization methods

are most suitable for a relatively low-dimensional prob-

lem like our two-resonance gyro (where about 5–10 DoFs

can be optimized). We will use Bayesian optimization,

a simple but powerful machine-learning-based optimiza-

tion algorithm which requires relatively few function eval-

uations (as compared to other heuristic methods such as

simulated annealing and evolutionary algorithms [92, 93])

and has been observed to be particularly eﬀective for op-

timizing ∼20 DoFs [42].

III. RESULTS

As noted above, the Fisher Information and the Min-

imum Detectable Rotation (MDR) of the gyroscope is

determined by various parameters and can be optimized

by judiciously adjusting their values. In our design, the

operational wavelengths (of the input/output light) are

ﬁxed at λ1= 1590 nm and at λ2=795 nm. The mate-

rial property of TFLN is taken from the literature [82]:

the refractive index is n= 2.2 while the second order

nonlinear susceptibility is χ(2) = 30 pm/V. The intrin-

sic quality factors are ﬁxed at Qi1= 107for the 1590

nm and Qi2= 106for the 795 nm. The Rayleigh scat-

tering rates for both cavity modes β1= 5.4×104Hz

and β2= 5.4×105Hz are inferred by adjusting the

literature-reported values [16] to the quality factors Qi

of our TFLN platform. We have also ﬁxed the radius of

the resonator at R= 20 mm as well as the cross-sectional

dimensions of the resonator waveguide (1.2µm width

×0.6µm thickness) and the fabrication side wall angle

of 75◦, leading to a cross-sectional area of 0.8µm2. The

two TE00 modes have phase mismatch of 1.354 µm−1,

which can be compensated by poling with a period of

4.64 µm. Based on Eqs. 19, the quasi-phase matched

χ(2) [54], and the numerically simulated modal overlap-

ping factor ζ= 1.18 /µm, the nonlinear coupling strength

is χ= 1.26×106Hz, which is independent of the injection

schemes. The rest of the parameters remain to be deter-

mined, including the injection power at the fundamental

and the second harmonics P1and P2, the quality factors

due to coupling to the waveguides for each cavity mode:

Qc1and Qc2. These four parameters will be determined

by Bayesian optimization. We will investigate gyroscopic

performance under diﬀerent injection schemes including

(1) coherent state input at the second harmonic (λ2), (2)

coherent state input at the sub-harmonic (λ1), and (3)

coherent state inputs at both second and sub-harmonics

(λ1and λ2).

A. Optical parametric oscillator gyro (coherent

injection at second harmonic)

First, we study the performance of an optical para-

metric oscillator gyroscope under the coherent injection

at the second harmonic frequency. As shown in Fig.

4(a), classical laser light with a wavelength λ2= 795

nm is injected from opposite directions from the waveg-

uide ports bin

2,cw and bin

2,ccw, while no light is injected from

the waveguide port bin

1,cw or bin

1,cw, i.e., P1= 0. The care-

fully phase-matched fundamental and second harmonic

modes facilitate parametric down conversion, in which

one photon with higher frequency (shorter wavelength

λ2breaks down into two photons with half the frequency

(longer wavelength λ1), generating phase-squeezed sig-

nals [47]. First we investigate the mininum detectable

rotation (MDR) as a function of the input power P2and

the coupling factor Qc2. Using Bayesian optimization,

we identify a high sensitivity regime, that is, low MDR,

as shown in the 2D density plot of MDR in terms of

P2and Qc2(Fig. 4b). We mapped over the parame-

ter space where Qc2ranges from 5 ×105to 6 ×105and

P2ranges from 20 mW to 30 mW. At the second har-

monic injection, the selection of P2is determined by the

critical power Pc, below which the steady-state solutions

of the system become unstable. This phenomenon has

been discussed in details by Drummond [56]. Here Pc=

14.05 mW such that P2should be larger than this value.

Within this region, small MDR (0 - 1.7 ◦/h) is observed

(indicated by rainbow colors), showing that high sensi-

tivity is achieved for a sizable parameter range such that

the enhanced sensitivity is not an isolated singularity.

The lowest MDR (Ωmin <0.25 ◦/h) appears in the nar-

row purple band. Outside this region, MDR gradually

increases as the color becomes more greenish and red-

dish, indicating reduced sensitivity. In principle, as the

rotation rate increases, the diﬀerence between the CW

and CCW mode becomes more signiﬁcant. To bench-

mark the gyroscope performance, we compared the sen-

sitivity of our optical parametric oscillator (OPO) gyro

under the second harmonic injection (solid blue line) with

the sensitivity of a standard linear gyroscope (solid red

line) in Fig. 4(c). The optimal coupling factors for the

OPO gyro are Qc1= 1.018 ×105and Qc2= 5.462 ×105,

both discovered by Bayesian optimization. Meanwhile,

the highest sensitivity of the linear gyroscope is found at

Qc1= 9.58 ×106≈107. Note that, in the limit of van-

ishing β1, the lowest linear gyro MDR is achieved when

κ1=γ1(see Eq. 13). Since Qi1is ﬁxed at 107, the op-

timal Qc1for the linear gyro is expected to be close but

not exactly equal to this value, considering the inﬂuence

of the small but non-zero back-scattering. In Fig. 4(d),

as the input power is increased from 20 mW to 30 mW,

the MDR of the OPO gyro drops from ∼0.42 ◦/h down

to near zero and rises back to ∼1.21 ◦/h with a local

minimum at 23.507 mW, corresponding to the optimal

sensitivity point (Ωmin = 4.465×10−3◦/h). To better vi-

sualize this local sensitivity minimum, a zoomed-in curve

is shown in the inset of Fig. 4(d). We also found that

the optimal-sensitivity point is associated with 9.9 dB

phase squeezing at an experimentally feasible value on a

TFLN platform [78]. Meanwhile, the MDR of the linear

gyroscope remains >0.49 ◦/h, which demonstrates that

the OPO gyro is ∼124.4 ×more sensitive than the lin-

ear gyro under the same injection power, resonator size

and intrinsic quality factors. In order to visualize this

eﬀect, we investigated the mean current values as a func-

tion of the rotation rate Ω at diﬀerent frequencies in Fig.

4(d). The mean values of the output diﬀerential currents

at the subharomonic (< i1>) and the second harmonic

(< i2>) are expressed by the solid blue and red line re-

spectively, while that of the linear gyroscope at the same

power consumption is expressed by the solid green line

for reference. It is shown that as Ω increases from 0 - 100

◦/h, < i1>increases from 0 - 0.43 nA, < i2>increases

from 0 - 0.4 nA while the output current of the linear

gyroscope increases from 0 - 0.17 nA. This result shows

that the nonlinear eigenmode dispersion, as discussed in

Section II B, produces stronger output signals (diﬀeren-

tial currents) at both wavelengths compared to the linear

gyroscope.

B. Coherent injection at the fundamental

frequency

Aside from injecting light at the second harmonic,

we also investigated the scheme of the fundamental fre-

quency (subharmonic) injection. As shown in Fig. 5(a),

the input light at the wavelength λ1= 1590 nm is in-

jected from the waveguide ports bin

1,cw and bin

1,cw while

P2= 0, stimulating intra-cavity up-conversion (two pho-

tons of lower energy are combined to one photon of higher

energy). We study the sensitivity (MDR) in terms of P1

and Qc1, as indicated by the 2D density plot Fig. 5(b),

in which variations in MDR again show up in rainbow

colors. In Fig. 5(b) we mapped over the parameter

space where Qc1ranges from 6 ×106to 7 ×106and P1

ranges from 0.9µW to 1 µW . Note that much lower

power injection is required at this injection scheme com-

pared to coherent injection at the second harmonic, re-

lated to the fact that the steady-state solutions of the

cavity modes are stable only when P1is below the criti-

cal power when the second harmonic injection is absent

[56]. Here the critical power is Pc= 3.24 mW. Only

solutions of P1smaller than this value are stable, result-

ing in orders of magnitude lower power consumption. In

FIG. 4. (a) The schematic of the second harmonic injection

scheme. The input light is only injected at the second har-

monic (blue arrow). (b) 2D density plot of the sensitivity

at second harmonic injection. The sensitivity (MDR) is ex-

pressed as a function of the input power P2and the qual-

ity factor due to coupling loss of the second harmonic cavity

mode Qc2. (c) 1D linear plot of the sensitivity of the second

harmonic injection scheme (blue) and the standard linear gy-

roscope (red) at optimal Qcin terms of the input power. (d)

1D linear plot of the mean diﬀerential current of the out-

put light at fundamental frequency (blue), second harmonic

(green) and the standard linear gyroscope (red).

Fig. 5(b), MDR varies from 0 - 5 ◦/h across the entire

parameter space. The lowest MDR is found in a nar-

row band-like region. The mean values of the output

currents are also studied here. To better evaluate the

gyroscope performance, we compared it with the linear

gyroscope at the same power consumption. Figure 5(c)

shows the sensitivity of the fundamental frequency injec-

tion (solid blue line) and the linear gyroscope (solid red

line) in terms of the injection power at ﬁxed optimal Q

factors (Qc1= 6.747 ×106, Qc2= 6.675 ×107), which

are determined by Bayesian optimization, and optimal

Qc= 9.58 ×106in the linear gyro. As P1increases from

0.9µW to 1 µW , MDR drops from ∼0.46 ◦/h down to

near zero at 0.945 µW , then rises back to ∼0.49 ◦/h.

Meanwhile, the sensitivity of the linear gyroscope mono-

tonically but slowly decreases from ∼89.78 ◦/h to ∼

85.17 ◦/h. The lowest MDR is found in a narrow band-

like region. The optimal sensitivity Ωmin = 0.093 ◦/h

is found at P1= 0.945 µW . At the optimal point, a

surprisingly high sensitivity improvement of ∼942.5 ×

is observed (0.093 ◦/h vs 87.62 ◦/h). To better visual-

ize the sensitivity of the fundamental frequency injection

scheme, a magniﬁed plot of the optimal sensitivity region

is shown in the inset of Fig. 5(d), clearly exhibiting the

local sensitivity minimum at 0.945 µW . We note that no

phase squeezing is observed in this scheme (although a

6.03 dB amplitude squeezing was observed at the second

harmonic (795 nm)); rather than squeezing, we posit that

the sensitivity improvement is largely due to nonlinear

eigenmode dispersion in the presence of critically sensi-

tive χ(2)-mediated wave mixing processes, not disimilar

to enlarged Sagnac shifts reported in dispersive materi-

als [94]. Most importantly, the fundamental frequency

injection scheme merges high sensitivity and low power

consumption together in a compact form, which shows

great potential in practical applications. Similar to the

analysis in Section III A, the mean diﬀerential currents

(< i1>,< i2>and linear) as a function of the rotation

rate Ω are shown in Fig. 5(d) in solid blue, red and green

lines. As Ω increases from 0 - 100 ◦/h, < i1>increases

from 0 - 0.05 pA, < i2>increases from 0 - 0.0062 pA and

the output diﬀerential current of the linear gyroscope in-

creases from 0 - 0.0069 pA. The diﬀerential currents are

in the pA range, not the nA range, again because of much

weaker input power. It is shown that the sensitivity of

< i1>is almost 10×larger than that of < i2>and that

of the output current in the linear gyroscope, suggesting

that, even though very little second harmonic power is

ultimately extracted, the presence of a non-linearly in-

teracting second harmonic mode critically enhances the

sensitivity of the fundamental mode.

FIG. 5. (a) The schematic of the fundamental frequency (sub-

harmonic) injection scheme. The input light is only injected

at the fundamental frequency (red arrow). (b) 2D density

plot of the sensitivity at the fundamental frequency injection

scheme. The sensitivity (MDR) is expressed as a function of

the input power P1and the quality factor due to coupling loss

of the cavity mode Qc1. (c) 1D linear plot of the sensitivity

of the fundamental frequency injection scheme (blue) and the

standard linear gyroscope (red) at optimal Qcin terms of the

input power. The inset shows the magniﬁed plot of the criti-

cal region. (d) 1D linear plot of the mean diﬀerential current

of the output light at fundamental frequency (blue), second

harmonic (green) and the standard linear gyroscope (red).

C. Dual frequency injection

For the sake of completeness, we also studied the dual

injection scheme (Fig. 6a) where coherent light is injected

at both fundamental and second harmonics (λ1= 1590

nm and λ2= 795 nm). Here, we optimize four indepen-

dent parameters: P1,P2,Qc1, and Qc2. The 2D density

plot of the sensitivity (MDR) in terms of P1and P2is

shown in Fig. 6(b). Here Qc1and Qc2are ﬁxed at the

optimal values of 4.353×105and 8.769×106, again discov-

ered by Bayesian optimization. In Fig. 6(b), P1ranges

from 1 mW to 2 mW and P2ranges from 1 mW to 2

mW. Similar to the scheme of fundamental frequency in-

jection, we found that low input power is also necessary

for high sensitivity, which is beneﬁcial for integrated op-

tical gyroscopes. As shown in the ﬁgure, MDR ranges

from 0 - 1.4 ◦/h across the entire parameter space. The

region of the lowest MDR is expressed by a narrow purple

band in the ﬁgure while outside this region MDR grad-

ually increases. We also compare the MDR of the dual

injection scheme (solid blue line) with the linear gyro-

scope (solid red line), as shown in Fig. 6(c). Note that

in the ﬁgure the x-axis denotes the total input power P,

which equals to P1+P2for the dual frequency injection

and P1for the linear gyroscope. Here Qc1,Qc2and P2

are ﬁxed at 4.353×105, 8.769×106and 1.873 mW, which

are the optimal values discovered by Bayesian optimiza-

tion. It is shown that the sensitivity of dual frequency

injection drops from ∼0.88◦/h down to near zero as P

increases from 3 mW to 3.373 mW, then rises back to

∼0.762 ◦/h as Pincreases to 4 mW while sensitivity of

the linear gyroscope drops from 1.555 ◦/h to 1.347 ◦/h

within the same range of injection power. At the opti-

mal power (3.373 mW), a high sensitivity of 0.013 ◦/h

is observed, leading to a substantial sensitivity improve-

ment of ∼113.1 ×over the linear gyroscope (0.013 ◦/h

vs 1.47 ◦/h). To better visualize this optimal point, a

magniﬁed plot of the optimal sensitivity region is shown

in the inset of Fig. 6(c). To better understand the in-

ﬂuence of rotation, the mean output currents are shown

in Fig. 6(d). As the rotation rate Ω increases from 0

- 100 ◦/h, < i1>,< i2>and the mean diﬀerential

current of the linear gyroscope monotonically increases

from 0 to 5.3 pA, 11.1 pA and 24.7 pA respectively. Note

that here neither the sub nor the second harmonic mode

shows stronger output current compared to the linear gy-

roscope. This is because the injection power of the linear

gyroscope equals to the summation of both the sub and

the second harmonic injection. Nonetheless, signiﬁcant

sensitivity improvement is still observed in this case due

to the combination of the nonlinear coupling and the gen-

eration of the phase-squeezed photons.

D. Discussion

Table I summarizes the maximal sensitivity enhance-

ment factors (over the linear baseline) that can obtained

FIG. 6. (a) The schematic of the dual (sub-second harmonic)

injection scheme. The input light is only injected at both

the second (blue arrow) and the fundamental frequency (red

arrow). (b) 2D density plot of the sensitivity at the dual

injection scheme. The sensitivity (MDR) is expressed as a

function of the input power P1and P2. (c) 1D linear plot

of the sensitivity of the dual injection scheme (blue) and the

standard linear gyroscope (red) at optimal Qcin terms of the

input power. The inset shows the magniﬁed plot of the critical

region. (d) 1D linear plot of the mean diﬀerential current of

the output light at the fundamental frequency (blue), second

harmonic (green) and the standard linear gyroscope (red).

in multiple operational regimes over a wide range of

critical power requirements. Under the optimal second-

harmonic injection at ≈23.5 mW, a 9.9 dB quadra-

ture squeezing is predicted, where our nonlinear multi-

resonant cavity quantum photonics gyro (or quantum-

optic nonlinear gyro QONG in short) can be nearly

124.4×more sensitive than an optimized linear gyro with

the same radius, intrinsic quality factor and power bud-

get, allowing for a minimum detectable rotation (MDR)

as small as 0.0044 ◦/h. Alternatively, even larger en-

hancement factors can be obtained at smaller powers

under the fundamental and the dual frequency injection

schemes. The dual frequency injection scheme achieves

near 7 ×sensitivity improvement over the fundamen-

tal frequency injection scheme (0.013 ◦/h vs 0.093 ◦/h),

which could be the result of the extra squeezed photons

generated by the process of parametric down conversion.

In fact, these two latter scenarios do not necessarily pro-

mote squeezing but rely on nonlinear eigenmode disper-

sion due to critically-sensitive three-wave mixing dynam-

ics between the two resonances—an eﬀect that resembles

the enlarged Sagnac shifts observed in suitably dispersive

materials [94]. In either of our fundamental or second-

harmonic injection scheme, we measured the output sig-

nals at both the fundamental and the second harmonic

frequencies in order to fully utilize the input pump power

(which gets converted into both harmonics), setting up

a fair comparison to a linear gyro under the same pump

TABLE I. Optimal sensitivities of various injection schemes

power. For a more conservative comparison, one may ar-

gue for using dual inputs and outputs in the linear case.

Aside from the fact that having to use two diﬀerent fre-

quency lasers can be disadvantageous, a simple calcula-

tion readily shows that measuring two non-interacting

resonances in a linear gyro can oﬀer only up to √2×

improvement (under the same power budgets)—in fact,

much less than √2 due to the smaller Qi2—highlighting

that nonlinear eﬀects are indeed indispensable for sig-

niﬁcant sensitivity enhancements. Most importantly, the

crucial insight we have drawn from our investigations is to

realize that multiple resonances in a nonlinear resonator

can be engineered to reinforce each other through nonlin-

ear wave mixing, and can be used as powerful degrees of

freedom to optimize sensitivities. This critical realization

suggests an exciting future direction: to generalize our

QONG from just two resonances to many more nonlin-

early interacting resonances (see also Section IV), which

may lead to even better sensitivities and functionalities

(approaching the ultimate Heisenberg limit).

IV. SUMMARY AND OUTLOOK

We have introduced a new type of quantum light gy-

roscopes based on nonlinear multi-resonant cavity quan-

tum photonics (quantum-optic nonlinear gyro or QONG

in short). Speciﬁcally, we analyzed and optimized the

quantum-enhanced gyroscopic sensitivity of a doubly res-

onant χ(2) cavity, revealing that, under quantum noise

conditions, ≳900×enhancement is possible over the

classical shot noise limit. We highlight that our cur-

rent design, which uses two resonances, represents only

an elementary step and a relatively simple example of

an QONG. In future works, we will develop a compre-

hensive QONG inertial sensing paradigm, where a syn-

ergistic amalgamation of both quadratic χ(2) and cu-

bic χ(3) nonlinearities, along with multiple intermixing

resonances, mutually reinforced Sagnac shifts, co-arising

quantum correlations, electro-optics dynamical control

and geometry-induced anomalous dispersion eﬀects, can

unleash extraordinary complexities and freedoms, which

can be fully exploited by state-of-the-art optimization

techniques [95–98] in order to identify unprecedented

regimes for gyroscopic operation and sensitivities. A

full incarnation of an QONG can be described by a

Heisenberg-Langevin system of the form (or an equiv-

alent density-operator Master equation [64]):

dˆaµ

dt =iωj+iδµ

j(Ω) −κj

2−γj

2ˆaµ

j+iβjˆaν

klαβ

f(2)

klαβ ˆaα

k,ˆaβ

l,(ˆaα

k)†,(ˆaβ

l)†

klmαβθ

f(3)

klmαβθ ˆaα

k,ˆaβ

l,ˆaθ

m,(ˆaα

k)†,(ˆaβ

l)†,(ˆaθ

m)†

kqκc

jk ˆaµ

jk,in +X

kqγr

jk ˆηµ

jk (30)

for a selected set of carefully phase-matched and

dispersion-engineered resonances {ωj, j = 1, ..., N }.

Here, δµ(Ω), µ ∈ {cw,ccw}, is the rotation-dependent

Sagnac shift in the CW or CCW mode at each res-

onance. The functions f(2) and f(3) are polynomials

of the annihilation and creation operators, represent-

ing all possible quantum-coherent three-wave mixing and

four-wave mixing interactions between the selected res-

onances; these processes include sum and diﬀerence fre-

quency generations of diﬀerent orders and combinations

as well as Kerr-variety self-phase and cross-phase mod-

ulation, and even cascaded processes [99]. It is impor-

tant to note that the strengths of diﬀerent f(2) and f(3)

terms are determined by nonlinear coupling factors [55]

which characterize the ﬁeld concentration and nonlin-

ear overlaps of the modes of the photonic resonator and

can be computed from nanophotonic simulations. There-

fore, on-chip structural parameters, ranging from a few

simple shape parameters to entire permittivity distribu-

tions, can serve as design degrees of freedom [100], by

which we can engineer and optimize the diﬀerent non-

linear processes (e.g. their relative contributions). The

outputs of this multi-resonance system are collected by

multiple waveguide ports and are set to passively inter-

fere with each other and/or go through active electro-

optics pulse processing (readily achievable on a TFLN

platform [71]) before arriving at multiple photodetectors

to yield multiple photocurrent signals i={ˆ

i1, ...,ˆ

iM}.

From these multi-variable (vector-valued) measurements,

one can perform deep inferential analysis (such as ad-

vanced Bayesian computing [101]) to deduce the underly-

ing non-inertial motion; the sensitivity of the entire pro-

cess can be characterized by an end-to-end computation

of Fisher Information, which will serve as an optimization

ﬁgure of merit. We recognize tremendous opportunity in

analyzing and optimizing such a system with increasing

levels of mathematical and computational vigor, start-

ing from steady-state analysis, small-signal modeling,

classical stochastic simulations, to the non-perturbative

quantum phase-space apparatus involving positive P-

representations, Fokker-Planck equations and stochastic

calculus [47, 64], from few-parameter deterministic global

optimization [96, 102], multi-parameter Bayesian opti-

mization [42] and evolutionary algorithms [103], machine-

learning assisted hybrid optimization [104], and Monte

Carlo gradient computations [98] to billion-voxel topol-

ogy optimization [105] and full end-to-end inverse design

[106] of the entire workﬂow from the underlying resonator

geometry to multi-variable inferential processes. Exper-

imentally, thin ﬁlm lithium niobate (TFLN) continues

to oﬀer the most suitable platform which features state-

of-the-art on-chip frequency combs, pulse shaping, fre-

quency shifting and ultra-fast signal processing capabili-

ties [79, 107–109].

ACKNOWLEDGMENTS

We thank Charles Roques-Carmes, Steven G. John-

son, Kiyoul Yang and Michael Larsen for informative

discussions. Financial support was provided by generous

gifts from the Virginia Tech Foundation. VK thanks the

late Stavros Katsanevas for pointing out the new runs of

VIRGO based on quantum mechanical states in Novem-

ber of 2018.

Appendix A: Steady state solutions

As discussed in Section II B, the nonlinear coupled

equations are solved by linearization. The classical scalar

valued amplitudes αare obtained from the steady state

analysis of Eqs. 14-17. To evaluate the steady state solu-

tions, the noise terms containing the quantum operators

are omitted. The equations are thus simpliﬁed as follows:

f1=κ1

2+γ1

2−iδ1a1,cw −iβ1a1,ccw

−χa∗

1,cwa2,cw −√κ1bin

1,cw

(A1)

f2=κ1

2+γ1

2+iδ1a1,ccw −iβ1a1,cw

−χa∗

1,ccwa2,ccw −√κ1bin

1,ccw

(A2)

f3=κ2

2+γ2

2−iδ2a2,cw −iβ2a2,ccw

2χa2

1,cw −√κ2bin

2,cw

(A3)

f4=κ2

2+γ2

2+iδ2a2,ccw −iβ2a2,cw

2χa2

1,ccw −√κ2bin

2,ccw

(A4)

Next, the steady state solutions are hence obtained by

solving the equations F(f1, f2, f3, f4) = 0. Aside from

the cavity modes an,cw/ccw, the input ﬁelds are also ex-

pressed as the steady states bn,cw/ccw . These values are

determined by the input power Pnat both waveguide

ports as discussed in Equation 3. Here we ﬁx bnas

real values, which reduces the phase noise as reported

by Dowling [12]. At diﬀerent injection schemes, diﬀerent

arrangements of bnare employed. For example, b2= 0

at when the input light is injected at the fundamental

frequency and b1= 0 at the second harmonic injec-

tion. Though the steady state solutions of a similar sys-

tem has been studied by Drummond [56], in which the

analytical solutions are given at each injection scheme.

In our system, however, the rotation-induced frequency

shift δnand the Rayleigh back-scattering βnwhich in-

troduces cross-coupling between the CW and the CCW

modes make it impossible to calculate the analytical solu-

tion, hence we calculated the numerical solutions instead.

Since these equations are nonlinear equations, multiple

solutions are expected at each set of parameters. In or-

der to discover the steady state solutions, linear stability

analysis is performed [110, 111]. By checking the eigen

values of the Jacobian matrix [112] associated with each

set of solution we can determine the stability of these so-

lutions. The Jacobian matrix J of Eqs A1-A4 is obtained

by taking the gradient over a vector of the unknown vari-

ables an,cw/ccw:

J=∇F|an,cw/ccw (A5)

Note that in Equation A5 the function system Fand

the variables an,cw/ccw are both vectors, hence the gradi-

ent operator ∇generates a matrix Jinstead of a vector.

When the real part of the eigen values of the matrix is

negative, the solution is stable and can be used for fur-

ther calculation.

Appendix B: The algebra of the quantum operators

The system is assumed to be quantum-limited, mean-

ing that the shot noise is considered as the main source

of noise. To this end, it is necessary to investigate the

statistical properties of the output light. As discussed in

Section II B, the output light are expressed by the op-

erators ˆ

bout

n,cw/ccw. As shown in Fig. 2, the output light

is measured by homodyne detection that ˆ

bout

n,cw and ˆ

bout

n,ccw

are coupled with each other before being detected by two

independent photodetectors:

b1+ =ˆ

bout

1,cwe−iϕ1+iˆ

bout

1,ccweiϕ1/√2 (B1)

b1−=iˆ

bout

1,cwe−iϕ1+ˆ

bout

1,ccweiϕ1/√2 (B2)

b2+ =ˆ

bout

2,cwe−iϕ2+iˆ

bout

2,ccweiϕ2/√2 (B3)

b2−=iˆ

bout

2,cwe−iϕ2+ˆ

bout

2,ccweiϕ2/√2 (B4)

Here ϕ1and ϕ2are the propagation phase shifts of the

output light at the sub and the second harmonics, which

can be arbitrarily selected. Here we set them to zero.

The resultant diﬀerential current current is given by:

i1=Rℏω1ˆ

b†

1+ˆ

b1+ −ˆ

b†

1−ˆ

b1−(B5)

i2=Rℏω2ˆ

b†

2+ˆ

b2+ −ˆ

b†

2−ˆ

b2−(B6)

Here R is the responsivity of the photodetectors, set as

0.58 A/W in our analysis. Setting A1=Rℏω1and A2=

Rℏω2, Eqs. B5-B6 can be further simpliﬁed as:

i1=iA1ˆ

bout†

1,cwˆ

bout

1,ccw −ˆ

bout†

1,ccwˆ

bout

1,cw(B7)

i2=iA2ˆ

bout†

2,cwˆ

bout

2,ccw −ˆ

bout†

2,ccwˆ

bout

2,cw(B8)

Following Maleki’s approach [7], the output operators are

linearized as ˆ

bout

n,cw/ccw =bout

n,cw/ccw +ˆ

δbout

n,cw/ccw. Here we

analyze the perturbation terms, such that Eqs. B7- B8

are simpliﬁed as:

δi1=iA1bout

1,ccw ˆ

δbout†

1,cw +bout∗

1,cw ˆ

δbout

1,ccw

−bout∗

1,ccw ˆ

δbout

1,cw −bout

1,cw ˆ

δbout†

1,ccw

(B9)

δi2=iA2bout

2,ccw ˆ

δbout†

2,cw +bout∗

2,cw ˆ

δbout

2,ccw

−bout∗

2,ccw ˆ

δbout

2,cw −bout

2,cw ˆ

δbout†

2,ccw

(B10)

Then quadrature basis expansion is performed to sepa-

rate the real and the imaginary parts (Xand Y) of the

operators:

bout

n,cw/ccw =Xout

n,cw/ccw +iY out

n,cw/ccw (B11)

bout∗

n,cw/ccw =Xout

n,cw/ccw −iY out

n,cw/ccw (B12)

δbout

n,cw/ccw =ˆ

δXout

n,cw/ccw +iˆ

δY out

n,cw/ccw (B13)

δbout†

n,cw/ccw =ˆ

δXout

n,cw/ccw −iˆ

δY out

n,cw/ccw (B14)

Hence, Eqs. B9-B10 are converted to:

δi1= 2A1Xout

1,ccw ˆ

δY out

1,cw −Yout

1,ccw ˆ

δXout

1,cw

−Xout

1,cw ˆ

δY out

1,ccw +Yout

1,cw ˆ

δXout

1,ccw(B15)

δi2= 2A2Xout

2,ccw ˆ

δY out

2,cw −Yout

2,ccw ˆ

δXout

2,cw

−Xout

2,cw ˆ

δY out

2,ccw +Yout

2,cw ˆ

δXout

2,ccw(B16)

Next we need to determine the statistical properties of

δi1and ˆ

δi2. Nevertheless, the output light is in com-

plex quantum states (squeezed vacuum/squeezed coher-

ent) which are diﬃcult to calculate. On the other hand,

these quadrature operators are nothing but linear com-

binations of the input light which is in relatively simple

quantum states (vacuum/coherent). To this end, we cal-

culate the mean values and the variances from the input

states. Rewrite Eqs. B15 and B16 in the form below:

δi1=

n=1

(b(1)

x,n,cw/ccwˆ

bin

X,n,cw/ccw +b(1)

y,n,cw/ccwˆ

bin

Y,n,cw/ccw

+c(1)

x,n,cw/ccwˆcin

X,n,cw/ccw +c(1)

y,n,cw/ccwˆcin

Y,n,cw/ccw)

(B17)

δi2=

n=1

(b(2)

x,n,cw/ccwˆ

bin

X,n,cw/ccw +b(2)

y,n,cw/ccwˆ

bin

Y,n,cw/ccw

+c(2)

x,n,cw/ccwˆcin

X,n,cw/ccw +c(2)

y,n,cw/ccwˆcin

Y,n,cw/ccw)

(B18)

In Eqs. B17-B18, ˆ

bin

X,n,cw/ccw,ˆ

bin

Y,n,cw/ccw, ˆcin

X,n,cw/ccw

and ˆcin

Y,n,cw/ccw are the quadrature operators (real and

imaginary parts) of the injection light light ˆ

bin

n,cw/ccw

and the intrinsic loss channels ˆcin

n,cw/ccw for sub/second

(n=1,2) harmonic light, where ˆ

bin

X,n,cw/ccw,ˆ

bin

Y,n,cw/ccw are

in coherent states and ˆcin

X,n,cw/ccw and ˆcin

Y,n,cw/ccw are in

vacuum states. b(1,2)

x,n,cw/ccw,b(1,2)

y,n,cw/ccw,c(1,2)

x,n,cw/ccw and

c(1,2)

y,n,cw/ccw are the corresponding coeﬃcients of these op-

erators at either sub or second harmonic. Assume ψ1/ψ2

and N1/N2are the initial phases and the numbers of the

injected photons of the input light ˆ

b1,cw/ccw and ˆ

b2,cw/ccw,

where N1=|bin

1|2and N2=|bin

2|2. With all these ingredi-

ents, now we can calculate mean values and the variances

of both diﬀerential currents. For a coherent state |α >,

the mean values of both quadrature operators and their

squares are given by [113]:

< α|ˆ

X|α >=√Ncos ψ(B19)

< α|ˆ

Y|α >=√Nsin ψ(B20)

< α|ˆ

X2|α >=4N(cos ψ)2+ 1

4(B21)

< α|ˆ

Y2|α >=4N(sin ψ)2+ 1

4(B22)

and the variances are deﬁned as the mean values of the

square minus the square of the mean values of the quadra-

ture operators:

< α|∆ˆ

X2|α >=< α|ˆ

X2|α > −(< α|ˆ

X|α >)2=1

(B23)

< α|∆ˆ

Y2|α >=< α|ˆ

Y2|α > −(< α|ˆ

Y|α >)2=1

(B24)

Note that when two quadrature operators do not share

the same eigen vectors, the deﬁnitions are diﬀerent. For

example, the inner product of a real quadrature opera-

tor at the second harmonic and an imaginary quadrature

operator at the fundamental frequency injection is given

by:

< α|ˆ

X2ˆ

Y1|α >=pN1pN2cos ψ2sin ψ2(B25)

< α|ˆ

X2ˆ

Y1|α > −< α|ˆ

X1|α >< α|ˆ

Y1|α >= 0 (B26)

Now we can obtain the mean values and the variances of

δi1and ˆ

δi2:

<ˆ

δi1>=

n=1b(1)

x,n,cw/ccwpNncos ψn

+b(1)

y,n,cw/ccwpNnsin ψn

(B27)

<ˆ

δi2>=

n=1b(2)

x,n,cw/ccwpNncos ψn

+b(2)

y,n,cw/ccwpNnsin ψn

(B28)

<∆ˆ

δi2

1>=1

n=1[(b(1)

x,n,cw/ccw)2+ (b(1)

y,n,cw/ccw)2

+ (c(1)

x,n,cw/ccw)2+ (c(1)

y,n,cw/ccw)2]

(B29)

<∆ˆ

δi2

2>=1

n=1[(b(2)

x,n,cw/ccw)2+ (b(2)

y,n,cw/ccw)2

+ (c(2)

x,n,cw/ccw)2+ (c(2)

y,n,cw/ccw)2]

(B30)

As discussed in Section II B, in order to determine the

covariance matrix, it is also necessary to calculate the

correlation between ˆ

δi1and ˆ

δi2following the rule of op-

erator calculation deﬁned above:

<ˆ

δi1ˆ

δi2>−<ˆ

δi1>< ˆ

δi2>=

n=1b(1)

x,n,cw/ccwb(2)

x,n,cw/ccw

+b(1)

y,n,cw/ccwb(2)

y,n,cw/ccw

+c(1)

x,n,cw/ccwc(2)

x,n,cw/ccw

+c(1)

y,n,cw/ccwc(2)

y,n,cw/ccw

(B31)

<ˆ

δi2ˆ

δi1>−<ˆ

δi2>< ˆ

δi1>=<ˆ

δi1ˆ

δi2>−<ˆ

δi1>< ˆ

δi2>

(B32)

With everything discussed in this section, particularly

Eqs. B27-B32, now we can calculate Eqs. 23-25 to de-

termine the Fisher information and the corresponding

sensitivity of the system.

[1] U. S. G. A. Oﬃce, Defense navigation capabilities: Dod

is developing positioning, navigation, and timing tech-

nologies to complement gps (2021).

[2] G. Sagnac, L’ether lumineux demontre par l’eﬀet du

vent relatif d’ether dans un interferometre en rotation

uniforme, Proceedings of the USSR Academy of Sciences

157, 708 (1913).

[3] E. J. Post, Sagnac eﬀect, Reviews of Modern Physics

39, 475 (1967).

[4] H. J. Arditty and H. C. Lef`evre, Sagnac eﬀect in ﬁber

gyroscopes, Opt. Lett. 6, 401 (1981).

[5] W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E.

Sanders, W. Schleich, and M. O. Scully, The ring laser

gyro, Rev. Mod. Phys. 57, 61 (1985).

[6] J. D. Cresser, W. H. Louisell, P. Meystre, W. Schleich,

and M. O. Scully, Quantum noise in ring-laser gyros.

i. theoretical formulation of problem, Phys. Rev. A 25,

2214 (1982).

[7] A. B. Matsko, W. Liang, A. A. Savchenkov, V. S.

Ilchenko, and L. Maleki, Fundamental limitations of sen-

sitivity of whispering gallery mode gyroscopes, Physics

Letters A 382, 2289 (2018), special Issue in memory of

Professor V.B. Braginsky.

[8] Z. Wang, Y. Yang, P. Lu, R. Luo, Y. Li, D. Zhao,

C. Peng, and Z. Li, Dual-polarization interferometric

ﬁber-optic gyroscope with an ultra-simple conﬁguration,

Opt. Lett. 39, 2463 (2014).

[9] H. Ma, W. Wang, Y. Ren, and Z. Jin, Low-noise low-

delay digital signal processor for resonant micro op-

tic gyro, IEEE Photonics Technology Letters 25, 198

(2013).

[10] G. A. Sanders, S. J. Sanders, L. K. Strandjord, T. Qiu,

J. Wu, M. Smiciklas, D. Mead, S. Mosor, A. Arrizon,

W. Ho, and M. Salit, Fiber optic gyro development

at Honeywell, in Fiber Optic Sensors and Applications

XIII , Vol. 9852, edited by E. Udd, G. Pickrell, and

H. H. Du, International Society for Optics and Pho-

tonics (SPIE, 2016) p. 985207.

[11] W. Z. Korth, A. Heptonstall, E. D. Hall, K. Arai, E. K.

Gustafson, and R. X. Adhikari, Passive, free-space het-

erodyne laser gyroscope, Classical and Quantum Grav-

ity 33, 035004 (2016).

[12] J. P. Dowling, Correlated input-port, matter-wave in-

terferometer: Quantum-noise limits to the atom-laser

gyroscope, Phys. Rev. A 57, 4736 (1998).

[13] W. Liang, V. S. Ilchenko, A. A. Savchenkov, E. Dale,

D. Eliyahu, A. B. Matsko, and L. Maleki, Resonant mi-

crophotonic gyroscope, Optica 4, 114 (2017).

[14] J. Zhang, H. Ma, H. Li, and Z. Jin, Single-polarization

ﬁber-pigtailed high-ﬁnesse silica waveguide ring res-

onator for a resonant micro-optic gyroscope, Opt. Lett.

42, 3658 (2017).

[15] P. P. Khial, A. D. White, and A. Hajimiri, Nanopho-

tonic optical gyroscope with reciprocal sensitivity en-

hancement, Nature Photonics 12, 671 (2018).

[16] Y.-H. Lai, M.-G. Suh, Y.-K. Lu, B. Shen, Q.-F. Yang,

H. Wang, J. Li, S. H. Lee, K. Y. Yang, and K. Vahala,

Earth rotation measured by a chip-scale ring laser gy-

roscope, Nature Photonics 14, 345 (2020).

[17] L. Ge, R. Sarma, and H. Cao, Rotation-induced evolu-

tion of far-ﬁeld emission patterns of deformed microdisk

cavities, Optica 2, 323 (2015).

[18] J. Scheuer, Direct rotation-induced intensity modula-

tion in circular bragg micro-lasers, Opt. Express 15,

15053 (2007).

[19] C. Ciminelli, D. D’Agostino, G. Carnicella, F. Dell’Olio,

D. Conteduca, H. P. M. M. Ambrosius, M. K. Smit, and

M. N. Armenise, A high-q inp resonant angular velocity

sensor for a monolithically integrated optical gyroscope,

IEEE Photonics Journal 8, 1 (2016).

[20] M. J. Grant, P.-B. Vigneron, A. Feshali, W. Jin,

N. Abrams, M. Paniccia, and M. Digonnet, Chip-scale

gyroscope using silicon-nitride waveguide resonator with

a Q factor of 100 million, in Optical and Quantum Sens-

ing and Precision Metrology II, Vol. 12016, edited by

J. Scheuer and S. M. Shahriar, International Society for

Optics and Photonics (SPIE, 2022) p. 120160S.

[21] J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow,

M. Soltani, D. Christodoulides, and M. Khajavikhan,

Ultrasensitive micro-scale parity-time-symmetric ring

laser gyroscope, Opt. Lett. 42, 1556 (2017).

[22] J. Scheuer and A. Yariv, Sagnac eﬀect in coupled-

resonator slow-light waveguide structures, Phys. Rev.

Lett. 96, 053901 (2006).

[23] L. Yan, Z. Xiao, X. Guo, and A. Huang, Circle-

coupled resonator waveguide with enhanced Sagnac

phase-sensitivity for rotation sensing, Applied Physics

Letters 95, 10.1063/1.3243456 (2009), 141104.

[24] D. D. Smith, H. Chang, L. Arissian, and J. C. Diels,

Dispersion-enhanced laser gyroscope, Phys. Rev. A 78,

053824 (2008).

[25] M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal,

M. Messall, and K. Salit, Ultrahigh enhancement in ab-

solute and relative rotation sensing using fast and slow

light, Phys. Rev. A 75, 053807 (2007).

[26] M. J. F. Digonnet and J. N. Chamoun, Recent develop-

ments in laser-driven and hollow-core ﬁber optic gyro-

scopes, in Fiber Optic Sensors and Applications XIII,

Vol. 9852, edited by E. Udd, G. Pickrell, and H. H. Du,

International Society for Optics and Photonics (SPIE,

2016) p. 985204.

[27] M. Larsen and M. Bulatowicz, Nuclear magnetic reso-

nance gyroscope: For darpa’s micro-technology for posi-

tioning, navigation and timing program, in 2014 Inter-

national Symposium on Inertial Sensors and Systems

(ISISS) (2014) pp. 1–5.

[28] G. T´oth and I. Apellaniz, Quantum metrology from

a quantum information science perspective, Journal of

Physics A: Mathematical and Theoretical 47, 424006

(2014).

[29] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum

sensing, Rev. Mod. Phys. 89, 035002 (2017).

[30] C. Roques-Carmes, Y. Salamin, J. Sloan, S. Choi,

G. Velez, E. Koskas, N. Rivera, S. E. Kooi, J. D.

Joannopoulos, and M. Soljaˇci´c, Biasing the quantum

vacuum to control macroscopic probability distribu-

tions, Science 381, 205 (2023).

[31] C. M. Caves, Quantum-mechanical noise in an interfer-

ometer, Phys. Rev. D 23, 1693 (1981).

[32] M. R. Grace, C. N. Gagatsos, Q. Zhuang, and

S. Guha, Quantum-enhanced ﬁber-optic gyroscopes us-

ing quadrature squeezing and continuous-variable en-

tanglement, Phys. Rev. Appl. 14, 034065 (2020).

[33] M. Fink, F. Steinlechner, J. Handsteiner, J. P. Dowl-

ing, T. Scheidl, and R. Ursin, Entanglement-enhanced

optical gyroscope, New Journal of Physics 21, 053010

(2019).

[34] A. et al. (Virgo Collaboration), Increasing the astro-

physical reach of the advanced virgo detector via the

application of squeezed vacuum states of light, Phys.

Rev. Lett. 123, 231108 (2019).

[35] T. et al., Quantum-enhanced advanced ligo detectors

in the era of gravitational-wave astronomy, Phys. Rev.

Lett. 123, 231107 (2019).

[36] W. Jin, Q.-F. Yang, L. Chang, B. Shen, H. Wang, M. A.

Leal, L. Wu, M. Gao, A. Feshali, M. Paniccia, K. J. Va-

hala, and J. E. Bowers, Hertz-linewidth semiconductor

lasers using cmos-ready ultra-high-q microresonators,

Nature Photonics 15, 346 (2021).

[37] I. Krasnokutska, J.-L. J. Tambasco, X. Li, and A. Pe-

ruzzo, Ultra-low loss photonic circuits in lithium niobate

on insulator, Opt. Express 26, 897 (2018).

[38] K. Liu, N. Jin, H. Cheng, N. Chauhan, M. W. Puck-

ett, K. D. Nelson, R. O. Behunin, P. T. Rakich, and

D. J. Blumenthal, Ultralow 0.034 db/m loss wafer-scale

integrated photonics realizing 720 million q and 380 µw

threshold brillouin lasing, Opt. Lett. 47, 1855 (2022).

[39] L. Li, H. Lin, Y. Huang, R.-J. Shiue, A. Yadav, J. Li,

J. Michon, D. Englund, K. Richardson, T. Gu, and

J. Hu, High-performance ﬂexible waveguide-integrated

photodetectors, Optica 5, 44 (2018).

[40] M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D.

Stone, and L. Jiang, Quantum noise theory of excep-

tional point amplifying sensors, Phys. Rev. Lett. 123,

180501 (2019).

[41] D. Anderson, M. Shah, and L. Fan, Clariﬁcation of

the exceptional-point contribution to photonic sensing,

Phys. Rev. Appl. 19, 034059 (2023).

[42] B. Shahriari, K. Swersky, Z. Wang, R. P. Adams, and

N. de Freitas, Taking the human out of the loop: A re-

view of bayesian optimization, Proceedings of the IEEE

104, 148 (2016).

[43] M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. Stone,

and L. Jiang, Quantum noise theory of exceptional point

amplifying sensors, Physical review letters 123, 180501

(2019).

[44] M. S. Stefszky, C. M. Mow-Lowry, S. S. Y. Chua,

D. A. Shaddock, B. C. Buchler, H. Vahlbruch, A. Kha-

laidovski, R. Schnabel, P. K. Lam, and D. E. McClel-

land, Balanced homodyne detection of optical quantum

states at audio-band frequencies and below, Classical

and Quantum Gravity 29, 145015 (2012).

[45] C. Ciminelli, F. Dell’Olio, and M. N. Armenise, High-q

spiral resonator for optical gyroscope applications: Nu-

merical and experimental investigation, IEEE Photonics

Journal 4, 1844 (2012).

[46] N. M. Kondratiev, V. E. Lobanov, A. E. Shitikov, R. R.

Galiev, D. A. Chermoshentsev, N. Y. Dmitriev, A. N.

Danilin, E. A. Lonshakov, K. N. Min’kov, D. M. Sokol,

S. J. Cordette, Y.-H. Luo, W. Liang, J. Liu, and I. A.

Bilenko, Recent advances in laser self-injection lock-

ing to high-q microresonators, Frontiers of Physics 18,

21305 (2023).

[47] P. D. Drummond, Squeezing with nonlinear optics, in

Quantum Squeezing (Springer Berlin Heidelberg, Berlin,

Heidelberg, 2004) pp. 99–139.

[48] H. Wang, Y.-H. Lai, Z. Yuan, M.-G. Suh, and K. Va-

hala, Petermann-factor sensitivity limit near an excep-

tional point in a brillouin ring laser gyroscope, Nature

Communications 11, 1610 (2020).

[49] Y. Zhao, Y. Okawachi, J. K. Jang, X. Ji, M. Lipson,

and A. L. Gaeta, Near-degenerate quadrature-squeezed

vacuum generation on a silicon-nitride chip, Phys. Rev.

Lett. 124, 193601 (2020).

[50] G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L.

O’Brien, and A. Furusawa, Continuous-variable entan-

glement on a chip, Nature Photonics 9, 316 (2015).

[51] C. Couteau, Spontaneous parametric down-conversion,

Contemporary Physics 59, 291 (2018).

[52] D. A. Kleinman, Theory of second harmonic generation

of light, Phys. Rev. 128, 1761 (1962).

[53] I. Breunig, Three-wave mixing in whispering gallery res-

onators, Laser and Photonics Reviews 10, 569 (2016).

[54] J. Lu, J. B. Surya, X. Liu, A. W. Bruch, Z. Gong, Y. Xu,

and H. X. Tang, Periodically poled thin-ﬁlm lithium nio-

bate microring resonators with a second-harmonic gen-

eration eﬃciency of 250,000%/w, Optica 6, 1455 (2019).

[55] X. Guo, C.-L. Zou, and H. X. Tang, Second-

harmonic generation in aluminum nitride microrings

with 2500%/w conversion eﬃciency, Optica 3, 1126

(2016).

[56] P. Drummond, K. McNeil, and D. Walls, Non-

equilibrium transitions in sub/second harmonic genera-

tion, Optica Acta: International Journal of Optics 27,

321 (1980).

[57] J. R. Johansson, P. D. Nation, and F. Nori, Qutip: An

open-source python framework for the dynamics of open

quantum systems, Computer Physics Communications

183, 1760 (2012).

[58] W. Tang, F. Lee, and R. Ridley, Small-signal modeling

of average current-mode control, IEEE Transactions on

Power Electronics 8, 112 (1993).

[59] L. Gillner, G. Bj¨ork, and Y. Yamamoto, Quantum noise

properties of an injection-locked laser oscillator with

pump-noise suppression and squeezed injection, Phys.

Rev. A 41, 5053 (1990).

[60] S. Machida, Y. Yamamoto, and Y. Itaya, Observation of

amplitude squeezing in a constant-current–driven semi-

conductor laser, Phys. Rev. Lett. 58, 1000 (1987).

[61] Y. K. Chembo, Quantum dynamics of kerr optical fre-

quency combs below and above threshold: Spontaneous

four-wave mixing, entanglement, and squeezed states of

light, Physical Review A 93, 033820 (2016).

[62] S. Pontula, J. Sloan, N. Rivera, and M. Soljacic,

Strong intensity noise condensation using nonlinear dis-

persive loss in semiconductor lasers, arXiv preprint

arXiv:2212.07300 (2022).

[63] P. J. Costa, The hartman–grobman theorem, in Select

Ideas in Partial Diﬀerential Equations (Springer Inter-

national Publishing, Cham, 2021) pp. 173–198.

[64] P. D. Drummond and C. W. Gardiner, Generalised p-

representations in quantum optics, Journal of Physics

A: Mathematical and General 13, 2353 (1980).

[65] S. L. Braunstein and C. M. Caves, Statistical distance

and the geometry of quantum states, Phys. Rev. Lett.

72, 3439 (1994).

[66] P. Drummond, K. McNeil, and D. Walls, Non-

equilibrium transitions in sub/second harmonic genera-

tion, Optica Acta: International Journal of Optics 28,

211 (1981).

[67] R. Kapral, Quantum dynamics in open quantum-

classical systems, Journal of Physics: Condensed Matter

27, 073201 (2015).

[68] M. Giglio, P. Patimisco, A. Sampaolo, G. Scamarcio,

F. K. Tittel, and V. Spagnolo, Allan deviation plot

as a tool for quartz-enhanced photoacoustic sensors

noise analysis, IEEE Transactions on Ultrasonics, Fer-

roelectrics, and Frequency Control 63, 555 (2016).

[69] J. H. Shapiro, The quantum theory of optical commu-

nications, IEEE Journal of Selected Topics in Quantum

Electronics 15, 1547 (2009).

[70] H. J. Carmichael, Dissipation in quantum mechanics:

The master equation approach, in Statistical Methods

in Quantum Optics 1: Master Equations and Fokker-

Planck Equations (Springer Berlin Heidelberg, Berlin,

Heidelberg, 1999) pp. 1–28.

[71] D. Zhu, L. Shao, M. Yu, R. Cheng, B. Desiatov,

C. J. Xin, Y. Hu, J. Holzgrafe, S. Ghosh, A. Shams-

Ansari, E. Puma, N. Sinclair, C. Reimer, M. Zhang, and

M. Lonˇcar, Integrated photonics on thin-ﬁlm lithium

niobate, Adv. Opt. Photon. 13, 242 (2021).

[72] C. Cui, L. Zhang, and L. Fan, In situ control of eﬀec-

tive kerr nonlinearity with pockels integrated photonics,

Nature Physics 18, 497 (2022).

[73] J. M. Silver, L. Del Bino, M. T. Woodley, G. N. Gha-

lanos, A. Ø. Svela, N. Moroney, S. Zhang, K. T. Grat-

tan, and P. Del’Haye, Nonlinear enhanced microres-

onator gyroscope, Optica 8, 1219 (2021).

[74] K. J. Peters and S. R. Rodriguez, Exceptional precision

of a nonlinear optical sensor at a square-root singularity,

Physical Review Letters 129, 013901 (2022).

[75] J. Pastrˇn´ak and L. Roskovcov´a, Refraction index mea-

surements on aln single crystals, physica status solidi

(b) 14, K5 (1966).

[76] S. Singh, J. R. Potopowicz, L. G. Van Uitert, and S. H.

Wemple, Nonlinear Optical Properties of Hexagonal Sil-

icon Carbide, Applied Physics Letters 19, 53 (2003).

[77] D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat,

Optical properties of alxga1x as, Journal of Applied

Physics 60, 754 (1986).

[78] R. Nehra, R. Sekine, L. Ledezma, Q. Guo, R. M. Gray,

A. Roy, and A. Marandi, Few-cycle vacuum squeezing

in nanophotonics, Science 377, 1333 (2022).

[79] M. W. McCutcheon, D. E. Chang, Y. Zhang, M. D.

Lukin, and M. Lonˇcar, Broadband frequency conversion

and shaping of single photons emitted from a nonlinear

cavity, Opt. Express 17, 22689 (2009).

[80] P.-K. Chen, I. Briggs, S. Hou, and L. Fan, Ultra-

broadband quadrature squeezing with thin-ﬁlm lithium

niobate nanophotonics, Opt. Lett. 47, 1506 (2022).

[81] J. Zhao, C. Ma, M. R¨using, and S. Mookherjea, High

quality entangled photon pair generation in periodically

poled thin-ﬁlm lithium niobate waveguides, Phys. Rev.

Lett. 124, 163603 (2020).

[82] M. Zhang, C. Wang, R. Cheng, A. Shams-Ansari, and

M. Lonˇcar, Monolithic ultra-high-q lithium niobate mi-

croring resonator, Optica 4, 1536 (2017).

[83] Z.-F. Bi, A. W. Rodriguez, H. Hashemi, D. Duch-

esne, M. Loncar, K.-M. Wang, and S. G. Johnson,

High-eﬃciency second-harmonic generation in doubly-

resonant χ(2) microring resonators, Opt. Express 20,

7526 (2012).

[84] A. Shams-Ansari, M. Yu, Z. Chen, C. Reimer,

M. Zhang, N. Picqu´e, and M. Lonˇcar, Thin-ﬁlm lithium-

niobate electro-optic platform for spectrally tailored

dual-comb spectroscopy, Communications Physics 5, 88

(2022).

[85] K. Nakamura, J. Kurz, K. Parameswaran, and

M. M. Fejer, Periodic poling of magnesium-oxide-doped

lithium niobate, Journal of Applied Physics 91, 4528

(2002).

[86] A. Rantamaki, E. Saarinen, J. Lyytik¨ainen, J. Kontio,

J. Heikkinen, K. Lahtonen, M. Valden, and O. Okhot-

nikov, Towards high power ﬂip-chip long-wavelength

semiconductor disk lasers, in Vertical External Cav-

ity Surface Emitting Lasers (VECSELs) V , Vol. 9349,

edited by M. Guina, International Society for Optics

and Photonics (SPIE, 2015) p. 934908.

[87] X. Guo, L. Shao, L. He, K. Luke, J. Morgan, K. Sun,

J. Gao, T.-C. Tzu, Y. Shen, D. Chen, B. Guo, F. Yu,

Q. Yu, M. Jafari, M. Lonˇcar, M. Zhang, and A. Beling,

High-performance modiﬁed uni-traveling carrier photo-

diode integrated on a thin-ﬁlm lithium niobate platform,

Photon. Res. 10, 1338 (2022).

[88] A. Guarino, G. Poberaj, D. Rezzonico,

R. Degl’Innocenti, and P. G¨unter, Electro–optically

tunable microring resonators in lithium niobate, Nature

Photonics 1, 407 (2007).

[89] M. Wang, J. Li, H. Yao, X. Li, J. Wu, K. S. Chiang,

and K. Chen, Thin-ﬁlm lithium-niobate modulator with

a combined passive bias and thermo-optic bias, Opt.

Express 30, 39706 (2022).

[90] R. Lu, Y. Yang, M.-H. Li, and S. Gong, Ghz low-

loss acoustic rf couplers in lithium niobate thin ﬁlm,

IEEE Transactions on Ultrasonics, Ferroelectrics, and

Frequency Control 67, 1448 (2020).

[91] A. Shams-Ansari, G. Huang, L. He, Z. Li, J. Holz-

grafe, M. Jankowski, M. Churaev, P. Kharel, R. Cheng,

D. Zhu, N. Sinclair, B. Desiatov, M. Zhang, T. J.

Kippenberg, and M. Lonˇcar, Reduced material loss in

thin-ﬁlm lithium niobate waveguides, APL Photonics 7,

081301 (2022).

[92] D. Bertsimas and J. Tsitsiklis, Simulated Annealing,

Statistical Science 8, 10 (1993).

[93] A. Eiben, R. Hinterding, and Z. Michalewicz, Parameter

control in evolutionary algorithms, IEEE Transactions

on Evolutionary Computation 3, 124 (1999).

[94] D. D. Smith, H. Chang, L. Arissian, and J. C. Diels,

Dispersion-enhanced laser gyroscope, Phys. Rev. A 78,

053824 (2008).

[95] S. H. Haji and A. M. Abdulazeez, Comparison of op-

timization techniques based on gradient descent algo-

rithm: A review, PalArch’s Journal of Archaeology of

Egypt / Egyptology 18, 2715 (2021).

[96] C. A. Floudas, Deterministic global optimization: the-

ory, methods and applications, Vol. 37 (Springer Science

and Business Media, 2013).

[97] A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M.

Siskind, Automatic diﬀerentiation in machine learning:

a survey, Journal of Marchine Learning Research 18, 1

(2018).

[98] S. Mohamed, M. Rosca, M. Figurnov, and A. Mnih,

Monte carlo gradient estimation in machine learning,

Journal of Machine Learning Research 21 (2020).

[99] L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M.

Murnane, and H. C. Kapteyn, Generation of broadband

vuv light using third-order cascaded processes, Phys.

Rev. Lett. 87, 013601 (2001).

[100] S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vuckovi´c,

and A. W. Rodriguez, Inverse design in nanophotonics,

Nature Photonics 12, 659 (2018).

[101] K. Csill´ery, M. G. Blum, O. E. Gaggiotti, and

O. Fran¸cois, Approximate bayesian computation (abc)

in practice, Trends in ecology and evolution 25, 410

(2010).

[102] R. Horst and H. Tuy, Global Optimization: Determinis-

tic Approaches (Springer Berlin Heidelberg, 2013).

[103] T. B¨ack and H.-P. Schwefel, An overview of evolution-

ary algorithms for parameter optimization, Evolution-

ary Computation 1, 1 (1993).

[104] Q. Wu, Y. Cao, H. Wang, and W. Hong, Machine-

learning-assisted optimization and its application to an-

tenna designs: Opportunities and challenges, China

Communications 17, 152 (2020).

[105] N. Aage, E. Andreassen, B. S. Lazarov, and O. Sigmund,

Giga-voxel computational morphogenesis for structural

design, Nature 550, 84 (2017).

[106] Z. Lin, R. Pestourie, C. Roques-Carmes, Z. Li, F. Ca-

passo, M. Soljaˇci´c, and S. G. Johnson, End-to-end

metasurface inverse design for single-shot multi-channel

imaging, Opt. Express 30, 28358 (2022).

[107] C. Wang, M. Zhang, M. Yu, R. Zhu, H. Hu, and M. Lon-

car, Monolithic lithium niobate photonic circuits for

kerr frequency comb generation and modulation, Na-

ture Communications 10, 978 (2019).

[108] M. Yu, D. Barton III, R. Cheng, C. Reimer, P. Kharel,

L. He, L. Shao, D. Zhu, Y. Hu, H. R. Grant, L. Jo-

hansson, Y. Okawachi, A. L. Gaeta, M. Zhang, and

M. Lonˇcar, Integrated femtosecond pulse generator on

thin-ﬁlm lithium niobate, Nature 612, 252 (2022).

[109] J. Lu, D. N. Puzyrev, V. V. Pankratov, D. V. Skryabin,

F. Yang, Z. Gong, J. B. Surya, and H. X. Tang, Two-

colour dissipative solitons and breathers in microres-

onator second-harmonic generation, Nature Communi-

cations 14, 2798 (2023).

[110] T. Erneux and P. Glorieux, Laser Dynamics (Cam-

bridge University Press, 2010).

[111] A. Gavrielides, V. Kovanis, and T. Erneux, Analytical

stability boundaries for a semiconductor laser subject

to optical injection, Optics Communications 136, 253

(1997).

[112] H. Sayama, Introduction to the modeling and analysis

of complex systems (2015).

[113] M. O. Scully and M. S. Zubairy, Quantum Optics (Cam-

bridge University Press, 1997).

ResearchGate has not been able to resolve any citations for this publication.

Recent advances in laser self-injection locking to high-Q microresonators

Article

Full-text available

Mar 2023

The stabilization and manipulation of laser frequency by means of an external cavity are nearly ubiquitously used in fundamental research and laser applications. While most of the laser light transmits through the cavity, in the presence of some back-scattered light from the cavity to the laser, the self-injection locking effect can take place, which locks the laser emission frequency to the cavity mode of similar frequency. The self-injection locking leads to dramatic reduction of laser linewidth and noise. Using this approach, a common semiconductor laser locked to an ultrahigh- Q microresonator can obtain sub-Hertz linewidth, on par with state-of-the-art fiber lasers. Therefore it paves the way to manufacture high-performance semiconductor lasers with reduced footprint and cost. Moreover, with high laser power, the optical nonlinearity of the microresonator drastically changes the laser dynamics, offering routes for simultaneous pulse and frequency comb generation in the same microresonator. Particularly, integrated photonics technology, enabling components fabricated via semiconductor CMOS process, has brought increasing and extending interest to laser manufacturing using this method. In this article, we present a comprehensive tutorial on analytical and numerical methods of laser self-injection locking, as well a review of most recent theoretical and experimental achievements.

Two-colour dissipative solitons and breathers in microresonator second-harmonic generation

Article

Full-text available

May 2023

Frequency conversion of dissipative solitons associated with the generation of broadband optical frequency combs having a tooth spacing of hundreds of giga-hertz is a topical challenge holding the key to practical applications in precision spectroscopy and data processing. The work in this direction is underpinned by fundamental problems in nonlinear and quantum optics. Here, we present the dissipative two-colour bright-bright and dark-dark solitons in a quasi-phase-matched microresonator pumped for the second-harmonic generation in the near-infrared spectral range. We also found the breather states associated with the pulse front motion and collisions. The soliton regime is found to be typical in slightly phase-mismatched resonators, while the phase-matched ones reveal broader but incoherent spectra and higher-order harmonic generation. Soliton and breather effects reported here exist for the negative tilt of the resonance line, which is possible only via the dominant contribution of second-order nonlinearity.

Integrated femtosecond pulse generator on thin-film lithium niobate

Article

Full-text available

Nov 2022
NATURE

Integrated femtosecond pulse and frequency comb sources are critical components for a wide range of applications, including optical atomic clocks1, microwave photonics2, spectroscopy3, optical wave synthesis4, frequency conversion5, communications6, lidar7, optical computing8 and astronomy9. The leading approaches for on-chip pulse generation rely on mode-locking inside microresonators with either third-order nonlinearity10 or with semiconductor gain11,12. These approaches, however, are limited in noise performance, wavelength and repetition rate tunability 10,13. Alternatively, subpicosecond pulses can be synthesized without mode-locking, by modulating a continuous-wave single-frequency laser using electro-optic modulators1,14–17. Here we demonstrate a chip-scale femtosecond pulse source implemented on an integrated lithium niobate photonic platform18, using cascaded low-loss electro-optic amplitude and phase modulators and chirped Bragg grating, forming a time-lens system19. The device is driven by a continuous-wave distributed feedback laser chip and controlled by a single continuous-wave microwave source without the need for any stabilization or locking. We measure femtosecond pulse trains (520-femtosecond duration) with a 30-gigahertz repetition rate, flat-top optical spectra with a 10-decibel optical bandwidth of 12.6 nanometres, individual comb-line powers above 0.1 milliwatts, and pulse energies of 0.54 picojoules. Our results represent a tunable, robust and low-cost integrated pulsed light source with continuous-wave-to-pulse conversion efficiencies an order of magnitude higher than those achieved with previous integrated sources. Our pulse generator may find applications in fields such as ultrafast optical measurement19,20 or networks of distributed quantum computers21,22. A femtosecond pulse generator is realized using an electro-optic time-lens system integrated on a lithium niobate photonic chip, capable of tunable repetition rates and wavelengths.

Thin-film lithium-niobate modulator with a combined passive bias and thermo-optic bias

Article

Full-text available

Oct 2022
OPT EXPRESS

It is essential to bias a thin-film lithium-niobate Mach-Zehnder electro-optic (EO) modulator at the desired operation condition to ensure optimal performance of the modulator. While thermo-optic (TO) control can solve the problem of bias drift, it consumes significant electric power. In this paper, we propose a technique to largely reduce bias power consumption by combining passive bias and TO bias. In our design, waveguide sections with different widths are introduced in the two arms of the MZ modulator to produce a desired phase difference of π/2 rad (the desired operation condition), and local heating with electrode heaters placed on the waveguides is employed to provide compensation for any phase drift caused by fabrication errors and other effects. As the TO control only serves to compensate for small errors, the electric power required is low and the response is fast. To demonstrate our technique experimentally, we fabricate several modulators of the same design on the same chip. Our experimental modulators can operate up to ∼40 GHz with a half-wave voltage of ∼2.0 V over a wide optical bandwidth, and the performances are insensitive to ambient temperature variations. The TO bias powers required range from 1 mW to 15 mW, and the thermal rise and fall times are 47 µs and 14 µs, respectively. Our technique can facilitate the development of practical high-speed EO modulators on the lithium-niobate-on-insulator platform.

Reduced material loss in thin-film lithium niobate waveguides

Article

Full-text available

Aug 2022

Thin-film lithium niobate has shown promise for scalable applications ranging from single-photon sources to high-bandwidth data communication systems. Realization of the next generation high-performance classical and quantum devices, however, requires much lower optical losses than the current state of the art resonator ( Q-factor of ∼10 million). Yet the material limitations of ion-sliced thin film lithium niobate have not been explored; therefore, it is unclear how high the quality factor can be achieved in this platform. Here, using our newly developed characterization method, we find out that the material limited quality factor of thin film lithium niobate photonic platform can be improved using post-fabrication annealing and can be as high as Q ≈ 1.6 × 10 ⁸ at telecommunication wavelengths, corresponding to a propagation loss of 0.2 dB/m.

Biasing the quantum vacuum to control macroscopic probability distributions

Article

Jul 2023
SCIENCE

Quantum field theory suggests that electromagnetic fields naturally fluctuate, and these fluctuations can be harnessed as a source of perfect randomness. Many potential applications of randomness rely on controllable probability distributions. We show that vacuum-level bias fields injected into multistable optical systems enable a controllable source of quantum randomness, and we demonstrated this concept in an optical parametric oscillator (OPO). By injecting bias pulses with less than one photon on average, we controlled the probabilities of the two possible OPO output states. The potential of our approach for sensing sub-photon-level fields was demonstrated by reconstructing the temporal shape of fields below the single-photon level. Our results provide a platform to study quantum dynamics in nonlinear driven-dissipative systems and point toward applications in probabilistic computing and weak field sensing.

Clarification of the Exceptional-Point Contribution to Photonic Sensing

Article

Mar 2023

Exceptional points, with simultaneous coalescence of eigenvalues and eigenvectors, can be realized with non-Hermitian photonic systems. With the enhanced response, exceptional points have been proposed to improve the performance of photonic sensing. Recently, there are intense debates about the actual sensing advantage of exceptional points. The major concern is that intrinsic noise is also increased at exceptional points. Here, we aim to clarify the contribution of exceptional points for photonic sensing. This is achieved by analyzing the condition to realize divergent quantum Fisher information in linear non-Hermitian photonic systems. We show that the divergence of quantum Fisher information is the result of lasing threshold, instead of exceptional points. However, exceptional points correspond to the condition that lasing threshold is simultaneously achieved across multiple photonic modes. Therefore, exceptional points can further improve the sensitivity on top of lasing threshold. On the other hand, exceptional points alone cannot provide sensing advantage.

Few-cycle vacuum squeezing in nanophotonics

Article

Sep 2022

One of the most fundamental quantum states of light is the squeezed vacuum, in which noise in one of the quadratures is less than the standard quantum noise limit. In nanophotonics, it remains challenging to generate, manipulate, and measure such a quantum state with the performance required for a wide range of scalable quantum information systems. Here, we report the development of a lithium niobate–based nanophotonic platform to demonstrate the generation and all-optical measurement of squeezed states on the same chip. The generated squeezed states span more than 25 terahertz of bandwidth supporting just a few optical cycles. The measured 4.9 decibels of squeezing surpass the requirements for a wide range of quantum information systems, demonstrating a practical path toward scalable ultrafast quantum nanophotonics.

Exceptional Precision of a Nonlinear Optical Sensor at a Square-Root Singularity

Article

Jun 2022

Exceptional points (EPs)-spectral singularities of non-Hermitian linear systems-have recently attracted interest for sensing. While initial proposals and experiments focused on enhanced sensitivities neglecting noise, subsequent studies revealed issues with EP sensors in noisy environments. Here we propose a single-mode Kerr-nonlinear resonator for exceptional sensing in noisy environments. Based on the resonator's dynamic hysteresis, we define a signal that displays a square-root singularity reminiscent of an EP. However, our sensor has crucial fundamental and practical advantages over EP sensors: the signal-to-noise ratio increases with the measurement speed, the square-root singularity is easily detected through intensity measurements, and both sensing precision and information content of the signal are enhanced around the singularity. Our sensor also overcomes the fundamental trade-off between precision and averaging time characterizing all linear sensors. All these unconventional features open up new opportunities for fast and precise sensing using hysteretic resonators.

The Hartman–Grobman Theorem

Chapter

Jan 2021

Peter J. Costa

By generalizing the Hartman–Grobman Theorem for the flows of vector fields, it is seen that a class of nonlinear evolution equations admit a local linearization. Under the restriction that the nonlinear part of the solution operator is compact, the linearization turns out to be global. This local linearization is applied to the periodic Korteweg de Vries, Burgers’, and cubic Schrödinger equations.

Nonlinear Multi-Resonant Cavity Quantum Photonics Gyroscopes Quantum Light Navigation

Abstract

Recommended publications

Nonlinear multi-resonant cavity QED gyroscope

Bayesian optimization of Fisher Information in nonlinear multiresonant quantum photonics gyroscopes

In Situ Control of Kerr Nonlinear Coefficient by Cascaded Pockels Effect in Micro-Ring Resonator

Chip-scale gyroscope using silicon-nitride waveguide resonator with a Q factor of 100 million