appendix

# Scroll Waves and Filaments in excitable Media of higher spatial Dimension

Marie Cloet1,2 Louise Arno1,2 Desmond Kabus1,3,2 Joeri Van der Veken4
Alexander V. Panfilov5,6,7 Hans Dierckx1,2<EMAIL_ADDRESS>1Department
of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk 8500, Belgium,
2iSi Health, Institute of Physics-based Modeling for In Silico Health, KU
Leuven, Leuven 3000, Belgium, 3Laboratory of Experimental Cardiology, Leiden
University Medical Center (LUMC), Leiden 2333 ZA, The Netherlands, 4Department
of Mathematics, KU Leuven, Leuven 3001, Belgium, 5Department of Physics and
Astronomy, Ghent University, Ghent 9000, Belgium, 6Laboratory of Computational
Biology and Medicine, Ural Federal University, Ekaterinburg 620002, Russia,
7World-Class Research Center “Digital biodesign and personalized healthcare”,
Sechenov University, Moscow 119991, Russia

(Received 2 May 2023; accepted 30 August 2023)

###### Abstract

Excitable media are ubiquitous in nature, and in such systems the local
excitation tends to self-organize in traveling waves, or in rotating spiral-
shaped patterns in two or three spatial dimensions. Examples include waves
during a pandemic or electrical scroll waves in the heart. Here we show that
such phenomena can be extended to a space of four or more dimensions and
propose that connections of excitable elements in a network setting can be
regarded as additional spatial dimensions. Numerical simulations are performed
in four dimensions using the FitzHugh-Nagumo model, showing that the vortices
rotate around a two-dimensional surface which we define as the superfilament.
Evolution equations are derived for general superfilaments of co-dimension two
in an $N$-dimensional space and their equilibrium configurations are proven to
be minimal surfaces. We suggest that biological excitable systems, such as the
heart or brain which have non-local connections can be regarded, at least
partially, as multidimensional excitable media and discuss further possible
studies in this direction.

††preprint: APS/123-QED

Introduction. Many real-world systems exhibit a large non-linear response to
external or internal stimuli, and are therefore called excitable systems. In
the continuum limit, several of these systems are reasonably understood, as
unifying geometric principles have been revealed that enable us to understand
and quantify their dynamics. As a first example, non-linear traveling waves of
constant amplitude have been observed as depolarization waves in neural [1]
and cardiac tissue [2], combustion [3] or oxidation [4] processes, Mexican
waves in crowds [5], and infection waves in a pandemic [6]. This extremely
diverse range of phenomena has been described mathematically in the continuum
limit using reaction-diffusion equations

$\partial_{t}\mathbf{u}=\mathbf{P}\Delta_{N}\mathbf{u}+\mathbf{F}(\mathbf{u})\,,$
(1)

where $\mathbf{u}(\vec{r},t)$ is a state vector with $S$ components that
varies in $N$-dimensional space and time, and $\mathbf{P}$ is a diagonal
$S\times S$ matrix containing the diffusivity of each state variable,
$\Delta_{N}$ the Laplacian operator in $N$ spatial dimensions and
$\mathbf{F}(\mathbf{u})$ a local excitation model. The state vector can
support waves and it was shown mathematically that their wave fronts undergo
curvature-driven dynamics, with the lowest order effect behaving like surface
tension [7, 8, 9]. These curvature effects are governed by the medium
characteristics, mathematically represented by the model parameters. Wave
fronts are known to be lines in two-dimensional (2D) media, and surfaces in 3D
media. This property can be summarized as wave fronts having spatial
codimension $C=1$, i.e., they are $(N-C)$-dimensional structures in an $N$-D
space.

In observations of excitable surfaces ($N=2$), excitation patterns were often
found to organize into rotating spiral-shaped patterns. Their study is largely
motivated by cardiological applications, since electrical rotors have been
observed on the surface of the heart during rhythm disorders. Like for wave
fronts, spiral waves drift is affected by geometric factors, such as the local
Gaussian curvature of the excitable surface [10], possibly incorporating
anisotropy of wave propagation within the surface [11]. Spiral waves on a
surface rotate around a core region, which is commonly idealized into a phase
singularity (PS) point, of co-dimension 2 [12]. The case where quasi-
periodically moving (meandering) PSs occur is more involved, and has been
described recently [13, 14, 15]. In 3D media, such as cardiac tissue, a stack
of spiral waves is known as a scroll wave. A scroll wave rotates around a
curve consisting of PSs, the filament which has dimension $N-C=3-2=1$. The
curvature-induced dynamics of filaments was first established numerically in
[16, 17, 18] for isotropic excitable media where they were shown to exhibit
so-called filament tension [19], a medium-dependent emerging parameter. For
positive filament tension, filaments straighten up [17, 18], while for
negative filament tension, filaments elongate and tend to break up [16], if
the medium is thick enough [20]. In the context of cardiac arrhythmias, the
topological configuration of filament curves in the cardiac wall (e.g., $O$,
$I$ or $U$-shaped) is being used as a classifier for arrhythmia patterns.
Moreover, spatio-temporal drift (due to internal interaction or gradients in
the medium) is thought to be responsible for transitions between arrhythmias
[21].

From the above overview on the current knowledge, a natural question arises
that is at first sight purely mathematical: Which other patterns of co-
dimension $C$ exist in $N$-dimensional media? In this paper, we provide some
answers to these questions. We confirm that indeed other patterns exist, more
specifically, superfilaments ($C=2$), i.e., the organizing center of
superscrolls in a space with $N>3$ spatial dimensions.

Research hypothesis.  We claim that such higher dimensional patterns are
significant for real-world excitable systems. The crucial point is that
simulations on an $N$-D grid are related to a regular network of connected
excitable elements. Indeed, when the aforementioned systems in $N=2$ or $N=3$
spatial dimensions are studied numerically, they are often simulated on a
Cartesian grid with lattice constant $h$. Using the second order accurate
Laplacian stencil (see Fig. 1), the discretization of Eq. (1) for the $i$-th
node in the grid becomes

$\partial_{t}\mathbf{u}_{i}=\mathbf{P}\sum_{j:A_{ij}=1}\frac{\mathbf{u}_{j}-\mathbf{u}_{i}}{h^{2}}+\mathbf{F}(\mathbf{u}_{i})\,.$
(2)

Here, $A_{ij}=1$ if the node $i$ is next to $j$ and $0$ otherwise. In
$\mathbb{R}^{1}$, every interior point will have $2$ such neighbors, in
$\mathbb{R}^{2}$, $4$ neighbors, etc. Thus, if one will discretize similarly
the Laplacian in $\mathbb{R}^{4}$, every interior point will have $8$
neighbors, and in $\mathbb{R}^{N}$, one obtains $2N$ such neighbors. Note,
however, that Eq. (2) can also be regarded as an excitable network [22], where
$A_{ij}$ is the adjacency matrix of the graph consisting of the edges that
form the network. In this view, going to $N>3$ actually makes sense, as an
idealized model for excitable networks in which each node is connected to $2N$
neighbors. In network language, such graph would be a limit of a
translationally invariant network with average degree $2N$, also called a
regular network. In view of real-world networks with unexplained emergent
pattern formation, $N$ can grow large. For example, on social networks, the
average degree is of magnitude $10^{2}-10^{3}$ [23]. In brain tissue,
cerebellar granule cells make about 675 synaptic connections with Purkinje
cells, which in their turn receive input from around 185,000 parallel fibers
[24]. If a network with high degree is realized in 3D space that surrounds us,
it implies that the network exhibits long-range, or global connections. Within
this work, analytical results are obtained for any integer $N>2$. However, we
will consider in simulations only one extra dimension, such that $N=4$, which
already shows interesting new physical phenomena.

(a) 1D

(b) 2D

$y$$z$$x$ (c) 3D

Figure 1:  Stencil for the second order accurate discretization of the
Laplacian. Nodes connected to the central point obtain a weight $1/h^{2}$, see
Eq. (2).

Analytical derivation. Motivated by multiple applications, as presented above,
and inspired by previous results in 2 and 3 dimensions [8, 19, 25], the
equation of motion for the superfilaments of co-dimension $C=2$ will be
derived here. In those works, a scroll wave was constructed by stacking a set
of spiral waves along a filament curve. Let $x^{i}$, $1\leq i\leq N$, be the
coordinates in an $N$-dimensional domain. By definition, the superfilament
$\mathcal{F}$ will be an $(N-2)$-dimensional submanifold, hence we can define
its coordinates $\sigma^{m}$, where $1\leq m\leq N-2$. The embedding of the
superfilament in the $N$-dimensional domain is then given by
$X^{i}(\sigma^{1},\ldots,\sigma^{N-2})$, $1\leq i\leq N$. In one point of
$\mathcal{F}$, we construct 2 orthogonal unit normal vectors $\vec{N}_{c}$,
with $c\in\\{1,2\\}$. The spatial change of these vectors between different
points on the submanifold is given by the Weingarten formula [26]:

$\displaystyle
D_{m}\vec{N}_{c}=-S_{\vec{N}_{c}}\left(\vec{e}_{m}\right)+\nabla_{m}^{\perp}\vec{N}_{c}\,.$
(3)

Here, $D_{m}$ is the derivative in $\mathbb{R}^{N}$ in the direction of
coordinate $\sigma^{m}$, $S_{\vec{N}_{c}}$ is the shape operator inherited
from the normal direction $\vec{N}_{c}$ and $\vec{e}_{m}$ is the tangent
vector to $\mathcal{F}$ corresponding to coordinate direction $\sigma^{m}$.
The second term $\nabla_{m}^{\perp}\vec{N}_{c}$ is the normal connection,
describing the relative rotation of both normals when one moves along the
superfilament. This term can be set to 0 by choosing the normal vectors
$\vec{N}_{1}$ and $\vec{N}_{2}$ to be parallel with respect to the normal
connection. This procedure is similar to choosing a relative parallel frame to
a curve in 3D [27].

Next, we introduce normal coordinates $\rho^{c}$ for points in the vicinity of
the superfilament:

$\displaystyle
x^{i}=X^{i}\left(\sigma^{m}\right)+\sum_{c=1}^{2}\rho^{c}N_{c}^{i}.$ (4)

This allows us to define the normal vectors $\vec{N}_{c}$ also in the vicinity
of the superfilament, by $N_{c}^{i}=\frac{\partial x^{i}}{\partial\rho^{c}}$.
The divergence of $\vec{N}_{c}$ is then related to the curvature of the
superfilament. Indeed, by construction, $D_{d}\vec{N}_{c}=\vec{0}$ for
$c\in\\{1,2\\}$, $d\in\\{N-1,N\\}$, such that with Eq. (3) we obtain [26, 28]:

$\operatorname{div}\left(\vec{N}_{c}\right)=\sum_{m=1}^{N-2}\vec{e}_{m}\cdot
D_{m}\vec{N}_{c}=-\operatorname{tr}\left(S_{\vec{N}_{c}}\right)=-\vec{N}_{c}\cdot\vec{H}\,,$
(5)

with $\vec{H}$ the (non-normalized) mean curvature of the superfilament.

With these preparations, we can now explicitly construct the superscroll as a
stack of spiral wave solutions. Say $\mathbf{U}(x,y;\phi)$ is a spiral wave
solution in the 2D plane, rotated around the origin over an angle $\phi$.
Then, we propose as a solution to Eq. (1) in $N$ dimensions:

$\displaystyle\mathbf{u}\left(x^{i},t\right)$
$\displaystyle=\mathbf{U}\left(\rho^{1},\rho^{2},\phi(\sigma^{m},t)\right)+\tilde{\mathbf{u}}\left(x^{i},t\right),$
$\displaystyle\dot{\phi}\left(\sigma^{m},t\right)$
$\displaystyle=\omega+\tilde{\omega}\left(\sigma^{m},t\right)$ (6)

where $\omega$ is the spiral’s natural rotation frequency in a plane. This
approximation is a gradient expansion around a planar superfilament, and the
$\tilde{\mathbf{u}}$ and $\tilde{\omega}$ terms are higher-order corrections
in the extrinsic curvature of the superfilament, similar to earlier work on
classical scroll wave filaments [8, 25]. Substituting the expressions in the
reaction diffusion equation (RDE) brings, in leading order,

$\dot{\tilde{\mathbf{u}}}-\hat{\mathbf{L}}\tilde{\mathbf{u}}=\sum_{c=1}^{2}\left[\mathbf{P}\operatorname{div}(\vec{N}_{c})+\left(\vec{N}_{c}\cdot\dot{\vec{X}}\right)\right]\frac{\partial\mathbf{U}}{\partial\rho^{c}}+\omega\frac{\partial\mathbf{U}}{\partial\phi},$
(7)

where
$\hat{\mathbf{L}}=\mathbf{P}\Delta_{2}+\omega\partial_{\phi}+\mathbf{F}^{\prime}(\mathbf{U})$.
This operator only depends on the standard 2D unperturbed rotating spiral
solution. From symmetry arguments, it is known to have critical eigenmodes
(Goldstone modes) [8, 29, 30, 31, 25, 32], as well as response functions [31].
They are critical modes of the adjoint operator
$\hat{\mathbf{L}}^{\dagger}=\mathbf{P}^{H}\Delta_{2}-\omega\partial_{\phi}+\mathbf{F}^{\prime
H}(\mathbf{U})$. Taking the overlap integral of Eq. (7) with the response
functions delivers, after averaging over one rotation period [19, 25]:

$\dot{\vec{X}}=-\gamma_{1}\sum_{c=1}^{2}\operatorname{div}\left(\vec{N}_{c}\right)\vec{N}_{c}-\gamma_{2}\sum_{c,d=1}^{2}\varepsilon_{cd}\operatorname{div}\left(\vec{N}_{d}\right)\vec{N}_{c},$
(8)

Here, $\gamma_{1}$ and $\gamma_{2}$ are the scalar and pseudoscalar filament
tension coefficients that were introduced in [19] and $\varepsilon_{cd}$ are
the components of the Levi-Civita symbol
$\bar{\bar{\mathbf{\varepsilon}}}=\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right)$. Using Eq. (5) we can express the superfilament
dynamics in terms of its mean curvature:

$\dot{\vec{X}}=\gamma_{1}\vec{H}+\gamma_{2}\bar{\bar{\mathbf{\varepsilon}}}\cdot\vec{H},$
(9)

Our result is consistent with the lower-dimensional case: For $N=3$, the
superfilament becomes a filament curve with arc length $s$, curvature $k$,
tangent vector $\vec{T}$ and unit normal $\vec{N}$, such that
$\vec{H}=k\vec{N}=\frac{\partial\vec{T}}{\partial s}$. Then, Eq. (9) reduces
to the classical result of Biktashev et al. [19].

From the equation of motion (9), it follows that stationary superfilaments
must have $\vec{H}=\vec{0}$, thus the total mean curvature must vanish. For
the case $N=4$, this means that a stationary superfilament must be a minimal
surface, generalizing Wellner’s minimal principle to superscrolls [33]. It
proves a property that we postulated before based on an action principle [34],
see also the discussion on Fig. 2c below.

Similar to wave fronts and classical filaments, the motion of superfilaments
in a homogeneous medium is driven by curvature. It is possible to explicitly
show that in the case of positive filament tension, the superfilament will
decrease its size, as follows. With $A$ the total area of a (hyper)surface
$\mathcal{S}$ and $\dot{\vec{X}}$ prescribing temporal evolution for all
points on that surface, Gauss’ first law for the variation of surface area
states $dA/dt=-\int_{\mathcal{F}}\vec{H}\cdot\dot{\vec{X}}\mbox{d}V$, with
$\mbox{d}V$ the volume form on the (hyper)surface [35]. For superfilaments
$\mathcal{F}$, combining this relation with Eq. (9) yields:

$\displaystyle\frac{dA}{dt}$
$\displaystyle=-\int_{\mathcal{F}}\vec{H}\cdot\left[\gamma_{1}\vec{H}+\gamma_{2}\bar{\bar{\mathbf{\varepsilon}}}\cdot\vec{H}\right]\mbox{d}V=-\gamma_{1}\int_{\mathcal{F}}\lVert\vec{H}\rVert^{2}\mbox{d}V$
(10)

with $\lVert\,.\,\rVert$ the Euclidean norm of a vector. Eq. (10) gives the
analogue of the change-of-length rate $dL/dt=-\gamma_{1}\int k^{2}ds$ for
classical filaments [19], which we retrieve as the case $N=3$. We have here
shown that superfilaments monotonically decrease their generalized surface
area in case of positive filament tension. As a corollary, stable superscrolls
can persist in excitable media as minimal surfaces (or their higher-
dimensional counterparts), e.g., by anchoring to opposite boundaries of the
medium, as in [33, 34]. Stable knotted structures may also exist [36], but
this investigation falls outside our present scope. Conversely, if
$\gamma_{1}<0$, classical filaments are known to destabilize, which is named
as a possible pathway from ventricular tachycardia in the heart to
fibrillation [37, 20]. Based on Eqs. (9),(10), this phenomenon can also occur
in higher-dimensional media.

a) |  c)
---|---
b)
Figure 2: Superscrolls rotating around superfilaments in 4 spatial dimensions.
Projections of superfilament points are shown centrally; the back left, back
right and bottom of the medium are colored according to the $u$-value at those
planes. (a) A superscroll rotating around a 2D curved superfilament
($n=61,h=1\,$mm). Four visualizations of the same timeframe are shown, each
frame with a different dimension rendered as color of the superfilament. (b) A
pierced superfilament ($n=31,h=0.5\,$mm), shown with projections as in (a).
See Supplemental Material [38] for interactive figures and movies of these
simulations. (c) Projection of superfilament points on the Y, Z, and A-axes
that spans a minimal surface between obstacles (shaded in grey) at opposite
domain boundaries ($n=50,h=0.5\,$mm).

Numerical results. To substantiate the analytical results, the reaction-
diffusion system (1) was integrated numerically using the Forward Euler
method, with an $N$-dimensional hypercube as the domain:
$\vec{r}\in{[0,L]}^{N}$ and no-flux boundary conditions at the edges. In each
dimension, space was discretized using $n$ points along each dimension. This
discretization results in a lattice with spacing $h=\frac{L}{n}$, chosen small
enough such that no discretization artifacts could be observed in the state
variable fields. For the local excitation dynamics $\mathbf{F}(\mathbf{u})$,
we chose the FitzHugh-Nagumo (FHN) kinetics [39]: $\mathbf{u}={[u,v]}^{T}$,
$\mathbf{F}={[f(u,v),g(u,v)]}^{T}$ with

$\displaystyle f$
$\displaystyle=\epsilon^{-1}\left(u+\frac{u^{3}}{3}-v\right),$ $\displaystyle
g$ $\displaystyle=\epsilon\left(u-av+b\right)$ (11)

and $\mathbf{P}=\operatorname{diag}(1,0)$, meaning that only $u$ is diffused.
We used $\epsilon=0.3$, $a=0.5$, and $b=0.68$, which corresponds to the
$\gamma_{1}>0$ regime in 3D. The FHN model was originally formulated for
neural conduction, but has been applied equally to continuous models of
cardiac excitation [40]. As such, it is well suited to explore the boundary
between discrete and continuous excitation models.

We here report numerical results for $N=4$ only. In a medium of $61\times
61\times 61\times 61$ voxels of size $h=1\,$mm, a traveling wave was initiated
by setting $u=1$ inside a ball around the point
$(-$20\text{\,}\mathrm{mm}$,-$20\text{\,}\mathrm{mm}$,$31\text{\,}\mathrm{mm}$,$45\text{\,}\mathrm{mm}$)$
of radius $40\text{\,}\mathrm{mm}$. After $15\text{\,}\mathrm{mm}$, $u=1$ was
set for $y\leq$10\text{\,}\mathrm{mm}$$. This protocol resulted in a curved
planar superfilament, to which we applied filament tracking in each pair of
coordinate planes [41] with threshold values $u=0$, $v=0$. We thus created the
first superscroll rotating around a superfilament surface, see Fig. 2a. We
show only 3D projections here. Other snapshots and movies are given in the
Supplemental Material [38].

Secondly, we created a non-trivial state with a pierced superfilament, see
panel 2b. We started from a stable superscroll with planar superfilament in
${[$0\text{\,}\mathrm{mm}$,$61\text{\,}\mathrm{mm}$]}^{4}$, limited the domain
to ${[$0\text{\,}\mathrm{mm}$,$15\text{\,}\mathrm{mm}$]}^{4}$, doubled spatial
resolution to $h=$0.5\text{\,}\mathrm{mm}$$ by linear interpolation, and set
$u=0$ in the hypercyilinder ${(z-8)}^{2}+{(a-8)}^{2}<25$. Remarkably, the
shrinking of the hole in the filament does not occur in the plane of the
superfilament, but according to Eq. (9) at an angle to it, since the
pseudoscalar filament tension component $\gamma_{2}\neq 0$ for this system,
reminiscent of the oblique drift of classical filaments under an external
perturbation [19, 42].

In a third simulation, we created a superfilament in 4D that is anchored to 2
obstacles at opposite domain boundaries, which shows indeed that a minimal
surface spanned between the obstacles is a stationary configuration. See
detailed set-up in the Supplemental Material [38].

Discussion. We have shown via both analytical calculations and simulations
that adding a fourth spatial dimension to a reaction-diffusion systems yields
new structures: a superscroll and a superfilament. It is expected that even
more complex structures will arise when the number of dimensions $N$, or the
co-dimension $C$ of the structure is further increased. Our motivation for
this research is the link between Laplacian stencils used for simulations
according to the Euler method (see Fig. 1) and regular networks [22]. When the
classical scroll waves and filament were introduced in cardiology, they
enabled a better understanding of the emergent patterns seen at the outer
surface of the heart. Also, the distance-weighed averaging of the excitation
patterns is reflected in the electrical recordings near the heart or on the
body surface, and the properties of the scroll wave are now known to affect
the frequencies present in the electrograms [43].

With this Letter, we aim to open up research on scrolls in more dimensions,
here called superscrolls, in relation to non-linear excitation dynamics in
networks. Several additional facts will have to be accounted for. First, we
assume here a regular, translationally invariant network. In realistic
networks, this condition is typically not met. However, locally there will be
closed loops along which the excitation dynamics can travel and re-excite; we
leave open the possibility that its description may require also a non-integer
number of dimensions [44]. Secondly, the properties of a network typically
vary between the excitable elements. Heterogeneity of excitable nodes can be
dealt with as a gradient as in cardiac modeling, whereas the local variations
in nodal degree (i.e., number of connections) may require a probabilistic
approach. Thirdly, in our derivation of curvature-driven dynamics, a linear
coupling between elements was assumed in Eq. (2). Such effective coupling is
also traditional in cardiac tissue modeling, although it is known that signal
conduction consists of a traveling wave along the cell membrane interleaved
with conduction over gap junctions.

Finally, we suggest to search for superscrolls in real-life systems, as they
may offer geometric insight in yet unknown mechanisms. In this search, the
notion of excitability can be taken very broadly, as any arrangement of
elements that responds non-linearly to stimuli. Examples in human society may
include spreading of opinions or epidemics. In technology, contamination,
combustions, (temporary) failure and overflow may provide examples. Also
oscillatory media may host superscrolls, but the distinction with phase waves
needs to be made.

Conclusion. In this work, we have introduced a mathematical generalization of
scroll waves and filaments to $N$-dimensional Euclidean space. Superfilaments
were shown to obey curvature-driven dynamics and an area-minimizing principle
in case of positive tension. Although the world around us is only three-
dimensional, we conjecture that superscrolls can occur in media where
oscillatory or excitable elements are connected on average to more than $2N$
neighbors, i.e., in networks with non-local connections, such as neural
tissues, and social, biochemical or technological networks.

###### Acknowledgements.

A. V. Panfilov is grateful to A.S. Mikhailov for helpful discussions. M. Cloet
was funded by KU Leuven grant STG/19/007 L. Arno was funded by a FWO-Flanders
fellowship, grant 117702N. D. Kabus is supported by KU Leuven grant
GPUL/20/012. H. Dierckx and computational infrastructure were funded by KU
Leuven grant STG/19/007. J. Van der Veken is supported by the Research
Foundation–Flanders (FWO) and the National Natural Science Foundation of China
(NSFC) under collaboration project G0F2319N, by the KU Leuven Research Fund
under project 3E210539 and by the Research Foundation- Flanders (FWO) and the
Fonds de la Recherche Scientifique (FNRS) under EOS Projects G0H4518N and
G0I2222N. A. V. Panfilov’s research at Sechenov University was financed by the
Ministry of Science and Higher Education of the Russian Federation within the
framework of state support for the creation and development of World-Class
Research Centers “Digital biodesign and personalized healthcare” (Grant No
075-15-2022-304).

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gbsn

# A table-top high-sensitivity gyroscope based on slow light and cavity
enhanced photon drag

Min She  College of Engineering and Applied Sciences, National Laboratory of
Solid State Microstructures, Nanjing University, Nanjing 210023, China
Jiangshan Tang  College of Engineering and Applied Sciences, National
Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210023,
China  Keyu Xia<EMAIL_ADDRESS>College of Engineering and Applied
Sciences, National Laboratory of Solid State Microstructures, Nanjing
University, Nanjing 210023, China

###### Abstract

A high-sensitivity gyroscope is vital for both investigation of the
fundamental physics and monitor of the subtle variation of Earth’s behaviors.
However, it is challenge to realize a portable gyroscope with sensitivity
approaching a small fraction of the Earth’s rotation rate. Here, we
theoretically propose a method for implementing a table-top gyroscope with
remarkably high sensitivity based on photon drag in a rotating dielectric
object. By inserting an $\text{Er}^{3+}$-doped glass rod in a Fabry-Pérot
optical cavity with only $20\leavevmode\nobreak\ \mathrm{c}\mathrm{m}$ length,
we theoretically show that the giant group refractive index and the narrowing
cavity linewidth due to slow light can essentially increase the nonreciprocal
phase shift due to the photon drag to achieve a rotation sensitivity of
$26\leavevmode\nobreak\ \mathrm{f}\rad/\mathrm{s}/\sqrt{\mathrm{Hz}}$. This
work paves the way to accurately detect tiny variations of the Earth’s
rotation rate and orientation, and even can test the geodetic and frame-
dragging effects predicted by the general relativity with a small-volume
equipment.

##  I Introduction

The sensitive gyroscope measuring rotation of an object or the frame can be
used to test fundamental physical effects Cerdonio  _et al._ (2022); Di
Virgilio  _et al._ (2020) and also promise important applications such as
directional positioning Prikhodko  _et al._ (2013); Xu  _et al._ (2023),
inertial navigation systems Zhang  _et al._ (2019); Ban  _et al._ (2022), and
sensing length-of-day variation of the Earth He  _et al._ (2020); Lai  _et
al._ (2020). Currently, various methods are developed to measure slow rotation
by using microelectromechanical gyroscopes Guo  _et al._ (2022); Weinberg  _et
al._ (2006); Tsukamoto  _et al._ (2018); Torunbalci  _et al._ (2015); Minotti
_et al._ (2018), surface acoustic wave gyroscopes Xu  _et al._ (2015); Mahmoud
_et al._ (2018); Wang  _et al._ (2011); Tian  _et al._ (2022), ring laser
gyroscopes (RLGs) Gebauer  _et al._ (2020); Hokmabadi  _et al._ (2021a); Liu
_et al._ (2021b); Wang  _et al._ (2020a), fiber optic gyroscopes (FOGs) Liu
_et al._ (2023); Yao  _et al._ (2019); Yertutanol  _et al._ (2021), atomic-
optical hybrid gyroscopes Moan  _et al._ (2020); Wu  _et al._ (2020), and
nanophotonic optical gyroscopes Khial  _et al._ (2018). Among them, optical
gyroscopes based on the Sagnac effect are most reliable and show the highest
sensitivity thus far. Its principle involves two counter-propagating waves in
the same closed loop, generating a beat frequency reflecting the rotation rate
Sagnac (2013).

A Sagnac-effect-based gyroscope can reach a high sensitivity but typically
requires a large size. The ROMY RLG with six $12\leavevmode\nobreak\
\mathrm{m}$ arms displays a sensitivity of $\sim 80\leavevmode\nobreak\
\mathrm{p}\rad/\mathrm{s}/\sqrt{\mathrm{Hz}}$ Igel
emphet al. (2021); Gebauer
emphet al. (2020). Thus, it can analyze small ground disturbances caused by
earthquakes and oceans. The GINGERINO with four $3.6\leavevmode\nobreak\
\mathrm{m}$ long sides shows a sensitivity up to the level of
$\mathrm{p}\rad/\mathrm{s}/\sqrt{\mathrm{Hz}}$ Di Virgilio  _et al._ (2023).
The ring gyroscope G using a $16$\- m -circumference ring cavity achieves
$12\leavevmode\nobreak\ \mathrm{p}\rad/\mathrm{s}/\sqrt{\mathrm{Hz}}$ and a
resolution of several milliseconds in detecting disturbances on the Earth’s
surface after $120$ days of continuous measurements Schreiber  _et al._
(2023). These RLGs utilize mirrors with ultra-low optical loss to construct
high finesse optical cavity and use two counter-propagating light beams to
reduce bias drift errors Hurst  _et al._ (2009); Di Virgilio  _et al._ (2022).
Despite a great success in rotation measurement, these RLGs are large in size
thus far, costly and fixed on a basis. In contrast, FOGs shrink the volume by
circling a long optical fiber to many-turn coil and therefore are portable,
but the obtained sensitivity is relatively lower. The state-of-the-art FOG can
only achieve a sensitivity of $9.7\times 10^{-8}\leavevmode\nobreak\
\rad/\mathrm{s}/\sqrt{\mathrm{Hz}}$ Cao  _et al._ (2023), several orders lower
than the RLG counterparts. Because of the long optical path, the application
of FOGs is constraint by the asymmetric excitation in the middle of the fiber
circuit caused by temporal and local perturbation, resulting in the non-
negligible dynamic errors Zhao  _et al._ (2022); Andronova  _et al._ (2007).
Such asymmetric perturbation is a common challenge in the Sagnac-effect-based
gyroscope with a big ring. Precise measurement of daily variations of the
Earth’s rotation rate ($\Omega_{\text{e}}=72.92\leavevmode\nobreak\
\mathrm{\SIUnitSymbolMicro}\rad/\mathrm{s}/\sqrt{\mathrm{Hz}}$) requires a
sensitivity approaching the level of tens
$\mathrm{p}\rad/\mathrm{s}\sqrt{\mathrm{Hz}}$, which has only been achieved by
the large-size RLG Di Virgilio  _et al._ (2020); Gebauer
emphet al. (2020); Di Virgilio  _et al._ (2023); Schreiber  _et al._ (2023).
Therefore, it is highly desirable to develop a portable gyroscope with such
sensitivity but remains a challenge Hand (2017).

Propagation of linearly-polarized light in a spinning medium results in the
change of polarization plan, known as the photon drag effect, because the
rotation of medium causes circular birefringence Player (1976). The slow-light
medium can significantly enhance the photon drag and narrow the linewidth of a
cavity by several orders Franke-Arnold  _et al._ (2011); Sabooni  _et al._
(2013). However, it is thus far yet to explore the interplay of slow light and
the photon drag effect for a sensitive gyroscope.

In this work, we propose a photon-drag-based gyroscope by embedding a slow-
light medium in an $20\leavevmode\nobreak\ \mathrm{c}\mathrm{m}$ long FP
cavity. By measuring the transmitted light power of the cavity, we can, in
principle, achieve the remarkable gyro sensitivity. Our proposal provides a
table-top device for measuring daily variations in the Earth’s rotation rate
and testing the effects of Lorentz Violation and General Relativity on earth.

The paper is organized as follows: The system and model are described and
developed in section II. Then, the results are presented in section III. The
feasibility of implementing our system is discussed in the section
“Implementation”. In the end, the paper is concluded in section V.

##  II System and model

Figure 1: (a) Schematic of gyroscope based on the slow-light enhanced photon
drag in a high-Q FP cavity embedded with an $\text{Er}^{3+}$-doped glass rod
spinning at a rate $\Omega$. A HP light is input to the cavity. The VP
transmitted laser field is detected by a photodetector (PD). (b) Group
refractive index versus the input photon flux under different mirror
reflectivity when $\delta=10\leavevmode\nobreak\ \mathrm{Hz}$. (c) Group
refractive index with input photon flux at various frequency detuning when the
mirror reflectivity $R=0.998$ and $\kappa=5$ M Hz . Other parameters:
$L=10\leavevmode\nobreak\ \mathrm{c}\mathrm{m}$ and
$L_{0}=20\leavevmode\nobreak\ \mathrm{c}\mathrm{m}$, yielding $\eta=0.5$.

The system of our gyroscope is depicted in Fig. 1(a). An
$\text{Er}^{3+}$-doped glass rod is inserted into the FP cavity form by two
highly reflective mirrors $\text{M}_{1}$ and $\text{M}_{2}$. By detecting the
field component orthogonal in polarization to the input field, this system can
work as a gyroscope. We assume that the two mirrors have the same reflectivity
$R$ and cause a decay rate $\kappa_{\text{e}}$ to the cavity each. We also
assume that the cavity has an intrinsic loss with rate $\kappa_{\text{i}}$.
The overall decay rate of the cavity is then given by
$\kappa=2\kappa_{\text{e}}+\kappa_{\text{i}}$. The two ends of the rod is
coated with anti-reflective film. A horizontally-polarized (HP) probe laser
beam, denoted as $\hat{a}_{\text{in}}$, is input to the cavity via the
$\text{M}_{1}$ mirror. $\hat{a}_{\text{out}}$ denotes the transmitted field.
When the frame or the glass rod spins at a rate $\Omega$, the photon drag
causes difference $\Delta n$ between the refractive indices $n_{\text{l}}$ and
$n_{\text{r}}$ “seen” by the left-hand circularly polarized (LCP) and the
right-hand circularly polarized (RCP) cavity eigenmodes, respectively. Because
of the unidirectional mechanical angular momentum, this difference $\Delta n$
leads to nonreciprocal phase shift and lifts the resonance frequencies of the
LCP and RCP eigenmodes to non-degenerate. As a result, the LCP and RCP
components of the input field accumulate a non-zero relative phase after
transmitting through the cavity. A vertically-polarized (VP) component
$\hat{a}^{V}_{\text{o}}$ appears in the transmitted field, denoted as
$\hat{a}_{\text{out}}$. It is then separated by a polarization beam splitter
(PBS). By detecting its power with a photodetector, we can determine the
spinning rate $\Omega$. Besides, quantum noise ($\hat{e}_{1}$ and
$\hat{e}_{2}$) also enters the photodetector and causes error.

We consider the slowly rotating frame, e.g. Earth’s rotation. Then, the photon
drag is very weak. To amplify the signal, another control laser beam
$\hat{b}_{\text{in}}$ is used to increase the group velocity index of the
probe field to $n_{\text{g}}\gg 1$ in the glass rod via the coherent
population oscillation (CPO) process of the $\text{Er}^{3+}$ ions. The
enhancement resulting from slow light is twofolds: enhancing the photon drag
by several orders Franke-Arnold  _et al._ (2011) and significantly narrowing
the cavity linewidth Sabooni  _et al._ (2013). The combination of both allows
us to make a table-top gyroscope with sensitivity exceeding the large RLGs.
Here, we neglected the influence of the photon drag due to the spin of the
mirrors and the PBS, because it is much smaller than the slow-light enhanced
photon drag in the $\text{Er}^{3+}$-doped glass rod.

Now we discuss how to enhance the gyroscope sensitivity by using slow light in
a FP cavity. In the absence of rotation, the control and probe laser beams act
on the $\text{Er}^{3+}$-doped glass rod together. At this time,
$\text{Er}^{3+}$ ions can be approximated as a two-level system with
transition ${}^{4}I_{15/2}\to{}^{4}I_{13/2}$ at $\lambda\approx
1536\leavevmode\nobreak\ \mathrm{n}\mathrm{m}$ Arrieta-Yáñez _et al._ (2011);
Melle  _et al._ (2007). The transition frequency is $\omega$. $T_{1}$ is the
relaxation time of the excited state. The burnt hole can appear in the probe
spectrum via the CPO process. It can cause a giant group refractive index
$n_{\text{g}}$ given by Bigelow  _et al._ (2003); Boyd  _et al._ (1981) (see
the Appendix A).

$n_{\text{g}}=n_{0}+\frac{\alpha_{0}cT_{1}}{2}\frac{I_{0}}{1+I_{0}}\left[\frac{1}{\left(T_{1}\delta\right)^{2}+\left(1+I_{0}\right)^{2}}\right],$
(1)

where $n_{0}$ is the refractive index of the host medium, $\alpha_{0}$ is the
unsaturated absorption coefficient,
$\delta=\omega_{\text{p}}-\omega_{\text{s}}$ with $\omega_{\text{p}}$ and
$\omega_{\text{s}}$ being the frequencies of the probe and control fields,
respectively. $I_{\text{sat}}$ is the saturated absorption intensity of the
ions. $I_{\text{p}}$ and $I_{\text{s}}$ denote the intensity of the probe and
control field, respectively. $n_{\text{g}}$ is crucially dependent on the
ratio $I_{0}=I_{\text{s}}/I_{\text{sat}}$. The saturated absorption intensity
of the ions is given by

$I_{\text{sat}}=\hbar\omega/[T_{1}(\sigma_{\text{12}}+\sigma_{\text{21}})]\;.$
(2)

The operators $\sigma_{12}$ and $\sigma_{21}$ are the absorption cross-section
and emission cross-section, respectively, $\sigma_{12}$ $\approx$
$\sigma_{21}$. The intensities of the control fields in the cavity is given by

$I_{\text{s}}=c\varepsilon_{0}E_{0}^{2}|b_{\text{in}}|^{2}\kappa_{e}/\kappa^{2}\;,$
(3)

where $E_{0}=\sqrt{\hbar\omega_{\text{s}}/\varepsilon_{0}V}$,
$\varepsilon_{0}$ is the dielectric constant, $V$ is the mode volume. The
input powers of the control and probe fields are
$P_{\text{s,in}}=\hbar\omega_{\text{s}}|b_{\text{in}}|^{2}$ and $P_{\text{p,
in}}=\hbar\omega_{\text{p}}|a_{\text{in}}|^{2}$, respectively. Throughout the
following investigation, we fix the power ratio to $P_{\text{p,
in}}/P_{\text{s, in}}=I_{\text{p}}/I_{\text{s}}=0.08$, which is accessible in
experiment Schweinsberg
emphet al. (2006). In experiment, one can modulate the input laser beam to
form the control and probe fields. The power or intensity ratio between the
control beam and the probe beam can be tuned via modulation. In our system,
the modulation frequency is very small such that
$\omega_{\text{p}}\approx\omega_{\text{s}}$ to guarantee a giant group
refractive index $n_{\text{g}}$. For an $\text{Er}^{3+}$-doped glass rob, we
have $n_{0}=1.48$, $T_{1}=10.5\leavevmode\nobreak\ \mathrm{m}\mathrm{s}$ and
$\alpha_{0}=0.16\leavevmode\nobreak\ \mathrm{c}\mathrm{m}^{-1}$ Melle  _et
al._ (2007).

Figure 1(b) reflects the group refractive index $n_{\text{g}}$ as a function
of the power of the control laser field for various mirror reflectivity. The
peak values of $n_{\text{g}}$ are very close for different reflectivities but
the required control laser power, $P_{\text{s, in}}$, increases as the
reflectivity reduces. When $R=0.99$, the peaked $n_{\text{g}}\approx
3.7\leavevmode\nobreak\ \times 10^{6}$ is attained at $P_{\text{s,
in}}=440\leavevmode\nobreak\ \mathrm{m}\mathrm{W}$. When the reflectivity is
improved to $R=0.999$, which is much harder, the peaked $n_{\text{g}}$ is
about $3.7\leavevmode\nobreak\ \times 10^{6}$ and the optimal power
$P_{\text{s, in}}$ decreases to $\sim 272\leavevmode\nobreak\
\mathrm{m}\mathrm{W}$, corresponding to $2\times 10^{18}$. Thus, it is
experimentally convenient to choose the reflectivity between $0.99-0.999$.
Fig. 1(c) shows the dependence of $n_{\text{g}}$ on the input photon flux of
the control light for various frequency difference $\delta$. The peak
$n_{\text{g}}$ first increases rapidly and then decreases after reaching the
maximum value as the control laser power increases. The peak value also
decreases as the modulation frequency $\delta$ increasing. From Figs. 1(b) and
(c), the group refractive index can reach to the level of $10^{6}$.

We learn from Ref. Sabooni  _et al._ (2013) that slow light can narrow the
linewidth of the cavity. The total loss can be approximated as
$\kappa^{\prime}\approx\kappa/n_{\text{g}}$ and
$\kappa_{\text{e}}^{\prime}\approx\kappa_{\text{e}}/n_{\text{g}}$. According
to the input-output relationship of the optical cavity Gardiner
emphet al. (1985); Xia
emphet al. (2013), the probe filed output operator can be expressed as

$\hat{a}_{\text{out}}=\frac{2\hat{a}_{\text{in}}\kappa_{\text{e}}}{i\Delta/n_{\text{g}}+\kappa}\;,$
(4)

where $\Delta=\omega_{\text{c}}-\omega_{\text{p}}$, $\omega_{\text{c}}$ is the
cavity frequency. In the case of rotation, the rotation of the polarization
state is caused by the refractive index difference between LCP and RCP light
Player (1976). The refractive index difference between LCP and RCP light is
given as

$\Delta
n=n_{\text{r}}-n_{\text{l}}=\left(n_{\text{g}}-\frac{1}{n_{\phi}}\right)\frac{\Omega}{\omega_{\text{c}}},$
(5)

where $n_{\phi}$ is the phase refractive index, $\Omega$ is the medium
rotation rate, $n_{\text{r}}$ and $n_{\text{l}}$ are the refractive indices
for RCP and LCP light, respectively. Based on the resonance condition of the
cavity, the phase change can be expressed as

$\Delta\omega_{\pm}=\mp\left(n_{g}-\frac{1}{n_{\phi}}\right)\Omega\eta,$ (6)

where $\eta=L/L_{0}$ represents the duty ratio, $L_{0}$ and $L$ are the cavity
length and medium length, respectively. At this point, the probe filed output
operator can be approximated as

$\hat{a}_{\text{out}}^{\pm}=\xi\hat{a}_{\text{in}}e^{-i\Delta\omega_{\pm}n_{\text{g}}/\kappa}\;,$
(7)

where $\xi=2\kappa_{\text{e}}/\kappa$. Subsequently, the output light from the
FP cavity is split into the HP and VP components after passing through the
PBS. Using the relationship between circularly and linearly polarized light,
we obtain the output of VP component is

$\hat{a}_{\text{o}}^{V}=-\xi\hat{a}_{\text{in}}\sin\left(\Delta\phi\right)+\hat{e}_{\text{in}},$
(8)

where
$\Delta\phi=\left(\Delta\omega_{+}-\Delta\omega_{-}\right)/2\kappa^{\prime}$,
$\hat{e}_{\text{in}}=\left(\hat{e}_{\text{in}}^{1}+\hat{e}_{\text{in}}^{2}\right)/\sqrt{2}$
is the quantum fluctuation. We define the HP component output photon number
operator as

$\hat{M_{\text{o}}}=\xi^{2}\sin^{2}(\Delta\phi)\hat{a}_{\text{in}}^{\dagger}\hat{a}_{\text{in}}+\hat{e}_{\text{in}}^{\dagger}\hat{e}_{\text{in}}-\xi\sin(\Delta\phi)(\hat{a}_{\text{in}}^{\dagger}\hat{e}_{\text{in}}+\hat{e}_{\text{in}}^{\dagger}\hat{a}_{\text{in}})\;.$
(9)

Based on the error propagation analysis Wu  _et al._ (2020); Chen  _et al._
(2016), the rotation sensitivity is defined as

$\Delta\Omega={\langle\Delta
M_{\text{o}}^{2}\rangle^{1/2}}/{|\partial\langle\hat{M_{\text{o}}}\rangle/\partial\Omega|}\;,$
(10)

where

$\displaystyle\langle\Delta M_{\text{o}}^{2}\rangle$
$\displaystyle=\langle\hat{M_{\text{o}}^{2}}\rangle-\langle\hat{M_{\text{o}}}\rangle^{2}\approx\xi^{2}\sin^{2}(\delta\phi)N_{\text{in}}\;,$
(11a)
$\displaystyle\left|{\partial\langle\hat{M_{\text{o}}}\rangle}/{\partial\Omega}\right|$
$\displaystyle=2\xi^{2}|\sin(\Delta\phi)\cos(\Delta\phi)|n_{\text{g}}^{2}N_{\text{in}}\eta/{\kappa},$
(11b)

with $N_{\text{in}}=|a_{\text{in}}|^{2}$ being the input probe photon flux.
According to the above formula, our system sensitivity $\Delta\Omega$ is given
by

$\Delta\Omega=\frac{\kappa}{2\xi\eta n_{\text{g}}^{2}\sqrt{N_{\text{in}}}}.$
(12)

The gyro sensitivity relies on the input photons flux $(N_{\text{in}})$, the
group refractive index $(n_{g})$, the duty ratio $(\eta)$, and the decay rate
$(\kappa)$. In order to obtain optimal sensitivity, the optimal operation is
discussed next.

Figure 2: (a) Photon flux (color bar) of the VP transmitted light as a
function of the rotation rate of the rod and the input photon flux. (b)
Sensitivity of the gyroscope as a function of the input photon flux. Other
parameters as in Fig. 1 except for $R=0.998$.

##  III Results

The performance of our gyro crucially relies on the power of input probe
light. Figure 2 shows the photon flux of the output VP component and the
gyroscope sensitivity. According to the photon drag effect, the deflection
phase caused by a rotating medium is proportional to the group refractive
index $n_{\text{g}}$ Leach  _et al._ (2008). Thus, the giant group refractive
index induced in the $\text{Er}^{3+}$-doped glass rod can essentially enhance
the polarization rotation of light due to the photon drag Franke-Arnold  _et
al._ (2011), greatly amplifying the output power to a detectable level. Figure
2(a) shows the output photon flux as a function of the probe power and the
rotation rate. Taking the experimentally accessible values for parameters
$R=0.998$, $L=10\leavevmode\nobreak\ \mathrm{c}\mathrm{m}$,
$L_{0}=20\leavevmode\nobreak\ \mathrm{c}\mathrm{m}$, $\eta=0.5$, the maximum
output photon flux increases with the rotation rate. It reaches
$P_{\text{out}}/\hbar\omega_{\text{p}}\approx 10^{11}\leavevmode\nobreak\
\text{photon/s}\leavevmode\nobreak\ (\approx 17\leavevmode\nobreak\
\text{nW})$ when $\Omega/\Omega_{\text{e}}\approx 10^{-9}$. This level of
photon flux can be well detected experimentally. The peak output photon flux
is attainable for $\Omega/\Omega_{\text{e}}<1.5\times 10^{-9}$. The required
input power less than $P_{\text{p,in}}/\hbar\omega_{\text{p}}<4\times
10^{17}\text{photons}\mathrm{/}\mathrm{s}$, corresponding to an probe field
power $<50\leavevmode\nobreak\ \mathrm{m}\mathrm{W}$. The performance of our
gyroscope is determined by the duty ratio $\eta$ but not the cavity length
$L_{0}$. Throughout the following investigation, we fix
$L_{0}=20\leavevmode\nobreak\ \mathrm{c}\mathrm{m}$.

According to Eq. (12), it can be seen that the gyro sensitivity is
proportional to $\kappa$ and inversely proportional to $\eta$,
$n_{\text{g}}^{2}$, and $\sqrt{N_{\text{in}}}$. Therefore, the sensitivity
tends to be improved as the input power increases, as shown in Fig. 2(b), but
reaches the optimal value of $\Delta\Omega_{\text{min}}=26\leavevmode\nobreak\
\mathrm{f}\rad/\mathrm{s}/\sqrt{\mathrm{Hz}}$ when the input photon flux
$P_{\text{p,in}}/\hbar\omega_{\text{p}}\approx 2\times
10^{17}\leavevmode\nobreak\ \text{photons\mathrm{/}\mathrm{ s
}}\leavevmode\nobreak\ (\approx 27\leavevmode\nobreak\ \mathrm{m}\mathrm{W})$,
and then reduces. The reason is that $n_{\text{g}}$ first increases rapidly
and then decreases after reaching the maximum value as the input power
increases, see Figs. 1(b) and (c). It implies that, to obtain the best
sensitivity, the gyro needs to operate at the optimal input power.

Figure 3: (a) Scaled sensitivity (color bar, in unit of
$\mathrm{f}\mathrm{rad}\mathrm{/}\mathrm{s}\sqrt{\mathrm{Hz}}$) as a function
of the mirror reflectivity and the probe photon flux. (b) Highest sensitivity
obtained at different mirror reflectivity when changing the probe light power.
Other parameters: $L=10\leavevmode\nobreak\ \mathrm{c}\mathrm{m}$,
$L_{0}=20\leavevmode\nobreak\ \mathrm{c}\mathrm{m}$, $\eta=0.5$, and
$\kappa=5$ M Hz .

According to Fig. 1(b), we are interested in $R=0.99-0.999$, which allows us
to achieve the maximum $n_{\text{g}}$ with low control laser power. Figure
3(a) shows the gyro sensitivity versus the mirror reflectivity $R$ and the
probe power $P_{\text{p, in}}$. The gyro is most sensitive when $R\approx
0.998$ and $P_{\text{p, in}}/\hbar\omega_{\text{p}}\approx 2\times
10^{17}\leavevmode\nobreak\ \text{photons\mathrm{/}\mathrm{ s }}$, see Fig. 3.
Figure 3(b) clearly shows the obtained optimal sensitivity of
$\Delta\Omega_{\text{min}}\approx 26\leavevmode\nobreak\
\mathrm{f}\mathrm{rad}\mathrm{/}\mathrm{s}\mathrm{/}\sqrt{\mathrm{Hz}}$ at
$R\approx 0.998$. A higher reflectivity of the cavity mirrors not only leads
to a lower cavity loss Tian  _et al._ (2019), but also significantly affects
$n_{\text{g}}$, as shown in Fig. 1(b). As a tradeoff, The mirror reflectivity
is optimal at $R\approx 0.998$. Note that the slow-light enhances
$n_{\text{g}}$ and thus the photon drag effect by a factor of $10^{6}$ Hand
(2017), and significantly narrows the intracavity linewidth Sabooni  _et al._
(2013). Thanks to this doublet improvement, the gyro sensitivity can reach 9
orders of magnitude higher than Earth’s rotation rate, enabling precise
detection of tiny variations of the Earth’s rotation and its wobbling
Schreiber  _et al._ (2023).

Figure 4 presents the dependence of the gyro sensitivity on the medium length
$L$. For a given $L$, the gyro sensitivity first improves as the probe photon
flux increases. After reaching an optimal point, its value increases. As
predicted by Eq. 12, the sensitivity is inversely proportional to the duty
ratio $\eta$. An example for $P_{\text{p, in}}/\hbar\omega_{\text{p}}=2\times
10^{17}\leavevmode\nobreak\ \text{photons\mathrm{/}\mathrm{ s }}$ is presented
in Fig. 4(b). Obviously, a larger duty ratio yields is preferable for
achieving high performance. However, considering practical experimental losses
and operational, we choose a spinning medium length of
$L=10\leavevmode\nobreak\ \mathrm{c}\mathrm{m}$. On this basis, the gyro
sensitivity can reach $26\leavevmode\nobreak\
\mathrm{f}\rad/\mathrm{s}/\sqrt{\mathrm{Hz}}$. This level of sensitivity is
accurate enough to detect small variations in the Earth’s rotation rate, the
effects of Lorentz Violation and Ggeneral Relativity in gravity research.

##  IV Implementation

In this work, we use a FP cavity embedded with a spinning medium to construct
a sensitive gyro. The medium is an $\text{Er}^{3+}$-doped glass rob and slow
light can be obtained with a laser beam. This laser field at wavelength $\sim
1536\leavevmode\nobreak\ \mathrm{n}\mathrm{m}$ is modulated in intensity with
frequency $\delta$, $I=I_{\text{s}}+I_{\text{p}}\cos(\delta t)$. We fix
$I_{\text{p}}/I_{\text{s}}=0.08$, which is experimentally accessible
Schweinsberg
emphet al. (2006). To measure the modulated sideband light, the output VP
component is measured via the homodyne detection. Specifically, the output VP
light is first interfered with a local light field to generate an electric
signal oscillating at frequency $\delta$. Then, this oscillating signal is
filtered by an electronic circuit and measured.

Figure 4: (a) Sensitivity (color bar, in unit of
$\mathrm{f}\mathrm{rad}\mathrm{/}\mathrm{s}\sqrt{\mathrm{Hz}}$) as a function
of the medium length and the probe photon flux. (b) Optimal sensitivity
obtained at different medium length by tuning the probe power. Other
parameters: $R=0.998$ and $\kappa=5$ M Hz .

##  V Conclusion

Vibration of the Earth’s rotation rate are currently measured mainly by using
a large ring-laser gyroscope or fiber optic gyroscopes based on the Sagnac
effect. However, these gyroscopes are large, costly, and subject to errors
induced by asymmetric perturbation and common-mode noise. Here, we
theoretically propose to implement a table-top gyroscope with unprecedented
high sensitivity up to $26\leavevmode\nobreak\
\mathrm{f}\rad/\mathrm{s}/\sqrt{\mathrm{Hz}}$ by exploring the nonreciprocity
of photon drag in a rotating medium. The proposed gyroscope is small in volume
and thus cost-effective. The two circularly-polarized components of laser beam
co-propagate in the system, therefore eliminating the effects of the
asymmetric perturbation and common-mode noise. This work opens up the door for
portable gyroscope measuring the fluctuation of Earth’s motion.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China
(Grants No. 92365107 and No. 12305020), National Key R&D Program of China
(Grants No. 2019YFA0308700), the Program for Innovative Talents and Teams in
Jiangsu (Grant No. JSSCTD202138), the China Postdoctoral Science Foundation
(Grant No. 2023M731613), and the Jiangsu Funding Program for Excellent
Postdoctoral Talent (Grant No. 2023ZB708). We thank the High Performance
Computing Center of Nanjing University for allowing the numerical calculations
on its blade cluster system.

## APPENDIX A: COHERENT POPULATION OSCILLATORY PROCESSES IN THE
$\text{Er}^{3+}$-DOPED GLASS ROD

Based on the CPO in a two-level system Bigelow  _et al._ (2003); Boyd  _et
al._ (1981), we can derive the density matrix equation for the
$\text{Er}^{3+}$-doped glass rod. $\text{Er}^{3+}$ can be excited by a laser
beam with wavelength $\lambda\approx 1536\leavevmode\nobreak\
\mathrm{n}\mathrm{m}$ from the ground state ${}^{4}I_{15/2}$ to the substable
state ${}^{4}I_{13/2}$ as a two-level system Arrieta-Yáñez _et al._ (2011);
Melle  _et al._ (2007). Due to the long lifetime of metastable level
${}^{4}I_{13/2}$ of $\text{Er}^{3+}$, the system exhibits a capacity for rapid
nonlinear response within a narrow spectrum. When the control and probe fields
with a small frequency difference are input simultaneously, the medium
exhibits phenomena of saturable absorption.-In this situation, the number of
ground state particles undergoes periodic oscillations, resulting in spectral
hole burning in the absorption spectrum. At the burning point, the refractive
index of light rapidly change. According to the Kramers-Kronig relationship,
the group velocity also undergoes drastic changes.

For analyzing the slow light, we denote the ground state level as
$\left|g\right\rangle$ and the metastable level as $\left|e\right\rangle$.
According to the calculation steps in Ref. Boyd  _et al._ (1981) , we can
write the motion equation of system in the density matrix as

$\displaystyle\dot{\rho}_{\text{eg}}=$
$\displaystyle-\left(i\omega_{\text{eg}}+\frac{1}{T_{2}}\right)\rho_{\text{eg}}+\frac{i}{\hbar}V_{\text{eg}}\left(\rho_{\text{ee}}-\rho_{\text{gg}}\right),$
(1a)
$\displaystyle\begin{split}\dot{\rho}_{\text{ee}}-\dot{\rho}_{\text{gg}}=&\left(-\frac{1}{T_{1}}\right)\left[\left(\rho_{\text{ee}}-\rho_{\text{gg}}\right)-\left(\rho_{\text{ee}}-\rho_{\text{gg}}\right)^{0}\right]+\frac{2i}{\hbar}\\\
&\left(V_{\text{ge}}\rho_{\text{eg}}-V_{\text{eg}}\rho_{\text{ge}}\right),\end{split}$
(1b)

where $T_{1}$ is the relaxation time of the excited state, and $T_{2}$ is the
decoherence time of off-diagonal elements in the density matrix;
$\left(\rho_{\text{ee}}-\rho_{\text{gg}}\right)^{0}$ is the inversion of
particle number in thermal equilibrium state.
$V_{\text{eg}}=V_{\text{ge}}^{*}=-\mu_{\text{eg}}\left(E_{\text{s}}e^{-i\omega_{\text{s}}t}+E_{\text{p}}e^{-i\omega_{\text{p}}t}\right)$
is the interaction Hamiltonian, where $E_{\text{s}}$ and $E_{\text{p}}$ are
the amplitudes of the control and probe light, $\omega_{\text{s}}$ and
$\omega_{\text{p}}$ are the frequencies of the control light and probe light,
respectively, $\mu_{\text{ba}}$ is the dipole matrix element. When $1/T_{2}$
is larger than both the detuning frequency
$\left(\delta=\omega_{\text{p}}-\omega_{\text{s}}\right)$ and the Rabi
frequency of the control laser
$\left(\Omega_{\text{s}}=2|\mu_{\text{eg}}||E_{\text{s}}|/\hbar\right)$, the
diagonal elements of the density matrix can be expressed as

$\rho_{\text{eg}}\approx-\frac{i\mu_{\text{eg}}}{\hbar}\frac{\left(E_{\text{s}}e^{-i\omega_{\text{s}}t}+E_{\text{p}}e^{-i\omega_{\text{p}}t}\right)}{i\left(\omega_{\text{eg}}-\omega_{\text{s}}\right)+1/T_{2}}\left[\rho_{\text{ee}}\left(t\right)-\rho_{\text{gg}}\left(t\right)\right]\;.$
(2)

Following the Ref. Boyd  _et al._ (1981), the overall particle number
inversion is given by

$\rho_{\text{ee}}-\rho_{\text{gg}}\approx\left(\rho_{\text{ee}}-\rho_{\text{gg}}\right)^{dc}+\left(\rho_{\text{ee}}-\rho_{\text{gg}}\right)^{\delta}e^{i\delta
t}+\left(\rho_{\text{ee}}-\rho_{\text{gg}}\right)^{-\delta}e^{-i\delta t},$
(3)

where $\left(\rho_{\text{ee}}-\rho_{\text{gg}}\right)^{dc}$ is the direct
current component of the overall inversion,
$\left(\rho_{\text{ee}}-\rho_{\text{gg}}\right)^{\pm\delta}$ are the amplitude
components oscillating at $\delta$. When $\delta\leq 1/T_{1}$ , particle
number oscillations are evident. We consider the resonance case:
$\omega_{\text{eg}}=\omega_{\text{s}}$. The response at the probe light
frequency can be expressed as:

$\begin{split}\rho_{\text{eg}}\left(\omega_{p}\right)=\frac{i\mu_{\text{eg}}T_{2}E_{\text{p}}}{\hbar}\times\left[\frac{1}{1+\Omega_{\text{s}}^{2}T_{1}T_{2}}-\Omega_{\text{s}}^{2}\frac{T_{2}}{T_{1}}\frac{1+i\delta/\beta}{\delta^{2}+\beta^{2}}\right]\;,\end{split}$
(4)

where $\beta={1}/{T_{1}}\left(1+\Omega_{\text{s}}^{2}T_{1}T_{2}\right)$. In
the two-level system, the linear susceptibility is given by
$\chi\left(\delta\right)=N\mu_{\text{eg}}\rho_{\text{eg}}\left(\omega_{\text{p}}\right)/E_{\text{p}}$.
The refractive index and absorption at the hole burning location are
respectively given by

$n\left(\delta\right)=1+\frac{\alpha_{0}cT_{1}}{2\omega_{\text{c}}}\frac{I/I_{\text{sat}}}{1+I/I_{\text{sat}}}\left(\frac{\delta}{\left(T_{1}\delta\right)^{2}+\left(1+I/I_{\text{sat}}\right)^{2}}\right),$
(5)
$\alpha\left(\delta\right)=\frac{\alpha_{0}}{1+I/I_{\text{sat}}}\left(1-\frac{I/I_{\text{sat}}\left(1+I/I_{\text{sat}}\right)}{\left(T_{1}\delta^{2}\right)+\left(1+I/I_{\text{sat}}\right)^{2}}\right),$
(6)

wwhere $I$ is the intensity of input power, $I_{\text{sat}}$ is the saturated
absorption intensity, $\alpha_{0}$ being the unsaturated absorption
coefficient. Using the formula
$n_{\text{g}}=n(\delta)+\omega_{\text{c}}dn/d\delta$, the group refractive
index $n_{\text{g}}$ can be calculated as Bigelow  _et al._ (2003)

$n_{\text{g}}=n_{0}+\frac{\alpha_{0}cT_{1}}{2}\frac{I/I_{\text{sat}}}{1+I/I_{\text{sat}}}\left(\frac{1}{\left(T_{1}\delta\right)^{2}+\left(1+I/I_{\text{sat}}\right)^{2}}\right),$
(7)

where $n_{0}$ is the refractive index of the host medium at $\delta=0$ when
only the control light passes through.

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