# Resolving the topology of encircling multiple exceptional points

Chitres Guria  Department of Physics, Yale University, New Haven, Connecticut
06520, USA  Qi Zhong  Department of Physics, Michigan Technological
University, Houghton, Michigan, 49931, USA  Department of Engineering Science
and Mechanics, and Materials Research Institute, The Pennsylvania State
University, University Park, Pennsylvania 16802, USA  Sahin Kaya Ozdemir
Department of Engineering Science and Mechanics, and Materials Research
Institute, The Pennsylvania State University, University Park, Pennsylvania
16802, USA  Yogesh S S Patil  Department of Physics, Yale University, New
Haven, Connecticut 06520, USA  Ramy El-Ganainy<EMAIL_ADDRESS>Department of
Physics, Michigan Technological University, Houghton, Michigan, 49931, USA
Henes Center for Quantum Phenomena, Michigan Technological University,
Houghton, Michigan, 49931, USA  and Jack Gwynne Emmet Harris
<EMAIL_ADDRESS>Department of Physics, Yale University, New Haven,
Connecticut 06520, USA  Department of Applied Physics, Yale University, New
Haven, Connecticut 06520, USA  Yale Quantum Institute, Yale University, New
Haven, Connecticut 06520, USA

###### Abstract

Non-Hermiticity has emerged as a new paradigm for controlling coupled-mode
systems in ways that cannot be achieved with conventional techniques. One
aspect of this control that has received considerable attention recently is
the encircling of exceptional points (EPs). To date, most work has focused on
systems consisting of two modes that are tuned by two control parameters and
have isolated EPs. While these systems exhibit exotic features related to EP
encircling, it has been shown that richer behavior occurs in systems with more
than two modes. Such systems can be tuned by more than two control parameters,
and contain EPs that form a knot-like structure. Control loops that encircle
this structure cause the system’s eigenvalues to trace out non-commutative
braids. Here we consider a hybrid scenario: a three-mode system with just two
control parameters. We describe the relationship between control loops and
their topology in the full and two-dimensional parameter space. We demonstrate
this relationship experimentally using a three-mode mechanical system in which
the control parameters are provided by optomechanical interaction with a high-
finesse optical cavity.

## Introduction

Modeling physical systems as collections of coupled oscillators is an
important strategy in many disciplines, including quantum field theory,
condensed matter, optics, acoustics, soft matter, and biology. Classical
linear coupled-oscillator models (COMs) are described by the system of
equations $\dot{\mathbf{x}}=-iH\mathbf{x}$ where the complex $N$-vector
$\mathbf{x}$ represents the phase space coordinates of $N$ oscillators, and
the $N\times N$ matrix $H$ encodes the oscillators’ properties. One important
feature of any COM is its spectrum of resonance frequencies $\bm{\lambda}$,
which is the (unordered) set of the eigenvalues of $H$. In many applications
it is desirable to tune $\bm{\lambda}$ by varying parameters that appear in
$H$. This tuning may be static, in the sense of setting the parameters in
order to fix specific resonances (for example, when using the COM as a
transducer for external signals with known frequencies). Alternatively, the
tuning may be dynamical in the sense of being carried out in real time to
manipulate excitations within the COM (as in adiabatic control schemes). In
both cases, it is important to understand how $\bm{\lambda}$ depends upon
parameters in $H$. While the details of this dependence will vary from one
system to another, it is known to possess a number of generic features.

One such feature that has attracted considerable attention recently is the
topological structure that emerges when $H$ is non-Hermitian Feng  _et al._
(2017); El-Ganainy  _et al._ (2018); Özdemir  _et al._ (2019); El-Ganainy  _et
al._ (2019); Miri and Alù (2019); Wiersig (2020). This structure is manifested
in the evolution of $\bm{\lambda}$ when the parameters of $H$ are varied
around a loop Berry and O’Dell (1998); Heiss (2000); Dembowski  _et al._
(2001); Keck  _et al._ (2003); Dembowski  _et al._ (2004); Uzdin  _et al._
(2011); Berry and Uzdin (2011). The simplest example occurs in a system of
$N=2$ oscillators with two control parameters ($b_{1}$ and $b_{2}$) chosen so
that the 2D space they span contains a single point at which the eigenvalues
are degenerate (for generic $b_{1}$ and $b_{2}$ this degeneracy will be an
exceptional point (EP) Gilmore (1993); Kato (2012)). If
$\mathbf{b}=(b_{1},b_{2})$ is varied smoothly around a loop (which does not
intersect the EP), the evolution of $\bm{\lambda}$ will result in a
permutation of the two eigenvalues if and only if the loop’s winding number
around the EP is odd.

For $N>2$, a generic 2D control space (which we refer to as $\mathcal{B}$)
will contain isolated two-fold EPs that correspond to degeneracies between
various mode pairs Gilmore (1993); Kato (2012). In such systems the
relationship between a control loop, the EPs, and the resulting eigenvalue
permutation is less intuitive than for $N=2$. In particular, there is no
simple correspondence between a loop’s topology (more precisely, its homotopy
class in $\bar{\mathcal{B}}$, defined as $\mathcal{B}$ with the EPs removed)
and the resulting eigenvalue permutation. For example, loops that are homotopy
equivalent in $\bar{\mathcal{B}}$ all give the same permutation, but homotopy
inequivalent loops do not necessarily give distinct permutations.

As shown in Ref. Zhong  _et al._ (2018), the permutation associated with a
given loop can be calculated by introducing into $\mathcal{B}$ branch cuts
(BCs) for $\bm{\lambda}$ and tracking the manner in which the loop crosses
these BCs, and the resulting eigenvalue permutation is less intuitive than for
$N=2$ Zhong  _et al._ (2018); Pap  _et al._ (2018). This approach has the
advantage of being straightforward to visualize, as it only involves
quantities that are defined in the 2D space $\mathcal{B}$. It is also relevant
to the many systems that in practice offer just two control parameters.
However, relying on the introduction of BCs can obscure the topological
relationship between control loops and EPs.

A different approach is to view the control loop in the space that is spanned
by all the coefficients of $p_{H}$, the characteristic polynomial of $H$ Patil
_et al._ (2022). These coefficients are simple functions of the elements of
$H$, and they provide a smooth parametrization of the eigenvalue spectra
Arnold (1971). If we take $H$ to be traceless (see Methods), then $p_{H}$ has
$N-1$ coefficients. As these are complex, the space they span (which we denote
as $\mathcal{L}_{N}$) is isomorphic to the Euclidean space
$\mathbb{R}^{2(N-1)}$.

Within $\mathcal{L}_{N}$, the degeneracies comprise a subspace (which we
denote as $\mathcal{V}_{N}$) having two dimensions fewer than
$\mathcal{L}_{N}$. Intuitively, this follows because degeneracy corresponds to
$\lambda_{i}=\lambda_{j}$ (i.e., equality of two eigenvalues), which can be
regarded as two real constraints. For a loop $\mathcal{C}$ that does not
intersect $\mathcal{V}_{N}$ (i.e., $\mathcal{C}$ lies entirely in the space of
non-degenerate spectra
$\mathcal{G}_{N}\equiv\mathcal{L}_{N}-\mathcal{V}_{N}$), it is straightforward
to show that the resulting evolution of $\bm{\lambda}$ is a braid of $N$
strands, where each strand of the braid represents the evolution of one of the
eigenvalues of $H$. Furthermore, there is a one-to-one correspondence between
the topology of the control loop in $\mathcal{G}_{N}$ and the topology of the
resulting eigenvalue braid (formally, the correspondence is between homotopy
equivalence classes of based loops in $\mathcal{G}_{N}$ and isotopy
equivalence classes of braids) Fox and Neuwirth (1962); Arnold (2014); Patil
_et al._ (2022).

The high dimension of these spaces and the non-trivial geometry of
$\mathcal{V}_{N}$ (for $N>2$) make it challenging to visualize this approach.
However, it has the advantage of giving the topological character of the
evolution of $\bm{\lambda}$ directly in terms of the manner in which
$\mathcal{C}$ encircles the EPs.

The rich behavior exhibited by COMs with $N>2$ has been shown to offer
considerable promise for a range of applications, including enhanced sensing,
topological control, and line-narrowing in lasers Am-Shallem  _et al._ (2015);
Schnabel  _et al._ (2017); Kullig  _et al._ (2018); Chakraborty and Sarma
(2019); Zhang and Chan (2019); Ryu  _et al._ (2019); Hodaei  _et al._ (2020);
Znojil (2020); Cui  _et al._ (2021); Xiong  _et al._ (2021); Xia  _et al._
(2021); Dey  _et al._ (2021); Sayyad and Kunst (2022); Tschernig  _et al._
(2022); Pick  _et al._ (2019). As a result, it is important to develop an
intuitive description of how $\bm{\lambda}$ can be tuned in such systems.
Here, we describe experiments that use a system of $N=3$ mechanical
oscillators to elucidate the connections between viewing $\bm{\lambda}$ using
a 2D control space ($\mathcal{B}$) and using its full control space
($\mathcal{L}_{3}$). Specifically, we measure $\bm{\lambda}$ in various
$\mathcal{B}$ throughout $\mathcal{L}_{3}$, and use this data to track the
evolution of $\bm{\lambda}$ around various loops in each $\bar{\mathcal{B}}$.
This data also provides the structure of $\mathcal{G}_{3}$ and
$\mathcal{V}_{3}$, and so allows us to illustrate a number of qualitative
features in the topological behavior of $\bm{\lambda}$. These measurements
demonstrate that loops which encircle different EPs in a given
$\bar{\mathcal{B}}$ produce the same permutation if they are homotopy
equivalent in $\mathcal{G}_{3}$. They also demonstrate that loops can encircle
the same EPs in a given $\bar{\mathcal{B}}$ and produce different permutations
if they are homotopy inequivalent in $\mathcal{G}_{3}$. Lastly, they
demonstrate the role of the loops’ basepoints in determining their homotopy
equivalence.

## Results

### Three-mode systems

To illustrate the view of control loops and EPs provided by the full control
space $\mathcal{L}_{N}$, we give an explicit description of the case $N=3$.
The characteristic polynomial of any $3\times 3$ traceless matrix can be
written as $p_{H}=\lambda^{3}-y\lambda-x$ where the complex numbers $x$ and
$y$ are the control parameters. The space they span is $\mathcal{L}_{3}$,
which is isomorphic to $\mathbb{R}^{4}$. This means that $\mathcal{V}_{3}$,
the subspace consisting of the degeneracies, will be two-dimensional (as
degeneracies comprise a subspace with two fewer dimensions than the full
control space).

To find the structure of $\mathcal{V}_{3}$, we use the fact that a
polynomial’s roots are degenerate if and only if its discriminant vanishes.
For $p_{H}$, this corresponds to the condition

$4y^{3}=27x^{2}$ (1)

so that the solutions of Eq. 1 are the coordinates of the degeneracies in
$\mathcal{L}_{3}$. The trivial solution $x=y=0$ (corresponding to the origin
of $\mathcal{L}_{3}$) corresponds to a three-fold degeneracy (which we denote
as $\mathrm{EP}_{3}$), with $\bm{\lambda}=\\{0,0,0\\}$. This is the only
$\mathrm{EP}_{3}$ in $\mathcal{L}_{3}$ (since $\mathrm{Tr}[H]=0$), so the
remainder of $\mathcal{V}_{3}$ must consist of two-fold degeneracies (which we
denote as $\mathrm{EP}_{2}$ or simply EP). These can be found by first
considering a hypersphere $\mathcal{S}_{r}$ with radius $r$ and centered at
the origin (i.e., it is defined by $|x|^{2}+|y|^{2}=r^{2}$). It is
straightforward to show that this constraint together with Eq. 1 fixes $|x|$
and $|y|$ while also requiring that $2\mathrm{arg}(x)=3\mathrm{arg}(y)$ (here
arg denotes the complex argument). This defines a $(2,3)$ torus knot
$\mathcal{K}$ (also known as a trefoil knot) within $\mathcal{S}_{r}$ Milnor
(1969). Since this reasoning holds for any $r>0$, the full space of
degeneracies $\mathcal{V}_{3}$ can be viewed as the result of “extruding”
$\mathcal{K}$ in the radial direction (i.e., so that it collapses to a point
at the origin). The resulting two-dimensional surface is known as the
topological cone of the trefoil knot, denoted as $C\mathcal{K}$.

In addition to giving the structure of $\mathcal{V}_{3}$, this reasoning also
provides the structure of $\mathcal{G}_{3}$ (the non-degenerate space in which
control loops are assumed to lie) as this is simply the complement of
$\mathcal{V}_{3}$ in $\mathcal{L}_{3}$. Lastly, the one-to-one correspondence
between loops in $\mathcal{G}_{3}$ and braids of the three eigenvalues
reflects the fact that the fundamental group of $\mathcal{G}_{3}$ is $B_{3}$,
the Artin braid group Artin (1947) on three strands.

Figure 1: Controlling the spectrum of a three-mode system. a The full space of
control parameters. The EPs are shown in yellow, and form a trefoil knot
$\mathcal{K}$. The coordinates $X,Y,Z$ are defined in Methods. The grey plane
shows an example of a 2D subspace ($\mathcal{B}$). The yellow discs are the
five intersections of $\mathcal{B}$ with $\mathcal{K}$. The red, green, and
blue loops all lie in $\mathcal{B}$ and have a common basepoint (black
circle). b The control plane $\mathcal{B}$, showing the five EPs it contains
(yellow circles) and the three control loops. The coordinate along each loop
is $\rho$. For each loop, the basepoint (black circle) corresponds to $\rho=0$
and $\rho=1$. c The eigenvalue spectrum $\bm{\lambda}$ calculated as a
function of $\rho$ along each of the loops. A detailed description of these
plots is in Methods.

This structure is illustrated in Fig. 1a, which shows a stereographic
projection of the unit hypersphere $\mathcal{S}_{1}$, along with the locations
of the EPs it contains. The latter can be seen to form a trefoil knot
$\mathcal{K}$ (yellow curve). Also shown in Fig. 1a is an example of a 2D
subspace $\mathcal{B}$ (gray plane). This particular choice for $\mathcal{B}$
intersects $\mathcal{K}$ at five locations. One of the intersections is
tangential (the one at the greatest $Y$ value), while the other four are
transverse Guillemin and Pollack (2010). Figure 1b depicts $\mathcal{B}$ and
the five EPs within it.

To illustrate the specific problem that we consider in the experiments
described below, Figs. 1(a,b) show three control loops. All three loops lie in
$\mathcal{B}$ and share a common basepoint. Viewed within $\mathcal{B}$ (as in
Fig. 1b) the loops enclose zero, one, and two EPs, and so are not homotopic in
$\bar{\mathcal{B}}$. However it is straightforward to see from Fig. 1a that
they are homotopic in $\mathcal{G}_{3}$. As a result, these three loops all
result in the same eigenvalue braid. In fact each of these loops is
contractible, and so result in the identity braid (and hence the identity
permutation). This agrees with the explicit calculation of $\bm{\lambda}$
around each loop, shown in Figure 1c. A detailed description of Fig. 1 is in
Methods.

### Experimental setup

To demonstrate these features experimentally, we used a COM consisting of
three vibrational modes of a 1 mm $\times$ 1 mm $\times$ 50 nm Si3N4 membrane.
The dynamical matrix $H$ for these modes is controlled by placing the membrane
inside a high-finesse optical cavity. When the cavity is driven by one or more
lasers, light inside the cavity exerts radiation pressure on the membrane,
altering $H$ via the well-known dynamical back-action (DBA) effect of cavity
optomechanics Kippenberg and Vahala (2007); Marquardt and Girvin (2009);
Aspelmeyer  _et al._ (2014).

Figure 2a shows a schematic of the experiment. The optical cavity is addressed
by two lasers, labelled control and probe. The control laser is split into
three tones (by an acousto-optic modulator (AOM)) which are used to tune $H$.
For each choice of $H$, the membrane’s spectrum $\bm{\lambda}$ is determined
from its frequency-dependent mechanical susceptibility $\chi(\tilde{\omega})$,
which is measured using the probe laser. A complete description of the setup
is given in Refs. Patil  _et al._ (2022); Henry (2022).

Figure 2: Experimental setup. a A Si3N4 membrane (red) is placed between the
mirrors of a Fabry-Pérot cavity (white) in a cryostat (blue). Three of the
membrane’s modes (pink box) are tuned using three tones generated from the
“control” laser via an AOM (cAOM). The membrane is driven by modulating the
“probe” laser’s intensity (via the amplitude modulation port (AM-in) of the
source that drives a second AOM (pAOM)). The probe laser also provides a local
oscillator (LO) which generates a signal $\tilde{V}$ that is proportional to
the membrane’s displacement and is monitored via a lock-in amplifier (LIA). b
The detunings $\Delta$ of the three control tones (with respect to the
cavity’s resonance). Dark blue: the magnitude of the cavity’s optical
susceptibility. The parameter $\eta=-2\pi\times 100$ Hz is chosen to provide
an optimal rotating frame Patil  _et al._ (2022). c A measurement of the
membrane’s mechanical susceptibility for $\Psi=(2\pi\times
46\;\mathrm{kHz},$109.4\text{\,}\mathrm{\SIUnitSymbolMicro
W}$,$376.8\text{\,}\mathrm{\SIUnitSymbolMicro
W}$,$77.0\text{\,}\mathrm{\SIUnitSymbolMicro W}$)$. For each frequency range,
the left panel shows $|\tilde{V}(\tilde{\omega}_{\mathrm{AM}})|$ and the right
panel shows a parametric plot of $\tilde{V}$. Each data point is colored
according to the value of $\tilde{\omega}_{\mathrm{AM}}$. The black lines are
a global fit to all the data shown. This fit returns the system’s eigenvalues
$\bm{\lambda}=2\pi\times\\{49.670-i\;{84.977},57.636-i\;{29.834},112.222-i\;{26.325}\\}\mathrm{Hz}$
in the rotating frame $\mathcal{R}$. The magnitude of each mode’s contribution
(as determined from the fit) is shown as the orange, green, and light blue
curves in the left-hand column.

Figure 2b shows the detuning of the three control tones relative to the cavity
resonance. The control tone indexed by $k\in\\{1,2,3\\}$ is detuned by
$\sim-\tilde{\omega}_{k}^{(0)}$ where the elements of
$\\{{\tilde{\omega}_{1}}^{(0)},{\tilde{\omega}_{2}}^{(0)},{\tilde{\omega}_{3}}^{(0)}\\}$
are the bare resonance frequencies of the membrane modes in the absence of
DBA. For this experiment, we vary the control tones’ common detuning $\delta$
(defined in Fig. 2b) and their powers $P_{1},P_{2}$ and $P_{3}$. The relative
detunings between the control tones are fixed, and define a rotating frame
$\mathcal{R}$ in which the three modes are nearly degenerate for
$P_{1}=P_{2}=P_{3}=0$. As shown in Patil  _et al._ (2022), the experimental
parameters $\Psi=(\delta,P_{1},P_{2},P_{3})$ provide sufficient control to
span $\mathcal{L}_{3}$ in the neighborhood of the $\mathrm{EP}_{3}$, and in
particular to measure the structure of $\mathcal{V}_{3}$ and
$\mathcal{G}_{3}$.

Figure 2c shows a representative measurement of $\chi(\tilde{\omega})$. The
drive is produced by modulating the probe beam’s intensity at a frequency
$\tilde{\omega}_{\mathrm{AM}}$. The resulting heterodyne signal $\tilde{V}$,
which is proportional to the membrane’s motion, is recorded for values of
$\tilde{\omega}_{\mathrm{AM}}$ in a window centered on
${\tilde{\omega}_{k}}^{(0)}$ for each $k$. Figure 2c shows $|\tilde{V}|$ in
the left panel and a parametric plot of $\tilde{V}$ in the right panel for
each $k$. This data is fit to the expected form of $\chi(\tilde{\omega})$
using $\bm{\lambda}$ as a fit parameter Patil  _et al._ (2022). Throughout
this paper $\bm{\lambda}$ is determined by data and fits such as those shown
in Figure 2c, and its values are given in the frame $\mathcal{R}$. Quantities
with a tilde are given in the lab frame, and without a tilde in the rotating
frame $\mathcal{R}$. The device parameters are given in Table 1 (Methods).

### Measurements of braids and permutations

The setup described above was used to measure $\bm{\lambda}$ at $\sim 3\times
10^{4}$ values of $\Psi$ Patil  _et al._ (2022). A small subset of this data
was taken for values of $\Psi$ ranging throughout $\mathcal{L}_{3}$ and was
used to determine the location of the $\mathrm{EP}_{3}$. The rest of the data
was taken for values of $\Psi$ on a hypersurface $\mathcal{S}$ surrounding
$\mathrm{EP}_{3}$. These measurements are described in detail in Ref. Patil
_et al._ (2022). Figure 3a shows part of this data. At each value of $\Psi$
the measured $\bm{\lambda}$ was converted to
$D=(\lambda_{1}-\lambda_{2})^{2}(\lambda_{2}-\lambda_{3})^{2}(\lambda_{3}-\lambda_{1})^{2}$,
which is the discriminant of $p_{H}$. The color scale in Figure 3a shows
$\mathrm{arg}(D)$. The yellow circles show the two locations identified in
this data by a vortex-finding algorithm Patil  _et al._ (2022). As described
in Ref. Patil  _et al._ (2022), a vortex in $\mathrm{arg}(D)$ is one signature
of an $\mathrm{EP}_{2}$. These locations agree well with those returned by
three other algorithmic means of identifying $\mathrm{EP}_{2}$s from the data
Patil  _et al._ (2022). Figure 3b displays a stereographic projection of
$\mathcal{S}$ in which the EPs identified via vortices in $\mathrm{arg}(D)$
are depicted by the yellow spline curve (see Methods).

Figure 3: Equivalent eigenvalue braids from different loops. a The 2D control
space $\mathcal{B}^{(1)}$. Color scale: $\mathrm{arg}(D)$. Yellow circles:
vortices in $\mathrm{arg}(D)$, which correspond to EPs. Green and blue curves:
control loops. Black circle: the loops’ basepoint. A view of
$\mathcal{B}^{(1)}$ in $\mathcal{L}_{3}$ is in the Methods. b Stereographic
projection of the hypersurface $\mathcal{S}$. The axes $X,Y,Z$ are defined in
Methods. Yellow curve: the measured EPs. The black circle, green curve, and
yellow circles are as in (a). The control loops consist of straight segments
in (a) but appear curved in (b) owing to the stereographic projection. Inset:
a simplified cartoon of the relationship between the EPs and the control loop.
c The same as in b, but for the blue control loop. d The eigenvalue spectrum
$\bm{\lambda}$ as a function of position along the green control loop. $\rho$
indexes the measurements of $\bm{\lambda}$ along the loop. e As in (d), but
for the blue control loop. Although the loops are not homotopic in
$\bar{\mathcal{B}}^{(1)}$, they are homotopic in $\mathcal{G}_{3}$, and so
produce isotopic braids.

To examine the effect of control loops (viewed in either the full control
space or in a 2D subspace) we take the region $\mathcal{B}^{(1)}$ shown in
Fig. 3a as our first example of a 2D control space, and consider the two loops
shown as green and blue in Fig. 3a. These loops share a common basepoint, and
are clearly not homotopic in $\bar{\mathcal{B}}^{(1)}$ (defined as
$\mathcal{B}^{(1)}$ without the two EPs). However Figs. 3(b,c) show that these
loops are homotopic in $\mathcal{G}_{3}$ and that each loop is contractible.
Measurements of $\bm{\lambda}$ at several positions along each loop are shown
in Figs. 3(d,e), and demonstrate that these loops produce isotopic braids (in
this case, the identity braid), as would be expected from Fig. 3(b,c). This
illustrates one of the striking features from Fig. 1: the braid traced out by
$\bm{\lambda}$ is determined by the loop’s homotopy class in
$\mathcal{G}_{N}$, and not by its homotopy class in any particular
$\bar{\mathcal{B}}$.

Another important feature of 2D control spaces is that control loops may
encircle the same EPs and yet give rise to distinct eigenvalue braids Zhong
_et al._ (2018). This scenario is demonstrated in Fig. 4, which shows the 2D
control space $\mathcal{B}^{(2)}$. Here the two loops (red and blue) share a
common basepoint and encircle the same EPs. Nevertheless, they are not
homotopic in either $\bar{\mathcal{B}}^{(2)}$ or $\mathcal{G}_{3}$ (Figs.
4(b,c)) and the braids they produce (Figs. 4(d,e))) are not isotopic.

Figure 4: Different braids by encircling the same EPs. a The 2D control space
$\mathcal{B}^{(2)}$. Color scale: $\mathrm{arg}(D)$. Yellow circles: vortices
in $\mathrm{arg}(D)$ corresponding to EPs. Blue and red curves: control loops.
Black circle: the loops’ basepoint. A view of $\mathcal{B}^{(2)}$ in
$\mathcal{L}_{3}$ is in Methods. b Stereographic projection of $\mathcal{S}$.
Yellow curve: the measured EPs. The black circle, blue curve, and yellow
circles are as in a. Inset: a simplified cartoon of the relationship between
the EPs and the control loop. c The same as in (b), but for the red control
loop. d The eigenvalue spectrum $\bm{\lambda}$ as a function of position along
the blue control loop. $\rho$ indexes the measurements of $\bm{\lambda}$ along
the loop. e As in (d), but for the red control loop. The braids produced by
the two loops are not isotopic (and do not produce the same permutation) even
though they encircle the same EPs in $\mathcal{B}^{(2)}$.

Next, we investigate the role of the loops’ basepoints in determining their
homotopy equivalence. To do so we use the data shown in Fig. 5, which depicts
a third 2D control space $\mathcal{B}^{(3)}$, along with two loops. These
loops are homotopic if their basepoint is the white circle; however, they are
non-homotopic if their basepoint is the black circle. This also holds if the
loops are viewed in $\mathcal{L}_{3}$, as shown in Figs. 5(b,c). As a result,
the loops generate isotopic braids when based at the white point (Figure
5(d,e)) and non-isotopic braids when based at the black point (Figure 5(f,g)).

Figure 5 illustrates another feature that is absent for $N=2$ but generic for
$N>2$: the noncommutativity of braids. Specifically, the braids in Fig. 5(f,g)
do not commute with each other: concatenating the braid in Fig. 5f and the
braid in Fig. 5g results in the leftmost eigenvalue (as it appears in the
figure) being transported to the middle one, while reversing the concatenation
causes the leftmost eigenvalue to be transported to the rightmost one.

Figure 5: Measuring the basepoint dependence of homotopy equivalence. a The 2D
control space $\mathcal{B}^{(3)}$. Color scale: $\mathrm{arg}(D)$. Yellow
circles: vortices in $\mathrm{arg}(D)$ corresponding to EPs. Blue and red
curves: control loops. Black and white circles: two choices for the loops’
basepoint. A view of $\mathcal{B}^{(3)}$ in $\mathcal{L}_{3}$ is in Methods. b
Stereographic projection of $\mathcal{S}$. Yellow curve: the measured EPs. The
black circle, white circle, red curve, and yellow circles are as in (a).
Inset: a simplified cartoon of the relationship between the EPs, the control
loop, and the basepoints. c As in (b), but for the blue control loop. d The
eigenvalue spectrum $\bm{\lambda}$ as a function of position along the red
control loop, using the white basepoint. $\rho$ indexes the measurements of
$\bm{\lambda}$ along the loop. e As in (d), but for the blue control loop. f
As in (d), but for the black basepoint. g As in (e), but for the black
basepoint. Thus, when the two loops shown in (a) are based at the white point,
they are homotopic and so produce isotopic braids; however, the same loops
based at the black point are not homotopic and produce non-isotopic braids.

## Discussion

In conclusion, we have considered two complementary ways of describing the
topological features that arise when the spectrum of a non-Hermitian system is
tuned. The first describes tuning with two control parameters Zhong  _et al._
(2018), while the second considers all of the system’s control parameters
Patil  _et al._ (2022). The former offers ease of visualization and
corresponds to many actual experiments, but may mask the features that
determine which control loops are topologically distinct from each other. The
latter provides a clear picture of topological equivalence but is more
challenging to visualize.

We have illustrated both of these approaches experimentally, using a three-
mode mechanical system that is tuned optomechanically. However we emphasize
that the results presented here are generic and can be applied to non-
Hermitian systems realized in any physical platform.

These results help to elucidate the complex topology of non-Hermitian systems
possessing many degrees of freedom. This insight may aid in efforts to realize
new types of mode switching via dynamical encircling of EPs Xu  _et al._
(2016); Doppler  _et al._ (2016); Zhang  _et al._ (2018); Choi  _et al._
(2023). It may also help to clarify which aspects of a linear system’s
topology remain relevant to its behavior in nonlinear regimes Ji  _et al._
(2023); Wang  _et al._ (2019). Lastly, it may find application in the fields
of non-Hermitian band structure Makris  _et al._ (2008); Shen  _et al._
(2018); Wojcik  _et al._ (2020); Wang  _et al._ (2021a); Wojcik  _et al._
(2022); Elbaz  _et al._ (2022); Hu  _et al._ (2022) and complex scattering
phenomena Elbaz  _et al._ (2022); Sweeney  _et al._ (2019); Wang  _et al._
(2021b), where closely related concepts arise.

## Methods

### Tracelessness of $H$

Throughout this paper we take $H$ to be traceless. This choice simplifies the
analysis, and does not alter the generality of the conclusions presented here.

Intuitively, this can be understood by considering a control loop
parameterized by $0\leq\rho\leq 1$ for which $H(\rho)$ is not assumed to be
traceless (and in which $\mathrm{tr}(H(\rho))$ may vary with $\rho$). The
eigenvalues of $H$, viewed as a function of $\rho$, will trace out a braid of
$N$ strands Fox and Neuwirth (1962); Arnold (2014); Patil  _et al._ (2022).
For each value of $\rho$ along the braid, $\mathrm{tr}(H(\rho))$ sets the
average of the $N$ eigenvalues. If for each value of $\rho$ we remove the
trace of $H$ (i.e., replace $H(\rho)$ with $H(\rho)-I\mathrm{tr}(H(\rho))$
where $I$ is the identity matrix), this amounts to translating (in the complex
plane) the eigenvalue spectrum as a whole. Even if this translation is
different for each value of $\rho$, the fact that it is applied to the
spectrum as a whole means that the topological character of the braid is
unchanged.

More formally, shifting the center of the eigenvalue braid by an amount that
varies with $\rho$ gives a family of homeomorphisms that defines an ambient
isotopy (the ambient space of the braids is $\mathbb{C}\times\mathbb{I}$,
where $\mathbb{I}$ is the unit interval). As a result, it leaves the braid’s
isotopy equivalence class unchanged Klajdzievski (2015).

### Detailed description of Figure 1

In this section we provide a detailed description of Fig. 1, which illustrates
the space spanned by the coefficients of $p_{H}$ (the characteristic
polynomial of the $3\times 3$ dynamical matrix $H$). It also shows the
structure of the degeneracies within this space, along with one particular
choice of a 2D subspace.

The characteristic polynomial of a matrix is monic, meaning that the
coefficient of its highest-order term is unity. The coefficient of the next-
highest-order term is simply the trace of the matrix. In COMs, the trace of
the dynamical matrix can be trivially absorbed into the definitions of the
oscillators’ coordinates, so we take $\mathrm{tr}(H)=0$ throughout this paper
without loss of generality. As a result, the most general form for $p_{H}$ is
$\lambda^{3}-y\lambda-x$, where the complex numbers $x$ and $y$ fully specify
the eigenvalue spectrum. Degeneracy between roots of $p_{H}$ corresponds to
the condition $D_{p_{H}}=0$, where $D_{p_{H}}=4y^{3}-27x^{2}$ is known as the
discriminant of $p_{H}$.

The space spanned by $x$ and $y$ (denoted $\mathcal{L}_{3}$) can be viewed as
$\mathbb{R}^{4}$ and can be labeled with the Cartesian coordinates
$(\mathrm{Re}(x),\mathrm{Im}(x),\mathrm{Re}(y),\mathrm{Im}(y))$. As described
in the main paper (as well as in Refs. Patil  _et al._ (2022); Milnor (1969);
Brauner (1928)), any hypersphere centered at the origin of $\mathcal{L}_{3}$
(i.e., at the point $(0,0,0,0)$) contains two-fold degeneracies
($\mathrm{EP}_{2}$s). These form a closed curve that is a trefoil knot.

Figure 1a shows this structure explicitly. It is a stereographic projection of
the unit hypersphere $\mathcal{S}_{1}$ (defined by $|x|^{2}+|y|^{2}=1$) onto
$\mathbb{R}^{3}$. In particular, this projection uses the point $(-1,0,0,0)$
as its pole, so that the coordinates $X,Y,Z$ of Fig. 1 are defined via:

$\displaystyle\mathrm{Re}(x)=\dfrac{1-X^{2}-Y^{2}-Z^{2}}{1+X^{2}+Y^{2}+Z^{2}}$
(2) $\displaystyle\mathrm{Im}(x)=\dfrac{2Z}{1+X^{2}+Y^{2}+Z^{2}}$
$\displaystyle\mathrm{Re}(y)=\dfrac{2X}{1+X^{2}+Y^{2}+Z^{2}}$
$\displaystyle\mathrm{Im}(y)=\dfrac{2Y}{1+X^{2}+Y^{2}+Z^{2}}$

The yellow curve in Fig. 1 shows the locations of the degeneracies within
$\mathcal{S}_{1}$. It is determined by the two constraints $|x|^{2}+|y|^{2}=1$
(which defines $\mathcal{S}_{1}$) and $4y^{3}-27x^{2}=0$ (which defines the
vanishing of $D_{p_{H}}$). It can be seen to form the trefoil knot
$\mathcal{K}$.

As described in the main text, there are many settings in which it is useful
to consider a 2D subspace of $\mathcal{L}_{3}$. The choice of such a subspace
(denoted $\mathcal{B}$) may be motivated by which pair of parameters are most
readily accessible in a particular device. There are infinitely many possible
choices for $\mathcal{B}$, so it is helpful to distinguish between features
that are generic (i.e., which would result for nearly any choice of
$\mathcal{B}$) and features that only result for finely-tuned choices of
$\mathcal{B}$. For example, a generic $\mathcal{B}$ will have a finite
(possibly zero) number of transverse intersections with $\mathcal{K}$, while
only fine-tuned choices for $\mathcal{B}$ will have non-transverse
intersections with $\mathcal{K}$ Guillemin and Pollack (2010).

Figure 1a shows a particular $\mathcal{B}$ that illustrates both generic and
non-generic features. This $\mathcal{B}$ is defined by $Y+m(Z-Z_{0})=0$, with
$m=\mathrm{tan}(-17\pi/36)$ and $Z_{0}\approx 0.284$. This $\mathcal{B}$
happens to be a plane in the coordinates $X,Y,Z$. Its flatness (in these
coordinates) is a non-generic feature, but is chosen to simplify the
visualization and does not impact the discussion. This particular
$\mathcal{B}$ has four transverse intersections with $\mathcal{K}$. As
mentioned above, such intersections are generic, though the specific number
depends on the choice of $\mathcal{B}$ Guillemin and Pollack (2010). It also
has one non-transverse intersection, which can be viewed as the result of
fine-tuning the parameters $m$ and $Z_{0}$. This non-generic intersection is
included for illustrative purposes, but is not central to this discussion.

Also shown in Figure 1a are three control loops, all of which lie in
$\mathcal{B}$ and share a common basepoint. Each loop is described by:

$\displaystyle X(\rho)$
$\displaystyle=X_{b}+r[\cos(\phi_{c})-\cos(2\pi\rho+\phi_{c})]$ (3)
$\displaystyle Y(\rho)$
$\displaystyle=Y_{b}+r[\sin(\phi_{c})-\sin(2\pi\rho+\phi_{c})]$ $\displaystyle
Z(\rho)$ $\displaystyle=Z_{0}-Y(\rho)/m$

where $\rho\in\left[0,1\right]$ parameterizes position along a circular path
of radius $r$ that starts and stops at the basepoint
$\left(X_{b},Y_{b},Z_{0}-Y_{b}/m\right)$. The angle $\phi_{c}$ sets the
orientation of the circle’s center with respect to the basepoint.

The three loops in Fig. 1a all have their basepoint at $X_{b}=0.25$,
$Y_{b}=0.3$, radii $r=\\{0.325,0.6,1.1\\}$, and orientations
$\phi_{c}=\left\\{-7\pi/18,15\pi/36,-13\pi/36\right\\}$ (for the green, red
and blue loop respectively).

Figure 1b shows the plane $\mathcal{B}$ (i.e., the space spanned by $X$ and
$Y$ with $Z=Z_{0}-Y/m$), along with the five intersections between
$\mathcal{B}$ and $\mathcal{K}$ and the three loops just described. These
loops can be seen to enclose zero, one, or two of the five intersections
between $\mathcal{B}$ and $\mathcal{K}$.

To plot the eigenvalue braids that result from each of these control loops, we
first use Eq. 2 to convert the loop coordinates from $(X,Y,Z)$ as given in Eq.
3 to the coordinates
$(\mathrm{Re}(x),\mathrm{Im}(x),\mathrm{Re}(y),\mathrm{Im}(y))$. We then find
the roots of $p_{H}$ numerically for 101 values of $\rho$ ranging from 0 to 1.
For each value of $\rho$, we show the three roots (which comprise the
eigenspectrum $\bm{\lambda}$ of $H$) in the complex plane (indicated by
$\mathrm{Re}(\lambda)$ and $\mathrm{Im}(\lambda)$ in Fig. 1 c). Their
evolution as a function of $\rho$ is shown in Fig. 1c by stacking a copy of
the complex plane for each $\rho$. The black crosses highlight $\bm{\lambda}$
at $\rho=0$ (the bottom of each plot), which by construction is identical to
$\bm{\lambda}$ at $\rho=1$ (the top of each plot).

### Representing the measured knot of degeneracies

The features shown in Fig. 1 are calculated using a general 3-mode COM, while
Figs. 3(b,c), 4(b,c), and 5(b,c) show the corresponding features measured in
the optomechanical system described in the main text (and in greater detail in
Ref. Patil  _et al._ (2022)). In the experiments described here, $H$ is tuned
via the parameters $(\delta,P_{1},P_{2},P_{3})$ defined in the main text. As
described in Ref. Patil  _et al._ (2022), these parameters span
$\mathcal{L}_{3}$ in the neighborhood of $\mathrm{EP}_{3}$. Specifically, the
Jacobian relating $(\delta,P_{1},P_{2},P_{3})$ to
$(\mathrm{Re}(x),\mathrm{Im}(x),\mathrm{Re}(y),\mathrm{Im}(y))$ has non-zero
determinant.

Degeneracies of $H$ were located by fixing two of these parameters (for
example, $\delta$ and $P_{1}$) and densely rastering the other two (in this
example, $P_{2}$ and $P_{3}$). At each point in this “sheet”, the mechanical
susceptibility $\chi(\tilde{\omega}$) was measured for three windows of
$\tilde{\omega}$ centered near $\tilde{\omega}^{(0)}_{1}$,
$\tilde{\omega}^{(0)}_{2}$, and $\tilde{\omega}^{(0)}_{3}$. A typical example
of such a measurement is shown in Fig. 2c. This data is fit to the expected
form of the mechanical susceptibility, which is the sum of nine Lorentzians.

The fit returns 13 complex fit parameters. First, there are the complex
frequencies of the three modes; these comprise $\bm{\lambda}$. (The center
frequencies of the other six Lorentzians are fixed by these values and the
frequencies of the intracavity beatnotes produced by the control laser tones).
Next, there is a complex constant background and a transduction factor in each
of the three windows. Lastly, there are the nine complex amplitudes $s_{i,j}$
of the Lorentzians (denoting the amplitude of the $i$th Lorentzian in the
window near $\tilde{\omega}^{(0)}_{j}$); however, only four of the $s_{i,j}$
are linearly independent. A detailed description of the fitting procedure is
in Ref. Patil  _et al._ (2022).

The quantity
$D=(\lambda_{1}-\lambda_{2})^{2}(\lambda_{2}-\lambda_{3})^{2}(\lambda_{3}-\lambda_{1})^{2}$
is calculated directly from the value of $\bm{\lambda}$ returned by this fit.
Fig. 3a, Fig. 4a, and Fig. 5a all show measurements of $\mathrm{arg}(D)$.

To locate the degeneracies of $H$, we make use of $D$ as well as the quantity
$E\equiv(\mathrm{det}(S))^{-2}$ where $S$ is the matrix whose elements are the
$s_{i,j}$. Both $D$ and $E$ are complex numbers. Both are expected to vanish
at an EP, and to exhibit a $2\pi$ phase winding around an EP. As a result, the
EPs within a given sheet were identified in four ways: by applying a minimum-
finding algorithm to both $|D|$ and $|E|$, and by applying a vortex-finding
algorithm to both $\mathrm{arg}(D)$ and $\mathrm{arg}(E)$ Patil  _et al._
(2022). This procedure was repeated for sixty-one distinct sheets, all lying
in a hypersurface $\mathcal{S}$, and resulted in the identification of 291
degeneracies.

As described in Ref. Patil  _et al._ (2022), $\mathcal{S}$ consists of eight
3D rectangular cuboids (“faces”). Each 3D face corresponds to fixing one of
the four experimental control parameters ($\delta,P_{1},P_{2},P_{3}$) to its
minimum (or its maximum) value, and allowing the other three parameters to
range from their minimum to maximum values.

Figs. 3(b,c), 4(b,c), and 5(b,c) all show the same stereographic projection
from $\mathcal{S}$ to $\mathbb{R}^{3}$. The details of this projection are
given in Ref. Patil  _et al._ (2022). These figures also show the
experimentally determined EPs within $\mathcal{S}$. In Ref. Patil  _et al._
(2022), all of the 291 EPs were displayed as individual points (e.g., in Fig.
3 of that paper). For ease of visualization, in the present paper we replace
the points with a curve. This curve is a cubic spline through the 67
degeneracies that were identified as vortices in $\mathrm{arg}(D)$. Producing
such a curve requires not only the locations of the points, but also their
ordering (i.e., it is necessary to know which degeneracies are “next to” each
other along the knot). Since $\mathcal{K}$ is a smooth curve, this ordering
could in principle be inferred directly from the locations of the degeneracies
(i.e., from their coordinates $X,Y,Z$). However, with a finite number of
experimentally determined degeneracies this approach can lead to ambiguities,
for example when distinct portions of the curve $\mathcal{K}$ happen to pass
close to each other. To remove this potential ambiguity, we note that each
degeneracy is also characterized by a parameter $\theta$ which runs from $0$
to $2\pi$ around $\mathcal{K}$. The formal definition of $\theta$ is given in
Ref. Patil  _et al._ (2022), but here we note that $\theta$ is simply the
complex argument of the non-degenerate member of $\bm{\lambda}$ (when viewed
in a rotating frame in which $H$ is traceless), and so is readily determined
from the data.

To place the experimentally identified degeneracies in order (i.e., as they
appear around $\mathcal{K}$), we define for the $i$th point the quantity
$\Omega_{i}=(X_{i},Y_{i},Z_{i},\cos[\theta_{i}],\sin[\theta_{i}])$. Then for
each point, we find the nearest (and the next nearest) neighbor based on the
Euclidean distance between the various $\Omega_{i}$. Using this information,
we sort the list of degeneracies so that the nearest neighbors are the
adjacent elements of the list. We then interpolate the sorted list with a
cubic spline.

### Description of the three 2D control spaces

This section describes the three 2D control spaces used in the main text. Each
of these consists of the union of two or three of the “sheets” described in
the previous section (and in Ref. Patil  _et al._ (2022)).

The 2D control space shown in Fig. 3 ($\mathcal{B}^{(1)}$) is the union of two
sheets. In the first, $P_{3}$ and $\delta$ are fixed to $78$ µW and
$2\pi\times 60$ kHz respectively. In the second, $P_{3}$ and $\delta$ are
fixed to $78$ µW and $2\pi\times 75.6$ kHz respectively. The 2D control space
shown in Fig. 4 ($\mathcal{B}^{(2)}$) is the union of three sheets. The first
two are the same as for $\mathcal{B}^{(1)}$, and the third consists of fixing
$P_{2}$ and $\delta$ to $289$ µW and $2\pi\times 75.6$ kHz respectively. The
2D control space shown in Fig. 5 ($\mathcal{B}^{(3)}$) is the union of two
sheets: the latter sheet from $\mathcal{B}^{(1)}$ and the lattermost sheet
from $\mathcal{B}^{(2)}$.

The three sheets used to form $\mathcal{B}^{(1)}$, $\mathcal{B}^{(2)}$, and
$\mathcal{B}^{(3)}$ are shown in Figure 6. One of the views in Figure 6 uses
the same stereographic projection as in Figs. 3(b,c), 4(b,c), and 5(b,c). The
other view uses the “rectilinear stereographic” projection described in Ref.
Patil  _et al._ (2022). The latter provides easier interpretation in terms the
experimental parameters ($\delta,P_{1},P_{2},P_{3}$).

Figure 6: The sheets used to form the 2D control spaces shown in Figs. 3a, 4a,
5a. Sheets ② and ③ are joined to make $\mathcal{B}^{(1)}$ in Fig. 3a. Sheets
①, ② and ③ are joined to make $\mathcal{B}^{(2)}$ in Fig. 4a. Sheets ① and ②
are joined to make $\mathcal{B}^{(3)}$ in Fig. 5a. a The three sheets, shown
in a rectilinear stereographic projection of the hypersurface $\mathcal{S}$. b
The same sheets, shown in the stereographic projection used in Figs. 3(b,c),
4(b,c), 5(b,c).

### Device parameters

This section gives the values of the various parameters in the optomechanical
model of the experimental three-mode system. The model is described in detail
in Ref. Patil  _et al._ (2022). The values (presented in Table 1) are
determined by standard optomechanical characterization measurements as
described in Refs. Patil  _et al._ (2022); Henry (2022).

Table 1:  Parameters of mechanical and optical modes, optomechanical coupling and laser source used for the experiment. $\tilde{\omega}_{i}^{(0)}$: the bare resonance frequency of the $i$th mechanical mode (i.e., in the absence of any optomechanical effects). $\tilde{\gamma}_{i}^{(0)}$: the bare energy damping rate of the $i$th mechanical mode. $\kappa$: the optical cavity linewidth. $\kappa_{\mathrm{in}}$: the optical cavity input coupling rate. $g_{i}$: the optomechanical coupling rate between the optical cavity and the $i$th mechanical mode. $\lambda$: the laser wavelength.  Parameter |  Value
---|---
${\tilde{\omega}_{1}}^{(0)}/2\pi$ |  352243.3 $\pm$ 0.1 Hz
${\tilde{\omega}_{2}}^{(0)}/2\pi$ |  557216.8 $\pm$ 0.1 Hz
${\tilde{\omega}_{3}}^{(0)}/2\pi$ |  704836.7 $\pm$ 0.1 Hz
${\tilde{\gamma}_{1}}^{(0)}/2\pi$ |  4.4 $\pm$ 0.1 Hz
${\tilde{\gamma}_{2}}^{(0)}/2\pi$ |  3.8 $\pm$ 0.1 Hz
${\tilde{\gamma}_{3}}^{(0)}/2\pi$ |  3.6 $\pm$ 0.1 Hz
$\kappa/2\pi$ |  173.8 kHz
$\kappa_{\mathrm{in}}/2\pi$ |  46.4 kHz
$g_{1}/2\pi$ |  0.1979 Hz
$g_{2}/2\pi$ |  0.3442 Hz
$g_{3}/2\pi$ |  0.3092 Hz
$\lambda$ |  1064 nm

### Scaling the experiment to larger $N$

In principle, the experimental approach described here can be extended to any
number of modes. The membrane hosts many mechanical modes that couple
optomechanically to the cavity, and in general any number $N$ of these can be
tuned in the manner used here. Specifically, the $N-1$ coefficients of the
characteristic polynomial of these modes’ dynamical matrix can be varied over
the complex plane via the detuning and power of $N$ laser tones. Roughly
speaking, each tone is detuned from the cavity resonance by an amount
$-\omega_{n}+\delta$, where $\omega_{n}$ is the frequency of the $n$th
mechanical mode and $\delta$ is a single parameter common to all of the tones.
In this picture, the beatnote between any two of these tones results in
coupling between the corresponding mechanical modes.

This specific approach to tuning a non-Hermitian system is generic to any
situation in which the $N$ modes all parametrically couple to an auxiliary
mode that can be driven externally. Tracing out this auxiliary mode from the
equations of motion leaves an $N$-mode non-Hermitian system whose parameters
are determined by the drive tones applied to the auxiliary mode. In the work
presented here, the $N$ modes are the membrane’s mechanical modes and the
auxiliary mode is the optical cavity.

In practice, a number of issues may limit the maximum value of $N$ that can be
tuned in this way. For example, if the ratio $\omega_{n}/\kappa$ (where
$\kappa$ is the cavity linewidth) is too small or too large, it becomes
challenging to tune the matrix elements of $H$ over a large region of the
complex plane. In addition, complications arise if multiple mode pairs share
the same frequency difference, as this means that there is no simple
correspondence between the laser beatnotes and the mechanical modes between
which they induce coupling.

### Data availability

The data generated in this study have been deposited in the Zenodo database
under accession code https://zenodo.org/records/10451386.

### References

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### Acknowledgements

This work is supported by Air Force Office of Scientific Research (AFOSR)
Multidisciplinary University Research Initiative (MURI) Award on Programmable
systems with non-Hermitian quantum dynamics, award No. FA9550-21-1-0202 (R.E.,
S.K.O, J.G.E.H); the Alexander von Humboldt Foundation (R.E.); AFOSR award no.
FA9550-18-1-0235 (S.K.O.); AFOSR award no. FA9550-15-1-0270 (J.G.E.H.); and
Vannevar Bush Faculty Fellowship no. N00014-20-1-2628 (J.G.E.H.). We thank P.
Henry, J. Höller, L. Jiang, N. Kralj, J. R. Lane, N. Read, and Y. Zhang for
helpful conversations.

### Author contributions

J.G.E.H. and R.E. conceived the project. Y.S.S.P. supervised the data
acquisition and analysis. C.G. and Y.S.S.P. performed the experiment and
analyzed the data with feedback from Q.Z. and S.O. All authors contributed to
the manuscript preparation.

### Competing interests

The authors declare no competing interests.