# Manifestation of the coupling phase in microwave cavity magnonics

Alan Gardin<EMAIL_ADDRESS>School of Physics, The University of
Adelaide, Adelaide SA 5005, Australia  IMT Atlantique, Technopole Brest-
Iroise, CS 83818, 29238 Brest Cedex 3, France  Lab-STICC (UMR 6285), CNRS,
Technopole Brest-Iroise, CS 83818, 29238 Brest Cedex 3, France  Jeremy
Bourhill  IMT Atlantique, Technopole Brest-Iroise, CS 83818, 29238 Brest Cedex
3, France  ARC Centre of Excellence for Engineered Quantum Systems and ARC
Centre of Excellence for Dark Matter Particle Physics,
Department of Physics, University of Western Australia, 35 Stirling Highway,
Crawley, Western Australia 6009, Australia  Vincent Vlaminck  IMT Atlantique,
Technopole Brest-Iroise, CS 83818, 29238 Brest Cedex 3, France  Lab-STICC (UMR
6285), CNRS, Technopole Brest-Iroise, CS 83818, 29238 Brest Cedex 3, France
Christian Person  IMT Atlantique, Technopole Brest-Iroise, CS 83818, 29238
Brest Cedex 3, France  Lab-STICC (UMR 6285), CNRS, Technopole Brest-Iroise, CS
83818, 29238 Brest Cedex 3, France  Christophe Fumeaux  School of Electrical
and Electronic Engineering, The University of Adelaide, Adelaide SA 5005,
Australia  Vincent Castel  IMT Atlantique, Technopole Brest-Iroise, CS 83818,
29238 Brest Cedex 3, France  Lab-STICC (UMR 6285), CNRS, Technopole Brest-
Iroise, CS 83818, 29238 Brest Cedex 3, France  Giuseppe C. Tettamanzi  School
of Physics, The University of Adelaide, Adelaide SA 5005, Australia  School of
Chemical Engineering and Advanced Materials, The University of Adelaide,
Adelaide SA 5005, Australia

###### Abstract

The interaction between microwave photons and magnons is well understood and
originates from the Zeeman coupling between spins and a magnetic field.
Interestingly, the magnon/photon interaction is accompanied by a phase factor
which can usually be neglected. However, under the rotating wave
approximation, if two magnon modes simultaneously couple with two cavity
resonances, this phase cannot be ignored as it changes the physics of the
system. We consider two such systems, each differing by the sign of one of the
magnon/photon coupling strengths. This simple difference, originating from the
various coupling phases in the system, is shown to preserve, or destroy, two
potential applications of hybrid photon/magnon systems, namely dark mode
memories and cavity-mediated coupling. The observable consequences of the
coupling phase in this system is akin to the manifestation of a discrete
Pancharatnam–Berry phase, which may be useful for quantum information
processing.

##  I Introduction

Magnons are quasi-particles associated with the collective excitation of spins
in magnetic materials. Magnons in ferrimagnetic insulators such as Yttrium-
Iron-Garnet (YIG) are promising for information transduction, due to their
ability to couple to a plethora of systems, such as mechanical resonators or
optical and microwave photonic modes [1, 2]. The emerging field of cavity
magnonics focuses on the photon/magnon interaction confined within cavities at
microwave or optical frequencies [3, 4, 5]. In the microwave domain, strong
and ultra-strong coherent coupling have been demonstrated [6, 7, 8, 9, 10],
with the latter even reachable at room temperature [11].

Owing to this flexibility, indirect coupling of two non-interacting systems by
an auxiliary mode has been investigated in [12, 13, 14, 15, 16, 17, 18]. For
instance, two macroscopically distant magnetic samples in a cavity can
indirectly couple by using the cavity photons as a bridge [15, 16, 17, 18, 19,
20, 21]. Notably, in this configuration some eigenmodes are “dark”, in the
sense that they do not lead to an experimental signature when probing the
system. This dark mode physics can be used to create dark mode gradient
memories as experimentally demonstrated in Zhang  _et al._. Another example of
cavity-mediated coupling is the coupling of a magnon with a superconducting
qubit, again mediated by a common cavity mode [22, 23]. This versatile
configuration motivated studies of quantum magnonics [24], with proposals for
single-magnon sources [25, 26], multi-magnon blockade [27], single-shot
detection of a single magnon [28] and quantum sensing of magnons [29].

These applications rely on the coupling between microwave photons and magnons,
which physically originates from the Zeeman coupling between the spins in the
ferrimagnetic material and the cavity’s RF magnetic field [30]. Adopting a
quantum formalism to describe this coupling allows for the precise computation
of the coupling strength based on microscopic parameters and the cavity
geometry [31]. Interestingly, the magnon/photon coupling term resulting from
the Zeeman coupling term is accompanied by a phase factor. To the authors’
knowledge, this phase has never been discussed explicitly in the literature,
probably due to its inconspicuous nature in the systems considered so far.

The main contribution of this paper is to highlight how the coupling phases
can become relevant in systems composed of several magnon and cavity modes. To
that effect, we begin by explaining the origin of the coupling phase between a
microwave photon and a magnon in section II. In section III, we introduce a
hybrid system composed of two YIG spheres, each coupling to two magnetic
eigenmodes of a microwave cavity. We show that the various magnon/photon
coupling phases in such a system lead to an observable quantity $\theta$
parametrising the Hamiltonian. To illustrate the impact of the physical phase
$\theta$, we study the physics of such a system in section IV, and find that
both cavity-mediated coupling and dark mode physics are $\theta$-dependent.

##  II Coupling phase between a microwave photon and a magnon

###  II.1 Free Hamiltonian

In this section, we consider the interaction between one YIG sphere and one
magnetic eigenmode of a microwave cavity. The geometry of the cavity fixes the
resonance frequency $\omega_{c}/2\pi$ of the magnetic mode $\mathbf{H}$. These
two quantities can be obtained by solving Maxwell’s equations, for instance
using an electromagnetic finite-element modelling software such as COMSOL
Multiphysics®.

The lowest-order magnetostatic mode in the YIG sphere (the Kittel mode,
$\mathbf{k}=0$) corresponds to a collective precession of the spins, which can
be described using a macroscopic spin [32, 33]. Whilst higher-order standing
wave modes (for $\mathbf{k}\neq 0$) exist [34], we focus on the fundamental
uniform mode in this work. For a spherical YIG sample, the ferromagnetic
resonance (FMR) frequency $\omega_{m}/2\pi$ can be tuned by an applied static
magnetic field $\mathbf{H}_{0}=H_{0}\hat{\mathbf{z}}$ as
$\omega_{m}=\gamma\absolutevalue{\mathbf{H}_{0}}$, with $\gamma$ the
gyromagnetic ratio.

After quantisation of the electromagnetic field, the cavity mode takes the
form of a quantised harmonic oscillator $\hbar\omega_{c}c^{\dagger}c$,
described using bosonic annihilation and creation operators $c$ and
$c^{\dagger}$ respectively. Provided the number of magnons is negligible
compared to the number of spins in the YIG, we can describe the magnon by a
bosonic annihilation operator $m$ after a Holstein-Primakoff transformation
[35], leading to a harmonic oscillator for the magnon mode [34]. The resulting
free Hamiltonian (i.e. without interactions) is the sum of two quantised
harmonic oscillators

$H_{\text{free}}=\hbar\omega_{c}c^{\dagger}c+\hbar\omega_{m}m^{\dagger}m,$ (1)

describing the cavity and magnon mode respectively. Note that these operators
commute.

###  II.2 Interaction term

The magnetic mode in the cavity, $\mathbf{H}$, couples to the macrospin of the
YIG sphere by a Zeeman interaction. Assuming the cavity mode $\mathbf{H}$ has
no $z$ dependence (for instance by considering a re-entrant cavity [30, 31,
36, 37]), the interaction term reads (details in appendix A)

$H_{I}=\hbar ge^{i\varphi}\quantity(c+c^{\dagger})m^{\dagger}+h.c,$ (2)

where the coupling strength $g/2\pi>0$ and coupling phase $\varphi>0$ are real
numbers, and $h.c$ denotes the omitted hermitian conjugate terms. The coupling
strength has the well-known expression [30, 31]

$\frac{g}{2\pi}=\eta\sqrt{\omega_{c}}\frac{\gamma}{4\pi}\sqrt{\frac{\mu}{g_{L}\mu_{B}}\mu_{0}\hbar
n_{s}}$ (3)

where $\mu_{B}$ is the Bohr magneton, $\mu=5\mu_{B}$ the magnetic moment of
YIG, $g_{L}=2$ the Landé $g$-factor, $\mu_{0}$ the magnetic permeability of
vacuum, $n_{s}=4.22\times 10^{27}\text{ m}^{-3}$ the spin density of YIG, and
the so-called form-factor

$\eta=\sqrt{\frac{\quantity(\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{x}}\differential[3]{r})^{2}+\quantity(\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{y}}\differential[3]{r})^{2}}{V_{m}\int_{V_{c}}\absolutevalue{\mathbf{H}}^{2}\differential[3]{r}}},$
(4)

with $V_{c}$ the volume of the cavity and $V_{m}$ the volume of the magnetic
sample. On the other hand, the coupling phase $\varphi$ reads

$\varphi=\arg\widetilde{H},$ (5)

where

$\widetilde{H}=\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{x}}\differential[3]{r}+i\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{y}}\differential[3]{r}.$
(6)

From the definition of $\widetilde{H}$, we see that the coupling phase depends
on the orientation of the cavity’s magnetic mode $\mathbf{H}$ traversing the
magnetic sample.

For moderate values of the coupling strength $g/2\pi$, we can perform the
rotating wave approximation (RWA) and neglect the counter-rotating terms
$cm+c^{\dagger}m^{\dagger}$ in eq. 2, so that the interaction simplifies to

$H_{I}=\hbar ge^{i\varphi}cm^{\dagger}+h.c,$ (7)

which is valid in the strong coupling regime ($g/\omega_{c}<0.1$), but not in
the ultrastrong coupling regime ($g/\omega_{c}\geqslant 0.1$) [11]. We refer
the reader to [38, 39, 40] for details about the ultrastrong coupling regime
and the applicability of the RWA. In the remainder of this paper, we will
always apply the RWA, and consider eq. 7 instead of eq. 2.

To conclude this section, we would like to mention that the coupling phase
$\varphi$ is fundamentally different from the phase factor used to model
dissipative couplings [41]. Indeed, the magnon/photon dissipative coupling can
be modelled as

$H_{I,\text{dissipative}}=\hbar J(cm^{\dagger}+e^{i\phi}c^{\dagger}m),$ (8)

where $J$ and $\phi$ are constants. We notice that this term is not hermitian,
contrary to eq. 7. As a result, a dissipatively coupled system can have
complex eigenvalues, which are at the origin of energy level attraction in the
spectrum. Conversely, a hermitian system guarantees real eigenvalues, and the
coupling manifests as the standard level repulsion instead, or in other words
an anti-crossing in the spectrum.

##  III Example with a physical phase

The previous section highlights the presence of a coupling phase $\varphi$ for
each magnon/photon coupling term. In this section, we consider a system under
the rotating wave approximation composed of two identical YIG spheres, each
coupling to two magnetic eigenmodes of the cavity. The free Hamiltonian of the
previous section is naturally generalised to

Figure 1: Illustration of the appearance of a physical phase $\theta$ when two
magnon modes $m_{0},m_{1}$ (blue circles) both couple to two cavity modes
$c_{0},c_{1}$ (red circles). The black lines between each circle represent a
coupling described by eq. 7 (the hermitian conjugate term is omitted for
readability). The coupling phase measures the difference in orientation of the
yellow arrows coming out of each circle. By moving to an appropriate reference
frame using a unitary transformation, we can align the yellow arrows of two
interacting modes, hence removing the coupling phase. We successively (from
left to right) rotate the cavity mode $c_{0}$, the cavity mode $c_{1}$, and
finally the magnon mode $m_{1}$ to align the yellow arrows with that of
$m_{0}$. After successive rotation of the operators, all yellow arrows point
in the same direction, and yet a physical phase
$\theta=\varphi_{11}-\varphi_{01}-\quantity(\varphi_{10}-\varphi_{00})$
remains. Note that the free Hamiltonian is unaffected by the successive
rotation of the modes, see appendix B.

$H_{\text{free}}=\hbar\omega_{c,0}c_{0}^{\dagger}c_{0}+\hbar\omega_{c,1}c_{1}^{\dagger}c_{1}+\hbar\omega_{m,0}m_{0}^{\dagger}m_{0}+\hbar\omega_{m,1}m_{1}^{\dagger}m_{1},$
(9)

where the $c_{k}$ describe the two magnetic eigenmodes of the cavity, and
$m_{k}$ the two magnon modes associated with each YIG sphere. Again, the
operators $c_{k}$ and $m_{k}$ all commute with each other, as they describe
independent harmonic oscillators. We introduce the notations $\omega_{o}$ and
$\delta_{o}$ for each operator $o\in\quantity{c,m}$ as

$\omega_{o}=\frac{\omega_{o,0}+\omega_{o,1}}{2},\quad\delta_{o}=\frac{\omega_{o,1}-\omega_{o,0}}{2},$
(10)

so that

$\omega_{o,0}=\omega_{o}-\delta_{o},\quad\omega_{o,1}=\omega_{o}+\delta_{o}.$
(11)

The microwave cavity design determines the two resonances of the cavity modes,
and hence the detuning $\delta_{c}$ between them and the average cavity
resonance $\omega_{c}$. For the magnons, the applied magnetic field
$\mathbf{H}_{0}$ tunes the average magnon frequency
${\omega}_{m}=\gamma\absolutevalue{\mathbf{H_{0}}}$, whilst the magnon
detuning $\delta_{m}$ can be created by a permanent magnet or a coil placed
near the YIG spheres (such a setup was used in [21] for instance).

Assuming that both magnon modes couple with the same coupling strength $g_{k}$
to the cavity mode $c_{k}$, the interaction Hamiltonian under the RWA is
generalised to

$\displaystyle H_{I}$ $\displaystyle=\hbar
g_{0}e^{i\varphi_{00}}c_{0}m_{0}^{\dagger}+\hbar
g_{0}e^{i\varphi_{01}}c_{0}m_{1}^{\dagger}+$ (12) $\displaystyle\quad\hbar
g_{1}e^{i\varphi_{10}}c_{1}m_{0}^{\dagger}+\hbar
g_{1}e^{i\varphi_{11}}c_{1}m_{1}^{\dagger}+h.c.$

This interaction Hamiltonian can be simplified by focusing all the coupling
phases into one term (see fig. 1) using unitary transformations (we recall
some elementary properties of unitary transformations in appendix B). The
transformed interaction Hamiltonian is

$\displaystyle H_{I,\theta}$ $\displaystyle=\hbar
g_{0}c_{0}m_{0}^{\dagger}+\hbar g_{0}c_{0}m_{1}^{\dagger}$ (13)
$\displaystyle\quad+\hbar g_{1}c_{1}m_{0}^{\dagger}+\hbar
g_{1}e^{i\theta}c_{1}m_{1}^{\dagger}+h.c,$

with $\theta=\varphi_{11}-\varphi_{01}-\quantity(\varphi_{10}-\varphi_{00})$.
The “physical phase” $\theta$ is reminiscent of a discrete Pancharatnam–Berry
phase, a gauge-invariant geometrical phase factor originating from the
existence of a local $U(1)$ gauge degree of freedom for each state involved in
a loop in some parameter space [42]. This $U(1)$ gauge is nothing but a choice
of phase, which in our system is the coupling phase located on the black lines
of fig. 1. The various coupling phases are linked to each other through the
circles, forming a loop – leading to an observable physical phase $\theta$.

To conclude this section, we note that most systems studied in the literature
consists of single magnon/photon systems, or several magnon modes coupling to
a single cavity mode. In particular, setups comprised of two magnon modes
coupling to one cavity mode have attracted interest for indirect coupling and
dark mode physics [21, 15, 16, 17, 19, 20], as well as the generation of
entangled states [43, 44, 45]. In these cases, the coupling phases do not lead
to a physical phase, as we show in appendix C. This is in stark contrast with
the system proposed here.

##  IV Influence of the physical phase

We now highlight the $\theta$-dependent physics of the system described in the
previous section, i.e.

$H_{\theta}=H_{\text{free}}+H_{I,\theta},$ (14)

with $H_{\text{free}}$ and $H_{I,\theta}$ given by eqs. 9 and 13. While
$\theta$ could in principle take any value in $\quantity[0,2\pi]$, we will
focus our analysis on $\theta=0$ and $\theta=\pi$, differing only in the sign
of the coupling strength of the last term.

###  IV.1 Diagonalisation method

The coupling between the photonic and matter degrees of freedom described by
eq. 13 leads to the appearance of quasiparticles known as polaritons.
Polaritons are hybridised light/matter states and, in the context of cavity
magnonics, they are known as cavity magnon-polaritons (CMP) [46, 3].
Diagonalisation of the system allows one to find the frequencies of the
polaritons, as well as their composition in terms of the cavity and magnon
mode operators $c_{k},m_{k}$. All the modes of the system being bosonic,
diagonalisation can be achieved by writing $H_{\theta}=\hbar{\bf\it
v}^{\dagger}A{\bf\it v}$ with ${\bf\it
v}=\matrixquantity(c_{0},c_{1},m_{0},m_{1})^{t}$ (t indicates matrix
transposition) and

$A=\matrixquantity(\omega_{c,0}&0&g_{0}&g_{0}\\\
0&\omega_{c,1}&g_{1}&g_{1}e^{-i\theta}\\\ g_{0}&g_{1}&\omega_{m,0}&0\\\
g_{0}&g_{1}e^{i\theta}&0&\omega_{m,1}).$ (15)

After diagonalisation of $A$, the eigenvectors give the polaritonic operators
$p_{\mu}$, while the eigenvalues give the polaritonic frequencies
$\omega_{\mu}/2\pi$. The polaritons can be expressed in terms of the initial
operators as

$p_{\mu}=u_{\mu 0}c_{0}+u_{\mu 1}c_{1}+v_{\mu 2}m_{0}+v_{\mu 3}m_{1},$ (16)

where the coefficients $u$ ($v$) are the first (last) two components of the
eigenvectors of $A$.

We note that in general, diagonalising $A$ amounts to solving a polynomial
equation of degree four, namely

$\det(A-\omega_{\mu}I_{4})=0$ (17)

where $I_{4}$ is the $4\times 4$ identity matrix. Whilst analytical solutions
exist, they are rather unpleasant and do not clarify the physics.

Figure 2: Numerical spectra of $H_{\theta}$ for $\theta=0$ and $\theta=\pi$ as
a function of the average magnon frequency $\omega_{m}/2\pi$, for two values
of the magnon detuning $\delta_{m}$. Parameters are $\omega_{c,0}=4$ GHz,
$\omega_{c,1}=6$ GHz and $g_{0}=g_{1}=0.03\omega_{c}$. The dashed line
corresponds to $\omega=\omega_{m}$. The legend in (a) is common to (b), (c)
and (d).

###  IV.2 Spectrum and cavity-mediated coupling

#### Spectral features.

We numerically diagonalised eq. 15 for $\theta=0$ and $\theta=\pi$ in fig. 2.
The coherent magnon/photon couplings manifest as two anti-crossings, each
located near the resonance of the associated cavity mode (i.e. at
$\omega_{c}\pm\delta_{c}$). We observe that the value of the physical phase
$\theta$ seems to have an impact only when $\delta_{m}=0$, since as
$\delta_{m}$ increases the cases $\theta=0$ and $\theta=\pi$ become less
distinguishable.

Interestingly, it appears that when $\delta_{m}=0$ and
$\omega_{m}=\omega_{c}$, the eigenvalues $\omega_{1}$ and $\omega_{2}$ cross
for $\theta=0$ (fig. 2), but do not for $\theta=\pi$ (fig. 2). Repulsion
between energy levels is usually the signature of a coupling phenomenon, while
level crossing is indicative of the absence of coupling. This suggests that at
$\omega_{m}=\omega_{c}$, the eigenmodes associated with $\omega_{1}$ and
$\omega_{2}$ couple for $\theta=\pi$, but do not for $\theta=0$.

#### Analysis in the dispersive regime.

To clarify this spectral feature, we consider the dispersive regime, where
both magnons are assumed to be far detuned from both cavity modes. Defining
the detuning $\Delta_{kk^{\prime}}=\omega_{m,k^{\prime}}-\omega_{c,k}$ between
the cavity mode $c_{k}$ and the magnon mode $m_{k^{\prime}}$, the dispersive
limit corresponds to $\absolutevalue{\Delta_{kk^{\prime}}}\gg g_{k}$ for all
$k,k^{\prime}\in\quantity{0,1}$. Under this approximation, we can employ a
Schrieffer-Wolff transformation to perform first-order perturbation theory in
$\absolutevalue{\frac{g_{k}}{\Delta_{kk^{\prime}}}}\ll 1$ (see appendix D). We
find that the magnon modes being significantly detuned from the cavity modes,
the photon/magnon couplings are negligible, and the cavity and magnon modes
decouple from each other (see eq. 57). However, virtual photons still mediate
a magnon/magnon interaction, which for $\omega_{m}=\omega_{c}$, is described
by the effective Hamiltonian (again, see appendix D for details)

$\displaystyle H^{\prime}_{\text{magnons}}$
$\displaystyle=\hbar\omega^{\prime}_{m,0}m_{0}^{\dagger}m_{0}+\hbar\omega^{\prime}_{m,1}m_{1}^{\dagger}m_{1}$
(18) $\displaystyle\quad+\hbar\quantity(G_{\theta}m_{0}m_{1}^{\dagger}+h.c),$

where the frequencies of the magnons are shifted due to the cavity modes as

$\omega^{\prime}_{m,k}=\omega_{c}+\frac{g_{0}^{2}}{\Delta_{0k}}+\frac{g_{1}^{2}}{\Delta_{1k}}$
(19)

and the indirect magnon-magnon coupling $G_{\theta}/2\pi$ is

$G_{\theta}=\frac{\delta_{c}}{\delta_{c}^{2}-\delta_{m}^{2}}\quantity(g_{0}^{2}-e^{i\theta}g_{1}^{2}).$
(20)

The magnitude of $G_{\theta}$ characterises the strength of the magnon-magnon
coupling, and predicts level repulsion of the eigenmodes of eq. 18. However,
for $g_{0}=g_{1}$ and $\theta=0$, the magnons do not interact since
$G_{\theta}$ vanishes, leading to crossing of the energy levels.

#### Cavity-mediated coupling.

To verify our analysis, we numerically plotted $\omega_{1}$ and $\omega_{2}$
by diagonalising eq. 15 at $\omega_{m}=\omega_{c}$ in fig. 3. We observe the
crossing of the solid lines (corresponding to $\theta=0$), while the dots
($\theta=\pi$) anti-cross. The situation changes if we set $g_{0}\neq g_{1}$
as shown in fig. 3, and now we observe level repulsion for both $\theta=0$ and
$\theta=\pi$. This behaviour is exactly the one we inferred in the dispersive
regime. Note that our analysis of the dispersive regime is valid if
$\absolutevalue{\frac{g_{k}}{\Delta_{kk^{\prime}}}}\ll 1$. At
$\omega_{m}=\omega_{c}$, this is equivalent to imposing
$g_{0},g_{1}\ll\absolutevalue{\delta_{c}-\delta_{m}}$, which the parameters
employed in fig. 3 satisfy. Regardless of the values of $g_{0}$ and $g_{1}$,
fig. 3 shows that the frequency gap between the eigenmodes of eq. 18 is
minimal when $\omega^{\prime}_{m,0}=\omega^{\prime}_{m,1}$, i.e
$\delta_{m}=0$. The location of this minimum gap is given by
$\omega_{c}^{\prime}=\omega_{c}+\frac{g_{0}^{2}-g_{1}^{2}}{\delta_{c}}$ due to
the frequency shift of the magnon modes described by eq. 19.

Physically, eq. 20 highlights an interference effect between the magnon-magnon
couplings contributed by each cavity mode. This interference can be
constructive for $\theta=\pi$, or destructive for $\theta=0$, and we always
have $\absolutevalue{G_{\pi}}>\absolutevalue{G_{0}}$. In particular, if
$g_{0}=g_{1}$, the cavity-mediated coupling between the magnons completely
vanishes for $\theta=0$. We conclude that the case $\theta=0$ is always
detrimental to indirectly couple two spatially distant magnons, while the case
$\theta=\pi$ is benificial. This conclusion highlights the importance of the
physical phase $\theta$ for multimode cavity-mediated coupling applications.

Figure 3: Numerical spectra of $H_{\theta}$ for $\theta=0$ (solid lines), and
$\theta=\pi$ (dots) as a function of the magnon detuning $\delta_{m}$ at
$\omega_{m}=\omega_{c}$. The dashed lines are at
$\omega_{c}^{\prime}\pm\absolutevalue{G_{\pi}}$ while the dotted lines are at
$\omega_{c}^{\prime}\pm\absolutevalue{G_{0}}$, with $G_{\theta}$ evaluated
using eq. 20 for $\delta_{m}=0$. We set $g_{0}=g_{1}=0.01\omega_{c}$ in (a),
$g_{0}=0.01\omega_{c,0}$ and $g_{1}=0.01\omega_{c,1}$ in (b). The definition
of $\omega_{c}^{\prime}$ is per the main text. Other parameters and legend are
as in fig. 2.

###  IV.3 Dark mode physics

#### Rotated magnon basis.

After having looked at the spectrum, we now turn our attention to the
eigenmodes of $H_{\theta}$. The physics will be more easily understood by
considering a rotated magnon basis ${M_{\theta},M_{\theta+\pi}}$, defined by

$M_{\theta}=\frac{m_{0}+e^{-i\theta}m_{1}}{\sqrt{2}}.$ (21)

It is easy to prove that, since $m_{0}$ and $m_{1}$ commute with each other,
so do $M_{\theta}$ and $M_{\theta+\pi}$. Note that $M_{0}=M_{2\pi}$
corresponds to co-rotating magnons (i.e. both magnons precess in-phase) while
$M_{\pi}$ corresponds to counter-rotating magnons (precessing out-of-phase).
In this basis, the free Hamiltonian eq. 9 now reads

$\displaystyle H_{\text{free}}$
$\displaystyle=\hbar\omega_{c,0}c_{0}^{\dagger}c_{0}+\hbar\omega_{c,1}c_{1}^{\dagger}c_{1}$
(22)
$\displaystyle\quad+\hbar\omega_{m}M_{\theta}^{\dagger}M_{\theta}+\hbar\omega_{m}M_{\theta+\pi}^{\dagger}M_{\theta+\pi}$
$\displaystyle\quad-\delta_{m}\quantity(M_{\theta}M_{\theta+\pi}^{\dagger}+h.c).$

while the interaction Hamiltonian eq. 13 reads

$\displaystyle H_{I,\theta}$
$\displaystyle=\hbar\widetilde{g}_{0}e^{-i\theta/2}c_{0}\quantity(\cos\frac{\theta}{2}M_{\theta}-i\sin\frac{\theta}{2}M_{\theta+\pi})^{\dagger}$
(23) $\displaystyle\quad+\hbar\widetilde{g}_{1}c_{1}M_{\theta}^{\dagger}+h.c,$

where we introduced the notation $\widetilde{g}_{k}=\sqrt{2}g_{k}$. The
different interaction pathways between the modes are illustrated in fig. 4.

Notably, if $\delta_{m}=0$, $H_{\theta}$ describes the interaction between
four different modes, while for $\theta=0$ it separates into a 3-mode system
$\quantity{c_{0},c_{1},M_{0}}$ and a one-mode system $\quantity{M_{\pi}}$, and
for $\theta=\pi$ it separates into two two-mode systems
$\quantity{c_{0},M_{0}}$ and $\quantity{c_{1},M_{\pi}}$. In particular, for
$\delta_{m}=0$ and $\theta=0$ (fig. 4), we see that $M_{\pi}$ does not couple
to any other mode.

Figure 4: Schematic of the interactions present in $H_{\theta}$ in the rotated
magnon basis (see eqs. 22 and 23) for (a) arbitrary $\theta$, (b) $\theta=0$,
(c) $\theta=\pi$. The black lines are labelled by the magnitude of the
couplings, and the dotted black line represents the possibility of vanishing
$\delta_{m}$. Note that $M_{0}=M_{2\pi}$.

#### Dark mode definition.

The presence of such an uncoupled eigenmode is reminiscent of dark mode
physics. Dark (bright) states are states that weakly (strongly) couple with
the readout mechanism. Due to their weak coupling, these states benefit from
an increased lifetime over their bright counterpart, and hence are interesting
candidates for storing information. Dark and bright modes are defined
similarly, but refer to the associated polaritonic mode in continuous variable
systems.

The definition of a dark mode given above is dependent on the mechanism used
to probe the system. Since here we are interested in photon/magnon
hybridisation, we consider a detection based on a microwave transmission
experiment mediated by photons. As a consequence, we expect that any
eigenmodes coupling with cavity modes will indirectly couple to the readout
medium, hence acquiring an experimental signature.

The input-output formalism [47] allows one to model this scenario. To
illustrate our discussion of dark modes, we now consider the transmission
through the cavity ($S_{21}$ parameter), experimentally obtained by attaching
a vector network analyser to the two ports of a cavity. The derivation of the
input-output theory is rather lengthy, so we refer the reader to appendix E
for more details. We consider that each cavity mode $c_{k}$ couples to a
photonic bath with coupling rate $\sqrt{\gamma/2\pi}$ and we set $\gamma=5$
MHz. We further introduce the intrinsic dissipation $\kappa=1$ MHz of the
cavity (due to its finite quality factor) and magnon modes (Gilbert damping)
through $\widetilde{\omega}_{o,k}=\omega_{o,k}-i\kappa$. Other parameters are
as in fig. 2. The numerical results are plotted in fig. 5.

Figure 5: Numerical calculation of the microwave transmission through the
system obtained using an input-output theory for $H_{\theta}$ at different
magnon detuning $\delta_{m}$. See the main text for the parameters. In (a) we
see that the spectral line associated to $\omega_{1}$ (see fig. 2) does not
lead to a maximum of transmission, contrary to (b), (c) and (d). This shows
that the eigenmode associated with $\omega_{1}$ is a dark mode.

#### Dark mode physics.

Given the definition of a dark mode given above, we conclude that for
$\theta=0$, $M_{\pi}$ is dark, since it is uncoupled to cavity modes. This is
indeed what we observe in fig. 5: comparing to the spectrum in fig. 2, the
polaritonic frequency $\omega_{1}/2\pi$ associated with $M_{\pi}$ does not
give a maximum of transmission, unlike the other frequencies. This is in
contrast with fig. 5, in which all the polaritonic frequencies lead to maxima
of transmissions. Indeed, for $\theta=\pi$, $M_{\pi}$ couples to $c_{1}$.
Hence, since all the eigenmodes for $\theta=\pi$ couple to photons, we
conclude that strictly speaking, there are no dark modes for $\theta=\pi$.

These observations rely on both magnons modes having identical resonance
frequencies, i.e. $\delta_{m}=0$. If this symmetry is broken by setting
$\delta_{m}\neq 0$, a new interaction pathway between $M_{\pi}$ and $M_{0}$
opens (see fig. 4), regardless of the value of $\theta$. Since $M_{0}$ couples
to the cavity modes, then $M_{\pi}$ will also couple to the cavity modes
through $M_{0}$. As a consequence, we expect $M_{\pi}$ to acquire an
experimental signature if $\delta_{m}\neq 0$. This theoretical prediction is
supported numerically by figs. 5 and 5. We note that this phenomenon of the
illumination of a dark mode by symmetry breaking was already discussed by [20]
in the interaction picture.

To conclude this section, we comment on the presence of two types of anti-
resonances. The first type corresponds to dark diagonal lines which follow the
FMR of the YIG spheres. They are the result of destructive interference
between the FMR and the RF magnetic mode of the cavity, as explained by [48].
The other type is due to destructive interference between the two cavity
modes, and corresponds to the dark horizontal lines centred around
$\omega_{c}=\frac{\omega_{c,0}+\omega_{c,1}}{2}$. These anti-resonances are
uncoupled for $\theta=0$ (straight line for all $\omega_{m}$), while for
$\theta=\pi$ an anti-crossing appears. This phenomenon can be explained by the
fact that for $\theta=0$, both cavity modes couple to $M_{0}$ with equal
coupling strengths, while the situation is different for $\theta=\pi$ (see
fig. 4). Had we set $g_{0}\neq g_{1}$, the anti-resonance due to the cavity
modes would have also coupled near the FMR for $\theta=0$.

##  V Conclusion

To summarise our results, we showed that the interaction between a microwave
photon and a magnon in a cavity is always associated with a coupling phase,
which is different from that in dissipatively coupled systems. In most cases
studied in the literature, this coupling phase can be omitted because of the
simple topology of the interaction diagrams. Indeed, we traced the
manifestation of the coupling phase to the presence of loops in the
interaction diagrams (as illustrated by fig. 6), a phenomenon similar to a
discrete Pancharatnam–Berry phase. To illustrate that the physics was
dependent on the coupling phase, we considered a model composed of two cavity
modes and two magnon modes, parametrised by the physical phase $\theta$. For
different values of $\theta$, we found different behaviours with respect to
two potential applications of cavity magnonics: one case is advantageous for
cavity-mediated coupling but cannot be used for dark mode memories
($\theta=\pi$), and vice-versa for the other case ($\theta=0$). We note that
while we showed results only for $\theta=0$ and $\theta=\pi$, our analytical
derivations were still parametrised by a general $\theta$, and hence apply to
intermediate cases. In fact, values of $\theta$ between 0 and $\pi$
continuously interpolate between the results presented in this paper (see the
cavity-mediated coupling strength of eq. 20 for example).

One could question whether there are simpler models in which there is a
physical phase. In theory, any system in which there are $n$ interactions
between $n$ bosonic modes, with one being a microwave photon/magnon
interaction, leads to a physical phase. One such example was considered by
Zhan  _et al._ , where two ferromagnetic bilayers couple to a common cavity
mode, but can also directly couple together thanks to the interlayer exchange
interaction. This results in a triangular-shaped interaction diagram, so that
in principle, a physical phase should parametrise their Hamiltonian. However,
it is unclear how the impact of the physical phase could be experimentally
verified in such a system. First, achieving different coupling phases for each
ferromagnetic bilayers is a challenging experimental task, since the two
layers need to be close to each other but be penetrated by the same cavity
mode with a different angle (see eq. 5). Furthermore, as noted by the authors
of Zhan  _et al._ , the experimental verification of their analysis requires
smaller magnon damping rates than currently reported in experiments.

Instead, the model introduced in section III should be experimentally
accessible using re-entrant cavity designs. Indeed, previous experimental
results demonstrated the tunability of the frequencies of the cavity modes,
with values of the coupling strengths compatible with the RWA [31, 36, 37].
Placing two YIG spheres in a 3-post re-entrant cavity should allow to
implement the two cases $\theta=0$ and $\theta=\pi$ discussed here, and will
be the subject of a separate investigation.

The engineering of cavities to achieve specific physical phases is an exciting
research direction. The design of such cavities will consists in controlling
the relative orientation of the magnetic modes traversing the magnetic
samples, as described by eq. 5. Finally, we note that we considered only a
rather simple model in which a physical phase manifests. For instance,
considering an additional cavity mode would lead to the appearance of two
physical phases, for which the physics is yet to explored. More generally, the
existence of discrete Pancharatnam–Berry phases in microwave cavity magnonics
may find applications in quantum information processing.

###### Acknowledgements.

We acknowledge financial support from Thales Australia and Thales Research and
Technology. We thank Tyler Whittaker, Thomas Kong and Ross Monaghan for
reading the manuscript and providing useful comments. The scientific colour
map oslo [49] is used in this study to prevent visual distortion of the data
and exclusion of readers with colourvision deficiencies [50].

##  Appendix A Photon/magnon coupling term

In this appendix, we show that the interaction term between a microwave cavity
mode $c$ and a magnon mode $m$ can be expressed as

$H_{I}=\hbar ge^{i\varphi}(c+c^{\dagger})m^{\dagger}+h.c,$ (24)

with $g$ and $\varphi$ positive real numbers. This derivation is valid in the
macrospin approximation [32, 33] and provided the number of magnon excitations
is small compared to the number of spins.

Our starting point is the supplementary material of ref. [30]. In particular,
it is shown that in the macrospin approximation the interaction term is given
by

$\displaystyle H_{I}$ $\displaystyle=\hbar
g^{x}\quantity(c+c^{\dagger})\quantity(m+m^{\dagger})+\hbar
ig^{y}\quantity(c+c^{\dagger})\quantity(m-m^{\dagger})$ (25)
$\displaystyle\quad+\hbar
g^{z}\quantity(c+c^{\dagger})m^{\dagger}m+\hbar\Omega^{z}\quantity(c+c^{\dagger})$

with

$\displaystyle g^{x}$
$\displaystyle=-\frac{\gamma}{2cV_{m}}\sqrt{\frac{\hbar\omega_{c}S}{\epsilon_{0}}}\frac{\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{x}}\differential[3]{r}}{\sqrt{\int_{V_{c}}\absolutevalue{\mathbf{H}}^{2}\differential[3]{r}}},$
(26) $\displaystyle g^{y}$
$\displaystyle=\frac{\gamma}{2cV_{m}}\sqrt{\frac{\hbar\omega_{c}S}{\epsilon_{0}}}\frac{\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{y}}\differential[3]{r}}{\sqrt{\int_{V_{c}}\absolutevalue{\mathbf{H}}^{2}\differential[3]{r}}},$
(27) $\displaystyle g^{z}$
$\displaystyle=\frac{\gamma}{cV_{m}}\sqrt{\frac{\hbar\omega_{c}}{2\epsilon_{0}}}\frac{\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{z}}\differential[3]{r}}{\sqrt{\int_{V_{c}}\absolutevalue{\mathbf{H}}^{2}\differential[3]{r}}},$
(28) $\displaystyle\Omega^{z}$ $\displaystyle=-\frac{\gamma
S}{cV_{m}}\sqrt{\frac{\hbar\omega_{c}}{2\epsilon_{0}}}\frac{\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{z}}\differential[3]{r}}{\sqrt{\int_{V_{c}}\absolutevalue{\mathbf{H}}^{2}\differential[3]{r}}},$
(29)

where $\gamma$ is the gyromagnetic ratio, $V_{m}$ the volume of the magnetic
sample, $V_{c}$ the volume of the cavity, $\omega_{c}$ the resonance of the
cavity, $S=\frac{\mu}{g_{L}\mu_{B}}N_{S}$ the total spin number of the
macrospin operator, $\mu$ the magnetic moment of the magnetic sample,
$g_{L}=2$ the Landé $g$-factor, $\mu_{B}$ the Bohr magneton, $N_{s}$ the
number of spins in the magnetic sample, $c$ the speed of light in vacuum,
$\epsilon_{0}$ the permittivity of vacuum, and $\mathbf{H}$ the magnetic mode
of the cavity.

Assuming that the magnetic mode has no $z$ dependence, $g^{z}=\Omega^{z}=0$
and eq. 25 can be written as
$H_{I}=\hbar\quantity(g^{x}-ig^{y})\quantity(c+c^{\dagger})m^{\dagger}+h.c$
with

$\displaystyle g^{x}-ig^{y}$
$\displaystyle=-\frac{\gamma}{2cV_{m}}\sqrt{\frac{S\hbar\omega_{c}}{\epsilon_{0}}}\frac{\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{x}}\differential[3]{r}+i\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{y}}\differential[3]{r}}{\sqrt{\int_{V_{c}}\absolutevalue{\mathbf{H}}^{2}\differential[3]{r}}}$
(30)
$\displaystyle=-\frac{\gamma}{2}\sqrt{\frac{\mu_{0}S\hbar\omega_{c}}{V_{m}}}\frac{\widetilde{H}}{\sqrt{V_{m}\int_{V_{c}}\absolutevalue{\mathbf{H}}^{2}\differential[3]{r}}}$
(31)

where we used $\mu_{0}\epsilon_{0}=1/c^{2}$ and we recognise the quantity
$\widetilde{H}=\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{x}}\differential[3]{r}+i\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{y}}\differential[3]{r}$
defined in eq. 6 of the main text. We introduced $\mathbf{H}$ as a cavity
mode, which means that it is the solution of an eigenvalue problem for the
electromagnetic field obeying the Maxwell’s equations in the cavity. By
definition, $-\mathbf{H}$ is also a solution to this eigenvalue problem, and
hence eq. 31 is still valid after replacing $\mathbf{H}\mapsto-\mathbf{H}$.
This removes the inconvenient minus sign eq. 31 so that we can now write

$\displaystyle H_{I}$
$\displaystyle=\frac{\hbar\gamma}{2}\sqrt{\frac{\mu_{0}S\hbar\omega_{c}}{V_{m}}}\frac{\widetilde{H}}{\sqrt{V_{m}\int_{V_{c}}\absolutevalue{\mathbf{H}}^{2}\differential[3]{r}}}\quantity(c+c^{\dagger})m^{\dagger}+h.c$
(32)
$\displaystyle=\frac{\hbar\gamma}{2}\sqrt{\frac{\mu_{0}S\hbar\omega_{c}}{V_{m}}}\eta
e^{i\theta}\quantity(c+c^{\dagger})m^{\dagger}+h.c$ (33)

where we defined

$\displaystyle\eta$
$\displaystyle=\frac{\absolutevalue{\widetilde{H}}}{\sqrt{V_{m}\int_{V_{c}}\absolutevalue{\mathbf{H}}^{2}\differential[3]{r}}}$
(34)
$\displaystyle=\sqrt{\frac{\quantity(\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{x}}\differential[3]{r})^{2}+\quantity(\int_{V_{m}}\mathbf{H}\cdot\hat{\mathbf{y}}\differential[3]{r})^{2}}{V_{m}\int_{V_{c}}\absolutevalue{\mathbf{H}}^{2}\differential[3]{r}}},$
(35) $\displaystyle\varphi$ $\displaystyle=\arg\widetilde{H}$ (36)

as announced in the main text.

To derive the expression of the coupling strength $g/2\pi$ as per the main
text, we follow [31] and write
$S=\frac{\mu}{g_{L}\mu_{B}}N_{s}=\frac{\mu}{g_{L}\mu_{B}}n_{s}V_{m}$ with
$n_{s}$ the spin density of the magnetic sample. Equation 33 is then rewritten

$H_{I}=\hbar ge^{i\varphi}\quantity(c+c^{\dagger})m^{\dagger}+h.c,$ (37)

with

$g/2\pi=\eta\sqrt{\omega_{c}}\frac{\gamma}{4\pi}\sqrt{\frac{\mu}{g_{L}\mu_{B}}\mu_{0}\hbar
n_{s}}.$ (38)

##  Appendix B Unitary transformations

A time-dependent unitary transformation $U$ maps a Hamiltonian $H$ to
$H^{\prime}$ such that

$H\mapsto H^{\prime}=UHU^{\dagger}+i\hbar\dot{U}U^{\dagger}.$ (39)

This corresponds to the so-called rotating frame transformations. For a time-
independent unitary transformation, $H^{\prime}=UHU^{\dagger}$, we simply
refer to it as a change of frame. The Baker–Campbell–Hausdorff formula is
useful for computing the transformed Hamiltonian, it reads

$e^{X}Ye^{-X}=\sum_{n=0}^{\infty}\frac{\quantity[X^{(n)},Y]}{n!}$ (40)

where $X$ and $Y$ are operators, and we note $\quantity[\quantity(X)^{(n)},Y]$
the $n$-th iterated commutator defined as

$\displaystyle\quantity[X^{(0)},Y]\equiv
Y,\quad\quantity[X^{(1)},Y]=\quantity[X,Y]=XY-YX$ (41)
$\displaystyle\quantity[X^{(2)},Y]=\quantity[X,\quantity[X,Y]],\quad\quantity[X^{(3)},Y]=\quantity[X,\quantity[X^{(2)},Y]]$
(42)

and so on.

For an annihilation operator $a$ following bosonic commutation relation
$\quantity[a,a^{\dagger}]=1$ and $\varphi\in\mathbb{R}$, the unitary
transformation $U=e^{i\varphi a^{\dagger}a}$ transforms $a$ and $a^{\dagger}$
as

$\displaystyle UaU^{\dagger}$ $\displaystyle=e^{i\varphi
a^{\dagger}a}ae^{-i\varphi a^{\dagger}a}=ae^{-i\varphi},$ (43) $\displaystyle
Ua^{\dagger}U^{\dagger}$ $\displaystyle=a^{\dagger}e^{i\varphi}.$ (44)

The associated number operator $a^{\dagger}a$ is unaffected by the
transformation, since

$\displaystyle Ua^{\dagger}aU^{\dagger}$
$\displaystyle=Ua^{\dagger}U^{\dagger}UaU^{\dagger}=a^{\dagger}e^{i\varphi}ae^{-i\varphi}$
(45) $\displaystyle=a^{\dagger}a$ (46)

where $U^{\dagger}U=I$ since $U$ is unitary.

Additionally, for any operator $O$ that commutes with $a^{\dagger}a$, i.e
$\quantity[a^{\dagger}a,O]=0$, then $O$ is unaffected by the transformation,
i.e $UOU^{\dagger}=O$, due to the vanishing of the iterated commutators in the
Baker–Campbell–Hausdorff formula (eq. 40).

##  Appendix C Relevance of the phase factor

###  C.1 Removal of the coupling phase for a magnon/photon system

In the simple case of a single cavity mode coupling to a single magnon mode,
the Hamiltonian is $H=H_{\text{free}}+H_{I}$ defined by eqs. 1 and 7. In light
of fig. 6, we can

Figure 6: Example diagrams showing coupling phase removal by unitary
transformations. The conventions used are that of fig. 1. (a) Rotation of the
cavity mode $c$ using $U_{c}=e^{i\varphi c^{\dagger}c}$. The magnon mode $m$
is unaffected. (b) Rotation of the magnon mode $m$ using $U_{m}=e^{-i\varphi
m^{\dagger}m}$. (c) Rotation of both magnon modes using
$U=e^{-i\varphi_{j}m_{j}^{\dagger}m_{j}}$ for $j\in\quantity{0,1}$. (d)
Rotation of the cavity mode $c$ using $U=e^{i\varphi_{0}c^{\dagger}c}$ and the
magnon mode $m_{1}$ using
$U=e^{-i\quantity(\varphi_{1}-\varphi_{0})m_{1}^{\dagger}m_{1}}$.

choose to rotate the cavity mode by $\varphi$ using $U_{c}=e^{i\varphi
c^{\dagger}c}$. Formally, $H_{\text{free}}\mapsto
H_{\text{free}}^{\prime}=H_{\text{free}}$ is unchanged while

$\displaystyle H_{I}\mapsto H_{I}^{\prime}$ $\displaystyle=e^{i\varphi
c^{\dagger}c}\quantity(\hbar ge^{i\varphi}cm^{\dagger}+h.c)e^{-i\varphi
c^{\dagger}c}$ (47) $\displaystyle=\hbar ge^{i\varphi}\quantity(e^{i\theta
c^{\dagger}c}ce^{-i\varphi c^{\dagger}c})\quantity(e^{i\theta
c^{\dagger}c}m^{\dagger}e^{-i\varphi c^{\dagger}c})+h.c$ (48)
$\displaystyle=\hbar ge^{i\varphi}\quantity(ce^{-i\varphi})m^{\dagger}+h.c$
(49) $\displaystyle=\hbar gcm^{\dagger}+h.c$ (50)

hence removing the coupling phase $\varphi$. Pictorially, this is illustrated
on the right of fig. 6, where we see that the yellow arrow of mode $c$ has
changed orientation, and is now pointing in the same direction as that of mode
$m$. Alternatively, as illustrated in fig. 6, we can rotate the magnon mode
instead, using $U_{m}=e^{-i\varphi m^{\dagger}m}$, which yields the same
result, albeit with a different final orientation of the yellow arrows.

The fact that $\varphi$ is absent after a change of frame implies that it
cannot be physical. Similarly, note that the two extremal cases $\varphi=0$
and $\varphi=\pi$ differ only by the sign of the coupling strength, and hence
we deduce that its polarity does not affect the physics.

###  C.2 Removal of the coupling phase for two magnons coupling to a common
cavity mode

We now assume that an additional magnon mode couples to the cavity mode, as
considered in many works [21, 15, 16, 17, 19, 20, 43, 44, 45]. We also note
that since all modes are bosonic, this scenario is equivalent to one where a
single magnon mode couples to two cavity modes, as considered e.g by [48]. The
Hamiltonian is $H=H_{\text{free}}+H_{I}$ with

$\displaystyle H_{\text{free}}$
$\displaystyle=\hbar\omega_{c}c^{\dagger}c+\hbar\omega_{m,0}m_{0}^{\dagger}m_{0}+\hbar\omega_{m,1}m_{1}^{\dagger}m_{1},$
(51) $\displaystyle H_{I}$ $\displaystyle=\hbar
g_{0}e^{i\varphi_{0}}cm_{0}^{\dagger}+\hbar
g_{1}e^{i\varphi_{1}}cm_{1}^{\dagger}+h.c,$ (52)

and is illustrated in figs. 6 and 6.

One simple possibility to remove the coupling phases $\varphi_{i}$ is to
successively rotate $m_{i}$ using
$U_{m_{i}}=e^{-i\varphi_{i}m_{i}^{\dagger}m_{i}}$ for $i\in\quantity{0,1}$
(fig. 6). While this is the simplest method, we could also rotate the cavity
mode $c$ (fig. 6), for instance to remove $\varphi_{0}$ using
$U_{c}=e^{i\varphi_{0}c^{\dagger}c}$, but then the coupling to the second
magnon is also affected and reads $\hbar
g_{1}e^{i\quantity(\varphi_{1}-\varphi_{0})}cm_{1}^{\dagger}+h.c$. Applying
$U_{m_{1}}=e^{-i\quantity(\varphi_{1}-\varphi_{0})m_{1}^{\dagger}m_{1}}$
removes the remaining phase.

Formally, it appears that the coupling phases are again unimportant for the
resulting physics. While the physics is unchanged, we would like to mention
that the _interpretation_ of the physics is: the rotated frame is not the
laboratory frame. As an example, [20] studied the specific case of
$\varphi_{0}=\varphi_{1}=0$ and $g_{0}=g_{1}=g$, and found that
$M_{-}=\frac{m_{0}-m_{1}}{\sqrt{2}}$ was the only dark mode of the system. If
we were to consider $g_{0}=-g_{1}=g$ instead (obtained by applying $U=e^{-i\pi
m_{1}^{\dagger}m_{1}}$ for instance), we would find that the dark mode is
transformed to $M_{+}=\frac{m_{0}+m_{1}}{\sqrt{2}}$ in this frame. Still, the
spectrum and other physical predictions of [20] would all hold.

###  C.3 Two cavity modes and two magnon modes

We note that the removal of the coupling phases in the previous examples was
facilitated by the fact that the interaction diagrams had open ends: the
rotation of the modes at the ends of the interaction chain would at most
affect one edge. Having a closed loop, however, removes this freedom,
resulting in an observable physical phase $\theta$. We believe the hybrid
system introduced in section III is the simplest system with an interaction
loop involving a magnon/photon interaction, which could be experimentally
accessible.

##  Appendix D Effective Hamiltonian in the dispersive regime

For the purpose of this section, we adopt different notations for convenience.
The model studied in the main text can be written as
$H_{\theta}=H_{\text{free}}+H_{I}$

$\displaystyle H_{\text{free}}$
$\displaystyle=\sum_{k=0}^{1}\hbar\omega_{c,k}c_{k}^{\dagger}c_{k}+\hbar\omega_{m,k}m_{k}^{\dagger}m_{k}$
(53) $\displaystyle H_{I}$ $\displaystyle=\sum_{k,k^{\prime}=0}^{1}\hbar
g_{kk^{\prime}}c_{k}m_{k^{\prime}}^{\dagger}+h.c$ (54)

with $g_{0k^{\prime}}=g_{0}$, $g_{10}=g_{1}$, and $g_{11}=g_{1}e^{i\theta}$.
We will hence assume that $g_{kk^{\prime}}\in\mathbb{C}$ in this section, even
though we will mostly be interested in the cases $\theta=0$ or $\pi$, for
which the $g_{kk^{\prime}}$ become reals. We also introduce the detuning
$\Delta_{kk^{\prime}}=\omega_{m,k^{\prime}}-\omega_{c,k}$ between the cavity
mode $c_{k}$ and the magnon mode $m_{k^{\prime}}$. In the rest of this
section, we assume that the magnons are significantly detuned from the both
cavity mode such that $\absolutevalue{\lambda_{kk^{\prime}}}\ll 1$ with
$\lambda_{kk^{\prime}}=\frac{g_{kk^{\prime}}}{\Delta_{kk^{\prime}}}$ for
$k,k^{\prime}\in\quantity{0,1}$.

To find the effective Hamiltonian in this regime, we employ a Schrieffer-Wolff
transformation [51] to treat $H_{I}$ as a perturbation of $H_{\text{free}}$.
Using the unitary transformation $U=e^{\Lambda}$ with

$\Lambda=\sum_{k,k^{\prime}}\quantity(\lambda_{kk^{\prime}}c_{k}m_{k^{\prime}}^{\dagger}-\lambda_{kk^{\prime}}^{*}c^{\dagger}_{k}m_{k^{\prime}}),$
(55)

we have $H_{I}+\quantity[\Lambda,H_{\text{free}}]=0$ and the
Baker–Campbell–Hausdorff formula (see eq. 40) gives
$H^{\prime}_{\theta}=e^{\Lambda}H_{\theta}e^{-\Lambda}=H_{\text{free}}+\frac{1}{2}\quantity[\Lambda,H_{I}]$
to first order in $\absolutevalue{\lambda}$.

After using commutator identities, we find the commutator

$\displaystyle=2\hbar\sum_{k}\quantity[\quantity(\sum_{l}\frac{\absolutevalue{g_{lk}}^{2}}{\Delta_{lk}})m_{k}^{\dagger}m_{k}-\quantity(\sum_{l}\frac{\absolutevalue{g_{kl}}^{2}}{\Delta_{kl}})c_{k}^{\dagger}c_{k}]$
(56) $\displaystyle\quad+\hbar\sum_{k\neq
k^{\prime}}\quantity[\quantity(\sum_{l}\lambda_{lk^{\prime}}g_{lk}^{*})m_{k}m_{k^{\prime}}^{\dagger}+h.c]$
$\displaystyle\quad-\hbar\sum_{k\neq
k^{\prime}}\quantity[\quantity(\sum_{l}\lambda_{kl}g_{k^{\prime}l}^{*})c_{k}c_{k^{\prime}}^{\dagger}+h.c]+\text{cst}.$

The constant originates from using the bosonic commutation relations for the
operators $c_{k},m_{k}$. Hence, after applying $U=e^{\Lambda}$, we obtain the
effective Hamiltonian

$\displaystyle H^{\prime}_{\theta}$
$\displaystyle=\sum_{k}\hbar\omega_{c,k}^{\prime}c_{k}^{\dagger}c_{k}+\hbar\omega_{m,k}^{\prime}m_{k}^{\dagger}m_{k}$
(57) $\displaystyle\quad+\sum_{k\neq
k^{\prime}}\quantity(\kappa_{kk^{\prime}}c_{k}c_{k^{\prime}}^{\dagger}+h.c)+\sum_{k\neq
k^{\prime}}\quantity(G_{kk^{\prime}}m_{k}m_{k^{\prime}}^{\dagger}+h.c)$

with shifted resonances

$\displaystyle\omega_{c,k}^{\prime}$
$\displaystyle=\omega_{c,k}-\sum_{k^{\prime}}\frac{\absolutevalue{g_{kk^{\prime}}}^{2}}{\Delta_{kk^{\prime}},}$
(58) $\displaystyle\omega_{m,k}^{\prime}$
$\displaystyle=\omega_{m,k}+\sum_{k^{\prime}}\frac{\absolutevalue{g_{k^{\prime}k}}^{2}}{\Delta_{k^{\prime}k}},$
(59)

photon/photon couplings

$\kappa_{kk^{\prime}}=-\sum_{l}\frac{g_{kl}g_{k^{\prime}l}^{*}}{2\Delta_{kl}},$
(60)

and magnon/magnon couplings

$G_{kk^{\prime}}=\sum_{l}\frac{g_{lk^{\prime}}g_{lk}^{*}}{2\Delta_{lk^{\prime}}}.$
(61)

The effective Hamiltonian highlights the decoupling of the photonic and matter
degrees of freedom in the dispersive regime: light and matter do not interact
anymore. However, we obtained a photon/photon coupling mediated by virtual
magnons, nad magnon/magnon couplings mediated by virtual photons.

Concentrating on the magnon modes, and setting $\omega_{m}=\omega_{c}$, we
obtain the effective Hamiltonian for the magnons

$H_{m}^{\prime}=\sum_{k}\hbar\omega_{m,k}^{\prime}m_{k}^{\dagger}m_{k}+\hbar\quantity(G_{\theta}m_{0}m_{1}^{\dagger}+h.c)$
(62)

with $G_{\theta}=G_{01}+G_{10}^{*}$ and shifted magnon frequencies

$\displaystyle\omega_{m,0}^{\prime}$
$\displaystyle=\omega_{c}+\frac{\quantity(g_{0}^{2}+g_{1}^{2})\delta_{m}+\quantity(g_{0}^{2}-g_{1}^{2})\delta_{c}}{\delta_{c}^{2}-\delta_{m}^{2}},$
(63) $\displaystyle\omega_{m,1}^{\prime}$
$\displaystyle=\omega_{c}-\frac{\quantity(g_{0}^{2}+g_{1}^{2})\delta_{m}-\quantity(g_{0}^{2}-g_{1}^{2})\delta_{c}}{\delta_{c}^{2}-\delta_{m}^{2}}.$
(64)

##  Appendix E Details of the input-output theory

In the main text, we consider the transmission of the cavity by microwave
photons. In this appendix, we derive an analytical expression of the
transmission of the cavity using the input-output formalism [47]. We note that
similar results can be obtained employing the loop theory introduced in [52].
In the following, we assume that we have replaced
$\omega_{o,k}\mapsto\omega_{o,k}-i\kappa$ for
$o_{k}\in\quantity{c,m},k\in\quantity{0,1}$ with $\kappa$ the intrinsic
dissipation rate.

#### Bath modelling.

We model the environment with a bosonic bath Hamiltonian $H_{\text{bath}}$
with operators $a_{\omega}$ and $b_{\omega}$ for the left and right side of
the cavity. It reads

$H_{\text{bath}}=\int_{-\infty}^{\infty}\differential{\omega}\quantity(\hbar\omega
a_{\omega}^{\dagger}a_{\omega}+\hbar\omega b_{\omega}^{\dagger}b_{\omega}).$
(65)

The interaction between the bath and the cavity is

$H_{\text{sys-
bath}}=i\hbar\int_{-\infty}^{\infty}\differential{\omega}\quantity(\sum_{k=0}^{1}c_{k}\quantity(\kappa_{a,k}a^{\dagger}_{\omega}+\kappa_{b,k}b^{\dagger}_{\omega})+h.c),$
(66)

where we define $\sqrt{\gamma_{o,k}/2\pi}=\kappa_{o,k}$ for the bath operators
$o\in\quantity{a,b}$. The Heisenberg equation of motion for the bath operators
$o\in\quantity{a,b}$ is

$\dot{o}_{\omega}=-\frac{i}{\hbar}\quantity[o_{\omega},H_{\text{bath}}+H_{\text{sys-
bath}}]=-i\omega o_{\omega}+\sum_{k}\kappa_{o,k}c_{k},$ (67)

with the formal solutions

$\displaystyle o_{\omega}(t)$
$\displaystyle=o_{\omega}(t_{0})e^{-i\omega(t-t_{0})}$ (68)
$\displaystyle\quad+\sum_{k}\kappa_{o,k}\int_{t_{0}}^{t}\differential{t^{\prime}}c_{k}(t^{\prime})e^{-i\omega(t-t^{\prime})}$

for $t_{0}<t$ and

$\displaystyle o_{\omega}(t)$
$\displaystyle=o_{\omega}(t_{1})e^{-i\omega(t-t_{1})}$ (69)
$\displaystyle\quad-\sum_{k}\kappa_{o,k}\int_{t}^{t_{1}}\differential{t^{\prime}}c_{k}(t^{\prime})e^{-i\omega(t-t^{\prime})}$

for $t<t_{1}$.

#### System for the cavity modes in frequency space.

We now define the input fields

$o^{\text{in}}(t)=\frac{1}{\sqrt{2\pi}}\int\differential{\omega}e^{-i\omega(t-t_{0})}o_{\omega}(t_{0})$
(70)

and using the solution of eq. 68 for $t_{0}<t$ we have

$\kappa_{o,k}^{*}\int_{\mathbb{R}}\differential{\omega}o_{\omega}(t)=\sqrt{\gamma_{o,k}^{*}}o^{\text{in}}(t)+\sum_{k^{\prime}}\frac{\sqrt{\gamma_{o,k}^{*}\gamma_{o,k^{\prime}}}}{2}c_{k^{\prime}}(t)$
(71)

We adopt the notations of appendix D and write the interaction term
$H_{I}=\sum_{k,k^{\prime}=0}^{1}g_{kk^{\prime}}c_{k}m_{k^{\prime}}^{\dagger}+h.c$.
The Heisenberg equation for the cavity modes at $t_{0}<t$ reads

$\displaystyle\dot{c}_{k}$
$\displaystyle=-\frac{i}{\hbar}\quantity[c_{k},H_{\text{sys}}+H_{\text{sys-
bath}}]$ (72)
$\displaystyle\begin{gathered}=-i\omega_{c,k}c_{k}-i\sum_{k^{\prime}}g_{kk^{\prime}}m_{k^{\prime}}-\sum_{o\in\quantity{a,b}}\sqrt{\gamma_{o,k}^{*}}o^{\text{in}}(t)\\\
-\sum_{o\in\quantity{a,b},k^{\prime}}\frac{\sqrt{\gamma_{o,k}^{*}\gamma_{o,k^{\prime}}}}{2}c_{k^{\prime}}(t)\end{gathered}$
(75)

while that for the magnon modes is

$\displaystyle\dot{m}_{k}$
$\displaystyle=-\frac{i}{\hbar}\quantity[m_{k},H_{\text{sys}}]=-i\omega_{m,k}m_{k}-i\sum_{k^{\prime}}g_{k^{\prime}k}c_{k^{\prime}}.$
(76)

In frequency space,

$-i\omega\widetilde{m}_{k}=-i\omega_{m,k}\widetilde{m}_{k}-i\sum_{k^{\prime}}g_{k^{\prime}k}\widetilde{c}_{k^{\prime}}$
(77)

which gives

$\widetilde{m}_{k}=\frac{\sum_{k^{\prime}}g_{k^{\prime}k}\tilde{c}_{k^{\prime}}}{\omega-\omega_{m,k}}.$
(78)

At this stage, it is convenient to introduce the detunings

$\Delta_{o,k}=\omega-\omega_{o,k},\quad
o\in\quantity{c,m},\,k\in\quantity{0,1}.$ (79)

The cavity modes in Fourier space then read

$\begin{gathered}\Delta_{c,k}\tilde{c}_{k}+\sum_{o\in\quantity{a,b},k^{\prime}}\quantity(i\frac{\sqrt{\gamma_{o,k}^{*}\gamma_{o,k^{\prime}}}}{2}-\sum_{l}\frac{g_{kl}g_{k^{\prime}l}}{\Delta_{m,l}})\tilde{c}_{k^{\prime}}\\\
+\sum_{o\in\quantity{a,b}}i\sqrt{\gamma_{o,k}^{*}}\tilde{o}^{\text{in}}=0.\end{gathered}$
(80)

_Separating the terms $k=k^{\prime}$ and $k\neq k^{\prime}$_, we obtain the
system

$\displaystyle
A_{0}\tilde{c}_{0}+B_{01}\tilde{c}_{1}+\sum_{o\in\quantity{a,b}}O_{o,0}\tilde{o}^{\text{in}}$
$\displaystyle=0$ (81) $\displaystyle
B_{10}\tilde{c}_{0}+A_{1}\tilde{c}_{1}+\sum_{o\in\quantity{a,b}}O_{o,1}\tilde{o}^{\text{in}}$
$\displaystyle=0$ (82)

with the solutions

$\displaystyle\tilde{c}_{0}$
$\displaystyle=\sum_{o\in\quantity{a,b}}\frac{O_{o,0}A_{1}-O_{o,1}B_{01}}{B_{10}B_{01}-A_{0}A_{1}}\tilde{o}^{\text{in}}$
(83) $\displaystyle\tilde{c}_{1}$
$\displaystyle=\sum_{o\in\quantity{a,b}}-\frac{O_{o,0}B_{10}-O_{o,1}A_{0}}{B_{10}B_{01}-A_{0}A_{1}}\tilde{o}^{\text{in}}$
(84)

and

$\displaystyle A_{k}$
$\displaystyle=\quantity(\Delta_{c,k}+i\sum_{o\in\quantity{a,b}}\frac{\absolutevalue{\gamma_{o,k}}}{2})\Delta_{m,0}\Delta_{m,1}$
$\displaystyle\quad-g_{k0}^{2}\Delta_{m,1}-g_{k1}^{2}\Delta_{m,0},$ (85)
$\displaystyle B_{kk^{\prime}}$
$\displaystyle=i\quantity(\sum_{o\in\quantity{a,b}}\frac{\sqrt{\gamma_{o,k}^{*}\gamma_{o,k^{\prime}}}}{2})\Delta_{m,0}\Delta_{m,1}$
$\displaystyle\quad-
g_{k0}g_{k^{\prime}0}\Delta_{m,1}-g_{k1}g_{k^{\prime}1}\Delta_{m,0},$ (86)
$\displaystyle O_{o,k}$
$\displaystyle=i\sum_{o\in\quantity{a,b}}\sqrt{\gamma_{o,k}^{*}}\Delta_{m,0}\Delta_{m,1}.$
(87)

#### Input-output relations.

We now define the output field

$o^{\text{out}}(t)=\frac{1}{\sqrt{2\pi}}\int\differential{\omega}e^{-i\omega(t-t_{1})}o_{\omega}(t_{1}),$
(88)

and using the solution at $t<t_{1}$ we have

$\displaystyle\kappa_{o,k}^{*}\int_{\mathbb{R}}\differential{\omega}o_{\omega}(t)$
$\displaystyle=\sqrt{\gamma_{o,k}^{*}}o^{\text{out}}(t)-\sum_{k^{\prime}}\frac{\sqrt{\gamma_{o,k}^{*}\gamma_{o,k^{\prime}}}}{2}c_{k^{\prime}}(t)$
(89)

which combined with eq. 71 leads to input-output relations

$\tilde{o}^{\text{out}}-\tilde{o}^{\text{in}}=\sum_{k}\sqrt{\gamma_{o,k}}\tilde{c}_{k}$
(90)

valid for each bath operator $o\in\quantity{a,b}$.

#### Transmission.

The transmission through the system can be defined as

$S_{21}=\left.\frac{\expectationvalue{\tilde{b}^{\text{out}}}}{\expectationvalue{\tilde{a}^{\text{in}}}}\right|_{\expectationvalue{\tilde{b}^{\text{in}}}=0}=\frac{\sum_{k}\sqrt{\gamma_{b,k}}\expectationvalue{\tilde{c}_{k}}}{\expectationvalue{\tilde{a}^{\text{in}}}}$
(91)

where we used the input-output relations. Using eqs. 83 and 84, we obtain

$S_{21}=\sqrt{\gamma_{b,0}}\frac{O_{a,0}A_{1}-O_{a,1}B_{01}}{B_{10}B_{01}-A_{0}A_{1}}-\sqrt{\gamma_{b,1}}\frac{O_{a,0}B_{10}-O_{a,1}A_{0}}{B_{10}B_{01}-A_{0}A_{1}}.$
(92)

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