# Magnon-Cooparons in magnet-superconductor hybrids

I. V. Bobkova  Institute of Solid State Physics, Chernogolovka, 142432 Moscow
region, Russia  Moscow Institute of Physics and Technology, Dolgoprudny,
141700 Moscow region, Russia  National Research University Higher School of
Economics, 101000 Moscow, Russia  A. M. Bobkov  Institute of Solid State
Physics, Chernogolovka, 142432 Moscow region, Russia  Akashdeep Kamra
Condensed Matter Physics Center (IFIMAC) and Departamento de Física Teórica de
la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
Wolfgang Belzig  Fachbereich Physik, Universität Konstanz, D-78457 Konstanz,
Germany

###### Abstract

Generation and detection of spinfull Cooper pairs in conventional
superconductors has been intensely pursued by designing increasingly complex
magnet-superconductor hybrids. We demonstrate theoretically that magnons with
nonzero wavenumbers universally induce a cloud of spinfull triplet Cooper
pairs around them in an adjacent conventional superconductor. The resulting
composite quasiparticle, termed magnon-cooparon, consists of a spin flip in
the magnet screened by a cloud of the spinfull superfluid condensate. Thus, it
inherits a large effective mass, which can be measured experimentally.
Furthermore, we demonstrate that two magnetic wires deposited on a
superconductor serve as a controllable magnonic directional coupler mediated
by the nonlocal and composite nature of magnon-cooparons. Our analysis
predicts a quasiparticle that enables generation, control, and use of spinfull
triplet Cooper pairs in the simplest magnet-superconductor heterostructures.

Introduction.—The widely available and used conventional superconductors
consist of spin-singlet Cooper pairs which are devoid of a net spin.
Unconventional superconductors, in contrast, host qualitatively distinct
phenomena and Cooper pair properties [1, 2]. Their limited experimental
availability, however, has driven the scientific community to try and engineer
heterostructures comprising conventional superconductors into effectively
unconventional ones [3, 4, 5, 6, 7], e.g., in achieving Majorana bound states
[8, 9]. In particular, the highly desired spinfull spin-triplet Cooper pairs
111In this Letter, we use the term “spinfull Cooper pairs” to describe the
equal-spin spin-triplet Cooper pairs (e.g.,
$\left|\uparrow\uparrow\right\rangle$) that bear a net spin along the $z$
direction. They are to be contrasted with the spinless spin-triplet Cooper
pairs
($\sim\left|\uparrow\downarrow\right\rangle+\left|\downarrow\uparrow\right\rangle$)
that do not exhibit a net spin. The latter kind of Cooper pairs are commonly
found in spin-active superconducting hybrids, but they are not capable of
delivering, for example, a nondissipative spin transfer torque. can be
generated from a conventional superconductor if the latter interacts with two
or more non-collinear magnetic moments [3, 4, 5, 6, 7]. With this design
principle, a wide range of magnet-superconductor hybrids with multiple
magnetic layers to generate and detect spinfull Cooper pairs have been
investigated [11, 12, 13, 14, 15, 16]. The challenge of detecting a spin or
its flow directly has resulted in the need for increasingly complex magnet-
superconductor hybrids rendering their direct detection a highly demanding,
debated, and pursued goal [17, 18, 19, 20].

The ambition is to go beyond detection, and towards exploiting the fascinating
physics of these unconventional Cooper pairs for phenomena that are otherwise
out of reach [21]. For example, the spinfull Cooper pairs could be used for
delivering nondissipative spin transfer torques and magnetic switching [22,
23, 24, 25, 26, 27]. Such goals face a similar challenge that even when a
complex heterostructure generates spinfull Cooper pairs, it becomes difficult
to steer them. An injected charge current predominantly converts into
conventional spinless supercurrent [22]. Due to such reasons, several
advantages of magnet-superconductor heterostructures realized in various
concepts and devices are still dominated by the quasiparticle properties [28,
29, 30, 7, 31, 32, 33, 34, 35]. The exploiting of spinfull Cooper pairs for
new physics and devices has been impeded by the complex hybrids needed to
generate them and the difficulty of controlling them.

In this Letter, we uncover a ubiquitous existence and control of spinfull
Cooper pairs in the simplest magnet-superconductor hybrid - a bilayer - that
has escaped attention thus far. We find that a magnon, the quasiparticle of
spin waves in a magnet, with nonzero wavevector induces a cloud of spinfull
Cooper pairs in the adjacent superconductor [Fig. 1(a)]. This accompanying
cloud screens the magnon spin giving rise to a composite heavy quasiparticle
with an enhanced effective mass, which is termed magnon-cooparon. This
induction of spinfull Cooper pairs in a conventional superconductor is caused
by the noncollinear magnetization profile of a spin wave with finite
wavevector [Fig. 1(b)], an effect not seen when considering ferromagnetic
resonance of the uniform magnon mode. Furthermore, we demonstrate
theoretically that magnon-cooparons enable a magnonic directional coupler [36,
37] composed of two separate ferromagnetic wires with coupling lengths shorter
than previously feasible. Thus, it enables a new application in magnon-based
logic and circuits [38, 39]. The magnon-cooparon is reminiscent of the
fermionic polaron quasiparticle created by screening of an electron by a
phonon cloud [40], although the magnon-cooparon is a bosonic excitation.
Considering the gradual discovery of polaron and its variants in a wide range
of phenomena [40, 41, 42], we expect magnon-cooparon to find a similar
important role in a broad range of magnet-superconductor hybrids.

Figure 1: (a) A localized spin-flip or magnon wave packet induces a
surrounding cloud of spinfull triplet Cooper pairs in an adjacent conventional
spin-singlet superconductor. The spatially varying magnetization or spin
profile (depicted via dashed line) associated with the excitation induces
spinfull condensate, that screens the magnon spin, in the otherwise spinless
superconductor. The resulting quasiparticle, termed magnon-cooparon, bears a
smaller spin and larger effective mass. (b) A spin wave with wavenumber $k$
propagates in an in-plane direction. The associated noncollinear magnetization
profile (red arrows) induces an analogous spatially varying exchange field
(black arrows) in the adjacent superconductor, resulting in spinfull triplet
condensate.

Emergence and effective mass of magnon-cooparons.—We consider a bilayer as
depicted in Fig. 1(b), in which a ferromagnetic insulator FI (e.g., yttrium
iron garnet) is interfaced with a conventional spin-singlet s-wave
superconductor S (e.g., Nb). The two layers with thicknesses $d_{\mathrm{FI}}$
and $d_{S}$ ($\ll\xi_{S}$, the superconducting coherence length) are
considered thin such that physical properties vary only in the in-plane
direction. In its ground state, the FI is assumed to be magnetized along the z
direction.

We wish to examine excitations of the hybrid. To this end, we assume existence
of a spin wave with wavevector $k\boldsymbol{e}_{z}$ in the FI [Fig. 1(b)]
such that the magnetization unit vector
$\boldsymbol{m}(\boldsymbol{r},t)=\boldsymbol{m}_{0}+\delta\boldsymbol{m}(\boldsymbol{r},t)$
consists of the equilibrium part $\boldsymbol{m}_{0}=\boldsymbol{e}_{z}$ and
the excitation part $\delta\boldsymbol{m}(\boldsymbol{r},t)=\delta
m\left[\cos(kz+\omega t)\boldsymbol{e}_{x}+\sin(kz+\omega
t)\boldsymbol{e}_{y}\right]\exp(-\kappa t)$. While we consider an excitation
with wavevector along $\boldsymbol{e}_{z}$, our analysis is general and valid
for any in-plane wavevector. The magnetization dynamics is described within
the Landau-Lifshitz-Gilbert framework [43] as

$\displaystyle\dot{\boldsymbol{m}}$
$\displaystyle=-\gamma\left(\boldsymbol{m}\times\boldsymbol{H}_{\mathrm{eff}}\right)+\alpha\left(\boldsymbol{m}\times\dot{\boldsymbol{m}}\right)+\tilde{J}\left(\boldsymbol{m}\times\boldsymbol{s}\right),$
(1)

where $\gamma~{}(>0)$ is the FI gyromagnetic ratio, $\alpha$ is the Gilbert
damping parameter, $\boldsymbol{H}_{\mathrm{eff}}$ is the effective magnetic
field in the FI, and $\tilde{J}\equiv J/d_{\mathrm{FI}}$ with $J$
parameterizing interfacial exchange interaction between FI and S. The last
term on the right hand side of Eq. (1) accounts for the spin torque exerted on
the magnetization by the spin density $\boldsymbol{s}$ it induces in S [44].
Expressing $\boldsymbol{s}=s_{0}\boldsymbol{m}_{0}+\delta
s_{\parallel}\delta\boldsymbol{m}+\delta
s_{\perp}(\delta\boldsymbol{m}\times\boldsymbol{m}_{0})$ and substituting the
expressions for $\boldsymbol{s}$ and $\boldsymbol{m}$ in Eq. (1) above, we
obtain

$\displaystyle\omega$ $\displaystyle=D_{m}k^{2}+\gamma K+\tilde{J}\left(\delta
s_{\parallel}-s_{0}\right),$ (2) $\displaystyle\kappa$
$\displaystyle=\alpha\omega-\tilde{J}\delta s_{\perp},$ (3)

where $D_{m}$ is the FI spin wave stiffness and $K$ parameterizes a uniaxial
anisotropy. Thus, the spin density $\boldsymbol{s}$ induced in the S may
renormalize both the excitation frequency and its lifetime.

We now evaluate the induced spin density $\boldsymbol{s}$ treating S using the
quasiclassical Green’s functions framework [45, 7, 46, 5]. Working in the
dirty limit, we need to solve the Usadel equation for the $8\times 8$ matrix
Green’s function $\check{g}$ in spin, particle-hole, and Keldysh spaces:

$\displaystyle
iD\boldsymbol{\nabla}\left(\check{g}\otimes\boldsymbol{\nabla}\check{g}\right)$
$\displaystyle=\left[\epsilon\hat{\mathbb{I}}\hat{\tau}_{z}\hat{\mathbb{I}}-\boldsymbol{h}\cdot\hat{\boldsymbol{\sigma}}\hat{\tau}_{z}\hat{\mathbb{I}}+i\Delta\hat{\mathbb{I}}\hat{\tau}_{y}\hat{\mathbb{I}},\check{g}\right]_{\otimes},$
(4)

where $\hat{\mathbb{I}}$ is the $2\times 2$ identity matrix, outer-product
between the $2\times 2$ matrices (decorated by overhead $\hat{}~{}$) in
obtaining an $8\times 8$ matrix (identified via an overhead $\check{}~{}$) is
implied, and we set $\hbar=1$ throughout this Letter. Further, working in the
mixed ($\epsilon,t$) representation, we employ the notation
$[A,B]_{\otimes}\equiv A\otimes B-B\otimes A$ with $A\otimes
B\equiv\exp[(i/2)(\partial_{\epsilon_{1}}\partial_{t_{2}}-\partial_{\epsilon_{2}}\partial_{t_{1}})]~{}A(\epsilon_{1},t_{1})B(\epsilon_{2},t_{2})|_{\epsilon_{1}=\epsilon_{2}=\epsilon,t_{1}=t_{2}=t}$.
$\hat{\tau}_{x,y,z}$ and $\hat{\sigma}_{x,y,z}$ are Pauli matrices in
particle-hole and spin spaces, respectively. A real $\Delta$ accounts for the
intrinsic conventional spin-singlet order parameter of the superconductor. The
term $\boldsymbol{h}\cdot\hat{\boldsymbol{\sigma}}$ accounts for the exchange
field induced by the adjacent FI layer [47, 48, 5, 49, 7, 50, 51, 52]. For the
magnetization profile associated with the excitation under consideration, we
obtain

$\displaystyle\boldsymbol{h}$ $\displaystyle=h_{0}\boldsymbol{e}_{z}+\delta
h\left[\cos(kz+\omega t)\boldsymbol{e}_{x}+\sin(kz+\omega
t)\boldsymbol{e}_{y}\right],$ (5)

where $h_{0}$ and $\delta h$ respectively capture the static and dynamic
components of the induced exchange field
$\boldsymbol{h}=JM_{s}\boldsymbol{m}/2\gamma d_{S}$, with $M_{s}$ the FI
saturation magnetization. The contribution of the superconductor dynamics to
the excitation under consideration can be evaluated by solving Eqs. (4) and
(5) up to the first order in $\delta h$. The desired spin density in the
superconductor is evaluated as [7]

$\displaystyle\boldsymbol{s}$ $\displaystyle=-\frac{N_{F}}{16}\int
d\epsilon~{}\mathrm{Tr}_{4}\left[\left(\hat{\boldsymbol{\sigma}}\hat{\tau}_{z}\right)\breve{g}^{K}\right],$
(6)

where $\mathrm{Tr}_{4}$ denotes trace over a $4\times 4$ matrix (decorated by
an overhead $\breve{}~{}$), $\breve{g}^{K}$ is the $4\times 4$ Keldysh
component of the full $8\times 8$ Green’s function $\check{g}$, and $N_{F}$ is
the normal state density of states at the Fermi level in S.

Figure 2: Relative change in the spin stiffness $\delta D_{m}/D_{m}$ as a
function of temperature $T$ for different values of the static exchange field
$h_{0}$ induced in S. A reduction of the spin stiffness signifies an increased
effective mass of the composite excitation. $T_{c}$ is the critical
temperature of the superconductor. $\Delta_{0}$ is the superconducting gap
when $T=h_{0}=0$. The dashed line plots the analytic result [Eq. (7)] obtained
in the limit $T\to T_{c}$.

Following the method outlined above and detailed in the Supplemental Material
(SM) [53], we obtain analytic expressions for $\boldsymbol{s}$ [Eq. (6)], and
thus, the excitation frequency [Eq. (2)] and lifetime [Eq. (3)]. These
unwieldy expressions simplify considerably in the adiabatic limit of
$\omega\ll\Delta_{0}$, that we discuss first. Further, this limit allows us to
disregard quasiparticles and focus on the superfluid condensate. We find
$\delta s_{\perp}\to 0$ in this limit such that the excitation decay rate [Eq.
(3)] is not influenced by S. This is consistent with the nondissipative nature
of the condensate. $\delta s_{\parallel}-s_{0}$ is found to scale as $\sim
k^{2}$ in this limit, such that the excitation frequency becomes
$\omega=\tilde{D}_{m}k^{2}+\gamma K$ with $\tilde{D}_{m}\equiv D_{m}+\delta
D_{m}$ and

$\displaystyle\delta D_{m}$ $\displaystyle=-\frac{\pi N_{F}\gamma
d_{S}D\Delta^{2}}{16T_{c}d_{\mathrm{FI}}M_{s}}\left[\frac{1}{x}\tanh
x-\frac{1}{\cosh^{2}x}\right],$ (7)

where $x=h_{0}/2T_{c}$, and $T_{c}$ is the superconducting critical
temperature taking into account the static exchange field $h_{0}$. In
obtaining Eq. (7), we further worked in the limit $|T-T_{c}|\ll T_{c}$. The
same stiffness renormalization [Eq. (7)] is obtained from purely energy
considerations within the Ginzburg-Landau framework [54]. The effective mass
$m_{\mathrm{eff}}$ of the composite quasiparticle is obtained as
$m_{\mathrm{eff}}=1/2\tilde{D}_{m}=1/(2D_{m}+2\delta D_{m})$. Since $\delta
D_{m}<0$ [Eq. (7)], the effective mass of the composite quasiparticle is
enhanced as compared to that of a magnon. Numerically evaluated $\delta D_{m}$
plotted in Fig. 2 versus temperature further shows the direct role of the
superconducting condensate and suggests temperature as a handle to control the
quasiparticle effective mass. The material parameters assumed in Fig. 2 are
detailed further below, together with the discussion on experimental
detection.

Thus, this composite quasiparticle shares some similarities with the polaron
[40]. The latter, predicted almost a century ago [55, 56] and having found
numerous applications throughout condensed matter physics [40, 41, 42], is
formed when an electron is screened by the phonon cloud leading to a heavy
fermionic excitation. The quasiparticle under consideration is a bosonic
magnon spin being screened by a superconducting condensate. Due to this
similarity (and yet many distinctions), we term this spin flip surrounded by a
spinfull Cooper pairs cloud [Fig. 1(a)] magnon-cooparon. Physically, a finite
wavenumber $k$ is needed to create spinfull Cooper pairs, in an otherwise
spinless conventional superconductor, via a noncollinear exchange field [Fig.
1(b)]. This explains the $\sim k^{2}$ dependence of the frequency
renormalization, as well as the previous studies investigating uniform ($k=0$)
magnon modes not encountering the magnon-cooparon.

Finally, going beyond the adiabatic approximation $\omega\ll\Delta_{0}$, we
find a nonzero renormalization of the $k=0$ mode frequency and the decay rate
$\sim k^{2}$ [53]. Similar effects are expected based on spin pumping into a
normal metal or quasiparticles in a superconductor [57, 58, 31]. Specifically,
when quasiparticle spin relaxation is disregarded, the increase in decay rate
requires a spin sink, which may be provided by the noncollinear magnetic
moment in a second magnet [59]. In our case, a spatially distinct part of the
same magnet provides the noncollinear spin absorption channel, thereby short-
circuiting the spin wave. Hence, in this case too, the unique dynamic
noncollinearity of the finite-$k$ spin wave results in novel effects.

Spin of magnon-cooparons.—The cloud of spinfull Cooper pairs screening the
magnon spin that increases its effective mass further implies that (i) the
total spin of magnon-cooparon is reduced from 1 222In this work, we have
considered spin-1 magnons and disregarded squeezing effects [78, 79], for
simplicity., and (ii) a magnon spin current $j_{m}\boldsymbol{e}_{z}$ in FI is
accompanied by a superfluid spin current $j_{S}\boldsymbol{e}_{z}$ in S. We
now address these effects and ascertain the net spin of the magnon-cooparon.

Since the dc spin current $j_{m}$ accompanying a spin wave or magnon scales as
$\delta m^{2}$, we anticipate $j_{S}$ to scale as $\delta h^{2}$, confirming
this via a rigorous calculation [53]. As a result, we now need to solve Eq.
(4) for the matrix Green’s function up to the second order in $\delta h$.
Since this is a more demanding calculation than that carried out above, we
restrict ourselves to the adiabatic approximation $\omega\ll\Delta_{0}$ and
the limit $|T-T_{c}|\ll T_{c}$ in the rest of our analysis. The spin current
flowing along the direction of magnon propagation ($\boldsymbol{e}_{z}$ here)
in S is then obtained as [7]

$\displaystyle\boldsymbol{j}_{S}\boldsymbol{e}_{z}$
$\displaystyle=\frac{N_{F}D}{16}\int
d\epsilon~{}\mathrm{Tr}_{4}\left[\left(\hat{\boldsymbol{\sigma}}\hat{\mathbb{I}}\right)\left(\check{g}\partial_{z}\check{g}\right)^{K}\right]~{}\boldsymbol{e}_{z},$
(8)

where the direction of $\boldsymbol{j}_{S}$ pertains to the spin space. On
explicit evaluation [53], $\boldsymbol{j}_{S}$ is found to bear only a z
component, as can be expected from its screening the magnon spin, which itself
bears only a z component. The total spin current may thus be expressed as

$\displaystyle\boldsymbol{j}_{m}+\boldsymbol{j}_{S}=Sv_{k}n_{k}~{}\boldsymbol{e}_{z}\equiv\left(1+\frac{j_{Sz}}{j_{m}}\right)v_{k}n_{k}~{}\boldsymbol{e}_{z},$
(9)

where $v_{k}=2\tilde{D}_{m}k$ is the magnon-cooparon group velocity, $n_{k}$
is the number of excitations, and $S$ becomes its net spin evaluated via

$\displaystyle\frac{j_{Sz}}{j_{m}}$
$\displaystyle=-\frac{8N_{F}D\gamma}{\tilde{D}_{m}M_{s}}~{}\sum_{\omega_{n}>0}\frac{\pi
T_{c}\Delta^{2}h_{0}^{2}~{}\omega_{n}^{2}}{\left(\omega_{n}^{2}+h_{0}^{2}\right)^{2}\left(2\omega_{n}+Dk^{2}\right)^{2}},$
(10)

where $\omega_{n}$ are the fermionic Matsubara frequencies. Since
$j_{Sz}/j_{m}<0$, the net spin of the magnon-cooparon is reduced from 1 as per
our expectation from the screening. Further, similar to the relative change in
the spin stiffness (Fig. 2), the spin reduction $|j_{Sz}/j_{m}|\lesssim 1$ for
typical material parameters, as estimated further below.

Figure 3: Schematic depiction of a magnonic directional coupler based on
magnon-cooparons. A spin wave propagating through one FI wire is controllably
transferred to the second FI wire.

Superfluid-mediated magnonic directional coupler.—Since the Cooper pairs cloud
comprising a magnon-cooparon extends over a length $\sim\xi_{S}$, it enables
transfer of energy from a spin wave in one FI wire to another, nonlocally
(Fig. 3) 333This is to be contrasted with the recent theoretical proposal [80]
in which a charge current flow through an unconventional and exotic spin-
triplet superconductor, which is inevitably spin polarized, induces magnon
spin current in an adjacent ferromagnet. As per our analysis, the effect
predicted there vanishes when one considers a conventional spin-singlet
superconductor, as we consider here.. Thus, two FI wires deposited on a
conventional superconductor within $\sim\xi_{S}$ from each other act as a
magnonic directional coupler [36, 37], proposed to be a key building block in
wave-based logic and computing [62, 63, 64, 65, 66, 67]. The magnon-cooparon
based design that we demonstrate here offers stronger coupling strengths,
smaller footprint, additional control (e.g., via temperature), and
universality (e.g., for antiferromagnets) as compared to the dipole-
interaction based designs considered previously [36, 37].

Considering the setup depicted in Fig. 3, we now assume existence of spin
waves with the wavevector $k\boldsymbol{e}_{z}$ in both FIs, assumed identical
for simplicity. As a result, there exists the same static exchange field
$h_{0}$ in S below both the FI wires. However, distinct dynamic exchange
fields $\delta h_{l,r}$, similar to Eq. (5), exist in S below each of the FI
wires. These are proportional to the respective spin wave amplitudes $\delta
m_{l,r}$ in the two FIs. Solving the Usadel equation (4) under this exchange
field profile as detailed in the SM, we obtain the spin density in S:
$\boldsymbol{s}(x)=\boldsymbol{s}_{l}(x)+\boldsymbol{s}_{r}(x)$ with

$\displaystyle\boldsymbol{s}_{l,r}(x)$
$\displaystyle=s_{0}(x)+s_{\mathrm{loc}}(x)\frac{\delta\boldsymbol{h}_{l,r}}{h_{0}}+s_{\mathrm{nl}}(x)\frac{\delta\boldsymbol{h}_{r,l}}{h_{0}}.$
(11)

Here, the contributions $s_{0}$ and $s_{\mathrm{loc}}$ are due to the static
and dynamic exchange fields induced by the FI directly above the S region.
Thus, these are identical to our analysis of the magnon-cooparon in a FI/S
bilayer. The nonlocal contribution $s_{\mathrm{nl}}(x)$ characterizes the spin
density generated in S below the left FI by the right one, and vice versa.
Relegating its detailed expression to the SM, we note that for
$d\lesssim\xi_{S}$, $s_{\mathrm{nl}}$ is comparable to the spin density
$s_{\mathrm{loc}}$ accompanying a magnon-cooparon.

The induced nonlocal spin density leads to a fieldlike spin torque with the
contribution
$\tilde{J}\bar{s}_{\mathrm{nl}}\boldsymbol{m}_{l,r}\times\boldsymbol{m}_{r,l}$
to the magnetization dynamics $\dot{\boldsymbol{m}}_{l,r}$ in the two FIs
[68], where $\bar{s}_{\mathrm{nl}}$ is $s_{\mathrm{nl}}(x)$ averaged over the
width $t$ of the FI wire (Fig. 3) [53]. The resulting eigenmodes are magnon-
cooparons distributed over the two FIs and the S layer with dispersion:
$\omega_{\pm}=\gamma K+\tilde{D}_{m}k^{2}\mp\tilde{J}\bar{s}_{\mathrm{nl}}$.
Hence, a pure spin wave injected with frequency $\omega$ into the left FI
transfers its energy via the spinfull superfluid to the right FI after
traveling the so-called coupling length $L$:

$\displaystyle L$
$\displaystyle=\frac{2\pi}{k_{+}-k_{-}}=\frac{\pi\tilde{D}_{m}(k_{+}+k_{-})}{\tilde{J}\bar{s}_{\mathrm{nl}}},$
(12)

where $k_{\pm}$ are the wavenumbers corresponding to the frequency $\omega$ of
the injected spin wave. A smaller $L$ allows transfer of energy and the
concomitant implementation of logic operations in smaller devices and thus is
advantageous.

Numerical estimates and experimental detection.—We now employ material
parameters pertinent to yttrium iron garnet [69] as FI and Nb as S in finding
the effects discussed above to be large. We consider $D_{m}=5\cdot
10^{-29}~{}\mathrm{erg}\cdot\mathrm{cm}^{2}$, $M_{s}=140$ G, $\gamma=1.76\cdot
10^{7}~{}\mathrm{G}^{-1}\mathrm{s}^{-1}$, $\gamma K=10^{-17}$ erg,
$D=3~{}\mathrm{cm}^{2}\mathrm{s}^{-1}$, $N_{F}=1.3\cdot
10^{35}~{}\mathrm{erg}^{-1}\mathrm{cm}^{-3}$, $\Delta_{0}=18$ K, and
$\xi_{S}=\sqrt{D/\Delta_{0}}\sim 10$ nm. Further, we consider
$d_{\mathrm{FI}}=d_{S}$. Figure 2 plotted with these values shows a large
enhancement of the effective mass with decreasing temperature. This can be
measured using, for example, the Brillouin Light Scattering technique [70, 71]
employed regularly in measuring magnon group velocities [72, 73]. Furthermore,
the enhanced effective mass, and thus a reduced mobility, will manifest itself
in the typical nonlocal magnonic spin transport experiment [74, 75, 76].
Besides the in-situ control via, for example, temperature, the FI thickness
can be used to engineer $\tilde{D}_{m}$ ex-situ. A negative value of
$\tilde{D}_{m}$ signifies that our assumed uniformly ordered magnetization is
no longer the ground state [77].

With the material parameters above, $h_{0}=0.61\Delta_{0}$, and $T=0.9T_{c}$,
the net spin of the magnon-cooparon [Eq. (9)] is evaluated as $0.4$, reduced
from spin 1 of the bare magnon. Further, assuming $t=10\xi_{S}$, $d=\xi_{S}$,
and $(k_{+}+k_{-})/2=10^{7}~{}\mathrm{m}^{-1}$, the coupling length $L$ [Eq.
(12)] of the magnon-cooparon based directional coupler is evaluated as $\sim
100$ nm. This is an order of magnitude smaller than the coupling length
afforded by dipolar-interaction based designs [36, 37]. The experimental
realization of the magnon-cooparon based directional coupler can follow the
procedure similar to its dipole-interaction based counterpart [62, 63, 64, 65,
66] with the FI layers deposited on a superconductor instead of a substrate.

Summary.—We have demonstrated theoretically the ubiquitous existence of a
quasiparticle, termed magnon-cooparon, comprising a spin flip in a magnetic
insulator screened by a spinfull Cooper pairs cloud in an adjacent
conventional superconductor. The nonlocal nature of the magnon-cooparon is
then exploited to propose a high performance magnonic directional coupler.
While we have focused on a uniformly ordered ferromagnetic insulator, our
analysis is general and anticipates an important role for magnon-cooparons in
a wide range of hybrids comprising different magnetic insulators with various
ground states.

We acknowledge financial support from the Spanish Ministry for Science and
Innovation – AEI Grant CEX2018-000805-M (through the “Maria de Maeztu”
Programme for Units of Excellence in R&D), from the Deutsche
Forschungsgemeinschaft (DFG; German Research Foundation) via a DFG-RSF project
(ID 443404566), the SFB 1432 (ID 425217212), the SPP 2244 (ID 443404566) and
from the Russian Science Foundation via the RSF-DFG project No.22-42-04408.

##  I Supplemental material

###  I.1 Quasiclassical green’s function description of the superconductor

In the superconductor the Usadel equation for the $8\times 8$ matrix Green’s
function $\check{g}$ in the direct product of Keldysh, spin and particle-hole
spaces takes the form:

$\displaystyle
iD\bm{\nabla}\bigl{(}\check{g}\otimes\bm{\nabla}\check{g}\bigr{)}=\bigl{[}\varepsilon\tau_{z}-\bm{h}\bm{\sigma}\tau_{z}+\Delta
i\tau_{y},\check{g}\bigr{]}_{\otimes},$ (13)

where $[A,B]_{\otimes}=A\otimes B-B\otimes A$ and we work in the mixed
$(\varepsilon,t)$ representation with $A\otimes
B=\exp[(i/2)(\partial_{\varepsilon_{1}}\partial_{t_{2}}-\partial_{\varepsilon_{2}}\partial_{t_{1}})]A(\varepsilon_{1},t_{1})B(\varepsilon_{2},t_{2})|_{\varepsilon_{1}=\varepsilon_{2}=\varepsilon;t_{1}=t_{2}=t}$.
In case $A[B](\varepsilon,t)=A[B](\varepsilon)\exp[i\Omega_{A[B]}t]$ the
$\otimes$-product is reduced to $A(\varepsilon,t)\otimes
B(\varepsilon,t)=A(\varepsilon-\Omega_{B}/2)B(\varepsilon+\Omega_{A}/2,t)\exp[i(\Omega_{A}+\Omega_{B})t]$.
$\tau_{x,y,z}$ and $\sigma_{x,y,z}$ are Pauli matrices in particle-hole and
spin spaces, respectively. $\Delta$ is the superconducting order parameter.
The explicit structure of the Green’s function in the Keldysh space takes the
form:

$\displaystyle\check{g}=\left(\begin{array}[]{cc}\check{g}^{R}&\check{g}^{K}\\\
0&\check{g}^{A}\end{array}\right),$ (16)

where $\check{g}^{R(A)}$ is the retarded (advanced) component of the Green’s
function and $g^{K}$ is the Keldysh component. Further we express the Keldysh
part of the Green’s function via the retarded, advanced Green’s function and
the distribution function $\check{\varphi}$ as follows:
$\check{g}^{K}=\check{g}^{R}\otimes\check{\varphi}-\check{\varphi}\otimes\check{g}^{A}$.

The exchange field is taken in the form of a time-independent component and a
circularly polarized magnon:

$\displaystyle\bm{h}=h_{0}\bm{e}_{z}+\delta h\cos(\bm{k}\bm{r}+\omega
t)\bm{e}_{x}+\delta h\sin(\bm{k}\bm{r}+\omega t)\bm{e}_{y}.$ (17)

Then

$\displaystyle\delta\bm{h}\bm{\sigma}=\delta he^{-i(\bm{k}\bm{r}+\omega
t)\sigma_{z}}\sigma_{x}.$ (18)

The quasiclassical Green’s function is to be found in the form:
$\check{g}=\check{g}_{0}+\delta\check{g}$, where $\check{g}_{0}$ is the
Green’s fucntion in the absence of the magnon and $\delta g$ is the first
order correction with respect to $\delta\bm{h}$. Taking into account that
$\bm{\nabla}g_{0}=0$ (we assume that in the absence of the magnon the bilayer
is spatially homogeneous along the interface) from Eq. (13) we obtain the
following equation for $\delta\check{g}$:

$\displaystyle
iD\check{g}_{0}\otimes\nabla^{2}\delta\check{g}=\bigl{[}\varepsilon\tau_{z}-h_{0}\bm{e}_{z}\sigma_{z}\tau_{z}+\tau_{z}\hat{\Delta},\delta\check{g}\bigr{]}_{\otimes}-\bigl{[}\delta
he^{-i(\bm{k}\bm{r}+\omega
t)\sigma_{z}}\sigma_{x}\tau_{z},\check{g}_{0}\bigr{]}_{\otimes}$ (19)

Introducing the unitary operator $\hat{U}=e^{-i(\bm{k}\bm{r}+\omega
t)\sigma_{z}/2}$ we can transform the Green’s function as follows:

$\displaystyle\delta\check{g}=U\otimes\delta\check{g}_{m}\otimes U^{\dagger}.$
(20)

In case if the system is spatially homogeneous except for the magnon,
$\delta\check{g}_{m}$ does not depend on coordinates. Then

$\displaystyle\nabla^{2}\delta\check{g}=-\frac{k^{2}}{2}\check{U}\otimes\bigl{(}\delta\check{g}_{m}-\sigma_{z}\delta\check{g}_{m}\sigma_{z}\bigr{)}\otimes
U^{\dagger}=-k^{2}\delta\check{g},~{}~{}~{}~{}$ (21)

where when passing to the second equality it is used that
$\delta\check{g}=\delta\hat{g}_{x}\sigma_{x}+\delta\hat{g}_{y}\sigma_{y}$ and
has no $z$-component in the spin space according to the spin structure of the
magnon exchange field $\delta\bm{h}$.

From the normalization condition $\check{g}\otimes\check{g}=1$ it follows that
$\check{g}_{0}\otimes\delta\check{g}=-\delta\check{g}\otimes\check{g}_{0}$. It
gives us
$\check{g}_{0}\otimes\delta\check{g}=(1/2)[\check{g}_{0},\delta\check{g}]_{\otimes}$.
Eq. (19) takes the form:

$\displaystyle\bigl{[}\varepsilon\tau_{z}+i\frac{Dk^{2}}{2}\check{g}_{0}-h_{0}\bm{e}_{z}\sigma_{z}\tau_{z}+\tau_{z}\hat{\Delta},\delta\check{g}\bigr{]}_{\otimes}-\bigl{[}\delta
he^{-i(\bm{k}\bm{r}+\omega
t)\sigma_{z}}\sigma_{x}\tau_{z},\check{g}_{0}\bigr{]}_{\otimes}=0.$ (22)

From Eq. (22) the following equation for $\delta\check{g}_{m}$ is obtained:

$\displaystyle\bigl{[}\hat{\Lambda}_{d}\tau_{z}+\hat{\Lambda}_{od}i\tau_{y},\delta\check{g}_{m}\bigr{]}=\delta
h\bigl{[}\sigma_{x}\tau_{z},\frac{1}{2}\bigl{(}(g_{0,+}+g_{0,-}\sigma_{z})\tau_{z}+(f_{0,+}+f_{0,-}\sigma_{z})i\tau_{y}\bigr{)}\bigr{]}.$
(23)

It does not contain time dependence and $\otimes$-products. In Eq. (23) we use
the following definitions

$\displaystyle g_{0,\pm}=g_{0,\uparrow}(\varepsilon+\frac{\omega}{2})\pm
g_{0,\downarrow}(\varepsilon-\frac{\omega}{2}),$ (24) $\displaystyle
f_{0,\pm}=f_{0,\uparrow}(\varepsilon+\frac{\omega}{2})\pm
f_{0,\downarrow}(\varepsilon-\frac{\omega}{2}),$ (25)

where $g_{0,\uparrow(\downarrow)}$ represent the bulk Green’s functions for
the superconductor in the exchange field $h_{0}$:

$\displaystyle g_{0,\uparrow(\downarrow)}^{R}=\frac{|\varepsilon\mp
h_{0}|}{\sqrt{(\varepsilon+i\delta\mp h_{0})^{2}-\Delta^{2}}}$ (26)
$\displaystyle f_{0,\uparrow(\downarrow)}^{R}=\frac{\Delta{\rm
sgn}(\varepsilon\mp h_{0})}{\sqrt{(\varepsilon+i\delta\mp
h_{0})^{2}-\Delta^{2}}},$ (27)

and $g(f)_{0,\uparrow(\downarrow)}^{A}=-g(f)_{0,\uparrow(\downarrow)}^{R*}$,
$g(f)_{0,\uparrow(\downarrow)}^{K}=[g(f)_{0,\uparrow(\downarrow)}^{R}-g(f)_{0,\uparrow(\downarrow)}^{A}]\tanh[\varepsilon/2T]$.

$\displaystyle\hat{\Lambda}_{d}=\Lambda_{d}^{0}+\Lambda_{d}^{z}\sigma_{z}=\varepsilon+\frac{iDk^{2}}{4}g_{0,+}+\bigl{(}\frac{\omega}{2}-h_{0}+\frac{iDk^{2}}{4}g_{0,-}\bigr{)}\sigma_{z},$
(28)
$\displaystyle\hat{\Lambda}_{od}=\Lambda_{od}^{0}+\Lambda_{od}^{z}\sigma_{z}=\Delta+\frac{iDk^{2}}{4}f_{0,+}+\frac{iDk^{2}}{4}f_{0,-}\sigma_{z}.$
(29)

Solving Eq. (23) we obtain:

$\displaystyle\delta\check{g}_{m}=\delta g_{mx}\sigma_{x}\tau_{z}+\delta
f_{mx}\sigma_{x}i\tau_{y},$ (30)

where

$\displaystyle\delta g_{mx}=\frac{\delta
h\bigl{[}f_{0,+}\Lambda_{od}^{z}-g_{0,-}\Lambda_{d}^{0}\bigr{]}}{2\bigl{[}\Lambda_{d}^{0}\Lambda_{d}^{z}-\Lambda_{od}^{0}\Lambda_{od}^{0}\bigr{]}}$
(31)

The distribution function also acquires a correction due to the magnon:
$\check{\varphi}=\check{\varphi}_{0}+\delta\check{\varphi}$. It is convenient
to work with the transformed distribution function
$\hat{U}^{\dagger}\check{\varphi}\hat{U}=\check{\varphi}_{m}+\delta\varphi_{m}$,
which does not depend on time and spatial coordinates. Here
$\check{\varphi}_{m}=(1/2)[\varphi_{m+}+\varphi_{m-}\sigma_{z}]$ with
$\varphi_{m\pm}=\tanh[(\varepsilon+\omega/2)/2T]\pm\tanh[(\varepsilon-\omega/2)/2T]$
is the result of the unitary transformation of the equilibrium distribution
function $\check{\varphi}_{0}=\tanh[\varepsilon/2T]$. From the Keldysh part of
the Usadel equation (23) we can obtain the following equation for the first
order correction to the distribution function $\delta\check{\varphi}_{m}$:

$\displaystyle
iDk^{2}[\delta\check{\varphi}_{m}-\check{g}_{m0}^{R}\delta\check{\varphi}_{m}\check{g}_{m0}^{A}]+\check{g}_{m0}^{R}[\check{K},\delta\check{\varphi}_{m}]-[\check{K},\delta\check{\varphi}_{m}]\check{g}_{m0}^{A}+\check{g}_{m0}^{R}[\check{\varphi}_{m},\delta
h\sigma_{x}\tau_{z}]-[\check{\varphi}_{m},\delta
h\sigma_{x}\tau_{z}]\check{g}_{m0}^{A}=0,$ (32)

where

$\displaystyle\check{K}=(\varepsilon+(\omega/2-h_{0})\sigma_{z})\tau_{z}+\Delta
i\tau_{y},$ (33)
$\displaystyle\check{g}_{m0}^{R,A}=\hat{U}^{\dagger}\otimes\check{g}_{0}^{R,A}\otimes\hat{U}=(1/2)[(g_{0,+}^{R,A}+g_{0,-}^{R,A}\sigma_{z})\tau_{z}+(f_{0,+}^{R,A}+f_{0,-}^{R,A}\sigma_{z})i\tau_{y}],$
(34)

Structure of Eq. (32) dictates that

$\displaystyle\delta\check{\varphi}_{m}=\left(\begin{array}[]{cc}0&\delta\varphi_{m}^{\uparrow}\\\
\delta\varphi_{m}^{\downarrow}&0\end{array}\right).$ (37)

Substituting Eq. (37) into Eq. (32) we obtain the following result:

$\displaystyle\delta\varphi_{m}^{\sigma}=-\delta
h\varphi_{m-}\frac{2h_{0\omega}G_{-,\sigma}+i\sigma
Dk^{2}(g_{m\sigma}^{R}-g_{m\bar{\sigma}}^{A})}{4h_{0\omega}^{2}G_{-,\sigma}+4h_{0\omega}i\sigma
Dk^{2}(g_{m\sigma}^{R}-g_{m\bar{\sigma}}^{A})-(Dk^{2})^{2}G_{+,\sigma}},~{}~{}~{}~{}$
(38)

where we introduce the spin subband index $\sigma=\uparrow(\downarrow)$ in the
subscripts/superscripts and $\sigma=\pm 1$ for spin-up(down) subbands,
respectively, if it is as a factor. $\bar{\sigma}=-\sigma$,
$h_{0\omega}=\omega/2-h_{0}$,
${g(f)}_{m\sigma}^{R,A}=g(f)_{0,\sigma}^{R,A}(\varepsilon+\sigma\omega/2)$ and
$G_{\pm,\sigma}=1-g_{m\sigma}^{R}g_{m\bar{\sigma}}^{A}\pm
f_{m\sigma}^{R}f_{m\bar{\sigma}}^{A}$.

###  I.2 Renormalization of the excitation dispersion and damping

Following the methodology detailed in the previous section, the electron spin
polarization $\bm{s}$ in the superconductor can be calculated as

$\displaystyle\bm{s}=-\frac{N_{F}}{16}\int d\varepsilon{\rm
Tr}_{4}\Bigl{[}\bm{\sigma}\tau_{z}\check{g}^{K}\Bigr{]}.$ (39)

It can be written in the form:

$\displaystyle\bm{s}=s_{0}\bm{m}_{0}+\delta s_{\parallel}\delta\bm{m}+\delta
s_{\perp}(\delta\bm{m}\times\bm{m}_{0}),$ (40)

where $s_{0}$ is the equilibrium value of the electron spin polarization in
the superconductor, corresponding to the absence of the magnon. $\delta
s_{\parallel}$ and $\delta s_{\perp}$ describe the dynamic corrections to the
spin polarization due to the magnon.

$\displaystyle
s_{0}=-\frac{N_{F}}{4}\int\limits_{-\infty}^{\infty}d\varepsilon\tanh\frac{\varepsilon}{2T}{\rm
Re}\bigl{[}g_{0,\uparrow}^{R}-g_{0,\downarrow}^{R}\bigr{]},$ (41)
$\displaystyle\delta
s_{\parallel}=-\frac{N_{F}}{8}\int\limits_{-\infty}^{\infty}d\varepsilon\Bigl{\\{}2\varphi_{m+}{\rm
Re}[\delta g_{mx}^{R}]+\frac{h_{0}}{\delta
h}\sum\limits_{\sigma}(g_{m\sigma}^{R}-g_{m\bar{\sigma}}^{A})\delta\varphi_{m}^{\sigma}\Bigr{\\}},$
(42) $\displaystyle\delta
s_{\perp}=\frac{N_{F}}{8}\int\limits_{-\infty}^{\infty}d\varepsilon\Bigl{\\{}2\varphi_{m-}{\rm
Im}[\delta g_{mx}^{R}]+i\frac{h_{0}}{\delta
h}\sum\limits_{\sigma}\sigma(g_{m\sigma}^{R}-g_{m\bar{\sigma}}^{A})\delta\varphi_{m}^{\sigma}\Bigr{\\}}.$
(43)

As per the main text, $\delta s_{\parallel}-s_{0}$ accounts for the
renormalization of the magnon dispersion. It can be shown that in the
adiabatic limit $\hbar\omega\ll{\rm max}[\Delta,T]$ if one neglects
$\hbar\omega$ with respect to the superconducting energies $\delta
s_{\parallel}-s_{0}\propto Dk^{2}$. Consequently, it only renormalizes the
magnon stiffness. The explicit expression for the stiffness correction in the
limit $T\to T_{c}$ takes the form

$\displaystyle\delta D_{m}=-\frac{\pi T_{c}N_{F}\hbar\gamma
d_{S}D}{d_{FI}M_{s}}\sum\limits_{\omega_{n}>0}\frac{\Delta^{2}h_{0}^{2}}{[\omega_{n}^{2}+h_{0}^{2}]^{2}}.$
(44)

The same result for the stiffness renormalization has been obtained from the
consideration of the total energy of the bilayer in the framework of the
Ginzburg-Landau theory [54].

If one does not make the adiabatic approximation, then the zero-momentum value
of the magnon energy is also slightly renormalized. The renormalization
correction takes the form:

$\displaystyle\hbar(\omega_{\bm{k}=0}-\gamma K)=\frac{J}{d_{FI}}\bigl{(}\delta
s_{\parallel}(\omega,\bm{k}=0)-\delta
s_{\parallel}(\omega=0,\bm{k}=0)\bigr{)},$ (45)

where

$\displaystyle\delta
s_{\parallel}(\omega,\bm{k}=0)=\frac{N_{F}}{4}\frac{h_{0}}{\omega/2-h_{0}}\int\limits_{-\infty}^{\infty}d\varepsilon[\tanh\frac{\varepsilon+\frac{\omega}{2}}{2T}{\rm
Re}[g_{0,\uparrow}(\varepsilon+\frac{\omega}{2})]-\tanh\frac{\varepsilon-\frac{\omega}{2}}{2T}{\rm
Re}[g_{0,\downarrow}(\varepsilon-\frac{\omega}{2})]].$ (46)

The renormalization of the zero-momentum magnon frequency in the limit of
$h_{0}\to 0$ has already discussed [58], and our result above is consistent
with this previous work in the relevant limit $h_{0}\to 0$. The general
results obtained above have been employed in plotting the solid lines of Fig.
2 in the main text.

For estimates and numerical calculations we take material parameters of YIG as
a FI and Nb as a superconductor. $\sigma_{S}=9\cdot 10^{16}s^{-1}$ is the Nb
conductivity and $D/\hbar=3cm^{2}\cdot s^{-1}$ is the Nb diffusion constant.
Then the DOS at the Fermi level in Nb is
$N_{F}=\hbar\sigma_{S}/(e^{2}D)=0.13\cdot 10^{36}erg^{-1}\cdot cm^{-3}$. The
YIG parameters are [69] $D_{m}=5\cdot 10^{-29}erg\cdot cm^{2}$,
$M_{s}=1.4\cdot 10^{2}G$, $\gamma=1.76\cdot 10^{7}G^{-1}s^{-1}$ and
$\hbar\gamma K=10^{-17}erg=4\cdot 10^{-3}\Delta_{0}$, where $\Delta_{0}=18K$
is the zero-temperature order parameter in Nb. With these parameters we obtain
that the renormalization of the zero-momentum magnon frequency
$(\omega_{\bm{k}=0}-\gamma K)/\gamma K\lesssim 10^{-2}$ due to the smallness
of the bare magnon frequency $\gamma K$ with respect to $\Delta_{0}$.

Figure 4: Fractional increase in the efffective Gilbert damping for the
magnon-cooparon due to the superconducting layer. This enhancement is mediated
by the quasiparticles, and not the superfluid condensate.

According to Eq. (3) in the main text, $\delta s_{\perp}$ determines the
correction to the magnon decay rate. It is zero in the framework of the
quasistatic approximation when we neglect $\hbar\omega$ with respect to the
superconducting energies. Beyond this approximation is it non-zero and the
correction to the Gilbert damping parameter $\delta\alpha=-(J/d_{FI})\delta
s_{\perp}/\omega$ is presented in Fig. 4 as a function of temperature for
different values of $h_{0}$. The normal state value of $\delta\alpha$ can be
obtained analytically:

$\displaystyle\delta\alpha_{N}=\frac{d_{S}}{d_{FI}}\frac{(2\hbar\gamma
N_{F}h_{0}^{2}/M_{s})Dk^{2}}{4(\frac{\omega}{2}-h_{0})^{2}+(Dk^{2})^{2}}.$
(47)

It is seen that $\delta\alpha_{N}$ is positive and vanishes for zero-momentum
magnons. The result is natural because in the framework of our model we
consider the only source of spin relaxation processes in superconductor - the
finite momentum of the magnon, which results in the spin relaxation rate
$Dk^{2}$. We do not consider any other spin relaxation processes such as spin-
orbit relaxation and relaxation at magnetic impurities. The corresponding
relaxation rates would additionally increase the correction to the Gilbert
damping. The correction to the Gilbert damping due to the spin-orbit
relaxation and relaxation at magnetic impurities has been investigated in Ref.
58 by calculation of the spin susceptibility of the superconductor. Eq. (47)
at $h_{0}\to 0$ coincides with the result obtained in Ref. 58 with the
substitution $Dk^{2}\to 2\Gamma+\tau_{s}^{-1}$, where $\Gamma$ is the Dynes
parameter and $\tau_{s}^{-1}$ is the spin-orbit relaxation rate. Our estimates
suggest that $\delta\alpha_{N}\sim 10^{-2}(\xi_{S}k^{2})$, where
$\xi_{S}=\sqrt{D/\Delta_{0}}$ is the superconducting coherence length, which
is of the order of $10$ nm for Nb films. Consequently, for short wave-length
magnons the superconducting correction is essential in comparison with the
extremely low values of the Gilbert damping parameter in YIG[69]
$\alpha\approx 5\cdot 10^{-5}$. The influence of superconductivity on the
correction to the Gilbert damping constant is presented in Fig. 4. The most
important feature of the temperature dependence is that in the presence of the
finite $h_{0}$ in the superconductor the correction in the superconducting
state demonstrates a sharp drop at a given temperature, which physically
corresponds to the value of the superconducting order parameter
$\Delta(T)=h_{0}$, where the gap in the superconducting state is closed. At
this point the number of thermal quasiparticles, which can flip their spin,
grows sharply. The results obtained at $h_{0}\to 0$ and, consequently, not
taking into account the Zeeman splitting of the superconducting gap, manifest
gradual decline of the correction at the superconducting state [58].

###  I.3 Spin current

Now we calculate the spin current carried by the triplet pairs, induced in the
superconductor by the magnon:

$\displaystyle\hat{j}_{s}=\frac{\hbar N_{F}D}{16}{\rm Tr}_{4}\int
d\varepsilon\bm{\sigma}\bigl{(}\check{g}\bm{\nabla}\check{g}\bigr{)}^{K}$ (48)

The first order contribution with respect to $\delta h$ to the spin current is
zero after the time averaging. Therefore, the spin current is of the second
order with respect to $\delta h$. It has the only non-zero component $j_{S,z}$
carrying the $z$-component of spin along the $\bm{k}$ direction. In general,
the calculation of second order terms with respect to $\delta h$ terms in the
Green’s function is rather cumbersome. Here we restrict ourselves by the
calculation at $T\to T_{c}$, where the Green’s functions can be linearized
with respect to the superconducting order parameter $\Delta$. Then the
solution for the retarded quasiclassical Green’s function takes the form
$\check{g}^{R}=\tau_{z}+\check{f}^{R}$. The linearized with respect to the
anomalous Green’s function version of Eq. (48) takes the form:

$\displaystyle\hat{j}_{s}=$ $\displaystyle\frac{\hbar N_{F}D}{16}{\rm
Tr}_{2}\int
d\varepsilon\bm{\sigma}\bigl{[}\frac{1}{2}\bigl{(}\hat{f}^{R}\bm{\nabla}\hat{\tilde{f}}^{R}+\hat{\tilde{f}}^{R}\bm{\nabla}\hat{f}^{R}-(\bm{\nabla}\hat{f}^{R})\hat{\tilde{f}}^{R}-(\bm{\nabla}\hat{\tilde{f}}^{R})\hat{f}^{R}-R\to
A\bigr{)}\tanh\frac{\varepsilon}{2T}$
$\displaystyle+2\bm{\nabla}\hat{\varphi}^{(2)}+2\bm{\nabla}\hat{\tilde{\varphi}}^{(2)}\bigr{]},$
(49)

where the distribution function
$\check{\varphi}=\tanh[\varepsilon/2T]+\hat{\varphi}^{(2)}(1+\tau_{z})/2+\hat{\tilde{\varphi}}^{(2)}(1-\tau_{z})/2$
contains the seconds order correction with respect to the anomalous Green’s
function due to the non-equilibrium generation of the triplet pairs. At first
let us calculate $\hat{\varphi}^{(2)}$. The equation for the distribution
function is obtained from the Keldysh part of Eq. (13) and up to the second
order with respect to the anomalous Green’s function takes the form:

$\displaystyle
iD\bm{\nabla}\Bigl{[}\frac{1}{2}\bigl{(}\hat{f}^{R}\bm{\nabla}\hat{\tilde{f}}^{R}-(\bm{\nabla}\hat{f}^{R})\hat{\tilde{f}}^{R}-R\to
A\bigr{)}\tanh\frac{\varepsilon}{2T}+2\nabla\hat{\varphi}^{(2)}\Bigr{]}=$
$\displaystyle\frac{h_{0}}{2}[\sigma_{z},(\hat{f}^{R}\hat{\tilde{f}}^{R}+\hat{f}^{A}\hat{\tilde{f}}^{A})]\tanh\frac{\varepsilon}{2T}+\frac{1}{2}[\delta\bm{h}\bm{\sigma},(\hat{f}^{R}\hat{\tilde{f}}^{R}+\hat{f}^{A}\hat{\tilde{f}}^{A})]\tanh\frac{\varepsilon}{2T}-2h_{0}[\sigma_{z},\hat{\varphi}^{(2)}]-2[\delta\bm{h}\bm{\sigma},\hat{\varphi}^{(2)}].$
(50)

In order to simplify the calculations further we work in the quasistatic
approximation, when we neglect the correction of the order of $\omega/T_{c}$
and higher orders of this parameter. In this limit all the $\otimes$-products
in Eq. (50) can be changed by the usual multiplication because the neglected
corrections are of the order of $\omega/T_{c}$, which is assumed to be small.
To this approximation the correction $\varphi_{z}^{(2)}$ to the distribution
function (which could contribute to the spin current) is zero.

Then the spin current is expressed by the first two lines of Eq. (49). In
order to calculate the spin current we need to calculate

$\displaystyle
I=\hat{f}^{R}\bm{\nabla}\hat{\tilde{f}}^{R}+\hat{\tilde{f}}^{R}\bm{\nabla}\hat{f}^{R}-(\bm{\nabla}\hat{f}^{R})\hat{\tilde{f}}^{R}-(\bm{\nabla}\hat{\tilde{f}}^{R})\hat{f}^{R}=-2[\hat{f}^{R}\bm{\nabla}\hat{f}^{R}-(\bm{\nabla}\hat{f}^{R})\hat{f}^{R}]$
(51)

the second order with respect to $\delta h$ contribution to $I$ takes the
form:

$\displaystyle
I^{(2)}=i\bm{k}U\Bigl{[}\delta\hat{f}_{m}^{R}[\sigma_{z},\delta\hat{f}_{m}^{R}]-[\sigma_{z},\delta\hat{f}_{m}^{R}]\delta\hat{f}_{m}^{R}\Bigr{]}U^{\dagger}$
(52)

If we linearize the Usadel equation with respect to the superconducting order
parameter and the anomalous Green’s functions, the solution takes the form
$\check{g}^{R}=\tau_{z}+\check{f}^{R}$. The anomalous Green’s function is
$\check{f}^{R}=\check{f}_{0}^{R}+\delta f^{R}$, where $\delta
f^{R}=\hat{U}\delta f_{m}^{R}\hat{U}^{\dagger}$. $\delta f_{m}^{R}$ can be
obtained from Eq. (23) and takes the form:

$\displaystyle\delta\hat{f}_{m}^{R}=\delta f_{m,x}^{R}\sigma_{x},$
$\displaystyle\delta f_{m,x}^{R}=\frac{\delta
hf_{0,s}^{R}}{\varepsilon+iDk^{2}/2}.$ (53)

In Eq. (53) $f_{0,s}^{R}=\Delta\varepsilon/(\varepsilon^{2}-h_{0}^{2})$ and
$\varepsilon$ has an infinitesimal positive imaginary part $\delta$.

Substituting Eq. (52) into Eq. (49) and taking into account that
$\hat{f}^{A}=-\hat{f}^{R*}$, we obtain:

$\displaystyle\bm{j}_{S,z}=\frac{\hbar N_{F}D\bm{k}}{2}\int
d\varepsilon\tanh\frac{\varepsilon}{2T}{\rm Im}\Bigl{[}\delta
f_{m,x}^{R,2}\Bigr{]}=2\pi T_{c}\hbar
N_{F}D\bm{k}\sum\limits_{\omega_{n}>0}\frac{\Delta^{2}\delta
h^{2}\omega_{n}^{2}}{(\omega_{n}^{2}+h_{0}^{2})^{2}(\omega_{n}+Dk^{2}/2)^{2}}.$
(54)

The spin current in the FI, carried by the magnons takes the form:

$\displaystyle\bm{j}_{m}=-\bm{k}\Bigl{(}\frac{\delta
h}{h_{0}}\Bigr{)}^{2}\frac{\tilde{D}_{m}M_{s}}{\hbar\gamma},$ (55)

where $\tilde{D}_{m}=D_{m}+\delta D_{m}$ is the exchange stiffness
renormalized by the superconductor and $M_{s}$ is the saturation
magnetization. Taking the YIG parameters [69] $D_{m}=5\cdot 10^{-29}erg\cdot
cm^{2}$, $\gamma=1.76\cdot 10^{7}G^{-1}\cdot s^{-1}$ and $M_{s}=1.4\cdot
10^{2}G$, we obtain

$\displaystyle\bm{j}_{m}\approx\left(\frac{\delta
h}{h_{0}}\right)^{2}\bm{k}\cdot 4\cdot 10^{-7}erg\cdot cm^{-1}.$ (56)

From the spin currents thus obtained, the net spin of the magnon-cooparon is
evaluated as discussed in the main text.

###  I.4 Nonlocal excitation of the magnon spin current via a superconductor

The system under consideration is depicted in Fig. 3 of the main text. It is
assumed that the width $t$ of each of the FIs is smaller than the typical
magnon wavelength in order to neglect inhomogeneities of the magnetization
distribution in each of the FIs along the $x$-direction. From the other hand
we assume $t\gg\xi_{S}$. In this case we can consider each of the FIs as semi-
infinite from the point of view of the superconductor. With these assumptions
we solve the linearized Usadel equation in the superconductor with the
following spatial profile of the exchange field:

$\displaystyle\bm{h}(x)=h_{0}(x)\bm{e}_{z}+\delta\bm{h}(x),$ $\displaystyle
h_{0}(x)=h_{0}[\Theta(-\frac{d}{2}-x)+\Theta(x-\frac{d}{2})],$
$\displaystyle\delta\bm{h}(x)=\delta\bm{h}_{l}\Theta(-\frac{d}{2}-x)+\delta\bm{h}_{r}\Theta(x-\frac{d}{2}),$
(57) $\displaystyle\delta\bm{h}_{l,r}\bm{\sigma}=\delta
h_{l,r}e^{-i(\bm{k}\bm{r}+\omega t)}\sigma_{x}.$ (58)

In order to catch the physical essence of the effect basing on the simplest
equations, we again work in the linearized with respect to $\Delta$ and
adiabatic approximation. Making use of the unitary transformation Eq. (20) we
come to the following equation for the dynamical triplet anomalous Green’s
function

$\displaystyle-i\frac{Dk^{2}}{4}[\delta\hat{f}_{m}^{R}-\sigma_{z}\delta\hat{f}_{m}^{R}\sigma_{z}]+i\frac{D\partial_{x}^{2}\delta\hat{f}_{m}^{R}}{2}=\varepsilon\delta\hat{f}_{m}^{R}-\frac{h_{0}}{2}\\{\sigma_{z},\delta\hat{f}_{m}^{R}\\}-\delta
h(x)\sigma_{x}f_{0,s}^{R}(x),$ (59)

where $\delta h(x)=\delta h_{l}\Theta(-\frac{d}{2}-x)+\delta
h_{r}\Theta(x-\frac{d}{2})$ and $f_{0,s}^{R}(x)$ is now also spatially
inhomogeneous and should be found from the system of coupled equations:

$\displaystyle i\frac{D\partial_{x}^{2}f_{0,s}^{R}}{2}=\varepsilon
f_{0,s}^{R}-h_{0}(x)f_{0,z}^{R}-\Delta,$ $\displaystyle
i\frac{D\partial_{x}^{2}f_{0,z}^{R}}{2}=\varepsilon
f_{0,z}^{R}-h_{0}(x)f_{0,s}^{R}.$ (60)

The singlet anomalous Green’s function obeying these equations takes the form:

$\displaystyle
f_{0,s}^{R}(x)=\left\\{\begin{array}[]{cc}\frac{\Delta\varepsilon}{(\varepsilon+i\delta)^{2}-h_{0}^{2}}+C_{+l}e^{\lambda_{+}(x+d/2)}+C_{-l}e^{\lambda_{-}(x+d/2)},&x<-d/2,\\\
\frac{\Delta_{0}}{\varepsilon}+C_{1}e^{\lambda x}+C_{2}e^{-\lambda
x},&-d/2<x<d/2,\\\
\frac{\Delta\varepsilon}{(\varepsilon+i\delta)^{2}-h_{0}^{2}}+C_{+r}e^{-\lambda_{+}(x-d/2)}+C_{-r}e^{-\lambda_{-}(x-d/2)},&x>d/2,\end{array}\right.$
(64)

where $\Delta_{0}$ and $\Delta$ are superconducting order parameters in the
middle superconducting region $-d/2<x<d/2$ and in the left and right regions
covered by the FIs, respectively. $\Delta<\Delta_{0}$ because of the order
parameter suppression by the exchange field $h_{0}$.
$\lambda=\sqrt{-2i\varepsilon/D}$, $\lambda_{\pm}=-2i(\varepsilon\pm
h_{0})/D$. Constants $C_{\pm l,r}$ take the form:

$\displaystyle C_{\pm l}=C_{\pm
r}=\frac{1}{2}\left(\frac{\Delta}{\varepsilon\pm
h_{0}}-\frac{\Delta_{0}}{\varepsilon}\right)\frac{\frac{\lambda_{\pm}}{\lambda}(1-\cosh\lambda
d)-\sinh\lambda d}{(1+\frac{\lambda_{\pm}^{2}}{\lambda^{2}})\sinh\lambda
d+2\frac{\lambda_{\pm}}{\lambda}\cosh\lambda d}.$ (65)

The solution of Eq. (59) takes the form $\hat{f}_{m}^{R}=f_{m,x}\sigma_{x}$
with

$\displaystyle\delta
f_{m}^{R}(x)=\left\\{\begin{array}[]{cc}C_{xl}e^{\lambda_{k}(x+d/2)}+\frac{\delta
h_{l}\Delta\varepsilon}{(\varepsilon+iDk^{2}/2)((\varepsilon+i\delta)^{2}-h_{0}^{2})}+\frac{2\delta
h_{l}C_{+l}e^{\lambda_{+}(x+d/2)}}{iD(\lambda_{k}^{2}-\lambda_{+}^{2})}+\frac{2\delta
h_{l}C_{-l}e^{\lambda_{-}(x+d/2)}}{iD(\lambda_{k}^{2}-\lambda_{-}^{2})},&x<-d/2,\\\
C_{x1}e^{\lambda_{k}x}+C_{x2}e^{-\lambda_{k}x},&-d/2<x<d/2,\\\
C_{xr}e^{-\lambda_{k}(x-d/2)}+\frac{\delta
h_{r}\Delta\varepsilon}{(\varepsilon+iDk^{2}/2)((\varepsilon+i\delta)^{2}-h_{0}^{2})}+\frac{2\delta
h_{r}C_{+r}e^{-\lambda_{+}(x-d/2)}}{iD(\lambda_{k}^{2}-\lambda_{+}^{2})}+\frac{2\delta
h_{r}C_{-r}e^{-\lambda_{-}(x-d/2)}}{iD(\lambda_{k}^{2}-\lambda_{-}^{2})},&x>d/2,\end{array}\right.$
(69)

where $\lambda_{k}=\sqrt{-2i(\varepsilon+iDk^{2}/2)/D}$ and constants take the
form:

$\displaystyle C_{xl,r}=-\frac{\delta
h_{l,r}\Delta\varepsilon}{2(\varepsilon+i\frac{Dk^{2}}{2})[(\varepsilon+i\delta)^{2}-h_{0}^{2}]}+\frac{\delta
h_{r,l}\Delta\varepsilon
e^{-\lambda_{k}d}}{2(\varepsilon+i\frac{Dk^{2}}{2})[(\varepsilon+i\delta)^{2}-h_{0}^{2}]}-$
$\displaystyle\frac{\delta
h_{l,r}}{iD}\Bigl{(}\frac{1\mp\frac{\lambda_{+}}{\lambda_{k}}}{\lambda_{k}^{2}-\lambda_{+}^{2}}C_{+l,r}+\frac{1\mp\frac{\lambda_{-}}{\lambda_{k}}}{\lambda_{k}^{2}-\lambda_{-}^{2}}C_{-l,r}\Bigr{)}+\frac{\delta
h_{r,l}e^{-\lambda_{k}d}}{iD}\Bigl{(}\frac{1\pm\frac{\lambda_{+}}{\lambda_{k}}}{\lambda_{k}^{2}-\lambda_{+}^{2}}C_{+r,l}+\frac{1\pm\frac{\lambda_{-}}{\lambda_{k}}}{\lambda_{k}^{2}-\lambda_{-}^{2}}C_{-r,l}\Bigr{)}.$
(70)

The spin polarization created by the triplet pairs is calculated from Eq.
(49). In the superconducting region under the left(right) FI it can be written
as follows:

$\displaystyle\bm{s}_{l,r}=\bm{s}_{0}(x)+s_{loc}(x)\frac{\delta\bm{h}_{l,r}}{h_{0}}+s_{nl}(x)\frac{\delta\bm{h}_{r,l}}{h_{0}},$
(71)

where the first term is the polarization induced by the equilibrium FI
magnetization, $s_{loc}$ is the polarization induced in the left (right)
covered superconducting region by the magnon travelling in the left (right) FI
and $s_{nl}$ is the nonlocal part of the polarization induced by the magnon
travelling in the right(left) covered superconducting region via the nonlocal
triplet correlations penetrating from the other covered superconducting
region. As it was mentioned above, we assume that the FI magnetization is
homogeneous along the $x$-direction in each of the FIs. Consequently, only the
averaged over the FI width value of the polarization enters the LLG equation.
Therefore, we need to average Eq. (71) over the FI width in each of the
superconducting regions under the FIs. Then $\overline{\bm{s}_{0}}$ and
$\overline{s_{loc}}$ contain terms of zero and first order with respect to the
parameter $\xi_{S}/t$ and we can neglect the terms $\sim\xi_{S}/t\ll 1$ for
simplicity without loss of important qualitative physics. At the same time
$\overline{s_{nl}}$ is $\sim\xi_{S}/t$ because it is entirely determined by
the correlations coming from the second FI region and decaying at $\xi_{S}$.
Under the described simplifying assumptions we obtain that $\bm{s}_{0}$ and
$s_{loc}\delta\bm{h}/h_{0}$ are described by the linearized versions of Eqs.
(41) and (42), respectively, and the nonlocal contribution to the polarization
takes the form:

$\displaystyle\overline{s_{nl}}=-N_{F}\int
d\varepsilon\tanh\frac{\varepsilon}{2T}{\rm
Re}\Bigl{[}\frac{h_{0}\Delta\varepsilon
e^{-\lambda_{k}d}}{\lambda_{k}t[(\varepsilon+i\delta)^{2}-h_{0}^{2}]}\Bigl{\\{}\frac{\Delta\varepsilon}{2[(\varepsilon+i\delta)^{2}-h_{0}^{2}](\varepsilon+iDk^{2}/2)}+$
$\displaystyle\frac{1}{iD}\Bigl{(}\frac{C_{+,l}}{\lambda_{k}(\lambda_{k}+\lambda_{+})}+\frac{C_{-,l}}{\lambda_{k}(\lambda_{k}+\lambda_{-})}\Bigr{)}\Bigr{\\}}\Bigr{]}.$
(72)

Taking into account the torque resulting from the nonlocal polarization, we
obtain the coupled system of the LLG equations for the both FIs:

$\displaystyle\frac{\partial\bm{m}_{l,r}}{\partial
t}=-\gamma\bm{m}_{l,r}\times\bm{H}_{eff}+\alpha\bm{m}_{l,r}\times\frac{\partial\bm{m}_{l,r}}{\partial
t}+\frac{J}{d_{FI}}\overline{s_{nl}}\bm{m}_{l,r}\times\bm{m}_{r,l}.$ (73)

The last term couples the both magnetizations. Then we express
$\bm{m}_{l,r}=\bm{m}_{0}+\delta\bm{m}_{l,r}$ as a sum of the equilibrium
magnetization $\bm{m}_{0}$ and the magnon contribution $\delta\bm{m}_{l,r}$.
Linearizing Eqs. (73) with respect to $\delta\bm{m}_{l,r}$ and solving them
for the magnon dispersion, we obtain:

$\displaystyle\omega_{\pm}(k)=\omega_{0}+\tilde{D}_{m}k^{2}\mp\frac{J}{d_{FI}}\overline{s_{nl}},$
(74)

where $\omega_{0}+\tilde{D}_{m}k^{2}$ is the magnon dispersion in a separate
FI/S bilayer with the renormalized stiffness $\tilde{D}_{m}=D_{m}+\delta
D_{m}$ and the last term accounts for the hybridization and splitting of the
magnon modes due to the interaction via the superconductor. The eigenvectors
of the linearized with respect to $\delta\bm{m}_{l,r}$ version of Eqs. (73)
corresponding to the eigenfrequencies Eq. (74) take the form:

$\displaystyle\left(\begin{array}[]{c}\delta\bm{m}_{l,\pm}\\\
\delta\bm{m}_{r,\pm}\end{array}\right)=\left(\begin{array}[]{c}1\\\ \pm
1\end{array}\right),$ (79)

that is, the eigenmodes of the system are represented by the symmetric and
antisymmetric combination of the uncoupled magnons. The value of the frequency
splitting $\Delta\omega=(J/d_{FI})\overline{s_{nl}}$ can be estimated as
$\Delta\omega\sim 10^{9}\exp(-d/\xi_{S})$ Hz for the material and geometric
parameters used above and $t/\xi_{S}\sim 10$.

The coupling between the uniform modes of two ferromagnetic insulators via a
superconductor layer has recently been investigated [68]. However, the
nonzero-wavevector excitations considered here allow for a realization of
magnon directional coupler based on a fundamentally different physical
principle than it has been proposed earlier [37, 66, 67]. Indeed magnons of a
given frequency injected into the coupled region have different wave vectors
$k_{\pm}\approx k_{0}\pm\Delta k$, where $\Delta
k=k_{0}\Delta\omega/\tilde{D}_{m}k_{0}^{2}$. The coupling length admitted by
this design is given in Eq. (12) of the main text. Among the other advantages
of this coupling principle we can mention more compactness of the proposed
setup, because the strength of the dipole-dipole coupling is strongly reduced
with lowering the thickness $d_{FI}$ of the FI layers along the $z$-direction
[37], while in the framework of the proposed mechanism the coupling strength
is $\propto d_{FI}^{-1}$, that is ultra-thin ferromagnetic films are more
favorable. Second, the superconducting coupling can be switched on/off by any
means, which are known to control superconductivity: temperature, magnetic
field, voltage. An interesting perspective is to investigate the possibility
to control the coupling strength via the superconducting phase. Our proposed
design also enables an analogous coupler for magnons in antiferromagnetic
lines, because the dynamical triplets should also be generated there, but the
stray fields are weak and, therefore, the dipole-dipole coupling principle
does not work well.

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