# A Platform for Braiding Majorana Modes with Magnetic Skyrmions

Shiva T. Konakanchi<EMAIL_ADDRESS>Department of Physics and Astronomy,
Purdue University, West Lafayette, Indiana 47907, USA  Jukka I. Väyrynen
Department of Physics and Astronomy, Purdue University, West Lafayette,
Indiana 47907, USA  Yong P. Chen  Department of Physics and Astronomy, Purdue
University, West Lafayette, Indiana 47907, USA  Elmore Family School of
Electrical and Computer Engineering, Purdue University, West Lafayette,
Indiana 47907, USA  Birck Nanotechnology Center, and Purdue Quantum Science
and Engineering Institute, Purdue University, West Lafayette, Indiana 47907,
USA  Pramey Upadhyaya  Elmore Family School of Electrical and Computer
Engineering, Purdue University, West Lafayette, Indiana 47907, USA  Leonid P.
Rokhinson<EMAIL_ADDRESS>Department of Physics and Astronomy, Purdue
University, West Lafayette, Indiana 47907, USA  Elmore Family School of
Electrical and Computer Engineering, Purdue University, West Lafayette,
Indiana 47907, USA  Birck Nanotechnology Center, and Purdue Quantum Science
and Engineering Institute, Purdue University, West Lafayette, Indiana 47907,
USA

###### Abstract

After a decade of intense theoretical and experimental efforts, demonstrating
braiding of Majorana modes remains an unsolved problem in condensed matter
physics due to platform specific challenges. In this work, we propose
topological superconductor – magnetic multilayer heterostructures with on-chip
microwave cavity readout as a novel platform for initializing, braiding and
reading out Majorana modes. Stray fields from a skyrmion in the magnetic
layers can nucleate a vortex in the topological superconducting layer. Such a
vortex is known to host Majorana bound states at its core. Through analytical
calculations within London and Thiele formalisms, and through micromagnetic
simulations, we show that our nucleation and braiding scheme can be
effectively realized with a variety of existing options for magnetic and
superconducting layers. Further, we show that the coupling of the Majorana
bound states to electric field of a resonator leads to an experimentally
observable parity-dependent dispersive shift of the resonator frequency. Our
work paves the way for realizing Majorana braiding in the near future.

††preprint: APS/123-QED

##  I Introduction

Demonstration of non-abelian exchange statistics is one of the most active
areas of condensed matter research and yet experimental realization of
braiding of Majorana modes remains elusive [1, 2]. Most efforts so far have
been focused on superconductor/semiconductor nanowire hybrids, where Majorana
bound states (MBS) are expected to form at the ends of a wire or at boundaries
between topologically trivial and non-trivial regions [3, 4, 5, 6]. Recently,
it became clear that abrupt interfaces may also host topologically trivial
Andreev states with experimental signatures similar to MBS [7, 8], which makes
demonstrating braiding in nanowire-based platforms challenging. Phase-
controlled long Josephson junctions (JJ) open much wider phase space to
realize MBS with a promise to solve some problems of the nanowire platform,
such as enabling zero-field operation to avoid detrimental flux focusing for
in-plane fields [9, 10]. However, MBSs in long JJs suffer from the same
problems as in the original Fu-Kane proposal for topological
insulator/superconductor JJs, such as poor control of flux motion along the
junction and presence of sharp interfaces in the vicinity of MBS-carrying
vortices which may host Andreev states and trap quasiparticles. For instance,
MBS spectroscopy in both HgTe and InAs-based JJs shows a soft gap [11],
despite a hard SC gap in an underlying InAs/Al heterostructure.

(a)

(b)

(c)

Figure 1:  (a) Schematic of the Majorana braiding platform. Magnetic
multilayer (MML) is patterned into a track and is separated from TSC by a thin
insulating layer. Green lines represent on-chip microwave resonators for a
dispersive parity readout setup. The left inset shows a magnified view of a
SVP and the right inset shows the role of each layer (b) Expanded view of the
composition of an MML (c) Process flow diagram for our Majorana braiding
scheme. Here, $T_{c}$ is superconducting transition temperature and $T_{BKT}$
is Berezinskii–Kosterlitz–Thouless transition temperature for the TSC.

In the search for alternate platforms to realize Majorana braiding,
spectroscopic signatures of MBS have been recently reported in STM studies of
vortex cores in iron-based topological superconductors (TSC) [12]. Notably, a
hard gap surrounding the zero-bias peak at a relatively high temperature of
$0.55$ K, and a $5$ K separation gap from trivial Caroli-de Gennes-Matricon
(CdGM) states were observed [13, 14]. Moreover, vortices in a TSC can be
field-coupled to a skyrmion in an electrically-separated magnetic multilayer
(MML) [15, 16], which can be used to manipulate the vortex. This allows for
physical separation of the manipulation layer from the layer wherein MBS
reside, eliminating the problem of abrupt interfaces faced by nanowire hybrids
and JJs. Finally, recent advances in the field of spintronics provide a
flexible toolbox to design MML in which skyrmions of various sizes can be
stabilized in zero external magnetic field and at low temperatures [16, 17,
18]. Under the right conditions, stray fields from these skyrmions alone can
nucleate vortices in the adjacent superconducting layer. In this paper, we
propose TSC–MML heterostructures hosting skyrmion-vortex pairs (SVP) as a
viable platform to realize Majorana braiding. By patterning the MML into a
track and by driving skyrmions in the MML with local spin-orbit torques (SOT),
we show that the SVPs can be effectively moved along the track, thereby
facilitating braiding of MBS bound to vortices.

The notion of coupling skyrmions (Sk) and superconducting vortices (Vx)
through magnetic fields has been studied before [15, 19, 20, 21, 22, 23, 24,
25, 26]. Menezes et al. [27] performed numerical simulations to study the
motion of a skyrmion–vortex pair when the vortex is dragged via supercurrents
and Hals et al. [28] proposed an analytical model for the motion of such a
pair where a skyrmion and a vortex are coupled via exchange fields. However,
the dynamics of a SVP in the context of Majorana braiding remains largely
unexplored. Furthermore, no in-situ non-demolition experimental technique has
been proposed to measure MBS in these TSC–MML heterostructures. In this paper,
through micromagnetic simulations and analytical calculations within London
and Thiele formalisms, we study the dynamics of a SVP subjected to external
spin torques. We demonstrate that the SVP moves without dissociation up to
speeds necessary to complete Majorana braiding within estimated quasiparticle
poisoning time. We further eliminate the problem of in-situ MBS measurements
by proposing a novel on-chip microwave readout technique. By coupling the
electric field of the microwave cavity to dipole-moments of transitions from
Majorana modes to CdGM modes, we show that a topological non-demolition
dispersive readout of the MBS parity can be realized. Moreover, we show that
our platform can be used to make the first experimental observations of
quasiparticle poisoning times in topological superconducting vortices.

The paper is organized as follows: in Section II we present a schematic and
describe our platform. In Section III we present the conditions for
initializing a skyrmion–vortex pair and discuss its equilibrium properties. In
particular, we characterize the skyrmion–vortex binding strength. In Section
IV we discuss the dynamics of a SVP in the context of braiding. Then in
Section V, we present details of our microwave readout technique. Finally, we
discuss the scope of our platform in Section VI.

(a)

(b)

(c)

Figure 2:  (a – b) Normalized energies and forces for Sk–Vx interaction
between a Pearl vortex and a Néel skyrmion of varying thickness. (c)
Attractive $F_{Vx-Avx}$ and repulsive $F_{Sk-Avx}$ (colored lines) for the
example materials in Appendix A: $M_{0}=1450$ emu/cc, $r_{sk}=35$ nm,
$d_{s}=50$ nm, $\Lambda=5$ $\mu$m and $\xi=15$ nm.

##  II Platform Description

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3:  (a) Schematic of our braiding process: manipulations of four
skyrmions in the MML track are shown. MBS at the centers of vortices bound to
each of these skyrmions are labeled $\gamma_{1}$–$\gamma_{4}$. Ohmic contacts
in HM layers of the MML are shown in brown and rf readout lines are shown in
green. II–VI show the steps involved in braiding $\gamma_{2}$ and
$\gamma_{4}$. In step II, $\gamma_{1}$ and $\gamma_{2}$ are brought close to
rf lines by applying charge currents from C to A and D to B, respectively.
$\gamma_{1}$ and $\gamma_{2}$ are then initialized by performing a dispersive
readout of their parity (see Section V). Similarly, $\gamma_{3}$ and
$\gamma_{4}$ are initialized after applying charge currents along P to R and Q
to S, respectively. In step III, $\gamma_{2}$ is moved aside to make room for
$\gamma_{4}$ by applying currents from B to X followed by applying currents
from X to C. In step IV, $\gamma_{4}$ is braided with $\gamma_{2}$ by applying
currents along S to X and X to B. Finally, in step V, the braiding process is
completed by bringing $\gamma_{2}$ to S by applying currents from A to X and
from X to S. Parities (i.e., fusion outcomes) of $\gamma_{1}$ and
$\gamma_{4}$, and $\gamma_{3}$ and $\gamma_{2}$ are then measured in step VI.
Fusion outcomes in each pair of MBS indicate the presence or absence of a
fermion corresponding to a parity of $\pm 1$ [29, 30]. (b) Initial position of
the skyrmions labeled A and B in the micromagnetic simulation for skyrmion
braiding (see Appendix. A) (c–h) Positions of the two skyrmions at the given
times as the braiding progresses. Charge current $j=2\times 10^{12}$ A/m2 was
applied.

Our setup consists of a thin TSC layer that hosts vortices grown on top of a
MML that hosts skyrmions as shown in Fig. 1(a). A thin insulating layer
separates the magnetic and superconducting layers ensuring electrical
separation between the two. Vortices in a TSC are expected to host MBS at
their cores [12, 13, 14]. Stray fields from a skyrmion in the MML nucleate
such a vortex in the TSC, forming a bound skyrmion–vortex pair under favorable
energy conditions (see Sec. III). This phenomenon has been recently
experimentally demonstrated in Ref. [16], where stray fields from Néel
skyrmions in Ir/Fe/Co/Ni magnetic multilayers nucleated vortices in a bare
Niobium superconducting film.

The MML consists of alternating magnetic and heavy metal (HM) layers, as shown
in Fig. 1(b). The size of a skyrmion in a MML is determined by a delicate
balance between exchange, magnetostatic, anisotropy and Dzyaloshinskii–Moriya
interaction (DMI) energies [31, 32] – and the balance is highly tunable,
thanks to advances in spintronics [17, 18, 33]. Given a TSC, this tunability
allows us to find a variety of magnetic materials and skyrmion sizes that can
satisfy the vortex nucleation condition [to be detailed in Eq. (1)]. In
Appendix A, we provide a specific example of FeTeSe topological superconductor
coupled with Ir/Fe/Co/Ni magnetic multilayers.

Due to large intrinsic spin-orbit coupling, a charge current through the heavy
metal layers of a MML exerts spin-orbit torques (SOT) on the magnetic moments
in the MML, which have been shown to drive skyrmions along magnetic tracks
[34, 35]. In our platform, to realize Majorana braiding we propose to pattern
the MML into a track as shown in Fig. 1(a) and use local spin-orbit torques to
move skyrmions along each leg of the track. If skyrmions are braided on the
MML track, and if skyrmion-vortex binding force is stronger than total pinning
force on the SVPs, then the MBS hosting vortices in TSC will closely follow
the motion of skyrmions, resulting in the braiding of MBS. We note here that
there is an upper threshold speed with which a SVP can be moved as detailed in
Sec. IV. By using experimentally-relevant parameters for TSC and MML in
Appendix A, we show that our Majorana braiding scheme can be realized with
existing materials.

We propose a non-demolition microwave measurement technique for the readout of
the quantum information encoded in a pair of vortex Majorana bound states
(MBS). A similar method has been proposed for the parity readout in
topological Josephson junctions [36, 37, 38, 39, 40] and in Coulomb blockaded
Majorana islands [41]. Dipole moments of transitions from MBS to CdGM levels
couple dispersively to electric fields in a microwave cavity, producing a
parity-dependent dispersive shift in the cavity resonator frequency. Thus by
probing the change in the resonator’s natural frequency, the state of the
Majorana modes can be inferred. Virtual transitions from Majorana subspace to
excited CdGM subspace induced due to coupling to the cavity electric field are
truly parity conserving, making our readout scheme a so-called topological
quantum non-demolition technique [40, 39]. The readout scheme is explained in
greater detail in Sec. V.

As discussed above, in our platform we consider coupling between a thin
superconducting layer and magnetic multilayers. We note that in thin
superconducting films, vortices are characterized by the Pearl penetration
depth, given by $\Lambda\ =\ \lambda^{2}/d_{s}$, where $\lambda$ is the London
penetration depth and $d_{s}$ is the thickness of the TSC film. Typically,
these penetration depths $\Lambda$ are much larger than skyrmion radii
$r_{sk}$ in MMLs of interest. Further, interfacial DMI in MML stabilizes a
Néel skyrmion as opposed to a Bloch skyrmion. So hereon, we only study
coupling between a Néel skyrmion and a Pearl vortex in the limit $\Lambda\gg
r_{sk}$.

##  III Initialization and SVP in Equilibrium

Fig. 1(c) illustrates the process flow of our initialization scheme. Skyrmions
can be generated individually in MML by locally modifying magnetic anisotropy
through an artificially created defect center and applying a current through
adjacent heavy metal layers [42]. Such defect centers have been experimentally
observed to act as skyrmion creation sites [43]. When the TSC–MML
heterostructure is cooled below the superconducting transition temperature (SC
$T_{C}$), stray fields from a skyrmion in the MML will nucleate a vortex and
an antivortex in the superconducting layer if the nucleation leads to a
lowering in overall free energy of the system [15]. An analytical expression
has been obtained for the nucleation condition in Ref. [44] ignoring
contributions of dipolar and Zeeman energies to total magnetic energy: a Néel
skyrmion nucleates a vortex directly on top of it if

$d_{m}\left[\alpha_{K}\frac{Kr_{sk}^{2}}{2}-\alpha_{A}A-M_{0}\phi_{0}\right]\geq\frac{{\phi_{0}}^{2}}{8\pi^{2}\lambda}\ln\left(\frac{\Lambda}{\xi}\right).$
(1)

Here, $d_{m}$ is the effective thickness, $M_{0}$ is the saturation
magnetization, $A$ is the exchange stiffness and $K$ is the perpendicular
anisotropy constant of the MML; $\alpha_{K}$ and $\alpha_{A}$ are positive
constants that depend on skyrmion’s spatial profile (see Appendix A), $r_{sk}$
is the radius of the skyrmion in the presence of a Pearl vortex 111The radius
of a skyrmion is not expected to change significantly in the presence of a
vortex [44]. We verified this claim with micromagnetic simulations. For the
materials in Appendix A, when vortex fields are applied on a bare skyrmion,
its radius increased by less than $10\%$. So, for numerical calculations in
this paper, we use bare skyrmion radius for $r_{sk}$., $\phi_{0}$ is the
magnetic flux quantum, and $\Lambda$ ($\xi$) is the Pearl depth (coherence
length) of the TSC. Although a complete solution of the nucleation condition
must include contributions from dipolar and Zeeman energies to total energy of
a MML, such a calculation can only be done numerically and Eq. (1) can still
be used as an approximate estimate. For the choice of materials listed in the
Appendix, the left side of the equation exceeds the right side by $400\%$,
strongly suggesting the nucleation of a vortex for every skyrmion in the MML.
Furthermore, skyrmions in Ir/Fe/Co/Ni heterostructures have also been
experimentally shown to nucleate vortices in Niobium superconducting films
[16].

We proceed to characterize the strength of a skyrmion (Sk) – vortex (Vx)
binding force as it plays a crucial role in determining the feasibility of
moving the skyrmion and the vortex as a single object. Spatial magnetic
profile of a Néel skyrmion is given by
$\bm{M}_{sk}=M_{0}[\zeta\sin\theta(r)\bm{\hat{r}}+\cos\theta(r)\bm{\hat{z}}]$,
where $\zeta=\pm$1 is the chirality and $\theta(r)$ is the angle of the
skyrmion. For $\Lambda\gg r_{sk}$, the interaction energy between a vortex and
a skyrmion is given by [44]:

$E_{Sk-
Vx}=\frac{M_{0}\phi_{0}r_{sk}^{2}}{2\Lambda}\int_{0}^{\infty}\frac{1}{q^{2}}(e^{-q\tilde{d}}-1)J_{0}(qR)m_{z,\theta}(q)\,dq,$
(2)

where $\tilde{d}=d_{m}/r_{sk}$, $J_{n}$ is the nth-order Bessel function of
the first kind, and $R=r/r_{sk}$ is the normalized horizontal displacement $r$
between the centers of the skyrmion and the vortex. $m_{z,\theta}(q)$ contains
information about skyrmion’s spatial profile and is given by [44]:
$m_{z,\theta}(q)=\int_{0}^{\infty}x[\zeta
q+\theta^{\prime}(x)]J_{1}(qx)\sin\theta(x)\,dx$, where $\theta(x)$ is
determined by skyrmion ansatz.

We now derive an expression for the skyrmion–vortex restoring force by
differentiating Eq. (2) with respect to $r$:

$F_{Sk-
Vx}=-\frac{M_{0}\phi_{0}r_{sk}}{2\Lambda}\int_{0}^{\infty}\frac{1}{q}(1-e^{-q\tilde{d}})J_{1}(qR)m_{z,\theta}(q)\,dq.$
(3)

For small horizontal displacements $r\ll r_{sk}$ between the centers of the
skyrmion and the vortex, we can approximate the Sk–Vx energy as:

$E_{Sk-Vx}=\frac{1}{2}kr^{2},$ (4)

with an effective spring constant

$k=-\frac{M_{0}\phi_{0}}{4\Lambda}\int_{0}^{\infty}(1-e^{-q\tilde{d}})m_{z,\theta}(q)\,dq.$
(5)

Figs. 2(a)–2(b) show binding energy and restoring force between a vortex and
skyrmions of varying thickness for the materials listed in Appendix A. Here we
used domain wall ansatz for the skyrmion with
$\theta(x)=2\tan^{-1}[\frac{\sinh(r_{sk}/\delta)}{\sinh(r_{sk}x/\delta)}]$,
where $r_{sk}/\delta$ is the ratio of skyrmion radius to its domain wall width
and $x$ is the distance from the center of the skyrmion normalized by
$r_{sk}$. As seen in Fig. 2(b), the restoring force between a skyrmion and a
vortex increases with increasing separation between their centers until it
reaches a maximum value, $F_{max}$, and then decreases with further increase
in separation. We note that $F_{max}$ occurs when Sk–Vx separation is equal to
the radius of the skyrmion, i.e. when $R=1$ in Eq. (3):

$F_{max}=-\frac{M_{0}\phi_{0}r_{sk}}{2\Lambda}\int_{0}^{\infty}\frac{1}{q}(1-e^{-q\tilde{d}})J_{1}(q)m_{z,\theta}(q)\,dq.$
(6)

As the size of the skyrmion increases, the maximum binding force $F_{max}$ of
the SVP increases. For a given skyrmion size, increasing the skyrmion
thickness increases the attractive force until the thickness reaches the size
of the skyrmion. Further increase in MML thickness does not lead to an
appreciable increase in stray fields outside the MML layer and, as a result,
the Sk–Vx force saturates.

It is important to note that stray fields from a skyrmion nucleate both a
vortex and an antivortex (Avx) in the superconducting layer [15, 45, 46, 47].
While the skyrmion attracts the vortex, it repels the antivortex. Eqs. (2) and
(3) remain valid for Sk–Avx interaction, but switch signs. The equilibrium
position of the antivortex is at the location where repulsive
skyrmion–antivortex force, $F_{Sk-Avx}$, is balanced by the attractive
vortex–antivortex force, $F_{Vx-Avx}$ [48, 49]. Fig. 2(c) shows $F_{Vx-Avx}$
against $F_{Sk-Avx}$ for the platform in the Appendix. We see that for thicker
magnets, the location of the antivortex is far away from that of the vortex,
where the Avx can be pinned with artificially implanted pinning centers [50,
51]. For thin magnetic films, where the antivortex is expected to be nucleated
right outside the skyrmion radius, we can leverage
Berezinskii–Kosterlitz–Thouless (BKT) transition to negate $F_{Vx-AVx}$ for
Vx-Avx distances $r<\Lambda$ [52, 53, 54, 55]. Namely, when a Pearl
superconducting film is cooled to a temperature below $T_{C}$ but above
$T_{BKT}$, vortices and antivortices dissociate to gain entropy, which
minimizes the overall free energy of the system [56]. While the attractive
force between a vortex and an antivortex is nullified, a skyrmion in the MML
still attracts the vortex and pushes the antivortex towards the edge of the
sample, where it can be pinned. Therefore we assume that the antivortices are
located far away and neglect their presence in our braiding and readout
schemes.

##  IV Braiding

Majorana braiding statistics can be probed by braiding a pair of MBS [1] which
involves swapping positions of the two vortices hosting the MBS. We propose to
pattern the MML into interconnected Y-junctions as shown in Fig. 3 to enable
that swapping. Ohmic contacts in HM layers across each leg of the Y-junctions
enable independent application of charge currents along each leg of the track.
These charge currents in-turn apply spin-orbit torques on the adjacent
magnetic layers and enable skyrmions to be moved independently along each leg
of the track. As long as skyrmion and vortex move as a collective object,
braiding of skyrmions in the MML leads to braiding of MBS hosting vortices in
the superconducting layer. Below we study the dynamics of a SVP subjected to
spin torques for braiding. We calculate all external forces acting on the SVP
in the process and discuss the limits in which the skyrmion and the vortex
move as a collective object.

For a charge current $\bm{J}$ in the HM layer, the dynamics in the magnetic
layer is given by the modified Landau–Lifshitz–Gilbert (LLG) equation [57,
58]:

$\partial_{t}\bm{m}=-\gamma(\bm{m}\times{{\bm{H}}_{eff}}+\eta J\
\bm{m}\times\bm{m}\times\bm{p})+\alpha\bm{m}\times\partial_{t}\bm{m}$ (7)

where we have included damping-like term from the SOT and neglected the field-
like term as it does not induce motion of Néel skyrmions for our geometry
[59]. Here, $\gamma$ is the gyromagnetic ratio, $\alpha$ is the Gilbert
damping parameter, and ${{\bm{H}}_{eff}}$ is the effective field from dipole,
exchange, anisotropy and DMI interactions.
$\bm{p}=sgn(\Theta_{SH})\bm{\hat{J}}\times\hat{\bm{n}}$ is the direction of
polarization of the spin current, where $\Theta_{SH}$ is the spin Hall angle,
$\bm{\hat{J}}$ is the direction of charge current in the HM layer and
$\hat{\bm{n}}$ is the unit vector normal to the MML.
$\eta=\hbar\Theta_{SH}/2eM_{0}d_{m}$ quantifies the strength of the torque,
$\hbar$ is the reduced Planck’s constant and $e$ is the charge of an electron.

Assuming skyrmion and vortex move as a collective object, semiclassical
equations of motion for the centers of mass of the skyrmion and the vortex can
be written using collective coordinate approach as done in Ref. [28]:

$\displaystyle m_{sk}\ddot{\bm{R}}_{sk}={\bf{F}}_{SOT}-\frac{\partial U_{sk,\
pin}}{\partial\bm{R}_{sk}}-$
$\displaystyle{\bm{G}}_{sk}\times\dot{\bm{R}}_{sk}-4\pi
s\alpha\dot{\bm{R}}_{sk}$ (8) $\displaystyle-k({\bm{R}}_{sk}-{\bm{r}}_{vx}),$

and

$\displaystyle m_{vx}\ddot{\bm{R}}_{vx}=-\frac{\partial U_{vx,\
pin}}{\partial\bm{R}_{vx}}-$
$\displaystyle{\bm{G}}_{vx}\times\dot{\bm{R}}_{vx}-{\alpha}_{vx}\dot{\bm{R}}_{vx}$
(9) $\displaystyle+k({\bm{R}}_{sk}-{\bm{r}}_{vx}),$

where ${\bm{R}}_{sk}$ (${\bm{R}}_{vx}$), $m_{sk}$ ($m_{vx}$) and $q_{sk}$
($q_{vx}$) are the position, mass and chirality of the skyrmion (vortex). $k$
is the effective spring constant of the Sk–Vx system, given in Eq. (5).
${\bm{F}}_{SOT}=\pi^{2}\gamma\eta r_{sk}s\bm{{J}}\times\hat{\bm{n}}$ is the
force on a skyrmion due to spin torques in Thiele formalism, where
$s=M_{0}d_{m}/\gamma$ is the spin density [60, 61]. The third term on the
right side of Eq. (8) gives Magnus force on the skyrmion, with
${\bm{G}}_{sk}=4\pi sq_{sk}\hat{\bm{z}}$, and the fourth term characterizes a
dissipative force due to Gilbert damping. Similarly, the second term on the
right side of Eq. (9) gives the Magnus force on the vortex with
${\bm{G}}_{vx}=2\pi sn_{vx}q_{vx}\hat{\bm{z}}$, with $n_{vx}$ being the
superfluid density of the TSC, and the third term characterizes viscous force
with friction coefficient ${\alpha}_{vx}$. $U_{sk,\ pin}$ ($U_{vx,\ pin}$)
gives the pinning potential landscape for the skyrmion (vortex). The last term
in Eq. (9) represents restoring force on a vortex due to its separation from a
skyrmion and is valid when $\mid{\bm{R}}_{sk}-{\bm{R}}_{vx}\mid<r_{sk}$. Here,
$k$ is the effective spring constant characterizing Sk–Vx force, as given by
Eq. (4).

We consider steady-state solutions of the equations of motion assuming that
the skyrmion and the vortex are bound. We discuss conditions for the
dissociation of a SVP later. For a given external current $\bm{J}$, velocity
$v$ of a SVP in steady state is obtained by setting
$\ddot{\bm{R}}_{sk}=\ddot{\bm{R}}_{vx}=0$ and
$\dot{\bm{R}}_{sk}=\dot{\bm{R}}_{vx}=\dot{\bm{R}}$ in Eqs. (8) and (9):

$v=|\dot{\bm{R}}|=\frac{\pi^{2}\gamma\eta
r_{sk}sJ}{\sqrt{(G_{sk}+G_{vx})^{2}+(4\pi s\alpha+\alpha_{vx})^{2}}}.$ (10)

In general, the SVP moves at an angle $\varphi$ relative to $\bm{{F}}_{SOT}$
due to Magnus forces on the skyrmion and the vortex, with:

$\displaystyle\tan\varphi=\frac{G_{sk}+G_{vx}}{4\pi s\alpha+\alpha_{vx}}.$
(11)

Armed with the above equations, we extract some key parameters that determine
the feasibility of our braiding scheme. First, if $\bm{{F}}_{SOT}$ from
external currents is unable to overcome the maximum pinning force on either
the skyrmion ($F_{pin,sk}$) or the vortex ($F_{pin,vx}$), the SVP will remain
stationary. This gives us a lower threshold $J^{-}$ on the external current
which is obtained by weighing $\bm{{F}}_{SOT}$ against the pinning forces:

$J^{-}=\frac{max(F_{pin,sk},F_{pin,vx})}{\pi^{2}\gamma\eta r_{sk}s}.$ (12)

Second, once the SVP is in motion, drag and Magnus forces that act on the
skyrmion and the vortex are proportionate to their velocity. If the net
external force on a vortex in motion is larger than the maximum force with
which a skyrmion can pull it ($F_{max}$), then the skyrmion and the vortex
dissociate and no longer move as a collective object. This sets an upper bound
$v^{+}$ on the SVP speed which can be obtained by balancing $F_{max}$ with the
net force from Magnus and drag forces on the vortex. This maximum speed plays
a key role in determining whether our braiding and readout scheme can be
completed within the quasiparticle poisoning time.

$v^{+}=\frac{F_{max}}{\sqrt{(\alpha_{vx})^{2}+(G_{vx})^{2}}}.$ (13)

An upper bound on the SVP speed implies an upper bound $J^{+}$ on the external
current which can be obtained by putting $v^{+}$ in Eq. (10):

$J^{+}=\frac{v^{+}\sqrt{(G_{sk}+G_{vx})^{2}+(4\pi
s\alpha+\alpha_{vx})^{2}}}{\pi^{2}\gamma\eta r_{sk}s}.$ (14)

Another parameter of critical importance, the distance of closest approach
between two skyrmion–vortex pairs ($r_{min}$) plays a crucial role in
achieving significant overlap of the MBS wavefunctions centered at the vortex
cores and is given by balancing Sk–Vx attractive force by Vx–Vx repulsive
force:

$r_{min}=\frac{\phi_{0}^{2}}{4\pi^{2}\Lambda}\frac{1}{F_{max}}.$ (15)

Finally, the power $P$ dissipated in heavy metal layers due to Joule heating
from charge currents has to be effectively balanced by the cooling rate of the
dilution refrigerator:

$P=n_{hm}LWt_{hm}\rho_{hm}J^{2},$ (16)

where $n_{hm}$ is the number of heavy metal layers, $L$ ($W$) is the length
(width) of the active segment of the MML track, $t_{hm}$ is the thickness of
each heavy metal layer and $\rho_{hm}$ is the resistivity of a heavy metal
layer.

By applying a current $J^{-}<J<J^{+}$ locally in a desired section of the MML
track, each SVP can be individually addressed. For the materials listed in
Appendix A, the maximum speed $v^{+}$ with which a SVP can be moved is over
$1000$ m/s. At this top speed, SVPs can cover the braiding distance (the sum
of the lengths of the track in steps I–VI of Fig. 3(a)) of $50r_{sk}$ in about
$0.15$ ns, but the process generates substantial Joule heating. At a reduced
speed of $0.25$ m/s, SVPs cover that distance in $7$ $\mu$s generating $30$
$\mu$W of heat during the process, which is within the cooling power of modern
dilution refrigerators. SVPs can be braided at faster speeds if the dilution
fridges can provide higher cooling power or if the resistivity of heavy metal
layers in the MML can be lowered. Although quasiparticle poisoning times in
superconducting vortices have not been measured yet, estimates in similar
systems range from hundreds of microseconds to seconds [62, 63, 64]. Our
braiding time falls well within such estimates for quasiparticle poisoning
times, indicating the viability of our platform. Furthermore, the ability to
easily tune braiding times in our platform by varying magnitude of currents in
heavy metal layers can be used to investigate the effects of quasiparticle
poisoning on the braiding protocol.

As will be shown in Section V, Vx–Vx distances $<10\xi$ should be sufficient
to perform a dispersive readout of MBS parity in adjacent vortices. For the
materials listed in Appendix A, the distance of closest approach between two
vortices is $r_{min}=40$ nm. The shape of the MML track further limits how
close two vortices can be brought together (see step II in Fig. 3(a)). With
the geometry of the track taken into account, Vx–Vx distance less than $10\xi$
can still be easily achieved, enough to induce a detectable shift in cavity’s
resonance frequency during the dispersive readout.

Figs. 3(b)–3(h) show the results of micromagnetic simulation of braiding
skyrmions in a smaller section of the MML (for computational reasons) for the
example platform. The details of the simulation are given in the Appendix A.
The simulation results demonstrate the effectiveness of using local SOT to
move individual skyrmions and realize braiding. Finally, as discussed in this
section, due to the strong skyrmion–vortex binding force, vortices hosting MBS
in the TSC will braid alongside the skyrmions.

Figure 4:  (a) Schematic of our readout process. When two vortices are brought
close, microwave transitions can be dispersively driven from the MBS to the
excited hybridized CdGM levels (only level $1$ is shown). Parity of the
Majorana mode can be inferred from the difference in the cavity frequency
shift produced by $\omega_{-\mathcal{M},1}$ and $\omega_{\mathcal{M},1}$
transitions (see Eq. (20)). The allowed fermion parity conserving transitions
are shown in both single-particle and many-particle representations. In the
latter, dashed and solid lines denote states in the two fermion parity
sectors. The transition of frequency $\omega_{-\mathcal{M},1}$ (blue arrows)
corresponds to breaking a Cooper pair and exciting the MZM and CdGM levels
(MZM being initially unoccupied). When the MZM is occupied, the transition of
frequency $\omega_{\mathcal{M},1}$ (red arrow) excites the MZM quasiparticle
into the CdGM level. The dipole transition matrix elements are different for
the two processes, enabling parity readout. (b) MZM-parity sensitive dipole
transition strength versus vortex pair separation. We denote
$g^{2}_{n}=(|\mathbf{E}_{0}\cdot\mathbf{d}_{n,-\mathcal{M}}|^{2}-|\mathbf{E}_{0}\cdot\mathbf{d}_{n,\mathcal{M}}|^{2})$
the dipole transition strength between the Majorana level and the $n$th CdGM
level. We plot the dimensionless strength normalized by
$U=e|\mathbf{E}_{0}|l$. As expected from MZM hybridization, $g_{n}$ decays
approximately exponentially in the distance between the two vortices.
Oscillations in $g^{2}_{n}$ represent oscillations in the wave functions of a
clean system. In a disordered (real) system the oscillations are expected to
be smeared out. The inset shows the probability density for the MZM hosted by
a vortex pair 400 nm apart. The simulation was done for an effective 2D model
(a $1000\times 600\mathrm{nm}^{2}$ rectangle) of a 3D topological insulator
surface, see Refs. [65, 66, 67]. We used $\xi=15\mathrm{nm}$, vortex radius
$r=\xi$, and $E_{F}=1.125\Delta$ in the vortex.

##  V Readout

Quantum information is encoded in the charge parity of MBS hosted in a pair of
vortices which we propose to readout with a dispersive measurement technique.
Fig. 4a summarizes our readout scheme - the top figure shows single particle
energy levels and the bottom figure shows many-body energy levels of a pair of
vortices brought close to each other. In the dispersive limit, a microwave
cavity electric field can drive virtual transitions from the ground state
Majorana manifold to excited CdGM manifold (only one CdGM level, labeled 1, is
considered in the figure). The transitions allowed by selection rules, labeled
$\omega_{-\mathcal{M},1}$ and $\omega_{\mathcal{M},1}$ are shown in the many-
body spectrum. Each of these virtual transitions causes a state-dependent
dispersive shift in the cavity’s natural frequency and the parity of the
vortex pair can be inferred from relative change in cavity frequency. Note
that each of the allowed transitions is truly parity conserving since
microwaves cannot change the number of fermions. Since parity states are true
eigenstates (as opposed to approximate eigenstates) of the measurement
operation, our readout scheme can be dubbed as a topological quantum non-
demolition technique [40, 39]. We now proceed to calculate dipole coupling
strengths of the allowed transitions to cavity electric field and the
corresponding dispersive shift.

In BCS mean-field theory, the coupling to electric field can be described by
the Hamiltonian

$\delta
H=-\mathbf{E}(t)\cdot\hat{\mathbf{d}}\,,\quad\hat{\mathbf{d}}=\frac{e}{2}\int
d^{2}\mathbf{r}\mathbf{r}\hat{\Psi}^{\dagger}\tau_{z}\hat{\Psi}\,,$ (17)

where $\mathbf{E}(t)=\mathbf{E}_{0}\cos\omega t$ is the microwave-induced
time-dependent electric field which is approximately uniform over the scale of
the vortices 222In Eq. (17), we assume a thin film superconductor that can be
approximated by a 2D system. This model can also describe a 3D superconductor
when the electric field $\mathbf{E}$ does not penetrate deep into its bulk. .
The electric field couples to the dipole operator $\hat{\mathbf{d}}$ of the
electronic states in the vortices. We have written it in terms of the electron
field operator in Nambu spinor notation,
$\hat{\Psi}=(\psi_{\uparrow},\psi_{\downarrow},\psi_{\downarrow}^{\dagger},-\psi_{\uparrow}^{\dagger})^{T}$;
The Pauli matrix $\tau_{z}$ acts on the particle-hole indices. At low
energies, we expand the field operators in terms of eigenstates as

$\hat{\Psi}(\mathbf{r})=\phi_{1}(\mathbf{r})\hat{\gamma}_{1}+\phi_{2}(\mathbf{r})\hat{\gamma}_{2}+\Phi_{1}(\mathbf{r})\hat{\Gamma}_{1}+\Phi_{-1}(\mathbf{r})\hat{\Gamma}_{1}^{\dagger}+\dots\,,$
(18)

where $\hat{\gamma}_{1,2}$ are the Majorana operators for vortices 1 and 2,
and $\hat{\Gamma}_{1}^{(\dagger)}$ is the annihilation (creation) operator for
the lowest CdGM state. The corresponding wave functions multiply the operators
in Eq. (18).

At low frequencies much below the level spacing $\delta E$ of the vortex
quasiparticle bound states, $\omega\ll\delta E/\hbar$, the microwave field
does not excite the quasiparticle states of the vortices. We shall also assume
that these quasiparticle states are not occupied, for example due to
quasiparticle poisoning. Under these conditions, the vortex pair stays in its
ground state manifold consisting of the two states of unoccupied/occupied non-
local MBS. With sufficiently weak microwave driving we can use dispersive
readout to measure the charge parity
$\sigma_{z}=i\hat{\gamma}_{1}\hat{\gamma}_{2}$ [68, 40]. The dispersive
Hamiltonian of the resonator-vortex pair system reads [40],

$H_{\text{resonator}}+\delta
H=\hat{a}^{\dagger}\hat{a}(\hbar\omega+\sigma_{z}\hbar\chi)\,,$ (19)

where $\hat{a},\hat{a}^{\dagger}$ are the harmonic oscillator annihilation and
creation operators for the resonator. The MBS parity-dependent dispersive
frequency shift is

$\hbar\chi=\frac{g_{1}^{2}}{\delta E}\left[\frac{\delta E^{2}}{\delta
E^{2}-(\hbar\omega)^{2}}\right]\,,$ (20)

where we denote
$g_{1}^{2}=|\mathbf{E}_{0}\cdot\mathbf{d}_{1,-\mathcal{M}}|^{2}-|\mathbf{E}_{0}\cdot\mathbf{d}_{1,\mathcal{M}}|^{2}$
and $\omega$ is the resonator bare frequency, $\mathbf{E}_{0}$ is the electric
field amplitude, and $\delta E$ is the energy gap separating the MBS from the
first excited CdGM mode. We ignore here the exponentially small energy
splitting between the MBS, which would give subleading corrections to $\chi$;
we will see that $\chi$ itself will be exponentially small in the vortex
separation (due to the parity-sensitive transition dipole matrix elements
$\mathbf{d}_{1,-\mathcal{M}}$ and $\mathbf{d}_{1,\mathcal{M}}$ being almost
equal). We denote here $\mathbf{d}_{1,\mathcal{M}}=\langle
1|\hat{\mathbf{d}}|\mathcal{M}\rangle$ and
$\mathbf{d}_{1,-\mathcal{M}}=\langle\mathcal{M},1|\hat{\mathbf{d}}|0\rangle$
where the relevant state are the ground state $|0\rangle$, the single-particle
excited states
$|\mathcal{M}\rangle=\hat{\Gamma}_{\mathcal{M}}^{\dagger}|0\rangle$ and
$|1\rangle=\hat{\Gamma}_{1}^{\dagger}|0\rangle$, and the two-particle excited
state
$|\mathcal{M},1\rangle=\hat{\Gamma}_{1}^{\dagger}\hat{\Gamma}_{\mathcal{M}}^{\dagger}|0\rangle$;
we introduced the annihilation operator
$\hat{\Gamma}_{\mathcal{M}}=(\hat{\gamma}_{1}+i\hat{\gamma}_{2})/2$ for the
non-local MBS.

Evaluating the dipole transition matrix elements
$\mathbf{d}_{1,\pm\mathcal{M}}$ microscopically is somewhat involved since
proper screening by the superconducting condensate needs to be carefully
accounted for and is beyond the BCS mean-field theory [69, 70, 71, 72, 73].
Nevertheless, to estimate $\mathbf{d}_{1,\pm\mathcal{M}}$ we can use Eq. (17)
by replacing $\mathbf{r}\approx l\hat{\mathbf{z}}$ in it, with $l\approx
a_{B}$ being the effective distance to the image charge in the superconductor
and $\hat{\mathbf{z}}$ the surface normal vector [69]. Here $a_{B}$ denotes
the Bohr radius. We evaluate the dimensionless matrix elements of the
effective dipole “charge” $\mathbf{d}\cdot\hat{\mathbf{z}}/l$ by using a
numerical simulation of the Majorana and CdGM states in a double vortex system
depicted in Fig. 4b. The numerical simulations will be detailed in a future
publication [67].

In Fig. 4b we plot the parity-sensitive term $g_{n}^{2}$ that largely
determines the dispersive shift $\chi$, Eq. (20). We find that even a
relatively distant vortex pair can provide a parity-dependent shift
$g_{n}^{2}\sim 10^{-2}(elE_{0})^{2}$. Since the relevant dipole moment is
normal to the superconductor surface, we can couple to the dipole by using a
microwave resonator above the surface, producing a large perpendicular
electric field. With a resonator zero-point voltage $V_{0}\sim
100\,\mu\mathrm{V}$ at a $\sim 10\mathrm{nm}$ distance from the vortices, we
obtain $elE_{0}\approx 1\mu\mathrm{eV}\cdot(l/\text{\AA})\approx 2.4\times
10^{2}h\mathrm{MHz}\cdot(l/\text{\AA})$. (We estimate such high zero-point
voltages can be achieved in high-inductance resonators [74].) Taking a low-
lying CdGM state with $\delta E\sim 10\mu eV$, we obtain $\chi/2\pi\sim
20\,\mathrm{MHz}\cdot(l/\text{\AA})^{2}$ where $l\sim a_{B}\gtrsim
1\text{\AA}$ is the typical dipole size [69]. We thus see that the MBS vortex
parity measurement is well within standard circuit QED measurement
capabilities [68]. We note that the above estimate does not include the
resonant enhancement, the second factor in Eq. (20), which may further
substantially increase the frequency shift.

Finally, we note that the dipole operator $\hat{\mathbf{d}}$ also has a non-
zero diagonal matrix element $\mathbf{d}_{\mathcal{M}}$ in the Majorana state
[72], leading to a term
$\mathbf{E}_{0}\cdot\mathbf{d}_{\mathcal{M}}\sigma_{z}(\hat{a}+\hat{a}^{\dagger})$
in Eq. (19). This term in principle allows one to perform longitudinal readout
of the MBS parity. However, making longitudinal readout practical may require
parametric modulation of the coupling, in our case $\mathbf{d}_{\mathcal{M}}$,
which may be difficult [39, 68].

##  VI Summary

Measuring braiding statistics is the ultimate method to conclusively verify
the existence of non-abelian excitations. We proposed a unified platform to
initialize, braid and readout Majorana modes, avoiding abrupt topological-
trivial interfaces at each stage. We derived general expressions for braiding
speeds with spin currents, distance of closest approach between two Majorana
modes and the resultant dispersive shift in cavity resonance frequency. We
showed that our setup can be readily realized with existing options for TSC
and MML materials.

###### Acknowledgements.

We would like to thank Axel Hoffman and Mohammad Mushfiqur Rahman for helpful
discussions. JIV thanks Dmitry Pikulin and Rafał Rechciński for helpful
discussions on 3D TI simulations. JIV and LPR acknowledge support from the
Office of the Under Secretary of Defense for Research and Engineering under
award number FA9550-22-1-0354. YPC and PU acknowledges partial support of the
work from US Department of Energy (DOE) Office of Science through the Quantum
Science Center (QSC, a National Quantum Information Science Research Center)
and NSF ECCS-1944635. STK acknowledges support from the Purdue research
foundation.

##  Appendix A Example Platform

While our platform can be realized with a variety of existing combinations of
MML and TSC, we choose Ir/Fe/Co/Ni and FeTe0.55Se0.45 for the respective
roles[16, 12]. For the TSC, we used $\lambda=500$ nm, $\xi=15$ nm and chose a
thickness of $d_{s}=50$ nm. This gives a Pearl length of $\Lambda=5$ $\mu$m.
We propose to use 10 layers of Ir/Fe/Co/Ni heterostructure with $0.5$ nm of Fe
and $0.5$ nm of Co per layer giving an effective magnet thickness of
$d_{m}=10$ nm.

We performed micromagnetic simulations using mumax [75, 76] with parameters
similar to those in Ref. [16]: $M_{0}=1450$ emu/cc, $A=13.9$ pJ/m, $K=2000$
kJ/m3, $D=3$ mJ/m2 and $B_{ext}=180$ mT, where $D$ is the
Dzyaloshinskii–Moriya interaction (DMI) constant and $B_{ext}$ is the external
magnetic field. We modified the values of uniaxial anisotropy constant and DMI
constant from those in Ref. [16] to stabilize skyrmions of large size. We also
used external magnetic field in the simulation to stabilize isolated skyrmions
in the simulation. In experimental setup, this magnetic field can be applied
via exchange interaction in magnetic multilayers. Thanks to versatile
spintronics tools [77, 33, 17, 18], these modifications are quite feasible to
realize. Using a grid size of $2\times 2\times 2$ nm3, isolated skyrmions of
size $r_{sk}=35$ nm were stabilized in the simulation.

To demonstrate skyrmion braiding, the MML was patterned into a Y-shaped track
and a spin current corresponding to a charge current of $j=2\times 10^{12}$
A/m2 in HM layers was locally applied in each active leg of the MML track. To
save computational resources, vacuum layers (representing heavy metal layers)
have not been included in mumax simulations. The inclusion of vacuum layers
does not affect the conclusions of this paper in any way. Here, we set spin-
Hall angle, $\Theta_{SH}=0.15$, and the ratio of field-like SOT term to
damping-like SOT term to $0.1$.

Analytical expression for nucleation condition of Eq. (1) was derived in Ref.
[44] by assuming a linear ansatz for skyrmion profile where:
$\alpha_{A}=2\pi\int_{0}^{\infty}dx\,x[\theta^{\prime
2}(x)+\sin^{2}\theta(x)/x^{2}]$,
$\alpha_{K}=4\pi\int_{0}^{\infty}dx\,x\sin^{2}\theta(x)$, and
$\theta^{\prime}(x)=d\theta(x)/dx$, with $\theta(x)=\pi(1-x)$ for $x<1$ and
$\theta(x)=0$ otherwise. When the example material parameters are substituted
in Eq. (1) with $r_{sk}=35$ nm, the left side of the equation exceeds the
right side by about $400\%$, suggesting nucleation of a vortex-antivortex pair
in the TSC for every skyrmion in the MML.

For quantitative analysis of Eqs. (12)–(16), we assume clean limit and neglect
pinning forces on skyrmion and vortex. We include drag and Magnus forces on
skyrmion, and drag force on vortex. We neglect Magnus force on the vortex as
it is expected to be small compared to its drag force [78]. We model drag
force on the vortex using friction coefficient $\alpha_{vx}=\eta_{vx}d_{s}$,
where $\eta_{vx}$ is the Bardeen-Stephen viscous drag coefficient given by
$\eta_{vx}=\phi_{0}^{2}/(2\pi\xi^{2}\rho_{n})$ [79, 80]. Here, $\rho_{n}$ is
the normal state resistivity of the TSC. Following parameters were used for
calculations: Gilbert damping parameter for the MML $\alpha=0.1$ and
$\rho_{n}=10^{-7}$ $\Omega$m giving a $\eta_{vx}=3\times 10^{-8}$ Ns/m2. We
propose that each leg of the MML track in Fig. 1(a) be $10r_{sk}$ wide and
$10r_{sk}$ long. This implies that skyrmions have to cover a total distance of
$50r_{sk}$ for the entirety of the braiding process in Fig. 3(a). We used
resistivity of Iridium $\rho_{hm}=5\times 10^{-7}$ $\Omega$m to calculate the
Joule heat dissipated during the braiding process.

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