Now we prove (b). The smooth pullback of vanishing cycles (Lemma 4.4 (c))
yields the canonical equivalence of the functors

(106)
$\mathrm{ev}_{p}^{*}\circ\varphi_{\mu_{1}}=\varphi_{\mu_{\pi,{\mathcal{M}}}}\circ\mathrm{ev}_{p}^{*}.$

Since the vanishing cycle complex

(107)
$\varphi_{\mu_{1}}{\underline{{\mathbb{C}}}}\in\mathrm{D}^{b}_{c}\left([\mathfrak{h}_{\pi}/H_{\pi}]\right)$

is the $H_{\pi}$-equivariant vanishing cycle complex
$\varphi_{\mu_{\pi}}{\underline{{\mathbb{C}}}}$ on $\mathfrak{h}_{\pi}$, we
see from Lemma 4.3 that (107) is the shifted skyscraper sheaf supported at
$0_{H}$ with the trivial $H_{\pi}$-action. Hence, applying (106) to the
trivial local system ${\underline{{\mathbb{C}}}}$, we deduce that the
vanishing cycle complex
$\varphi_{\mu_{\pi,{\mathcal{M}}}}{\underline{{\mathbb{C}}}}$ is canonically
isomorphic to ${\underline{{\mathbb{C}}}}$ on ${\mathcal{M}}^{D}_{r,L}(\pi)$
with a shift. Since the functor $\varphi_{\mu_{\pi,{\mathcal{M}}}}$ preserves
the perverse t-structures, (105) is concluded.

For (c), the proper base change of vanishing cycles [13, Proposition 4.2.11]
implies the canonical equivalence of the functors

${\mathrm{Rh^{D}_{\pi}}}_{*}\circ\varphi_{\mu_{\pi,{\mathcal{M}}}}=\varphi_{\mu_{\pi,{\mathcal{A}}}}\circ{\mathrm{Rh^{D}_{\pi}}}_{*}.$

As a consequence, we obtain

(108)
$\varphi_{\mu_{\pi,{\mathcal{A}}}}\left(\mathrm{Rh^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)=\mathrm{Rh^{D-p}_{\pi}}_{*}{\underline{{\mathbb{C}}}}[-r_{0}]$

by applying the pushforward functor $\mathrm{Rh^{D}_{\pi}}_{*}$ to (105) and
the fact that $h^{D}_{\pi}$ coincides with $h^{D-p}_{\pi}$ restricting to
${\mathcal{M}}^{D-p}_{r,L}$. Noticing that the regular function

$\mu_{\pi,{\mathcal{M}}}:{\mathcal{M}}^{D}_{r,L}(\pi)\to{\mathbb{A}}^{1}$

passes through the Hitchin base ${\mathcal{A}}^{D}(\pi)$, it admits a
fiberwise $\Gamma$-action. By Lemma 4.4 (a), the vanishing cycle functor
$\varphi_{\mu_{\pi,{\mathcal{M}}}}$ is $\Gamma$-equivariant. Hence the
isomorphism (105) is compatible with the $\Gamma$-equivariant structures on
the shifted trivial local systems on ${\mathcal{M}}^{D}_{r,L}(\pi)$ and
${\mathcal{M}}^{D-p}_{r,L}(\pi)$. After pushing forward, we see that (108)
matches for any $\kappa$-isotypic components with respect to the
$\Gamma$-action. This completes the proof of (c). ∎

The value of the function
$\mu_{\pi,{\mathcal{A}}}:{\mathcal{A}}^{D}(\pi)\rightarrow{\mathbb{A}}^{1}$ is
constant along each orbit of the Galois group $G_{\pi}$-action on
${\mathcal{A}}^{D}(\pi)$. Hence $\mu_{\pi,{\mathcal{A}}}$ induces a regular
function on the $G_{\pi}$-quotient of ${\mathcal{A}}^{D}(\pi)$:

(109)
$\mu_{\gamma,{\mathcal{A}}}:{\mathcal{A}}^{D}_{\gamma}\to{\mathbb{A}}^{1}$

where the element $\gamma\in\Gamma$ corresponds to $\pi:C^{\prime}\to C$.
Alternatively, (109) is the restriction of the function on the
$\mathrm{SL}_{n}$-Hitchin base

$\mu_{\pi=\mathrm{id},{\mathcal{A}}}:{\mathcal{A}}^{D}\to{\mathbb{A}}^{1}$

to the closed subvariety
$i^{D}_{\gamma}:{\mathcal{A}}^{D}_{\gamma}\hookrightarrow{\mathcal{A}}^{D}$.

Recall the notation from the diagram (29). We have the following corollary of
Theorem 4.5.

######  Corollary 4.6.

Theorem 4.5 (c) induces for any character $\kappa\in\hat{\Gamma}$ a natural
isomorphism

$\varphi_{\mu_{\gamma,{\mathcal{A}}}}\left(\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}=\left(\mathrm{Rh^{D-p}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-r_{0}].$

Here $r_{0}$ is the same as in Theorem 4.5 (b,c).

###### Proof.

We consider the quotient map
$q_{\mathcal{A}}:{\mathcal{A}}^{D}(\pi)\to{\mathcal{A}}^{D}_{\gamma}$. By the
proper base change [13, Proposition 4.2.11], we have

(110)
$\varphi_{\mu_{\gamma,{\mathcal{A}}}}\left({q_{\mathcal{A}}}_{*}\mathrm{Rh^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)={q_{\mathcal{A}}}_{*}\varphi_{\mu_{\pi,{\mathcal{A}}}}\left(\mathrm{Rh^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right).$

Similar as in the proof of Theorem 4.5 (c), the identity (110) is compatible
with the $G_{\pi}$\- and $\Gamma$-actions on both sides. By taking the
$G_{\pi}$-invariant and the $\kappa$-isotypic parts, we obtain that

$\begin{split}\varphi_{\mu_{\gamma,{\mathcal{A}}}}\left(({q_{\mathcal{A}}}_{*}\mathrm{Rh^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}})^{G_{\pi}}\right)_{\kappa}=\left({q_{\mathcal{A}}}_{*}\varphi_{\mu_{\pi,{\mathcal{A}}}}\left(\mathrm{Rh^{D}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\right)^{G_{\pi}}=\left({q_{\mathcal{A}}}_{*}\mathrm{Rh^{D-p}_{\pi}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}^{G_{\pi}}[-r_{0}]\end{split}$

where we use Theorem 4.5 (c) in the last identity. Hence Lemma 1.7 implies
that

$\varphi_{\mu_{\gamma,{\mathcal{A}}}}\left(\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}=\left(\mathrm{Rh^{D-p}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-r_{0}].\qed$

######  Remark 4.7.

A different choice of trivialization (86) affects the functions (103), and
therefore the vanishing cycle sheaves, by a scalar
$\lambda\in{\mathbb{C}}^{*}$. Since all the isomorphisms in our main theorems
are only constructed up to scaling, choices of (86) do not matter for our
purpose.

###  4.4. Proof of Theorem 3.2

In this section, we prove Theorem 3.2 by constructing the operator

(111)
$c^{D}_{\kappa}:\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i^{D}_{\gamma}}_{*}\left(\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}_{\gamma}].$

This recovers the $G_{\pi}$-equivariant operator

$q_{\mathcal{A}}^{*}\left(c^{D}_{\kappa}\right):q_{\mathcal{A}}^{*}\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}\xrightarrow{~{}~{}\simeq~{}~{}}\left(\mathrm{Rh_{\pi}^{D}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\Big{|}_{{\mathcal{A}}^{D}(\pi)^{*}}[-2d^{D}_{\gamma}].$

for Theorem 3.1 as explained in Section 3.1.

We construct (111) in the following three cases:

  1. Case 1.

The effective divisor $D$ satisfies that $\mathrm{deg}(D)$ is even and greater
than $2g-2$. This is Theorem 3.3 already proven in Section 3, where the
operator comes from Ngô’s endoscopic correspondence and the support theorems.

  2. Case 2.

The effective divisor $D$ satisfies that $\mathrm{deg}(D)$ is odd and greater
than $2g-2$. In this case, the support theorems still hold, but the
isomorphism of Proposition 3.13 is missed due to parity reasons; see Remark
3.14. We apply Theorem 4.5 to reduce this case to Case 1 as follows.

Let $p$ be a closed point of $C$. Then the effective divisor

$D_{p}=D+p$

satisfies the assumption of Case 1. Hence we have an isomorphism

(112)
$c^{D_{p}}_{\kappa}:\left(\mathrm{Rh^{D_{p}}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i^{D_{p}}_{\gamma}}_{*}\left(\mathrm{Rh^{D_{p}}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D_{p}}_{\gamma}]\in
D^{b}_{c}({\mathcal{A}}^{D_{p}})$

which is canonical up to scaling constructed from Ngô’s endoscopic
correspondence. Applying to (112) the vanishing cycle functor
$\varphi_{\mu_{{\mathcal{A}}}}$ associated with the function

$\mu_{\mathcal{A}}:=\mu_{\mathrm{id},{\mathcal{A}}}:{\mathcal{A}}^{D_{p}}\to{\mathbb{A}}^{1},$

we obtain

(113)
$\varphi_{p}(c^{D_{p}}_{\kappa}):=\varphi_{\mu_{\mathcal{A}}}(c^{D_{p}}_{\kappa}):\varphi_{\mu_{{\mathcal{A}}}}\left(\mathrm{Rh^{D_{p}}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}\varphi_{\mu_{{\mathcal{A}}}}\left({i^{D_{p}}_{\gamma}}_{*}\left(\mathrm{Rh^{D_{p}}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\right)[-2d^{D_{p}}_{\gamma}].$

Since (112) is canonical up to scaling, so is (113) by Lemma 4.4 (b). In fact,
the operator (113) gives the desired operator $c^{D}_{\kappa}$. To justify
this, we calculate both sides of (113).

For the left-hand side, Theorem 4.5 (c) applied to the special case
$\pi=\mathrm{id}$ yields

$\varphi_{\mu_{{\mathcal{A}}}}\left(\mathrm{Rh^{D_{p}}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}=\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-r_{1}],\quad
r_{1}=\mathrm{codim}_{{\mathcal{M}}^{D_{p}}_{n,L}}({\mathcal{M}}^{D}_{n,L}).$

For the right-hand side, we have by the proper base change ([13, Proposition
4.2.11]) and Corollary 4.6 that

$\begin{split}\varphi_{\mu_{{\mathcal{A}}}}\left({i^{D_{p}}_{\gamma}}_{*}\left(\mathrm{Rh^{D_{p}}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\right)&={i^{D_{p}}_{\gamma}}_{*}\varphi_{\mu_{\gamma,{\mathcal{A}}}}\left(\mathrm{Rh^{D_{p}}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}\\\
&={i^{D}_{\gamma}}_{*}\left(\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-r_{2}]\end{split}$

where

$r_{2}=\mathrm{codim}_{{\mathcal{M}}^{D_{p}}_{r,L}(\pi)}({\mathcal{M}}^{D}_{r,L}(\pi)).$

In conclusion, (113) gives an isomorphism

$\varphi_{p}(c^{D_{p}}_{\kappa}):\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i_{\gamma}}_{*}\left(\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D_{p}}_{\gamma}+r_{1}-r_{2}].$

By the dimension formulas in [6, Section 6.1], we have

$\begin{split}r_{1}-r_{2}&=\left(\mathrm{dim}({\mathcal{M}}^{D_{p}}_{n,L})-\mathrm{dim}({\mathcal{M}}^{D}_{n,L})\right)-\left(\mathrm{dim}({\mathcal{M}}^{D_{p}}_{r,L}(\pi)-\mathrm{dim}({\mathcal{M}}^{D}_{r,L}(\pi)\right)\\\
&=2d^{D_{p}}_{\gamma}-2d^{D}_{\gamma}.\end{split}$

Hence $-2d^{D_{p}}_{\gamma}+r_{1}-r_{2}=-2d^{D}_{\gamma}$, and the operator

(114) $c^{D}_{\kappa}=\varphi_{p}(c^{D_{p}}_{\kappa})$

induces an isomorphism (111) as desired.

  3. Case 3.

We now treat the last case $D=K_{C}$. In this case the support theorems fail
for the Hitchin fibrations (_c.f._ [7]). However, we are able to construct the
operator (111) following the same strategy as in Case 2.

Let $p,q$ be 2 closed points on the curve $C$. Then the divisor

$K_{p,q}=K_{C}+p+q$

satisfies the assumption of Case 1, and we have

$c^{K_{p,q}}_{\kappa}:\left(\mathrm{Rh^{K_{p,q}}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa}\xrightarrow{\simeq}{i^{K_{p,q}}_{\gamma}}_{*}\left(\mathrm{Rh^{K_{p,q}}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{K_{p,q}}_{\gamma}].$

Similar to (114), we define

(115)
$c^{K_{C}}_{\kappa}:=\varphi_{q}\left(\varphi_{p}\left(c^{K_{p,q}}_{\kappa}\right)\right)$

which gives the desired isomorphism (111) for $D=K_{C}$. ∎

We have completed the construction of (111) which proves Theorems 3.1 and 3.2.
In Cases 2 and 3 above, the construction of the operator $c^{D}_{\kappa}$ _a
priori_ depends on the choice of the closed points $p$ and $q$. We finish this
section by showing in the following proposition that $c^{D}_{\kappa}$ in Case
2 or 3 is in fact independent of the choice of the points.

######  Proposition 4.8.

The operators (114) and (115) do not depend on the choice of $p\in C$ and
$p,q\in C$ respectively.

###### Proof.

We prove the independence for (114). By varying the point $p$, we have a
family of Hitchin fibrations

$h^{D_{p}}:{\mathcal{M}}^{D_{p}}_{n,L}\to{\mathcal{A}}^{D_{p}}$

over a base $T$. The construction of the correspondence (112) works relatively
over the base which gives a family of operators $c^{D_{p}}_{\kappa}$. By
applying the vanishing cycle functor relatively over $T$, we obtain a family
of operators $\phi_{p}(c^{D_{p}}_{\kappa})$ which form a section of the
trivial local system

${\underline{{\mathbb{C}}}}\otimes\mathrm{Hom}_{{\mathcal{A}}^{D}}({\mathcal{F}}_{1},{\mathcal{F}}_{2})$

on $T$. Here

${\mathcal{F}}_{1}=\left(\mathrm{Rh^{D}_{*}}{\underline{{\mathbb{C}}}}\right)_{\kappa},\quad{\mathcal{F}}_{2}={i_{\gamma}}_{*}\left(\mathrm{Rh^{D}_{\gamma}}_{*}{\underline{{\mathbb{C}}}}\right)_{\kappa}[-2d^{D}].$

are independent of the point $p$. Hence $\phi_{p}(c^{D_{p}}_{\kappa})$ is
constant over $T$.

An identical argument works for (115) which we omit. ∎

######  Remark 4.9.

Applying the vanishing cycle functors as in Cases 2 and 3 above, we obtain
that Proposition 2.11 also holds also for $D=K_{C}$.

##  5\. The P=W conjecture and the Hausel-Thaddeus conjecture

Throughout Section 5, we assume that the curve $C$ has genus $g\geq 2$. We
assume that $D$ is an effective divisor of degree $\mathrm{deg}(D)>2g-2$ or
$D={K}_{C}$. For a cyclic Galois cover $\pi:C^{\prime}\to C$, we denote by
$D^{\prime}$ the divisor $\pi^{*}D$ on $C^{\prime}$.

We discuss some applications of Theorems 3.1 and 3.2.

###  5.1. Perverse filtrations

We briefly recall the definition of perverse filtrations [10, 5].

Let $f:X\rightarrow Y$ be a proper morphism with $X$ a nonsingular algebraic
variety. The perverse $t$-structure on the constructible derived category
$D_{c}^{b}(Y)$ induces an increasing filtration on the cohomology
$H^{*}(X,{\mathbb{C}})$,

(116) $P_{0}H^{\ast}(X,{\mathbb{C}})\subset
P_{1}H^{\ast}(X,{\mathbb{C}})\subset\dots\subset
P_{k}H^{\ast}(X,{\mathbb{C}})\subset\dots\subset H^{\ast}(X,{\mathbb{C}}),$

called the _perverse filtration_ associated with $f$.

The perverse filtration (116) can be described via the decomposition theorem
[3]. In fact, applying the decomposition theorem to the map $f:X\to Y$, we
obtain an isomorphism

$\mathrm{Rf}_{*}{\underline{{\mathbb{C}}}}[\dim(X)-l]\simeq\bigoplus_{i=0}^{2l}\mathcal{P}_{i}[-i]\in
D^{b}_{c}(Y)$

with $\mathcal{P}_{i}$ a perverse sheaf on $Y$ and $l$ the defect of semi-
smallness:

$l=\mathrm{dim}\left(X\times_{Y}X\right)-\mathrm{dim}(X).$

The $k$-th piece of the perverse filtration is

$P_{k}H^{j}(X,{\mathbb{Q}})=\mathrm{Im}\Big{\\{}H^{j-(\dim(X)-l)}(Y,\bigoplus_{i=0}^{k}\mathcal{P}_{i}[-i])\to
H^{j}(X,{\mathbb{Q}})\Big{\\}}.$

###  5.2. The P=W conjecture

Perverse filtrations appear naturally in studying the topology of Hitchin
fibrations. For notational convenience, we let

$h:{\mathcal{M}}\to{\mathcal{A}}$

be the Hitchin fibration with ${\mathcal{M}}={\mathcal{M}}_{n,L}$ or
$\widetilde{{\mathcal{M}}}_{n,d}$.555Here the divisor $D$ is chosen as the
canonical divisor $K_{C}$. We denote by ${\mathcal{M}}^{B}$ the Betti moduli
space associated with ${\mathcal{M}}$. There is a diffeomorphism
${\mathcal{M}}\cong{\mathcal{M}}^{B}$ induced by non-abelian Hodge theory [36,
37, 22], which identifies the cohomology

(117)
$H^{*}({\mathcal{M}},{\mathbb{C}})=H^{*}({\mathcal{M}}^{B},{\mathbb{C}}).$

A central question concerning the cohomological aspect of the non-abelian
Hodge theory is _the P=W conjecture_ formulated by de
Cataldo–Hausel–Migliorini [5], connecting the perverse filtration associated
with the Hitchin fibration $h$ to the weight filtration

$W_{0}H^{*}({\mathcal{M}}^{B},{\mathbb{C}})\subset
W_{1}H^{*}({\mathcal{M}}^{B},{\mathbb{C}})\subset\cdots\subset
W_{k}H^{*}({\mathcal{M}}^{B},{\mathbb{C}})\subset\cdots\subset
H^{*}({\mathcal{M}}^{\mathrm{B}},{\mathbb{C}})$

associated with the mixed Hodge structure on ${\mathcal{M}}^{\mathrm{B}}$.

######  Conjecture 5.1 (”P=W” [5]).

Under the non-abelian Hodge correspondence (117), we have

$P_{k}H^{i}({\mathcal{M}},{\mathbb{C}})=W_{2k}H^{i}({\mathcal{M}}^{B},{\mathbb{C}}).$

For the $\mathrm{GL}_{n}$-case, the P=W conjecture was proven for $n=2$ in
[5], and recently, for $g=2$ [8]. Furthermore, [8] reduces the full P=W
conjecture to the multiplicativity of the perverse filtration; see [8,
Introduction] for the precise statement. In either situation, the way for
attacking the P=W conjecture is to analyze the location of the _tautological
classes_ in both the perverse and the weight filtrations.

The case of $\mathrm{SL}_{n}$ is more complicated due to the lack of
tautological classes accessing the $\Gamma$-variant cohomology. When $n$ is a
prime number, the shapes of the perverse and the weight filtrations on the
$\Gamma$-variant parts are of simpler forms, and therefore the P=W conjecture
was verified for the $\Gamma$-variant cohomology via direct calculations; see
[5] for $n=2$ and [9] for any prime number $n$.

When $n$ is not a prime number, numerical evidence from the Hausel–Thaddeus
conjecture suggests that the P=W conjecture for $\mathrm{SL}_{n}$ should rely
on the P=W conjecture for a sequence of moduli spaces of stable
$\mathrm{GL}_{*}$-Higgs bundles on different curves with different ranks. In
particular, we expect that the P=W conjecture for $\mathrm{SL}_{n}$ can be
eventually reduced to the P=W conjecture for $\mathrm{GL}_{r}$.

As a first step towards this direction, we introduce the operator (125) below
connecting $H^{*}({\mathcal{M}}_{n,L},{\mathbb{C}})_{\kappa}$ and the
cohomology of the moduli space of stable $\mathrm{GL}_{r}$-Higgs bundles on
another curve $C^{\prime}$, where $r$ and $C^{\prime}$ are determined by
$\kappa\in\hat{\Gamma}$. Then we prove Theorem 5.4 on the compatibility of the
perverse filtrations.

###  5.3. The stable cohomology

Let $\pi:C^{\prime}\to C$ be a cyclic Galois cover of degree $m$. Let
$L\in\mathrm{Pic}^{d}(C)$ be a fixed line bundle with $\mathrm{gcd}(n,d)=1$.
We allow $D$ to be any effective divisor with $\mathrm{deg}(D)>2g-2$ or
$D=K_{C}$.

We recall the moduli spaces
$\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime})$ and
${\mathcal{M}}_{r,L}^{D}(\pi)$ as well as their Hitchin fibrations (10) and
(16) respectively. The group scheme

$\widetilde{{\mathcal{M}}}^{D}_{1,0}(C)=\mathrm{Pic}^{0}(C)\times
H^{0}(C,{\mathcal{O}}_{C}(D))$

acts on the moduli space
$\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime})$ inducing

(118)
$\widetilde{q}:\widetilde{{\mathcal{M}}}^{D}_{1,0}(C)\times{\mathcal{M}}_{r,L}^{D}(\pi)\to\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),\quad\left(({\mathcal{E}}_{1},\theta_{1}),({\mathcal{E}}_{r},\theta_{r})\right)\mapsto(\pi^{*}{\mathcal{E}}_{1}\otimes{\mathcal{E}}_{r},\pi^{*}\theta_{1}+\theta_{r}).$

Here $\theta_{1}\in H^{0}(C,{\mathcal{O}}_{C}(D))$ and its pullback gives a
section $\pi^{*}\theta_{1}\in
H^{0}(C^{\prime},{\mathcal{O}}_{C^{\prime}}(D^{\prime}))$. The finite group
$\Gamma$ acts on the left-hand side of (118) diagonally,

${\mathcal{L}}\cdot\left(({\mathcal{E}}_{1},\theta_{1}),({\mathcal{E}}_{r},\theta_{r})\right)=\left(({\mathcal{E}}_{1}\otimes{\mathcal{L}}^{-1},\theta_{1}),({\mathcal{E}}_{r}\otimes{\mathcal{L}},\theta_{r})\right),\quad{\mathcal{L}}\in\Gamma.$

The morphism (118) factors through this $\Gamma$-quotient and the fibers of
(118) are given by $\Gamma$-orbits. By dimension reasons, the right-hand side
of (118) coincides with the $\Gamma$-quotient of the left-hand side. We have
the following canonical isomorphisms of the cohomology

(119)
$\begin{split}H^{*}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})&\xrightarrow{\simeq}H^{*}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C)\times{\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})^{\Gamma}\\\
&=\left(H^{*}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C),{\mathbb{C}})\otimes
H^{*}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})\right)^{\Gamma}\\\
&=H^{*}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C),{\mathbb{C}})\otimes
H^{*}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})^{\Gamma}\end{split}$

where the first isomorphism is induced by the $\Gamma$-quotient map
$\widetilde{q}^{*}$, the second identity is the Künneth decomposition, and the
last identity follows from the triviality of the $\Gamma$-action on
$H^{*}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C),{\mathbb{C}})$.

For any Hitchin-type moduli space $\widetilde{{\mathcal{M}}}^{D}_{n,d}$,
${\mathcal{M}}^{D}_{n,L}$, or ${\mathcal{M}}^{D}_{r,L}(\pi)$, we consider the
perverse filtrations on the cohomology defined via the corresponding Hitchin
fibration (10), (12), or (16) respectively. The following proposition provides
a description of the perverse filtration on the stable part of
$H^{*}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})$.

######  Proposition 5.2.

The quotient map (118) induces a canonical isomorphism

(120)
$H^{*}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})=H^{*}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C),{\mathbb{C}})\otimes
H^{*}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})_{\mathrm{st}}$

satisfying that

(121)
$P_{k}H^{*}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})=\bigoplus_{i+j=k}H^{i}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C),{\mathbb{C}})\otimes
P_{j}H^{*}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})_{\mathrm{st}}.$

Here the stable part $(-)_{\mathrm{st}}$ denotes the $\Gamma$-invariant part
of the cohomology.

###### Proof.

The first isomorphism is induced by (119). It suffices to show the
compatibility (121) of the perverse filtrations.

We notice that the quotient map (118) is compatible with the Hitchin
fibrations, and we have the commutative diagram

${\widetilde{{\mathcal{M}}}^{D}_{1,0}(C)\times{\mathcal{M}}_{r,L}^{D}(\pi)}$${\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime})}$${H^{0}(C,{\mathcal{O}}_{C}(D))\times{\mathcal{A}}^{D}(\pi)}$${\widetilde{{\mathcal{A}}}^{D^{\prime}}(C^{\prime})}$$\scriptstyle{\widetilde{q}}$$\scriptstyle{\widetilde{h}^{D}(C)\times
h_{\pi}^{D}}$$\scriptstyle{\widetilde{h}^{D^{\prime}}}$$\scriptstyle{\simeq}$

where the bottom arrow is a canonical identification. The pullback morphism
$\widetilde{q}^{*}$ for the cohomology is induced sheaf-theoretically by the
canonical morphism

(122)
${\underline{{\mathbb{C}}}}\to\widetilde{q}_{*}{\underline{{\mathbb{C}}}}$

where the first and the second ${\underline{{\mathbb{C}}}}$ denote the trivial
local systems on the target and the source of $\widetilde{q}$ respectively. By
applying the perverse truncation functor to the pushforward of (122) along
$\widetilde{h}^{D^{\prime}}$, we obtain that the first map of (119) satisfies

(123)
$P_{k}H^{*}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})\xrightarrow{\simeq}P_{k}H^{*}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C)\times{\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})^{\Gamma}.$

We conclude (121) from (123), the Künneth decomposition, and the fact that the
perverse filtration on

$H^{*}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C),{\mathbb{C}})=H^{*}(\mathrm{Pic}^{0}(C),{\mathbb{C}})$

coincides with the cohomological filtration $H^{*\leq k}$. ∎

As a consequence of Proposition 5.2, we obtain a canonical operator given by
the projection

(124)
$\mathfrak{p_{1}}:H^{i}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})\rightarrow
H^{i}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})_{\mathrm{st}},\quad\forall
i\geq 0$

sending a class in
$H^{i}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})$
to its projection to the direct summand component

$H^{0}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C),{\mathbb{C}})\otimes
H^{i}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})_{\mathrm{st}}=H^{i}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})_{\mathrm{st}}.$

with respect to the decomposition (120). The identity above is induced by the
fundamental class $1\in
H^{0}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C),{\mathbb{C}})$.

######  Corollary 5.3.

The operator (124) respects the perverse filtrations:

$\mathfrak{p_{1}}\left(P_{k}H^{i}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})\right)=P_{k}H^{i}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})_{\mathrm{st}}.$

###### Proof.

Since the fundamental class $1\in
H^{0}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C),{\mathbb{C}})$ lies in
$P_{0}H^{0}(\widetilde{{\mathcal{M}}}^{D}_{1,0}(C),{\mathbb{C}})$, the
corollary follows from (121). ∎

###  5.4. Operators

Let $\pi:C^{\prime}\to C$ be the cyclic Galois cover given by
$\gamma\in\Gamma$, which corresponds to $\kappa\in\hat{\Gamma}$ via (3). We
define the operator

(125)
$\mathfrak{p}_{\kappa}:H^{i-2d^{D}_{\gamma}}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})\twoheadrightarrow
H^{i}({\mathcal{M}}^{D}_{n,L},{\mathbb{C}})_{\kappa}$

as the following composition:

(126)
$\begin{split}H^{*}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})\xrightarrow{\mathfrak{p}_{1}}&H^{*}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})_{\mathrm{st}}=H^{*}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})_{\kappa}\\\
\xrightarrow{\mathrm{proj.}}H^{*}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})^{G_{\pi}}_{\kappa}=&H^{*}({\mathcal{M}}_{\gamma}^{D},{\mathbb{C}})_{\kappa}\xrightarrow[\simeq]{\mathrm{Thm}~{}~{}3.2}H^{*+2d^{D}_{\gamma}}({\mathcal{M}}^{D}_{n,L},{\mathbb{C}})_{\kappa}.\end{split}$

Here the first morphism is (124), the second isomorphism is given by
Proposition 2.10, the third morphism is the projection to the
$G_{\pi}$-invariant part, the fourth isomorphism is given by Lemma 1.7, and
the last isomorphism follows from Theorem 3.2. Hence we obtain that (125) is
surjective and canonically defined up to scaling.

######  Theorem 5.4.

We have

$\mathfrak{p}_{\kappa}\left(P_{k}H^{i}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})\right)=P_{k+d^{D}_{\gamma}}H^{i+2d^{D}_{\gamma}}({\mathcal{M}}_{n,L}^{D},{\mathbb{C}})_{\kappa}.$

###### Proof.

By Corollary 5.3, the morphism $\mathfrak{p}_{1}$ preserves the perverse
filtrations. All the other morphisms in (126) except the last one are deduced
from sheaf-theoretic morphisms which clearly preserve the perverse
filtrations. Hence we have

$P_{k}H^{i}(\widetilde{{\mathcal{M}}}_{r,d}^{D^{\prime}}(C^{\prime}),{\mathbb{C}})\twoheadrightarrow
P_{k}H^{i}({\mathcal{M}}^{D}_{\gamma},{\mathbb{C}})_{\kappa}.$

The last morphism of (126) is given by the sheaf-theoretic isomorphism (65).
Taking account of the shift, we have

$P_{k}H^{i}({\mathcal{M}}^{D}_{\gamma},{\mathbb{C}})_{\kappa}\xrightarrow{\simeq}P_{k+d^{D}_{\gamma}}H^{i+2d^{D}_{\gamma}}({\mathcal{M}}_{r,L}^{D}(\pi),{\mathbb{C}})_{\kappa}.\qed$

Now we consider the special case $D=K_{C}$. Passing through the isomorphisms
(117) induced by the non-abelian Hodge theory, we obtain an operator for the
corresponding Betti moduli spaces

$\mathfrak{p}^{B}_{\kappa}:H^{i-2d_{\gamma}}(\widetilde{{\mathcal{M}}}_{r,d}^{B}(C^{\prime}),{\mathbb{C}})\twoheadrightarrow
H^{i}({\mathcal{M}}^{B}_{n,L},{\mathbb{C}})_{\kappa}.$

Here $\widetilde{{\mathcal{M}}}_{r,d}^{B}(C^{\prime})$ is the Betti moduli
space associated with the curve $C^{\prime}$, the group $\mathrm{GL}_{r}$, and
the degree $d$, and ${\mathcal{M}}^{B}_{n,L}$ stands for the Betti moduli
space associated with the curve $C$, the group $\mathrm{SL}_{n}$, and the line
bundle $L$. We refer to [22] for more details on these moduli spaces.

######  Question 5.5.

Is it true that

$\mathfrak{p}^{B}_{\kappa}\left(W_{2k}H^{i}(\widetilde{{\mathcal{M}}}_{r,d}^{B}(C^{\prime}),{\mathbb{C}})\right)=W_{2k+2d_{\gamma}}H^{i+2d_{\gamma}}({\mathcal{M}}_{n,L}^{B},{\mathbb{C}})_{\kappa}?$

If Question 5.5 has an affirmative answer, then Theorem 5.4 implies that, if
the P=W conjecture holds for $\mathrm{GL}_{r}$ for any $r$ dividing $n$, then
the P=W conjecture holds for $\mathrm{SL}_{n}$. However, the construction of
the operator $\mathfrak{p}_{\kappa}$ relies heavily on the topology of Hitchin
fibrations, which is mysterious on the Betti side. A better understanding of
the operator $\mathfrak{p}_{\kappa}$ may be needed.

###  5.5. The Hausel–Thaddeus conjecture

We explain in the last section that Theorem 3.2 implies Theorem 0.5. Here for
Higgs bundles, we again work with any effective divisor $D$ with
$\mathrm{deg}(D)>2g-2$ or $D=K_{C}$.

###### Proof of Theorem 0.5.

We first note that for two line bundles $L_{1}$ and $L_{2}$ with
$L_{1}=L_{2}\otimes N^{\otimes n}$, there is a natural identification of the
moduli spaces

(127)
${\mathcal{M}}^{D}_{n,L_{1}}\xrightarrow{\simeq}{\mathcal{M}}^{D}_{n,L_{2}},\quad({\mathcal{E}},\theta)\mapsto({\mathcal{E}}\otimes
N,\theta)$

compatible with the Hitchin fibrations. Hence we obtain

(128)
$\begin{split}[P_{k}H^{i}({\mathcal{M}}^{D}_{n,L},{\mathbb{C}})_{\kappa}]=[P_{k}H^{i}({\mathcal{M}}^{D}_{n,L^{\prime\otimes
de}},{\mathbb{C}})_{\kappa}]&=[P_{k-d^{D}_{\gamma}}H^{i-2d^{D}_{\gamma}}(({\mathcal{M}}^{D}_{n,L^{\prime\otimes
de}})_{\gamma},{\mathbb{C}})_{\kappa}]\\\
&=[P_{k-d^{D}_{\gamma}}H^{i-2d^{D}_{\gamma}}(({\mathcal{M}}^{D}_{n,L^{\prime}})_{\gamma},{\mathbb{C}})_{de\kappa}]\end{split}$

in the Grothendieck group $K_{0}(\mathrm{Vect})$ of ${\mathbb{C}}$-vector
spaces. Here the first identity is induced by (127) since
$\mathrm{deg}(L)=\mathrm{deg}(L^{\prime\otimes de})~{}~{}\textup{mod}~{}~{}n$,
the second identity follows from (65), and the third identity is given by
Remark 4.9 and Proposition 2.11.

This proves the Betti number version of the refined Hausel–Thaddeus conjecture
(7). To get the enhanced version concerning Hodge structures, we follow [11,
Section 2.1] to work with the category of mixed Hodge modules [35] which
refines the category of perverse sheaves. Identical arguments show that
Theorems 3.1 and 3.2 actually hold in the derived category of mixed Hodge
modules, which gives the enhanced version of (128) in $K_{0}(\mathrm{HS})$.
This completes the proof of (7).

Finally, we note that (7) implies (8). This follows from taking the summation
over all $\gamma\in\Gamma$ and the natural identification of the fixed loci

$({\mathcal{M}}^{D}_{n,L^{\prime}})_{\gamma}=({\mathcal{M}}^{D}_{n,L^{\prime}})_{q\gamma}$

for any $q\in{\mathbb{Z}}$ coprime to $n$. ∎

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# Low-field microwave-free sensors using dipolar spin relaxation of quartet
spin states in silicon carbide

Oscar Bulancea-Lindvall  Department of Physics, Chemistry and Biology,
Linköping University, SE-581 83 Linköping, Sweden  Matthew T. Eiles  Max-
Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187
Dresden, Germany  Nguyen Tien Son  Department of Physics, Chemistry and
Biology, Linköping University, SE-581 83 Linköping, Sweden  Igor A. Abrikosov
Department of Physics, Chemistry and Biology, Linköping University, SE-581 83
Linköping, Sweden  Viktor Ivády<EMAIL_ADDRESS>Department of Physics,
Chemistry and Biology, Linköping University, SE-581 83 Linköping, Sweden  Max-
Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187
Dresden, Germany  Wigner Research Centre for Physics, PO Box 49, H-1525,
Budapest, Hungary

###### Abstract

Paramagnetic defects and nuclear spins are the major sources of magnetic
field-dependent spin relaxation in point defect quantum bits. The detection of
related optical signals has led to the development of advanced relaxometry
applications with high spatial resolution. The nearly degenerate quartet
ground state of the silicon vacancy qubit in silicon carbide (SiC) is of
special interest in this respect, as it gives rise to relaxation rate extrema
at vanishing magnetic field values and emits in the first near-infra-red
transmission window of biological tissues, providing an opportunity for
developing novel sensing applications for medicine and biology. However, the
relaxation dynamics of the silicon vacancy center in SiC have not yet been
fully explored. In this paper, we present results from a comprehensive
theoretical investigation of the dipolar spin relaxation of the quartet spin
states in various local spin environments. We discuss the underlying physics
and quantify the magnetic field and spin bath dependent relaxation time
$T_{1}$. Using these findings we demonstrate that the silicon vacancy qubit in
SiC can implement microwave-free low magnetic field quantum sensors of great
potential.

##  I Introduction

Due to their robustness, sensitivity and versatility, point defect quantum
bits exhibit enormous potential for quantum sensing. The stringent
requirements of emerging multidisciplinary applications, for example room
temperature operation, sensing at low or zero magnetic field, microwave-free
and all-optical control, and bioinertness, pose numerous challenges for such
devices. Novel point defect sensors which can meet some or all of these
criteria are continuously sought after.

The NV center in diamondDoherty  _et al._ (2013) is the leading contender in
quantum sensing applicationsBarry  _et al._ (2020) realized by optically
addressable point defect qubits. Degen  _et al._ (2017) Recent developments in
NV relaxometryTetienne  _et al._ (2013); Schmid-Lorch  _et al._ (2015); van
der Sar  _et al._ (2015); Hall  _et al._ (2016); Rendler  _et al._ (2017);
Finco  _et al._ (2021), where the variation of the longitudinal spin
relaxation time is detected by optical means, have made high temperature
microwave-free sensing applications possible. Sensing at low magnetic field
with such sensors is, however, not possible due to the large zero-field-
splitting of the triplet ground state, which requires a bias field of
approximately 100 mT to bring the spin states to near degeneracy. The presence
of the bias field is often undesirable as it may perturb the sample and
influence the measurement.Zheng  _et al._ (2019); Lenz  _et al._ (2020); Fu
_et al._ (2020); Wang  _et al._ (2021)

Relaxometry-based sensing has not been explored for other point defect qubits,
such as the divacancyKoehl  _et al._ (2011); Bulancea-Lindvall  _et al._
(2021) and the negatively charged silicon vacancySoltamov  _et al._ (2012);
Widmann  _et al._ (2015a); Simin  _et al._ (2016) in silicon carbide (SiC).
Bulancea-Lindvall  _et al._ (2021) The latter defect is, however, attractive
for low magnetic field relaxometry applications owing to its small zero-field
splitting value and resulting quasi degenerate electron spin states. For such
applications, a detailed understanding of the relaxation processes of this
defect is crucial.

The negatively charged silicon-vacancy in SiC provides an optically
addressable point defect whose quartet ground state spinRiedel  _et al._
(2012) has a long room temperature coherence timeWidmann  _et al._ (2015b);
Simin  _et al._ (2017). The unusual high spin state has been utilized in
various applications including quantum sensingLee  _et al._ (2015); Simin  _et
al._ (2016); Niethammer  _et al._ (2016); Anisimov  _et al._ (2016), room
temperature maserKraus  _et al._ (2014), and near infra-red quantum
information processingNagy  _et al._ (2019); Son  _et al._ (2020); Babin  _et
al._ (2022). In recent years, considerable attention has been paid to
experimental characterization of the relaxation dynamics of the quartet
silicon vacancy spin states in a broad temperature range in various SiC
samples.Simin  _et al._ (2017); Soltamov  _et al._ (2019); Ramsay and Rossi
(2020) The external field dependence of the relaxation processes have received
little attention thus far, but must be understood in detail for various
relaxometry applications. Recent theoretical developments have enabled
parameter-free calculations of the two major contributions to the longitudinal
spin relaxation, namely the temperature dependent spin-lattice
relaxationGugler  _et al._ (2018); Park  _et al._ (2020); Xu  _et al._ (2020)
and the magnetic field dependent dipolar spin relaxation induced by local
environmental spinsIvády (2020); Ivády  _et al._ (2021); Bulancea-Lindvall
_et al._ (2021).

In this paper, we study the magnetic field and local spin environment
dependence of the dipolar spin relaxation of the V1 and V2 silicon vacancy
centers in the 4H polytype of SiC (4H-SiC). Ivády  _et al._ (2017) We consider
various environmental spin species, such as the naturally abundant 13C and
29Si nuclear spins and spin-1/2 and spin-1 point defects over a wide range of
concentrations. We identify the most relevant level anti-crossings (LACs),
where environmental spins efficiently relax the quartet spin states, and
quantify the spin relaxation time $T_{1}$. We find several narrow resonances,
where the spin relaxation time can vary over several orders of magnitude
within a small magnetic field interval close to $B=0$. Utilizing these
observations we propose novel quantum sensors for biological use and estimate
their sensitivity.

The paper is organized as follows. In section II, we detail the models and the
simulation technique used in this article. In section III, we present our
computational results, while in section IV we discuss our proposal for low-
field refractometry with the silicon vacancy center in SiC. Finally, in
section V, we summarize our findings.

##  II Methodology

We model the relaxation dynamics of many-spin systems consisting of a quartet
silicon vacancy electron spin and a number environmental spins of different
kinds. We consider spin-1/2 13C and 29Si nuclear spins and doublet and triplet
electron spin environments. More information on relevant paramagnetic defects
in SiC can be found in Ref. [Bulancea-Lindvall  _et al._ , 2021]. Nuclear and
electron baths are considered independently and calculated separately.

The spin Hamiltonians of the many-body system is given by

$H=H_{0}+\sum_{i}H^{\text{x}}_{i}+\sum_{i}H_{0i}^{\text{x}}+\sum_{i,j}H_{ij}\text{,}$
(1)

where the Hamiltonian $H_{0}$ of the quartet silicon vacancy spin includes the
zero-field splitting (ZFS) and the Zeeman term (ZE),

$H_{0}=H_{\text{ZFS}}+H_{\text{ZE}}=D\left(S_{z}^{2}-\frac{5}{4}\right)+\left(g_{e}S_{z}+g_{3\parallel}\frac{S_{+}^{3}-S_{-}^{3}}{4i}\right)\mu_{B}B_{z}\text{.}$
(2)

In Eq. (2), the ZFS parameter $D=2.6$ MHz ($D=35.0$ MHz) for the V1 center (V2
center) in 4H-SiC(Ivády  _et al._ , 2017). The second term on the r.h.s. of
Eq. (2) accounts for the linear and nonlinear Zeeman interactions of the
silicon vacancy center, where $g_{e}=2.0$, $g_{3\parallel}=0.6$, and $\mu_{B}$
is the Bohr magneton.Simin  _et al._ (2016) The non-linear Zeeman term is a
direct consequence of the three-fold rotation symmetry and the high spin state
of the defect.(Ivády  _et al._ , 2018; Simin  _et al._ , 2016)

The Hamiltonian $H^{\text{x}}_{i}$ in Eq. (1) depends on the considered spin
bath and accordingly $\text{x}=\left\\{\text{n},\text{de},\text{te}\right\\}$.
For nuclear spins, it includes only the nuclear Zeeman interaction term,

$H_{i}^{\text{n}}=-g_{\text{N}}\mu_{N}I_{i,z}B_{z}\text{,}$ (3)

where $g_{\text{N}}$ is the nuclear g-factor of either the 13C or the 29Si
isotope, $\mu_{N}$ is the nuclear magneton, and $I_{i,z}$ is the doublet
nuclear spin $z$ operator. For doublet electron (de) spins, the Hamiltonian
includes only the linear Zeeman term,

$H_{i}^{\text{de}}=g_{e}\mu_{B}S_{i,z}B_{z}\text{,}$ (4)

while for triplet electron (te) spins the Hamiltonian includes the ZFS and the
linear Zeeman term,

$H_{i}^{\text{te}}=D_{i}\left(S_{i,z}^{2}-\frac{1}{3}\right)+\frac{E_{i}}{2}\left(S_{+}^{2}+S_{-}^{2}\right)+g_{e}\mu_{B}S_{i,z}B_{z}\text{,}$
(5)

where $D_{i}$ and $E_{i}$ are ZFS parameters of triplet spin defect $i$. In
our study, we the triplet spin bath consists divacancy defects, whose ZFS
parameters can be found in Ref. [Falk  _et al._ , 2013].

The Hamiltonian $H_{0i}^{\text{x}}$ terms in Eq. (1) account for the
interaction between the silicon vacancy center and the environmental spins.
For nuclear spins the coupling term can be written as,

$H_{0i}^{\text{n}}=\mathbf{S}A_{i}\mathbf{I}_{i}\text{,}$ (6)

where $\mathbf{S}$ and $\mathbf{I}_{i}$ are the quartet electron spin and
spin-1/2 nuclear spin vector operators, respectively, and $A_{i}$ is the
hyperfine tensor determined from ab initio density functional theory (DFT)
calculations. The details of the DFT hyperfine calculations can be found in
Ref. [Ivády, 2020]. For an electron spin bath, the coupling Hamiltonian term
is equal to the magnetic dipole-dipole coupling and can be written as

$H_{0i}^{\text{de}}=H_{0i}^{\text{te}}=-\frac{\mu_{0}g_{e}^{2}\mu_{B}^{2}}{4\pi
r^{3}}\left(3\left(\mathbf{S}\mathbf{r}\right)\left(\mathbf{S}_{i}\mathbf{r}\right)-\left(\mathbf{S}\mathbf{S}_{i}\right)\right)\text{,}$
(7)

where $\mu_{0}$ is the vacuum permeability, $\mathbf{S}_{i}$ is either the
double or the triplet electron spin operator vector of environmental spin
defect $i$, $\mathbf{r}$ is the distance vector of the silicon vacancy and the
paramagnetic defect, and $r=\left|\mathbf{r}\right|$. Finally, the last term
on the right hand side of Eq. (1) accounts for intra-spin bath couplings.
According to our numerical tests, this term does not significantly contribute
to spin relaxation phenomena and it is neglected hereinafter.

In the numerical simulations we investigate the relaxation dynamics of a
central electron spin by utilizing a clustering-based computational method
recently developed in Ref. [Ivády, 2020] and summarized in Ref. [Bulancea-
Lindvall  _et al._ , 2021]. This method uses an extended Lindbladian to
facilitate effective interactions. In this work, we use first-order cluster
approximation, i.e. a system of $N$ spins is divided into $N$ subsystems. Each
subsystem includes the quartet silicon vacancy electron spin and one spin from
the surrounding spin bath. The spin Hamiltonian of the first order cluster
systems can be written as

$\widetilde{H}_{i}^{1}=H_{0}+H_{i}+H_{0i}\text{.}$ (8)

Our spin bath models include either $N=31$ (spin-1/2 or spin-1) paramagnetic
defects or $N=127$ (13C or 29Si) nuclear spins. According to our previous
calculations in 4H-SiCBulancea-Lindvall  _et al._ (2021) and diamondIvády
(2020); Ivády  _et al._ (2021), these values ensures that our simulations are
converged with respect to the bath size. In order to obtain ensemble averaged
quantities we carry out averaging over different spin bath configurations. In
all cases, an ensemble of 200 random spin bath configurations are considered.
All configurations correspond to either a given isotope abundance or a given
electron spin defect concentration. We note that in the nuclear spin bath
calculations we do not consider those random configurations that contain 13C
nuclear spin in the first neighbour shell of the defect. In the electron spin
bath calculations we exclude those spins whose coupling strength exceed 100
MHz, since in such cases the energy level structure is completely mixed by the
strong interaction and the center does not function as a regular silicon
vacancy qubit. These restrictions affect only a marginal part of the random
ensemble.

For the different spin environments, we carry out two different time-dependent
studies. First, starting from a highly polarized state of the quartet spin and
a thermal state of the bath, we simulate the time propagation of the many-spin
system over a short period of time, $t=1$ $\mu$s, and study the amount of
population transferred from the initial state to the other states of the
quartet silicon vacancy. These qualitative studies shed light on the external
parameter dependence of the silicon vacancy-environment couplings.
Furthermore, the results of such calculations may be compared with
photoluminescence (PL) studies.Bulancea-Lindvall  _et al._ (2021); Ivády  _et
al._ (2021) Second, starting from similar initial states, we carry out long
time evolution simulations to quantitatively study the spin relaxation time
$T_{1}$. The simulation time $t$ is tested and optimized for all the
considered magnetic field values and spin bath concentrations. Close to the
LACs, we use 0.1 ms, while far away from the LACs, we use up to 1 ms
simulation time. Similarly to the simulation time, the time step $dt$ of the
time propagation is also optimized. As a general rule, $dt$ is selected in a
way that even the fastest coherent oscillation is well-resolved in the
simulations. In a sufficiently large spin bath the initial population of the
spin states relaxes exponentially, from which the time scale of the decay
($T_{1}$) can be obtained. Here, we note that the simulation time is often
much shorter than the $T_{1}$ time. Due to the extrapolation, the uncertainty
of the calculated $T_{1}$ time is expectedly larger for a weakly coupled spin
bath, where the decay time may reach seconds. Ensemble spin relaxation times
are obtained by averaging the time-dependent population data over the random
spin bath configurations and then fitting an exponential decay curve to the
obtained ensemble-averaged population data.

For high spin defects, the dipolar spin relaxation depends on the initial spin
state.Ramsay and Rossi (2020) For the silicon vacancy, we calculate the spin
relaxation effect starting from two different spin states. The initial
population is either evenly distributed in the
$m_{\text{S}}=\left\\{-1/2,+1/2\right\\}$ subspace or set completely in the
$m_{\text{S}}=-1/2$ state. Since the quartet electron spin is polarized in the
$m_{\text{S}}=\left\\{-1/2,+1/2\right\\}$ subspace in the optical excitation
cycle, the former initial state is natural for microwave-free applications.
High fidelity initialization in a selected spin state can be achieved by
applying a resonant microwave pulse.Nagy  _et al._ (2019) In all cases, the
initial state of the bath spins is set to a thermal state.

##  III Results

Figure 1:  Energy levels of the V2 center and nuclear spin bath induced
population transfer between the electron spin states of the silicon vacancy.
(a) Spin energy levels of an isolated V2 center in 4H-SiC as a function of the
external magnetic field. Due to the zero-field splitting ($2D$) and the Zeeman
shift of the states, three crossings can be observed, labelled as LAC A, LAC
B, and LAC C. The hyperfine interaction with nearby nuclear spins opens a gap
between the crossing states and gives rise to LACs . LACs enable fast spin
flip-flops at the corresponding magnetic field values. (b) Population transfer
from the initial spin state to the rest of the silicon vacancy electron spin
states. The inset in (b) depicts the close-up of the zero magnetic field
region. The electron spin is initialized either in the
$m_{\text{S}}=\left\\{+1/2,-1/2\right\\}$ subspace with equal population
(labelled as ”1/2 ss”) or solely in the $m_{\text{S}}=-1/2$ spin state
(labelled as ”-1/2 o”). The LACs induce efficient spin relaxation that rapidly
weakens with increasing magnetic field beyond the positions of the LAC C. The
depicted population transfer curves are obtained after a 1 $\mu$s long time
evolution of our model system.

First, we report on spin relaxation effects caused by the surrounding nuclear
spin bath of the V1 and V2 centers in 4H-SiC. In order to understand the
results of this section, we briefly discuss the magnetic field dependence of
the quartet energy levels beforehand. As shown in Fig. 1(a) for the V2 center,
the doubly degenerate $m_{\text{S}}=\left\\{-1/2,+1/2\right\\}$ and
$m_{\text{S}}=\left\\{-3/2,3/2\right\\}$ subspaces are split by $2D$ at $B=0$,
due to the ZFS interaction of the $C_{3v}$ symmetric quartet ground state. The
magnetic field applied along the three-fold rotation axis of the defect lifts
the initial degeneracies and gives rise to nearly linear Zeeman shifts. For a
positive magnetic field, the $m_{\text{S}}=-3/2$ level crosses both the
$m_{\text{S}}=+1/2$ and the $m_{\text{S}}=-1/2$ levels. Inclusion of a weakly
coupled spin-1/2 nuclear spin does not undermine the depicted level structure,
but most importantly its hyperfine interaction gives rise to LACs at the
crossings of electronic spin states of $\Delta m_{\text{S}}=\pm 1$. The
positions of these LACs are labelled by LAC A and LAC C in Fig. 1(a).
Furthermore, second order effects due to couplings that include two nuclear
spins, or a nuclear spin and the non-linear Zeeman terms enable quantum jumps
of $\Delta m_{\text{S}}=\pm 2$. Such second order effects give rise to an
additional LAC midway between LAC A and LAC C, which is labelled as LAC B in
Fig. 1(a). At all of these LACs, enhanced electron spin relaxation is expected
due to the nuclear spin couplings.

In order to qualitatively study the magnetic field dependence of nuclear spin
bath induced spin relaxation, we carry out time evolution simulations up to a
fixed time and investigate the amount of population transferred from the
initial state to the rest of the electron spin states. Fig. 1(b) depicts the
magnetic field dependence of the obtained population variations for the V1 and
the V2 centers for two different initialization conditions denoted by “1/2 ss”
and “-1/2 o”. The former means initialization in the
$m_{\text{S}}=\left\\{-1/2,+1/2\right\\}$ subspace while the latter means
initialization in the $m_{\text{S}}=-1/2$ state only. In all cases, the
population lost from the initial state is polynomially reduced as the magnetic
field exceeds $B_{\text{LAC C}}$. Note that different initialization
conditions give rise to different relaxation pathways. In fact, since
initialization in the $m_{\text{S}}=-1/2$ enables relaxation to the
$m_{\text{S}}=+1/2$ state also, at larger magnetic fields the corresponding
population transfer is twice as large as for 1/2 ss subspace initialization.

For low magnetic field values, the decay of the initial population is sizeable
due to the presence of several LACs in the fine energy level structure in this
region. In case of the V2 center, whose ZFS parameter is $D=35$ MHz, all three
LACs can be identified in the population transfer plot in Fig. 1(b). Note that
LAC A is observed exclusively for the $m_{\text{S}}=-1/2$ initialization
condition, while LAC B is observed exclusively when the $m_{\text{S}}=+1/2$
state is also populated. Since the ZFS is $2D=5.2$ MHz for the V1 center, the
LACs can be found very close to $B=0$ for this center. Due to the width of the
LACs, the individual peaks in the population transition curve cannot be
resolved.

Figure 2:  Magnetic field dependence of the ensemble averaged spin relaxation
time T1 of the V1 and V2 centers in 4H-SiC due to hyperfine coupling. The 13C
and 29Si nuclear spin bath induced relaxation are provided separately for the
V1 center. Solid and dotted lines depict relaxation time curves that
correspond to different initialization conditions. For the 13C nuclear spin
bath, the spin is initialized in the $m_{\text{S}}=-1/2$ state only (labelled
as ”-1/2 o”), while for the 29Si nuclear spin bath the spin is initialized in
the $m_{\text{S}}=\left\\{1/2,-1/2\right\\}$ subspace with equal population in
the two states (labelled as 1/2 ss). The inset highlights relaxation times
near zero magnetic field.

In order to quantitatively analyze the relaxation mechanism of a silicon
vacancy qubit in the nuclear spin bath of natural and 29Si depleted 4H-SiC, we
carry out large-scale numerical simulations to determine the corresponding
$T_{1}$ times. Fig. 2 summarizes the results of these calculations. For
natural samples of $4.68$% 29Si and $1.07$% 13C isotope content, the
relaxation time is determined by the 29Si nuclear spin and the contribution of
the paramagnetic 13 spins can be neglected. Indeed, the latter gives rise to a
relaxation time which is two orders of magnitude larger than that of 29Si,
which is the majority nuclear spin source in the sample. Consequently,
depletion of the 29Si isotope may give rise to a substantial increase in the
relaxation time, when no other relaxation generators, such as electron spins
and spin-phonon coupling, are present.

The magnetic field dependence of the $T_{1}$ time clearly shows that the LACs
drastically reduce the lifetime of the quartet spin in the $B=0-60$ Gauss
interval for both the V1 and the V2 centers. Within this critical region, the
$T_{1}$ expectedly limits the coherence time, while outside of this region the
$T_{1}$ times exceeds 1 ms, and the dipolar spin relaxation does not limit the
coherence time anymore.

Figure 3:  (a) Energy levels of a quartet-doublet two electron spin system.
The inset shows a close-up of the zero magnetic field region. Altogether five
level anti crossings, marked by LAC 1-5, can be found at distinct magnetic
field values. The solid pink and the dashed black lines indicates the up and
down states of the spin-1/2 defect. (b) Electron spin defect induced
population transfer from the highly polarized initial state to the empty
quartet spin states of the V1 and V2 centers. Solid and dashed lines depict
curves that correspond to different initialization conditions. For the former,
the spin is initialized in the $m_{\text{S}}=-1/2$ state only (labelled as
”-1/2 o”), while for the latter the spin is initialized in the
$m_{\text{S}}=\left\\{1/2,-1/2\right\\}$ subspace with equal population in the
two states (labelled as 1/2 ss). The concentration for the spin-1/2 defect is
set to 1018 cm-3 and an overall simulation time of 1 $\mu$s is used.

Next, we investigate spin relaxation effects induced by spin-1/2 electron
spins (the case of spin-1 defect environment is briefly discussed in the
Appendix). In SiC, there are several different common paramagnetic defects
whose concentration is determined by the growth conditions and after growth
treatments. Ref. [Bulancea-Lindvall  _et al._ , 2021] summarizes the most
relevant defects and their expected concentrations in connection with the spin
relaxation processes of the divacancy qubits in 4H-SiC.

As depicted in Fig. 3(a), the fine energy level structure of a quartet-doublet
electron spin pair reveals important differences compared to the energy level
structure seen in Fig. 1(a) for the quartet electron spin system. Since both
electron spins exhibit Zeeman splittings with $g\approx 2$ and the zero-field
splitting value of the silicon vacancy qubit is already suppressed at small
magnetic field values, the magnetic field dependence of the coupled pair’s
energy levels can be interpreted as the sum of a quintet (spin-2) and a
triplet (spin-1) subspaces. The triplet subspace is nearly degenerate with the
three innermost states of the quintet subspace, see Fig. 3(a). To be able to
keep track of the quartet silicon vacancy spin states, hereinafter we use the
$\left|m_{S}m_{S^{\prime}}\right\rangle$ notation to label the states. Due to
the small but non-zero ZFS of the silicon vacancy center, four LACs (LAC1-4)
can be found in the vicinity of $B=0$, see the inset of Fig. 3(a).
Furthermore, an additional LAC is found at larger magnetic field value (LAC5).
It occurs due to the non-linearity of the quartet spin states of the silicon
vacancy. We note that the position of LAC5 may not be well-defined in a
natural sample, as any inhomogeneity, for example due to the local hyperfine
fields of the two electron spins, may significantly shift the position of
LAC5.

Our qualitative study for spin-1/2 electron spin bath induced spin relaxation
is summarized in Fig. 3(b). When the silicon vacancy spin is initialized in
the $m_{\text{S}}=\left\\{1/2,-1/2\right\\}$ subspace, three narrow resonance
peaks correspond to LAC 2-4, while a wide peak at around $B=380$ Gauss marks
the position of LAC 5. The oscillations observable on the sides of the
resonance peak of LAC 5 is a side effect of the finite simulation time and the
slow coherent oscillations between states $\left|-3/2,+1/2\right\rangle$ and
$\left|-1/2,-1/2\right\rangle$. For the V1 center, all LAC related resonances
occur much closer to $B=0$.

When the initial population is set in the $m_{S}=-1/2$ state only, we observe
an unexpectedly high, nearly magnetic field independent population transfer to
the empty states, see the dashed lines in Fig. 3(b). To understand this
observation, we refer to Fig. 3(a) which shows a degenerate pair of states at
the zero value of the energy scale. This magnetic field independent degenerate
subspace includes the $\left|+1/2,-1/2\right\rangle$ and the
$\left|-1/2,+1/2\right\rangle$ states that can be coupled by the spin flip-
flop operator of the dipole-dipole interaction. Since the states are
degenerate, even a weak coupling between the two electron spins can give rise
to a sizeable population transfer within the doublet manifold, explaining the
constant high value in Fig. 3(b). We note that recent measurements have
demonstrated fast
$\left|-1/2\right\rangle\leftrightarrow\left|+1/2\right\rangle$
relaxationRamsay and Rossi (2020) in accordance with our findings. The
phenomena of efficient spin relaxation between the $\left|\pm
1/2\right\rangle$ states and its effects on the decoherence of the V2 center
is further investigated in Ref. [Bulancea-Lindvall  _et al._ , 2022].

Figure 4:  Magnetic field and electron spin defect concentration dependence of
the ensemble averaged spin relaxation time T1 of (a) the V1 center and (b) the
V2 center. The quartet spin state is initialized in the
$m_{\text{S}}=\left\\{1/2,-1/2\right\\}$ subspace. Dashed horizontal line in
(b) highlights the typical coherence time for the V2 center.

Next, we quantitatively investigate the magnetic field dependence of the
longitudinal spin relaxation time of quartet silicon vacancies surrounded by
spin-1/2 paramagnetic defects. Fig. 4 presents the calculated relaxation time
of the V1 and the V2 centers for various spin bath concentrations. As can be
seen, the relaxation time ranges from 100 s to 20 $\mu$s depending on the
concentration of the defects and the magnetic field. As expected, the LACs
drastically shorten the lifetime of the silicon vacancy spins, however, the
$T_{1}$ time increases rapidly beyond LAC 5.

##  IV Discussion

The careful examination of the most relevant spin relaxation mechanisms of the
silicon vacancy qubit presented in section III allows us to propose a low
magnetic field relaxometry application, relying on the variation of the spin
state lifetime of the silicon vacancy center in SiC measured through its
fluorescence intensity. Such a microwave-free sensor is desirable for various
biological applications. Furthermore, the V1 and V2 silicon vacancy centers
with 862 nm and 917 nm zero-phonon photo luminescence emit in the first near
infra-red transmission window of biological tissue (650 nm and 950 nm), and
thus such a fluorescence silicon vacancy sensor would also be suitable for in
vivo applications.

Since the photoluminescence signal of the silicon vacancy depends on the
population of the spin states, spin relaxation-induced population transfers
have a direct signature in the optical signal. Under continuous optical
excitation, the silicon vacancies are probabilistically excited and the time
$t$ spent in the ground state between two excitations follows an exponential
distribution. When the center is in the ground state, the initial high degree
of polarization in the $m_{\text{S}}=\pm 1/2$ state exponentially decays.
Therefore, the PL signal of an ensemble of silicon vacancy centers is equal to

$\mathcal{S}=\mathcal{C}\mathcal{I}_{0}\int_{0}^{\infty}\varrho_{T_{d}}\\!\left(t\right)e^{-t/T_{1}}dt$
(9)

where $\varrho_{T_{d}}\\!\left(t\right)$ is the probability density function
of exponential distribution, $T_{d}$ is the average dwell time in the ground
state for a given excitation power, $\mathcal{C}$ is the spin-dependent
contrast of the optical signal, and $\mathcal{I}_{0}$ is the fluorescence
intensity of the defect. Here we assume that the initialization of the spin
state through the optical cycle requires negligible time in comparison with
$T_{d}$. After integration, the continuous wave signal is equal to

$\mathcal{S}=\mathcal{C}\mathcal{I}_{0}\frac{T_{1}}{T_{1}+T_{d}}\text{,}$ (10)

and hence its derivative with respect to the spin relaxation time is

$\frac{d\mathcal{S}}{dT_{1}}=\mathcal{C}\mathcal{I}_{0}\left(-\frac{T_{1}}{\left(T_{1}+T_{d}\right)^{2}}+\frac{1}{T_{1}+T_{d}}\right)\text{,}$
(11)

which takes its maximal value at $T_{1}/T_{d}\rightarrow 0$. It is therefore
advisable to set the parameters of silicon vacancy sensors, excitation laser
power and spin defect concentration, in such a way that the $T_{1}<T_{d}$
relation is ensured.

Magnetic field sensing can be realized utilizing the magnetic field dependence
of the spin relaxation time $T_{1}\\!\left(B\right)$ studied in this paper. It
is important to note, however, that the magnetic field independent spin-orbit
and electron-phonon interactions induced spin lattice relaxation
$T_{1}\\!\left(\mathcal{T}\right)$, where $\mathcal{T}$ is the temperature,
need to be taken into consideration as well, especially at room temperature.
The net relaxation rate can be written as a sum of the two terms,

$\frac{1}{T_{1}\\!\left(B,\mathcal{T}\right)}=\frac{1}{T_{1}\\!\left(\mathcal{T}\right)}+\frac{1}{T_{1}\\!\left(B\right)}\text{.}$
(12)

Based on recent low magnetic field measurements on the silicon vacancy center
in SiC, $T_{1}\\!\left(\mathcal{T}\right)$ is in the range of 100 $\mu$s at
room temperature. Simin  _et al._ (2017); Singh  _et al._ (2020) In order to
obtain a measurable magnetic field dependent signal, the dipole-dipole
interaction induced spin relaxation time $T_{1}\\!\left(B\right)$ needs to be
at least in the same order of magnitude. For the V1 (V2) center this can be
achieved at low magnetic field strength by using $C=10^{17}$ cm-3 ($C=3\times
10^{17}$ cm-3 ) spin-1/2 defect concentration in the host material.

In order to achieve the highest sensitivity, the derivative of the relaxation
time with respect to the magnetic field $dT_{1}\\!\left(B\right)/dB$ should be
maximal. In this respect, the hyperfine interaction that gives rise to local
inhomogeneities, variation of the LAC positions, and broadening of the
resonance signal is undesirable. Therefore, for high performance sensing
applications 29Si (and ${}^{13}C$) depleted samples are needed. In our
calculations, we consider relaxation mechanisms in nuclear and electron spin
bath separately, thus our results for spin-1/2 electron spin environments
correspond to isotope purified samples.

Figure 5:  Spin relaxation curve of the V1 center at room temperature
($T_{1}\\!\left(300~{}K\right)=100$ $\mu$s) for $C=10^{17}$ cm-3 spin-1/2
point defect concentration in isotope purified 4H-SiC sample.

The highest gradients can be found close to zero magnetic field at the sharp
resonances of the spin relaxation curves that are associated with LACs in the
energy level structure. In Fig. 5, we depict the $T_{1}\\!\left(B,300\text{
K}\right)$ curve of the V1 center close to zero magnetic field for $C=10^{17}$
cm-3 spin-1/2 defect concentration in paramagnetic isotope purified 4H-SiC
sample. The largest derivative of 0.6 msGauss-1 is obtained at an external
magnetic field value of 1.75 G. Inserting this gradient into Eq. (9) we obtain
the variation of the PL signal through the following formula

$\frac{1}{\mathcal{C}\mathcal{I}_{0}}\left|\frac{d\mathcal{S}}{dB}\right|=4.2\text{
G}^{-1}\text{.}$ (13)

This implies that the PL signal changes by 4.2% of the spin contrast
$\mathcal{C}$ for $\Delta B=1$ $\mu$T at $B=0.175$ mT external field. Such a
high gradient may lead to the realization of sensitive DC magnetic field
sensors in SiC at very small magnetic field values.

##  V Summary

In summary, we comprehensively investigated the dipolar spin relaxation of
quartet silicon vacancy qubits in 4H-SiC. The observed relaxation phenomena
significantly differ from the case of triplet qubits in wide band gap
semiconductors, such as the NV-center in diamond and the divacancy in SiC. Due
to the small zero-field splitting, the LACs and most of the strong relaxation
processes squeeze into a small magnetic field interval close to $B=0$. While
this region is generally avoided in experiments due to the strong couplings,
here we show that one may utilize these effects in room-temperature microwave-
free magnetic field sensing applications. Based on our results we propose that
the most suitable 4H-SiC sample for this purpuse is a paramagnetic isotope
purified sample that contains spin-1/2 defects in $\approx 10^{17}$ cm-3
concentration.

## Acknowledgments

We acknowledge support from the Knut and Alice Wallenberg Foundation through
WBSQD2 project (Grant No. KAW 2018.0071). Support from the Swedish Government
Strategic Research Area SeRC and the Swedish Government Strategic Research
Area in Materials Science on Functional Materials at Linköping University
(Faculty Grant SFO-Mat-LiU No. 2009 00971) is gratefully acknowledged. VI
acknowledges the support from the MTA Premium Postdoctoral Research Program.
N. T. S. acknowledges the support from the Swedish Research Council (Grant No.
VR 2016-04068), the EU H2020 project QuanTELCO (Grant No. 862721). The
calculations were performed on resources provided by the Swedish National
Infrastructure for Computing (SNIC) and Liu local at the National
Supercomputer Centre (NSC).

## Appendix: Spin relaxation due to spin-1 defects

Figure 6:  Population transfer between a silicon vacancy and a bath of
divacancy qubits in 4H-SiC. (a) and (b) depict the case of V1 and V2 centers,
respectively. Dark and light color curves show different spin initialization
conditions. In both cases the figure integrates the contributions from all of
the four different divacancy configurations.

In this appendix, we qualitatively investigate the coupling of the quartet
silicon vacancy spin states to a bath of divacancies. The concentration of
these spin-1 defects is not substantial in commonly studied 4H-SiC samples,
thus their contribution to the spin relaxation time may be negligible in most
cases. On the other hand, when the electron spin states are resonant, even a
few neighbouring divacancy centers can give rise to sizeable relaxation
effects that may be undesirable from the application point of view. To reveal
the magnetic field values of the enhanced relaxation due to divacancy spins,
we study spin bath coupling induced polarization transfer between the
initially polarized and the initially empty states over a 1 $\mu$s simulation
time, see Figs. 6(a) and (b) for the V1 and V2 centers. Note that the spin
bath includes all four possible divacancy configurations, thus Fig. 6 shows
the integrated effect of all the different divacancies. As can be seen in Fig.
6, there are numerous resonances due to the multiple crossings between
electronic states and the variance of the $D_{i}$ and $E_{i}$ ZFS parameters.
Furthermore, the relative amplitude of certain resonance peaks may vary
depending on the initial spin state of the silicon vacancy and divacancy
qubits.

As can be seen in Fig. 6, the number of resonance peaks drops as the magnetic
field increases. In particular, for 300 G $<B<$ 400 G and for 500 G $<B$ no
resonance peaks can be found. These magnetic field regions may be advantageous
for silicon vacancy applications when both silicon vacancies and divacancy
defects are created by positive ion implantation and subsequent annealing.

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