# Field-Enhanced Critical Current and Tunable Skewness in Altermagnetic
Josephson Junctions

Hai-Peng Sun<EMAIL_ADDRESS>Institute for Theoretical Physics
and Astrophysics, University of Würzburg, 97074 Würzburg, Germany  Würzburg-
Dresden Cluster of Excellence ct.qmat, 97074 Würzburg, Germany  Song-Bo Zhang
<EMAIL_ADDRESS>Hefei National Laboratory, Hefei, Anhui, 230088, China
International Center for Quantum Design of Functional Materials (ICQD),
University of Science and Technology of China, Hefei, Anhui 230026, China
Chang-An Li  Institute for Theoretical Physics and Astrophysics, University of
Würzburg, 97074 Würzburg, Germany  Würzburg-Dresden Cluster of Excellence
ct.qmat, 97074 Würzburg, Germany  Björn Trauzettel  Institute for Theoretical
Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany
Würzburg-Dresden Cluster of Excellence ct.qmat, 97074 Würzburg, Germany

###### Abstract

We study the influence of external electric and Zeeman fields on the Josephson
response of a planar superconductor/altermagnet/superconductor junction. We
show that the critical Josephson current across the junction oscillates as the
altermagnetic field strength increases. The current-phase relation can be
forward- or backward-skewed due to significant second-order Josephson
response. It decisively depends on the altermagnetic field strength. We
discover that the skewness can be manipulated by electric gating. We reveal
two additional impacts of external fields on the Josephson current: (i) Fields
can induce 0-$\pi$ transitions. (ii) Fields can substantially enhance the
critical current.

_Introduction._ —Altermagnism is a newly discovered magnetic phase categorized
by spin-group symmetries [1, 2, 3, 4, 5, 6, 7, 8, 9]. It is an unconventional
collinear antiferromagnet, where opposite spins at different sublattices are
related by crystal rotation or reflection symmetry [1, 2], in contrast to
classical antiferromagnets, where the spins are related via inversion or
lattice translation. Hence, the combined time-reversal and translation (or
inversion) symmetry is broken in altermagnets, leading to anisotropic spin-
split energy bands in momentum space. This phenomenon has been predicted in
various candidate materials [10, 7, 8, 6, 11, 2, 9, 12, 1, 13, 14, 15], such
as RuO2 [10, 7, 8], MnTe [2, 13], CrSb [2, 14], FeSb2 [11, 2], Mn5Si3 [12],
and CrNb4S8 [14], and confirmed experimentally in Mn5Si3 [16], MnTe [17, 18,
19, 20] and CrSb [21, 22]. One notable feature of altermagnets is spin-
momentum locking, protected by spin groups [2, 1, 23]. Similar spin-momentum
locking can be achieved by twisting antiferromagnetic bilayers [24, 25, 26,
27]. In these systems, the altermagnetic field strength (AFS) depends
crucially on the twist angle.

Figure 1: (a) Schematic of the AMJJ under external electric ($E_{z}$) or
magnetic ($B_{y}$) fields. Yellow and purple regions indicate $s$-wave
superconducting leads and altermagnet (AM), respectively. Black and blue
arrows indicate external fields applied to AM. (b) Skewness change of the CPR
with increasing AFS $t_{J}$. $\phi_{m}$ is the phase difference of the maximum
Josephson current. (c) CPRs of the AMJJ with $t_{J}=0.4t$ for various gate
potentials $V_{z}$. Supercurrents are in units of $e\Delta_{0}/h$. (d)
Absolute value of the critical current $I_{c}$ as a function of in-plane
Zeeman field $B_{y}$ for different $t_{J}$.

Recently, the interplay of altermagnetism and superconductivity has triggered
intensive interest [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38] due to its
potential for uncovering novel phenomena and applications in superconducting
spintronics and quantum computing. Particularly, Andreev reflection has been
investigated at superconductor/altermagnet interfaces [30, 31, 32] and
proposed to function as a memory device [39]. Altermagnetic Josephson
junctions (AMJJs), such as planar superconductor/altermagnet/superconductor
junctions, have been predicted to exhibit 0-$\pi$ transitions by modulating
the junction length despite the vanishing net magnetization [33, 34, 35]. It
has further been suggested that AMJJs support the formation of
$\varphi$-junctions [40] and exhibit the superconducting diode effect [41].

In this Letter, we study the effects of external electric or Zeeman fields on
the Josephson response of AMJJs as shown in Fig. 1(a). We find that the
critical current of the AMJJ oscillates as the AFS grows. Remarkably, the
current-phase relation (CPR) can be either forward or backward-skewed due to a
large second-order Josephson response. The skewness changes with AFS parameter
$t_{J}$ [Fig. 1(b)]. Near 0-$\pi$ transitions [plateau in Fig. 1(b)],
supercurrent reversal emerges as the second-order Josephson response
dominates. When an electric field is applied to the altermagnetic region, for
significant AFS, the critical current starts to oscillate around zero, leading
to $0$-$\pi$ transitions. Interestingly, the magnitude of the critical current
can be substantially enhanced [Fig. 1(c)]. In addition, external electric
fields can also modify the CPR skewness [Fig. 1(c)]. Notably, the magnitude of
the critical current can be substantially enhanced with increasing field
strength for in-plane Zeeman fields[Fig. 1(d)]. These are characteristic
features of AMJJs.

_Model._ —The altermagnet manifests non-relativistic spin splitting while
having zero net magnetization due to magnetic rotational symmetry. We model
the altermagnet by a tight-binding Hamiltonian with $d$-wave symmetry on a
square lattice [12, 1]

$\displaystyle H({\bf k})=t(\cos k_{x}+\cos k_{y})\sigma_{0}+t_{J}(\cos
k_{x}-\cos k_{y})\sigma_{z}$ (1)

where ${\bf k}=(k_{x},k_{y})$ is the 2D momentum. The first term is the normal
kinetic energy and the second term parameterized by $t_{J}$ describes the
altermagnetic order. In Eq. (1), $\sigma_{z}$ and $\sigma_{0}$ are Pauli spin
and identity matrices, respectively. The model obeys $[C_{2}||C_{4z}]$ spin
symmetry, i.e., a combination of four-fold rotation in real space and two-fold
rotation in spin space, which enforces $d$-wave magnetism in momentum space.
Note that there exist several candidate materials with $d$-wave planar
altermagnetism, such as KRu4O8, La2CuO4 and FeSb2 [2, 1]. Transforming into
real space, the Hamiltonian reads

$\displaystyle\mathcal{H}_{\text{AM}}=\sum_{\bf{r}}\left(\psi_{{\bf
r}}^{\dagger}T_{x}\psi_{{\bf r}+{\bm{\delta}}_{x}}+\psi_{{\bf
r}}^{\dagger}T_{y}\psi_{{\bf r}+{\bm{\delta}}_{y}}\psi_{{\bf r}}+H.c.\right),$
(2)

where $\psi_{\bf r}=(c_{{\bf r}\uparrow},c_{{\bf r}\downarrow})^{T}$ and
$c_{{\bf r}\uparrow}\;(c_{{\bf r}\downarrow})$ is the electron annihilation
operator at site $\bf r$ with spin $\uparrow(\downarrow)$, ${\bm{\delta}}_{x}$
(${\bm{\delta}}_{y}$) is the unit vector in the $x$ ($y$) direction, and
$T_{x}=t\sigma_{0}/2+t_{J}\sigma_{z}/2$ and
$T_{y}=t\sigma_{0}/2-t_{J}\sigma_{z}/2$ are the hopping matrices in the $x$
and $y$ directions, respectively.

We consider planar Josephson junctions constructed by sandwiching the
altermagnet with two conventional $s$-wave superconductors, as depicted in
Fig. 1(a). The junction length and width are $L$ and $W$, respectively. This
setup can be described by the Bogoliubov-de-Gennes Hamiltonian

${\cal H}_{\text{BdG}}({\bf r})=\begin{pmatrix}\mathcal{H}({\bf r})-\mu({\bf
r})&i\Delta({\bf r})\sigma_{y}\\\ -i\Delta^{\dagger}({\bf
r})\sigma_{y}&-\mathcal{H}^{\ast}({\bf r})+\mu({\bf r})\end{pmatrix},$ (3)

where the position-dependent pairing potential is $\Delta({\bf
r})=\Delta_{0}e^{-i\;\text{sgn}(x)\phi/2}\Theta(|x|-L/2)$ and the chemical
potential is $\mu({\bf
r})=\mu_{\text{AM}}\Theta(L/2-|x|)+\mu_{\text{S}}\Theta(|x|-L/2)$.
$\sigma_{y}$ is the Pauli matrix in spin space. ${\cal H}({\bf r})$ is given
by Eq. (2) with finite $t_{J}$ for the altermagnetic region ($|x|<L/2$) while
$t_{J}=0$ for the superconducting regions ($|x|\geqslant L/2$). $\phi$ is the
superconducting phase difference between the superconductors, $\Delta_{0}$ is
the pairing potential of the superconductors, $\Theta(x)$ is the Heaviside
function, and $\text{sgn}(x)$ is the sign function. To obtain the Josephson
supercurrent, we adopt the recursive Green’s function approach [42, 43]. For
simplicity, periodic boundary conditions along $y$-direction have been
applied. We set parameters $L=20$, $t=1$, $\mu_{\rm{S}}=\mu_{\rm{AM}}=0.2t$,
$\Delta_{0}=0.01t$, and temperature $k_{B}T=0.02\Delta_{0}$ throughout the
paper unless specified differently.

Figure 2: (a) CPRs of the AMJJ for different $t_{J}$. Blue, green, and purple
curves correspond to the AFS marked as stars in (b), respectively. (b)
Critical current $I_{c}$ as a function of AFS $t_{J}$. (c) Harmonic expansion
for $t_{J}$=0.03$t$. Dominant terms are first and second harmonics marked by
blue and red colors, respectively. (d) Harmonic expansion for $t_{J}$=0.44$t$.
Note the sign change of the second harmonic with respect to (c). (e) Higher
harmonics $I_{n}$ compared to first harmonic $I_{1}$ as a function of AFS
$t_{J}$ with $a_{n}\equiv I_{n}/I_{1}$. (f) Upper panel shows the CPR and
lower panel illustrates dominant second harmonic for $t_{J}=0.13t$.

_Altermagnetism induced tunable CPR skewness._ —We first analyze the response
of the Josephson current by changing AFS $t_{J}$. Figure 2(a) presents several
typical CPRs for various $t_{J}$. All the CPRs are $2\pi$-periodic, i.e.,
$J_{s}(\phi)=J_{s}(\phi+2\pi)$, and odd in the phase difference $\phi$, i.e.,
$J_{s}(\phi)=J_{s}(-\phi)$, due to the inversion symmetry of the junction. The
supercurrent $J_{s}$ generally has either one maximum or minimum in the
positive phase region $\phi\in(0,\pi)$, which we define as the critical
supercurrent $I_{c}$. When the first harmonic dominates (e.g., in case of low-
transparent junction interfaces), the supercurrent has sinusoidal form
$J_{s}=I_{c}\sin\phi$. Thus, the maximum or minimum is located at
$\phi=\pi/2$. However, when the junction interfaces become more transparent,
higher harmonics contribute, leading to skewed CPRs. The supercurrent is
essentially determined by the derivative of the free energy $F$ of the
junction to the phase difference $\phi$, $J_{s}\propto\partial
F/\partial\phi$. For positive $I_{c}$, the ground state (corresponding to the
free energy minimum) of the junction occurs at $\phi=0$. In contrast, for
negative $I_{c}$, the ground state occurs at $\phi=\pi$. In Fig. 2(b), we
calculate the critical current $I_{c}$ as a function of $t_{J}$. We observe
that $I_{c}$ oscillates around zero as $t_{J}$ increases. These oscillations
indicate that the AMJJ undergoes 0-$\pi$ transitions [Fig. 2(a)]. They can be
attributed to the proximity-induced finite-momentum pairing correlations in
the altermagnet [33]. The amplitude of $I_{c}$ first quickly decreases as the
altermagnetism is turned on. Then, it oscillates with a reduced amplitude and
finally vanishes for $t_{J}>0.9t$. Note that $t_{J}$ is different from
material to material and may be varied, e.g., by the twist angle in twisted
bilayer platforms [24, 25, 26, 27].

Altermangetism can not only induce the 0-$\pi$ transitions but also affect the
skewness of the CPR where the impact of the second-order Josephson response is
vital. The CPR with dominant first harmonic term has the form
$J_{s}=I_{c}\sin\phi$. A skewed CPR implies the presence of higher harmonic
terms contributing to the supercurrent, leading to a non-sinusoidal form. The
skewness can be characterized by the deviation of the maximum/minimum position
$\phi_{m}$ of $J_{s}$ from $\pi/2$, i.e., $\delta\phi\equiv\phi_{m}-\pi/2$.
Thus, a positive $\delta\phi$ corresponds to a forward-skewed CPR, while a
negative $\delta\phi$ corresponds to a backward-skewed CPR. In Fig. 1(b), we
calculate the skewness of the CPR as a function of altermgnetic strength
$t_{J}$. Strikingly, the CPR can be either forward- (pink area) or backward-
skewed (green area). Examples of CPRs with forward and backward skewness are
explicitly shown in Fig. 2(a). When the transparency of the junction changes
because of the presence of altermagnetism, the second-order Josephson response
can not be ignored, leading to the non-sinusoidal form of Josephson
supercurrent $J_{s}=I_{1}\sin(\phi)\pm I_{2}\sin(2\phi)$ with $I_{1(2)}>0$.
Note that the skewness of the CPR is mainly determined by the sign of the
second harmonic term. For a negative second harmonic term, the CPR is forward-
skewed with $J_{s}=I_{1}\sin(\phi)-I_{2}\sin(2\phi)$ as illustrated in Figs.
2(a) and 2(c) with $t_{J}=0.03t$ and $I_{2}/I_{1}=0.37$. For a positive second
harmonic term, the CPR is backward-skewd with
$J_{s}=I_{1}\sin(\phi)+I_{2}\sin(2\phi)$ as shown in Figs. 2(a) and 2(d) with
$t_{J}=0.44t$ and $I_{2}/I_{1}=0.64$. Note that a similar ratio of
$I_{2}/I_{1}$ has recently been observed in Al/InAs-quantum well planar
Josephson junctions [44]. The second-order Josephson response in AMJJs can be
detected in SQUID devices flux-biased to $\pi$ or in tunnel junctions [45].
More details of the analysis on the skewness can be found in the Supplemental
Material (SM) [46]. Notably, the skewness $\delta\phi$ can be as large as $\pm
0.3\pi$. This particular feature of tunable skewness of CPRs in AMJJs can be
exploited to realize a large Josephson diode effect, e.g, in supercurrent
interferometers [47]. We also note that multiple maxima can appear in the CPR
due to dominant higher harmonics. Figure 2(e) shows the higher harmonic
contribution compared to the first harmonic as a function of AFS $t_{J}$ with
$I_{n}$ the amplitude of $n$th harmonic and $a_{n}\equiv I_{n}/I_{1}$. When
higher harmonics are dominant as illustrated in the grey shaded area in Fig.
2(e), the AMJJ turns into a $\phi$-Josephson junction. An example for
$a_{2}>1$ is shown in Fig. 2(f). In such cases, the skewness is not well-
defined. Therefore, we discard these cases in Fig. 1(b).

Figure 3: (a) Phase diagram of the AMJJ as functions of AFS $t_{J}$ and
applied gate potential $V_{z}$. The color bar stands for the magnitude of the
critical current $I_{c}$ in units of $e\Delta_{0}/h$. (b) $I_{c}$ as a
function of $V_{z}$ for different $t_{J}$. The blue and cyan curves are for
$t_{J}=0.18t$ and 0.34$t$, respectively. The light blue and pink stars
indicate the $V_{z}$ values used in panel (c). (c) CPRs for different $V_{z}$
with $t_{J}=0.34t$. (d) CPRs of $\phi$-junctions for different $V_{z}$ with
$t_{J}=0.4t$.

_Tuning CPR skewness and critical current by gating._ —Now, we investigate the
influence of an external gate potential $V_{z}$ on the Josephson current.
Figure 3(a) shows the critical current $I_{c}$ of the AMJJ as functions of AFS
$t_{J}$ and gate potential $V_{z}$ for $\mu_{\text{S}}=\mu_{\text{A}M}$. For
small $t_{J}$ ($t_{J}<0.05t$), $I_{c}$ is always positive with only small
oscillations. These oscillations appear due to Fabry-Pérot interferences.
Thus, there is no 0-$\pi$ transition. Increasing $t_{J}$, we observe $0-\pi$
transitions and a tilted interference pattern. The larger $t_{J}$, the more
inclined the interference pattern becomes, illustrated in the phase diagram
[Fig. 3(a)]. Hence, $I_{c}$ oscillates around zero with increasing $V_{z}$ for
larger AFS ($t_{J}>0.1t$). Figure 3(b) presents $I_{c}$ as a function of
$V_{z}$ for $t_{J}=0.18t$ and $0.34t$, respectively. This plot indicates the
switching of the AMJJ between 0 and $\pi$ junctions by changing the gate
potential as illustrated in Fig. 3(c). Meanwhile, the magnitude of the
critical current is substantially enhanced for a wide range of gate potential
as shown in Fig. 3(b). Corresponding CPRs are plotted in Fig. 3(c). The
observed $I_{c}$ enhancement is primarily governed by finite-momentum pairing,
depending on both the gate potential and the AFS [33]. Furthermore, the CPR
skewness can also be manipulated by electric gating. For instance, the CPR
changes from forward- to backward-skewed as shown in Fig. 1(c). Additionally,
a 0-junction can be transformed into a $\pi$\- or $\phi$-junction by adjusting
the gate potential as illustrated in Figs. 1(c) and 3(d). These gate-tunable
characteristics of AMJJs enrich their potential applications in
superconducting circuits and spintronics.

Figure 4: (a) Phase diagram of the AMJJ as functions of $t_{J}$ (in units of
$t$) and Zeeman field $B_{y}$ (in units of $\Delta_{0}$). The color bar
indicates the magnitude of $I_{c}$ in units of $e\Delta_{0}/h$. (b) $I_{c}$
(in units of $e\Delta_{0}/h$) as a function of $B_{y}$ in absence of
altermagnetism. (c) $I_{c}$ as a function of $B_{y}$. The blue and red curves
are for $t_{J}=0.16t$ and $0.48t$, respectively. (d) $I_{c}$ as a function of
$B_{y}$. The cyan and pink curves are for $t_{J}=0.12t$ and $0.42t$,
respectively.

_Enhancement of supercurrent by Zeeman fields._ —Now, we discuss the influence
of an in-plane magnetic field on the supercurrent in AMJJs. We model it by the
Zeeman term $H_{Z}=B_{y}\cdot\sigma_{y}$. The particular in-plane direction of
the magnetic field does not matter qualitatively for our results. Figure 4(a)
illustrates the phase diagram of the AMJJ against $t_{J}$ and $B_{y}$. We find
that $0$-$\pi$ transitions emerge as $B_{y}$ changes. These oscillations stem
from the Fermi surface spin splitting by the Zeeman field. In absence of
altermagnetism, the Zeeman field results in 0-$\pi$ transition, but the
magnitude of $I_{c}$ decreases while $B_{y}$ increases as illustrated in Fig.
4(b). In addition, the stripes bend downward as $t_{J}$ grows, shown in Fig.
4(a). This demonstrates the competition between altermagnetism and Zeeman
effect. Strikingly, this competition gives rise to the enhancement of the
critical current as shown in Figs. 1(d) and 4(c). Notably, the periodicity is
nearly unchanged by the Zeeman effect [Figs. 4(c)-(d)]. The 0-$\pi$ transition
is diminished for large AFS. Nevertheless, the enhancement of the magnitude of
$I_{c}$ remains.

The enhancement of the critical current in the presence of an exchange field
is uncommon because the critical current generally decreases with increasing
field strength in both ferromagnetic [48] and altermagnetic [Fig. 2(b)]
Josephson junctions. Nevertheless, in a superconductor/ferromagnet bilayer
separated by an insulating film, the critical current can also be enhanced
when the exchange fields in the two ferromagnets are anti-parallel, resulting
in no net magnetization [49]. Notably, the mechanism underlying the
enhancement of the critical current in AMJJs differs from this scenario, as
AMJJs exhibit net magnetization in presence of Zeeman fields. The enhancement
of the critical current in AMJJs is due to the competition between
altermagnetic and Zeeman exchange fields. This phenomenon is a characteristic
signature of altermagnetic superconducting heterostructures.

_Conclusion_.—We investigate the influence of external electric and Zeeman
fields on AMJJs. We reveal that changing the AFS can not only induce 0-$\pi$
transitions but also significantly alter the shape of the CPR, enabling both
forward- and backward-skewed CPRs in the junction. Applying either an electric
or a Zeeman field can significantly enhance the magnitude of the critical
current across the junction. Remarkably, 0-$\pi$ transitions can be controlled
by either electric gating or Zeeman fields. Moreover, we discover that the CPR
skewness can be manipulated by electric gating.

###### Acknowledgements.

We thank Lun-Hui Hu for helpful discussions. This work was supported by the
Würzburg-Dresden Cluster of Excellence ct.qmat (Project-id 390858490), the DFG
(SFB 1170), and the Bavarian Ministry of Economic Affairs, Regional
Development and Energy for ﬁnancial support within the High-Tech Agenda
Project “Bausteine für das Quanten Computing auf Basis topologischer
Materialen”. S.B.Z. acknowledges the support of the start-up fund at HFNL, the
Innovation Program for Quantum Science and Technology (Grant No.
2021ZD0302800), and Anhui Initiative in Quantum Information Technologies
(Grant No. AHY170000).

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Supplemental Material

In the following, we derive the real space lattice model and introduce the
recursive Green’s function approach. Additionally, we analyse the sknewness
change by a simple model.

##  SI Model

We use the following continuum model [12, 1]

$\displaystyle
H(k)=t(k_{x}^{2}+k_{y}^{2})\sigma_{0}+t_{J}(k_{y}^{2}-k_{x}^{2})\sigma_{z}.$
(S1)

By substituting $k_{j}^{2}\rightarrow\frac{2}{a^{2}}[1-\cos(k_{j}a)]$ with
$j=x,y$ and $a=1$, we map the continuum model to a square lattice, which reads

$\displaystyle H(k_{x},k_{y})$ $\displaystyle=$ $\displaystyle
4t\sigma_{0}-2t[\cos(k_{x})+\cos(k_{y})]\sigma_{0}+2t_{J}[\cos(k_{x})-\cos(k_{y})]\sigma_{z}.$
(S2)

Ignoring the energy shift $4t$ and substituting $t^{\prime}=-2t$ and
$t_{J}^{\prime}=2t_{J}$, we obtain the lattice model in the main text. After
Fourier transformation, we arrive at the real space Hamiltonian with periodic
boundary condition along $y$-direction as

$\displaystyle\mathcal{H}\left(x,k_{y}\right)$ $\displaystyle=$
$\displaystyle\sum_{x,x^{\prime}}\\{\Psi_{x+a}^{\dagger}[-t\sigma_{0}+t_{J}\sigma_{z}]\Psi_{x^{\prime}}+\Psi_{x}^{\dagger}[4t\sigma_{0}-2t\cos
k_{y}\sigma_{0}-2t_{J}\cos k_{y}\sigma_{z}]\Psi_{x^{\prime}}+H.c.\\}.$ (S3)

##  SII Recursive Green’s function approach

We calculate the Josephson current exploiting the recursive Green’s function
approach [42, 43]. Since we assume translation symmetry along $y$-direction,
$k_{y}$ is a good quantum number. The planar Josephson junction we study is a
quasi-1D system along $x$-direction. In the calculation, we put the Boltzmann
constant $k_{B}=1$.

Employing the lattice model in the main text, the dc Josephson current can be
calculated as

$\displaystyle I=\frac{ieT}{\hbar}\sum_{\omega_{n}}\sum_{k=0}^{n_{y}}{\rm
Tr}[\tau_{z}T_{j,j+1}G_{\omega_{n}}(j,k;j+1,k)-\tau_{z}T_{j+1,j}G_{\omega_{n}}(j+1,k;j,k)],$
(S4)

where $G_{\omega_{n}}(j,k;j^{\prime},k^{\prime})$ is the Green’s function in
Nambu space, $\tau_{z}$ is the third Pauli matrix in particle-hole space,
$\omega_{n}=(2n+1)\pi T$ is the Matsubara frequency, and $T_{j,j^{\prime}}$ is
the hopping matrix. The Nambu Green’s function for each Matsubara frequency
and momentum $k_{y}$ can be obtained by the iterative method

$\displaystyle
G_{\omega_{n}}(j,k;j^{\prime},k^{\prime})=[i\omega_{n}\delta_{k,k^{\prime}}I-H_{0}-H_{1}G_{\omega_{n}}(j-1,k;j-1,k^{\prime})H_{1}]^{-1},$
(S5)

where $I$ is a $4\times 4$ unit matrix, $H_{0}$ is the on-site energy matrix,
and $H_{1}$ is the hopping matrix between two neighbouring sites. The
superconducting leads are modeled using self-energy terms attached to the
altermagnetic region. The self-energy terms incorporate the superconducting
gap and the phase difference between the superconductors. By attaching the
superconductor to the altermagnetic region, the Green’s function of the
connected system can be obtained as

$\displaystyle
G_{\omega_{n}}(j,k;j+1,k^{\prime})=\sum_{l=0}^{n_{y}}\\{G_{\omega_{n}}(j,k;j,l)^{-1}-H_{1}G_{\omega_{n}}(j+1,k;j+1,l)H_{1}\\}^{-1}H_{1}G_{\omega_{n}}(j+1,l;j+1,k^{\prime}),$
$\displaystyle
G_{\omega_{n}}(j+1,k;j,k^{\prime})=\sum_{l=0}^{n_{y}}G_{\omega_{n}}(j+1,k;j+1,l)H_{1}\\{G_{\omega_{n}}(j,l;j,k^{\prime})^{-1}-H_{1}G_{\omega_{n}}(j+1,l;j+1,k^{\prime})H_{1}\\}^{-1}.$
(S6)

By substituting the above two equations into Eq.(S4) for each Matsubara
frequency and then summing over them, we obtain the dc Josephson current.

##  SIII Skewness analysis

In the main text, we show that the skewness varies as the altermagnetic field
strength grows (Fig.1 in the main text). Here, we propose a simple model to
simulate the skewness change. The sinusoidal form of the Josephson
supercurrent reads $J_{s}=I_{c}\sin(\phi)$ with $I_{c}$ the critical current,
shown in Fig. S1(a). When the transparency of the junction changes because of
the introduction of altermagnetism, the second harmonic contribution can not
be ignored leading to the nonsinusoidal form of Josephson supercurrent
$J_{s}=I_{1}\sin(\phi)\pm I_{2}\sin(2\phi)$ with $I_{2}>0$. For the case with
negative second harmonic term, the skewness is forward with
$J_{s}=I_{1}\sin(\phi)-I_{2}\sin(2\phi)$ as illustrated in Fig. S1(a). For the
case with positive second harmonic term, the skewness is backward with
$J_{s}=I_{1}\sin(\phi)+I_{2}\sin(2\phi)$. The first harmonic and second
harmonic terms are depicted in Fig. S1(b). The nonsinusoidal behavior without
altermagnetism ($t_{J}=0$) can be captured by expressing the CPR as Fourier
series

$\displaystyle
J_{s}(\phi)=\sum_{n}(-1)^{n-1}I_{n}\sin(n\phi)=\sum_{n}(-1)^{n-1}I_{1}a_{n}\sin(n\phi)$
(S7)

where $n$ is an integer, $a_{n}\equiv I_{n}/I_{1}$ and $I_{n}>0$. It is
illustrated in Figs. S1 (c)-(d) with $a_{2}=I_{2}/I_{1}=0.46$. When the
altermagnetism is introduced, the higher harmonic terms change sign. For
instance, as the sign of the second harmonic term ($n=2$) changes from minus
to plus the skewness of the CPR becomes backward as illustrated in Figs. S1
(e)-(f) with $a_{2}=I_{2}/I_{1}=0.53$.

Figure S1: (a) Josephson current $J_{s}$ as a function of superconducting
phase difference $\phi$. Green curve indicates the sinusoidal case. Blue and
red curves indicate the non-sinusoidal cases with negative and positive second
harmonic terms $\sin(2\phi)$, respectively. (b) First and second harmonic
terms, that contribute to the total Josephson current $J_{s}$, as a function
of $\phi$. (c) CPR for $t_{J}=0$ with forward skewness. Red curve and blue
circles represent the CPR data and harmonic fit, respectively. (d) Harmonic
components for $t_{J}$=0. Dominant terms are first and second harmonic
contribution marked by blue and red colors, respectively. (e) The harmonic
components for the case with $t_{J}$=0.1$t$. The dominant terms are first and
second harmonic contribution marked by blue and red colors, respectively. (f)
CPR for $t_{J}=0.1t$ with backward skewness. Red curve and blue circles
represent the CPR data and harmonic fit, respectively.


# SAPPHIRE: Search for exotic parity-violation interactions with quantum spin
amplifiers

Yuanhong Wang  These authors contributed equally to this work CAS Key
Laboratory of Microscale Magnetic Resonance and School of Physical Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, China  CAS
Center for Excellence in Quantum Information and Quantum Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China  Ying Huang
These authors contributed equally to this work CAS Key Laboratory of
Microscale Magnetic Resonance and School of Physical Sciences, University of
Science and Technology of China, Hefei, Anhui 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, China  Chang Guo  CAS Key
Laboratory of Microscale Magnetic Resonance and School of Physical Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, China  CAS
Center for Excellence in Quantum Information and Quantum Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China  Min Jiang
<EMAIL_ADDRESS>CAS Key Laboratory of Microscale Magnetic Resonance and
School of Physical Sciences, University of Science and Technology of China,
Hefei, Anhui 230026, China  CAS Center for Excellence in Quantum Information
and Quantum Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China  Xiang Kang  CAS Key Laboratory of Microscale Magnetic
Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei, Anhui 230026, China  CAS Center for Excellence in
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei, Anhui 230026, China  Haowen Su CAS Key Laboratory of
Microscale Magnetic Resonance and School of Physical Sciences, University of
Science and Technology of China, Hefei, Anhui 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, China  Yushu Qin CAS Key
Laboratory of Microscale Magnetic Resonance and School of Physical Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, China  CAS
Center for Excellence in Quantum Information and Quantum Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China  Wei Ji
Helmholtz-Institut, GSI Helmholtzzentrum für Schwerionenforschung, Mainz
55128, Germany  Johannes Gutenberg University, Mainz 55128, Germany  Dongdong
Hu  State Key Laboratory of Particle Detection and Electronics, University of
Science and Technology of China, Hefei, Anhui 230026, China  Xinhua Peng
<EMAIL_ADDRESS>CAS Key Laboratory of Microscale Magnetic Resonance and
School of Physical Sciences, University of Science and Technology of China,
Hefei, Anhui 230026, China  CAS Center for Excellence in Quantum Information
and Quantum Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China  Dmitry Budker  Helmholtz-Institut, GSI Helmholtzzentrum
für Schwerionenforschung, Mainz 55128, Germany  Johannes Gutenberg University,
Mainz 55128, Germany  Department of Physics, University of California,
Berkeley, CA 94720-7300, USA

Quantum sensing provides sensitive tabletop tools to search for exotic spin-
dependent interactions beyond the Standard Model, which have attracted great
attention in theories and experiments[1]. Here we develop a technique based on
quantum Spin Amplifier[2] for Particle PHysIcs REsearch (SAPPHIRE) to
resonantly search for exotic interactions, specifically parity-odd spin-spin
interactions. The present technique effectively amplifies the pseudomagnetic
field generated by exotic interactions by a factor of about 200 while being
insensitive to spurious external magnetic fields. Our studies, using such a
quantum amplification technique, open the doors to exploring the parity-
violation interactions mediated by $Z^{\prime}$ bosons[3, 4] in the
challenging parameter space (force range between $3\,\text{mm}$ and
$0.1\,\text{km}$) and set the most stringent constraints on
$Z^{\prime}$-mediated electron-neutron couplings, significantly improving
previous limits[5] by up to five orders of magnitude. Moreover, our bounds on
$Z^{\prime}$-mediated couplings between nucleons reach into a hitherto
unexplored parameter space (force range below $1\,\text{m}$), complementing
the existing astrophysical and laboratory studies[3, 6, 7].

Numerous theories have predicted the existence of exotic interactions beyond
the Standard Model that are distinct from the four known interactions [8, 3].
These exotic interactions were first predicted to be mediated by hypothetical
spin-0 bosons such as axions and axion-like particles [9, 10], whose existence
may explain many central puzzles in physics, including the strong charge-
parity problem [9, 10] and the nature of dark matter [2, 11]. Subsequently,
the interaction mediators were further extended to other new gauge bosons such
as massive spin-1 bosons and paraphotons [3], resulting in sixteen types of
potentials under the frame of quantum field theory. A series of astrophysical
and laboratory searches are presented to constrain such exotic interactions
[12, 1]. Since Lee and Yang first discussed the use of gravitational
equivalence tests (Eötvös-type experiments) to constrain a long-range
interaction [13], recent progress in exploring a broad range of exotic
interactions has been enabled by advances in precision techniques [14, 15, 16,
17, 18, 6, 19, 20, 21, 7, 22, 23].

After parity violation was discovered in beta-decay of 60Co [24], various
searches for parity violation are performed, including different parity non-
conservation tests [5, 12, 25] and static electric dipole moment searching
experiments (see a recent review [26] and references therein). The
manifestation of the parity-violation nature of exotic interactions kindles an
interest in the search for parity-odd interactions, as the study on such
exotic interactions has expanded the area of parity-violation research, which
could be a harbinger of potential new physics effects beyond the Standard-
Model expectations. In the present study, we investigate the parity-odd spin-
spin (POSS) interaction induced by the exchange of hypothetical spin-1
$Z^{\prime}$ bosons. Due to the potential loopholes left by astrophysical
observations on the constraints of this interaction [3, 27], it would be
valuable to perform direct tests with tabletop experiments. However, the
$Z^{\prime}$-mediated POSS interaction has been explored by only a few
laboratory experiments [7, 6, 5]. In particular, it remains challenging to
extract the sought-after signal at short (meter scale and shorter) distances
due to interference from usual magnetic interactions.

We focus on the exotic POSS interaction between electrons and neutrons (Fig.
1), whose corresponding potential $V_{11}$ following the notation in Refs.
[3,4] is given as ($\hbar=c=1$)

$\displaystyle
V_{11}=-f_{11}\left[(\hat{\sigma}_{n}\times\hat{\sigma}_{e})\cdot\hat{r}\right]\left(\frac{1}{\lambda
r}+\frac{1}{r^{2}}\right)\frac{e^{-r/\lambda}}{4\pi m_{e}},$ (1)

where $\hat{\sigma}_{e}$ ($\hat{\sigma}_{n}$) is electron (neutron) unit spin
vector, $\hat{r}$ is the unit vector directed from electrons to neutrons, $r$
is the distance between them, $\lambda$ is the force range and $f_{11}$ is the
dimensionless coupling constant. The potential can be generated by the
exchange of an ultralight spin-1 boson $Z^{\prime}$ [3,4] whose mass
$m_{Z^{\prime}}=\lambda^{-1}$ determines the force range of the interaction.
Experimental searches for exotic spin-spin interactions are typically
performed by positioning two polarized objects in close proximity to each
other, with one acting as a magnetic detector (“spin sensor”) to measure the
exotic field produced by the other polarized object (“spin source”). The POSS
interaction induces an energy shift of the nuclei in the spin sensor $\Delta
E=V_{11}$, which is due to a pseudomagnetic field
$\textbf{B}_{11}\hat{\sigma_{n}}=-V_{11}/\mu_{\text{N}}$ with $\mu_{\text{N}}$
being the magnetic moment of nuclei. It is worthy noting that the exotic field
$\textbf{B}_{11}\propto-\hat{\sigma}_{e}\times\hat{r}$ acted on the mirror
symmetrical spin sensor is in the opposite direction to the one before parity
transformation (Fig. 1), illustrating that the parity of the $V_{11}$
interaction is broken.

Figure 1: Illustration of parity-odd spin-spin (POSS) interaction. The exotic
interaction between electrons and neutrons involving the exchange of
$Z^{\prime}$ bosons can be described by the Feynman diagrams [3]. As a parity-
odd interaction, the induced exotic field changes sign under a parity
transformation as shown in the mirror.

In our experiment, the spin sensor uses polarized neutron spins of
isotopically enriched polarized 129Xe gas and the spin source uses polarized
electron spins of 87Rb atoms. As shown in Fig. 2A, the spin source is
positioned $50.7$ mm above the center of the spin-sensor vapor cell. The
corresponding number of polarized 87Rb electron spins in the spin source is
calibrated about $(2.1\pm 0.2)\times 10^{14}$ (see Methods and Supplementary
Section I for details).

Figure 2: Spin amplification setup and magnetic-field sensitivity. (A) The
87Rb atoms (the spin source) contained in a vapor cell with N2 buffer gas are
polarized with a circularly polarized laser beam. The spin-sensor cell
containing enriched 129Xe, buffer gas N2 and a droplet of 87Rb is placed below
the spin source. A circularly polarized beam along $\hat{z}$ and a linearly
polarized beam along $\hat{x}$ are used to, respectively, pump and probe the
spin-sensor. (B) The sensitivity of the spin amplifier along $\hat{x}$ is
measured to be $33.9$ fT/Hz1/2 at the resonance frequency and $6.4$ pT/Hz1/2
off resonance. In contrast, the magnetic sensitivity along $\hat{z}$ is only
$257.5$ pT/Hz1/2.

A quantum spin amplifier [2, 28] is introduced to resonantly amplify the
effect of the exotic field $\textbf{B}_{11}$. The spin amplifier employs
spatially overlapping hyperpolarized long-lived 129Xe nuclear spins and
optically-pumped polarized 87Rb spins (Supplementary Section II.A). The 129Xe
spins are polarized by spin-exchange collisions with polarized 87Rb spins [2,
28]. A constant field $B_{0}\hat{z}$ is applied in the sensor, along with an
oscillatory transverse field $\textbf{B}_{\text{ac}}$ to be measured. When the
oscillation frequency of $\textbf{B}_{\text{ac}}$ matches the 129Xe Larmor
frequency, the polarized 129Xe spins are tilted away from their polarization
axis, and generate a transverse oscillatory magnetization. Due to Fermi-
contact interactions [28, 2], an effective field $\textbf{B}^{n}_{\text{eff}}$
generated by the transverse magnetization is read out by the embedded 87Rb
spins that act as an $in$-$situ$ magnetometer, as shown in Expended Data Fig
1A. Importantly, we find that the 129Xe spins can act as a quantum
preamplifier, which converts the external resonant field into an amplified
field $|\textbf{B}^{n}_{\text{eff}}|\gg|\textbf{B}_{\text{ac}}|$ without
introducing additional noise.

The amplification effect can be quantitatively described by introducing an
amplification factor [2, 28]

$\eta=|\textbf{B}^{n}_{\textrm{eff}}|/|\textbf{B}_{\textrm{ac}}|=\dfrac{4\pi}{3}\kappa_{0}\gamma_{N}M^{N}_{z}T_{2}^{N},$
(2)

where $\kappa_{0}\approx 540$ is the Femi-contact enhancement factor,
$\gamma_{N}$ is the gyromagnetic ratio of 129Xe nucleus; $M^{N}_{z}$ is the
initial longitudinal magnetization along the polarization axis and
$T_{2}^{N}\approx 20\,$s is the transverse relaxation time. For the 129Xe-87Rb
system reported in this work, the amplification factor is measured to be
$\eta\approx$ 187.4. This shows that the spin amplifier can enhance the effect
of the exotic field by more than two orders of magnitude, whose magnetic
sensitivity reaches $33.9$ fT/Hz1/2 at the resonance frequency, as shown in
Fig. 2B. See Methods and Supplementary Section II.C for more details. In
addition, the spin amplifier is only sensitive to a transverse oscillatory
field, since fields along the polarization axis cannot induce a measurable
transverse magnetization.

Similar resonant techniques based on nuclear magnetic resonance have been
recently introduced to search for exotic interactions [18, 29, 17], and most
of their experimental demonstrations are still ongoing. Similar to the quantum
spin amplifier, these resonant techniques measure exotic fields through
measuring the response of nuclear spins with an auxiliary magnetometer.
However, 87Rb atoms in the spin amplifier act as an embedded magnetometer,
enabling $in$-$situ$ measurement and continuous polarization of the 129Xe
atoms. A significant advantage offered by $in$-$situ$ measurements is the
enhancement of the nuclear resonance signals due to the large Fermi-contact
amplification factor. In addition, since 129Xe nuclear spins are polarized
continuously by spin-exchange collisions with polarized 87Rb, the spin
amplifier can search for exotic fields with a 100% duty cycle, which is
suitable for ultrasensitive continuous wave detection of exotic interactions
[14, 15].

In order to resonantly search for the POSS interaction, we modulate the
polarization of the spin source, thus modulating the pseudomagnetic field
$\textbf{B}_{11}$. As shown in Extended Data Fig. 1A, the exotic signal is
modulated by periodically blocking the pump laser beam, with a $50\%$ duty
cycle, with an optical chopper. Once the chopping frequency matches the
resonance frequency (i.e. $\nu\approx\nu_{0}\approx 10.0\,$Hz), the spin
amplifier enables to resonantly search for the exotic field with a sensitivity
improvement of about 200 (see Methods for details).

The polarized spin source inevitably generates an usual magnetic field in
addition to the exotic field, whose existence poses a challenge for exotic
spin-spin interaction searches. As a result, most experimental studies on
spin-spin interactions have focused on meter-scale or kilometer-scale searches
[6, 7]. In our case, taking advantage of sensitivity anisotropy and small-size
magnetic shields, the magnetic field is suppressed by about five orders of
magnitude in total, enabling to test the POSS interaction at millimeter-scale
distance. See Methods and Supplementary Section III for details.

Table 1:  Summary of calibrated parameters and systematic errors. The corrections to $f_{11}$ for $\lambda=0.1$ m are listed. Parameter | Value | $\Delta f_{11}\left(\times 10^{-22}\right)$
---|---|---
Position of spin source $x$ (mm) | $-1.41\pm 0.40$ | $<0.01$
Position of spin source $y$ (mm) | $50.67\pm 0.71$ | $\pm 0.07$
Position of spin source $z$ (mm) | $3.19\pm 0.01$ | $<0.01$
Num of polarized Rb | $(2.14\pm 0.24)\times 10^{14}$ | ${}^{-0.17}_{+0.20}$
Phase delay $\phi_{a}$ (deg) | $13.20\pm 0.54$ | ${}^{+0.71}_{-0.45}$
Calib. const. $\alpha$ (V/nT) | $1.99^{<0.01}_{-0.17}$ | ${}^{<0.01}_{+0.19}$
Final $f_{11}(\times 10^{-22})$ | $2.12$ | $\pm 5.87\ (\text{stat})$
$(\lambda=0.1\,\text{m})$ |  | $\pm 0.77\ (\text{syst})$

The experimental search for the POSS interaction consists in calibration
experiments and search experiments. We perform calibration experiments to
determine the required parameters, which are summarized in Table 1. Calibrated
parameters are used to obtain a reference signal of the pseudomagnetic field
for subsequent data analysis (see Methods and Supplementary Section IV.A).
During search experiments, data are collected in records of one hour. Between
every two records, we re-optimized the magnetic sensitivity of the spin
amplifier. The total duration of search data is 24 hours.

Figure 3: Constraints on POSS coupling strengths. The red solid line
represents the experimental limits at the 95$\%$ confidence level as a
function of the force range $\lambda$ and the boson mass (top axis). The red
dashed line is projected limits obtained by combining the proposal 3He-K spin
amplifier with a highly polarized solid-state spin source. (A) The vertical
axes on the left and right represent the constraints on $g^{e}_{V}g^{n}_{A}$
and $g^{e}_{A}g^{n}_{V}$, respectively. The black solid [7] and green dashed
[22] lines indicate the constraints from the measurement of $V_{11}$ and
$V_{16}$, respectively, using polarized electrons in the Earth. The blue line
stands for the constraints on $g^{e}_{A}g^{n}_{V}$ from combining results of
the Cs and Yb parity-violation experiments [30, 5]. (B) The vertical axis on
the left and right represent the limits on $g^{n}_{V}g^{p}_{A}$ and
$g^{n}_{A}g^{p}_{V}$, respectively.

Using the search data, we can constrain the gauge coupling of $Z^{\prime}$
bosons as mediators of the parity-odd spin-spin interaction [3], whose
existence arises in numerous Standard-Model extensions such as supersymmetric
theories and general hidden portal models [31]. The POSS interaction can be
induced by exchange of a $Z^{\prime}$ boson following the effective Lagrangian
as depicted by the Feynman diagrams in Fig. 1 [3,4]

$\displaystyle\mathcal{L}_{Z^{\prime}}=Z^{\prime}_{\mu}\sum_{\psi=e,n}\left[\overline{\psi}\gamma^{\mu}(g^{\psi}_{V}+g^{\psi}_{A}\gamma_{5})\psi\right]\,,$
(3)

where $\psi$ denotes fermion field (electron or neutron),
$g^{\psi}_{A}(g^{\psi}_{V})$ stands for the axial (vector) coupling to the
fermion $\psi$, $\gamma^{\mu}$ and
$\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ are Dirac matrices. The
dimensionless coefficient $f_{11}$ related to couplings constants
$g^{\psi}_{A,V}$ following the notation of Eq. (5.29) in Ref. [3] is given as

$\displaystyle
f^{en}_{11}=\frac{1}{2}g^{e}_{V}g^{n}_{A}+\frac{m_{e}}{2m_{n}}g^{e}_{A}g^{n}_{V},$
(4)

where 129Xe and 87Rb respectively contribute the neutrons and electrons in our
experiment. Here the contribution of nucleon in 87Rb is ignored (it is
discussed below).

Figure 3A shows the experimental limits on the POSS couplings between
electrons and neutrons. For the force range $\lambda>10^{3}$ m, the most
stringent constraints were set by Hunter et al. [7] as shown with the black
line, where the polarized electrons of the Earth were used as the spin source.
However, the work did not constrain the couplings for $\lambda<10^{3}$ m, as
their limits are exponentially suppressed at short distances because the
relative contribution to polarized electrons from the Earth’s outermost crust
is only $2\%$ [7]. For the force range $\lambda<3\times 10^{-3}$ m, the most
stringent limits on $g^{e}_{A}g^{n}_{V}$ (blue line) were set from atomic
parity-violation tests by detecting the induced electric-dipole transition
[5]. Our searches rely on centimeter-scale detection technique, bridging the
gap between atomic-scale and Earth-scale experiments. In consequence, our
result yields the most stringent constraints for $3\times
10^{-3}\,\text{m}<\lambda<10^{3}$ m (red solid line), improving over previous
limits by up to five orders of magnitudes.

Our experiment can also constrain the POSS coupling constants between
nucleons. We consider valence protons within the Rb nuclei in the spin source.
The valence proton is nearly fully polarized ($\sim 90\%$) due to the high
polarization of 87Rb. The proton-neutron coupling strength $f_{11}$ can be
redefined as

$\displaystyle
f^{np}_{11}=\frac{m_{e}}{2m_{p}}g^{n}_{A}g^{p}_{V}+\frac{m_{e}}{2m_{n}}g^{n}_{V}g^{p}_{A},$
(5)

where the coupling of 87Rb electrons is ignored. Therefore, our work also
provides new experimental limits on $g^{n}_{A}g^{p}_{V}$ (and
$g^{n}_{V}g^{p}_{A}$), as shown by the red solid line in Fig. 3B. For the
force range below 1 m, our work sets the best constraints on the POSS coupling
and reaches into the hitherto unexplored parameter space to test parity-odd
interaction between nucleons.

Astrophysical and cosmological observations can usually be used to constrain
the coupling strength between hypothetical bosons and Standard-Model
particles. The coupling constant of spin-0 bosons to electrons and nucleons
are tightly constrained by astrophysical limits, such as those from star
cooling. It is the opposite for spin-1 exchanged interactions, where potential
loopholes may exist in the relevant theories [3, 27]. As a result, our
constraints from laboratory direct search for the spin-1 exchanged POSS
interaction are the most stringent, where astrophysical limits are relaxed.

There is still room for improvement in the present experiment. We currently
modulate spin polarization by chopping circularly polarized pump light; if
instead we reverse the direction of the circular polarization, this will yield
a factor of two gain in the exotic-field amplitude and, moreover, would allow
to better subtract possible systematic effect due to the apparatus
imperfections. Such systematics would be further suppressed by adding an
additional symmetrically located spin sensor and/or spin source (Fig. 1). Such
an arrangement would allow taking full advantage of all possible reversals
implied by the form of the POSS interaction, Eq. (1).

Further improvement in experimental sensitivity can be anticipated by
replacing the 129Xe-Rb system in the spin amplifier with a 3He-K system [2,
14, 16] and using solid-state spin sources, such as nitrogen-vacancy centers
in diamonds [32, 33] or pentacene-doped terphenyl crystal [34, 35]. Combining
such 3He-K spin amplifier system and solid-state spin sources, the projected
constraints are more stringent than our present limits by about eight orders
of magnitude (see Methods), as shown by red dashed lines in Fig 3. Over a wide
force range, the projected limits would improve on the existing bounds,
including the limits from the Earth-geoelectron experiments [7, 22]. This
makes it possible to discover new physics on $Z^{\prime}$ bosons with the
ultrasensitive SAPPHIRE experiments.

References

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Methods

Apparatus. Objects with large polarization number density are preferentially
chosen as the spin source and placed close enough to the spin sensor for
maximizing exotic signals. As shown in Fig. 2A, a 0.58-cm3 cubic glass cell
containing polarized 87Rb vapor with 600 torr of N2 buffer gas is used as the
spin source, and is positioned $50.7$ mm above the center of the spin-sensor
vapor cell. The spin-source vapor cell is heated to 180∘C to obtain 87Rb
number density ($\approx 4.0\times 10^{14}$ cm-3). A laser beam generated with
a tapered amplifier system with a power of 0.34 W is tuned to the D1
transition of 87Rb and delivered to the spin-source vapor cell along $\hat{z}$
with a single-mode fiber. The laser beam is circularly polarized and expanded
to pump all the 87Rb in the spin-source cell to an average electron
polarization of $>0.9$ and the corresponding number of polarized 87Rb electron
spins is about $(2.1\pm 0.2)\times 10^{14}$. Here, the electron polarization
along $\hat{z}$ defined as $P_{z}=2\langle S_{z}\rangle$ is determined by
numerical calculations, where all the required parameters (such as 87Rb number
density and pump laser power etc.) are experimentally calibrated. The
uncertainty of the electron polarization is estimated from the uncertainties
of the relevant parameters (see Supplementary Section IV.A for details). The
spin sensor and the spin source are both enclosed within a single-layer
magnetic shield (withing the five-layer shield).

In contrast to our recent work [14] using an unpolarized source for detecting
spin-mass velocity-dependent interactions, the polarized spin source in this
work inevitably generates a real magnetic field in addition to the exotic
field. The existence of this magnetic fields poses a challenge for spin-spin
interaction searches. Large-size magnetic shields [6] or long-range detection
[7] was usually used in the previous search experiments, where the spin source
was far from the sensor. This sets a bottleneck for the sensitivity for short
force ranges ($\sim 1\,$cm) due to the exponential fall-off of the signal. As
a result, most experimental studies on spin-spin interactions have focused on
meter-scale searches, corresponding to mediator masses less than
$10^{-4}\,$meV. In our case, a significant dipole magnetic field of about 1.5
pT would be produced by the polarized 87Rb spins acting on the spin sensor if
the source and the sensor were not enclosed in additional shielding. We use
single-layer $\mu$-metal shielding for both the source and the sensor with the
total combined shielding factor experimentally determined to be about
$10^{4}$. In addition, we arrange the spin source directly above the spin
sensor (Fig. 2A), where the dipole magnetic field is oriented along the
insensitive axis of the spin amplifier. The component of the magnetic field
experienced by the spin sensor along the sensitive directions is estimated to
be about $10$ aT (see Supplementary Section III), which can be negligible with
respect to the 0.1 fT magnetic field detectable by the spin amplifier for
twenty-four hours of operation. Taking advantage of sensitivity anisotropy and
small-size magnetic shields, the magnetic field is suppressed by about five
orders of magnitude in total, enabling to test the POSS interaction at short
distance, down to about the mm range.

Amplification factor and magnetic-field sensitivity. We calibrate the
amplification factor and magnetic sensitivity of the quantum spin amplifier
which is important for determining the sensitivity to the POSS interaction. By
comparing the response to external resonant and off-resonant fields, the
amplification factor is measured to be $\eta\approx$ 187.4. This shows that
the spin amplifier can enhance the effect of the exotic field by more than two
orders of magnitude. By scanning the frequency of the applied oscillatory
field, we obtain the frequency dependence of the sensitivity to magnetic
fields along $\hat{x}$ and $\hat{z}$, respectively depicted in Fig. 2B. The
129Xe Lamor frequency is tuned to $\nu_{0}\approx 10.0$ Hz by setting the
constant magnetic field $B_{0}\hat{z}$ at 847 nT to avoid low-frequency noise.
The optimal magnetic sensitivity is $33.9$ fT/Hz1/2 at the resonance
frequency, with the oscillatory field applied along $\hat{x}$, while the off-
resonance sensitivity is only $6.4$ pT/Hz1/2. The spin amplifier is only
sensitive to a transverse oscillatory field, since fields along the
polarization axis cannot induce a measurable transverse magnetization. As a
result, the residual sensitivity to $\hat{z}$-fields (owing to the Rb
magnetometer) of ($\approx 257.5\,\text{pT/Hz}^{1/2}$) is about two orders of
magnitude worse than that for $\hat{x}$ fields. Taking advantage of the
sensitivity anisotropy of the spin amplifier, it is possible to distinguish
spurious magnetic fields from the pseudomagnetic field.

Modulation technique and resonant search. In order to resonantly search for
the exotic POSS interaction, we modulate the polarization of the spin source,
thus modulating the pseudomagnetic field $\textbf{B}_{11}$. As shown in
Extended Data Fig. 1A, the exotic signal is modulated by periodically blocking
the pump laser beam, with a $50\%$ duty cycle, with an optical chopper. The
modulated pseudomagnetic field as a function of time can be represented as

$\displaystyle\\!\textbf{B}_{11}(t)$
$\displaystyle\\!=\\!\textbf{B}_{11}^{0}\cdot\left\\{\frac{1}{2}+\frac{\mathcal{S}\left[\text{sin}(2\pi\nu
t+\phi)\right]}{2}\right\\},$
$\displaystyle=\sum\limits_{N=\text{odd}}\textbf{B}^{(N)}_{11}\frac{\sin(2\pi
N\nu t+\phi_{N})}{2}+C,$ (6)

where $\textbf{B}_{11}^{0}$ is the pseudomagnetic field produced by the spin
source under pumping; $\mathcal{S}$ stands for the signum function; $\phi$ is
the initial phase of the exotic signal;
$|\textbf{B}^{(N)}_{11}|=|\textbf{B}_{11}^{0}|\cdot 4/(\pi N)$ is the peak-to-
peak amplitude of the $N$th harmonic, $\phi_{N}$ is the corresponding phase
and $C=\textbf{B}_{11}^{0}/2$ is the DC component of the signal. Here, we
ignore the rising edge and falling edge of the exotic square-wave signals
because the transient of the source polarization which is faster than the
relaxation time of electrons ($\sim$1 ms) is much shorter than the pumping
period ($\sim$100 ms). Once the optical chopping frequency matches the
resonance frequency (i.e. $\nu\approx\nu_{0}\approx 10.0\,$Hz), the spin
amplifier enables to resonantly search for the exotic field with a sensitivity
improvement of about 200.

We experimentally test the sensitivity of the resonant measurement by applying
a square-wave (real) field to the spin amplifier to simulate the modulated
exotic field with an amplitude of $|\textbf{B}_{11}^{0}|$, whose Fourier-
transform spectrum contains only odd harmonics (Extended Data Fig. 1B). As the
primary component of simulated exotic fields, the first harmonic with peak-to-
peak amplitude of $|\textbf{B}^{(1)}_{11}|$ is about 1.27 times larger than
the signal $\textbf{B}_{11}^{0}$. The resonance frequency of spin amplifier is
tuned to match the frequency of the modulated field. Only the first harmonic
of the field is amplified by the spin sensor and the effects of other
harmonics are negligible (Extended Data Fig. 1C).

Data analysis. In data analysis, the exotic reference signal of pseudomagnetic
fields must be obtained first in order to extract weak signals from noisy
environments [16, 14, 15]. Because only the first harmonic is amplified, we
obtain the reference signal as $|\textbf{B}^{(1)}_{11}|/2\cdot\sin(2\pi\nu
t+\phi_{1}-\phi_{a})$, where the frequency is equal to the modulation
frequency $\nu=10.0$ Hz and the phase is obtained as $\phi_{1}-\phi_{a}$ due
to the existence of phase delay. The amplitude of reference signal
$|\textbf{B}^{(1)}_{11}|/2$ can be determined by the input exotic field
$\textbf{B}_{11}^{0}$ [see Eq. (1)]

$\displaystyle\\!\textbf{B}_{11}^{0}(f_{11})$
$\displaystyle\\!=\\!\frac{-f_{11}}{4\pi
m_{e}\mu_{\textrm{Xe}}}\\!\int_{V}\\!\rho(\bm{r})(\\!\hat{\sigma}_{e}\\!\times\\!\hat{r})\left(\\!\frac{1}{\lambda
r}\\!+\\!\frac{1}{r^{2}}\right)\\!e^{-r/\lambda}d\bm{r},$ (7)

where $V$ represents the integration space of the internal volume for the
spin-source cell and $\rho(\bm{r})$ is the number density of the polarized
electron spins. Here the volume shape and polarization gradient of the spin
source should be considered (see Supplementary Section IV.B for details).

To obtain the constraints on coupling strength $f_{11}$, a “lock-in” analysis
method [14, 15, 16] is introduced to extract exotic signals. This method
derives the wanted reference signal with a certain frequency and phase from
the experimental signal $S(t)$ and determines its amplitude. In order to
obtain statistical errors of the coupling strength within each one-hour data
set, the experimental signal $S(t)$ is separated into segments $S(t)_{i}$ with
a single period $T=0.1$ s. The coupling strength $f^{i}_{11}$ for every period
is calculated as

$f^{i}_{11}=\frac{\pi}{2\alpha|\textbf{B}^{0}_{11}(1)|}\frac{\int^{T}_{0}\sin{(2\pi\nu
t+\phi_{1}-\phi_{a})}S(t)_{i}\,dt}{\int^{T}_{0}\sin^{2}{(2\pi\nu
t+\phi_{1}-\phi_{a})}\,dt},$ (8)

where $\alpha$ is the calibration constant which converts the output signal
(V) of the spin amplifier to the exotic field (nT) and
$\textbf{B}^{0}_{11}(1)$ is the input pseudomagnetic field corresponding to
$f_{11}=1$. The experimentally measured coupling strengths $f^{i}_{11}$ are
fitted with a Gaussian distribution to determine the mean value and
statistical error for each one-hour data set, as shown in the inset of
Extended Data Fig. 2. For the force range $\lambda=0.1$ m, the coupling
strength for total 24 h is obtained as $f_{11}\approx(2.1\pm
5.9_{\text{stat}})\times 10^{-22}$ with a weighted reduced $\chi^{2}\approx
1.34$. The corresponding systematic errors are summarized in the third column
of Table 1, in which the major systematic error comes from the phase delay of
the spin amplifier. By combining all the systematic errors in quadrature,
which are assumed to be independent of each other, the overall systematic
uncertainty is derived. Accordingly, for the force range $\lambda=0.1$ m, we
quote the final coupling strength as $f_{11}\approx(2.1\pm
5.9_{\text{stat}}\pm 0.8_{\text{syst}})\times 10^{-22}$ and the constraint can
be determined as $1.5\times 10^{-21}$ at the $95\%$ confidence (see
Supplementary Section V). By varying the force range $\lambda$ and repeating
the above process, constraints for the entire explored force range were
obtained.

Projected improvement in SAPPHIRE. Further improvement in experimental
sensitivity to the parity-odd spin-spin interactions can be anticipated by
replacing the Xe-Rb system in the spin amplifier with a 3He-K system [2, 14,
16]. As seen in Eq. (2), to realize larger amplification, a prerequisite is to
achieve long transverse relaxation time and high polarized spin density of
noble gas. Specifically, the predicted amplification factor of a 3He-K system
can be better than the current one by at least two orders of magnitude [6]. In
addition to the amplification factor, K magnetometers have higher magnetic
sensitivity (a few fT/Hz1/2) than Rb magnetometers [6]. Overall, the projected
magnetic sensitivity of 3He-K spin amplifier can be improved by four orders of
magnitude over the current spin amplifier.

The number density of polarized spins in the source can be greatly increased
by using solid-state spin sources, such as nitrogen-vacancy centers in
diamonds [32, 33] and pentacene-doped terphenyl crystal [34, 35]. Near-unity
spin polarization can be realized with optical pumping. For example, a crystal
of $p$-terphenyl doped with $0.1\,\text{mol}\%$ fully deuterated pentacene can
be a good candidate for high-polarization spin sources, with polarized-
electron density reaching $\sim 1.5\times 10^{18}$ $\text{cm}^{-3}$ under
optical pumping [34]. The high nucleon polarization of synthesized deuterated
$p$-terphenyl-2′,3′,5′,6′-$d_{4}$ ($\sim 34\%$) can be created by dynamic
nuclear polarization (DNP) in 0.4 T at room temperature, which transfers the
polarization from the electrons of deuterated pentacene $0.05\,\text{mol}\%$
to the protons of terphenyl, with the total number of polarized protons
achieved exceeding $3\times 10^{18}$ [35]. Overall, using optically pumped
pentacene-doped terphenyl crystals, one could increase the strength of exotic
field by about four orders of magnitude. Combining the enhanced spin amplifier
based on 3He-K system discussed above and highly polarized solid-state spin
sources, the projected constraints are more stringent than our present limits
by about eight orders of magnitude (red ashed lines in Fig 3).

Data availability.

Source data are provided with this paper. All other data that support the
plots in this paper and other findings of this study are available from the
corresponding author upon reasonable request.

Code availability.

The code that supports the plots in this paper is available from the
corresponding author upon reasonable request.

Acknowledgements.

We thank Hao Wu and Dong Sheng for valuable discussions. This work was
supported by National Key Research and Development Program of China (grant no.
2018YFA0306600), National Natural Science Foundation of China (grants nos.
11661161018, 11927811, 12004371), Anhui Initiative in Quantum Information
Technologies (grant no. AHY050000), USTC Research Funds of the Double First-
Class Initiative (grant no. YD3540002002) and the Deutsche
Forschungsgemeinschaft (DFG) - Project ID 423116110. This work was also
supported by the Cluster of Excellence “Precision Physics, Fundamental
Interactions, and Structure of Matter” (PRISMA+ EXC 2118/1) funded by the
German Research Foundation (DFG) within the German Excellence Strategy
(Project ID 39083149).

Author contributions.

Y.H.W. and Y.H. designed experimental protocols, performed experiments,
analyzed the data and wrote the manuscript. C.G., X.K., and Y.S.Q. performed
experiments and edited the manuscript. H.W.S., W.J., and D.D.H. analyzed the
data and edited the manuscript. M.J., X.H.P., and D.B. proposed the
experimental concept, designed experimental protocols and proofread and edited
the manuscript. All authors contributed with discussions and checking the
manuscript.

Competing interests.

The authors declare no competing interests.

Extended Data Fig 1: Demonstration of resonance-sensitive search for exotic
signal based on spin amplifier. (A) Under a high-power ($>$0.34 W) modulated
optical pumping along $\hat{z}$, the spin source generates a
$\hat{x}$-directed square-wave pseudomagnetic field on the spin senor with
50$\%$ duty cycle at a frequency of $\nu\approx 10.0$ Hz. The exotic signal
$\textbf{B}_{11}$ is enhanced by the 129Xe atoms, yielding an effective field
$\textbf{B}^{n}_{\text{eff}}$ read out by 87Rb atoms. (B) Ratios of the peak-
to-peak amplitude of square-wave’s Nth harmonic relative to the amplitude of
original signal $|\textbf{B}^{0}_{11}|$. Right top inset: Experimental
measured time-domain square-wave signal of simulated exotic field
$\textbf{B}_{11}$. (C) Experimental response in frequency domain of the spin
amplifier to the square-wave magnetic field. Only the first harmonic of the
input exotic signal $\textbf{B}^{(1)}_{11}$ can be amplified and detected.
Extended Data Fig 2: Experimental coupling strength $f_{11}$ for the force
range $\lambda=0.1\,$m. Each point and error bar stands for mean value and
statistical error for one hour respectively. Left top inset: Graph of values
for the first-hour record ($\\#$1) follows a Gaussian distribution.