#  Axion Magnetic Resonance:
A Novel Enhancement in Axion-Photon Conversion

Hyeonseok Seong<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron
DESY, Notkestr. 85, 22607 Hamburg, Germany  Chen Sun<EMAIL_ADDRESS>Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA  Seokhoon Yun<EMAIL_ADDRESS>Particle Theory and Cosmology Group,
Center for Theoretical Physics of the Universe, Institute for Basic Science
(IBS), Daejeon, 34126, Korea  Dipartimento di Fisica e Astronomia, Università
degli Studi di Padova, Via Marzolo 8, 35131 Padova, Italy  INFN, Sezione di
Padova, Via Marzolo 8, 35131 Padova, Italy

###### Abstract

We identify a new resonance, axion magnetic resonance (AMR), that can greatly
enhance the conversion rate between axions and photons. A series of axion
search experiments rely on converting them into photons inside a constant
magnetic field background. A common bottleneck of such experiments is the
conversion amplitude being suppressed by the axion mass when $m_{a}\gtrsim
10^{-4}~{}$eV. We point out that a spatial or temporal variation in the
magnetic field can cancel the difference between the photon dispersion
relation and that of the axion, hence greatly enhancing the conversion
probability. We demonstrate that the enhancement can be achieved by both a
helical magnetic field profile and a harmonic oscillation of the magnitude.
Our approach can extend the projected ALPS II reach in the axion-photon
coupling ($g_{a\gamma}$) by two orders of magnitude at
$m_{a}=10^{-3}\;\mathrm{eV}$ with moderate assumptions.

††preprint: DESY-23-118††preprint: LA-UR-23-29378††preprint: CTPU-PTC-23-39

Introduction While there are a plethora of experimental agendas to search for
axions utilizing its coupling to photon directly in a laboratory CAST:2007jps
; CAST:2017uph ; Ehret:2010mh ; Bahre:2013ywa ; Ortiz:2020tgs ; OSQAR:2015qdv
; DellaValle:2015xxa ; Betz:2013dza  (see Refs. Agrawal:2017cmd ;
Irastorza:2018dyq ; Choi:2020rgn  for the comprehensive review), the bounds
get considerably weakened at $m_{a}\gtrsim 10^{-4}-10^{-2}~{}\mathrm{eV}$. In
this mass range, experiments that rely on an axion-to-photon conversion in a
magnetic field battle the so-called small-mixing-non-linear regime, where the
conversion probability is suppressed by $m_{a}^{-4}$ much before it is limited
by the length scale of the experiments. Therefore, in this regime, a longer
conversion baseline will not improve the bound. It is of great interest to
study ways of enhancing the axion-photon conversion in this regime.

The usual axion-photon oscillation formula assumes a coherent magnetic field
that is constant in both the amplitude and its orientation Anselm:1985obz ;
VanBibber:1987rq . In this Letter, we report for the first time that a spatial
variation of the magnetic field profile can compensate the fast axion-photon
oscillation due to a large axion mass.As a result, a spatially- or temporally-
varying magnetic field can enhance the conversion probability, which leads to
a great improvement in model-independent axion searches that do not rely on an
axion dark matter abundance, such as the Light-Shining-Through-Walls (LSTW)
experiments.

The spatial variation of the magnetic field necessitates a more comprehensive
treatment of the dynamics of the system. One way to understand this resonance
is to use the language of parametric resonance. It occurs when the frequency
of the spatial variation coincides with the axion-photon oscillation
frequency. This resonance phenomenon holds the potential to significantly
amplify the conversion probability, providing an exciting avenue for
experimental exploration.

Alternatively, the enhancement can be understood if we change to the basis
where the magnetic field becomes constant. An extra mass-like term for the
photon states is generated during this transformation. The enhancement is a
result of an avoided level crossing. 111Similar effect due to a varying matter
potential in the neutrino Mikheyev–Smirnov–Wolfenstein (MSW) effect is well
studied in e.g. Kuo:1989qe . This contribution vanishes for a constant mixing
matrix, hence, it has been safely neglected in the well-established
oscillation formula (e.g. Raffelt:1987im ; Mirizzi:2005ng ; Mirizzi:2006zy ;
Mirizzi:2009nq ; Masaki:2017aea ; Buen-Abad:2020zbd ).

Lastly, we provide a third approach by noting that there is a mismatch between
the dispersion relations of axions and photons. The resonance happens when the
magnetic field variation frequency coincides with this gap, i.e., the momentum
transfer. For this reason, we dub the new resonance axion magnetic resonance
(AMR), by analogy with the nuclear magnetic resonance (NMR).

As a demonstration of the experimental relevance of the AMR, we take the LSTW
experiment ALPS II Bahre:2013ywa ; Ortiz:2020tgs  as an example that a helical
magnetic profile can enhance the experimental reach by $1$-$2$ orders of
magnitude where the search remains model-independent and does not require a
local axion abundance.

Helical Magnetic Background  One of the most intriguing axion interactions in
terms of phenomenological searches is its anomalous coupling with the photon

$\displaystyle\mathcal{L}_{a\gamma}$
$\displaystyle=\frac{g_{a\gamma}}{4}aF_{\mu\nu}\tilde{F}^{\mu\nu}=-g_{a\gamma}a\,\mathbf{E}\cdot\mathbf{B},$
(1)

where $\tilde{F}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}/2$. We
denote the propagation direction as $z$, and the photon-axion system in the
interaction basis to be
$\tilde{\Psi}=[\gamma_{\perp}~{}\gamma_{\parallel}~{}a]^{T}$. The
$\perp(\parallel)$ is for the direction of photon polarization perpendicular
(parallel) to the magnetic field at the initial point, $z=0$, at rest with the
lab.

Let us start with $\mathbf{B}$ of a constant magnitude and a changing
orientation along the $z$-direction, i.e., a helical profile. Since the
$\perp$ and $\parallel$ directions are defined in terms of the $\mathbf{B}$
field direction at $z=0$, when $\mathbf{B}$ changes direction it will have
both $\perp$ and $\parallel$ components, i.e.
$\mathbf{B}=[B_{\perp},B_{\parallel},0]=[B\sin\theta,B\cos\theta,0]$, with
$\theta(z=0)=0$. The equation of motion (EOM) reads

$\displaystyle
i\partial_{z}\tilde{\Psi}=\frac{1}{2\omega}H^{2}(\theta)\tilde{\Psi}$ (2)
$\displaystyle=\frac{1}{2\omega}\begin{bmatrix}0&0&g_{a\gamma}\omega B\,{\rm
s}_{\theta}\\\ 0&0&g_{a\gamma}\omega B\,{\rm c}_{\theta}\\\ g_{a\gamma}\omega
B\,{\rm s}_{\theta}&g_{a\gamma}\omega B\,{\rm c}_{\theta}&\Delta
m^{2}_{a\gamma}\end{bmatrix}\begin{bmatrix}\gamma_{\perp}\\\
\gamma_{\parallel}\\\ a\end{bmatrix},$

where ${\rm c}_{\theta}\equiv\cos\theta$, ${\rm s}_{\theta}\equiv\sin\theta$,
and $\Delta m^{2}_{a\gamma}\equiv m_{a}^{2}-m_{\gamma}^{2}$. We have
subtracted the diagonal term $m_{\gamma}^{2}$ since it only generates an
overall phase for $\tilde{\Psi}$.

To compute the probability of axion converting with either photon
polarization, let us first make a rotation in the
$\gamma_{\perp}$-$\gamma_{\parallel}$ direction:

$\displaystyle H^{2}(\theta)=U(\theta)H^{2}(0)U^{\dagger}(\theta)$ (3)
$\displaystyle=\begin{bmatrix}{\rm c}_{\theta}&{\rm s}_{\theta}&0\\\ -{\rm
s}_{\theta}&{\rm c}_{\theta}&0\\\ 0&0&1\end{bmatrix}\begin{bmatrix}0&0&0\\\
0&0&g_{a\gamma}\omega B\\\ 0&g_{a\gamma}\omega B&\Delta
m^{2}_{a\gamma}\end{bmatrix}\begin{bmatrix}{\rm c}_{\theta}&-{\rm
s}_{\theta}&0\\\ {\rm s}_{\theta}&{\rm c}_{\theta}&0\\\ 0&0&1\end{bmatrix}.$

We define an auxiliary basis, $\hat{\Psi}=U^{\dagger}\tilde{\Psi}$. Since the
rotation matrix $U$ is $z$-dependent, it can only diagonalize the Hamiltonian
instantaneously. As a result, at different locations, we need different
$U(\theta)$ matrices to transform $\tilde{\Psi}$ to $\hat{\Psi}$. This
generates extra off-diagonal terms Kuo:1989qe ; Wang:2015dil  similar to a
gauge transformation, while a $\theta(z)$ profile corresponds to a specific
gauge fixing imposed by the external magnetic field:

$\displaystyle i\partial_{z}\hat{\Psi}$
$\displaystyle=\bigg{[}U^{\dagger}\frac{H^{2}}{2\omega}U-iU^{\dagger}\partial_{z}U\bigg{]}\hat{\Psi}$
(4) $\displaystyle=\begin{bmatrix}0&-i\dot{\theta}&0\\\
i\dot{\theta}&0&g_{a\gamma}B/2\\\ 0&g_{a\gamma}B/2&\Delta
m_{a\gamma}^{2}/2\omega\end{bmatrix}\hat{\Psi}\,,$ (5)

where $\dot{\theta}\equiv d\theta/dz$. To diagonalize the 1-2 component, we
perform a $45^{\circ}$ rotation in the 1-2 direction followed by a
$90^{\circ}$ phase shift for the first component:

$\displaystyle V=\begin{bmatrix}i&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix}\begin{bmatrix}\cos(\pi/4)&-\sin(\pi/4)&0\\\
\sin(\pi/4)&\cos(\pi/4)&0\\\ 0&0&1\end{bmatrix},$ (6)

which gives us the equation of motion for
$\Psi=V^{\dagger}U^{\dagger}\tilde{\Psi}$

$\displaystyle i\partial_{z}\Psi$
$\displaystyle=\begin{bmatrix}-\dot{\theta}&0&g_{a\gamma}B/2\sqrt{2}\\\
0&\dot{\theta}&g_{a\gamma}B/2\sqrt{2}\\\
g_{a\gamma}B/2\sqrt{2}&g_{a\gamma}B/2\sqrt{2}&\Delta
m_{a\gamma}^{2}/2\omega\end{bmatrix}\Psi\,.$ (7)

In this basis, we observe that a resonance can happen when
$|\dot{\theta}|=|\Delta m_{a\gamma}^{2}|/2\omega$, which leads to an amplified
conversion probability. We denote this resonance as the axion magnetic
resonance (AMR).

In the presence of a constant $\dot{\theta}$, and under the limit of
$m_{a}^{2}/2\omega\gg g_{a\gamma}B$, the $a$-to-$\gamma$ conversion
probability is given by 222This is an approximation in the small-mixing limit,
so the unitarity is preserved up to the order of $g_{a\gamma}B/(\Delta
m_{a\gamma}^{2}/2\omega)$ level.

$\displaystyle P_{a{\rightarrow}\gamma}$
$\displaystyle\simeq\sum_{i=\pm}\frac{(g_{a\gamma}B/\sqrt{2})^{2}}{\Delta_{i}^{2}}\sin^{2}\left(\frac{\Delta_{i}l}{2}\right)\,$
(8)

with

$\displaystyle\Delta_{\pm}=\sqrt{\left(\Delta
m_{a\gamma}^{2}/2\omega\pm\dot{\theta}\right)^{2}+\left(g_{a\gamma}B/\sqrt{2}\right)^{2}}\,.$
(9)

Interestingly, $\pm\dot{\theta}$ can compensate the $\Delta
m_{a\gamma}^{2}/2\omega$ term. Such a cancellation can enhance the oscillation
amplitude or bring the conversion back to the linear regime, i.e.,
$\Delta_{i}l\ll 1$. In both cases, it lifts the $m_{a}^{-4}$ mass suppression.
In Fig. 1, we show the numerical analysis of Eq. (2) with a rotating magnetic
field and compare it with the conventional constant magnetic field setup
(i.e., $\dot{\theta}=0$). We find good agreement between the numerical results
and Eq. (8). In the limit $\dot{\theta}\rightarrow 0$, Eq. (8) approximately
reduces to the usual oscillation formula; see Appendix A for a brief review.
While there is a small difference in the wavenumber, it is relevant only in
the maximal-mixing non-linear regime, which the LSTW experiments never enter.

Figure 1: The axion-to-photon conversion probability with the following
benchmark: $m_{a}=10^{-3}\,\mathrm{eV}$,
$\omega=1.16\,\mathrm{eV}~{}(1064\,\mathrm{nm}$),
$g_{a\gamma}=10^{-9}\,\mathrm{GeV}^{-1}$, $B=5.3\,\mathrm{T}$. The magnetic
field is chosen to be constant (green), rotating as in Eq. (2) with angular
velocity $\dot{\theta}=m_{a}^{2}/2\omega$ (blue), and harmonic as in Eq. (10)
with the same angular velocity $\dot{\theta}=m_{a}^{2}/2\omega$ (orange). The
solid curves are solved numerically, while the dotted curves are from the
analytic expression Eq. (8) with the wave number modified according to Eq.
(9).

The $\gamma$-to-$a$ conversion is more subtle because it depends on the
initial polarization of the photon. Let us look at the $a\rightarrow\gamma$
process instead and utilize the CPT theorem. In the case of
$\dot{\theta}=\Delta m_{a\gamma}^{2}/2\omega$, ${\Psi=}[0~{}1~{}0]^{\rm T}$
will be the dominant photon component produced in the non-linear regime. After
rotating back with $U$ and $V$, it corresponds to
$[-i/\sqrt{2}~{}1/\sqrt{2}~{}0]^{\rm T}$ in the
$[\gamma_{\perp}~{}\gamma_{\parallel}~{}a]^{\rm T}$ lab frame. This is
expected since the axion field has no preferred direction. Heuristically, the
external magnetic field determines the polarization of the daughter photons
converted from axions. Since only the photons that are parallel to
$\mathbf{B}$ couple to axions, with a rotating $\mathbf{B}$, the polarization
vector of the signal photons corotates with $\mathbf{B}$. This feature could
potentially open up new search strategies. We leave a study of this to future
work.

Without modifying the linearly polarized laser setup of LSTW, the axion
production rate $P_{\gamma{\rightarrow}a}\simeq P_{a{\rightarrow}\gamma}/2$
holds in the non-linear regime, regardless of the initial photon polarization;
see Appendix B for more details. We report that
$P_{\gamma{\rightarrow}a}<P_{a{\rightarrow}\gamma}$ in the linear regime
unless we start with the initial polarization ${\Psi=}[0~{}1~{}0]^{\rm T}$, in
which case $P_{\gamma{\rightarrow}a}=P_{a{\rightarrow}\gamma}$. The dependence
on the initial photon polarization does not affect our main result since the
major improvement for ALPS II is in the non-linear regime.

As a cross check of the formalism, we provide a derivation using the approach
of parametric resonance in Appendix C and another heuristic approach by
changing of reference frame in Appendix D. All of these methods lead to
consistent resonance conditions.

Harmonic Magnetic Background  We now turn to a scenario where the orientation
is fixed but its amplitude varies along the propagation direction. Since the
$\gamma_{\perp}$ state perpendicular to the background magnetic field now
remains inert and unaffected by oscillations, we take the following two-
component EOM:

$\displaystyle i\frac{\partial}{\partial
z}\begin{bmatrix}\gamma_{\parallel}\\\ a\end{bmatrix}=\frac{1}{2\omega}H_{\rm
A}^{2}(\theta)\begin{bmatrix}\gamma_{\parallel}\\\ a\end{bmatrix}$ (10)
$\displaystyle=\frac{1}{2\omega}\begin{bmatrix}0&g_{a\gamma}\omega B\,{\rm
c}_{\theta}\\\ g_{a\gamma}\omega B\,{\rm c}_{\theta}&\Delta
m^{2}_{a\gamma}\end{bmatrix}\begin{bmatrix}\gamma_{\parallel}\\\
a\end{bmatrix}\,.$

The diagonal $\Delta m_{a\gamma}^{2}/2\omega$ term can be factored out by
$a\rightarrow e^{-i(\Delta m_{a\gamma}^{2}/2\omega)z}a$, then the equation
above is rewritten as

$\displaystyle i\frac{\partial}{\partial
z}\begin{bmatrix}\gamma_{\parallel}\\\ a\end{bmatrix}$ (11) $\displaystyle=$
$\displaystyle\begin{bmatrix}0&\frac{g_{a\gamma}B}{2}\,{\rm
c}_{\theta}\,e^{-i(\Delta m_{a\gamma}^{2}/2\omega)z}\\\
\frac{g_{a\gamma}B}{2}\,{\rm c}_{\theta}\,e^{i(\Delta
m_{a\gamma}^{2}/2\omega)z}&0\end{bmatrix}\begin{bmatrix}\gamma_{\parallel}\\\
a\end{bmatrix}\,.$

In the case where the frequency of $\theta$ variation exactly matches with
$\Delta m_{a\gamma}^{2}/2\omega$, denoted as $|\dot{\theta}|=\Delta
m_{a\gamma}^{2}/2\omega$, the off-diagonal term retains a constant value of
$g_{a\gamma}B/4$. This constant term accumulates the oscillation phase with
the additional factor of $1/2$ stemming from
$\cos\theta=(e^{i\theta}+e^{-i\theta})/2$; the other fast-oscillating term
with $2\theta$ would be cyclic-averaged to effectively vanish. This specific
condition corresponds to a parametric resonance with respect to the strength
of the background magnetic field. As in a rotating magnetic field profile, the
resonance phenomenon arises when the frequency of $\theta$ variation matches
the characteristic frequency associated with the a-$\gamma$ system.
Consequently, the oscillations experience significant amplification, leading
to an enhanced conversion probability. The conversion probability based on Eq.
(11) is compared with the numerical result in Fig. 1.

We show in Appendix H an alternative approach where the harmonic magnetic
profile is decomposed into two helical modes. We find the same result.

Figure 2: We show the ALPS II projection based on Ortiz:2020tgs ;
Bahre:2013ywa  in black. Two benchmarks for a rotating magnetic field are
achieved by fixing $\dot{\theta}=m_{a}^{2}/2\omega$ at $m_{a}=5\times
10^{-4}\,{\rm eV}$ (blue) and $m_{a}=10^{-3}\,{\rm eV}$ (orange). By scanning
over $\dot{\theta}$ we achieve the improved contour (green). Note that we
assume a linearly polarized photon beam. In the combined contour, the linear
regime $m_{a}\lesssim 10^{-4}\;\mathrm{eV}$ depends on the photon linear
polarization direction, while the higher mass region is $2^{3/4}$ higher than
the original ALPS II plateau regardless of the photon linear polarization
direction. For comparison, CAST CAST:2017uph  and globular cluster
Dolan:2022kul  are shown in grey dashed lines.

Experimental Applications  We take ALPS II as an example and show the amount
of improvement we expect at $m_{a}\gtrsim 10^{-4}~{}\mathrm{eV}$. We first
show in Fig. 2 the reach in $g_{a\gamma}$ by fixing the magnetic rotating
frequency $\dot{\theta}$ to two different values. Next, we show the combined
improvement if the magnet period is adjusted between $100~{}\mathrm{m}$ to
$1~{}\mathrm{m}$. The combined improved constraint on $g_{a\gamma}$ is given
by the green curve in Fig. 2.

All the above assumes a constant rotating frequency of the magnetic field.
Next, let us demonstrate the robustness of AMR against noises in the magnetic
frequency. We take the fluctuation of the rotation frequency to be within a
certain value of the central value, e.g. $1\;\%$, or $10\;\%$. More precisely,
we assume

$\displaystyle\dot{\theta}=\bar{\dot{\theta}}+\delta\dot{\theta}\,,$ (12)

where $\delta\equiv\delta\dot{\theta}/\bar{\dot{\theta}}$ is a Gaussian random
variable centered around zero, with a standard deviation of 0.01 or 0.1.

We take two benchmarks of the noise frequency, 28 MHz (corresponding to
$\delta$ change 10 times during the propagation of the 106 meter baseline) and
5.7 GHz (corresponding to 2000 changes of $\delta$). We approximate that
$\delta$ stays constant after each time it changes to a different value until
the next change. The values of $\delta$ in different time intervals are
uncorrelated. We refer to Appendix E for more discussion on the impact of the
irregularity of the helical profile. We make $100$ polls of the Gaussian
distribution of $\delta$ for each $m_{a}$ point. We show in Fig. 3 the
averaged contour of the repeated $100$ scans in $m_{a}$ as well as the
$2\,\sigma$ uncertainty band of the contour. We observe that great improvement
in ALPS II can still be achieved even with this very conservative assumption
on the regularity of the helical profile.

Figure 3: We add noises to the magnetic field rotation frequency. We make 100
scans and show the $2\,\sigma$ uncertainty of the contour due to low-frequency
noise (upper) and high-frequency noise (lower). For each case, we take the
Gaussian noise amplitude $\delta$ to be 1% (blue) and 10% (orange).

There are some proposals of using magnetic pulses for food preservation
zotero-21784  and medical treatment zotero-21782 ; zotero-21787 ; zotero-21789
that utilizes $\sim\mathcal{O}(T)$ field with frequency up to MHz.
Interestingly, the Relativistic Heavy Ion Collider (RHIC) at Brookhaven
National Laboratory (BNL) uses helical dipole magnets to collide polarized
protons beams zotero-21781 ; zotero-21778 ; zotero-21777 ; Anerella:2003se .
From Anerella:2003se , the technology of magnet assembly in 2003 reached
$4\,\mathrm{T}$ dipole field that rotates $2\pi$ in a length of
$2.4~{}\mathrm{m}$. The angular error of the rotation is $\sim 1.9^{\circ}$
per $360^{\circ}$, below the error level that we assume. From Fig. 3, we can
see that the rotation rate of $\dot{\theta}\sim 2\pi/(2.4\,\mathrm{m})$ allows
to cover a significant part of new parameter space, corresponding to the axion
mass around $10^{-3}\,$eV.

Since ALPS II utilizes an optical cavity to enhance both the laser power and
the detectivity, one may be worried about whether the enhancement we discuss
applies to the optical cavity photons. We show in Appendix F that this is
indeed the case: all the analysis can be carried over for the cavity photons.

Aside from a helical magnet profile design, we note that it is also possible
to achieve equivalent enhancement through the modulation of lasers. In
Appendix G we show extra methods of modulating the laser beam that can achieve
comparable enhancement.

Conclusion  In the small-mixing non-linear regime of axion-photon conversion,
the conversion probability is independent of the baseline length but
suppressed by the axion mass. Put differently, once the axion-photon
oscillation enters the non-linear regime, not the full length of the
conversion baseline is fully used. We propose an experimental setup to
alleviate this limitation with a spatially varying magnetic field. We showed
that one way to think about the effect of a varying magnetic field is that,
with a carefully chosen profile, its spacial variation can lower the axion
oscillation wave number, which increases the conversion amplitude (i.e. the
prefactor in Eq. (8)). As a result, it delays the onset of the nonlinear
regime and makes better use of the full baseline.

We use ALPS II projection as an example to show that, in the axion mass range
$m_{a}\gtrsim 10^{-4}\,{\rm eV}$ the sensitivity of the experiment can be
greatly extended. We show that if the magnetic field rotation frequency can be
maintained to be constant within $1\%$ error, this setup deepens the reach in
$g_{a\gamma}$ by about two orders of magnitude at
$m_{a}=10^{-3}\;\mathrm{eV}$. As the signal photons will be have a rotating
polarization vector in the rotating $\mathbf{B}$ setup, it can potentially
open up new experimental searching strategies. Although the scanning of axion
mass will require longer operation time, we stress that this has great
advantage over simply increasing the running time with a constant $\mathbf{B}$
field, where the reach in $g_{a\gamma}$ can only increase as (time)1/8 at
best.

We highlight the relation and difference with previous work VanBibber:1987rq ;
Jaeckel:2007gk ; Arias:2010bha ; Arias:2010bh ; Zarei:2019sva ;
Sharifian:2021vsg , where the magnetic field or the photon (electric field) is
manipulated to enhance the experimental sensitivity in the heavy axion regime,
in more detail in Appendix I. The code that reproduces all results are
released on GitHub.

Acknowledgement We would like to thank Daniele Alves for useful comments on
unitarity and the enhancement related to the during of the experiment; Michael
Graesser for the effect of an induced electric field; Marianne Francois, James
Colgan, and Christopher Lee for comments on the experimental setup; Weiyao Ke
for discussions on the helical magnet; Tomer Volansky for the access to Tel
Aviv Univeristy High-Performance Computing, where part of the work was
performed. We thank Gilad Perez, Andreas Ringwald, and Axel Lindner for useful
communications. HS is supported by the Deutsche Forschungsgemeinschaft under
Germany Excellence Strategy — EXC 2121 “Quantum Universe” — 390833306. This
work was supported by the U.S. Department of Energy through the Los Alamos
National Laboratory. Los Alamos National Laboratory is operated by Triad
National Security, LLC, for the National Nuclear Security Administration of
U.S. Department of Energy (Contract No. 89233218CNA000001). Research presented
in this article was supported by the Laboratory Directed Research and
Development program of Los Alamos National Laboratory under project number
3W200A-XXG900, task SUN00000. SY was supported by IBS under the project code,
IBS-R018-D1.

Supplemental Material

##  Appendix A Axion-Photon Dynamics in a Constant Magnetic Field

The equation of motion in the relativistic limit is given by
$i\partial_{z}\tilde{\Psi}=\frac{H^{2}}{2\omega}\tilde{\Psi}$. In components,
it reads Raffelt:1987im ; Mirizzi:2006zy ; Mirizzi:2009nq

$\displaystyle i\frac{\partial}{\partial z}\begin{bmatrix}\gamma_{\perp}\\\
\gamma_{\parallel}\\\ a\end{bmatrix}$
$\displaystyle=\frac{1}{2\omega}\begin{bmatrix}m_{\gamma}^{2}&0&0\\\
0&m_{\gamma}^{2}&g_{a\gamma}\omega B\\\ 0&g_{a\gamma}\omega
B&m_{a}^{2}\end{bmatrix}\begin{bmatrix}\gamma_{\perp}\\\ \gamma_{\parallel}\\\
a\end{bmatrix}\,.$ (13)

In the conventional setup of LSTW experiments, $\dot{\theta}=0$, the
$a$-to-$\gamma$ conversion probability is given by

$\displaystyle{P_{a{\rightarrow}\gamma_{\parallel}}}$
$\displaystyle=\frac{(g_{a\gamma}B)^{2}}{\Delta_{\rm
osc}^{2}}\sin^{2}\left(\frac{\Delta_{\rm osc}l}{2}\right)$ (14)
$\displaystyle\approx\begin{cases}(g_{a\gamma}Bl)^{2}/4&~{}~{}\Delta_{\rm
osc}l\ll 1\,,\\\ (g_{a\gamma}B)^{2}/2\Delta_{\rm osc}^{2}&~{}~{}\Delta_{\rm
osc}l\gg 1\,,\end{cases}$

where $\Delta_{\rm osc}=\sqrt{(\Delta
m_{a\gamma}^{2}/2\omega)^{2}+(g_{a\gamma}B)^{2}}$, and $l$ denotes the domain
length of the background magnetic field. Ignoring the photon refractive index
$m_{\gamma}=0$, the conversion probability becomes independent of the axion
mass if the domain length is sufficiently short to impose a small phase
retardation condition $\Delta_{\rm osc}l\ll 1$; this is dubbed as the “linear”
regime. On the other hand, when the oscillation length $2\pi\Delta_{\rm
osc}^{-1}$ is significantly shorter than the domain length as the so-called
“non-linear” regime, a rapid oscillation of conversion probability is averaged
out over fluctuations of the parameters, e.g., $\omega$ and $l$ , and the
conversion probability is controlled by the mixing angle, which is suppressed
by the axion mass squared in the small-mixing limit, $m_{a}^{2}/2\omega\gg
g_{a\gamma}B$.

##  Appendix B Initial Photon Polarization

The production of axions from photon initial states has a dependence on the
photon polarization. We further make some clarifications here and discuss the
applicability of Eq. (8).

Let us start with Eq. (7) and look at the case
$\dot{\theta}=m_{a}^{2}/2\omega$. We always assume a small axion-photon
mixing, $g_{a\gamma}B\ll m_{a}^{2}/\omega$. In the linear regime, we have
$z\,m_{a}^{2}/\omega\lesssim 1$. Therefore, the conversion amplitude between
$\Psi_{1}$ and $a$ is approximated by

$\displaystyle\mathcal{M}_{1a}$
$\displaystyle\approx\begin{bmatrix}0&0&1\end{bmatrix}\left(\mathbb{1}-i\begin{bmatrix}-\dot{\theta}&0&g_{a\gamma}B/2\sqrt{2}\\\
0&\dot{\theta}&g_{a\gamma}B/2\sqrt{2}\\\
g_{a\gamma}B/2\sqrt{2}&g_{a\gamma}B/2\sqrt{2}&\dot{\theta}\end{bmatrix}{z}\right)\begin{bmatrix}1\\\
0\\\ 0\end{bmatrix}\approx\frac{-i}{2\sqrt{2}}g_{a\gamma}B\,z.$ (15)
$\displaystyle\mathcal{M}_{2a}$
$\displaystyle\approx\begin{bmatrix}0&0&1\end{bmatrix}\left(\mathbb{1}-i\begin{bmatrix}-\dot{\theta}&0&g_{a\gamma}B/2\sqrt{2}\\\
0&\dot{\theta}&g_{a\gamma}B/2\sqrt{2}\\\
g_{a\gamma}B/2\sqrt{2}&g_{a\gamma}B/2\sqrt{2}&\dot{\theta}\end{bmatrix}{z}\right)\begin{bmatrix}0\\\
1\\\ 0\end{bmatrix}\approx\frac{-i}{2\sqrt{2}}g_{a\gamma}B\,z.$ (16)

If we start with a linear initial state $[1~{}0~{}0]^{T}$ in the
$[\gamma_{\perp}~{}\gamma_{\parallel}~{}a]^{T}$ lab frame, after the $V$
rotation as shown in Eq. (6), we have
$\Psi=[-i/\sqrt{2}~{}i/\sqrt{2}~{}0]^{T}$. The amplitudes between $\Psi_{1}$
component converting to axion and $\Psi_{2}$ to axion will interfere
destructively, as $\mathcal{M}_{1a}/\sqrt{2}-\mathcal{M}_{2a}/\sqrt{2}\approx
0$. This is indeed what we observe numerically.

If, on the other hand, we start with $[0~{}1~{}0]^{T}$ initial state lab
frame, we will overcome this destructive interference. In the rotating basis
we have $\Psi=[1/\sqrt{2}~{}1/\sqrt{2}~{}0]^{T}$. The axion appearance
probability is given by $P_{\gamma
a}=|\mathcal{M}_{1a}/\sqrt{2}+\mathcal{M}_{2a}/\sqrt{2}|^{2}\approx\frac{1}{4}(g_{a\gamma}Bz)^{2}$,
which is the same as Eq. (8) in the linear regime.

In the nonlinear regime, the two amplitudes have very different magnitudes and
phases. In our example, $\mathcal{M}_{1a}\sim g_{a\gamma}B\omega/\Delta
m_{a\gamma}^{2}$, while $\mathcal{M}_{2a}\sim\mathcal{O}(1)$
$[i\sin\left(g_{a\gamma}Bz/2\sqrt{2}\right)]$ . Therefore, the aforementioned
two choices in the initial states will lead to the same axion production rate
$P_{\gamma a}\approx|\mathcal{M}_{2a}/\sqrt{2}|^{2}\approx P_{a\gamma}/2$.

This is of relevance to the experimental setup as we show that the dependence
of the axion production only relies on the initial photon polarization in the
linear regime. Since our main improvement is the non-linear regime – for ALPS
II this corresponds to $m_{a}\gtrsim 10^{-4}$ – the result is not affected by
the initial photon polarization relative to the magnetic field direction.

##  Appendix C Alternative Approach – Parametric Resonance

We show that the AMR with a helical magnetic field can also be understood
using the parametric resonance language. Let us again start from the EOM:

$\displaystyle
i\partial_{z}\tilde{\Psi}=\frac{1}{2\omega}H^{2}(\theta)\tilde{\Psi}$ (17)
$\displaystyle=\frac{1}{2\omega}\begin{bmatrix}0&0&g_{a\gamma}\omega B\,{\rm
s}_{\theta}\\\ 0&0&g_{a\gamma}\omega B\,{\rm c}_{\theta}\\\ g_{a\gamma}\omega
B\,{\rm s}_{\theta}&g_{a\gamma}\omega B\,{\rm c}_{\theta}&\Delta
m^{2}_{a\gamma}\end{bmatrix}\begin{bmatrix}\gamma_{\perp}\\\
\gamma_{\parallel}\\\ a\end{bmatrix},$

After factorizing out the fast oscillation of $a$, we get

$\displaystyle i\partial_{z}\tilde{\Psi}$
$\displaystyle\approx\frac{1}{2\omega}\begin{bmatrix}0&0&g_{a\gamma}\omega
B\,{\rm s}_{\theta}\,e^{{-}i(\Delta m_{a\gamma}^{2}/2\omega)z}\\\
0&0&g_{a\gamma}\omega B\,{\rm c}_{\theta}\,e^{{-}i(\Delta
m_{a\gamma}^{2}/2\omega)z}\\\ g_{a\gamma}\omega B\,{\rm
s}_{\theta}\,e^{{+}i(\Delta m_{a\gamma}^{2}/2\omega)z}&g_{a\gamma}\omega
B\,{\rm c}_{\theta}\,e^{{+}i(\Delta
m_{a\gamma}^{2}/2\omega)z}&0\end{bmatrix}\begin{bmatrix}\gamma_{\perp}\\\
\gamma_{\parallel}\\\ a\end{bmatrix}.$

The off-diagonal term has an oscillation that is faster than the $\gamma-a$
mixing caused by $g_{a\gamma}\omega B$, where it can be cycle-averaged to
zero, unless $\dot{\theta}=\pm\Delta m_{a\gamma}^{2}/2\omega$, in which case
we have a constant mixing term.

##  Appendix D Alternative Approach – Dispersion Relation and Reference Frame

Figure 4: Cartoon demonstrating the observed rotation of the photon
polarization vector in the inertial lab frame (upper panel) and a non-inertial
frame that rotates with frequency $\dot{\theta}$ with respect to the lab
(lower panel). The colored arrows represent the polarization vector. The blue
ones are from the snapshot taken at $t=0$, red ones are the snapshot at
$t=\Delta t$. In the lower panel, because the reference frame is rotating in
the same direction as the polarization vector, the observed phase after the
same amount of time $\Delta t$ is smaller, as represented by the angle between
the red and blue arrows. However, since there the phase dependence in the $z$
direction is not affected, one can find the same phase at $z=0$ and
$z=L_{0}(=\lambda)$ regardless of the reference frame.

The key point of the axion magnetic resonance is to modify the photon
dispersion relation effectively such that it matches the axion dispersion
relation. Let us verify this using a more heuristic approach. Since
$\dot{\theta}\ll\omega$, we take the classical picture to illustrate the
effect.

We start with the inertial frame of the lab. Inside a vacuum, the photon
dispersion relation tells us $\omega=k$. Without loss of generality, let us
look at circularly polarized photons that have helicity $-1$. The polarization
vector of the gauge field at each point rotates with the frequency $\omega$.

Suppose the magnetic field rotates in the same direction as the photon
helicity with a frequency of $\dot{\theta}$. Now, let us go to the frame where
the magnetic field constantly points to the $y$ direction. In this frame, an
observer will find the photons with a frequency
$\omega^{\prime}=\omega-\dot{\theta}$, while $k$ stays the same.

This can be understood as follows. At each point, because the observer is
rotating in the same direction as does the photon polarization vector, to the
observer the rotation becomes slower, i.e. it takes longer time to complete
the rotation of $2\pi$. As a result, $\omega^{\prime}<\omega$. On the other
hand, since there is no transformation in the $z$ direction, one can still
find the same distance between nearest points where the same phase is shared,
regardless of being in the rotating frame or the lab frame, i.e.
$k^{\prime}=k$. A cartoon demonstrating this point is shown in Fig. 4.

One might get the impression that only a mere change of reference frame would
do the job while no rotating magnetic field is needed. That is not the case.
Let us go back to the static magnetic field along the $y$ axis of the lab
frame. If we repeat the change of frame without an actual rotating magnetic
field, Eqs. (4)-(6) gives us

$\displaystyle i\partial_{z}\Psi$
$\displaystyle=\begin{bmatrix}-\dot{\theta}&0&\frac{g_{a\gamma}B}{2\sqrt{2}}\mathrm{e}^{+i\theta}\\\
0&\dot{\theta}&\frac{g_{a\gamma}B}{2\sqrt{2}}\mathrm{e}^{-i\theta}\\\
\frac{g_{a\gamma}B}{2\sqrt{2}}\mathrm{e}^{-i\theta}&\frac{g_{a\gamma}B}{2\sqrt{2}}\mathrm{e}^{+i\theta}&\frac{\Delta
m_{a\gamma}^{2}}{2\omega}\end{bmatrix}\Psi\,.$ (18)

After factorizing out the phase $\mathrm{e}^{i\dot{\theta}\,z}$,
$\mathrm{e}^{-i\dot{\theta}\,z}$, and $\mathrm{e}^{-i\Delta
m^{2}_{a\gamma}/2\omega}$, the mixing term only has an oscillation of
frequency $\Delta m^{2}_{a\gamma}/2\omega$:

$\displaystyle i\partial_{z}\Psi$
$\displaystyle=\begin{bmatrix}0&0&\frac{g_{a\gamma}B}{2\sqrt{2}}\mathrm{e}^{-i\frac{\Delta
m_{a\gamma}^{2}}{2\omega}z}\\\
0&0&\frac{g_{a\gamma}B}{2\sqrt{2}}\mathrm{e}^{-i\frac{\Delta
m_{a\gamma}^{2}}{2\omega}z}\\\
\frac{g_{a\gamma}B}{2\sqrt{2}}\mathrm{e}^{+i\frac{\Delta
m_{a\gamma}^{2}}{2\omega}z}&\frac{g_{a\gamma}B}{2\sqrt{2}}\mathrm{e}^{+i\frac{\Delta
m_{a\gamma}^{2}}{2\omega}z}&0\end{bmatrix}\Psi\,,$ (19)

which is independent of $\dot{\theta}$, the parameter we used for the change
of reference frame that otherwise does not correspond to any physical
quantities. In the small-mixing non-linear regime, i.e. $g_{a\gamma}B\ll\Delta
m_{a\gamma}^{2}/2\omega$, the mixing term is averaged out, which leads to the
usual suppression of the conversion probability.

Put differently, the rotating magnetic field chooses a preferred frame, in
which the photon dispersion relation is modified to be the same as that of the
axion. This is somewhat similar to the method we used in Eq. (4), where
choosing a given helical profile $\theta(z)$ is equivalent to fixing the
gauge.

##  Appendix E Noise in the Magnetic Helical Profile

We quantify how the noise impacts the sensitivity reach in this section. Let
us define the frequency of the noise as

$\displaystyle\omega_{\delta}$
$\displaystyle\equiv\frac{1}{\delta\dot{\theta}}\frac{d\delta\dot{\theta}}{dz}.$
(20)

When the noise frequency is very low, i.e.
$\omega_{\delta}\ll(100~{}\mathrm{m})^{-1}$, $\dot{\theta}$ can be treated as
constant throughout the propagation of most of the whole baseline: the matched
$m_{a}$ is simply not the one we expect,
$m_{a}^{2}=2\omega\bar{\dot{\theta}}$, but rather
$m_{a}^{2}=2\omega(\bar{\dot{\theta}}+\delta\dot{\theta})$. The reach in
$g_{a\gamma}$ is otherwise not impacted.

On the other hand, when the noise is at high frequency, the effect is likely
to average out as we demonstrate in Fig. 5. Therefore, the most damaging noise
comes from $1\,\mathrm{m}\lesssim\omega_{\delta}^{-1}\lesssim
100\,\mathrm{m}$. Therefore, we show in Fig. 3 the reach of $g_{a\gamma}$ with
$1\,\%$ and $10\,\%$ irregularities in the helical profile with a noise
frequency given by $\omega_{\delta}L\approx 10$ and $\omega_{\delta}L\approx
2000$.

Figure 5: We take the benchmark point $m_{a}=0.001\,\mathrm{eV}$,
$\omega=1.16\,\mathrm{eV}~{}(1064\,\mathrm{nm}$),
$g_{a\gamma}=10^{-11}\,\mathrm{GeV}^{-1}$, $B=5.3\,\mathrm{T}$,
$z=106\,\mathrm{m}$. We include a 10 % fluctuation in $\dot{\theta}$. We show
the noise benchmarks with a fixed noise frequency 23 MHz (blue) and 5.7 GHz
(red). We show 100 runs of each point. Every red and black curve corresponds
to a realization of the Gaussian distributed $\dot{\theta}$ profile. We
include the conversion probability with a constant $B$ field (grey).

##  Appendix F The Optical Cavity

In the previous computation, we studied the laser beam directly shining
through the wall. In reality, ALPS II adopts an optical cavity Sikivie:2007qm
; Bahre:2013ywa ; Ortiz:2020tgs  to enhance the effective beam power. In this
section, we show that the AMR at $\dot{\theta}=m_{a}^{2}/2\omega$ persists in
the cavity setup.

The optical cavity is designed to impose a Dirichlet boundary condition such
that the photons with the following wave number are enhanced

$\displaystyle k$ $\displaystyle=\frac{2\pi}{\lambda}=\frac{n\pi}{L},$ (21)

where $L$ is the length of the cavity. It is straightforward to understand the
enhanced photon-to-axion conversion in the axion production cavity. For a
laser with a certain wavelength, say $\lambda=1064\,\mathrm{nm}$, we can tune
the cavity length by moving the mirrors such that $L=n\,\lambda/2$ for an
integer $n$. This will coherently enhance the wave from the laser power
$P_{0}$ to $P_{0}/\eta$, where $R=1-\eta$ is the reflectivity of the mirrors
Hoogeveen:1990vq . Other than this enhancement of photon power, all the
discussions above still hold.

The photon regeneration, on the other hand, is trickier. Let us start with an
incident axion beam

$\displaystyle a(x,t)={a_{0}}\sin(k_{a}z-\omega t)\,.$ (22)

Based on the context, we use $a(x,t)$ to denote the propagating field at
$(x,t)$ while ${a_{0}}$ is the amplitude of the wave. Note that due to the
cavity boundary condition only the following photons modes can stably exist:

$\displaystyle\vec{A}_{n}(z,t)=\vec{A}_{n}(t)\sin\left(\frac{n\pi}{L}z\right).$
(23)

We use $\vec{A}(z,t)$ and $\vec{A}(t)$ to denote the field and the amplitude
of the wave when there is no ambiguity. Let us quickly review the field
dynamics in a static magnetic field, $\mathbf{B}=[0~{}B_{0}~{}0]^{T}$.
Electromagnetic waves are sourced by the axion field with the following
equation of motion.

$\displaystyle\nabla\times\vec{B}-\partial_{t}\vec{E}=-g\vec{B}_{0}\partial_{t}a,$
(24)

where $B_{0}$ is the external magnetic field. With the assumption of no free
charge and under Coulomb gauge, this can be rewritten as Sikivie:2007qm

$\displaystyle(\partial_{t}^{2}+\gamma\partial_{t}+\omega_{n}^{2})A_{n,y}(t)\sin\left(\omega_{n}z\right)$
$\displaystyle=gB_{0}\omega{a_{0}}\cos(k_{a}z-\omega t),$ (25)

where $\omega_{n}=n\pi/L$. Since there is a power leakage from the mirror,
which causes the photon amplitude to decrease. This is effectively captured by
the friction term $\gamma$. Using the orthogonality of the basis, we get

$\displaystyle\left[\int_{0}^{L}dz\,\sin(\omega_{n}z)\sin\left(\omega_{n}z\right)\right](\partial_{t}^{2}+\gamma\partial_{t}+\omega_{n}^{2})A_{n,y}(t)$
(26)
$\displaystyle=gB_{0}\omega{a_{0}}\left[\int_{0}^{L}dz\,\sin(\omega_{n}z)\,\cos(k_{a}z-\omega
t)\right],$ (27)

which leads to

$\displaystyle(\partial_{t}^{2}+\gamma\partial_{t}+\omega_{n}^{2})A_{n,y}(t)$
(28)
$\displaystyle=gB_{0}\omega{a_{0}}\;\frac{2}{L}\left[\int_{0}^{L}dz\,\sin(\omega_{n}z)\,\cos(k_{a}z-\omega
t)\right]$ (29)
$\displaystyle=gB_{0}\omega{a_{0}}\;\frac{1}{L}\bigg{[}\int_{0}^{L}dz\,\sin((k_{a}+\omega_{n})z-\omega
t)$ (30)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\sin((-k_{a}+\omega_{n})z+\omega
t)\bigg{]}.$

Let us adjust the regeneration cavity that $\omega_{n}=\omega$. The last line
contains a fast-oscillating mode that averages out to zero and a slow-
oscillating mode. Finally, we find

$\displaystyle(\partial_{t}^{2}+\gamma\partial_{t}+\omega^{2})A_{n,y}(t)$ (31)
$\displaystyle\approx\frac{gB_{0}\omega{a_{0}}}{(k_{a}-\omega)L}\;[\cos(\omega
t-(k_{a}-\omega)L)-\cos(\omega t)]$ (32)
$\displaystyle=gB_{0}\omega{a_{0}}\left[\frac{2/L}{k_{a}-\omega}\sin(\frac{k_{a}-\omega}{2}L)\right]\;\sin(\omega
t-\frac{k_{a}-\omega}{2}L).$

This will result in the resonant enhancement imposed by $\gamma$ as described
in Ref. Sikivie:2007qm .

When the magnetic field is rotating, Eq. (25) is replaced by two coupled
equations

$\displaystyle(\partial_{t}^{2}+\gamma\partial_{t}+\omega_{n}^{2})A_{n,x}(t)\sin\left(\omega_{n}z\right)$
$\displaystyle=gB_{0}\sin(\dot{\theta}z)\omega{a_{0}}\cos(k_{a}z-\omega t),$
(33)
$\displaystyle(\partial_{t}^{2}+\gamma\partial_{t}+\omega_{n}^{2})A_{n,y}(t)\sin\left(\omega_{n}z\right)$
$\displaystyle=gB_{0}\cos(\dot{\theta}z)\omega{a_{0}}\cos(k_{a}z-\omega t),$
(34)

Going through a similar procedure, we reach the equation of motion for
$[A_{n,x}(t)~{}A_{n,y}(t)~{}0]^{T}$ as follows:

$\displaystyle(\partial_{t}^{2}+\gamma\partial_{t}+\omega^{2})A_{n,x}(t)$ (35)
$\displaystyle=\sum_{\alpha=\pm}\frac{\alpha}{2}gB_{0}\omega{a_{0}}\left[\frac{2/L}{q_{\alpha}}\sin(\frac{q_{\alpha}}{2}L)\right]\cos(\omega
t-\frac{q_{\alpha}}{2}L)$
$\displaystyle(\partial_{t}^{2}+\gamma\partial_{t}+\omega^{2})A_{n,y}(t)$ (36)
$\displaystyle=\sum_{\alpha=\pm}\frac{1}{2}gB_{0}\omega{a_{0}}\left[\frac{2/L}{q_{\alpha}}\sin(\frac{q_{\alpha}}{2}L)\right]\sin(\omega
t-\frac{q_{\alpha}}{2}L)\,,$

where

$\displaystyle q_{\pm}=k_{a}-\omega\pm\dot{\theta}\,.$ (37)

In the limit of $\dot{\theta}$ reducing to zero, it reduces to Eq. (31). In
the case $\dot{\theta}\approx\omega-k_{a}\approx m_{a}^{2}/2\omega$, only one
of the two source terms is significant. Since $A_{n,x}$ and $A_{n,y}$ differ
by $\pi/2$ phase during propagation, we only need to consider one component
for the purpose of computing the signal power. Taking $A_{n,y}$ as an example,
Eq. (36) is the same as Eq. (31), modulo a factor of two, after correcting the
wave number with $\dot{\theta}$. Therefore, we show that the resonance still
applies to the LSTW experiments that are enhanced by an optical cavity.

After this substitution, one can proceed to solve the axion-to-photon
conversion probability as in Sikivie:2007qm .

##  Appendix G Additional Experimental Setups

As we discussed in the main text, the enhancement of an LSTW experiment can
come from a harmonic magnetic field or a rotating magnetic field setup, both
varying either in space or in time. Aside from designing a specific magnetic
profile, we demonstrate similar enhancements that can originate from a
modulation of the laser.

Using the change of reference frame shown in Sec. D let us go to the rotating
frame where the magnetic field is constant. The laser that is linearly
polarized in the lab frame now has a polarization vector that rotates with
$\dot{\theta}$ frequency.

$\displaystyle\vec{A}(t,z)$
$\displaystyle=\begin{bmatrix}\cos(\dot{\theta}z)\\\ \sin(\dot{\theta}z)\\\
0\end{bmatrix}\mathrm{e}^{-i(\omega t-kz)}$ (38)
$\displaystyle=\frac{1}{2}\begin{bmatrix}1\\\ -i\\\
0\end{bmatrix}\mathrm{e}^{-i(\omega
t-(k+\dot{\theta})z)}+\frac{1}{2}\begin{bmatrix}1\\\ i\\\
0\end{bmatrix}\mathrm{e}^{-i(\omega t-(k-\dot{\theta})z)}.$

To mimic what the rotating magnetic field does, one can directly modify the
photon dispersion relation in the lab frame using a material with a refractive
index $n\sim 1+\dot{\theta}/\omega$, which we refer to as the frequency
modulation (FM) method.

In addition, a similar effect can be achieved by modulating the laser
amplitude with a frequency $\dot{\theta}$, which we refer to as the amplitude
modulation (AM) method. Suppose one prepares a laser as
$\vec{A}(t,z)=[\cos(\dot{\theta}t)~{}0~{}0]^{T}\mathrm{e}^{-i(\omega t-kz)}$.
This can be decomposed into two plane waves

$\displaystyle\vec{A}(t,z)$ $\displaystyle=\frac{1}{2}\begin{bmatrix}1\\\ 0\\\
0\end{bmatrix}\left[\mathrm{e}^{-i((\omega
t-(k-\dot{\theta})z)}+\mathrm{e}^{-i((\omega t-(k+\dot{\theta})z)}\right],$
(39)

where the dispersion of the photons is modified according to $\dot{\theta}$.

Lastly, let us also comment on why simply rotating the laser does not work.
One might get the impression that the relative rotation between the magnetic
field and the laser machine is the key. As a result, it is intriguing to think
of using an optical device to modulate the laser such that the laser machine
is effectively rotating while the magnetic field is constant. However, if one
rotates the laser machine, both the frequency of the photons and the
wavelength will be modified. Without introducing the complication due to the
ellipticity, let us assume the laser machine has a quarter wave plate so that
the photons it emits are circularly polarized. Suppose that, at some point in
time, the laser emitter outputs a photon whose polarization vector points to
the $x$ direction. The wavelength can be determined by the time interval after
which another photon pointing to $x$ direction is emitted. Admittedly,
rotating the laser machine can lower the frequency of the photons seen by an
observer in the lab, $\omega\rightarrow\omega-\dot{\theta}$, but it will also
increase the time interval one has to wait to find another photon that shares
the same phase, $k\rightarrow k-\dot{\theta}$. As a result, the photons’
dispersion relation is not modified by a rotating laser machine.

##  Appendix H Harmonic vs Helical

We can understand the resonant enhancement by a harmonic magnetic profile in
terms of two helical magnetic fields. We decompose the harmonic magnetic field
into two helical magnetic fields with opposite rotating directions. Let us
denote the Hamiltonian with a helical profile in Eq. (2) to be
$H_{E}^{2}(\theta;~{}g_{a\gamma}B,\Delta m_{a\gamma}^{2})$. We can decompose
$H_{A}^{2}$ into

$\displaystyle H_{A}^{2}=H_{E}^{2}(\theta;\frac{g_{a\gamma}B}{2},\Delta
m_{a\gamma}^{2})+H_{E}^{2}(-\theta;\frac{g_{a\gamma}B}{2},0),$ (40)

where we used the shorthand $\theta$ but both $H_{A}^{2}$ and $H_{E}^{2}$
depend on $\theta(z)$ explicitly. Now we can treat the first term the same way
as we did with Eq. (2). The resulted EOM reads

$\displaystyle i\partial_{z}\Psi$
$\displaystyle=\begin{bmatrix}-\dot{\theta}&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\\\
0&\dot{\theta}&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\\\
\frac{g_{a\gamma}B/2}{2\sqrt{2}}&\frac{g_{a\gamma}B/2}{2\sqrt{2}}&\frac{\Delta
m_{a\gamma}^{2}}{2\omega}\end{bmatrix}\Psi$ (41)
$\displaystyle+\begin{bmatrix}0&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{+i2\theta}\\\
0&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{-i2\theta}\\\
\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{-i2\theta}&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{+i2\theta}&0\end{bmatrix}\Psi\,.$

We can factorize out the phase $\mathrm{e}^{+i\theta}$,
$\mathrm{e}^{-i\theta}$, $\mathrm{e}^{-i\Delta m_{a\gamma}^{2}/2\omega}$,
after which the EOM reads

$\displaystyle i\partial_{z}\Psi$
$\displaystyle=\begin{bmatrix}0&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\mathrm{e}^{-i\Omega_{+}\,z}\\\
0&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\mathrm{e}^{-i\Omega_{-}\,z}\\\
\frac{g_{a\gamma}B/2}{2\sqrt{2}}\mathrm{e}^{+i\Omega_{+}\,z}&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\mathrm{e}^{+i\Omega_{-}\,z}&0\end{bmatrix}\Psi$
(42)
$\displaystyle+\begin{bmatrix}0&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{-i\Omega_{-}\,z}\\\
0&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{-i\Omega_{+}\,z}\\\
\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{+i\Omega_{-}\,z}&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{+i\Omega_{+}\,z}&0\end{bmatrix}\Psi\,,$
(43)

where $\Omega_{\pm}=\Delta m^{2}_{a\gamma}/2\omega\pm\dot{\theta}$. Without
loss of generality, we choose $\Omega_{-}=\Delta
m_{a\gamma}^{2}/2\omega-\dot{\theta}=0,\Omega_{+}=\Delta
m_{a\gamma}^{2}/\omega$. Therefore, the EOM can be rearranged as

$\displaystyle i\partial_{z}\Psi$
$\displaystyle=\begin{bmatrix}0&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\\\
0&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\\\
\frac{g_{a\gamma}B/2}{2\sqrt{2}}&\frac{g_{a\gamma}B/2}{2\sqrt{2}}&0\end{bmatrix}\Psi$
(44)
$\displaystyle+\begin{bmatrix}0&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{-i\Omega_{+}\,z}\\\
0&0&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{-i\Omega_{+}\,z}\\\
\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{+i\Omega_{+}\,z}&\frac{g_{a\gamma}B/2}{2\sqrt{2}}\,\mathrm{e}^{+i\Omega_{+}\,z}&0\end{bmatrix}\Psi\,,$

In the small-mixing nonlinear regime, we have $\Delta
m_{a\gamma}^{2}/2\omega\gg g_{a\gamma}B$. Therefore, the second term can be
safely averaged out. This is the same as Eq. (11) up to a $45^{\circ}$
rotation.

##  Appendix I Comparison with Previous Works

We highlight a few key differences between our work and previous works where
the variation of a magnetic field is suggested.

In VanBibber:1987rq , a segmented magnetic field configuration of alternating
polarity, the so-called the Wiggler, was suggested to scan the heavier axion
masses in the original paper. The paper cast the separated magnetic dipoles in
terms of a form factor and sketched out this configuration’s potential in
extending the reach of axion searches. In Arias:2010bha ; Arias:2010bh  more
systematic studies were performed for the alternating dipole design. In
addition, a magnet array with gaps in-between was studied. However, the
resonance was not identified in any of these papers nor were the optimal
magnetic configurations discussed. We note that the AMR studied in this Letter
is more generic. It not only explains the Wiggler/magnet-with-gap setup using
approaches different from the original work, but also points to the optimal
setup that can create the resonance, namely a magnetic field with a sinusoidal
modulation in either or both directions perpendicular to the photon
propagation direction. Besides, the AMR applies to a rotating/oscillating
magnetic field temporally/spatially, as well as more generic setups such as
the laser modulations as we show. We also discuss how the magnetic field
induced enhancement is related to the usual method, where the cavity is filled
with gas/plasma, in the context of the AMR formalism.

In Jaeckel:2007gk , an idea using phase-shift plates to extend the
experimental reach in the LSTW setup is also suggested instead of varying
magnetic fields. This modifies the photon fields directly. We note that the
phase modulation at the wave plate serves as an “external kick” that
introduces extra phase between the photon field and the external magnetic
field. This, in turn, can be understood using the resonance within the AMR
framework.

In the experiment PVLAS zotero-21715  a rotating magnetic field is used.
However, the rotation of the magnetic field in zotero-21715  is not to create
a resonance with the axion-photon momentum transfer but to reduce measurement
noise.

The work in Zarei:2019sva ; Sharifian:2021vsg  discusses how a varying
magnetic profile affects the vacuum birefringence searches, i.e. PVLAS. First
of all, our approach focuses on the luminosity of the photon-axion conversion,
hence is relevant to LSTW and solar experiments, while Zarei:2019sva ;
Sharifian:2021vsg  studies the axion-induced phase shift in the context of
PVLAS.

In addition, Zarei:2019sva ; Sharifian:2021vsg  studies the pulsed magnetic
field setup while we discussed both the pulsed (harmonic) setup and the
helical profile, which turns out to be even more effective than the harmonic
profile.

While the authors of Zarei:2019sva ; Sharifian:2021vsg  use second quantized
fields to compute the effect, we establish that a simple way to understand
that such a resonance can be achieved through the pure classical approach. In
addition, we verify it using different pictures including the avoided level
crossing (Eq. (6)), the parametric resonance (Eq. (C)), and the preferred non-
inertial reference frame (Fig. 4). We provide a simple way to understand the
resonance in the harmonic magnetic profile by decomposing it into two helical
profiles in Eq. (44).

We also demonstrate the technological feasibility of a helical magnetic
profile with the specification of the RHIC helical magnets.

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