KEK-QUP-2023-0029

KEK-TH-2568

KEK-Cosmo-0332

Quantum entanglement of ions for light dark matter detection

Asuka Ito1,2, Ryuichiro Kitano2,3, Wakutaka Nakano2 and Ryoto Takai2,3

1International Center for Quantum-field Measurement Systems for Studies of the
Universe and Particles (QUP), KEK, Tsukuba 305-0801, Japan
2KEK Theory Center, Tsukuba 305-0801, Japan
3The Graduate University for Advanced Studies, SOKENDAI, Tsukuba 305-0801,
Japan

###### Abstract

A detection scheme is explored for light dark matter, such as axion dark
matter or dark photon dark matter, using a Paul ion trap system. We first
demonstrate that a qubit, constructed from the ground and first excited states
of vibrational modes of ions in a Paul trap, can serve as an effective sensor
for weak electric fields due to its resonant excitation. As a consequence, a
Paul ion trap allows us to search for weak electric fields induced by light
dark matter with masses around the neV range. Furthermore, we illustrate that
an entangled qubit system involving $N$ ions can enhance the excitation rate
by a factor of $N^{2}$. The sensitivities of the Paul ion trap system to
axion-photon coupling and gauge kinetic mixing can reach previously unexplored
parameter space.

###### Contents

  1. 1 Introduction
  2. 2 Overview of trapped ions
  3. 3 Electric fields induced by light dark matter
  4. 4 Detecting light dark matter with single ion
  5. 5 Quantum enhancement with entangled qubits
  6. 6 Conclusion
  7. A Heating noises
  8. B Phase damping
  9. C Excitation of spin qubits in 171Yb+

##  1 Introduction

Ion traps are innovative technology that utilizes electromagnetic fields to
capture and control ions, in particular, which has been applied for the
implementation of qubits within a quantum computer. Several trap
configurations such as Penning [1, 2] and Paul traps [3] have been proposed
and developed toward the achievement of quantum operation with high fidelity.
Interestingly, such quantum operation, like quantum entanglement or squeezing,
enables us to detect very small electric or magnetic field even beyond the
standard quantum limit. Therefore, potentially, ion traps can be utilized as
quantum sensors for probing tiny signals from unexplored physics like dark
matter.

Dark matter is one of the unsolved puzzles in physics. The existence of dark
matter is supported through the gravitational interaction by various
observations such as the velocity distribution of the disc galaxies [4, 5],
bullet clusters of galaxies [6, 7], the cosmic microwave background [8], and
the large scale structure [8]. In addition to the gravitational interaction,
dark matter is expected to have interactions with the Standard Model
particles. For instance, the QCD axion was proposed originally by Peccei and
Quinn as the pseudo Nambu–Goldstone boson of broken U(1) symmetry to resolve
the strong CP problem [9, 10, 11, 12]. The QCD axion interacts with photons as

$-\frac{1}{4}g_{a\gamma}aF^{\mu\nu}\tilde{F}_{\mu\nu},$ (1)

where $a$ denotes the axion, $F^{\mu\nu}$ is the field strength tensor of the
electromagnetic field, $\tilde{F}_{\mu\nu}$ is its dual, and $g_{a\gamma}$
represents the coupling constant. Note that, for the QCD axion, $g_{a\gamma}$
is not an independent parameter but is related to its mass $m_{a}$:
${g_{a\gamma}}/{m_{a}}\sim(0.01\text{--}1)~{}\text{GeV}^{-2}$. This ratio
depends on the QCD axion models [13, 14, 15, 16]. One can also consider the
axion-like particle (ALP), which also has the interaction with photons as in
Eq. (1) with a coupling strength independent from its mass, while ALPs would
not provide a resolution of the strong CP problem. Both of the QCD axion and
ALP can be good dark matter candidates. In this paper, we will use the term
“axion” to encompass both QCD axion and ALP for the sake of simplicity. The
dark photon, which is a hidden U(1) gauge field, is also a candidate for dark
matter. It can mix with the Standard Model U(1) gauge field, i.e. photon,
through the kinetic mixing term [17]

$\frac{\epsilon}{2}F^{\mu\nu}F^{\prime}_{\mu\nu},$ (2)

where $\epsilon$ is the mixing parameter and $F^{\prime}_{\mu\nu}$ denotes the
field strength tensor of the dark photon. There have been many works to search
for axion and dark photon through the coupling in Eq. (1) and in Eq. (2),
respectively, and they have searched for wide regions in the parameter space.
However, still large parameter space is remained unexplored [18, 19].

In this paper, we explore the possibility of probing axion dark matter and
dark photon dark matter with quantum sensing of Paul traps. Quantum sensing is
known to be useful in searching for light dark matter such as axion dark
matter and dark photon dark matter, and there have been various recent
proposals [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. A trapped ion
in a Paul trap has the qubits consist of its spin states and of its
vibrational states, independently. Utilizing this property, we point out that
the ion trap qubit system enables the enhancement of the signal from light
dark matter through qubit operations. As a consequence, it turns out that Paul
traps can be outstanding quantum sensors for light dark matter detection,
which allows us to probe previously unexplored parameter space of axion-photon
coupling and gauge kinetic mixing.

The axion and dark photon detection by a Penning trap has been discussed in
Ref. [20]. There, two dimensional crystal configurations of ions are used. It
has been discussed that the use of the entanglement between the vibrational
and spin states provides quantum enhancement of the signal. Our discussion is
closely related, but we discuss more on Paul trap configurations with which
various qubit operations have already been realized with a good fidelity.
Which scheme is better depends on technical difficulties in realizing the set-
up, such as boundary conditions of the electric field as well as the size and
strength of the magnetic fields. We do not try to compare the sensitivities,
while we discuss how quantum manipulations of qubits with high fidelity are
useful in the experiments.

This paper is organized as follows. The basis of a Paul trap system is
reviewed briefly in Sec. 2. In Sec. 3, we explain that electric fields are
induced by axion dark matter or dark photon dark matter. In Sec. 4, we show
that the small electric field induced by the light dark matter can be detected
with single ion in a Paul trap. In Sec. 5, the operation for amplifying the
signal of the small electric field caused by the light dark matter in Paul
traps with the use of entangled multiple ions is discussed, and conclusion is
given in Sec. 6. In Appendix A, effects of heating noises on a vibrational
qubit is evaluated. Appendix B discusses the effect of the decoherence in the
$N$-ion set-up. The potential use of spin qubits, especially in 171Yb+,
instead of vibrational qubits for light dark matter search is studied in
Appendix C.

##  2 Overview of trapped ions

Quantum computers utilize qubits for computation, which is similar to how
classical computers use bits. However, they differ significantly in that
qubits can exist in superposition states. This phenomenon is unique to quantum
mechanics. Various physical platforms have been proposed, such as
superconductors, photons, and trapped ions [33], to implement quantum systems
with two states that can blend and interact with each other. Indeed, quantum
computers have been realized using techniques like the Paul trap [34] or the
Penning trap [35], following the foundational concept of trapped ion quantum
computers proposed in Ref. [36].

This paper focuses on the Paul trap system. In this setup, single ionized
atoms, such as beryllium, calcium, and ytterbium, are particularly
advantageous due to their ease of ionization. The electronic states serve as
the basis for computational qubits, referred to as optical, fine structure,
hyperfine, and Zeeman qubits, depending on the selected two levels [37]. We
refer to such two level states as a spin qubit regardless of its type for the
sake of simplicity, and represent the ground and excited states as $\ket{g}$
and $\ket{e}$, respectively. A distinctive feature of Paul ion traps is the
utilization of quantum oscillations of ions. In the Paul trap system, ions are
confined using both static and oscillating electric fields within an ultra-
high vacuum environment. As a result, a quantum system of coupled harmonic
oscillators is realized by the chain of ions. The ions are aligned with an
approximate interval of $\mathcal{O}(1\text{--}10)\ \mu\text{m}$ by balance
between the Coulomb force and the externally applied electric fields. Let us
consider $z$-axis as the alignment direction for the $N$ ions, along which
their oscillations are employed. Following laser cooling, the ions exhibit
behavior resembling quantum harmonic oscillators. Interactions between ions
are facilitated through the Coulomb force, necessitating the consideration of
collective vibrational degrees of freedom. Specifically, only two states come
into play as a qubit: either all $N$ ions oscillate collectively ($\ket{n=1}$,
that is a superposition of the first excited state for one of $N$ oscillators)
or they are all at the ground state ($\ket{n=0}$). We call the two level
states a vibrational qubit in this paper.

Let us illustrate the Paul trap system more specifically. To this end, we
first consider the single ion case for simplicity. The Hamiltonian is

$H_{0}=\omega_{z}a^{\dagger}a+\frac{\omega_{0}}{2}\sigma_{z}.$ (3)

Here $\omega_{z}$ is the angular frequency of the vibrational mode,
$\omega_{0}$ is the energy gap between the two states $|g\rangle$ and
$|e\rangle$, and $\sigma_{z}$ is the $z$ component of the Pauli matrix to act
on the spin state vector. Note that usually there is a hierarchy between
$\omega_{z}\sim 10$ MHz and $\omega_{0}\sim 10$ GHz. The four levels
$\ket{g,0}$, $\ket{g,1}$, $\ket{e,0}$, and $\ket{e,1}$ are utilized for the
implementation. We set $a^{\dagger}$ and $a$ as the ladder operators of the
vibrational mode and $\sigma_{+}$ and $\sigma_{-}$ as the ladder operators of
the spin qubit, such as $a^{\dagger}\sigma_{-}\ket{e,0}=\ket{g,1}$. The
operation controlling the ion states is performed by applying lasers with
frequencies suitable for each ion species. In the interaction picture, the
general Hamiltonian which describes the interaction between an ion and a laser
with an angular frequency $\omega$ and phase is $\phi$ [38]

$H_{\text{laser}}=\frac{1}{2}\Omega\sigma_{+}e^{-i(\omega-\omega_{0})t+i\phi}+\frac{i}{2}\eta\,\Omega\,a^{\dagger}\left(\sigma_{+}e^{-i(\omega-\omega_{0}-\omega_{z})t+i\phi}-\sigma_{-}e^{i(\omega-\omega_{0}+\omega_{z})t-i\phi}\right)+\text{h.c.},$
(4)

where $\Omega$ is the Rabi frequency, and $\eta$ is the Lamb-Dicke parameter,
which is typically sufficiently small such as of
$\mathcal{O}(0.01\text{--}0.1)$ and thus we omit the higher order terms of
$\eta$ in the Hamiltonian [39]. When $\omega=\omega_{0}$, the first term takes
precedence due to resonance, called the carrier resonance. This phenomenon
facilitates the transition of the ion’s state between $\ket{g,n}$ and
$\ket{e,n}$ ($n=0,1$). Similarly, lasers with $\omega=\omega_{0}-\omega_{z}$
induce the red sideband resonance. The red sideband resonance is responsible
for the transition between $\ket{g,1}$ and $\ket{e,0}$, which is described as
the third term. Another condition, $\omega=\omega_{0}+\omega_{z}$, corresponds
to the blue sideband resonance. This resonance involves the mixing between
$\ket{g,0}$ and $\ket{e,1}$ and is expressed as the second term. These three
resonances enable all operations on the qubits, encompassing not only single
qubit gates but also multiple-qubit operations.

Extending the above discussion to multiple ions is straightforward. The state
for a chain of $N$ ions consists of the tensor product of $N$ individual spin
qubits and one vibrational qubit like $\ket{s_{1},s_{2},\dots,s_{N},n}$, where
$s_{j}=g,e$ and $n=0,1$ as explained above. The fact that all $N$ ions share
the vibrational qubit provides the basis for constructing entangled states, as
will be discussed in Sec. 5.

##  3 Electric fields induced by light dark matter

In this section, we explain that both of axion dark matter and dark photon
dark matter can generate electric fields in a Paul trap. The oscillating
electric fields can drive the excitation of the vibrational mode, and that can
be detected by coupling to the spin qubit. For axion dark matter, the electric
field is generated resonantly through the axion-photon coupling under an
external magnetic field [40]. For dark photon dark matter, the electric field
is produced resonantly through the gauge kinetic mixing. In both cases, the
point is that dark matter around $m_{X}=\mathcal{O}$(neV) behaves as the
classical wave since the number of particles contained in space with the size
of the de Broglie wavelength is much larger than unity. Moreover, the
distribution of dark matter is uniform within the de Broglie wavelength, which
is typically much longer than the size of a Paul trap system. Thus, one can
neglect the spatial dependence of the classical wave.

For the case of axion dark matter, the axion field is given by

$a(t)=a_{0}\cos(m_{a}t-\phi_{a}),$ (5)

where it oscillates with the amplitude $a_{0}$ and the frequency $m_{a}$.
$\phi_{a}$ is the phase of the axion dark matter field and is randomly
distributed. The phase does not change during the coherence time of the dark
matter field $T_{a}=2\pi/m_{a}v_{\text{DM}}^{2}$, where $v_{\text{DM}}\sim
10^{-3}$ is the relative speed of dark matter which is determined by the
virial velocity of our galaxy [30]. Also, since dark matter is non-
relativistic, i.e., $v_{\text{DM}}\ll 1$, the amplitude $a_{0}$ is related to
the local dark matter density as $\rho_{\text{DM}}=m_{a}^{2}a_{0}^{2}/2\sim
0.45~{}\text{GeV cm}^{-3}$. In a Paul trap, the system of qubits is under a
uniform external magnetic field $\vec{B}$ along the axis direction of a
cylinder with a radius of $R$ ($\ll 1/m_{a}$) [41]. Then, axion dark matter
can convert to electric field in the presence of the magnetic field. Taking
the direction of the applied magnetic field to be $z$-axis, the amplitude of
electric field induced by axion dark matter is [42]

$E_{a,z}=\epsilon_{a}\sqrt{2\rho_{\text{DM}}}\sin\left(m_{a}t-\phi_{a}\right),$
(6)

where

$\epsilon_{a}=\frac{B_{z}}{2}\frac{g_{a\gamma}}{m_{a}}(m_{a}R)^{2}\left[\left(\log\frac{m_{a}R}{2}+\gamma-\frac{1}{2}\right)^{2}+\left(\frac{\pi}{2}\right)^{2}\right]^{1/2},$
(7)

for $m_{a}R\ll 1$ and $\gamma=0.5772\dots$ is the Euler–Mascheroni constant.

The longitudinal mode of dark photon dark matter with mass of
$\mathcal{O}$(neV) also oscillates in time as the non-relativistic classical
field. The dark electric field component of dark photon dark matter is given
by

$\vec{E}^{\prime}(t)=\vec{E}^{\prime}_{0}\sin\left(m_{\text{DP}}t-\phi_{\text{DP}}\right),$
(8)

where $m_{\text{DP}}$ is the mass of dark photon dark matter and
$\phi_{\text{DP}}$ is the random phase. As in the case of axion dark matter,
the amplitude is related to the local dark matter density,
$\rho_{\text{DM}}=m_{\text{DP}}^{2}|\vec{E}^{\prime}_{0}|^{2}/2$, and the
coherence time is given by $T_{\rm DP}=2\pi/m_{\rm DP}v_{\text{DM}}^{2}$. The
dark electric field generates the electric field through the gauge kinetic
mixing (2). Especially, for comparison with Eq. (6), we pay attention to
$z$-component of the electric field induced by dark photon dark matter, that
is

$E_{\text{DP},z}=\epsilon_{\text{DP}}\sqrt{2\rho_{\text{DM}}}\sin\left(m_{\text{DP}}t-\phi_{\text{DP}}\right),$
(9)

where $\epsilon_{\text{DP}}=\epsilon\cos\theta$. $\theta\in[0,\pi]$ is the
angle between the polarization vector of the longitudinal mode and the
$z$-axis. In the derivation, we assumed that the vacuum chamber is made of
glass as is usual for linear Paul traps so that one do not need to put
boundary conditions for the electric and magnetic fields, which usually gives
rise to a suppression factor determined by the ratio between the size of the
chamber to the Compton wavelength of dark matter. On the other hand, there is
the suppression factor of $(m_{a}R)^{2}$ in Eq. (6) in the case of axion dark
matter. This is because axion dark matter can convert to the electric field
only in the region of the existence of an external magnetic field, which is
now parameterized by $R$. However, potentially, we expect that the region
where external magnetic field are applied can be extended independently from
the Paul trap system by using such as superconducting magnets. Then, one may
be able to identify $R$ as the size of the magnet, whose typical size is
around meters. Therefore, in the following discussion, we refer to $R$ as the
radius of cylindrical magnet.

In the next section, we will show how the electric field induced by the light
dark matter can be detected with Paul traps.

##  4 Detecting light dark matter with single ion

The vibrational qubit, introduced in Sec. 2, potentially works as a sensor to
detect small oscillating electric fields induced by light dark matter, namely
Eqs. (6) and (9). We demonstrate this by evaluating the excitation rate of the
vibrational qubit.

The interaction Hamiltonian between the vibrational qubit and electric field
induced by dark matter is given by

$\begin{split}H_{X}&=\frac{eE_{X,z}}{\sqrt{2m_{\text{ION}}\omega_{z}}}(a^{\dagger}e^{i\omega_{z}t}+ae^{-i\omega_{z}t})\\\
&=e\epsilon_{X}\sin(m_{X}t-\phi_{X})\sqrt{\frac{\rho_{\text{DM}}}{m_{\text{ION}}\omega_{z}}}(a^{\dagger}e^{i\omega_{z}t}+ae^{-i\omega_{z}t}),\end{split}$
(10)

where $X=a$ or DP represents the case of axion or dark photon and
$m_{\text{ION}}$ is the mass of the ion. This shows that the resonance occurs
at $\omega_{z}=m_{X}$. Then, using the rotational wave approximation, the time
evolution of the vibrational qubit system is described by the displacement
operator:

$\displaystyle D(\beta)=\exp\left(\beta a^{\dagger}-\beta^{*}a\right),$ (11)

where

$\beta=\alpha_{X}T,\quad\alpha_{X}=\frac{e\epsilon_{X}e^{i\phi_{X}}}{2}\sqrt{\frac{\rho_{\text{DM}}}{m_{\text{ION}}m_{X}}}.$
(12)

$T$ is time duration of the observation which needs to be within the coherence
time of light dark matter $T_{X}\sim 0.4~{}{\rm s}\times(10~{}{\rm
neV}/m_{X})$, where again $X=a$ or DP. Starting from $\ket{n=0}$, this
operator develops the vibrational state into

$D(\beta)\ket{0}=\cos|\beta|\ket{0}+\frac{\beta}{|\beta|}\sin|\beta|\ket{1}\simeq\ket{0}+\alpha_{X}T\ket{1},$
(13)

which gives the probability of the excitation, $|\alpha_{X}|^{2}T^{2}$.

The projection into the vibrational eigenstate can be performed by the
following sequence. First, we bring up to the vibtation state into the spin
state by a red sideband $\pi$ pulse, $c_{0}|g,0\rangle+c_{1}|g,1\rangle\to
c_{0}|g,0\rangle+c_{1}|e,0\rangle$. The spin state is then observed by the
protocol of the spin-qubit measurements such as by the observation of the
fluorescence after illuminating a laser to bring one of the state into some
short-lived state.

In reality, the vibrational qubit is disturbed by thermal photons mainly from
the electrodes [43]. Taking into account of the excitation of the vibrational
qubit by such thermal photons, as discussed in Appendix A, the total
probability of the excitation is given by

$P_{1}(T)=\dot{\bar{n}}T+|\alpha_{X}|^{2}T^{2}.$ (14)

The contribution of the noise increases linearly over the duration of
measurement $T$, and the coefficient $\dot{\bar{n}}$ is called a heating rate.

In addition, preparing the ground state of the vibrational qubit cannot be
performed perfectly. Such imperfection is represented by the density matrix
$\rho_{0}$ with the infidelity $\bar{\mathcal{F}}_{0}$ as

$1-\bar{\mathcal{F}}_{0}=\bra{g,0}\rho_{0}\ket{g,0}.$ (15)

If the preparation is perfect, the fidelity $1-\bar{\mathcal{F}}_{0}$ is equal
to one. We further assume that the density matrix of the initial state is
depolarized, i.e.,

$\rho_{0}=\left(1-\frac{4}{3}\bar{\mathcal{F}_{0}}\right)\ket{g,0}\bra{g,0}+\frac{\bar{\mathcal{F}_{0}}}{3}{\bm{1}},$
(16)

on average during observations. The cooling time for preparing the initial
state is expected to be $\mathcal{O}(100)\,\mu$s [44]. Again, the displacement
operator describes the time evolution of the density matrix due to the
electric field induced by light dark matter:

$\rho_{1}=D(\beta)\rho_{0}D^{\dagger}(\beta)=\left(1-\frac{4}{3}\bar{\mathcal{F}_{0}}\right)\left(\ket{g,0}+\beta
T\ket{g,1}\right)\left(\bra{g,0}+\bra{g,1}\beta^{*}\right)+\frac{\bar{\mathcal{F}_{0}}}{3}{\bm{1}}$
(17)

within $T_{X}$. Here we omitted the heating effect. One observes the signal of
light dark matter by projecting the time evolved density matrix onto the
excited vibrational qubit state, i.e., $\bra{g,1}\rho_{1}\ket{g,1}$. However,
the readout process also contains imperfection, which can be parameterized as

$P_{\mathcal{F}}(T)=(1-\bar{\mathcal{F}}_{1})\bra{g,1}\rho_{1}\ket{g,1}+(T\text{-independent
terms}),$ (18)

where $\bar{\mathcal{F}}_{1}$ represents the infidelity of readout. The
$T$-independent terms are $\mathcal{O}(\bar{\mathcal{F}}_{0,1})$.

From Eqs. (14), (17) and (18), we obtain the readout probability including the
heating effect and the infidelity as

$P(T)=P(0)+\mathcal{F}(\dot{\bar{n}}T+|\alpha_{X}|^{2}T^{2}),$ (19)

where $\mathcal{F}=(1-\mathcal{F}_{1})(1-4\mathcal{F}_{0}/3)$. It is reported
that the value of infidelity $\bar{\mathcal{F}}_{0}$ and
$\bar{\mathcal{F}}_{1}$ can reach around $10^{-4}$ for a single ion [37].
Therefore, $\mathcal{F}\approx 1$ and $P(0)\ll\dot{\bar{n}}T$ can be taken for
the calculation.

We observe the state after waiting for an observation time $T$ ($<T_{X},\
\dot{\bar{n}}^{-1}$) and repeat the observation by $N_{\rm shot}$ times, so
that the total time duration of the experiment is $T_{\rm total}\simeq N_{\rm
shot}T$. The number of signals induced by dark matter is given by $S=N_{\rm
shot}|\alpha_{X}|^{2}T^{2}$, while that due to noise is $B=N_{\rm
shot}\dot{\bar{n}}T$. By defining the sensitivity by $S/\sqrt{B}>1.645$ (95%
confidence level), we obtain the sensitivity to the electric field with an
angular frequency $\omega_{z}$ as

$E_{z}=3.6\ {\rm
nV/m}\times\left(\frac{\dot{\bar{n}}}{0.1~{}\text{s}^{-1}}\right)^{1/4}\left(\frac{T_{\text{total}}}{1~{}\text{day}}\right)^{-1/4}\left(\frac{T}{0.4~{}\text{s}}\right)^{-1/2}\left(\frac{m_{\text{ION}}}{37~{}\text{GeV}}\right)^{1/2}\left(\frac{\omega_{z}}{10~{}\text{neV}}\right)^{1/2}.$
(20)

Here we assumed the use of 40Ca+ ion, and ignored the term of $\log m_{a}R$ in
Eq. (7), which just gives an $\mathcal{O}(1)$ factor. Eq. (20) shows that Paul
trap systems with a single ion can be good sensors for very weak oscillating
electric fields at the level of nV/m. The reference value of the heating rate,
$0.1~{}\text{s}^{-1}$, is a few times better than the best ones reported in
Ref. [43]. Such a low heating rate is required by the condition of
$\dot{\bar{n}}T\ll 1$ if we take the waiting time $T$ to be maximal $T\sim
T_{X}$. The heating rate is generically smaller for larger trap sizes and
lower temperatures. For an actual experimental set-up, one should optimize the
size and temperature in order to maximize the sensitivity.

By using this sensor, Paul trap systems can be utilized for probing light dark
matter, which produce weak electric fields, namely Eqs. (6) and (9). Using
Eqs. (6), (9) and (20), sensitivities of a single ion system to the axion-
photon coupling and the gauge kinetic mixing are obtained as follows,

$\begin{split}g_{a\gamma}&=4.4\times
10^{-11}~{}\text{GeV}^{-1}\times\left(\frac{\dot{\bar{n}}}{0.1~{}\text{s}^{-1}}\right)^{1/4}\left(\frac{T_{\text{total}}}{1~{}\text{day}}\right)^{-1/4}\left(\frac{T}{0.4~{}\text{s}}\right)^{-1/2}\left(\frac{R}{3~{}\text{m}}\right)^{-2}\\\
&\quad\times\left(\frac{B_{z}}{100~{}\text{mT}}\right)^{-1}\left(\frac{m_{\text{ION}}}{37~{}\text{GeV}}\right)^{1/2}\left(\frac{m_{a}}{10~{}\text{neV}}\right)^{-1/2}\left(\frac{\rho_{\text{DM}}}{0.45~{}\text{GeV
cm}^{-3}}\right)^{-1/2}\end{split}$ (21)

and

$\begin{split}\epsilon&=6.4\times
10^{-12}\times\left(\frac{\dot{\bar{n}}}{0.1~{}\text{s}^{-1}}\right)^{1/4}\left(\frac{T_{\text{total}}}{1~{}\text{day}}\right)^{-1/4}\left(\frac{T}{0.4~{}\text{s}}\right)^{-1/2}\\\
&\quad\times\left(\frac{m_{\text{ION}}}{37~{}\text{GeV}}\right)^{1/2}\left(\frac{m_{\text{DP}}}{10~{}\text{neV}}\right)^{1/2}\left(\frac{\rho_{\text{DM}}}{0.45~{}\text{GeV
cm}^{-3}}\right)^{-1/2}\end{split}$ (22)

at 95 % confidence level, respectively. In deriving Eq. (22), we took average
on $\theta$. One can also scan the probed masses of light dark matter by
adjusting the resonance frequency of the vibrational mode. Potentially, the
resonance frequency may be tunable in the range of 100 kHz–10 MHz. The
obtained sensitivities to the axion-photon coupling and the gauge kinetic
mixing are depicted in Fig. * ‣ 2 and Fig. 3 in pink, respectively.

##  5 Quantum enhancement with entangled qubits

Paul ion traps are designed to be useful for creating quantum entanglement
among qubits, particularly for their implementation in quantum computers. In
this section, we demonstrate that a maximally entangled state of $N$
vibrational qubits, created through several quantum operations in a Paul trap,
enhances the excitation rate by a factor of $N^{2}$ compared to the single ion
case.

As was explained before, the time evolution of the vibrational qubit caused by
light dark matter is described by the displacement operator $D(\beta)$ defined
as Eq. (11). Here, $\beta=\beta_{\text{r}}+i\beta_{\text{i}}$ is an arbitrary
complex number small enough that its quadratic terms can be ignored. The
displacement operator can be rewritten as

$D(\beta)=\left(1-i\beta_{\text{r}}\sigma^{2}_{\text{vib}}\right)e^{i\beta_{\text{i}}\sigma^{1}_{\text{vib}}}=\left(1+i\beta_{\text{i}}\sigma^{1}_{\text{vib}}\right)e^{-i\beta_{\text{r}}\sigma^{2}_{\text{vib}}}$
(23)

by using the first and second Pauli matrices acting onto vibrational qubits
$\sigma^{1}_{\text{vib}}=a^{\dagger}+a$ and
$\sigma^{2}_{\text{vib}}=ia^{\dagger}-ia$.

We first outline a method for measuring the imaginary part,
$\beta_{\text{i}}$, whose flow is shown in Fig. 1.

Figure 1:  The figure illustrates the operational flow for utilizing an
entangled $N$-ion system to achieve the $N^{2}$ enhancement discussed in this
section.

The Mølmer–Sørenson (MS) gate realizes the transformation

$\ket{g,g,\dots,g,0}\to|\Psi_{\rm{GHZ}}\rangle=\frac{1}{\sqrt{2}}\left(\ket{g,g,\dots,g,0}+\ket{e,e,\dots,e,0}\right)$
(24)

by using the phase rotation operator only onto $\ket{e}$ [45]. In this state,
so-called the Greenberger–Horne–Zeilinger (GHZ) state, the $N$ spin qubits are
maximally entangled. On the other hand, to access the entanglement of $N$
vibrational qubits, the ions should be apart from each other so that the each
ion system is considered as an isolated system. After isolating the ions, the
ions do not share the vibrational modes any more. We then perform operations
to transform the GHZ state into

$\ket{\Psi_{2}}=\frac{1}{\sqrt{2}}\left[\left(\frac{\ket{g,0}+\ket{g,1}}{\sqrt{2}}\right)^{\otimes
N}+\left(\frac{\ket{g,0}-\ket{g,1}}{\sqrt{2}}\right)^{\otimes N}\right].$ (25)

This is done by acting the Hadamard gate for each ion,
$\ket{g,0}\to(\ket{g,0}+\ket{e,0})/\sqrt{2}$ and
$\ket{e,0}\to(\ket{g,0}-\ket{e,0})/\sqrt{2}$, and the red sideband resonance
to each ion to drive
$(\ket{g,0}\pm\ket{e,0})/\sqrt{2}\to(\ket{g,0}\pm\ket{g,1})/\sqrt{2}$. The
electric field induced by light dark matter acts on the state, yielding

$\begin{split}\ket{\Psi_{3}}=\frac{1}{\sqrt{2}}&\left[e^{iN\beta_{\text{i}}}\left(\frac{\ket{g,0}+\ket{g,1}}{\sqrt{2}}+\beta_{\text{r}}\frac{\ket{g,0}-\ket{g,1}}{\sqrt{2}}\right)^{\otimes
N}\right.\\\
&\left.+e^{-iN\beta_{\text{i}}}\left(\frac{\ket{g,0}-\ket{g,1}}{\sqrt{2}}+\beta_{\text{r}}\frac{\ket{g,0}+\ket{g,1}}{\sqrt{2}}\right)^{\otimes
N}\right].\end{split}$ (26)

Since the states $(\ket{g,0}\pm\ket{g,1})/\sqrt{2}$ are the eigenstates of
$\sigma^{1}_{\text{vib}}$ corresponding to the eigenvalues $\pm 1$, the new
phases $\exp(\pm iN\beta_{\text{i}})$ appear in each terms. By applying the
inverse operation of the red sideband transition and the Hadamard gate, one
obtains the state

$\ket{\Psi_{4}}=\frac{1}{\sqrt{2}}\left[e^{iN\beta_{\text{i}}}\big{(}\ket{g,0}+\beta_{\text{r}}\ket{e,0}\big{)}^{\otimes
N}+e^{-iN\beta_{\text{i}}}\big{(}\ket{e,0}+\beta_{\text{r}}\ket{g,0}\big{)}^{\otimes
N}\right].$ (27)

Note that we implicitly assumed that the coherence time of the system such as
the lifetime of the excited states of the qubits, i.e., $\ket{e}$ (usually
denoted as $T_{1}$) and the decoherence time of the entanglement ($T_{2}$),
were much longer than the coherence time of light dark matter $T_{X}$. We
discuss the effect of $T_{2}$ in Appendix B.

Since the $\beta_{\text{r}}$-dependent terms in Eq. (27) are orthogonal to the
$\beta_{\text{r}}$-independent terms, we can put them aside. We now couple all
$N$ ions again by bringing them back to the original positions, so that they
share the global vibrational qubit. The operations of the CNOT (Controlled-
NOT) gates between the first qubit and other qubits
$\text{CNOT}_{1,j}\,(j=2,\dots,N)$ bring the component of
$\ket{e,e,\dots,e,0}$ to $\ket{e,g,\dots,g,0}$ while $\ket{g,\dots,g,0}$
untouched. The final operation is the Hadamard gate only for the first qubit
which transfers $(\ket{g,g,\dots,g,0}\pm\ket{e,g,\dots,g,0})/\sqrt{2}$ into
$\ket{g,g,\dots,g,0}$ and $\ket{e,g,\dots,g,0}$, respectively, so that we
obtain

$\ket{\Psi_{5}}\approx\ket{g,g,g,\dots,g,0}-iN\beta_{\rm
i}\,\ket{e,g,g,\dots,g,0}+\cdots.$ (28)

The series of the operations brings back to the original $\ket{g,g,\dots,g,0}$
state if $\beta=0$. The signal of dark matter is the excitation of the first
qubit with the probability of

$|\braket{e,g,g,\dots,g,0}{\Psi_{5}}|^{2}=N^{2}\beta_{\text{i}}^{2}.$ (29)

It is amusing to find that the probability is enhanced by $N^{2}$ due to the
maximal entanglement of the GHZ state. Similarly, the signal of
$\beta_{\text{r}}$ can be detected with an amplification factor of $N^{2}$ by
following almost the same procedure by replacing $\ket{\Psi_{2}}$ by

$\ket{\Psi_{2^{\prime}}}=\frac{1}{\sqrt{2}}\left[\left(\frac{\ket{g,0}+i\ket{g,1}}{\sqrt{2}}\right)^{\otimes
N}+\left(\frac{\ket{g,0}-i\ket{g,1}}{\sqrt{2}}\right)^{\otimes N}\right].$
(30)

Note that this state is composed of the eigenstates of
$\sigma^{2}_{\text{vib}}$. In our situation the averages of the real and
imaginary parts of signal are expected to be the same, i.e.,
$\overline{\beta_{r}^{2}}=\overline{\beta_{i}^{2}}=|\alpha_{X}|^{2}T^{2}/2$.

Figure 2: Sensitivities at 95 % C.L. to the axion-photon coupling
$g_{a\gamma}$ as a function of the axion mass $m_{a}$ are shown. The pink line
represents the sensitivity using a single ion. The red and orange lines
represent the sensitivities using unentangled and entangled twenty ions,
respectively. The total observation time is set to 1 day and the frequency
$\omega_{z}$ is set as 10 neV in the estimation. The pink, red, and orange
dashed lines show the expected sensitivities for the scan of axion masses by
adjusting $\omega_{z}$. For comparison, limits by laboratory experiments are
shown; The black, dark green, blue, and purple regions are excluded by CAST
[46], SHAFT [47], ABRACADABRA [48], and BASE [49], respectively***While there
also exists astrophysical constraints on the axion-photon coupling [50, 51,
52, 53, 54], they are not shown in the figure because the results might be
affected by astrophysical uncertainties..

The signal enhancement with entangled ions relies on the assumption that light
dark matter excites the vibrational qubits spatially coherently. This
assumption is valid because light dark matter, whose coherence length is
$(m_{X}v_{\rm DM})^{-1}\sim 100$ km for $m_{X}=10$ neV, homogeneously interact
with the ions. However, whether the heating noise, discussed in Appendix A,
disturbs ions coherently or not depends on the specific configuration of a
Paul ion trap system. If the heating noise excites ions incoherently, the
excitation rate is merely $N$ times of that of a single ion. Therefore, the
significance of the signal scales as

$\frac{S}{\sqrt{B}}\propto N^{3/2}.$ (31)

In the case that the heating noise excites ions coherently, the excitation
rate becomes $N^{2}$ times of that of a single ion as well as the signal
amplification. Thus, the significance would be $S/\sqrt{B}\propto N$. In both
cases, it is evident that the use of entanglement is advantageous in detecting
the dark matter signals.

Figure 3: Sensitivities at 95 % C.L. to the mixing parameter $\epsilon$ as a
function of the dark photon mass $m_{\text{DP}}$ are shown. The pink line
represents the sensitivity using a single ion. The red and orange lines
represent the sensitivities using unentangled and entangled twenty ions,
respectively. The total observation time is set to 1 day and the frequency
$\omega_{z}$ is set as 10 neV in the estimation. The pink, red, and orange
dashed lines show the expected sensitivities for the scan of dark photon
masses by adjusting $\omega_{z}$. The two gray regions are excluded by the
cosmological observations [55, 56].

Improved sensitivities to $g_{a\gamma}$ and $\epsilon$ with the use of twenty
entangled qubits are depicted in Figs. * ‣ 2 and 3, respectively. In the
figures, the sensitivity curves depicted in orange represent the case that
ions are entangled and the heating noise interact with them incoherently. The
background due to heating noise may grow up with increasing the number of
qubits. It should be noted that the effect has to be kept small.

One needs to take into account of the fidelity of $N$-ion detector, which
usually tends to get worse compared to the single ion case. The typical value
of the fidelity of GHZ state for multi-ions is found in Ref. [41]. In the
estimation of the sensitivities in Figs. * ‣ 2 and 3, however, as in the case
of the single ion, we assumed that the infidelity for multiple ions is small
enough compared to the background due to the heating rate. Although high
fidelity could be achieved for multiple ions in a Paul trap [57], maintaining
it for a large number of ions (greater than $\sim$ 20) is challenging. The
situation could change when considering a network of connected Paul traps,
with each trap containing a single ion [58, 59, 60, 61].

##  6 Conclusion

We investigated the potential use of Paul traps as light dark matter
detectors. We first showed that a vibrational qubit of a single ion in the
Paul trap can serve as an effective sensor for electric fields. Therefore, it
can be utilized for probing weak electric fields induced by axion dark matter
and dark photon dark matter with masses in the range of
$\mathcal{O}(0.1\text{--}10)$ neV. Furthermore, in Sec. 5, we demonstrated the
scheme for improving the sensitivity by employing $N$ entangled vibrational
qubits to get closer to the Heisenberg limit. Using this strategy, the number
of dark matter signal can be enhanced by a factor of $N^{2}$. Consequently,
Paul traps can be good quantum detectors for investigating previously
unexplored parameter space of axion-photon coupling and gauge kinetic mixing,
as illustrated in Figs. * ‣ 2 and 3.

It should be mentioned that our sensitivity calculations rely on several
assumptions. First, the coherence time of the qubit system parameterized by
$T_{2}$ are assumed to be longer than the coherence time of light dark matter
$T_{X}$. This assumption would hold true if $T_{2}\gtrsim 5$ s for masses of
light dark matter considered in this paper ($\geq 0.1$ neV). The second
assumption is about the infidelity for $N$ multiple ions, which are assumed to
be much smaller than heating noise. This may remain valid for $N\simeq 20$
even when considering multiple ions in a Paul trap [57], while it may be
difficult for larger $N$. The situation could change when considering a
network connected Paul traps [58, 59, 60, 61], potentially enabling high
fidelity operations with a large number of ions in the near future. In
particular, ion traps have been extensively studied and developed for the
implementation of qubits within a quantum computer. It is intriguing to
investigate other potential applications of ion traps as proposed in this
paper.

## Acknowledgements

We would like to thank Mitsuhiro Yoshida and Shusei Kamioka for discussions.
We also would like to thank Utako Tanaka for providing us with useful
information on ion traps. This work was in part supported by World Premier
International Research Center Initiative (WPI), MEXT, Japan, and JSPS KAKENHI
Grant Numbers JP22K14034 (A.I.), JP23KJ2173 (W.N.), JP22K21350 (R.K.),
JP21H01086 (R.K.), and JP19H00689 (R.K.).

## Note added

While finalizing this manuscript, Ref. [62] appeared. There the authors
discuss the possible quantum enhancement of dark matter signals by using
entangled qubits in general quantum computing setup. The discussion is closely
related to the one in Sec. 5 of our paper.

##  Appendix A Heating noises

A vibrational qubit is disturbed by thermal photons mainly from the electrodes
[43]. In the interaction picture, the interaction Hamiltonian between the
vibrational mode of a single ion and the thermal photons is given by

$H_{\rm
heat}=\sum_{j}g_{j}\left(a^{\dagger}b_{j}e^{-i(\omega_{z}-\omega(k_{j}))t}+ab_{j}^{\dagger}e^{i(\omega_{z}-\omega(k_{j}))t}\right),$
(A.1)

where $b_{j}$ ($b_{j}^{\dagger}$) represents the annihilation (creation)
operator of the thermal photons with a momentum $k_{j}$, and $g_{j}$ denotes
the coupling strength. Then, the total Hamiltonian in the presence of light
dark matter is

$H_{{\rm
total}}=\omega_{z}a^{\dagger}a+\sum_{j}\omega(k_{j})b_{j}^{\dagger}b_{j}+H_{X}+H_{\rm
heat}.$ (A.2)

By using the quantum master equation for the density matrix of the vibrational
mode, which is obtained by tracing out the noise [63], following equations are
obtained:

$\displaystyle\frac{d}{dt}\braket{a^{\dagger}a}$
$\displaystyle=-\gamma\braket{a^{\dagger}a}-i|\alpha_{X}|e^{i\Delta
t}\braket{a^{\dagger}}+i|\alpha_{X}|e^{-i\Delta t}\braket{a}+\gamma\bar{N},$
$\displaystyle\frac{d}{dt}\braket{a}$ $\displaystyle=-i|\alpha_{X}|e^{i\Delta
t}-\frac{\gamma}{2}\braket{a},$ (A.3)
$\displaystyle\frac{d}{dt}\braket{a^{\dagger}}$
$\displaystyle=i|\alpha_{X}|e^{-i\Delta
t}-\frac{\gamma}{2}\braket{a^{\dagger}},\ $

where $\Delta=\omega_{z}-m_{X}$ is the detuning between the vibrational mode
and the light dark matter. $\bar{N}$ is the mean particle number of the
thermal photons at $\omega_{z}$ and
$\gamma=2\pi\sum_{j}\delta(\omega_{z}-\omega(k_{j}))g_{j}^{2}$. One can solve
the above equations with initial conditions,
$\braket{a^{\dagger}a}=\braket{a}=\braket{a^{\dagger}}=0$, and obtain

$\braket{a^{\dagger}a}=-\frac{8|\alpha_{X}|^{2}\cos\Delta
t}{4\Delta^{2}+\gamma^{2}}e^{-\gamma t/2}+\frac{4|\alpha_{X}|^{2}(1+e^{-\gamma
t})}{4\Delta^{2}+\gamma^{2}}+\bar{N}(1-e^{-\gamma t}).$ (A.4)

When $\gamma t<\Delta t\ll 1$, Eq.(A.4) can be approximated to

$\braket{a^{\dagger}a}\simeq\gamma\bar{N}t+|\alpha_{X}|^{2}t^{2}.$ (A.5)

One can define a heating rate $\dot{\bar{n}}=\gamma\bar{N}$. When
$\braket{a^{\dagger}a}\ll 1$, Eq.(A.5) can be regarded as the excitation rate
of the vibrational qubit by the light dark matter.

##  Appendix B Phase damping

The interaction between ions and environment causes the decoherence of
entangled qubits, namely, exponential damping of off-diagonal components of
density matrix. This phenomenon is referred to phase damping, and $T_{2}$ is
defined as the lifetime. It is impossible to remove the phase damping since
qubits can interact even with the vacuum environment, however, it is possible
to avoid its effect on signals up to its linear order. In this Appendix B, we
show that our method to extract maximally enhanced signals, explained in Sec.
5, conducts this mission in long $T_{2}$ limit.

The effect of phase damping without dissipation can be expressed by the phase
kick,

$R_{z}(\theta)=\begin{pmatrix}e^{-i\theta/2}&0\\\
0&e^{i\theta/2}\end{pmatrix}=\exp\left(-\frac{i\theta}{2}\sigma^{3}_{\rm
vib}\right)$ (B.1)

on the bases $\ket{0}$ and $\ket{1}$, where $\sigma^{3}_{\rm vib}$ is the
third Pauli matrix onto the vibrational qubit. The phase $\theta$ is normally
distributed with the average zero and the variance $2\delta$, which is related
to $T_{2}$ as $\delta\simeq T/T_{2}$. In the multi-ion case, each ion
$j=1,\dots,N$ is kicked with individual phase $\theta_{j}$ according to the
same distribution. As a result, the damping rate of coherence is multiplied up
to a factor of $N$. (If the damping effect acts to all qubits homogeneously,
the maximum multiplication factor can be $N^{2}$ [64].)

At the linear order of the signal $\beta$ and the kick $\theta$, these two
effects can be divided. Starting the maximally entangled state (25), the
density matrix including the time evolution by $\beta$ is

$\rho_{3}=\ket{\Psi_{3}}\bra{\Psi_{3}},$ (B.2)

which is in Eq. (26). Additionally the ions are kicked so that their state is
changed into

$\rho^{\prime}_{3}=\int^{+\infty}_{-\infty}\prod_{j}\left(\frac{{\rm
d}\theta_{j}}{\sqrt{4\pi\delta}}\
e^{-\theta_{j}^{2}/4\delta}\right)R_{z}(\theta)\rho_{3}R_{z}^{\dagger}(\theta),$
(B.3)

where $R_{z}(\theta)=R_{z}(\theta_{1})\otimes\cdots\otimes R_{z}(\theta_{N})$.
The state vector $R_{z}(\theta)\ket{\Psi_{3}}$ is given as the replacement
$\beta_{\rm r}\to\beta_{\rm r}-i\theta_{j}/2$ for the first term and
$\beta_{\rm r}\to\beta_{\rm r}+i\theta_{j}/2$ for the second term in Eq. (26),
with higher order terms neglected. Following the logic in Sec. 5, these terms
do not disturb the state which we collect, and then the final probability does
not change, assuming that $T_{2}$ is long enough.

There is also phase damping accompanying dissipation originating from factors
such as the Hamiltonian (A.1). In this case, the lifetime of coherence is
$T_{2}\simeq\dot{\bar{n}}^{-1}$. Furthermore, for multiple ions, the damping
rate of coherence would be multiplied up to a factor of $N$ for a spatially
incoherent noise and $N^{2}$ for a spatially coherent noise, as discussed
above. Therefore, it is essential to mitigate the heating noise in order to
realize the procedure explained in Sec. 5 with high fidelity. We note that
$\dot{\bar{n}}$ and $T_{2}$ are treated as independent parameters when
estimating sensitivities of entangled qubits to light dark matter in Fig. * ‣
2 and Fig. 3.

##  Appendix C Excitation of spin qubits in 171Yb+

In the main body, we focused only on the vibrational qubits and demonstrated
their ability as dark matter detectors. For comprehensive study of Paul traps
for light dark matter detection, let us also consider the excitation of spin
qubits due to the dark matter. Especially, we consider 171Yb+ ion whose spin
qubit is composed by hyperfine splitting in $6^{2}$S1/2. The ground state is
the spin singlet state and the excited state is one of the spin triplet state
as follows,

$\ket{g}=\frac{1}{\sqrt{2}}\left(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow}\right),\quad\ket{e}=\frac{1}{\sqrt{2}}\left(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow}\right),$
(C.1)

where the former and the latter spin symbols represent those of the outermost
electron and the ytterbium nucleus, respectively. The direct excitation from
$\ket{g}$ to $\ket{e}$ can be caused by applying magnetic fields. Notably, the
lifetime of $\ket{e}$ is considerably long $\sim 5\times 10^{11}$ s. For later
convenience, the spin up and down states of an electron and a nucleus are
taken to be the eigenstates of the $z$ component of the spin operators.

In general, axion can also couple to electron field $\psi$ as

$\mathcal{L}_{\rm int}=ig_{ae}a\bar{\psi}\gamma_{5}\psi,$ (C.2)

where $g_{ae}$ represents the coupling constant. Axion dark matter behaves as
an effective magnetic field through Eq. (C.2) by taking the non-relativistic
limit [65]:

$\vec{B}_{a}=\frac{g_{ae}}{e}\vec{v}_{\text{DM}}\sqrt{2\rho_{\text{DM}}}\sin\left(m_{a}t-\phi_{a}\right),$
(C.3)

which is coupling to the spin of electrons. On the other hand, in addition to
the dark electric field (8), the longitudinal mode of dark photon dark matter
also has the component of the dark magnetic field. It induces the magnetic
field through the gauge kinetic mixing (2):

$\vec{B}_{\text{DP}}=\epsilon\vec{v}_{\text{DM}}\times\vec{E}^{\prime}_{0}\sin\left(m_{\text{DP}}t-\phi_{\text{DP}}\right).$
(C.4)

Such magnetic fields from the light dark matter can interact with the spin of
electrons of ions,

$H_{s,X}=\frac{e}{m_{e}}\vec{B}_{X}\cdot\vec{S}_{e},$ (C.5)

where $X=a$ or DP. When the mass of the dark matter coincides with the energy
level difference between $\ket{g}$ and $\ket{e}$, that is 12.6 GHz for 171Yb+
ion, the dark matter induced magnetic fields resonantly excite the spin qubit
through the $z$ component. At the resonance point, the excitation probability
is approximately given by $(e|\vec{B}_{X}|T\cos\theta/4m_{e})^{2}$, where
$\theta$ is the angle between the directions of $\vec{B}_{X}$ and the
$z$-axis.

The observation of this excitation is contaminated by the imperfection of
preparing the ground state and reading out the signal, as discussed in Sec. 4.
Following the discussion, the sensitivity of the spin qubit to the light dark
matter can be evaluated as

$\begin{split}g_{ae}&=7.9\times
10^{-3}\times\left(\frac{\mathcal{F}}{1}\right)^{-1/2}\left(\frac{P(0)}{10^{-4}}\right)^{1/4}\left(\frac{T_{\text{total}}}{1~{}\text{day}}\right)^{-1/4}\\\
&\quad\times\left(\frac{T}{8\times
10^{-5}~{}\text{s}}\right)^{-3/4}\left(\frac{\rho_{\text{DM}}}{0.45~{}\text{GeV
cm}^{-3}}\right)^{-1/2}\left(\frac{v_{\text{DM}}}{10^{-3}}\right)^{-1}\end{split}$
(C.6)

and

$\begin{split}\epsilon&=3.2\times
10^{-2}\times\left(\frac{\mathcal{F}}{1}\right)^{-1/2}\left(\frac{P(0)}{10^{-4}}\right)^{1/4}\left(\frac{T_{\text{total}}}{1~{}\text{day}}\right)^{-1/4}\\\
&\quad\times\left(\frac{T}{8\times
10^{-5}~{}\text{s}}\right)^{-3/4}\left(\frac{\rho_{\text{DM}}}{0.45~{}\text{GeV
cm}^{-3}}\right)^{-1/2}\left(\frac{v_{\text{DM}}}{10^{-3}}\right)^{-1}\end{split}$
(C.7)

at 95 % confidence level, respectively. In the derivation we assumed the
randomness of the velocity direction and the polarization of dark photon. One
measurement time $T$ is taken to be the coherence time of the light dark
matter $T_{X}$, as is done in the main body. However, it should be noted that
this choice is now close the typical duration of quantum operations $\sim
100~{}\mu$s.

It may be useful to give the sensitivity to $B_{X,z}$ ($z$ component of
$\vec{B}_{X}$) as

$B_{X,z}=7.9\times
10^{-11}\,\text{T}\times\left(\frac{\mathcal{F}}{1}\right)^{-1/2}\left(\frac{P(0)}{10^{-4}}\right)^{1/4}\left(\frac{T_{\text{total}}}{1~{}\text{day}}\right)^{-1/4}\left(\frac{T}{8\times
10^{-5}~{}\text{s}}\right)^{-3/4}.$ (C.8)

171Yb+ ion can detect oscillating magnetic field of this amplitude at $12.6$
GHz. If we adopt the setup used in ABRACADABRA [66] where magnetic fields are
induced by the axionic effective current, one can also measure the axion-
photon coupling in principle, although the sensitivity with a single ion would
be weaker than SQUID sensors.

As discussed in Sec. 5, we can improve sensitivities with entangled $N$ ions.
For spin qubits, we can use the GHZ state (24) as an initial state to be
stimulated by the light dark matter rather than Eq. (26).

## References

  * [1] G. Gabrielse and F. C. MacKintosh, Cylindrical penning traps with orthogonalized anharmonicity compensation, International Journal of Mass Spectrometry and Ion Processes, vol. 57, no. 1, pp. 1–17, 1984.
  * [2] J. Tan and G. Gabrielse, One electron in an orthogonalized cylindrical penning trap, Applied physics letters, vol. 55, no. 20, pp. 2144–2146, 1989.
  * [3] W. Paul, Electromagnetic traps for charged and neutral particles, Reviews of modern physics, vol. 62, no. 3, p. 531, 1990.
  * [4] F. Zwicky, Die Rotverschiebung von extragalaktischen Nebeln, Helv. Phys. Acta, vol. 6, pp. 110–127, 1933, doi:10.1007/s10714-008-0707-4.
  * [5] F. Zwicky, On the Masses of Nebulae and of Clusters of Nebulae, Astrophys. J., vol. 86, pp. 217–246, 1937, doi:10.1086/143864.
  * [6] D. Clowe, A. Gonzalez, and M. Markevitch, Weak lensing mass reconstruction of the interacting cluster 1E0657-558: Direct evidence for the existence of dark matter, Astrophys. J., vol. 604, pp. 596–603, 2004, doi:10.1086/381970, arXiv:astro-ph/0312273.
  * [7] M. Markevitch, A. H. Gonzalez, D. Clowe, A. Vikhlinin, L. David, W. Forman, C. Jones, S. Murray, and W. Tucker, Direct constraints on the dark matter self-interaction cross-section from the merging galaxy cluster 1E0657-56, Astrophys. J., vol. 606, pp. 819–824, 2004, doi:10.1086/383178, arXiv:astro-ph/0309303.
  * [8] N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys., vol. 641, p. A6, 2020, doi:10.1051/0004-6361/201833910, arXiv:1807.06209 [astro-ph.CO].  [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * [9] R. D. Peccei and H. R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett., vol. 38, pp. 1440–1443, 1977, doi:10.1103/PhysRevLett.38.1440.
  * [10] R. D. Peccei and H. R. Quinn, Constraints Imposed by CP Conservation in the Presence of Instantons, Phys. Rev. D, vol. 16, pp. 1791–1797, 1977, doi:10.1103/PhysRevD.16.1791.
  * [11] S. Weinberg, A New Light Boson?, Phys. Rev. Lett., vol. 40, pp. 223–226, 1978, doi:10.1103/PhysRevLett.40.223.
  * [12] F. Wilczek, Problem of Strong $P$ and $T$ Invariance in the Presence of Instantons, Phys. Rev. Lett., vol. 40, pp. 279–282, 1978, doi:10.1103/PhysRevLett.40.279.
  * [13] J. E. Kim, Weak Interaction Singlet and Strong CP Invariance, Phys. Rev. Lett., vol. 43, p. 103, 1979, doi:10.1103/PhysRevLett.43.103.
  * [14] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Can Confinement Ensure Natural CP Invariance of Strong Interactions?, Nucl. Phys. B, vol. 166, pp. 493–506, 1980, doi:10.1016/0550-3213(80)90209-6.
  * [15] A. R. Zhitnitsky, On Possible Suppression of the Axion Hadron Interactions. (In Russian), Sov. J. Nucl. Phys., vol. 31, p. 260, 1980\.
  * [16] M. Dine, W. Fischler, and M. Srednicki, A Simple Solution to the Strong CP Problem with a Harmless Axion, Phys. Lett. B, vol. 104, pp. 199–202, 1981, doi:10.1016/0370-2693(81)90590-6.
  * [17] B. Holdom, Two U(1)’s and Epsilon Charge Shifts, Phys. Lett. B, vol. 166, pp. 196–198, 1986, doi:10.1016/0370-2693(86)91377-8.
  * [18] C. O’Hare, “cajohare/axionlimits: Axionlimits.” https://cajohare.github.io/AxionLimits/, July 2020.
  * [19] A. Caputo, A. J. Millar, C. A. J. O’Hare, and E. Vitagliano, Dark photon limits: A handbook, Phys. Rev. D, vol. 104, no. 9, p. 095029, 2021, doi:10.1103/PhysRevD.104.095029, arXiv:2105.04565 [hep-ph].
  * [20] K. A. Gilmore, M. Affolter, R. J. Lewis-Swan, D. Barberena, E. Jordan, A. M. Rey, and J. J. Bollinger, Quantum-enhanced sensing of displacements and electric fields with two-dimensional trapped-ion crystals, Science, vol. 373, no. 6555, pp. 673–678, 2021, doi:10.1126/science.abi5226, arXiv:2103.08690 [quant-ph].
  * [21] X. Fan, G. Gabrielse, P. W. Graham, R. Harnik, T. G. Myers, H. Ramani, B. A. D. Sukra, S. S. Y. Wong, and Y. Xiao, One-Electron Quantum Cyclotron as a Milli-eV Dark-Photon Detector, Phys. Rev. Lett., vol. 129, no. 26, p. 261801, 2022, doi:10.1103/PhysRevLett.129.261801, arXiv:2208.06519 [hep-ex].
  * [22] S. Chen, H. Fukuda, T. Inada, T. Moroi, T. Nitta, and T. Sichanugrist, Detection of hidden photon dark matter using the direct excitation of transmon qubits, 12 2022, arXiv:2212.03884 [hep-ph].
  * [23] G. Afek, D. Carney, and D. C. Moore, Coherent Scattering of Low Mass Dark Matter from Optically Trapped Sensors, Phys. Rev. Lett., vol. 128, no. 10, p. 101301, 2022, doi:10.1103/PhysRevLett.128.101301, arXiv:2111.03597 [physics.ins-det].
  * [24] D. Budker, P. W. Graham, H. Ramani, F. Schmidt-Kaler, C. Smorra, and S. Ulmer, Millicharged Dark Matter Detection with Ion Traps, PRX Quantum, vol. 3, no. 1, p. 010330, 2022, doi:10.1103/PRXQuantum.3.010330, arXiv:2108.05283 [hep-ph].
  * [25] A. Berlin et al., Searches for New Particles, Dark Matter, and Gravitational Waves with SRF Cavities, 3 2022, arXiv:2203.12714 [hep-ph].
  * [26] A. J. Brady, C. Gao, R. Harnik, Z. Liu, Z. Zhang, and Q. Zhuang, Entangled Sensor-Networks for Dark-Matter Searches, PRX Quantum, vol. 3, no. 3, p. 030333, 2022, doi:10.1103/PRXQuantum.3.030333, arXiv:2203.05375 [quant-ph].
  * [27] A. J. Brady et al., Entanglement-enhanced optomechanical sensor array for dark matter searches, 10 2022, arXiv:2210.07291 [quant-ph].
  * [28] H. Shi and Q. Zhuang, Ultimate precision limit of noise sensing and dark matter search, npj Quantum Inf., vol. 9, no. 1, p. 27, 2023, doi:10.1038/s41534-023-00693-w, arXiv:2208.13712 [quant-ph].
  * [29] A. O. Sushkov, Quantum Science and the Search for Axion Dark Matter, PRX Quantum, vol. 4, no. 2, p. 020101, 2023, doi:10.1103/PRXQuantum.4.020101, arXiv:2304.11797 [hep-ph].
  * [30] T. Ikeda, A. Ito, K. Miuchi, J. Soda, H. Kurashige, and Y. Shikano, Axion search with quantum nondemolition detection of magnons, Phys. Rev. D, vol. 105, no. 10, p. 102004, 2022, doi:10.1103/PhysRevD.105.102004, arXiv:2102.08764 [hep-ex].
  * [31] S. Chigusa, M. Hazumi, E. D. Herbschleb, N. Mizuochi, and K. Nakayama, Light Dark Matter Search with Nitrogen-Vacancy Centers in Diamonds, 2 2023, arXiv:2302.12756 [hep-ph].
  * [32] S. Chigusa, D. Kondo, H. Murayama, R. Okabe, and H. Sudo, Axion detection via superfluid 3He ferromagnetic phase and quantum measurement techniques, 9 2023, arXiv:2309.09160 [hep-ph].
  * [33] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Quantum computers, nature, vol. 464, no. 7285, pp. 45–53, 2010, arXiv:1009.2267 [quant-ph].
  * [34] W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, Localized visible ba+ mono-ion oscillator, Physical Review A, vol. 22, no. 3, p. 1137, 1980\.
  * [35] M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, High-fidelity quantum control using ion crystals in a penning trap, 2009, arXiv:0906.0398 [quant-ph].
  * [36] J. I. Cirac and P. Zoller, Quantum Computations with Cold Trapped Ions, Phys. Rev. Lett., vol. 74, pp. 4091–4094, 1995, doi:10.1103/PhysRevLett.74.4091.
  * [37] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Applied Physics Reviews, vol. 6, no. 2, p. 021314, 2019, arXiv:1904.04178 [quant-ph].
  * [38] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, Experimental issues in coherent quantum state manipulation of trapped atomic ions, J. Res. Natl. Inst. Stand. Tech., vol. 103, p. 259, 1998, arXiv:quant-ph/9710025.
  * [39] R. Ozeri, W. M. Itano, R. Blakestad, J. Britton, J. Chiaverini, J. D. Jost, C. Langer, D. Leibfried, R. Reichle, S. Seidelin, et al., Errors in trapped-ion quantum gates due to spontaneous photon scattering, Physical Review A, vol. 75, no. 4, p. 042329, 2007.
  * [40] P. Sikivie, Experimental Tests of the Invisible Axion, Phys. Rev. Lett., vol. 51, pp. 1415–1417, 1983, doi:10.1103/PhysRevLett.51.1415.  [Erratum: Phys.Rev.Lett. 52, 695 (1984)].
  * [41] I. Pogorelov, T. Feldker, C. D. Marciniak, L. Postler, G. Jacob, O. Krieglsteiner, V. Podlesnic, M. Meth, V. Negnevitsky, M. Stadler, B. Höfer, C. Wächter, K. Lakhmanskiy, R. Blatt, P. Schindler, and T. Monz, Compact ion-trap quantum computing demonstrator, PRX Quantum, vol. 2, p. 020343, Jun 2021, doi:10.1103/PRXQuantum.2.020343.
  * [42] J. Ouellet and Z. Bogorad, Solutions to Axion Electrodynamics in Various Geometries, Phys. Rev. D, vol. 99, no. 5, p. 055010, 2019, doi:10.1103/PhysRevD.99.055010, arXiv:1809.10709 [hep-ph].
  * [43] M. Brownnutt, M. Kumph, P. Rabl, and R. Blatt, Ion-trap measurements of electric-field noise near surfaces, Rev. Mod. Phys., vol. 87, pp. 1419–1482, Dec 2015, doi:10.1103/RevModPhys.87.1419.
  * [44] L. Feng, W. L. Tan, A. De, A. Menon, A. Chu, G. Pagano, and C. Monroe, Efficient ground-state cooling of large trapped-ion chains with an electromagnetically-induced-transparency tripod scheme, Phys. Rev. Lett., vol. 125, p. 053001, Jul 2020, doi:10.1103/PhysRevLett.125.053001.
  * [45] K. Mølmer and A. Sørensen, Multiparticle entanglement of hot trapped ions, Physical Review Letters, vol. 82, no. 9, p. 1835, 1999, arXiv:quant-ph/9810040.
  * [46] V. Anastassopoulos et al., New CAST Limit on the Axion-Photon Interaction, Nature Phys., vol. 13, pp. 584–590, 2017, doi:10.1038/nphys4109, arXiv:1705.02290 [hep-ex].
  * [47] A. V. Gramolin, D. Aybas, D. Johnson, J. Adam, and A. O. Sushkov, Search for axion-like dark matter with ferromagnets, Nature Phys., vol. 17, no. 1, pp. 79–84, 2021, doi:10.1038/s41567-020-1006-6, arXiv:2003.03348 [hep-ex].
  * [48] C. P. Salemi et al., Search for Low-Mass Axion Dark Matter with ABRACADABRA-10 cm, Phys. Rev. Lett., vol. 127, no. 8, p. 081801, 2021, doi:10.1103/PhysRevLett.127.081801, arXiv:2102.06722 [hep-ex].
  * [49] J. A. Devlin et al., Constraints on the Coupling between Axionlike Dark Matter and Photons Using an Antiproton Superconducting Tuned Detection Circuit in a Cryogenic Penning Trap, Phys. Rev. Lett., vol. 126, no. 4, p. 041301, 2021, doi:10.1103/PhysRevLett.126.041301, arXiv:2101.11290 [astro-ph.CO].
  * [50] D. Noordhuis, A. Prabhu, S. J. Witte, A. Y. Chen, F. Cruz, and C. Weniger, Novel Constraints on Axions Produced in Pulsar Polar-Cap Cascades, 9 2022, arXiv:2209.09917 [hep-ph].
  * [51] C. Dessert, D. Dunsky, and B. R. Safdi, Upper limit on the axion-photon coupling from magnetic white dwarf polarization, Phys. Rev. D, vol. 105, no. 10, p. 103034, 2022, doi:10.1103/PhysRevD.105.103034, arXiv:2203.04319 [hep-ph].
  * [52] M. Ajello et al., Search for Spectral Irregularities due to Photon–Axionlike-Particle Oscillations with the Fermi Large Area Telescope, Phys. Rev. Lett., vol. 116, no. 16, p. 161101, 2016, doi:10.1103/PhysRevLett.116.161101, arXiv:1603.06978 [astro-ph.HE].
  * [53] S. Hoof and L. Schulz, Updated constraints on axion-like particles from temporal information in supernova SN1987A gamma-ray data, JCAP, vol. 03, p. 054, 2023, doi:10.1088/1475-7516/2023/03/054, arXiv:2212.09764 [hep-ph].
  * [54] M. Escudero, C. K. Pooni, M. Fairbairn, D. Blas, X. Du, and D. J. E. Marsh, Axion Star Explosions: A New Source for Axion Indirect Detection, 2 2023, arXiv:2302.10206 [hep-ph].
  * [55] P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel, J. Redondo, and A. Ringwald, WISPy Cold Dark Matter, JCAP, vol. 06, p. 013, 2012, doi:10.1088/1475-7516/2012/06/013, arXiv:1201.5902 [hep-ph].
  * [56] S. J. Witte, S. Rosauro-Alcaraz, S. D. McDermott, and V. Poulin, Dark photon dark matter in the presence of inhomogeneous structure, JHEP, vol. 06, p. 132, 2020, doi:10.1007/JHEP06(2020)132, arXiv:2003.13698 [astro-ph.CO].
  * [57] T. Choi, S. Debnath, T. A. Manning, C. Figgatt, Z.-X. Gong, L.-M. Duan, and C. Monroe, Optimal quantum control of multimode couplings between trapped ion qubits for scalable entanglement, Phys. Rev. Lett., vol. 112, p. 190502, May 2014, doi:10.1103/PhysRevLett.112.190502.
  * [58] Britton et al., Engineered two-dimensional ising interactions in a trapped-ion quantum simulator with hundreds of spins, Nature, vol. 484, no. 7395, pp. 489–492, 2012, doi:10.1038/nature10981.
  * [59] D. Kielpinski et al., Architecture for a large-scale ion-trap quantum computer, Nature, vol. 417, no. 6890, pp. 709–711, 2002, doi:10.1038/nature00784.
  * [60] L. J. Stephenson et al., High-rate, high-fidelity entanglement of qubits across an elementary quantum network, Phys. Rev. Lett., vol. 124, p. 110501, Mar 2020, doi:10.1103/PhysRevLett.124.110501.
  * [61] C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects, Phys. Rev. A, vol. 89, p. 022317, Feb 2014, doi:10.1103/PhysRevA.89.022317.
  * [62] S. Chen, H. Fukuda, T. Inada, T. Moroi, T. Nitta, and T. Sichanugrist, Quantum Enhancement in Dark Matter Detection with Quantum Computation, 11 2023, arXiv:2311.10413 [hep-ph].
  * [63] D. F. Walls and G. J. M. Ostriker, Quantum Optics.  Springer, 1994.
  * [64] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems.  Oxford University Press, 2007.
  * [65] R. Barbieri, M. Cerdonio, G. Fiorentini, and S. Vitale, AXION TO MAGNON CONVERSION: A SCHEME FOR THE DETECTION OF GALACTIC AXIONS, Phys. Lett. B, vol. 226, pp. 357–360, 1989, doi:10.1016/0370-2693(89)91209-4.
  * [66] Y. Kahn, B. R. Safdi, and J. Thaler, Broadband and Resonant Approaches to Axion Dark Matter Detection, Phys. Rev. Lett., vol. 117, no. 14, p. 141801, 2016, doi:10.1103/PhysRevLett.117.141801, arXiv:1602.01086 [hep-ph].