Quantum entanglement of ions for light dark matter detection

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.


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 where a denotes the axion, F µν is the field strength tensor of the electromagnetic field, Fµν is its dual, and g aγ represents the coupling constant.Note that, for the QCD axion, g aγ is not an independent parameter but is related to its mass m a : g aγ /m a ∼ (0.01-1) 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] where ϵ is the mixing parameter and F ′ µν 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][33].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 171 Yb + , instead of vibrational qubits for light dark matter search is studied in Appendix C.

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 [34], 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 [35] or the Penning trap [36], following the foundational concept of trapped ion quantum computers proposed in Ref. [37].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 [38].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 |g⟩ and |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 O(1-10) µm by bal-ance 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 (|n = 1⟩, that is a superposition of the first excited state for one of N oscillators) or they are all at the ground state (|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 Here ω z is the angular frequency of the vibrational mode, ω 0 is the energy gap between the two states |g⟩ and |e⟩, and σ z is the z component of the Pauli matrix to act on the spin state vector.Note that usually there is a hierarchy between ω z ∼ 10 MHz and ω 0 ∼ 10 GHz.The four levels |g, 0⟩, |g, 1⟩, |e, 0⟩, and |e, 1⟩ are utilized for the implementation.We set a † and a as the ladder operators of the vibrational mode and σ + and σ − as the ladder operators of the spin qubit, such as a † σ − |e, 0⟩ = |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 ω and phase ϕ is [39] where Ω is the Rabi frequency, and η is the Lamb-Dicke parameter, which is typically sufficiently small such as of O(0.01-0.1)and thus we omit the higher order terms of η in the Hamiltonian [40].When ω = ω 0 , the first term takes precedence due to resonance, called the carrier resonance.This phenomenon facilitates the transition of the ion's state between |g, n⟩ and |e, n⟩ (n = 0, 1).Similarly, lasers with ω = ω 0 − ω z induce the red sideband resonance.The red sideband resonance is responsible for the transition between |g, 1⟩ and |e, 0⟩, which is described as the third term.Another condition, ω = ω 0 + ω z , corresponds to the blue sideband resonance.This resonance involves the mixing between |g, 0⟩ and |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 |s 1 , s 2 , . . ., 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.

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 [41].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 = 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 where it oscillates with the amplitude a 0 and the frequency m a .ϕ 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 = 2π/m a v 2 DM , where v DM ∼ 10 −3 is the relative speed of dark matter which is determined by the virial velocity of our galaxy [31].Also, since dark matter is non-relativistic, i.e., v DM ≪ 1, the amplitude a 0 is related to the local dark matter density as ρ DM = m 2 a a 2 0 /2 ∼ 0.45 GeV cm −3 .In a Paul trap, the system of qubits is under a uniform external magnetic field ⃗ B along the axis direction of a cylinder with a radius of R (≪ 1/m a ) [42].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 [43] where for m a R ≪ 1 and γ = 0.5772 . . . is the Euler-Mascheroni constant.
The longitudinal mode of dark photon dark matter with mass of 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 where m DP is the mass of dark photon dark matter and ϕ DP is the random phase.As in the case of axion dark matter, the amplitude is related to the local dark matter density, , and the coherence time is given by T DP = 2π/m DP v 2 DM .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 where ϵ DP = ϵ cos θ. θ ∈ [0, π] 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.

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 where X = a or DP represents the case of axion or dark photon and m ION is the mass of the ion.This shows that the resonance occurs at ω z = m X .Then, using the rotational wave approximation, the time evolution of the vibrational qubit system is described by the displacement operator: where T is time duration of the observation which needs to be within the coherence time of light dark matter T X ∼ 0.4 s × (10 neV/m X ), where again X = a or DP.Starting from |n = 0⟩, this operator develops the vibrational state into which gives the probability of the excitation, |α X | 2 T 2 .
The projection into the vibrational eigenstate can be performed by the following sequence.
First, we bring up to the vibtational state into the spin state by a red sideband π pulse, . The spin state is then observed by the protocol of the spin-qubit measurements 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 [44].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 The contribution of the noise increases linearly over the duration of measurement T , and the coefficient ṅ 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 ρ 0 with the infidelity F0 as If the preparation is perfect, the fidelity 1 − F0 is equal to one.We further assume that the density matrix of the initial state is depolarized, i.e., on average during observations.The cooling time for preparing the initial state is expected to be O(100) µs [45].Again, the displacement operator describes the time evolution of the density matrix due to the electric field induced by light dark matter: 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., ⟨g, 1| ρ 1 |g, 1⟩.However, the readout process also contains imperfection, which can be parameterized as where F1 represents the infidelity of readout.The T -independent terms are O( F0,1 ).
From Eqs. ( 14), ( 17) and (18), we obtain the readout probability including the heating effect and the infidelity as where It is reported that the value of infidelity F0 and F1 can reach around 10 −4 for a single ion [38].Therefore, F ≈ 1 and P (0) ≪ ṅT can be taken for the calculation.
We observe the state after waiting for an observation time T (< T X , ṅ−1 ) and repeat the observation by N shot times, so that the total time duration of the experiment is T total ≃ N shot T .The number of signals induced by dark matter is given by S = N shot |α X | 2 T 2 , while that due to noise is B = N shot ṅT .By defining the sensitivity by S/ √ B > 1.645 (95% confidence level), we obtain the sensitivity to the electric field with an angular frequency ω z as Here we assumed the use of 40 Ca + ion, and ignored the term of log m a R in Eq. (7), which just gives an 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 s −1 , is a few times better than the best ones reported in Ref. [44].
Such a low heating rate is required by the condition of ṅT ≪ 1 if we take the waiting time T to be maximal T ∼ 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,  (22) at 95% confidence level, respectively.In deriving Eq. ( 22), we took average on θ.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.As was explained before, the time evolution of the vibrational qubit caused by light dark matter is described by the displacement operator D(β) defined as Eq. ( 11).Here, β = β r + iβ i is an arbitrary complex number small enough that its quadratic terms can be ignored.The displacement operator can be rewritten as by using the first and second Pauli matrices acting onto vibrational qubits σ 1 vib = a † + a and σ 2 vib = ia † − ia.
We first outline a method for measuring the imaginary part, β i , whose flow is shown in by using the phase rotation operator only onto |e⟩ [46].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 Since the states (|g, 0⟩±|g, 1⟩)/ √ 2 are the eigenstates of σ 1 vib corresponding to the eigenvalues ±1, the new phases exp(±iN β i ) appear in each terms.By applying the inverse operation of the red sideband transition and the Hadamard gate, one obtains the state 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., |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 β r -dependent terms in Eq. ( 27) are orthogonal to the β 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 CNOT The series of the operations brings back to the original |g, g, . . ., g, 0⟩ state if β = 0.The signal of dark matter is the excitation of the first qubit with the probability of 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 β r can be detected with an amplification factor of N 2 by following almost the same procedure by replacing |Ψ 2 ⟩ by Figure 2: Sensitivities at 95% C.L. to the axion-photon coupling g aγ 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 ω 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 ω z .For comparison, limits by laboratory experiments are shown; The black, dark green, blue, and purple regions are excluded by CAST [47], SHAFT [48], ABRACADABRA [49], and BASE [50], respectively † .
Note that this state is composed of the eigenstates of σ 2 vib .In our situation the averages of the real and imaginary parts of signal are expected to be the same, i.e., 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 DM ) −1 ∼ 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 † While there also exists astrophysical constraints on the axion-photon coupling [51][52][53][54][55], they are not shown in the figure because the results might be affected by astrophysical uncertainties.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 ω 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 ω z .The two gray regions are excluded by the cosmological observations [56,57].
N times of that of a single ion.Therefore, the significance of the signal scales as 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/ √ B ∝ N .In both cases, it is evident that the use of entanglement is advantageous in detecting the dark matter signals.
Improved sensitivities to g aγ and ϵ 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. [42].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 [58], maintaining it for a large number of ions (greater than ∼ 20) is challenging.The situation could change when considering a network of connected Paul traps, with each trap containing a single ion [59][60][61][62].

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 O(0.1-10) 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 ≳ 5 s for masses of light dark matter considered in this paper (≥ 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 ≃ 20 even when considering multiple ions in a Paul trap [58], while it may be difficult for larger N .The situation could change when considering a network connected Paul traps [59][60][61][62], 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.
where ∆ = ω z − m X is the detuning between the vibrational mode and the light dark matter.N is the mean particle number of the thermal photons at ω z and γ = 2π j δ(ω z − ω(k j ))g 2 j .One can solve the above equations with initial conditions, ⟨a † a⟩ = ⟨a⟩ = ⟨a † ⟩ = 0, and obtain When γt < ∆t ≪ 1, Eq.(A.4) can be approximated to One can define a heating rate ṅ = γ N .When ⟨a † a⟩ ≪ 1, Eq.(A.5) can be regarded as the excitation rate of the vibrational qubit by the light dark matter.

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, on the bases |0⟩ and |1⟩, where σ 3 vib is the third Pauli matrix onto the vibrational qubit.The phase θ is normally distributed with the average zero and the variance 2δ, which is related to T 2 as δ ≃ T /T 2 .In the multi-ion case, each ion j = 1, . . ., N is kicked with individual phase θ 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 [65].)At the linear order of the signal β and the kick θ, these two effects can be divided.Starting the maximally entangled state (25), the density matrix including the time evolution by β is which is in Eq. (26).Additionally the ions are kicked so that their state is changed into The state vector R z (θ) |Ψ 3 ⟩ is given as the replacement β r → β r − iθ j /2 for the first term and β r → β r + iθ 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).For this factor, the lifetime of coherence would be T 2 ≃ 2 ṅ−1 .(This decoherence disappears at linear order, according to the above discussion.)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 ṅ and T 2 are treated as independent parameters when estimating sensitivities of entangled qubits to light dark matter in Fig. 2 and Fig. 3.
C Excitation of spin qubits in 171 Yb + 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 171 Yb + ion whose spin qubit is composed by hyperfine splitting in 6 2 S 1/2 .The ground state is the spin singlet state and the excited state is one of the spin triplet state as follows, where the former and the latter spin symbols represent those of the outermost electron and the ytterbium nucleus, respectively.The direct excitation from |g⟩ to |e⟩ can be caused by applying magnetic fields.Notably, the lifetime of |e⟩ is considerably long ∼ 5 × 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 ψ as L int = ig ae a ψγ 5 ψ, (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 [66]: 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 ∼ 100 µs.(C.8) 171 Yb + ion can detect oscillating magnetic field of this amplitude at 12.6 GHz.If we adopt the setup used in ABRACADABRA [67] 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).

Figure 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.

Figure 3 :
Figure 3: Sensitivities at 95% C.L. to the mixing parameter ϵ as a function of the dark photon mass m 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 ω 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 ω z .The two gray regions are excluded by the cosmological observations[56,57].

1 − 1 / 2 P
g ae e ⃗ v DM 2ρ DM sin (m a t − ϕ a ) , (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):⃗ B DP = ϵ⃗ v DM × ⃗ E ′ 0 sin (m DP t − ϕ DP ) .(C.4)Such magnetic fields from the light dark matter can interact with the spin of electrons of ions, X = a or DP.When the mass of the dark matter coincides with the energy level difference between |g⟩ and |e⟩, that is 12.6 GHz for 171 Yb + 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| ⃗ B X |T cos θ/4m e ) 2 , where θ is the angle between the directions of ⃗ 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 g ae = 7.9 × 10 −3 × F

1 − 1 / 2 P
It may be useful to give the sensitivity to B X,z (z component of ⃗ B X ) as B X,z = 7.9 × 10 −11 T × F