Searching spin-mass interaction using a diamagnetic levitated magnetic resonance force sensor

Axion-like particles (ALPs) are predicted to mediate exotic interactions between spin and mass. We propose an ALP-searching experiment based on the levitated micro-mechanical oscillator, which is one of the most sensitive sensors for spin-mass forces at a short distance. The experiment tests the spin-mass resonant interaction between the polarized electron spins and a diamagnetically levitated microsphere. By periodically flipping the electron spins, the contamination from non-resonant background forces can be eliminated. The levitated micro-oscillator can prospectively enhance the sensitivity by nearly 3 orders over current experiments for ALPs with mass of 7 meV.


I. INTRODUCTION
Light pseudoscalars exist in a number of Beyond the Standard Model theories. One well-motivated example is the axion [1,2], which is introduced via spontaneously broken the Peccei-Quinn (PQ) U (1) symmetry [3,4] to solve the strong CP problem, and also a low-mass candidate for the dark matter in the universe [5]. Generalized axion-like particles (ALPs) rise from dimensional compactification in string theory, which share similar interaction with electromagnetic fields, and share a similar phenomenological role with the axions [6][7][8]. Motivated by axion and ALP's potential role in particle physics and cosmology, a number of experimental methods and techniques have been developed over the past few decades, such as the method proposed by Moody and Wilczek to detect cosmic axion [9], the photon-axion-photon conversion light shining through wall experiments [10,11], the axion emission from the Sun [12,13], the dichroism and birefringence effects in external fields [14,15], and the light pseudoscalar mediated macroscopic mass-mass [16], spin-mass [17][18][19][20][21][22][23][24] and spin-spin [25,26] forces.
The pseudoscalar exchange between fermions results in spin-dependent forces [27]. Most prior works detecting exotic spin-dependent forces [19-22, 28, 29] are focused on the so-called axion window [30], where the interaction range is 200 µm -20 cm. Due to the interest in nonzero mass, it is desired to find experimental techniques to search for such anomalous spin-dependent interactions at even shorter distances [31]. * kongxi@nju.edu.cn † hp@nju.edu.cn ‡ djf@ustc.edu.cn The levitated micro-mechanical and nano-mechanical oscillators have been demonstrated as one of the ultrasensitive force sensors [32][33][34][35][36][37][38] due to its ultra-low dissipation and small size. It is one of the ideal methods to measure short-range force [39][40][41][42][43][44] with high precision. However, in short-ranged force measurements, surface noises from the electric static force fluctuation, the Casimir force and magnetic force, limit the final sensitivity.
Here we propose a new method to investigate the spinmass interaction using an ensemble of electron spins and a levitated diamagnetic microsphere mechanical oscillator. By periodically flip the electron spin at the resonant frequency with the mechanical oscillator, the postulated force between electron spins and the microsphere mass is preserved while the spin-independent force noise from the surface is eliminated.

II. SCHEME
We use a levitated diamagnetic microsphere mechanical oscillator to investigate the spin-mass interaction ( Fig. 1(a)). The microsphere is trapped in the magnetogravitational trap and levitated stably in high vacuum. The diamagnetic-levitated micro-mechanical oscillator achieves the best sensitivity in micro-and nanomechanical systems to date, orders of magnitude improvement over the reported state-of-the-art systems based on different principles. The cryogenic diamagneticlevitated oscillator described here is applicable to a wide range of mass, making it a good candidate for measuring force with ultra-high sensitivity [45]. The position of the microsphere is mainly determined by the equilibrium between the gravity force and the main magnetic force of the trap. A uniform magnetic field is applied to tune the The red and blue parts of the N and S poles represent the profile of the permanent magnet, and the green part represents the profile of the spin source. We use a microsphere as the force sensor, which is placed in the magneto-gravitational trap above the surface of the spin source (see Appendix A for the descripton of its motion). The geometry is sophisticated designed to eliminate the spin-induced magnetic force on the levitated microsphere (see Appendix C for details). (b) Flipping of the electron-spins by a microwave π. The spin mass force Fsm flips with these spins, while those spinindependent forces, for example, the Casimir force Fcas, are independent of the spins and therefore do not flip with the spins. (c) Microwave (MW) pulse sequences. The spins flip at 2ωz, twice the resonance frequency of the levitated oscillator. This leads to a periodical force Fsm of the frequency ωz, while the spin-independent forces such as Casimir force Fcas remains constant during the measurement. levitation position (see Appendix A). A groove-shaped electron spin ensemble (see Appendix C for detail) is located below the mass source as a spin source.
The spin-mass interaction between a polarized electron and an unpolarized nucleon is: [27]: where g N s and g e p are the coupling constants of the interaction, with g N s representing the axion scalar coupling constant to an unpolarized nucleon and g e p representing the axion pseudoscalar coupling constant to the electron spin, λ = /(m a c) is the interaction range, m a is the ALP mass, m e is the mass of electron,σ is electron spin operator, r is the displacement between the electron and nucleon, and e r is the direction. The spin-mass force along z axis is calculated by integrating the force element between microsphere and spin-source based on Eq.(1) as: where ρ e (t) is the time dependent net electron spin density along z axis, ρ m is the nucleon density of the microsphere ζ sm (R, d, λ) is the effective volume for spin-mass interaction that depends on geometry parameters (see Appendix D), R is the microsphere radius and d is the surface distance between the mass and the spin-source. The electrons spins are initially polarized along the magnetic field under high field and low temperature, so that ρ e (0) = ρ e0 , where ρ e0 is the electron density of the spin-source. Then they are flipped periodically in resonance with the microsphere mechanical oscillator (see Fig. 1(b)). On one hand, the spin-independent interactions, such as the Casimir force, will be off-resonance and become eliminated ( Fig. 1(c)). On the other hand, the spin-mass interaction is preserved on the resonance condition. The spin autocorrelation function is defined as ρ e (t)|ρ e (0) = ρ e (0) 2 P (t) = ρ e (0) 2 e −t/T1 ξ(t), where T 1 is the electron spin-lattice relaxation time and ξ(t) is the modulation function (see Appendix B). The microwave π pulses flip the electron spin periodically with frequency 2ω z . ξ(t) jump between -1 and +1 every time the electron spins are flipped. The corresponding power spectral density (PSD) of the spin-related force is proportional to G (ω), which is the Fourier transform of P (t). The PSD of spin-mass force is then : If spin-mass interaction signal is observed on resonance (ω = ω z ), the coupling g N s g e p can be derived as Apart from the spin-mass force, spin-induced magnetic force F s between electron spins and the diamagnetic microsphere is recorded during the measurement. Fortunately, well designed spin-source geometry can eliminate most of the force (see Appendix C). Then the residual spin-induced magnetic force is where ζ s (R, d) is the effective volume for spin-induced force. Similarly, the PSD of F s is Considering the fluctuating noise, the equation of motion for the system center of mass is where m is the mass of the microsphere, ω z /2π is the resonance frequency, γ/2π is the intrinsic damping rate and F flu (t) is the fluctuating noise force that includes the thermal Langevin force F th (t) and the radiation pressure fluctuations F ba (t) [46]. The total detected displacement PSD is given by: where χ (ω) is the mechanical susceptibility given by Due to these noises, the detection limit of spin-mass coupling strength g N s g e p is thus:

III. RESULTS
As a reasonable example we consider a microsphere with mass m = 1.5 × 10 −13 kg and radius R = 3.2 µm of density 1.1 × 10 3 kg/m −3 . Thus, the corresponding nucleon density is ρ m = 6.7 × 10 29 m −3 . The magnetic susceptibility of the microsphere is −9.1 × 10 −6 . The whole system is placed in a cryostat with temperature T = 20 mK and external uniform magnetic field B ext = 1.85 T. A permanent magnet provides 0.15 T magnetic field and correspondingly the z direction magnetic gradient ∂B 0z /∂z = 750 T/m. The microsphere is then levitated with a surface distance d = 1.46 µm above the spin source. The whole mechanical oscillator system have a typical frequency of 24 Hz [47] and the electron density of the spin-source is ρ e0 = 2.3 × 10 27 m −3 . The direction of the electron spins is initially prepared along the external magnetic field, which in our design is approximately along the z axis, with a maximum tilted angle of 4 • .
The total measurement time is set as 1s. We take the experimental sensitivity limited by the total fluctuation noise as S flu ff (ω z ) = S th ff + S ba ff + m 2 S imp zz |χ(ω z )| −2 . Here S th ff is estimated to be 5.14 × 10 −43 N 2 /Hz according to S th ff = 4mγk B T , with γ/2π = 10 −6 Hz [45].
limit for the force range of λ = 2µm as an example. The green line denotes (g N s g e p ) limit calculated from the total fluctuation noise, which decreases as T1 increases; the red line denotes the correction of (g N s g e p ) limit by taking the residual spin-induced magnetic force into account, which is independent of T1; the blue curve denotes their sum. The correction from spin-induced magnetic force (red curve) is dominant when T1 > 1 ms.
Imprecision noise and backaction noise are related, when they contribute equally, the sum has a minimum S sum In a practical condition, the measurement efficiency η ≥ 0.001 [48], which imply S sum ff (ω z ) = 9.36 × 10 −49 N 2 /Hz. Thus, the total fluctuation noise is dominated by the thermal noise, with S flu ff (ω z ) ≈ 5.14 × 10 −43 N 2 /Hz. Under such an experimental sensitivity, (g N s g e p ) limit = 8πm e (S flu ff (ω z )/ G(ω z )) 1 2 /ζ sm 2 ρ m ρ e0 . As G(ω z ) is proportional to the electron spin-lattice relaxation time, (g N s g e p ) limit decreases as T 1 increases, which is shown in green in Fig. 2.
Practically, it is not feasible to completely eliminate the spin-induced magnetic force due to fabrication imperfection of the spin-source geometry (see Appendix C). A correction for (g N s g e p ) limit is introduced as follows. Since the spin-induced magnetic noise is spin-dependent while the G(ω z ) has the same scaling, its contribution to (g N s g e p ) limit is constant (blue curve in Fig. 2). For T 1 > 1ms, the (g N s g e p ) limit is dominated by the spininduced magnetic force and approaches to the minimum 8πm e (S s ff (ω z )/ G(ω z )) 1 2 /ζ sm 2 ρ m ρ e0 .
Finally, Fig. 3 shows the calculated (g N s g e p ) limit (see Appendix E) set by this work at λ = 0.5µm -50 µm together with reported experimental results for the constraints of spin-mass coupling. Here our result is estimated through supposing T 1 = 1s, for spin-lattice relaxation time can be longer than the scale of seconds at low temperature [49,50]. The limitation for our pro-  [17,23,26,[51][52][53][54][55]. The estimated bound of our method is plotted in red for the spin-mass force range λ = 0.5 µm -50 µm. Our result is nearly 3 orders of magnitude more stringent at the ALP mass of 7 meV compared with those from ref. [51][52][53].
posal is the residual spin-induced magnetic force, which can not be eliminated by spin flip procedures. For λ = 2 µm, the minimum detectable spin-mass coupling is (g N s g e p ) limit = 4.3 × 10 −22 ( Table I) due to the spininduced magnetic noise S s ff under reasonable fabrication imperfection ∆ζ s = 1.38 × 10 −21 m 3 (see Appendix C). In conclusion, compared to those from ref. [51][52][53], our result shows an improvement of nearly 3 orders of magnitude more stringent at the ALP mass range of 7 meV.

IV. DISCUSSION
The nearly 3 orders of magnitude enhancement in our scheme comes from the following two aspects. Firstly, the magnetic resonance spin flipping is applied to suppress the short-range force noise which limits the precision of probing spin-mass coupling. Secondly, the diamagnetic levitation realizes an ultra-low dissipation in comparison with other reported mechanical systems, and this together with low temperature condition provides an ultra-low detection noise. The main limitation of our method comes from the spin-induced magnetic force that evolves in accord with the spin-mass interaction, which cannot be eliminated with finite size of the force sensor and imperfect geometric symmetry in the layout of the electron spins. Such a magnetic background could be measured by a sensitive magnetometer with high spatial resolution, such as a single NV center, and then be subtracted from the measured signal, leading to more stringent constraints of the spin-mass coupling strength g N s g e p .

Appendix A: DYNAMICS OF MICRO-SPHERE OSCILLATOR
For the microsphere, the dynamics in the z direction of its center of mass (CM) in our system reads: where mγż is the residual air damping force, F sm is spinmass force, F s is the spin-induced magnetic force, F flu is the fluctuating noise force. E p is the trap potential subject to gravitational field, main magnetic field, spininduced magnetic field, and Casimir attractive force, i.e., where mgz is the gravity of microsphere, m dV represents the volume integral over the microsphere, µ 0 is permeability of vacuum, and χ m is magnetic susceptibility of the microsphere. B 0z is the main magnetic field at the center-mass (CM) of the microsphere, which is the sum of the magnetic field generated by permanent magnet and the uniform external magnetic field B ext . B sz accounts for the spin-induced magnetic field, and V cas is the Casimir potential [56][57][58][59] between the surface of microsphere and the surface of spin-source, reads as, where R is the radius of the microsphere, z corresponds to the displacement of the microsphere, d = 1.46 µm is the surface distance between the microsphere and the spin source when the microsphere locates in equilibrium (Fig. 4), η c = 0.059 characterizes the reduction in the Casimir force, depending on the dielectric functions of the microsphere and the spin-source. The value of E p versus the displacement of the microsphere is shown in Fig. 4. Thus our mechanical system can be described as a damping harmonic oscillator subject to F sm , F s and F flu , i.e., where ω z is the resonant frequency of the microsphere, The equilibrium position of the microsphere can be derived by ∂E p /∂z = 0. The spin-induced magnetic field and V cas are so weak that they have negligible influence on this trap, so that the equilibrium position is mainly determined by the gravity field and the main magnetic field B 0z . Thus we can indirectly tune it by the uniform external magnetic field B ext .

Appendix B: AUTOCORRELATION FUNCTION OF NET ELECTRON-SPINS DENSITY
The autocorrelation function of electron polarization is ρ e (t)ρ e (0) . Suppose these electron spins are indepen- Microwave π pulses carried on with a frequency of 2ωz. The time interval between two adjacent π pulses is τ0 = π/ωz. Spin flips with a frequency of ωz and its amplitude varies slowly over time due to the spin-lattice relaxation. dent of each other, we have where P (t) is the autocorrelation function of a single spin, i.e., Here p ↑,↓ (t) represents the spin population on | ↑ or | ↓ . Every time when a π pulse is applied to flip the electron spin, where τ ∈ (0, τ 0 ), τ 0 corresponds to the period between two adjacent π pulses (Fig. 5), p 1 (τ ) = 1 − e −τ /T1 is the spin flip probability during τ 0 , T 1 is spin-lattice relaxation time. The evolution of P (t) is shown in Fig. 6(a). P (t) presents a sawtooth-like wave of frequency 2ω z for t = kτ 0 + τ T 1 , (k=0, 1, 2, . . . ), Only the signal with resonant frequency ω z needs to be collected. After dropping the sawtooth-like signal whose frequency is 2ω z , the resonant signal is shown in Fig. 6(b). The resonant signal is a square wave with a exponential decay, i.e., where ξ(t) is the modulation function of the following form Here ς(ω z t + π 2 ) is a square wave of frequency ω z . According to the Wiener-Khinchine theorem, its single side PSD is: .

Appendix C: PSD OF SPIN INDUCED MAGNETIC FORCE
Apart from the desired magnetic trap, the spin-source can induce a magnetic force F s on the microsphere as follows This force can be eliminated by deliberately designing the configuration of spin source (in Fig. 7). The z-direction component of magnetic field produced by a single spin is where θ is the polar angle and l is the distance from the microsphere to the spin. The magnetic field of a spinsource cylinder at z axis is then Here i = 1 and 2 correspond to cylinder1 and cylinder 2πr i dr i , and ρ e (r i , z i , t) is net spin density along the z axis in the cylinder.
The microsphere is assumed to be right above the center of the cylinder, so that the magnetic field in the microsphere is approximately uniform in the transverse direction. Thus, the magnetic force produced by a cylinder on this microsphere is where m dV is the integral over the microsphere. Therefore, the magnetic force produced by the spin-source on the microsphere is as follows: where ζ s (d, R) is the effective volume for F s (t), reads: In the cylindrical coordinate system, we have m dV = The geometry shape and the imperfections on fabrications are considered. The geometric parameters are optimized to make F s as small as possible. Table II lists the optimized geometric parameters and their standard deviations according to the practical condition. Here we exaggerate the ρ e (t) to be ρ e0 . From the table, we can see that the value of optimized F s is 4.2 × 10 −22 N, while the total uncertainty of ∆F s is ∆F s = 5.03 × 10 −20 N. More generally, the variation of ∆F s versus the standard deviations of geometric parameters is plotted in Fig. 8. The PSD of the spin-induced magnetic force reads:

Appendix D: PSD OF SPIN-MASS FORCE
The spin-mass effective magnetic field generated by a polarized spin on an unpolarized nucleon is: The spin-mass effective magnetic field generated by the microsphere on a polarized spin is obtained by integrating the volume of the microsphere with Eq. (D1), i.e., where ρ m is the nucleon density of microsphere, e and are the unit vector and distance between the microsphere and the spin, respectively. From Eq. (D2) and Eq. (D3) , we can find that in the calculation of spin-mass effective magnetic field, the microsphere is completely equivalent to a center mass. Therefore, Eq. (D2) is equivalent to the effective magnetic field produced by the CM of the microsphere. The spin-mass potential between the microsphere and the spin-source is obtained by integrating the volume of spin-source with Eq. (D2): V sm (t) = cy1 dV ρ e (r 1 , z 1 , t)µ B · B m − cy2 dV ρ e (r 2 , z 2 , t)µ B · B m , and ρ e (r i , z i , t) represents the net electron spin density along the z axis in the spin-source.
Consequently, the spin-mass force between the micro-sphere and the spin-source is Appendix E: CALCULATION OF (g N s g e p ) limit To observe the spin-mass signal, g N s g e p needs to be no less than .
For the worst situation, (g N s g e p ) limit takes its upper bound: sup (g N s g e p ) limit = sup χmµB and min 2 ρm 8πme ζ sm (R, d, λ) is numerically calculated with parameters R and d taken within the uncertainty ranges (see Table II). Combined with Eq. (E1) and Eq. (E2), the estimated (g N s g e p ) limit in the worst situation is shown in red in Fig. 3 in the main text.