1. Introduction

The role of spin in interactions between elementary particles has been a central question in physics since its discovery [1]. As we know, the concept of spin originates from the study of the anomalous Zeeman effect [2] as a consequence of the interaction of the electron’s magnetic dipole moment with an external magnetic field. Due to this interaction, it is reasonable for us to think that other sorts of (the so called exotic) spin-dependent interactions might exist between fermions [1]. These exotic interactions concern the relationship between spin and spacetime torsion in general relativity [1], the hypothetical electric dipole moments for elementary particles [3], axions [4,5] which is a dark matter candidate [6]and axionlike particles, etc. Because of this, many stimulated laboratory searches have been performed to study and constrain many of them [1,7]. In these search experiments, various atomic, molecular, and optical (AMO) physics based experimental methods such as trapped ions experiments [8], nitrogen-vacancy centers in diamond [9,10], atomic spectroscopy [11], and molecular spectroscopy [12,13]etc. have been employed [14], and more and new AMO based methods are expected to be utilized in the hereafter exotic spin-dependent interactions search experiments because of the unprecedented accuracy of AMO precision measurements [1].

In this paper, we theoretically propose a new AMO based method which can be used to search for an exotic spin-dependent interaction: axial-vector dipole-dipole interaction. Due to the convenient room-temperature optical detection and stability in very small crystals and nanostructures, nitrogen-vacancy (NV) centers have received much attention for sensing purposes [15]. A NV center can be coupled to a mechanical resonator [16,17], forming a quantum system can be coherently manipulated typically by time-dependent fields [15]. In our scheme, a NV center is magnetically coupled to a cantilever resonator, resulting a hybrid mechanical system [18,19]. Then with a static magnetic field, a pump beam and a probe beam applied, a probe absorption spectrum could be obtained. Here how this absorption spectrum will be affected by the hypothetical exotic dipole-dipole interaction between electrons is studied. Based on this study, we propose our detection principle and then provide a prospective constraint. And we expect our method can be realized experimentally.

The remainder of the paper is organized as follows: In Sec. 2 we present the theoretical model and derive essential dynamical equations, in Sec. 3 we present and describe the numerical results, in Sec. 4 we propose the detection principle and establish a prospective constraint, in Sec. 5 we summarize the paper.

2. Theoretical model and dynamical equations

Figure 1(a) shows our setup. Here we propose a hybrid device composed of a cantilever resonator and a [111]-oriented NV center, the symmetry axis of which is assumed as z direction. A single crystal of p-terphenyl doped with pentacene-$d_{14}$, the thickness of which is $h\approx 50nm$, is placed on the resonator. The spin density of the crystal is estimated to be $\rho =1.62\times {10}^{-3} {nm}^{-3}$ [10,20] and the long axis of the pentacene molecule is along [111](see the upper right inset in Fig. 1(a)). A laser pulse is applied to the crystal, resulting a cylindrical bulk of polarized electrons [10,20] with a length of $h$ and a radius of $R\approx 500nm$ (see the right inset in Fig. 1(a)). The NV is positioned at a distance $d\approx 80nm$ under the cylinder, the symmetry axis of which coincides with the one of the NV. For the purpose of isolating the laser and fluorescence from the crystal, we place one $5nm$ thick layer of barrier material made of silver and PMMA between the single crystal and the resonator [10,20].

Now we demonstrate the interactions between the cylindrical bulk and NV. There are two interactions between a polarized electron spin and NV: magnetic dipole-dipole interaction and axial-vector mediated interaction. The first interaction can be written as [10]

 

Fig. 1. Setup and the energy-level diagram. (a) Schematic setup. A p-terphenyl crystal with a thickness of $h$ is placed on a cantilever resonator and there is one layer of barrier material between the two. A [111]-oriented NV center, the symmetry axis of which is assumed as z direction, is positioned at a distance $d$ under the crystal. A laser pulse is applied to the crystal, resulting a cylindrical bulk of polarized electrons with a radius of $R$ as shown in the right inset. And the symmetry axes of the bulk and NV coincide with each other. A static magnetic field is applied along the z direction. A pump beam and a probe beam are applied to the NV center at a specific moment. (b) The energy-level diagram of the NV center spin coupled to the resonator.$\Delta _{pu}$ and $\Delta _{pr}$ are pump-spin detuning and probe-spin detuning respectively.

Download Full Size | PDF

Let us focus on the NV center first. The ground state of it is an S=1 spin triplet with three substates $|0\rangle$ and $|\pm 1\rangle$. These substates are separated by a zero-field splitting of $\omega _s/2\pi \simeq 2.88GHz$. We apply a moderate static magnetic field ${\overrightarrow B}_{ext}$ along the z direction to remove the degeneracy of the $|\pm 1\rangle$ spin states, causing the NV spin restricted to a two-level subspace spanned by $|0\rangle$ and $|-1\rangle$ [22]. Consequently, the Hamiltonian of the NV center in the magnetic field ${\overrightarrow B}_{ext}$ can be written as $H_{NV}=\hbar \omega _s S_z$. $S_z$ together with $S^{\pm }$ characterize the spin operator. Next, how the NV center is coupled to the resonator is demonstrated.

As mentioned above, NV feels an effective magnetic field $B(t)$ from the bulk of polarized electrons on the resonator. Then a magnetic coupling between NV and the resonator could be achieved through magnetic field gradient [16,18]. And the Hamiltonian of the whole system, which consists of NV, the resonator, the single crystal and the barrier material, can be described as [16]

We have mentioned that $P(t)$ have a maximum value when $t=t_0$. At this moment, we apply a pump beam and a probe beam to the NV center (see Fig. 1(a)). As a result of that, the vibration mode of the resonator can be treated as phonon mode and some nonlinear optical phenomena occur. The energy level diagram of the NV coupled to the resonator is illustrated in Fig. 1(b). In the following we attempt to derive the first order linear optical susceptibility.

The Hamiltonian of the NV center spin in ${\overrightarrow B}_{eff}$ coupled with two optical fields reads as follows [23]:

Applying the Heisenberg equations of motion for operators $S_z , S^-$ and $\tau =b^+ +b$, and introducing the corresponding damping and noise terms, three quantum Langevin equations can be derived as follows [23,24]:

3. Numerical results

We consider a Si cantilever resonator of dimensions $(l,w,t)$ $=(10,1.2,0.05)\mu m$ with a fundamental frequency of $\omega _r \sim 2MHz$ [26], where l, w, t denote length, width, and thickness respectively. The quality factor of the cantilever resonator is assumed as $Q\sim {10}^6$, resulting a decay rate $\Theta ={\omega _r}/Q =2Hz$. The mass $m$ is estimated to be $m\sim 10^{-15} kg$. We also assume $\Delta _s =0$, the Rabi frequency of the pump field is $\Omega = 1kHz$, and the induced electric dipole moment is $\mu =10D$. The NV electron spin dephasing time $T_2$ can be selected as $T_2 =1ms$ [27]. Thus the corresponding dephasing rate is $\Upsilon _2 = 1/T_2 =1kHz$ and the electron spin relaxation rate is $\Upsilon _1 =2\Upsilon _2 =2kHz$. With these parameters, according to Eqs. (18)–(21), we plot the probe absorption spectrums ( probe absorption, i.e.,the imaginary part of $\chi ^ {(1)}$, as a function of pump-probe detuning $\delta$ ) around $\delta =\pm \omega _r$ for different values of $g$ in Fig. 2. Next we describe this plot in detail.

 

Fig. 2. Graph of different probe absorption spectrums around $\delta =\pm \omega _r$, the relationship between height of peak and the value of $g$, and a dressed-state picture to interpret the peaks. (a)-(b) We plot three probe absorption spectrums corresponding to $g=0, 0.5, 1Hz$ respectively around $\delta =-\omega _r$ ((a)) and $\delta =\omega _r$ ((b)). In these two subfigures, the colors of black, red, and blue are used to differentiate three values of $g$. The parameters used are: $\omega _r \sim 2MHz$, $Q\sim {10}^6$, $\gamma _n =2Hz$, $\Delta _s =0$, $\Omega = 1KHz$, $\mu =10D$, $\Upsilon _1 =2\Upsilon _2 =2KHz$. (c) The height of positive (negative) peak in the probe absorption spectrum as a function of $g$. (d) Both the negative peaks in (a) and the positive ones in (b) are identified by the corresponding transition between dressed states of the NV electron spin. TP denotes the three-photon resonance and AC denotes the ac-Stark-shifted resonance.

Download Full Size | PDF

In Fig. 2(a), at $-\omega _r-50Hz\leq \delta \leq -\omega _r+50Hz$ we plot three probe absorption spectrums corresponding to $g=0, 0.5Hz, 1Hz$ respectively. In the case of $g=0$ (black), there is only a straight line. When the value of $g$ grows to $0.5Hz$ (red), a negative sharp peak centered at $\delta =-\omega _r$ appears in the corresponding probe absorption spectrum, the rest of which coincides with the black straight line. When the value of $g$ continues grows to $1Hz$ (blue), the corresponding probe absorption spectrum is the same as the case of $g=0.5Hz$, except the negative peak is larger. In Fig. 2(b), we plot the same three probe absorption spectrums at $\omega _r-50Hz\leq \delta \leq \omega _r+50Hz$. There is only a black straight line in the case of $g=0$. And a positive steep peak centered at $\delta =\omega _r$ appears for the cases of the red curve ($g=0.5Hz$ ) and the blue ( $g=1Hz$), while the rest of these two curves coincides with the straight line. In addition, the blue peak is larger than the red, which is the same as the situation in the Fig. 2(a). We can also take other values of $g$ into consideration, though not illustrated in Fig. 2. Then We summarize the probe absorption spectrums around $\delta =\pm \omega _r$, i.e. $\delta \in [-\omega _r -50Hz , -\omega _r +50Hz]\cup [\omega _r -50Hz , \omega _r +50Hz]$. For the probe absorption spectrum of $g=0$, there is no peak at $\delta =\pm \omega _r$, around each of which is only a straight line. On the contrary, when the value of $g$ grows to $0.3 Hz$ from $0$, two evident peaks including a negative and a positive one appear in the corresponding probe absorption spectrum. Here the negative peak is centered at $\delta =-\omega _r$ while the positive one at $\delta =\omega _r$. And the rest of the spectrum coincides with the case of $g=0$. If the value of $g$ continues to grow from $0.3Hz$ but not exceed $1kHz$, both two peaks centered at $\delta =\pm \omega _r$ would keep becoming larger while the rest of the corresponding spectrums will coincide with the case of $g=0$. Furthermore, It is found that the heights of the two peaks in the same probe absorption spectrum corresponding to $0.3Hz\leq g \leq 1kHz$ are equal to each other. The height of the positive(negative) peak can be considered as a function of $g$ whose domain is $0.3Hz\leq g \leq 1kHz$. And the graph of this function is plotted in Fig. 2(c). In addition, our detection is based on the relative height between the probe absorption at $\pm \omega _r$ and the one around $\pm \omega _r$. Generally speaking, if the dipole-dipole interaction is strong, the value of $g$ will be big, then this relative height (the height of the peak) will be large, as shown in Fig. 2(a), (b) and (c). Two peaks in a probe absorption spectrum for any positive value of $g$ can be interpreted by a dressed-state picture, in which the original energy levels of the NV electron spin $|-1\rangle$ and $|0\rangle$ have been dressed by the phonon mode of the cantilever resonator. Consequently, $|-1\rangle$ and $|0\rangle$ split into dressed states $|-1,n\rangle$ and $|0,n\rangle$, where $|n\rangle$ denotes the number states of the phonon mode (see part (1) of Fig. 2(d)). The feature of the negative peak can be interpreted by TP, which denotes the three-photon resonance. Here the NV spin makes a transition from the lowest dressed state $|0,n\rangle$ to the highest dressed state $|-1,n+1\rangle$ by the simultaneous absorption of two pump photons and the emission of a photon at $\omega _1 -\omega _r$ (see part (2) of Fig. 2(d)). Meanwhile, the feature of the positive peak corresponds to the usual absorptive resonance of the NV spin as modified by the ac Stark effect, shown by part (3) of Fig. 2(d).

Till now, the relationship between the probe absorption spectrum around $\delta =\pm \omega _r$ and the value of $g$ has been demonstrated. Besides, the relationship between $g$ and the coupling constant ${g_A^e g_A^e}/{4\pi \hbar c}$ has been made clear through Eq. (22). Based on these two points, we propose our principle of detection of the axial-vector mediated dipole-dipole interaction and then a prospective constraint in the following. In Fig. 2(a), at $-\omega _r-50Hz\leq \delta \leq -\omega _r+50Hz$ we plot three probe absorption spectrums corresponding to $g=0, 0.5Hz, 1Hz$ respectively. In the case of $g=0$ (black), there is only a straight line. When the value of $g$ grows to $0.5Hz$ (red), a negative sharp peak centered at $\delta =-\omega _r$ appears in the corresponding probe absorption spectrum, the rest of which coincides with the black straight line. When the value of $g$ continues grows to $1Hz$ (blue), the corresponding probe absorption spectrum is the same as the case of $g=0.5Hz$, except the negative peak is larger. In Fig. 2(b), we plot the same three probe absorption spectrums at $\omega _r-50Hz\leq \delta \leq \omega _r+50Hz$. There is only a black straight line in the case of $g=0$. And a positive steep peak centered at $\delta =\omega _r$ appears for the cases of the red curve ($g=0.5Hz$ ) and the blue ( $g=1Hz$), while the rest of these two curves coincides with the straight line. In addition, the blue peak is larger than the red, which is the same as the situation in the Fig. 2(a). We can also take other values of $g$ into consideration, though not illustrated in Fig. 2. Then We summarize the probe absorption spectrums around $\delta =\pm \omega _r$, i.e. $\delta \in [-\omega _r -50Hz , -\omega _r +50Hz]\cup [\omega _r -50Hz , \omega _r +50Hz]$. For the probe absorption spectrum of $g=0$, there is no peak at $\delta =\pm \omega _r$, around each of which is only a straight line. On the contrary, when the value of $g$ grows to $0.3 Hz$ from $0$, two evident peaks including a negative and a positive one appear in the corresponding probe absorption spectrum. Here the negative peak is centered at $\delta =-\omega _r$ while the positive one at $\delta =\omega _r$. And the rest of the spectrum coincides with the case of $g=0$. If the value of $g$ continues to grow from $0.3Hz$ but not exceed $1kHz$, both two peaks centered at $\delta =\pm \omega _r$ would keep becoming larger while the rest of the corresponding spectrums will coincide with the case of $g=0$. Furthermore, It is found that the heights of the two peaks in the same probe absorption spectrum corresponding to $0.3Hz\leq g \leq 1kHz$ are equal to each other. The height of the positive(negative) peak can be considered as a function of $g$ whose domain is $0.3Hz\leq g \leq 1kHz$. And the graph of this function is plotted in Fig. 2(c).

Two peaks in a probe absorption spectrum for any positive value of $g$ can be interpreted by a dressed-state picture, in which the original energy levels of the NV electron spin $|-1\rangle$ and $|0\rangle$ have been dressed by the phonon mode of the cantilever resonator. Consequently, $|-1\rangle$ and $|0\rangle$ split into dressed states $|-1,n\rangle$ and $|0,n\rangle$, where $|n\rangle$ denotes the number states of the phonon mode (see part (1) of Fig. 2(d)). The feature of the negative peak can be interpreted by TP, which denotes the three-photon resonance. Here the NV spin makes a transition from the lowest dressed state $|0,n\rangle$ to the highest dressed state $|-1,n+1\rangle$ by the simultaneous absorption of two pump photons and the emission of a photon at $\omega _1 -\omega _r$ (see part (2) of Fig. 2(d)). Meanwhile, the feature of the positive peak corresponds to the usual absorptive resonance of the NV spin as modified by the ac Stark effect, shown by part (3) of Fig. 2(d).

Till now, the relationship between the probe absorption spectrum around $\delta =\pm \omega _r$ and the value of $g$ has been demonstrated. Besides, the relationship between $g$ and the coupling constant ${g_A^e g_A^e}/{4\pi \hbar c}$ has been made clear through Eq. (22). Based on these two points, we propose our principle of detection of the axial-vector mediated dipole-dipole interaction and then a prospective constraint in the following.

4. Detection principle and prospective constraint

Our detection procedure consists of two steps. Firstly, we perform the experiment based on our model and then obtain a probe absorption spectrum. Then through this spectrum around $\delta =\pm \omega _r$ we estimate or constrain the value of $g$. Secondly, We specify the interaction range $\lambda$. Then according to the relationship between $g$ and ${g_A^e g_A^e}/{4\pi \hbar c}$ (Eq. (22)), we determine the value of ${g_A^e g_A^e}/{4\pi \hbar c}$ or establish an upper bound of it. In order to perform the second step well, we analyze Eq. (22) in the following.

The Eq. (22) is rewritten here:

Now focus on Eq. (24). According to it, ${g_A^e g_A^e}/{4\pi \hbar c}$ could be considered as a function of $g$ if $\lambda$ is reasonable specified and $g$ satisfies $g\ge {10}^{-2}Hz$. Then if $\lambda$ takes values of ${10}^{-n}m$ where $n=4,5,6,7,8$ respectively, five relating functions with ${10}^{-2}Hz\leq g \leq {10}^{2}Hz$ would be obtained. And the graphs of them are plotted in Fig. 3. Then, Several findings of this figure denoted by (I)-(III) are listed: (I) The graphs corresponding to $\lambda =10^{-n} m$ where $n=4,5,6,$ overlap completely. (II) One value of $g$ possibly corresponds to different values of ${g_A^e g_A^e}/{4\pi \hbar c}$ when $\lambda$ takes different values. (III) For any one of these five graphs, the value of ${g_A^e g_A^e}/{4\pi \hbar c}$ keeps decreasing as the value of $g$ decreases continuously. Furthermore, the finding (III) can be generalized to a easily proved conclusion: If we assume ${10}^{-10}m\leq \lambda \leq {10}^{-1}m$ and $g\ge {10}^{-2}Hz$, a smaller value of $g$ would correspond to a smaller value of ${g_A^e g_A^e}/{4\pi \hbar c}$. Till now the demonstration of our detection method has just been finished. Then according to this method, we set a prospective constraint for ${g_A^e g_A^e}/{4\pi \hbar c}$ in the following.

 

Fig. 3. Graphs of functions. We assume $\lambda$ takes five values of ${10}^{-n}m$ where $n=4,5,6,7,8$. Consequently, according to Eq. (24), ${g_A^e g_A^e}/{4\pi \hbar c}$ can be considered as five functions of $g$ with ${10}^{-2}Hz\leq g \leq {10}^{2}Hz$. We plot the graphs of them with different colors. The blue and green lines correspond to the cases of $\lambda ={10}^{-8}m,{10}^{-7}m$ respectively, while the carmine line simultaneously denotes the cases of $\lambda ={10}^{-4}m,{10}^{-5}m, {10}^{-6}m$.

Download Full Size | PDF

We assume that an evident peak could be observed in a probe absorption spectrum. Based on this and the numerical result that there are two evident peaks centered at $\delta =\pm \omega _r$ respectively in the absorption spectrum corresponding to $g=0.3Hz$, we assume that the minimum value of $g$ could be identified is $0.3Hz$. We also assume that in the relating experiment the exotic dipole-dipole interaction would not be observed if $\lambda$ is assumed as ${10}^{-10}m\leq \lambda \leq {10}^{-1}m$. Combining the second assumption and the third one we can conclude that the value of $g$ corresponding to the experimentally generated probe absorption spectrum would satisfy

 

Fig. 4. Upper limits on the dimensionless coupling constant ${g_A^e g_A^e}/4\pi \hbar c$ as a function of the interaction range $\lambda$ and mass of the exchanged boson $m_b$. The red dashed curve represents our work, while the blue and pink ones correspond to the results from Refs. [4] and [22] respectively. The yellow region is excluded parameter space.

Download Full Size | PDF

5. Summary

In this paper we have developed an AMO based method which could be utilized to detect axial-vector dipole-dipole interaction between polarized electrons. In our scheme, with two beams and a static magnetic field applied to a spin-resonator hybrid mechanical system, we could obtain a probe absorption spectrum through which we determine the exotic dipole-dipole interaction exists or not. Furthermore, we provide a prospective constraint, which is most stringent at about $3\times {10}^{-8}m<\lambda <2\times {10}^{-7}m$. Certainly, we hope our method could be realized experimentally in the near future. Finally, we hope our method can be enlightening in the area of experimental searches for exotic spin-dependent interactions.