Searching for an Exotic Spin-Dependent Interaction between Electrons at the Nanometer Scale with Molecular Rulers

TL;DR

This study uses molecular rulers with adjustable electron spins to set new, more stringent bounds on exotic spin-dependent interactions at nanometer scales, improving previous limits significantly.

Contribution

It introduces a novel method employing molecular rulers to constrain electron spin interactions at nanometer distances, achieving more precise bounds than prior experiments.

Findings

Established new upper bounds on exotic dipole-dipole interactions between electrons.

Improved the laboratory limit at 200 nm by a factor of 20.

Demonstrated the feasibility of molecular rulers for probing fundamental spin interactions.

Abstract

We propose that a type of molecular rulers, which contains two electron spins with precisely adjustable distance by varying the length of the shape-persistent polymer chains, can be utilized to constrain the axial-vector mediated interaction between electron spins at the nanometer scale. With measurements of the coupling strengths between two electron spins within different molecular rulers, an improved laboratory bound of exotic dipole-dipole interaction between electrons is established within the force range from 3 to 220 nm. The upper bound of the coupling $g_A^e g_A^e/4\pi\hbar c $ at 200 nm is $|g_A^e g_A^e/4\pi\hbar c| \leq 4.9\times10^{-13}$, which is about 20 times more stringent than previous limits.

Equations

$$H_d=-\frac{{\mu }_0{\gamma }_1{\gamma }_2{\hslash }^2}{16\pi r^3}[3(\overrightarrow{{\sigma }_1}\cdot \hat{r})(\overrightarrow{{\sigma }_2}\cdot \hat{r})-(\overrightarrow{{\sigma }_1}\cdot \overrightarrow{{\sigma }_2})],$$

$$H_2=\frac{g_A^e{g}_A^e}{4\pi \hslash c}\frac{\hslash c}{r}(\overrightarrow{{\sigma }_1}\cdot \overrightarrow{{\sigma }_2})e^{-\frac{r}{\lambda }},$$

$$\{\begin{array}{rl}& H_d^{secular}=-\frac{{\mu }_0{\gamma }_{Gd}^2{\hslash }^2}{16\pi r^3}(3{\cos }^2\theta -1){\sigma }_1^z{\sigma }_2^z,\\ & H_2^{secular}=\frac{g_A^e{g}_A^e}{4\pi \hslash c}\frac{\hslash c}{r}{e}^{-\frac{r}{\lambda }}{\sigma }_1^z{\sigma }_2^z.\end{array}$$

$${\omega }_{\perp }=\frac{{\mu }_0{\gamma }_{G{d}_{(\mathrm{III})}}^2{\hslash }^2}{4\pi h}\frac{1}{r^3}+\frac{g_A^e{g}_A^e}{4\pi \hslash c}\frac{4\hslash c}{h}\frac{e^{-\frac{r}{\lambda }}}{r}.$$

$$\begin{array}[]{ c }{V_{AA}(\boldsymbol{r})=-\frac{g_{A}^{e}g_{A}^{e}}{4\pi m^{2}}m^{2}\hbar c\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}\frac{e^{-mr}}{r}}{-\frac{g_{A}^{e}g_{A}^{e}}{4\pi m^{2}}\hbar c\left[\left(\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}\right)\left(\frac{1}{r^{3}}+\frac{m}{r^{2}}+\frac{4\pi}{3}\delta(r)\right)\right.}{-\left(\boldsymbol{\sigma}_{1}\hat{\boldsymbol{r}}\right)\left(\boldsymbol{\sigma}_{2}\hat{\boldsymbol{r}}\right)\left(\frac{3}{r^{3}}+\frac{3m}{r^{2}}+\frac{m^{2}}{r}\right)]e^{-mr}}\end{array}$$

Full text

Light bosons such as pseudoscalar fields (axion and the axion-like particles (ALPs)weinberg1978new ; moody1984new ) and axial-vector fields (paraphotons and extra Z bosons) are hypothetically expected to address mysteries of fundamental sciences, namely, the microscopic origins of dark matter and dark energy, the resolution to the strong CP issue in quantum chromodynamics, as well as the possible connection with the hierarchy problemfeng2010dark . The exchange of such bosons may mediate exotic spin-dependent interactions between ordinary fermions, which enables laboratorial searches on new particles via possible detections of new interactionsmoody1984new ; dobrescu2006spin ; PhysRevA.99.022113 ; safronova2018search . The exotic spin-dependent interactions are characterized by the dimensionless coupling constants between new bosons and fermions and the force range associated with the reduced Compton wavelength $\lambda$ of mediating bosons of mass $m$ dobrescu2006spin ; PhysRevA.99.022113 . There are experiments searching for the axial-vector dipole-dipole interaction between polarized electrons dobrescu2006spin , ranging from atomic scale to the radius of earth safronova2018search . For the force range shorter than a millimeter, there are experiments such as Nitrogen-Vacancy (NV) centers in diamondsPhysRevLett.121.080402 , trapped ionskotler2015constraints , single-atom Electron Spin Resonance with Scanning Tunneling Microscope (ESR-STM) choi2017atomic ; luo2017constraints and atomic spectroscopy ficek2017constraints , providing stringent upper limit of the exotic interaction.

In this paper, we propose a novel method to search for exotic spin-dependent interaction between electrons at the nanometer scale. A type of molecular rulers, which contains two electron spins, has been proposed to constrain the exotic spin-dependent interaction. The distance between two electron spins can be precisely set by varying the length of the shape-persistent polymer chains of the molecular rulers. The coupling strengths between the two electron spins can be obtained by Double Electron Electron Resonance (DEER) experiments. Then we established a constraint on the axial-vector mediated interaction between electrons at the nanometer scale, which considerably improves on previous experimental bounds.

Figure 1a shows that two electron spins interact with each other in an external magnetic field. The magnetic dipole-dipole interaction between two electron spins is described by the following Hamiltonian:

[TABLE]

where $\vec{\sigma_{1}}$ and $\vec{\sigma_{2}}$ stand for Pauli vectors of the two electron spins, $\gamma_{1},\gamma_{1}$ are gyromagnetic ratios of the two electron spins, $r=|\vec{r}|$ and $\hat{r}=\vec{r}/r$ are the displacement and the unit displacement vector between two electron spins. $\mu_{0}$ is the magnetic constant, and $\hbar$ is Plank’s constants divided by 2$\pi$. The axial-vector dipole-dipole interaction mediated by hypothetical axial-vector bosons can be written as dobrescu2006spin ,

[TABLE]

where $g_{A}^{e}g_{A}^{e}/4\pi\hbar c$ is dimensionless axial-vector coupling constant between electrons, $\lambda=\hbar/(mc)$ is the force range, $m$ is the mass of the hypothetical particle and $c$ is the speed of light.

To constrain the exotic interaction described by equation 2 within micrometer scale, several state-of-art methods have been developed to place two electron spins with precise distance and to measure the coupling strength between them. The coupling between two ${}^{88}S_{r}^{+}$ ions, each with a single electron spin, has been measured with high resolution Kotler_2014 , and the interaction can be constrained with micrometer scale kotler2015constraints . Single NV centers can be utilized as quantum sensors to measure the magnetic signal from polarized electrons, and the constraint was improved for the force range from 10 to 900 $\mu$m PhysRevLett.121.080402 . Recently, ESR-STM have been combined so that individual atoms can be precisely placed on a surface with atomic control Baumann417 and the coupling between them can be detected choi2017atomic . With experimental data obtained from ESR-STM measurements on pairs of metal ions with tunable distances choi2017atomic , constraint on the axial-vector dipole-dipole interaction at the nanometer scale can be established luo2017constraints .

Our method is to measure the coupling between two electron spins located on two ends of a molecular ruler, and to constrain the exotic dipole-dipole interaction at the nanometer scale. Molecular rulers is a type of shape-persistent molecules, which contains several repeating units, such as p-phenylene (PP) and ethynylene (E). The PP and E units have geometrically unambiguous structure and high stiffness jeschke2010flexibility . The lengths of molecular rulers can be precisely tuned by changing the number of repeating units. A pair of spin labels with electron spins, can be attached on both ends of the molecular ruler. Thus the distance between electron spins can be adjusted in the nanometer scale by varying the length of shape-persistent polymer chain. DEER experiments can be utilized to measure the coupling between the electron spins.

A type of molecular rulers connecting two $Gd_{(\rm III)}$ labels (Gd-rulers) have been chosen as the system for constraining the exotic interaction. $Gd_{(\rm III)}$ complex is a type of spin label with half integral high-spin(S=7/2), while the electron spin is in a ${}^{8}S$ ground state with 7 unpaired electrons in its f orbital dalaloyan2015gd . The chemically synthesized Gd-spacer-Gd compounds have a rodlike moiety consisting of PP and E units, acting as spacers which connect the two $Gd_{(\rm III)}$ spin labels as shown in Figure 1b. The synthesis of these geometrically well-defined Gd-rulers is reportedqi2016spacers . Precise DEER measurements of a series of Gd-rulers with different Gd-Gd distances are reported dalaloyan2015gd . As shown in Figure 1b, five Gd-rulers with lengths ranging from 4.25-8.78 nm are chosen to constraint the exotic interaction, and the $Gd_{(\rm III)}$-$Gd_{(\rm III)}$ pair can be treated as a pair of weakly coupled spin-half electron spins dalaloyan2015gd . These molecules are denoted as Gd-rulers $\mathbf{1_{n}}$ (n = 5,7,9,11) and $\mathbf{2_{2}}$, which have different sequences of repeating units of PP and E as shown in Figure.1b.

In the experiments, we consider the secular terms of the interaction Hamiltonians due to the large zeeman terms in the strong magnetic field. The magnetic dipole-dipole interaction and the exotic interaction between two $Gd_{(III)}$ spin labels have the following form,

[TABLE]

The magnetic dipole-dipole interaction $H^{secular}_{d}$ is dependent on the angle $\theta$ between external magnetic field and the displacement vector between of two spin labels in Gd-rulers. Because the Gd-rulers are randomly oriented, the splitting due to the magnetic dipole-dipole interaction is the Pake patterns Pake_1948 . If the additional splitting due to the possible $H^{secular}_{2}$ is taken into account, the strength of the coupling $\omega_{\perp}$ ($\theta=90^{\circ}$) is half of the splitting of the two prominent features,

[TABLE]

Figure 2 shows the DEER results in frequency domain. The value of the coupling, $\omega_{\perp}$, increases when the distance of two electron spins decreases. The linewidths of the spectrum are partly due to the distributions of the distance, which are from the flexibility of the molecular building blocksjeschke2010flexibility ; dalaloyan2015gd . The linewidths caused by flexibilities of the molecular rulers are less than one tenth of those of the experimental results SM . The decoherence of the electron spin labels also contributes to the experimental linedwiths.

The experimental results and a fitting of the couplings, $\omega_{\bot}$, between spin-labels with distance from 4.25-8.78 nm are presented in Figure 3. Black squares are the experimental values of the couplings $\omega_{\perp}$, which are half of the splitting of the peaks in Figure. 2. Due to the geometrically unambiguous and high stiffness of the Gd-ruler’s backbone at low temperatures, the distance between $Gd_{(\rm III)}$ can be estimated from density functional theory (DFT). We used scalar relativistic DFT as implemented in the Gaussian software packageg16 and performed full geometry optimizations for all structures. In our DFT calculations, the three-parameter hybrid exchange functional of Becke and the exchange functional of Lee CYang CParr (B3LYP) PhysRevB.37.785 ; Becke1993 ; PhysRevA.38.3098 was employed. More precisely, to ensure a correct approach for the open-shell species, we employed the spin-unrestricted open-shell version of this functional (UB3LYP). The standard basis set 6-31G(d) is used for C, H and N atoms while the Stuttgart RSC 1997 ECP basis set Andrae1990 is used for Gd atom. The DFT results shows that Gd-Gd distance for Gd-rulers $\mathbf{2_{2}}$, $\mathbf{1_{5}}$, $\mathbf{1_{7}}$, $\mathbf{1_{9}}$ and $\mathbf{1_{11}}$, are $4.25$, $4.70$, $6.05$, $7.41$ and $8.78$ nm, respectively. The red line in Figure 3 is a fitting to the experimental data with Equation 4 when the force range $\lambda$= 200 nm. The gyromagnetic ratio of the $Gd_{(\rm III)}$ spin obtained by this fit is $\gamma_{Gd_{(\rm III)}}=(0.92\pm 0.01)\gamma_{e}$, where $\gamma_{e}=2\pi\times 2.8~{}$MHz/Gauss. The coupling constant of the exotic axial-vector-mediated interaction for $\lambda=200~{}$nm is obtained to be $g_{A}^{e}g_{A}^{e}/4\pi\hbar c=(-1.42\pm 1.80)\times 10^{-13}$. The value of the axial-vector field induced interaction is less than its errors showing no evidence of the exotic interaction observed in these measurements. The constraint for other given force range $\lambda$ can be obtained with the same procedure.

Figure 4 shows the new constraint obtained from this work together with recent constraints from experimental searches for axial-vector-mediated dipole-dipole interactions. The horizontal axis shows the force range of the interaction, inversely proportional to the mass of the boson communicating the interaction. The vertical axis shows the dimensionless coupling parameter $g_{A}^{e}g_{A}^{e}/4\pi\hbar c$ between electrons. Filled areas correspond to excluded values. For the force range 0.22$<\lambda<$ 10 $\mu$m, the constraint was established by Kotler et al.kotler2015constraints . For the force range $\lambda<$ 3 nm, the upper limit was obtained from Ref.ficek2017constraints . For the force range from 10 to 200 nm, a previous limit was set by Luo et al.luo2017constraints with data from ESR-STM experiments choi2017atomic . The linewidths of the ESR-STM experiments are about several Megahertz, which are due to the short coherence time of the ions. Electron spin labels in molecular rulers have longer coherence times than those of metal ions in ESR-STM experiments. The linewidths of the DEER experiments for Gd-rulers are about sub-Megahertz. The method here can provide better precision of detection for coupling between two electrons compared with ESR-STM experiments. The red line is the constraint established in this work, which clearly shows that more stringent constraints in the range from 3 to 220 nm. Specifically, the obtained upper limit of the exotic dipole-dipole interaction at 200 nm is about a factor of 20 more stringent than the one obtained from Ref. luo2017constraints .

In summary, we provide a novel method to search for an exotic spin-dependent interaction at the nanometer scale. An improved constraint on axial-vector mediated interaction between electron spins has been established at the nanometer scale. DEER measurements have been carried on molecule with triple spin labeling with $Gd^{3+}$, $Mn^{2+}$, and a nitroxide Wu_2017 , these measurements based on chemically synthesized molecules may provide new possibilities for searching for exotic spin-dependent interactions. We also expect that spin labels of molecular rulers with longer coherence times can be synthesized for this study. Since longer coherence times ensure longer observation times, better frequency resolutions can be achieved and the constraint can be further improved.

Authors thank Prof. Dieter Suter for helpful discussion. We are also grateful to Prof. Daniella Goldfarb for sharing us the experimental data of the DEER experiments. We do appreciate Xiaojing Liu for the DFT calculations. This work was supported by the National Key R$\&$D Program of China (Grants No. 2018YFA0306600 and No. 2016YFB0501603), the Chinese Academy of Sciences (Grants No. GJJSTD20170001, No.QYZDY-SSW-SLH004 and No.QYZDB-SSW-SLH005), and Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000). X.R. thanks the Youth Innovation Promotion Association of Chinese Academy of Sciences for support. Y.F.C is supported in part by the NSFC (Nos. 11653002, 11722327), by National Youth Thousand Talents Program of China, by the CAST Young Elite Scientists Sponsorship Program (2016QNRC001), and by the Fundamental Research Funds for the Central Universities.

We noticed that a revision of spin-dependent interactions have recently been proposed PhysRevA.99.022113 . We provide the constraints on the revisited interactions in the Supplemental Material SM .

I 1. Distance distribution of the molecular rulers

In ESR experiments where the signal is contributed from ensemble molecules, it’s necessary to take the flexibility of molecules into account. The flexility of molecules can be estimated by the distance distribution of the molecular backbone. The distance distribution of the backbone containing repeating units of p-phenylene (PP) and ethynylene (E) can be estimated by harmonic segmented chain model(HSC), where the distance distribution is determined by the bending potential F and the thermal energy $k_{B}T$ jeschke2010flexibilitySM . Here we take Gd-ruler $1_{11}$ as an example, which contains 11 repeating units of PP and E. The distance distribution derived from HSC model is shown in Figure S1a with the FWHM of the distance distribution being $0.397nm$.

The linewidths of DEER spectrum caused by flexibilities can be estimated by the first term of Eqn.4 in the main text. As shown in Figure S1b, the FWHM of the coupling strength caused by flexibilities is $0.008~{}$MHz, which is much smaller than that of the experimental result which is about $0.1054~{}$MHz. It shows that the molecular rulers with high stiffness are good platform to investigate the spin-dependent interactions.

II 2. Constraints on Revisited spin-dependent interactions

Recently, a revision of spin-dependent interactions has been proposed, which gives nine non-relativistic potentials by the exchange of a spin-0 or spin-1 boson in coordinate-space PhysRevA.99.022113SM . Different from the spin-dependent interactions derived in a mixed momentum and coordinate-space proposed in 2006 dobrescu2006spinSM , the revisited spin-dependent interactions in coordinate-space sort the spin-dependent interactions in a different way. Our result can also provide constraints on the revisited spin-dependent interactions. We consider the axial-vector potential mediated by a massive spin-1 boson of the revisited interactions, the interaction between electrons can be written as PhysRevA.99.022113SM ,

[TABLE]

where $m$ is the mass of the hypothetical particle, $g_{A}^{e}g_{A}^{e}$ is dimensionless axial-vector coupling constant between electrons and $c$ is the speed of light. We take the combination of parameters $|g_{A}^{e}g_{A}^{e}|/4\pi m^{2}$ as a constant and give constraints on it (Fig. S2).