All-optical quantum sensing of rotational Brownian motion of magnetic molecules

TL;DR

This paper introduces a novel all-optical method using NV centers in nanodiamonds to detect fast rotational Brownian motions of magnetic molecules, enabling high-resolution sensing of local viscosity and molecular dynamics.

Contribution

The work demonstrates a new all-optical technique for detecting rapid rotational motions of small magnetic molecules using NV centers, surpassing conventional optical methods in spatial resolution.

Findings

Successfully detected motions of Gd molecules with NV centers.

Differentiated solutions with varying viscosities through relaxation signals.

Established a new approach for high-resolution molecular motion sensing.

Abstract

Sensing local environment through the motional response of small molecules lays the foundation of many fundamental technologies. The information of local viscosity, for example, is contained in the random rotational Brownian motions of molecules. However, detection of the motions is challenging for molecules with sub-nanometer scale or high motional rates. Here we propose and experimentally demonstrate a novel method of detecting fast rotational Brownian motions of small magnetic molecules. With electronic spins as sensors, we are able to detect changes in motional rates, which yield different noise spectra and therefore different relaxation signals of the sensors. As a proof-of-principle demonstration, we experimentally implemented this method to detect the motions of gadolinium (Gd) complex molecules with nitrogen-vacancy (NV) centers in nanodiamonds. With all-optical measurements of the NV centers' longitudinal relaxation, we distinguished binary solutions with varying viscosities. Our method paves a new way for detecting fast motions of sub-nanometer sized magnetic molecules with better spatial resolution than conventional optical methods. It also provides a new tool in designing better contrast agents in magnetic resonance imaging.

Equations

$$\frac{1}{T_1}=\frac{1}{T_{1,bulk}}+\sum _{k}3{\gamma }_k^2{B}_{\perp ,k}^2\frac{{\tau }_{c,k}}{1+{\omega }_0^2{\tau }_{c,k}^2},$$

$$R_{Gd,tot}=R_{Gd,dip}+R_{vib}+R_{trans}+R_{rot}$$

$$R_{rot}=\frac{k_{B}T}{8\pi a^3\eta f_r},f_r={\left(\frac{6a_s}{a}+\frac{1+\frac{3a_s}{a+2a_s}}{(1+\frac{2a_s}{a}{)}^3}\right)}^{-1}$$

$$\delta R_{Gd}^{min}\sqrt{T}\approx \frac{1}{C\sqrt{\mathcal{D}{T}_D}}\sqrt{\frac{2e{R}_{Gd,tot}}{3{\gamma }_e^2{B}_{\perp ,Gd}^2}}\frac{(R_{Gd,tot}^2+{\omega }_0^2{)}^{3/2}}{|R_{Gd,tot}^2-{\omega }_0^2|},$$

Full text

1 Introduction.

Characterizing local environments and capturing dynamical variations of local quantities can provide important information about physical and biological processes. In particular, detecting variations in molecules’ Brownian motions would yield information about particle size and local viscosity in biological sensing [1]. For example, the random rotations of small molecules can respond to environmental changes and reveal local dynamics and biological functions. Conventional optical microscopy methods are capable of measuring the rotational Brownian motions (RBM) of particles in the sub-micron scale or larger, using dark-field microscopy [2, 3], microrheological techniques based on particle’s anisotropy [4, 5] or fluorescence polarization spectroscopy [6]. However, it is very difficult to extend these techniques to the nanometer regime: the rotational motion of nanometer-size molecules is typically much smaller than the optical diffraction limit, therefore it cannot be directly captured; in addition, their motional rates are typically in the GHz range, which are beyond the detection rates of most optical techniques. To this end, an all-optical method capable of capturing the fast rotations is still lacking.

Quantum sensors [7] have recently emerged as a powerful tool to explore properties at the nanoscale. For example, nitrogen-vacancy (NV) color centers in diamond have shown the potential to study magnetic [8, 9, 10, 11] and electric [12] fields, temperature [13, 14, 15, 16], strain [17, 18, 19] and other quantities, displaying an exceptional sensitivity and spatial resolution even at room temperature, thanks to the control on their spin states. In particular, NV centers in nanodiamonds (NDs) have been employed to sense various properties [20, 21, 22, 23] that affect, and reveal, chemical and biological processes. NV centers in NDs have many favorable properties, ranging from very high photo- and thermal-stability, to bio-compatibility. All these desirable properties make NDs preferable over organic dyes in biosensing applications [24]. Furthermore, the size of NDs containing stable NV centers can reach down to just several nanometers [25], promising nanometer spatial resolution.

Here we propose a novel method that exploits the electronic spins of NV centers in NDs to detect the fast RBMs of sub-nanometer sized magnetic molecules, overcoming the limitations of conventional optical approaches. The key idea is that fast RBM rates modify the spectrum of the magnetic noise generated by the magnetic molecules, slowing down the relaxation of our spin sensors in a way analogous to motional narrowing in NMR [26]. The all-optical measurement of the relaxation rates of our spin sensors captures the RBMs of the target magnetic molecules, while providing a simple and accessible experimental technique. We point out that while we consider RBM of magnetic molecules here, this method can be extended to detect other molecules’ RBM by attaching magnetic labels to them.

We demonstrate this novel method in a proof-of-principle experiment. The longitudinal relaxation of NV centers in NDs is monitored in the presence of gadolinium (Gd) complex molecules, which are among the most commonly used contrast agents in magnetic resonance imaging (MRI). We observed a difference in the relaxation rate corresponding to changes in the RBM rate when varying the local viscosity, thus demonstrating the feasibility of the proposed method.

Conventionally, the rotational motion of Gd(III) complex molecules can be indirectly extracted from EPR spectra with nuclear magnetic relaxation dispersion (NMRD) analysis [27, 28, 29]. However, this technique not only entails demanding experimental conditions, but also it does not provide the complete picture of Gd molecules RBM. For example, the rotational correlation time obtained from proton’s or 17O’s relaxation measurements represents, instead of the rotation information for the whole molecule, the local rotational correlation time of specific bond vectors. Compared with the NMRD technique, our method is more robust as it measures the effective rotational motion rates of the whole molecule. Moreover, in principle, the spatial resolution is only limited by ND size, which is in the nanometer regime. The experimental apparatus is simpler, as NV centers allow all-optical initialization and readout, without the need for a strong external magnetic field. Finally, our method is fully compatible with microfluidic techniques, supporting study of RBM in living cells where sample volume is constrained. Our technique would then provide a versatile tool to understand the dynamics of magnetic molecules and design better contrast agents in (functional) MRI [29, 30, 31]. For instance, magnetic molecules with long rotational correlation times are usually preferred for higher relaxivity. Biologically relevant ligand coatings or bindings in synthesized magnetic nanoparticles could increase the particles’ size and therefore increase their relaxivity. The method we propose here provides an independent study of rotational correlation time and would be helpful in quantifying the performance of contrast agents.

2 Theoretical principle

The fast RBMs of small magnetic particles contribute to high frequency fluctuations in the magnetic noise spectrum. This effect can be detected by nearby spins, whose relaxation times are sensitive to external magnetic noise. Counterintuitively, the added motion actually often lengthens the spin relaxation time. To understand how, consider a Lorentzian spectrum as a function of the noise fluctuation rate $R$ (Fig. 1.(a)). When the fluctuation rate is larger than the resonant frequency of the spin sensors $R>\omega_{0}$, a further increase in fluctuation rate results in a smaller noise spectrum intensity at the spin resonance frequency. This effect is similar to motional narrowing in NMR [26], and leads to a longer relaxation time of the spin sensors.

To provide a more quantitative description, we analyze all the effects contributing to the longitudinal relaxation time T1. In the presence of magnetic particles, including paramagnetic impurities on the surface and in the surrounding solution, the T1 time of a single NV electronic spin is given by [21]:

[TABLE]

where $T_{1,bulk}$ is the relaxation time of NV in bulk diamond and $\omega_{0}=(2\pi)2.87$ GHz the NV energy level splitting between the $|m_{s}=0\rangle$ and $|m_{s}=\pm 1\rangle$ states at zero external magnetic field. For each magnetic particle $k$, $\gamma_{k}$ is the gyromagnetic ratio; $B_{\perp,k}$ is the rms transverse magnetic noise strength (see the supplementary material [32]); and $\tau_{c,k}=1/R_{k}$ is the noise correlation time, the inverse of the noise fluctuation rate. As already mentioned, when $R_{k}>\omega_{0}$, any further increase in $R_{k}$ results in longer T1.

As an example, we consider using T1 relaxation to detect the RBM of Gd(III) chelators. The total noise fluctuation rate from Gd(III) molecules $R_{Gd,tot}$ is given by [8]:

[TABLE]

where $R_{Gd,dip}$ represents the dipolar interaction rates between Gd molecules, $R_{vib}$ the intrinsic vibrational rate between the Gd ion’s vibrational energy levels, and $R_{trans(rot)}$ the translational (rotational) Brownian motion rates. Variations in the fluctuation rate $R_{Gd,tot}$ induce changes in the T1 signal decay time of NV electronic spins via Eq. (1). It is then possible to detect RBM and sense the local environment through T1 relaxometry of NV sensors in an all-optical fashion.

3 Experimental demonstration

Next, we demonstrate our proposed method in a proof-of-principle experiment. We use NDs hosting NV centers as sensors to detect the RBM rates of one type of Gd(III) complex, gadolinium 1,4,7,10-tetraazacyclododecane-N,N’,N”,N”’-tetraacetate (Gd-DOTA). At zero magnetic field, the NV center has an electronic spin-1 ground state with resonance frequency $\omega_{0}=(2\pi)2.87$ GHz (Fig. 1(c)). The spin can be optically polarized and read out with $532$ nm laser excitation. In the absence of laser illumination, the spin state population will reach its thermal equilibrium with a time scale T1.

We measured this longitudinal relaxation time T1 of an ensemble of NV centers with a home-built confocal microscope at room temperature, without any applied external magnetic field. The NDs we used have an average size of around $25$ nm and are terminated with carboxyl groups. In the absence of Gd-DOTA, we measured a relaxation time of about $130~{}\mu$s, which is significantly shorter than T1,bulk (typically a few ms, even in the presence of other bulk paramagnetic impurities). We attribute this difference to unpaired paramagnetic spins on the ND surface [33, 34, 35, 36]. The magnetic noise induced by these surface spins adds a new depolarization channel and yields a decrease in T1 according to Eq. (1). Based on this model, we can further deduce the surface spin density to be about 1 nm*-2* [32]. The T1 relaxometry can then provide information in estimating surface spin densities of ND samples.

We first show the strong T1 quenching effect induced by Gd molecules in the solution. According to Eq. (1), the presence of Gd molecules will induce a strong magnetic noise in addition to that arising from the surface spins, and significantly increase the sensor’s spin relaxation rate $1/T_{1}$. We prepared solutions of NDs and varying densities of Gd-DOTAs, and measured the corresponding T1 of the NV centers. Higher densities of Gd-DOTA molecules lead to stronger magnetic noise thus larger T1 quenching ratio. The experimental results quantitatively match our theoretical predictions if we assume a higher Gd density than the average density in solution (Fig. 2). This is likely due the tendency of Gd molecules to accumulate close to the bottom of the microfluidic channel. Since we measured NV spins at this location, this accumulation yields higher Gd density close to detection spots than the average concentration in the solution. Then, as the sensors only detect the local rather than global averaged environment, they see an effective higher Gd molecule density.

We next show the capability of detecting the RBMs of Gd-DOTA molecules with our sensors. In particular, we are interested in demonstrating the ability to distinguish variations in the magnetic particle fluctuation rates due to changes in the solution viscosity. We dissolved a fixed density of Gd-DOTA molecules in solutions with varying concentrations of water and acetone. Varying the ratio between water and acetone in the binary solution changes the local viscosity felt by Gd-DOTA molecules. We perform T1 relaxation measurements by loading the sample in a microfluidic channel to mimic biological environment, while preventing the binary solution from quickly evaporating. We observed a clear difference (4.34 $\sigma$) in the measured T1 times for pure water with respect to (nearly) pure acetone (Fig. 3 (a)). To understand more quantitatively these results, we develop an analytical model of the RBM rate in our experiments.

The RBM of a molecule is influenced by its local viscosity. For a molecule with hydrodynamic radius $a$ in a solvent of viscosity $\eta$ and molecular radius $a_{s}$, the RBM rate is expressed by the Stokes-Einstein equation [27, 29]:

[TABLE]

Here $f_{r}$ is the microviscosity factor that takes into account the discrete nature of solvent molecules [27, 37]. As we can see, a lower viscosity corresponds to faster motions, with a RBM rate $R_{rot}$ as large as $14$ GHz in our experiment. We plot in in Fig. 3 (b) the dynamical viscosity $\eta$ and the expected RBM rate of Gd-DOTA. The RBM rate of Gd-DOTA molecules varies between around $2$ to $14$ GHz in our water-acetone binary solution: This large range is attributed to the contrast in viscosities as well as different molecule sizes of water and acetone. Since $R_{Gd,tot}\gg\omega_{0}$ due to the large $R_{Gd,dip}$ contribution, we expect longer relaxation time of the NV sensors for higher RBM rates, as seen in Fig. 3 (a).

We further verified that our assumption that the microviscosity factor is essential to describe the RBM of the (nanoscale) Gd-DOTA molecules is correct. We compared the estimated RBM rate using Eq. (3), either with and without the microviscosity factor (Fig. 3 (c,d)), and found the predicted RBM rate with the microviscosity factor agrees better with experimental results (Fig. 3 (a)). This matches our expectation, since the discrete nature of solvent molecules should come into play when sub-nanometer sized Gd-DOTA molecules are concerned. Finally, from the experimental data and considering a $10\%$ variation in Gd molecule density, we estimate that the RBM rate of Gd molecules in nearly pure acetone is $20.2\pm 10.6$ GHz, consistent with the $14.2$ GHz calculated from Eq. 3 at room temperature.

Since our measurements were performed with an ensemble of NDs, one concern was the inhomogeneous density distribution of Gd molecules when one changes solutions. To investigate this issue, we compared the distribution of relaxation times of spatially separated sensing spots for the water and nearly pure acetone cases. As one can see in Fig. 4 (a), the two distributions spread broadly, which might be a result of the spatial inhomogeneity of Gd molecule density or spatially varying local charge environments (see next section). Nevertheless, the two distributions are clearly distinct form each other, as shown in Fig. 4 (b), with Gaussian fittings. This further demonstrates that our relaxation measurements can distinguish different RBM rates influenced by local viscosities.

4 Discussions

To quantify the performance of our proposed method, we can estimate the minimal detectable value of the total magnetic noise fluctuation rate $\delta R^{min}_{Gd}\sqrt{T}$ per unit time, for a single NV center located at the center of a single ND. We find

[TABLE]

where $T_{D}$ is the detection window, $\mathscr{D}$ the photon counting rate, and $C$ the contrast. In our experiments we have a detection window $T_{D}=500$ ns. We assume a signal contrast $C=0.2$ and a photon counting rate $\mathscr{D}=1\times 10^{5}$ counts/s for a single NV. At optimized Gd density (corresponding to total Gd fluctuation rate $R_{Gd,tot}\approx 60.2$ GHz), we get a sensitivity of $\delta R^{min}_{Gd}$=6.9 (9.6) GHz for a single ND with diameter $d_{0}$=20 (25) nm in a $T=10$ s data acquisition time, as presented in Fig. 4 (c). Note that here the calculation only takes the photon shot-noise into consideration, as it is the main source of detection noise.

To extend our method to single NV measurements, which can yield superior spatial resolutions, we need to have well-characterized NV centers with spatially homogeneous distributed magnetic molecules around them. In addition, the charge state conversion between NV*-* and undesired NV0 states should be well characterized. Indeed, the charge conversion rate might vary both due to different chemical solutions and across NV centers, thus masking the real relaxation differences. As we averaged over many NDs, this was not a main concern here. We point out that this issue can be addressed by characterizing or mitigating the charge instabilities beforehand with the use, for example, of additional lasers to control the photoionization process, or microwave control to better select the NV- dynamics [38, 39, 40].

We note that the presence of carboxyl termination group on the surface of NDs might naturally compete and interact with the DOTA chelators. Careful surface treatment should be performed to reduce this effect and increase the colloidal stability of NDs to make the technique more suitable for practical applications. For example, fluorination of NDs can suppress detrimental hydrophobic interactions and make NDs’ aqueous solution colloidally stable [41, 42]. The method we proposed here is capable of determining the RBM rate (thus viscosity and particle hydrodynamic size) and, subject to careful evaluations of all assumptions in our model [32], we can turn our qualitatively measurements into quantitative ones, while improving the sensitivity. Future studies that independently determine particle accumulation in the microfluidic channel could also help improve the performance of our sensor.

We have shown that, in the context of quantum sensing, the T1 relaxometry provides a versatile tool to detect the RBM rate changes in response to variations of local environments. At the same time, we point out that the relation between sensor relaxation and rate variation can be employed in MRI techniques, where the fluctuation rates of contrast agents (usually magnetic molecules such as Gd(III) chelators) are modulated to increase their relaxivity and improve their performance [29].

5 Conclusions

In conclusion, we proposed a novel method to detect the rotational Brownian motion of nanometer-sized magnetic particles, and experimentally demonstrated the protocol using NV centers in nanodiamonds. The technique is capable of detecting fast rotation (GHz-rate) of magnetic molecules with size down to sub-nanometer scale. The experiment is fully optical, requiring no microwave control or any external magnetic field. Our quantum sensing technique provides a new way of extracting local viscous information with high spatial resolution, and could also contribute to the design of contrast agents for MRI.

**Acknowledgements **

We thank Tingtao Zhou and Yixiang Liu for fruitful discussions, Liyuan Zhang and Yinan Shen for help with DLS measurement and Kurt Broderick for assistance with the fabrication of microfluidic channels. This work was supported in part by the U.S. Army Research Office through Grants No. W911NF-11-1-0400 and No. W911NF-15-1-0548. DL acknowledges financial support from the German Research Foundation (DFG) with a postdoctoral fellowship.