Diamond based single molecule magnetic resonance spectroscopy

The detection of a nuclear spin in an individual molecule represents a key challenge in physics and biology whose solution has been pursued for many years. The small magnetic moment of a single nucleus and the unavoidable environmental noise present the key obstacles for its realization. Here, we demonstrate theoretically that a single nitrogen-vacancy (NV) center in diamond can be used to construct a nano-scale single molecule spectrometer that is capable of detecting the position and spin state of a single nucleus and can determine the distance and alignment of a nuclear or electron spin pair. The proposed device will find applications in single molecule spectroscopy in chemistry and biology, such as in determining protein structure or monitoring macromolecular motions and can thus provide a tool to help unravelling the microscopic mechanisms underlying bio-molecular function.

The detection of a nuclear spin in an individual molecule represents a key challenge in physics and biology whose solution has been pursued for many years. The small magnetic moment of a single nucleus and the unavoidable environmental noise present the key obstacles for its realization.
Here, we theoretically demonstrate that a single nitrogen-vacancy (NV) center in diamond can be used to construct a nano-scale single molecule spectrometer that is capable of detecting the position and spin state of a single nucleus and can determine the distance and alignment of a nuclear or electron spin pair. In combination with organic spin labels, this device will find applications in single molecule spectroscopy in chemistry and biology, such as in determining protein structure or monitoring macromolecular motions and can thus provide a tool to help unravelling the microscopic mechanisms underlying bio-molecular function.
Introduction.-As single nuclei have exceedingly small magnetic moments, a large ensemble (typically 10 18 nuclear spins) is necessary to obtain an observable signal employing methods such as magnetic resonance spectroscopy (NMR). As a consequence, chemical and biological processes have thus usually been tracked with ensemble measurements, which only provide ensemble averages and distribution information. Single-molecule studies can instead allow one to learn structural information and time trajectories of individual molecules free of natural disorder [1][2][3][4]. The detection of a single nucleus can thus provide various new possibilities for single molecule spectroscopy. Furthermore, single nuclear spins have long coherence time due to their weak coupling with the environment which makes them promising candidates for a qubit or a quantum register. For example, it has been proposed to engineer nitrogen nuclear spin in the molecule of 14 N@C 60 [5,6] or phosphorus donor [7] as a qubit candidate. For these purposes efficient readout of the nitrogen nuclear spin state of a single molecule is crucial.
Nitrogen-vacancy (NV) defects in diamond have been used to construct ultrasensitive nano-scale magnetometers [8][9][10][11][12][13], and have led to interesting applications in nano imaging [14,15] and biology as well [16][17][18]. NV centers in diamond benefit from long coherence times at room temperature and highly developed techniques for coherent control and optical readout of their electron spins. A single NV centre can be used to detect single external electron spin [19] and a proximal nuclear spin (within a few atom shells) [20][21][22]. It has also been shown that it is possible to detect a strongly coupled nuclear spin pair [23] and measure noise spectra [24,25] by applying dynamical decoupling pulses. The scheme nevertheless is not applicable for sensing single nuclei [23][24][25], as the effect of a single nucleus is completely removed by the decoupling pulses and is featureless in the dynamics of NV spin. As a consequence the detection of individual nuclear spins in the presence of realistic environments remains an unsolved challenge. Key obstacles originate firstly from the requirement that coherence times are sufficiently long in order to observe the effect of single nuclei on NV spin. The second key obstacle lies in the difficulty of distinguishing a distant nuclear spin from other environmental nuclei that couple to the NV spin.
In this work, we address both challenges by continuously driving a single NV spin and use it as a probe to measure a specific transition frequency of the target system. The role of continuous driving is two-fold: firstly, it decouples the NV spin from the unwanted influence of a spin bath (which is particularly strong for NV centers located close to the surface [19]) to achieve sufficiently long coherence times [26][27][28]; secondly, by changing the Rabi frequency of the external driving field we can tune the NV center to match the target frequency and thus selectively enhance the sensitivity for this specific frequency and thus single out the target system. With this scheme, we are able to determine the position of a single nucleus in the presence of a realistic environment. We demonstrate how to use this mechanism to implement quantum non-demolition(QND) measurement of a single nitrogen nuclear spin state in a cage molecule of fullerene ( 14 N@C 60 ). This is achieved by exploiting the fact that the flip-flop between the NV spin and the target system will either be allowed or prohibited dependent on the nuclear spin state for a specific NV spin initial state. Our proposed detector can be applied straightforwardly to the determination of the distance and alignment of a spin pair. We show that the present model can find applications in single molecule spectroscopy with organic spin labels. It can also be used to witness the creation and recombination of charge separate state in radical pair reactions. We expect that the present diamond-based single molecule probe will have more potential applications in determining protein structure and monitoring chemical (biological) processes, see the examples of molecular motors, ATPase and RNA folding in [29][30][31]. We apply a continuous field to drive the electronic transition |ms = 0 ↔ |ms = −1 . When the Rabi frequency Ω is on resonance with one specific transition frequency of the target system, the flip-flop process will happen and lead to the decay of the NV dressed state population.
Basic model of NV center spectrometer-The probe in our model is a nitrogen-vacancy (NV) center spin in diamond, whose ground state is a spin-1 with a zerofield splitting of 2.87GHz. By applying an addition magnetic field, one can lift the degeneracy of |m s = −1 and |m s = +1 and thus allow selective driving with continuous microwave field of one specific electronic transition, e.g. |m s = 0 ↔ |m s = +1 (henceforth denoted by |0 and |+1 respectively) . Within the |0 , |+1 subspace the Hamiltonian of the driving field can be written as H N V = Ωσ x , where σ x is spin-1 2 operator and the eigenstates represent the dressed states of the system. We remark that different driving schemes may be used depending on the properties of target systems as we will discuss later. Our goal is to use such a NV dressed spin as a probe to detect a specific frequency in the target system. The interaction between the NV spin and another spin is H N V −S = N g N [3 (S ·r) (I N ·r) − S · I N ], where S, I N are the NV spin and target spin operators, g N = −( µ 0 γ N γ e )/(4πR 3 N ), γ e and γ N are the gyromagnetic ratio of electron spin and target spin respectively,r is the unit vector that connects NV center and the target spin. The large zero-field splitting leads to energy mismatch which prohibits direct NV spin flip-flop dynamics and allows for the secular approximation to simplify the NV-target spin interaction to (1) The coupling operator S z leads to dephasing type interaction in the bare basis but induces flips of the NV spin in the dressed states basis. The flip-flop process will be most efficient when the Rabi frequency Ω, i.e. the energy splitting of the dressed states, matches the the transition frequency of the system spin that we wish to probe. Therefore, one can initialize the NV spin in one of the dressed states and tune the Rabi frequency to determine the transition frequency of the target system by mea-suring the probability that the NV spin remains in the initial state. The target system distance can be inferred from the flip-flop rate of the NV center if the magnetic moment is known.
Measure position of a single nucleus.-We first apply our idea to detect the position of a single nucleus. This is particularly interesting in biology, for example in determining where certain important nuclei (e.g. 14 N, 31 P) are located in the protein complex. We consider the combined system of NV spin under driving and a target nuclear spin with the following effective Hamiltonian where σ z is the spin-1 2 operator, and the hyperfine vectorĥ is determined by the position vector asĥ x = 3r x r z / 3r 2 z + 1,ĥ y = 3r y r z / 3r 2 z + 1 andĥ z = (3r 2 z − 1)/ 3r 2 z + 1. The effective nuclear spin Larmor frequency ω N is determined by We choose the Rabi frequency Ω on resonance with the nuclear Larmor frequency to match the Hartmann-Hahn double resonance condition as Ω = ω N [32]. After preparing the NV spin in the initial state |+ = 1 √ 2 (|0 + |+1 ) and assuming that the nuclear spin is in the thermal state namely ρ N (0) = 1 2 I at room temperature, the signal where J = 1 4 g 3r 2 z + 1 1 −ĥ ·b is the effective spin flip-flop rate. The effect of the 13 C spin bath in diamond can be greatly suppressed if the 13 C environmental spectral density is low at the Rabi frequency Ω of the driving field. We can use an ultra pure diamond sample, and thus the intra-interaction between 13 C spins is weak. The relevant frequency of the spin-bath is mainly determined by the applied magnetic field, namely ω = γ13 c B. If the mismatch between Ω and ω is sufficiently large, the NV spin will be effectively decoupled from the 13 C spin bath. There is a possibility to grow isotropically pure 12 C diamond [33]. The sensitivity for measuring the target nucleus is then determined by the relaxation time T 1 which is very long for an NV spin (T 1 ∼ 10 ms at room temperature) [34]. We demonstrate our idea by showing how to use a NV spin to measure the position of 31 P in a single molecule 1 H 3 31 PO 4 . Due to its half-integer nuclear spin and high abundance of 31 P, NMR spectroscopy based on phosphorus-31 has become a useful tool in studies of biological samples. Phosphorus is commonly found in organic compounds, coordination complexes and proteins, such as phosphatidyl choline which is the major component of lecithin. In 1 H 3 31 PO 4 , 31 P interacts with a few adjacent hydrogen atoms, which captures generic features of the protein. Thus it can serve as a paradigmatic example for practical applications of our model. We apply a magnetic field such that the Larmor frequencies of 31 P, 1 H and 13 C are different. The Rabi frequency of the NV spin is tuned to match the (effective) Larmor frequency of 31 P and is sufficiently strong to eliminate the effect from the 13 C spin bath, see SI for details. The line broadening caused by the adjacent 1 H is relatively large when compared with the coupling between the NV spin and 31 P. To overcome this problem, we propose to continuously drive 1 H nuclei, which effectively averages out the line broadening. The difference of Larmor frequencies among different nuclei allows us to selectively drive 1 H while not affecting the other nuclei. With this, we can obtain information of the hyperfine vector (and thereby the relative direction of 31 P), see Fig.2(a). One can then measure the flip-flop rate, from which the distance from 31 P to NV center can be obtained from Eq.(3), see SI for more details. We remark that shallow (below 5 nm) implantation was realized experimentally [35].
Quantum non-demolition measurement of a single nuclear spin state.-In the above model, the flip-flop process happens under the resonance condition but also requires that the nuclear spin state is complementary to NV spin state. As nuclear spins have long coherence time and can serve as robust qubit we would like to show that it is possible to perform QND measurement on the system of 14 N@C 60 [5,6], i.e. a nitrogen atom in a C 60 cage. The 14 N@C 60 molecule has an electron spin- 3 2 coupled to the 14 N nuclear spin-1. The hyperfine interaction is isotropic and the spin Hamiltonian is given by (Color online) (a) Scheme for quantum nondemolition measurement of the nitrogen nuclear spin state in 14 N@C60 with a NV center at a distance of 8nm. For the nuclear spin state |mI = 0 , the allowed electron transition frequency in 14 N@C60 is ωe and will not be on resonant with ωNV , while for |mI = +1 , the resonant condition is satisfied when ωe + a = 2Ω 2 + ω 2 e 1/2 . The signal S and the nuclear spin state fidelity as a function of time t for the nuclear spin state |mI = +1 (b) and |mI = 0 (c). The Rabi frequency Ω is chosen as Ω = 70.7MHz corresponding to the resonant frequency ΩNV = 316.2MHz for the nuclear spin state |mI = +1 . The applied magnetic field is 300MHz. The electron spin of 14 N@C60 is not polarized, and its initial state is approximated by the maximally mixed state.
where ω e = −γ e B, ω N = γ N B and the quadrupole splitting is ∆ Q = 5.1MHz, the hyperfine coupling is a = 15.88MHz. We can encode a qubit in the nuclear spin state |0 I = |m I = 0 and |1 I = |m I = +1 . As the electron and nuclear Zeeman splitting exhibit a large mismatch due to a strong magnetic field, the main effect of hyperfine coupling is to modulate the electronic transition frequencies dependent on the nuclear spin state. When the nuclear spin state is |m I = 0 , the allowed electronic transition frequency is ω e , while for the nuclear spin state |m I = +1 , it is ω e + a. We note that under an additional magnetic field, the energy separation of NV spin state |m s = +1 and |m s = −1 is ∆ = 2ω e , which means that we will inevitably drive both electron spin transitions (namely |0 ↔ |±1 ) if the driving amplitude Ω is around ω e in order to satisfy the resonance condition. We thus apply a driving field at the frequency ω 0 with the detuning ±∆ for two allowed electronic transitions. The effective electronic transition frequencies become ω 1 = 2Ω 2 + ω 2 e 1/2 and ω 2 = 2 2Ω 2 + ω 2 e 1/2 , see SI for details.
We prepare NV spin in the dressed state |D = (Ω |+1 + ω e |0 − Ω |−1 ) /(2Ω 2 + ω 2 e ) 1/2 and then tune the Rabi frequency Ω to be on resonance with the allowed electron transition frequency in the system 14 N@C 60 corresponding to the nuclear spin state |m I = +1 , namely 2Ω 2 + ω 2 e 1/2 = ω e + a. It is important to note that we do not require the electron spin of 14 N@C 60 to be polarized and assume it is in the maximally mixed state, i.e. ρ e = I 2 at room temperature. After a time t, if the nuclear spin is in the state |m I = 0 , the resonance condition will not be satisfied and the NV spin will remain in its initial state |D ; if the nuclear spin is in state |m I = +1 the population of state |D will change with time. The nuclear spin state remains unaffected throughout. Thus the readout represents a non-demolition measurement that can be repeated. The feasibility of this idea is verified by our numerical simulation, see Fig.3. It can be seen that the required time for one readout is much shorter than the coherence time T 2 of the electron spin in 14 N@C 60 (which is 20µs at room temperature [5]) when the distance between 14 N@C 60 and NV center is 8nm. We remark that the present mechanism can be exploited to polarize nuclear spins, see SI for details. Resonance of a pair of nitroxide spin labels with a distance of 8nm. The magnetic field along the vector which connects two spin labels. Continuous driving field with Rabi frequency of 20MHz is applied on the spin labels.
Measure distance and alignment of a spin pair.-For the tracking of molecular conformations or the determination of molecular structure it can be of advantage to determine the distance between two specific spins. To this end, we consider a target system which consists of two spins interacting with each other. To determine their intra-distance R and the orientation of the alignment vectorr, we apply a strong magnetic field ω N = γ N B, as the energy shift will depend on the relative orientation of distance vector and applied magnetic field. In the mean time make the NV spectrometer will be operated with a strong driving field to decouple from the 13 C spin bath in diamond. We tune the amplitude of driving on the NV spin around the Zeeman energy ω N , so that the dominant resonant frequencies are Ω = ω N ± 3g 4 1 − 3(r ·b(θ, φ)) 2 , see SI for more details. The exact values of g andr can be inferred by applying magnetic fields in nine directions and measuring the resonant frequencies respectively (see SI for more details).
As an example, we apply our idea to measure the distance between two hydrogens in a water molecule. In Fig.4, we plot the resonance frequencies with a magnetic field applied in thex direction. The results for other magnetic field directions can be found in supplementary information. The 1 H 2 O molecule is 5nm from the NV center, and the vector that connects two hydrogen atoms is described by (r, θ 0 , φ 0 ) = (0.1515nm, 118.2 o , 288.85 o ). With the resonance frequencies, we can infer that g = 34.684kHz, which implies an the intra-molecular distance of d = 0.1518nm and the alignment direction is (θ, φ) = (118.29 o , 288.82 o ), see SI for more details. The difference from the exact values is due to the slight Larmor frequency shift induced by the NV spin. If the spin pair is rotating randomly for example in solution, the resonance frequency will only dependent on the intradistance.
The above idea can be combined with spin labels and have potential applications in chemistry and biology. Spin labels are organic molecules with a stable unpaired electron [36]. They can be attached to the protein (covalent or as a ligand) via a functional group. It was proposed to use a spin label to amplify the sensitivity of a diamond manometer [12]. Electron spin resonance (ESR) on an ensemble based on spin labels has widely been used as a spectroscopic ruler to determine protein structure and monitor macromolecular assembly processes. However, it is very hard to go beyond a distance of 5nm between spin labels, due to e.g. inhomogeneous line broadening. We consider the widely used nitroxide spin labels and show that the resonance linewidth (which mainly depends on the coupling strength between NV spin and spin labels) assisted by dynamical nuclear polarization and continuous drivings can be smaller than the resonance frequency splitting for a pair of spin labels with a distances larger than 5nm (more details are present in supplementary information), see Fig.4 (b) for an example of 8nm. In a similar way, one can monitor the creation and recombination of radical pairs, e.g. in reaction centers, see supplementary information for more details.
Summary.-We have proposed a scheme to construct a nano-scale single molecule spectrometer based on NV centers in diamond under continuous driving. This spectrometer is tunable by changing the Rabi frequency. We demonstrate its application to the detection of a single nucleus, including its position and spin state. The idea can also be used to measure the distance and alignment of a spin pair. This opens a novel route to determine the structure of proteins and monitor conformational changes and other processes of relevance to biology, chemistry and medicine. We expect that our result and its extension can greatly enrich the diamond-based quantum technologies and their applications in chemistry and biology.

SUPPLEMENTARY MATERIAL
Continuous dynamical decoupling of the 13 C spin bath The NV center spin is coupled with a 13 C spin bath, the interaction is described as follows where R m is the distance from NV center to the nuclei, andr is the unit vector that connects NV center and nuclei. The Hamiltonian of the spin bath itself is where R mn is the distance between two nuclei, andr is the unit vector that connects two nuclei. The mass density of diamond is ρ = 3.5g · cm −3 , so the number density of carbon is n = 3.5/(12 × 1.66 × 10 −24 )cm −3 = 176nm −3 . For a sample with 0.01% 13 C, the number of 13 C in a sphere with the radius 4nm centered at NV center is 176 × 4π 3 × 4 3 × 0.01% ≈ 5. As we want to single out the effect of the target system, we apply continuous driving filed on NV spin. Here, we use exact numerics to show that our continuous dynamical decoupling is very efficient for the parameters that we use for the examples in the main text, see Fig.5. We consider a diamond sample with 8 13 C spins in a 4 nm sphere. We remark that, in a similar way, continuous dynamical decoupling will also suppress the noise from the surface of diamond if it is due to spins that are different from the target spin. The effective hyperfine interaction between NV spin and the target spin is We introduce the hyperfine vectorĥ(θ 0 , φ 0 ), which is determined by the vectorr that connects the NV center and the target spin as sin (θ 0 ) sin φ 0 = 3r y r z 3r 2 z + 1 .
The effective Hamiltonian is thus rewritten as We note that S z = |+1 +1| = 1 2 I + σ z where σ z is the spin-1 2 operator, thus the above Hamiltonian is where the effective nuclear spin Larmor frequency is determined by B e = B(θ, φ) − 1 2 g 3r 2 z + 1 ĥ (θ 0 , φ 0 ). We choose the driving amplitude Ω 0 on resonance with the Larmor frequency of the target nuclear spin namely Ω = γ N B e . To measure the position of the target spin, we first prepare NV spin in the initial state |+ = 1 √ 2 (|0 + |1 ) and assume that the nuclear spin is in the thermal state which is well approximated by ρ N (0) = 1 2 I. If the Zeeman energy is much larger than the coupling (ω N = Ω g), the flip-flip between the NV spin and the nuclear spin is very unlikely, and the flip-flop is the dominant process. After time t, we measure the probability that the NV spin remains in the state |+ which is a function of the coupling strength J as follows From the signal S(t) one can infer the value of J that depends on the distance and direction of the nuclear spin to NV center The spin flip-flop rate J as in Eq.(14) depends on the direction of the effective magnetic field (θ e , φ e ). In the case that the applied magnetic field is much larger than the nuclear energy shift induced by NV spin, the direction (θ e , φ e ) can be approximated well by the applied magnetic field directionb(θ, φ). It can be seen that if and only if the magnetic field direction (θ, φ) is in parallel with the hyperfine vector, namely θ = θ 0 and φ = φ 0 , the effective flip-flop rate vanishes J = 0 even though the Hartmann-Hahn resonance condition is still satisfied. If the magnetic field is orthogonal to the hyperfine vector, the flip-flop rate is maximal as J m = 1 4 g 3r 2 z + 1 . By choosing a series of magnetic field directionsb(θ, φ), one can obtain both the values of the coupling strength g and the directionĥ(θ 0 , φ 0 ), from which one can obtain the direction of the target single spin (r x , r y , r z ) with respect to the NV center as follows The relative sign among r x , r y , r z can be determined from Eqs. (8)(9)(10).
We remarked that the fluctuation in the driving field (Ω) will limit the measurement precision. One way to overcome this problem is by using the dressed qubit from concatenated dynamical decoupling as propose in [1] (instead of a bare spin qubit) as a probe. For example, we can use a second-order sequential driving scheme with H d1 = Ω 1 cos(ωt)σ x and H d2 = Ω 2 cos [(ω + Ω)t] σ y , and get a qubit encoded in the second-order dressed states. In the second order interaction picture, we can tune the Rabi frequencies to satisfy the resonant condition Ω 1 + Ω 2 = ω N , and the coupling operator of NV spin with the target system becomes The energy gap of the dressed qubit {|+ y , |− y } is more robust against the error in the driving fields, see [1] for more details, and can thus make our scheme work with a higher accuracy. For 31 P and 1 H, the coupling strength is g = 48.6(6.075)kHz for the distance R m = 0.1(0.2)nm, which is relatively strong as compared the coupling strength between NV spin and the target 31 P spin. The sole information about the position of 31 P is blurred by such a strong interaction, see Fig.6 (a). We can apply a strong magnetic field such that (γ 1 H − γ 31 P )B g m , which suppresses the flip-flip (flip-flop) process between the target nuclear spins and its neighbouring hydrogen nuclei. The effective Hamiltonian of the target system thus becomes To overcome the line broadening (i.e. the last term in the above equation), we propose to drive the hydrogen nuclei continuously. Since the Larmor frequencies can be made sufficiently different for 31 P and 1 H, it is possible for us to selectively drive the hydrogen nuclear spins while not affecting the target 31 P nucleus due to a large detuning. This leads to the system Hamiltonian as follows If the Rabi frequency Ω 1 is large enough, the interaction term I z m I z N can be effectively averaged out and eliminated. We remark that one can also drive the target 31 P nuclear spin which can also suppress the line broadening caused by hydrogen nuclei.
In our numerical simulation, we apply a magnetic field such that the Larmor frequencies of 31 P, 1 H and 13 C are quite different, namely 500kHz, 1235kHz and 310.6kHz respectively. The driving amplitude of the NV spin is tuned to match the effective Larmor frequency of 31 P, thus its interaction with the 13 C spin bath is eliminated. The difference of Larmor frequencies also allow us to selectively drive the nuclei 1 H (with the driving amplitude 20kHz) while not affecting the other two types of nuclei. The distance between the nucleus 31 P and 1 H is about 0.2nm, so that the coupling constant is g ≈ 6KHz. The line broadening is much larger than the coupling between NV spin and the target nucleus 31 P (which is smaller than 1kHz at a distance of 5nm). In Fig.6(a), it can be seen that if we do not drive the nuclei 1 H, the interaction between 31 P and 1 H will smear the measured signal and we can hardly get information about the position of 31 P. By applying radio frequency driving field on resonance with the Larmor frequency of 1 H, we can then clearly obtain information of the hyperfine vector (and thereby the relative direction) of 31 P with respect to NV spin, see Fig.6(b).

Initial state preparation of a nuclear spin bath
Nuclear spins have long coherence times and can be used as robust quantum register [2]. It has been shown how to initialize the state of a nuclear spin proximal to the NV spin [3] and prepare a spin environment by optical cooling [4]. In the following, we demonstrate that our idea can also be exploited to initialize a nuclear spin bath with the distance from the NV center as far as a few nanometers with high efficiency and fidelity. We assume that the initial state of the nuclear spin is ρ N (0) = I 2 . The procedure of one cycle consists of two steps: first one prepares the NV spin in the state |+ = 1 √ 2 (|0 + |1 ) and then drives the NV spin for time τ with the Rabi frequency on resonance with the Larmor frequency of the nuclear spin. We define the nuclear spin up and down state along the direction of the applied magnetic field. After one cycle the nuclear spin down state |↓ population P ↓ increases from P ↓ (k) to P ↓ (k +1) = P ↓ (k)+ 1+cos (2Jτ ) 2 [1 − P ↓ (k)]. The formula will become more complicated if there are other nuclei. In fact one can prepare the initial state of several nuclear spins with the same procedure. The time τ is constrained by the spin-lattice relaxation time T 1 . To overcome such a limit, we can choose a small time τ and then repeat the above cycle.
The nuclear spin state |↓ population P ↓ after k cycles can be approximately written as P ↓ (k) = 1 − We consider a sample diamond with dilute carbon-13 nuclei (e.g. 0.01% 13 C) and show how one can initialize the quantum state of a single 13 C, which may later be used for quantum memory. It can be seen from Fig.7 that the 13 C can efficiently be initialized to a very high polarization (M = P ↓ − P ↑ ). As the 13 C is very dilute in such a diamond sample, the flip-flop process between different 13 C is very slow, and thus each 13 C is mainly directly polarized by the NV center. The required time is then dependent on the distance and alignment between the individual 13 C and the NV center.
Quantum non-demolition measurement of the nitrogen nuclear spin state in 14 N@C60 Our idea can be used to readout quantum state of a single nuclear spin. We have shown that it is possible to perform quantum non-demolition measurement on the system of 14 N@C 60 , which has a nitrogen atom in a C 60 cage. The molecule has an electron spin S = 3 2 coupled to the 14 N nuclear spin I = 1. The hyperfine interaction is isotropic and the spin Hamiltonian can be simplified as where ω e = −γ e B, ω N = γ N B and the quadrupole splitting is ∆ Q = 5.1MHz, and the hyperfine coupling strength is a = 15.88MHz. We here consider a qubit in the nuclear spin state |0 I = |m I = +1 and |1 I = |m I = 0 . If the difference between the electron and nuclear Zeeman splitting ω e − ω N is large, the non-secular terms in the hyperfine coupling can be neglected, and thus the system Hamiltonian is written as When the nuclear spin is in the state of |m I = 0 , the allowed electron spin energies is equidistant with ω e , while for the nuclear spin state |m I = +1 , it is ω e + a. By applying an additional magnetic field, the energy separation of the NV spin state |ms = +1 and |ms = −1 is ∆ = 2ω e , namely This means that we will inevitably driving both spin transitions |0 ↔ |±1 if the driving amplitude is around ω e in order to satisfy the resonant condition between NV spin and the electron spin in 14 N@C 60 . Thus, we have to take all three sublevels of the NV spin into account. Our idea is to apply a continuous driving field at frequency ω 0 = 2.87GHz as In the interaction picture with respect to H N V , we have where we have three dressed states as where η ± = (2Ω 2 + ω 2 e ) 1/2 ± ω e /Ω. We note that the NV spin coupling operator S z with the other spins will induce transitions among the dressed states with two transition frequencies ω 1 = (2Ω 2 + ω 2 e ) 1/2 and ω 2 = 2(2Ω 2 + ω 2 e ) 1/2 . Therefore, we first prepare NV spin in the state |D and then tune the Rabi frequency to be on resonance with the allowed electron transition frequency in the system 14 N@C 60 corresponding to the nitrogen nuclear spin state |m I = +1 , namely After time τ , we measure the probability that NV spin remains in the state |D . If the nuclear spin is in the state |m I = 0 , it can be seen that the resonant condition would never be satisfied once we drive the NV spin (namely The signal S measured at time t = 6µs as function of the effective NV transition frequency ωNV = (2Ω 2 + ω 2 e ) 1/2 for two nuclear spin state |mI = 0 (red) and |mI = +1 (blue). When |mI = 0 , the allowed electron transition frequency in 14 N@C60 is ωe and will not be on resonant with ωNV . If |mI = +1 , the resonant condition is satisfied when ωe + a = (2Ω 2 + ω 2 e ) 1/2 . The splitting of the resonant frequency stems from the virtual transitions between the electron and nuclear spin in 14 N@C60 caused by the non-secular hyperfine coupling terms.The applied magnetic field is 300MHz. The electron spin of 14 N@C60 is not polarized, and its initial state is approximated by the maximally mixed state. Ω = 0), NV spin will thus stays in the initial state |D ; otherwise if the nuclear spin state is |m I = +1 , the flip-flop process can happen and thus the state population of |D will change with time at the resonant frequencies. In Fig.8, we see that there are actually three resonant frequencies. The splitting comes from the virtual electronic transition caused by the nuclei (∼ a 2 /ω e = 0.84MHz). In the mean time, the nuclear spin state populations are not affected by the readout procedure, and one can thus implement repetitive measurement of the nitrogen nuclear spin state. In our numerical simulation, we apply a magnetic field |γ e B| = 300MHz, and the corresponding Larmor frequencies of 13 C and 14 N are 114.75kHz, 32.97kHz respectively. The amplitude of driving on the NV spin corresponding to the central resonant frequency ω N V = 316.2MHz is Ω = 70.7MHz, which is strong enough to suppress the effect of the 13 C spin bath. We remark that if we first polarize the electron spin to the state − 3 2 (e.g. by using NV center), it is possible to improve the readout efficiency.

Measure distance between two hydrogen nuclei in a water molecule
We consider a target system which consists of two spins interacting with each other. The system Hamiltonian is written as The interaction strength g = 4π µ 0 γ 2 N /R 3 depends on the intra-distance R between two spins. The unit vectorr connects two spins and characterizes the alignment of spin pair. If we do not apply an additional magnetic field, the system energy is only dependent on the value of g, thus it is not possible to obtain information about the direction r. On the other hand, when the interaction strength is not very strong, the amplitude of driving on NV spin (which should matche the target system transition frequency) will not be strong enough to suppress the effect of the 13 C spin bath in diamond. Therefore, we propose to apply a strong magnetic field γ N B g. Under such a condition, the flip-flip process is unlikely and the system Hamiltonian can be rewritten as 9. (Color online) Eigenstates of two interacting spins under a strong magnetic field. We use NV spin to detect the transition frequencies of Ω1 and Ω2, the difference between which provides the information about the spin-spin coupling strength and the relative direction of the spin pair with respect to the applied magnetic field.
where the spin operators I x , I y , I z are defined in the quantization axis induced by the applied magnetic field. The eigenstates can be written as The transition from the singlet eigen state |E 2 to the states |E 0 and |E 3 is determined by the difference of the coupling operators of two spins to the NV center, namelyÂ 1 −Â 2 . Therefore, if we tune the Rabi frequency of the NV spin around the Zeeman energy ω N , the dominant flip-flop process from the transition from |E 1 to |E 0 and |E 3 with two corresponding resonant frequencies, see Fig.9, The difference between these two resonant frequencies is thus gives us information about the coupling strength g and the alignment vectorr. To determine these exact values, we propose to apply magnetic fields in nine directions, and get the resonant frequencies respectively as follows After some calculations, we can obtain From Eq. (37-48), we obtain the value of coupling strength g as Furthermore, we get Finally, with the obtained values of g 2 , r 2 x , r 2 y , r 2 z we compare Eq.(40-48,50-52) and know the relative signs between r x , r y , r z and thereby we get the align directionr of the spin pair.
In the main text, we have applied our ideas to measure the distance between two hydrogen nuclei in a water molecule. In Fig.10, we plot the resonance frequencies with a magnetic field in nine different directions. The strength of the magnetic field is γ 1 H B = 500kHz. The amplitude of driving on NV spin is thus strong enough to suppress the effect of the 13 C spin bath, see Fig.5 (b). The 1 H 2 O molecule is 5nm from the NV center, and the vector that connects two hydrogen atoms is described by (r, θ 0 , φ 0 ) = (0.1515nm, 118.2 o , 288.85 o ). With the resonant frequencies as in Fig.10, we calculate based on Eq.(49-52) and obtain that g = 34.684kHz, and thereby the intra-molecular distance is d = 0.1518nm, and the alignment direction is (θ, φ) = (118.29 o , 288.82 o ).

Measure distance between two organic spin labels
Spin labels are organic molecules with a stable unpaired electron. They can be attached to the protein (covalent or as a ligand) via a functional group. Electron spin resonance (ESR) based on spin labels has been used as a spectroscopic ruler to determine protein structure and monitor macromolecular assembly processes. It is however hard to go beyond a distance of 5nm, due to e.g. (in)homogeneous line broadening. Here, we propose to use the NV center as a probe and combine with the application of spin labels, which opens possibilities of measuring a intraor inter-molecular distance larger than 5nm at a single molecule level [5]. We consider the widely used nitroxide spin labels, in which the electron spin interacts with the nitrogen nuclear spin. The coupling is relatively strong as compared with the interaction between two distant spin labels. To suppress the effect of the nitrogen nuclear spin and enhance the signal, one can use the NV spin to first polarize the electron spin and then use the polarized electron spin to prepare the nuclear spin into the state |m I = 0 . The coupling with the other nuclear spins is weak and can be suppressed by continuous driving the electron spin. If the driving amplitude is much stronger than the hyperfine coupling, the system Hamiltonian can be written as where the quantization axis of the spin labels are induced by the applied magnetic field. In the case that Ω 1 g, the system part Hamiltonian can be approximated as The eigenstates are similar to Fig.9 as follows The transition from the state |E 1 to the states |E 0 and |E 3 with two corresponding resonant frequencies and the difference between these two resonant frequencies is When the distance between two spin labels is large, such that the coupling operators A 1 and A 2 are sufficiently inhomogeneous. The transition from the singlet state |E 2 to the states |E 0 and |E 3 may also be observed with two corresponding resonant frequency and the difference between these two resonant frequencies is Monitor the charge recombination of radical pair Many chemical reactions involves radical pair intermediate, which consists of two unpaired electrons [6]. The radicals are in a charge separated state and interact with each other via exchange and dipole interactions. For the  . 12. (Color online) (a) Resonance of a model radical pair reaction with a distance between two radicals of d = 2nm, the recombination rate is k = 1µs −1 . The magnetic field is assumed to be along the vector which connects two radicals. The results are similar for other magnetic field directions. Continuous driving field with Rabi frequency of 100MHz is applied on the radical pair. (b) The Rabi frequency of NV spin is set at the resonant frequency. As the radical pair recombines, the effective flip-flop rate decreases and the slope of the signal S becomes smooth. distance larger than 1nm, the exchange interaction is negligible, and the main contribution comes from the dipole interaction. Here, we consider a simple model radical pair reaction, namely the radical pair is created in the singlet state and recombine at the same rate k for both singlet and triplet states, the dynamics of which can be described by the following master equation where H is the system Hamiltonian, where the interaction between two radicals and the coupling between NV spin and radicals exists only when it is in the charge separated state. The Lindblad operators L S , L T describe the recombination of the singlet and triplet radical pair into the product state, which are written as with Q S and Q T are the projectors into the singlet and triplet subspace, |S and |P represent the charge separate state and product state respectively. The above master equation is equivalent to the conventional Haberkorn approach [7]. We apply the same idea as in the model of spin labels by applying an additional magnetic field and also continuously driving the radical spins to suppress the effect of surrounding nuclei. We assume the radical pair is created in the singlet state, so the resonant frequency near the driving Rabi frequency Ω is see the above section of spin labels for more details. The charge recombination leads to the decay of the effective flip-flop rate. If the recombination rate is comparable or smaller than the coupling between NV spin and the radicals, it is possible to observe the resonance frequency, see an example in Fig.12 (a). Therefore, if we tune the Rabi frequency of NV spin at the resonant frequency, we can monitor the recombination of the radical pairs by observing the decay of the flip-flop rate. In other words, it can serve as an evidence for the charge recombination. We remark that this is very interesting as it provides possibility to see chemical reaction process at a single molecule level and may give insights into how radicals recombine into the product states.