All-optical magnetic resonance of high spectral resolution using a nitrogen-vacancy spin in diamond

We propose an all-optical scheme to prolong the quantum coherence of a negatively charged nitrogen-vacancy (NV) center in diamond at cryogenic temperatures. Optical control of the NV spin suppresses energy fluctuations of the 3 A 2 ?> ground states and forms an energy gap protected subspace. By optical control, the spectral linewidth of magnetic resonance is much narrower and the measurement of the frequencies of magnetic field sources has higher resolution. The optical control also improves the sensitivity of the magnetic field detection and can provide measurement of the directions of signal sources.

ensembles, which currently limits spatial resolution to the micrometer scale [2]. Recently considerable attention has focused on the application of negatively charged nitrogen-vacancy (NV) centers in diamond as an atomic-sized magnetic field sensor [3][4][5][6][7][8][9][10][11][12][13], where the NV centers are initialized and read out by optical fields [14,15]. To suppress the effect of environmental noise and to obtain magnetic resonance signals of the sensing targets, microwaves are used to implement pulsed [16][17][18][19][20] and continuous [21,22] dynamical decoupling control. The magnetic resonance signals represent important pieces of information to identify target spins and their relative positions of high spatial resolution [6,7,[9][10][11][12]. The magnetic resonance frequencies are determined by the pulse rates of pulsed dynamical decoupling [6,7,9] or the Rabi frequencies of continuous dynamical decoupling [10][11][12]. Increasing the pulse rates in the case of pulsed dynamical decoupling and their Rabi frequencies in the case of continuous dynamical decoupling beyond the GHz regime is highly challenging. The requirement of realizing microwaves (which have wavelengths of centimeters) imposes limitations on the setup and individual microwave control on NV centers is difficult. Hence there has been a major effort to desirable to develop methods to overcome these shortcomings. All-optical control was recently shown to be possible [23], and an all-optical scheme for sensing the amplitudes of magnetic fields was demonstrated by electromagnetically induced transparency in an NV ensemble [24]. However to date, there are no all-optical methods to measure the frequencies of the magnetic fields that provide rich magnetic resonance information about the signal sources.
It is well-know that by optical driving fields one can realize a dark state which is decoupled from the excited states [25]. The fundamental principle underlying dark states is destructive interference which can be perturbed by external dephasing noise. If the dephasing noise has a finite bandwidth and when the dark state is separated in energy from all other bright states by more than this bandwidth, the sensitivity of the dark state to dephasing noise can be reduced considerably. Indeed, a qubit encoded in a dark state and an auxiliary stable state has been shown to be resilient to low frequency environmental noise and amplitude fluctuations in the driving field [26,27]. The resulting enhanced coherence time enables longer coherent interaction with sensing targets and better sensing signals.
Here, we demonstrate the working principle by considering a negatively charged NV center at cryogenic temperatures. Specifically, we propose an all-optical magnetic resonance scheme using a negatively charged NV center to measure the frequencies of magnetic fields (see figure 1(a)). For the case that the magnetic resonance frequencies are determined by the pulse rates of pulsed dynamical decoupling [6,7,9] or the Rabi frequencies of continuous dynamical decoupling [10][11][12], errors in the dynamical decoupling control can broaden the resonant signal peaks. In contrast, in our scheme fluctuations in optical control do not broaden the resonant signal peaks, and the frequency of magnetic resonance is determined by the energy gap of the A 3 2 ground sublevels, which can easily extend the sensing frequencies to the GHz range. The optical control of the NV center suppresses the energy fluctuations of the A 3 2 ground sublevels and significantly extends the coherence times of the NV centers. Since the magnetic resonance linewidth broadening by dephasing is eliminated through the optical control, magnetic resonance of high spectral resolution with NV centers becomes possible. Higher spectral resolution can also improve measurement of the spatial positions of target spins when the schemes utilize magnetic resonance signals [6,7,12]. The all-optical magnetic resonance scheme may also have applications in solid-state GHz frequency standards and in all-optical quantum information processing with NV centers.

A negatively charged NV center under optical control
In applications of negatively charged NV centers in quantum technologies, it is important to prolong the quantum coherence of the A driven by H dep will evolve to Ψ 〉 = . The coherence between 〉 |0 g and ± 〉 | 1 g is described by the average of the relative random phase factor = φ ± ± L e t 0, 1 i ( ) , which vanishes when the random phase is large. For Gaussian noise, To suppress the dephasing using only optical control, we use two laser fields resonantly coupling the triplet ground states ± 〉 | 1 g to the E 3 excited state = + + − Figure 1. (a) NV level structure with optical control; a Λ system is formed by the lasers resonantly driving the transitions between ± 〉 | 1 g and 〉 A | 2 . (b) Energy diagram of a NV center in the dressed-state picture. The optical control induces a coherence protected space spanned by 〉 d | g and 〉 |0 g (in yellow shadow), which is protected by the energy gaps ± Ω. eigenenergies of the E 3 levels at low temperatures can be found in the review paper [14]. To have well-resolved excited states, we put the NV center at cryogenic temperatures (≲10 K). Using 〉 + 〉 = 〉 + 〉 k k k g 1 g g 0 g g g g g g g the system Hamiltonian reads H t H t sig i sig i g g For accurate numerical simulations, we model the NV spin with six levels: three ground states ± 〉 | 1 g and 〉 |0 g , the two excited states 〉 A | 1 and 〉 A | 2 , and a singlet state 〉 s | to describe the intersystem crossing transitions. The dynamics of the NV center spin described by a density matrix ρ t ( ) is governed by the Lindblad master equation [32] ⎜ ⎟ , where the Lindblad operators σ β α ≡ 〉〈 βα | |, γ βα are the decay rates, and the Hamiltonian H is given by equation (5).

Noise suppression by optical control
To illustrate the fundamental idea of noise suppression by optical control, we consider a simplified model without contributions from spontaneous decay (taken into account in detailed numerical simulations to present subsequently). When the energy gap δ ≫ ± Ω , the coupling to 〉 A | 1 is negligible in equation (5), and we have the Λ-type Hamiltonian by dropping out terms related to the state 〉 L2 d e p s i g The laser driving fields form a Λ system with an excited state 〉 A | 2 and two ground states ± 〉 | 1 g (see figure 1). The dark state decoupled from the laser is The subspace spanned by the two states 〉 d | g and 〉 |0 g are separated from the other eigenstates 〉 ± l | by the energy gaps ± Ω. Therefore the transitions from this subspace to 〉 ± l | are suppressed by an energy penalty that is proportional to the Rabi frequency of the optical driving fields (see figure 1 (b) and [26]).
The spin operator S z in the basis of 〉 b | g and 〉 d | g reads κ = + To suppress the dephasing of the NV spin using the energy penalty Ω, we choose the laser fields that satisfy Under such a condition, the spin operator S z becomes L * is a tunable relative phase. With a relatively large Ω, the spectral power density of the magnetic field fluctuations β t ( ) z at the frequencies around the energy gap Ω is negligible. The off-resonant fluctuations β t ( ) z cannot induce the transitions from 〉 d | g to 〉 ± l | , which are strongly suppressed by the energy penalty ± Ω (see figure 1(b)). Because fluctuations in the effective Rabi frequency Ω only cause small changes in the magnitudes of the energy gap, our scheme is stable against the fluctuations of Ω [26], as long as the magnitudes of the energy gap are still much larger than the fluctuation frequencies of β t ( ) z . When there are relative fluctuations in ± Ω and ± c , the second line in equation (14) does not vanish, and a fraction κ ∼ of the noise will not be suppressed by the energy gaps ± Ω made by the optical control. The decoherence time caused by this fraction of noise is estimated to be κ ∼ T T frac 2 * , and the fluctuations only cause negligible effects if T frac is much larger than the controlled evolution time. The relative amplitude fluctuations in the driving fields can be made very small if the fields are obtained from the same laser.
To manifest the effects of dephasing, we set = H 0 sig in equation (9). The quantum coherence between, e.g., 〉 |0 g and 〉 d | g is approximately described by the average The coherence between 〉 |0 g and 〉 d | g decreases when the absolute value of L t ( ) Without optical driving fields, for Gaussian noise and vanishes when the random phase is large. From the expression of equation (23), we can see that when the noise is relatively slow compared to the frequency Ω, the effect of the noise β t ( ) z is averaged out by the oscillating functions Ωt cos and Ωt sin . Dynamical decoupling also uses similar modulation functions to average out unwanted noise [19][20][21][22]. A more detailed analysis in frequency domain is given in appendix A.
To demonstrate our scheme, we performed numerical simulations by implementing the master equation (8), including the spontaneous decay and dephasing. We used the parameters in the experimental paper [23]  MHz. In the simulation, the noise fluctuations β t ( ) z were simulated by the Ornstein-Uhlenbeck process [33], which is Gaussian.
Generated by the Ornstein-Uhlenbeck processes, the expectation value of β t ( ) where t 0 is the starting time to generate an Ornstein-Uhlenbeck process and β c is a diffusion coefficient [34]. These quantities converge to stationary values after a time larger than the correlation time τ β . In the simulation, we chose and we had an exponentiallydecaying correlation function β β ′ ≈ can be treated as stationary stochastic process. We chose the correlation time τ = β 25 μs in the simulation [34]. The realizations of Ornstein-Uhlenbeck processes were generated by the exact simulation algorithm in [36] β Δ β τ which requires the generation of a unit Gaussian random number n G at each time step. The algorithm is exact because the update algorithm equation (24) is valid for any finite time step Δt [36]. We chose the value of the diffusion coefficient τ ≈

μs.
To demonstrate the coherence protection by optical control, we prepared the quantum state in the superposition With this initial state and using the master equation (8) in the simulations, the quantum coherence in the simplified model (9) without contributions from spontaneous decay). We had examined L t | ( )| d 0, g as a function of time t under optical driving fields of different Rabi frequencies Ω. We found that there is an optimal working point of Ω for coherence protection. A plot of L t | ( )| d 0, g as a funtion of Ω was shown in figure 2, where the coherence was measured at t = 50 μs. From simulations for other choices of t, we observed that the optimal Ω is independent of the measurement times t (which were chosen to be larger than a few μs for a clear inspection). The existence of an optimal Ω is a result of the leakage to the excited states. When the optical driving fields are turned on, the dephasing Hamiltonian H dep in equation (5)   . This effect can be seen from the reduction of coherence when Ω increases from 0 to ∼1 MHz in figure 2. Large driving field amplitudes Ω lead to a recovery of the quantum coherence by suppressing the noise effects (see the results in figure 2 for 1 MHz ≲ ≲ Ω 7 MHz).However, increasing Ω also increases the coupling to the state 〉 A | 1 (see equation (6)), which degrades the approximated Λtype system (see equation (9)) and leads to population leakage to 〉 A | 1 . At zero static field, the energy gap δ ≈ 2 GHz and the mixing coefficients = + − c c | | | | in equation (6), and therefore 〉 d | g only directly couples to , estimated by time-independent perturbation theory. Because of relatively large decay rates on the excited states 〉 A | 2 and 〉 A | 1 , we expect an optimal > Ω 0 in our scheme when the leakage + 〉 〉

P P
is a minimum. In figure 2(a),the optimal control with ∼ Ω 10 MHz gives the best coherence protection. We can increase δ by applying a static bias magnetic field B z,bias along the axis of the NV center (z direction). However, non-zero B z,bias also changes the mixing coefficients ± c . For example, is strongly reduced (see figure 2( T, the effective Rabi frequency ∼ Ω 7 MHz yields the best coherence protection. Having obtained the optimal driving amplitudes of Ω, in figure 3 we plotted the quantum coherence as a function of time for cases of free induction decay ( = Ω 0) and with optical driving fields ( = Ω 10 MHz) at zero static fields. With an optical Rabi frequency = Ω 10 MHz, the coherence is significantly prolonged and exceeds 50 μs, which is over 16-fold improvement in the coherence time ≈ T 3 2 * μs. Since the coherence can be preserved up to ∼50 μs and a longer evolution time provides better spectral resolution, in the following simulations, we will perform measurement at the time t = 50 μs.
Although the scheme is robust to the fluctuations of Ω [26], equation (15) implies that independent fluctuations δ ± Ω t ( ) in the amplitudes of ± Ω change the bright and dark states in equations (17) and (18). To demonstrate that the scheme is not sensitive to independent fluctuations δ ± Ω t ( ), we modelled δ ± Ω t ( ) by Ornstein-Uhlenbeck processes. We selected a diffusion coefficient with a correlation time τ = 100 Ω μs and a variance of relative . When the dephasing (i.e., the effect of energy broadening) is suppressed, we achieve a narrower linewidth, and hence spectroscopy with higher accuracy.
Before giving the details, we summarize the steps of the sensing scheme. We first initialize the NV center in the state 〉 |0 g by optical pumping. To start the sensing, we turn off the pumping lasers and turn on the driving fields to form the dark state. After some evolution time, we turn off the driving fields and measure the population at 〉 |0 g by optical readout. The population at 〉 |0 g gives the magnetic resonance signals.
We assume that the signal fields have negligible frequency components around the energy gap Ω. The signal Hamiltonian in equation (9)  where η t ( ) sig is the magnetic signal field with zero mean η = t ( ) 0 sig and θ sig is the direction of the magnetic field in the x-y plane normal to the NV axis. We initialize the NV center in the state 〉 |0 g by optical pumping. The population that remains in the initial state 〉 |0 g is approximately governed by the dynamics induced by Λ H given by equation (9) In the simulation, we generated single-frequency sources η η ω φ = + t t ( ) cos ( ) sig 0 s s with initial random phases φ s at each run of the simulation. To have the accurate resonance frequency ω Res , we diagonalized the Hamiltonian H given by equation (6), as ω Res is the energy between 〉 |0 g and the state 〉 d |˜g which is approximately 〉 d | g and has a little mixing with the excited states 〉 A | 2 and 〉 A | 1 . For δ ≪ Ω , ω ϵ ≈ − Res 0, 1 , up to a correction δ ∼Ω 2 . We consider the magnetic resonance signal where the difference between the signal frequency and the resonant frequency is relatively small, i.e., ω

Res Res
. This enables the application of a rotating wave approximation by neglecting oscillating terms with frequencies ω ∼2 Res in equation (25). The simulations used the master equation (8) with the Hamiltonian equation (5).

Measurement of signal frequencies
When there is no static bias field , δ ≈ 2 GHz, the states ± 〉 | 1 g are degenerate, and By choosing the laser phase ϕ θ = + π 2 L sig , we achieve the largest sensitivity as the signal The states 〉 d | g and 〉 |0 g are coherence protected. We obtain the magnetic resonance signal by tuning the resonant frequency. When the resonant frequency is tuned to the frequency of the signal fields, the state 〉 |0 g will transit to 〉 d | g and the change of 〉 P t ( ) |0 g gives the signal. In figure 5, we plot the magnetic resonance signal at the evolution time t = 50 μs. The laser phase is ϕ θ = + π 2  To reduce the resonant frequency ω Res within the MHz range, we apply a static bias magnetic field along the axis of the NV center to narrow the energy gap ϵ − 0, 1 . When the magnetic field ≈ B 0.1 z,bias T, δ ≈ 5.7 GHz, the ground states 〉 |0 g and − 〉 | 1 g have an energy gap within the MHz range, and the energy gap between 〉 |0 g and + 〉 | 1 g is large (≳2.9 GHz). For large energy gaps between 〉 |0 g and + 〉 | 1 g , the transition from 〉 |0 g to + 〉 | 1 g induced by signal fields is negligible and we have is negligible. The transitions between 〉 |0 g and 〉 b | g are suppressed, and we get  , the signal Hamiltonian equation (27) depends on the direction θ sig of the signal source. Note that the transition from 〉 |0 g to 〉 b | g is suppressed when there is optical control. If ϕ θ = 2 L sig in equation (27), the effect of the signal Hamiltonian is suppressed by the energy gap between 〉 |0 g and 〉 b | g states, and the population change Δ 〉 P |0 g is small. We have shown that the signal is large when the laser phase ϕ θ = + π 2 L sig . We use this phase dependence to determine the direction θ sig of the signal field.
In figure 7, we applied a resonant signal field at the frequency ω ω =  figure 7, we can infer the direction θ sig of the signal source in the x-y plane. For the case without optical control, x and y directions are equivilent and it is obvious that we cannot infer the direction of the signal source in the x-y plane. Note that if the laser phase ϕ L has a small deviation in the region , the signal Δ 〉 P |0 g is also large with little change ≈ ( 1%). Therefore, our scheme is robust to fluctuations of laser phase.

Sensitivity enhanced by optical control
Note that in figures 5 and 6, the signal peaks with optical control are more pronounced. This implies that the sensitivity is also improved when the dephasing noise is suppressed by the optical control. In figure 8, we plot the one-trial sensitivity [36]  is the standard deviation of the population 〉 P |0 g in one measurement. Averaging the data by repeating the measurement N times improves the sensitivity by a factor of α = N 1 s . If we perform the experiment for a given time T all , we have = + N T T t ( ) all init , where T init is the time for both initialization and readout and t is the evolution time of the NV spin. With a large bias field, the transition from 〉 |0 g to + 〉 | 1 g is suppressed by the energy gap between 〉 |0 g and + 〉 | 1 g (see equation (28)). Therefore the effective signal strength is reduced and the sensitivity with optical control is reduced in figure 8 at a short sensing time ≲ t 10 μs. At a longer sensing time, the benefits from decoherence suppression by optical control manifest and the sensitivity is improved. At zero bias field, the transition from 〉 |0 g to + 〉 | 1 g is kept under optical control, and the one-trial sensitivity is improved for a wide range of times ≲ t 45 μs in the figure.
With the chosen values of η 0 in figure 8, we can see that the case of optical control has significantly lower sensitivity around t = 50 μs (compared with the high performance around ≈ t 30 μs). To analyze the reduction of the one-trial sensitivity, we simplified the Lindblad master equation by using a Hamiltonian η 〉〈 + ′ d | 0 | h.c. μs. Therefore, with optical control, the low sensitivity around t = 50 μs in figure 8 is caused by the population leakage out of the subspace spanned by the sensing levels 〉 d | g and 〉 |0 g . In figure 8, we chose the values of η 0 to show the effect of population leakage on the sensitivity, and these values of η 0 were also used in figures 5 and 6, where relatively larger η 0 provided more pronounced resonance signals. The sensitivity at t = 50 μs can be improved if the strength of the signal sources η 0 is smaller.