All-optical high-resolution magnetic resonance 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. Optical control of the NV spin suppresses energy fluctuations of the 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.
High-resolution magnetic resonance is currently one of the most important tools in many areas of science and technology, including analytical chemistry, materials science, structural biology, neuroscience, and medicine Ernst:1994:OxfordUniversityPress (). However, the sensitivity of conventional techniques is restricted to large spin ensembles, which currently limits spatial resolution to the micrometer scale Ciobanu:2002:178 (). Recently considerable attention has focused on the application of negatively charged nitrogen-vacancy (NV) centers in diamond as an atomic-sized magnetic field sensor to detect nuclear magnetic resonance (NMR) signals by quantum control with both laser fields and microwaves Degen:2008:243111 (); Maze:2008:644 (); Balasubramanian:2008:648 (); Zhao:2011:242 (); Zhao:2012:657 (); Mamin:2013:557 (); Staudacher:2013:561 (); London:2013:067601 (); Muller2013Draft (); Cai:2013:013020 (). In these works, the NV centers are initialized and readout by optical fields Doherty:2013:1 (); Dobrovitski:2013:23 (); but the noise protection is based on pulsed Hahn:1950:580 (); Carr:1954:630 (); Naydenov:2011:081201 (); Yang:2010:2 (); Lidar:2013:QEC () and continuous Cai:2012:113023 (); cai2012long () dynamical decoupling techniques employing microwave control. 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 Yale:2013:7595 (), and an all-optical scheme for sensing the amplitudes of magnetic fields was demonstrated by electromagnetically induced transparency in an NV ensemble Acosta:2013:213605 (). However to date, there are no all-optical methods to measure the frequencies of the magnetic fields which provides rich magnetic resonance information about the signal sources.
In this work, we propose an all-optical magnetic resonance scheme using a negatively charged NV center to measure the frequencies of magnetic fields [see Fig. 1(a)]. Unlike the magnetic resonance supported by dynamical decoupling, 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 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 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, high-resolution magnetic resonance with NV centers becomes possible. 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.
Ii 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 triplet spin ground states and , where refers to the orbital state with 0 orbital angular momentum projections along the NV axis. The relaxation time of an NV spin can approach 200 s at low temperatures (10 K) Jarmola:2012:197601 (), whereas the dephasing time is relatively short, with typical values for the inhomogeneous dephasing time of 0.5 to 5 Dobrovitski:2013:23 (); Doherty:2013:1 (). With a large energy difference between and , dephasing is the main source of decoherence and limits the overall decoherence time.
For simplicity, we model the dephasing by magnetic field fluctuations on the NV axis (along direction), which couple to the NV spins through the Zeeman interaction ()
with the spin operator . We assume that has zero-mean , where the overline denotes ensemble averaging. The random field fluctuations induce broadening of the states . An initial state of the center spin driven by will evolve to , where the accumulated random phase . The coherence between and is described by the average of the relative random phase factor , 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 to the excited state
with (see Fig. 1). The lasers also couple the states and but with a large detuning , which is the energy gap between the states and . The optical transitions between and are spin conserving Batalov:2009:195506 (); Togan:2010:730 (); Maze:2011:025025 (). The state properties of the NV centers, such as the parameters and , depend on electric, magnetic, and strain fields. The effective Hamiltonian to obtain the eigenstates and eigenenergies of the levels at low temperatures can be found in the review paper Doherty:2013:1 (). To have well-resolved excited states, we put the NV center at cryogenic temperatures ( K). Using and , we have the Hamiltonian under constant optical control
where are the Rabi frequencies and are the energies of the ground states. and are the phases and frequencies of the lasers, respectively. We also include an interaction Hamiltonian for possible signal sources. The energy of is set as the reference energy. The laser fields resonantly drive the transitions between and with the laser detuning . In the rotating frame of with
the system Hamiltonian reads
For accurate numerical simulations, we model the NV spin with 6 levels: three ground states and , the two excited states and , and a singlet state to describe the intersystem crossing transitions. The dynamics of the NV center spin described by a density matrix is governed by the Lindblad master equation Rivas:2012:OpenQuantumSystems (),
where the Lindblad operators , are the decay rates, and the Hamiltonian is given by Eq. (5).
Iii 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 , the coupling to is negligible in Eq. (5), and we have the -type Hamiltonian by dropping out terms related to the state ,
with the effective Rabi frequency
and the bright state
The laser driving fields form a system with an excited state and two ground states (see Fig. 1). The dark state decoupled from the laser is
with a global phase . The laser Hamiltonian has the eigenstates , , and with energies . The subspace spanned by the two states and are separated from the other eigenstates by the energy gaps . Therefore the transitions from this subspace to are suppressed by an energy penalty that is proportional to the Rabi frequency of the optical driving fields (see Fig. 1 and Ref. Timoney:2011:185 ()).
The spin operator in the basis of and reads
where . To suppress the dephasing of the NV spin using the energy penalty , we choose the laser fields that satisfy
Under such a condition, the spin operator becomes
where and is a tunable relative phase. With a relative large , the spectral power density of the magnetic field fluctuations at the frequencies around the energy gap is negligible. The off-resonant fluctuations cannot induce the transitions from to , which are strongly suppressed by the energy penalty [see Fig. 1(b)]. Because fluctuations in the effective Rabi frequency only cause small changes in the magnitudes of the energy gap, our scheme is stable against the fluctuations of Timoney:2011:185 (), as long as the magnitudes of the energy gap are still much larger than the fluctuation frequencies of . When there are relative fluctuations in and , the second line in Eq. (14) does not vanish, and a fraction of the noise will not be suppressed by the energy gaps made by the optical control. The decoherence time caused by this fraction of noise is estimated to be , and the fluctuations only cause negligible effects if 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 in Eq. (9). The quantum coherence between, e.g., and is described by the average
where with the time-ordering operator . The coherence can be simplified as
by using the transformation with
The coherence between and decreases when the absolute value of decreases. Without optical driving fields, for Gaussian noise and vanishes when the random phase is large. From the expression of Eq. (22), we can see that when the noise is relatively slow compared to the frequency , the effect of the noise is averaged out by the oscillating functions and . Dynamical decoupling also uses similar modulation functions to average out unwanted noise Yang:2010:2 (); Lidar:2013:QEC (); Cai:2012:113023 (); cai2012long (). 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 Yale:2013:7595 () for the numerical simulation. The decay rate from the excited states and to the ground sates and is MHz; the rate for intersystem crossing from the excited states to the singlet is MHz; the inverse intersystem crossing rate from to the ground states is MHz. In the simulation, the noise fluctuations were simulated by the Ornstein-Uhlenbeck process Wang:1945:323 (), which is Gaussian. Generated by the Ornstein-Uhlenbeck processes, the expectation value of is ; the two-point correlation , where is the starting time to generate an Ornstein-Uhlenbeck process and is a diffusion coefficient Bibbona:2008:S117 (). These quantities converge to stationary values after a time larger than the correlation time . In the simulation, we chose for . We used and simulated the spin dynamic at , so that and we had an exponentially-decaying correlation function . At , can be treated as stationary stochastic process. We chose the correlation time in the simulation de2010universal (). The realizations of Ornstein-Uhlenbeck processes were generated by the exact simulation algorithm in Ref. Gillespie:1996:2084 (),
which requires the generation of a unit Gaussian random number at each time step. The algorithm is exact because the update algorithm Eq. (23) is valid for any finite time step Gillespie:1996:2084 (). We chose the value of the diffusion coefficient so that without optical driving fields at the dephasing time .
To demonstrate the coherence protection by optical control, we prepared the quantum state in the superposition of and . To obtain the optimal amplitude of driving fields, we plotted the coherence at s at different in Fig. 2. By increasing the driving amplitude , the quantum coherence was recovered by suppressing the noise effects. However, increasing also increases the coupling to the state [see Eq. (6)], which degrades the approximated -type system [see Eq. (9)] and leads to population leakage out of the subspace spanned by and . At zero static field, the energy gap GHz and the mixing coefficients in Eq. (6), and therefore only directly couples to . For , the population to state is , estimated by time-independent perturbation theory. The value of at a moment decreases when the leakage increases. Here is the estimated decay rate from the excited states. In Fig. 2(a), the optimal control with MHz gives the best coherence protection. We can increase by applying a static bias magnetic field along the axis of the NV center (z direction). However, non-zero also changes the mixing coefficients . For example, T gives GHz, , and . When , the dark state can transit to both excited states and , and is strongly reduced [see Fig. 2(b)]. At T, the effective Rabi frequency MHz yields the best coherence protection.
In Fig. 3, we plot the quantum coherence as a function of time for cases of free induction decay () and with optical driving fields ( MHz) at zero static fields. With an optical Rabi frequency MHz, the coherence is significantly prolonged and exceeds 50 s, which is over 16-fold improvement in the coherence time s.
Although the scheme is robust to the fluctuations of Timoney:2011:185 (), Eq. (15) shows that independent fluctuations in the amplitudes of change the bright and dark states in Eqs. (17) and (18). To demonstrate that the scheme is not sensitive to independent fluctuations , we modelled by Ornstein-Uhlenbeck processes. We selected a diffusion coefficient with a correlation time s and a variance of relative fluctuations . The impact of independent intensity fluctuations on the coherence is shown in Fig. 4. The scheme still provides good performance even though the standard deviation of relative fluctuations reaches .
Iv High-resolution magnetic resonance by optical control
In magnetic resonance spectroscopy, when the energy gap, e.g., between and , coincides with the frequency of the magnetic field, resonance transitions induce a signal peak in the frequency domain. The linewidth and depth of the peak determine the resolution and sensitivity of the spectroscopy. Because of the energy broadening induced by the random fluctuations , the minimum linewidth of the signal peak is limited by the deviation of the fluctuations . When the dephasing (i.e., the effect of energy broadening) is suppressed, we achieve a narrower linewidth, and hence spectroscopy with higher accuracy.
We assume that the signal fields have negligible frequency components around the energy gap . The signal Hamiltonian in Eq. (9) is written as
where is the magnetic signal field with zero mean and is the direction of the magnetic field in the plane normal to the NV axis. We initialize the NV center in the state by optical pumping. The population that remains in the initial state is approximately governed by the dynamics induced by given by Eq. (9),
In the simulation, we generated single-frequency sources with initial random phases at each run of the simulation. To have the accurate resonance frequency , we diagonalized the Hamiltonian given by Eq. (6), as is the energy between and the state which is approximately and has a little mixing with the excited states and . For , , up to a correction . We consider the magnetic resonance signal where the difference between the signal frequency and the resonant frequency is small, i.e., . This enables the application of a rotating wave approximation by neglecting oscillating terms with frequencies in Eq. (24). The simulations used the master equation (8) with the Hamiltonian Eq. (5).
iv.1 Measurement of signal frequencies
When there is no static bias field , , the states are degenerate, and
By choosing the laser phase , we achieve the largest sensitivity as the signal Hamiltonian The states and 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 will transit to and the change of gives the signal.
In Fig. 5, we plot the magnetic resonance signal at the evolution time s. The laser phase is , and the signal fields have an amplitude MHz. Without optical control, the signal fields with frequency lead to a maximum resonant population change with a linewidth MHz (defined by the full width at half maximum); while with optical control MHz, the signal induces a much larger population dip with a much narrower linewidth MHz, which is limited by the evolution time (within the coherence time range).
To reduce the resonance frequency within the MHz range, we apply a static bias magnetic field along the axis of the NV center to narrow the energy gap . When the magnetic field T, , the ground states and have an energy gap within the MHz range, and the energy gap between and is large ( GHz). For large energy gaps between and , the transition from to induced by signal fields is negligible and we have
Under optical control with a large , we can ensure that the spectral density of around and around is negligible. The transitions between and are suppressed, and we get
Therefore, when the field is on resonant with the transition frequency around , the population of decreases. In this way, the Fourier components of the signal source can be measured in high resolution. In Fig. 6, MHz, T, and the evolution time s. Without optical control, the signal fields lead to a maximum population dip with a linewidth MHz; while with optical control MHz, the population has a larger peak with a much narrower linewidth MHz limited by finite evolution time.
iv.2 Measurement of the directions of signal sources
For the case of , the signal Hamiltonian Eq. (26) depends on the direction of the signal source. Note that the transition from to is suppressed when there is optical control. If in Eq. (26), the effect of the signal Hamiltonian is suppressed by the energy gap between and states, and the population change is small. We have shown that the signal is large when the laser phase . We use this phase dependence to determine the direction of the signal field.
In Fig. 7, we applied a resonant signal field at the frequency and tuned the laser phase with different angles . The control strength MHz and the amplitude of the signal field MHz. From the lowest point in Fig. 7, we can infer the direction of the signal source in the plane. For the case without optical control, and directions are equivilent and it is obvious that we cannot infer the direction of the signal source in the plane. Note that if the laser phase has a small deviation in the region , the signal is also large with little change . Therefore, our scheme is robust to fluctuations of laser phase.
iv.3 Sensitivity enhanced by optical control
Note that in Figs. 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 Fig. 8, we plot the one-trial sensitivity for the cases of the bias magnetic field and T, respectively. The parameters are the same as those for the optical sensing in Figs. 5 and 6 with . Here is the standard deviation of the population in one measurement. Averaging the data by repeating the measurement times improves the sensitivity by a factor of . If we perform the experiment for a given time , we have , where is the time for both initialization and readout and is the evolution time of the NV spin. With a large bias field, the transition from to is suppressed by the energy gap between and [see Eq. (27)]. Therefore the effective signal strength is reduced and the sensitivity with optical control is reduced in Fig. 6 at a short sensing time s. At a longer sensing time, the benefits from decohernce suppression by optical control manifest and the sensitivity is improved. At zero bias field, the transition from to is kept under optical control, and the one-trial sensitivity is improved for a wide range of times s in the figure. The reduction of the one-trial sensitivity with optical control at s is caused by the leakage of the population out of the -type system.
V Discussion and conclusion
We have proposed an all-optical scheme to prolong the quantum coherence of a negatively charged NV center in diamond. With the quantum coherence extended and the energy fluctuations in the ground sublevels suppressed by the optical driving fields, we have achieved magnetic resonance with much narrower spectral linewidth and higher detection sensitivity. Unlike magnetic resonance by pulse sequences or more generally by dynamical decoupling, in our scheme driving field fluctuations do not broaden the resonant signal peaks. The sensing frequency by optical control is determined by the energy gap of the NV ground sublevels, which can easily reach the GHz range and enables stable GHz frequency standards in solids. At zero field, the magnetic resonance spectrum also enables measurement of the direction of signal sources in the plane perpendicular to the NV symmetry axis. The performance of the all-optical scheme has been confirmed by numerical simulations, by selecting the driving amplitudes within a range where the transitions to the off-resonant excited state are small and the -type optical transition is a good approximation. Although we apply the optical driving fields on NV centers in diamond, the method is general and is applicable to other systems where a -type optical transition can be formed.
Acknowledgements.This work is supported by an Alexander von Humboldt Professorship, the EU Integrating projects SIQS and DIADEMS and the DFG via SPP 1601. A.R. acknowledges the support of ISF grant no. 1281/12.
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Appendix A Decoherence function in the frequency domain
Here we outline the decoherence suppression by optical control in the frequency domain. For simplicity, we consider a simplified model without contributions from spontaneous decay and examine the effect of up to the second order:
where the modulation function
With and the symmetry for classical noise, we have
We assume stationary noise; i.e., noise with time translation symmetry, . We write
in terms of the spectral power density
and the filter function
Without optical control, i.e., , the function cannot filter out low-frequency fluctuations, which are dominant sources of decoherence. With large optical driving fields, the filter function with has a power-law decay with the deviation , and the low frequency fluctuations are filtered out for large . In this simplified model, when the spectral power density around the frequency is negligible, and the decoherence is strongly suppressed.