# Extension of the Coherence Time by Generating MW Dressed States in a Single NV Centre in Diamond

## Abstract

Nitrogen-vacancy (NV) centres in diamond hold promise in quantum sensing applications. A major interest in them is an enhancement of their sensitivity by the extension of the coherence time (T2). In this report, we experimentally generated more than four dressed states in a single NV centre in diamond based on Autler-Townes splitting (ATS). We also observed the extension of the coherence time to T2 ~ 1.5 ms which is more than two orders of magnitude longer than that of the undressed states. As an example of a quantum application using these results we propose a protocol of quantum sensing, which shows more than an order of magnitude enhancement in the sensitivity.

## Introduction

A single spin of a nitrogen-vacancy (NV) centre in diamond has a long coherence time (T2) under ambient conditions, and hence it is a promising candidate for classical- and quantum-sensing applications1,2,3,4,5,6,7,8,9,10,11,12,13. Various approaches to enhance the sensitivity of the NV based sensors have been experimentally demonstrated so far. For example, hybrid quantum sensors, which are inspired by quantum memory effects12,13, are of great concern. They use nuclear spins of carbon or nitrogen around the NV centre as quantum memories. If paramagnetic impurities such as substitutional nitrogen (P1) centres and 13C nuclear spins are sufficiently suppressed14, we can keep T2 long and the sensitivity can be enhanced15. While, when we consider the situation to increase the number of nearby nuclear spins as quantum memories to enhance the sensitivity more, T2 of both the electron and nuclear spins become shorter16. Here, we focus on the microwave (MW) dressed state based on Autler-Townes splitting (ATS)17,18,19,20,21,22. Using ATS, a large number of the dressed states can be generated17. It has also been reported that T2 of the MW dressed states is longer than that of undressed states17.

In this report, we experimentally demonstrate the generation of the MW dressed states in the single NV centre in diamond by ATS at ambient conditions in order to analyse fundamental phenomena. Next, we show the extension of T2 under the generation of the dressed states. Finally, we propose a protocol for the quantum sensing with the dressed states for AC-field sensing and numerically estimate the sensitivity as functions of the number of the dressed states and T2.

## Mechanism of Autler-Townes Splitting

Dressed states based on the ATS are experimentally generated using the single NV centre in diamond by irradiation of an MW drive field. Figure 1(a) shows the energy level of the NV electron spin coupled with the 14N nuclear spin of the NV centre, where $$|{m}_{s},{m}_{I}\rangle$$ is defined as the electron and the 14N nuclear spin of the NV centre, respectively. After laser illumination, the NV centre is equally polarised in $$|0,0\rangle$$, $$|0,1\rangle$$, and $$|0,-1\rangle$$ depicted by the open circles under the application of a static magnetic field (B0). Figure 1(a) also depicts the irradiation of an unperturbed drive field whose frequency is close to a resonant frequency of a transition between $$|0,1\rangle$$ and $$|-1,1\rangle$$. When the drive field is considered as a classical mw mode, the NV centre can be coupled to the mode of the drive field. Then, each $$|0,1\rangle$$ and $$|-1,1\rangle$$ is split into two levels described in Fig. 1(b). It should be noted that Fig. 1(b) depicts the minimum number of dressed states by the ATS as an example. Thus, Fig. 1(b) depicts the generation of four dressed states of $$|0,1\rangle |1\rangle$$, $$|-1,1\rangle |0\rangle$$, $$|0,1\rangle |2\rangle$$, and $$|-1,1\rangle |1\rangle$$ in the presence of coupling between the NV centre ($$|0,1\rangle$$ and $$|-1,1\rangle$$) and the mode of the drive field ($$|0\rangle$$, $$|1\rangle$$, and $$|2\rangle$$). This phenomenon is called (weak) ATS. Figure 1(b) also shows the energy levels of the dressed states which are characterised by the Rabi frequency of an NV electron spin (Ω) and frequency of the drive field (ω), and its spectrum is given by the following equation23:

$$f(\nu )=\frac{\frac{1}{4}\kappa }{{({\rm{\nu }}-{\rm{\omega }})}^{2}+\frac{1}{4}{\kappa }^{2}}+\frac{\frac{3}{16}\kappa }{{({\rm{\nu }}-{\rm{\omega }}-\Omega )}^{2}+\frac{9}{16}{\kappa }^{2}}+\frac{\frac{3}{16}\kappa }{{({\rm{\nu }}-{\rm{\omega }}+\Omega )}^{2}+\frac{9}{16}{\kappa }^{2}},$$
(1)

where ν is the incident probe-frequency. κ is the inverse of the dephasing time. ∆ω = ω0ω. Here ω0 means the resonant frequency of an NV electron spin. The first term of Eq. (1) shows the resonant frequency depends on just ω and the second and third terms of Eq. (1) show the resonant frequencies depend on not only ω but also Ω. Thus, Eq. (1) is described as a Mollow-triplet spectrum. In the case of ∆ω = 0, the resonant frequency of the central peak, which is described by the first term in Eq. (1), does not depend on the Ω, while the side peaks, which are described by the second and third terms in Eq. (1), have linear dependences on Ω. In the case of $$\Delta \omega \ne 0$$, the resonant frequency of the central peak depends on ∆ω according to the following relation24: $$\Omega =\sqrt{{{\Omega }_{0}}^{2}+{(\Delta {\rm{\omega }})}^{2}}$$, where Ω0 is the Rabi frequency in the on-resonance condition. On the other hand, the changes in the resonant frequencies of the side peaks satisfy the following relation24: $$\Omega ={\Omega }_{0}\pm \sqrt{{{\Omega }_{0}}^{2}+{(\Delta {\rm{\omega }})}^{2}}$$.

## Results and Discussion

### The information of our sample and experimental setup

Figure 2(a) shows our experimental setup (See Methods). The sample is a high-temperature high-pressure (HTHP) type IIa (111) diamond. The second-order autocorrelation function, g(2)(τ), was measured using the Hanbury-Brown-Twiss (HBT) setup25 to confirm whether NV centre indicated by the circle depicted in Fig. 2(b) is a single centre or not. The power of the 532-nm laser is 100 µW. Figure 2(c) shows the measured g(2)(0) is 0.1. Therefore, this NV centre is a single centre.

In our experiment, we chose an NV centre that is weakly coupled to other nuclear spins (e.g., 13C nuclear spin). We measured the optically detected magnetic resonance (ODMR) spectrum with a 1-µs pulsed laser by sweeping the frequency of a 5.5-µs pulsed probe MW (Pmw) pulse (π pulse) depicted top of Fig. 3. In Fig. 3(a), the ODMR spectrum has three dips with 2.1 MHz splitting, which corresponds to the hyperfine splitting of the 14N nuclear spin of the NV centre26.

### Experimental generation of dressed states by ATS

First, we measured the change in the dressed-state resonant frequencies by changing the power of continuous drive MW (Dmw) with pulse sequence depicted in the top of Fig. 3. These experiments use the three Dmw frequencies of 2834.75 MHz (Dmw1), 2837.05 MHz (Dmw2), and 2839.18 MHz (Dmw3) to generate dressed states. The results are shown in Fig. 3(b). The signals for each Dmw frequency split into three above 10 µT.

Here we focus on the 2834.75 MHz of Dmw1. It should be noted that all three Dmw frequencies have the same dependences on the power of the continuous Dmw. The ODMR spectrum under continuous irradiation at Dmw1 with the power of 33 µT is shown in Fig. 3(c). It shows the increase of PL intensity was observed around 2835 MHz with continuous irradiation of Dmw1 (Fig. 3(c)) while the decrease of PL intensity was observed around 2835 MHz without continuous irradiation of Dmw1 (see Fig. 3(a)). The peaks are inverted, and the reason may come from the pulse sequence depicted in the top of Fig. 3. It shows the pulsed laser and the continuous Dmw field are simultaneously applied to the NV centre in the initialisation process. When the NV centre can be initialised into $$|-1\rangle$$ by the pulsed laser and continuous Dmw field, the increase of PL intensity may be observed at magnetic resonance conditions of the NV electron spin as discussed in ref.27. In addition, Fig. 3(c) shows the ODMR spectrum at 2834.75 MHz splits into three peaks under the irradiation of the Dmw. Figure 3(b) shows that the resonant frequencies of the dressed states as a function of Bdrive. The solid lines show the linear fitting for each observed data. The absolute values of these slopes in Fig. 3(b) agree well with the gyromagnetic ratio of the NV electron spin (γNV)28, so that means the resonant frequencies of side peaks are linearly proportional to the Rabi frequencies of the NV electron spin. This result is in agreement with the theory of the change of resonant frequencies according to the second and third terms in Eq. (1) with Ω = γBdrive.

Next, we measured the change in the resonant frequencies of the dressed states by changing the continuous Dmw frequency while fixing the Dmw power at 33 µT with the pulse sequence depicted in the top of Fig. 4. The Dmw frequency was changed by the step of 0.2 MHz. The result is shown in Fig. 4(a). It shows that the dressed states are generated when the Dmw frequencies are close to resonant frequencies of the NV electron spin.

Figure 4(b) illustrates the dressed-state resonant frequencies as a function of the Dmw frequency around the centre of 2835 MHz to understand more details of the results depicted in Fig. 4(a). The black squares in Fig. 4(b) show the change of resonant frequencies of the centre peaks as a function of Dmw frequency. The dependence can be fitted by a linear function shown in solid black line in Fig. 4(b), and hence this is in good agreement with the theoretical prediction of $$\Omega =\sqrt{{{\Omega }_{0}}^{2}+\,{(\Delta \omega )}^{2}}$$ described in Eq. (1). The red circles and the blue triangles in Fig. 4(b) also show the resonant frequencies of low- and high-resonant frequencies of the side peaks, respectively. Their resonant frequencies can be fitted by using the relation of the $$\Omega ={\Omega }_{0}\pm \sqrt{{{\Omega }_{0}}^{2}+{(\Delta \omega )}^{2}}$$24. Thus, all results are consistent with the theoretical prediction in ATS, demonstrating the generation of more than four dressed states by the ATS.

### Coherence time of dressed states

First, we show whether the magnetic moment of the dressed states is the same with that of the undressed NV electron spin or not, since the dressed states were generated due to the coupling between the NV electron spins and mode of the Dmw. In order to investigate the magnetic moments of the dressed and undressed states, we measured Rabi oscillations of the dressed states and the NV electron spin with the pulse sequence depicted in the top of Fig. 5. It noted that while the Rabi oscillation of the dressed state was measured with continuous Dmw, the Rabi oscillation of the NV electron spin was measured without continuous Dmw. Moreover, we kept the pulse sequence time (Tseq) constant adjusting interval between the pulsed Pmw and the readout laser pulse depicted in Fig. 5, in order to perform the initialisation of the dressed states by the simultaneous irradiation of the pulse laser and the continuous Dmw in the Rabi measurements. The bottom of Fig. 5 shows the results of the Rabi measurements, and it indicates the Rabi frequency of the dressed spin states is the same with that of undressed spin states. Consequently, the magnetic moment of the NV electron spin and the dressed states are the same with each other.

Next, we experimentally measured coherence time of the dressed states (T2ρ) and coherence time of a single NV centre (T2) in a 12C enriched diamond, since a coherent oscillation due to 13C nuclear spins on echo measurements29 can be suppressed using the NV centre in 12C enriched diamond. The top of Fig. 6 shows the pulse sequence for the T2ρ and T2 measurements. It is noted that while T2ρ was measured with the continuous Dmw irradiation, T2 was measured without continuous Dmw irradiation. Since the pulsed laser and continuous Dmw were simultaneously irradiated to the NV centre during the T2ρ measurements, we kept the pulse sequence time (Tseq) constant adjusting interval between the final π/2 Pmw pulse and the readout laser pulse depicted in Fig. 6. Then, the dressed spin states can be initialised by simultaneous irradiation of the pulse laser and the continuous Dmw in the T2ρ measurements. Additionally, a phase cycling technique was applied to T2ρ measurements in order to remove common-mode noise from laser fluctuations30. It is noted that the phase of the last π/2 Pmw pulse is indicated by ± signs depicted in the top of Fig. 6. In the case of a Pmw and Dmw strength of ~0.43 MHz and ~1.2 MHz, respectively, the result of T2ρ (black plots) and T2 (red plots) measurements fitted with exponential decay curves are shown in Fig. 6. The results show that we observed a coherence time of T2ρ ~ 1.5 ms of the dressed states, which is more than two orders of magnitude longer than T2 ~ 4.2 μs of the undressed states. While such an extension can also be demonstrated by a dynamical decoupling technique, e.g., a Carr-Purcell-Meiboom-Gill (CPMG) sequence in the NV centres6,31,32,33,34, an extension of two orders of magnitude by the ATS is much larger than the extension of about one order of T2 in the dynamical decoupling techniques6,30,31,32,33. The extended T2 by the ATS is also close to the longest T2 of a single NV centre in a 12C enriched diamond15.

### Estimation of the sensitivity of the quantum sensing with dressed states

We propose a quantum sensing protocol with the MW dressed states for AC magnetic field sensing and discuss the numerical estimation of the sensitivity for this sensing under a simple and ideal case. First, we shortly explain how to generate an arbitrary number of MW dressed states. Here we consider the NV centre is coupled with the mode of a strong Dmw, as illustrated in Fig. 7. The first energy is ħω/2, and the other states are separated by ħω, where ħ and ω are the reduced Planck constant and the frequency of Dmw, respectively. If such a mode of the Dmw ($$|n\rangle$$) is coupled with two states of the NV centre of $$|1\rangle$$ and $$|2\rangle$$, dressed states of $$|{(D(n)\rangle }_{\pm }$$ appear. The $$|D\,(n){\rangle }_{\pm }$$ and their energies (E±(n)) are described by the following equations18,35:

$$|{D(n)\rangle }_{\pm }={c}_{1}|1,n\rangle \pm {c}_{2}|2,n-1\rangle ,$$
(2)
$${E}_{\pm }(n)=(n-\frac{1}{2})\hslash \omega \pm \frac{1}{2}\hslash ,$$
(3)

respectively. c1 and c2 are the coefficients, which satisfy $${|{c}_{1}|}^{2}$$ + $${|{c}_{2}|}^{2}$$ = 1. Since a number of photons is large in the classical mw field mode17, a large number of dressed states can be generated. Then, we discuss the sensitivity of the AC magnetic field sensing using the arbitrary number of dressed states. The detections of the AC magnetic field using the undressed states and the dressed states discussed in Supplementary Informaiton are demonstrated based on a magnetic resonance technique. Thus, the sensitivity of the dressed states can be proportional to $$1/\sqrt{M{T}_{2{\rm{\rho }}}}$$, where 2M corresponds to the number of dressed states (Details are disucssed in Supplementary Information). In addition, our experimental demonstrations show the magnetic moments of dressed states are the same with that of undressed states and the coherence time of the dressed states is more than two orders of magnitude longer that of the undressed states. Based on the above discussion, the ratio of the sensitivities with and without the dressed states is taken by $$\sqrt{M{T}_{2\rho }/{T}_{2}}$$. In the case of T2ρ ~ 1.5 ms and T2 ~ 4.2 μs, the sensitivity is approximately enhanced 27 times with M = 2. Thus, the sensitivity can be effectively enhanced by using the dressed states. In particular, our protocol is very useful for an ensemble system which has very short T2, because of two effects: ‘the extension of T2’ and ‘the addition of the sensing information measured by generated dressed states’. Although we, here, do not consider the number of operations of the integration is limited by T2, our proposal opens up a new way for higher sensitivity on NV based AC magnetic field sensing.

## Conclusion

In conclusion, we have experimentally demonstrated the generation of more than four dressed states of an NV centre in diamond with irradiating the continuous Dmw based on ATS. Our experimental results were good agreement with the theoretical prediction. Additionally, we have proposed a new quantum sensing protocol with the dressed states for AC magnetic fields sensing. Numerical estimations show the sensitivity of the quantum sensing with the dressed states can be enhanced at least one-order of magnitude with experimentally observed T and T2. Thus, we believe that the quantum sensing with the dressed states can be applicable for improving the sensitivity of a quantum sensing.

Note: Recently, we have become aware of related works on quantum sensing with a one-time measurement based on a combination of the Mollow triplet and dynamical decoupling under sensing of a weak AC field with GHz frequencies36,37. Our work has three differences: 1) the frequency range for the sensing target, 2) the effect of integration of dressed states for a higher sensitivity, and 3) robustness against environmental noises (extension of T2). Our work can realise sensing of a weak low-frequency AC field by dressed states generated by ATS.

## Methods

### Sample preparation

To generate dressed states by ATS, we used high-temperature and high-pressure (HTHP) type IIa (111) diamond. After the nitrogen (14N) was implanted into the diamond with a 30-keV accelerating energy, the sample was annealed at 750 °C for 30 min for the generation of NV centres in diamond. To measure T2 and T2ρ, we used a single NV centre in a CVD-grown 12C enriched diamond layer on a type Ib (111) diamond substrate. The NV centres were generated during the growth of the diamond layer.

All experiments were performed by a homemade confocal microscope with an electromagnetic field (emf) irradiation system at room temperature depicted in Fig. 2(a). A 532-nm laser focused by an objective lens illuminates an NV centre in diamond. The detection system is composed of a 50:50 beam splitter (BS) and two avalanche photodiodes (APDs) in order to detect the photoluminescence and measure g(2)(τ). Two high frequencies with ~2.8 GHz irradiate to the NV centre by a by a thin copper wire with a diameter of 10 µm to manipulate the electron spin of an NV centre under the application of a static magnetic field generated by a neodymium magnet.

## Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work is supported by KAKENHI (No. 15H05868, 16H02088) and the Collaborative Research Program of ICR, Kyoto University (Grant # 2018-90). HM is supported by a Grant-in-Aid for Young Scientists (B), Grant No. 16K17484 and by the Future Development Funding Program of Kyoto University Research Coordination Alliance.

## Author information

H.M., T.T. and D.M. performed the measurements and the data analysis. H.K. synthesised the 12C-enriched diamond layers. All the authors (H.M., T.T., D.M., H.K., T.M., S.Y., M.F. and N.M.) contributed to the data analysis, discussion, and manuscript preparation.

Correspondence to H. Morishita or T. Tashima or N. Mizuochi.

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Morishita, H., Tashima, T., Mima, D. et al. Extension of the Coherence Time by Generating MW Dressed States in a Single NV Centre in Diamond. Sci Rep 9, 13318 (2019). https://doi.org/10.1038/s41598-019-49683-z