Direct Readout of Nitrogen-Vacancy Hybrid-Spin Quantum Register in Diamond by Photon Arrival Time Analysis
Jingyan He Yu Tian Zhiyi Hu Runchuan Ye Xiangyu Wang Dawei Lu Nanyang Xu

    





School of Physics & School of Microelectronics, Hefei University of Technology, Hefei, Anhui 230009, China
Research Center for Quantum Sensing, Zhejiang Lab, Hangzhou, 311000, China


Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China


School of Physics & School of Microelectronics, Hefei University of Technology, Hefei, Anhui 230009, China
Research Center for Quantum Sensing, Zhejiang Lab, Hangzhou, 311000, China


School of Physics & School of Microelectronics, Hefei University of Technology, Hefei, Anhui 230009, China
Research Center for Quantum Sensing, Zhejiang Lab, Hangzhou, 311000, China


Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China


ludw@sustech.edu.cn
Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China


nyxu@zhejianglab.edu.cn
Research Center for Quantum Sensing, Zhejiang Lab, Hangzhou, 311000, China





             
             
§ ABSTRACT 
Quantum state readout plays a pivotal role in quantum technologies, spanning applications in sensing, computation, and secure communication. In this work, we introduce a new approach for efficiently reading populations of hybrid-spin states in the nitrogen-vacancy center of diamond using a single laser pulse, which utilizes the excited state level anti-crossing mechanism at around 500 Gs. Reading spin state populations through this approach achieves the same outcome as traditional quantum state diagonal tomography but significantly reduces the experimental time by an order of magnitude while maintaining fidelity. Moreover, this approach may be extended to encompass full-state tomography, thereby obviating the requirement for a sequence of spin manipulations and mitigating errors induced by decoherence throughout the procedure.
   
          
Direct Readout of Nitrogen-Vacancy Hybrid-Spin Quantum Register in Diamond by Photon Arrival Time Analysis
    Nanyang Xu
    September 9, 2024
==========================================================================================================





§ INTRODUCTION


Owing to its outstanding optical and electron-spin properties, the negatively charged nitrogen-vacancy (NV) center in diamond has emerged as a highly promising platform for solid-state qubits <cit.>. The NV center provides both an electron spin and a substitutional nitrogen nuclear spin (^14 N or ^15 N), and these spins are coupled via a hyperfine coupling <cit.>. A notable feature of this system is its suitability for manipulation as a hybrid-spin quantum register. On one hand, the electron spin exhibits properties such as optical pumping and electron-spin-state-dependent fluorescence <cit.>. This enables feasible initialization and readout of the electron spin <cit.>, even at room temperature <cit.>. On the other hand, nuclear spins offer significantly longer spin lifetimes in comparison to electrons <cit.>, making them a valuable resource for applications such as quantum memories <cit.>, computational nodes for quantum error correction <cit.>, and quantum communication <cit.>. 


However, initialization and readout of nuclear spins with high fidelity remains a challenge due to the small magnetic moments of nuclear spins. To address this challenge, the electron spin often serves as an ancillary qubit, interacting with individual nuclear spins. Then the nuclear spin can be polarized by coherently transferring the state from the electron spin, and read out through the reverse process <cit.>. Achieving this necessitates the implementation of intricate quantum-control pulse sequences, operating at both microwave (MW) and radio frequencies (RF), to manipulate electron and nuclear spins, respectively. Nonetheless, due to its weak coupling to magnetic field, executing nuclear-spin operations takes roughly three orders of magnitude longer than the electron spin. As a result, controlling the nuclear-spin state is quite inefficient, and the microwave-induced heating effect hinders its application in thermally-sensitive sciences.

Due to the excited state level anti-crossing (ESLAC) phenomenon of NV center under a magnetic field around 500 Gs, the substitutional nitrogen spin can be automatically initialized via a dynamical nuclear polarization (DNP) process during the optical pumping <cit.>. This mechanism is frequently employed as a standard NV hybrid-spin initialization protocol in various applications <cit.>, while the readout method remains unchanged. In this work, we introduce a new avenue to directly obtain the spin state by analyzing the arrival time of the emitting photons in a single laser pulse. The difference in photon arrival time is also due to the ESLAC mechanism, which forms different photon time traces between the states. Combined with the ESLAC-based DNP protocol, we demonstrate a efficiency state initialization and readout experiment accompanied by exceeding an order of magnitude increase, this method can be extended to full-state tomography and reducing decoherence-induced errors.






§ PHYSICAL SYSTEM

The NV center in diamond is composed of a nitrogen atom substitution for a carbon atom, with a neighboring carbon vacancy as illustrated in Fig. <ref>(a). There exist two charge states, namely the neutral (NV^0) and the negatively charged (NV^-) NV color centers <cit.>. This investigation is primarily concerned with the negatively charged NV color center, characterized by a ground state in the form of a spin triplet state denoted as ^3A. This state displays a zero-field splitting of 2.87 GHz between its spin sublevels, specifically m_s=0 and m_s=±1. Additionally, there is an excited state denoted as ^3E, also a spin triplet, with a zero-field splitting of 1.4 GHz. The energy level diagram of the NV center is depicted in Fig. <ref>(b).
The NV center in diamond demonstrates a spin-dependent cycling transition of the electron spin under the influence of laser excitation <cit.>.

In Fig. <ref>(b), the spin state m_s=0 exhibits a notably low probability of undergoing intersystem crossing (ISC) <cit.> and a heightened likelihood of experiencing excitation to the corresponding m_s=0 state within the ^3E state (with a lifetime of 12 ns) before reverting to its initial state. Conversely, for the spin state m_s=±1, there is a substantial likelihood of transitioning to the intermediate ^1A state (with a lifetime of 250 ns), and an excited state lifetime of 7.8 ns for m_s=± 1. Once the NV center is in its metastable state, the decay back into the triplet ground state preferably occurs into the m_s=0 state with high fidelity. Under optical pumping, the electron spin state will ultimately reach m_s=0, regardless of its initial state.


Although laser illumination initializes the electron spin state, studying fluorescence evolution over time for each state (see Fig. <ref>(c)) is insightful. If the electron spin state was m_s=0 before laser illumination, it remains there, continually undergoing optical excitation and fluorescence emission cycles. This results in a sustained high level of fluorescence, except initially, where a small ISC rate, non-luminous and long-lived, leads to the initial fluorescence decrease. However, if the spin state was m_s=± 1, the higher ISC rate causes the electron spin to eventually transition from the excited state triplet to the excited singlet state during laser illumination. This transition leads to a breakdown in the fluorescence count rate until the NV returns to its ground state with m_s=0.
	
	
 The crucial aspect for optical readout of the electron spin state is that the NV center in the m_s=0 state emits a higher number of photons within the first few hundred nanoseconds compared to when it is in the m_s=±1 state. This leads to approximately 30 % more total photon counts, which we refer to as the “signal photon”. This difference in emitted photons for m_s=0 and m_s=±1 states allows for the electron spin state of the NV center to be efficiently read out at room temperature <cit.>. The two polarized electron spin states, m_s=0 and m_s=1, correspond to the upper and lower fluorescence boundaries of the NV system, denoted as L_0 and L_1, respectively. Any superposition of these two states results in a general photon count L that falls between these two fluorescence boundaries when measured repeatedly. L can be expressed as a function of L_0 and L_1 by L=c_0 L_0+c_1 L_1, 
where c_0 and c_1 denote the respective probabilities of the states m_s=0 and m_s=1, and c_0+c_1=1. Consequently, the electron spin state can be ascertained through the measurement of fluorescence counts.



The achievement of single-shot readout for an individual nuclear spin has been successfully demonstrated utilizing a single NV center in diamond <cit.>. This work was realized within the NV hybrid-spin system, where an NV center is coupled with ^14N nuclear spins, forming a spin-triplet state. The hybrid-spin system is illustrated in Fig. <ref>(b). When a magnetic field B is applied along the NV defect axis, the Hamiltonians for both the ground state and the excited state share the same form, given by H=DŜ^2_z+γ_eBŜ_z+γ_nBŜ_z+AŜÎ. Here, D represents the zero-field splitting, γ_e and γ_n denote the gyromagnetic ratios for the electron spin and nuclear spin, respectively, while Ŝ and Î correspond to the operators for the electron and nuclear spins. Additionally, A signifies the strength of the hyperfine interaction. At a magnetic field strength of 500 Gs, corresponding to the ESLAC region, electron-nuclear-spin flip-flops occur, leading to polarization into the m_s=0 state with m_I=+1.

Although the electron spin is a spin-1 system, a subspace containing only m_s = 0 and m_s=+1 (or -1) is utilized to form a two-level quantum bit (i.e., a qubit) in most cases of quantum information processing <cit.>. Specifically for quantum sensing, a spin-1 state (m_s=±1) has a relatively higher mangnetic moment, which can enhance magnetic sensing performance. However, the readout of m_s=+1 and m_s=-1 states still relies on the help of m_s=0 state <cit.>. Here, we consider only the subspace of m_s=0 and m_s=-1 within the ground spin-triplet state and denote the states with m_s=0 and m_s=-1 within the ground spin-triplet state as |0⟩_e and |1⟩_e, respectively, corresponding to the electron spin qubit. For the nuclear spin qubit of the ^14N nuclei, we use |↑⟩_n and |↓⟩_n to represent the states with m_I=0 and m_I=+1, as depicted in Fig. <ref>(b). The resulting polarized spin state achieved in this system is |0,↓⟩.
In a similar vein, the total fluorescence counts, denoted as L, are directly related to the population of the four basis states i.e., |0,↑⟩, |0,↓⟩, |1,↑⟩, and |1,↓⟩ as follows:

    L=c_0↑ L_0↑+c_0↓ L_0↓+c_1↑ L_1↑+c_1↓ L_1↓,

where c_0↑, c_0↓, c_1↑ and c_1↓ denote the respective probabilities of the corresponding states , and c_0↑+c_0↓+c_1↑+c_1↓=1.

However, Eq. (<ref>) is inadequate for the complete readout of the hybrid-spin states. To achieve spin state readout, the conventional approach involves the application of a set of unitary operations designed to transform the population, followed by another round of fluorescence count measurements. The specific experimental sequence is shown in Table <ref>.



The experimental process described above can be succinctly summarized by the following matrix equation  

    [[ L_0↑ L_0↓ L_1↑ L_1↓; L_0↑ L_1↓ L_1↑ L_0↓; L_0↓ L_0↑ L_1↑ L_1↓; L_0↑ L_1↑ L_0↓ L_1↓ ]]·[
    	[ c_0↑; c_0↓; c_1↑; c_1↓ ]]=[
    	[ L_0; L_1; L_2; L_3 ]].

Utilizing Eq. (<ref>), we can calculate the probabilities of different states and consequently determine the state being read out. 




§ DIRECT SPIN-READOUT SCHEME


Here, our primary focus centers on the photon time trace, which offers a detailed account of how fluorescence evolves over time for each state. We have effectively harnessed this approach to successfully read out the electron spin state, as previously demonstrated <cit.>.

At a magnetic field strength of 500 Gs, a noteworthy phenomenon called ESLAC emerges, as depicted in Fig. <ref> (a). Since the state |0,↑⟩ and |1,↓⟩ exchanges in the excited states due to the ESLAC mechanism, the nuclear states become unbalanced and also been polarized in the same process <cit.>. Crucially, the transition from |0,↓⟩ to the excited state conserves the nuclear spin, resulting in sustained high fluorescence until a small ISC rate leads to fluorescence decrease. Meanwhile, the transition from |0,↑⟩ has a high probability of transitioning to  |1,↓⟩ with a high ISC rate, thus fluorescence rapidly drops until the spin in metastable state returns to the ground state of |0,↓⟩. Consequently, the hybrid-spin system in the state |0,↓⟩ emits a higher number of photons compared to the  state |0,↑⟩. By incorporating the optically polarized electron spin mechanism, as described in the second part, where the electron spin m_s= ± 1 is lower than m_s= 0, it becomes evident that there are differences in fluorescence intensities among the four energy levels, namely |0,↑⟩, |0,↓⟩, |1,↑⟩, and |1,↓⟩.



The nuclear spin-dependent photon time traces are depicted in Fig. <ref>(b), and these traces are acquired after activating the readout laser. They are recorded using a custom-built time tagger with a resolution of 2 ns, implemented on a commercial field-programmable gate array module <cit.>. If we assume there are n time bins within the detection window, and m^i (where i ranges from 1 to n) represents the photon counts residing in the i-th time bin, then an arbitrary photon time trace can be expressed as a one-dimensional vector M=[m^1, ⋯, m^i, ⋯, m^n]^T. The fluorescence count m^i obtained from the readout is intricately linked to the population of basis states as 

    m^i=l^i_0↑c_0↑+l^i_0↓c_0↓+l^i_1↑c_1↑+l^i_1↓c_1↓.

In this context, l^i_0↑, l^i_0↓, l^i_1↑ and l^i_1↓ denote the photon counts present in the i-th time bin for the four basis states, respectively. Since there are n time bins, we can derive n functions, which can be represented in the matrix form

    [[ l^1_0↑ l^1_0↓ l^1_1↑ l^1_1↓; l^2_0↑ l^2_0↓ l^2_1↑ l^2_1↓;      ⋮      ⋮      ⋮      ⋮; l^i_0↑ l^i_0↓ l^i_1↑ l^i_1↓;      ⋮      ⋮      ⋮      ⋮; l^n_0↑ l^n_0↓ l^n_1↑ l^n_1↓;        ]]·[
    	[ c_0↑; c_0↓; c_1↑; c_1↓ ]]=[
    	[ m^1; m^2;   ⋮; m^i;   ⋮; m^n ]].


This function describes how an unknown state, represented as a vector M=[m^1, ⋯  m^i, ⋯  m^n]^T, can be expressed using a set of basis vectors L_0↑,0↓,1↑,1↓=[l^1_0↑,0↓,1↑,1↓,  ⋯, l^i_0↑,0↓,1↑,1↓, ⋯, l^n_0↑,0↓,1↑,1↓]^T within a Hilbert space. We use matrix form to represented the basis vectors as L=[L_0↑, L_0↓, L_1↑, L_1↓]. The coefficients c=[c_0↑, c_0↓, c_1↑, c_1↓]^T correspond to the probabilities associated with the four coefficients.


As an alternative to linear inversion methods, maximum likelihood estimation has been widely employed <cit.>. In this work, we utilize optimization techniques to perform a direct analysis of the photon time trace, with the objective of determining the optimal values c^ optimal that provide the best approximation to c^ theoretical with respect to a specific matrix norm. The optimization process is subject to a constraint condition, which is defined as



    min    L· c-M_2,
       s.t.     c^T · c=1.








§ EXPERIMENTAL REALIZATION

In our experimental setup, we conducted tests using an NV center embedded in a bulk chemical vapor deposition diamond. These experiments were carried out on a custom-built confocal microscopy system, all of which took place at room temperature. The specific NV center utilized in these experiments was positioned at a depth of 10 μ m within the diamond material. This NV center was addressed and manipulated using our in-house optically detected magnetic resonance system. The experimental setup involves a 532 nm green laser beam that is passed through an acoustic-optic modulator to facilitate switching. After passing through an oil objective lens and focusing on the NV center, the fluorescence emitted is collected by an avalanche photo diode (APD). The output signal from the APD is then detected by a custom-built time tagger with a resolution of 2 ns. To create a static magnetic field, approximately 500 Gs, a columnar neodymium magnet is employed. This magnetic field is applied parallel to the NV axis, which is in the z-direction. The purpose of this magnetic field is to exclusively affect the z-terms of the electron spin Ŝ_z, leading to the splitting of the m_s=±1 sublevels and enabling the ESLAC. This ESLAC phenomenon facilitates the optical polarization of the nuclear spin with ease. To manipulate the electron spin and nuclear spin for general-purpose measurements, MW and RF sources are utilized, respectively. The frequencies of RF_1 (5.102067 MHz) and RF_2 (2.941124 MHz) have been calibrated through electron-nuclear double resonance experiments.


To validate the procedure, our initial step involved calibrating the photon time traces of the four basis states through a substantial number of measurements, totaling 10^9. These calibrated traces are collectively represented as a set of basis vectors, denoted as {L_0↑, L_0↓, L_1↑, L_1↓}. Subsequently, we prepared test states, namely |0,↑⟩, |0,↓⟩, |1,↑⟩, and |1,↓⟩, through different operations. And the fidelity of test sate are over 0.99 (see Appendix <ref>). We then carried out laser-based readouts for these test states, totaling 10^7 measurements, and the results were documented as M.
Utilizing the optimal method, we successfully obtained the four coefficients: c_0 ↑, c_0 ↓, c_1 ↑, c_1 ↓, and readout the spin state populations with fidelity 0.9972 ± 0.00042, 0.9963 ± 0.00043, 0.9981± 0.00038, and 0.9982 ± 0.00038, respectively. Here, we characterize the fidelity of population readout as Equ. (<ref>), obtained from the full state tomography caculation <cit.>.

    F_p=(c^ th, c^ exp)/√((c^ th, c^ th)(c^ exp, c^ exp)),

where c^ th (c^ exp) is theoretical( experimentally reconstructed) population vectors formed by [c_0↑, c_0↓, c_1↑, c_1↓]^T.

Now, we conside the population readout of an arbitrary state, in Fig. <ref>.(b) the time trace of basis state are calibrated and we think the time trace of arbitrary state is straightforwardly the superposition of the four basic states, which has be verified in Supplementary Materials. Here, we perform one more experiment employing the superposition states of electron or nuclear spins and the results are shown in Table <ref>. 
 


The obtained fidelity values, denoting the similarity between the input states 1/√(2)(|0,↑⟩+|0,↓⟩), 1/√(2)(|0,↓⟩+|1,↓⟩) and 1/√(2)(|1,↑⟩+|1,↓⟩) are  0.99145 ± 0.00065, 0.99206 ± 0.00050 and 0.98845 ± 0.00083 respectively. The observed differences between the experimental and theoretical results can primarily be attributed to factors such as statistical variations in photon detection and control errors arising from pulse imperfections during state preparation and tomography implementation.



§ TIME-COST ANALYSIS

The direct readout method sets itself apart from the conventional approach by eliminating the need for a spin operation sequence. In the general method, reading out the nuclear spin state of the NV hybrid-spin quantum register necessitates at least four operations and laser-based readout. However, given the effective isolation of nuclear spins from their surrounding environment, manipulating nuclear spins entails executing a lengthy RF sequence lasting approximately 100 μs.

 

In practical experiments, the most significant factor contributing to the fidelity loss is the shot noise σ_M. Due to the photon counts follow a Poisson distribution, it is correlated with the photon signal, i.e σ_M=[√(m^1), ⋯ √(m^i), ⋯ √(m^n)]^T. This loss can be mitigated by increasing the number of measurement sweeps. Here, we conducted simulations based on experimental data to analyze the relationship between fidelity, the number of sweeps, and the total experiment time. The specific simulation process is as follows.


 Initially, we perform measurements to obtain the photon time traces for the four basis states through 10^9 measurements. These traces are denoted as {L_0↑, L_0↓, L_1↑, L_1↓}, and we assume them to be error-free. The total number of measurements for this step is marked as S1.  In the second step, we randomly generate a set of target samples represented as (c_0↑^i, c_0↓^i, c_1↑^i, c_1↓^i). Next, we generate theoretical data, m^i_ th, using the following equation: m^i_ th=S1/S2c_0↑^i l_0↑^i+c_0↓^i l_0↓^i+c_1↑^i l_1↑^i+c_1↓^i l_1↓^i. Here, S2  represents the number of experiment sweeps. To simulate the real experimental scenario, we add photon counting statistics noise, δ m^i_ th, to the theoretical data, m^i_ th, resulting in experimental data, m^i_ exp. The noise, δ m^i_ th, is drawn from a Gaussian distribution and falls within the range [-√(m^i_ th), √(m^i_ th)]. By changing the value of S1, we can generate experimental data at different numbers of sweeps, which allows us to study the relationship between fidelity, sweeps, and experiment time. Finally, we use the generated photon time trace and experimental data, which have been normalized based on their maximum values, as inputs to an optimization model for analysis.

The evolution of fidelity F_p as a function of the number of sweeps under a magnetic field strength of 500 Gs is depicted in Fig. <ref>(a). When the fidelity level is not very high, the direct readout method requires fewer sweeps, especially when fidelity is less than 0.90. As the number of sweeps increases, the operation time of the general method grows linearly. In contrast, the direct readout method has a shorter readout time because it only includes the laser component and no additional operational time. Consequently, for the same number of sweeps, the direct readout method requires less time.



To further elucidate the relationship between fidelity and readout time, Fig. <ref>(b) presents the corresponding data. When compared to the conventional general method, our approach demonstrates a substantial enhancement in time efficiency. For instance, when striving for a fidelity of 0.95, the direct readout method requires only 6.83× 10^8 ns, whereas the general method demands a significantly longer time of 2.24×10^10 ns. Therefore, in this particular scenario, our method accelerates the experimental process by a factor of 32. This time-saving achievement is primarily realized by reducing the number of essential spin operations, making it particularly advantageous for systems characterized by shorter decoherence times.




§ DEPENDENCE ON MAGNETIC FIELD
  
The ESLAC phenomenon is a crucial aspect of our method. Optical excitation at ESLAC results in state-selective spin mixing between the NV electron and nuclear spin. To gain deeper insights into our approach, we conducted measurements of the photon time trace under five different magnetic fields, comprising 10^8 measurements each. Fig. <ref> presents simulation data for these various magnetic field strengths, illustrating the relationship between fidelity F_p and sweeps. The evolution of  F_p as a function of sweeps can be well-fitted using the functions F_p=1-e^(as^2+bs+c) and F_p=1-e^(a(ι-δ)^2+b(ι-δ)+c). In the field regime close to 500 Gs (ESLAC point), we observe an optimal experimental effect where achieving the same fidelity level requires the fewest sweeps and the shortest time. This phenomenon arises due to the excited state electron-nuclear spin flip-flop process, induced by hyperfine interactions.


  

The spin flip frequency is closely related to the magnetic field strength, and it reaches an extreme value at approximately 500 Gs. Under the ESLAC conditions, the non-eigenstate |0,↓⟩ completely transforms into |1,↑⟩ and then back again, resulting in the most substantial differences in the photon time traces among the four initial states. In fact, even minor state mixing in the excited state can lead to noticeable distinctions in the photon time traces.

In the evaluation of photon time traces within the spin readout mechanism, we examine the worst-case noise magnification ratio κ<cit.>, which exhibits a quadratic correlation with the magnetic field, as shown in Fig. <ref>. This observed change pattern aligns with the experimental results.




§ DISCUSSION AND CONCLUSION

In summary, we have effectively showcased a novel approach for achieving single-shot readout of nuclear spins linked to room-temperature NV centers in diamond. Our method centers around the direct analysis of photon time traces, with a specific emphasis on the discernible characteristics present during the ESLAC. When contrasted with conventional techniques, our approach introduces a substantial time-saving advantage, reducing the necessary time by a factor of 32, thanks to the elimination of spin operations prior to laser readout. While the ESLAC point represents the optimal condition for our method, it can also be applied to read out spin states whenever discernible differences in photon time traces are present.

 This method bestows benefits akin to those derived from diagonal element tomography experiments, and it can also be effectively applied to full-state tomography experiments. In state tomography experiments, it mitigates the requirement for intricate nuclear spin manipulation, offering a clear advantage in scenarios where preserving high fidelity poses challenges due to short decoherence times. This approach streamlines the experimental setup while retaining its capacity to uphold fidelity, rendering it a valuable tool for applications in quantum information processing and other domains that involve nuclear spin readout and manipulation.
 


§ ACKNOWLEDGMENTS

This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 226-2023-00137), the National Natural Science Foundation of China (Grant Nos. 92265114, 92265204, 12104213), the National Key Research and Development Program of China (2019YFA0308100) and the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302200).



§ SIMULATION OF TIME TRACE


At a magnetic field strength of 500 Gs, a noteworthy phenomenon called ESLAC emerges (see Fig. <ref>(a)), which will influence the fluorescence intensities among the four energy levels, namely |0,↑⟩, |0,↓⟩, |1,↑⟩, and |1,↓⟩. Here, we provided numerical simulations of the time trace with different ESLAC rates (see Fig. <ref>). Furthermore, from the simulation of arbitrary superposition states in Fig. <ref>, it can be observed that an arbitrary state is straightforwardly the superposition of the basic states.






    

	
	
  
	



§ TWO-QUBIT FULL STATE TOMOGRAPHY

The density matrix ρ of two-qubit entanglement is as followes:

   
    ρ_0=[[                      c_0↑                         … a_|0↑⟩⟨ 1↓|+j b_|0↑⟩⟨ 1↓|; a_|0↑⟩⟨ 0↓|-j b_|0↑⟩⟨ 0↓|                         … a_|0↓⟩⟨ 1↓|+j b_|0↓⟩⟨ 1↓|; a_|0↑⟩⟨ 1↑|-j b_|0↑⟩⟨ 1↑|                         … a_|1↑⟩⟨ 1↓|+j b_|1↑⟩⟨ 1↓|; a_|0↑⟩⟨ 1↓|-j b_|0↑⟩⟨ 1↓|                         …                      c_1↓;                           ]].
 


The setup for measure diagonal elements c_0↑, c_0↓, c_1↑ and c_1↓ are explanated detaily in the second part of the main text. The  a_|0↑⟩⟨ 0↓|, |0↑⟩⟨ 1↑|, |0↑⟩⟨ 1↓|, |0↓⟩⟨ 1↑|, |0↓⟩⟨ 1↓|, |1↑⟩⟨ 1↓| and b_|0↑⟩⟨ 0↓|, |0↑⟩⟨ 1↑|, |0↑⟩⟨ 1↓|, |0↓⟩⟨ 1↑|, |0↓⟩⟨ 1↓|, |1↑⟩⟨ 1↓| are denote the real and imaginary part of off-diagonal elements |0↑⟩⟨ 0↓|, |0↑⟩⟨ 1↑|, |0↑⟩⟨ 1↓|, |0↓⟩⟨ 1↑|, |0↓⟩⟨ 1↓| and |1↑⟩⟨ 1↓| respectively.    
	

For measure off-diagonal element, we need apply microwave(MW) or radio frequencies(RF) sequence to transform off-diagonal element into diagonal, then readout by laser. The off-diagonal experimental sequence is shown in <ref>.

Next, take off-diagonal element |0↑⟩⟨ 1↓| as an example for a detailed introduction. Firstly, we apply a MW_2 π pulse, and the partial density matrix evolves as follows:

    
    ρ   =[[  1  0  0  0;  0  0  0 -j;  0  0  1  0;  0 -j  0  0;    ]] ρ_0 [[ 1 0 0 0; 0 0 0 j; 0 0 1 0; 0 j 0 0;   ]]
       =[[                      c_0↑ ja_|0↑⟩⟨ 1↓|- b_|0↑⟩⟨ 1↓|                         …                         …; -ja_|0↑⟩⟨ 1↓|+b_|0↑⟩⟨ 1↓|                      c_1↓                         …                         …;                         …                         …                      c_1↑                         …;                         …                         …                         …                      c_0↓;                           ]].




Then applying four RF_1 π/2 pulse with four different phases, the evolution of 2 × 2 density matrices ρ_1,2,3,4^sub go as follows:


    ρ_0^sub   =[[                      c_0↑ ja_|0↑⟩⟨ 1↓|- b_|0↑⟩⟨ 1↓|; -ja_|0↑⟩⟨ 1↓|+b_|0↑⟩⟨ 1↓|                      c_1↓;                           ]],
    ρ_1^sub   =[[  1 -j; -j  1;    ]]
            ρ_0^sub[[ 1 j; j 1;   ]]
       =[[     c_0 ↑+c_1 ↓/2-a_|0↑⟩⟨ 1↓|  j/2(c_0 ↑-c_1 ↓)-b_|0↑⟩⟨ 1↓|; -j/2(c_0 ↑-c_1 ↓)+b_|0↑⟩⟨ 1↓|     c_0 ↑+c_1 ↓/2+a_|0↑⟩⟨ 1↓|;                               ]],
    ρ_2^sub   =[[ 1 j; j 1;   ]]ρ_0^sub[[  1 -j; -j  1;    ]]
       =[[     c_0 ↑+c_1 ↓/2+a_|0↑⟩⟨ 1↓| -j/2(c_0 ↑-c_1 ↓)-b_|0↑⟩⟨ 1↓|;  j/2(c_0 ↑-c_1 ↓)-b_|0↑⟩⟨ 1↓|     c_0 ↑+c_1 ↓/2-a_|0↑⟩⟨ 1↓|;                               ]],
    ρ_3^sub   =[[  1 -1;  1  1;    ]]ρ_0^sub[[  1  1; -1  1;    ]]
       =[[      c_0 ↑+c_1 ↓/2+b_|0↑⟩⟨ 1↓| 1/2(c_0 ↑-c_1 ↓)+j a_|0↑⟩⟨ 1↓|;   (c_0 ↑-c_1 ↓)/2-ja_|0↑⟩⟨ 1↓|      c_0 ↑+c_1 ↓/2-b_|0↑⟩⟨ 1↓|;                                ]],
    ρ_4^sub   =[[  1  1; -1  1;    ]]ρ_0^sub[[  1 -1;  1  1;    ]]
       =[[     c_0 ↑+c_1 ↓/2-b_|0↑⟩⟨ 1↓| -(c_0 ↑-c_1 ↓)/2+ja_|0↑⟩⟨ 1↓|; -(c_0 ↑-c_1 ↓)/2-ja_|0↑⟩⟨ 1↓|     c_0 ↑+c_1 ↓/2+b_|0↑⟩⟨ 1↓|;                               ]].



And now, the off-diagonal elements have been transfered to the elements, which can be optical readout. Following a laser readout, we get:

   
    X_1   =(c_0↑+c_1↓/2-a_|0↑⟩⟨ 1↓|)L_0↑
       +(c_0↑+c_1↓/2+a_|0↑⟩⟨ 1↓|)L_0 ↓+c_1↑L_1 ↑+c_0 ↓L_1 ↓,



    X_2   =(c_0↑+c_1↓/2+a_|0↑⟩⟨ 1↓|)L_0↑
       +(c_0↑+c_1↓/2-a_|0↑⟩⟨ 1↓|)L_0 ↓+c_1↑L_1 ↑+c_0 ↓L_1 ↓,



    Y_1   =(c_0↑+c_1↓/2+b_|0↑⟩⟨ 1↓|)L_0↑
       +(c_0↑+c_1↓/2-b_|0↑⟩⟨ 1↓|)L_0 ↓+c_1↑L_1 ↑+c_0 ↓L_1 ↓,



    Y_2   =(c_0↑+c_1↓/2-b_|0↑⟩⟨ 1↓|)L_0↑
       +(c_0↑+c_1↓/2+b_|0↑⟩⟨ 1↓|)L_0 ↓+c_1↑L_1 ↑+c_0 ↓L_1 ↓,



    a_|0↑⟩⟨ 1↓|=-X_1+X_2/2(L_0 ↑-L_0 ↓),


    b_|0↑⟩⟨ 1↓|=Y_1-Y_2/2(L_0 ↑-L_0 ↓).
 


In this work, the all off-diagonal elements are calculated by the similar method, and the results of tomography as shown in Fig. <ref>.
   
	
unsrt









50
fxundefined [1]
 ifx#1 

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Doherty et al.(2016)Doherty, Meriles, Alkauskas, Fedder, Sellars, and Manson]PhysRevX.6.041035
  author author M. W. Doherty, author C. A. Meriles, author A. Alkauskas,
  author H. Fedder, author M. J. Sellars,  and author N. B. Manson, 10.1103/PhysRevX.6.041035 
  journal journal Phys.Rev. X volume 6, pages 041035
  (year 2016)NoStop
[Nemoto et al.(2014)Nemoto,
  Trupke, Devitt, Stephens,
  Scharfenberger, Buczak, Nöbauer, Everitt, Schmiedmayer, and  Munro]PhysRevX.4.031022
  author author K. Nemoto, author M. Trupke,
  author S. J. Devitt, author A. M. Stephens, author
  B. Scharfenberger, author
  K. Buczak, author T. Nöbauer, author M. S.  Everitt, author J. Schmiedmayer,  and author W. J.  Munro, 10.1103/PhysRevX.4.031022
  journal journal Phys. Rev. X volume 4, pages 031022 (year
  2014)NoStop
[Hanson et al.(2008)Hanson,
  Dobrovitski, Feiguin, Gywat,  and Awschalom]science.1155400
  author author R. Hanson, author V. V. Dobrovitski, author A. E. Feiguin, author O. Gywat,   and author D. D. Awschalom,  10.1126/science.1155400 journal journal Sci. volume 320, pages 352
  (year 2008)NoStop
[Harley et al.(1984)Harley,
  Henderson, and Macfarlane]R.T.Harley_1984
  author author R. T. Harley, author M. J. Henderson,  and author R. M. Macfarlane, 10.1088/0022-3719/17/8/002 journal journal J. Phys. C: Solid State Phys. volume 17, pages L233 (year
  1984)NoStop
[Lovchinsky et al.(2016)Lovchinsky, Sushkov, Urbach, de Leon, Choi, Greve, Evans, Gertner, Bersin, Müller, McGuinness, Jelezko, Walsworth, Park, and Lukin]science.aad8022
  author author I. Lovchinsky, author A. O. Sushkov, author E. Urbach,
  author N. P. de Leon, author S. Choi, author
  K. D. Greve, author
  R. Evans, author R. Gertner, author E. Bersin, author C. Müller, author L. McGuinness, author F. Jelezko, author R. L.  Walsworth, author H. Park,  and author M. D. Lukin, 10.1126/science.aad8022 journal
  journal Sci. volume 351, pages 836 (year 2016)NoStop
[Staudacher et al.(2013)Staudacher, Shi, Pezzagna, Meijer, Du, Meriles, Reinhard, and Wrachtrup]science.1231675
  author author T. Staudacher, author F. Shi,
  author S. Pezzagna, author J. Meijer, author
  J. Du, author C. A. Meriles, author F. Reinhard,  and author J. Wrachtrup, 10.1126/science.1231675
  journal journal Sci. volume 339, pages 561 (year
  2013)NoStop
[Glenn et al.(2018)Glenn,
  Bucher, Lee, Lukin,
  Park, and Walsworth]nature25781
  author author D. R. Glenn, author D. B. Bucher,
  author J. Lee, author
  M. D. Lukin, author
  H. Park,  and author
  R. L. Walsworth, 10.1038/nature25781
  journal journal Nat.  volume 555, pages 351 (year
  2018)NoStop
[Suter and Jelezko(2017a)]j.pnmrs.2016.12.001
  author author D. Suter and author F. Jelezko, https://doi.org/10.1016/j.pnmrs.2016.12.001
  journal journal Prog. Nucl. Magn. Reson.
  Spectrosc. volume 98-99, pages 50
  (year 2017a)NoStop
[Cramer et al.(2016a)Cramer, Kalb,
  Rol, Hensen, Blok,
  Markham, Twitchen, Hanson,  and Taminiau]ncomms11526
  author author J. Cramer, author N. Kalb,
  author M. A. Rol, author B. Hensen, author
  M. S. Blok, author
  M. Markham, author D. J. Twitchen, author R. Hanson,  and author T. H.  Taminiau, 10.1038/ncomms11526 journal journal Nat. Commun. volume
  7, pages 11526 (year
  2016a)NoStop
[Mamin et al.(2013)Mamin,
  Kim, Sherwood, Rettner,
  Ohno, Awschalom, and Rugar]mamin_nanoscale_2013
  author author H. J. Mamin, author M. Kim, author M. H. Sherwood, author
  C. T. Rettner, author
  K. Ohno, author D. D. Awschalom,  and author D. Rugar, 10.1126/science.1231540
  journal journal Sci. volume 339, pages 557 (year
  2013)NoStop
[Oort et al.(1988)Oort,
  Manson, and Glasbeek]oort_optically_1988
  author author E. V. Oort, author N. B. Manson,   and author M. Glasbeek, 10.1088/0022-3719/21/23/020 journal journal J. Phys. C: Solid State Phys. volume
  21, pages 4385 (year 1988)NoStop
[Terblanche et al.(2001)Terblanche, Reynhardt, Rakitianski, and  Van Wyk]TERBLANCHE2001107
  author author C. J. Terblanche, author E. C. Reynhardt, author S. A. Rakitianski,  and author J. A. Van Wyk, https://doi.org/10.1006/snmr.2001.0025
  journal journal Solid State Nucl. Magn. Reson.  volume 19, pages 107 (year
  2001)NoStop
[Dutt et al.(2007)Dutt,
  Childress, Jiang, Togan,
  Maze, Jelezko, Zibrov,
  Hemmer, and Lukin]science.1139831
  author author M. V. G.  Dutt, author L. Childress, author L. Jiang,
  author E. Togan, author J. Maze, author
  F. Jelezko, author A. S. Zibrov, author P. R.  Hemmer,  and author M. D.  Lukin, 10.1126/science.1139831 journal journal Sci. volume 316,  pages 1312 (year 2007)NoStop
[Maurer et al.(2012)Maurer,
  Kucsko, Latta, Jiang,
  Yao, Bennett, Pastawski,
  Hunger, Chisholm, Markham,
  Twitchen, Cirac, and Lukin]science.1220513
  author author P. C. Maurer, author G. Kucsko,
  author C. Latta, author L. Jiang, author
  N. Y. Yao, author S. D. Bennett, author F. Pastawski, author D. Hunger, author N. Chisholm, author M. Markham, author D. J.  Twitchen, author J. I.  Cirac,  and author M. D.  Lukin, 10.1126/science.1220513 journal journal Sci. volume 336,  pages 1283 (year 2012)NoStop
[Childress et al.(2006)Childress, Gurudev Dutt, Taylor,
  Zibrov, Jelezko, Wrachtrup,
  Hemmer, and Lukin]science.1131871
  author author L. Childress, author M. V. Gurudev Dutt, author J. M. Taylor, author A. S. Zibrov,
  author F. Jelezko, author J. Wrachtrup, author
  P. R. Hemmer,  and author
  M. D. Lukin, 10.1126/science.1131871 
  journal journal Sci.  volume 314, pages 281 (year
  2006)NoStop
[Cramer et al.(2016b)Cramer, Kalb,
  Rol, Hensen, Blok,
  Markham, Twitchen, Hanson,  and Taminiau]cramer_repeated_2016
  author author J. Cramer, author N. Kalb,
  author M. A. Rol, author B. Hensen, author
  M. S. Blok, author
  M. Markham, author D. J. Twitchen, author R. Hanson,  and author T. H.  Taminiau, 10.1038/ncomms11526 journal journal Nat. Commun. volume
  7, pages 11526 (year
  2016b)NoStop
[Reiserer et al.(2016)Reiserer, Kalb, Blok, Van Bemmelen, Taminiau, Hanson,
  Twitchen, and Markham]reiserer_robust_2016
  author author A. Reiserer, author N. Kalb,
  author M. S. Blok, author K. J. M. van Bemmelen, author T. H. Taminiau, author
  R. Hanson, author D. J. Twitchen,  and author M. Markham, 10.1103/PhysRevX.6.021040
  journal journal Phys. Rev. X volume 6, pages 021040 (year
  2016)NoStop
[Kalb et al.(2017)Kalb,
  Reiserer, Humphreys, Bakermans, Kamerling, Nickerson,
  Benjamin, Twitchen, Markham,  and Hanson]kalb_entanglement_2017
  author author N. Kalb, author A. A. Reiserer,
  author P. C. Humphreys, author J. J. W. Bakermans, author S. J. Kamerling, author
  N. H. Nickerson, author
  S. C. Benjamin, author
  D. J. Twitchen, author
  M. Markham,  and author
  R. Hanson, 10.1126/science.aan0070
  journal journal Sci.  volume 356, pages 928 (year
  2017)NoStop
[Pfender et al.(2019a)Pfender, Wang,
  Sumiya, Onoda, Yang,
  Dasari, Neumann, Pan,
  Isoya, Liu, and Wrachtrup]pfender_high-resolution_2019
  author author M. Pfender, author P. Wang,
  author H. Sumiya, author S. Onoda, author
  W. Yang, author D. B. R. Dasari, author P. Neumann, author X.-Y.  Pan, author J. Isoya, author R.-B. Liu,   and author J. Wrachtrup, 10.1038/s41467-019-08544-z journal journal Nat. Commum. volume 10, pages 594 (year 2019a)NoStop
[Lvovsky(2004)]lvovsky_iterative_2004
  author author A. I. Lvovsky, 10.1088/1464-4266/6/6/014 journal journal J. Opt. B: Quantum Semiclassical Opt.  volume 6, pages S556 (year
  2004)NoStop
[Pfender et al.(2019b)Pfender, Wang,
  Sumiya, Onoda, Yang,
  Dasari, Neumann, Pan,
  Isoya, Liu, and Wrachtrup]s41467-019-08544-z
  author author M. Pfender, author P. Wang,
  author H. Sumiya, author S. Onoda, author
  W. Yang, author D. B. R. Dasari, author P. Neumann, author X. Y.  Pan, author J. Isoya, author R. B. Liu,   and author J. Wrachtrup, 10.1038/s41467-019-08544-z journal journal Nat. Commun. volume 10, pages 594 (year 2019b)NoStop
[Hopper et al.(2018)Hopper,
  Shulevitz, and Bassett]hopper_spin_2018
  author author D. Hopper, author H. Shulevitz,   and author L. Bassett, 10.3390/mi9090437 journal journal
  Micromachines volume 9, pages 437
  (year 2018)NoStop
[Fuchs et al.(2008)Fuchs,
  Dobrovitski, Hanson, Batra,
  Weis, Schenkel, and Awschalom]PhysRevLett.101.117601
  author author G. D. Fuchs, author V. V. Dobrovitski, author R. Hanson,
  author A. Batra, author C. D. Weis, author
  T. Schenkel,  and author
  D. D. Awschalom, 10.1103/PhysRevLett.101.117601 journal journal
  Phys. Rev. Lett. volume 101, pages
  117601 (year 2008)NoStop
[Jacques et al.(2009)Jacques, Neumann, Beck, Markham, Twitchen, Meijer, Kaiser, Balasubramanian, Jelezko, and  Wrachtrup]PhysRevLett.102.057403
  author author V. Jacques, author P. Neumann,
  author J. Beck, author
  M. Markham, author D. Twitchen, author J. Meijer, author F. Kaiser, author G. Balasubramanian, author F. Jelezko,  and author J. Wrachtrup, 10.1103/PhysRevLett.102.057403 journal journal
  Phys. Rev. Lett. volume 102, pages
  057403 (year 2009)NoStop
[Dréau et al.(2013)Dréau, Spinicelli, Maze, Roch, and Jacques]PhysRevLett.110.060502
  author author A. Dréau, author P. Spinicelli,
  author J. R. Maze, author J.-F. Roch,  and author V. Jacques, 10.1103/PhysRevLett.110.060502 journal journal
  Phys. Rev. Lett. volume 110, pages
  060502 (year 2013)NoStop
[Zangara et al.(2019)Zangara, Dhomkar, Ajoy, Liu, Nazaryan, Pagliero, Suter, Reimer, Pines, and Meriles]pnas.1811994116
  author author P. R. Zangara, author S. Dhomkar,
  author A. Ajoy, author
  K. Liu, author R. Nazaryan, author D. Pagliero, author D. Suter, author J. A.  Reimer, author A. Pines,  and author C. A. Meriles, 10.1073/pnas.1811994116 journal
  journal Proc. Natl. Acad. Sci. U.S.A. volume 116, pages 2512 (year 2019)NoStop
[Qian et al.(2022)Qian,
  Zhai, Hu, Zheng,
  Chen, and Xu]PhysRevA.106.033506
  author author P. Qian, author Y. Zhai, author J. Hu, author
  R. Zheng, author B. Chen,  and author N. Xu, 10.1103/PhysRevA.106.033506
  journal journal Phys. Rev. A volume 106, pages 033506 (year
  2022)NoStop
[Jelezko and Wrachtrup(2006)]pssa.200671403
  author author F. Jelezko and author J. Wrachtrup, https://doi.org/10.1002/pssa.200671403
  journal journal Phys. Status Solidi A volume 203, pages 3207 (year
  2006)NoStop
[Goldman et al.(2015)Goldman, Sipahigil, Doherty, Yao, Bennett, Markham, Twitchen, Manson, Kubanek, and  Lukin]PhysRevLett.114.145502
  author author M. L. Goldman, author A. Sipahigil,
  author M. W. Doherty, author N. Y. Yao, author
  S. D. Bennett, author
  M. Markham, author D. J. Twitchen, author N. B.  Manson, author A. Kubanek,  and author M. D. Lukin, 10.1103/PhysRevLett.114.145502 journal journal Phys. Rev. Lett. volume 114, pages 145502 (year
  2015)NoStop
[Lin et al.(2023)Lin,
  Fan, Ye, Zhou, Song, Lu, and Xu]s11467-022-1235-5
  author author X. Lin, author J. Fan, author R. Ye, author
  M. Zhou, author Y. Song, author D. Lu,  and author N. Xu,  10.1007/s11467-022-1235-5 journal journal Front. Phys. volume 18, pages 21301 (year 2023)NoStop
[Suter and Jelezko(2017b)]suter_single-spin_2017
  author author D. Suter and author F. Jelezko, 10.1016/j.pnmrs.2016.12.001 journal journal Prog. Nucl. Magn. Reson. Spectrosc.  volume 98-99, pages 50 (year
  2017b)NoStop
[Gulka et al.(2021)Gulka,
  Wirtitsch, Ivády, Vodnik,
  Hruby, Magchiels, Bourgeois,
  Gali, Trupke, and Nesladek]gulka_room-temperature_2021
  author author M. Gulka, author D. Wirtitsch,
  author V. Ivády, author J. Vodnik, author
  J. Hruby, author G. Magchiels, author E. Bourgeois, author A. Gali, author M. Trupke,  and author M. Nesladek, 10.1038/s41467-021-24494-x journal journal Nat. Commun. volume
  12, pages 4421 (year 2021)NoStop
[Warren(1997)]warren_usefulness_1997
  author author W. S. Warren, 10.1126/science.277.5332.1688 journal journal Sci. volume 277,  pages 1688 (year 1997)NoStop
[Song et al.(2020)Song,
  Tian, Hu, Zhou, Xing, Lu, Chen, Wang,
  Xu, and Du]song_pulse-width-induced_2020
  author author Y. Song, author Y. Tian, author Z. Hu, author
  F. Zhou, author T. Xing, author D. Lu, author B. Chen, author Y. Wang, author
  N. Xu,  and author
  J. Du, 10.1364/PRJ.386983
  journal journal Photonics Res. volume 8, pages 1289 (year
  2020)NoStop
[Neumann et al.(2010)Neumann, Beck, Steiner, Rempp, Fedder, Hemmer, Wrachtrup, and Jelezko]science.1189075
  author author P. Neumann, author J. Beck,
  author M. Steiner, author F. Rempp, author
  H. Fedder, author P. R. Hemmer, author J. Wrachtrup,  and author F. Jelezko, doi:10.1126/science.1189075
  journal journal Sci. volume 329, pages 542 (year
  2010)NoStop
[Liu et al.(2017)Liu,
  Xing, Ma, Wang, Li, Po, Zhang, Fan,
  Liu, and Pan]PhysRevLett.118.150504
  author author G.-Q. Liu, author J. Xing, author W.-L. Ma, author
  P. Wang, author C.-H. Li, author H. C.  Po, author Y.-R. Zhang, author H. Fan, author R.-B. Liu,  and author X.-Y. Pan, 10.1103/PhysRevLett.118.150504 journal journal
  Phys. Rev. Lett. volume 118, pages
  150504 (year 2017)NoStop
[Gillard et al.(2022)Gillard, Clarke, and Chekhovich]gillard_harnessing_2022
  author author G. Gillard, author E. Clarke,   and author E. A. Chekhovich,  10.1038/s41467-022-31618-4 journal journal Nat. Commun. volume 13, pages 4048 (year 2022)NoStop
[Xu et al.(2019)Xu,
  Tian, Chen, Geng,
  He, Wang, and Du]PhysRevApplied.12.024055
  author author N. Xu, author Y. Tian, author B. Chen, author
  J. Geng, author X. He, author Y. Wang,  and author J. Du,  10.1103/PhysRevApplied.12.024055 journal
  journal Phys. Rev. Appl. volume 12,  pages 024055 (year 2019)NoStop
[Qian et al.(2021)Qian,
  Lin, Zhou, Ye, Ji, Chen, Xie, and Xu]qian_machine-learning-assisted_2021
  author author P. Qian, author X. Lin, author F. Zhou, author
  R. Ye, author Y. Ji, author B. Chen, author G. Xie,  and  author N. Xu, 10.1063/5.0038590 journal journal Appl. Phys.
  Lett. volume 118, pages 084001
  (year 2021)NoStop
[Smeltzer et al.(2009)Smeltzer, McIntyre, and Childress]PhysRevA.80.050302
  author author B. Smeltzer, author J. McIntyre,
   and author L. Childress,  10.1103/PhysRevA.80.050302 journal journal Phys. Rev. A volume 80, pages 050302(R)NoStop
[Steiner et al.(2010)Steiner, Neumann, Beck, Jelezko, and Wrachtrup]PhysRevB.81.035205
  author author M. Steiner, author P. Neumann,
  author J. Beck, author
  F. Jelezko,  and author
  J. Wrachtrup, 10.1103/PhysRevB.81.035205 journal journal Phys.
  Rev. B volume 81, pages 035205
  (year 2010)NoStop
[Dréau et al.(2012)Dréau, Maze, Lesik, Roch, and Jacques]PhysRevB.85.134107
  author author A. Dréau, author J.-R. Maze,
  author M. Lesik, author J.-F. Roch,  and author V. Jacques, 10.1103/PhysRevB.85.134107 journal journal Phys.
  Rev. B volume 85, pages 134107
  (year 2012)NoStop
[Ye et al.(2022)Ye,
  Lin, Zhou, Dai, Hu, Li, Xie, and Xu]ye_synchronized_2022
  author author R. Ye, author X. Lin, author F. Zhou, author
  Y. Dai, author Q. Hu, author X. Li, author G. Xie,  and  author N. Xu, 10.1063/5.0086943 journal journal Rev. Sci.
  Instrum. volume 93, pages 063102
  (year 2022)NoStop
[Banaszek et al.(1999)Banaszek, D’Ariano, Paris, and  Sacchi]PhysRevA.61.010304
  author author K. Banaszek, author G. M. D’Ariano, author M. G. A. Paris,  and author M. F. Sacchi, 10.1103/PhysRevA.61.010304 journal journal Phys. Rev. A volume
  61, pages 010304(R)NoStop
[Zhang et al.(2023)Zhang,
  Hegde, and Suter]PhysRevLett.130.090801
  author author J. Zhang, author S. S. Hegde,   and author D. Suter, 10.1103/PhysRevLett.130.090801 journal journal Phys. Rev. Lett. volume 130, pages 090801 (year 2023)NoStop
[Wang et al.(2008)Wang,
  Yu, and Yi]j.physleta.2008.10.083
  author author X. Wang, author C.-S. Yu,  and  author X. X. Yi, https://doi.org/10.1016/j.physleta.2008.10.083 journal journal Phys. Lett. A volume
  373, pages 58 (year 2008)NoStop
[Bogdanov et al.(2010)Bogdanov, Brida, Genovese, Kulik, Moreva, and Shurupov]PhysRevLett.105.010404
  author author Y. I. Bogdanov, author G. Brida,
  author M. Genovese, author S. P. Kulik, author
  E. V. Moreva,  and author
  A. P. Shurupov, 10.1103/PhysRevLett.105.010404 journal journal
  Phys. Rev. Lett. volume 105, pages
  010404 (year 2010)NoStop
[Shen et al.(2016)Shen,
  Heeres, Reinhold, Jiang,
  Liu, Schoelkopf, and Jiang]PhysRevA.94.052327
  author author C. Shen, author R. W. Heeres,
  author P. Reinhold, author L. Jiang, author
  Y.-K. Liu, author R. J. Schoelkopf,  and author L. Jiang, 10.1103/PhysRevA.94.052327
  journal journal Phys. Rev. A volume 94, pages 052327 (year
  2016)NoStop
[Nguyen et al.(2019)Nguyen,
  Sukachev, Bhaskar, Machielse,
  Levonian, Knall, and Stroganov, Riedinger, Park, Lonččar,  and Lukin]PhysRevLett.123.183602
  author author C.T. Nguyen, author D.D. Sukachev,author M.K. Bhaskar,
  author B. Machielse, author D.S. Levonian, author
  E.N. Knall, author P. Stroganov, author R. Riedinger, author H. Park,author M. Lonččar, and author M.D. Lukin, 10.1103/PhysRevLett.123.183602
  journal journal Phys. Rev. Lett. volume 123, pages 183602 (year
  2019)NoStop
[Bradley et al.(2019)Bradley,
  Randall, Abobeih, Berrevoets,
  Degen, Bakker, Markham, Twitchen,  and Taminiau]PhysRevX.9.031045
  author author C.E. Bradley, author J. Randall,author M.H. Abobeih,
  author R.C. Berrevoets, author M.J. Degen, author
  M.A. Bakker, author M. Markham, author D.J. Twitchen , and author T.H. Taminiau, 10.1103/PhysRevX.9.031045
  journal journal Phys. Rev. X volume 9, pages 031045 (year
  2019)NoStop
[Mamin et al.(2014)Mamin,
  Sherwood, Kim, Rettner,
  Ohno, Awschalom,  and Rugar]PhysRevLett.113.030803
  author author H.J. Mamin, author M.H. Sherwood,author M. Kim,
  author C.T. Rettner, author K. Ohno, author
  D.D. Awschalom, and author D. Rugar, 10.1103/PhysRevLett.113.030803
  journal journal Phys. Rev. Lett. volume 113, pages 030803 (year
  2014)NoStop
[Myers et al.(2014)Myers,
  Ariyaratne, Jayich,  and Bleszynski]PhysRevLett.118.197201
  author author B.A. Myers, author A. Ariyaratne,author  Jayich, and author A.C. Bleszynski, PhysRevLett.118.197201
  journal journal Phys. Rev. Lett. volume 118, pages 197201 (year
  2017)NoStop
[Bauch et al.(2014)Bauch,
	Hart, Schloss, Turner, Barry, Kehayias, Singh, and Walsworth]PhysRevX.8.031025
author author E. Bauch, author C.A. Hart,author J.M. Schloss,author M.J. Turner, author J.F. Barry,author P. Kehayias,author S. Singh, and author R.L. Walsworth, 10.1103/PhysRevX.8.031025
journal journal Phys. Rev. X volume 8, pages 031025 (year
	2018)NoStop







