A method of electron and nuclear spin resonance based on a non-uniform driving field

Abstract

The application relates to an electron and nuclear spin resonance method based on a non-uniform driving field. The electron and nuclear spin resonance method comprises the following steps: a plurality of groups of periodic driving fields are sequentially applied to a spin electron in an initial state; wherein the application phase angle of each group of periodic driving fields is alternately changed between the initial state of the spin electron and the orthogonal state of the initial state, the application time of each group of periodic driving fields is the same, and the change frequency of the application phase angle of the next group of periodic driving fields is different from that of the previous group of periodic driving fields; and the condition that the change frequency of the application phase angle of the driving field satisfies the formula (1) indicates that the electron and nuclear spins are in resonance. The method can realize the resonance coupling of the electron and nuclear spins, enhance the resonance coupling signal of the electron and nuclear spins, and be suitable for the electron and nuclear spin system under a strong magnetic field.

Description

Technical Field

[0001] This invention relates to the field of quantum technology, and in particular to a method for electron-nuclear spin resonance based on a non-uniform driving field. Background Technology

[0002] In spin quantum technology, spin resonant coupling is a fundamental condition. When coupled spins have the same energy, state flipping can be achieved, which can be used for quantum information processing and nanoscale nuclear magnetic resonance (NMR). Spin quantum technology can be implemented in diamond systems, where electrons at the nitrogen-vacancy (NV) centers of diamond can be used to detect, polarize, and control neighboring spins through resonant coupling.

[0003] Currently, various methods for achieving resonant coupling between electrons and nuclear spins have been proposed both domestically and internationally, including:

[0004] One approach is to achieve resonance between electron and nuclear spins through a dynamic decoupling (DD) π-pulse sequence. This involves generating a resonance signal when the frequency of the pulses in the π-pulse sequence matches the Larmor frequency of the nuclear spin. However, due to limitations in the application speed of the π-pulses, this method is only applicable to nuclear spin systems under low magnetic fields, where the Larmor frequency of the nuclear spin is typically within a few MHz.

[0005] Another approach is to continuously drive the spin electrons with a constant-amplitude driving field, causing the electron spins to flip at a Rabi frequency proportional to the amplitude of the driving field. When this Rabi frequency equals the Larmor frequency of the nuclear spin, i.e., when the Harman-Hahn resonance condition is satisfied, the electron and nuclear spins will resonantly couple. However, limited by the driving power of the driving field, this method is only applicable to nuclear spin systems under low magnetic fields.

[0006] Therefore, most existing methods for achieving resonant coupling between electron and nuclear spins are only applicable to nuclear spin systems under low magnetic fields. Furthermore, while existing technologies have developed methods for achieving resonant coupling between electron and nuclear spins by rapidly modulating a low-power driving field, such as the existing technology "Modulated Continuous Wave Control for Energy-Efficient Electron-Nuclear Spin Coupling" (Physical Review Letters 122, 010407 (2019)), although this method can be applied to nuclear spin systems under strong magnetic fields, in this existing technology, because the driving field applied according to its method does not meet the Harman-Hahn resonance condition, the driving field can only accumulate an effect on the electron-nuclear system for a maximum of half a cycle during the application time of the driving field. Therefore, the ultimately detected resonant coupling signal between electron and nuclear spins is not strong. Summary of the Invention

[0007] Based on this, the purpose of the present invention is to provide a new method for electron-nuclear spin resonance in electron-nuclear spin systems under strong magnetic fields, which utilizes a non-uniform driving field to achieve resonant coupling between electron and nuclear spins.

[0008] A method for electron-nuclear spin resonance based on a non-uniform driving field, characterized by comprising the following steps:

[0009] Multiple sets of periodic driving fields are sequentially applied to a spin electron in its initial state; wherein the applied phase angle of each set of driving fields alternates between the initial state of the spin electron and the orthogonal state of that initial state, the application time of each set of driving fields is the same, and the frequency of change of the applied phase angle of the driving field in the next set of driving fields is different from the frequency of change of the applied phase angle of the driving field in the previous set of driving fields; wherein the initial state is an eigenstate of the X-axis or Y-axis.

[0010] When the frequency of the change in the applied phase angle of the driving field satisfies the following formula, it indicates that the electron and the nuclear spin resonate:

[0011] ω n =k D ν+r D (1-k D )Ω

[0012] Where, ω n The Larmor frequency, k, represents the nuclear spin. D Let ν represent an integer, and r represent the frequency of the change in the applied phase angle of the driving field. D Ω represents the non-uniformity of the electron spin driven by the positive and negative X-axis directions, and Ω represents the Rabi frequency of the driving field.

[0013] The method for electron-nuclear spin resonance based on a non-uniform driving field described in this invention uses a non-uniformly modulated periodic continuous driving field to drive the spin electrons. By changing the frequency of the applied phase angle of the driving field, the resonant coupling of electron and nuclear spin is achieved.

[0014] Furthermore, for the driving field of each cycle, a duration of [duration missing] is first applied in the positive X-axis direction. A driving field with a Rabi frequency of Ω is applied, followed by a driving field with a duration of τ in the negative X-axis direction. - A driving field with a Rabi frequency of Ω is applied again in the positive x-axis direction for a duration of Ω. A driving field with a Rabi frequency of Ω is generated, thus forming an applied field with a duration of τ. + +τ - The period of τ, where τ + τ represents the time during which the driving field is applied in the positive X-axis direction. - This indicates the application time of the driving field in the negative X-axis direction.

[0015] Furthermore, Where τ represents the duration of one cycle of the driving field.

[0016] Furthermore, k D =1,τ + =π / (ν-Ω), τ - =π / (ν+Ω), r D =Ω / ν, to enhance the resonant coupling signal between the electron and the nuclear spin.

[0017] Furthermore, the electron spin operator σ of the spin electron corresponding to the frequency of change in the applied phase angle of the driving field for each cycle is obtained. z The average value < σ z This is used to determine whether the frequency of change of the applied phase angle of the driving field satisfies the above formula.

[0018] Furthermore, the present invention also provides a method for obtaining the Larmor frequency of the nuclear spin based on electron-nuclear spin resonance in a non-uniform driving field, characterized by comprising the following steps:

[0019] S1. Multiple sets of periodic driving fields are sequentially applied to a spin electron in its initial state; wherein the applied phase angle of each set of driving fields alternates between the initial state of the spin electron and the orthogonal state of that initial state, the application time of each set of driving fields is the same, and the frequency of change of the applied phase angle of the next set of driving fields is different from that of the previous set of driving fields; thereby obtaining the electron spin operator σ of the spin electron corresponding to the frequency of change of the applied phase angle of the driving field in each set of driving fields within a frequency range. z The average value < σ z >;

[0020] The initial state is an eigenstate of the X-axis or the Y-axis;

[0021] S2. Based on the frequency of the change in the applied phase angle of the driving field in each cycle and the corresponding electron spin operator σ of the spin electron. z The frequency change and the electron spin operator σ are obtained. z The average value < σ z Correspondence diagram between >;

[0022] S3. Obtain the frequency of change of the applied phase angle of the driving field corresponding to the peak position in the correspondence diagram.

[0023] S4. Based on the frequency change obtained in step S3, calculate the Larmor frequency of the nuclear spin using the following formula:

[0024] ω n =k D ν+r D (1-k D )Ω,

[0025] Where, ω n The Larmor frequency, k, represents the nuclear spin. D Let ν represent an integer, and r represent the frequency of the change in the applied phase angle of the driving field. D Ω represents the non-uniformity of the electron spin driven by the positive and negative X-axis directions, and Ω represents the Rabi frequency of the driving field.

[0026] To better understand and implement this invention, the following detailed description is provided in conjunction with the accompanying drawings. Attached Figure Description

[0027] Figure 1 This is a schematic diagram showing the application point of the driving field according to the present invention;

[0028] Figure 2 The driving field of the present invention with a Rabi frequency of Ω is applied for one period (i.e., τ = τ). + +τ - A schematic diagram illustrating the changes within the area;

[0029] Figure 3 To illustrate the electron spin operator σ under different driving field amplitude errors, this invention... z The average value < σ z >Schematic diagram showing the relationship between the frequency of phase angle change of the driving field;

[0030] Figure 4 To illustrate the electron spin operator σ under different driving field detuning errors, this invention... z The average value < σ z >A schematic diagram showing the relationship between the frequency of the phase angle change of the driving field. Detailed Implementation

[0031] This invention provides a method for electron-nuclear spin resonance based on a non-uniform driving field. This method can achieve and enhance the resonant coupling of electron and nuclear spins under low-power driving fields, and is applicable to electron-nuclear spin systems under strong magnetic fields. Specifically, in this invention, a non-uniformly modulated, periodic, and continuous driving field is used to drive the spin electrons, enabling the electron spin to generate a resonance frequency higher than the Rabi frequency of the driving field. The phase angle of the driving field alternates between the initial state of the spin electron and its orthogonal state in each cycle. Furthermore, the frequency at which the phase angle of the driving field alternates between the initial state and its orthogonal state in the next cycle is different from the frequency of the phase angle change in the previous cycle. This achieves non-uniform modulation of the driving field, enabling electron-nuclear spin resonance under low-power driving field control and enhancing the resonance signal of the electron-nuclear spins.

[0032] To better illustrate the present invention, an embodiment of the electron-nuclear spin resonance method based on a non-uniform driving field is provided below. Please also refer to... Figure 1-4 It includes: sequentially applying multiple sets of periodic driving fields to a spin electron in its initial state; wherein the applied phase angle of each set of periodic driving fields alternates between the initial state of the spin electron and the orthogonal state of the initial state, the applied time of each set of periodic driving fields is the same, and the change frequency of the applied phase angle of the next set of periodic driving fields is different from the change frequency of the applied phase angle of the previous set of periodic driving fields.

[0033] Furthermore, for a system composed of electrons and nuclear spins, one having a Larmor frequency of ω... n When the nuclear spin satisfies the condition expressed by the following formula, the nuclear spin will resonate with the electron spin in a quantum resonance:

[0034] ω n = k D ν + r D (1 - k D )Ω (1)

[0035] Where, ω n The Larmor frequency, k, represents the nuclear spin. D Let ν represent an integer ≥ 1, where ν represents the frequency of change of the applied phase angle θ(t) of the driving field. proportionality coefficient r D This indicates the non-uniformity of the spin electrons driven by the positive and negative X-axis directions. τ represents the duration of the driving field applied over one period. + τ represents the time during which the driving field is applied in the positive X-axis direction. - Ω represents the application time of the driving field in the negative X-axis direction, and Ω represents the Rabi frequency of the driving field.

[0036] Specifically, in this invention, for systems formed by electrons and nuclear spins, please refer to [link to relevant documentation]. Figure 1 We choose the line connecting any two opposite points on the Bloch sphere from its center to the equatorial surface as the X-axis, and select the initial state of the electron spin |ψ initial > represents the eigenstate |+> on the X-axis, and multiple sets of periodic driving fields are applied to the spin electrons on the X-axis.

[0037] Please refer to Figure 2 The application period of the driving field is expressed as Where τ represents the duration of the driving field applied for one period, τ + τ represents the time during which the driving field is applied in the positive X-axis direction. - This represents the application time of the driving field in the negative X-axis direction. Specifically, first, an application time of [duration to be filled in] is applied in the positive X-axis direction (corresponding to the initial phase angle θ = 0 of the driving field). A driving field with a Rabi frequency of Ω is applied, followed by an application of a duration of τ in the negative X-axis direction (corresponding to adjusting the phase angle of the driving field to θ = π). - Furthermore, a driving field with a Rabi frequency of Ω is applied again in the positive x-axis direction (corresponding to adjusting the phase angle of the driving field to θ = 0) for a duration of [missing information]. Furthermore, a driving field with a Rabi frequency of Ω is formed, thus creating a period with a duration of τ = τ + +τ - The applied period, wherein the Rabi frequency is proportional to the amplitude of the driving field.

[0038] In this scheme, the same set of periodic driving fields are applied to the spin electron multiple times. Based on the fundamental principles of quantum mechanics, the population of the electron's quantum state can be obtained through methods such as projection measurement, specifically through nuclear magnetic resonance or electron paramagnetic resonance experiments. Then, based on the population, the electron spin operator σ corresponding to the driving field of that period is obtained. zFurthermore, by changing the frequency of the applied phase angle of the driving field, the driving field for the next set of cycles is applied again, and the corresponding electron spin operator σ is obtained. z .

[0039] In this system, the driving field is applied the same number of times in each cycle, and the total application time for each cycle is T = Nτ, where T represents the total application time of the driving field in each cycle, N represents the number of times the driving field in that cycle is applied, and N ≥ 2. Furthermore, to obtain the electron spin operator σ... z After each application of a periodic driving field, the quantum state signal of the electron spin is obtained at the initial state position of the electron spin, such as the eigenstate |+> on the X-axis in this invention, i.e., at phase angle θ = 0. This is achieved using methods such as projection measurement, and specifically through nuclear magnetic resonance or electron paramagnetic resonance experiments. This allows the determination of the population of the quantum state of the spin electron, and further, based on this population, the quantum state of the spin electron, determined by the electron spin operator σ, is obtained. z The quantum state represented.

[0040] Of course, in other embodiments, the initial state |ψ of the electron spin can also be set. initial > represents the eigenstate |+> on the Y-axis, and multiple sets of periodic driving fields are applied to the spin electrons on the Y-axis.

[0041] In this invention, when the electron spin and nuclear spin do not resonate, i.e., when the resonance condition expressed by formula (1) is not satisfied, the electron spin state will not flip. When the two resonate, i.e., when the resonance condition expressed by formula (1) is satisfied, the single nuclear spin flips between the |↓> and |↑> states, while simultaneously forcing the electron spin to deviate from the initial state |ψ. initial And a spin state flip occurs. Therefore, based on electron spin, micro / nano nuclear magnetic resonance can obtain the initial state |ψ of the electron spin. initial The changes in > can be used to determine whether electrons and nuclear spins resonate.

[0042] Furthermore, in this invention, in order to enhance the resonant coupling signal between electrons and nuclear spins, τ is set... + =π / (ν-Ω), τ - =π / (ν+Ω), r D =Ω / ν、k D =1. At this point, formula (1) can be expressed as ω n =ν, thus, ν is changed, and the electron spin operator σ is measured. z The signal, when ν=ω n When this occurs, the electrons and the target nucleus spins can resonate, and the detected resonant coupling signal becomes stronger.

[0043] In this invention, after scanning a frequency range by continuously changing the frequency of the applied phase angle of the driving field, the frequency of the applied phase angle of the driving field and the electron spin operator σ can be used to determine the relationship between the frequency of the applied phase angle of the driving field and the electron spin operator σ. z The average value < σ z >, thus obtaining the correspondence diagram between the two, as shown below. Figure 3-4 As shown. The frequency of the applied phase angle of the driving field is not required to change. It can be set to increase, decrease or otherwise change in sequence according to the specific operation needs, as long as it changes within the specified frequency range and is different from the set frequency value. The frequency range of the phase angle change frequency includes the frequency value corresponding to the electron and nuclear spin resonance condition shown in formula (1).

[0044] exist Figure 3-4 The diagram shows the enhancement of the electron-nuclear spin resonance coupling signal, i.e., satisfying τ. + =π / (ν-Ω), τ - =π / (ν+Ω), r D =Ω / ν、k D Under the condition that = 1, the frequency of the change in the applied phase angle of the driving field is related to the electron spin operator σ. z The average value < σ z The graph shows the correspondence between ν and 2π, where the horizontal axis represents ν / 2π and the vertical axis represents the electron spin operator σ. z The average value < σ z > and the electron spin operator σ z The average value < σ z >=Tr[ρ(T)σ z ], Tr represents the summation of the diagonal terms of the matrix, ρ(T) represents the density operator of electron and nuclear spin after the driving field is applied for a duration T, and the initial state of electron spin is σ. z eigenstates At room temperature, the nuclear spin is in the hot state ρ n ≈(|↑><↑|+|↓><↓|) / 2.

[0045] Specifically, when the frequency of the applied phase angle of the driving field is far from the Larmor frequency of the nuclear spin, the electron spin and the nuclear spin do not resonate, and the electron spin remains essentially in its initial state |ψ initial >, thus in Figure 3-4 In the above, the electron spin operator σ z The average value < σ z The values ​​are essentially the same and close to or equal to 1; when the frequency of the applied phase angle of the driving field changes close to the Larmor frequency of the nuclear spin, the nuclear spin affects the electron spin, thereby causing the electron spin state to deviate to some extent from the initial state |ψ.initial However, no spin state flip occurs at this time, thus... Figure 3-4 In this case, the electron spin operator σ z The average value < σ z The frequency of the applied phase angle of the driving field gradually decreases; when the frequency of the change in the applied phase angle is equal to the Larmor frequency of the nuclear spin, the electron spin and the nuclear spin resonate, and the electron spin deviates from the initial state |ψ> due to the influence of the nuclear spin flipping between the |↓> and |↑> states. initial >and an electron spin state reversal occurs, thereby in Figure 3-4 In this case, the electron spin operator σ z The average value < σ z Located at the lowest point, i.e., point A in the diagram. Therefore, the frequency of the applied phase angle change is related to the electron spin operator σ of the spin electron. z In the corresponding graph, a signal peak appears, and the peak point of this signal peak is as follows: Figure 3-4 Point A in the diagram represents the resonance between the electron and nuclear spin. This means that the frequency of the change in the applied phase angle of the driving field and the Larmor frequency of the nuclear spin satisfy the resonance condition set by this invention. Furthermore, the stronger the signal corresponding to this peak, the stronger the resonant coupling between the electron and spin. Simultaneously, in the electron-nuclear spin resonance method of this invention, the driving field can have a cumulative effect on the electron and nuclear spin system throughout the application time. Therefore, the cumulative effect of the driving field during the application time can be greater than half a cycle, thus strengthening the resonant coupling signal.

[0046] Furthermore, in this scheme, the frequency of the change in the applied phase angle of the driving field is plotted in relation to the electron spin operator σ. z When obtaining the correspondence diagram, one can connect the corresponding points in the coordinate system to form a curve. Alternatively, after connecting the corresponding points in the coordinate system to form a curve, to make the result more accurate, a function can be used to fit the points in the coordinate system corresponding to the data, thereby obtaining a more precise curve. This function is the frequency of the applied phase angle change and the electron spin operator σ. z The functional relationship between the average values ​​of the two.

[0047] Furthermore, to improve the accuracy of obtaining the abscissa data of the peak position A, it is preferable to obtain the data points of the entire signal peak, thereby improving the accuracy of function fitting. After obtaining the above function, in addition to directly reading the abscissa data corresponding to the peak position of the signal peak from the correspondence image, it can also be obtained by finding the extreme value of the function.

[0048] In addition, Figure 3In the experiment, when the amplitude error of the driving field is 1%, the obtained curve almost coincides with the ideal curve, and the resonance signal peak almost coincides with the signal peak in the ideal curve. Furthermore, even when the amplitude error of the driving field is 5%, the position of the resonance signal peak of the curve remains essentially unchanged, indicating that the method in this invention has good tolerance to amplitude errors. Figure 4 In the driving field, the detuning error is 0.05MHz, meaning the detuning error is less than r. D When Ω, the obtained curve basically coincides with the ideal curve, indicating that the scheme in this invention can offset a certain degree of detuning error, and only needs to satisfy the detuning error Δ << r D The effects of these static noises are negligible.

[0049] Of course, the frequency of the applied phase angle change is related to the electron spin operator σ of the spin electron. z In the correspondence graph, the horizontal axis can be ν, and the vertical axis can be the electron spin operator σ. z The average value < σ z The resulting correspondence image also contains a signal peak, the peak of which represents the resonance between electrons and nuclear spins.

[0050] In one specific embodiment, a system consisting of electrons and nuclear spins formed by the NV center and its neighboring nuclear spins is selected. A constant magnetic field, such as B, is applied to the system along the axial direction of NV (e.g., the z-axis). z = 1T. Furthermore, the Hamiltonian H of this system under the driving field is (where, ):

[0051]

[0052] in, Let S represent the Hamiltonian of electron spin, D represent the NV zero-field splitting energy and D = (2π) × 2.87 GHz, S z S x Both represent electron spin operators and S z =|1><1|-|-1><-1|+0|0><0|、 γ e The gyromagnetic ratio, B, represents the electron spin. z It is a constant magnetic field applied along the NV axis (z-axis); ∑ j γ j B z I z The Hamiltonian, γ, represents nuclear spin. j I represents the gyromagnetic ratio of the j-th nuclear spin. z Operator in the z-axis direction of nuclear spin; The term representing the interaction between electrons and nuclear spin is given by the following expression: Operators representing the x and z axes of the j-th nuclear spin. and and are the coupling coefficients of the j-th nuclear spin with the electron in the x and z directions, respectively, under low power or strong magnetic field conditions. and Much smaller than the Zeeman energy γ induced by the magnetic field j B z ; ω represents the applied non-uniform driving field term. mw Let ω represent the angular frequency of the driving field applied to the electrons, Ω represent the Rabi frequency of the driving field, θ(t) represent the phase angle of the driving field, and let ω represent the angular frequency of the driving field. mw With electronic energy level m s =0 and m s The energy frequencies between ω and 1 are equal, i.e., ω mw =D+γ e B z .

[0053] Furthermore, considering the high magnetic field conditions, γ e B z It is much larger than Ω, so the driving field will not cause the quantum state to leak into |m. s =-1> state, therefore |m can be ignored s =-1> The influence of the state.

[0054] exist In the interaction scenario, the electron is located at |m s =0> and |m s =1> The Hamiltonian of the system in the corresponding subspace can be simplified to:

[0055]

[0056] Among them, the Pauli operator σ for electrons z =|1><0|+|0><1|=|+><+|-|-><-|, where Δ represents the frequency error of the driving field (i.e., detuning error), and Δ=D+γ e B z -ω mw This frequency error will cause the driving field to deviate from the equatorial plane, σ x =|1><1|-|0><0|=|+><-|+|-><+|, Larmor frequency of nuclear spin Frequency ω under the electron spin pendant state phenomenon e (t)=(1+δ)Ω(t), where δ represents the potential control error of the driving field (i.e., amplitude error), and the Rabi frequency Ω(t)≡Ωeiθ It switches between the maximum value Ω and the minimum value -Ω over time.

[0057] By applying the same periodic driving field to NV multiple times under the above conditions using the method of this invention, while varying the frequency of the applied phase angle of the driving field, and measuring the electron spin state signal through nuclear magnetic resonance or electron paramagnetic resonance experiments, the electron spin operator σ corresponding to the driving field of that period is obtained. z The electron spin operator is used to determine whether the electron spin state has flipped, thereby determining whether resonance occurs between the electron and the nuclear spin. Alternatively, the frequency of the change in the applied phase angle of the driving field and the electron spin operator σ can be obtained. z The corresponding relationship diagram is used to determine whether the electron and nuclear spin resonate.

[0058] Furthermore, according to the above scheme, since the peak position A of the signal peak in the corresponding relationship image indicates that the electron and the nuclear spin have resonated at this time, after obtaining the horizontal coordinate value corresponding to the peak position, it is possible to use formula (1) or formula ω n =ν, quickly obtaining the Larmor frequency of the nuclear spin. Furthermore, since the scheme of the present invention has a certain tolerance for amplitude error and detuning error of the driving field, the Larmor frequency of the nuclear spin obtained by the above-mentioned resonance method of electron and nuclear spin is highly accurate.

[0059] Compared with existing technologies, the solution in this invention manipulates the Larmor frequency ω of the nucleus spin. n The Rabi frequency Ω of a much smaller electron, i.e., Ω << ω n This invention achieves resonant coupling between electron and nuclear spins under low-power driving field manipulation, obtaining a stronger resonant coupling signal between the electron and nuclear spins. It can be applied to micro- and nano-scale nuclear spin detection and quantum control based on electron spin under strong magnetic fields. Furthermore, the scheme of this invention exhibits good tolerance to amplitude errors of the external driving field, and also has a certain tolerance to static noise acting on electron spin (which can manifest as detuning errors of the external driving field), as long as Δ << r D The effects of these static noises are negligible, therefore, the solution of the present invention can improve the accuracy of Larmor frequency detection of nuclear spin.

[0060] The embodiments described above are merely examples of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and the present invention also intends to include these modifications and variations.

Claims

1. A method for electron-nuclear spin resonance based on a non-uniform driving field, characterized in that, Includes the following steps: Multiple sets of periodic driving fields are sequentially applied to a spin electron in its initial state; wherein the applied phase angle of each set of driving fields alternates between the initial state of the spin electron and the orthogonal state of that initial state, the application time of each set of driving fields is the same, and the frequency of change of the applied phase angle of the driving field in the next set of driving fields is different from the frequency of change of the applied phase angle of the driving field in the previous set of driving fields; wherein the initial state is an eigenstate of the X-axis or Y-axis. When the frequency of the change in the applied phase angle of the driving field satisfies the following formula (1), it indicates that the electron and the nuclear spin resonate: （1） in, Larmor frequency, representing nuclear spin Represents an integer. This indicates the frequency of change in the applied phase angle of the driving field. This indicates the non-uniformity of the spin electrons driven by the positive and negative X-axis directions. This represents the Rabi frequency of the driving field. , ,in, This indicates the application time of the driving field in the positive X-axis direction. This indicates the application time of the driving field in the negative X-axis direction. This indicates the duration of application of the driving field for one cycle. .

2. The method for electron-nuclear spin resonance according to claim 1, characterized in that: For each cycle of the driving field, first apply a duration of [duration to be filled in] in the positive X-axis direction. Rabi frequency is The driving field is then applied in the negative X-axis direction for a duration of... Rabi frequency is The driving field is then applied again in the positive X-axis direction for a time of... Rabi frequency is The driving field, thus forming an application time of The cycle.

3. The method for electron-nuclear spin resonance according to claim 2, characterized in that: =1， ， 、 。 4. The method for electron-nuclear spin resonance according to any one of claims 1-3, characterized in that: Obtain the electron spin operator of the spin electron corresponding to the frequency of change in the applied phase angle of the driving field for each cycle. average This is used to determine whether the frequency of the change in the applied phase angle of the driving field satisfies the above formula (1).

5. A method for obtaining the Larmor frequency of the nuclear spin based on electron-nuclear spin resonance in a non-uniform driving field, characterized in that, Includes the following steps: S1. Multiple sets of periodic driving fields are sequentially applied to a spin electron in its initial state; wherein the applied phase angle of each set of driving fields alternates between the initial state of the spin electron and its orthogonal state, the application time of each set of driving fields is the same, and the frequency of change of the applied phase angle of the next set of driving fields is different from that of the previous set of driving fields; thereby obtaining the electron spin operator of the spin electron corresponding to the frequency of change of the applied phase angle of the driving fields in each set of driving fields within a frequency range. average ; The initial state is an eigenstate of the X-axis or the Y-axis; S2. Based on the frequency of the change in the applied phase angle of the driving field in each cycle and the corresponding electron spin operator of the spin electron. The change frequency and electron spin operator are obtained. average Correspondence diagram between them; S3. Obtain the frequency of change of the applied phase angle of the driving field corresponding to the peak position in the correspondence diagram. S4. Based on the frequency change obtained in step S3, calculate the Larmor frequency of the nuclear spin using the following formula: ， in, Larmor frequency, representing nuclear spin Represents an integer. This indicates the frequency of change in the applied phase angle of the driving field. This indicates the non-uniformity of the spin electrons driven by the positive and negative X-axis directions. This represents the Rabi frequency of the driving field. , ,in, This indicates the application time of the driving field in the positive X-axis direction. This indicates the application time of the driving field in the negative X-axis direction. This indicates the duration of application of the driving field for one cycle. .

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