Quantitative method and system for double electron quantum correlation based on bohmian mechanics

Abstract

The application relates to the technical field of strong-field high-harmonic generation, and particularly discloses a double-electron quantum correlation quantitative method and system based on Bohm mechanics, which comprises the following steps: establishing a time-dependent Schrodinger equation containing an external laser field, obtaining a system ground state wave function and generating N pairs of Bohm particles initial coordinates, and performing time evolution on the wave function; according to the evolved wave function, the accurate velocity field and the accurate motion trajectory of each Bohm particle are calculated; based on the marginal probability density and the marginal probability flow density, the marginal velocity field not containing instantaneous quantum correlation effects is constructed, and the corresponding correlation-free reference trajectory is generated; the accurate trajectory is compared with the reference trajectory, and the correlation model of the correlation quantum potential and the high-harmonic radiation intensity is established. Through comparison of the accurate Bohm trajectory and the correlation-free reference trajectory constructed based on the marginal velocity field, the quantum correlation effects in a strong-field double-electron system are quantitatively separated and visually output.

Description

Technical Field

[0001] This invention relates to the field of strong field high-order harmonic modulation technology, specifically to a quantitative method and system for two-electron quantum correlation based on Bohmian mechanics. Background Technology

[0002] With advancements in ultrafast and ultra-intense laser technology, research on the interaction between light and matter has delved into the dynamics of multi-electron correlations. The generation of higher harmonics, a core physical process in attosecond science, extreme ultraviolet light sources, and molecular orbital imaging, is traditionally understood primarily based on a single-electron "three-step model." However, in multi-electron systems, quantum correlations between electrons become the core physical factor determining the intensity, spectral structure, and dynamic path of harmonic radiation. For example, experimental observations have shown that correlations between inner-shell electrons in atoms can induce "giant resonances" and double-plateau characteristics in the harmonic spectrum, highlighting the importance of correlation effects.

[0003] Currently, a deep understanding and characterization of multi-electron correlation dynamics under strong fields faces theoretical challenges. Traditional mean-field methods (such as Hartree-Fock) cannot handle strong correlation effects; time-dependent density functional theory (TDDFT) has approximate limitations in describing violent non-equilibrium correlation fluctuations and long-range correlations; while multi-configuration methods (such as MCTDHF) or full numerical solutions, although highly accurate, fail to provide an intuitive picture of single-particle dynamics and cannot clearly reveal how correlations at the femtosecond scale regulate the motion of each electron in real time.

[0004] Bohmian mechanics provides a feasible theoretical framework for the aforementioned technical problems. This framework describes quantum systems as deterministic particle trajectories, whose intrinsic quantum potential directly reflects the nonlocality of many-body correlations. However, existing research based on Bohmian mechanics has not yet developed a complete and computable method for quantitatively decomposing and visualizing correlation effects, especially in the generation of higher harmonics, where there is a lack of "trajectory-potential field" analysis tools that can clearly separate correlation contributions and guide harmonic modulation. Summary of the Invention

[0005] The purpose of this invention is to provide a quantitative method and system for two-electron quantum correlation based on Bohmian mechanics, so as to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A quantitative method for two-electron quantum correlation based on Bohmian mechanics, the method comprising:

[0008] A time-dependent Schrödinger equation with an external laser field is established, the ground state wave function of the system is obtained through imaginary time evolution, and N pairs of initial coordinates of Bohm particles are generated. The wave function is then subjected to time evolution.

[0009] Based on Bohm's mechanical formula, the precise velocity field and precise trajectory of each Bohm particle are calculated according to the evolved wave function; the total quantum potential of the system is decomposed into the marginal quantum potential of the first electron, the marginal quantum potential of the second electron, and the correlated quantum potential.

[0010] Based on the marginal probability density and marginal probability flux density, a marginal velocity field without instantaneous quantum correlation effects is constructed to generate the corresponding uncorrelated reference trajectory.

[0011] The precise trajectory is compared with the reference trajectory, and the trajectory offset caused by quantum correlation is calculated and output.

[0012] Analyze the spatiotemporal evolution characteristics of the trajectory offset and the associated quantum potential, establish a correlation model between the associated quantum potential and the higher harmonic radiation intensity, and output the control parameters for optimizing the higher harmonic yield.

[0013] As a further aspect of the present invention, the total subpotential of the system The expression is:

[0014] ;

[0015] in, For correlation quantum potential, marginal quantum potential Defined solely by marginal probability density The constructed quantum potential.

[0016] As a further embodiment of the present invention, the marginal velocity field The expression:

[0017] ;

[0018] in, Let be the marginal probability flow density.

[0019] As a further embodiment of the present invention, the uncorrelated reference trajectory is determined by integrating the marginal velocity field:

[0020] ;

[0021] in, Let be the initial position of the k-th Bohm particle in the i-th electron. This represents the marginal velocity of the k-th Bohm particle within the i-th electron; This represents the reference trajectory of the k-th Bohm particle in the i-th electron.

[0022] As a further aspect of the present invention, the step of analyzing the spatiotemporal evolution characteristics of the trajectory offset and the associated quantum potential specifically includes: analyzing the evolution behavior of the associated quantum potential in the two-electron re-collision stage and near the nucleus region.

[0023] As a further embodiment of the present invention, the control parameters include at least laser intensity, frequency, and carrier envelope phase.

[0024] As a further aspect of the present invention, it also includes calculating and outputting the amplitude, range of action, and corresponding optimized laser parameter comparison table of the associated quantum potential under each atomic system based on the potential energy parameters of different atomic systems.

[0025] This invention also provides a two-electron quantum correlation quantitative system based on Bohmian mechanics, the system comprising:

[0026] The model building module is used to establish the time-dependent Schrödinger equation containing the external laser field, obtain the system ground state wave function through imaginary time evolution, generate the initial coordinates of N pairs of Bohm particles, and perform time evolution on the wave function;

[0027] The precise trajectory generation module is used to calculate the precise velocity field and precise trajectory of each Bohm particle based on the Bohm mechanics formula and the evolved wave function; it decomposes the total quantum potential of the system into the marginal quantum potential of the first electron, the marginal quantum potential of the second electron, and the correlated quantum potential.

[0028] The reference trajectory generation module is used to construct a marginal velocity field that does not contain instantaneous quantum correlation effects based on the marginal probability density and marginal probability flux density, and generate the corresponding uncorrelated reference trajectory.

[0029] The offset calculation module is used to compare the precise trajectory with the reference trajectory, calculate and output the trajectory offset caused by quantum correlation;

[0030] The comparison output module is used to analyze the spatiotemporal evolution characteristics of the trajectory offset and the associated quantum potential, establish a correlation model between the associated quantum potential and the higher harmonic radiation intensity, and output control parameters for optimizing the higher harmonic yield.

[0031] Compared with the prior art, the beneficial effects of the present invention are: by decomposing the total quantum potential into marginal quantum potential and correlated quantum potential, not only is the correlation term separated mathematically, but more importantly, the core mechanism of the correlated quantum potential in the two-electron re-collision process is revealed.

[0032] By comparing the precise Bohm trajectory with the uncorrelated reference trajectory constructed based on the marginal velocity field, the quantum correlation effect in a strong-field two-electron system was quantitatively separated and visualized. Attached Figure Description

[0033] To more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention.

[0034] Figure 1 This is a flowchart of the method of the present invention.

[0035] Figure 2 The figures show a comparison between the marginal trajectory and the marginal probability density. (a) The figure shows the evolution of the Bohm trajectory of the first electron calculated from the marginal velocity over time; (b) The figure shows the evolution of the marginal probability density of the first electron over time.

[0036] Figure 3 This study demonstrates the overall distribution of Bohm trajectories in a real two-electron system under strong field conditions, and selects three pairs of typical trajectories that generate higher harmonics.

[0037] Figure 4 for Figure 3 A magnified view of the trajectory of a typical high-order harmonic generation in the re-collision stage.

[0038] Figure 5 Figure (a) shows the Bohm trajectory plot calculated from the marginal velocity and the true velocity, and Figure (b) shows the quantum potential change plot. Detailed Implementation

[0039] To make the technical problems to be solved, the technical solutions, and the beneficial effects of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the present invention and are not intended to limit the present invention.

[0040] In this embodiment of the invention, a quantitative method for two-electron quantum correlation based on Bohmian mechanics is provided, the method comprising:

[0041] A time-dependent Schrödinger equation with an external laser field is established, the ground state wave function of the system is obtained through imaginary time evolution, and N pairs of initial coordinates of Bohm particles are generated. The wave function is then subjected to time evolution.

[0042] Based on Bohm's mechanical formula, the precise velocity field and precise trajectory of each Bohm particle are calculated according to the evolved wave function; the total quantum potential of the system is decomposed into the marginal quantum potential of the first electron, the marginal quantum potential of the second electron, and the correlated quantum potential.

[0043] Based on the marginal probability density and marginal probability flux density, a marginal velocity field without instantaneous quantum correlation effects is constructed to generate the corresponding uncorrelated reference trajectory.

[0044] The precise trajectory is compared with the reference trajectory, and the trajectory offset caused by quantum correlation is calculated and output.

[0045] Analyze the spatiotemporal evolution characteristics of the trajectory offset and the associated quantum potential, establish a correlation model between the associated quantum potential and the higher harmonic radiation intensity, and output the control parameters for optimizing the higher harmonic yield.

[0046] like Figure 1 As shown, in this embodiment, a time-dependent Schrödinger equation describing a one-dimensional xenon atom two-electron system is defined, and a linearly polarized laser electric field is introduced as the external field; the ground state wave function of the system is obtained using the imaginary time evolution method, and the initial coordinates of N pairs of Bohm particles are randomly sampled from the ground state probability density using the discard sampling method;

[0047] The wave function is obtained by using the second-order symmetric difference method. We perform time evolution, taking into account the effect of the laser electric field, to ensure that the wave function satisfies spatial exchange symmetry;

[0048] According to Bohm's mechanical formula, the velocity of each electron is calculated from the phase gradient of the evolving wave function, and the exact Bohm trajectory is obtained by integration; the total quantum potential of the system is decomposed into marginal quantum potentials characterizing the average effect of single particles and correlated quantum potentials characterizing nonlocal correlations.

[0049] Based on marginal probability density and marginal probability flow, a marginal velocity field containing only conditional average information is constructed, and a set of reference trajectories without instantaneous correlation effects are generated for comparison by integrating the same initial conditions.

[0050] By comparing the exact Bohm trajectory with the uncorrelated reference trajectory, the trajectory spatial shift ΔX(t) caused by quantum correlation is identified and quantified, and the dynamic effect of the correlation is separated.

[0051] This study focuses on typical trajectory pairs that undergo re-collision (ionized electrons returning to the nucleus and bound electrons approaching each other), analyzing their motion characteristics. Combining the spatiotemporal evolution of the correlated quantum potential, it analyzes how this potential modulates and triggers synchronous vibrations of the two electrons near the nucleus when they approach each other. The study analyzes the spatiotemporal evolution characteristics of the correlated quantum potential during the re-collision process, assesses its modulating effect on the synchronous vibration behavior of the two electrons, and establishes a correlation model between the correlated quantum potential and the intensity of higher harmonic radiation, providing a quantifiable dynamic basis for harmonic modulation.

[0052] Under one-dimensional conditions, considering external field effects, the time-dependent Schrödinger equation (TDSE) satisfied by electrons in a two-electron atom can be expressed as:

[0053] ;

[0054] In the formula: i represents the i-th electron; This represents the first-order time partial derivative with respect to the wave function; x1 represents the two-electron wavefunction satisfied by the Bohm particle; x2 represents the position of the first electron; x1 represents the position of the second electron. This represents the square of the velocity of the first electron; This represents the square of the velocity of the second electron;

[0055] Formula for calculating the Coulomb attraction potential energy between nuclei:

[0056] ;

[0057] Formula for calculating the Coulomb repulsion potential energy between two electrons: The expression for the laser electric field:

[0058] ;

[0059] Where E0 = 0.13au, T = 110.23au, τ = 31.49au, , ;

[0060] The two-electron wavefunction satisfied by a Bohm particle can be expressed as:

[0061] ;

[0062] In the formula: R and S are both real numbers; This represents the real part of the wave function; Indicates the amount of action;

[0063] By extracting the imaginary and real parts from both sides of the equation, we obtain the following equation:

[0064] ;

[0065] Quantum Hamilton-Jacobi equation: ;

[0066] Where, x i This indicates the position of the i-th electron.

[0067] , , , These represent the kinetic energy, quantum potential energy, classical Coulomb potential energy, and electric potential energy of two electrons, respectively. This represents the total energy. The expression for quantum potential energy is:

[0068] ;

[0069] The velocity of the i-th electron is v i(t) can also be represented by the action S:

[0070] ;

[0071] In the formula: x1(t) represents the position of the first electron at time t; x2(t) represents the position of the second electron at time t;

[0072] The velocity of the i-th electron is v i (t) can also be represented by the wave function:

[0073] ;

[0074] In the formula: Im represents taking the imaginary part of the complex number;

[0075] Therefore, the position x of the i-th electron can be obtained at any time based on the given wave function. i (t):

[0076] ;

[0077] By using the imaginary time evolution method, the probability density function of the xenon atom's ground state at the initial time was obtained. Then, the initial spatial coordinates of N pairs of Bohm particles (BPs) are randomly sampled using a discarding method, and these coordinates are used to represent the potential initial positions of the two electrons, i.e.: and , where represents the position of the k-th Bohm particle, k=1,2,3...,N.

[0078] Using a second-order symmetric difference method, the wavefunction of a one-dimensional xenon atom under the action of a laser electric field is analyzed. Perform time evolution. The initial state of the wave function satisfies spatial exchange symmetry to ensure the accuracy of the simulation.

[0079] The velocities and trajectories of paired particles are calculated using the many-body Bohmian mechanics (BM) formulas:

[0080] ;

[0081] ;

[0082] ;

[0083] ;

[0084] In the formula: This represents the velocity of the k-th Bohm particle in the first electron; This represents the velocity of the k-th Bohm particle in the second electron; This indicates the position of the k-th Bohm particle in the first electron; This indicates the position of the k-th Bohm particle in the second electron;

[0085] As a preferred embodiment of the present invention, the total subpotential of the system The expression is:

[0086] ;

[0087] in, For correlation quantum potential, marginal quantum potential Defined solely by marginal probability density The constructed quantum potential.

[0088] In this embodiment, the effective quantum potential experienced by electron i, which is equivalent to a statistical average of all possible positions of electron j, is expressed as:

[0089] ;

[0090] Related quantum potential:

[0091] ;

[0092] The above formula is the remainder obtained by subtracting two marginal quantum potentials from the total quantum potential. It is defined as the remainder after subtracting the two marginal quantum potentials from the total quantum potential, and it explicitly depends on the two-electron coordinates, characterizing the correlation effect beyond the mean field.

[0093] As a preferred embodiment of the present invention, the marginal velocity field The expression:

[0094] ;

[0095] in, Let be the marginal probability flow density.

[0096] In this embodiment, the marginal velocity field is defined as the statistical average velocity (marginal velocity) obtained by weighting all possible positions of another electron by instantaneous conditional probabilities, given the position of one electron.

[0097] This velocity field does not contain instantaneous correlation effects, and its marginal probability current density is:

[0098] ;

[0099] By integrating the marginal velocity field at the same initial position, With precise velocity field The reference trajectory can be obtained by integration.

[0100] In a preferred embodiment of the present invention, the uncorrelated reference trajectory is determined by integrating the marginal velocity field:

[0101] ;

[0102] in, Let be the initial position of the k-th Bohm particle in the i-th electron. This represents the marginal velocity of the k-th Bohm particle within the i-th electron; This represents the actual velocity of the k-th Bohm particle within the i-th electron. This represents the reference trajectory of the k-th Bohm particle in the i-th electron.

[0103] In this embodiment, the two-electron wavefunction is obtained by numerically solving the time-dependent Schrödinger equation, and the trajectories of hundreds of Bohm particle pairs are calculated based on the Bohmian mechanics framework. To verify the reliability of the trajectory method, the spatial distribution of electrons obtained from Bohm trajectory statistics is compared with the probability density of the wavefunction. The results show that the two are highly consistent in spatiotemporal evolution (e.g., ...). Figure 2 As shown in the figure, this demonstrates that Bohmian mechanics can accurately describe the dynamics of two electrons in a strong field.

[0104] As a preferred embodiment of the present invention, the step of analyzing the spatiotemporal evolution characteristics of the trajectory offset and the associated quantum potential specifically includes: analyzing the evolution behavior of the associated quantum potential in the two-electron re-collision stage and near the nucleus region.

[0105] In this embodiment, in order to reveal the role of quantum correlation in the re-collision process, a trajectory comparison method is introduced: by constructing a marginal velocity field containing only mean field information, a set of reference trajectories is generated; these are then compared with the precise Bohm trajectory containing full correlation information. Figure 3 and Figure 4 The distribution of several typical trajectories is shown, among which multiple pairs of trajectories that collided again were identified (marked in red, green, and blue). Figure 3 The medium gray background represents a set of Bohm particle trajectories, while the red / green / blue pairs represent typical trajectories that produce HHG. Analysis reveals that after the re-collision, the two electrons exhibit significant synchronous vibrations near the nucleus (|x| < 5 au), and this phenomenon only appears in the actual two-electron trajectory, not observed in the reference trajectory. Figure 5 As shown in (a), Figure 5 (a) The black curves represent the precise trajectory; the blue curves represent the baseline trajectory.

[0106] To further explore the relationship between this synchronous vibration and quantum correlation, the evolution of various quantum potentials was analyzed. For example... Figure 5As shown in (b), before the re-collision occurs (t < 100 au), the marginal quantum potential of the bound electron has already begun to oscillate, but at this time the correlation quantum potential (Q) is still low. corr The anti-phase oscillations cancel out the driving effect of the marginal quantum potential, thus suppressing the premature vibration of the bound electrons. Figure 5 In (b), the blue trajectory represents the marginal quantum potential, and the magenta trajectory represents the correlated quantum potential of the two particles. When the spatial distance between the two electrons approaches the critical range (approximately Δx < 5a.u.), (Q corr The mechanism of action changes, coordinating the two electrons into a synchronous vibration phase. This indicates that the correlated quantum potential directly regulates the cooperative motion after the two electrons collide again.

[0107] By comparing trajectories and decomposing potential fields, the modulation effect of correlated quantum potentials on the synchronous vibration of two electrons can be quantified, and a correlation model between this effect and higher harmonic radiation can be established. This "trajectory-potential field" analysis framework provides a computational and visual theoretical tool for optimizing the performance of strong-field harmonics.

[0108] This invention proposes a computational method based on Bohmian mechanics for trajectory comparison and quantum potential decomposition, achieving quantitative separation and visualization of quantum correlation effects in strong-field two-electron systems. By comparing precise Bohmian trajectories with uncorrelated reference trajectories constructed based on marginal velocity fields, this method reveals, for the first time at the single-particle motion level, the real-time trajectory shift caused by non-local correlations, thus clearly separating the dynamic effects of correlation from the mean-field background. The practical significance of this quantitative separation lies in: through the trajectory shift ΔX(t) and the correlation quantum potential... The numerical output allows for the establishment of a quantitative model of the relationship between quantum correlation strength and harmonic radiation efficiency, accurately identifying key time windows (e.g., the re-collision stage t=80-120 au) and spatial regions (e.g., the core region |x|<5 au) where correlation effects dominate. Experimentally, this can be used to optimize the gating time and spatial focusing position of attosecond pulses, and to quantitatively evaluate the contribution weight of correlation effects to harmonic yield under different laser parameters based on the output correlation coefficient matrix. This avoids the high-cost screening of traditional trial-and-error methods and directly locks in the optimal parameter combination.

[0109] In a preferred embodiment of the present invention, the control parameters include at least laser intensity, frequency, and carrier envelope phase.

[0110] It also includes calculating and outputting the amplitude, range of action, and corresponding optimized laser parameter comparison table of the associated quantum potential for each atomic system based on the potential energy parameters of different atomic systems.

[0111] In this embodiment, the correlated quantum potential When the inter-electron spacing Δx < 5 au, the mode of action changes, enhancing the intensity of higher harmonic radiation by coordinating the entry of the two electrons into the synchronous vibration phase (e.g., Figure 5 (b) shows that its essence is to modulate the trajectory of electrons in real time through non-local correlation terms, and to construct a quantum coherent state that is conducive to cooperative radiation at the moment of re-collision.

[0112] In experiments, regulation can be achieved through the following pathways:

[0113] Laser parameter control: Adjust the laser intensity I0 (e.g., 0.03-0.08 au) and frequency. (e.g., 0.05-0.07 au) and carrier envelope phase By altering the electron return kinetic energy and the residence time in the nuclear region, the regulation of The spatiotemporal evolution model.

[0114] Atomic system selection: The potential energy parameters of different atoms (xenon / krypton / argon) will affect Regarding amplitude and range of action, this method can output a table comparing the optimal parameters for each system.

[0115] Initial state preprocessing: The initial wavefunction shape is altered through pre-excitation or polarization modulation to guide... Evolves along a specific path.

[0116] The optimal parameters output by this method (e.g., I0 = 0.05 au, λ = 800 nm, τ = 20 fs) are all within the range achievable by existing femtosecond laser technology. Numerical verification shows that, compared with the single-electron approximation model, the yield of the 25th-45th harmonics can be improved by 2-3 orders of magnitude, and the harmonic cutoff energy is extended by about 15 eV, proving that the manipulation is experimentally feasible.

[0117] Breaking through the yield limit of the traditional single-electron model, the intensity of the extreme ultraviolet light source is significantly improved by utilizing the dual-electron cooperative radiation mechanism. This provides a higher signal-to-noise ratio light source for applications such as molecular orbital imaging and attosecond pump-probe experiments, reduces the stringent requirements on detector sensitivity, and promotes the transition of strong-field correlated quantum control from theory to experimental application.

[0118] The results show that the spatiotemporal evolution of the correlated quantum potential directly modulates the synchronous vibrational behavior of the two electrons in the nucleus, thereby triggering coordinated high-order harmonic radiation. This discovery elucidates the microscopic mechanism by which quantum correlations influence the radiation process from the perspective of "potential field-trajectory" coupling.

[0119] This invention also provides a two-electron quantum correlation quantitative system based on Bohmian mechanics, the system comprising:

[0120] The model building module is used to establish the time-dependent Schrödinger equation containing the external laser field, obtain the system ground state wave function through imaginary time evolution, generate the initial coordinates of N pairs of Bohm particles, and perform time evolution on the wave function;

[0121] The precise trajectory generation module is used to calculate the precise velocity field and precise trajectory of each Bohm particle based on the Bohm mechanics formula and the evolved wave function; it decomposes the total quantum potential of the system into the marginal quantum potential of the first electron, the marginal quantum potential of the second electron, and the correlated quantum potential.

[0122] The reference trajectory generation module is used to construct a marginal velocity field that does not contain instantaneous quantum correlation effects based on the marginal probability density and marginal probability flux density, and generate the corresponding uncorrelated reference trajectory.

[0123] The offset calculation module is used to compare the precise trajectory with the reference trajectory, calculate and output the trajectory offset caused by quantum correlation;

[0124] The comparison output module is used to analyze the spatiotemporal evolution characteristics of the trajectory offset and the associated quantum potential, establish a correlation model between the associated quantum potential and the higher harmonic radiation intensity, and output control parameters for optimizing the higher harmonic yield.

[0125] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A quantitative method for two-electron quantum correlation based on Bohmian mechanics, characterized in that, The method includes: A time-dependent Schrödinger equation with an external laser field is established, the ground state wave function of the system is obtained through imaginary time evolution, and N pairs of initial coordinates of Bohm particles are generated. The wave function is then subjected to time evolution. Based on Bohm's mechanical formula, the precise velocity field and precise trajectory of each Bohm particle are calculated according to the evolved wave function; the total quantum potential of the system is decomposed into the marginal quantum potential of the first electron, the marginal quantum potential of the second electron, and the correlated quantum potential. Based on the marginal probability density and marginal probability flux density, a marginal velocity field without instantaneous quantum correlation effects is constructed to generate the corresponding uncorrelated reference trajectory. The precise trajectory is compared with the reference trajectory, and the trajectory offset caused by quantum correlation is calculated and output. Analyze the spatiotemporal evolution characteristics of the trajectory offset and the associated quantum potential, establish a correlation model between the associated quantum potential and the higher harmonic radiation intensity, and output the control parameters for optimizing the higher harmonic yield.

2. The quantitative method for two-electron quantum correlation based on Bohmian mechanics according to claim 1, characterized in that, Total subpotential of the system The expression is: ； in, For correlation quantum potential, marginal quantum potential Defined solely by marginal probability density The constructed quantum potential.

3. The quantitative method for two-electron quantum correlation based on Bohmian mechanics according to claim 2, characterized in that, The marginal velocity field The expression: ； in, Let be the marginal probability flow density.

4. The quantitative method for two-electron quantum correlation based on Bohmian mechanics according to claim 3, characterized in that, The uncorrelated reference trajectory is determined by integrating the marginal velocity field: ； in, Let be the initial position of the k-th Bohm particle in the i-th electron. This represents the marginal velocity of the k-th Bohm particle within the i-th electron; This represents the reference trajectory of the k-th Bohm particle in the i-th electron.

5. The quantitative method for two-electron quantum correlation based on Bohmian mechanics according to claim 1, characterized in that, The steps for analyzing the trajectory offset and the spatiotemporal evolution characteristics of the associated quantum potential specifically include: analyzing the evolution behavior of the associated quantum potential in the two-electron re-collision stage and near the nucleus region.

6. The quantitative method for two-electron quantum correlation based on Bohmian mechanics according to claim 1, characterized in that, The control parameters include at least laser intensity, frequency, and carrier envelope phase.

7. The quantitative method for two-electron quantum correlation based on Bohmian mechanics according to claim 1, characterized in that, It also includes calculating and outputting the amplitude, range of action, and corresponding optimized laser parameter comparison table of the associated quantum potential for each atomic system based on the potential energy parameters of different atomic systems.

8. A two-electron quantum correlation quantitative system based on Bohmian mechanics, used to implement the two-electron quantum correlation quantitative method based on Bohmian mechanics as described in any one of claims 1-7, characterized in that, The system includes: The model building module is used to establish the time-dependent Schrödinger equation containing the external laser field, obtain the system ground state wave function through imaginary time evolution, generate the initial coordinates of N pairs of Bohm particles, and perform time evolution on the wave function; The precise trajectory generation module is used to calculate the precise velocity field and precise trajectory of each Bohm particle based on the Bohm mechanics formula and the evolved wave function; it decomposes the total quantum potential of the system into the marginal quantum potential of the first electron, the marginal quantum potential of the second electron, and the correlated quantum potential. The reference trajectory generation module is used to construct a marginal velocity field that does not contain instantaneous quantum correlation effects based on the marginal probability density and marginal probability flux density, and generate the corresponding uncorrelated reference trajectory. The offset calculation module is used to compare the precise trajectory with the reference trajectory, calculate and output the trajectory offset caused by quantum correlation; The comparison output module is used to analyze the spatiotemporal evolution characteristics of the trajectory offset and the associated quantum potential, establish a correlation model between the associated quantum potential and the higher harmonic radiation intensity, and output control parameters for optimizing the higher harmonic yield.

Images