Qualitative analysis method for electron re-collision process under laser pulse based on bomb mechanics

Abstract

The invention is suitable for the technical field of laser pulses, and provides a bomb mechanics-based qualitative analysis method for an electron re-collision process under laser pulses, which comprises the following steps of: constructing a model and initial conditions; carrying out wave function time evolution; calculating the speed, the track and the stress of the bom particles; performing ionization judgment; designing a composite state and initializing the composite state; and analyzing the re-collision process and drawing a conclusion. According to the method, firstly, it is proved that the double-ionization process can be accurately described through bom mechanics; secondly, a composite model is established to research the re-collision process in an external field, and the research finds that the probability of non-order double ionization is increased along with the reduction of the ground state probability, the reduction of the kinetic energy of incident bomb particles and the lightening of an ion potential well; wherein the influence of the ground state probability on non-sequential double ionization is particularly significant. In addition, it is found that quantum force plays an important role in the non-sequential double ionization process. And finally obtaining a conclusion that the interference effect of the complex wave packet and the ground state wave packet is a main factor influencing non-order double ionization.

Description

Technical Field

[0001] The invention belongs to the technical field of laser pulses, and in particular relates to a qualitative analysis method of an electron re-collision process under a laser pulse based on Bohm mechanics. Background Art

[0002] With the rapid development of laser technology, laser pulses are tending to be ultra-strong and ultra-short. Under such extreme physical conditions, many nonlinear phenomena have been observed, such as single ionization, double ionization, molecular dissociation and high-order harmonic generation of atoms and molecules in strong laser fields. The study of the microscopic processes behind these phenomena has greatly deepened our understanding of the physical world. Among them, the double ionization of atoms or molecules under strong fields, especially non-sequential double ionization, is particularly eye-catching. It not only reveals the interaction between electrons and external fields, but also further demonstrates the correlation properties between electrons, which is one of the most basic multi-electron interactions. The two main mechanisms of non-sequential double ionization have been widely recognized: one is "return impact ionization" (RII), that is, the tunneling electron collides with the second electron after obtaining enough energy in the laser field, so that it also obtains enough energy and is ionized; the other is the "re-collision excitation and subsequent ionization" mechanism (RESI), that is, the energy obtained by the first electron is not enough to ionize the second electron immediately, but it will cause it to transition to a higher excited state and then be ionized again. Here we mainly focus on the RII mechanism.

[0003] The study of ionization processes has spawned a series of computational methods, such as the strong field approximation and the simple-man model. However, these methods have defects in describing the re-collision process. The classical method is based on Newtonian mechanics equations and ignores quantum effects; while the method of solving the Schrödinger equation is accurate, it cannot provide a clear physical picture. Therefore, we need more effective theoretical and computational methods.

[0004] In recent years, the Bohm mechanics theory has been widely used in the study of strong field physical phenomena. This is a causal interpretation of orthodox quantum mechanics, which believes that electrons have trajectories and can accurately describe their motion. Bohm mechanics can reproduce all phenomena in traditional quantum mechanics and provide a clearer physical picture. However, the calculation of multi-body Bohm mechanics requires solving the multi-body Schrödinger equation. Especially when dealing with two-body problems, solving the Schrödinger equation in three-dimensional space is not only cumbersome but also inefficient. In addition, the dynamic behavior of Bohm particles is also easily restricted by the sparsity of the spatial grid. Under limited computing resources, in order to ensure the accuracy of the calculation, it is inevitable to reduce the spatial boundary, but this may cause the calculation results to be affected in the case of relatively large field strength. Therefore, there are still many technical difficulties in calculating the three-dimensional two-electron Bohm mechanics. This has not been reported in related studies at home and abroad. To this end, the present invention proposes a qualitative analysis method for the electron re-collision process under laser pulses based on Bohm mechanics. Summary of the invention

[0005] The purpose of the present invention is to provide a qualitative analysis method for the electron re-collision process under laser pulse based on Bohm mechanics, aiming to solve the problems raised in the above background technology.

[0006] The purpose of the present invention is achieved through the following technical solutions:

[0007] The qualitative analysis method of the electron re-collision process under laser pulse based on Bohm mechanics includes the following steps:

[0008] Step 1: Construct the model and initial conditions;

[0009] Define the time-dependent Schrödinger equation describing the two-electron atom and introduce the laser electric field as the external field;

[0010] The probability density function of the ground state of the xenon atom is obtained using the imaginary time evolution method, and the initial spatial coordinates of N pairs of Bohm particles are randomly sampled using the rejection method to represent the potential initial positions of the two electrons.

[0011] Step 2: Time evolution of wave function;

[0012] The second-order symmetric difference method is used to perform the time evolution of the wave function ψ(x1,x2,t), taking into account the effect of the laser electric field to ensure that the wave function satisfies the spatial exchange symmetry;

[0013] Step 3: Calculate the velocity, trajectory and force of the Bohm particle;

[0014] Calculate the speed and trajectory of paired particles according to the multi-body Bohm mechanics formula; calculate the forces acting on each Bohm particle according to Bohm mechanics theory, including Coulomb force, quantum force and electric field force;

[0015] Step 4: Ionization judgment;

[0016] The method of dividing the electron space region is used as the basis for judging ionization, and the number of unionized, singly ionized and doubly ionized Bohm particle pairs is counted to calculate the corresponding ionization probability;

[0017] Step 5: Design the composite state and initialize it;

[0018] The superposition of the two-electron ground state and the singly ionized state is constructed as the initial state to simulate the recombination of the ionized electron with its paired electron under the action of the laser field;

[0019] Step 6: Analyze the re-collision process and draw conclusions;

[0020] Analyze the probability density image of the electron re-collision process and observe the characteristics of the re-collision process;

[0021] Compare the results of Bohm mechanics method with the numerical solution of the time-dependent Schrödinger equation to verify the accuracy of the method;

[0022] Explore the effects of different parameters on non-sequential double ionization and use Bohm trajectories for in-depth analysis;

[0023] Observe the initial positions and re-collision trajectories of non-sequential doubly ionized Bohm particle pairs, and summarize the laws of the initial positions;

[0024] Compare the forces acting on composite particles with the same initial position but different ground state probabilities, and analyze the role of quantum forces in the re-collision process;

[0025] Based on the analysis results, conclusions are drawn about the probability of non-sequential double ionization, the law of the initial positions of paired particles, and the role of quantum mechanics in the re-collision process.

[0026] Furthermore, the specific process of step 1 is as follows:

[0027] Under one-dimensional conditions, when considering the external field effect, the time-dependent Schrödinger equation satisfied by the electrons in the two-electron atom is expressed as:

[0028] Formula 1:

[0029] Where: i represents the i-th electron; represents the first-order time partial derivative of the wave function; ψ(x1,x2,t) represents the two-electron wave function satisfied by the Bohm particle; x1 represents the position of the first electron; x2 represents the position of the second electron; represents the velocity square of the first electron; represents the speed squared of the second electron; represents the Coulomb attraction potential energy between nuclei; represents the mutual Coulomb repulsion potential between two electrons; is the expression of the laser electric field, where E0 = 0.13 au, T = 110.23 au, τ = 31.49 au, φ = 3.15π, ω = 0.057 au;

[0030] The two-electron wave function satisfied by the Bohm particle is expressed as:

[0031] Formula 2: ψ(x1,x2,t)=R(x1,x2,t)exp(iS(x1,x2,t))

[0032] Where: R and S are real numbers; R(x1,x2,t) represents the real part of the wave function; S(x1,x2,t) represents the action;

[0033] Substituting Equation 2 into Equation 1 and extracting the imaginary and real parts on both sides of the equation yields the following equation:

[0034] Formula 3: Formula 4:

[0035] Where: represents the total energy; x i represents the position of the i-th electron; Q(x1,x2,t),V a (x1,x2), (x1+x2)E(t) represent the kinetic energy, quantum potential energy, classical Coulomb potential energy and electric potential energy of the two electrons respectively; the expression of quantum potential energy is as follows:

[0036] Formula 5:

[0037] The velocity v of the i-th electron i (t) is expressed in terms of action S:

[0038] Formula 6:

[0039] Where: x1(t) represents the position of the first electron at time t; x2(t) represents the position of the second electron at time t;

[0040] According to Equation 2, the velocity v of the i-th electron i (t) is represented by the wave function:

[0041] Formula 7:

[0042] Where: Im represents the imaginary part of a complex number;

[0043] According to the given wave function, we can obtain the position x of the i-th electron at any time. i (t):

[0044] Formula 8:

[0045] The probability density function of the xenon atom ground state at the initial time is obtained using the imaginary time evolution method ψ0(x1,x2,t=0) 2 , and then use the elimination method to randomly sample N pairs of initial spatial coordinates of Bohm particles to represent the potential initial positions of the two electrons, that is: and Where k = 1, 2, 3…, N represents the position of the kth Bohm particle.

[0046] Furthermore, the specific process of step 3 is as follows:

[0047] The speed and trajectory of paired particles are calculated according to the multi-body Bohm mechanics formula:

[0048] Formula 9:

[0049] Formula 10:

[0050] Formula 11:

[0051] Formula 12:

[0052] Where: represents the speed of the kth Bohm particle in the first electron; represents the speed of the kth Bohm particle in the second electron; represents the position of the kth Bohm particle in the first electron; represents the position of the kth Bohm particle in the second electron;

[0053] According to Bohm mechanics theory, each Bohm particle is subject to the following forces:

[0054] The net force acting on each Bohm particle is:

[0055] Formula 13:

[0056] Where: represents the speed of the kth Bohm particle in the ith electron;

[0057] The Coulomb force of the nucleus on the kth Bohm particle in the ith electron is:

[0058] Formula 14:

[0059] Where: represents the position of the kth Bohm particle in the ith electron;

[0060] The Coulomb force of the kth Bohm particle of the jth electron on the kth Bohm particle of the ith electron is:

[0061] Formula 15:

[0062] Where: x j represents the position of the jth electron; represents the position of the kth Bohm particle in the jth electron;

[0063] The quantum force acting on each Bohm particle is:

[0064] Formula 16: F Q =F R -F nucleus Coulomb -F e-e Coulomb -F E

[0065] Where: F E represents the electric force acting on each Bohm particle;

[0066] The electric force acting on each Bohm particle is:

[0067] Formula 17: F E =-E(t).

[0068] Furthermore, the specific process of step 4 is as follows:

[0069] When paired Bohm particles satisfy the following conditions in Bohm mechanics, namely:

[0070] exist In the case of , no ionization occurred, and N1 times occurred; or In the case of single ionization, N2 occurred; In the case of double ionization, N3 occurred; represents the position of the kth Bohm particle in the first electron, represents the position of the kth Bohm particle in the second electron;

[0071] The probabilities of unionization, singly ionization, and doubly ionization are calculated using the following formulas:

[0072] Formula 18: P1(t) = N1(t) / Ntotal, P2(t) = N2(t) / Ntotal and P3(t) = N3(t) / Ntotal

[0073] Where: P1(t), P2(t), P3(t) represent the probabilities of unionized, singly ionized and doubly ionized, respectively; N1(t), N2(t), N3(t) represent the number of Bohm particle pairs of unionized, singly ionized and doubly ionized, respectively; Ntotal represents the total number of Bohm particle pairs;

[0074] For the time-dependent Schrödinger equation, the probability that both electrons are in the bound state is:

[0075] Formula 19:

[0076] The probability of single ionization is:

[0077] Formula 20:

[0078] The probability of double ionization is:

[0079] Formula 21: P3(t)=1-P1(t)-P2(t).

[0080] Furthermore, the specific process of step 5 is as follows:

[0081] The two-electron ground state ψ0(x1,x2) and the singly ionized state ψ s The initial state is a superposition of (x1, x2), where the singly ionized state ψ s (x1, x2) is used to simulate the recombination of an ionized electron with its paired electron under the action of a laser field; assuming that the single ionization state satisfies the spatial exchange symmetry, the expression is:

[0082] Equation 22: ψ s (x1,x2)=ψ(x1)g(x2)+ψ(x2)g(x1)

[0083] Where: g(x1), g(x2) represent the ground state wave functions of the first electron and the second electron respectively; ψ(x1), ψ(x2) represent the composite wave functions of the first electron and the second electron respectively;

[0084] ψ(x) represents the single electron composite wave function, which is expressed as:

[0085] Formula 23: ψ(x) = exp[-4log2×(x+100) 2 / λ0 2 ]×exp(ipx)

[0086] Where: x represents the coordinate of the wave function space; ipx represents the action; λ0 = 40 a.u., so the initial state is expressed as:

[0087] Formula 24: ψ(x1,x2)=C1ψ0(x1,x2)+C2ψ s (x1,x2)

[0088] Where: C1, C2 represent ψ0(x1, x2) and ψ s The probability amplitude of (x1,x2);

[0089] The initial state of the two electrons also satisfies the spatial exchange symmetry.

[0090] Compared with the prior art, the present invention has the following beneficial effects:

[0091] The present invention first obtains the double ionization process by calculating 10,000 Bohm trajectories, and the result obtained by numerically solving the time-dependent Schrödinger equation is qualitatively the same, proving that Bohm mechanics can accurately describe the double ionization process. Secondly, the present invention establishes a composite model to study the re-collision process under an external field, and the study finds that the probability of non-sequential double ionization increases with the decrease of the ground state probability, the decrease of the kinetic energy of the incident Bohm particle, and the shallowing of the ion potential well. Among them, the influence of the ground state probability on non-sequential double ionization is particularly significant. In addition, the analysis of the external force on Bohm particles with the same initial position but different ground state probabilities shows that quantum force plays an important role in the non-sequential double ionization process. It is finally concluded that the interference effect between the composite wave packet and the ground state wave packet is the main factor affecting non-sequential double ionization. BRIEF DESCRIPTION OF THE DRAWINGS

[0092] Figure 1 The figure is a flow chart of the method of the present invention.

[0093] Figure 2 The time evolution of the laser field during the re-collision process and the probability density image of the first electron obtained by numerically solving the time-dependent Schrödinger equation; (a) the time evolution of the laser field during the re-collision process; (b) the probability density image of the first electron obtained by numerically solving the TDSE.

[0094] Figure 3 The graph of 100,000 pairs of Bohm particles (BPs) used to represent the time-varying probabilities of single and double ionization of xenon atoms, where the solid black and dotted blue (dashed red and dash-dotted magenta) curves represent the single and double ionization probabilities calculated by the TDSE (time-dependent Schrödinger equation) and BM (Bohm mechanics) methods, respectively; Figure 3 In (a), (b), (c) and (d), the ground state probability, composite wave packet kinetic energy and atomic potential parameters are:

[0095]

[0096] Figure 4 The number of non-sequential doubly ionized particles and doubly ionized particle pairs under different parameters are calculated by Bohm mechanics method. Figure 4 a, b, c, and d correspond to Figure 3 At the end of the time evolution in (a)-(d), the number of doubly ionized particle pairs calculated by the Bohm mechanics method; black represents the number of doubly ionized particle pairs, and red represents the number of doubly ionized particle pairs produced by re-collision (non-sequential double ionization).

[0097] Figure 5 is the initial position of the Bohm particle when the ground state probability is 0.1.

[0098] Figure 6 Figure 3 shows the influence of different ground state probabilities on the re-collision trajectory of the Bohm particle under the same initial position; (a) the re-collision trajectory when the ground state probability is 0.1; (b) the re-collision trajectory when the ground state probability is 0.4, and the initial position is the same as (a).

[0099] Figure 7 It is the Bohm trajectory in the composite wave packet with the same initial position and the force exerted on the black Bohm particle during the re-collision process when the ground state probability is 0.4; (a) is the Bohm trajectory in the composite wave packet with the same initial position (red is the case when the ground state probability is 0.1, and black is the case when the ground state probability is 0.4); (b) is the force exerted on the black Bohm particle in (a). DETAILED DESCRIPTION

[0100] In order to have a clearer understanding of the technical features, purposes and beneficial effects of the present invention, the technical solution of the present invention is now described in detail below, but it should not be construed as limiting the applicable scope of the present invention.

[0101] The specific implementation of the present invention is described in detail below in conjunction with specific embodiments.

[0102] One embodiment of the present invention provides a qualitative analysis method for the electron re-collision process under laser pulse based on Bohm mechanics, and its flow chart is as follows: Figure 1 As shown, the method comprises the following steps:

[0103] Step 1: Construct the model and initial conditions; the details are as follows:

[0104] Under one-dimensional conditions, when considering the external field effect, the time-dependent Schrödinger equation (TDSE) satisfied by the electrons in the two-electron atom can be expressed as:

[0105] Formula 1:

[0106] Where: i represents the i-th electron; represents the first-order time partial derivative of the wave function; ψ(x1,x2,t) represents the two-electron wave function satisfied by the Bohm particle; x1 represents the position of the first electron; x2 represents the position of the second electron; represents the velocity square of the first electron; represents the speed squared of the second electron; represents the Coulomb attraction potential energy between nuclei; represents the mutual Coulomb repulsion potential between two electrons; is the expression of the laser electric field, where E0 = 0.13 au, T = 110.23 au, τ = 31.49 au, φ = 3.15π, ω = 0.057 au;

[0107] The two-electron wave function satisfied by the Bohm particle can be expressed as:

[0108] Formula 2: ψ(x1,x2,t)=R(x1,x2,t)exp(iS(x1,x2,t))

[0109] Where: R and S are real numbers; R(x1,x2,t) represents the real part of the wave function; S(x1,x2,t) represents the action;

[0110] Substituting Equation 2 into Equation 1 and extracting the imaginary and real parts on both sides of the equation, we get the following equation:

[0111] Formula 3: Formula 4:

[0112] Where: x i represents the position of the i-th electron; Equation 4 is the quantum Hamilton-Jacobi equation. Q(x1,x2,t),V a (x1,x2),(x1+x2)E(t) respectively represent the kinetic energy, quantum potential energy, classical Coulomb potential energy and electric potential energy of the two electrons; represents the total energy. The expression of quantum potential energy is:

[0113] Formula 5:

[0114] The velocity v of the i-th electron i (t) can also be expressed in terms of action S:

[0115] Formula 6:

[0116] Where: x1(t) represents the position of the first electron at time t; x2(t) represents the position of the second electron at time t;

[0117] According to Equation 2, the velocity v of the i-th electron i (t) can also be expressed by a wave function:

[0118] Formula 7:

[0119] Where: Im represents the imaginary part of a complex number;

[0120] Therefore, we can obtain the position x of the i-th electron at any time based on the given wave function i (t):

[0121] Formula 8:

[0122] By using the imaginary time evolution method, we obtain the probability density function of the ground state of the xenon atom at the initial time ψ0(x1,x2,t=0) 2 Then, the initial spatial coordinates of N pairs of Bohm particles (BPs) are randomly sampled using the elimination method and used to represent the potential initial positions of the two electrons, namely: and (k=1,2,3…,N represents the position of the kth Bohm particle).

[0123] Step 2: Time evolution of wave function; details are as follows:

[0124] The second-order symmetric difference method is used to perform the time evolution of the wave function ψ(x1,x2,t) of a one-dimensional xenon atom under the action of a laser electric field. The initial state of the wave function satisfies the spatial exchange symmetry to ensure the accuracy of the simulation.

[0125] Step 3: Calculate the velocity, trajectory, and force of the Bohm particle; details are as follows:

[0126] The speed and trajectory of paired particles are calculated according to the multi-body Bohm mechanics (BM) formula:

[0127] Formula 9:

[0128] Formula 10:

[0129] Formula 11:

[0130] Formula 12:

[0131] Where: represents the speed of the kth Bohm particle in the first electron; represents the speed of the kth Bohm particle in the second electron; represents the position of the kth Bohm particle in the first electron; represents the position of the kth Bohm particle in the second electron;

[0132] According to the Bohmian Mechanics (BM) theory, each Bohm particle (BP) is subject to the following forces:

[0133] The net force acting on each Bohm particle is:

[0134] Formula 13:

[0135] Where: represents the speed of the kth Bohm particle in the ith electron;

[0136] The Coulomb force of the nucleus on the kth Bohm particle in the ith electron is:

[0137] Formula 14:

[0138] Where: represents the position of the kth Bohm particle in the ith electron;

[0139] The Coulomb force of the kth Bohm particle of the jth electron on the kth Bohm particle of the ith electron is:

[0140] Formula 15:

[0141] Where: x j represents the position of the jth electron; represents the position of the kth Bohm particle in the jth electron;

[0142] The quantum force acting on each Bohm particle is:

[0143] Formula 16: F Q =F R -F nucleus Coulomb -F e-e Coulomb -F E

[0144] Where: F E represents the electric force acting on each Bohm particle;

[0145] The electric force acting on each Bohm particle is:

[0146] Formula 17: F E =-E(t)

[0147] Step 4: Ionization judgment; details are as follows:

[0148] When calculating ionization, we use the method of dividing the electron space region as the basis for judging ionization. When the paired Bohm particles meet the following conditions in Bohm mechanics, that is:

[0149] exist In the case of , no ionization occurs, and this happens N1 times; or In the case of single ionization, N2 occurred; In the case of double ionization, N3 occurred; represents the position of the kth Bohm particle in the first electron, represents the position of the kth Bohm particle in the second electron.

[0150] The probabilities of unionization, singly ionization, and doubly ionization can be calculated using the following formulas:

[0151] Formula 18: P1(t) = N1(t) / Ntotal, P2(t) = N2(t) / Ntotal and P3(t) = N3(t) / Ntotal

[0152] Where: P1(t), P2(t), P3(t) represent the probabilities of unionized, singly ionized and doubly ionized, respectively; N1(t), N2(t), N3(t) represent the number of Bohm particle pairs of unionized, singly ionized and doubly ionized, respectively; Ntotal represents the total number of Bohm particle pairs;

[0153] For the time-dependent Schrödinger equation, the probability that both electrons are in the bound state is:

[0154] Formula 19:

[0155] The probability of single ionization is:

[0156] Formula 20:

[0157] The probability of double ionization is:

[0158] Formula 21: P3(t)=1-P1(t)-P2(t)

[0159] Step 5: Design the composite state and initialize it; the details are as follows:

[0160] The initial states we designed are the two-electron ground state ψ0(x1,x2) and the single ionized state ψ s The superposition of (x1,x2), where the singly ionized state ψ s (x1, x2) is used to simulate the recombination of an ionized electron with its paired electron under the action of a laser field. Assuming that the single ionization state satisfies the spatial exchange symmetry (spin antisymmetry), its expression is:

[0161] Equation 22: ψ s (x1,x2)=ψ(x1)g(x2)+ψ(x2)g(x1)

[0162] Where: g(x1), g(x2) represent the ground state wave functions of the first electron and the second electron respectively; ψ(x1), ψ(x2) represent respectively; g(x) represents the single electron ground state, and ψ(x) represents the single electron composite wave function, which we call the composite state, and its expression is:

[0163] Formula 23: ψ(x) = exp[-4log2×(x+100) 2 / λ0 2 ]×exp(ipx)

[0164] Where: x represents the coordinate of the wave function space; ipx represents the action; λ0 = 40 a.u., so the initial state can be expressed as:

[0165] Formula 24: ψ(x1,x2)=C1ψ0(x1,x2)+C2ψ s (x1,x2)

[0166] Where: C1, C2 represent ψ0(x1, x2) and ψ s The probability amplitude of (x1,x2).

[0167] Such a design assumes that the initial states of the two electrons also satisfy spatial exchange symmetry.

[0168] Step 6: Analyze the re-collision process and draw conclusions; the details are as follows:

[0169] Figure 2 (b) shows a probability density image of an electron re-collision process, where the probability of the ground state of the two electrons is and the probability of a singly ionized state The initial position of the composite wave packet is at X = -100 atomic units (au), and it moves toward the nucleus with momentum p = 1 a.u. It is worth noting that this re-collision process requires an external electric field, because without an electric field, the wave function will always be in a steady state. In order to reasonably simulate the re-collision process, we deliberately only provide a half-cycle electric field to simulate the ionized electrons returning to the nucleus under the action of the laser field, such as Figure 2 As shown in (a).

[0170] In our study, we used 10,000 pairs of Bohm particles to calculate the probabilities of single and double ionization and compared the results with the numerical solution of the time-dependent Schrödinger equation. The results showed that the calculation results of the two methods were highly consistent when the ground state probability, the kinetic energy of the composite wave packet and the atomic Coulomb potential were different (e.g. Figure 3 In order to analyze the influence of different parameters on non-sequential double ionization (double ionization caused by two electron collisions), we used Bohm trajectory to explore it, because this method can identify the particle pairs that undergo non-sequential double ionization. Figure 3 (a) is used as a reference. Figure 3 In (b), the ground state probability decreases from 0.4 to 0.1. Figure 3 In (c), the kinetic energy of the re-collision wave packet is reduced from 1 to 0.5. Figure 3 The atomic Coulomb parameter in (d) increases from 4.26 to 5. The results show that the double ionization probability increases slightly with the decrease of the ground state probability, the kinetic energy of the composite wave packet and the atomic Coulomb parameter, as shown in Figure 4 As shown in the black histogram in Figure 4a, b, c, and d correspond to Figure 3 At the same time, we counted the number of non-sequential double ionization tracks caused by re-collision from the double ionization tracks, such as Figure 4 It can be found that as the ground state probability decreases, the proportion of non-sequential double ionization probability increases significantly, as shown in the red histogram in Figure 4 This is shown in the red histogram in (b), which indicates that the ground state probability has the most obvious effect on non-sequential double ionization.

[0171] Figure 5 and Figure 6 (a) shows the probability of the ground state C1 2 = 0.1 for the initial positions and re-collision trajectories of non-sequential doubly ionized Bohm particle pairs. Figure 5 The gray area in the figure represents the initial positions of 10,000 pairs of Bohm particles, while the black and red areas represent the initial positions of the Bohm particle pairs that undergo non-sequential double ionization (red and black are a pair of Bohm particles, satisfying the spatial exchange symmetry). We observed the regularity of the initial positions of non-sequential double ionization Bohm particle pairs: the particles that collide in the ground state are distributed in the lower half of the ground state wave packet, while the particles that re-collide in the composite wave packet are distributed in the upper half of the wave packet. To simplify the analysis, we selected the Bohm particle representing the first electron in state ψ(x) and the Bohm particle representing the second electron in state g(x), and plotted the trajectories of the non-sequential double ionization of these two electrons in the ψ(x1,x2) state, as shown in Figure 2. Figure 6 As shown in (a).

[0172] By comparison Figure 4 In a and b, we find that the proportion of non-sequential double ionization decreases significantly with the increase of the ground state probability. To further study the effect of the ground state probability on non-sequential double ionization, we selected a pair of Bohm particles with the same initial position as when the ground state probability is 0.1 to study the behavior of the Bohm trajectory when the ground state probability is 0.4. The results are shown in Figure 6 As shown in (b), its initial position is Figure 6 The same as (a). Figure 6 In (b), we can see that the black Bohm particle in the composite state ψ(x) remains almost near the nucleus after the collision, while the red Bohm particle in the ground state g(x) is only partially knocked out. This shows that as the ground state probability increases, particle pairs with the same initial position are less likely to undergo non-sequential double ionization.

[0173] To better understand this phenomenon, we selected Figure 6 A pair of trajectories from a composite wave packet with the same initial position in (a) and (b) are compared, as shown in Figure 7 As shown in (a), the red track comes from Figure 6 (a), the black trace comes from Figure 6 (b). In the re-collision process, the black Bohm particle in the ground state with a probability of 0.4 is captured by the nucleus. In order to explore the root cause, we use formula 13-17 to calculate the electric field force F exerted on the black Bohm particle in the re-collision process. E (electric field force), nuclear attraction F nucleus Coulomb (Coulomb force from nucleus), the repulsive force F of another electron e-e Coulomb (Coulomb force from the second BP) and the quantum force F generated by electron interference Q The strength of the quantum force is determined by analyzing the strength of these forces to determine which force is dominant in the re-collision process. Figure 7 The calculation results show that the black re-collision Bohm particle with a ground state probability of 0.4 is kept in the nucleus region mainly because of the strong quantum force (such as Figure 7 This quantum force reflects the interference of the electrons themselves, indicating that the interference between the recollision electrons and their ground state is the main cause of the non-order double ionization. This shows that when the atomic Coulomb potential is not very large (such as the case of xenon atoms), the main cause of the non-order double ionization is the interference between the composite wave packet and the ground state wave packet.

[0174] Conclusion: We used many-body Bohm mechanics to study the re-collision process of two electrons in xenon atoms. First, we found that the probability of non-sequential double ionization increases with the decrease of the ground state probability, the kinetic energy of the composite wave packet and the atomic softening Coulomb parameter, among which the ground state probability has the most obvious effect on the probability of non-sequential double ionization. Secondly, we discovered the regularity of the initial positions of paired particles that undergo non-sequential double ionization. Finally, by comparing the forces on composite particles with the same initial positions but different ground state probabilities, we found that the quantum force directly related to the interference effect plays a key role in the re-collision process. Therefore, we conclude that the interference effect between the composite wave packet and the ground state wave packet is the main factor affecting non-sequential double ionization.

[0175] The above are only preferred embodiments of the present invention. It should be pointed out that, for those skilled in the art, several modifications and improvements can be made without departing from the concept of the present invention. These should also be regarded as the protection scope of the present invention. These will not affect the effect of the implementation of the present invention and the practicality of the patent.

Claims

1. A qualitative analysis method for the electron re-collision process under laser pulses based on Bohm mechanics, characterized in that: The following steps are involved: Step 1: Construct the model and initial conditions; Define the time-dependent Schrödinger equation describing the two-electron atom and introduce the laser electric field as the external field; The probability density function of the ground state of the xenon atom is obtained using the imaginary time evolution method, and the initial spatial coordinates of N pairs of Bohm particles are randomly sampled using the rejection method to represent the potential initial positions of the two electrons. Step 2: Time evolution of wave function; The second-order symmetric difference method is used to perform the time evolution of the wave function ψ(x1,x2,t), taking into account the effect of the laser electric field to ensure that the wave function satisfies the spatial exchange symmetry; Step 3: Calculate the velocity, trajectory and force of the Bohm particle; Calculate the speed and trajectory of paired particles according to the multi-body Bohm mechanics formula; calculate the forces acting on each Bohm particle according to Bohm mechanics theory, including Coulomb force, quantum force and electric field force; Step 4: Ionization judgment; The method of dividing the electron space region is used as the basis for judging ionization, and the number of unionized, singly ionized and doubly ionized Bohm particle pairs is counted to calculate the corresponding ionization probability; Step 5: Design the composite state and initialize it; The superposition of the two-electron ground state and the singly ionized state is constructed as the initial state to simulate the recombination of the ionized electron with its paired electron under the action of the laser field; Step 6: Analyze the re-collision process and draw conclusions; Analyze the probability density image of the electron re-collision process and observe the characteristics of the re-collision process; Compare the results of Bohm mechanics method with the numerical solution of the time-dependent Schrödinger equation to verify the accuracy of the method; Explore the effects of different parameters on non-sequential double ionization and use Bohm trajectories for in-depth analysis; Observe the initial positions and re-collision trajectories of non-sequential doubly ionized Bohm particle pairs, and summarize the laws of the initial positions; Compare the forces acting on composite particles with the same initial position but different ground state probabilities, and analyze the role of quantum forces in the re-collision process; Based on the analysis results, conclusions are drawn about the probability of non-sequential double ionization, the law of the initial positions of paired particles, and the role of quantum mechanics in the re-collision process.

2. The method for qualitative analysis of electron re-collision process under laser pulse based on Bohm mechanics according to claim 1, characterized in that: The specific process of step 1 is as follows: Under one-dimensional conditions, when considering the external field effect, the time-dependent Schrödinger equation satisfied by the electrons in the two-electron atom is expressed as: Equation 1: Where: i represents the i-th electron; represents the first-order time partial derivative of the wave function; ψ(x1,x2,t) represents the two-electron wave function satisfied by the Bohm particle; x1 represents the position of the first electron; x2 represents the position of the second electron; represents the velocity square of the first electron; represents the speed squared of the second electron; represents the Coulomb attraction potential energy between nuclei; represents the mutual Coulomb repulsion potential between two electrons; is the expression of the laser electric field, where E0 = 0.13 au, T = 110.23 au, τ = 31.49 au, φ = 3.15π, ω = 0.057 au; The two-electron wave function satisfied by the Bohm particle is expressed as: Formula 2: ψ(x1,x2,t)=R(x1,x2,t)exp(iS(x1,x2,t)) Where: R and S are real numbers; R(x1,x2,t) represents the real part of the wave function; S(x1,x2,t) represents the action; Substituting Equation 2 into Equation 1 and extracting the imaginary and real parts on both sides of the equation yields the following equation: Formula 3: Formula 4: Where: represents the total energy; x i represents the position of the i-th electron; Q(x1,x2,t),V a (x1,x2), (x1+x2)E(t) represent the kinetic energy, quantum potential energy, classical Coulomb potential energy and electric potential energy of the two electrons respectively; the expression of quantum potential energy is as follows: Formula 5: The velocity v of the i-th electron i (t) is expressed in terms of action S: Formula 6: Where: x1(t) represents the position of the first electron at time t; x2(t) represents the position of the second electron at time t; According to Equation 2, the velocity v of the i-th electron i (t) is represented by the wave function: Formula 7: Where: Im represents the imaginary part of a complex number; According to the given wave function, we can obtain the position x of the i-th electron at any time. i (t): Formula 8: The probability density function of the xenon atom ground state at the initial time is obtained using the imaginary time evolution method ψ0(x1,x2,t=0) 2 , and then use the elimination method to randomly sample N pairs of initial spatial coordinates of Bohm particles to represent the potential initial positions of the two electrons, that is: and Where k = 1, 2, 3…, N represents the position of the kth Bohm particle.

3. The method for qualitative analysis of electron re-collision process under laser pulse based on Bohm mechanics according to claim 1, characterized in that: The specific process of step 3 is as follows: The speed and trajectory of paired particles are calculated according to the multi-body Bohm mechanics formula: Formula 9: Formula 10: Formula 11: Formula 12: Where: represents the speed of the kth Bohm particle in the first electron; represents the speed of the kth Bohm particle in the second electron; represents the position of the kth Bohm particle in the first electron; represents the position of the kth Bohm particle in the second electron; According to Bohm mechanics theory, each Bohm particle is subject to the following forces: The net force acting on each Bohm particle is: Formula 13: Where: represents the speed of the kth Bohm particle in the ith electron; The Coulomb force of the nucleus on the kth Bohm particle in the ith electron is: Formula 14: Where: represents the position of the kth Bohm particle in the ith electron; The Coulomb force of the kth Bohm particle of the jth electron on the kth Bohm particle of the ith electron is: Formula 15: Where: x j represents the position of the jth electron; represents the position of the kth Bohm particle in the jth electron; The quantum force acting on each Bohm particle is: Formula 16: F Q =F R -F nucleusCoulomb -F e-eCoulomb -F E Where: F E represents the electric force acting on each Bohm particle; The electric force acting on each Bohm particle is: Formula 17: F E =-E(t).

4. The method for qualitative analysis of electron re-collision process under laser pulse based on Bohm mechanics according to claim 1, characterized in that: The specific process of step 4 is as follows: When paired Bohm particles satisfy the following conditions in Bohm mechanics, namely: exist In the case of , no ionization occurred, and N1 times occurred; or In the case of single ionization, N2 occurred; In the case of double ionization, N3 occurred; represents the position of the kth Bohm particle in the first electron, represents the position of the kth Bohm particle in the second electron; The probabilities of unionization, singly ionization, and doubly ionization are calculated using the following formulas: Formula 18: P1(t) = N1(t) / Ntotal, P2(t) = N2(t) / Ntotal and P3(t) = N3(t) / Ntotal Where: P1(t), P2(t), P3(t) represent the probabilities of unionized, singly ionized and doubly ionized, respectively; N1(t), N2(t), N3(t) represent the number of Bohm particle pairs of unionized, singly ionized and doubly ionized, respectively; Ntotal represents the total number of Bohm particle pairs; For the time-dependent Schrödinger equation, the probability that both electrons are in the bound state is: Formula 19: The probability of single ionization is: Formula 20: The probability of double ionization is: Formula 21: P3(t)=1-P1(t)-P2(t).

5. The method for qualitative analysis of electron re-collision process under laser pulse based on Bohm mechanics according to claim 1, characterized in that: The specific process of step 5 is as follows: The two-electron ground state ψ0(x1,x2) and the singly ionized state ψ s The initial state is a superposition of (x1, x2), where the singly ionized state ψ s (x1, x2) is used to simulate the recombination of an ionized electron with its paired electron under the action of a laser field; assuming that the single ionization state satisfies the spatial exchange symmetry, the expression is: Equation 22: ψ s (x1,x2) = ψ(x1)g(x2) + ψ(x2)g(x1) Where: g(x1), g(x2) represent the ground state wave functions of the first electron and the second electron respectively; ψ(x1), ψ(x2) represent the composite wave functions of the first electron and the second electron respectively; ψ(x) represents the single electron composite wave function, which is expressed as: Formula 23: ψ(x) = exp[-4log2×(x+100) 2 / λ0 2 ]×exp(ipx) where: x represents the coordinate of the wave function space; ipx represents the action; λ0 = 40 a.u., so the initial state is expressed as: Equation 24: ψ(x1,x2) = C1ψ0(x1,x2) + C2ψ s (x1,x2) Where: C1, C2 represent the electron at ψ0(x1, x2) and ψ s The probability amplitude of (x1,x2); The initial state of the two electrons also satisfies the spatial exchange symmetry.

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