Bohmian Particles Research Articles

Electron correlation in two-electron atoms: A Bohmian analysis of high-order harmonic generation in high-frequency domain

Abstract In studying interactions between intense laser fields and atoms or molecules, the role of electron correlation effects on the dynamical response is an important and pressing issue to address. Utilizing Bohmian mechanics (BM), we have theoretically explored the two-electron correlation characteristics while generating high-order harmonics in xenon atoms subjected to intense laser fields. We initially employed Bohmian trajectories to reproduce the dynamics of the electrons and subsequently utilized time-frequency analysis spectra to ascertain the emission time windows for high-order harmonics. Within these time windows, we classified the nuclear region Bohmian trajectories and observed that intense high-order harmonics are solely generated when paired Bohmian particles (BPs) concurrently appear in the nuclear region and reside there for a duration within a re-collision time window. Furthermore, our analysis of characteristic trajectories producing high-order harmonics led us to propose a two-electron re-collision model to elucidate this phenomenon. The study demonstrates that intense high-order harmonics are only generated when both electrons are in the ground state within the re-collision time window. This work discusses the implications of correlation effects between two electrons and offers valuable insights for studying correlation in multi-electron high-order harmonic generation.

Order, Chaos and Born’s Distribution of Bohmian Particles

We study order, chaos and ergodicity in the Bohmian trajectories of a 2D quantum harmonic oscillator. We first present all the possible types (chaotic, ordered) of Bohmian trajectories in wavefunctions made of superpositions of two and three energy eigenstates of the oscillator. There is no chaos in the case of two terms and in some cases of three terms. Then, we show the different geometries of nodal points in bipartite Bohmian systems of entangled qubits. Finally, we study multinodal wavefunctions and find that a large number of nodal points does not always imply the dominance of chaos. We show that, in some cases, the Born distribution is dominated by ordered trajectories, something that has a significant impact on the accessibility of Born’s rule P=|Ψ|2 by initial distributions of Bohmian particles with P0≠|Ψ0|2.

Chaos in Bohmian Quantum Mechanics: A Short Review

This is a short review in the theory of chaos in Bohmian Quantum Mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically and numerically) the different kinds of Bohmian trajectories where, in general, chaos and order coexist. Finally we explore the effect of quantum entanglement on the evolution of the Bohmian trajectories and study chaos and ergodicity in qubit systems which are of great theoretical and practical interest. We find that the chaotic trajectories are also ergodic, i.e. they give the same final distribution of their points after a long time regardless of their initial conditions. In the case of strong entanglement most trajectories are chaotic and ergodic and an arbitrary initial distribution of particles will tends to Born's rule over the course of time. On the other hand, in the case of weak entanglement the distribution of Born's rule is dominated by ordered trajectories and consequently an arbitrary initial configuration of particles will not tend, in general, to Born's rule, unless it is initially satisfied. Our results shed light on a fundamental problem in Bohmian Mechanics, namely whether there is a dynamical approximation of Born's rule by an arbitrary initial distribution of Bohmian particles.

Ionization of two-electron atom (xenon) studied by Bohmian mechanics theory* *Project supported by the Jilin Provincial Science and Technology Development Plan Program for Excellent Youth Talents, China (Grant No. 20180520174JH) and the National Natural Science Foundation of China (Grant Nos. 11704145, 11904050, 11774129, 11747007, 11534004, and 12074145).

The ionization dynamics of two-electron atom in an intense laser field is studied by the Bohmian mechanics (BM) theory, and the xenon atomic potential function is used as a model. The single ionization process and double ionization process are calculated by the BM theory and their results are in good agreement with those calculated by numerically solving the time-dependent Schrödinger equation. The analyses of the types, trajectories, and forces of Bohmian particles (BPs) undergoing the single and double ionizations indicate that the re-collision process accounts for a considerable proportion in the singly ionized cases. Furthermore, the analysis of the work done by the external force acting on the BPs shows that the quantum force plays an important role in the re-collision process. This work is helpful in understanding the ionization of two-electron atom in an intense laser field.

How Humean is Bohumianism?

An important part of the influential Humean doctrine in philosophy is the supervenience principle (sometimes referred to as the principle of separability). This principle asserts that the complete state of the world supervenes on the intrinsic properties of its most fundamental components and their spatiotemporal relations (the so-called Humean mosaic). There are well-known arguments in the literature purporting to show that in quantum mechanics the Humean supervenience principle is violated, due to the existence of entangled states. Recently, however, arguments have been presented to the effect that the supervenience principle can be defended in Bohmian mechanics. The key element of this strategy lies in the observation that according to Bohmian mechanics the fundamental facts about particles are facts about their spatial locations, and moreover, for any proper subsystem of the world its state may non-trivially depend on the spatial configuration of the rest of the universe. Thus quantum-mechanical states of subsystems do not represent their intrinsic properties but rather characterize their relations with the environment. In this paper we point out the worry that this Bohmian strategy—known as Bohumianism—saves the letter but not the spirit of the Humean doctrine of supervenience, since it prima facie violates another seemingly important Humean principle, which we call Strong Supervenience and whose denial implies the existence of necessary connections among distinct individuals. We argue that the best defense for Bohumians is to question the fundamental existence of complex physical systems and their states by treating any reference to them as a convenient description of the underlying collection of Bohmian particles. We consider several pros and cons of this strategy.

Resonance-enhanced two-photon ionization of hydrogen atom in intense laser field investigated by Bohmian-mechanics**Project supported by Jilin Province Science and Technology Development Plan Project–Excellent Youth Talents Fund Project, China (Grant No. 20180520174JH) and the National Natural Science Foundation of China (Grant Nos. 11704145 11904050, 11774129, 11747007, and 11534004).

Resonance enhanced two-photon ionization process of hydrogen atom via the resonant laser pulse is studied by Bohmian mechanics (BM) method. By analyzing the trajectories and energies of Bohmian particles (BPs), we find that under the action of high frequency and low intensity multi-circle resonant laser pulses, the ionized BPs first absorb one photon completing the excitation, and then absorb another photon, completing the ionization after staying in the first excited state for a period of time. The analysis of work done by the forces shows that the electric field force and quantum force play a major role in the whole ionization process. At the excitation moment and in the excitation-ionization process, the effect of the quantum force is greater than that of the electric field force. Finally, we discuss the principle of work and energy for BPs, and find that the electric field force and quantum force are non-conservative forces whose work is equal to the increment of mechanical energy of the system. In addition, it is proved that the quantum potential energy actually comes from the kinetic energy of the system and the increment of kinetic energy is equal to that of the kinetic energy of the system.

Multi-field and Bohm’s theory

In the recent literature, it has been shown that the wave function in the de Broglie–Bohm theory can be regarded as a new kind of field, i.e., a “multi-field”, in three-dimensional space. In this paper, I argue that the natural framework for the multi-field is the original second-order Bohm’s theory. In this context, it is possible: (i) to construe the multi-field as a real-valued scalar field; (ii) to explain the physical interaction between the multi-field and the Bohmian particles; and (iii) to clarify the status of the energy–momentum conservation and the dynamics of the theory.

Bohmian mechanics at space–time singularities. I. Timelike singularities

We develop an extension of Bohmian mechanics to a curved background space–time containing a singularity. The present paper focuses on timelike singularities. We use the naked timelike singularity of the super-critical Reissner–Nordström geometry as an example. While one could impose boundary conditions at the singularity that would prevent the particles from falling into the singularity, we are interested here in the case in which particles have positive probability to hit the singularity and get annihilated. The wish for reversibility, equivariance, and the Markov property then dictates that particles must also be created by the singularity, and indeed dictates the rate at which this must occur. That is, a stochastic law prescribes what comes out of the singularity. We specify explicit equations of a model involving an interior–boundary condition on the wave function at the singularity, which can be used also in other versions of quantum theory besides Bohmian mechanics. As the resulting theory involves particle creation and annihilation, it can be regarded as a quantum field theory, and the stochastic process for the Bohmian particles is analogous to Bell-type quantum field theories.

Entangled Quantum Dynamics of Many-Body Systems using Bohmian Trajectories

Bohmian mechanics is an interpretation of quantum mechanics that describes the motion of quantum particles with an ensemble of deterministic trajectories. Several attempts have been made to utilize Bohmian trajectories as a computational tool to simulate quantum systems consisting of many particles, a very demanding computational task. In this paper, we present a novel ab-initio approach to solve the many-body problem for bosonic systems by evolving a system of one-particle wavefunctions representing pilot waves that guide the Bohmian trajectories of the quantum particles. In this approach, quantum entanglement effects arise due to the interactions between different configurations of Bohmian particles evolving simultaneously. The method is used to study the breathing dynamics and ground state properties in a system of interacting bosons.

Revisiting Entanglement within the Bohmian Approach to Quantum Mechanics.

We revisit the concept of entanglement within the Bohmian approach to quantum mechanics. Inspired by Bohmian dynamics, we introduce two partial measures for the amount of entanglement corresponding to a pure state of a pair of quantum particles. One of these measures is associated with the statistical correlations exhibited by the joint probability density of the two Bohmian particles in configuration space. The other partial measure corresponds to the correlations associated with the phase of the joint wave function, and describes the non-separability of the Bohmian velocity field. The sum of these two components is equal to the total entanglement of the joint quantum state, as measured by the linear entropy of the single-particle reduced density matrix.

Why Bohmian Mechanics? One- and Two-Time Position Measurements, Bell Inequalities, Philosophy, and Physics.

In Bohmian mechanics, particles follow continuous trajectories, so two-time position correlations have been well defined. However, Bohmian mechanics predicts the violation of Bell inequalities. Motivated by this fact, we investigate position measurements in Bohmian mechanics by coupling the particles to macroscopic pointers. This explains the violation of Bell inequalities despite two-time position correlations. We relate this fact to so-called surrealistic trajectories that, in our model, correspond to slowly moving pointers. Next, we emphasize that Bohmian mechanics, which does not distinguish between microscopic and macroscopic systems, implies that the quantum weirdness of quantum physics also shows up at the macro-scale. Finally, we discuss the fact that Bohmian mechanics is attractive to philosophers but not so much to physicists and argue that the Bohmian community is responsible for the latter.

Conditions for the classicality of the center of mass of many-particle quantum states

We discuss the conditions for the classicality of quantum states with a very large number of identical particles. By defining the center of mass from a large set of Bohmian particles, we show that it follows a classical trajectory when the distribution of the Bohmian particle positions in a single experiment is always equal to the marginal distribution of the quantum state in physical space. This result can also be interpreted as a single experiment generalization of the well-known Ehrenfest theorem. We also demonstrate that the classical trajectory of the center of mass is fully compatible with a quantum (conditional) wave function solution of a classical non-linear Schrödinger equation. Our work shows clear evidence for a quantum–classical inter-theory unification, and opens new possibilities for practical quantum computations with decoherence.

Electron excitation from ground state to first excited state: Bohmian mechanics method**Project supported by the Doctoral Research Start-up Funding of Northeast Dianli University, China (Grant No. BSJXM-201332), the National Natural Science Foundation of China (Grant Nos. 11547114, 11534004, 11474129, 11274141, 11447192, and 11304116), and the Graduate Innovation Fund of Jilin University, China (Grant No. 2015091).

The excitation process of electrons from the ground state to the first excited state via the resonant laser pulse is investigated by the Bohmian mechanics method. It is found that the Bohmian particles far away from the nucleus are easier to be excited and are excited firstly, while the Bohmian particles in the ground state is subject to a strong quantum force at a certain moment, being excited to the first excited state instantaneously. A detailed analysis for one of the trajectories is made, and finally we present the space and energy distribution of 2000 Bohmian particles at several typical instants and analyze their dynamical process at these moments.

Mixed quantal-semiquantal dynamics with stochastic particles for backreaction.

A mixed quantal-semiquantal theory is presented in which the semiquantal squeezed-state wave packet describes the heavy degrees of freedom. Starting from the mean-field equations of motion that are naturally derived from the time-dependent variational principle, we introduce the stochastic particle description for both the quantal and semiquantal parts in an aim to take into account the interparticle correlation, in particular the "quantum backreaction" beyond the mean-field approximation. A numerical application on a model of O2 scattering from a Pt surface demonstrates that the proposed scheme gives correct asymptotic behavior of the scattering probability, with improvement over the mixed quantum-classical scheme with Bohmian particles, which is comprehended by comparing the Bohmian and the stochastic trajectories.

Dynamic stabilization of ionization for an atom irradiated by high-frequency laser pulses studied with the Bohmian-trajectory scheme

We study the atomic ionization process in high-frequency laser pulses by the Bohmian-trajectory scheme. Combined with the analysis of the trajectories of Bohmian particles and the time-dependent potential well based on the Kramers-Henneberger transformation, we find that there are two kinds of forces that act on each Bohmian particle (BP): the classical force from the combined potential of the atomic Coulomb potential and the laser field and the quantum force. By investigating the forces acting on different BPs, we find that the quantum force plays an important role in the atomic ionization process in the laser pulse.

Convergence characteristics of a domain decomposition scheme for approximation of quantum forces

We investigate the convergence characteristics of a domain decomposition scheme to approximately compute quantum forces in the context of semiclassical Bohmian mechanics. The study is empirical in nature. Errors in the approximate quantum forces are compiled while relevant parameters in the numerical scheme are systematically changed. The compiled errors are analyzed to extract underlying trends. Our analysis shows that the number of Bohmian particles used in discretization has relatively weak influence on the error, while the length scale of interaction among subdomains controls the error in most cases. More precisely, the overall numerical error decreases as the length scale of interaction among subdomains decreases, if the error due to the truncation of the tail of the probability density distribution is adequately controlled. Our results suggest that it may be necessary to develop an efficient method to effectively control the error due to the truncation of the tail. Further studies, especially rigorous mathematical ones, should follow to understand and improve the behavior of the scheme.

Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons

Bohmian mechanics reproduces all statistical predictions of quantum mechanics, which ensures that entanglement cannot be used for superluminal signaling. However, individual Bohmian particles can experience superluminal influences. We propose to illustrate this point using a double double-slit setup with path-entangled photons. The Bohmian velocity field for one of the photons can be measured using a recently demonstrated weak-measurement technique. The found velocities strongly depend on the value of a phase shift that is applied to the other photon, potentially at spacelike separation.

Position measurements in the de Broglie–Bohm interpretation of quantum mechanics

The de Broglie–Bohm interpretation of quantum mechanics assigns positions and trajectories to particles. We analyze the validity of a formula for the velocities of Bohmian particles which makes the analysis of these trajectories particularly simple. We apply it to particle detectors of four different types and show that the detectors of three of these types lead to “surrealistic trajectories”, i.e., leave a trace where the Bohmian particle was not present.

Are all particles real?

In Bohmian mechanics elementary particles exist objectively, as point particles moving according to a law determined by a wavefunction. In this context, questions as to whether the particles of a certain species are real—questions such as, Do photons exist? Electrons? Or just the quarks?—have a clear meaning. We explain that, whatever the answer, there is a corresponding Bohm-type theory, and no experiment can ever decide between these theories. Another question that has a clear meaning is whether particles are intrinsically distinguishable, i.e., whether particle world lines have labels indicating the species. We discuss the intriguing possibility that the answer is no, and particles are points— just points.

On Peres' Statement “Opposite Momenta Lead to Opposite Directions,” Decaying Systems and Optical Imaging

We re-examine Peres' statement “opposite momenta lead to opposite directions”. It will be shown that Peres' statement is only valid in the large distance or large time limit. In the short distance or short time limit an additional deviation from perfect alignment occurs due to the uncertainty of the location of the source. This error contribution plays a major role in Popper's orginal experimental proposal. Peres' statement applies rather to the phenomenon of optical imaging, which was regarded by him as a verification of his statement. This is because this experiment can in a certain sense be seen as occurring in the large distance limit. We will also reconsider both experiments from the viewpoint of Bohmian mechanics. In Bohmian mechanics particles with perfectly opposite momenta will move in opposite directions. In addition it will prove particularly useful to use Bohmian mechanics because the Bohmian trajectories coincide with the conceptual trajectories drawn by Pittman et al. In this way Bohmian mechanics provides a theoretical basis for these conceptual trajectories.