Bohmian Trajectories Research Articles

Massive Gravitons on Bohmian Congruences

Taking a quantum corrected form of Raychaudhuri equation in a geometric background described by a Lorentz-violating massive theory of gravity, we go through investigating a time-like congruence of massive gravitons affected by a Bohmian quantum potential. We find some definite conditions upon which these gravitons are confined to diverging Bohmian trajectories. The respective behaviour of those quantum potentials are also derived and discussed. Additionally, and through a relativistic quantum treatment of a typical wave function, we demonstrate schematic conditions on the associated frequency to the gravitons, in order to satisfy the necessity of divergence.

On the correctness of cosmology from quantum potential

We examine in detail the cosmology based on quantal (Bohmian) trajectories as suggested in a recent study [A. F. Ali and S. Das, Phys. Lett. B 741, 276 (2014)]. We disagree with the conclusions regarding predicting the value of the cosmological constant [Formula: see text] and evading the Big Bang singularity. Furthermore, we show that the approach of using a quantum corrected Raychaudhuri equation (QRE), as suggested in A. F. Ali and S. Das, Phys. Lett. B 741, 276 (2014), is unsatisfactory, because, essentially, it uses the Raychaudhuri equation (RE), which is a kinematical equation, in order to predict dynamics. In addition, even within this inconsistent framework, the authors have adopted unjustified assumptions and carried out incorrect steps leading to doubtful conclusions.

Experimental nonlocal and surreal Bohmian trajectories.

Weak measurement allows one to empirically determine a set of average trajectories for an ensemble of quantum particles. However, when two particles are entangled, the trajectories of the first particle can depend nonlocally on the position of the second particle. Moreover, the theory describing these trajectories, called Bohmian mechanics, predicts trajectories that were at first deemed "surreal" when the second particle is used to probe the position of the first particle. We entangle two photons and determine a set of Bohmian trajectories for one of them using weak measurements and postselection. We show that the trajectories seem surreal only if one ignores their manifest nonlocality.

Quantum Dynamics with Gaussian Bases Defined by the Quantum Trajectories.

Development of a general approach to construction of efficient high-dimensional bases is an outstanding challenge in quantum dynamics describing large amplitude motion of molecules and fragments. A number of approaches, proposed over the years, utilize Gaussian bases whose parameters are somehow-usually by propagating classical trajectories or by solving coupled variational equations-tailored to the shape of a wave function evolving in time. In this paper we define the time-dependent Gaussian bases through an ensemble of quantum or Bohmian trajectories, known to provide a very compact representation of a wave function due to conservation of the probability density associated with each trajectory. Though the exact numerical implementation of the quantum trajectory dynamics itself is, generally, impractical, the quantum trajectories can be obtained from the wave function expanded in a basis. The resulting trajectories are used to guide compact Gaussian bases, as illustrated on several model problems.

Synthesizing quantum probability by a single chaotic complex-valued trajectory

The current trajectory interpretation of quantum mechanics is based on an ensemble viewpoint that the evolution of an ensemble of Bohmian trajectories guided by the same wavefunction Ψ converges asymptotically to the quantum probability |Ψ|2. Instead of the Bohm's ensemble-trajectory interpretation, the present paper gives a single-trajectory interpretation of quantum mechanics by showing that the distribution of a single chaotic complex-valued trajectory is enough to synthesize the quantum probability. A chaotic complex-valued trajectory manifests both space-filling (ergodic) and ensemble features. The space-filling feature endows a chaotic trajectory with an invariant statistical distribution, while the ensemble feature enables a complex-valued trajectory to envelop the motion of an ensemble of real trajectories. The comparison between complex-valued and real-valued Bohmian trajectories shows that without the participation of its imaginary part, a single real-valued trajectory loses the ensemble information contained in the wavefunction Ψ, and this explains the reason why we have to employ an ensemble of real-valued Bohmian trajectories to recover the quantum probability |Ψ|2. © 2015 Wiley Periodicals, Inc.

Pilot-Wave Quantum Theory with a Single Bohm’s Trajectory

The representation of a quantum system as the spatial configuration of its constituents evolving in time as a trajectory under the action of the wave-function, is the main objective of the Bohm theory. However, its standard formulation is referred to the statistical ensemble of its possible trajectories. The statistical ensemble is introduced in order to establish the exact correspondence (the Born's rule) between the probability density on the spatial configurations and the quantum distribution, that is the squared modulus of the wave-function. In this work we explore the possibility of using the pilot wave theory at the level of a single Bohm's trajectory. The pilot wave theory allows a formally self-consistent representation of quantum systems as a single Bohm's trajectory, but in this case there is no room for the Born's rule at least in its standard form. We will show that a correspondence exists between the statistical distribution of configurations along the single Bohm's trajectory and the quantum distribution for a subsystem interacting with the environment in a multicomponent system. To this aim, we present the numerical results of the single Bohm's trajectory description of the model system of six confined rotors with random interactions. We find a rather close correspondence between the coordinate distribution of one rotor along its trajectory and the time averaged marginal quantum distribution for the same rotor. This might be considered as the counterpart of the standard Born's rule. Furthermore a strongly fluctuating behavior with a fast loss of correlation is found for the evolution of each rotor coordinate. This suggests that a Markov process might well approximate the evolution of the Bohm's coordinate of a single rotor and it is shown that the correspondence between coordinate distribution and quantum distribution of the rotor is exactly verified.

Relativistic Bohmian Mechanics Without a Preferred Foliation

In non-relativistic Bohmian mechanics the universe is represented by a probability space whose sample space is composed of the Bohmian trajectories. In relativistic Bohmian mechanics an entire class of empirically equivalent probability spaces can be defined, one for every foliation of spacetime. In the literature the hypothesis has been advanced that a single preferred foliation is allowed, and that this foliation derives from the universal wave function by means of a covariant law. In the present paper the opposite hypothesis is advanced, i.e., no law exists for the foliations and therefore all the foliations are allowed. The resulting model of the universe is basically the "union" of all the probability spaces associated with the foliations. This hypothesis is mainly motivated by the fact that any law defining a preferred foliation is empirically irrelevant. It is also argued that the absence of a preferred foliation may reduce the well known conflict between Bohmian mechanics and Relativity.

Calculation of Bohmian quantum trajectories for STEM.

We present in this work the calculation of Bohmian quantum trajectories representing the wave function propagation in a crystal for a focused electron probe in a scanning transmission electron microscope (STEM). The wave function and quantum trajectories are obtained from the calculation of time-dependent Schrödinger equation by fast Fourier transformation multislice algorithm. In our work, the Bohmian quantum trajectories of a scanning probe penetrating a Cu crystal are studied as an example of this calculation scheme. The results help us to better understand the electron diffraction process in a microscopic imaging from a trajectory-based point of view. This Bohmian quantum trajectory method can be used to extend the application of classical Monte Carlo method from the study of electron interaction with amorphous solid to crystalline structure.

Scattering by a reflectionless modified Pöschl–Teller potential: Bohmian trajectories and arrival times

A nonreflecting wavepacket is constructed by the superposition of reflectionless eigenstates of modified Pöschl–Teller potential. Free propagation and propagation in the presence of the above potential of such a wavepacket is considered using the concept of arrival time. Comparison is made with the free evolving Gaussian wavepacket. Mean arrival time at a detector behind the well is given as a function of mass for separate cases. A selection of Bohmian trajectories in the interacting case are computed and compared to the trajectories of a free particle guided by a Gaussian packet.

Weak measurement from the electron displacement current: new path for applications

The interest on weak measurements is rapidly growing during the last years as a unique tool to better understand and predict new quantum phenomena. Up to now many theoretical and experimental weak-measurement techniques deal with (relativistic) photons or cold atoms, but there is much less investigation on (non-relativistic) electrons in up-to-date electronics technologies. We propose a way to perform weak measurements in nanoelectronic devices through the measurement of the total current (particle plus displacement component) in such devices. We study the interaction between an electron in the active region of a electron device with a metal surface working as a sensing electrode by means of the (Bohmian) conditional wave function. We perform numerical (Monte Carlo) simulations to reconstruct the Bohmian trajectories in the iconic double slit experiment. This work opens new paths for understanding the quantum properties of an electronic system as well as for exploring new quantum engineering applications in solid state physics.

Investigation of the characteristic properties of high-order harmonic spectrum in atoms using Bohmian trajectories

We study the electron quantum dynamics of high harmonic generation (HHG) processes of atomic hydrogen under intense near infrared (NIR) laser fields by means of the De Broglie–Bohm’s framework of Bohmian mechanics. The proposed accurate 3D numerical scheme is utilized to discuss the mechanism of the multiple plateau generation and the cut-off extension, as the main characteristic features of HHG spectrum. One-color (1600 nm) and two-color (1600 nm + 800 nm) laser fields with different time delays are used to investigate the effect of pulse shape on electron dynamics and HHG process. The presented results on Bohmian trajectories and their energy content, along with the analysis of the emission time period of different groups of trajectories, provide a comprehensive and fresh electron dynamical picture and uncover novel mechanisms of the HHG processes and power spectra.

Trajectory approach to the Schrödinger–Langevin equation with linear dissipation for ground states

The Schrödinger–Langevin equation with linear dissipation is integrated by propagating an ensemble of Bohmian trajectories for the ground state of quantum systems. Substituting the wave function expressed in terms of the complex action into the Schrödinger–Langevin equation yields the complex quantum Hamilton–Jacobi equation with linear dissipation. We transform this equation into the arbitrary Lagrangian–Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation is simultaneously integrated with the trajectory guidance equation. Then, the computational method is applied to the harmonic oscillator, the double well potential, and the ground vibrational state of methyl iodide. The excellent agreement between the computational and the exact results for the ground state energies and wave functions shows that this study provides a synthetic trajectory approach to the ground state of quantum systems.

Weak measurements of trajectories in quantum systems: classical, Bohmian and sum over paths

Weak values, obtained from weak measurements, attempt to describe the properties of a quantum system as it evolves from an initial to a final state, without practically altering this evolution. Trajectories can be defined from weak measurements of the position, or inferred from weak values of the momentum operator. The former can be connected to the asymptotic form of the Feynman propagator and the latter to Bohmian trajectories. Employing a time-dependent oscillator as a model, this work analyzes to what extent weak measurements can shed light on the underlying dynamics of a quantum system expressed in terms of trajectories, in particular by comparing the two approaches.

Calculation of the Duration of the Atomic Tunneling Ionization in a Strong Electrostatic Field by Using Bohmian Trajectories Approach**Supported by the National Basic Research Program of China under Grant No 2013CB922200, the National Natural Science Foundation of China under Grant Nos 11274141, 11304116, 11274001 and 11247024, and the Jilin Provincial Research Foundation for Basic Research under Grant No 20140101168JC.

The duration of a bound electron tunneling through the barrier formed by atomic potential and electrostatic field is calculated by the Bohmian trajectories scheme. The time of the tunneling ionization decreases with the increase of the amplitude of the electrostatic field. By using the information about the position, velocity and force of the Bohmian trajectories, the dynamical process of tunneling through the barrier is investigated.

Complex-valued derivative propagation method with approximate Bohmian trajectories for quantum barrier scattering

The complex quantum Hamilton–Jacobi equation for the complex action is approximately solved by propagating individual Bohmian trajectories in real space. Equations of motion for the complex action and its spatial derivatives are derived through use of the derivative propagation method. We transform these equations into the arbitrary Lagrangian–Eulerian version with the grid velocity matching the flow velocity of the probability fluid. Setting higher-order derivatives equal to zero, we obtain a truncated system of equations of motion describing the rate of change in the complex action and its spatial derivatives transported along approximate Bohmian trajectories. A set of test trajectories is propagated to determine appropriate initial positions for transmitted trajectories. Computational results for transmitted wave packets and transmission probabilities are presented and analyzed for a one-dimensional Eckart barrier and a two-dimensional system involving either a thick or thin Eckart barrier along the reaction coordinate coupled to a harmonic oscillator.

Exploration of the subcycle multiphoton ionization dynamics and transient electron density structures with Bohmian trajectories

An accurate three-dimensional numerical scheme for the De Broglie--Bohm framework of Bohmian mechanics is presented. This method is utilized to explore the subcycle multiphoton ionization dynamics of the hydrogen atom subject to intense near-infrared laser fields on the subfemtosecond time scale. The analysis of the time-dependent electron density reveals that several distinct density portions can be shaped and detached from the core within a half cycle of the laser field. As a complementary perspective, we identify several distinct groups of the Bohmian trajectories which represent the multiple detachments of the electron density at different times. The method presented provides very accurate electron densities and Bohmian trajectories that allow to uncover the origin of the formation of the transient and distinct electron structures seen in the multiphoton ionization processes.

Time-dependent potential barriers and superarrivals

Scattering of a Gaussian wavepacket from rectangular potential barriers with increasing widths or heights is studied numerically. It is seen that during a certain time interval the time-evolving transmission probability increases compared to the corresponding unperturbed cases. In the literature this effect is known as superarrival in transmission probability. We present a trajectory-based explanation for this effect by using the concept of quantum potential energy and computing a selection of Bohmian trajectories. Relevant parameters in superarrivals are determined for the case that the barrier width increases linearly during the dispersion of the wavepacket. Nonlinearity in time perturbation is also considered.

Quantum noise from a Bohmian perspective: fundamental understanding and practical computation in electron devices

In the literature, the study of electron transport in quantum devices is mainly devoted to DC properties. The fluctuations of the electrical current around these DC values, the so-called quantum noise, are much less analyzed. The computation of quantum noise is intrinsically linked (by temporal correlations) to our ability to understand/compute the time-evolution of a quantum system that is measured several times. There are several quantum theories that provide different (but empirically equivalent) ways of understanding/computing the perturbation of the wave function when it is measured. In this work, quantum noise associated to an electron impinging upon a semitransparent barrier is explained using Bohmian mechanics (which deals with wave functions and point-like particles). From this result, the fundamental understanding and practical computation of quantum noise with Bohmian trajectories are discussed. Numerical simulations of low and high frequency features of quantum shot noise in a resonant tunneling diode are presented (through the BITLLES simulator), showing the usefulness of the Bohmian approach.

Numerical Calculation of the Bohmian Trajectory: Ionization of a One-dimensional Atom in an Intense Laser Field

Numerical Calculation of the Bohmian Trajectory: Ionization of a One-dimensional Atom in an Intense Laser Field

Cosmology from quantum potential

It was shown recently that replacing classical geodesics with quantal (Bohmian) trajectories gives rise to a quantum corrected Raychaudhuri equation (QRE). In this article we derive the second order Friedmann equations from the QRE, and show that this also contains a couple of quantum correction terms, the first of which can be interpreted as cosmological constant (and gives a correct estimate of its observed value), while the second as a radiation term in the early universe, which gets rid of the big-bang singularity and predicts an infinite age of our universe.