X. Oriols

Applied Bohmian mechanics

Abstract Bohmian mechanics provides an explanation of quantum phenomena in terms of point-like particles guided by wave functions. This review focuses on the use of nonrelativistic Bohmian mechanics to address practical problems, rather than on its interpretation. Although the Bohmian and standard quantum theories have different formalisms, both give exactly the same predictions for all phenomena. Fifteen years ago, the quantum chemistry community began to study the practical usefulness of Bohmian mechanics. Since then, the scientific community has mainly applied it to study the (unitary) evolution of single-particle wave functions, either by developing efficient quantum trajectory algorithms or by providing a trajectory-based explanation of complicated quantum phenomena. Here we present a large list of examples showing how the Bohmian formalism provides a useful solution in different forefront research fields for this kind of problems (where the Bohmian and the quantum hydrodynamic formalisms coincide). In addition, this work also emphasizes that the Bohmian formalism can be a useful tool in other types of (nonunitary and nonlinear) quantum problems where the influence of the environment or the nonsimulated degrees of freedom are relevant. This review contains also examples on the use of the Bohmian formalism for the many-body problem, decoherence and measurement processes. The ability of the Bohmian formalism to analyze this last type of problems for (open) quantum systems remains mainly unexplored by the scientific community. The authors of this review are convinced that the final status of the Bohmian theory among the scientific community will be greatly influenced by its potential success in those types of problems that present nonunitary and/or nonlinear quantum evolutions. A brief introduction of the Bohmian formalism and some of its extensions are presented in the last part of this review.

Time-Dependent Many-Particle Simulation for Resonant Tunneling Diodes: Interpretation of an Analytical Small-Signal Equivalent Circuit

A full many-particle (beyond the mean-field approximation) electron quantum-transport simulator, which is named BITLLES, is used to analyze the transient current response of resonant tunneling diodes (RTDs). The simulations have been used to test an analytical (free-fitting parameters) small-signal equivalent circuit for RTDs under stable direct-current-biased conditions. The comparison provides an excellent agreement and furnishes a way to physically interpret each circuit element. In addition, a nonlinear novel RTD behavior in the negative differential conductance region has been established, i.e. asymmetric time constants in the RTD current response when high-low or low-high voltage steps are considered.

Self-consistent simulation of quantum shot noise in nanoscale electron devices

An approach for studying shot noise in mesoscopic systems that explicitly includes the Coulomb interaction among electrons, by self-consistently solving the Poisson equation, is presented. As a test, current fluctuations on a standard resonant tunneling diode are simulated in agreement with previous predictions and experimental results. The present approach opens a new path for the simulation of nanoscale electron devices, where pure quantum mechanical and Coulomb blockade phenomena coexist.

Can the Wave Function in Configuration Space Be Replaced by Single-Particle Wave Functions in Physical Space?

The ontology of Bohmian mechanics includes both the universal wave function (living in 3N-dimensional configuration space) and particles (living in ordinary 3-dimensional physical space). Proposals for understanding the physical significance of the wave function in this theory have included the idea of regarding it as a physically-real field in its 3N-dimensional space, as well as the idea of regarding it as a law of nature. Here we introduce and explore a third possibility in which the configuration space wave function is simply eliminated—replaced by a set of single-particle pilot-wave fields living in ordinary physical space. Such a re-formulation of the Bohmian pilot-wave theory can exactly reproduce the statistical predictions of ordinary quantum theory. But this comes at the rather high ontological price of introducing an infinite network of interacting potential fields (living in 3-dimensional space) which influence the particles’ motion through the pilot-wave fields. We thus introduce an alternative approach which aims at achieving empirical adequacy (like that enjoyed by GRW type theories) with a more modest ontological complexity, and provide some preliminary evidence for optimism regarding the (once popular but prematurely-abandoned) program of trying to replace the (philosophically puzzling) configuration space wave function with a (totally unproblematic) set of fields in ordinary physical space.

Robust weak-measurement protocol for Bohmian velocities

We present a protocol for measuring Bohmian - or the mathematically equivalent hydrodynamic - velocities based on an ensemble of two position measurements, defined from a Positive Operator Valued Measure, separated by a finite time interval. The protocol is very accurate and robust as long as the first measurement uncertainty divided by the finite time interval between measurements is much larger than the Bohmian velocity, and the system evolves under flat potential between measurements. The difference between the Bohmian velocity of the unperturbed state and the measured one is predicted to be much smaller than 1% in a large range of parameters. Counter-intuitively, the measured velocity is that at the final time and not a time-averaged value between measurements.

Computation of many-particle quantum trajectories with exchange interaction: application to the simulation of nanoelectronic devices

Following Oriols (2007 Phys. Rev. Lett. 98 066803), an algorithm to deal with the exchange interaction in non-separable quantum systems is presented. The algorithm can be applied to fermions or bosons and, by construction, it exactly ensures that any observable is totally independent of the interchange of particles. It is based on the use of conditional Bohmian wave functions which are solutions of single-particle pseudo-Schrödinger equations. The exchange symmetry is directly defined by demanding symmetry properties of the quantum trajectories in the configuration space with a universal algorithm, rather than through a particular exchange-correlation functional introduced into the single-particle pseudo-Schrödinger equation. It requires the computation of N(2) conditional wave functions to deal with N identical particles. For separable Hamiltonians, the algorithm reduces to the standard Slater determinant for fermions (or permanent for bosons). A numerical test for a two-particle system, where exact solutions for non-separable Hamiltonians are computationally accessible, is presented. The numerical viability of the algorithm for quantum electron transport (in a far-from-equilibrium time-dependent open system) is demonstrated by computing the current and fluctuations in a nano-resistor, with exchange and Coulomb interactions among electrons.

Improving the intrinsic cut-off frequency of gate-all-around quantum-wire transistors without channel length scaling

Progress in high-frequency transistors is based on reducing electron transit time, either by scaling their lengths or by introducing materials with higher electron mobility. For gate-all-around quantum-wire transistors with lateral dimensions similar or smaller than their length, a careful analysis of the displacement current reveals that a time shorter than the transit time controls their high-frequency performance. Monte Carlo simulations of such transistors with a self-consistent solution of the 3D Poisson equation clearly show an improvement of the intrinsic cut-off frequency when their lateral areas are reduced, without length scaling.

Bohm trajectories for the Monte Carlo simulation of quantum-based devices

A generalization of the classical ensemble Monte Carlo (MC) device simulation technique is proposed to simultaneously deal with quantum-mechanical phase-coherence effects and scattering interactions in quantum-based devices. The proposed method restricts the quantum treatment of transport to the regions of the device where the potential profile significantly changes in distances of the order of the de Broglie wavelength of the carriers (the quantum window). Bohm trajectories associated to time-dependent Gaussian wave packets are used to simulate the electron transport in the quantum window. Outside this window, the classical ensemble MC simulation technique is used. Classical and quantum trajectories are smoothly matched at the boundaries of the quantum window according to a criterium of total-energy conservation. A self-consistent one-dimensional simulator for resonant tunneling diodes has been developed to demonstrate the feasibility of our proposal.

COMPUTATION OF QUANTUM ELECTRICAL CURRENTS THROUGH THE RAMO–SHOCKLEY–PELLEGRINI THEOREM WITH TRAJECTORIES

Motivated by a recent approach to solve quantum dynamics with full Coulomb correlations [X. Oriols, Phys. Rev. Lett.98 (2007) 066803], we present here an extension of the Ramo–Shockley–Pellegrini theorem for quantum systems to compute the total (conduction plus displacement) current in terms of quantum (Bohmian) trajectories. By way of test, we derive an extension of the Ramo-Shockley-Pellegrini theorem using standard quantum mechanics and we compare it to our former result. As expected, both formulations give identical results, however we emphasize the numerical viability of computing self-consistently the total current by means of quantum trajectories in front of the difficulties to do it in terms of standard quantum mechanics.

High frequency components of current fluctuations in semiconductor tunneling barriers

The power spectral density of current noise in phase-coherent semiconductor tunneling scenarios is studied in terms of Bohm trajectories associated to time-dependent wave packets. In particular, the influence of the particles reflected by the barrier on the noise spectrum is analyzed. An enhancement of the power spectral density of the current fluctuations is predicted for very high frequencies. The experimental measurement of this high frequency effect is discussed as a possible test of Bohm trajectories.