Arbitrary Time-dependent Fields Research Articles

An approach for a polychromatic generalized Lorenz–Mie theory

In this work a polychromatic version of the generalized Lorenz–Mie theory stricto sensu (GLMT) is derived. In this new formalism, arbitrary time-dependent fields are expanded into partial waves using Bromwich scalar potentials and, instead of the usual expansion coefficients – the beam shape coefficients (BSCs) – found in the monochromatic GLMT, now one finds field shape spectra (FSSs) which are intrinsically frequency-dependent. Expressions for the incident, scattered and internal fields are presented, and it is shown how physical quantities defined and expressed in the monochromatic GLMT in terms of the BSCs are modified and redefined in terms of the FSSs in polychromatic light scattering problems, like scattered intensities and phase angles, absorption, extinction and scattering cross-sections, and radiation pressure cross-sections.

Invariant-Parameterized Exact Evolution Operator for SU(2) Systems with Time-Dependent Hamiltonian.

We report the step-by-step construction of the exact, closed and explicit expression for the evolution operator U(t) of a localized and isolated qubit in an arbitrary time-dependent field, which for concreteness we assume to be a magnetic field. Our approach is based on the existence of two independent dynamical invariants that enter the expression of SU(2) by means of two strictly related time-dependent, real or complex, parameters. The usefulness of our approach is demonstrated by exactly solving the quantum dynamics of a qubit subject to a controllable time-dependent field that can be realized in the laboratory. We further discuss possible applications to any SU(2) model, as well as the applicability of our method to realistic physical scenarios with different symmetry properties.

Quantum transport for electron quasiparticles interacting with a disordered binary alloy: A Wigner approach

Quantum transport is developed in the Wigner function representation for a Bloch electron quasiparticle interacting with a disordered binary alloy in the presence of a homogeneous electric field of arbitrary time dependence and amplitude. The electron quasiparticle is described by a single-band effective Hamiltonian, and the homogeneous electric field is treated in the vector potential gauge. The methodology for the quantum transport analysis proceeds by first transforming the Liouville equation to one in which the interaction Hamiltonian (the binary alloy Hamiltonian) appears quadratically. The basis states employed in evaluating the requisite matrix elements are the instantaneous eigenstates of the electron quasiparticle Hamiltonian in the presence of the electric field. The Wigner quantum transport equations are derived, and the binary alloy collision term is suitably ensemble averaged over the disordered binary alloy matrix elements. In addition, the general drift and diffusion terms are exactly obtained, resulting in the complete Wigner-Boltzmann equation for the binary alloy system. In approximating the collision term for the two separate cases of parabolic energy dispersion and the long-wavelength limit, it is found that the reduced Wigner-Boltzmann equation includes the manifestation of the intracollisional field effect and other quantum generalities. As a contrast to the actual random alloy treatment, attention is given to the canonical problem of quantum transport for a virtual crystal (VC). As an alternative to the random binary alloy scattering problem, the VC Hamiltonian adopted for this treatment is derived by ensemble averaging the random binary alloy Hamiltonian. The resulting Wigner transport equation for the VC case is descriptive of Bloch dynamics in a graded semiconductor alloy.

Bloch-electron dynamics in homogeneous electric fields: Application to multiphoton absorption in semiconductors and insulators

The theory of Bloch-electron dynamics for carriers in a homogeneous electric field of arbitrary time dependence is developed in consideration of the properties of multiphoton absorption (MPA) in semiconductors and insulators. The general approach is to utilize the accelerated Bloch-state representation (ABR) as a basis thereby treating the electric field exactly; also, the electric field is described in the vector potential gauge. In developing the ABR, the instantaneous eigenstates for the central Hamiltonian are obtained as the accelerated Bloch states and utilized to find the time-dependent solution to the Schr\"odinger equation. In introducing the Wigner-Weisskopf approximation (WWA), the transition probability amplitude between the states of the system is obtained and used in the application to MPA in semiconductors and insulators. The probability amplitude for MPA is derived for a plane-polarized radiation field. The periodicity of the time-dependent electric field serves as a principle time constant in developing the probability amplitude per period. Utilizing the WWA, the general MPA transition probability is derived for transitions between an arbitrary set of valence and conduction bands for a time-varying electric field in the $x$ direction. Within the WWA, the MPA transition probability, equivalent to the well-known Keldysh transition amplitude, is derived and expressed in terms of infinite-variable generalized Bessel functions and modified Bessel functions. The exact MPA result is analyzed in the small ($\ensuremath{\lesssim}{10}^{6}$ V/cm) and large ($\ensuremath{\gtrsim}{10}^{8}$ V/cm) electric field amplitude limits.

The Bloch Electron Response to Electric Fields: Application to Graphene

The theory of Bloch electron dynamics for carriers in homogeneous electric fields of arbitrary time dependence is developed in consideration of the electronic transport properties in graphene. The general approach is to use the accelerated Bloch state representation (ABR) as a basis so that the dependence upon the electric field, including Zener tunneling, is treated exactly; also, the electric field is described in the vector potential gauge. Within the ABR, the instantaneous eigenstates for the central Hamiltonian are described and utilized to develop both the time‐dependent wave functions and the single‐particle density matrix pictures with explicit application to intrinsic graphene. The Bloch electron analysis for graphene in a constant electric field reveals the explicit manifestation of electron–hole pair creation, Bloch oscillations, and Wannier–Stark localization. The average velocity and acceleration are established as an explicit function of the electric field, and their behavior is characterized on short and long time scales. Further, it is shown that the average acceleration in a nonvanishing electric field gives rise to an electric field induced dynamical effective mass tensor in the direction of the field.

Quantum transport and the Wigner distribution function for Bloch electrons in spatially homogeneous electric and magnetic fields

The theory of Bloch electron dynamics for carriers in homogeneous electric and magnetic fields of arbitrary time dependence is developed in the framework of the Liouville equation. The Wigner distribution function (WDF) is determined from the single particle density matrix in the ballistic regime, i.e., collision effects are excluded. The single particle transport equation is established with the electric field described in the vector potential gauge, and the magnetic field is treated in the symmetric gauge. The general approach is to employ the accelerated Bloch state representation (ABR) as a basis so that the dependence upon the electric field, including multiband Zener tunneling, is treated exactly. In the formulation of the WDF, we transform to a new set of variables so that the final WDF is gauge invariant and is expressed explicitly in terms of the position, kinetic momentum, and time. The methodology for developing the WDF is illustrated by deriving the exact WDF equation for free electrons in homogeneous electric and magnetic fields. The methodology is then extended to the case of electrons described by an effective Hamiltonian corresponding to an arbitrary energy band function. In treating the problem of Bloch electrons in a periodic potential, the methodology for deriving the WDF reveals a multiband character due to the inherent nature of the Bloch states. In examining the single-band WDF, it is found that the collisionless WDF equation matches the equivalent Boltzmann transport equation to first order in the magnetic field. These results are necessarily extended to second order in the magnetic field by employing a unitary transformation that diagonalizes the Hamiltonian using the ABR to second order. The work includes a discussion of the multiband WDF transport analysis and the identification of the combined Zener-magnetic field induced tunneling.

Exactly solvable model for nonlinear light-matter interaction in an arbitrary time-dependent field

Exact analytic expressions are derived for the dipole moment and nonlinear current of a one-dimensional quantum particle subject to a short-range attractive potential and an arbitrary time-dependent electric field. An efficient algorithm for the current evaluation is described and a robust implementation suitable for numerical simulations is demonstrated.

Equivalence of sweep-rate and magnetic-viscosity dynamics

The irreversible response of magnetic materials to magnetic fields of arbitrary time dependence is investigated by a master-equation approach. The magnetization reversal is expressed in terms of renormalized magnetization modes, and the resulting set of two-level master equations is solved by direct integration. The theory applies not only to linear energy-barrier laws but also to the physically more reasonable case where the activation energy is a nonlinear function of the applied field. Particular emphasis is on the relation between sweep-rate and magnetic-viscosity dynamics. Other regimes, such as oscillating magnetic fields, can be mapped onto sweep-rate dynamics. Magnetic-viscosity and sweep-rate experiments reflect the same fundamental magnetization processes, but energy barriers probed by dynamic experiments are smaller by about 20%.

Retarded Green's function for an electron in a parabolic quantum dot subject to a constant uniform magnetic field and an electric field of arbitrary time dependence and orientation

Abstract We have examined the retarded Schrodinger Green's function for electron propagation in a parabolically confined quantum dot subject to a constant uniform magnetic field and an electric field having arbitrary time dependence and orientation with respect to the magnetic field. This analysis was carried out using the operator equation-of-motion method of Schwinger. Our result for the Green's function of a quantum dot confined in three dimensions differs from the corresponding path integral propagator given by Jones and Papadopoulos, in that we treat an anisotropic confinement potential with anisotropic mass and include electron spin.

Retarded Green's function for an electron in a parabolic quantum dot subject to a constant uniform magnetic field and an electric field of arbitrary time dependence and

Retarded Green's function for an electron in a parabolic quantum dot subject to a constant uniform magnetic field and an electric field of arbitrary time dependence and

Diffracton of Focus Wave Mades At Plane Apertures

The Kirchhoff-Huyghens diffraction integral is generalized to arbitrary time-dependent fields and used to analyse the diffraction of focus wave modes by circular and rectangular plane apertures. It is shown that their diffraction by a slit is still an open problem. The physical implications of these results are discussed.

Driven two-level atom.

The two-level atom in an external field of arbitrary time dependence is solved formally using projection techniques. As a particular result of the general solution, an exact expression is presented for the density matrix of a two-level atom in a sinusoidal field.

Dipole Shielding Tensor

An exact formula is derived for the dipole shielding tensor of an atom in an arbitrary time dependent field. Conditions are given under which a variational calculation will yield similar results.

The Quantum Theory of Optical Coherence

The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The $n\mathrm{th}$ order function expresses the correlation of values of the fields at $2n$ different points of space and time. Certain values of these functions are measurable by means of $n$-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.