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Physica Scripta

Quantum sources in external fields

Wilhelm Becker, M Kleber

Published Oct 30, 2018• 5 citations
Publisher

Abstract

Abstract The Schrödinger equation is homogeneous reflecting the linearity of quantum mechanics. However, we show that adding a source term provides a very convenient and flexible tool for the description of processes where an electron is generated by detachment or by ionization and subsequently propagates in the presence of an external field, which may include the detaching or the ionizing field. The source term provides the mathematical formulation of a separation of the process into two steps, detachment and subsequent propagation. We present the source formalism and illustrate it by various examples: photodetachment in the presence of a uniform electric field, the conductivity of an ideal wire, scanning tunneling microscopy, the velocity map of an electron liberated from a bound state in an atom or ion by an intense laser field and its features such as the carpet pattern, and a novel realization of the time-honored double-slit experiment by means of temporal slits afforded by an atom in a few-cycle laser pulse. All examples compare the theoretical source formulation with experimental data.

DOI
10.1088/1402-4896/aaecae

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Physical Review A•Feb 13, 2018•0 citations
Abstract The Schrödinger equation is homogeneous reflecting the linearity of quantum mechanics. However, we show that adding a source term provides a very convenient and flexible tool for the description of processes where an electron is generated by detachment or by ionization and subsequently propagates in the presence of an external field, which may include the detaching or the ionizing field. The source term provides the mathematical formulation of a separation of the process into two steps, detachment and subsequent propagation. We present the source formalism and illustrate it by various examples: photodetachment in the presence of a uniform electric field, the conductivity of an ideal wire, scanning tunneling microscopy, the velocity map of an electron liberated from a bound state in an atom or ion by an intense laser field and its features such as the carpet pattern, and a novel realization of the time-honored double-slit experiment by means of temporal slits afforded by an atom in a few-cycle laser pulse. All examples compare the theoretical source formulation with experimental data.
Localized scattering phenomena may result in the formation of stationary matter waves originating from a compact region in physical space. Mathematically, such waves are advantageously expressed in terms of quantum sources that are introduced into the Schrödinger equation. The source formalism yields direct access to the scattering wavefunction, particle distribution and total current. As an example, we study emission from three-dimensional Gaussian sources into a homogeneous force field. This model describes the behaviour of an atom laser supplied by an ideal Bose-Einstein condensate under the influence of gravity. We predict a strong dependence of the beam profile on the condensate size and the presence of interference phenomena recently observed in photodetachment experiments.
The Aharonov–Bohm (AB) effect in the photodetachment microscopy of the H− ions in an electric field has been studied on the basis of the semiclassical theory. After the H− ion is irradiated by a laser light, they provide a coherent electron source. When the detached electron is accelerated by a uniform electric field, two trajectories of a detached electron which run from the source to the same point on the detector, will interfere with each other and lead to an interference pattern in the photodetachment microscopy. After the solenoid is electrified beside the H− ion, even though no Lorentz force acts on the electron outside the solenoid, the photodetachment microscopy interference pattern on the detector is changed with the variation in the magnetic flux enclosed by the solenoid. This is caused by the AB effect. Under certain conditions, the interference pattern reaches the macroscopic dimensions and could be observed in a direct AB effect experiment. Our study can provide some predictions for the future experimental study of the AB effect in the photodetachment microscopy of negative ions.
The Schrödinger equation is modified. A source term, or \emph{source}, is added, making it non-homogeneous. A time-L2L^2L2 constraint ∫−∞∞t∣ψ(t)∣2∈R\int_{-\infty}^\infty t |\psi(t)|^2 \in \R∫−∞∞​t∣ψ(t)∣2∈R and an unphysical decay are also added. The result is called the \emph{blue equation} because ``non-homogeneous Schrödinger equation with unphysical decay'' is too long. The source term is a way to set the initial condition of a system, thus a source term can be used to specify a time-dependent quantum state. The ramifications of representing quantum states with sources are explored throughout. Measurement, in the form of probability amplitudes between sources, is used to normalise the sources. Sources are used to simplify scattering theory. The ``in'' and ``out'' states of the Lippmann-Schwinger equation are normalised. The Neumann series is used to do time-dependent perturbation theory. A new kind of SSS-matrix, the red operator, is defined, which works the same way, only on sources. This red operator is used to interpret sources as physical quantities of their own. Applications of sources to quantum field theory are considered. It is shown that a source is dual to a field operator, thus sources can be contracted directly with the field operators in a correlation function. The result is the probability amplitude between sources. In this way, one can avoid the use of the LSZ reduction formula. A path integral is defined and used to demonstrate that the duality between source and field operators exists in the context of path integrals as well. The source reveals a causal nature which was hidden or absent before. There is a causal connection from a source to the generated quantum state. A naïve definition of causality based off of this connection is considered. This definition cannot be used, however, due to special relativity. Epstein and Glaser's definition of causality, as used in causal perturbation theory, is used instead. It is proven that a \emph{time-local} system is causal, according to this definition.
A model which describes the time evolution of strong-field photoionization of atoms is presented. Based on the numerical solution of the non-stationary Schrodinger equation, the model allows one to predict and to interpret the results of experiments on double photoionization of atoms by a combined action of a very short (attosecond) XUV pulse and a few-cycle IR pulse of a powerful laser at various delay times between the two pulses. Depending on the binding energy of the ionized electron, two types of processes are considered. If the electron is tightly bound (Ne case), the XUV pulse ionizes the atom and shakes up another (outer) electron to an excited state, which is subsequently ionized by the strong IR field. For an atom with a weakly bound outer electron (Li case), the IR field ionizes the latter, while the XUV pulse, ionizing the inner shell, terminates (or suppresses) the strong-field ionization. In both cases the yield of doubly ionized ions strongly depends on the delay time between the two pulses, revealing 'steps', oscillations and other features which characterize the time evolution of the ionization process. The presented model describes qualitatively the results of recent experiments on Ne.
An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms.
Four-photon double detachment of the helium negative ion is investigated experimentally and theoretically for photon energies where the transient helium atom is in the 1s2s3S1s2s\phantom{\rule{0.16em}{0ex}}^{3}S1s2s3S or 1s2p3Po1s2p\phantom{\rule{0.16em}{0ex}}^{3}P^{\mathrm{o}}1s2p3Po states, which subsequently ionize by absorption of three photons. Ionization is enhanced by intermediate resonances, giving rise to series of peaks in the He+{\mathrm{He}}^{+}He+ spectrum, which we study in detail. The He+{\mathrm{He}}^{+}He+ yield is measured in the wavelength ranges from 530 to 560 nm and from 685 to 730 nm and for various polarizations of the laser light. Double detachment is treated theoretically as a sequential process, within the framework of RRR-matrix theory for the first step and effective Hamiltonian theory for the second step. Experimental conditions are accurately modeled, and the measured and simulated yields are in good qualitative and, in some cases, quantitative agreement. Resonances in the double detachment spectra can be attributed to well-defined Rydberg states of the transient atom. The double detachment yield exhibits a strong dependence on the laser polarization which can be related to the magnetic quantum number of the intermediate atomic state. We also investigate the possibility of nonsequential double detachment with a two-color experiment but observe no evidence for it.