Atoms In Magnetic Fields Research Articles

Regular and stochastic motion of the electron in the hydrogen atom in a magnetic field

The influence of the term ∝B2 on the electron in the hydrogen atom in a weak uniform magnetic field is investigated. The separation of fast and slow motion is carried out and the analytical solution of the averaged problem is constructed. The equations of fixed trajectories for arbitrary Lz are obtained. The influence of the fast phase term in the Hamiltonian on the stability of these trajectories is discussed. The origin of the stochastic layer is discussed and the formulae for its width is derived.

Hartree-Fock mesh calculations of the energy levels of the helium atom in magnetic fields

The mesh method of solving the Hartree-Fock equations for atoms in magnetic fields of arbitrary strength is presented. The method is applied for the ground and a series of excited states of the He atom. The energy levels of the ion He+ in the states with n=1 and 2 are also calculated for the same strengths of the magnetic field as for the He atom. The use of the mesh method results in reaching the Hartree-Fock energy limit for fields from weak to strong, including the intermediate field region.

Recurrence spectroscopy: Observation and interpretation of large-scale structure in the absorption spectra of atoms in magnetic fields.

Measurements were made of the absorption spectrum of hydrogen atoms to levels near the ionization threshold in a strong magnetic field. Taking advantage of a classical scaling law, we varied the photon energy and the magnetic-field strength simultaneously, and measured absorption versus ${\mathit{B}}^{\mathrm{\ensuremath{-}}1/3}$ at fixed scaled energy, \ensuremath{\varepsilon}=E/(B/${\mathit{B}}_{\mathrm{o}}$${)}^{2/3}$. The absorption rate has sinusoidal fluctuations which are correlated with closed classical orbits of the electron. Fourier transformation of this signal gives peaks, which we interpret as ``recurrence strength,'' as a function of the classical action of the closed orbit. Closed-orbit theory gives formulas for these recurrence strengths. We find that the formulas are in good agreement with the measurements. As the scaled energy is increased, observed recurrences proliferate, consistent with a change from orderly to chaotic motion of the electron. Bifurcation theory provides organizing principles for understanding this proliferation and for interpreting the data. New ``exotic'' orbits suddenly appear through saddle-node bifurcations. The ``main sequence'' of orbits is produced from an orbit parallel to B through a sequence of pitchfork and period-doubling bifurcations. Other recurrences are created by period-tripling and higher-order bifurcations of existing orbits. These bifurcations can have ``generic'' structure, or sometimes the structures are modified by symmetries of the system. Focusing effects associated with these bifurcations cause some recurrences to be particularly strong.

Phase-space distributions and spectral properties for non-hydrogenic atoms in magnetic fields

The authors present the first study of the quantum phase-space behaviour (Wigner functions) for non-hydrogenic atoms in magnetic fields as well as a comprehensive study of spectral properties. They consider primarily an energy regime (scaled energy -0.5) where hydrogen is near-integrable and hydrogenic wavefunctions would be localized on tori. They find that the quantum energy level statistics for non-hydrogenic atoms are at the 'chaotic' (Wigner) limit. However, the quantum phase space distributions, contrary to what one would expect if the underlying classical motion were chaotic, remain dominated by torus-like structures. But the wavefunctions do explore a larger fraction of phase-space than in the hydrogenic case where, in the integrable regime, Wigner wavefunctions are generally localized on a single torus. Due to the non-semiclassical nature of the core they are localized on more than one torus; additional structures other than the tori are also present. Possible interpretations of the results in terms of models of the underlying classical dynamics are discussed.

The diamagnetic Kepler problem

It has turned out that the analysis ofclassical periodic orbits is the key to understanding the modulations in thequantal spectra of hydrogen Rydberg atoms in magnetic fields. Inspired by this quantum chaological connection, we analyse some fundamental periodic orbits of the diamagnetic Kepler problem, abandoning the condition of vanishing azimuthal angular momentuml z used in the literature so far. We report the bifurcation and confluence schemes of the orbits in their dependence onl z and discuss the structural changes in terms of catastrophe theory.

Dynamic Stark effect action on optical pumping of atoms in an external magnetic field

The influence of the dynamic Stark effect on the optical pumping of atoms in a magnetic field, using the broad band approximation, is examined. It is demonstrated that the dynamic Stark effect can lead to a nonlinear effect on the light intensity conversion of alignment produced by linearly polarized light in the orientation of the angular momentum of atoms.

Regularity and irregularity in the centre of mass motion of the positronium atom in a magnetic field

The author investigates the classical centre of mass motion of the positronium atom in a homogeneous magnetic field. For the case of vanishing pseudomomentum and angular momentum component along the magnetic field axis the centre of mass moves on a straight line. Its behaviour changes drastically from, for example, quasiperiodic oscillations to random fluctuations if the internal motion changes from regularity to chaos. The author shows that the one-dimensional spreading of an ensemble of trajectories obeys a linear diffusion law. Some particularly interesting examples of the centre of mass motion for Lz not=0 are also discussed.

From regularity to chaos: a direct experimental test using atoms in magnetic field

From regularity to chaos: a direct experimental test using atoms in magnetic field

Hamiltonian bifurcation theory of closed orbits in the diamagnetic Kepler problem.

Classically chaotic systems possess a proliferation of periodic orbits. This phenomenon was observed in a quantum system through measurements of the absorption spectrum of a hydrogen atom in a magnetic field. This paper gives a theoretical interpretation of the bifurcations of periodic or closed orbits of electrons in atoms in magnetic fields. We ask how new periodic orbits can be created out of existing ones or ``out of nowhere'' as the energy changes. Hamiltonian bifurcation theory provides the answer: it asserts the existence of just five typical types of bifurcation in conservative systems with two degrees of freedom. We show an example of each type. Every case we have examined falls into one of the patterns described by the theory.

Oscillator approach to the problem of the hydrogen atom in a magnetic field and Feneuille scaling

An attempt has been made to derive an expression for the energy for the hydrogen atom in a uniform magnetic field having a dependence on only one variable as conjectured by Feneuille (1982). The Schrodinger equation is transformed by a complex Kustaanheimo-Stiefel transformation to that for a four-dimensional oscillator with a constraint. The equivalent oscillator problem is solved approximately to obtain an expression for the quantity n2E (n being the principal quantum number, E the energy) as a function of a single variable beta c=n3wc (wc being the cyclotron frequency). Agreement with experimental observations is discussed.

Density-functional theory of atoms in strong magnetic fields.

A density-functional model of atoms in strong magnetic fields is proposed along lines parallel to Hohenberg-Kohn-Sham density-functional theory. We derive a Thomas-Fermi-Weiz\acker-like (TFW) functional by using the classical-path approximation to the relevant one-particle Green's function. We also address the correlation effect between the effective potential and the magnetic field. The TFW-like model provides a simple way of calculating ground-state energies and charge densities of atoms in magnetic fields, and can be systematically improved.

Lyapunov exponents for classical orbits of the hydrogen atom in a magnetic field.

Lyapunov characteristic exponents are calculated for classical trajectories of the Hamiltonian describing a hydrogen atom in a uniform magnetic field, and particular attention is given to periodic orbits. As the magnetic field is turned on, instability grows around the almost circular orbit which is a precise circle in the integrable limit \ensuremath{\varepsilon}=-\ensuremath{\infty}, \ensuremath{\varepsilon} being the scaled energy of the system. The Lyapunov exponent of the almost circular orbit is proportional to \ensuremath{\Vert}\ensuremath{\varepsilon}${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}3/2}$ near the integrable limit, and this is consistent with a square-root law found by G. Benettin [Physica D 13, 211 (1984)] for the onset of instability in certain billiards.

The hydrogen atom in a magnetic field of intermediate strength

The grid solution of the Schrodinger equation allows one to obtain highly accurate values of the binding energy EB of low-lying states of the hydrogen atom in a homogeneous magnetic field B of arbitrary strength. The relative precision of EB in the grid calculations presented was 10-3-10-6 for all the states and fields that appeared in the work by Rosner et al. For some states the precision reaches 10-7-10-8. The results of the calculations allows the author to fill all the gaps in the results of the work mentioned above for B>or=4.7*102 T and to improve the precision of a number of results in the intermediate field region. The computational method is described. The form of the results, set of states and field strengths are as in Rosner et al.

Hydrogenic Rydberg atoms in strong magnetic fields: Theoretical and experimental spectra in the transition region from regularity to irregularity

For deuterium Rydberg atoms in a magnetic field of ∼6 T we compare the complete experimental spectrum in the range −190 cm−1 to −20 cm−1 with the positions and oscillator strengths of the corresponding quantum theoretically calculated photoabsorption lines. The agreement is excellent. The range of energy covered extends from the end of thel-mixing regime up to the regions where the approximate integrability of the problem is completely lost, and the corresponding classical system undergoes a transition to chaos.

Atoms in strong magnetic fields

We review recent results of investigations of hydrogen-like systems at magnetic field strengths where the Lorentz forces are comparable to, or larger than, the Coulomb binding forces. This situation is realized for low-lying states at field strengths typical of magnetic white dwarfs and neutron stars, while for Rydberg states already laboratory field strengths are sufficient. We discuss the wavelength spectrum of the hydrogen atom in magnetic fields of arbitrary strength, and describe in which way the spectroscopy of "stationary lines", which appear in this spectrum, has made possible the detection of the largest magnetic field strength ever found in a white dwarf star to date. For Rydberg states in strong laboratory fields we perform a quantitative comparison between experimental and theoretical spectra, and demonstrate that symptoms of "quantum stochasticity" are recovered in the spectra of magnetized Rydberg atoms. In particular we point out that the breakdown of quasi-separability in the quantal problem is closely related to the disappearance of regular orbits in the classical problem. We conclude that magnetized Rydberg atoms lend themselves as ideal objects in which to study, theoretically and experimentally, manifestations of quantum stochasticity.

Effect of closed classical orbits on quantum spectra: Ionization of atoms in a magnetic field.

Measurements of the absorption spectrum near the ionization threshold for an atom in a strong magnetic field showed that the spectrum is a superposition of many oscillatory terms (``quasi-Landau oscillations''). We have developed a quantitative theory which shows that each classical closed electron orbit which begins and ends near the nucleus contributes an oscillatory term to the average oscillator strength. The theory gives new understanding of the behavior under combined Coulomb and Lorentz forces, and it elucidates the roles of isolated closed orbits in chaotic systems. The first results of this theory are shown to be in good agreement with experimental results.

Rigorous analytical lower bound on the ground-state energies of hydrogenic atoms in high magnetic fields

By employing the virial theorem and Hellmann-Feynman relation in combination with the pertinent uncertainty-principle inequality, one can derive a rigorous lower bound on the ground- state energies of hydrogenic atoms in magnetic fields of arbitrarily large intensity, ${E}_{0}$>\ensuremath{\lambda}(3t-${t}^{\mathrm{\ensuremath{-}}1}$-2.00232) Ry, where the magnetic field enters through the parameter \ensuremath{\lambda}\ensuremath{\equiv}B/(4.701\ifmmode\times\else\texttimes\fi{}${10}^{9}$ G) and t is defined implicitly by t(1-${t}^{2}$${)}^{\mathrm{\ensuremath{-}}2}$=\ensuremath{\lambda}${Z}^{\mathrm{\ensuremath{-}}2}$ for atomic number Z. With straightforward modifications, the method presented here can also be employed to derive rigorous analytical lower bounds on the ground-state energies of other parameter-dependent nonseparable quantum systems.

Zeeman diamagnetism in hydrogen at arbitrary field strengths

The non-relativistic energy spectrum of the ground state of hydrogen-like atoms in magnetic fields of arbitrary strength is investigated by using variational functions. We present four- and five-parameter functions which produce energies almost as good as the best ones available in the literature.

"Circular" states of Rydberg atoms in strong magnetic fields.

Using the numerical techniques that have been successful in elucidating the low-lying part of the level scheme of atoms in magnetic fields pertaining to white dwarf stars and neutron stars, we have computed the energies and wave functions of ``circular'' Rydberg states (i.e., with \ensuremath{\Vert}m\ensuremath{\Vert}=n-1) with \ensuremath{\Vert}m\ensuremath{\Vert}=24--35 for laboratory fields up to several tens of teslas. We can, therefore, follow a number of physical properties, such as transition frequencies and radiative decay times, of circular states as functions of the magnetic field strength and make predictions that should be verifiable experimentally.

Hydrogen atom in a strong magnetic field: calculation of the energy levels by quantising the normal form of the regularised Kepler Hamiltonian

The energy levels of the hydrogen atom in a magnetic field are obtained by quantising the Birkhoff-Gustavson normal form of the diamagnetically perturbed, regularised Kepler Hamiltonian. The method enables the authors to go beyond the approximations quadratic in the field and to calculate quartic and higher terms in the power expansion for the energy levels. (Only the case m=0 is considered). The result is equivalent to finding the field-dependent third integral of motion.