Recurrence Spectra Research Articles

Absorption and recurrence spectra of hydrogen in crossed electric and magnetic fields

The absorption spectrum of the hydrogen atom in crossed electric and magnetic fields was measured. The magnetic field was set at 6.002 T, and the electric field was fixed at several values from 750 to 1000 V/cm. The observations require several different theoretical methods for their interpretation: (1) The spectrum is unscaled, the recurrences are weak, and their periods vary with energy over the interval of the observations. This observation caused us to develop a chirped Fourier transform method to extract closed orbits from the measured spectra. (2) The absorption spectrum consists of quasidiscrete peaks superposed on a smoothly rising continuum. To interpret this observation, a theoretical model of the continuum absorption is created, and we get results consistent with the measurements. (3) The experiments distinguish between prompt and delayed electrons, corresponding to lifetimes of the excited hydrogen atoms that are, respectively, less than or greater than about 100 ns. (4) At high energies, the measured absorption spectrum contains some regular quasidiscrete states. We use an Einstein-Brillouin-Keller quantization method to identify these as states that lie close to the plane perpendicular to the magnetic field.

Scaled-energy spectroscopy of strontiumM = 0 Rydberg atoms in an electric field

Scaled-energy spectroscopy for strontium M = 0 Rydberg atoms inan electric field is studied. Fourier transformedrecurrence spectra are compared with a hydrogenicclosed-orbit calculation with the uniform approximation. The experimental M = 0 excitation configuration enables the largecore effect on the recurrence spectra to be investigated in detail.Extra peaks induced by ionic core scattering as well as the coreshadowing effect are observed in the experimental recurrencespectra.

DYNAMIC MEMORY MODEL BASED ON THE FERMI-PASTA-ULAM INTERACTING SPECTRA GENERATED BY INTERCONNECTED RING LASERS

A model of a memory system consisting of three neuron networks based on three interacting ring lasers–neurons is suggested. The neurons are considered as carriers of the Fermi–Pasta–Ulam recurrence spectra described within the framework of the modified Zabusky and Kruskal approach. The interaction of the FPU spectra is studied using both fluid and optical models. Physical evaluation shows that such a memory system does not have apparent quantity limits for the information stored.

LONG-RANGE CORRELATIONS IN QUANTUM SPECTRA

Starting From Berry-Tabor trace formula, Einstein-Brillouin-Keller (EBK) quantization condition is adopted to derive action quantization conditions of periodic orbits in two dimensional uncoupled oscillators. The correspondence relations between quantum levels and periodic orbits are found. They indicate that two levels' cotributions to recurrence function coherently interfere with each other if the two levels correspond to periodic orbits with identical topology. The peaks in recurrence spectra results from coherent interference among levels whose corresponding periodic orbits have identical topology. There are long-range correlations among these levels.

SCALED-ENERGY SPECTROSCOPY OF STRONTIUM |M|=1 RYDBERG ATOMS IN AN ELECTRIC FIELD

The strontium atoms in ground states were excited to higher |M|=1 Rydberg states while the laser wavelength and the electric field strength are scanned simultaneously to keep the scaled-energy ε constant. Spectra were recorded for ε=-3.00, -2.50 and -1.88. The experimental Fourier transform recurrence spectra were compared with the hydrogenic closed-orbit calculation. The positions of the experimental recurrence peaks were well coincided with the theory, but the strengths of the peaks were strongly affected by the strontium core effect.

Comment on “Recurrences without closed orbits”

In a recent paper Robicheaux and Shaw [Phys. Rev. A 58, 1043 (1998)] calculate the recurrence spectra of atoms in electric fields with non-vanishing angular momentum <L_z> not equal to 0. Features are observed at scaled actions ``an order of magnitude shorter than for any classical closed orbit of this system.'' We investigate the transition from zero to nonzero angular momentum and demonstrate the existence of short closed orbits with L_z not equal to 0. The real and complex ``ghost'' orbits are created in bifurcations of the ``uphill'' and ``downhill'' orbit along the electric field axis, and can serve to interpret the observed features in the quantum recurrence spectra.

Reply to “Comment on ‘Recurrences without closed orbits’ ”

We discuss some of the difficulties in using the orbits proposed by Main [preceding paper, Phys. Rev. A 60, 1726 (1999)] to calculate the recurrence spectra in Robicheaux and Shaw [Phys. Rev. A 58, 1043 (1998)].

Bifurcations of closed orbits in singlet helium Stark spectra

We measured constant-scaled-energy $M=0$ spectra of helium Rydberg atoms $(n=55$ to 80) in an electric field, allowing for a test of closed-orbit theory under conditions where bifurcations from the uphill and downhill orbits are important. Recent advances in the theory (uniform approximation) were implemented for a comparison with experimental data. The observed recurrence spectra could be explained in detail as far as the actions of all hydrogenic orbits and the intensities of most recurrences are concerned. Additional peaks in the recurrence spectra that correspond to orbits with an action equal to the sum of two hydrogenic orbits were attributed to scattering at the ${\mathrm{He}}^{+}$ core. This was confirmed in a closed-orbit calculation including the quantum defects in helium. Small changes in the recurrence strength of the hydrogenic orbits were found both in experimental recurrence spectra and in the closed-orbit calculation for helium, and could be ascribed to the effect of core shadowing.

General Rydberg Atoms in Crossed Electric and Magnetic Fields: Calculation of Semiclassical Recurrence Spectra with Special Emphasis on Core Effects

The photoabsorption spectra of highly excited Rydberg atoms in external fields are known to be related to closed classical orbits of the excited electron. For atoms in a pure magnetic field, scaled recurrence spectra have successfully been calculated from classical orbits using the semiclassical closed-orbit theory. Rydberg atoms in crossed fields are more complicated as regards their theoretical and numerical treatment, as the Hamiltonian is nonseparable in three degrees of freedom, in contrast to only two nonseparable degrees of freedom in a pure magnetic field. In this paper, we present an extension of the closed-orbit theory to three degrees of freedom, taking into account arbitrary quantum defects. Motivated by nonhydrogenic resonances discovered in experimental scaled recurrence spectra of rubidium atoms, we investigate the influence of the ionic core on the three-dimensional closed orbits, using a simple model potential. We find that the introduction of a core potential results in an extreme increase of the number of closed orbits as compared to hydrogen. The novel orbits appear to be composed of hydrogenic orbits and are created through scattering by the core potential. Investigating the classical deflection function of the model potential, general properties of the new orbits can be explained. With the closed-orbit theory extended to three nonseparable degrees of freedom, we are able to calculate scaled recurrence spectra for Rydberg atoms in crossed fields with arbitrary quantum defects. Our results are in good agreement with the experimental spectra. In particular, the nonhydrogenic resonances can be explained in terms of the new orbits created by classical core scattering.

Recurrence spectroscopy of atoms in electric fields: Failure of classical scaling laws near bifurcations

The photoabsorption spectra of atoms in a static external electric field shows modulations from {ital recurrences}: electron waves that go out from and return to the vicinity of the atomic core. Closed-orbit theory predicts the amplitudes and phases of these modulations in terms of closed classical orbits. A classical scaling law relates the properties of a closed orbit at one energy and field strength to its properties at another energy and field strength at fixed scaled energy {epsilon}=EF{sup {minus}1/2}. The scaling law states that the recurrence strength of orbits along the electric field axis scale as F{sup 1/4}. We show how this law fails near bifurcations when the effective Planck constant {cflx {h_bar}}{equivalent_to}{h_bar}F{sup 1/4} increases with increasing field at fixed {epsilon}. The recurrences of orbits away from the axis scale as F{sup 1/8} in accordance with the classical prediction. These deviations from the classical scaling law are important in interpreting the recurrence spectra of atoms in current experiments. This leads to an extension of the uniform approximation developed by Gao and Delos [Phys. Rev. A {bold 56}, 356 (1997)] to complex momenta. {copyright} {ital 1998} {ital The American Physical Society}

Recurrences without closed orbits

The results of quantum photoabsorption calculations are presented for H, K, and Cs atoms in static electric fields. The recurrence spectra for {l_angle}L{sub z}{r_angle}{ne}0 show features at scaled actions an order of magnitude shorter than for any classical closed orbit of this system. Two interesting manifestations are presented, and some of the systematics of the peak strengths are explored. These features suggest that closed-orbit theory may need to be generalized to account for these effects. A heuristic formula is presented that reproduces some of these effects. {copyright} {ital 1998} {ital The American Physical Society}

Scaled-Energy Spectroscopy of the Rydberg-Stark Spectrum of Helium: Influence of Exchange on Recurrence Spectra

We report the first observation of the manifestation of exchange effects in recurrence spectra. Using fast-atom--laser-beam scaled-energy spectroscopy, we have measured the Rydberg-Stark recurrence spectrum of the two spin forms of helium in strong fields. Singlets and triplets are produced by excitation of metastable $1s2s{}^{1,3}S$ states. We observe pronounced modulations in the triplet spectrum due to interference between hydrogenic and core-scattered combination orbits. The modulations are phase correlated with repeating hydrogenic orbits and not present in singlets or in hydrogen.

Scaled-energy Floquet spectroscopy in a strong electric field: A semiquantal calculation of the recurrence spectrum

We consider a hydrogen atom in a strong static electric field with a weak parallel radio-frequency (rf) field. We compute the photoabsorption spectrum by calculating the spectrum of Floquet states, including their quasienergies and their oscillator strengths. Our calculation is based upon ``semiquantal'' formulas: we calculate the discrete spectrum of quasienergy states by using a quantum adiabatic approximation combined with semiclassical (Bohr-Sommerfeld) quantization rules. We express this spectrum in a manner consistent with the method of scaled-variable spectroscopy, and then calculate the Fourier transform. These calculated absorption spectra and recurrence spectra are in good agreement with experiments on Li atoms. Additional approximations show that the recurrence spectrum is approximately equal to the product of the recurrence spectrum in a static field times an envelope function. That envelope function is the Fourier transform of a cluster of sidebands surrounding a progenitor level in the rf field. The resulting formula agrees with the low-frequency limit of a formula obtained from a semiclassical treatment.

Angle-resolved spin-orbit autoionization dynamics of Rydberg electron wave packets in Ar: A time-dependent MQDT approach

The spin-orbit autoionization dynamics of a Rydberg electron wave packet in Ar have been investigated using time-dependent multichannel quantum defect theory (TD-MQDT). The time-dependent photoionization cross sections reveal interesting interference patterns in the recurrence spectra which can be accounted for in terms of the quantum defects of the autoionizing channels. An expression for the time- and energy-dependent asymmetry parameter β(E,t) has been derived to calculate time-resolved photoelectron angular distributions (PADs) for the first time. An oscillation in the time-resolved PAD is observed, with a period which is inversely proportional to the difference between the quantum defects of the autoionizing channels.

Numerical simulation of electronic wave-packet evolution

We present an accurate and flexible method for the numerical simulation of the evolution of electronic wave packets in alkali atoms. As a testing ground for our approach, we calculate the dynamics of Stark wave packets in cesium for different electric-field strengths. An agreement with recent experimental results is demonstrated, and especially the influence of the electric-field strength and the core scattering on the recurrence spectra is reproduced accurately.

Recurrence Spectroscopy of a Time-Dependent System: A Rydberg Atom in an Oscillating Field

We report the results of an experimental and theoretical investigation of the recurrence spectra of Rydberg atoms in a static plus weak oscillating electric field. Experiments reveal the systematic weakening of orbits in a recurrence spectrum as the oscillating field strength and frequency are changed. We describe a generalization of closed orbit theory to time-dependent systems and show that it provides a qualitative and quantitative description of the phenomena.

High Resolution Quantum Recurrence Spectra: Beyond the Uncertainty Principle

Highly resolved recurrence spectra are obtained by harmonic inversion of quantum spectra of classically chaotic systems and compared in detail to the results of semiclassical periodic orbit and closed orbit theory. Our analysis is sensitive to separate orbits with nearly degenerate recurrence periods and uncovers complex (``ghost'') orbits even when they are hidden behind close-by real orbits. The method is demonstrated on an example of the hydrogen atom in external magnetic and electric fields, for both the density of states and the quantum photoabsorption cross section.

Hydrogen atom in a magnetic field: Ghost orbits, catastrophes, and uniform semiclassical approximations

Applying closed-orbit theory to the recurrence spectra of the hydrogen atom in a magnetic field, one can interpret most, but not all, structures semiclassically in terms of closed classical orbits. In particular, conventional closed-orbit theory fails near bifurcations of orbits where semiclassical amplitudes exhibit unphysical divergences. Here we analyze the role of ghost orbits living in complex phase space. The ghosts can explain resonance structures in the spectra of the hydrogen atom in a magnetic field at positions where no real orbits exist. For three different types of catastrophes, viz. fold, cusp, and butterfly catastrophes, we construct uniform semiclassical approximations and demonstrate that these solutions are completely determined by classical parameters of the real orbits and complex ghosts.

Nonhydrogenic Rydberg atoms in a magnetic field: A rigorous semiclassical approach.

Multielectron atoms in external fields are essentially more complicated than hydrogen with regard to theoretical treatments. Experimental spectra of helium as well as R-matrix quantum-defect calculations revealed discrepancies between the diamagnetic hydrogen atom and general Rydberg atoms. They appeared most transparent as novel resonance structures in constant scaled-energy recurrence spectra of nonhydrogenic atoms at positions where no hydrogenic resonances exist. To reveal the physical origin of these resonances we performed a rigorous semiclassical investigation of nonhydrogenic atoms in magnetic fields. The ionic core is introduced into the Hamiltonian via a short-ranged core potential. For this Hamiltonian we analyze in detail the classical dynamics of closed orbits. Classical core-scattering results in the creation of a huge number of new closed orbits. They appear to be composed of a sequence of slightly different hydrogenic orbits, interconnected by the core-scattering, and can be grouped into families accordingly. With a semiclassical closed-orbit theory generalized to arbitrary quantum defects of the ionic core and with the closed orbits at hand we are able to calculate photoabsorption spectra of nonhydrogenic atoms. Although each of the new orbits has a low amplitude, the interference of all members of a family results in clearly visible resonances in the Fourier transform recurrence spectra, in good agreement with experiment and quantum calculations. The novel structures in nonhydrogenic spectra are now identified and semiclassically interpreted in terms of families of core-scattered classical orbits. \textcopyright{} 1996 The American Physical Society.

The role of the quantum defect and of high-order dispersion in rydberg wave packets

Experimental and theoretical recurrence spectra of Rydberg electron wave packets are presented. Fractional revivals up to the seventh order are observed. In addition we observe a new interference effect: a forerunner wave packet in the first full revival, due to second-order dispersion. Also it is shown that the quantum defect of rubidium shifts the peaks in the interference pattern by the quantum defect modulo one times the classical orbiting period, while it leaves the envelope of the recurrence spectrum unaffected. All these quantum mechanical effects are explained by a Taylor expansion of the phase factors of the stationary states of which the wave packet is composed.