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arXiv:physics/9905019 · Canadian Journal of Physics

The evolution and revival structure of angular momentum quantum wave packets

Marcis P. Auzinsh

Published Jan 1, 1999• 6 citations
arXivHTMLPDF

Abstract

In this paper, a coherent superposition of angular-momentum states created by absorption of polarized light by molecules is analyzed. Attention is paid to the time evolution of wave packets representing the spatial orientation of the internuclear axis of a diatomic molecule. Two examples are considered in detail. Molecules absorbing light in a permanent magnetic field experiencing the Zeeman effect and molecules absorbing light in a permanent electric field experiencing the quadratic Stark effect. In a magnetic field, we have a wave packet that evolves in time exactly as a classical dipole oscillator in a permanent magnetic field (classical-physics picture of the Zeeman effect). In the second case, we have a wave packet that goes through periodical changes of shape of the packet with revivals of the initial shape. This is pure quantum behavior. The classical motion of angular momentum in an electric field in the case of a quadratic Stark effect is known to be a periodic. Solutions obtained for wave packet evolution are briefly compared with Rydberg-state coherent wave packets and harmonic-oscillator wave packets. Zeeman and Stark effects in small molecules continuously attract the attention of researchers, theoreticians, as well as experimentalists. These investigations allow us to obtain a deeper understanding of the interaction of molecules with stationary external fields and also can be used as a practical tool to measure different molecular characteristics, such as permanent electric or magnetic dipole moments, intramolecular perturbations, etc. It is worthwhile analyzing these effects as an evolution of wave packets. All this motivates a comparison of the quantum and classical picture of Zeeman and Stark effects in molecules.PACS No.: 33.55.Be

DOI
10.1139/cjp-77-7-491

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arXiv:quant-ph/9608040•Physical Review A•Jan 1, 1997•36 citations
In this paper, a coherent superposition of angular-momentum states created by absorption of polarized light by molecules is analyzed. Attention is paid to the time evolution of wave packets representing the spatial orientation of the internuclear axis of a diatomic molecule. Two examples are considered in detail. Molecules absorbing light in a permanent magnetic field experiencing the Zeeman effect and molecules absorbing light in a permanent electric field experiencing the quadratic Stark effect. In a magnetic field, we have a wave packet that evolves in time exactly as a classical dipole oscillator in a permanent magnetic field (classical-physics picture of the Zeeman effect). In the second case, we have a wave packet that goes through periodical changes of shape of the packet with revivals of the initial shape. This is pure quantum behavior. The classical motion of angular momentum in an electric field in the case of a quadratic Stark effect is known to be a periodic. Solutions obtained for wave packet evolution are briefly compared with Rydberg-state coherent wave packets and harmonic-oscillator wave packets. Zeeman and Stark effects in small molecules continuously attract the attention of researchers, theoreticians, as well as experimentalists. These investigations allow us to obtain a deeper understanding of the interaction of molecules with stationary external fields and also can be used as a practical tool to measure different molecular characteristics, such as permanent electric or magnetic dipole moments, intramolecular perturbations, etc. It is worthwhile analyzing these effects as an evolution of wave packets. All this motivates a comparison of the quantum and classical picture of Zeeman and Stark effects in molecules.PACS No.: 33.55.Be
High angular momentum Rydberg wave packets are studied. Application of carefully tailored electric fields to low angular momentum, high- n ( n ∼ 300) Rydberg atoms creates coherent superpositions of Stark states with near extreme values of angular momentum, [cursive l]. Wave packet components orbit the parent nucleus at rates that depend on their energy, leading to periods of localization and delocalization as the components come into and go out of phase with each other. Monitoring survival probability signals in the presence of position dependent probing leads to observation of characteristic oscillations based on the composition of the wave packet. The discrete nature of electron energy levels is observed through the measurement of quantum revivals in the wave packet localization signal. Time-domain spectroscopy of these signals allows determination of both the population and phase of individual superposition components. Precise manipulation of wave packets is achieved through further application of pulsed electric fields. Decoherence effects due to background gas collisions and electrical noise are also detailed. Quantized classical trajectory Monte-Carlo simulations are introduced and agree remarkably well with experimental results.
The well-known Zeeman and Stark effects can lead to unusual atomic state dynamics when both magnetic and electric fields are present. In this paper we analyze quantum mechanical and classical models of the time evolution of the angular momentum of an atom in the presence of weak, orthogonal electric and magnetic fields. The anisotropic electric polarizability of the atom plays a crucial role in the dynamics. The classical model leads to nonlinear evolution equations, and we investigate how quantum mechanics ‘‘avoids’’ the nonlinearity. Finally, we note that the classical equations of motion are identical to those used to describe the time evolution of Stokes vectors for polarized light propagating in an optically nonlinear medium. The treatment is appropriate for an undergraduate course in quantum mechanics.
The aim of this paper is to present the interference effects that occur during the time evolution of simple angular wave packets (WP's) which can be associated with a diatomic rigid molecule (heteronuclear) or with a quantum rigid body with axial symmetry like a molecule or a nucleus. The time evolution is understood entirely within the framework of fractional revivals discovered by Averbukh and Perelman (Phys. Lett. A 39, 449 (1989); Usp. Fiz. Nauk 161, 41 (1991) [Sov. Phys. Usp. 37, 572 (1991)]), since the energy spectrum is exactly quadratic. Our objectives are to study how these interference effects differ when there is a change of the initial WP. For this purpose we introduce a two-parameter set of angular-momentum coherent states. On the one hand, this set emerges quite naturally from the three-dimensional coherent states of the harmonic oscillator; on the other hand, this set is shown to be built from intelligent spin states. By varying one parameter $(\ensuremath{\eta}),$ a scenario of interferences occurs on the sphere at fractional parts of the revival time that strongly depend on $\ensuremath{\eta}.$ For $\ensuremath{\eta}=\ifmmode\pm\else\textpm\fi{}1$ the WP, which coincides with a WP found by Mostowski [Phys. Lett. A 56, 369 (1976)], is a superposition of Bloch [Phys. Rev. 70, 460 (1946)] or Radcliffe [J. Phys. A 4, 313 (1971)] states, and clone exactly in time according to a scenario found for the infinite square well in one dimension, and also for a two dimensional rotor. In the context of intelligent spin states it is also natural to study the evolution by changing $\ensuremath{\eta}.$ For $\ensuremath{\eta}=0$ the WP is called linear, and in time produces a set of rings with axial symmetry over the sphere. The WP's for other values of $\ensuremath{\eta}$ are called elliptic, and sets of fractional waves are generated which make a transition between two symmetries. We call these fractional waves ``mutants.'' For specific times a clone is produced that stands among the mutants. Therefore the change in $\ensuremath{\eta}$ produces a change in the quantum spread on the sphere. We have also constructed simple coherent states for a symmetric rotor which are applicable to molecules and nuclei. Their time evolution also shows a cloning mechanism for the rational ratio of moments of inertia. For irrational values of this ratio, the scenario of partial revivals completed by Bluhm, Kostelecky, and Tudose [Phys. Lett. A 222, 220 (1996)] is valid.
We examine the revival structure of Rydberg wave packets. These wave packets exhibit initial classical periodic motion followed by a sequence of collapse, fractional/full revivals, and fractional/full superrevivals. The effects of quantum defects on wave packets in alkali-metal atoms and a squeezed-state description of the initial wave packets are also described. We then examine the revival structure of Rydberg wave packets in the presence of an external electric field, i.e., the revival structure of Stark wave packets. These wave packets have energies that depend on two quantum numbers and exhibit new types of interference behavior.
Optical pumping of the electronic ground state of molecules has been investigated under conditions of plane, elliptically, and circularly polarized light broadband excitation. The analysis has been performed assuming arbitrary angular momentum values, and applying the apparatus of polarization moments, the latter forming the coefficients in the expansion of the density matrix over irreducible tensor operators, including simplifications due to asymptotic limits for large momentum values. In addition to accounting for the external magnetic field, attention has been also given to the dynamical Stark effect arising from the absence of coincidence between the centers of the exciting line and the absorption line. An alternative classical description is also proposed using multiple moments as coefficients of the expansion of a classical quasidensity of states over spherical harmonics. The equivalence of both approaches of description in the limit of infinitely large angular moments is demonstrated, and the meaning of the origin of coherence is clarified, as well as its destruction by the magnetic field in the classical description of the system.
The revival structure of Stark wave packets is considered. These wave packets have energies depending on two quantum numbers and are characterized by two sets of classical periods and revival times. The additional time scales result in revival structures different from those of free Rydberg wave packets. We show that Stark wave packets can exhibit fractional revivals. We also show that these wave packets exhibit particular features unique to the Stark effect. For instance, the wave functions can be separated into distinct sums over even and odd values of the principal quantum number. These even and odd superpositions interfere in different ways, resulting in unexpected periodicities in the interferograms of Stark wave packets.