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arXiv:quant-ph/0305171 · Physical Review A

Elliptic Rydberg states as direction indicators

Netanel H. Lindner, Asher Peres, Daniel R. Terno

Published Oct 9, 2003• 20 citations
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Abstract

The orientation in space of a Cartesian coordinate system can be indicated by the two vectorial constants of motion of a classical Keplerian orbit: the angular momentum and the Laplace-Runge-Lenz vector. In quantum mechanics, the states of a hydrogen atom that mimic classical elliptic orbits are the coherent states of the SO(4) rotation group. It is known how to produce these states experimentally. They have minimal dispersions of the two conserved vectors and can be used as direction indicators. We compare the fidelity of this transmission method with that of the idealized optimal method.

DOI
10.1103/physreva.68.042308

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The orientation in space of a Cartesian coordinate system can be indicated by the two vectorial constants of motion of a classical Keplerian orbit: the angular momentum and the Laplace-Runge-Lenz vector. In quantum mechanics, the states of a hydrogen atom that mimic classical elliptic orbits are the coherent states of the SO(4) rotation group. It is known how to produce these states experimentally. They have minimal dispersions of the two conserved vectors and can be used as direction indicators. We compare the fidelity of this transmission method with that of the idealized optimal method.
We show how to build atomic states that mimic the classical Bohr-Sommerfeld elliptic orbits with minimum quantum fluctuations. These elliptic states are uniquely defined from symmetry considerations. They are the coherent states of the SO(4) symmetry group of the Coulomb interaction in three dimensions and are superpositions of the usual spherical states with well-defined weights and phases. They can be experimentally produced from laser excitation of Rydberg atoms in crossed electric and magnetic fields. We finally indicate how to build Coulomb wave packets localized both in space and time.
In the problem of inverse-square force, there exists a hidden SO(4) symmetry, which is manifested as a conserved quantity the Laplace-Runge-Lenz vector in the classical Kepler problem, and as the degeneracy of hydrogen atom energy levels beyond the three-dimensional rotational symmetry in quantum mechanics. This paper presents a more intuitive representation of this hidden symmetry through geometric imagery. Additionally, the paper derives in detail the relevant formulas for the raising and lowering operators corresponding to this symmetry in quantum mechanics.
The quantum theory of atoms is based on a Hamiltonian which is just that of the classical Kepler theory of planetary orbits. In spite of this common foundation, the predictions of quantum theory ordinarily do not much resemble those of classical theory. Even in the limit of large quantum numbers the energy eigenstates of quantum theory do not look at all like the classical elliptical orbits. Recently, Parker and Stroud1 have shown that it is possible to excite a Rydberg atom wave packet whose radial dependence resembles the classical ideal of a particle traveling in an elliptical orbit. In this paper we discuss the formation of a wave packet in the polar and azimuthal angles by the coherent excitation of Rydberg states with the same principal quantum, but with a distribution of the angular momentum quantum numbers. When this wave packet is formed of the Rydberg states of an alkali atom the connection with classical theory is again striking. The wave packet precesses due to the core polarization, just as the classical elliptical orbit precesses due to an added r–4 perturbing potential.
The usual explicit expressions for spherical harmonics are derived by a method which does not involve the solution of differential equations or the use of angular momentum operators in spherical polar form. Instead a raising operator constructed from an arbitrary vector is used together with the usual angular momentum raising and lowering operators. The method is first presented as simply as possible in Schrödinger representation using position as the vector. Then it is generalized to an arbitrary vector and arbitrary integral angular momentum without using a particular representation. Finally it is shown how a conserved vector (other than angular momentum) of a central system can be used to reveal properties of the energy eigenstates. As an example, eigenstates of the hydrogen atom are found using the Runge-Lenz vector.