# Rydberg Atoms/Quantum Defect Theory

#### The Hydrogen Atom[edit | edit source]

Rydberg atoms are excited states of atoms with a large principle quantum number, where the Rydberg electron is only weakly bound to the ionic core. This weak binding makes Rydberg atoms very sensitive to external perturbations and results in a wide range of unique features. To understand the basic properties of Rydberg atoms, it is instructive to first look at the solution of the hydrogen atom. In atomic units, the Hamiltonian is given by

where position and momentum operators are canonically conjugated and is the reduced math according to the proton mass . To solve the hydrogen problem, we first introduce the angular momentum operators

and its square . Then, we can write the Hamiltonian as

The angular momentum part can be solved independently from the radial part, using the angular momentum algebra. The eigenvalues of are given by , with being a non-negative integer. Each eigenvalue of is -fold degenerate. This degenerate manifold can be expressed in terms of eigenvectors of , having integer eigenvalues with .

The radial part can be solved by expressing the radial momentum term in its coordinate representation, resulting in

The resulting Schrödinger equation can then be solved using standard techniques ^{[1]}. We then obtain the solution

where is the principal quantum number and is the reduced Bohr radius. The associated Laguerre polynomials are normalized according to

The eigenvalues do not depend on , a fact that follows from the conservation of the Runge-Lenz vector, and are given by

- .

Rydberg states are atomic excitation with a large principal quantum number . We can call being large if the properties of the Rydberg state are drastically different from the ground state. For practical purposes, this usually means . For example, consider the expectation value of the radius ^{[2]},

Already at relatively modest values of , the spatial extension of the Rydberg state is already orders of magnitude larger than that of the ground state, see Fig. 1. Such an enhanced scaling with the principal quantum number is characteristic for Rydberg states and leads to strongly exaggerated properties of Rydberg atoms. We will see many examples of such a scaling in the following.

#### Quantum defect[edit | edit source]

^{[3]}
| |
---|---|

0 | 3.13 |

1 | 2.64 |

2 | 1.35 |

3 | 0.016 |

>3 | 0 |

Of course, we are not only interested in the properties of hydrogen atoms. However, for Rydberg states with a single excited electron, the eigenenergies can be well described by a simple phenomenological extension of the expression for hydrogen, as was first noted by Rydberg himself ^{[4]}. This can be done by introducting a *quantum defect* that depends (in leading order) only on the angular momentum quantum number, yielding

The quantum defect accounts for the corrections to the Coulomb potential by the core electrons. We can introduce an effective quantum number . Already on the basis of the hydrogen wave functions we see that the probability to find the Rydberg electron within the core decreases with , therefore the quantum defect should also decrease with . Using laser spectroscopy, the eigenenergies of atoms can be measured very accurately, and as an example, the quantum defect for rubidium is shown in Tab. 1. For , the eigenstates are essentially hydrogenic. Experimentally, the quantum defect can be measured with much higher accuracy up to a relative uncertainty of ^{[5]}, but requires the treatment of the electron spin and introducing an -dependence of the quantum defect.

However, it is evident that the Hamiltonian has to be modified to yield the desired eigenvalues. The easiest way to achieve this is to add an additional term to the radial part such that

where the effective quantum number is given by

with being an integer ^{[6]}. The radial eigenfunction can then be expressed using an extension of the associated Laguerre polynomials for non-integer ,

Then, we obtain for the radial eigenfunctions

Note the different normalization factor compared to the hydrogen case due to the different definition of the associated Laguerre polynomials. The value of the integer can be fixed by an appropriate choice of the number of nodes, , e.g., the ground state should not have any nodes. Alternatively, it is possible to improve the wave functions by choosing such that transition matrix elements to match experimentally observed values ^{[6]}.

#### Polarizability of Rydberg states[edit | edit source]

One important property of quantum defect theory is that the scaling relations we already saw in the hydrogen case remain valid, with the principal quantum number being replaced by . The key difference is that the degeneracy found in the hydrogen atom is lifted. This means that we no longer have atoms with a permanent electric dipole moment (linear Stark effect) ^{[7]}, but induced dipole moments (quadratic Stark effect). The strength of the quadractic Stark shift is captured in the polarizability , according to

where is the energy shift from the Stark effect and is the applied electric field. Within the electic dipole approximation, the perturbation from the electric field is given by the Hamiltonian , where is the electric dipole operator. In second order perturbation theory, we obtain for the Stark shift

Using this expression, we can identify the polarizability as

Note that while the polarizability is always positive for the ground state (since ), the polarizability of Rydberg states can actually be negative.

We are now interested in the scaling behavior of the polarizability with the effective quantum number . For this, we first assume that the main contribution to the polarizability comes from the coupling to a single state, namely the one which is closest in energy for any dipole-allowed transition (i.e., ). The matrix element is proportional to a length and therefore has the same scaling as the expectation value for the orbital radius, . The scaling for the energy difference in the denominator can be calculated as

If we combine these two scaling relations, we obtain for the polarizability an asymptotic behavior according to . This dramatic scaling with the seventh power of the principal quantum number makes Rydberg atoms very sensitive to external electric fields and therefore very good candidates for the realization of electric field sensors.

If one wants to go beyond simple scaling relations, it is often appropriate to use semiclassical approximations for the dipole matrix element ^{[8]}. This allows to express the dipole matrix elements in an analytical form,

where is the Anger function ^{[9]} and are integrals of spherical harmonics representing the angular part ^{[10]}. For example, this semiclassical expression predicts for the dipole moment between the 43s and the 43p state in rubidium a value of , whereas the exact value is ^{[3]}, i.e., less than 3% discrepancy.

#### References[edit | edit source]

- ↑ Merzbacher, Eugen (December 1997).
*Quantum Mechanics*(3 edition ed.). New York: Wiley. ISBN 9780471887027.CS1 maint: extra text (link) - ↑ Bethe, Hans A.; Salpeter, Edwin E. (1957).
*Quantum Mechanics of One- And Two-Electron Atoms*. Springer. ISBN 9781614276227. - ↑
^{3.0}^{3.1}Löw, Robert; Weimer, Hendrik; Nipper, Johannes; Balewski, Jonathan B.; Butscher, Björn; Büchler, Hans Peter; Pfau, Tilman (2012-06-14). "An experimental and theoretical guide to strongly interacting Rydberg gases".*Journal of Physics B: Atomic, Molecular and Optical Physics***45**(11): 113001. doi:10.1088/0953-4075/45/11/113001. ISSN 0953-4075. http://iopscience.iop.org/0953-4075/45/11/113001. Retrieved 2015-03-25. - ↑ Rydberg, J. R. (1890-04-01). "XXXIV. On the structure of the line-spectra of the chemical elements".
*Philos. Mag. Ser. 5***29**(179): 331–337. doi:10.1080/14786449008619945. ISSN 1941-5982. http://dx.doi.org/10.1080/14786449008619945. Retrieved 2015-03-25. - ↑ Mack, Markus; Karlewski, Florian; Hattermann, Helge; Höckh, Simone; Jessen, Florian; Cano, Daniel; Fortágh, József (2011-05-23). "Measurement of absolute transition frequencies of $^{87}\mathrm{Rb}$ to $\mathit{nS}$ and $\mathit{nD}$ Rydberg states by means of electromagnetically induced transparency".
*Physical Review A***83**(5): 052515. doi:10.1103/PhysRevA.83.052515. http://link.aps.org/doi/10.1103/PhysRevA.83.052515. Retrieved 2015-03-25. - ↑
^{6.0}^{6.1}Kostelecký, V. Alan; Nieto, Michael Martin (1985-12-01). "Analytical wave functions for atomic quantum-defect theory".*Physical Review A***32**(6): 3243–3246. doi:10.1103/PhysRevA.32.3243. http://link.aps.org/doi/10.1103/PhysRevA.32.3243. Retrieved 2015-03-25. - ↑ In reality, hydrogen atoms also do not have a permanent electric dipole moment because the interaction with the vacuum of the radiation field results in the Lamb shift that also lifts the degeneracy.
- ↑ Kaulakys, B (1995-12-14). "Consistent analytical approach for the quasi-classical radial dipole matrix elements".
*Journal of Physics B: Atomic, Molecular and Optical Physics***28**(23): 4963–4971. doi:10.1088/0953-4075/28/23/008. ISSN 1361-6455 0953-4075, 1361-6455. http://stacks.iop.org/0953-4075/28/i=23/a=008?key=crossref.eff9f25dcefdd5ba5a146f72d5e0e030. Retrieved 2015-04-21. - ↑ Milton Abramowitz; Irene A. Stegun, eds. (1972).
*Hadbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. New York, NY: Dover. ISBN 9780486612720. - ↑ Arfken, George B.; Weber, Hans J. (1995-10-04).
*Mathematical Methods for Physicists, Fourth Edition*(4 edition ed.). San Diego: Academic Press. ISBN 9780120598151.CS1 maint: extra text (link)