Atoms in External Fields: From Perturbation Theory to Rydberg Ionization and Light Interaction

Overview

This lecture examines atomic responses to external electric fields, starting with weak DC fields and the quadratic Stark shift derived from perturbation theory, and moving to strong fields where perturbation theory breaks down and field ionization occurs. The discussion highlights Rudberg (Rydberg) atoms, quantum defects, and the sensitivity of highly excited states to electric fields. The instructor also demonstrates practical aspects of state-selective field ionization used in detection, and sets the stage for extending the discussion to AC fields and light interaction.

Key Takeaways

• Perturbation theory explains DC polarizability and quadratic Stark shifts for weak fields
• In excited states perturbation theory can fail due to small energy denominators and near degenerate manifolds
• Rydberg atoms exhibit strong field sensitivity, enabling field ionization and precise detection
• AC Stark shift introduces frequency dependent polarizability and connects to light-matom interactions

Introduction to Stark Effects and Perturbation Theory

The lecturer reviews how external static (DC) electric fields polarize atoms, deriving the polarizability α from second-order perturbation theory and showing energy level shifts that scale quadratically with the field. A central point is the interpretation of this shift as an internal energy cost to mix states and create a dipole, akin to the energy stored in a stretched spring. The discussion emphasizes the regime of validity for perturbation theory and the importance of the energy denominator in the perturbative expansion, pointing out that only states of opposite parity can mix with an electric field due to selection rules.

Limitations and Scaling for Highly Excited States

For a single electron (Hydrogen-like) atom, the excitation energy to the first excited state is around one Rydberg. The matrix element for dipole transitions scales as n^2 for highly excited states, while the energy spacing scales as n^-3, leading to a dramatic growth of the critical field with principal quantum number n. When quantum defects are included for real atoms, the energy differences between states with different angular momentum scale roughly as n^-3, causing the required field to reach perturbation theory breakdown to become far smaller for excited states. This explains why highly excited states, or Rydberg atoms, are extremely sensitive to even modest fields and why perturbation theory can fail at comparatively low fields for these states.

Field Ionization and the Saddle-Point Model

At sufficiently strong fields the external potential lowers the barrier keeping the electron bound. A one-dimensional saddle-point analysis of the Coulomb potential plus the external field yields a simple criterion for stability, predicting a classical threshold field E_crit that scales as 1/n^4 in atomic units. This leads to a widely quoted relation E_crit ~ 1/(16 n^4) in atomic units, which maps to extremely large laboratory fields for light atoms, but becomes accessible for Rydberg states. The breakdown of stability is described as a tunneling process in a quantum treatment, with corrections that refine the simple classical threshold. Hydrogen and alkali atoms differ due to degeneracies and quantum defects, with hydrogen showing more complex behavior because of exact degeneracies and parabolic coordinates that allow some states to remain stable above the classical barrier.

Spectroscopy and Field Ionization Experiments

Experiments in lithium and hydrogen demonstrate the spectra of Stark-mixed manifolds as the field is scanned. In the Rudberg regime, states are distributed across manifolds with strong Stark mixing; at still higher fields, ionization signals appear as the states dissolve into the continuum. The historical context includes pioneering spectroscopic studies by Kleppner and Haroche and the demonstration of state-selective field ionization in the context of quantum optics and cavity QED experiments with highly excited states.

AC Stark Effect and Time-Dependent Perturbation Theory

The transition to AC fields introduces a time-dependent dipole interaction with the field. The speaker derives the dipole moment induced by a driven field, showing that the response is governed by a frequency dependent polarizability. The result features two contributions, one associated with near resonance (co-rotating term) and another with the counter-rotating term (Bloch-Siegert shift). In the rotating-wave approximation the near-resonant term dominates, but both contributions are needed for the exact DC limit. The discussion connects the semi-classical picture to a fully quantum one by noting that the AC Stark shift can also be obtained by quantizing the field and treating the light as photons in coherent states, yielding the same results as time-dependent perturbation theory.

Oscillator Strengths, Test-Atom Picture, and Classical Analogy

To gain intuition, the lecturer introduces the oscillator strength formalism, where the atom behaves like an ensemble of classical oscillators with the oscillator strength summing to unity (Thomas-Reiche-Kuhn sum rule). The D-line of alkali atoms is highlighted as a near-unit oscillator strength, making the polarizability essentially determined by the resonant transition frequency and oscillator strength. The classical harmonic oscillator model is shown to reproduce the same frequency structure as the quantum atom, providing a powerful correspondence that underpins absorption imaging and dispersive imaging with light.

Outlook and Next Topics

The session closes by linking perturbation theory results to practical atomic manipulation, detection, and imaging with light, and by foreshadowing the more complete quantum electrodynamics treatment of atoms in light, including spontaneous emission and multi-mode field quantization, to be developed in subsequent lectures.