Wave-packet coherence of a harmonic oscillator radiating beyond the electric-dipole approximation: the quadrupole and higher-multipole extension of the 1948 Bauer–Jensen radiation-damping problem

A charged quantum harmonic oscillator coupled to the quantized electromagnetic field is the simplest system in which the emergence of classical radiation damping from quantum dynamics can be tested. In 1948, Hans Bauer and J. Hans D. Jensen proved, in the electric-dipole (long-wavelength) approximation and keeping only the resonant Weisskopf–Wigner (rotating-wave) interaction terms, that the Schrödinger wave packet built from oscillator eigenfunctions with Poisson-distributed weights does not spread under radiation damping: it remains a Gaussian of ground-state width whose centroid follows the classical damped trajectory with the Abraham–Lorentz damping constant, and the oscillator–field wave function factorizes exactly. They stated they could not extend this decoupling to quadrupole and higher multipole radiation and suspected that there the wave packet would spread and the classical correspondence would break down; they also left the common phase factor of the solution uncomputed and the long-time neglect of non-resonant terms unjustified. No published work extends this decoupling to the multipole channels. Modern open-quantum-systems theory exactly solved the non-resonant dipole model and showed that the exact non-spreading property is strictly a rotating-wave property, but its diagonalization and Langevin methods rely on the bilinear form of the dipole coupling and do not apply to the nonlinear multipole coupling. The problem therefore remains open: determine the quantum dynamics of a harmonic oscillator radiating through quadrupole and higher multipole channels — in particular whether a factorized non-spreading solution exists, whether the wave packet spreads, and whether the classical damped correspondence survives — together with a controlled account of the long-time validity of the resonant approximation in the presence of the multipole channels.

Background

A classical charged particle bound in a harmonic potential radiates electromagnetic waves, and the emitted radiation reacts back on the particle as a damping force. For weak damping this is the Abraham–Lorentz friction, which for an oscillator of mass $m_0$, charge $e_0$ and angular frequency $\omega_0$ damps the amplitude exponentially with the classical radiation-damping constant $\gamma = 2e_0^2\omega_0^2/(3m_0c^3)$.

Quantum mechanically, the undamped harmonic oscillator admits the non-spreading wave packet introduced by Schrödinger in 1926: a Gaussian whose width stays at the ground-state value $1/(\alpha\sqrt{2})$ with $\alpha=\sqrt{m_0\omega_0/\hbar}$, while its center follows the classical trajectory. In modern language this is the coherent state, a superposition of energy eigenstates with Poisson-distributed weights (formalized by Glauber in 1963 in quantum optics). Whether such a 'classical' packet survives quantization of the radiation field — spontaneous emission being a stochastic quantum-jump process that could destroy the phase relations between the eigenfunctions — is a sharp version of the quantum-classical correspondence problem for a radiating charge.

Bauer and Jensen (1948) addressed exactly this question, in the tradition of classical electron theory (the paper is dedicated to Richard Becker). Working with the oscillator coupled to the quantized electromagnetic field in the electric-dipole approximation — the long-wavelength limit in which the field is evaluated at the position of the charge, so that the interaction is linear in the oscillator coordinate — and retaining only the resonant, energy-conserving interaction terms (the rotating-wave or Weisskopf–Wigner approximation, introduced by Weisskopf and Wigner in 1930 for natural line widths) — they constructed an exact solution of the coupled Schrödinger equation in which the total wave function factorizes into an oscillator part and a field part: the oscillator remains in a Poisson-weighted packet whose amplitude is damped, $x_0(t)=x_0^0 e^{-\gamma t/2}$; the probability density stays a Gaussian of unchanged ground-state width whose centroid follows the classical damped trajectory with exactly the classical $\gamma$; and the system decays non-ergodically (the oscillator ends in its ground state while the field retains phase relations, without thermal mixing). This proved a thesis attributed to F. Bloch that the packet 'stays together' even when radiation is included.

The result, however, rests on two idealizations that the authors themselves flagged. First, the dipole interaction is linear in the oscillator coordinate, so it connects only adjacent oscillator levels and emits photons at $\omega_0$; the next terms of the multipole expansion of the coupling — the electric quadrupole (E2) and higher multipoles — couple through $x^2$-type and higher operators, connect levels differing by two or more quanta, and radiate at the harmonics $2\omega_0, 3\omega_0, \ldots$; the authors report being unable to achieve the analogous decoupling for these channels and suspect that the packet then spreads and the classical correspondence breaks down. Second, even within the dipole coupling, the non-resonant (counter-rotating) terms were dropped without a rigorous long-time justification, and the common phase factor $e^{i\varphi(t)}$ of the solution was not computed (it cancels from the probability density and was deemed physically unimportant).

In modern language the dipole version of the problem is textbook: the zero-temperature amplitude-damping (linear-loss) channel maps coherent states to coherent states, so the packets of Bauer and Jensen are precisely the pointer states — the states most robust against environment-induced decoherence — of the damped oscillator. The exactly solvable model of a particle bilinearly coupled to a bath of oscillators (Caldeira and Leggett, 1983; Ford, Lewis and O'Connell's quantum Langevin equation of 1988 — an exact Heisenberg-picture equation of motion for the particle in which the bath enters as a fluctuating force) contains the full non-resonant dipole interaction, and Rosenau da Costa, Caldeira, Dutra and Westfahl (2000) diagonalized it with and without the rotating-wave approximation, showing that 'dissipative coherent states' — states that remain pure coherent states while decaying — exist only within the rotating-wave approximation, which is recovered as the weak-dissipation limit. No corresponding treatment exists for the multipole-coupled oscillator: because the beyond-dipole coupling is nonlinear in the oscillator operators, the diagonalization and Langevin methods that solve the dipole problem do not apply, and the authors' suspicion about packet spreading and the breakdown of classical correspondence has remained untested in the literature.

Problem Statement

Determine the quantum dynamics of a charged harmonic oscillator (frequency $\omega_0$, mass $m_0$, charge $e_0$) coupled to the quantized electromagnetic field beyond the electric-dipole approximation — including at least the electric-quadrupole interaction and, ideally, the general multipole expansion of the coupling — and thereby answer, for the multipole radiating oscillator, the questions that Bauer and Jensen (1948) could answer only in the dipole, rotating-wave case:

(a) Does the oscillator's wave packet remain non-spreading? Starting from the coherent (Poisson-weighted) packet of the undamped oscillator, does the reduced oscillator state remain a minimum-width Gaussian of unchanged ground-state width, as in the dipole rotating-wave solution, or does the packet spread — and if it spreads, at what rate and in what manner?

(b) Does the classical correspondence survive? Does the packet centroid follow the classical motion of an oscillator damped by its multipole radiation (including the nonlinear multipole radiation-reaction corrections to the Abraham–Lorentz damping), and under what conditions on coupling strength, amplitude and timescales?

(c) Does a factorizing solution of the coupled oscillator–field Schrödinger equation still exist, decoupling the oscillator motion from the radiation field as in the dipole case, and can the common phase factor $e^{i\varphi(t)}$, left uncomputed in the 1948 solution, be given explicitly?

(d) Provide a controlled account of the approximations: the validity, for arbitrarily long times, of neglecting the non-resonant interaction terms — which the 1948 treatment dropped without rigorous justification — in the presence of the multipole channels.

An acceptable answer may be an exact solution, a controlled approximation with explicit error bounds, or a rigorous proof that the factorized non-spreading property fails; it must address the multipole-coupled dynamics itself rather than restating the dipole rotating-wave case, and must state its multipole truncation and coupling conventions explicitly. All four parts (a)–(d) are required: a result that settles one of them (for example, a rigorous proof that the non-spreading property fails, resolving (a)) does not by itself complete the task.

References

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