Super Radiance in Multi-Atom Systems: Dicke States, Symmetry, and Extended Samples

In this MIT OpenCourseWare lecture, the phenomenon of super radiance is developed from a single excited atom to two atoms, revealing how symmetry and collective coupling to the electromagnetic field enhance emission. The instructor shows that with two atoms, the initial emission rate matches the single-atom case but decays faster, and the ground-state projection of the two-atom system corresponds to a half-photon per atom. The talk then generalizes to N atoms, introducing symmetric and antisymmetric Dicke states and an angular-momentum framework that explains why certain collective states radiate much more strongly than isolated atoms. The class ties the two-atom picture to extended samples, laser interactions, and real molecular and atomic experiments.

Overview of Superradiance

The lecture introduces super radiance as a collective radiative phenomenon where many atoms radiate coherently into the same electromagnetic field mode. The key idea is that when atoms are localized within a wavelength, their coupling to the field is not independent; instead, the photons interfere constructively, yielding an emission that can be enhanced by the number of atoms involved. The speaker emphasizes that the phenomenon is primarily about coherent photons rather than coherence among the atoms alone, and draws connections to lasers in the sense of a single common radiation mode, or a laser without mirrors in certain perspectives.

Two-Atom Case and Symmetry

Starting with two atoms, one excited and one in the ground state, the initial emission rate is the same as for a single atom, yet the decay is faster. The probability that the second atom is found in the ground state asymptotically approaches one half, illustrating the profound way in which correlation changes the emission dynamics. The correct quantum mechanical description uses a symmetrized and an antisymmetrized two-atom wave function. The symmetric combination couples to the radiation field with an enhanced coupling, while the antisymmetric combination (the subradiant state) does not couple and becomes a dark state. The coupling enhancement for the symmetric state scales as the square root of the number of atoms in the manifold, producing an accelerated emission process that is the hallmark of super radiance.

Angular Momentum Formalism and Dicke States

The discussion then moves to treating the two-atom system in terms of total spin S and its projection M, identifying the four two-atom basis states with the Dicke states: the symmetric triplet with S=1 and M=1, 0, and -1, and the antisymmetric singlet with S=0, M=0. The interaction Hamiltonian, expressed as a sum of spin raising and lowering operators, acts within the manifold of fixed total spin S. This framework shows that only the symmetric states participate in the emission process, and the single-excitation Dicke state radiates with an enhanced rate proportional to the number of atoms, culminating in a dramatic N-fold increase in the spontaneous emission rate for the last photon in the cascade.

N-Atom Generalization

Extending to N atoms, each atom is treated as a pseudospin-1/2, and the total spin S can range from N/2 down to 0 or 1/2 depending on N. The maximum spin state, where all atoms are excited, forms a fully symmetric ladder with many closely spaced levels. The emission rate from the highest excited state scales in a way that, for spontaneous emission, leads to an overall intensity enhancement proportional to N. Conversely, the lowest excited state in the symmetric manifold still exhibits collective behavior, with the single excitation radiating faster due to perfect symmetry. The discussion introduces the Dicke states as a natural labeling for these collective excitations and emphasizes that the enhancement arises from the indistinguishability of the emission pathway and the constructive interference of the emitted photons.

Classical and Quantum Intuition

A classical antenna picture complements the quantum formalism: aligned spins or atoms create a giant oscillating dipole moment that radiates with an intensity proportional to the square of the total dipole moment. When half the spins are excited, the dipole moment is maximized in the transverse plane, yielding the strongest radiative response. The lecturer also discusses a subtle point: in driven, externally driven (stimulated) situations, the field acts coherently on each atom, leading to no net enhancement of Rabi oscillations; the enhancement is a feature of spontaneous emission where the emission itself is a shared, interference-enabled process rather than a unitary drive alone.

Extended Samples and Optical Density

The analysis then moves to extended samples whose length greatly exceeds the optical wavelength. In such geometries, the collective enhancement is tied to the optical density of the medium rather than to a simple N-fold increase. The light emitted into a preferred mode along the long axis of an elongated sample can be coherently amplified, yielding a laser-like gain mechanism described by an effective gain proportional to the optical density. The concept of phase factors e^{ikx} is used to show how emission from different atoms can be phase-matched to a single propagation direction, maintaining permutation symmetry for the preferred mode while allowing other modes to couple to different Dicke ladders.

From Dicke States to Real Experiments

The lecture closes with connections to experiments, including sodium dimer molecules where subradiant and superradiant channels are observed as molecules form from two atoms. The talk highlights how super radiance can persist in extended samples under the right conditions and how density and mode structure influence the observed radiative properties. It also touches on cavity QED, the idea of using giant collective coupling to enhance light-matter interactions, and the synergy between superradiant effects and optical density in modern quantum optics experiments led by researchers such as Vladan Vuletić.

Key Takeaways

• Super radiance is a collective, coherent phenomenon in which many atoms radiate into a common field mode, producing an enhanced emission rate.
• Symmetric Dicke states couple to the field and exhibit enhanced emission, while antisymmetric states can be dark or subradiant.
• Total spin labeling provides a powerful framework for N-atom systems, with a ladder of Dicke states governing transitions.
• In extended samples, the enhancement is governed by optical density and coherent mode selection, linking to lasing phenomena and cavity QED.
• Spontaneous emission exhibits enhanced dynamics in collective systems, whereas driven (stimulated) processes do not necessarily show the same enhancement.