Superradiant phase transition

In quantum optics, a superradiant phase transition is a phase transition that occurs in a collection of fluorescent emitters (such as atoms), between a state containing few electromagnetic excitations (as in the electromagnetic vacuum) and a superradiant state with many electromagnetic excitations trapped inside the emitters. The superradiant state is made thermodynamically favorable by having strong, coherent interactions between the emitters.

The superradiant phase transition was originally predicted by the Dicke model of superradiance, which assumes that atoms have only two energetic levels and that these interact with only one mode of the electromagnetic field.^[1][2] The phase transition occurs when the strength of the interaction between the atoms and the field is greater than the energy of the non-interacting part of the system. (This is similar to the case of superconductivity in ferromagnetism, which leads to the dynamic interaction between ferromagnetic atoms and the spontaneous ordering of excitations below the critical temperature.) The collective Lamb shift, relating to the system of atoms interacting with the vacuum fluctuations, becomes comparable to the energies of atoms alone, and the vacuum fluctuations cause the spontaneous self-excitation of matter.

The transition can be readily understood by the use of the Holstein-Primakoff transformation^[3] applied to a two-level atom. As a result of this transformation, the atoms become Lorentz harmonic oscillators with frequencies equal to the difference between the energy levels. The whole system then simplifies to a system of interacting harmonic oscillators of atoms, and the field known as Hopfield dielectric which further predicts in the normal state polarons for photons or polaritons. If the interaction with the field is so strong that the system collapses in the harmonic approximation and complex polariton frequencies (soft modes) appear, then the physical system with nonlinear terms of the higher order becomes the system with the Mexican hat-like potential, and will undergo ferroelectric-like phase transition.^[4] In this model, the system is mathematically equivalent for one mode of excitation to the Trojan wave packet, when the circularly polarized field intensity corresponds to the electromagnetic coupling constant. Above the critical value, it changes to the unstable motion of the ionization.

The superradiant phase transition was the subject of a wide discussion as to whether or not it is only a result of the simplified model of the matter-field interaction; and if it can occur for the real physical parameters of physical systems (a no-go theorem).^[5][6] However, both the original derivation and the later corrections leading to nonexistence of the transition – due to Thomas–Reiche–Kuhn sum rule canceling for the harmonic oscillator the needed inequality to impossible negativity of the interaction – were based on the assumption that the quantum field operators are commuting numbers, and the atoms do not interact with the static Coulomb forces. This is generally not true like in case of Bohr–van Leeuwen theorem and the classical non-existence of Landau diamagnetism. The negating results were also the consequence of using the simple Quantum Optics models of the electromagnetic field-matter interaction but not the more realistic Condenced Matter models like for example the superconductivity model of the BCS but with the phonons replaced by photons to first obtain the collective polaritons. The return of the transition basically occurs because the inter-atom dipole-dipole or generally the electron-electron Coulomb interactions are never negligible in the condensed and even more in the superradiant matter density regime and the Power-Zienau unitary transformation eliminating the quantum vector potential in the minimum-coupling Hamiltonian transforms the Hamiltonian exactly to the form used when it was discovered and without the square of the vector potential which was later claimed to prevent it. Alternatively within the full quantum mechanics including the electromagnetic field the generalized Bohr–van Leeuwen theorem does not work and the electromagnetic interactions cannot be eliminated while they only change the ${\displaystyle \mathbf {p} \cdot \mathbf {A} }$ vector potential coupling to the electric field ${\displaystyle \mathbf {x} \cdot \mathbf {E} }$ coupling and alter the effective electrostatic interactions. It can be observed in model systems like Bose–Einstein condensates^[7] and artificial atoms.^[8][9]