Superradiant phase transition
The superradiant state is made thermodynamically favorable by having strong, coherent interactions between the emitters.
[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.
As a result of this transformation, the atoms become Lorentz harmonic oscillators with frequencies equal to the difference between the energy levels.
[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.
[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.
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.
[8][9] A superradiant phase transition is formally predicted by the critical behavior of the resonant Jaynes-Cummings model, describing the interaction of only one atom with one mode of the electromagnetic field.
Starting from the exact Hamiltonian of the Jaynes-Cummings model at resonance Applying the Holstein-Primakoff transformation for two spin levels, replacing the spin raising and lowering operators by those for the harmonic oscillators one gets the Hamiltonian of two coupled harmonic-oscillators: which readily can be diagonalized.
The simplified Hamiltonian of the Jaynes-Cummings model, neglecting the counter-rotating terms, is and the energies for the case of zero detuning are where
The normal approach is that the latter integral is calculated by the Gaussian approximation around the maximum of the exponent: This leads to the critical equation This has the solution only if which means that the normal, and the superradiant phase, exist only if the field-atom coupling is significantly stronger than the energy difference between the atom levels.
The better insight on the nature of the superradiant phase transition as well on the physical value of the critical parameter which must be exceeded in order for the transition to occur may be obtained by studying the classical stability of the system of the charged classical harmonic oscillators in the 3D space interacting only with the electrostatic repulsive forces for example between electrons in the locally harmonic oscillator potential.
The condition is almost identical to this obtained in the original discovery of the superradiant phase transition when replacing the harmonic oscillators with two level atoms with the same distance between the energy levels, dipole transition strength, and the density which means that it occurs in the regime when the Coulomb interactions between electrons dominate over locally harmonic oscillatory influence of the atoms.
