In quantum optics, a superradiant phase transition is a phase transition that occurs in a collection of fluorescent emitters (such as atoms), between an 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 favourable by having strong, coherent interactions between the emitters.

The superradiant phase transition was originally predicted in so called Dicke model of superradiance when it is assumed that atoms have only two energetic levels and they are interacting only with 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 larger than the energy of the non-interacting part of the system which similarly to the case of superconductivity and ferromagnetism leads to the effective dynamical interactions between atoms of the ferromagnetic type and the spontaneous ordering of excitations below the critical temperature. It means that the collective Lamb shift in the system of atoms interacting with the vacuum fluctuations becomes comparable with the energies of atoms alone and the vacuum fluctuations cause the spontaneous self-excitation of matter.

The transition can be readily understood with the use of Holstein-Primakoff transformation[3] applied to two level atom. As the result of this transformation the atoms become the Lorentz harmonic oscillators with the frequency equal to the difference between the energy levels and the whole system becomes the system of the 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 now 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 and above the critical value it changes to the unstable motion of the ionization.

The superradiant phase transition was the subject of a wide discussion if 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 (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 which generally is not true. It currently can be observed in model systems like Bose-Einstein condensates and artificial atoms [7] .[8]

## Theory

### Criticality of linearized Jaynes-Cummings model

Superradiant phase transition is formally predicted already by the critical behavior of the resonant Jaynes-Cummings model describing the interaction of not ${\displaystyle N}$ but only one atom with one mode of the electromagnetic field. Starting from the exact Hamiltonian of the Jaynes-Cummings model at resonance

${\displaystyle {\hat {H}}_{\text{JC}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+\hbar \omega {\frac {{\hat {\sigma }}_{z}}{2}}+{\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}+{\hat {a}}{\hat {\sigma }}_{-}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{+}\right),}$

Applying the Holstein-Primakoff transformation for two spin levels and replacing the spin raising and lowering operators by those for the harmonic oscillators

${\displaystyle {\hat {\sigma }}_{-}\approx {\hat {b}}}$
${\displaystyle {\hat {\sigma }}_{+}\approx {\hat {b}}^{\dagger }}$
${\displaystyle {\hat {\sigma }}_{z}\approx 2{\hat {b}}^{\dagger }{\hat {b}}}$

one gets the Hamiltonian of two coupled harmonic oscillators:

${\displaystyle {\hat {H}}_{\text{JC}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+\hbar \omega {\hat {b}}^{\dagger }{\hat {b}}+{\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {b}}^{\dagger }+{\hat {a}}^{\dagger }{\hat {b}}+{\hat {a}}{\hat {a}}+{\hat {a}}^{\dagger }{\hat {b}}^{\dagger }\right),}$

which readily can be diagonalized. Postulating its normal form

${\displaystyle {\hat {H}}_{\text{JC}}=\Omega _{+}{\hat {A_{+}}}^{\dagger }{\hat {A_{+}}}+\Omega _{-}{\hat {A_{-}}}^{\dagger }{\hat {A_{-}}}+C}$

where

${\displaystyle {\hat {A_{\pm }}}=c_{\pm 1}{\hat {a}}+c_{\pm 2}{\hat {a}}^{\dagger }+c_{\pm 3}{\hat {b}}+c_{\pm 4}{\hat {b}}^{\dagger }}$

one gets the eigenvalue equation

${\displaystyle [{\hat {A_{\pm }}},{\hat {H}}_{\text{JC}}]=\Omega _{\pm }A}$

with the solutions

${\displaystyle \Omega _{\pm }=\omega {\sqrt {1\pm {\frac {\Omega }{\omega }}}}}$

The system collapses when one of the frequencies becomes imaginary i.e. when

${\displaystyle \Omega >\omega }$

or when the atom-field coupling is stronger than the frequency of the mode and atom oscillators. While there are physically higher terms in the true system the system in this regime will therefore undergo the phase transition.

### Criticality of James-Cummings model

The simplified Hamiltonian of the Jaynes-Cummings model neglecting the counter-rotating terms is

${\displaystyle {\hat {H}}_{\text{JC}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+\hbar \omega {\frac {{\hat {\sigma }}_{z}}{2}}+{\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right),}$

and the energies for the case of zero detuning are

${\displaystyle E_{\pm }(n)=\hbar \omega \left(n+{\frac {1}{2}}\right)\pm {\frac {1}{2}}\hbar \Omega (n),}$
${\displaystyle \Omega (n)=\Omega {\sqrt {n+1}}}$

where ${\displaystyle \Omega _{n}}$ is the Rabi frequency. One can approximately calculate the canonical partition function

${\displaystyle Z=\sum _{\pm ,n}{\mathrm {e} }^{-\beta E_{\pm }(n)}\approx \sum _{\pm }\int {\mathrm {e} }^{-\beta E_{\pm }(n)}dn=\int {\mathrm {e} }^{\Phi (n)}dn}$,

where the discrete sum was replaced by the integral.

The normal approach is that the later integral is calculated by the Gaussian approximation around the maximum of the exponent:

${\displaystyle {\frac {\partial \Phi (n)}{\partial n}}=0}$
${\displaystyle \Phi (n)=-\beta \hbar \omega \left(n+{\frac {1}{2}}\right)+\log 2\cosh {\frac {\hbar \Omega (n)\beta }{2}}}$

This leads to the critical equation

${\displaystyle \tanh {\frac {\hbar \Omega (n)\beta }{2}}=4{\frac {\omega }{\Omega }}{\sqrt {n+1}}}$

This has the solution only if

${\displaystyle \Omega >4\omega }$

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. When the condition is fulfilled the equation gives the solution for the order parameter ${\displaystyle n}$ depending on the inverse of the temperature ${\displaystyle 1/\beta }$ which means non-vanishing ordered field mode. Similar considerations can be done in true thermodynamic limit of the infinite number of atoms.

## References

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