Criticality of linearized Jaynes-Cummings model

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

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

Applying the Holstein-Primakoff transformation for two spin levels, 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:

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

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 Jaynes-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 ${\displaystyle \Omega }$ 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 latter 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.

Instability of the classical electrostatic model

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. Despite of the original model of the superradiance the quantum electromagnetic field is totally neglected here. The oscillators may be assumed to be placed for example on the cubic lattice with the lattice constant ${\displaystyle a}$ in the analogy to the crystal system of the condensed matter. The worse scenario of the defect of the absence of the two out-of-the-plane motion-stabilizing electrons from the 6-th nearest neighbors of a chosen electron is assumed while the four nearest electrons are first assumed to be rigid in space and producing the anti-harmonic potential in the direction perpendicular to the plane of the all five electrons. The condition of the instability of motion of the chosen electron is that the net potential being the superposition of the harmonic oscillator potential and the quadratically expanded Coulomb potential from the four electrons is negative i.e.

${\displaystyle {\frac {m\omega ^{2}}{2}}-{\frac {1}{2}}\times 4\times {\frac {e^{2}}{4\pi \epsilon _{0}}}{\frac {1}{a^{3}}}<0}$

or

${\displaystyle {\frac {e^{2}}{\pi \epsilon _{0}m\omega ^{2}}}{\frac {1}{a^{3}}}>1}$

Making it artificially quantum by multiplying the numerator and the denominator of the fraction by the ${\displaystyle \hbar }$ one obtains the condition

${\displaystyle {\frac {2}{\pi }}{\frac {|D_{12}|^{2}}{E_{12}\epsilon _{0}}}\left({\frac {N}{V}}\right)>1}$

where

${\displaystyle |D_{12}|^{2}={\frac {e^{2}\hbar }{2m\omega }}}$

is the square of the dipole transition strength between the ground state and the first excited state of the quantum harmonic oscillator,

${\displaystyle E_{12}=\hbar \omega }$

is the energy gap between consecutive levels and it is also noticed that

${\displaystyle {\frac {1}{a^{3}}}={\frac {N}{V}}}$

is the spatial density of the oscillators. 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. It that sense the free electron gas with ${\displaystyle \omega =0}$ is also purely superradiant.

The critical inequality rewritten yet differently

${\displaystyle \omega <{\sqrt {{\frac {e^{2}}{m\pi \epsilon _{0}}}{\frac {N}{V}}}}\approx {\sqrt {{\frac {e^{2}}{m\epsilon _{0}}}{\frac {N}{V}}}}}$

expresses the fact that superradiant phase transition occurs when the frequency of the binding atomic oscillators is lower than so called electron gas plasma frequency.