Hopfield dielectric explained

Hopfield dielectric – in quantum mechanics, a model of dielectric consisting of quantum harmonic oscillators interacting with the modes of the quantum electromagnetic field. The collective interaction of the charge polarization modes with the vacuum excitations, photonsleads to the perturbation of both the linear dispersion relation of photons and constant dispersion of charge waves by the avoided crossing between the two dispersion lines of polaritons.^[1] Similar to the acoustic and the optical phonons and far from the resonance one branch is photon-like while the other charge is wave-like.Mathematically the Hopfield dielectric for the one mode of excitation is equivalent to the Trojan wave packet in the harmonicapproximation. The Hopfield model of the dielectric predicts the existence of eternal trapped frozen photons similar to the Hawking radiation inside the matter with the density proportional to the strength of the matter-field coupling.

Theory

The Hamiltonian of the quantized Lorentz dielectric consisting of

N harmonic oscillators interacting with the quantum electromagnetic field can be written in the dipole approximation as:H = \sum\limits ^{2 \over} _{A}} 2 m} + {{m{\omega} ^{2- e{x} _{A} ⋅} E(r _{A) +\sum\limits} ^{2 \int} ^{λ=1} d ^{+} _{λ k} a _{λ} \hbar c k where E(r _{A)={i\over} L ^{2 \int} _{λ=1} d ^{3k} [{{c k}\over { 2 \epsilon ^{1 \over 2} _{0 }}]} [e _{λ(k)a} _{λ(k)\exp(ikr} _{A)-H.C.]} is the electric field operator acting at the position r _{A} .

Expressing it in terms of the creation and annihilation operators for the harmonic oscillators we get

H = \sum\limits ^{+ ⋅} _{A} a _{A)\hbar} \omega - {e\over{{\sqrt{ 2 }}\beta}}(a _{A+{a} ^{+ ) ⋅} ^{A}} E(r _{A)+\sum} _{λ\sum} _{k a} ^{+ a} ^{λ k} \hbar c k

Assuming oscillators to be on some kind of the regular solid lattice and applying the polaritonic Fourier transform

B ^{N\exp(ikr} _{A)a} ^{+ ,} ^{A} B _{k={1\over{\sqrt{N}}}\sum\limits} ^{N\exp( - ikr} ^{A)a} _{A} and defining projections of oscillator charge waves onto the electromagnetic fieldpolarization directionsB ^{+= e} _{λ} (k) ⋅ B ^{+} _{k} B _{λ} = e _{λ} (k) ⋅ B _{k,} after dropping the longitudinal contributions not interacting with the electromagnetic field one may obtain the Hopfield Hamiltonian H = \sum _{λ} \sum _{k(B} ^{+ B} ^{λ k} + { 1 \over 2 })\hbar \omega + \hbar cka ^{+ a} _{λ k} + {ie\hbar\over {\sqrt {\epsilon _{0} m\omega}}}\sqrt{N\over V} {\sqrt {ck}} [B _{λ} a _{λ -k} + B ^{+ a} _{λ k} - B ^{+ - B} _{λ k} a ^{+ ]} _{λ k} Because the interaction is notmixing polarizations this can be transformed to the normal form with the eigen-frequencies of two polaritonic branches:H = \sum _{λ} \sum _{k} \left[ \Omega _{+} (k)C ^{+ C} _{λ+ k} + \Omega _{-} (k)C ^{+ C} _{λ- k} \right] + const with the eigenvalue equation[C _{λ} ,H] = \Omega _{\pm} (k) C _{λ} C _{λ} = c _{1} a _{λ} + c _{2} a _{λ} + c _{3} a ^{+} _{λ k} + c _{4} a ^{++ c} _{5} B _{λ} + c _{6} B _{λ} + c _{7} B ^{+} _{λ k} + c _{8} B ^{+} _{λ- k} where\Omega _{-} (k) ^{2={} {\omega ^{2+\Omega} ^{2-\sqrt} { {(\omega ^{2-\Omega} ^{2)}} ^{2+4{g}\omega} ^{2\Omega} ^{2} }\over 2 }}, \Omega _{+} (k) ^{2={{\omega} ^{2+\Omega} ^{2+\sqrt{{(\omega} ^{2-\Omega} ^{2)}} ^{2+4{g}\omega} ^{2\Omega} ^{2}\over} 2 }} ,with\Omega(k) = ck, (vacuum photon dispersion)andg = {{Ne ^{2 }}} _{0 \omega} is the dimensionless coupling constant proportional to the density N/V of the dielectric withthe Lorentz frequency \omega (tight-binding charge wave dispersion).One may notice that unlike in the vacuum of the electromagnetic field without matter the expectationvalue of the average photon number <a ^{+ a} _{λ k} > is non zero in the ground state of the polaritonic HamiltonianC _{k} | 0>=0 similarly to the Hawking radiation in the neighbourhood of the black hole because of the Unruh-Davies effect. One may readily notice that the lower eigenfrequency \Omega _{-} becomes imaginary when the coupling constant becomes criticalat g> 1 which suggests that Hopfield dielectric will undergo the superradiant phase transition.

Notes and References

1. Hopfield . J. J. . 1958 . Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals . Physical Review . 112 . 5 . 1555–1567 . 10.1103/PhysRev.112.1555 . 1958PhRv..112.1555H.