Hopfield dielectric

The Hamiltonian of the quantized Lorentz dielectric consisting of ${\displaystyle N}$ harmonic oscillators interacting with the quantum electromagnetic field can be written in the dipole approximation as:

${\displaystyle H=\sum \limits _{A=1}^{N}{{p_{A}}^{2} \over 2m}+{{m{\omega }^{2}} \over 2}{x_{A}}^{2}-e{x_{A}}\cdot E(r_{A})+\sum \limits _{\lambda =1}^{2}\int d^{3}ka_{\lambda k}^{+}a_{\lambda k}\hbar ck}$

where

${\displaystyle E(r_{A})={i \over L^{3}}\sum \limits _{\lambda =1}^{2}\int d^{3}k[{{ck} \over {2\epsilon _{0}}}]^{1 \over 2}[e_{\lambda }(k)a_{\lambda }(k)\exp(ikr_{A})-H.C.]}$

is the electric field operator acting at the position ${\displaystyle r_{A}}$.

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

${\displaystyle H=\sum \limits _{A=1}^{N}(a_{A}^{+}\cdot a_{A})\hbar \omega -{e \over {{\sqrt {2}}\beta }}(a_{A}+{a_{A}}^{+})\cdot E(r_{A})+\sum _{\lambda }\sum _{k}a_{\lambda k}^{+}a_{\lambda k}\hbar ck}$

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

${\displaystyle B_{k}^{+}={1 \over {\sqrt {N}}}\sum \limits _{A=1}^{N}\exp(ikr_{A})a_{A}^{+},}$

${\displaystyle B_{k}={1 \over {\sqrt {N}}}\sum \limits _{A=1}^{N}\exp(-ikr_{A})a_{A}}$

and defining projections of oscillator charge waves onto the electromagnetic field polarization directions

${\displaystyle B_{\lambda k}^{+}=e_{\lambda }(k)\cdot B_{k}^{+}}$

${\displaystyle B_{\lambda k}=e_{\lambda }(k)\cdot B_{k},}$

after dropping the longitudinal contributions not interacting with the electromagnetic field one may obtain the Hopfield Hamiltonian

${\displaystyle H=\sum _{\lambda }\sum _{k}(B_{\lambda k}^{+}B_{\lambda k}+{1 \over 2})\hbar \omega +\hbar cka_{\lambda k}^{+}a_{\lambda k}+{ie\hbar \over {\sqrt {\epsilon _{0}m\omega }}}{\sqrt {N \over V}}{\sqrt {ck}}[B_{\lambda k}a_{\lambda -k}+B_{\lambda k}^{+}a_{\lambda k}-B_{\lambda k}^{+}a_{\lambda -k}^{+}-B_{\lambda k}a_{\lambda k}^{+}]}$

Because the interaction is not mixing polarizations this can be transformed to the normal form with the eigen-frequencies of two polaritonic branches:

${\displaystyle H=\sum _{\lambda }\sum _{k}\left[\Omega _{+}(k)C_{\lambda +k}^{+}C_{\lambda +k}+\Omega _{-}(k)C_{\lambda -k}^{+}C_{\lambda -k}\right]+const}$

with the eigenvalue equation

${\displaystyle [C_{\lambda \pm k},H]=\Omega _{\pm }(k)C_{\lambda \pm k}}$

${\displaystyle C_{\lambda \pm k}=c_{1}a_{\lambda k}+c_{2}a_{\lambda -k}+c_{3}a_{\lambda k}^{+}+c_{4}a_{\lambda -k}^{+}+c_{5}B_{\lambda k}+c_{6}B_{\lambda -k}+c_{7}B_{\lambda k}^{+}+c_{8}B_{\lambda -k}^{+}}$

where

${\displaystyle \Omega _{-}(k)^{2}={\omega ^{2}+\Omega ^{2}-{\sqrt {{(\omega ^{2}-\Omega ^{2})}^{2}+4{g}\omega ^{2}\Omega ^{2}}} \over 2},}$

${\displaystyle \Omega _{+}(k)^{2}={\omega ^{2}+\Omega ^{2}+{\sqrt {{(\omega ^{2}-\Omega ^{2})}^{2}+4{g}\omega ^{2}\Omega ^{2}}} \over 2}}$,

with

${\displaystyle \Omega (k)=ck,}$

(vacuum photon dispersion) and

${\displaystyle g={{Ne^{2}} \over {Vm\epsilon _{0}\omega ^{2}}}}$

is the dimensionless coupling constant proportional to the density ${\displaystyle N/V}$ of the dielectric with the Lorentz frequency ${\displaystyle \omega }$ (tight-binding charge wave dispersion). One may notice that unlike in the vacuum of the electromagnetic field without matter the expectation value of the average photon number ${\displaystyle <a_{\lambda k}^{+}a_{\lambda k}>}$ is non zero in the ground state of the polaritonic Hamiltonian ${\displaystyle C_{k\pm }|\mathbf {0} >=0}$ 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 ${\displaystyle \Omega _{-}}$ becomes imaginary when the coupling constant becomes critical at ${\displaystyle g>1}$ which suggests that Hopfield dielectric will undergo the superradiant phase transition.