Canonical Quantization of Electromagnetic Field in the Presence of Nonlinear Anisotropic Magnetodielectric Medium with Spatial-Temporal Dispersion

Abstract

Modeling a nonlinear anisotropic magnetodielectric medium with spatial-temporal dispersion by two continuum collections of three dimensional harmonic oscillators, a fully canonical quantization of the electromagnetic field is demonstrated in the presence of such a medium. Some coupling tensors of various ranks are introduced that couple the magnetodielectric medium with the electromagnetic field. The polarization and magnetization fields of the medium are defined in terms of the coupling tensors and the oscillators modeling the medium. The electric and magnetic susceptibility tensors of the medium are obtained in terms of the coupling tensors. It is shown that the electric field satisfy an integral equation in frequency domain. The integral equation is solved by an iteration method and the electric field is found up to an arbitrary accuracy.

Appendices

Appendix A: Derivation of $\underline {L}_{int}(t)$ in the Relation (9)

We express the real valued fields A(r,t),φ(r,t),X _ω (r,t),Y _ω (r,t) and the real valued coupling tensors f ⁽ⁿ⁾,g ⁽ⁿ⁾,n=1,2,3,... in terms of their spatial Fourier transforms. Then we substitute them in the interaction Lagrangian (4) and integrate over the position space. This leads to the interaction Lagrangian in the reciprocal space as

$$\begin{array}{@{}rcl@{}} \underline{L}_{int}&=&{\int}_{0}^{\infty} d\omega_{1}{\int}^{\prime} d^{3}k {\int}^{\prime} d^{3}k_{1} \left[\underline{f}^{(1)}_{ij}(\omega_{1},\textbf{k}, \textbf{k}_{1})\underline{E}^{*^{i}}(\textbf{k},t)\underline{X}^{j}_{\omega_{1}}(\textbf{k}_{1},t)+H.C\right]\\ &&+{\int}_{0}^{\infty} d\omega_{1}{\int}^{\prime} d^{3}k {\int}^{\prime} d^{3}k_{1} \left[\underline{f}^{(1)}_{ij}(\omega_{1},-\textbf{k}, \textbf{k}_{1})\underline{E}^{i}(\textbf{k},t)\underline{X}^{j}_{\omega_{1}}(\textbf{k}_{1},t)+H.C\right]\\ &&+{\int}_{0}^{\infty} d\omega_{1} {\int}_{0}^{\infty} d\omega_{2}{\int}^{\prime} d^{3}k {\int}^{\prime} d^{3}k_{1}{\int}^{\prime} d^{3}k_{2} [\underline{f}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k}, \textbf{k}_{1}, \textbf{k}_{2}) \\ &&\times\underline{E}^{*^{i}}(\textbf{k},t)\underline{X}^{j}_{\omega_{1}}(\textbf{k}_{1},t)\underline{X}^{k}_{\omega_{2}}(\textbf{k}_{2},t)+H.C]\\ &&+{\int}_{0}^{\infty} d\omega_{1} {\int}_{0}^{\infty} d\omega_{2}{\int}^{\prime} d^{3}k {\int}^{\prime} d^{3}k_{1}{\int}^{\prime} d^{3}k_{2}[\underline{f}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k}, \textbf{k}_{1}, -\textbf{k}_{2})\\ &&\times\underline{E}^{*^{i}}(\textbf{k},t)\underline{X}^{j}_{\omega_{1}}(\textbf{k}_{1},t)\underline{X}^{*^{k}}_{\omega_{2}}(\textbf{k}_{2},t)+H.C]\\ &&+{\int}_{0}^{\infty} d\omega_{1} {\int}_{0}^{\infty} d\omega_{2}{\int}^{\prime} d^{3}k {\int}^{\prime} d^{3}k_{1}{\int}^{\prime} d^{3}k_{2} [\underline{f}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k},- \textbf{k}_{1}, \textbf{k}_{2})\\ &&\times\underline{E}^{*^{i}}(\textbf{k},t)\underline{X}^{*^{j}}_{\omega_{1}}(\textbf{k}_{1},t)\underline{X}^{k}_{\omega_{2}}(\textbf{k}_{2},t)+H.C]\\ &&+{\int}_{0}^{\infty} d\omega_{1} {\int}_{0}^{\infty} d\omega_{2}{\int}^{\prime} d^{3}k {\int}^{\prime} d^{3}k_{1}{\int}^{\prime} d^{3}k_{2} [\underline{f}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k},- \textbf{k}_{1},- \textbf{k}_{2})\\ &&\times\underline{E}^{*^{i}}(\textbf{k},t)\underline{X}^{*^{j}}_{\omega_{1}}(\textbf{k}_{1},t)\underline{X}^{*^{k}}_{\omega_{2}}(\textbf{k}_{2},t)+H.C]+....\\ &&+{\int}_{0}^{\infty} d\omega_{1}{\int}^{\prime}d^{3}k {\int}^{\prime} d^{3}k_{1} [\underline{g}^{(1)}_{ij}(\omega_{1},\textbf{k}, \textbf{k}_{1})\underline{B}^{*^{i}}(\textbf{k},t)\underline{Y}^{j}_{\omega_{1}}(\textbf{k}_{1},t)+H.C]\\ &&+{\int}_{0}^{\infty} d\omega_{1}{\int}^{\prime}d^{3}k {\int}^{\prime} d^{3}k_{1} [\underline{g}^{(1)}_{ij}(\omega_{1},-\textbf{k}, \textbf{k}_{1})\underline{B}^{i}(\textbf{k},t)\underline{Y}^{j}_{\omega_{1}}(\textbf{k}_{1},t)+H.C]\\ &&+{\int}_{0}^{\infty} d\omega_{1} {\int}_{0}^{\infty} d\omega_{2}{\int}^{\prime} d^{3}k {\int}^{\prime} d^{3}k_{1}{\int}^{\prime} d^{3}k_{2}[\underline{g}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k}, \textbf{k}_{1}, \textbf{k}_{2})\\ &&\times\underline{B}^{*^{i}}(\textbf{k},t)\underline{Y}^{j}_{\omega_{1}}(\textbf{k}_{1},t)\underline{Y}^{k}_{\omega_{2}}(\textbf{k}_{2},t)+H.C]\\ &&+{\int}_{0}^{\infty} d\omega_{1} {\int}_{0}^{\infty} d\omega_{2}{\int}^{\prime} d^{3}k {\int}^{\prime} d^{3}k_{1}{\int}^{\prime} d^{3}k_{2}[\underline{g}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k}, \textbf{k}_{1}, -\textbf{k}_{2})\\ &&\times\underline{B}^{*^{i}}(\textbf{k},t)\underline{Y}^{j}_{\omega_{1}}(\textbf{k}_{1},t)\underline{Y}^{*^{k}}_{\omega_{2}}(\textbf{k}_{2},t)+H.C]\\ &&+{\int}_{0}^{\infty} d\omega_{1} {\int}_{0}^{\infty} d\omega_{2}{\int}^{\prime} d^{3}k {\int}^{\prime}d^{3}k_{1}{\int}^{\prime} d^{3}k_{2}[\underline{g}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k},- \textbf{k}_{1}, \textbf{k}_{2})\\ \end{array} $$

$$\begin{array}{@{}rcl@{}} &&\times\underline{B}^{*^{i}}(\textbf{k},t)\underline{Y}^{*^{j}}_{\omega_{1}}(\textbf{k}_{1},t)\underline{Y}^{k}_{\omega_{2}}(\textbf{k}_{2},t)+ H.C]\\ &&+{\int}_{0}^{\infty} d\omega_{1} {\int}_{0}^{\infty} d\omega_{2}{\int}^{\prime} d^{3}k {\int}^{\prime} d^{3}k_{1}{\int}^{\prime} d^{3}k_{2}[\underline{g}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k},- \textbf{k}_{1}, -\textbf{k}_{2})\\ &&\times\underline{B}^{*^{i}}(\textbf{k},t)\underline{Y}^{*^{j}}_{\omega_{1}}(\textbf{k}_{1},t)\underline{Y}^{*^{k}}_{\omega_{2}}(\textbf{k}_{2},t)+H.C]+... \end{array} $$

(45)

where the relation (7) for the field X _ω (r,t) and similar relations for the other fields and also the relations (8) have been used. In (45) the symbol ${\int }^{\prime } d^{3}k$ denote the integration on the half space k _z ≥0 and three points ....in the end of the formula indicate the terms which are dependent on the coupling tensors more than the third rank.

Appendix B: The Expressions for the Electric and Magnetic Susceptibility Tensors in Constitutive Relations (30) and (31)

Insertion the solutions (25) and (26) in the definitions of polarization and magnetization densities given by (13) and (14) one can easily obtain the constitutive relations (30) and (31), where the electric and magnetic susceptibility tensors χ ⁽ⁿ⁾ and ζ ⁽ⁿ⁾ for n=1,2,3,... are as

$$\begin{array}{@{}rcl@{}} &&\chi^{(n)}_{ii_{1}i_{2}...i_{n}}(t_{1},t_{2},...,t_{n},\textbf{k},\textbf{k}_{1},\textbf{k}_{2},...,\textbf{k}_{n})=\\ &&{\Theta}(t_{1}){\Theta}(t_{2})....{\Theta}(t_{n}){\int}_{0}^{\infty} d\omega_{1}{\int}_{0}^{\infty} d\omega_{2}....{\int}_{0}^{\infty} d\omega_{n} \frac{\sin \omega_{1} t_{1}}{\omega_{1}}\frac{\sin \omega_{2} t_{2}}{\omega_{2}}....\frac{\sin \omega_{n} t_{n}}{\omega_{n}}\\ &&\times\int d^{3}p_{1}\int d^{3}p_{2}....\int d^{3}p_{n} [\underline{f}^{(n)}_{ij_{1}j_{2}...j_{n}}(\omega_{1},\omega_{2},...,\omega_{n},\textbf{k},\textbf{p}_{1},\textbf{p}_{2},...\textbf{p}_{n})\\ &&\times \underline{f}^{\dag^{(1)}}_{j_{1}i_{1}}(\omega_{1},\textbf{k}_{1},\textbf{ p}_{1}) \underline{f}^{\dag^{(1)}}_{j_{2}i_{2}}(\omega_{2},\textbf{k}_{2},\textbf{ p}_{2})....\underline{f}^{\dag^{(1)}}_{j_{n}i_{n}}(\omega_{n},\textbf{k}_{n},\textbf{ p}_{n})] \end{array} $$

(46)

$$\begin{array}{@{}rcl@{}} &&\zeta^{(n)}_{ii_{1}i_{2}...i_{n}}(t_{1},t_{2},...,t_{n},\textbf{k},\textbf{k}_{1},\textbf{k}_{2},...,\textbf{k}_{n})=\\ &&{\Theta}(t_{1}){\Theta}(t_{2})....{\Theta}(t_{n}){\int}_{0}^{\infty} d\omega_{1}{\int}_{0}^{\infty} d\omega_{2}....{\int}_{0}^{\infty} d\omega_{n} \frac{\sin \omega_{1} t_{1}}{\omega_{1}}\frac{\sin \omega_{2} t_{2}}{\omega_{2}}....\frac{\sin \omega_{n} t_{n}}{\omega_{n}}\\ &&\times\int d^{3}p_{1}\int d^{3}p_{2}....\int d^{3}p_{n} [\underline{g}^{(n)}_{ij_{1}j_{2}...j_{n}}(\omega_{1},\omega_{2},...,\omega_{n},\textbf{k},\textbf{p}_{1},\textbf{p}_{2},...\textbf{p}_{n})\\ &&\times \underline{g}^{\dag^{(1)}}_{j_{1}i_{1}}(\omega_{1},\textbf{k}_{1},\textbf{ p}_{1}) \underline{g}^{\dag^{(1)}}_{j_{2}i_{2}}(\omega_{2},\textbf{k}_{2},\textbf{ p}_{2})....\underline{g}^{\dag^{(1)}}_{j_{n}i_{n}}(\omega_{n},\textbf{k}_{n},\textbf{ p}_{n})] \end{array} $$

(47)

where Θ(t) is the step function and the summation over the repeated indices should be done.

Appendix C: Derivation of the Noise Polarization and Magnetization Densities in the Constitutive Relations (30) and (31)

If One substitute the solutions (25) and (26) in the definitions (13) and (14), Then the terms which are dependent on the fields X _{N ω} (k,t) and Y _{N ω} (k,t) are called the noise electric and magnetic polarization densities, respectively. Straightforwardly it can be shown that the electric and magnetic noise polarization densities are given, respectively, by the expressions

$$\begin{array}{@{}rcl@{}} &&\underline{P}_{Ni}(\textbf{k},t)={\int}_{0}^{\infty} d\omega_{1}\int d^{3}k_{1} \underline{f}^{(1)}_{ij}(\omega_{1},\textbf{k},\textbf{k}_{1})\underline{X}_{N\omega_{1}}^{j}(\textbf{k}_{1},t)\\ &&+{\int}_{0}^{\infty} d\omega_{1}{\int}_{0}^{\infty} d\omega_{2}\int d^{3}k_{1}\int d^{3} k_{2} [\underline{f}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k},\textbf{k}_{1},\textbf{k}_{2})\underline{X}_{N\omega_{1}}^{j}(\textbf{k}_{1},t)\underline{X}_{N\omega_{2}}^{k}(\textbf{k}_{2},t)]\\ && +{\int}_{0}^{\infty} d\omega_{1}{\int}_{0}^{\infty} d\omega_{2}\int d^{3}k_{1}\int d^{3} k_{2} \left[\underline{f}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k},\textbf{k}_{1},\textbf{k}_{2}) {{\int}_{0}^{t}} dt^{\prime}\frac{\sin\omega_{2}(t-t^{\prime})}{\omega_{2}}\right.\\ &&\times\left. \int d^{3}p_{2}\underline{f}^{\dag^{(1)}}_{kl}(\omega_{2},\textbf{p}_{2},\textbf{k}_{2}) \left( \underline{X}_{N\omega_{1}}^{j}(\textbf{k}_{1},t)\underline{E}^{l}(\textbf{p}_{2},t^{\prime})+\underline{E}^{l}(\textbf{p}_{2},t^{\prime})\underline{X}^{j}_{N\omega_{1}}(\textbf{k}_{1},t)\right)\right]+...\\ && \end{array} $$

(48)

$$\begin{array}{@{}rcl@{}} &&\underline{M}_{Ni}(\textbf{k},t)={\int}_{0}^{\infty} d\omega_{1}\int d^{3}k_{1} \underline{g}^{(1)}_{ij}(\omega_{1},\textbf{k},\textbf{k}_{1})\underline{Y}_{N\omega_{1}}^{j}(\textbf{k}_{1},t)\\ &&+{\int}_{0}^{\infty} d\omega_{1}{\int}_{0}^{\infty} d\omega_{2}\int d^{3}k_{1}\int d^{3} k_{2} [\underline{g}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k},\textbf{k}_{1},\textbf{k}_{2})\underline{Y}_{N\omega_{1}}^{j}(\textbf{k}_{1},t)\underline{Y}_{N\omega_{2}}^{k}(\textbf{k}_{2},t)]\\ && +{\int}_{0}^{\infty} d\omega_{1}{\int}_{0}^{\infty} d\omega_{2}\int d^{3}k_{1}\int d^{3} k_{2} \left[\underline{g}^{(2)}_{ijk}(\omega_{1},\omega_{2},\textbf{k},\textbf{k}_{1},\textbf{k}_{2}) {{\int}_{0}^{t}} dt^{\prime}\frac{\sin\omega_{2}(t-t^{\prime})}{\omega_{2}}\right.\\ &&\times\left. \int d^{3}p_{2}\underline{g}^{\dag^{(1)}}_{kl}(\omega_{2},\textbf{p}_{2},\textbf{k}_{2}) \left( \underline{Y}_{N\omega_{1}}^{j}(\textbf{k}_{1},t)\underline{B}^{l}(\textbf{p}_{2},t^{\prime})+\underline{B}^{l}(\textbf{p}_{2},t^{\prime})\underline{Y}^{j}_{N\omega_{1}}(\textbf{k}_{1},t)\right)\right]+...\\ && \end{array} $$

(49)

where the symmetry relations (34) and (35) for the coupling tensors f ⁽²⁾,g ⁽²⁾ have been used and the three points ... denote the terms which are dependent on the coupling tensors more than the third rank.