Abstract
We consider the radiation field operators in a cavity with varying dielectric medium in terms of solutions of Heisenberg’s equations of motion for the most general one-dimensional quadratic Hamiltonian. Explicit solutions of these equations are obtained and applications to the radiation field quantization, including randomly varying media, are briefly discussed.
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Notes
The standard form of Heisenberg’s equations can be obtained by the time reversal t→−t.
See also Koutschan.nb [51]. This notebook is also available from the website: http://iopscience.iop.org/0953-4075/46/10/104007/media.
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Acknowledgements
We thank Michael Berry for bringing Ref. [5] to our attention and Victor Dodonov and Luc Vinet for valuable discussions. This research was partially supported by an AFOSR grant FA9550-11-1-0220 and by the Austrian Science Foundation FWF, grants Z130-N13 and S50-N15, the latter in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”.
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Appendix: Factorized Media
Appendix: Factorized Media
For the phenomenological Maxwell equations in linear, dispersive, time-varying media, namely
the continuity equation,
has the stationary solution ρ≡0 under the condition \(\operatorname{grad} (\widetilde{\sigma }/\widetilde{\varepsilon } )=0\).
With the help of the vector A and scalar φ potentials,
the Maxwell equations can be reduced to the gauge condition
and the single second-order generalized wave equation
Here, we consider the factorized (real-valued) dielectric permittivity, the magnetic permeability, and the conductivity (tensors)
(the case \(\widetilde{\sigma }\equiv 0\) was discussed in [20]). Under the imposed condition of \(\operatorname{grad} (\widetilde{\sigma }/\widetilde{\varepsilon } )=0\), one can choose \(4\pi\overline{\sigma }=\overline{\varepsilon }\) without loss of generality.
The solution of the classical problem for a given single mode, υ say, has the form
(k is a constant), and
provided that
and certain required boundary conditions are satisfied on the boundary of the cavity (see [20, 25, 32, 50, 85] for more details).
As a result, we can choose c=d=f=g=0 and
in Theorem 1 for the quantization of the mode of the electromagnetic field under consideration. (See also [2, 13, 67, 77, 78].)
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Krattenthaler, C., Kryuchkov, S.I., Mahalov, A. et al. On the Problem of Electromagnetic-Field Quantization. Int J Theor Phys 52, 4445–4460 (2013). https://doi.org/10.1007/s10773-013-1764-3
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DOI: https://doi.org/10.1007/s10773-013-1764-3