Thermal balance and photon-number quantization in layered structures

The quantization of the electromagnetic field in lossy and dispersive dielectric media has been widely studied during the last few decades. However, several aspects of energy transfer and its relation to consistently defining position-dependent ladder operators for the electromagnetic field in nonequilibrium conditions have partly escaped the attention. In this work we define the position-dependent ladder operators and an effective local photon-number operator that are consistent with the canonical commutation relations and use these concepts to describe the energy transfer and thermal balance in layered geometries. This approach results in a position-dependent photon-number concept that is simple and consistent with classical energy conservation arguments. The operators are formed by first calculating the vector potential operator using Green's function formalism and Langevin noise source operators related to the medium and its temperature, and then defining the corresponding position-dependent annihilation operator that is required to satisfy the canonical commutation relations in arbitrary geometry. Our results suggest that the effective photon number associated with the electric field is generally position dependent and enables a straightforward method to calculate the energy transfer rate between the field and the local medium. In particular, our results predict that the effective photon number in a vacuum cavity formed between two lossy material layers can oscillate as a function of the position suggesting that also the local field temperature oscillates. These oscillations are expected to be directly observable using relatively straightforward experimental setups in which the field-matter interaction is dominated by the coupling to the electric field. The approach also gives further insight on separating the photon ladder operators into the conventional right and left propagating parts and on the anomalies reported for the commutation relations of the corresponding operators within optical cavities.

Figure 1

(a) The effective photon number, (b) electric field fluctuation, (c) electric LDOS, (d) energy density, (e) Poynting vector, and (f) net emission rate in the vicinity of an interface separating a lossy medium with refractive index $n_1=2.5+0.5i$ and temperature 300 K from vacuum at 0 K. The solid lines correspond to the photon energy $\hslash \omega =0.10$ eV ($\lambda =12.4$ $\mu$m) and the dashed lines correspond to the photon energy $\hslash \omega =0.14$ eV ($\lambda =8.86$ $\mu$m). The electric field fluctuation is given in the units of $\hslash \omega /\left(2\pi {\varepsilon }_{0}cS\right)$, the electric LDOS in the units of $2/\left(\pi cS\right)$, the energy density in the units of $\hslash \omega /\left(2\pi cS\right)$, the Poynting vector in the units of $\hslash \omega /\left(1000\pi S\right)$, and the net emission rate in the units of $\hslash {\omega }^2/\left(1000\pi cS\right)$.

Figure 2

(a) The effective photon number, (b) electric field fluctuation, (c) electric LDOS, (d) energy density, (e) Poynting vector, and (f) net emission rate in the vicinity of a vacuum gap separating lossy media with refractive indices $n_1=1.5+0.3i$ and $n_2=2.5+0.5i$ at temperatures 400 and 300 K. The width of the cavity is 10 $\mu$m. The solid lines correspond to the second resonant photon energy $\hslash \omega =0.118$ eV ($\lambda =10.5$ $\mu$m) and the dashed lines correspond to the off-resonant energy $\hslash \omega =0.140$ eV ($\lambda =8.86$ $\mu$m). The electric field fluctuation is given in the units of $\hslash \omega /\left(2\pi {\varepsilon }_{0}cS\right)$, the electric LDOS in the units of $2/\left(\pi cS\right)$, the energy density in the units of $\hslash \omega /\left(2\pi cS\right)$, the Poynting vector in the units of $\hslash \omega /\left(1000\pi S\right)$, and the net emission rate in the units of $\hslash {\omega }^2/\left(1000\pi cS\right)$.