Casimir Light in Dispersive Nanophotonics

Time-varying optical media, whose dielectric properties are actively modulated in time, introduce a host of novel effects in the classical propagation of light, and are of intense current interest. In the quantum domain, time-dependent media can be used to convert vacuum fluctuations (virtual photons) into pairs of real photons. We refer to these processes broadly as “dynamical vacuum effects” (DVEs). Despite interest for their potential applications as sources of quantum light, DVEs are generally very weak, presenting many opportunities for enhancement through modern techniques in nanophotonics, such as using media which support excitations such as plasmon and phonon polaritons. Here, we present a theory of weakly modulated DVEs in arbitrary nanostructured, dispersive, and dissipative systems. A key element of our framework is the simultaneous incorporation of time-modulation and “dispersion” through time-translation-breaking linear response theory. As an example, we use our approach to propose a highly efficient scheme for generating entangled surface polaritons based on time-modulation of the optical phonon frequency of a polar insulator.

Figure 1

Photon pair emission from arbitrary time-dependent dielectric media. (a) A dispersive dielectric $ϵ(\mathbf{r},\omega )$ subject to an arbitrary time modulation can be described as having the more general dielectric function $ϵ(\mathbf{r},\omega ,{\omega }^{\prime })$ which encodes both time dependence and dispersion. (b) A schematic of a thin film of polar insulator which has a small top layer which undergoes a time modulation. As a result, surface phonon-polariton pairs are produced with frequencies $\omega ,{\omega }^{\prime }$, and wave vectors $q$, $q^{\prime }$.

Figure 2

Dynamical Casimir effect for silicon carbide phonon polaritons. (a) Dispersion relation of phonon polaritons on $d_{\mathrm{slab}}=100\mathrm{nm}$ thick slab of SiC. Dashed lines mark the edges of the RS band. Inset shows the Lorentz oscillator permittivity around ${\omega }_0=1.49\times {10}^{14}\mathrm{rad}/\mathrm{s}$. (b) Schematic representation of nondispersive time modulations of the permittivity, versus dispersive modulations of the transverse optical phonon frequency ${\omega }_0$. (c)–(f) Differential rate per unit area $(1/A)d\mathrm{\Gamma }/d\omega$ for phonon-polariton pairs production for various values of ${\mathrm{\Omega }}_0/{\omega }_0=\{2.01,2.1,2.3,2.4\}$, as well as short ($T=80\mathrm{fs}$) and long ($T=5\mathrm{ns}$) pulses. The modulated region is assumed to be $d=10\mathrm{nm}$ thick. (c),(d) Nondispersive modulation with $\delta ϵ={10}^{-3}$. (e),(f) Dispersive modulation with $\delta \omega ={10}^{-3}$.

Figure 3

Achieving strong DVEs through dispersive modulations. (a) Dispersion of surface phonon-polaritons on a thin layer of hBN ($d_{\mathrm{slab}}=100\mathrm{nm}$) in the upper RS band (${\omega }_0=2.56\times {10}^{14}\mathrm{rad}/\mathrm{s}$). (b),(c) Differential rate per unit area $(1/A)d\mathrm{\Gamma }/d\omega$ for phonon-polariton pairs production for various values of ${\mathrm{\Omega }}_0/{\omega }_0=\{2,2.05,2.1,2.15\}$, as well as short ($T=80\mathrm{fs}$) and long ($T=5\mathrm{ns}$) pulses. (d) Total emission rate per area of phonon-polariton pairs as a function of pulse duration $T$ and frequency ${\mathrm{\Omega }}_0$. (e)–(g) Same as (b)–(d), except that the modulation is dispersive. (g) Strong enhancement which occurs for monochromatic modulations when ${\mathrm{\Omega }}_0/{\omega }_0=2$, corresponding to enhancement of DVEs by dispersive parametric amplification.