Approach for describing spatial dynamics of quantum light-matter interaction in dispersive dissipative media

Solving the challenging problem of the amplification and generation of an electromagnetic field in nanostructures enables us to implement many properties of the electromagnetic field at the nanoscale in practical applications. A first-principles quantum-mechanical consideration of such a problem is sufficiently restricted by the exponentially large number of degrees of freedom and does not allow the electromagnetic-field dynamics to be described if it involves a high number of interacting atoms and modes of the electromagnetic field. Conversely, the classical description of electromagnetic fields is incorrect at the nanoscale due to the high level of quantum fluctuations connected to high dissipation and noise levels. In this paper, we develop a framework with a significantly reduced number of degrees of freedom, which describes the quantum spatial dynamics of electromagnetic fields interacting with atoms. As an example, we consider the interaction between atoms placed in a metallic subwavelength groove and demonstrate that a spontaneously excited electromagnetic pulse propagates with the group velocity. The developed approach may be exploited to describe nonuniform amplification and propagation of electromagnetic fields in arbitrary dispersive dissipative systems.

Figure 1

(a) Dispersion curve of the eigenmodes of the metallic groove with the profile $\zeta \left(x\right)=-Aexp\left(-x^2/R^2\right)$. Inset: schematically illustrated metallic groove. (b) Dependence of population inversion of the first (blue solid line) and second (green dashed line) atoms and the dependence of the population inversion of the first atom with time in the absence of the second atom (red dot-dashed line) obtained from Eqs. (2, 3, 4). The distance between the atoms is $l=500{\lambda }_{\sigma }$, the groove length is $L=2000{\lambda }_{\sigma }$, and the atoms are placed symmetrically with respect to the groove center. Here $t_0$ is equal to ${10}^2{\lambda }_{\sigma }/v_g$, where ${\lambda }_{\sigma }$ is the wavelength of the atom transition; $v_g={\left.\left(\partial \omega /\partial k\right)\right|}_{\omega ={\omega }_{\sigma }}$ is group velocity.

Figure 2

(a) Dependence of the population inversion of the second atom on time and distance between atoms obtained from Eqs. (2, 3, 4). The blue dashed line is determined by group velocity $z=v_{g}t$. Here $z_0$ is equal to ${10}^2{\lambda }_{\sigma }$, and $t_0$ is equal to ${10}^2{\lambda }_{\sigma }/{\nu }_g$. (b) Dependence of the population inversion of the second atom on time and the transition frequency of the atoms obtained from Eqs. (2, 3, 4). The solid line is the curve $t=l/v_g[v_g={\left.\left(\partial \omega /\partial k\right)\right|}_{\omega ={\omega }_{\sigma }}$ is group velocity], the dashed line is the curve $t=l/v_{\varphi }[v_{\varphi }={\left.\left(\omega /k\right)\right|}_{\omega ={\omega }_{\sigma }}$ is phase velocity], and the dashed-dot line is the curve $t=l/c$.

Figure 3

Dependence of the population inversion of the second atom on time and distance between atoms obtained from Eqs. (5, 6, 7). The blue dashed line is determined by group velocity $z=v_{g}t$. Here $z_0$ is equal to ${10}^2{\lambda }_{\sigma }$ where ${\lambda }_{\sigma }$ is the wavelength of the atom radiation; $v_g={\left.\left(\partial \omega /\partial k\right)\right|}_{\omega ={\omega }_{\sigma }}$ is group velocity.

Figure 4

Dependence of the intensity of the electromagnetic field in the cavity on time and coordinates obtained from Eqs. (2, 3, 4). The white dashed lines are determined by group velocity $z=v_{g}t$. The first atom is located at $z=0$, and the second atom is located at $z=400{\lambda }_{\sigma }$. Here $z_0$ is equal to ${10}^2{\lambda }_{\sigma }$, and $t_0$ is equal to ${10}^2{\lambda }_{\sigma }/v_g$, where ${\lambda }_{\sigma }$ is the wavelength of the atom radiation; $v_g={\left.\left(\partial \omega /\partial k\right)\right|}_{\omega ={\omega }_{\sigma }}$ is group velocity.

Figure 5

(a) The dependence of the number of photons on the pumping obtained from Eqs. (2, 3, 4) (blue solid line) and from Eqs. (8) and (9) (red dashed line). The black vertical line corresponds to the lasing threshold. (b) The dependence of the normalized electromagnetic-field intensity on the coordinate obtained from Eqs. (2)–(4) (blue solid line) and from rate equations (8) and (9) (red dashed line). Vertical dashed lines correspond to the edges of the gain medium.