Computational Nanophotonics

6.4.3.5  Hilbert Spaces

Two composite Hilbert spaces—one for the transmit node and the other for the receive node—are considered. In the transmit node, separate Hilbert spaces are constructed for the atom, each of the cavity modes and each of the laser modes. The composite Hilbert space is constructed by taking the tensor product of these individual Hilbert spaces and is defined as

$H_T=H_{atom}\otimes H_{cav}^{\left(0\right)}\otimes H_{cav}^{\left(1\right)}\otimes H_{L01}^{\left(0\right)}\otimes H_{L12}^{\left(0\right)}\otimes H_{L01}^{\left(1\right)}\otimes H_{L12}^{\left(1\right)}$

(6.60)

where

H_atom is the Hilbert space of ⁸⁷Rb atom

$H_{cav}^{\left(0\right)}$ and $H_{cav}^{\left(1\right)}$ are the Hilbert spaces of the two cavity modes

$H_{L01}^{\left(0\right)}$, $H_{L12}^{\left(0\right)}$, $H_{L01}^{\left(1\right)}$, and $H_{L12}^{\left(1\right)}$ are the Hilbert spaces for the laser modes $L_{01}^{\left(0\right)}$, $L_{01}^{\left(1\right)}$, $L_{12}^{\left(0\right)}$, and $L_{12}^{\left(1\right)}$, respectively

Similarly, the Hilbert space for the receive node is defined ...