Direction-dependent coupling between a nanofiber-guided light field and a two-level atom with an electric quadrupole transition

We study the directional dependence of the coupling between a nanofiber-guided light field and a two-level atom with an electric quadrupole transition. We examine the situation where the nanofiber is aligned along the $z$ axis, the atom lies on the fiber transverse $x$ axis, the quantization axis for the atomic internal states is the other orthogonal transverse $y$ axis, the atomic upper and lower levels are the magnetic sublevels $M^{\prime }$ and $M$ of hyperfine-structure levels of an alkali-metal atom, and the field is in a quasilinearly polarized fundamental guided mode ${\mathrm{HE}}_{11}$ with the polarization $\xi =x$ or $y$. We find that the absolute value of the quadrupole Rabi frequency depends on the propagation direction of the light field in the cases of ($M^{\prime }-M=\pm 1, \xi =y$) and ($M^{\prime }-M=\pm 2, \xi =x$). We show that the directional dependence of the coupling leads to the directional dependence of spontaneous emission into guided modes. We find that the directional dependence of the atom-field coupling in the case of quadrupole transitions is not entirely due to spin-orbit coupling of light: there are some other contributions resulting from the gradient of the spatial phase factor of the field.

Figure 1

(a) Atom with the local quantization coordinate system $\{x_1,x_2,x_3\}$ in the vicinity of an optical nanofiber with the fiber-based Cartesian coordinate system $\{x,y,z\}$ and the corresponding cylindrical coordinate system $\{r,\phi ,z\}$. (b) Schematic of a two-level atom whose upper and lower levels are the magnetic sublevels $|n^{\prime }{F}^{\prime }{M}^{\prime }\rangle$ and $|nFM\rangle$, respectively, of hfs levels of a realistic alkali-metal atom. Under appropriate conditions, quadrupole coupling between the atom and the nanofiber-guided light field depends on the propagation direction.

Figure 2

Radial-distance dependence of the absolute value $|{\mathrm{\Omega }}_q^{\left(f\xi \right)}|$ of the Rabi frequency for the quadrupole transition between the sublevel $M=2$ of the hfs level $5S_{1/2}F=2$ and the sublevel $M^{\prime }=M+q$ of the hfs level $4D_{5/2}{F}^{\prime }=4$ of a ${}^{87}\mathrm{Rb}$ atom in the case where the quantization axis is $x_3\parallel z$. The fiber radius is $a=180\mathrm{nm}$. The wavelength of the atomic transition is ${\lambda }_0=516.5\mathrm{nm}$. The refractive indices of the fiber and the vacuum cladding are $n_1=1.4615$ and $n_2=1$, respectively. The field in the fundamental guided mode ${\mathrm{HE}}_{11}$ is quasilinearly polarized along the $x$ axis (solid lines) or the $y$ axis (dashed lines) and propagates in the direction $f=+1$ or $-1$ of the fiber $z$ axis. The power of the guided light field is 1 nW. The atom is located on the positive side of the $x$ axis and outside the fiber.

Figure 3

Radial-distance dependence of the absolute value $|{\mathrm{\Omega }}_q^{\left(f\xi \right)}|$ of the Rabi frequency for the quadrupole transition between the sublevel $M=2$ of the hfs level $5S_{1/2}F=2$ and the sublevel $M^{\prime }=M+q$ of the hfs level $4D_{5/2}{F}^{\prime }=4$ of a ${}^{87}\mathrm{Rb}$ atom in the case where the quantization axis is $x_3\parallel y$. The guided field is quasilinearly polarized along the (a),(c),(e) $\xi =x$ axis and along the (b),(d) $\xi =y$ axis, and propagates in the positive direction $f=+1$ (solid red lines) or the negative direction $f=-1$ (dashed blue lines) of the fiber $z$ axis. Other parameters are as for Fig. 2. The Rabi frequencies for $\xi =y$ and $q=0,\pm 2$ and for $\xi =x$ and $q=\pm 1$ are vanishing and are therefore not plotted.

Figure 4

Asymmetry parameter ${\eta }_q^{\left(\xi \right)}$ for the directional dependence of the absolute value of the Rabi frequency as a function of the radial distance $r$. The parameters used are as for Fig. 3.

Figure 5

Asymmetry parameters ${\eta }_1^{\left(y\right)}$ and ${\eta }_2^{\left(x\right)}$ (solid lines) for the radial distance $r$ in the region $10\le r/a\le 30$. The parameters used are as for Fig. 3. The dotted lines indicate the limiting values ${\eta }_1^{\left(y\right)}\left(\infty \right)$ and ${\eta }_2^{\left(x\right)}\left(\infty \right)$, calculated from Eqs. (21).

Figure 6

Absolute value $|{\mathrm{\Omega }}_q^{\left(f\xi \right)}|$ of the Rabi frequency for the quadrupole transition between the sublevel $M=2$ of the hfs level $5S_{1/2}F=2$ and the sublevel $M^{\prime }=M+q$ of the hfs level $4D_{5/2}{F}^{\prime }=4$ of a ${}^{87}\mathrm{Rb}$ atom with the quantization $x_3\parallel y$ axis as a function of the fiber radius $a$. The atom is located on the fiber surface ($r=a$). Other parameters are as for Fig. 3. The vertical dashed lines indicate the single-mode cutoff value $a_{\mathrm{cutoff}}\cong 185.5\mathrm{nm}$ of the fiber radius.

Figure 7

Asymmetry parameter ${\eta }_q^{\left(\xi \right)}$ for the directional dependence of the absolute value of the Rabi frequency as a function of the fiber radius $a$. The parameters used are as for Fig. 6. The vertical dashed lines indicate the single-mode cutoff value $a_{\mathrm{cutoff}}\cong 185.5\mathrm{nm}$ of the fiber radius.

Figure 8

Asymmetry parameters (a) ${\eta }_1^{\left(\xi \right)}$ and (b) ${\eta }_2^{\left(\xi \right)}$ as functions of the azimuthal angle $\phi$ for the position of the atom in the fiber transverse $xy$ plane. The quantization axis is $x_3\parallel y$ and the guided field is quasilinearly polarized along the $\xi =x$ axis (solid lines) or the $\xi =y$ axis (dashed lines). The radial distance from the atom to the fiber surface is $r-a=50\mathrm{nm}$. Other parameters are as for Fig. 3.