Enhancement of the quadrupole interaction of an atom with the guided light of an ultrathin optical fiber

We investigate the electric quadrupole interaction of an alkali-metal atom with guided light in the fundamental and higher-order modes of a vacuum-clad ultrathin optical fiber. We calculate the quadrupole Rabi frequency, the quadrupole oscillator strength, and their enhancement factors. In the example of a ${}^{87}\mathrm{Rb}$ atom, we study the dependencies of the quadrupole Rabi frequency on the quantum numbers of the transition, the mode type, the phase circulation direction, the propagation direction, the orientation of the quantization axis, the position of the atom, and the fiber radius. We find that the root-mean-square (rms) quadrupole Rabi frequency reduces quickly but the quadrupole oscillator strength varies slowly with increasing radial distance. We show that the enhancement factors of the rms Rabi frequency and the oscillator strength do not depend on any characteristics of the internal atomic states except for the atomic transition frequency. The enhancement factor of the oscillator strength can be significant even when the atom is far away from the fiber. We show that, in the case where the atom is positioned on the fiber surface, the oscillator strength for the quasicircularly polarized fundamental mode ${\mathrm{HE}}_{11}$ has a local minimum at the fiber radius $a\simeq 107$ nm, and is larger than that for quasicircularly polarized higher-order hybrid modes, transverse electric modes, and transverse magnetic modes in the region $a<498.2$ nm.

Figure 1

(a) Atom with the local quantization coordinate system $\{x_1,x_2,x_3\}$ in the vicinity of an ultrathin optical fiber with the fiber-based Cartesian coordinate system $\{x,y,z\}$ and the corresponding cylindrical coordinate system $\{r,\phi ,z\}$. (b) Schematic of the hyperfine-structure (hfs) levels of the $4D_{5/2}$ and $5S_{1/2}$ states of a ${}^{87}\mathrm{Rb}$ atom.

Figure 2

Absolute value of the Rabi frequency ${\mathrm{\Omega }}_{FM{F}^{\prime }{M}^{\prime }}$ for the quadrupole transition between the sublevel $M=2$ of the level $5S_{1/2}F=2$ and a sublevel $M^{\prime }$ of the level $4D_{5/2}{F}^{\prime }=4$ as a function of the radial distance $r$ for different magnetic quantum numbers $M^{\prime }=0,1,2,3,4$ and different guided mode types $N={\mathrm{HE}}_{11}$, ${\mathrm{TE}}_{01}$, ${\mathrm{TM}}_{01}$, and ${\mathrm{HE}}_{21}$. The fiber radius is $a=280$ nm. The wavelength of the atomic transition is ${\lambda }_0=516.5$ nm. The refractive indices of the fiber and the vacuum cladding are $n_1=1.4615$ and $n_2=1$, respectively. The power of the guided light field is 2.5 nW. The field propagates in the $+z$ direction. The hybrid modes are counterclockwise quasicircularly polarized. The quantization axis is $x_3=z$. The azimuthal angle for the position of the atom in the fiber cross-section $xy$ plane is arbitrary.

Figure 3

Radial dependencies of the absolute value of the Rabi frequency ${\mathrm{\Omega }}_{FM{F}^{\prime }{M}^{\prime }}$ for the quadrupole transition between the sublevels $|F=2,M=2\rangle$ and $|F^{\prime }=4,M^{\prime }=4\rangle$ for different choices of the quantization axis $x_3$ and different guided modes. The atom is positioned on the positive side of the $x$ axis ($\phi =0$) and the hybrid modes are counterclockwise quasicircularly polarized. Other parameters are as for Fig. 2.

Figure 4

Radial dependencies of the absolute value of the Rabi frequency ${\mathrm{\Omega }}_{FM{F}^{\prime }{M}^{\prime }}$ for the opposite phase circulation directions $p=\pm 1$ of the circularly polarized hybrid modes ${\mathrm{HE}}_{11}$ and ${\mathrm{HE}}_{21}$. The lower and upper levels of the transition are $|F=2,M=2\rangle$ and $|F^{\prime }=4,M^{\prime }=4\rangle$ and the quantization axis is $x_3=z$. Other parameters are as for Fig. 2.

Figure 5

Radial dependencies of the absolute value of the Rabi frequency ${\mathrm{\Omega }}_{FM{F}^{\prime }{M}^{\prime }}$ for the opposite propagation directions $f=\pm 1$ of different guided modes. The lower and upper levels of the transition are $|F=2,M=2\rangle$ and $|F^{\prime }=4,M^{\prime }=4\rangle$ (left column) and $|F=2,M=2\rangle$ and $|F^{\prime }=4,M^{\prime }=3\rangle$ (right column). The quantization axis is $x_3=y$, the atom is positioned on the positive side of the $x$ axis, and the hybrid modes are counterclockwise quasicircularly polarized. Other parameters are as for Fig. 2.

Figure 6

Radial dependencies of the rms Rabi frequency ${\overline{\mathrm{\Omega }}}_{F{F}^{\prime }}$ for different guided modes. The hfs levels are $F=2$ and $F^{\prime }=4$. The hybrid modes are quasicircularly polarized and the quantization axis is arbitrary. Other parameters are as for Fig. 2.

Figure 7

Radial dependencies of the oscillator strength $f_{F{F}^{\prime }}$ for different guided modes. Parameters used are as for Fig. 6.

Figure 8

The rms Rabi frequency ${\overline{\mathrm{\Omega }}}_{F{F}^{\prime }}$ as a function of the fiber radius $a$ for different guided modes. The atom is positioned on the fiber surface. The hybrid modes are quasicircularly polarized and the quantization axis is arbitrary. Other parameters are as for Fig. 2. The vertical dotted lines indicate the positions of the cutoffs for higher-order modes.

Figure 9

Oscillator strength $f_{F{F}^{\prime }}$ as a function of the fiber radius $a$ for different guided modes. Parameters used are as for Fig. 8.

Figure 10

Radial dependencies of the oscillator-strength enhancement factor ${\eta }_{\mathrm{osc}}$ for different guided modes. The hybrid modes are quasicircularly polarized and the quantization axis is arbitrary. Other parameters are as for Fig. 2.

Figure 11

Oscillator-strength enhancement factor ${\eta }_{\mathrm{osc}}$ as a function of the fiber radius $a$ for different guided modes. The atom is positioned on the fiber surface. The hybrid modes are quasicircularly polarized and the quantization axis is arbitrary. Other parameters are as for Fig. 2. The vertical dotted lines indicate the positions of the cutoffs for higher-order modes.

Figure 12

Oscillator-strength enhancement factors ${\eta }_{\mathrm{osc}}$ for the quasilinearly polarized ${\mathrm{HE}}_{11}$ and ${\mathrm{HE}}_{21}$ modes as functions of the radial distance $r$ at different azimuthal angles $\phi$. The orientation angle of the quasilinear polarization axis is ${\phi }_{\mathrm{pol}}=0$ and the quantization axis is arbitrary. Other parameters are as for Fig. 2. For comparison, the results for the corresponding quasicircularly polarized hybrid modes are shown by the dotted black curves.

Figure 13

Oscillator-strength enhancement factors ${\eta }_{\mathrm{osc}}$ for the quasilinearly polarized ${\mathrm{HE}}_{11}$ and ${\mathrm{HE}}_{21}$ modes as functions of the fiber radius $a$. The atom is positioned on the fiber surface at different azimuthal angles $\phi$. The orientation angle of the quasilinear polarization axis is ${\phi }_{\mathrm{pol}}=0$ and the quantization axis is arbitrary. Other parameters are as for Fig. 2. The vertical dotted line indicates the position of the cutoff for the ${\mathrm{HE}}_{21}$ mode. For comparison, the results for the corresponding quasicircularly polarized hybrid modes are shown by the dotted black curves.

Figure 14

Oscillator-strength enhancement factors ${\eta }_{\mathrm{osc}}$ for the quasilinearly polarized ${\mathrm{HE}}_{11}$ and ${\mathrm{HE}}_{21}$ modes as functions of the azimuthal angle $\phi$ for the position of the atom in the fiber cross-section plane. The orientation angle of the quasilinear polarization axis is ${\phi }_{\mathrm{pol}}=0$ and the quantization axis is arbitrary. Other parameters are as for Fig. 2.

Figure 15

Oscillator-strength enhancement factors ${\eta }_{\mathrm{osc}}$ for the quasilinearly polarized ${\mathrm{HE}}_{11}$ and ${\mathrm{HE}}_{21}$ modes as functions of the position of the atom in the fiber cross-section plane. The orientation angle of the quasilinear polarization axis is ${\phi }_{\mathrm{pol}}=0$ and the quantization axis is arbitrary. Other parameters are as for Fig. 2.