Propagation of nanofiber-guided light through an array of atoms

We study the propagation of nanofiber-guided light through an array of atomic cesium, taking into account the transitions between the hyperfine levels $6S_{1/2}F=4$ and $6P_{3/2}{F}^{\prime }=5$ of the $D_2$ line. We derive the coupled-mode propagation equation, the input-output equation, the scattering matrix, the transfer matrix, and the reflection and transmission coefficients for the guided field in the linear, quasistationary, weak-excitation regime. We show that, when the initial distribution of populations of atomic ground-state sublevels is independent of the magnetic quantum number, the quasilinear polarizations along the principal axes $x$ and $y$, which are parallel and perpendicular, respectively, to the radial direction of the atomic position, are not coupled to each other in the linear coherent scattering process. When the guided probe field is quasilinearly polarized along the major principal axis $x$, forward and backward scattering have different characteristics. We find that, when the array period is far from the Bragg resonance, the backward scattering is weak. Under the Bragg resonance, most of the guided probe light can be reflected back in a broad region of field detunings even though there is an irreversible decay channel into radiation modes. When the atom number is large enough, two different band gaps may be formed, whose properties depend on the polarization of the guided probe field.

Figure 1

Probing an array of atoms by a guided light field propagating along a thin optical fiber.

Figure 2

Dependencies of the optical depth per atom $\mathcal{D}$ on the radial position of the atom (a) and the detuning of the guided probe field (b) in the cases where the guided probe field is quasicircularly polarized (solid black lines), quasilinearly polarized along the $x$ direction (dashed red lines), and quasilinearly polarized along the $y$ direction (dotted green lines). The fiber radius is $a=250$ nm and the light wavelength is $\lambda =852$ nm. The atom is located on the $x$ axis. In (a), the field is tuned to exact resonance with the atom. In (b), the radial position of the atom is $r/a=1.8$.

Figure 3

Dependencies of the single-atom reflectivity ${\left|R\right|}^2$ (upper row) and transmittivity ${\left|T\right|}^2$ (lower row) on the radial position of the atom (left column) and the detuning of the guided probe field (right column) in the cases where the guided probe field is quasilinearly polarized along the $x$ direction (solid blue lines) and the $y$ direction (dashed red lines). The atom is located on the $x$ axis. In (a) and (c), the field is tuned to exact resonance with the atom. In (b) and (d), the radial position of the atom is $r/a=1.8$. Other parameters are as in Fig. 2.

Figure 4

Powers $P_{fl}$ of quasicircularly guided modes $fl$, normalized to the input-field power $P_{\mathrm{input}}$, as functions of the number $N$ of atoms in the array. The input guided probe field propagates along the fiber in the positive direction $+z$ and is counterclockwise $\left(l=+\right)$ quasicircularly polarized. The distance from the atoms to the fiber surface is $r-a=200$ nm. The period of the array is $\Lambda =498$ nm. The detuning of the field is $\delta =0$. Other parameters are as in Fig. 2.

Figure 5

Powers $P_{fx}$ of quasilinearly guided modes $fx$, normalized to the input-field power $P_{\mathrm{input}}$, as functions of the number $N$ of atoms in the array. The input guided probe field propagates along the fiber in the positive direction $+z$ and is quasilinearly polarized along the major principal axis $x$, which is the radial direction of the atomic position. The distance from the atoms to the fiber surface is $r-a=200$ nm. The period of the array is $\Lambda =498$ nm. The detuning of the field is $\delta =0$. Other parameters are as in Fig. 2. The powers $P_{fy}$ are zero in the considered case and are therefore not plotted.

Figure 6

Powers $P_{fy}$ of quasilinearly guided modes $fy$, normalized to the input-field power $P_{\mathrm{input}}$, as functions of the number $N$ of atoms in the array. The input guided probe field propagates along the fiber in the positive direction $+z$ and is quasilinearly polarized along the minor principal axis $y$, which is perpendicular to the radial direction of the atomic position. The distance from the atoms to the fiber surface is $r-a=200$ nm. The period of the array is $\Lambda =498$ nm. The detuning of the field is $\delta =0$. Other parameters are as in Fig. 2. The powers $P_{fx}$ are zero in the considered case and are therefore not plotted.

Figure 7

As Fig. 4 but the period of the array is $\Lambda =2\pi /{\beta }_L=745$ nm and the field detuning is $\delta =0$ (solid blue curves) or $2\pi \times 10$ MHz (dashed red curves).

Figure 8

As Fig. 5 but the period of the array is $\Lambda =2\pi /{\beta }_L=745$ nm and the field detuning is $\delta =0$ (solid blue curves) or $2\pi \times 10$ MHz (dashed red curves).

Figure 9

As Fig. 6 but the period of the array is $\Lambda =2\pi /{\beta }_L=745$ nm and the field detuning is $\delta =0$ (solid blue curves) or $2\pi \times 10$ MHz (dashed red curves).

Figure 10

Normalized total power $P_{\mathrm{tot}}/P_{\mathrm{input}}$ of the guided fields as a function of the number $N$ of atoms in the array in the cases where the probe field is quasicircularly polarized (a), $x$ polarized (b), and $y$ polarized (c). The distance from the atoms to the fiber surface is $r-a=200$ nm. The period of the array is $\Lambda =498$ nm (dashed red curves) or $\Lambda =745$ nm (solid blue curves). The field detuning is $\delta =0$. Other parameters are as in Fig. 2.

Figure 11

Powers $P_{f\xi }$ of the transmitted (left column) and reflected (middle column) guided fields and their total power $P_{\mathrm{tot}}$ (right column), normalized to the input-field power $P_{\mathrm{input}}$, as functions of the array period $\Lambda$. The guided probe field is quasilinearly polarized along the major principal direction $x$ (upper row) or the minor principal direction $y$ (lower row). The number of atoms in the array is $N=200$. The distance from the atoms to the fiber surface is $r-a=200$ nm. The field detuning is $\delta =0$. Other parameters are as in Fig. 2.

Figure 12

Powers $P_{f\xi }$ of the transmitted (left column) and reflected (middle column) guided fields and their total power $P_{\mathrm{tot}}$ (right column), normalized to the input-field power $P_{\mathrm{input}}$, as functions of the field detuning $\delta$. The guided probe field is quasilinearly polarized along the major principal direction $x$ (upper row) or the minor principal direction $y$ (lower row). The number of atoms in the array is $N=200$. The distance from the atoms to the fiber surface is $r-a=200$ nm. The array period is $\Lambda =745$ nm, which satisfies the Bragg resonance condition for the field frequency ${\omega }_L={\omega }_0$. Other parameters are as in Fig. 2.

Figure 13

Reflectivity $|R_N{|}^2$ of the atomic array as a function of the field detuning $\delta$ for three different values of the atom number $N$. The guided probe field is quasilinearly polarized along the major principal direction $x$ (left column) or the minor principal direction $y$ (right column). The number of atoms in the array is $N=400$ (solid red lines), 800 (dashed green lines), and 1600 (dotted blue lines). The distance from the atoms to the fiber surface is $r-a=200$ nm. The array period is $\Lambda =745$ nm, which satisfies the Bragg resonance condition for the field frequency ${\omega }_L={\omega }_0$. Other parameters are as in Fig. 2.

Figure 14

Reflectivity $|R_N{|}^2$ (a) and transmittivity $|T_N{|}^2$ (b) of the atomic array as functions of the field detuning $\delta$ for the atom number $N=150000$. The guided probe field is quasilinearly polarized along the major principal direction $x$. The distance from the atoms to the fiber surface is $r-a=200$ nm. The array period is $\Lambda =745$ nm. Other parameters are as in Fig. 2.

Figure 15

As Fig. 14 but the guided probe field is quasilinearly polarized along the minor principal direction $y$. The insets show the narrow peak structures around the point $\delta =0$.

Figure 16

Reflectivity $|R_{\infty }{|}^2$ (upper row) and phase ${\phi }_{R_{\infty }}$ of the reflection coefficient $R_{\infty }$ (lower row) of an infinite atomic array as functions of the field detuning $\delta$. The guided probe field is quasilinearly polarized along the major principal direction $x$ (left column) or the minor principal direction $y$ (right column). The distance from the atoms to the fiber surface is $r-a=200$ nm. The array period is $\Lambda =745$ nm. Other parameters are as in Fig. 2.