Electromagnetically induced transparency for guided light in an atomic array outside an optical nanofiber

We study the propagation of guided light along an array of three-level atoms in the vicinity of an optical nanofiber under the condition of electromagnetically induced transparency. We examine two schemes of atomic levels and field polarizations where the guided probe field is quasilinearly polarized along the major or minor principal axis, which is parallel or perpendicular, respectively, to the radial direction of the atomic position. Our numerical calculations indicate that 200 cesium atoms in a linear array with a length of 100 $\mu \mathrm{m}$ at a distance of 200 nm from the surface of a nanofiber with a radius of 250 nm can slow down the speed of guided probe light by a factor of about $3.5\times {10}^6$ (the corresponding group delay is about 1.17 $\mu \mathrm{s})$. In the neighborhood of the Bragg resonance, a significant fraction of the guided probe light can be reflected back with a negative group delay. The reflectivity and the group delay of the reflected field do not depend on the propagation direction of the probe field. However, when the input guided light is quasilinearly polarized along the major principal axis, the transmittivity and the group delay of the transmitted field substantially depend on the propagation direction of the probe field. Under the Bragg resonance condition, an array of atoms prepared in an appropriate internal state can transmit guided light polarized along the major principal in one specific direction even in the limit of infinitely large atom numbers. The directionality of transmission of guided light through the array of atoms is a consequence of the existence of a longitudinal component of the guided light field as well as the ellipticity of both the field polarization and the atomic dipole vector.

Figure 1

Left column: An array of atoms outside an optical nanofiber with a plane-wave control field ${\mathcal{E}}_c$, a guided probe field ${\mathcal{E}}_p$, and a guided reflected field ${\mathcal{E}}_b$. The atomic array is aligned parallel to the fiber axis $z$ and lies in the $zx$ plane of the Cartesian coordinate frame $\{x,y,z\}$. The probe field ${\mathcal{E}}_p$ is quasilinearly polarized along the major principal axis $x$ in scheme (a) and the minor principal axis $y$ in scheme (b). The control field ${\mathcal{E}}_c$ propagates along the axis $x$ and is linearly polarized along the axis $y$ in scheme (a), and propagates along the axis $y$ and is circularly polarized in scheme (b). Right column: The diagram of atomic energy levels and electric dipole transitions for the analysis. Atomic cesium is used. The quantization axis $z_Q$ is the axis $y$. In both schemes (a) and (b), the upper level is $|e\rangle =|6P_{3/2},F^{\prime }=4,M^{\prime }=\pm 4\rangle$. In scheme (a), the lower levels are $|g\rangle =|6S_{1/2},F=3,M=\pm 3\rangle$ and $|h\rangle =|6S_{1/2},F=4,M=\pm 4\rangle$. In scheme (b), the lower levels are $|g\rangle =|6S_{1/2},F=4,M=\pm 4\rangle$ and $|h\rangle =|6S_{1/2},F=3,M=\pm 3\rangle$.

Figure 2

EIT of an $x$-polarized guided probe field propagating along an array of cesium atoms with a $\sigma$-type probe transition. The optical depth per atom ${\mathcal{D}}_f$ (upper row), the phase shift per atom ${\mathrm{\Theta }}_f$ (middle row), and the group-velocity reduction factor $c/V_g^{\left(f\right)}$ (lower row) are plotted as functions of the detuning $\mathrm{\Delta }$. The working levels are $|e\rangle =|6P_{3/2},F^{\prime }=4,M^{\prime }=4\rangle ,|g\rangle =|6S_{1/2},F=3,M=3\rangle$, and $|h\rangle =|6S_{1/2},F=4,M=4\rangle$. The lower-level decoherence rate is ${\mathrm{\Gamma }}_{hg}=2\pi \times 50$ kHz. The probe field propagates in the positive (left column) or negative (right column) direction of the fiber axis $z$. The fiber radius is $a=250$ nm. The distance from the atoms to the fiber surface is $r-a=200$ nm. The atom-number density is $n_A=1/\mathrm{\Lambda }$ with $\mathrm{\Lambda }=498.13$ nm. The control field is at exact resonance with the atomic transition $|e\rangle \leftrightarrow |h\rangle$ and is a plane wave propagating along the $x$ direction and polarized along the $y$ direction. The intensity of the control field is $I_c=1$ mW/${\mathrm{cm}}^2$ (the corresponding Rabi frequency is ${\mathrm{\Omega }}_c=0.46{\gamma }_0)$.

Figure 3

EIT of a $y$-polarized guided probe field propagating along an array of cesium atoms with a $\pi$-type probe transition. The optical depth per atom $\mathcal{D}$ (a), the phase shift per atom $\mathrm{\Theta }$ (b), and the group-velocity reduction factor $c/V_g$ (c) are plotted as functions of the detuning $\mathrm{\Delta }$. The working levels are $|e\rangle =|6P_{3/2},F^{\prime }=4,M^{\prime }=4\rangle ,|g\rangle =|6S_{1/2},F=4,M=4\rangle$, and $|h\rangle =|6S_{1/2},F=3,M=3\rangle$. The lower-level decoherence rate is ${\mathrm{\Gamma }}_{hg}=2\pi \times 50$ kHz. The atom-number density is $n_A=1/\mathrm{\Lambda }$ with $\mathrm{\Lambda }=498.13$ nm. The control field is at exact resonance with the atomic transition $|e\rangle \leftrightarrow |h\rangle$ and is a plane wave propagating along the $y$ direction with the counterclockwise circular polarization. The intensity of the control field is $I_c=1$ mW/${\mathrm{cm}}^2$ (the corresponding Rabi frequency is ${\mathrm{\Omega }}_c=0.43{\gamma }_0)$. Other parameters are as for Fig. 2. The results do not depend on the propagation direction of the guided probe field.

Figure 4

Transmittivity $|T_A^{\left(f\right)}{|}^2$ (a) and reflectivity $|R_A{|}^2$ (b) of an $x$-polarized guided probe field as functions of the detuning $\mathrm{\Delta }$ in the homogeneous-medium approximation. The atom-number density is $n_A=1/\mathrm{\Lambda }$ with $\mathrm{\Lambda }=498.13$ nm. The medium length is $L=(N-1)\mathrm{\Lambda }$ with $N=200$. Other parameters are as for Fig. 2.

Figure 5

Transmittivity $|T_A{|}^2$ (a) and reflectivity $|R_A{|}^2$ (b) of a $y$-polarized guided probe field as functions of the detuning $\mathrm{\Delta }$ in the homogeneous-medium approximation. The atom-number density is $n_A=1/\mathrm{\Lambda }$ with $\mathrm{\Lambda }=498.13$ nm. The medium length is $L=(N-1)\mathrm{\Lambda }$ with $N=200$. Other parameters are as for Fig. 3.

Figure 6

Transmittivity $|T_A^{\left(f\right)}{|}^2$ (a) and reflectivity $|R_A{|}^2$ (b) of an $x$-polarized guided probe field as functions of the detuning $\mathrm{\Delta }$ in the phase-matching approximation. The array period is $\mathrm{\Lambda }=498.13$ nm and the array length is $L=(N-1)\mathrm{\Lambda }$, where $N=200$. Other parameters are as for Fig. 2.

Figure 7

Transmittivity $|T_A{|}^2$ (a) and reflectivity $|R_A{|}^2$ (b) of a $y$-polarized guided probe field as functions of the detuning $\mathrm{\Delta }$ in the phase-matching approximation. The array period is $\mathrm{\Lambda }=498.13$ nm and the array length $L=(N-1)\mathrm{\Lambda }$, where $N=200$. Other parameters are as for Fig. 3.

Figure 8

Transmittivity $|T_N^{\left(f\right)}{|}^2$ (a) and reflectivity $|R_N{|}^2$ (b) of an $x$-polarized guided probe field as functions of the detuning $\mathrm{\Delta }$. The array period is $\mathrm{\Lambda }=498.13$ nm, which is far from the Bragg resonance. The atom number is $N=200$. Other parameters are as for Fig. 2.

Figure 9

Group delays ${\tau }_T^{(+)}$ (a), ${\tau }_T^{(-)}$ (b), and ${\tau }_R$ (c) of an $x$-polarized guided probe field as functions of the detuning $\mathrm{\Delta }$. The array period is $\mathrm{\Lambda }=498.13$ nm, which is far from the Bragg resonance. The atom number is $N=200$. Other parameters are as for Fig. 2. The dotted red line is for the zero group delay and is a guide to the eye.

Figure 10

Transmittivity $|T_N{|}^2$ (a) and reflectivity $|R_N{|}^2$ (b) of a $y$-polarized guided probe field as functions of the detuning $\mathrm{\Delta }$. The array period is $\mathrm{\Lambda }=498.13$ nm, which is far from the Bragg resonance. The atom number is $N=200$. Other parameters are as for Fig. 3.

Figure 11

Group delays ${\tau }_T$ (a) and ${\tau }_R$ (b) of a $y$-polarized guided probe field as functions of the detuning $\mathrm{\Delta }$. The array period is $\mathrm{\Lambda }=498.13$ nm, which is far from the Bragg resonance. The atom number is $N=200$. Other parameters are as for Fig. 3. The dotted red line is for the zero group delay and is a guide to the eye.

Figure 12

Time dependencies of the normalized intensities $|{\mathcal{A}}_{\mathrm{out},\mathrm{ref}}/{\mathcal{A}}_0{|}^2$ of the transmitted and reflected pulses in the case of the $x$-polarization scheme. The input pulse is of the Gaussian form, with the central frequency ${\omega }_p={\omega }_0$, the pulse length ${\tau }_0=2\mu \mathrm{s}$, and the peak time $t=0$. The number of atoms is $N=200$. The array period is $\mathrm{\Lambda }=498.13$ nm. Other parameters are as for Fig. 2. The dotted red line is for the input pulse peak time $t=0$ and is a guide to the eye.

Figure 13

Time dependencies of the normalized intensities $|{\mathcal{A}}_{\mathrm{out},\mathrm{ref}}/{\mathcal{A}}_0{|}^2$ of the transmitted and reflected pulses in the case of the $y$-polarization scheme. The input pulse is of the Gaussian form, with the central frequency ${\omega }_p={\omega }_0$, the pulse length ${\tau }_0=2\mu \mathrm{s}$, and the peak time $t=0$. The number of atoms is $N=200$. The array period is $\mathrm{\Lambda }=498.13$ nm. Other parameters are as for Fig. 3. The dotted red line is for the input pulse peak time $t=0$ and is a guide to the eye.

Figure 14

Transmittivity $|T_N^{\left(f\right)}{|}^2$ (a) and reflectivity $|R_N{|}^2$ (b) of the atomic array in the $x$-polarization scheme as functions of the detuning $\mathrm{\Delta }$. The period of the array is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. The number of atoms in the array is $N=200$. Other parameters are as for Fig. 2.

Figure 15

Group delays ${\tau }_T^{(+)}$ (a), ${\tau }_T^{(-)}$ (b), and ${\tau }_R$ (c) in the $x$-polarization scheme as functions of the detuning $\mathrm{\Delta }$. The period of the array is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. The atom number is $N=200$. Other parameters are as for Fig. 2. The dotted red line is for the zero group delay and is a guide to the eye.

Figure 16

Transmittivity $|T_N{|}^2$ (a) and reflectivity $|R_N{|}^2$ (b) of the atomic array in the $y$-polarization scheme as functions of the detuning $\mathrm{\Delta }$. The period of the array is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. The atom number is $N=200$. Other parameters are as for Fig. 3.

Figure 17

Group delays ${\tau }_T$ (a) and ${\tau }_R$ (b) in the $y$-polarization scheme as functions of the detuning $\mathrm{\Delta }$. The period of the array is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. The atom number is $N=200$. Other parameters are as for Fig. 3. The dotted red line is for the zero group delay and is a guide to the eye.

Figure 18

Dependencies of the transmittivities $|T_N^{(+)}{|}^2$ and $|T_N^{(-)}{|}^2$, the reflectivity $|R_N{|}^2$, and the group delays ${\tau }_T^{(+)},{\tau }_T^{(-)}$, and ${\tau }_R$ on the atom number $N$ in the case of the $x$-polarization scheme. The period of the array is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. The detuning of the probe field is $\mathrm{\Delta }=0$. Other parameters are as for Fig. 2.

Figure 19

Dependencies of the transmittivity $|T_N{|}^2$, the reflectivity $|R_N{|}^2$, and the group delays ${\tau }_T$ and ${\tau }_R$ on the atom number $N$ in the case of the $y$-polarization scheme. The period of the array is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. The detuning of the probe field is $\mathrm{\Delta }=0$. Other parameters are as for Fig. 3.

Figure 20

Dependencies of the transmittivities $|T_N^{(+)}{|}^2$ and $|T_N^{(-)}{|}^2$, the reflectivity $|R_N{|}^2$, and the group delays ${\tau }_T^{(+)},{\tau }_T^{(-)}$, and ${\tau }_R$ on the array period $\mathrm{\Lambda }$ in the case of the $x$-polarization scheme. The detuning of the probe field is $\mathrm{\Delta }=0$. The number of atoms in the array is $N=200$. Other parameters are as for Fig. 2.

Figure 21

Dependencies of the transmittivity $|T_N{|}^2$, the reflectivity $|R_N{|}^2$, and the group delays ${\tau }_T$ and ${\tau }_R$ on the array period $\mathrm{\Lambda }$ in the case of the $y$-polarization scheme. The detuning of the probe field is $\mathrm{\Delta }=0$. The number of atoms in the array is $N=200$. Other parameters are as for Fig. 3.

Figure 22

Dependencies of the transmittivities $|T_N^{(+)}{|}^2$ and $|T_N^{(-)}{|}^2$, the reflectivity $|R_N{|}^2$, and the group delays ${\tau }_T^{(+)},{\tau }_T^{(-)}$, and ${\tau }_R$ on the control field intensity $I_c$ in the case of the $x$-polarization scheme. The period of the array is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. The detuning of the probe field is $\mathrm{\Delta }=0$. The number of atoms in the array is $N=200$. Other parameters are as for Fig. 2.

Figure 23

Dependencies of the transmittivity $|T_N{|}^2$, the reflectivity $|R_N{|}^2$, and the group delays ${\tau }_T$ and ${\tau }_R$ on the control field intensity $I_c$ in the case of the $y$-polarization scheme. The period of the array is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. The detuning of the probe field is $\mathrm{\Delta }=0$. The number of atoms in the array is $N=200$. Other parameters are as for Fig. 3.

Figure 24

Time dependencies of the normalized intensities $|{\mathcal{A}}_{\mathrm{out},\mathrm{ref}}/{\mathcal{A}}_0{|}^2$ of the transmitted and reflected pulses in the case of the $x$-polarization scheme. The input pulse is of the Gaussian form, with the central frequency ${\omega }_p={\omega }_0$, the pulse length ${\tau }_0=2\mu \mathrm{s}$, and the peak time $t=0$. The number of atoms is $N=200$. The array period is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. Other parameters used are as for Fig. 2. The dotted red line is for the input-pulse peak time $t=0$ and is a guide to the eye.

Figure 25

Time dependencies of the normalized intensities $|{\mathcal{A}}_{\mathrm{out},\mathrm{ref}}/{\mathcal{A}}_0{|}^2$ of the transmitted and reflected pulses in the case of the $y$-polarization scheme. The input pulse is of the Gaussian form, with the central frequency ${\omega }_p={\omega }_0$, the pulse length ${\tau }_0=2\mu \mathrm{s}$, and the peak time $t=0$. The number of atoms is $N=200$. The array period is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. Other parameters used are as for Fig. 3. The dotted red line is for the input-pulse peak time $t=0$ and is a guide to the eye.

Figure 26

Photonic band gaps in the case of the $x$-polarization scheme with $\mathrm{\Delta }\mathrm{\Lambda }=0$. The transmittivities $|T_N^{(+)}{|}^2$ (a) and $|T_N^{(-)}{|}^2$ (b) and the reflectivity $|R_N{|}^2$ (c) are plotted as functions of the detuning $\mathrm{\Delta }$. The period of the array is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. The number of atoms in the array is $N=200000$. The intensity of the control field is $I_c=20$ mW/${\mathrm{cm}}^2$ (the corresponding Rabi frequency is ${\mathrm{\Omega }}_c=2.06{\gamma }_0)$. Other parameters are as for Fig. 2. The insets show the narrow structures around the point $\mathrm{\Delta }=0$.

Figure 27

Photonic band gaps in the case of the $y$-polarization scheme with $\mathrm{\Delta }\mathrm{\Lambda }=0$. The transmittivity $|T_N{|}^2$ (a) and the reflectivity $|R_N{|}^2$ (b) are plotted as functions of the detuning $\mathrm{\Delta }$. The period of the array is $\mathrm{\Lambda }=745.16$ nm, which satisfies the second-order Bragg resonance condition. The number of atoms in the array is $N=200000$. The intensity of the control field is $I_c=20$ mW/${\mathrm{cm}}^2$ (the corresponding Rabi frequency is ${\mathrm{\Omega }}_c=1.94{\gamma }_0)$. Other parameters are as for Fig. 3. The insets show the narrow structures around the point $\mathrm{\Delta }=0$.

Figure 28

As Fig. 26 but for $\mathrm{\Delta }\mathrm{\Lambda }=0.3$ nm.

Figure 29

As Fig. 27 but for $\mathrm{\Delta }\mathrm{\Lambda }=0.3$ nm.