Nanofiber-mediated chiral radiative coupling between two atoms

We investigate the radiative coupling between two two-level atoms with arbitrarily polarized dipoles in the vicinity of a nanofiber. We present a systematic derivation for the master equation, the single- and cross-atom decay coefficients, and the dipole-dipole interaction coefficients for the atoms interacting with the vacuum of the field in the guided and radiation modes of the nanofiber. We study numerically the case where the atomic dipoles are circularly polarized. In this case, the rate of emission depends on the propagation direction along the fiber axis and, hence, the radiative interaction between the atoms is chiral. We examine the time evolution of the atoms for different initial states. We calculate the fluxes and mean numbers of photons spontaneously emitted into guided modes in the positive and negative directions of the fiber axis. We show that the chiral radiative coupling modifies the collective emission of the atoms. We observe that the modifications strongly depend on the initial state of the atomic system, the radiative transfer direction, the distance between the atoms, and the distance from the atoms to the fiber surface.

Figure 1

(a) Two two-level atoms in the vicinity of a nanofiber. (b) Chiral emission of an individual atom with a dipole rotating in the meridional plane. (c) Chiral coupling between two atoms with dipoles rotating in the meridional plane.

Figure 2

Decay rate ${\gamma }_{jj}^{\left(\mathrm{g}\right)}$ into guided modes (solid black lines) and its directional components ${\gamma }_{jj}^{\left(\mathrm{g}\right)f}$ for $f=+$ (dashed red lines) and $f=-$ (dotted blue lines), relative to the free-space spontaneous decay rate ${\gamma }_0$, as functions of (a) the radial position $r_j-a$ and (b) the axial position $z_j$ of atom $j$. One coordinate of the atom is varied, while the two others are fixed as (a) ${\phi }_j=0$ and $z_j=0$ and (b) $r_j=a$ and ${\phi }_j=0$. The dipole matrix element of the atom is ${\mathbf{d}}_j=(d/\sqrt{2})(i,0,-1)$, corresponding to the ${\sigma }_{+}$-polarized transition with respect to the $y$ quantization axis. The fiber radius is $a=250\mathrm{nm}$. The refractive indices of the fiber and the surrounding vacuum are $n_1=1.45$ and $n_2=1$, respectively. The wavelength of the atomic transition is ${\lambda }_0=852\mathrm{nm}$.

Figure 3

Decay rate ${\gamma }_{jj}^{\left(\mathrm{r}\right)}$ into radiation modes (solid black lines) and its directional components ${\gamma }_{jj}^{\left(\mathrm{r}\right)f}$ for $f=+$ (dashed red lines) and $f=-$ (dotted blue lines), relative to the free-space spontaneous decay rate ${\gamma }_0$, as functions of (a) the radial position $r_j-a$ and (b) the axial position $z_j$ of atom $j$. Other parameters are as for Fig. 2.

Figure 4

Absolute value $|{\gamma }_{12}^{\left(\mathrm{g}\right)}|$ of the coefficient of the cross-atom decay into guided modes (solid black lines) and its directional components $|{\gamma }_{12}^{\left(\mathrm{g}\right)f}|$ for $f=+$ (dashed red lines) and $f=-$ (dotted blue lines), relative to the free-space spontaneous decay rate ${\gamma }_0$, as functions of (a) the radial position $r_2-a$ and (b) the axial position $z_2$ of atom 2. The position of atom 1 is fixed at $r_1=a, {\phi }_1=0$, and $z_1=0$. One coordinate of atom 2 is varied, while the two others are fixed as (a) ${\phi }_2=0$ and $z_2=0$ and (b) $r_2=a$ and ${\phi }_2=0$. The dipole matrix elements of both atoms are ${\mathbf{d}}_1={\mathbf{d}}_2=(d/\sqrt{2})(i,0,-1)$, corresponding to the ${\sigma }_{+}$-polarized transitions with respect to the $y$ quantization axis. Other parameters are as for Fig. 2.

Figure 5

Absolute value $|{\gamma }_{12}^{\left(\mathrm{r}\right)}|$ of the coefficient of the cross-atom decay into radiation modes (solid black lines) and its directional components $|{\gamma }_{12}^{\left(\mathrm{r}\right)f}|$ for $f=+$ (dashed red lines) and $f=-$ (dotted blue lines), relative to the free-space spontaneous decay rate ${\gamma }_0$, as functions of (a) the radial position $r_2-a$ and (b) the axial position $z_2$ of atom 2. Other parameters are as for Figs. 2 and 4.

Figure 6

Real (solid red lines) and imaginary (dashed blue lines) parts of the coefficient ${\gamma }_{12}^{\left(\mathrm{g}\right)}$ for cross-atom decay into guided modes, relative to the free-space spontaneous decay rate ${\gamma }_0$, as functions of (a) the radial position $r_2-a$ and (b) the axial position $z_2$ of atom 2. Other parameters are as for Figs. 2 and 4.

Figure 7

Real (solid red lines) and imaginary (dashed blue lines) parts of the coefficient ${\gamma }_{12}^{\left(\mathrm{r}\right)}$ for cross-atom decay into radiation modes, relative to the free-space spontaneous decay rate ${\gamma }_0$, as functions of (a) the radial position $r_2-a$ and (b) the axial position $z_2$ of atom 2. Other parameters are as for Figs. 2 and 4.

Figure 8

Absolute value $|{\gamma }_{12}|$ (top row) and phase ${\phi }_{12}$ (bottom row) of the total cross-atom decay coefficient ${\gamma }_{12}$ as functions of the radial position $r_2-a$ (left column) and the axial position $z_2$ (right column) of atom 2. Other parameters are as for Figs. 2 and 4.

Figure 9

Real (solid red lines) and imaginary (dashed blue lines) parts of the guided-mode-mediated dipole-dipole interaction coefficient ${\mathrm{\Omega }}_{12}^{\left(\mathrm{g}\right)}$, relative to the free-space spontaneous decay rate ${\gamma }_0$, as functions of (a) the radial position $r_2-a$ and (b) the axial position $z_2$ of atom 2. Parameters used are as for Figs. 2 and 4. In (a), we formally take the limit $z_2\to z_1$ under the condition $z_2>z_1$.

Figure 10

Free-space dipole-dipole interaction coefficient ${\mathrm{\Omega }}_{12}^{\left(\mathrm{vac}\right)}$, relative to the free-space spontaneous decay rate ${\gamma }_0$, as a function of the distance $z_{21}=z_2-z_1$ between the atoms along the axis $z$. The other coordinates of the atoms are $r_1=r_2$ and ${\phi }_1={\phi }_2$. The dipole matrix elements of the atoms are ${\mathbf{d}}_1={\mathbf{d}}_2=(d/\sqrt{2})(i,0,-1)$, corresponding to the ${\sigma }_{+}$-polarized transitions with respect to the $y$ quantization axis. The dotted line is the zero horizontal axis and is a guide to the eye.

Figure 11

Absolute value $|{\mathrm{\Omega }}_{12}|$ (top row) and phase ${\theta }_{12}$ (bottom row) of the total dipole-dipole interaction coefficient ${\mathrm{\Omega }}_{12}$ as functions of the radial position $r_2-a$ (left column) and the axial position $z_2$ (right column) of atom 2. Other parameters are as for Figs. 2 and 4.

Figure 12

Time evolution of the populations (a) ${\rho }_{\mathrm{exc}}^{\left(1\right)}$ and (b) ${\rho }_{\mathrm{exc}}^{\left(2\right)}$ of the upper levels of atoms 1 and 2, respectively, in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_1\rangle$ (solid red lines) or $|{\psi }_2\rangle$ (dashed blue lines). The coordinates of the atoms are $r_1-a=r_2-a=200\mathrm{nm}, {\phi }_1={\phi }_2=0$, and $z_2-z_1=150\mathrm{nm}$. The dipole matrix elements of the atoms are ${\mathbf{d}}_1={\mathbf{d}}_2=(d/\sqrt{2})(i,0,-1)$, corresponding to ${\sigma }_{+}$-polarized transitions with respect to the $y$ quantization axis. Other parameters are as for Figs. 2 and 4.

Figure 13

Time evolution of the photon fluxes (a) $P_{\mathrm{gd}}^{(+)}$, (b) $P_{\mathrm{gd}}^{(-)}$, and (c) $P_{\mathrm{gd}}$ in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_1\rangle$ (solid red lines) or $|{\psi }_2\rangle$ (dashed blue lines). Other parameters are as for Figs. 2, 4, and 12. The fluxes are normalized to the decay rate ${\gamma }_0$ of a single atom in free space. For comparison, the results for the case of a single excited atom are shown by the dotted black lines.

Figure 14

Time evolution of the photon fluxes (a) $P_{\mathrm{rad}}$ and (b) $P_{\mathrm{tot}}$ in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_1\rangle$ (solid red lines) or $|{\psi }_2\rangle$ (dashed blue lines). Other parameters are as for Figs. 2, 4, and 12. The fluxes are normalized to the decay rate ${\gamma }_0$ of a single atom in free space. The dotted black lines are for the case of a single excited atom.

Figure 15

Same as Fig. 12 except for $z_2-z_1=100\mathrm{nm}$.

Figure 16

Same as Fig. 13 except for $z_2-z_1=100\mathrm{nm}$.

Figure 17

Same as Fig. 12 except for $z_2-z_1=960\mathrm{nm}$.

Figure 18

Same as Fig. 13 except for $z_2-z_1=960\mathrm{nm}$.

Figure 19

Dependencies of the mean emitted photon numbers (a) $N_{\mathrm{gd}}^{(+)}$, (b) $N_{\mathrm{gd}}^{(-)}$, and (c) $N_{\mathrm{gd}}$ on the axial separation $z_{21}$ between the atoms in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_1\rangle$ (solid red lines) or $|{\psi }_2\rangle$ (dashed blue lines). The radial and azimuthal coordinates of the atoms are $r_1-a=r_2-a=200\mathrm{nm}$ and ${\phi }_1={\phi }_2=0$, respectively. Other parameters are as for Figs. 2, 4, and 12. The dotted black lines are for the case of a single excited atom.

Figure 20

Dependencies of the mean emitted photon numbers (a) $N_{\mathrm{gd}}^{(+)}$, (b) $N_{\mathrm{gd}}^{(-)}$, and (c) $N_{\mathrm{gd}}$ on the distance $r-a$ from the atoms to the fiber surface in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_1\rangle$ (solid red lines) or $|{\psi }_2\rangle$ (dashed blue lines). The coordinates of the atoms are $r_1=r_2=r, {\phi }_1={\phi }_2=0$, and $z_2-z_1=150\mathrm{nm}$. Other parameters are as for Figs. 2, 4, and 12. The dotted black lines are for the case of a single excited atom.

Figure 21

Time evolution of the concurrence in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_1\rangle$ (solid red lines) or $|{\psi }_2\rangle$ (dashed blue lines). The coordinates of the atoms are $r_1-a=r_2-a=200\mathrm{nm}, {\phi }_1={\phi }_2=0$, and $z_2-z_1=100\mathrm{nm}$ (a), 150 nm (b), and 960 nm (c). Other parameters are as for Figs. 2 and 4. The dotted black lines are for the case of two atoms in free space.

Figure 22

Same as Fig. 21 except for $z_2-z_1=100\mu \mathrm{m}$.

Figure 23

Time evolution of the populations (a) ${\rho }_{\mathrm{exc}}^{\left(1\right)}$ and (b) ${\rho }_{\mathrm{exc}}^{\left(2\right)}$ of the upper levels of atoms 1 and 2, respectively, in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_{\mathrm{sym}}\rangle$ (solid red lines) or $|{\psi }_{\mathrm{asym}}\rangle$ (dashed blue lines). The coordinates of the atoms are $r_1-a=r_2-a=200$ nm, ${\phi }_1={\phi }_2=0$, and $z_2-z_1=125\mathrm{nm}$. The dipole matrix elements of the atoms are ${\mathbf{d}}_1={\mathbf{d}}_2=(d/\sqrt{2})(i,0,-1)$, corresponding to the ${\sigma }_{+}$-polarized transitions with respect to the $y$ quantization axis. Other parameters are as for Figs. 2 and 4.

Figure 24

Time evolution of the photon fluxes (a) $P_{\mathrm{gd}}^{(+)}$, (b) $P_{\mathrm{gd}}^{(-)}$, and (c) $P_{\mathrm{gd}}$ in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_{\mathrm{sym}}\rangle$ (solid red lines) or $|{\psi }_{\mathrm{asym}}\rangle$ (dashed blue lines). Other parameters are as for Figs. 2, 4, and 23. The fluxes are normalized to the decay rate ${\gamma }_0$ of a single atom in free space. The dotted black lines are for the case of a single excited atom.

Figure 25

Time evolution of the photon fluxes (a) $P_{\mathrm{rad}}$ and (b) $P_{\mathrm{tot}}$ in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_{\mathrm{sym}}\rangle$ (solid red lines) or $|{\psi }_{\mathrm{asym}}\rangle$ (dashed blue lines). Other parameters are as for Figs. 2, 4, and 23. The fluxes are normalized to the decay rate ${\gamma }_0$ of a single atom in free space. The dotted black lines are for the case of a single excited atom.

Figure 26

Same as Fig. 23 but for $z_{21}=300\mathrm{nm}$.

Figure 27

Same as Fig. 24 but for $z_{21}=300\mathrm{nm}$.

Figure 28

Same as Fig. 25 but for $z_{21}=300\mathrm{nm}$.

Figure 29

Dependencies of the mean emitted photon numbers (a) $N_{\mathrm{gd}}^{(+)}$, (b) $N_{\mathrm{gd}}^{(-)}$, and (c) $N_{\mathrm{gd}}$ on the axial separation $z_{21}$ between the atoms in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_{\mathrm{sym}}\rangle$ (solid red lines) or $|{\psi }_{\mathrm{asym}}\rangle$ (dashed blue lines). The radial and azimuthal coordinates of the atoms are $r_1-a=r_2-a=200\mathrm{nm}$ and ${\phi }_1={\phi }_2=0$, respectively. Other parameters are as for Figs. 2, 4, and 23. The dotted black lines are for the case of a single excited atom.

Figure 30

Dependencies of the mean emitted photon numbers (a) $N_{\mathrm{gd}}^{(+)}$, (b) $N_{\mathrm{gd}}^{(-)}$, and (c) $N_{\mathrm{gd}}$ on the distance $r-a$ from the atoms to the fiber surface in the cases where the initial state of the two-atom system is $|\psi \left(0\right)\rangle =|{\psi }_{\mathrm{sym}}\rangle$ (solid red lines) or $|{\psi }_{\mathrm{asym}}\rangle$ (dashed blue lines). The coordinates of the atoms are $r_1=r_2=r, {\phi }_1={\phi }_2=0$, and $z_2-z_1=150\mathrm{nm}$. Other parameters are as for Figs. 2, 4, and 23. The dotted lines are for the case of a single excited atom.