Spontaneous emission and energy shifts of a Rydberg rubidium atom close to an optical nanofiber

In this paper, we report on numerical calculations of the spontaneous emission rates and Lamb shifts of a ${}^{87}\mathrm{Rb}$ atom in a Rydberg-excited state $\left(n\le 30\right)$ located close to a silica optical nanofiber. We investigate how these quantities depend on the fiber's radius, the distance of the atom to the fiber, the direction of the atomic angular momentum polarization, as well as the different atomic quantum numbers. We also study the contribution of quadrupolar transitions, which may be substantial for highly polarizable Rydberg states. Our calculations are performed in the macroscopic quantum electrodynamics formalism, based on the dyadic Green's function method. This allows us to take dispersive and absorptive characteristics of silica into account; this is of major importance since Rydberg atoms emit along many different transitions whose frequencies cover a wide range of the electromagnetic spectrum. Our work is an important initial step toward building a Rydberg atom-nanofiber interface for quantum optics and quantum information purposes.

Figure 1

A ${}^{87}\mathrm{Rb}$ atom located at a distance, $R$, from the axis of an optical nanofiber of radius, $a$. The refractive index $n_1\left(\omega \right)$ for silica is obtained by a numerical fit of the experimental data taken from [35]. Outside the fiber, the refractive index is $n_2=1$. The axis of the nanofiber is arbitrarily chosen as the $z$-axis. The cylindrical coordinates $\left(\rho ,\varphi ,z\right)$ and frame $\left({\vec{e}}_{\rho },{\vec{e}}_{\varphi },{\vec{e}}_z\right)$ are introduced in the inset.

Figure 2

Spontaneous emission rates of an ${}^{87}\mathrm{Rb}$ atom in the state $\left|n{S}_{1/2}\right.〉$ (with $n=7,10,20,30$) dependent on the distance, $R$, from the atom to the nanofiber. We represent the ratios $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$ (left), ${\mathrm{\Gamma }}_g/\mathrm{\Gamma }$ (right) as functions of $R$. ${\mathrm{\Gamma }}_g$ and ${\mathrm{\Gamma }}_r$ denote the spontaneous emission rates toward the guided and radiative modes, respectively, $\mathrm{\Gamma }\equiv {\mathrm{\Gamma }}_g+{\mathrm{\Gamma }}_r$ is the total spontaneous emission rate, and ${\mathrm{\Gamma }}_0$ is the spontaneous emission rate in vacuum. The radius of the nanofiber is fixed at $a=150$ nm.

Figure 3

Spontaneous emission rates of an ${}^{87}\mathrm{Rb}$ atom in the state $\left|n{P}_{3/2},F=3,M_F=3\right.〉$ (with $n=7,10,20,30$) dependent on the distance, $R$, from the atom to the nanofiber. We represent the ratios $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$ (left) and ${\mathrm{\Gamma }}_g/\mathrm{\Gamma }$ (right) as functions of $R$. ${\mathrm{\Gamma }}_g$ and ${\mathrm{\Gamma }}_r$ denote the spontaneous emission rates into the guided and radiative modes, respectively, $\mathrm{\Gamma }\equiv {\mathrm{\Gamma }}_g+{\mathrm{\Gamma }}_r$ is the total spontaneous emission rate, and ${\mathrm{\Gamma }}_0$ is the spontaneous emission rate in vacuum. The radius of the nanofiber is fixed at $a=150\mathrm{nm}$.

Figure 4

Spontaneous emission rates of an ${}^{87}\mathrm{Rb}$ atom in the state $\left|n{D}_{5/2},F=4,M_F=4\right.〉$ (with $n=7,10,20,30$) dependent on the distance, $R$, from the atom to the nanofiber. We represent the ratios $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$ (left) and ${\mathrm{\Gamma }}_g/\mathrm{\Gamma }$ (right) as functions of $R$. ${\mathrm{\Gamma }}_g$ and ${\mathrm{\Gamma }}_r$ denote the spontaneous emission rates into the guided and radiative modes, respectively, $\mathrm{\Gamma }\equiv {\mathrm{\Gamma }}_g+{\mathrm{\Gamma }}_r$ is the total spontaneous emission rate, and ${\mathrm{\Gamma }}_0$ is the spontaneous emission rate in vacuum. The radius of the nanofiber is fixed at $a=150\mathrm{nm}$.

Figure 5

Spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber dependent on the fiber radius, $a$. We represent the ratio ${\mathrm{\Gamma }}_g/\mathrm{\Gamma }$, for an ${}^{87}\mathrm{Rb}$ atom in the states $\left|n{S}_{1/2}\right.〉$ (left), $\left|n{P}_{1/2}\right.〉$ (middle), and $\left|n{D}_{5/2},F=4,\left|M_F\right|=4\right.〉$ (right), with $n=\left(7,10,20,30\right)$, as a function of $a$. The atom is located 50 nm from the fiber (i.e., $R=a+50\mathrm{nm}$).

Figure 6

Spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber – dependence on the fiber radius, $a$. We represent the ratio ${\mathrm{\Gamma }}_g/\mathrm{\Gamma }$ for an ${}^{87}\mathrm{Rb}$ atom in the states $\left|30P_{3/2},F=0,2,3,\left|M_F\right|=0\cdots F\right.〉$ (left) and $\left|30P_{3/2},F=1,\left|M_F\right|=0\cdots F\right.〉$ (right) as a function of $a$. The atom is located $50\mathrm{nm}$ from the fiber (i.e., $R=a+50\mathrm{nm}$).

Figure 7

Spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber – dependence on the fiber radius, $a$. We represent the ratio ${\mathrm{\Gamma }}_g/\mathrm{\Gamma }$ for an ${}^{87}\mathrm{Rb}$ atom in the states $\left|30D_{5/2};F=1,\cdots ,4;\left|M_F\right|=0\cdots F\right.〉$ as a function of $a$. The atom is located at $50\mathrm{nm}$ from the fiber (i.e., $R=a+50\mathrm{nm}$).

Figure 8

Spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber: Contribution of the electric quadrupolar transitions. We represent the contribution of the electric quadrupolar transitions to the spontaneous emission rates into the radiative modes, ${\mathrm{\Gamma }}_r^Q$ (left), the first guided modes, ${\mathrm{\Gamma }}_g^Q$ (middle), and in vacuum, ${\mathrm{\Gamma }}_0^Q$ (right), for an ${}^{87}\mathrm{Rb}$ atom in the state $\left|n{S}_{1/2}\right.〉$ as a function of the principal quantum number, $n$. To get the highest possible value, we assume the atom is located on the nanofiber surface, i.e., $R=a$. In the case of the radiative modes (left), we considered two values for the fiber radius, $a=100$ and $200\mathrm{nm}$, while $a=200\mathrm{nm}$ for the other two plots.

Figure 9

Definition of the angles $\left(\mathrm{\Theta },\mathrm{\Phi }\right)$ characterizing the quantization axis directed along the unitary vector ${\vec{e}}_{\text{q}}\equiv sin\mathrm{\Theta }cos\mathrm{\Phi }{\vec{e}}_x+sin\mathrm{\Theta }sin\mathrm{\Phi }{\vec{e}}_y+cos\mathrm{\Theta }{\vec{e}}_z$.

Figure 10

Spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber with quantization axis in the $\left(Oxy\right)$ plane. We plot the spontaneous emission rates, ${\mathrm{\Gamma }}_g$ (left) and ${\mathrm{\Gamma }}_r$ (right), into the first guided and radiative modes, respectively, for an ${}^{87}\mathrm{Rb}$ atom in the state $\left|30D_{5/2},F=4,M_F=4\right.〉$ as functions of the angle $\mathrm{\Phi }$ (cf. Fig. 9), with $\mathrm{\Theta }=\pi /2$. The contributions to ${\mathrm{\Gamma }}_g$ of the first four guided modes, ${\text{HE}}_{11},{\text{TE}}_{01}$, ${\text{TM}}_{01}$, and ${\text{HE}}_{21}$, are displayed separately. The radius of the fiber is $a=250\mathrm{nm}$ and the atom is located $50\mathrm{nm}$ from the fiber (i.e., $R=a+50\mathrm{nm}$).

Figure 11

Spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber with quantization axis in the $\left(Oxz\right)$ plane. We represent the spontaneous emission rates, ${\mathrm{\Gamma }}_g$ (left) and ${\mathrm{\Gamma }}_r$ (right), into the first guided and radiative modes, respectively, for an ${}^{87}\mathrm{Rb}$ atom in the state $\left|30D_{5/2},F=4,M_F=4\right.〉$ as functions of the angle $\mathrm{\Theta }$ (cf. Fig. 9), with $\mathrm{\Theta }=\pi /2$. The contributions to ${\mathrm{\Gamma }}_g$ of the first four guided modes, ${\text{HE}}_{11},{\text{TE}}_{01}$, ${\text{TM}}_{01}$, and ${\text{HE}}_{21}$ are displayed separately. The radius of the fiber is $a=250\mathrm{nm}$ and the atom is located $50\mathrm{nm}$ from the fiber (i.e., $R=a+50\mathrm{nm}$).

Figure 12

Spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber with quantization axis in the $\left(Oyz\right)$ plane. We represent the spontaneous emission rates, ${\mathrm{\Gamma }}_g$ (left) and ${\mathrm{\Gamma }}_r$ (right), into the first guided and radiative modes, respectively, for an ${}^{87}\mathrm{Rb}$ atom in the state $\left|30D_{5/2},F=4,M_F=4\right.〉$ as functions of the angle $\mathrm{\Theta }$ (cf. Fig. 9), with $\mathrm{\Phi }=\pi /2$. The contributions to ${\mathrm{\Gamma }}_g$ of the first three guided modes, ${\text{HE}}_{11},{\text{TE}}_{01}$, and ${\text{TM}}_{01}$, are displayed separately. The radius of the fiber is $a=250\mathrm{nm}$ (i.e., $R=a+50\mathrm{nm}$).

Figure 13

Spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber. We represent the proportion of spontaneous emission into the guided modes, ${\mathrm{\Gamma }}_g/\mathrm{\Gamma }$, for an ${}^{87}\mathrm{Rb}$ atom in the state $|30{\mathrm{D}}_{5/2},F=4,M_F=4\rangle$ as a function of the angles $\left(\mathrm{\Theta },\mathrm{\Phi }\right)$ (cf. Fig. 9).

Figure 14

Directionality of the spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber. We represent the coefficient, ${\alpha }_n$ (see main text for definition), characterizing the directionality of the spontaneous emission with respect to the $z$ axis, of an ${}^{87}\mathrm{Rb}$ atom in the states $|{\mathrm{nS}}_{1/2},F=2,M_F=2\rangle$ (left panel) and $|{\mathrm{nS}}_{1/2},F=\left(2,1\right),M_F=1\rangle$ (right panel) for $n=6,10,20,30$, close to an optical nanofiber of radius $a=200\mathrm{nm}$ as a function of the distance, $R$, from the atom to the fiber axis.

Figure 15

Directionality of the spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber. We represent the coefficient, ${\alpha }_n$ (see main text for definition), characterizing the directionality with respect to the $z$ axis, of the spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom in the states $|{nP}_{3/2},F=3,M_F=3\rangle$ (left panel) and $|{nD}_{5/2},F=4,M_F=4\rangle$ (right panel) for $n=5,6,10,30$, close to an optical nanofiber of radius $a=200\mathrm{nm}$ as a function of the distance, $R$, from the atom to the fiber axis.

Figure 16

Directionality of the spontaneous emission of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber into the guided modes – We represent the ratio, $\left({\mathrm{\Gamma }}_g^{\left(+\right)}-{\mathrm{\Gamma }}_g^{\left(-\right)}\right)/{\mathrm{\Gamma }}_g$ (see main text for definitions), characterizing the directionality with respect to the $z$ axis, of the spontaneous emission into the guided modes of an ${}^{87}\mathrm{Rb}$ atom in the state $|{\mathrm{nD}}_{5/2},F=4,M_F=4\rangle$ for $n=7,10,20,30$, close to an optical nanofiber of radius $a=200\mathrm{nm}$ as a function of the distance, $R$, from the atom to the fiber axis.

Figure 17

Lamb shift of an ${}^{87}\mathrm{Rb}$ atom in the state $\left|n{S}_{1/2}\right.〉$, for $n=27,\cdots ,30$ near an optical nanofiber. We represent the energy difference, $E\left(n{S}_{1/2}\right)-E\left(5S_{1/2}\right)$, of the states $\left|n{S}_{1/2}\right.〉$ ($n=27\cdots 30$) and $\left|5S_{1/2}\right.〉$ of an ${}^{87}\mathrm{Rb}$ atom near an optical nanofiber of radius, $a=200\mathrm{nm}$ as a function of the distance, $R$, from the fiber. Energies are given in eV.

Figure 18

Lamb shift of an ${}^{87}\mathrm{Rb}$ atom in the states $\left|n{P}_{3/2}F=3,M_F=-F\cdots F\right.〉$ and $\left|n{D}_{5/2}F=4,M_F=-F\cdots F\right.〉$, for $n=29,30$ near an optical nanofiber. We represent the energy difference, $E-E\left(5S_{1/2}\right)$, of the states of interest with respect to $\left|5S_{1/2}\right.〉$ as a function of the distance, $R$, from the fiber. The radius of the nanofiber is $a=200\mathrm{nm}$. Energies are given in eV.

Figure 19

Casimir-Polder force felt by an ${}^{87}\mathrm{Rb}$ atom in the state $\left|30S_{1/2}\right.〉$ near an optical nanofiber. We represent the radial Casimir-Polder force, $F_{\text{vdW}}=-{\partial }_{R}U\left(R\right)$, felt by an ${}^{87}\mathrm{Rb}$ atom in the state $\left|30S_{1/2}\right.〉$ close to an optical nanofiber of radius $a=200\mathrm{nm}$ as a function of the distance, $R$, from the fiber. The total force, electric dipole, and quadrupole coupling contributions are represented by (blue) full, (red) dashed, and (green) dash-dotted lines, respectively.

Figure 20

Electric dipole and quadrupole contributions to the Lamb shift of an ${}^{87}\mathrm{Rb}$ atom in the state $\left|n{S}_{1/2}\right.〉$ near an optical nanofiber. Electric dipole and quadrupole components are represented as functions of the principal quantum number, $n$, by (blue) dots and (red) crosses, respectively. The radius of the optical nanofiber is $a=200\mathrm{nm}$ and the atom is located at $R=250\mathrm{nm}$ from the fiber axis.

Figure 21

Relative contributions to the Lamb shift of electric dipole and quadrupole couplings for an ${}^{87}\mathrm{Rb}$ atom in the state $\left|n{S}_{1/2}\right.〉$ close to an optical nanofiber. Electric dipole and quadrupole components are represented as functions of the principal quantum number, $n$, by (blue) dots and (red) crosses, respectively, for an atom located at $R=250\text{(top}\text{left)},300\text{(top}\text{right)},350\text{(bottom}\text{left)}$, and $400\mathrm{nm}$ (bottom right) from the fiber axis. The radius of the optical nanofiber is $a=200\mathrm{nm}$.