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 $^{87}\text{Rb}$ atom in a Rydberg-excited state $\left(n\leq30\right)$ 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 towards building a Rydberg atom-nanofiber interface for quantum optics and quantum information purposes.

In this paper, we report on numerical calculations of the spontaneous emission rates and Lamb shifts of a 87 Rb atom in a Rydberg-excited state (n ≤ 30) 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 towards building a Rydberg atomnanofiber interface for quantum optics and quantum information purposes.

I. INTRODUCTION
Within the last two decades, the strong dipole-dipole interaction experienced by two neighbouring Rydberg-excited atoms [1] has become the main ingredient for many atom-based quantum information protocol proposals [2]. This interaction can be so large as to forbid the simultaneous resonant excitation of two atoms if their separation is less than a specific distance, called the blockade radius [3], which typically depends on the intensity of the laser excitation and the interaction between the Rydberg atoms [4]. The discovery of this "Rydberg blockade" phenomenon [3,[5][6][7][8][9] paved the way for a new encoding scheme using atomic ensembles as collective quantum registers [5,[10][11][12] and repeaters [13][14][15].
Scalability is one of the crucial requirements for quantum devices [16] and interfacing atomic ensembles into a quantum network is a possible way to reach this goal. Photons naturally appear as ideal information carriers and the photon-based protocols considered so far include free-space [17], or guided propagation through optical fibers [13]. The former has the advantage of being relatively easy to implement, but presents the drawback of strong losses. The latter requires a cavity quantum electrodynamics setup, which is experimentally more involved. An alternative option would be to use optical nanofibers. Such fibers have recently received much attention [18,19] because the coupling to the evanescent guided modes of a nanofiber allows for easy-to-implement atom trapping [20][21][22] and detection [23][24][25]. This coupling increases in strength as the fiber diameter reduces and the atoms approach the fiber surface. It has also been shown that energy could be exchanged between two distant atoms via the guided modes of the fiber [26]. This suggests that optical nanofibers could play the role of a communication channel between the nodes of an atomic quantum network consisting of Rydberg-excited atomic ensembles.
In the perspective of building a quantum network based on Rydberg-blockaded atomic ensembles linked via an optical nanofiber, we recently studied the spontaneous emission of a highly-excited (Rydberg) sodium atom in the neighbourhood of an optical nanofiber made of silica [27]. To be more specific, we investigated how the atomic emission rates into the guided and radiative fiber modes are influenced by the radius of the fiber, the distance of the atom to the fiber and the symmetry of the Rydberg state. In the spirit of Ref. [28], we used the so-called mode function description of the nanofiber which does not allow one to take absorption and dispersion of the fiber into account. This point is critical with highly excited atoms since they can de-excite along many transitions of different frequencies for which the fiber index is different and potentially complex. This forced us, in Ref. [27], to restrict ourselves to Rydberg levels of moderate principal numbers so that the frequencies of the transitions involved remain in a nondispersive and nonabsorptive window of the silica spectrum. By contrast, here, we resort to the framework of macroscopic quantum electrodynamics based on the dyadic Green's function [29,30]. This formalism enables us to take the exact refractive index of silica into account and relaxes all constraints on the transitions we can address. This framework also offers a natural way to compute not only spontaneous emission rates, but also Lamb shifts and (resonant and nonresonant) electromagnetic forces the atom is subject to.
In this article, we present the numerical results we obtained with this approach for a rubidium atom prepared in a Rydberg-excited state |n ≤ 30; L = S, P, D; JF M F in the vicinity of a multimode silica optical nanofiber. We chose 87 Rb as it is commonly used in Rydberg atom experiments, like in the recent experimental work on Rydberg generation next to a nanofiber [31]. In particular, we show that a non-negligible fraction of spontaneously emitted light is guided along the fiber and study how it depends on principal quantum number, n, the radius of the nanofiber, a, the distance of the atom to the nanofiber axis, R, and the direction of angular momentum polarization. Interestingly, when the quantum and fiber axes do not coincide, spontaneous emission becomes directional, as already noticed for low-excited atoms [32,33] due to the peculiar polarization structure of the field in the neighbourhood of the fiber. As shown by our calculations, this effect is particularly strong for photons emitted into the fiber-guided modes and persists even for high principal quantum numbers, n. This is promising in view of potential applications in chiral quantum information protocols [34] based on a Rydberg-atom-nanofiber interface. We also address Lamb shifts and associated dispersion forces that arise. In particular, we show that, as n increases, the contribution of quadrupolar transitions becomes more and more important. This contrasts with spontaneous emission rates for which quadrupolar transitions have negligible influence.
The article is organized as follows. In Sec. II we present the system and introduce the important formulae used in our calculations. In Sec. III we present and interpret our numerical results for spontaneous emission rates, Lamb shifts and forces. We conclude in Sec. IV and give perspectives of our work. More technical details of our work can be found in Appendices.

II. SYSTEM AND METHODS
In this article, we consider a rubidium atom, 87 Rb, initially prepared in a highly-excited (Rydberg) level n ≤ 30, located at a distance R from the axis of a silica nanofiber of radius a. Our goal is to investigate how the fiber modifies the atomic spontaneous emission rates, the Lamb shifts, and the forces on the atom. To be more specific, we want to study the influence of: i) the radius of the fiber, ii) the distance of the atom to the fiber, iii) the different quantum numbers of the Rydberg state |nLJF M F , in particular the principal quantum number n, and iv) the direction of angular momentum polarization on these properties. On Fig. 1, we define the reference frame (Oxyz) and the associated unitary basis ( e x , e y , e z ). The origin O is chosen as the projection of the atomic center of mass onto the fiber axis, the z-axis coincides with the fiber axis, and the x-axis joins the origin O and the center of mass of the atom. In this basis, the position vector of the atom is R = R e x . For future reference we also introduce the cylindrical basis ( e ρ , e φ , e z ) on Fig. 1, defined by e ρ = cos φ e x + sin φ e y , e φ = − sin φ e x + cos φ e y .
We shall resort to the theoretical framework of macroscopic quantum electrodynamics [29,30], which allows one to consider the exact frequency-dependent form of the electric susceptibility of silica, obtained through a fit of experimental data given in [35]. This formalism is based on the dyadic Green's function G ( r, r ′ , ω), which is the solution to the Helmholtz equation where ε ( r, ω) is the relative electric permittivity of the medium at the position r and frequency ω while I is the unit dyadic [36]. The solution of Eq. (1) in the case of a cylindrical nanofiber is given in Appendix A. There exist two useful decompositions of G : i) G = G 0 + G sc where G 0 is the vacuum component, and G sc the scattering contribution due to the presence of the nanofiber and ii) G = G g + G r where G g,r are the respective contributions of the guided and radiative modes. We summarize below the main formulae we used to obtain the results presented in the next section, the derivation of which can be found in [30,37]. The spontaneous emission rate, Γ n , from an excited state, |n , is given by the sum, Γ n = k<n Γ nk , of rates Figure 1. A 87 Rb atom located at a distance, R, from the axis of an optical nanofiber of radius, a. The refractive index n1 (ω) for silica is obtained by a numerical fit of the experimental data taken from [35]. Outside the fiber, the refractive index is n2 = 1. The axis of the nanofiber is arbitrarily chosen as the z-axis. The cylindrical coordinates (ρ, φ, z) and frame ( eρ, e φ , ez) are introduced in the inset.
relative to the different transitions |n → |k for k < n, where ω nk and d nk ≡ n ˆ d k denote the bare frequency and the dipole matrix element of the transition |k → |n , respectively. In the same way, the Lamb shift, δω n , of an excited state, |n , is given by the sum, δω n = k δω nk , of all energy shifts induced by the different transitions |n → |k , for arbitrary k = n, with where P denotes the Cauchy principal value. Here, we shall use the non-retarded approximation [38] where Γ 0 R = lim ω→0 ω 2 c 2 G R, R, ω . This approximation is particularly suited for Rydberg atoms, since the main contributions to the Lamb shift are due to transitions to neighbouring states, therefore of long wavelengths.
Finally, the average resonant and nonresonant forces on an atom initially in the state |n , evaluated at t = 0, are given by (see Appendix B) where ∇ r acts on the spatial variable, r.

III. NUMERICAL RESULTS AND DISCUSSION
In this section we present and interpret the numerical results we obtained for spontaneous emission rates and Lamb shifts of a 87 Rb atom in the vicinity of a silica optical nanofiber. In particular, we investigate the effect of the distance, R, from the atom to the fiber axis, the fiber radius, a, and the atomic quantum numbers. We also study the influence of the direction of angular momentum polarization on the strength and directionality of spontaneous emission from a Rydberg level, specifically towards the guided modes. Finally, we address quadrupolar transitions, which, a priori, may have a substantial influence on Rydberg atom emission properties in view of their high polarizability.

A. Spontaneous emission rates
We start the discussion with the results we obtained for spontaneous emission rates. In Secs. III A 1-III A 3, the quantization axis is implicitly chosen along the fiber axis (Oz). In contrast, in Secs. III A 4-III A 5, we investigate the changes induced by other quantization axis choices. In some places, for pedagogical reasons, we shall resort to the so-called mode function approach (widely used in the works by F. Le Kien, see, e.g. [21]) as it offers a simple and illustrative way to physically interpret our results. However, we wish to emphasise that our calculations were performed using the (more general) Green's function formalism, which allows one to account for dispersive and absorptive characteristics of the fiber.  2, 3 and 4 show the variations with the distance, R, from the atom to the nanofiber axis of: i) the ratio Γ /Γ0 of the total spontaneous emission rate of the atom to the spontaneous emission rate in vacuum, ii) the ratio Γg /Γ of the spontaneous emission rate of the atom only into the guided modes to the total spontaneous emission rate for the states nS1 /2 , nP3 /2 , F = 3, M F = 3 , nD5 /2 , F ′ = 4, M F ′ = 4 , respectively, with n = 7, 10, 20, 30, and for a nanofiber radius a = 150 nm.
In all cases, close to the nanofiber, the total spontaneous emission is amplified when compared with its value in vacuum. This amplification vanishes as R increases. The small Drexhage-like oscillations observed [39] are due to the oscillatory behavior of the radiative modes themselves.
Close to the fiber, a non-negligible fraction of the spontaneous emission is captured by the guided modes. The strongest effect is obtained for S and D states, as already noted and interpreted in [27]. As R increases, the guided modes are (quasi-)exponentially damped, hence the damping of Γ g itself.
The dependence with n is less easy to interpret. Let us first note that Γ, Γ g and Γ 0 substantially decrease when the principal quantum number increases (see Table I for theoretical values of Γ 0 ). The -dependence on the distance, R, from the atom to the nanofiber. We represent the ratios Γ /Γ 0 (left), Γg /Γ (right) as functions of R. Γg and Γr denote the spontaneous emission rates towards the guided and radiative modes, respectively, Γ ≡ Γg +Γr is the total spontaneous emission rate and Γ0 the spontaneous emission rate in vacuum. The radius of the nanofiber is fixed at a = 150 nm.  -dependence on the distance, R, from the atom to the nanofiber. We represent the ratios Γ /Γ 0 (left), Γg /Γ (right) as functions of R. Γg, Γr denote the spontaneous emission rates into the guided and radiative modes, respectively, Γ ≡ Γg + Γr is the total spontaneous emission rate and Γ0 the spontaneous emission rate in vacuum. The radius of the nanofiber is fixed at a = 150 nm.  ratios Γ /Γ0 and Γg /Γ, however, keep the same order of magnitude and, therefore, the plots in Figs. 2, 3 and 4 for n = 7, 10, 20, 30 remain close to each other. In particular, for high values of n, the plots seem to tend to an asymptotic curve. This observation can be qualitatively understood as follows.
We first note that, for high n, only a few transitions substantially contribute to the spontaneous emission rate. In the crude but practical two-level approximation, we assume the spontaneous emission rate is dominated by one transition |n → |k whose total spontaneous emission rate, spontaneous emission rate towards guided modes and spontaneous emission rate in vacuum are, respectively, given by For increasing n, ω nk saturates, i.e., Rydberg levels are closer and closer in energy as the principal quantum number grows, and the terms ω 2 nk Im G R, R, ω nk and ω 2 nk Im G g R, R, ω nk , therefore, also saturate. Finally, since Γ, Γ g , Γ 0 ∝ d nk

2
, the ratios Γ /Γ0 and Γg /Γ do not (substantially) depend on the dipole and saturate as n increases.
2. Dependence on the fiber radius, a Figure 5 shows the dependence on the fiber radius, a, of the ratio Γg /Γ for an 87 Rb atom in the states nS1 /2 (left), nP1 /2 (middle) and nD5 /2 , F = 4, |M F | = 4 (right), with n = (7, 10, 20, 30). The atom is located at a distance d = 50 nm from the fiber surface, i.e., R = a + 50 nm from the fiber axis. Note that the contributions of all guided modes are summed.
The ratio Γg /Γ exhibits the same qualitative behavior with respect to a for S and D states, and ( Γg /Γ) S,D ≈ 10 ( Γg /Γ) P . Note that, for the states nS1 /2 and nP1 /2 , the hyperfine states (recall I = 3 2 for 87 Rb) have the same Γ g . This is not the case for nD5 /2 and in Fig. 5, we chose to represent the specific "edge" hyperfine state nD5 The abrupt slope changes observed in all plots originate from the appearance of additional guided modes as a increases. To be more explicit, the successive maxima of Γg /Γ can be interpreted as follows: i) As a function of the fiber radius, the amplitude of a specific guided mode at the location of the atom, i.e. at a distance d from the fiber surface, exhibits a maximum for a specific value, denoted by a max (ω, d), which depends both on the frequency of the mode and the distance, d.
(Note that a max actually also depends on other characteristics of the mode such as polarization, and wavevector). ii) For a given atomic transition, of frequency, ω 0 , the coupling to a given mode reaches its maximum when a = a max (ω 0 , d), hence a peak in Γg /Γ. Figure 6 shows the dependence on the fiber radius, a, of the ratio Γg /Γ for an 87 Rb atom in the states 30P3 /2 , F = 0 · · · 3, |M F | = 0 · · · F located at a distance d = 50 nm from the fiber surface, i.e., R = a + 50 nm from the fiber axis. As can be observed in the figure, though the different hyperfine magnetic sublevels for a given F show the same qualitative behavior, the spontaneous emission towards the guided modes is stronger for states of higher |M F |. This can be qualitatively understood as follows: i) Guided modes have a large (though not exclusive) transverse component, i.e., orthogonal to the fiber axis (Oz) (see Fig. 1); ii) High coupling to the guided modes is, therefore, obtained for transitions corresponding to dipoles in the transverse plane (Oxy); iii) The quantization axis being along the fiber axis, dipoles in the plane (Oxy) correspond to σ-transitions: therefore, the stronger the weight of σ-transitions in the de-excitation of an excited state, the higher the spontaneous emission rate towards guided modes; iv) The higher the value of |M F |, the stronger the weight of σ-transitions in the de-excitation of the state (this can be directly checked on 3j-coefficients), therefore, the higher |M F |, the higher the spontaneous emission rate towards guided modes.

Role of quadrupolar transitions
Because of their polarizability, Rydberg atoms are very sensitive to electric fields and electric field inhomogeneities. It is, therefore, reasonable to expect quadrupolar transitions to play a role in the de-excitation of a Rydberg atom in the vicinity of an optical nanofiber where spatial variations of the field are very rapid. Following [40][41][42], we calculate the correction due to electric quadrupolar transitions on the spontaneous emission rates of an 87 Rb atom in the state nS1 /2 located close to a silica optical nanofiber (see Appendix C for more details). Figure 8 (left) shows the dependence on n of the electric quadrupolar transition correction, Γ Q r , to the spontaneous emission rate into the radiative modes, for two values of the nanofiber radius, a = 100 and 200 nm. To obtain the strongest effect, we fixed R = a, corresponding to the unrealistic situation in which the atom is located at the fiber surface. As expected, for smaller values of a, the field inhomogeneities are more pronounced and the effect of electric quadrupolar transitions is higher. Moreover, the contribution Γ Q r decreases with increasing n, in the same way as the coupling to ground states that is responsible for the spontaneous emission.
The same observations can be made from Fig. 8 (middle, right), which show the dependence on n of the electric quadrupolar transition corrections Γ Q g and Γ Q 0 to the spontaneous emission rate into the first guided modes and vacuum, respectively. To obtain the strongest effect, we again fixed R = a. We, moreover, note that Γ Q r ≫ Γ Q g ≈ Γ Q 0 . Generally speaking, a comparison to the values calculated in the previous section shows that the quadrupolar contribution is negligible. In contrast, quadrupolar transitions play an important role in the Lamb shift, as we shall see below.

Influence of the quantization axis
Until now, the quantization axis was implicitly fixed along the fiber axis (Oz). Here, in the spirit of the experimental work in Ref [43], we study how the spontaneous emission rate of an atom close to an optical nanofiber depends on the direction of the quantization axis chosen to define its state, and therefore the direction of its angular momentum polarization. The angles (Θ, Φ) characterizing the quantization axis are specified in Fig. 9.
To be more specific, Figs. 10, 11 and 12 show the variations of the spontaneous emission rates towards the first four guided modes, Γ g (left), and towards the radiative modes, Γ r (right), for an 87 Rb atom prepared in the state 30D5 /2 , F = 4, M F = 4 and located at a distance R = 300 nm from the axis of a silica optical nanofiber of radius a = 250 nm when the quantization axis rotates in the planes (Oxy), (Oxz) and (Oyz), respectively.
Guided modes Before discussing our results on Γ g let us make a few remarks : A. Owing to our choice of initial atom state, 30D5 /2 , F = 4, M F = 4 , and the value of fiber radius considered here, a = 250 nm, the only transitions along which the atom can decay by emitting a photon into a guided mode are σ + -transitions towards P states, whose dipole is contained in the plane orthogonal to the quantization axis. B. A guided mode is characterized by its type (K=TE, TM, HE, EH), its frequency ω, two integers l ≥ 0 and m ≥ 0 called the azimuthal and radial mode orders, respectively, and two numbers f = ±1 and p = ±1, which characterize the propagation direction of the mode (f = ±1 Figure 9. Definition of the angles (Θ, Φ) characterizing the quantization axis directed along the unitary vector eq ≡ sin Θ cos Φ ex + sin Θ sin Φ ey + cos Θ ez.
conventionally corresponds to a mode propagating along (Oz) towards increasing/decreasing z) and the counterclockwise or clockwise phase circulation of the mode, respectively [44].
C. Because of field confinement, a guided mode µ ≡ (K lm , ω, f, p) possesses a non-vanishing longitudinal component, E is then purely imaginary. Moreover, the mode field components can be written in the form  Figure 10 corresponds to the configuration Θ ≡ π 2 , i.e., the quantization axis is chosen in the plane (Oxy) and directed along the vector e q ≡ cos Φ e x + sin Φ e y . The dipole, d kn , associated with the σ + -de-excitation, |n → |k , of frequency ω nk , can, therefore, be written in the form According to the remarks above, the coupling factor d kn · E (µ) of a given transition |n → |k to the (resonant) guided mode µ ≡ (K lm , ω nk , f, p) is propor- cos Φ and the associated contribution to the spontaneous emission rate is, therefore, itself proportional to f E  Figure 10. Spontaneous emission of an 87 Rb atom near an optical nanofiber with quantization axis in the (Oxy) plane. We plot the spontaneous emission rates, Γg (left) and Γr (right), into the first guided and radiative modes, respectively, for an 87 Rb atom in the state 30D5 /2 , F = 4, MF = 4 as functions of the angle Φ (c.f. Fig. 9), with Θ = π /2. The contributions to Γg of the first four guided modes, HE11, TE01, TM01 and HE21, are displayed separately. The radius of the fiber is a = 250 nm and the atom is located 50 nm from the fiber (i.e., R = a + 50 nm).
that the spontaneous emission rate, Γ (K lm ) g , into the first four guided modes K lm = HE 11 , TE 01 , TM 01 and HE 21 , is proportional to x 2 sin 2 Φ + E (K lm ,ω nk ) y 2 cos 2 Φ (Note that cross-terms between E z and E x compensate each other when summing over f ). In agreement with Fig. 10, we conclude that: i) Γ (K lm ) g is a π-periodic function of Φ and reaches its extrema when Φ = 0 π 2 . ii) For the modes TE 01 , since E x = E z = 0, Γ (TE01) g (Φ) ∝ cos 2 Φ is maximal for Φ = 0 [π], minimal for Φ = π 2 [π] and its minimum is zero. iii) For the modes TM 01 , since E y = 0, Γ , minimal for Φ = 0 [π] and its minimum is different from zero. For other modes (K lm = HE 11 , HE 21 ), Fig.  10 shows that minima and maxima of Γ , respectively. This can be explained by the inequality |E x | ≥ |E y | valid for these modes and the values (a, R) considered.
The same arguments can be used to interpret Fig. 11. This time, the quantization axis is chosen in the plane (Oxz), i.e., Φ ≡ 0, and e q ≡ sin Θ e x +cos Θ e z , whence d kn = d kn The contribution to the spontaneous emission rate into the resonant guided mode µ ≡ (K lm , ω nk , f, p) of a given transition |n → |k is proportional to cos ΘE  Figure 11. Spontaneous emission of an 87 Rb atom near an optical nanofiber with quantization axis in the (Oxz) plane. We represent the spontaneous emission rates, Γg (left) and Γr (right), into the first guided and radiative modes, respectively, for an 87 Rb atom in the state 30D5 /2 , F = 4, MF = 4 as functions of the angle Θ (c.f. Fig. 9), with Θ = π /2. The contributions to Γg of the first four guided modes, HE11, TE01, TM01 and HE21 are displayed separately. The radius of the fiber is a = 250 nm and the atom is located 50 nm from the fiber (i.e., R = a + 50 nm).
Radiative modes Our results on the spontaneous emission rate into the radiative modes are displayed in the right-hand panels of Figs. 10, 11 and 12. In the three different configurations, one observes a π-periodicity in (Φ, Θ). Moreover, the three figures seem to indicate that, for the values of (a, R) considered, radiative modes contributing to Γ r are mainly radial, i.e., their component along (Ox) dominates. Due to the variety and complexity of the structure of radiative modes, it is, however, difficult to go further into the interpretation of our results. Proportion of spontaneously emitted light towards the guided modes Figure 13 displays a 3D "summary" of Figs. 10, 11 and 12. To be more explicit, it shows the ratio Γg /Γ characterizing the proportion of spontaneous emitted light captured by guided modes. Note that the contribution of HE 11 to Γ g dominates. Besides π-periodicity in Φ and Θ, one observes maxima for Γg /Γ for e q = e z and saddle points for e q = e y .

Anisotropic spontaneous emission
Througout this section, the quantization axis is chosen along (Oy). Using the same notations as in the previous section, this corresponds to e q = e y . In this configuration, the atomic dipole associated with, e.g., a σ + -transition |n → |k lies in the plane (Oxz) and, more explicitly, d kn = d kn √ 2 [i e x + e z ]. Using, as in the previous section, the simplistic mode function approach, we conclude that the contribution of this transition to the spontaneous emission rate into a specific guided and clearly depends on the propagation direction, f . This heuristic argument cannot be straightforwardly transposed to radiative modes, but the same phenomenon is observed. The anisotropic spontaneous emission leads to a non-vanishing average lateral force on the atom whose order of magnitude is 0.5 zN (5 zN) for a rubidium atom in a 5D (5P ) state located at a distance d = 50 nm from a fiber of radius a = 200 nm. This force corresponds to the resonant part of the average Lorentz force, [F res ] z , Eq. (5) [30], and can be calculated in the Green's function approach. In particular, for an atom initially in a state |n , one can decompose [F res ] z as the sum of contributions F res nk,ν z relative to the transition |n → |k coupled to the (guided or radiative) mode, ν.
In order to quantitatively characterize the anisotropy of emission, we introduce the factor where the sum runs over all (radiative and guided) modes, ν, and final states, k. In this expression, Γ nk,ν represents the spontaneous emission rate for the transition |n → |k into the mode ν, Γ n is the total spontaneous emission rate from the state |n , k ν,z is the projection onto (Oz) of the wavevector for the (guided or radiative) mode (ν) and k ν = ων /c is its norm. With these definitions, ( Γ nk,ν /Γn) can be interpreted as the probability for a photon to be emitted from the state |n via the transition |n → |k and into the mode ν, whileh kν,z /hkν characterizes the inclination of the momentum of the photon emitted into the mode ν with respect to the fiber axis. Identifying −hk ν,z Γ nk,ν , i.e., the atomic recoil along (Oz) induced by the emission of a photon into the mode, ν, via the transition |n → |k , with the force F res nk,ν z , one can write α n = − k,ν [F res nk,ν ] z /Γnhkν (see [33] and Appendix B). Figs. 14 and 15 show the coefficient α n for an 87 Rb atom prepared in an excited S, P or D state decaying via σ + -transitions located close to an optical nanofiber of radius a = 200 nm as a function of the distance R from the atom to the fiber axis, Oz. The observed Drexhage-like oscillations are due to radiative modes [39]. Remarkably, though very weak, the spontaneous emission anisotropy for the S states is nonzero, at around 0.4% at most (see Fig. 14). For S states, α decreases for increasing n and vanishes when R → +∞ as expected (equivalent to the free-space configuration). As seen in Fig. 15, for P and D states, the spontaneous emission anisotropy, at around 20% on the surface of the nanofiber, is much stronger than for S states. When R → +∞, α n tends to zero as expected. For P states, α n decreases with n, while it only slightly varies for D states. Anisotropic emission is, therefore, observable for D states even at high values of n.
Anisotropic spontaneous emission into the guided modes of the nanofiber For guided modes, the anisotropy can be further characterized by the ratio, (Γ (+) denotes the spontaneous emission rate into forward/backward propagating guided modes and Γ g ≡ Γ + g +Γ − g . Using the same arguments as above, one can write this factor in the following form: − k,µ [F res nk,µ ] z /Γgh|kµ,z|, where now the sum runs over the guided modes, µ, only (see [32] and Appendix B). Figure 16 shows the ratio (Γ (+) g −Γ (−) g ) /Γg calculated for an 87 Rb atom prepared in the state |nD 5/2 , F = 4, M F = 4 , with n = 7, 10, 20, 30, and located near an optical nanofiber of radius a = 200 nm, as a function of the distance, R, from the atom to the fiber axis. The directionality of the guided emitted light remains strong even for high values of n and R. Note, however, that for large R > 300 nm the absolute value of Γ g itself is so small that the directionality has little practical meaning. Figure 17 displays the energy difference, E nS1 /2 −E 5S1 /2 , of the states nS1 /2 (n = 27 · · · 30) and 5S1 /2 for an 87 Rb atom near an optical nanofiber of radius a = 200 nm as a function of the distance, R, from the fiber axis. The Lamb shift of the ground state is assumed to be negligible with respect to that of the excited levels. When R decreases, E nS1 /2 − E 5S1 /2 itself decreases, though more rapidly for higher n. At shorter distances from the fiber, energy curves cross (not shown on Fig. 17) and the perturbative approach fails. The treatment of this area requires the diagonalization of the full Hamiltonian in the relevant degenerate Hilbert subspace. This will be investigated in future work.      Figure 17. Lamb shift of an 87 Rb atom in the state nS1 /2 , for n = 27, · · · , 30 near an optical nanofiber -We represent the energy difference, E nS1 /2 − E 5S1 /2 , of the states nS1 /2 (n = 27 · · · 30) and 5S1 /2 of an 87 Rb atom near an optical nanofiber of radius, a = 200 nm as a function of the distance, R, from the fiber. Energies are given in eV.  Figure 18.

B. Lamb shift and van der Waals force
Lamb shift of an 87 Rb atom in the states nP3 /2 F = 3, MF = −F · · · F and nD5 /2 F = 4, MF = −F · · · F , for n = 29, 30 near an optical nanofiber -We represent the energy difference, E − E 5S1 /2 , of the states of interest with respect to 5S1 /2 as a function of the distance, R, from the fiber. The radius of the nanofiber is a = 200 nm. Energies are given in eV. Figure 18 shows the same quantity for states nD5 /2 F = 4, m F = −F · · · F and nP3 /2 F = 3, m F = −F · · · F for n = 29, 30. Though the order of magnitude is comparable to that obtained for states nS1 /2 , one observes a degeneracy lift of the hyperfine components of different |M F | very close to the fiber; to be more explicit, the Lamb shift is stronger for states of higher |M F |. This can be qualitatively justified as follows: i) Radiative and guided modes have a strong -though not exclusive -transverse component, i.e., orthogonal to the fiber axis (Oz) (see Fig. 1); ii) High coupling to the guided modes is, therefore, obtained for transitions corresponding to dipoles in the transverse plane, (Oxy); iii) The quantization axis being along the fiber axis, dipoles in the plane (Oxy) correspond to σ transitions: therefore, the stronger the weight of σ transitions in the de-excitation of an excited state, the higher the spontaneous emission rate into guided modes; iv) The higher |M F |, the stronger the weight of σ transitions in the de-excitation of the state (this can be directly checked on 3j-coefficients): therefore, the higher |M F |, the higher the spontaneous emission rate into guided modes.
The R-dependence of the Lamb shift results in a radial van der Waals force, −∂ R U n (R), represented in Fig. 19 for the state 30S1 /2 as a function of R. Note the negative sign and, therefore, the attractive character of the force, as well as its order of magnitude of 10 −14 N, much larger than spontaneous emission recoil induced forces. Aside from the total force, we represented the contributions of the electric dipole and quadrupole couplings. Though the dipole contribution dominates, the quadrupolar component is far from negligible, especially close to the nanofiber when field inhomogeneities are magnified. Figure 20 displays the electric dipole and quadrupole components of the Lamb shift calculated for an 87 Rb atom in the state nS1 /2 located at a distance, R = 250 nm from an optical nanofiber of radius a = 200 nm. One observes that the higher the principal quantum number, n, the stronger One observes the same trend with n in Fig. 21, which displays the relative contributions of the electric dipole and quadrupole couplings to the Lamb shift calculated for an 87 Rb atom in the state nS1 /2 located at four different distances R = 250, 300, 350, and 400 nm from the optical nanofiber axis, as functions of n. As expected, the influence of quadrupolar transitions is lowered when the distance, R, increases, since the effect of the fiber on the electromagnetic field is less pronounced.

IV. CONCLUSION
The influence of a nanofiber near an 87 Rb atom prepared in a Rydberg-excited state, |n ≤ 30; L = S, P, D; JF M F , on the spontaneous emission rates and Lamb shift was investigated numerically in detail. In particular, the dependence of the spontaneous emission rates on the fiber radius, the distance of the atom to the fiber, the principal quantum number, n, orbital momentum, fine and hyperfine structures of the state considered, and the direction of angular momentum polarization were addressed. Close to the nanofiber, a non-negligible fraction of the emitted light can be captured by guided modes. This fraction is higher for larger |M F | but saturates for high n. When the quantum and fiber axes do not coincide, spontaneous emission into guided modes becomes strongly directional. This directionality persists even for high n. The contribution of quadrupolar transitions was shown to be negligible for spontaneous emission rates, while they may dominate Lamb shifts and van der Waals associated forces for high n. Our calculations were performed in the multimode fiber case, including all atomic transitions, using the general framework  of macroscopic quantum electrodynamics and this allowed us to account for the dispersive and absorptive characteristics of silica.
Our work is a preliminary step towards the building of a Rydberg-atom-optical-nanofiber platform. In particular, the collection and guidance of a substantial part of the spontaneous emitted light along the nanofiber suggests the possibility of constructing a network of Rydberg atomic ensembles in the same spirit as described in [13]. The strong directionality of spontaneous emission observed for specific Rydberg states and quantization axis is also very promising in view of potential applications in chiral quantum information protocols [34]. In future works, we will address the case of several Rydberg atoms in the neighbourhood of an optical nanofiber. In particular, we shall be interested in studying how the nanofiber modifies the Rydberg blockade phenomenon and whether the geometric arrangement of atoms can be used to enhance the coupling to guided modes.
where Γ n is the spontaneous emission from the excited state |n , p n (t) is the population of state |n at time t, d nk ≡ n|ˆ d|k . We neglect broadening in the denominator of the integrand in Eq. (B1), i.e., ω kn + i 2 (Γ n + Γ k ) ≈ ω kn . Then, by application of the residue theorem, we split this force into a resonant and a nonresonant part, i.e., F n = F res n + F nres n , with F res n = k<n 2µ 0 ω 2 nk Re ∇ r d nk · G sc r, R, ω nk · d kn r= R F nres n = − µ 0 π +∞ 0 dξ ξ 2 ω kn ω 2 kn + ξ 2 ∇ r d nk · G sc r, R, iξ | r= R · d kn .
We emphasize that the nonresonant part is summed over all transitions, while the resonant part takes into account only radiative transitions towards states |k of lower energy than |n . From the symmetry properties of g n , one deduces Finally, using ∇ r G ij r, R | r= R = 1 2 ∇ r G ij ( r, r) | r= R and noticing that 2Re [i∂ k G ij ] = −2Im [∂ k G ij ], we can get the resonant force projection in the ( e x , e y , e z ) basis (which corresponds to the cylindrical basis ( e ρ , e φ , e z ) at the location of the atom, see (2k + 1) (2l + 1) (2l ′ + 1) (2J + 1) (2J ′ + 1) (2F + 1) (2F ′ + 1) Finally, we can compute the spontaneous emission rates along the transition |n → |k due to dipole and quadrupole terms, respectively, to be given by nk α,β=x,y,z Q nk αβ Q kn γδ ∂ α ∂ ′ γ Im G βδ R, R ′ , ω nk and the van der Waals potential in the non-retarded approximation is given by