Excitation enhancement in electric multipole transitions near a nanoedge

Kosuke Shibata; Satoshi Tojo; Daniel Bloch

doi:10.1364/OE.25.009476

1. Introduction

A local field in the vicinity of a nanostructured surface is one of the most powerful tools in recent nanotechnologies due to high intensity and accurate localization below the diffraction limit of light in the far field [1]. In nanophotonics, a local field is often further strengthened by plasmonic media, which increase light emission by enhanced electric dipole moments. In particular, surface plasmon resonance has been exploited for various applications including the detection of single molecules with the use of surface-enhanced Raman processes [2] and plasmon-based optical trapping [3]. The plasmon resonance may also be used for cold atom experiments. Creation of a nanoscale optical lattice of cold atoms using plasmon resonance is proposed in [4]. Nanoantennas can also be used to enhance a local field. The enhancement in emissions around nanoantennas via classical [5] and quantum cavity effects [6] has drawn a great deal of attentions.

The localization of the light also leads to a large field gradient. Although the large field gradient has been out of target in photonic applications, it is of particular importance in higher-order or optical forbidden transitions, which are in general related to field gradients [7–13]. While the gradient of a plane far field is the wavenumber k = 2π/λ where λ is the wavelength, the gradient of a local field near a nanostructure can be far larger than k. As a result, extreme enhancement of a higher-order transition rate is expected in a local field even without plasmonic resonances or antenna effects.

The enhancement can open novel possibilities of the higher-order transitions, which are usually more than 5 orders of magnitude weaker than the allowed electric dipole (E1) transitions in the far-field and have not been exploited extensively due to their weakness. The enhanced higher-order transitions have possible applications to precise spectroscopy [14, 15], and to experimental simulations of higher-order multipole transitions in other research fields including solid state physics [16], nuclear physics [17], and molecular physics [11, 18]. The enhanced higher order transitions also play important roles in nano-scale engineering as demonstrated in [19].

In this work, we investigate the excitation efficiencies of electric multipole optical transitions in an optical near field in the vicinity of a nanostructure. We simulate the field distribution around an object with an edge of a nanoscale curvature radius using the finite-difference time domain (FDTD) method and evaluate the excitation efficiencies of transitions. We also estimate the emission rates accompanied with multipole excitations in alkali (Rb and Cs) atoms and discuss the possibility for observing the enhancement in the multipole excitations in cold atoms.

2. Method

We first briefly review optical transitions in an elementary quantum system (an atom or a molecule) in an optical field with an arbitrary structure [7]. For simplicity, a monochromatic optical field resonant to an atomic transition frequency is assumed in the following. We also neglect mixing of other atomic transitions because their energies are usually well separated from that of the considered one. With these assumptions, atom-field interaction Hamiltonian [20] is written as

\begin{array}{l} ℋ & = & d \cdot E (r_{0}) - m \cdot B (r_{0}) + e Q_{i j} \nabla_{i} E_{k} (r_{0}) + \dots, \\ = & ℋ_{E 1} + ℋ_{M1} + ℋ_{E 2} + \dots \end{array}

where r₀ is the position vector of atomic center mass, d is the electric dipole moment operator, m is the magnetic dipole moment operator, and Q_ij are the matrix elements of a rank 2 tensor Q = rr [7, 9]. The convention of implicit summation over repeated indices (i, j = x, y, z) representing Cartesian coordinates are used throughout the paper. For convenience, Q is replaced by an irreducible tensor

Q = rr - \frac{1}{3} r^{2} δ_{i j}

where δ_ij is the Kronecker symbol, using δ_ij∇_i E_k (r₀) = ∇ · E(r₀) = 0.

Higher order multipole transitions are derived in a similar way. The electric octupole (E3) Hamiltonian is given by

ℋ_{E 3} = e O_{i j k} \nabla_{i} \nabla_{j} E_{k} (r_{0})

with

O_{i j k} = r_{i} r_{j} r_{k} - \frac{1}{5} r^{2}

(r_iδ_jk + r_jδ_kl + r_kδ_ij) being the matrix elements of the electric octupole operator [18]. The electric hexadecapole (E4) Hamiltonian is given by

ℋ_{E 4} = e H_{i j k l} \nabla_{i} \nabla_{j} \nabla_{k} E_{l} (r_{0})

where

H_{i j k l} = r_{i} r_{j} r_{k} r_{l} - \frac{r^{2}}{7} (r_{i} r_{j} δ_{k l} + r_{i} r_{k} δ_{j l} + r_{i} r_{l} δ_{j k} + r_{j} r_{k} δ_{i l} + r_{j} r_{l} δ_{i k} + r_{k} r_{l} δ_{i j}) + \frac{3}{35} r^{4} δ_{i j k l}

is the electric hexadecapole operator [18]. The transition rates are given by Fermi’s golden rule [21] with these interaction Hamiltonians.

We consider excitations in an atom near a silver object with a nano-scale edge of a curvature radius R (see Fig. 1). Similar nano-edge will appear, for example, in micro-fabricated wires in an atom chip [22] and submicron-scale stripes [23]. The object treated here may be regarded as a simplified model of more complex nanostructures. The simplification makes our calculations tractable. We neglect the plasmon resonance to focus on the enhancement resulted from the local electromagnetic field distribution around a nanostructure. The object is assumed to be elongating infinitely along the axis y and we investigate a two-dimensional field in the x-z plane. The excitation light is a plane wave propagating along the +z direction and is linearly x-polarized.

Fig. 1 Schematic illustration of the simulated situation. A silver object with an edge of a curvature radius R is illuminated by an incident light of wavelength λ = 389 nm.

Download Full Size | PDF

One generally needs to calculate not only the excitation rate but also the emission rate to evaluate the quantum efficiency, defined by the ratio between number of photons emitted in a radiative state over the total number of photons, which is relevant to understand how the radiation is coupled to the far-field. In this paper, we consider the excitations with strong (E1) decay channels [7,24], which include the excitations of alkali atoms from the ground S state to the D state as described below. The enhancement of such a multipole excitation is observed via the spontaneous emission in the deexcitation process. Because the configuration here resembles to that of the nanoparticle antenna [25] the emission should rather increase than decrease and a high quantum efficiency is expected. Therefore, the photon detection rate will be approximately equal to the excitation rate, while the (enhanced) excitation rate is smaller than the E1 spontaneous emission rate. The transitions with no strong decay channels (for example, ¹P₁ – ³P₂ M2 transitions in alkali-earth atoms) are out of our scope.

We here derive the E2 excitation rates in the considered configuration. The E2 Hamiltonian for the considered configuration can be written using E_y = 0, ∇_y E_i = 0 and the Maxwell’s equations as

ℋ_{E 2} = e [\frac{Q^{(+ 2)} + Q^{(- 2)} - \sqrt{6} Q^{(0)}}{2} \nabla_{x} E_{x} + (Q^{(+ 1)} - Q^{(- 1)}) \nabla_{x} E_{x}]

where Q^(q) (q = 0, ±1, ±2) are the spherical components of the irreducible tensor Q, which are related to Q_ij by

Q_{11} - Q_{22} = Q_{2}^{(+ 2)} + Q_{2}^{(- 2)},

Q_{11} + Q_{22} - 2 Q_{33} = - \sqrt{6} Q_{2}^{(0)},

Q_{13} + Q_{31} = Q_{2}^{(- 1)} - Q_{2}^{(+ 1)} .

For an unpolarized atom in which all magnetic sublevels are equally populated, the total excitation rate between the state |a〉 and |b〉 ( $\propto \sum_{a, b} {| 〈 a | Q^{(q)} | b 〉 |}^{2}$ ) must be independent of q and the total E2 excitation rate is given by

R_{E 2} = C_{E 2} [{| \nabla_{x} E_{x} |}^{2} + {| \nabla_{z} E_{x} |}^{2}],

where C_E2 a factor independent of the field distribution.

The total E3 excitation rate is similarly calculated as

\begin{array}{l} R_{E 3} = C_{E 3} [\frac{1}{175} {| - \nabla_{x} \nabla_{z} E_{x} - \nabla_{x} \nabla_{x} E_{x} + 2 \nabla_{z} \nabla_{z} E_{z} |}^{2} \\ + \frac{1}{1050} {| 3 \nabla_{x} E_{x} - 4 \nabla_{z} \nabla_{z} E_{x} - 4 \nabla_{x} \nabla_{z} E_{z} - 4 \nabla_{z} \nabla_{x} E_{z} |}^{2} \\ + \frac{1}{105} {| \nabla_{x} \nabla_{z} E_{x} + \nabla_{z} \nabla_{x} E_{x} + \nabla_{x} \nabla_{x} E_{z} |}^{2} + \frac{1}{70} {| \nabla_{x} \nabla_{x} E_{x} |}^{2}] . \end{array}

Equations (9) and (10) are valid regardless of the field distribution. In particular, for the excitation by a x-polarized plane wave e_x E exp(ikz) in the free space (with no structure), they give R_E2 ∝ E²k² and R_E3 ∝ E²k⁴, respectively. As is expected, k^2(l−1) dependence of the excitation efficiency in an El transition is derived.

We simulate light propagation and scattering induced by the object with the finite-difference time-domain (FDTD) method, using a free software package [26, 27], in a volume of 1600 nm × 0.125 nm × 2000 nm (x × y × z) with a cubic mesh grid of Δx = Δy = Δz = 0.125 nm. The time step satisfies the Courant stability condition [26,27]. We assume the object is infinitely long in the +x direction and adopt the PML (perfect matched layer) conditions at the −x and ±y edges. Perfectly matched layers absorbing light at ±z edges are adopted to eliminate undesirable back-scattering and radiation from the top and bottom boundaries.

The wavelength λ is fixed at 389 nm, corresponding to the Rb 5²S_1/2 − 5²D_5/2 (E2) transition. The complex dielectric constant of the object is defined as ∊₂ = −3.142 + 0.518i based on the complex refractive index of silver n₂ = 0.146 + 1.778i [28]. We estimate excitation rates of El (l = 1, 2, 3, 4) transitions using the field distribution for the same λ. Although the field distribution depends on the excitation wavelength and the excitation rates change accordingly, the estimation gives a reasonable comparison of transition rates among different multipole transitions, because the wavelengths of other multipole transitions are also located in the ultraviolet region close to λ; for example, the wavelengths of 5²S_1/2 − 4²F_7/2 (E3) transition and 5²S_1/2 − 5²G_9/2 (E4) transition are 373 and 341 nm, respectively [29–31]. Although the plasmon resonance frequency for a nanostructured object can locate in the ultraviolet region, we neglect the plasmonic resonances in this work as already mentioned. This neglect does not affect the discussions below on the enhancement by local field structure. We also note that the above E3 and E4 wavelengths are still longer than the plasma wavelength of silver [32,33] and the incident fields with these wavelengths should be scattered by the object. Thus the results in this work give reasonable comparison between transitions with different ranks.

We derive the amplitude of the field and the field gradient at each spatial point using the simulation results during three cycles after the field approaches to its asymptotic value, with confirming the stable convergence of the calculated results. To eliminate possible artificial edge-enhancement effects due to insufficient spatial resolution of the simulation, two-dimensional median filtering is applied to the simulated values. We also avoid using 4 points, corresponding to 0.5 nm, from the object surface.

3. Results and discussion

3.1. Excitation efficiencies in total

Figure 2 shows two-dimensional plots of the excitation efficiencies of E1, E2, and E3 transitions in the xz plane in the vicinities of the nano-edges with the curvatures of R = 2, 10, and 50 nm, where the excitation efficiencies are defined as the ratio of the excitation rates over the E1 excitation rate by a plane-wave irradiation. For the comparison between El excitations with different l, we assume the El excitation efficiencies in regions far from the surface to be ((kr)^l−1/l!)² where r = 0.265 nm is the radius of Rb [34]. They are 4.57×10⁻⁶, 9.32×10⁻¹², for E2 and E3 transitions, respectively. The calculation gives a reasonable estimation for excitation rates in most transitions, because the El transitions in the far field corresponds to the l-th term of the plane wave expansion of the interaction Hamiltonian ∝ (p · e_λ)Ee^ik·r, where p is the momentum of the atom, e_λ and E are the polarization and the amplitude of the electric field, respectively. For more precise calculation of the excitation rates in real atoms, we use the values of the transition strength as described later.

Fig. 2 Two-dimensional plots of the excitation efficiencies of (a)–(c) E1, (d)–(f) E2, and (g)–(i) E3 transitions in the vicinities of nanostructures with curvatures of (a), (d), (g) R = 2 nm, (b), (e), (h) 10 nm, and (c), (f), (i) 50 nm.

Download Full Size | PDF

The excitation efficiencies of multipole transitions (E2 and E3) are considerably enhanced in the vicinity of the small edges of R = 2 nm (Figs. 2(a), 2(d), 2(g)) and 10 nm (Figs. 2(b), 2(e), 2(h)), whereas the efficiencies are only weakly enhanced for the edge of R = 50 nm (Figs. 2(c), 2(f), 2(i)). Near the edge of R = 2 nm, the excitation efficiency of the E2 transition is extremely enhanced and is almost similar to that of the E1 transition. The enhancement of E3 excitation efficiency is even greater. It is enhanced by eight orders of magnitude and reaches to 10⁻³. The greater enhancement for the higher transition indicates that these enhancements are due to extremely large field gradients produced near the object, rather than antenna effects caused by the sharp edge. We note that the enhancements of excitation efficiencies are considerably reduced for y–polarized incident light. The E2 excitation is enhanced by only several times in that case. This is because the light field is not very scattered since the incident light is parallel to the ridge line of the nanostructure [8].

We analyze the dependence of the excitation efficiency on the distance from the surface d, adopting the average of the excitation efficiencies at d as the measure of the excitation efficiency. Figure 3 shows traces in the excitation efficiencies for (a) R = 2 nm and (b) R = 10 nm, respectively. The excitation efficiencies of the E2, E3, and E4 transitions monotonically increases as d becomes smaller in the both case of R = 2 and 10 nm. The higher order excitation efficiencies change more drastically, which influences the fact that they are related to the higher order derivative of the field. In the region of d < R, the efficiency enhancements seem to be saturated, probably because the light scattered by the straight segment of the object hardly reaches the vicinity of the surface.

Fig. 3 The dependence of the excitation efficiencies on the distance from the surface d for the edge with the curvature radius R of (a) 2 nm and (b) 10 nm. The excitation efficiencies of E1 (black, dotted line), E2 (red, solid line), E3 (blue, dashed line), and E4 (green, dotted-dashed line) are shown.

Download Full Size | PDF

For a quantitative evaluation, we apply d^−a fitting to the results for R = 2 nm in the range from d = 4 to 40 nm and obtain a = 2.50, 4.90, and 6.83 for the E2, E3, and E4 transitions, respectively. The dependence can be explained as below. A plane wave is scattered to a spherical wave at an edge acting like a point source. Since the radial function of the scattered light is proportional to e^ikr/r, the multipole El excitation efficiency is expected be proportional to the (l − 1)-th spatial derivative of the field $\frac{\partial^{l - 1}}{\partial r^{l - 1}} \frac{e^{i k r}}{r}$ which is approximately $\frac{e^{i k r}}{r^{l}} (i k r - 1)$ for kr ≪ 1. Therefore, 2l − 2 < a < 2l is expected for the El transition, which is consistent with the result for d = 4 to 40 nm.

The large multipole excitation efficiencies near an edge can also be understood in the frame of the spherical basis. In general, a scattered field is given as the solution of the Helmholtz equation (∇² + k²) f(r) = 0 and the solution f(r) can be expanded to

f (r) = \sum_{L M} R_{L} (r) Y_{L M} (θ, ϕ),

where R_L (r) and Y_LM (θ, ϕ) are the radial function of the field and the spherical harmonics with L and M being the angular momentum and the magnetic quantum number, respectively [35,36]. R_L (r) is determined to fulfill the boundary condition at the surface. In the case of a plane wave, the radial part is expressed by the spherical Bessel function j_L. Since it is known that an El transition is brought by j_l (kr) and

j_{n} (x) \to \frac{x^{n}}{(2 n + 1)!!} (x \to 0)

, the E1 excitation is dominant for a plane wave irradiation. In other words, the atom-field interaction operator of a plane wave leads to efficient E1 coupling between the initial and final atomic (radial) wavefunctions, while the E2 and higher order components give poor matching. The light scattered by a nano-edge behaves like a spherical wave, the radial part of which is represented by the spherical Hankel function of the first kind

h_{n}^{(1)}

. Because an El transition corresponds to

h_{l}^{(1)} (k r)

and

h_{n}^{(1)} (x) \propto x^{- (n + 1)}

for x ≪ 1, the scattered near field can lead to extreme enhancement of the excitation efficiencies for higher order multipole transitions. Although one should take into account the interference between the scattered waves generated at different points for precise evaluation, this explanation gives an insight into multipole excitation enhancement in a local field.

We also investigate the R dependence of the excitation efficiency. Figure 4(a) – 4(d) shows a two-dimensional map of the R and d-dependences of the excitation efficiencies in El (l = 1, 2, 3, 4) transitions. In the region of R < 10 nm, the excitation efficiency of E2 transition can be larger than 10⁻² and the E3 transition is also considerably enhanced. The enhancement is more pronounced in the region of R < 3 nm, where the efficiencies of the E2, E3, and E4 transitions reach 10⁻¹, 10⁻³, and 10⁻⁵, respectively. They correspond to enhancement factors of 10⁴, 10⁷, and 10¹⁰. The R dependences for fixed distance of d = 10 and 2 nm are shown in Fig. 4(e) and 4(f), respectively. While the efficiencies in multipole transitions for d = 10 nm increases only slightly as R is reduced, the efficiencies increase considerably in the d = 2 nm case. The R^−b fitting in the range of R = 4 to 40 nm for d = 2 nm gives b = 1.89, 3.42, and 4.58 for E2, E3 and E4 transitions, respectively. The R dependence in the excitation efficiencies is mainly because the size of the scattered field, within which from the boundary the scattered field is as strong as the incident field, is reduced when R becomes small. The small size leads to a large gradient of the field. The decay exponent b in the R dependence is slightly smaller than a in the d dependence. This is possibly because the increase of the amount of scattered light due to the extension of the edge line mediates the R dependence.

Fig. 4 The dependence of excitation efficiencies on R and d of (a) E1, (b) E2, (c) E3 and (d) E4 transitions. The excitation efficiencies of E1 (black, dotted line), E2 (red, solid line), E3 (blue, dashed line), and E4 (green, dotted-dashed line) for d = 2 nm and d = 10 nm are shown in (e) and (f), respectively.

Download Full Size | PDF

3.2. Excitation rates with Δm resolved

In the calculations of Sec. 3.1, the rates of transitions between all magnetic sublevels are summed over. The excitation rate of each transition, however, depends on the magnetic quantum numbers in general. In order to reveal the magnetic quantum number dependence of the interaction between an atom and the local field, we investigate the excitation efficiencies with the change in the magnetic quantum number Δm resolved. The excitation rates are calculated using the decomposition of the El Hamiltonian to the spherical components $T_{E l}^{(q)}$ . The E2 Hamiltonian is decomposed as shown in Eq. (5). The expressions for higher order Hamiltonians are shown in the Appendix. The z axis is defined as the quantization axis in the calculations below. We assume the magnetic field existing around the object is sufficiently weak and the Zeeman shifts in the magnetic sublevels are negligible compared with the linewidths of the excitations.

Figure 5 shows the efficiencies of the El (l = 1, 2, 3, 4) excitations of an atom in the m = 0 state with Δm resolved. The excitation laser configuration is the same as described before and the edge curvature radius R is 2 nm. The distribution in the excitation efficiency for every resolved multipole transition is “flower-like”, that is, the bright and dark areas appear alternatively as the angle around the edge changes. The resolved excitation efficiencies with the same parity (odd or even) of Δm have similar distributions in each El transition. In addition, the distributions of excitation efficiencies with odd and even Δm seem to be reverse; the bright area for odd Δm corresponds to the dark area for even Δm and vice versa. The distributions are reminiscent of those of the spherical harmonics.

Fig. 5 Two-dimensional plots of the excitation efficiencies of E1, E2, E3, and E4 transitions with Δm resolved. The color scales of excitation efficiencies are the same in each El transition but vary among transitions with different l. The color bar for each El transition is shown at the right side of each row.

Download Full Size | PDF

We further investigate the angular dependence of the excitation efficiencies. Figure 6 (a) – 6(d) plots the excitation efficiencies of El transitions at d = 2 nm against the angle α around the edge (see Fig. 1). Each multipole El transition with Δm resolved seems to have around (l + 1) peaks. The odd and even transitions are complementary and the sum of all transitions is smooth.

Fig. 6 Angular dependence of the excitation efficiencies of (a) E1, (b) E2, (c) E3, and (d) E4 transitions at d = 2 nm with Δm resolved.

Download Full Size | PDF

Figure 7(a) – 7(d) illustrates the d and α dependence of excitation efficiencies for E1, E2, E3, and E4 transitions with Δm = 0 (corresponding to the first column in Fig. 5), respectively. The peak intervals at the near and far region are different and they are smoothly connected. Figure 7(e) plots the angular intervals between nearest peaks at d = 2, 5, 10 nm for E1, E2, E3, E4 and E5 (not shown in Fig. 5) transitions. The averages of the intervals in the transitions with Δm = 0 and 1 are plotted. The angular intervals at d = 10 nm and 2 nm are well represented by 360°/(2l) and 270°/(2l), respectively. At the intermediate distance of d = 5 nm, the intervals are between 360°/(2l) and 270°/(2l) as expected. We suppose that the interval of 360°/(2l) and 270°/(2l) result from rotational symmetries of order 2l. In the vicinity of the edge, a quadrant is filled with the object and the angle dependence is modified.

Fig. 7 The intervals between the peaks in the excitation efficiency. (a)–(d) shows d and α-dependence in the excitation efficiency of the excitation efficiencies of E1, E2, E3 and E4 transitions with Δm = 0, respectively. The color scales vary among figures. (e) plots the angular intervals between nearest peaks at d = 2 nm (filled red circles), d =5 nm (open green rectangles), and d = 10 nm (filled blue triangles) as a function of the rank of the transition l. The dot-dashed (blue) line and the dashed (brown) line correspond to 360°/(2l) and 270°/(2l), respectively.

Download Full Size | PDF

The distribution of the resolved excitation has a tiny structure beyond subwavelength. For example, when d = 2 nm and R = 2 nm, the intervals of the excitation region given by $\frac{3 π}{4}$ (d + R)/l are 4.71, 3.14, 2.36 and 1.88 nm (or λ/83, λ/129, λ/165, and λ/206) for the E2, E3, E4, and E5 transitions, respectively. More localized excitation may be realized through changes in the shape of a nanostructure. The multipole transitions in the near-field may be exploited for manipulation of cold atoms with ultra-subwavelength resolution and nanoscale trap of atoms via multipole interaction.

3.3. Emission rates of real atoms

We estimate the emission rates accompanied with multipole excitations in alkali (Rb and Cs) atoms to confirm the possibility of the experimental observation of the enhancement. We calculate the quadrupole moments $Q_{a b} = \int_{0}^{\infty} ϕ_{a} (r) ϕ_{b} (r) r^{2} d r$ using the absorption oscillator strengths (f_ab) in [37–40] and the energy difference of the atomic states to evaluate the emission rate. The calculated Q_ab are shown in Table 1. We first consider the 5²S_1/2 − 5²D_5/2 transition of Rb, the wavelength of which is adopted in the calculations of Sec. 3.1 and Sec. 3.2. This transition is exceptionally weak in E2 transitions; three or four orders of magnitude weaker than 5S − 4D transitions in Rb, transitions in Cs listed in Table 1, and the virtual E2 transition used in the excitation efficiency calculations.

Table 1. The properties of several E2 lines in Rb and Cs.

View Table

Since the 5D state has strong (E1) decay processes with a rate γ_d on the order of MHz, the excitation should lead to the E1 emission and the photon emission rate will be approximately equal to the excitation rate $γ_{e} ~ \frac{Ω^{2}}{2 γ_{d}}$ where Ω is the Rabi frequency. The fluorescence accompanied with the decaying processes (for example, the 776 nm (5D − 5P) line, the 780 nm (5P − 5S) line and the 420 nm (6P − 5S) line) are to be observed when atoms are excited. In general, a nanostructure leads to the modification of the emission rates, which can influence the photon detection rate. We, however, believe such modification does not change the situation under consideration drastically. The E2 (and higher order) emission rates would remain smaller than the E1 decay rate even if they are considerably enhanced near the object. The E1 emission rates can also be changed. Nonetheless, the change in an E1 emission rate near a surface is usually at most a small factor [43] and should be minor, as long as no surface resonances occur. Thus the excitation will be detected through the E1 spontaneous emission as discussed above.

In spite of the weakness of the 5²S_1/2 − 5²D_5/2 transition, γ_e can reach 2π × 10 Hz in the vicinity of a nanostructure when the excitation beam of 500 mW with the waist of w = 2 μm is applied. If we prepare a dense and cold atom cloud with the density 10¹⁵ cm⁻³ and the temperature T = 1 μK near a nanoedge with R = 5 nm and the length along y of 1 μm, the expected atom flux into the local field is ∼ 2 × 10⁴ s⁻¹. Since an atom scatters $~ \frac{γ_{e}}{2 π} \frac{R}{v} = 10^{- 5}$ photons (v: most probable velocity corresponding to T) on average in the local field, the atoms are expected to emit a few photons in ten seconds, which can be detected with a sensitive photodetector. The requirements for the detection of the E2 excitations other than the Rb 5S−5D transitions would be moderate. The emission rate will be several hundreds of Hz. The trapping of atoms in the vicinity of the object via, for example, the quantum vacuum force [44] may assist the detection.

4. Conclusion

We investigate the excitation efficiencies of electric multipole transitions in an atom in the vicinity of a nano-scale edge resulted from the spatial gradient of the electromagnetic field around it. The total excitation efficiencies are enhanced by several orders of magnitude. The angular dependence of excitation efficiencies with the change in the magnetic sublevels resolved are also examined. Based on the calculation on the excitation enhancement, we estimate the emission rates accompanied with the E2 excitations for alkali atoms in the vicinity of the edge to discuss the possibility for observing the enhancement in multipole excitations using cold atoms.

Appendix: Spherical components of electric multipoles

The matrix elements of arbitrary order electric multipole transitions are shown in [18]. In the present work, we expand the electric multipole transitions up to the fifth rank with the spherical basis.

The spherical components of the electric octupole (E3) operator given in Eq. (2) are expressed as

\begin{matrix} O^{(0)} = [[O_{x}^{(0)}, O_{y}^{(0)}, O_{z}^{(0)}]] \\ = \frac{1}{5 \sqrt{7}} [\begin{matrix} 0 & 0 & - 1 & 0 & 0 & 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 0 \\ - 1 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 2 \end{matrix}]; \\ O^{(\pm 1)} = \frac{1}{10 \sqrt{21}} [\begin{matrix} \pm 3 & - i & 0 & - i & \pm 1 & 0 & 0 & 0 & \mp 4 \\ - i & \pm 1 & 0 & \pm 1 & - 3 i & 0 & 0 & 0 & 4 i \\ 0 & 0 & \mp 4 & 0 & 0 & 4 i & \mp 4 & 4 i & 0 \end{matrix}]; \\ O^{(\pm 2)} = \frac{1}{\sqrt{210}} [\begin{matrix} 0 & 0 & 1 & 0 & 0 & \mp i & 1 & \mp i & 0 \\ 0 & 0 & \mp i & 0 & 0 & - 1 & \mp i & - 1 & 0 \\ 1 & \mp i & 0 & \mp i & - 1 & 0 & 0 & 0 & 0 \end{matrix}]; \\ O^{(\pm 3)} = \frac{1}{2 \sqrt{35}} [\begin{matrix} \mp 1 & i & 0 & i & \pm 1 & 0 & 0 & 0 & 0 \\ i & \pm 1 & 0 & \pm 1 & - i & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] . \end{matrix}

The spherical components for the E4 transition operator in Eq. (3) are written in a similar form as

\begin{matrix} H^{(0)} = [[\begin{matrix} H_{x x}^{(0)} & H_{x y}^{(0)} & H_{x z}^{(0)} \\ H_{y x}^{(0)} & H_{y y}^{(0)} & H_{y z}^{(0)} \\ H_{z x}^{(0)} & H_{z y}^{(0)} & H_{z z}^{(0)} \end{matrix}]] \\ = \frac{1}{105} [\begin{matrix} 3 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & - 4 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 4 & 0 & 0 & 0 & - 4 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & - 4 \\ 0 & 0 & 0 & 0 & 0 & - 4 & 0 & - 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 4 & - 4 & 0 & 0 \\ 0 & 0 & - 4 & 0 & 0 & 0 & 0 & - 4 & 0 \\ 0 & - 4 & 0 & - 4 & 0 & 0 & 0 & 0 & 8 \end{matrix}]; \\ H^{(\pm 1)} = \frac{1}{42 \sqrt{5}} [\begin{matrix} 0 & 0 & \pm 3 & 0 & 0 & - i & - i & \pm 1 & 0 \\ 0 & 0 & - i & 0 & 0 & \pm 1 & \pm 1 & - 3 i & 0 \\ \pm 3 & - i & 0 & - i & \pm 1 & 0 & 0 & 0 & 4 i \\ 0 & 0 & - i & 0 & 0 & \pm 1 & 0 & 0 & 0 \\ 0 & 0 & \pm 1 & 0 & 0 & - 3 i & 0 & 0 & 0 \\ 0 & 0 & \pm 1 & 0 & 0 & - 3 i & 0 & 0 & - 4 \\ - i & \pm 1 & 0 & \pm 1 & - 3 i & 0 & 0 & - 4 & 0 \\ \pm 3 & - i & 0 & - i & \pm 1 & 0 & 0 & 0 & \mp 4 \\ - i & \pm 1 & 0 & \pm 1 & - 3 i & 0 & 0 & 0 & 4 i \\ 0 & 0 & \mp 4 & 0 & 0 & 4 i & \mp 4 & 4 i & 0 \end{matrix}]; \\ H^{(\pm 2)} = \frac{1}{21 \sqrt{10}} [\begin{matrix} - 2 & \pm i & 0 & \pm i & 0 & 0 & 0 & 0 & 2 \\ \pm i & 0 & 0 & 0 & \pm i & 0 & 0 & 0 & \mp 2 i \\ 0 & 0 & 2 & 0 & 0 & \mp 2 & 2 & \mp 2 & 0 \\ \pm i & 0 & 0 & 0 & \pm i & 0 & 0 & 0 & \mp 2 i \\ 0 & \pm i & 0 & \pm i & 2 & 0 & 0 & 0 & - 2 \\ 0 & 0 & \mp 2 i & 0 & 0 & - 2 & \mp 2 & - 2 & 0 \\ 0 & 0 & 2 & 0 & 0 & \mp 2 i & 2 & \mp 2 i & 0 \\ 0 & 0 & \mp 2 i & 0 & 0 & - 2 & \mp 2 i & - 2 & 0 \\ 2 & \mp 2 i & 0 & \mp 2 i & - 2 & 0 & 0 & 0 & 0 \end{matrix}]; \\ H^{(\pm 3)} = \frac{1}{6 \sqrt{35}} [\begin{matrix} 0 & 0 & \mp 1 & 0 & 0 & i & \mp 1 & i & 0 \\ 0 & 0 & i & 0 & 0 & \pm 1 & i & \pm 1 & 0 \\ \mp 1 & i & 0 & i & \pm 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & i & 0 & 0 & \pm 1 & i & \pm 1 & 0 \\ 0 & 0 & \pm 1 & 0 & 0 & - i & \pm 1 & - i & 0 \\ i & \pm 1 & 0 & \pm 1 & - i & 0 & 0 & 0 & 0 \\ \mp 1 & i & 0 & i & \pm 1 & 0 & 0 & 0 & 0 \\ i & \pm 1 & 0 & \pm 1 & - i & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]; \\ H^{(\pm 4)} = \frac{1}{3 \sqrt{70}} [\begin{matrix} 1 & \mp i & 0 & \mp i & - 1 & 0 & 0 & 0 & 0 \\ \mp i & - 1 & 0 & - 1 & \pm i & 0 & 0 & 0 & 0 \\ - 1 & \mp i & 0 & \pm i & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] . \end{matrix}

The electric triakontadipole (E5) transition Hamiltonian is given by

ℋ_{E 5} = e T_{i j k l m} \nabla_{i} \nabla_{j} \nabla_{k} \nabla_{l} E_{m} (r, ω_{0}),

where

\begin{array}{l} T_{i j k l m} = r_{i} r_{j} r_{k} r_{l} r_{m} - \frac{1}{9} r^{2} (r_{i} r_{j} r_{k} δ_{l m} + r_{i} r_{j} r_{l} δ_{k m} \\ + r_{i} r_{j} r_{m} δ_{k l} + r_{i} r_{k} r_{l} δ_{j m} + r_{i} r_{k} r_{m} δ_{j l} + r_{i} r_{l} r_{m} δ_{j k} \\ + r_{j} r_{k} r_{l} δ_{i m} + r_{j} r_{k} r_{m} δ_{i l} + r_{j} r_{l} r_{m} δ_{i k} + r_{k} r_{l} r_{m} δ_{i j}) \\ + \frac{1}{21} r^{4} (r_{i} δ_{j k l m} + r_{j} δ_{i k l m} + r_{k} δ_{i j l m} + r_{l} δ_{i j k m} + r_{m} δ_{i j k l}), \end{array}

is a matrix element of the electric triakontadipole (E5) operator. The spherical components for the E5 transition can be written as

T^{(Δ m)} = [[T_{x}^{(Δ m)}, T_{y}^{(Δ m)}, T_{z}^{(Δ m)}]],

with

\begin{array}{l} T_{x}^{(Δ m)} = [[\begin{matrix} T_{x x x}^{(Δ m)} & T_{x x y}^{(Δ m)} & T_{x x z}^{(Δ m)} \\ T_{x y x}^{(Δ m)} & T_{x y y}^{(Δ m)} & T_{x y z}^{(Δ m)} \\ T_{x z x}^{(Δ m)} & T_{x z y}^{(Δ m)} & T_{x z z}^{(Δ m)} \end{matrix}]] \\ T_{y}^{(Δ m)} = [[\begin{matrix} T_{y x x}^{(Δ m)} & T_{y x y}^{(Δ m)} & T_{y x z}^{(Δ m)} \\ T_{y y x}^{(Δ m)} & T_{y y y}^{(Δ m)} & T_{y y z}^{(Δ m)} \\ T_{y z x}^{(Δ m)} & T_{y z y}^{(Δ m)} & T_{y z z}^{(Δ m)} \end{matrix}]] \\ T_{z}^{(Δ m)} = [[\begin{matrix} T_{z x x}^{(Δ m)} & T_{z x y}^{(Δ m)} & T_{z x z}^{(Δ m)} \\ T_{z y x}^{(Δ m)} & T_{z y y}^{(Δ m)} & T_{z y z}^{(Δ m)} \\ T_{z z x}^{(Δ m)} & T_{z z y}^{(Δ m)} & T_{z z z}^{(Δ m)} \end{matrix}]] . \end{array}

Funding

The Matsuo Foundation; The Research Foundation for Opto-Science and Technology; Grants-in-Aid for Scientific Research (15K05234, 26887033) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan; The NIFS Collaboration Research program (NIFS16KBAF025); Chuo University Joint Research Grant.

Acknowledgments

We would like to thank Itsuki Banno for early discussion of higher order transitions of atoms and Takashi Yatsui and Motoichi Ohtsu for fruitful discussion of enhanced multipole moments of nanoparticles.

References and links

1. M. Ohtsu and H. Hori, Near-Field Nano-Optics (Kluwer/Plenum, 1999).

2. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).

3. M. L. Juan, M. Righini, and R. Quidant, “’Plasmon nano-optical tweezers,” Nature Photon. 5, 349–356 (2011).

4. M. Gullans, T. G. Tiecke, D. E. Chang, J. Feist, J. D. Thompson, J. I. Cirac, P. Zoller, and M. D. Lukin, “Nanoplasmonic lattices for ultracold atoms,” Phys. Rev. Lett. 109, 235309 (2012).

5. H. Raether, Surface plasmons on smooth and rough surfaces and on gratings (Springer, 1988).

6. G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photon. 8, 835–840 (2014).

7. V. V. Klimov, D. Bloch, M. Ducloy, and J. R. Rios Leite, “Mapping of focused Laguerre-Gauss beams: The interplay between spin and orbital angular momentum and its dependence on detector characteristics,” Phys. Rev. A 85, 053834 (2012).

8. K. Deguchi, M. Okuda, A. Iwamae, H. Nakamura, K. Sawada, and M. Hasuo, “Simulation of electric quadrupole and magnetic dipole transition efficiencies in optical near fields generated by a subwavelength slit array,” J. Phys. Soc. Jpn. 78, 024301 (2009).

9. A. M. Kern and O. J. F. Martin, “Strong enhancement of forbidden atomic transitions using plasmonic nanostructures,” Phys. Rev. A 85, 022501 (2012).

10. P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14, 093033 (2012).

11. N. Yang and A. E. Cohen, “Local geometry of electromagnetic fields and its role in molecular multipole transitions,” J. Phys. Chem. B 115, 5304–5311 (2011).

12. S. Tojo, M. Hasuo, and T. Fujimoto, “Absorption enhancement of an electric quadrupole transition of cesium atoms in an evanescent field,” Phys. Rev. Lett. 92, 053001 (2004).

13. S. Tojo and M. Hasuo, “Oscillator-strength enhancement of electric-dipole-forbidden transitions in evanescent light at total reflection,” Phys. Rev. A 71, 012508 (2005).

14. M. Roberts, P. Taylor, G. P. Barwood, P. Gill, H. A. Klein, and W. R. C. Rowley, “Observation of an electric octupole transition in a single ion,” Phys. Rev. Lett. 78, 1876–1879 (1997).

15. K. Hosaka, S. A. Webster, A. Stannard, B. R. Walton, H. S. Margolis, and P. Gill, “Frequency measurement of the ²S_1/2−²F_7/2 electric octupole transition in a single ¹⁷¹Yb⁺ ion,” Phys. Rev. A 79, 033403 (2009).

16. S. Huotari, E. Suljoti, C. J. Sahle, S. Radel, G. Monaco, and F. M. F. de Groot, “High-resolution nonresonant x-ray Raman scattering study on rare earth phosphate nanoparticles,” New J. Phys. 17, 043041 (2015).

17. N. A. Jelley, Fundamentals of Nuclear Physics (Cambridge University, 1990).

18. S. Bernadotte, A. J. Atkins, and C. R. Jacob, “Origin-independent calculation of quadrupole intensities in X-ray spectroscopy,” J. Chem. Phys. 137, 204106 (2012).

19. T. Yatsui, T. Tsuboi, M. Yamaguchi, K. Nobusada, S. Tojo, F. Stehlin, O. Soppera, and D. Bloch, “Optically controlled magnetic-field etching on the nano-scale,” Light: Science & Applications 5, e16054 (2016).

20. W. Heitler, The quantum theory of radiation (Clarendon, 1954).

21. E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. 4, 87–132 (1932).

22. J. Fortágh and C. Zimmermann, “Magnetic microtraps for ultracold atoms,” Rev. Mod. Phys. 79, 235–289 (2007).

23. C. Stehle, H. Bender, C. Zimmermann, D. Kern, M. Fleischer, and S. Slama, “Plasmonically tailored micropotentials for ultracold atoms,” Nat. Photon. 5, 494–498 (2011).

24. V. Klimov, D. Bloch, M. Ducloy, and J. R. Rios Leite, “Detecting photons in the dark region of Laguerre-Gauss beams,” Opt. Express 17, 9718–9723 (2009).

25. P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007).

26. A. Taflove and S. C. Hagness, Computational Electrodynamics: The finite-difference time-domain method (Artech, 2000).

27. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm. 181, 687 (2010).

28. A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998).

29. J. E. Sansonetti, “Wavelengths, transition probabilities, and energy levels for the spectra of rubidium (Rb I through Rb XXXVII),” J. Phys. Chem. Ref. Data 35, 301–421 (2006).

30. I. Johansson, “Spectra of the alkali metals in the lead sulfide region,” Ark. Fys. 20, 135 (1961).

31. U. Litzén, “The 5 g Levels of the alkali metals,” Phys. Scr. 1, 253 (1970).

32. H. Ehrenreich and H. R. Philipp, “Optical Properties of Ag and Cu,” Phys. Rev. 128, 1622 (1962).

33. H. U. Yang, J. D’Archangel, M. L. Sundheimer, E. Tucker, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of silver,” Phys. Rev. B 91, 235137 (2015).

34. E. Clementi, D. L. Raimondi, and W. P. Reinhardt, “Atomic screening constants from SCF functions. II. Atoms with 37 to 86 electrons,” J. Chem. Phys. 471300–1307 (1967).

35. T. Inoue and H. Hori, “Theoretical treatment of electric and magnetic multipole radiation near a planar dielectric surface based on angular spectrum representation of vector field,” Opt. Rev. 5, 295–302 (1998).

36. M. Born and E. Wolf, Principles of optics, 7th ed. (Cambridge University, 1999).

37. K. Niemax, “Oscillator strength of Rb quadrupole lines,” J. Quant. Spectrosc. Radiat. Transfer , 17, 747–750 (1977).

38. B. Warner, “Atomic oscillator strength–III,” Mon. Not. R. Astr. Soc. 139, 115–128 (1968).

39. S. Tojo, T. Fujimoto, and M. Hasuo, “Precision measurement of the oscillator strength of the cesium 6 ²S_1/2 → 5 ²D_5/2 electric quadrupole transition in propagating and evanescent wave fields,” Phys. Rev. A , 71, 012507 (2005).

40. K. Niemax, “Broadening and oscillator strength of Cs quadrupole lines,” J. Quant. Spectrosc. Radiat. Transfer , 17, 125–129 (1977).

41. K.-H. Weber and C. J. Sansonetti, “Accurate energies of nS, nP, nD, nF, and nG levels of neutral cesium,” Phys. Rev. A 35, 4650–4660 (1987).

42. K. B. Eriksson, I. Johansson, and G. Norlén, “Precision wavelength measurements connecting the Cs I 6s, 6p, and 6d levels, with a study of the correction for phase change in infrared interferometry,” Ark. Fys. 28, 233 (1964).

43. W. Lukosz and R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 20, 195–199 (1977).

44. D. E. Chang, K. Shinha, J. M. Taylor, and H. J. Kimble, “Trapping atoms using nanoscale quantum vacuum forces,” Nat. Commun. 5, 4343 (2014).

	f_ab	ΔE[cm⁻¹]	Q_ab ( $a_{0}^{2}$ )
Rb 5S_1/2 − 4D_5/2	1.35 × 10^{−6 a}	19355.203^e	47.1
Rb 5S_1/2 − 4D_3/2	8.98 × 10^{−7 a}	19355.649^e	38.4
Rb 5S_1/2 − 5D_5/2	2.91 × 10^{−10 b}	25703.498^e	0.452
Rb 5S_1/2 − 5D_3/2	1.94 × 10^{−10 b}	25700.536^e	0.368
Cs 6S_1/2 − 5D_5/2	4.69 × 10^{−7 c}	14596.8423^f	42.4
Cs 6S_1/2 − 5D_3/2	3.28 × 10^{−7 d}	14499.2584^f	35.8
Cs 6S_1/2 − 6D_5/2	3.68 × 10^{−7 d}	22588.8210^g	19.5
Cs 6S_1/2 − 6D_3/2	2.95 × 10^{−7 d}	22631.6863^g	17.4

Excitation enhancement in electric multipole transitions near a nanoedge

Abstract

1. Introduction

2. Method

3. Results and discussion

3.1. Excitation efficiencies in total

3.2. Excitation rates with Δm resolved

3.3. Emission rates of real atoms

4. Conclusion

Appendix: Spherical components of electric multipoles

Funding

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (17)

Optics Express