Selection rule engineering of forbidden transitions of a hydrogen atom near a nanogap

1 Introduction

Long wavelength approximation is at the heart of well-known selection rules in atomic spectroscopy. The wavelength of light is much larger than the atom size, so that the light-atom interaction Hamiltonian can safely ignore the spatial variation in the scale of the wavelength, resulting in an effective Hamiltonian in the form of −p⇀⋅E⇀=−ex→⋅ε⌢E0e−iωt where p⇀ is the dipole moment operator, E⇀ the electric field of light, ε⌢ the polarization direction unit vector, E₀ the amplitude of the electric field, and ω the angular frequency of light. Thereby, spontaneous emission, stimulated emission, and absorption are all proportional to the matrix element 〈ψf|x→|ψi〉 (ψ_f , ψ_i =final and initial state wavefunctions, respectively), from which all selection rules follow. The most important selection rule Δl=±1; Δm=±1, 0 originates from the position operator x→ being represented by the spherical harmonics of order 1. While this selection rule can be broken by magnetic dipole transition, electric quadrupole transition, or by two photon transitions, these forbidden transitions are typically several orders of magnitudes weaker. For instance, the 2P-1S allowed transition lifetime of 2 ns for a hydrogen atom compares favorably with 4.6 days for the forbidden magnetic dipole transition lifetime of 2S to 1S. On the other hand, the electric quadrupole transition between 3D and 2S is somewhat less forbidden, taking 20 ms [1], [2]. Of some practical importance, this one-photon lifetime of the forbidden magnetic dipole transition between 2S and 1S is so impractically long that it is easily superseded by the two-photon lifetime of 0.15 s [3], which played an important role in the measurements of the Lamb shift [4].

While the spatial variation of electromagnetic waves in free space occurs within the wavelength scale, close to the induced sources such as surface current and surface charges which naturally occur in metallic nano objects, electric field vectors can vary in length scale much smaller than their vacuum wavelength, in the length scale of the nano objects themselves or the gap size between the metallic objects [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. Of particular interest in the present paper is the one-dimensional metallic nano- and sub-nanogaps whose widths can be in the 1–0.1 nm regime [17], [18], [19], comparable to the spatial extents of hydrogen atom wavefunctions while maintaining a macroscopic length of 1 mm to 1 cm. Electric fields emanating from these gaps possess rapidly varying electric fields, both in magnitude and in direction, in the length scale of the gap itself, creating a potentially very useful field configuration for the purpose of breaking down well-known selection rules, thereby facilitating forbidden transitions in large enough volumes to be experimentally detectable.

2 Materials and methods

To model spatial variation of the electric field emanating from nano- and sub-nanometer gaps, we first consider a line dipole with a line charge density λ and a gap width of w, fed by an alternating current source of angular frequency ω and surface current density Ke^−iωt with the charge conservation relationship K=iωλ. In the extreme subwavelength regime of our interest, we can ignore the retarded time, so that the electric field is approximated by the near-field term only [20], [21], [22], [23], [24], [25], [26]:

as plotted in Figure 1A for w=1 nm. By replacing λ with σdz and integrating over a film thickness of h=100 nm, we obtain a realistic field profile of a capacitative nanogap of surface charge density σ, as shown in Figure 1B. For large enough h<<w, this field is well approximated by a simple form:

Figure 1:

(A) Electric field lines above a line dipole of gap width 1 nm. Line charges are located at x=±0.5 nm, z=0. (B) Electric field profile obtained by an integration of the line dipoles from z=−100 to 0 nm, well fitted with an analytical form E→(x,z)=σw2πε0(z,−x)x2+z2 when far away from the gap. (C) Electric field lines E→(x,z)−E→(x0,z0) around a center position of (x₀, z₀)=(3 nm, 3 nm) together with |ψ₃₂₀(x, 0, z)|². The color scale is in arbitrary units and we use the integrated field profile of Eq. (2). (D) Contour plot of an xz cross section of the overlap integrand ψ320Hint(2)ψ200 around the center position of (x₀, z₀) (3 nm, 3 nm). The color scale is in arbitrary units.

for distances larger than w but smaller than h. Unless otherwise indicated, calculations are performed using the integrated field profile of Eq. (2) using Mathematica and Matlab.

3 Results

We now place a hydrogen atom at a position (x₀, z₀) in a field profile given as Eq. (3), and ask how the transition rates between different states will change relative to the case of a plane wave excitation. Near the source, say, at a position of (x₀, z₀)=(3 nm, 3 nm), the field lines are curved in the scale of the 3D wavefunctions of a hydrogen atom. It behooves us to examine the behavior of the electric field and the interaction Hamiltonian at the near field. Plotted in Figure 1C at (x₀, z₀)=(3 nm, 3 nm) are the electric field lines minus their central value, E→(x,z)−E→(x0,z0), with an xz cross section of the 3D wavefunction |ψ₃₂₀|² (n=3; l=2; m=0) in an area of 2 nm by 2 nm squared. The field lines approximate those of a vector field (x′, −z′), with (x, z)≡(x₀+x′, z₀+z′), producing an interaction Hamiltonian with a symmetry of x^′2−z^′2, as can be seen clearly when we expand around (x₀, z₀):

where e is the electron charge.

The first term is the dipole approximation Hamiltonian that gives rise to the usual selection rules, whereas the second term contains all the salient features:

having taken advantage of the divergence relation ∂Ex∂x+∂Ez∂z=0. To see how this Hamiltonian consisting of two-dimensional quadrutic polynomials can be taken advantage of by the D waves, we resume our interest in the 2S to 3D transition. With the photon energy of 13.6 eV(14−19)=1.89 eV (656 nm) well within the visible range, we can produce the essentially cylindrical field profile near the nanogap using common transition metals. In Figure 1D, the y cross section of the overlap integrand ψ320Hint(2)ψ200 is shown in a 2 nm by 2 nm area with the hydrogen nucleus at (x₀, z₀)=(3 nm, 3 nm). The integrand ψ320Hint(2)ψ200 stays mostly positive, because the symmetries of z^′2−x^′2 from the Hamiltonian and 2z^′2−x^′2 from ψ₃₂₀ are quite similar. This result suggests that there will be a significant transition matrix element between 2S and 3D states, especially at the vicinity of the gap.

Note that ψ322≡ψ32+2+ψ32−22 also couples to ψ₂₀₀ through z^′2−x^′2, with a matrix element smaller by a factor 3, whereas the −ex′z′∂Ez∂x|(x=x0,z=z0) part of the Hamiltonian does not participate significantly along the x=z line since |∂Ez∂x|≈0 along this line. Clearly, for general directions we also need to consider excitations into ψ₃₂₊₁ and ψ₃₂₋₁ through x′ z′.

To quantify how strong the forbidden transition matrix elements are between 2S and 3D states, we recall transition dipole moments of allowed excitations. Choosing a local electric field orientation as the z direction, a relevant dipole moment is defined as

dallowed=|〈ψf|Hint|ψi〉|E0=|〈ψf|eE0z|ψi〉|E0=|〈ψf|ez|ψi〉|~eaB,

where E₀ is the electric field strength at the hydrogen nucleus and a_B is the Bohr radius. Analogously, we define the transition dipole moment of a forbidden 2S to 3D excitation such that

d200→320=|〈ψ320|Hint|ψ200〉|E0(x0,z0).

In Figure 2, we quantify forbidden dipole moments of the 2S to 3D transitions. We calculate the total forbidden dipole/transition moment d2S→3D=∑md200→32m2 along the x=z line, as shown in Figure 2A. Calculations using the full H_int are represented by blue squares, while those using only Hint(2) are represented by a blue line, displaying near perfect agreement. Finally, taking advantage of |∂Ez∂z||E(x,z)|≈1|x| and |∂Ez∂x|≈0 along this line, we reach the simple closed form approximation for the total forbidden dipole moment:

Figure 2:

(A) Total forbidden dipole moment along the x=z line, calculated from the second-order Hamiltonian (blue line), the full interaction Hamiltonian (blue squares), and a fit to the analytical expression 0.719/2z (red line). (B) Contour plot of the forbidden transition moment d200→320=|〈ψ320|Hint|ψ200〉|E0(x0,z0) in units of ea_B where a_B is the Bohr radius, in a 20 nm by 20 nm area starting from z=1 nm. (C) Contour plot of the forbidden transition moment d200→321=|〈ψ321|Hint|ψ200〉|E0(x0, z0). (D) Contour plot of the total forbidden transition moment d2S→3D=d200→3202+d200→3222+d200→3212. The cylindrical symmetry is largely restored.

represented by a red line. In unit of ea_B the red line corresponds to 0.7192x0=0.719ρ0, where x₀ and the distance from the origin ρ₀ are in nanometers. All three results agree rather well.

To see the angular dependences of various forbidden excitations, we plot d200→3202+d200→3222 in Figure 2B, demonstrating that indeed excitations into ψ₃₂₂ and ψ₃₂₀ are maximum along the x=z line. Transition dipole moments into ψ321=ψ32+1+ψ32−12,d_200→321, are plotted in Figure 2C, showing an almost orthogonal angle dependence from that of d200→3202+d200→3222. Adding all the forbidden dipoles, an almost isotropic d2S→3D=d200→3202+d200→3212+d200→3222 is obtained (Figure 2D), fitted rather well with the simple analytical expression of 0.719ρ0 now applicable to all directions within an error of 0.2–2%, as we move away from the z-axis towards the x-axis.

We now study the width dependence of the forbidden transitions. Since the cylindrical symmetry of the total forbidden dipole moment is only approximate in our geometry, we expect to find deviations as we increase the gap width. Figure 3A depicts the case of d_2S→3D for w=3 nm. Away from the gap, the cylindrical symmetry is recovered, whereas for distances less than 5 nm, angular deviations and weaker moments are evident. For w=10 nm in Figure 3B, the deviations are more pronounced, but again, at distances larger than 10 nm, the forbidden dipole moment converges to those of narrower gaps. Figure 3C plots the forbidden dipole moment along the z-axis for several gap widths. For gap widths of 3, 5, and 10 nm, the forbidden dipole moments eventually converge to the 1/z line at z~w. Scanning along the x-direction for a fixed z=1 nm, the behavior is very different. At x=0, forbidden dipole moments are smaller mainly because along the middle of the gap, field curvatures are less. For w=3, 5, and 10 nm, d_2S→3D peaks at x≈w2 because sharp edges of the charge distributions are located at (x=±w2,z=0). Again, at all instances d_2S→3D recovers the 0.719ρ0 dependence for x>10 nm.

Figure 3:

Total forbidden dipole moment d_2S→3D plotted (A) when w=3 nm and (B) when w=10 nm. (C) Total forbidden dipole moment along the z-axis when w=1, 3, 5, and 10 nm. (Inset: a log-log plot). (D) Total forbidden dipole moment when the observation is along the x-axis keeping z=1 nm for w=1, 3, 5, and 10 nm.

We now consider the excited state wavefunction

at various locations and conditions. The near-perfect cylindrical symmetry of d_2S→3D for the 1 nm gap case suggests a strong symmetry for |ψex〉 as well. Figure 4A displays the excited state wavefunction squared using the full interaction Hamiltonian at three different locations. The excited state wavefunctions faithfully reproduce a pure |ψ321〉 state within a coordinate system defined by the local field orientation. While physically intuitive, mathematically it is because (x2−z2)2∂Ez∂z=xz∂Ez∂x when using the field profile of Eq. (2). We found that even with larger gap widths, excited wavefunctions’ orientations following the local field orientation remain largely unaffected except for right above the sharp edge (Figure 4B). In stark contrast, as shown in Figure 4C, the single wire case described by Eq. (1) displays wavefunctions not rotating with the local field orientation in a simplistic way. In spite of this complication, the excited wavefunctions remain a pure |ψ321〉 state at a properly rotated coordinate system, which directly follows from the Hamiltonian of Eq. (4) containing only two quadratic terms: (x²−z²); 2xz. Replacing our source by a simple point dipole and using Eq. (6) with three-dimensional Hamiltonian produce a whole combination of D wavefunctions evident in Figure 4D. On the equator relative to the dipole orientation, pure |ψ321〉 still get excited, whereas at most of other directions, all three states contribute.

Figure 4:

Plot of the xz cross section of the excited state wavefunction as defined in Eq. (6), at strategic locations under various conditions.

(A) w=1 nm with a cylindrical field profile described in Figure 1B, generated by integrating line dipoles from z=−100 to 0 nm, approximated by Eq. (2). (B) Same as (A) except w=4 nm. (C) w=1 nm using the line dipole field of Eq. (1), depicted in Figure 1A. (D) Excited state wavefunctions with a point dipole source. Unlike those excited by the infinite line sources, the wavefunctions are not of a uniform shape.

4 Discussion and conclusion

With the 2S to 3D transition being essentially allowed near the gap, we estimate the spontaneous decay lifetime from 3D to 2S states. An effective dipole moment of one Bohr radius gives rise to a lifetime of 44 ns, six orders of magnitudes faster than the quadrupole transition in the vacuum. The physics of this spontaneous emission modification by nanostructures [27], [28] is clear in our case: in vacuum, the quadrupole transition is weaker than the dipole transition by ∼(2πaBλ0)2, where λ₀ is the wavelength of light; on the other hand, near the nanogap with a distance of ρ, we replace this factor by ∼(2πaBρ)2, resulting in fast lifetimes comparable to those of the allowed transitions. Our infinite nanogaps have the advantage over point gaps in that it stretches to millimeter to centimeter length scale in the y-axis [11], [12], offering more robustness and million times larger effective areas than point gaps [29]. In attempts to break selection rules by going to shorter wavelength light, for example, X-rays [30], the transitions necessarily involve core orbitals of comparable short length scale, so that the effect is less dramatic than that presented here. Finally, while our technique applies to any S to D transitions, it may not apply to 1S to 3S transitions such as described in [31]. This is because the quadratic potential still gives rise to zero matrix element between two S states because of symmetry.

Our two-dimensional quadratic potentials have, in addition to the obviously larger volume, another advantage over point source dipoles that also give rise to forbidden transitions in surface-enhanced Raman scattering and infrared absorption [23], [24], [25], [26] in molecules. The excited wavefunctions are all of one nature, as shown in Figure 4A–C, which can give rise to constructive interference of quadrupole radiations. Finally, while an analytical field profile has been used throughout our paper, a COMSOL calculation assuming a 1 nm gap sandwiched by aluminum layers of 100 nm thickness at 656 nm produces a field profile of a cylindrical symmetry well described by Eq. (2). Finite-difference-time-domain calculations as well as vector field mapping experiments also support this picture [5], [32], [33], [34]. We therefore expect similar quantum mechanical results under finite elements electromagnetic simulations.

In conclusion, we have shown that the 2S-3D forbidden transition is allowed for all practical purposes, near the vicinity of a metallic nanogap. The relevant scale of this quadrupole transition becomes not the wavelength of light but the gap width and the distance of the atom from the gap. With million times larger effective volume than point gaps, together with the highly symmetric excited state wavefunctions, we foresee an intimate interaction between atomic spectroscopy and now mature nanogap technology in the near future, especially with free standing gaps. With the advantage of metallic nanogaps of infinite length with an ultimate field enhancement [32] whereby electromagnetic waves from microwaves to ultraviolet have all the same near-field profile [33], [34], up to the plasma frequency of metal, selection rule-free spectroscopy of atoms, molecules, and quantum dots will become of wide use.