Systematic design of a robust half-W1 photonic crystal waveguide for interfacing slow light and trapped cold atoms

A periodic modulation of the refractive index has deep consequences on light propagation. It enables a coupling between forward and backward propagating waves, opening photonic band gaps where light propagation is forbidden within a given frequency range. At the edge of these band gaps, the group velocity vanishes [42] and the Purcell factor diverges (see appendix A).

Motivated by the proposal in [28], we study a similar structure with a different material: GaInP. This material has been chosen for its advantageous optical and electronic properties. GaInP has a wide electronic band gap below 1.85 eV [43], and as such is transparent for a wide range of wavelengths (from 670 nm up), meaning it could be used with several alkali. At 780 nm, its refractive index is n = 3.35, reaching 3.55 at the electronic band edge. This large index contrast with the air gives rise to band gaps that are wider and further away from the light line [42], allowing for more flexibility in the design of the dispersion curves (both for the slow and the trapping modes). Finally, this material has attracted some attention in recent years as it is very convenient to operate in the telecom band due to its low two-photon absorption [44], and growth and fabrication processes have therefore been developed and well mastered.

As shown in figure 1(b), the holes etched in the GaInP slab do not go up to the edge, leaving a few hundreds of nanometers of unperturbed slab where the light can propagate. Being based on a 2D slab rather than a 1D structure, this geometry should be quite rigid and prevent detrimental effects from low frequency mechanical modes. The introduced symmetry breaking in the transverse direction allows for a more precise control on the dispersion properties of the waveguide since it offers extra degrees of freedom [45], while significantly improving the optical access. Transverse asymmetry has been harnessed in [46] to create exotic dispersion bands such as Dirac cones, multivalleys, or flat bands. Arrays of ⁸⁷Rb will then be trapped near the edge of the waveguide thanks to a two-color dipole trap, at 116 nm from the surface. For comparison, in tapered nanofiber platforms, atoms sit more than 200 nm away from the silica fiber.

For a given thickness t, chosen here to be t = 150 nm, the first step to determine the geometrical parameters consists in finding the lattice period a and hole radius r of the bulk 2D photonic crystal that allow for a band gap at the ⁸⁷Rb D₂ transition. Indeed the width and position of the band gap is entirely determined by these values [42]. The band gap has to be wide enough to allow for at least two guided modes, one that crosses 780 nm, and a blue-detuned one for trapping, as described later. Guided bands appear when introducing the defect at the edge, and we can align the band of interest with respect to the D₂ line by adjusting the width L.

Given these constraints, the geometrical parameters of the waveguide are found to be: a = 212 nm, r = 63 nm and L = 337 nm for t = 150 nm. The corresponding band structure, computed with the 3D FDTD software Lumerical ⁷ is displayed in figure 1(c) (represented over the 1st Brillouin zone, by application of Bloch's theorem). Three guided bands can be found inside the band gap of the 2D photonic crystal between 365 and 435 THz. The bulk modes are guided in the slab (k_z imaginary) but can propagate in any direction in the plane, even inside the 2D array of holes ($k_x, k_y$ real). Above the light cone, radiative modes have a real k vector in all directions and are therefore not guided.

Nanofabrication inherently leads to imperfections, even if errors below 2 nm can be reached [47]. A specific effort has been put in our design process to minimize the impact of such imperfections, thereby facilitating an experimental realization.

As the Purcell factor diverges at a band edge, one naive approach could be to align the D₂ line frequency $\omega_\mathrm{a}$ to any band edge of the band structure. However, fabrication imperfections, to first order, lead to a shift of the energy of the band [48]. The flatter the band, i.e. the smaller the group velocity, the more it is vulnerable to a shift in frequency [49]. If the D₂ line is aligned with the band edge, an infinitesimal shift to a lower energy will bring the atomic transition in the band gap of the 2D photonic crystal, impeding the propagation of the emitted light. In addition, the disorder in the frequency shift along the waveguide (from the randomness of the imperfections) can lead to strong localization of light inside the crystal [48, 50, 51].

Following [28], two main criteria are to be considered when assessing the robustness of a structure: the group velocity has to be independent of frequency around the atomic transition, i.e. $\partial v_\mathrm{g}/\partial\omega\rvert_{\omega_\mathrm{a}} \sim 0$, and the distance of the atomic transition to the band edge $\Delta \omega = |\omega_\mathrm{a} - \omega_\mathrm{e}|$ has to be as large as possible.

Designing slow modes with linear bands (i.e. an almost constant, large group index $n_{\mathrm{g}}$ over the widest range of ω possible) allows us to fulfill these two criteria. First, a linear dispersion corresponds to a vanishing group velocity dispersion (GVD) and the atom–photon coupling is proportional to the group index. Moreover, as shown in [30], it is possible to design a slow and linear band over a wide spectral range. As most fabrication imperfections lead to a shift $\Delta \omega$ of the guided bands, both these constraints aim at placing the relevant frequency at a position on the band where a small shift will affect the dispersion at the given frequency only slightly. It has been shown that linear bands can be achieved in at least two types of asymmetric PCWs [46]. Achieving such vanishing GVD has been extensively studied in the context of W1 waveguides, by tuning the position of rows of holes [30, 52, 53], chirping the waveguide properties [54], or changing the size of the holes [31].

We note that such engineering is only possible for non-fundamental modes, which have a faster decay outside the structure [55]. These modes come with non-trivial impedance and mode matching, which have been efficiently addressed in the literature using tapered regions [56, 57].

Inspired by these previous strategies, we optimize the shape of the slow-mode band by tuning the geometry of the 2D photonic crystal. As depicted in figure 1(b), the 6 independent optimization parameters are the radius of the first three rows of holes as well as their position along the y axis ($\delta r_i$, $\delta y_i$), $i \in \{1,2,3\}$. As full 3D FDTD simulations are computationally intensive, we use the approximate method of Guided Mode Expansion (GME) [58] thanks to the legume solver [59] to compute faster the shape of the guided band. At each iteration, a cost function enforcing the minimization of the GVD (averaged over the wave vector interval) while setting a target $n_\mathrm{g}$ value is evaluated and the ($\delta r_i$, $\delta y_i$) varied thanks to automatic differentiation. After a few hundred iterations we obtain the optimal shifts for achieving this target $n_\mathrm{g}$ value over the widest possible spectral range. Finally, the optimized structure is simulated in full 3D FDTD to validate the results from the approximate GME method.

In order for this optimization to give relevant results, $n_\mathrm{g}$ has to be set to an experimentally realistic value, ideally below 60. Indeed, experiments have shown that it is extremely challenging to reach higher values for the group index without losses [60]. The most concluding optimization results are obtained for a target around $n_\mathrm{g} = 30$. The shifts in position and radius after optimization are given in table 1 and the corresponding band structure is presented in figure 1(c). Figure 2(a) shows that we engineered a band with a constant group index $n_\mathrm{g} \simeq 28$ over a 9 nm range, and hence reach similar performance than a previous optimization of a W1 waveguide [30]. This feature offers a two-fold advantage. In addition to making it robust to shifts caused by fabrication imperfections, the optimization enables using the half-W1 waveguide in a large bandwidth regime (${\unicode{x2A7E}} 4$ THz) with very little dispersion.

Zoom In Zoom Out Reset image size

Table 1. Optimal changes in row positions and holes radii via automatic differentiation optimization. All the rows after the third one are unperturbed.

Row	Position $\delta y$ (nm)	Radius $\delta r$ (nm)
1	+42.7	+14.2
2	+53.8	−11.2
3	−3.7	−10.8

Finally, since the Purcell factor is proportional to the group index $n_\mathrm{g}$ (see appendix A), we obtain with our optimization strategy an almost constant Purcell factor in this frequency range as seen in figure 2(b). The residual variation arises from small changes in the structure of the electric field when moving along the guided band.

Given the optimized design, we now turn to the interaction between the slow mode and the ⁸⁷Rb atoms in the vicinity of the waveguide. Taking into account the multilevel character of Rubidium, we defined a transition-dependent Purcell factor in equation (A.4) given in appendix A. The group velocity $v_\mathrm{g}$ is evaluated from the simulated band structure, while the other terms are computed from the field map of the guided mode (as the Purcell factor is proportional to the slow-mode intensity). Figures 3(a) and (b) show that in order to have the maximum coupling, the atoms should be trapped close to the edge of the waveguide, aligning them to the holes of the first row.

Zoom In Zoom Out Reset image size

Figure 3(c) shows that the guided mode has a strong circular polarization along the whole structure quantified by the ellipticity vector $\mathbf{C} = \operatorname{Im}\left[(\mathbf{E} \times \mathbf{E}^*)/|\mathbf{E}|^2\right]$. This is reminiscent of the polarization of the light around nanofibers and comes from the longitudinal component of the electric field that appears when light is strongly confined. We note that this is also allowed by the slow-mode being far from the edge of the Brillouin zone, preventing time-reversal symmetry to cancel out the circularly polarized component of the slow mode [61]. By choosing a quantization axis perpendicular to our waveguide (along z), the polarization of the forward propagating mode seen by the atoms is predominantly $\sigma^+$. The selectivity is further enhanced by the Clebsch–Gordan coefficients which favor transitions at larger m_F . This becomes explicit in figure 4(a) which shows the excitation rate $\gamma_{\mathrm{exc}}^+/\Gamma_0$ of an atom in the state $|F = 2, m_F = + 2\rangle$ with a resonant guided mode propagating along increasing x. Decay and excitation rates are distinct as atoms can be excited with a mode propagating in either direction while decay happens in both, see appendix A. The $\sigma^+$ transition is stronger than the σ⁻ transition by two orders of magnitude, while the π transition is completely suppressed as the mode does not have any E_z component in the symmetry plane. The excitation of an atom by a guided mode is thus highly chiral. Once in $|F^{^{\prime}} = 3, m_{F^{^{\prime}}} = + 3\rangle$, guided mode emission will occur preferably in forward propagation, with a probability given by the polarization fraction, $92 \%$ at the position of the atoms.

Zoom In Zoom Out Reset image size

Besides, figure 4(a) shows that for atoms in state $|F = 2, m_F = + 2\rangle$ and trapped at 116 nm from the edge, the Purcell factor reaches a value of 0.71. As shown in figures 4(b) and (c), a small modulation in the x direction exists and the value of the Purcell factor decays rapidly as a function of the distance to the surface. To quantify the coupling of the atoms to the guided mode we also define the β factor, $\beta = \Gamma_{\mathrm{1D}}/\Gamma_{\mathrm{tot}}$ with $\Gamma_{\mathrm{tot}} = \Gamma_{\mathrm{1D}} + \Gamma^{^{\prime}}$, and $\Gamma^{^{\prime}}$ the decay rate in all the radiation modes other than the guided slow mode. Because of the complex shape of the local density of states accessible to the atoms, the behavior of $\Gamma^{^{\prime}}$ is hard to infer. We performed a numerical calculation of $\Gamma^{^{\prime}}$ using the same method as in [27] and found that $\Gamma^{^{\prime}} \simeq 0.8 \Gamma_0$, at the position of the trap minimum, i.e. at 116 nm from the surface. We thus obtain β = 0.47, very close to the spatially averaged value $\tilde{\beta} = 0.46$ for a thermal distribution (at a temperature of one half of the trap depth). This is at least 50 times better than the current systems involving nanofibers ($\beta = 10^{-2}$) [11] and in the same range of current PCW-based platforms (β = 0.45) [22].

Other waveguides combining chiral and strong coupling to emitters exist, however with a common hurdle in maximizing and overlapping spatially both features [62, 63]. In our case, this overlap is readily excellent, due to the absence of symmetry protection from chirality and to the limited spatial structure of the mode (see figure 4(b)), from which both chirality and Purcell factor inherit. In other words, with our design, we recover a behavior akin to the case of a nanofiber, with the increased group index enhancing the coupling.

In contrast with optical nanofibers, the guided modes in the half-W1 waveguide are structured along the propagation direction due to the Bloch wave structure of the light field. The intensity of the modes, which is an important quantity when looking at dipole trapping, is periodic with period a, as shown in figure 3(a) for the slow mode. This feature constrains the position of the trapped atoms to the maxima of intensity of the red-detuned mode. It makes the search for a blue detuned mode more challenging as this one will also be structured, while a uniform one would work perfectly well to repel the atoms from the surface [5]. A blue-detuned beam with an intensity pattern completely out of phase with the red-detuned one is needed. Fortunately, modes separated by a band gap usually have intensity maxima shifted by $a/2$ [65]. We then use the highest available guided band for the blue-detuned trap between 400 and 420 THz (figure 1(c)).

In order to have a full description of the potential seen by the atoms, we take into account the Casimir–Polder (CP) interaction [66] between the atoms and the surface. Vacuum fluctuations can polarize the atoms, even if they are not charged. When put in proximity to structures, the vacuum-induced dipole moment creates a mirror charge that acts on the original dipole, leading to an additional light shift. The CP potential $U_{\mathrm{CP}}$ is only significant at very close distances (${\unicode{x2A7D}} 150$ nm) but is crucial as it acts as an attractive potential close to the surface. For these systems, the approximation of an atom in the proximity of an infinite dielectric half space is often used $U^{\text{plane}}_{\textrm{CP}} = - C_3/d^3$ [65], where d is the distance to the surface. As a slab, our structure deviates significantly from a half space. Hence, we computed a more realistic, space-dependent CP potential, based on the pairwise summation technique (PWS) [67], as described in appendix B.

The trapping potentials were obtained via nanotrappy [68], a Python package developed by some of us, to design, calculate and optimize dipole traps around nanoscale waveguides, making the search process faster and more systematic.

Figure 5 shows the total trapping potential $U_\mathrm{tot}$ in 3 directions for an atom in the $|F = 2, m_F = + 2\rangle$ hyperfine level. $U_\mathrm{tot}$ is the sum of the contributions of the blue potential $U_\mathrm{blue}$, the red potential $U_\mathrm{red}$ and the CP potential $U_\mathrm{CP}$. Figures 5(a)–(c) show the trap along the x and y axis, in the symmetry plane of the waveguide. As shown in figure 5(d), trapping out of this plane is less obvious as the confinement of the atom is low in the azimuthal direction. The curvilinear coordinate is computed as the total distance the atom can travel in the valley of minimal potential shown in the inset with the origin corresponding to z = 0. The real depth of the trap is given by the potential barriers that appear in figure 5(d), and are physically around y = 0. The trap in this direction is about 100 µK deep. The low trapping in this direction comes from balancing opposing requirements needed for trapping. Indeed, a stable trap position with a repulsive barrier in the y direction, requires the blue evanescent mode to decay faster than the red one. This criterion, once met, also manifests itself in the azimuthal direction which leads to a predominant attractive trapping almost up to the surface, and the formation of a relatively flat valley in this direction. Even if the atoms can be spread over this valley, the averaged beta factor $\tilde{\beta}$ for a thermal cloud in this trap is still close to the peak value (see above).

Zoom In Zoom Out Reset image size

For this trap, a beam red-detuned from the D${_2}$ line of ⁸⁷Rb at 784.45 nm and a beam blue-detuned at 735.86 nm are used. For each color, another beam detuned by respectively 280 and 385 GHz at the same power is counterpropagated for vector shift cancellation. An absolute trap depth (relative to the atoms being infinitely far away) of 2.6 mK is obtained with a minimum at 116 nm from the surface. The total powers are $P_\mathrm{blue} = 2 \times 3.1$ mW and $P_\mathrm{red} = 2 \times 93~\mu$W, but a stable trap can be obtained over a wide range of powers, with trapping positions ranging typically from 115 to a few hundreds of nm. The main limitation can be the power handling of the structure which is still to be determined. In [44], power densities up to 1 GW cm⁻² were coupled to similar GaInP PCWs with group index 8.8. For our structure which has a cross section 10 times smaller and a group index 3 times bigger, this would be equivalent to coupling ${\simeq} 100$ mW into our waveguide. The proposed powers for the trap fall well below this bound.

The trapping frequencies are large in the x and y directions, with $\omega_x = 2 \pi \times 1.75$ MHz and $\omega_y = 2 \pi \times 2.00$ MHz. Out of the symmetry plane however, there is an important anharmonicity of the trap in the azimuthal direction. The trap is less constrained in that direction and we extract $\omega_z = 2 \pi \times 83$ kHz from the curvature at the bottom of the trap, which is to be taken with caution given the unusual shape of the trap. The addition of a blue-detuned beam with crossed polarization could be used to mitigate this problem, as extensively used in the context of nanofiber traps [4].

Importantly, we also verified that we can achieve a stable trap in the three directions for a wide range of wavelengths, which is a valuable feature for finding the right trade-off between heating the atoms with off-resonant scattering and power handling of the waveguide. If we allow the blue power to go up to 3 mW, we can find a stable trap for blue wavelengths ranging from 724 nm to 738 nm. Pushing the maximum allowed power to 5 mW we can find a trap for the full available blue-detuned air band, i.e. $\Delta \lambda = 21~\mathrm{nm}$. For the red-detuned laser, we have an available range from 780.5 nm up to 786 nm. Laser diodes are easily available on these wavelengths, reinforcing the feasibility of our platform.

Finally, as briefly noted before, we used counter-propagating beams here instead of single beams, albeit standing waves are not needed for periodic intensity modulation. The strong ellipticity of the guided modes, as shown in figure 3(c), acts as a fictitious magnetic field on the atoms, splitting the Zeeman levels [69]. If we start from atoms evenly distributed in all the m_F states, this effect would lead to a large inhomogeneous broadening up to a few GHz. It can be mitigated by using counter-propagating trapping beams slightly detuned from each other, as used for blue detuned beams in some compensated nanofiber traps [5]. Via nanotrappy, we estimated that adding a red-detuned laser at 280 GHz from the first one and a blue detuned at 385 GHz from the other reduces this broadening by 90$\%$. Counterpropagation creates a running wave at a velocity given by $\delta \omega/k$. This pattern propagates but at a speed so large the atoms only see the average of the potential.