Silicon cavities for colour centres and fast electrons, modelled with quasinormal modes.

The tasks of the call and where each is addressed

Task wording paraphrased from the call; the split into modules is mine.

Built for this study

• A 2D finite-element Maxwell solver (Nédélec elements of any order, NGSolve [12]) with perfectly matched layers, a shift-and-invert search for the QNMs, and their PML normalisation $\mathbf e^{T}\mathsf M\mathbf e = 1$ [8].
• QNM expansions of the transmission of a barrier, of the Purcell factor and of the energy loss of a passing electron, each compared with a direct solution.
• First-order QNM perturbation theory for moving boundaries (hole radii and positions, bowtie bridge), used for fabrication tolerances.
• A Lindblad master equation for a two-level emitter and one cavity mode, with the coupled-mode frequencies; unit tests run in CI on every push.

How each step would grow in the PhD

• 3D: the same QNM workflow for the full membrane, with its out-of-plane radiation and a real Q, and mode volumes without the effective-index step.
• Design: the perturbation sensitivities of module B are the gradients that inverse design needs; topology-optimised cavities as in [4].
• Electrons: 3D trajectories near and through the membrane, coupling constants for the electron-microscopy experiments of the network, and their statistics.
• Emitters: phonon sideband and spectral diffusion of the G centre, several QNMs, and the coupled QNM frequencies of [5] computed from the 3D field.

The dielectric barrier of [1] ($n = \pi$)

With the three QNMs $n$ = 3, 4, 5 the error of the transmission has its minimum, , at $\omega L/c$ = 4 (one QNM: ); [1] reports about 2% and 7%. With $2N+1$ QNMs the error falls as $(2N+1)^{-2}$ for the expansion of the total field and as $(2N+1)^{-1}$ for the scattered field. My fits, and , match those printed in Fig. 22 of [1]; with QNMs the errors are and .

A 2D QNM solver with perfectly matched layers

Silicon cylinder ($n$ = 3.48, $R$ = 0.5 µm) in air, in-plane electric field. The exact QNMs are the roots of $J_m'(nkR)/[nJ_m(nkR)] = H_m'(kR)/H_m(kR)$. The finite-element QNMs converge exponentially with the element order, to at order ( unknowns), for modes with $Q$ from 10 to 14 000.

Purcell factor from a QNM pair

A dipole 80 nm inside the rim of the cylinder, near the $m$ = 4 QNM ($Q$ = ). The QNMs of the cylinder come in degenerate pairs, so the single-resonance model uses both. At the peak, the pair reproduces the direct solution to (radial) and (azimuthal); away from it, the azimuthal dipole also couples to modes the pair leaves out.

Method and result

• 1D: analytic QNMs $\tilde\omega_mL/c = [2\pi m + i\ln((n-1)^2/(n+1)^2)]/2n$, with field $e^{in\tilde k_mx} + (-1)^m e^{-in\tilde k_mx}$ inside and norm $(-1)^m2n^2L$ [1]. Total field: $E \approx \sum_m a_m\tilde f_m/(\omega - \tilde\omega_m)$, with $a_m$ from the incoming field at the two surfaces; scattered field: $b_m = -\tfrac{\omega}{2}\int\tilde f_m\,\Delta\varepsilon\,E_{\rm in}\,dx$.
• 2D: curl–curl equation in Nédélec elements, a radial PML (absorption α = 2i), shift-and-invert Arnoldi on $(\mathsf A - \sigma\mathsf M)^{-1}\mathsf M$; spurious PML modes are rejected by their weight in the PML.
• Purcell factor: power of a 4 nm source disc, relative to the same source in homogeneous silicon; QNM model $\mathbf e = \sum_n \mathbf e_n(\mathbf e_n^T\mathbf f)/(\tilde k_n^2 - k^2)$.

Field of the cavity QNM

$|\mathbf E|^2$ of the normalised QNM, scaled to its maximum (colour scale saturated at 15% for the bowtie, whose maximum sits in the bridge).

The bowtie

The field is concentrated in the silicon bridge between the two air tips, along $y$, the direction of the dominant field of the mode.

Bridge width

The mode area falls roughly in proportion to $w$; the in-plane Q falls from about 10⁵ to about 3 300 because the bowtie scatters light in the plane. Mesh check at $w$ = 10 nm: halving the elements at the bridge or raising the order from 3 to 4 ( unknowns) changes A by at most and Q by .

Fabrication errors

Random errors of every hole radius and position (σ = 0.5–3 nm, samples each). At σ = nm the re-simulated spread of λ is nm (nanobeam) and nm (bowtie), against and nm from first-order theory, the approach of [6] for random surface deformations.

Coupling versus electron energy

The coupling peaks at keV (nanobeam) and keV (bowtie), for all three distances, and falls as $b$ grows. Above the peak the bowtie keeps much more coupling: at 300 keV, of its maximum against for the nanobeam. $C(v)$ is the spatial spectrum of the field at $q = \omega/v$, and the field concentrated at the bowtie has a broader spectrum.

Phase matching

The spatial spectrum of $\tilde f_x$ along the trajectory peaks at $q$ = π/a (nanobeam), next to the band edge of the mirror. Phase matching $v = \omega a/\pi$ gives keV, close to the computed optimum.

Energy loss near the resonance

At the energy of best coupling and $b$ = 50 nm. At the peak the single QNM agrees with the direct solution to (nanobeam) and (bowtie). The direct spectrum is positive at every frequency, as it must be for a passive structure.

Method and result

• Electron field: in vacuum, $\phi \propto e^{iqx-\kappa|y-y_e|}$ with $\kappa = q/\gamma$; it drives the scattered field through $k^2(\varepsilon-1)\mathbf E_0$ inside the silicon.
• Loss: $\Gamma(\omega) \propto -{\rm Re}\int dx\,e^{-iqx}E^{\rm ind}_x(x, y_e)$ along the trajectory inside the computational domain.
• Limits: a line charge, not a point electron; an aloof path in the plane, not through or across the membrane.

Weak or strong coupling

With this dephasing the nanobeam cannot reach strong coupling at any Q: $4g$ stays below $2\gamma^*$. The 10 nm bowtie can, for Q above , about five times its 2D in-plane Q. Vertical line: Q = 4 600, measured in [2].

Emitter decay in the bowtie cavity

At Q = 4 600 the master-equation decay rate, ns⁻¹, is close to the Purcell rate with dephasing, ns⁻¹. At Q = 10⁵ and 10⁶ it is and ns⁻¹, while the formula gives and : the excitation goes back and forth between emitter and cavity (dotted, without dephasing) and the Purcell formula no longer applies.

Anticrossing (bowtie, Q = 10⁶)

Spectrum of the light leaving the cavity (log scale) versus emitter detuning. Dotted: real parts of the coupled-mode frequencies, the quantity [5] obtains from a single QNM of the bare cavity. At zero detuning the spectrum peaks at g, inside the coupled-mode frequencies g, because the two broad lines overlap.

Reading the measurement of [2]

• The Purcell factor of [2], $F_P = \tau_0(1/\tau_{\rm on} - 1/\tau_{\rm off})/({\rm DW}\,\varepsilon)$, with its lifetimes τ₀ = 5.97, τ_on = 0.97 and τ_off = 4.27 ns and DW = 15%, gives F_P = /ε: their lower bound of 31 for ε = 1.
• F_P cannot exceed (3/4π²)Q/V = for their cavity, so the measurement implies ε ≥ : for ε = 1% the cavity would have to enhance the zero-phonon line by more than it can.
• A check of the code: the coupled-mode eigenvalues equal those of the Liouvillian in the one-excitation coherences (unit test).

Simplifications

• The cavity is modelled in 2D (top view of the membrane, effective index of the TE₀ slab mode, constant silicon index); Q is the in-plane value, and V = A h_eff is an estimate.
• The designs are scaled in-plane to put the mode at 1278 nm in this model; the scale factor would differ in 3D.
• The electron is a line charge on an aloof path in the plane of the membrane.
• The G centre is a two-level system with published lifetime, Debye–Waller factor and a range of quantum efficiencies; no phonon sideband dynamics, no spectral diffusion.

Reproduce

python -m venv .venv && .venv/bin/pip install -e . ngsolve pytest
cd scripts
../.venv/bin/python run_moduleA.py      # QNM checks (~25 min)
OMP_NUM_THREADS=1 ../.venv/bin/python run_moduleB.py   # designs, ensemble (~30 min)
../.venv/bin/python run_moduleC.py      # electrons (~7 min)
../.venv/bin/python run_moduleD.py      # colour centre (seconds)
../.venv/bin/python plots.py && ../.venv/bin/python build_site.py
../.venv/bin/pytest ../tests

Publication figures (PDF)