Theory and design of quantum light sources from quantum dots embedded in semiconductor-nanowire photonic-crystal systems

We introduce a platform for realizing on-chip quantum electrodynamics using photonic-crystal waveguide structures composed of periodic nanowire arrays with embedded semiconductor quantum dots to act as quantum light sources. These nanowire-based structures, which can now be fabricated with excellent precision, are found to produce waveguide Purcell factors exceeding 100 and on-chip $\beta$ factors up to 99%. We investigate the fundamental optical properties of photonic-crystal waveguides and finite-size structures, using both photonic band structure calculations and rigorous Green function computations which allow us to obtain the modal properties and the local density of photon states. A comparison with slab-based photonic crystals is also made and we highlight a number of key advantages in the nanowire system, including the potential to reduce extrinsic scattering losses and produce high theoretical Purcell factors and $\beta$ factors on-chip. We also demonstrate that these structures exhibit rich photonic Lamb shifts over broadband frequencies.

Figure 1

(a) Nominally identical GaAs NWs grown in an organized structure from Ref. [33]. (b) Proposed NW waveguide, formed by reducing the radius of a single column of NWs in an array. Guiding is done in the higher index (darker) upper portion of the NWs, while the lower index bottom separates the PC structure from the substrate below.

Figure 2

(a) Top view schematic of two unit cells of a general NW waveguide, showing structural parameters and coordinate convention used throughout this paper. (b) Band structure on left and spontaneous emission enhancement factor on right, as calculated with MPB, of a NW waveguide with $r_d=0.14a$. The bulk PC bands are shaded in blue, and the region above the light line in gray, with the guided band lying in the band gap. $F_z$ is calculated for an emitter in the center of the waveguide NW, and can be seen to diverge as one approaches the mode edge. The band gap is bounded with dotted orange lines. (c) Profile $|{\mathbf{E}}_{\lambda }{|}^2$, in arbitrary units, of the indicated waveguide mode perpendicular to the waveguide direction. The monopole-like profile and strong confinement to the waveguide channel are clearly visible.

Figure 3

(a) Spontaneous emission enhancement factors for QDs embedded in the center of the finite-size waveguide NW PC with length $15a$. Thick (dark) blue indicates heterogeneous design, and (light) orange and (dark) green correspond to thick and thin homogeneous NWs. Dotted lines indicate the mode edges computed for corresponding infinite structures in MPB. (b) Band structure from FDTD of homogeneous NW waveguide with $r_d=0.140a$. The light line is indicated in gray, waveguide band highlighted in black, and modal strength is shown on a logarithmic scale. Above the light line, the waveguide band broadens, as it couples to radiation modes and becomes leaky.

Figure 4

(a) Purcell factors and $\beta$ factors of a finite-size NW PC waveguide, for various waveguide lengths as a function of increasing length; Purcell $(\beta$ factor) for $15a$ waveguide in blue dashed line $(\times )$, $21a$ in light orange $(\circ )$, and $41a$ in dark green $(*)$. $\beta$ factors are calculated at discrete frequency points, and the dotted lines are provided only to guide the eye. The $41a$ ${\lambda }_0$ resonance peak $F_z$ increases to 384, although the axis terminates at 120. (b) and (c) $|{\mathbf{E}}_{\lambda }{|}^2$, in arbitrary units for $\lambda ={\lambda }_{\mathrm{FP}}$ in (b) and $\lambda ={\lambda }_0$ in (c) on the $y=0$ plane of the $21a$ waveguide.

Figure 5

(a) Purcell and $\beta$ factors of $15a$ waveguides with top-mounted QDs. Substrate-free structure $F_z$ $(\beta$ factor) in thick blue $(\times )$, structure with a simple substrate in dashed dark green $(*)$, and elevated NW design (described in text) in light orange $(\circ )$. (b) $|{\mathbf{E}}_{{\lambda }_0}{|}^2$ in $x=0$ plane of elevated NW waveguide, in arbitrary units.

Figure 6

Lamb shift from a 30-debye QD in various PC structures. Top: QD at antinode of simple high-$Q$ Lorentzian cavity. Middle: QD in center of the $15a$ substrate-free waveguide. Bottom: QD at top of the $15a$ waveguide with elevated substrate design.

Figure 7

(a) Purcell and $\beta$ factors of realistic and substrate-free photon gun in light orange and dark blue, respectively. Solid lines denote $F_z$; markers correspond to calculated $\beta$ factor values. (b) and (c) $S_x$ in arbitrary units calculated $1.4a$ from the terminus of substrate-free and standard photon gun, respectively, corresponding to power flow out of the structure. (d) $|{\mathbf{E}}_{{\lambda }_0}{|}^2$ in the $z=0$ plane of the proposed single-photon gun.