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Abstract
Improving the photon–spin conversion efficiency without polarization dependence is a major challenge in realizing quantum interfaces gate-defined quantum dots (QDs) for polarization-encoded photonic quantum network systems. Previously, we reported the design of an air-bridge bull’s-eye cavity that enhances the photon absorption efficiency of an embedded gate-defined QD regardless of the photon polarization. Here, we numerically demonstrate that a further 1.6 times improvement in efficiency is possible by simply adjusting the distance of the substrate from the semiconductor slab where the bull’s-eye structure is formed. Our analysis clarifies that the upward-preferred coupling and narrow far-field emission pattern realized by substrate-induced asymmetry enable the improvement.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A quantum network [1,2] providing coherent links of distant stationary qubits via flying qubits can advance quantum informatic apparatuses for communications [3], computing [4], and sensing [5]. Photonic quantum interfaces [6] that convert quantum states between stationary matter qubits and flying photonic qubits are essential building blocks in such quantum networks. Several platforms are candidates for hosting stationary qubits that enable photonic quantum interfaces, including superconducting circuits [7], Rydberg atoms [8], trapped ions [9], color centers in diamonds [10,11], and semiconductor quantum dots (QDs) [12,13]. Gate-defined QDs [14–16], which are formed by laterally confining the carriers in semiconductor quantum wells (QWs) with an applied gate voltage, are also attractive solid-state hosts for stationary spin qubits. Using spins in Si-based gate-defined QDs [17], two-qubit gate operations with high fidelity above the error correction threshold have been demonstrated recently [18,19]. These Si QDs can be realized with complementary-metal–oxide–semiconductor-compatible fabrication processes [20], which is a powerful advantage, particularly for the scalable integration of qubits. As for implementing photonic quantum interfaces, GaAs-based gate-defined QDs are suitable. Utilizing the polarization-dependent optical absorption in III-V semiconductor QWs [21,22], angular momentum transfer from a photon to an electron in a gate-defined GaAs QD has been experimentally realized [23,24]. However, the photon-to-spin conversion efficiency is currently low (∼10−5 [25]) because of the limited photon absorption efficiency of the QDs, where the interaction volume with incident photons is small. This efficiency problem is a major bottleneck for realizing quantum repeaters that exploit photon-to-spin conversion and becomes even more significant for entanglement distribution requiring the use of multiple repeaters [26].
A promising approach for the improving efficiency is to embed QDs in optical cavities. Because polarization–spin correspondence is essential for polarization-encoded qubits, the optical cavity must be polarization-independent. Although using time-bin qubits [27] may alleviate this polarization independence, the conservation of polarization information provides additional degrees of freedom in quantum network configurations. The so-called bull’s-eye structure [28–32] can exhibit polarization-independent features because of its centrosymmetric nature. Previously, we designed an air-bridge bull’s-eye optical cavity and numerically demonstrated that it could improve the photon absorption efficiency of an embedded gate-defined QD by a factor of more than 400 [33]. Recently, we reported the fabrication and preliminary proof-of-principle experiments demonstrating polarization-independent photon absorption enhancement using the designed air-bridge bull’s-eye cavities [34].
In this study, we discuss the possibility of further absorption enhancement due to directional coupling induced by out-of-plane asymmetry in the cavity. The coupling rates for the out-of-plane directions are unbalanced in such an asymmetric cavity; therefore, enhanced or suppressed coupling rates can be obtained in the desired direction. Several methods, such as controlling etching depth [28,35] and employing one-side reflectors, including distributed Bragg reflectors [36] or buried metal layers [31,37,38], have been reported for realizing optical cavities with out-of-plane asymmetry. However, these methods may limit productivity because of the requirement for precise control of the fabrication process or for a wafer with a complicated layer structure. Here, we numerically investigate the effect of asymmetry induced by the presence of substrate underneath the bull’s-eye structure; the substrate-induced asymmetry can be easily realized in real device structures where the device is suspended in the air by removing a sacrificial layer on the substrate [34]. Although the effect of the substrate on the radiation properties of photonic crystal cavities has been discussed in a previous study [39], we discuss its application in the case of cavity-enhanced absorption. Compared to the free-standing air-bridge bull’s-eye cavity, the absorption efficiency of the gate-defined QDs in the asymmetric bull’s-eye cavity can be enhanced by about 1.6 times by adjusting the air-gap distance between the substrate and the slab where the bull’s-eye structure is formed. The theoretical analysis based on the equations representing the cavity-enhanced QD light absorption derived in our previous study [33] shows that the directionality realized by the asymmetry serves as a key factor for increased efficiency. This result suggests that utilizing substrate-induced asymmetry is a promising approach for realizing efficient photon–spin interfaces using gate-defined QDs.
2. Model and key factors for absorption enhancement
2.1. Cavity structure and mode characteristics
The cavity structure we considered is schematically shown in Fig. 1(a). A bull’s eye structure with metal electrodes for creating a gate-defined QD is formed in a 165-nm-thick Al0.33Ga0.67As/GaAs/Al0.33Ga0.67As slab (${T_{\textrm{slab}}} = $ 165 nm) comprising a 15-nm-thick GaAs QW (${T_{\textrm{QW}}} = $ 15 nm, z = 0 at the center of the QW layer). A GaAs substrate, which is ${T_{\textrm{gap}}}$ apart from the slab, is included in the model. The presence of the substrate causes the cavity to be vertically asymmetric. We used a commercially available software (Ansys Lumerical) performing the three-dimensional finite difference time domain (3D-FDTD) method to calculate the light absorption efficiency of the QD (denoted as $\eta _{\textrm{abs}}^{\textrm{QD}}$) and related features. For the calculation of $\eta _{\textrm{abs}}^{\textrm{QD}}$, a gate-defined QD was modeled as a cuboid with a lateral size ${S_{\textrm{QD}}}$ of 50 nm ${\times} $ 50 nm and a thickness of ${T_{\textrm{QW}}}$, and a tightly focused [40,41] vertically incident Gaussian beam (beam waist diameter w = 1 $\mu $m) was used as the excitation source. We used refractive indices ${n_{\textrm{GaAs}}}$ = 3.659 + i$\alpha $ and ${n_{\textrm{AlGaAs}}}$ = 3.346 [42] in FDTD calculations, where $\alpha $ represents the absorption coefficient of the GaAs QW layer. $\alpha $ = 0.06 [43] is used for simulations considering a QW absorption, and $\alpha $ = 0 is used for calculations considering only passive (radiative) decay.
Fig. 1. (a) Schematic of vertically asymmetric air-bridge bull’s-eye cavity. GaAs substrate is ${T_{\textrm{gap}}}$ apart from the cavity. The decay rates of the cavity mode due to the top-emitted, bottom-emitted, and QW absorption are denoted by ${\gamma _\textrm{t}}$, ${\gamma _\textrm{b}}$, and ${\gamma _{\textrm{abs}}}$, respectively. An excitation beam with a Gaussian distribution is focused at the center of the QW layer (z = 0) with a beam waist of w. (b) Schematic of bull’s-eye optical cavity used in this study. (c) Distributions of main electric field of doubly degenerate modes supported in the cavity. The resonant wavelength ${\lambda _0}$, passive quality factor, and mode volume of the degenerate modes were 806 nm, 90.2, and 0.36 (${\lambda _0}/{n_{s\textrm{lab}}}$)3, respectively. The green squares at the cavity center represent the gate-defined QD formed by the electrodes.
As the bull’s-eye structure, we used our previously designed air-bridge bull’s-eye cavity [33] (Fig. 1(b)). The bull’s-eye structure comprised a six-period circular Bragg grating (period a = 340 nm, air-gap filling factor G = 0.4) surrounding a circular cavity with diameter ${A_\textrm{c}}$ = a. Two straight channels (width W = 100 nm) crossing the center provided space for gate formation and enabled the air-bridge structure [34]. Note that in the experimental demonstration presented in [34], the asymmetry caused by the substrate appears naturally without any additional processes. The channel width was selected to provide sufficient space for electrode formation [34,44] while keeping a reasonable absorption enhancement [33]. The electrodes consisted of four gold metal lines, which were 36 nm wide and 36 nm thick. The C4v symmetry of the entire structure guarantees that the cavity supports doubly degenerate modes, which are essential for achieving a polarization-independent response. Figure 1(c) shows the field distribution of the doubly degenerate dipole-like modes supported in bull’s-eye structure without the substrate. For obtaining the cavity mode properties such as resonant frequencies, field distributions, and quality factors, we used an internal dipole source positioned within the bull’s-eye cavity. The resonant wavelength ${\lambda _0}$, passive quality factor, and mode volume of the degenerate modes were 806 nm, 90.2, and 0.36(${\lambda _0}/{n_{s\textrm{lab}}}$)3, respectively, where ${n_{\textrm{slab}}} = \frac{{{T_{\textrm{QW}}}}}{{{T_{s\textrm{lab}}}}}{n_{\textrm{GaAs}}} + \frac{{{T_{\textrm{slab}}} - {T_{\textrm{QW}}}}}{{{T_{s\textrm{lab}}}}}{n_{\textrm{AlGaAs}}}$. The employed second-order Bragg condition (a = ${\lambda _0}/{n_{\textrm{eff}}}$, where ${n_{\textrm{eff}}}\; \cong \; G + ({1 - G} ){n_{s\textrm{lab}}}$) causes the radiation pattern of the mode to be concentrated in the vertical direction, resulting in a larger far-field overlap with the Gaussian excitation beam. The degenerate modes overlap at the cavity center; thus, the optical absorption of the QD (shown as green squares in Fig. 1(c)) is polarization-independent.
2.2. Analytical expression for cavity-enhanced light absorption efficiency
In our previous study, we derived an analytical expression for the cavity-enhanced light absorption efficiency of a gate-defined QD using the temporal coupled-mode theory. Following the result, the absorption efficiency of the QD (lateral size ${S_{\textrm{QD}}}$ = 50 nm ${\times} $ 50 nm) in a cavity at the resonant wavelength of ${\lambda _0}$ is obtained as [33]
We rewrite Eq. (4) in terms of $\mathrm{\Phi } \equiv {\gamma _\textrm{t}}/{\gamma _{\textrm{rad}}}$, which describe the asymmetry in radiation direction, and total decay rate of the cavity mode due to the radiation, ${\gamma _{\textrm{rad}}} = {\gamma _\textrm{t}} + {\gamma _\textrm{b}}$, as follows:
Equations (1) and (6) provide useful information for absorption enhancement; cavity modes that are well localized (large ${S_{\textrm{QD}}}/{S_{\textrm{eff}}}$: large spatial overlap between the QD and the near field of mode), spatially matched to the excitation source (large ${\eta _{\textrm{far}}}$: large far-field overlap between the excitation source and the mode for the efficient coupling), and satisfy the absorption-radiation rate matching (${Q_{\textrm{rad}}} = {Q_{\textrm{abs}}}$: critical coupling condition in the temporal coupled-mode theory) are beneficial. Furthermore, achieving a large $\mathrm{\Phi }$ has a linear effect on M, indicating the importance of directional coupling to the top.
3. Result and discussion
Figure. 2(a) shows the dependence of $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ on ${T_{\textrm{gap}}}$, calculated using Eq. (1). Each term in Eq. (1) was calculated using the corresponding equation in Sec. 2.2 with the field distributions and quality factors of the cavity mode obtained from the 3D-FDTD simulation. We also show the results calculated by independent 3D-FDTD simulations, which directly evaluate $\eta _{\textrm{abs}}^{\textrm{QD}}(\lambda )$. The results based on Eq. (1) and from direct calculations show a reasonable agreement, which demonstrates the validity of the analytical expression discussed in 2.2. The relatively large deviation between the theory and direct FDTD calculations, especially for ${T_{\textrm{gap}}}/{\lambda _0}\sim $0.55–0.8, is possibly due to the nonzero contributions (non-resonant absorption) which are not included in the formula. $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ oscillates with a period of approximately half of the resonant wavelength (${T_{\textrm{gap}}}/{\lambda _0}$∼0.5). Therefore, we intuitively understand that this phenomenon is related to the interference with the light reflected from the substrate. To analyze this phenomenon further, we calculated the approximate phase accumulation $\mathrm{\Delta }\phi $ after a round trip in the air gap and half of the slab by using Eq. (7) [39]:
Fig. 2. (a) Calculated absorption efficiency $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ as a function of ${T_{\textrm{gap}}}$. Results obtained by Eq. (1) and individual 3D-FDTD simulations are represented by the black curve and circles, respectively. Accumulated phase after a round trip in the air gap and half of the slab calculated by Eq. (7) is plotted as the dashed line in the right y-axis. (b) Dependence of rate matching coefficient (M), far-field overlap between the excitation source and the cavity mode (${\eta _{\textrm{far}}}$), ratio of the QD absorption cross section to the cavity mode near field (${S_{\textrm{QD}}}/{S_{\textrm{eff}}}$), and the absorption efficiency ($\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$) on ${T_{\textrm{gap}}}$. Dashed lines represent the corresponding values for the structure without considering the substrate.
Figure 2(b) shows the dependence of ${S_{\textrm{QD}}}/{S_{\textrm{eff}}}$, ${\eta _{\textrm{far}}}$, M, and $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ on ${T_{\textrm{gap}}}$. The corresponding values for the structure without considering the substrate (free-standing cavity) are shown as dashed lines for comparison. The change in ${S_{\textrm{QD}}}/{S_{\textrm{eff}}}$ was relatively small, which can be understood from the fact that the near-field distributions of the cavity modes change less when ${T_{\textrm{gap}}}$ changes. On the other hand, both ${\eta _{\textrm{far}}}$ and M were significantly affected by the variation in ${T_{\textrm{gap}}}$. Given the fact that the trend of $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ is almost identical to that of M, the improvement in M is the primary important where the higher efficiency $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ is obtained compared to the free-standing cavity. This interpretation will be also confirmed through a quantitative comparison shown later (see Table 1).
Table 1. Comparison of M, $\eta_{\textrm{far}}$, ${S_{\textrm{QD}}}/{S_{\textrm{eff}}}$, and $\eta _{\textrm{abs}}^{\textrm{QD}}$ for bull’s-eye cavities with/without considering the substrate
As shown in Eq. (5), M is composed of $\mathrm{\Phi }$ and the remaining part. The latter, $M/\mathrm{\Phi }$, represents how perfectly the rate matching (critical coupling) condition is satisfied. To investigate which factor dominates the change in M, we plot the dependence of M, $\mathrm{\Phi }$, and $M/\mathrm{\Phi }$ on in Fig. 3(a). Although both $\mathrm{\Phi }$ and $M/\mathrm{\Phi }$ change as ${T_{\textrm{gap}}}$ changes, M roughly follows $\mathrm{\Phi }$, meaning that the increase in asymmetry in the radiation leakage of the cavity mode plays the dominant role rather than the variation in $M/\mathrm{\Phi }$ in the present cavity design. While $\mathrm{\Phi }$ varies from ∼0.2 to ∼0.6, the variation in $M/\mathrm{\Phi }$ is less significant. Although ${Q_{\textrm{rad}}}$ varies more than twice from 52 to125 (Fig. 3(a)), $M/\mathrm{\Phi }$ changes only in the range from ∼0.7 to ∼0.95 with ${Q_{\textrm{abs}}}$ = 193 (Fig. 3(b)). At ${T_{\textrm{gap}}}/{\lambda _0}$ = 0.4, where M and $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ show their maxi, the critical coupling condition is not fully satisfied. Further improvement in efficiency would be attainable if we could increase ${Q_{\textrm{rad}}}$ while keeping $\mathrm{\Phi }$.
Fig. 3. (a) Dependence of M, $\mathrm{\Phi }$, and $M/\mathrm{\Phi }$ on ${T_{\textrm{gap}}}$. Variation of is also shown as black circles. ${Q_{\textrm{rad}}}$ for the free-standing cavity and ${Q_{\textrm{abs}}}$ = 193 are shown as black dashed and blue lines, respectively. (b) Value of $M/\mathrm{\Phi } = 4{Q_{\textrm{abs}}}{Q_{\textrm{rad}}}/{({{Q_{\textrm{abs}}} + {Q_{\textrm{rad}}}} )^2}$ when ${Q_{\textrm{rad}}}$ is varied from 20 to 370. The variation range of ${Q_{\textrm{rad}}}$ for the structures discussed are represented as dashed lines.
The dependence of ${\eta _{\textrm{far}}}$ on ${T_{\textrm{gap}}}$ is similar to that of $\mathrm{\Phi }$, with a slight difference as it reaches a maximum at approximately ${T_{\textrm{gap}}}/{\lambda _0}$ = 0.52 instead of 0.4. The variation in ${\eta _{\textrm{far}}}$ (Fig. 4(a)) comes from the change in the overlap between the radiation pattern form the cavity mode toward the top direction and the beam pattern of the excitation Gaussian beam (Fig. 4(b)). In Figs. 4(c)-(f), we show the far-field intensity (${|{\boldsymbol E} |^2}$) distributions for the structures with no substrate (the reference free-standing structure), ${T_{\textrm{gap}}}/{\lambda _0}$ = 0.4, 0.52, and 0.65, respectively. Each ${|{\boldsymbol E} |^2}$ was normalized such that the total field intensity integrated over the upper hemisphere was unity (Note that the color scales for Fig. 4(b)-(f) are the same). The far-field patterns for ${T_{\textrm{gap}}}/{\lambda _0}$ = 0.4 (Fig. 4(d)) and 0.52 (Fig. 4(e)) are more concentrated within the region of NA < 0.5 (corresponding to the half divergence angle < 30°), resulting in higher peak intensities than that of the free-standing case (Fig. 4(c)). In contrast, the far-field pattern of ${T_{\textrm{gap}}}/{\lambda _0}$ = 0.65 case distributes more uniformly over the larger divergence angle, decreasing ${\eta _{\textrm{far}}}$ significantly.
Fig. 4. (a) Dependence of ${\eta _{\textrm{far}}}$ on ${T_{\textrm{gap}}}$. Dashed line represents ${\eta _{\textrm{far}}}$ for free-standing cavity. Three marked spots (${T_{\textrm{gap}}}/{\lambda _0}$ = 0.4,0.5, and 0.65) represent ${T_{\textrm{gap}}}$ yielding the maximum $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$, maximum ${\eta _{\textrm{far}}}$, and minimum ${\eta _{\textrm{far}}}$, respectively. (b) Far-field intensity patterns of the Gaussian beam (w = 1 $\mu $m), (c)-(f) Far-field intensity patterns of the x-polarized mode in the cavities with no substrate, ${T_{\textrm{gap}}}/{\lambda _0}$ = 0.4,0.5, and 0.65, respectively.
Table 1 summarizes the values of ${S_{\textrm{QD}}}/{S_{\textrm{eff}}}$, ${\eta _{\textrm{far}}}$, and M that contribute to the largest and smallest absorption efficiencies. The absorption efficiency at ${T_{\textrm{gap}}}/{\lambda _0}$ = 0.4, $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ = 0.0195, is approximately 1.64 times enhanced than that of the free-standing situation ($\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ = 0.0119), where M and ${\eta _{\textrm{far}}}$ show an improvement of around 50% and 10%, respectively, whereas the reduction in ${S_{\textrm{QD}}}/{S_{\textrm{eff}}}$ is less than 1%. These results confirm again that the improvement in M dominates the improvement in absorption efficiency. At ${T_{\textrm{gap}}}/{\lambda _0}$ = 0.65, $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ is 0.0042, which is ∼0.35 times of $\eta _{\textrm{abs}}^{\textrm{QD}}({{\lambda_0}} )$ for no substrate case. The reductions of M and ${\eta _{\textrm{far}}}$ are responsible for the efficiency drop.
A proper cavity design can improve M and ${\eta _{\textrm{far}}}$ to further improve the absorption efficiency. Achieving ${Q_{\textrm{rad}}}$ close to ${Q_{\textrm{abs}}}$ allows M to be close to the maximally achievable $\mathrm{\Phi }$. ${\eta _{\textrm{far}}}$ can be as large as 1 if the emission profile of the cavity mode is spatially matched to that of the excitation source. Field-based design methods [46,47] can be applied to precisely design the emission profile. For bull’s-eye cavities, a simpler approach is to control the divergence of the emission profile by adjusting the size of the circular cavity. By increasing the cavity size, a narrower far-field emission pattern can be obtained. In addition, the increase in the quality factor due to the gentle spatial confinement in large-cavity bull’s-eye structures can be beneficial for achieving a larger absorption-radiation matching $M/\mathrm{\Phi }$. However, because a large cavity also causes a decrease in ${S_{\textrm{QD}}}/{S_{\textrm{eff}}}$, a proper optimization that considers the trade-off between M ${\times} {\eta _{\textrm{far}}}$ and ${S_{\textrm{QD}}}/{S_{\textrm{eff}}}$ is required in the future.
4. Conclusion
We numerically investigated the photon absorption efficiency of a gate-defined QD embedded in a bull’s eye cavity with a substrate underneath. Our results demonstrated that the absorption efficiency can be improved more than 1.6 times compared to that without the substrate when the distance between the bull's-eye structure and the substrate is approximately 0.4 times the resonant wavelength. The substrate acted as a partial mirror that reflected light, thereby modifying the vertical coupling constant of the cavity. At the distance, the constructive interference condition for increasing upward coupling is satisfied, which resulted in a 1.64 times improvement of the absorption efficiency. In experiments, the distance between the bull's-eye structure and the substrate can be controlled precisely by adjusting the thickness of the sacrificial layer.
The analysis of each factor determining the absorption efficiency clarified that the upward directionality, $\mathrm{\Phi } = {\gamma _t}/{\gamma _{\textrm{rad}}}$, plays a dominant role in the improvement. We also found that the far-field spatial overlap between the cavity mode and the incident Gaussian beam, ${\eta _{\textrm{far}}}$, is also favored by the presence of substrate, suggesting that further improvement is possible by tailoring the cavity radiation profile through careful tuning of the cavity design in the future. The current performance of our bull's-eye cavity is restricted by the ${S_{\textrm{QD}}}/{S_{\textrm{eff}}}$ factor, which cannot be substantially improved through regular device parameter optimization. Consequently, it may be worthwhile to explore special design features with extremely small ${S_{\textrm{eff}}}$, for example, a bowtie design [48] or utilizing the plasmonic effect [49], in future research.
The substrate-induced asymmetry can improve absorption in a simple manner. This simple method would be useful for improving the collection efficiency of photons emitted from a gate-defined QD [50]. We would also emphasize that this substrate effect is applicable to other host materials of solid-state qubits, such as bulk single-crystal diamonds [51].
Funding
Core Research for Evolutional Science and Technology (JPMJCR15N2); Japan Society for the Promotion of Science (22H04962, 23H05458); Moonshot Research and Development Program (JPMJMS2066).
Acknowledgements
The authors thank Yasutomo Ota for fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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