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Abstract
One of the main steps towards large-scale quantum photonics consists of the integration of single photon sources (SPS) with photonic integrated circuits (PICs). For that purpose, the PICs should offer an efficient light coupling and a high preservation of the indistinguishability of photons. Therefore, optimization of the indistinguishability through waveguide design is especially relevant. In this work we have developed an analytical model that uses the Green’s Dyadic of a 3D unbounded rectangular waveguide to calculate the coupling and the indistinguishability of an ideal point-source quantum emitter coupled to a photonic waveguide depending on its orientation and position. The model has been numerically evaluated through finite-difference time-domain (FDTD) simulations showing consistent results. The maximum coupling is achieved when the emitter is embedded in the center of the waveguide but somewhat surprisingly the maximum indistinguishability appears when the emitter is placed at the edge of the waveguide where the electric field is stronger due to the surface discontinuity.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Indistinguishability of single photons generated by point defects is the central topic of quantum photonic integrated circuits for quantum information applications like quantum simulation [1], quantum teleportation [2] or quantum networks [3]. Indistinguishable photons are usually generated by parametric down-conversion [4] or alternatively from a single two-level quantum emitter in a solid-state environment [5]. Over the last years several on-chip integration of different SPS material systems have been demonstrated: III-V quantum dots [6], carbon nanotubes [7], NV [8] or SiV centers in diamond [9] and 2D layered materials [10]. For most of those solid-state quantum emitters the intrinsic indistinguishability at room temperature is almost zero because pure dephasing rates are orders of magnitude larger than the population decay rate [11]. Improvement of the indistinguishability can be achieved by low temperature operation and by reducing the radiative lifetime of the SPS using an optical cavity that takes advantage of the Purcell effect [12]. The balance between dephasing and population decay rates varies significantly depending on the material system. Whereas for specific single self-assembled GaAs quantum dots the emission at low temperature can be radiative lifetime limited [13], defects in 2D materials can exhibit several orders of magnitude of difference between radiative decay and pure dephasing rates [14]. Purcell enhancement using photonic resonators permits on-chip control of light−matter interaction to enhance collection efficiency and generation of indistinguishable photons [15] that can be used for on-chip processing of quantum information [16–18]. Therefore, it is important to explore the coupling of SPS to PICs and its effect on the indistinguishability. In this work we use an analytical treatment of light radiation from a point source placed at an arbitrary location and with arbitrary orientation on a waveguide. The refractive indexes of the waveguide correspond to materials commonly used in silicon photonics (SiO2, Si3N4, Si) besides other high-index materials like WeS2 or WO3 [19]. It is worth to note that other specially designed nanomaterials with ultra high refractive index can be designed [20]. We explore how the position of the source and its orientation affects the coupling to the waveguide modes and the indistinguishability of the photons. We also explore how the dimensions of the waveguide impact the indistinguishability. We perform FDTD simulations to validate the analytical model and to calculate the Purcell effect. The results show remarkable differences depending on the orientation of the SPS and provide maximum indistinguishability when the source is placed at the edge of the waveguide, in contrast to the maximum coupling efficiency position at the center of the waveguide. The indistinguishability is expressed in terms of the pure dephasing value of the emitter, so that the effects of the waveguide can be compared between strong and weak dissipative emitters. Depending on the waveguide geometry and the position of the source the indistinguishability can either increase or decrease, showing non-negligible enhancements for weak dissipative emitters placed at optimum positions.
Several works deal with the radiation of a point source embedded in bounded dielectric slabs and square waveguides through Green’s function methods [21–26]. Also, the problem of the unbounded dielectric slab is treated in [27] from a classical perspective and in [28] from a quantum perspective. However, in those cases the description of the source comes from the macroscopic expression of the dipole moment, without computing the Green’s Dyadic. The Green’s function of the unbounded 2D dielectric slab is covered in [29] and the same for the 3D cylindrical fiber in [30–32] through the development of a transform theory. As far as we know, the Green’s Dyadic of a 3D unbounded rectangular waveguide has not been treated until this work. Here we develop a generalization of the transform theory from the 2D case [29] to obtain the solution of the 3D version of the problem for an unbounded rectangular waveguide. The obtention of the Green’s Dyadic allows us to directly connect the value of the indistinguishability with the geometrical parameters of the waveguide, which also has not been covered neither in the previously mentioned works.
2. Methods, results and discussion
2.1 Indistinguishability for different SPS
In an isolated two-level system, the emission rate can be fully described by its population decay rate Γ0. However, a solid-state quantum emitter has an interaction with the mesoscopic environment. The two-level system is affected by random fluctuations of its energy that can be described by a stationary stochastic process characterized by a dephasing rate Γ* [33]. In this situation the indistinguishability (I) is reduced to [34]:
In general, for any practical implementation in quantum information processing I ≥ 0.5 [33]. The pure dephasing rates at room temperature of solid-state quantum emitters like color centers, quantum dots or organic molecules are about 3 to 6 orders of magnitude larger than their radiative decay rates [34]. Improvement of this efficiency can be achieved by working at cryogenic temperatures. For example, for excitons weakly confined in GaAs quantum dots the dot ground-state transition at low temperature is near radiative life-time limited [13] which would provide a balance of about ${\mathrm{\Gamma }^{\ast }}$/${\mathrm{\Gamma }_0}$≈1 and I ≈ 0.5. There are recent reports of even better performance with strain free GaAs/AlGaAs quantum dots without the need of Purcell enhancement [35]. For those highly efficient emitters the ratio ${\mathrm{\Gamma }^\ast }$/${\mathrm{\Gamma }_0}$→0 and the intrinsic indistinguishability tends to the unity. As an example of an intermediate situation, InAs quantum dots have decay and pure dephasing rates ${\mathrm{\Gamma }^{\ast }}$/${\mathrm{\Gamma }_0}$ = 2.6 [36,37] and the indistinguishability is only I≈0.19. On the opposite side, strain-induced defects in 2D materials have typical radiative lifetimes in the order of nanoseconds with dephasing lifetimes in the order of picoseconds [14]. For those emitters the ${\mathrm{\Gamma }^\ast }$/${\mathrm{\Gamma }_0}$ balance reaches 103 with almost zero indistinguishability. However, recent works related to defects created in transition metal dichalcogenides (like MoS2) by local helium ion irradiation [38] show radiative lifetimes < 150 ps. Also, a lifetime < 100 ps has been observed recently in regular strain induced defects in WSe2 layers deposited on metallic surfaces [39, 40]. More examples of quantum emission demonstrations in 2D materials can be found in [41]. Therefore, emitters with a certain ${\mathrm{\Gamma }^{\ast }}$/${\mathrm{\Gamma }_0}$ ratio may enhance significantly their indistinguishability when properly integrated inside photonic waveguides due to the change in their radiative decay rate. We will show that for certain geometries and emitter positions I can be greatly reduced whereas optimal configurations can maintain or even enhance I significantly, especially for emitters with a certain ${\mathrm{\Gamma }^\ast }$/${\mathrm{\Gamma }_0}$ ratio.2.2 Analytic model for pure dephasing
We can assume that for a two-level emitter coupled to a waveguide the coupling (g) and the cavity decay rate (κ) are in the incoherent limit (2g ≪$\; {\mathrm{\Gamma }_0}$+${\mathrm{\Gamma }^\ast }$+κ) and “bad cavity” regime (κ ≫ ${\Gamma _0}$+${\Gamma ^\ast }$) [34]. In that limit the cavity can be adiabatically eliminated so the dynamics of the coupled system are described by an effective quantum emitter with decay rate (Γ+R) where R is the population transfer between the emitter and the cavity [34]:
The calculation of the Green’s Dyadic is based on the development of a 3D transform theory applied to the unbounded Helmholtz equation. Details of the calculation and the explicit dependence with the waveguide width and the position/orientation of the source (for each contributing guided mode) can be found in the Supplement 1. Using the Green’s dyadic we can obtain the Purcell enhancement as a function of the waveguide width for a point dipolar source that can be oriented parallel to the x-axis (s) or to the y-axis (p). The source is placed initially at the center of the waveguide cross-section (x0 = 0, y0 = 0). Initially, the waveguide thickness is arbitrarily fixed at b = 200 nm and we will change the width (a) and the refractive index of the waveguide (n1) using n1 = 1.44, 2 and 3.4 corresponding to SiO2, SiN and Si respectively. This will provide some initial hints on how the system actually behaves.
Figure 2 shows the value of the Purcell enhancement, $\Gamma $/${\Gamma _0}$, as a function of the normalized waveguide width, $a/\lambda $, for the mentioned values of n1 (1.44, 2, 3.4). Solid lines show the calculation of $\Gamma $/${\Gamma _0}$ using Eq. (5) and open dots show the values obtained through FDTD simulations. Details of the FDTD simulations can be found in the Supplement 1. Figure 2(a) shows the Purcell enhancement for the s-source. In general, it almost vanishes before the width reaches the cut-off of the TE10 mode, which happens for $a/\lambda $ = 0.13, 0.1 and 0.05 for n1 = 3.4, 2 and 1.44, respectively. Since the cut-off increases with n1, the vanishing threshold also increases with n1. After the cut-off, for increasing $a/\lambda ,$ The Purcell enhancement increases as the propagation constant decreases (with 1/a) and the mode gets more confined. The maximum values for $\Gamma $/${\Gamma _0}$ are 0.83, $\; $1 and 1 when $a/\lambda $ = 0.23, $\; $0.42 and 0.64 respectively and the light confinement is maximum. If the waveguide becomes wider the modes spread out with lower intensity at the position of the source producing a decrease in $\Gamma $/${\Gamma _0}$ that scales with 1/a, until the cut-off with the second order mode is reached at $a/\lambda $=0.43, $\; $0.8 and 1.2 for the same values of n1. At this point, the same mechanism takes place showing the second maxima and second decay. The process is repeated for each contributing mode. We note that there is no contribution from the lowest TM00 mode because the components of the Green dyadic vanish at the position of the source for this orientation. This is expected since the x-components of the fundamental modes are antisymmetric with respect to the source when it is placed at the center. For the p-source [Fig. 2(a)] the situation is somewhat opposite and the components do not vanish at the position of the source for the lowest order TM00. Since b = 200 nm, in the case of n1 = 2 and n1 = 1.44 the cut-off condition is already reached at $a/\lambda $ = 0. For n1 = 3.4 the cut-off is reached at $a/\lambda $ = 0.05. The Purcell enhancement for the s-source shows maximum values of $\Gamma $/${\Gamma _0}$ = 1.2, $\; $0.51 and 0.6 when $a/\lambda $ = 0.13, $\; $0.27 and 1 for the same values of n1 than before. For both s and p orientations the Purcell enhancement decays asymptotically with the width, although in a different trend due to the different (m, n) values for each contributing mode. The maxima located at $a/\lambda $=0 are accidentally generated by the model due to the unphysical divergence of the Green function at the origin. The maximum values of $\Gamma $/${\Gamma _0}$ for the s-source are about 40% higher than for the p-source with n1 = 2 and n1 = 1.44. The reason is the transverse electric field component of the TE10, which is higher than the TM00 at the position of the source (x0 = 0, y0 = 0) [44]. Nevertheless, for n1 = 3.4 the maximum $\Gamma $/${\Gamma _0}\; $is about 40% higher for the p-source. This happens because when n1 = 2 and n1 = 1.44 the TE10 mode is well confined for b=200 nm, but when n1 = 3.4 the TE10 mode is not optimally confined and the source has a better overlap with the TM00. Therefore, high modal confinement and good spatial overlapping to waveguide modes are key ingredients for Purcell enhancement, as one could intuitively expect. The indistinguishability should show its maximum value when the Purcell enhancement is maximum, according to Eq. (3). We note that a deviation in the optimal width of about 20 nm can decrease the Purcell enhancement, and therefore the indistinguishability, about 10%.
Fig. 2. Purcell enhancement of the radiative decay rate as a function of the wavelength-normalized waveguide width obtained from analytical calculations (lines) and FDTD simulations (open dots). n1 = 3.4 (black), n1 = 2 (blue), n1 = 1.44 (red). (a) Source orientation parallel to x-axis (s); (b) Parallel to y-axis (p).
Since the position of the emitter is very relevant, we explore now its effect keeping fixed the waveguide widths in $a/\lambda $=0.23, $\; $0.42 and 0.64 (respectively to the n1 values as before) and for the s orientation. We change the position of the source along the x-axis, from the center of the waveguide (x0/a=0) to far away from the edge (x0/a>${\pm} 1$).
First we focus on the region inside the waveguide core (x0/a<${\pm} 1$). Figure 3(a) shows the Purcell enhancement depending on the position of the s-source. As the s-source is separated from the center the overlapping to symmetric modes decreases and the enhancement decreases. The maximum enhancement happens at the center of the waveguide. A deviation from that optimal position of about x0/a=${\pm} $0.5 leads to a decrease of the Purcell enhancement of about 20%. The opposite behavior is obtained for a p-source [Fig. 3(b)]. In this case the minimum overlapping is obtained at the center of the waveguide and the maximum enhancement is for about x0/a=${\pm} $0.75, where the overlap with the antisymmetric modes is maximum. FDTD simulations provide a maximum value for the enhancement of 1.42 matching the analytical calculations within an error of 0.2% for the Purcell enhancement and 0.3% for x0.
Fig. 3. Enhancement of the radiative decay rate as a function of the position of the point source (normalized with respect to the waveguide width, (a) obtained from the analytical model (lines) and from FDTD simulations (open dots). The origin in the x-axis corresponds to the center of the waveguide and the edges to x0/a=${\pm} 1.$ n1 = 3.4 (black), n1 = 2 (blue), n1 = 1.44 (red). (a) s-source (b); p-source.
The variation of the coupling efficiency with the position of the source inside the waveguide follows a similar trend than the Purcell enhancement. Details of coupling definition and its calculation can be found in the Supplement 1. Figure 4 shows the coupling efficiency depending on the position of the source for both s and p orientations. At the center of the waveguide, the s-source achieves a maximum coupling of Pc/P0 = 0.88, 0.6 and 0.25 for n1 = 3.4, 2 and 1.44 respectively, where Pc is the emitted power coupled to guided modes. As expected, the coupling decreases with decreasing n1. When the s-source is separated from the center, the coupling to symmetric modes decreases. Again, the opposite behavior is obtained for the p-source, which shows minimum coupling at the center of the waveguide. Some discrepancies between analytic and FDTD results arise from the discretization of space in the FDTD simulations. Also, the slight asymmetries shown in the x0/a dependence in Fig. 4(b) are due to small misalignments between the simulation cells and the dielectric waveguide.
Fig. 4. Coupling efficiency versus normalized position of the source with respect to waveguide width a. The origin in the x-axis corresponds to the center of the waveguide. The edges of the waveguide correspond to x0/a=${\pm} 1.$ Figure shows results from analytical model (lines) and from FDTD simulations (open dots). n1 = 3.4 (black), n1 = 2 (blue), n1 = 1.44 (red). (a) s-source; (b) p-source.
Since there are recent experimental works that use heterogeneous integration of SPS and waveguides [45,46] it is worth to explore the dependence of the enhancement with the position of the source in the region outside the waveguide but close to its edge (x0/a>${\pm} 1$). Due to the index contrast between air and waveguide the electric field shows a strong discontinuity at the interface with an amount comparable to the square of the index ratio at the interface [47]. This effect can lead to a dramatic alteration of the mode profile in the vicinity of the edge that drastically increments the Purcell enhancement. This effect has been used, for example, to achieve ultrasmall cavity mode volumes of the order of $7 \times{10^{ - 5}}{\lambda ^3}$ that enable ultra-strong Kerr nonlinearities at the single-photon level [48]. When the source is placed at the edge the enhancement is $\Gamma $/${\Gamma _0}$=4.2, $\; $2.6 and 1.9 for the s-source, and $\Gamma $/${\Gamma _0}$=4.6, 1.2 and 0.87 for the p-source. The cost of the increase in the Purcell enhancement is a decrease in the coupling efficiency, which for the position at the edge is about Pc/P0 = 0.5, 0.37 and 0.15 for the s-source and about Pc/P0 = 0.5, 0.3 and 0.12 for the p-source. Details about this calculation can be seen in the Supplement 1. At the points x0/a=${\pm} 1$ (i.e., the edges of the waveguide) the mode field shows its highest contrast according to ${E_{\textrm{clad}}} = {({{n_1}/{n_2}} )^2}{E_{\textrm{core}}}$ where ${E_{\textrm{core}}}$ is the field inside the waveguide and ${E_{\textrm{clad}}}$ is the field outside the waveguide. For that reason, the maximum value of the Purcell enhancement lies in the edges of the waveguide, especially for high n1. Since the Purcell enhancement is strictly dependent on the field value at the position of the source, its maximum value is achieved at the edge of the waveguide. On the other hand, the coupling is proportional to the guided-mode field value divided by the non-guided modes field value. Despite the guided-mode field value is maximum at the edge, the value of non-guided modes is also maximum at the edge. In consequence, the coupling at the edge is weaker than in the center of the core (where the coupling to non-guided modes is smaller).
Now that we have a better understanding of the physical meaning of the model we can explore simultaneously both degrees of freedom (i.e., a and b) in order to find the optimal configurations in terms of the figures of merit. The source is placed initially at the center of the waveguide cross-section (x0 = 0, y0 = 0) in horizontal orientation (i.e., parallel to x-axis) but this time both a and b are varied from 0 to 0.7$\; \lambda $. The results are obtained for four different values of the refractive index of the waveguide n1 = 1.44, 2, 3.4, and 4.
Figure 5 shows the value of $\Gamma $/${\Gamma _0}$ as a function of the normalized waveguide width, $a/\lambda $, and normalized thickness, $b/\lambda $, calculated for the four different refractive indexes. The blue areas in the plots correspond to values of a and b below the first cut-off. The subsequent maxima and minima correspond to the activation of the TEmn and TMmn modes. For low refractive indexes (i.e., n1 = 1.44, 2) the two first modes appear. As the refractive index increases the source starts to overlap effectively with the rest of higher order modes. The absolute maxima of $\Gamma $/${\Gamma _0}$ increases with the refractive index, since the area of the spatial distribution of the modes decreases with n1, so the field intensity gets higher at the position of the source. We obtain maximum values of $\Gamma $/${\Gamma _0}$=1, $\; $1.1, $\; $1.6 and 1.9 for n1 = 1.44, 2, 3.4 and 4, respectively. Due to the symmetry of the system the plots for the vertical source show the same rotated 90 degrees (see Supplement 1).
Fig. 5. Purcell enhancement as a function of the normalized width, $a/\lambda ,$ and thickness, $b/\lambda $, when the s-source is placed at the center of the waveguide calculated with the analytical model. (a) n1 = 1.44; (b) n1 = 2; (c) n1 = 3.4; (d) n1 = 4.
Next, we place the source outside the waveguide, 10 nm away from the edge and oriented horizontally (i.e., parallel to x-axis). Figure 6 shows the value of $\Gamma $/${\Gamma _0}$ as a function of the normalized waveguide width, $a/\lambda $, and normalized thickness, $b/\lambda $, for different n1. As we saw before, the field discontinuity generates a dramatic enhancement when the source is placed near the evanescent region of the mode. We observe that in all of the cases the maxima are located in the bottom left region, where both a and b have reached the cut-off but the first mode has not reached the maximum confinement. For that geometry, the mode is not optimally confined inside the core and the field gets accumulated at the edges of the waveguide so the overlap is more efficient. We obtain maximum values of $\Gamma $/${\Gamma _0}$ = 2, $\; $3.5, $\; $8 and 10 for n1 = 1.44, 2, 3.4 and 4 respectively. At this time the orientation of the source matters, since we can arrange two different configurations: (a) Parallel to the larger side of the waveguide (i.e., parallel to the x-axis if the source is placed on top of the core, or parallel to the y-axis if the source is placed on one side of the core); (b) Perpendicular to the larger side of the waveguide (i.e., parallel to the y-axis if the source is placed on top of the core, or parallel to the x-axis if the source is placed on one side of the core). The plots in Fig. 6 correspond to the second case. When the source is parallel we obtain lower values for the maximum enhancements: $\Gamma $/${\Gamma _0}$ = 0.9, $\; $1.5, $\; $6.8 and 7.1 for the different values of n1. For emitters with orientations other than s or p one should decompose the projection of the orientation on the x-axis and the y-axis and treat the emitter as two separated emitters with corresponding s and p contributions. The total enhancement is given by the addition of those two contributions.
Fig. 6. Purcell enhancement as a function of $a/\lambda $ and $b/\lambda $ when the s-source is placed outside the core calculated with the analytical model. (a) n1 = 1.44, (b) n1 = 2, (c) n1 = 3.4, (d) n1 = 4.
The maximum enhancements obtained for the source at the edge can be used in Eq. (3) to obtain the maximum values for the indistinguishability. Figure 7 shows I for an s-emitter placed at the edge of the waveguide versus the intrinsic emitter normalized dephasing ratio, ${\Gamma ^\ast }$/${\Gamma _0}$. Results for the p-source show an analogous behavior. From Fig. 7 we see that for low dissipative emitters with ${\Gamma ^\ast }$/${\Gamma _0}\sim 1$ (like weakly confined GaAs dots of Ref. [13]) the expected indistinguishability can reach a value up to 0.8 when n1 = 4, which makes an enhancement of I of about 30% with respect to the same dots without coupling to a waveguide. For InAs quantum dots with ${\Gamma ^\ast }$/${\Gamma _0} = 2.6$ [36,37] we obtain I≈0.6, an enhancement of 40%. As the pure dephasing rate increases the indistinguishability decays asymptotically reaching 0.2 when ${\Gamma ^\ast }$/${\Gamma _0} = 50.$ Therefore for strong dissipative systems with ${\Gamma ^\ast }$/${\Gamma _0} > 50$ (like quantum emitters in 2D materials) the effect of the waveguide in the indistinguishability is very small. For emitters with lower dephasing ratio, ${\Gamma ^\ast }$/${\Gamma _0} < 1$, and high intrinsic indistinguishability (I >> 0.5) the effect of the waveguide becomes again negligible since I$\to $1 when ${\Gamma ^\ast }$/${\Gamma _0} \to 0$. As n1 increases the maximum ${\Gamma ^\ast }$/${\Gamma _0}$ for I >> 0.5 also increases reaching values up to 12 for n1 = 4.
Fig. 7. Indistinguishability of an s-source quantum emitter placed at the edge of the waveguide versus its $\frac{{{\Gamma ^{\ast }}}}{{{\Gamma _0}}}$ value. Green- n1 = 4, Black– n1 = 3.4, Blue- n1 = 2, Red- n1 = 1.44. (a) Perpendicular-source. (b) Parallel-source.
Despite a SPS can achieve much stronger Purcell enhancements inside optical cavities [49], it is often required small mode volumes or high quality factors, or both. Small mode volumes difficult the deposition of the emitter at the field maxima [50], and on the other hand, high quality factors reduce the extraction efficiency [34]. In the case of waveguide integration, deposition at the field maxima (i.e., edge of the waveguide) becomes trivial, and as it has been shown, high extraction efficiency can be achieved. However, our results also reveal that for strong dissipative SPS integrated in a waveguide the indistinguishability does not reach the standard requirements for quantum information applications. For those cases the use of an optical cavity is mandatory.
From a practical perspective, the heterogeneous integration of the emitter with a waveguide may be performed by placing the emitter inside a material with non-unity refractive index (i.e., n2$\ne 1$) [45], it is worth to explore how the Purcell enhancement is affected by different index contrasts. As we mentioned above, the field discontinuity at the edge of the waveguide is proportional to ${({n_1}/{n_2})^2}$, so we can expect a significant reduction of the enhancement depending on ${n_2}$.
Figure 8(a) shows the dependence of the Purcell enhancement with ${n_2}$ for an s-source placed at the edge of the waveguide with optimal (a,b) geometry. Since the Purcell enhancement in the edge depends on the index contrast ${n_1}/{n_2}$, it decreases asymptotically with n2. In the limit where n2${\sim} 1$ the Purcell value approaches to that shown in Fig. 6, and when n2${\sim} $ n1 we obtain the emitter decay rate corresponding to a homogeneous material (i.e., Pf = 1). For cladding materials like SiO2 (n2 = 1.44) the Purcell enhancement is reduced about 50% with respect to the value with n2$= 1$. On the other hand, Fig. 8(b) shows the reduction in the indistinguishability due to this lower Purcell enhancement for an emitter with ${\mathrm{\Gamma }^{\ast }}/{\mathrm{\Gamma }_0}$=10. Similarly to what happened to the Purcell enhancement, when n2${\sim} 1$ the indistinguishability approaches to that shown in Fig. 7, and when n2${\sim} $ n1 we obtain the value corresponding to the specific ${\mathrm{\Gamma }^{\ast }}/{\mathrm{\Gamma }_0}$ ratio in a homogeneous material (i.e., I = 0.1). Again, for n2 = nSiO2 the indistinguishability is reduced about 15% with respect to n2${\sim} 1$. Regarding the optimization sweep for finding the optimal waveguide height and width for the case of n2 in the range (1–1.6) the results are almost equivalent to that shown in Fig. 6 (case of n1 $= 1.44$). Therefore Fig. 6 can be used as a guide for waveguide design when n2 is inside that range.
Fig. 8. (a) Purcell enhancement of an s-source at the edge of the waveguide as a function of n2. (b) Indistinguishability of an s-source at the edge of the waveguide as a function of n2.
It is important to highlight that the positions for maximum indistinguishability differ from those of maximum coupling efficiency. Indistinguishability depends strongly on Purcell enhancement, which for the case of a waveguide achieves its maximum value at the edge, where the field is strongest. On the other side, the coupling efficiency on the edge is not as high as in the center, but still may have a value useful for some experiments or even some applications An interesting figure of merit for the geometrical optimization of the waveguide is the I ${\cdot}\beta $ product (with $\beta $ the coupling efficiency) that can be explored as a function of a and b. In the same way we did for the optimization of the Purcell enhancement, a and b change from 0 to 0.7 λ, and the emitter is placed at the edge of the waveguide. We set n2 = 1.44 this time and we vary n1 = 2, 3.4, 4 and 4.4. For the estimation of the indistinguishability we set ${\mathrm{\Gamma }^{\ast }}/{\mathrm{\Gamma }_0}$=10 as before.
Figure 9 shows the I$\; \beta $ value as a function of the normalized waveguide width, a/λ, and normalized thickness, b/λ, calculated for n1 = 2, 3.4, 4 and 4.4. As expected, for the four refractive indexes the highest I $\beta $ happens for the geometry that maximizes the Purcell enhancement. Maximum I$\; \beta $ = 0.35 is found for the highest waveguide index (n1 = 4.5) and minimum I$\; \beta $ = 0.15 for n1 = 2. Also, in the four cases the I $\beta $ product is significantly higher than the obtained when the source is at the center of the waveguide (0.06 for n1 = 2, 0.09 n1 = 3.4, and 0.13 for n1 = 4).
Fig. 9. I$\; \beta $ value as a function of the normalized width, a/λ, and thickness, b/λ, when the s-source is placed at the edge of the waveguide with n2 = 1.44 calculated with the analytical model. (a) n1 = 2, (b) n1 = 3.4, (c) n1 = 4, (d) n1 = 4.5.
2.3 Analytic model for spectral diffusion
We have explored so far the effect of pure dephasing in the indistinguishability. In addition, the effect of spectral diffusion needs to be treated separately. Whereas the dynamics of pure dephasing evolve at shorter time scales than the emitter decay rate ${\Gamma _0}$, spectral diffusion is related to processes with significantly larger time scales [51] so it is characterized by a statistical average over the different center frequencies associated with the emitter [52]. In this context, the spectral broadening of the emission is given by ${\Gamma _2} = \; {\Gamma _0} + \; \Gamma ^{\prime}$, where $\Gamma ^{\prime}$ represents the FWHM of the distribution associated with spectral diffusion. The indistinguishability reads $I = \; {\Gamma _0}/{\Gamma _2}$ [51]. Being $\Delta \omega$ the intrinsic width of each center frequency, and $\Delta \delta $ the extrinsic width due to the entanglement with the extrinsic environment, for a Lorentzian distribution the ratio $\theta = \Delta \delta /\Delta \omega $ is equal to $2\Gamma ^{\prime}/{\Gamma _0}$ [52]. The indistinguishability can be written in terms of $\theta $ as:
Figure 10 shows I for s- and p-emitters placed at the edge of the waveguide versus the normalized ratio $\theta $. From Fig. 1 we see that for emitters with with $\theta $∼1 [13] the expected indistinguishability is above 0.9 for the four refractive indexes. As the extrinsic width of the emitter increases respect to the intrinsic width the indistinguishability decays asymptotically reaching 0.3 when $\theta $ = 50 for n1 = 4 and $\theta $ = 10 for n1 = 1.44. Therefore, for emitters with extrinsic width much larger than intrinsic width the effect of the waveguide in the indistinguishability is negligible. As n1 increases the maximum $\theta $ for I >> 0.5 also increases reaching values up to 21 for n1 = 4. We can say that in general we observe a similar behavior to pure dephasing although with a slower asymptotic decay of I.
Fig. 10. Indistinguishability of an s-source quantum emitter placed at the edge of the waveguide versus its $\theta $ < value. Green- n1 = 4, Black– n1 = 3.4, Blue- n1 = 2, Red- n1 = 1.44. (a) Parallel-source. (b) Perpendicular-source.
3. Conclusions
We have calculated the indistinguishability of a point-source quantum emitter coupled to a waveguide because its technological implications in future quantum photonic integrated circuits. The emitter has arbitrary orientation and location with respect to the waveguide. We have obtained the results for different index of refraction of the waveguide (SiO2, Si3N4, Si, and other high index materials like WeS2 or WO3). The analytical model used permits a fast computing of the indistinguishability from a set of simple expressions derived from the same solution of the dyadic Helmholtz equation. The model has been numerically evaluated through 3D-FDTD simulations with excellent agreement. Maximum indistinguishability for an optimal waveguide width is found for a source placed outside the core, at the edge of the waveguide, in contrast to maximum coupling efficiency position at the center of the waveguide. For strong dissipative emitters with ${\Gamma ^\ast }$/${\Gamma _0} > 50$ (like transition metal dichalcogenides) the effects of the waveguide in the indistinguishability are negligible but for low dissipative emitters with ${\Gamma ^\ast }$/${\Gamma _0} \approx 1$ (like GaAs quantum dots) the indistinguishability can be enhanced up to a 30% and reach values around I ≈ 0.8 when dots are coupled to a waveguide. We hope this work can help for an optimized design of PIC waveguides in quantum photonic circuits.
Funding
Horizon 2020 Framework Programme (820423).
Acknowledgments
We gratefully acknowledge financial support from the European Union's Horizon 2020 research and innovation program under grant agreement No. 820423 (S2QUIP).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
All the data generated and analyzed in this work are available upon request.
Supplemental document
See Supplement 1 for supporting content.
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