Compact design of a gallium phosphide nanobeam cavity for coupling to diamond germanium-vacancy centers

Abdulrahman Alajlan; Ivan Cojocaru; Alexey V. Akimov

doi:10.1364/OME.9.001678

1. Introduction

Quantum information processing is a thriving field in quantum optics that requires manipulating light at a single photon level. One of the most powerful approaches to achieve that is by coupling spin qubits to solid-state cavity QED systems. Such system allows for a reversible interaction between the cavity and the qubit in the strong-coupling regime. Artificial atoms in the solid state have been exploited as active elements to realize such systems. Among these, color centers (optically active defects) in diamond.

Although diamond nitrogen-vacancy (NV) center was demonstrated as a promising quantum memory [1,2], quantum register [3,4], and an ideal nanoscale sensor [5,6], it is very sensitive to surface defects and damages created by the fabrication process. Moreover, NV center has a relatively weak zero phonon line (ZPL), at room temperature, and a strong phonon sideband leading to a small probability of interaction with single photons. On the other hand, silicon-vacancy (SiV) center has a small phonon sideband and strong coherent optical transitions in nanostructures [7,8]. Recently, SiV center has been observed and artificially produced in sub 10 nm diamonds [9,10]. The interesting properties of SiV centers stem from its inversion symmetry, making it more stable and resistant to the fluctuations in the local environment. The brightness of SiV centers is, however, very low as a result of the low quantum efficiency. Alternatively, negatively charged germanium-vacancy (GeV) centers are new color centers that have an inversion symmetry and optical properties similar to SiV centers [11–13]. The main advantage is that GeV centers have a very high, nearly unitary, quantum efficiency and a small phonon sideband, leading to a high cross-section of interaction between individual GeV centers and single photons [14]. Our goal is to use single GeV centers as an optical interface integrated into photonic devices to overcome inherently weak light–matter coupling and allow for transferring quantum states in a quantum network.

Over the past two decades, experiments have evolved in diverse physical systems, including cold atoms trapped in high finesse Fabry-Perot microcavities [15,16], quantum dots coupled to toroidal microresonators [17,18], and superconducting qubits coupled to superconducting resonators [19,20]. More recently, there have been a substantial progress in coupling qubits into photonic crystal cavities [21–24]. Although some groups have successfully integrated diamond color centers into photonic devices [25–32], strong coupling between single color centers and cavity modes remains a big challenge.

In this paper, we consider a scheme of a single GeV center on top a photonic crystal nanobeam cavity to achieve strong coupling between the cavity mode and the zero-phonon line (ZPL) transition of the GeV center. We first demonstrate a unique design for Gallium Phosphide (GaP) nanobeam cavity with $Q / V > 10^{8}$ and a small footprint, and illustrate our systematic approach to reduce 50% of the nanobeam footprint while maintaining the cavity $Q / V$ value and transmission. We then analyze cavity QED figures of merit for a single diamond nanocrystal (DNC) incorporated a single GeV center placed on top of the nanobeam cavity. We then evaluate the effect of the DNC size on cavity QED figures of merit and show that our strategy is capable for realizing the strong-coupling limit, even with relatively large DNC sizes.

2. Cavity design

Our design is based on the quadratic tapering method that combines ultra-high Q-factors and near maximum cavity transmission [33–35]. That method, however, requires a large number of holes in each side of the cavity, making it challenging for integrating a large number of on-chip devices. Using only 3D-FDTD numerical simulations free software (MEEP), we built a new design that is capable to shrink the nanobeam cavity by more than 50%. The design process consists of optimizing three elements: (a) cavity mirror strength, (b) width, and (c) number of holes. We consider a free-standing GaP with following parameters: refractive index ( $n = 3.36$ ), thickness ( $d / a = 1$ ), hole radius ( $r / a = 0.3$ ), and a single TE mode.

Modes localized within a photonic bandgap have a complex wavevector $k = (1 + i δ) π / a$ . The imaginary component $δ$ denotes the so-called “mirror strength”, the field decay due to the mirror reflection. The mirror strength can be calculated using this relation $δ = \sqrt{{(ω_{2} - ω_{1})}^{2} / {(ω_{2} + ω_{1})}^{2} - {(ω_{r e s} - ω_{0})}^{2} / ω_{0}^{2}}$ , where $ω_{2}$ , $ω_{1}$ , $ω_{0}$ and $ω_{r e s}$ are the air band edge, dialectric band edge, middle frequency and resonant frequency, respectively [34]. In order to increase the mirror strength, semicircular holes are added on the right and left side of the central holes, see Fig. 1(a). The mirror strength is found sensitive not only to the hole radius and nanobeam width, but also to the position of the semicircular holes. The maximum mirror strength is achieved when semicircular holes are adjusted such that they are shifted horizontally from the central holes by $a / 2$ , where $a$ is the lattice constant, Fig. 1(b). Interestingly, this idea works not only for designs with tapered widths, but also for other designs with tapered holes radii (See Appendix for further details).

Fig. 1 (a) A sketch of tapered nanobeam cavity with semicircular holes and $E_{y}$ field distribution of the TE fundamental mode. (b) Mirror strength as a function of nanobeam final width $W_{f}$ for three configurations: only central holes (green), central holes with in-phase semicircular holes (orange), central holes with out-of-phase semicircular holes (blue). (c) Mirror strength as a function of the hole segment number after parabolic tapering using this formula: $W_{f} = W_{0} + 2 R_{w} - 2 R_{w} \sqrt{1 - {(N / R_{w})}^{2}}$ , where $W_{0} / a = 1.5$ (1st hole) and $W_{f} / a = 1.8$ (15th hole). The dashed line is a linear fit with $R^{2} = 0.98$ . (d) TE band structure for the proposed design. The dashed line marks the cavity resonant frequency.

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A linear increase of the mirror strength as a function of the hole’s position along the cavity has been shown to form a Gaussian-like field profile [34], which suppresses greatly out-of-plane scattering losses estimated by Fourier space analysis [36–38]. To achieve such a profile, optimizing the nanobeam width is necessary. We first determine the initial width $W_{0}$ at the center of the cavity. In principle, a small initial width shifts the dielectric band towards the light line, and thus causes in-plane radiative losses, whereas a large width pulls high order modes into the bandgap and decreases its size. Examining the band structure for different initial widths, we found that $W_{0} / a = 1.5$ is a good compromise. Then, we select the final width $W_{f}$ at the corner of the cavity such that the corresponding mirror strength is high enough and not saturated. For example, $W_{f} / a = 1.8$ satisfies the previous conditions. Figure 1(c) shows that the mirror strength for our design increases linearly as a function of the hole’s segment number, providing a proof of Gaussian-like field profile. That is also verified by fitting a Gaussian function to the field distribution along the cavity axis (See Appendix for further details).

The cavity resonance is always the same as the first TE mode of the initial width for cavities based on the quadratic tapering method. TE modes were obtained from the corresponding band structure, Fig. 1(d). The bandgap for our design has slightly increased compared to the original geometry (without semicircular holes) indicating lower scattering losses (See Appendix for further details). In order to set the cavity resonance to a certain frequency, we choose the value of the lattice constant such that $a / λ$ matches the value of the first TE mode. In our case, $λ = 602 nm,$ the lattice constant is 135 nm.

Although we have selected the final nanobeam width based on the field analysis, we verify that more accurately by studying the cavity $Q / V$ value and transmission dependence on the nanobeam initial and final width. We first examine $Q / V$ value and transmission dependence for three final widths: $W_{f} / a = 1.7$ , $W_{f} / a = 1.8$ , and $W_{f} / a = 1.9$ . Results reveal that the $Q / V$ value enhances for small final widths due to the sharp linear increase of the associated mirror strength as a function of the hole segment number, Fig. 2(a). Nevertheless, the enhancement of the $Q / V$ value may not be as significant as the nanobeam footprint. Figure 2(b) shows that the number of holes for different final widths increases as a function of $Q / V$ value. We found that $W_{f} / a = 1.8$ is a good trade-off between the number of holes and the $Q / V$ value. Considering the previous parameters, the $Q / V$ value can reach up to $10^{8}$ with transmission higher than 70%.

Fig. 2 (a) & (b) The cavity transmission and number of holes as a function of $Q / V$ value for different nanobeam final widths, respectively. The nanobeam initial width in (a) & (b) is fixed, $W_{0} / a = 1.5$ . (c) The dependence of $Q / V$ value taken at 90% cavity transmission on the nanobeam initial width. The difference between the final and initial width is fixed, $Δ W / a = 0.3$ . (d) Number of holes as a function of $Q / V$ value for different nanobeam initial widths.

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The nanobeam initial width is also an important parameter that plays a crucial role in the $Q / V$ value and nanobeam footprint. In Fig. 2(c), we examine the $Q / V$ value for different nanobeam initial widths and show that the $Q / V$ value changes slightly except for $W_{0} / a = 1.4$ . The cavity resonance for such small initial widths shifts distinctly towards the light line, causing in-plane radiative losses. More importantly, the changes in the number of holes for different initial widths are remarkable, see Fig. 2(d). By comparing both the $Q / V$ value and the number of holes, we found that $W_{0} / a = 1.5$ is a good trade-off.

The reduction of the nanobeam footprint due to the semicircular holes is so significant. Results show that the cavity footprint is reduced by 50%, in agreement with the enhancement of the mirror strength calculated above. We noted that the reduction factor depends to some extent on the nanobeam width. To achieve a great reduction factor, a relatively small initial width is recommended (See Appendix for further details).

So far, we have considered only structures with an even total number of holes, i.e. the dielectric region lies in $x = 0$ symmetry plane. The electric field $E_{y}$ is even with respect to $x = 0$ symmetry plane and so the maximum energy density locates at the center of the nanobeam cavity. If an atom is placed at the cavity center it will create strong coupling between the atom and the cavity. In the next section we discuss a single GeV center placed at the center of the top nanobeam surface, Fig. 3(a).

Fig. 3 (a) A sketch of DNC placed on top of a nanobeam cavity. (b) Relevant cavity QED rates as a function of number of holes. $κ / 2 π$ is the cavity decay rate, $γ / 2 π$ is the atomic decay rate of the GeV center, and $g / 2 π$ is the single photon coupling rate. (c) Cooperativity (dots) and weak/strong coupling index (open circles) as a function of number of holes. (d) cooperativity as a function of number of holes for different DNC sizes.

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3. Cavity QED analysis

As we are particularly interested in studying the interaction between the single photon field in the nanobeam cavity and the GeV center, it is necessary to calculate the parameters governing the dynamics of cavity QED systems. The key parameters are: atom-photon interaction rate $g$ (vacuum Rabi frequency), atomic decay rate $γ_{s p}$ (spontaneous emission rate), and cavity field decay rate $κ$ . The Rabi frequency is defined as $g = \vec{d} \cdot \vec{E} / ℏ,$ where $d$ and $E$ are the dipole moment of the GeV center and the single photon electric field amplitude, respectively.

The dipole moment of the GeV center can be derived from its spontaneous emission lifetime, τ ∼6 ns [14], using perturbation theory [39], which gives:

d = \frac{1}{2} \sqrt{\frac{3 γ_{s p} ε_{0} λ^{3} ℏ}{2 π^{2} n_{d n c}}},

where

n_{d n c} = 2.4

is the refractive index of the diamond nanocrystal and

λ = 602 nm

is the ZPL optical transition of GeV center. Since the ZPL optical transition contributes about 60% of the total emission [11], the dipole moment must be scaled by

η = \sqrt{0.6}

.

The single photon electric field amplitude for a given cavity mode can be written as:

E = \sqrt{\frac{π ℏ c}{ε_{0} λ n_{G a P}^{2} V}},

where

n_{G a P} = 3.36

is the refractive index of GaP,

λ

is the cavity mode tuned to ZPL optical transition of the GeV center, and

V

is the mode volume defined as

V = \int d^{3} r ε {| E |}^{2} / {[ε {| E |}^{2}]}_{m a x}

. Since the DNC is assumed to be on top of the nanobeam cavity, the mode volume

V

must be scaled by

α^{2} = {| n_{G a P}^{2} E_{G a P}^{2} |}_{m a x} / {| n_{0}^{2} E_{0}^{2} |}_{m a x}

, ratio of the maximum energy density in the GaP nanobeam cavity to the maximum energy density at the DNC location. Here

n_{0} = 1

is refractive index of the surrounding medium (air in our case),

E_{G a P}

is the field inside the GaP nanobeam cavity,

E_{0}

is the field in the surrounding medium. Based on FDTD simulations, we found that

α^{2} = 4.5

if the DNC is placed at the center of the cavity. Though placing the DNC at the center is experimentally impossible, our simulations show that

α^{2} = 4.8

for the case where the DNC is placed away from the center and close to either cavity edge. It is important to note that

α^{2}

for thinner nanobeams approaches unity (See Appendix for further details).

Then, the Rabi frequency can be evaluated to give:

g = \frac{η}{2 α} \sqrt{\frac{3 c λ^{2}}{τ n_{d n c} n_{0}^{2} V}},

The relevant cavity QED figures of merit is plotted in Fig. 3(b). The cavity field decay rate

κ

is the only parameter that depends notably on the number of holes. The Rabi frequency has a weak dependence due to the changes in the mode volume. The strong-coupling criterion is satisfied when the atom-photon coupling rate exceeds the decay rate of the system,

g > max [κ, γ_{s p}]

. That condition is satisfied for nanobeam cavities with 12 holes and above, Fig. 3(c). The cooperativity, defined as

C = g^{2} / κ γ_{s p}

, in the strong-coupling regime varies from

10^{2}

up to

10^{5}

depending on the number of holes. Furthermore, cooperativity depends also on the nanobeam thickness. Results show that the cooperativity can enhance one order of magnitude for thinner nanobeam cavities mainly because of the gain in

α^{2}

value (See Appendix for further details).

Techniques have been developed in recent years to yield sub-10-nm DNCs at low costs [9,10]. However, such sizes might not be attainable in standard tools for many color centers. Therefore, it is important to quantify the effect of the DNC size on our system. We performed simulations of a GaP nanobeam cavity and a cube on the top surface with a refractive index of 2.4 and different sizes ranging from 10 nm to 50 nm. Interestingly, there is no effect for DNC with 10 nm size on the cavity $Q / V$ value. Thus, for DNCs with sizes around 10 nm the interaction between the color center and the cavity should not depend on the size of the DNC nor on its shape. For DNC with sizes larger than 10 nm, there is no a considerable effect on cavities with a low $Q / V$ value, whereas the ultrahigh $Q / V$ value drops exponentially as function of the DNC size, Fig. 3(d). In spite of that drop, the condition for strong coupling is still satisfied. In that case, the cooperativity value will be limited within a small range as Q-factor reaches a saturation limit at a certain number of holes.

In conclusion, we have illustrated a compact design for nanobeam cavities with an ultrahigh $Q / V$ value for the purpose of strong coupling between the cavity mode and a single GeV center. We have shown that the cooperativity for the first time can reach up to five orders of magnitude in the strong-coupling regime. This work will facilitate the design of ultrahigh $Q / V$ photonic devices, and thus support integrated photonic structures used in a broad range of quantum networks.

4 Appendix

Mirror strength for designs with tapered radii

Mirror strength changes based on the geometry of each design. Therefore, we tested additional designs based on the quadratic tapering method. Figure 4 shows that adding semicircular holes enhances the mirror strength, and thus reduces the nanobeam footprint. We noticed that mirror strength enhances more for designs with small nanobeam thicknesses and widths, Fig. 4(a).

Fig. 4 Mirror strength as a function of hole radius for two nanobeam designs with (blue) and without (orange) semicircular holes. The design in (a) is obtained from Notomi et al. [40], and the design in (b) is obtained from Qimin et al. [35]. The enhancement of the mirror strength (a) and (b) is 70% and 40%, respectively.

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Comparison between ultrahigh Q/V nanobeam cavities with and without semicircular holes

The design with only central holes is based on the previous design proposed by Ahn et al. [33]. In Fig. 5, we apply optimizations to that design and compare it to our design with semicircular holes. First, TE modes shifts slightly to a lower frequency, and so the bandgap becomes smaller compared to that of our design. Second, optimizations for both designs results in a Gaussian-like field profile as indicated by the linear mirror strength plotted in Fig. 5(b). Therefore, both designs can be used to achieve an ultrahigh $Q / V$ value within the same range. Nevertheless, Fig. 5(c) shows that our design surpasses the original design for large initial widths. Interestingly, both designs have very similar $Q / V$ value at $W_{0} / a = 1.5$ . The large cavity footprint is the main challenge for the design with only central holes. By means of semicircular holes, one can shrink the cavity footprint by at least 50% without affecting the cavity $Q / V$ value or its transmission, Fig. 5(d). Moreover, the reduction depends on the overall geometry, similar to the case of the mirror strength discussed above. We found that the reduction can be achieved mostly in the same range for different initial and final widths, Fig. 6. In general, small widths show more reduction due the enhancement in the corresponding mirror strength. This evidence that our design has some tolerance and not very sensitive to the slight change in the main parameters. Also, by varying hole size at moderate $Q / V = 5 \cdot 10^{6}$ we found, that semicircular holes do not introduce any extra sensitivity of $Q / V$ to the hole size except for the natural rise due to the overall increase of Q/V values in our design. The latter design leads to approximately twice faster shift of $Q / V$ with the hole radius, compared to the case without semicircular holes.

Fig. 5 (a) TE band structure for a tapered nanobeam cavity with only central holes. (b) Mirror strength as a function of hole position after parabolic tapering. (c) The dependence of $Q / V$ value with 90% transmission on the nanobeam initial width $W_{0}$ . (d) Number of holes as a function of $Q / V$ value. We considered the same initial and final width in both designs.

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Fig. 6 Reduction in the number of holes gained by the design demonstrated in the main text as a function of $Q / V$ value for different initial widths (a) and final widths (b).

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Electric field distribution analysis

_{$E_{y}$} field distribution obtained from FDTD simulation shows a Gaussian profile inside the cavity fitted with an analytical formula $E_{y} = \cos (π \frac{x - x_{0}}{a}) e x p (- σ^{2} (x - x_{0})^{2})$ , Fig. 7(a). Boundary conditions allow the normal field component to enhance in the semicircular holes, Fig. 7(b). In the main text we considered a geometry with an even $E_{y}$ symmetry. The alternative odd symmetry can also be proposed. The $Q / V$ value and transmission dependence show a similar trend in both symmetries, Fig. 7(c), in agreement with the previous design of an ultrahigh Q-factor nanobeam cavity [35]. Even though both symmetries have the same $Q / V$ value and transmission dependence, the even symmetry is more suitable for quantum applications.

Fig. 7 (a) $E_{y}$ field distribution for the TE fundamental mode obtained from FDTD simulations (blue dots) and from an analytical formula (red line). (b) $E_{y}$ cross sections of the TE fundamental mode. $α^{2}$ is the ratio of the energy density in the GaP nanobeam cavity to the maximum energy density at the DNC calculated for different nanobeam thicknesses. (c) The cavity transmission as a function of $Q / V$ value for dielectric-centered cavity (blue) and air-centered cavity (orange).

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Nanobeam thickness

The effect of the nanobeam thickness on the cavity $Q / V$ value and transmission is so crucial. Figure 8(a) shows that thick nanobeam cavities have higher transmission and $Q / V$ value. The field on top of the nanobeam surface, however, is low for thick nanobeams. Calculating the ratio of the energy density in the GaP nanobeam cavity to the maximum energy density at the DNC, we found that the cooperativity increases for thinner nanobeams up to one order of magnitude, Fig. 8(b).

Fig. 8 (a) The cavity transmission as a function of $Q / V$ value for different nanobeam thicknesses. (b) Cooperativity (dots) and weak/strong coupling index (open circles) as a function of nanobeam thickness. We considered cavities with $Q / V = 10^{6}$ for all thicknesses in (b) to maintain the cavity transmission.

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Funding

National Science Foundation (NSF) (PHY-1820930).

Acknowledgments

The authors acknowledge the Texas A&M University Brazos HPC cluster (brazos.tamu.edu) that contributed to the research reported here.

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Compact design of a gallium phosphide nanobeam cavity for coupling to diamond germanium-vacancy centers

Abstract

1. Introduction

2. Cavity design

3. Cavity QED analysis

4 Appendix

Mirror strength for designs with tapered radii

Comparison between ultrahigh Q/V nanobeam cavities with and without semicircular holes

Electric field distribution analysis

Nanobeam thickness

Funding

Acknowledgments

References

Cited By

Figures (8)

Equations (3)

Optical Materials Express