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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 8 — Apr. 17, 2006
  • pp: 3556–3562
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Diamond based photonic crystal microcavities

S. Tomljenovic-Hanic, M. J. Steel, C. Martijn de Sterke, and J. Salzman  »View Author Affiliations


Optics Express, Vol. 14, Issue 8, pp. 3556-3562 (2006)
http://dx.doi.org/10.1364/OE.14.003556


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Abstract

Diamond based technologies offer a material platform for the implementation of qubits for quantum computing. The photonic crystal architecture provides the route for a scalable and controllable implementation of high quality factor (Q) nanocavities, operating in the strong coupling regime for cavity quantum electrodynamics. Here we compute the photonic band structures and quality factors of microcavities in photonic crystal slabs in diamond, and compare the results with those of the more commonly-used silicon platform. We find that, in spite of the lower index contrast, diamond based photonic crystal microcavities can exhibit quality factors of Q=3.0×104, sufficient for proof of principle demonstrations in the quantum regime.

© 2006 Optical Society of America

1. Introduction

N-V centers in unstructured diamond have already been proposed as qubits in a quantum computer [16

16. M. S. Shahriar, P.R. Hemmer, S. Lloyd, P.S. Bhatia, and A. E. Craig, “Solid-state quantum computing using spectral holes,” Phys. Rev. A 66, 032301 (2002). [CrossRef]

]. The method uses an ensemble of spectrally-resolved N-V centers randomly distributed in the bulk of a piece of crystalline diamond. The diamond crystal is placed between two mirrors (a “classical” Fabry-Perot resonator) to form a cavity mode around each N-V center. The addressing is performed by optical beams, but in this proposed scheme, it seems rather difficult to construct an optical cavity that provides photon lifetimes sufficiently long to have each N-V center coupled to the optical field in the strong coupling regime. Among other factors, Fresnel losses at the diamond faces may degrade drastically the cavity Q-factor.

The quality factors needed for quantum computing depend on the particular approach that is considered, varying between low values in non-deterministic schemes [17–18

17. Y. L. Lim, A. Beige, and C. Kwek, “Repeat-until-success linear optics distributed quantum computing,” Phys. Rev. Lett 95, 030505 (2005). [CrossRef] [PubMed]

], to higher values in schemes that involve strong coupling. For example, estimates show that Q values of Q~105 are desirable for QIT experiments in the PCS microcavities proposed by Greentree et al in Ref. [19

19. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A. 73, 013818 (2006). [CrossRef]

]. However, as pointed out by these authors, proof of principle experiments will be possible with significantly lower Q values. One problem common to most coupled atom-photon experiments for QIT is that the main decoherence channel is the bus that couples the different subsystems, which is strongly perturbed by the environment. If the qubit is localized in an ultra-high-Q cavity, say Q~105, there is a difficulty in out-coupling the excitation from the cavity. However, in a photonic crystal cavity, the possibility exists to hold photons in a high-Q localized mode until a control device changes the cavity from high Q to low Q (a Q-switch) [19

19. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A. 73, 013818 (2006). [CrossRef]

]. With this “quantum gate for Q-switching” one can maintain high coherence until the cavity Q is deliberately lowered for read-out of the optical mode. Even though there have been experimental demonstrations of fairly efficient coupling to high-Q cavities such as in Ref. [20

20. P. Barclay, O. Painter, and K. Srinivasan, “Nonlinear responseof silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” 13, 801–820 (2005).

], the total fiber-to-cavity efficiency needs to be close to 100% for QIT experiments, and this is best accomplished by an active out-coupling.

Our design procedure consists of two steps: We first determine PBGs of diamond based crystal slabs and compare them with PBGs of the silicon PCS. Subsequently, we evaluate quality factors of microcavities in diamond-based PCS.

2. PC geometry and method

The model is a PCS composed of a hexagonal array of cylindrical air holes in a dielectric slab, [see Fig. 1(a)]. Diamond has smaller refractive index, n=2.4, than semiconductors, leading to weaker TIR guidance in the vertical direction and a cavity mode that is less well confined in the vertical direction. .

Fig. 1. (a) Schematic of bulk photonic crystal slab (PCS) and modifications of the geometry around the cavity described (b) by Zhang et al in Ref [7] and (c) by Song et al in Ref. [6].

In our design we first choose the parity of the dominant field component in the slab with respect to the mirror plane parallel to the slab, the thickness of the slab and radius of the holes for the bulk PCS without a cavity. Once the bulk parameters are optimized, we introduce the cavities. We then optimize the quality factor by modifying the geometry of the holes neighbouring the cavity [2

2. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003). [CrossRef] [PubMed]

, 7

7. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express 12, 3988–3995 (2004). [CrossRef] [PubMed]

].

3. Results

3.1 PBG calculations

We start with a PCS structure that is infinite in the plane and finite in the vertical direction, surrounded by air. The total quality factor, Q, can be separated into the in-plane value, Q , and out-of-plane value, Q . Q can, in principle, be made arbitrarily high by increasing the number of periods. As the vertical confinement is ruled by TIR, the out-of-plane factor is crucial in designing high quality factor cavities [4

4. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compresses hexagonal lattice photonic crystals,” Opt. Express 11, 579–593 (2003). [CrossRef] [PubMed]

]. As the structure has symmetry in the vertical direction, its modes may be classified as even or odd with respect to the parity of the dominant field component, E x. These modes have different bands and band gaps and the vertical radiation losses can be significantly reduced by choosing modes of a specific parity [5

5. O. Painter and K. Srinivasan, “Localised defect states in two-dimensional photonic crystal slab waveguides: A simple method based upon symmetry analysis,” Phys. Rev. B 68, 035110 (2003). [CrossRef]

]. Even modes were chosen because the first gap appears at lower frequencies than that for the odd modes.

We begin by selecting the properties of the bulk lattice. In Fig. 2, we show reduced band diagrams that carry information needed for the optimization: the width and relative frequency of the gap. The width and fraction of the band gap below the light line are important indicators in the optimization of the cavity, since frequency gaps tend to have high Q , and wide gap tend to have high Q [4

4. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compresses hexagonal lattice photonic crystals,” Opt. Express 11, 579–593 (2003). [CrossRef] [PubMed]

, 24

24. R. K. Lee, O. Painter, B. Kitzke, A. Schrerer, and A. Yariv, “Emission properties of a defect cavity in a two-dimensional photonic bandgap crystal slab,” JOSA B 17, 629–633 (2000). [CrossRef]

]. In order to obtain a large fraction of the PBG below the light line the first gap should occur at the lowest possible frequency.

Fig. 2. Lowest photonic band gap as a function of the relative hole radius of even parity fields for silicon (grey) and diamond (black).

3.2 Quality factors and modal volume

We consider a M1 cavity as described by Zhang et al in Ref. [7

7. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express 12, 3988–3995 (2004). [CrossRef] [PubMed]

]. We use the same parameters, hole radius is R=0.29a, and the thickness of the slab is h=0.7a for silicon and a thicker slab, h=0.91a, is used for diamond. The cavity is surrounded by 11 periods in all directions. Increasing the number of periods to 14 changes the quality factor by less than 2%. This means that the total Q approaches Q . Numerical parameters for the calculations are presented in Table 1. At first we use a computational window of including the entire structure to find the resonances and their symmetries. Then the computational window is reduced eight times using the field symmetry properties. Satisfactory convergence is obtained by using 20 points per period when calculating quality factors up to a few thousand, and 32 points per period for higher quality factors. The perfectly-matched layer width and the height of the computational window are chosen to be quite large as they strongly affect convergence (see Table 1).

Table 1. Numerical parameters in the calculations

table-icon
View This Table

For both materials, FDTD simulations indicate that there is one distinct resonance peak within the lowest band gap of the un-optimized structure consisting of two degenerate dipole-like modes. As expected, the quality factor in diamond Q dia=144 is smaller than the silicon value Q Si=400. However, both values are too small to satisfy QIT requirements [19

19. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A. 73, 013818 (2006). [CrossRef]

].

Fig. 3. (a) Quality factor of the diamond PCS as a function of radius of the holes above and below the cavity with fixed displacement d=0.21a; and (b) electric field amplitude E x of the resonant mode in the center of the slab. The coordinate axes are defined in Fig. 1(a).

The major electric field component, E x, at the centre of the diamond slab is shown in Fig. 3(b). It is symmetric in the x- and y-directions and anti-symmetric in the z-direction as for the corresponding mode of the silicon slab.

It is interesting to notice that the optimum occurs for both materials at the same parameters (displacement in the horizontal direction and radius of the neighbouring holes in the vertical direction). Varying the thickness of the diamond slab from h=0.7a to h=1.1a we find that the optimum does appear at h=0.91a for the given holes radius.

4. Conclusion

Acknowledgments

The authors thank Dr Andrew Greentree for interesting and helpful discussions. This work was produced with the assistance of the Australian Research Council (ARC) under the ARC Centers of Excellence Program. CUDOS (the Centre for Ultrahigh-bandwidth Devices for Optical Systems) is an ARC Centre of Excellence. J. Salzman acknowledges support of the Fund for Promotion of Research at the Technion.

References and links

1.

B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science 300, 1537 (2003). [CrossRef] [PubMed]

2.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003). [CrossRef] [PubMed]

3.

H-G. Park, J-K. Hwang, J. Huh, H-Y Ryu, Y-h. Lee, and J-S. Kim, “Nondegenerate monopole-mode two-dimensional photonic band gap laser,” Appl. Phys. Lett. 79, 3032–3034 (2001). [CrossRef]

4.

K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compresses hexagonal lattice photonic crystals,” Opt. Express 11, 579–593 (2003). [CrossRef] [PubMed]

5.

O. Painter and K. Srinivasan, “Localised defect states in two-dimensional photonic crystal slab waveguides: A simple method based upon symmetry analysis,” Phys. Rev. B 68, 035110 (2003). [CrossRef]

6.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13, 1202–1214 (2005). [CrossRef] [PubMed]

7.

Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express 12, 3988–3995 (2004). [CrossRef] [PubMed]

8.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016608 (2001). [CrossRef]

9.

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, “Optical-fiber-based measurement of an ultrasmall volume high-Q photonic crystal microcavity,” Phys. Rev. B 70, 081306® (2004). [CrossRef]

10.

H. Mabuchi and A. C. Doherty, “Cavity Quantum Electrodynamics: Coherence in Context,” Science 298, 1372–1377 (2002). [CrossRef] [PubMed]

11.

J. Hendrickson, B. C. Richards, J. Sweet, S. Mosor, C. Christenson, D. Lam, G. Khitrova, H. M. Gibbs, T. Yoshie, A. Scherer, O. B. Shchekin, and D. G. Deppe, “Quantum dot photonic-crystal-slab nanocavities: Quality factors and lasing,” Phys. Rev. B 72, 193303 (2005). [CrossRef]

12.

S. Y. Kilin, “Entangled states and nanoojects in quantum optics,” Opt. and Spectr. 94, 709–710 (2003).

13.

E. van Oort, N.B. Manson, and M. Glasbeek, “Optically detected spin coherence of the diamond N-V centre in its triplet ground state,” J. Phys. C 21, 4385–4391 (1988). [CrossRef]

14.

F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett 92, 076401 (2004). [CrossRef] [PubMed]

15.

F. Jelezko, T. Gaebel, I. Popa, M. Domham, A. Gruber, and J. Wrachtrup, , “Observation of oherent oscillation of a single nuclar spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004). [CrossRef] [PubMed]

16.

M. S. Shahriar, P.R. Hemmer, S. Lloyd, P.S. Bhatia, and A. E. Craig, “Solid-state quantum computing using spectral holes,” Phys. Rev. A 66, 032301 (2002). [CrossRef]

17.

Y. L. Lim, A. Beige, and C. Kwek, “Repeat-until-success linear optics distributed quantum computing,” Phys. Rev. Lett 95, 030505 (2005). [CrossRef] [PubMed]

18.

S. D. Barrett and P. Kok, “Efficient high-fidelity quantum computation using matter qubits and linear optics,” Phys. Rev. A 71, 060310 (2005). [CrossRef]

19.

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A. 73, 013818 (2006). [CrossRef]

20.

P. Barclay, O. Painter, and K. Srinivasan, “Nonlinear responseof silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” 13, 801–820 (2005).

21.

J. Salzman, S. Prawer, and D. Jamieson, Photonic crystal devices and systems in diamond, Provisional Patent, CCID 131000480.

22.

D. F. Edwards and H. R. Philipp, Handbook of optical constants of solids, (Academic Press,1985).

23.

M. Qiu, “Micro-cavities in silicon-on-insulator photonic crystal slabs: determing resonant frequencies and quality factor accurately,” Microw. … Opt. Techn. Lett. 45, 381–385 (2005). [CrossRef]

24.

R. K. Lee, O. Painter, B. Kitzke, A. Schrerer, and A. Yariv, “Emission properties of a defect cavity in a two-dimensional photonic bandgap crystal slab,” JOSA B 17, 629–633 (2000). [CrossRef]

25.

H-Y Ryu, M. Notomi, and Y-H Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett 83, 4294–4296 (2003). [CrossRef]

26.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005). [CrossRef]

OCIS Codes
(230.3990) Optical devices : Micro-optical devices
(230.5750) Optical devices : Resonators

ToC Category:
Photonic Crystals

History
Original Manuscript: February 21, 2006
Revised Manuscript: March 17, 2006
Manuscript Accepted: April 12, 2006
Published: April 17, 2006

Citation
S. Tomljenovic-Hanic, M. J. Steel, C. Martijn de Sterke, and J. Salzman, "Diamond based photonic crystal microcavities," Opt. Express 14, 3556-3562 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3556


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References

  1. B. S. Song, and S. Noda, T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300,1537 (2003). [CrossRef] [PubMed]
  2. Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003). [CrossRef] [PubMed]
  3. H-G. Park, J-K. Hwang, J. Huh, H-Y Ryu, Y-h. Lee, J-S. Kim, "Nondegenerate monopole-mode two-dimensional photonic band gap laser," Appl. Phys. Lett. 79, 3032-3034 (2001). [CrossRef]
  4. K. Srinivasan, and O. Painter, "Fourier space design of high-Q cavities in standard and compresses hexagonal lattice photonic crystals," Opt. Express 11, 579-593 (2003). [CrossRef] [PubMed]
  5. O. Painter and K. Srinivasan, "Localised defect states in two-dimensional photonic crystal slab waveguides: A simple method based upon symmetry analysis," Phys. Rev. B 68, 035110 (2003). [CrossRef]
  6. Y. Akahane, T. Asano, B. S. Song, and S. Noda, "Fine-tuned high-Q photonic-crystal nanocavity," Opt. Express 13, 1202-1214 (2005). [CrossRef] [PubMed]
  7. Z. Zhang, and M. Qiu, "Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs," Opt. Express 12, 3988-3995 (2004). [CrossRef] [PubMed]
  8. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608 (2001). [CrossRef]
  9. K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, "Optical-fiber-based measurement of an ultrasmall volume high-Q photonic crystal microcavity," Phys. Rev. B 70, 081306 (2004). [CrossRef]
  10. H. Mabuchi, and A. C. Doherty, "Cavity Quantum Electrodynamics: Coherence in Context," Science 298, 1372 -1377 (2002). [CrossRef] [PubMed]
  11. J. Hendrickson, B. C. Richards, J. Sweet, S. Mosor, C. Christenson, D. Lam, G. Khitrova, H. M. Gibbs, T. Yoshie, A. Scherer, O. B. Shchekin, and D. G. Deppe, "Quantum dot photonic-crystal-slab nanocavities: Quality factors and lasing," Phys. Rev. B 72, 193303 (2005). [CrossRef]
  12. S. Y. Kilin, "Entangled states and nanoojects in quantum optics," Opt. and Spectr. 94, 709-710 (2003).
  13. E. van Oort, N.B. Manson and M. Glasbeek, "Optically detected spin coherence of the diamond N-V centre in its triplet ground state," J. Phys. C 21, 4385-4391 (1988). [CrossRef]
  14. F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, "Observation of coherent oscillations in a single electron spin," Phys. Rev. Lett 92,076401 (2004). [CrossRef] [PubMed]
  15. F. Jelezko, T. Gaebel, I. Popa, M. Domham, A. Gruber, and J. Wrachtrup, "Observation of coherent oscillation of a single nuclar spin and realization of a two-qubit conditional quantum gate," Phys. Rev. Lett. 93, 130501 (2004). [CrossRef] [PubMed]
  16. M. S. Shahriar, P.R. Hemmer, S. Lloyd, P.S. Bhatia, and A. E. Craig, "Solid-state quantum computing using spectral holes," Phys. Rev. A 66,032301 (2002). [CrossRef]
  17. Y. L. Lim, A. Beige, and C. Kwek, "Repeat-until-success linear optics distributed quantum computing," Phys. Rev. Lett 95, 030505 (2005). [CrossRef] [PubMed]
  18. S. D. Barrett, and P. Kok, "Efficient high-fidelity quantum computation using matter qubits and linear optics," Phys. Rev. A 71, 060310 (2005). [CrossRef]
  19. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, "Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms," Phys. Rev. A. 73, 013818 (2006). [CrossRef]
  20. P. Barclay, O. Painter and K. Srinivasan, "Nonlinear responseof silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper," Opt. Express 13, 801-820 (2005).
  21. J. Salzman, S. Prawer, and D. Jamieson, Photonic crystal devices and systems in diamond, Provisional Patent, CCID 131000480.
  22. D. F. Edwards, and H. R. Philipp, Handbook of optical constants of solids, (Academic Press, 1985).
  23. M. Qiu, "Micro-cavities in silicon-on-insulator photonic crystal slabs: determing resonant frequencies and quality factor accurately," Microw. & Opt. Techn. Lett. 45, 381-385 (2005). [CrossRef]
  24. R. K. Lee, O. Painter, B. Kitzke, A. Schrerer, and A. Yariv, "Emission properties of a defect cavity in a two-dimensional photonic bandgap crystal slab," JOSA B 17, 629-633 (2000). [CrossRef]
  25. H-Y Ryu, M. Notomi, and Y-H Lee, "High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities," Appl. Phys. Lett 83, 4294-4296 (2003). [CrossRef]
  26. B. S. Song, S. Noda, T. Asano and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nature Mater. 4, 207-210 (2005). [CrossRef]

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