Abstract

By offering effective modal volumes significantly less than a cubic wavelength, slot-waveguide cavities offer a new in-road into strong atom-photon coupling in the visible regime. Here we explore two-dimensional arrays of coupled slot cavities which underpin designs for novel quantum emulators and polaritonic quantum phase transition devices. Specifically, we investigate the lateral coupling characteristics of diamond-air and GaP-air slot waveguides using numerically-assisted coupled-mode theory, and the longitudinal coupling properties via distributed Bragg reflectors using mode-propagation simulations. We find that slot-waveguide cavities in the Fabry-Perot arrangement can be coupled and effectively treated with a tight-binding description, and are a suitable platform for realizing Jaynes-Cummings-Hubbard physics.

© 2011 OSA

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J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED based photon lattices,” Phys. Rev. A 82(4), 043811 (2010).
[CrossRef]

M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. 10(3), 891–895 (2010).
[CrossRef] [PubMed]

2009

2008

A. D. Greentree, B. A. Fairchild, F. Hossain, and S. Prawer, “Diamond integrated quantum photonics,” Mater. Today 11(9), 22–31 (2008).
[CrossRef]

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[CrossRef]

K.-M. C. Fu, C. Santori, P. E. Barclay, I. Aharonovich, S. Prawer, N. Meyer, A. M. Holm, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers in diamond to a GaP waveguide,” Appl. Phys. Lett. 93(23), 234107 (2008).
[CrossRef]

J. Mu, H. Zhang, and W.-P. Huang, “A theoretical investigation of slot waveguide Bragg gratings,” IEEE J. Quantum Electron. 44(7), 622–627 (2008).
[CrossRef]

C.-H. Su, A. D. Greentree, W. J. Munro, K. Nemoto, and L. C. L. Hollenberg, “High-speed quantum gates with cavity quantum electrodynamics,” Phys. Rev. A 78(6), 062336 (2008).
[CrossRef]

A. Gondarenko and M. Lipson, “Low modal volume dipole-like dielectric slab resonator,” Opt. Express 16(22), 17689–17694 (2008).
[CrossRef] [PubMed]

M. P. Hiscocks, K. Ganesan, B. C. Gibson, S. T. Huntington, F. Ladouceur, and S. Prawer, “Diamond waveguides fabricated by reactive ion etching,” Opt. Express 16(24), 19512–19519 (2008).
[CrossRef] [PubMed]

2007

M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “Integrated waveguide Fabry-Perot microcavities with silicon/air Bragg mirrors,” Opt. Lett. 32(5), 533–535 (2007).
[CrossRef] [PubMed]

F. Dell’Olio and V. M. N. Passaro, “Optical sensing by optimized silicon slot waveguides,” Opt. Express 15(8), 4977–4993 (2007).
[CrossRef] [PubMed]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76(3), 031805 (2007).
[CrossRef]

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007).
[CrossRef]

2006

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2(12), 849–855 (2006).
[CrossRef]

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2(12), 856–861 (2006).
[CrossRef]

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2(6), 408–413 (2006).
[CrossRef]

Y. O. Barmenkov, D. Zalvidea, S. Torres-Peiró, J. L. Cruz, and M. V. Andrés, “Effective length of short Fabry-Perot cavity formed by uniform fiber Bragg gratings,” Opt. Express 14(14), 6394–6399 (2006).
[CrossRef] [PubMed]

2005

C. A. Barrios and M. Lipson, “Electrically driven silicon resonant light emitting device based on slot-waveguide,” Opt. Express 13(25), 10092–10101 (2005).
[CrossRef] [PubMed]

J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95(14), 143901 (2005).
[CrossRef] [PubMed]

K.-M. C. Fu, C. Santori, C. Stanley, M. C. Holland, and Y. Yamamoto, “Coherent population trapping of electron spins in a high-purity n-type GaAs semiconductor,” Phys. Rev. Lett. 95(18), 187405 (2005).
[CrossRef] [PubMed]

2004

2002

A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J.-P. Poizat, and P. Grangier, “Single photon quantum cryptography,” Phys. Rev. Lett. 89(18), 187901 (2002).
[CrossRef] [PubMed]

2000

C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, “Stable solid-state source of single photons,” Phys. Rev. Lett. 85(2), 290–293 (2000).
[CrossRef] [PubMed]

1999

1994

1984

1983

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).
[CrossRef]

1982

R. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21(6-7), 467–488 (1982).
[CrossRef]

Aharonovich, I.

K.-M. C. Fu, C. Santori, P. E. Barclay, I. Aharonovich, S. Prawer, N. Meyer, A. M. Holm, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers in diamond to a GaP waveguide,” Appl. Phys. Lett. 93(23), 234107 (2008).
[CrossRef]

Aichele, T.

M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. 10(3), 891–895 (2010).
[CrossRef] [PubMed]

Alasaarela, T.

Almeida, V. R.

Andrés, M. V.

Angelakis, D. G.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76(3), 031805 (2007).
[CrossRef]

Asano, T.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).
[CrossRef]

Atatüre, M.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

Badolato, A.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

Barclay, P. E.

K.-M. C. Fu, C. Santori, P. E. Barclay, I. Aharonovich, S. Prawer, N. Meyer, A. M. Holm, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers in diamond to a GaP waveguide,” Appl. Phys. Lett. 93(23), 234107 (2008).
[CrossRef]

Barmenkov, Y. O.

Barrios, C. A.

Barth, M.

M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. 10(3), 891–895 (2010).
[CrossRef] [PubMed]

Beausoleil, R. G.

K.-M. C. Fu, C. Santori, P. E. Barclay, I. Aharonovich, S. Prawer, N. Meyer, A. M. Holm, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers in diamond to a GaP waveguide,” Appl. Phys. Lett. 93(23), 234107 (2008).
[CrossRef]

Becker, J.

M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. 10(3), 891–895 (2010).
[CrossRef] [PubMed]

Benson, O.

M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. 10(3), 891–895 (2010).
[CrossRef] [PubMed]

Beveratos, A.

A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J.-P. Poizat, and P. Grangier, “Single photon quantum cryptography,” Phys. Rev. Lett. 89(18), 187901 (2002).
[CrossRef] [PubMed]

Bose, S.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76(3), 031805 (2007).
[CrossRef]

Brandão, F. G. S. L.

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2(12), 849–855 (2006).
[CrossRef]

Brouri, R.

A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J.-P. Poizat, and P. Grangier, “Single photon quantum cryptography,” Phys. Rev. Lett. 89(18), 187901 (2002).
[CrossRef] [PubMed]

Buluta, I.

I. Buluta and F. Nori, “Quantum simulators,” Science 326(5949), 108–111 (2009).
[CrossRef] [PubMed]

Chen, L.

J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. 95(14), 143901 (2005).
[CrossRef] [PubMed]

Cole, J. H.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2(12), 856–861 (2006).
[CrossRef]

Cooper, M. L.

Cruz, J. L.

Dell’Olio, F.

Domhan, M.

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2(6), 408–413 (2006).
[CrossRef]

F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. 93(13), 130501 (2004).
[CrossRef] [PubMed]

Fairchild, B. A.

A. D. Greentree, B. A. Fairchild, F. Hossain, and S. Prawer, “Diamond integrated quantum photonics,” Mater. Today 11(9), 22–31 (2008).
[CrossRef]

Fält, S.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

Feynman, R.

R. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21(6-7), 467–488 (1982).
[CrossRef]

Fischer, S.

M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. 10(3), 891–895 (2010).
[CrossRef] [PubMed]

Fu, K.-M. C.

K.-M. C. Fu, C. Santori, P. E. Barclay, I. Aharonovich, S. Prawer, N. Meyer, A. M. Holm, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers in diamond to a GaP waveguide,” Appl. Phys. Lett. 93(23), 234107 (2008).
[CrossRef]

K.-M. C. Fu, C. Santori, C. Stanley, M. C. Holland, and Y. Yamamoto, “Coherent population trapping of electron spins in a high-purity n-type GaAs semiconductor,” Phys. Rev. Lett. 95(18), 187405 (2005).
[CrossRef] [PubMed]

Fujita, M.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007).
[CrossRef]

Gacoin, T.

A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J.-P. Poizat, and P. Grangier, “Single photon quantum cryptography,” Phys. Rev. Lett. 89(18), 187901 (2002).
[CrossRef] [PubMed]

Gaebel, T.

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2(6), 408–413 (2006).
[CrossRef]

F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. 93(13), 130501 (2004).
[CrossRef] [PubMed]

Ganesan, K.

Gerace, D.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[CrossRef] [PubMed]

Gibson, B. C.

Girvin, S. M.

J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED based photon lattices,” Phys. Rev. A 82(4), 043811 (2010).
[CrossRef]

Gondarenko, A.

Grangier, P.

A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J.-P. Poizat, and P. Grangier, “Single photon quantum cryptography,” Phys. Rev. Lett. 89(18), 187901 (2002).
[CrossRef] [PubMed]

Greentree, A. D.

M. P. Hiscocks, C.-H. Su, B. C. Gibson, A. D. Greentree, L. C. L. Hollenberg, and F. Ladouceur, “Slot-waveguide cavities for optical quantum information applications,” Opt. Express 17(9), 7295–7303 (2009).
[CrossRef] [PubMed]

C.-H. Su, A. D. Greentree, W. J. Munro, K. Nemoto, and L. C. L. Hollenberg, “Pulse shaping by coupled cavities: single photons and qudits,” Phys. Rev. A 80(3), 033811 (2009).
[CrossRef]

A. D. Greentree, B. A. Fairchild, F. Hossain, and S. Prawer, “Diamond integrated quantum photonics,” Mater. Today 11(9), 22–31 (2008).
[CrossRef]

C.-H. Su, A. D. Greentree, W. J. Munro, K. Nemoto, and L. C. L. Hollenberg, “High-speed quantum gates with cavity quantum electrodynamics,” Phys. Rev. A 78(6), 062336 (2008).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematics of 2D coupled slot-waveguide cavity (SWC) arrays, where the lateral light transfer along x direction is realized through (a) a cladding-separated arrangement or (b) shared-rod arrangement. Their cross-sections in the xy-plane are depicted (c,d). (e) In both cases, the longitudinal coupling along z direction is realized by partial transmission at the distributed Bragg reflecting (DBR) boundary. Red arrows indicate nearest-neighbor couplings between cavities.

Fig. 2
Fig. 2

Ex -field distribution of the TE-supermodes of a coupled diamond-air slot-waveguide system with dimensions {wS , wR , h} = {20, 140, 110} nm for (a,b) wG = 200 nm or d = 500 μm) and (c,d) 1 μm (d = 1.3 μm). Strong E-field localization is found within the slots. The even supermodes are shown in (a,c) and the odd in (b,d). The predicted coupling strength between the waveguides is shown in Fig. 3.

Fig. 3
Fig. 3

Lateral coupling rate JL between two slot waveguides in cladding-separated configuration, plotted as a function of centre-to-centre separation between the waveguides. Circles (Crosses) denote the calculated results for coupled diamond-air (GaP-air) slots. The coupling strength exponentially decreases with the separation.

Fig. 4
Fig. 4

Cross-sectional Ex -field distribution (black curves) of the TE-modes of a five coupled diamond-air slot-waveguides (blue) with separated by a 200 nm wide cladding (i.e. d = 500 nm). The M-matrix for this structure is given in Eq. (6). The SWC dimensions follow Fig. 2(a).

Fig. 5
Fig. 5

(a) Lateral coupling rate JL between two different slot waveguides in shared-rod configuration. (b–f) E-field distribution of the five supermodes of a 5-waveguide array with separation d of 220 nm. The M-matrix for this structure is given in Eq. (7).

Fig. 6
Fig. 6

(a) Schematic for a distributed Bragg reflector with 4.5 periods that separates two SWCs. The grating alternates between solid rectangular guide and slot regions. The respective E-field distributions of their fundamental modes are shown in (b) and (c).

Fig. 7
Fig. 7

(a) Reflection coefficient R of a diamond-air DBR [Fig. 6(a)] appended to a diamond-air slot waveguide. The coefficient is associated to its fundamental TE-mode and is calculated for different grating periods P and number of periods Np . The maximal reflectivity is observed at P = 220 nm and these specifications are selected for (b) plotting longitudinal end-to-end coupling rate JE . In both figures, markers indicate simulation data points, and the lines are guides and are used to predict the coupling rate at larger Np .

Equations (10)

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E ( x , y , z ) = m = 1 N A m E m ( x , y )   exp ( i χ z ) x ^ .
M m n = { β m 2 + κ m n ( β n 2 χ 2 ) I m n + κ m n ,   m = n ,   m n ,
κ m n = k 2 l = 1 , l n N [ n l ( x , y ) 2 n c l 2 ] E m ( x , y ) E n ( x , y ) d x d y
M = AX A 1
J L = 2 c κ m n n eff k
M = ( 0.1574 2.6420 0.1775 0.0196 2.6284 0.3032 2.6262 0.1756 0.1754 2.6267 0.3046 2.6263 0.0192 0.1761 2.6269 0.3041      0.0025 0.0191 0.1757 2.6276    0.0026    0.0196   0.1780    2.6427     0.1560 ) + β 2 I ,
M = ( 241.578 21.076 0.820 2.350 27 .712 228.497 29 .854 5 .124 5.026 30.684 226 .753 30 .678 1.184 5.198 29 .803 228 .677      1.309 1 .168 5 .102 27 .676    1.201    2.110    0.429     21.461    241.212 ) .
L eff = L gr R 2 tanh 1 ( R )
V = n ( r ) 2 | E ( r ) | 2 d 2 r n ( r max ) 2 | E ( r max ) | 2 0 L ^ c sin 2 ( 2 π z / L ^ c ) d z
J E / 2 π = ln R τ .

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