Abstract

The quantum coupling of spatially distant spins via optical photons using cavity quantum electrodynamic (cQED) methods has proved experimentally challenging due to the large spin-photon coupling strengths required. To achieve such coupling strengths using traditional cQED methods requires either individual spins and ultra-small cavities or an ensemble of identical spins coupled to larger cavities. In this work we describe a method to couple distant spins via the collective enhanced coupling to a large ensemble ∼ N, of degenerate optical Whispering Gallery Modes (WGM) in a spherical resonator where the spins are spatially located at the antipodes. The setup can be scaled-up to build 1D, 2D and 3D cQED lattices to enable quantum simulation or computing.

© 2015 Optical Society of America

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J. Michl, T. Teraji, S. Zaiser, I. Jakobi, G. Waldherr, F. Dolde, P. Neumann, M. W. Doherty, N. B. Manson, J. Isoya, and J. Wrachtrup, “Perfect alignment and preferential orientation of nitrogen-vacancy centers during chemical vapor deposition diamond growth on (111) surfaces,” Appl. Phys. Lett. 104, 102407 (2014).
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2013 (7)

C. Junge, D. OShea, J. Volz, and A. Rauschenbeutel, “Strong coupling between single atoms and nontransversal photons,” Phys. Rev. Lett. 110, 213604 (2013).
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C. Phelan, T. Hennessy, and T. Busch, “Shaping the evanescent field of optical nanofibers for cold atom trapping,” Opt. Express 21, 27093 (2013).
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H.-Q. Zhao, M. Fujiwara, M. Okano, and S. Takeuchi, “Observation of 1.2-GHz linewidth of zero-phonon-line in photoluminescence spectra of nitrogen vacancy centers in nanodiamonds using a fabry-perot interferometer,” Opt. Express 21, 29679–29686 (2013).
[Crossref]

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref] [PubMed]

W. Chen, K. M. Beck, R. Bucker, M. Gullans, M. D. Lukin, H. Tanji-Suzuki, and V. Vuletic, “All-optical switch and transistor gated by one stored photon,” Science 341, 768–770 (2013).
[Crossref] [PubMed]

K. Nemoto, M. Trupke, S. J. Devitt, A. M. Stephens, K. Buczak, T. Nöbauer, M. S. Everitt, J. Schmiedmayer, and W. J. Munro, “Photonic architecture for scalable quantum information processing in NV-diamond,” Phys. Rev. X. 4, 031022 (2013).

A. Wickenbrock, M. Hemmerling, G. R. M. Robb, C. Emary, and F. Renzoni, “Collective strong coupling in multimode cavity QED,” Phys. Rev. A. 87, 043817 (2013).
[Crossref]

2012 (4)

X.-C. Yu, Y.-C. Liu, M.-Y. Yan, W.-L. Jin, and Y.-F. Xiao, “Coupling of diamond nanocrystals to a high-q whispering-gallery microresonator,” Phys. Rev. A. 86, 043833 (2012).
[Crossref]

Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, and Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A. 85, 031805 (2012).
[Crossref]

W. J. Munro, A. M. Stephens, S. J. Devitt, K. A. Harrison, and K. Nemoto, “Quantum communication without the necessity of quantum memories,” Nat. Photon. 6, 777–781 (2012).
[Crossref]

S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref] [PubMed]

2011 (2)

A. Faraon, P. E. Barclay, C. Santori, K.-M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301–305 (2011).
[Crossref]

J. T. Rubin and L. Deych, “On optical forces in spherical whispering gallery mode resonators,” Opt. Express 19, 22337 (2011).
[Crossref] [PubMed]

2010 (3)

W. J. Munro, K. A. Harrison, A. M. Stephens, S. J. Devitt, and K. Nemoto, “From quantum multiplexing to high-performance quantum networking,” Nat. Photon. 4, 792–796 (2010).
[Crossref]

O. Tsyplyatyev and D. Loss, “Classical and quantum regimes of the inhomogeneous dicke model and its ehrenfest time,” Phys. Rev. B 82, 024305 (2010).
[Crossref]

D. Englund, B. Shields, K. Rivoire, F. Hatami, J. Vuckovic, H. Park, and M. D. Lukin, “Deterministic coupling of a single nitrogen vacancy center to a photonic crystal cavity,” Nano Lett. 10, 3922–3926 (2010).
[Crossref] [PubMed]

2009 (3)

S. Schietinger and O. Benson, “Coupling single NV-centres to high-q whispering gallery modes of a preselected frequency-matched microresonator,” J. Phys. B. 42, 114001 (2009).
[Crossref]

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

S. J. Devitt, A. G. Fowler, A. M. Stephens, A. D. Greentree, L. C. L. Hollenberg, W. J. Munro, and K. Nemoto, “Architectural design for a topological cluster state quantum computer,” New J. Phys. 11, 083032 (2009).
[Crossref]

2008 (5)

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref] [PubMed]

A. Stephens, Z. Evans, S. Devitt, A. Greentree, A. Fowler, W. Munro, J. O’Brien, K. Nemoto, and L. Hollenberg, “Deterministic optical quantum computer using photonic modules,” Phys. Rev. A. 78, 032318 (2008).
[Crossref]

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

Y.-C. Liu, Y.-F. Xiao, B.-B. Li, X.-F. Jiang, Y. Li, and Q. Gong, “Coupling of a single diamond nanocrystal to a whispering-gallery microcavity: Photon transport benefitting from Rayleigh scattering,” Science 319, 1062–1065 (2008).

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Quantum many-body phenomena in coupled cavity arrays,” Laser Photonics Rev. 2, 527–556 (2008).
[Crossref]

2007 (6)

D. Rossini and R. Fazio, “Mott-insulating and glassy phases of polaritons in 1D arrays of coupled cavities,” Phys. Rev. Lett. 99, 186401 (2007).
[Crossref] [PubMed]

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

K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdiskquantum dot system,” Nature 450, 862–865 (2007).
[Crossref] [PubMed]

V. V. Klimov, V. S. Letokhov, and M. Ducloy, “Quantum theory of radiation of an excited atom placed near a microresonator containing a single-photon wavepacket: Photon correlation properties,” Laser Physics 17, 912– 926 (2007).
[Crossref]

A. Mazzei, S. Gtzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single rayleigh scatterer: A classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
[Crossref] [PubMed]

R. Ma, A. Schliesser, P. Del’Haye, A. Dabirian, G. Anetsberger, and T. J. Kippenberg, “Radiation-pressure-driven vibrational modes in ultrahigh-q silica microspheres,” Opt. Lett. 32, 2200 (2007).
[Crossref] [PubMed]

2006 (6)

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6, 557–561 (2006).
[Crossref] [PubMed]

S. Götzinger, L. de S Menezes, A. Mazzei, S. Kühn, V. Sandoghdar, and O. Benson, “Controlled photon transfer between two individual nanoemitters via shared high-q modes of a microsphere resonator,” Nano Lett. 6, 1151– 1154 (2006).
[Crossref] [PubMed]

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

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

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

Y.-S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006).
[Crossref] [PubMed]

2005 (1)

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A. 71, 013817 (2005).
[Crossref]

2002 (1)

A. N. Oraevsky, “Whispering-gallery waves,” Quantum Electron. 32, 377 (2002).
[Crossref]

1998 (1)

D. W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-q whispering gallery modes,” Phys. Rev. A. 57, 2293–2296 (1998).
[Crossref]

1996 (1)

1995 (1)

T. Junno, K. Deppert, L. Montelius, and L. Samuelson, “Controlled manipulation of nanoparticles with an atomic force microscope,” Appl. Phys. Lett. 66, 3627 (1995).
[Crossref]

1993 (1)

1992 (1)

J. Dalibard, Y. Castin, and K. Mölmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett. 68, 580–583 (1992).
[Crossref] [PubMed]

1989 (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A. 137, 393–397 (1989).
[Crossref]

1987 (1)

M. A. Morrison and G. A. Parker, “A guide to rotations in quantum mechanics,” Aust. J. Phys. 40, 465–497 (1987).
[Crossref]

1968 (1)

M. Tavis and F. W. Cummings, “Exact solution for an n-moleculeradiation-field hamiltonian,” Phys. Rev. 170, 379–384 (1968).
[Crossref]

Akimov, A. V.

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref] [PubMed]

Anetsberger, G.

Aoki, T.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

Artemyev, M. V.

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6, 557–561 (2006).
[Crossref] [PubMed]

Atature, M.

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

Badolato, A.

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

Banin, U.

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Lett. 6, 557–561 (2006).
[Crossref] [PubMed]

Barclay, P. E.

A. Faraon, P. E. Barclay, C. Santori, K.-M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301–305 (2011).
[Crossref]

Beausoleil, R. G.

A. Faraon, P. E. Barclay, C. Santori, K.-M. C. Fu, and R. G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity,” Nat. Photon. 5, 301–305 (2011).
[Crossref]

Beck, K. M.

W. Chen, K. M. Beck, R. Bucker, M. Gullans, M. D. Lukin, H. Tanji-Suzuki, and V. Vuletic, “All-optical switch and transistor gated by one stored photon,” Science 341, 768–770 (2013).
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Figures (8)

Fig. 1
Fig. 1

(a) A spherical resonator supporting many degenerate rotated WGMs (green tubes) coupled to two antipodal spins (red spheres) and (b) a depiction of the extension of the model into one and (c) two dimensional arrays.

Fig. 2
Fig. 2

The norm of the TM and TE field components for a 32.72 μm sphere with ns = 1.46 suspended in air which supports a WGM of wavelength λ = 637 nm. A special radial position is depicted, ‘Transversal Point’, where the azimuthal component of the TM mode electric field is zero and the WGM is completely transversal.

Fig. 3
Fig. 3

Depiction of the rotations which are performed to obtain expressions for the rotated WGMs. First the fundamental WGM, which lays in the xy plane, (green tube) is rotated about the x-axis by π 2 (red tube). The mode now intersects the spin (black sphere) which is located on the z-axis. Next, rotations about the z-axis by angle ηi are performed which generate the ith rotated WGM of the ensemble (blue tube).

Fig. 4
Fig. 4

The north pole of the spherical resonator where a single spin (red sphere) is located. The energy level diagram of the spin is presented where the emission of a π transition into a super position of σ+ and σ circularly polarised light is depicted. The two circular polarization correspond to two counter propagating fundamental WGMs.

Fig. 5
Fig. 5

(a) The latitudinal variation of the TM WGM field intensity and (b) of the polarisation about the maximum intensity. The calculations were performed using a 32.72 μm fused-silica microsphere supporting a WGM of wavelength 637 nm. (c) The enhanced coupling rate as a function of microsphere radius.

Fig. 6
Fig. 6

Ultra-strong coupling of a single spin to the collective optical modes as a function of the microsphere radius (a) n0 photon saturation number; (b) L denotes the visibility of the vacuume Rabi splitting; (c) P Purcell factor; (d) C Cooperativity. For strong coupling we require gE > κ, γ; P ≫ 1; L ≫ 1; n0 ≪ 1; C ≫ 1. The calculations were performed using a fused-silica microsphere with γ = 2π × 200 MHz at cryogenic temperatures.

Fig. 7
Fig. 7

Simulations of the total spin occupation probability, Tr [ ( i N σ ^ z i ) ρ ^ ], for two antipodal spins coupled to a single WGM (green curve) and an ensemble of WGMs in a fused-silica resonator with homogeneous/inhomogeneous coupling (black/red curves) and two clusters of 2070 antipodal spins within a diamond resonator (blue curve). The simulations for the silica resonator were performed using g = 2π × 250 MHz, κ = 2π × 157 kHz, γ = 2π × 200 MHz and a 2% random coupling inhomogeneity. For the diamond resonator g = 2π × 334 MHz, κ = 2π × 109 kHz and homogeneous coupling was considered.

Fig. 8
Fig. 8

The Rabi-oscillations of the mode in the JC, TC, MM and MMTC models. These simulations were performed with ΔD = 0 in units of g under ideal conditions (κi = γi = 0). The simulations of the TC/MM models were performed using 9 identical spins/modes and the initial states | ψ ( 0 ) TC = 1 3 i = 1 9 | 0 , e i and |ψ(0)〉MM = |0, e〉 respectively. In the MMTC simulation 3 identical spins and 3 identical modes were used with the initial state | ψ ( 0 ) MMTC = 1 3 i = 1 3 | 0 | e i .

Equations (39)

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H ^ JC = ω c a ^ a + 2 ω a σ ^ z + g ( σ ^ + a ^ + σ ^ a ^ ) ,
H ^ = ω c a ^ a ^ + 2 i = 1 N ω i σ ^ z i + i = 1 N g i ( σ ^ + i a ^ + σ ^ i a ^ ) .
H ^ TC I = g ¯ N ( Σ ^ + a ^ + Σ ^ a ^ ) ,
H ^ M M = i = 1 N ω i a ^ i a ^ i + 2 ω a σ ^ z + i = 1 N g i ( σ ^ + a ^ i + σ ^ a ^ i ) ,
H ^ C = i = 1 N ( ω i 1 2 i κ i ) a ^ i a ^ i + 2 ω a σ ^ z 2 i γ | e e | + H ^ M M I .
Δ ω ω ± s 2 ( l 2 m 2 ) 4 l 2 ,
Q S 1 Q S S 1 + Q Abs 1 8 π 2 3 σ 2 ζ 2 λ 4 l 1 / 3 + λ 2 π N α 4.3 × 10 3 ,
Q NP = 2 π n s V M M λ σ NP ,
σ NP = 8 π 3 k 4 r NP 6 ( s 2 1 s 2 + 2 ) 2 ,
| ψ ( η i ) = m = l l | l , m D m , l l ( η i , π 2 , 0 ) .
| ψ ( η i ) | ψ ( η j ) | 2 1 2 4 l | e i η i + e i η j | 4 l ,
N 2 π W 1 2 = π l max 2 log 2 ,
g = d ^ E ^ = μ ξ ω 2 0 V M | E | E max d ^ e ^ ,
H ^ I = g 2 odd i N ( σ ^ + ( + 1 ) a ^ i + σ ^ ( + 1 ) a ^ i ) + g 2 even i N ( σ ^ + ( 1 ) a ^ i + σ ^ ( 1 ) a ^ i ) ,
M 2 = λ 3 d 3 2070 ,
H ^ MMTC = i = 1 N ω i a ^ i a ^ i + 1 2 i = 1 M Ω i σ ^ z i + j = i M i = 1 N g i j ( σ ^ + j a ^ i + σ ^ j a ^ i ) .
ρ ^ ˙ = i [ H ^ MMTC , ρ ^ ] + k = 1 M γ k [ σ ^ k ρ ^ σ ^ + k 1 2 [ σ ^ + k σ ^ k , ρ ^ } ] + i = 1 N κ i [ a ^ i ρ ^ a ^ i + 1 2 { a ^ i a ^ i , ρ ^ } ] ,
H ^ C = j = 1 N ( ω j i 2 κ j ) a ^ j a ^ j + 2 j = 1 M ( Ω j σ ^ z j i γ j σ ^ e e j ) + j = 1 N k = 1 N g i j ( σ ^ + j a ^ k + σ ^ j a ^ k ) ,
| 0 | e k | 0 1 , , 0 N | g 1 , g 2 , , g k 1 , e k , g k + 1 , , g M ,
| 1 k | g | 0 1 , 0 2 , , 0 k 1 , 1 k , 0 k + 1 , , 0 N | g 1 , , g M ,
| 0 | g | 0 1 , , 0 N | g 1 , , g M .
| k | 1 k | g ,
| N + k | 0 | e k ,
| 0 | 0 | g .
ρ ^ ˙ = i [ H ^ C ρ ^ ρ ^ H ^ C ] + j = 1 N κ j a ^ j ρ ^ a j + j = 1 M γ j σ ^ j ρ ^ σ ^ + j = ^ C ρ ^ + 𝒥 ^ ρ ^ ,
^ C ρ ^ = i [ H ^ C ρ ^ ρ ^ H ^ C ] and 𝒥 ^ ρ ^ = j = 1 N 𝒥 ^ j ρ ^ ,
𝒥 ^ j ρ ^ = κ j a ^ j ρ ^ a ^ j and 𝒥 ^ N + j ρ ^ = γ j σ ^ j ρ ^ σ ^ + j .
ρ ^ ( 0 ) = j 1 , j 2 = 1 N + M ρ j 1 j 2 | j 1 j 2 | .
ρ ^ ( t ) = e ^ C t ρ ^ ( 0 ) = j 1 , j 2 = 1 N + M ρ j 1 j 2 ( t ) | j 1 j 2 | ,
𝒥 ^ ρ ^ = ( j = 1 N + M ρ i j Γ j ) | 0 0 | ,
Γ j = { κ j for j N , γ j for N < j N + M ,
ρ ^ ( t ) = e ^ C t ρ ^ ( 0 ) + 0 t d t 1 e ^ C ( t t 1 ) 𝒥 ^ e ^ C t 1 ρ ^ ( 0 ) + 0 t d t 2 0 t 2 d t 1 e ^ C ( t t 1 ) 𝒥 ^ e ^ C ( t 2 t 1 ) 𝒥 ^ e ^ C t 1 ρ ^ ( 0 ) +
ρ ^ ( t ) = e ^ C t ρ ^ ( 0 ) + 0 t d t 1 e ^ C ( t t 1 ) 𝒥 ^ e ^ C t 1 ρ ^ ( 0 ) ,
ρ ^ ( t ) = e ^ C t ρ ^ ( 0 ) + | 0 0 | 0 t j = 1 N + M Γ j j | e ^ C t 1 ρ ^ ( 0 ) | j .
ρ ^ ( t ) = e ^ C t ρ ^ ( 0 ) + | 0 0 | ( 1 Tr [ e ^ C t ρ ^ ( 0 ) ] ) ,
H ^ C I = j = 1 M ( Ω j i γ j 2 ω ¯ ) σ ^ g g j i 2 j = 1 N κ j a ^ j a ^ j + j = 1 M k = 1 N g k j ( σ ^ + j a ^ k + σ ^ j a ^ k ) ,
| ψ ( t ) = k = 1 N α k ( t ) | 0 | e k + k = 1 N C k ( t ) | 1 k | g ,
i α ˙ k ( t ) = j = 1 M ( Ω j i γ j 2 ω ¯ ) α k ( t ) + ( Ω k i γ k 2 ω ¯ ) α k ( t ) + j = 1 N g j k C j ( t ) ,
i C ˙ k ( t ) = j = 1 M ( Ω j i γ j 2 ω ¯ ) C k ( t ) i κ k 2 C k ( t ) + j = 1 M g k j α j ( t ) ,

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