Abstract

We propose a hybrid architecture for quantum information processing based on magnetically trapped ultracold atoms coupled via optical fields. The ultracold atoms, which can be either Bose-Einstein condensates or ensembles, are trapped in permanent magnetic traps and are placed in microcavities, connected by silica based waveguides on an atom chip structure. At each trapping center, the ultracold atoms form spin coherent states, serving as a quantum memory. An all-optical scheme is used to initialize, measure and perform a universal set of quantum gates on the single and two spin-coherent states where entanglement can be generated addressably between spatially separated trapped ultracold atoms. This allows for universal quantum operations on the spin coherent state quantum memories. We give detailed derivations of the composite cavity system mediated by a silica waveguide as well as the control scheme. Estimates for the necessary experimental conditions for a working hybrid device are given.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. D. Press, T. Ladd, B. Zhang, Y. Yamamoto, “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature 456, 218–221 (2008).
    [CrossRef] [PubMed]
  31. J. Matthews, A. Politi, A. Stefanov, J. OBrien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
    [CrossRef]
  32. M. Malak, N. Gaber, F. Marty, N. Pavy, E. Richalot, T. Bourouina, “Analysis of Fabry-Pérot optical micro-cavities based on coating-free all-Silicon cylindrical Bragg reflectors,” Opt. Express 21, 2378–2392 (2013).
    [CrossRef] [PubMed]
  33. M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250μm-gap silicon Fabry-Perot cavities,” J. Microelectromech. Syst. 21(1), 171–180 (2012).
    [CrossRef]
  34. M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, T. Bourouina, “Micromachined Fabry-Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett. 98(21), 211113 (2011).
    [CrossRef]
  35. S. Nolte, M. Will, J. Burghoff, A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A 77, 109–111 (2003).
    [CrossRef]
  36. G. Lepert, M. Trupke, E. Hinds, H. Rogers, J. Gates, P. Smith, “Demonstration of UV-written waveguides, Bragg gratings and cavities at 780 nm, and an original experimental measurement of group delay,” Opt. Express 19, 24933–24943 (2011).
    [CrossRef]
  37. J. Lepert, M. Trupke, M. Hartmann, M. Plenio, E. Hinds, “Arrays of waveguide-coupled optical cavities that interact strongly with atoms,” New J. Phys. 13, 113002 (2011).
    [CrossRef]
  38. M. Hijlkema, B. Weber, H. Specht, S. Webster, A. Kuhn, G. Rempe, “A single-photon server with just one atom,” Nature 3, 253–255 (2007).

2013 (3)

A. Pyrkov, T. Byrnes, “Entanglement generation in quantum networks of Bose-Einstein condensates,” New J. Phys. 15093019 (2013).
[CrossRef]

T. Byrnes, “Fractality and macroscopic entanglement in two-component Bose-Einstein condensates,” Phys. Rev. A 88, 023609 (2013).
[CrossRef]

M. Malak, N. Gaber, F. Marty, N. Pavy, E. Richalot, T. Bourouina, “Analysis of Fabry-Pérot optical micro-cavities based on coating-free all-Silicon cylindrical Bragg reflectors,” Opt. Express 21, 2378–2392 (2013).
[CrossRef] [PubMed]

2012 (6)

M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250μm-gap silicon Fabry-Perot cavities,” J. Microelectromech. Syst. 21(1), 171–180 (2012).
[CrossRef]

T. Byrnes, “Quantum computation using two component Bose-Einstein condensates,” World Acad. Sci. Eng. Technol. 63, 542 (2012).

G. Wilpers, P. See, P. Gill, A. Sinclair, “A monolithic array of three-dimensional ion traps fabricated with conventional semiconductor technology,” Nat. Nanotechnol. 7, 572–576 (2012).
[CrossRef] [PubMed]

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

C. Eichler, C. Lang, J. Fink, J. Govenius, S. Filipp, A. Wallraff, “Observation of entanglement between itinerant microwave photons and a superconducting qubit,” Phys. Rev. Lett. 109, 240501 (2012).
[CrossRef]

T. Byrnes, K. Wen, Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012).
[CrossRef]

2011 (5)

M. Kohnen, M. Succo, P. Petrov, R. Nyman, M. Trupke, E. Hinds, “An array of integrated atom-photon junctions,” Nat. Photonics 5, 35–38 (2011).
[CrossRef]

M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106, 210503 (2011).
[CrossRef] [PubMed]

M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, T. Bourouina, “Micromachined Fabry-Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett. 98(21), 211113 (2011).
[CrossRef]

J. Lepert, M. Trupke, M. Hartmann, M. Plenio, E. Hinds, “Arrays of waveguide-coupled optical cavities that interact strongly with atoms,” New J. Phys. 13, 113002 (2011).
[CrossRef]

G. Lepert, M. Trupke, E. Hinds, H. Rogers, J. Gates, P. Smith, “Demonstration of UV-written waveguides, Bragg gratings and cavities at 780 nm, and an original experimental measurement of group delay,” Opt. Express 19, 24933–24943 (2011).
[CrossRef]

2010 (3)

A. Abdelrahman, M. Vasiliev, K. Alameh, P. Hannford, “Asymmetrical two-dimensional magnetic lattices for ultracold atoms,” Phys. Rev. A 82, 012320 (2010).
[CrossRef]

M. Riedel, P. Böhi, Y. Li, T. Hänsch, A. Sinatra, P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef] [PubMed]

D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[CrossRef]

2009 (4)

P. Böhi, M. Riedel, J. Hoffrogge, J. Reichel, T. Haensch, P. Treutlein, “Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip,” Nat. Phys. 5, 592–597 (2009).
[CrossRef]

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[CrossRef]

S. Whitlock, R. Gerritsma, T. Fernholz, R. Spreeuw, “Two-dimensional array of microtraps with atomic shift register on a chip,” New J. Phys. 11, 023021 (2009).
[CrossRef]

J. Matthews, A. Politi, A. Stefanov, J. OBrien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
[CrossRef]

2008 (3)

D. Press, T. Ladd, B. Zhang, Y. Yamamoto, “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature 456, 218–221 (2008).
[CrossRef] [PubMed]

M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B At. Mol. Opt. Phys. 41, 065301 (2008).
[CrossRef]

T. Fernholz, R. Gerritsma, S. Whitlock, I. Barb, R. Spreeuw, “Fully permanent magnet atom chip for Bose-Einstein condensation,” Phys. Rev. A 77, 033409 (2008).
[CrossRef]

2007 (3)

M. Hijlkema, B. Weber, H. Specht, S. Webster, A. Kuhn, G. Rempe, “A single-photon server with just one atom,” Nature 3, 253–255 (2007).

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

2006 (2)

P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Mandel, A. Widera, T. Rom, I. Bloch, T. Hänsch, “Quantum information processing optical lattices and magnetic microtraps,” Fortschr. Phys. 54, 702 (2006).
[CrossRef]

A. Boozer, A. Boca, R. Miller, T. Northup, H. Kimble, “Cooling to the ground state of axial motion for one atom strongly coupled to an optical cavity,” Phys. Rev. Lett. 97, 083602 (2006).
[CrossRef] [PubMed]

2004 (1)

A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. Girvin, R. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

2003 (1)

S. Nolte, M. Will, J. Burghoff, A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A 77, 109–111 (2003).
[CrossRef]

2001 (1)

B. Julsgaard, A. Kozhekin, E. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413, 400–403 (2001).
[CrossRef] [PubMed]

2000 (2)

A. Sørensen, L. Duan, J. Cirac, P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature 409, 63–66 (2000).
[CrossRef]

P. Pinkse, T. Fischer, P. Maunz, G. Rempe, “Trapping an atom with single photons,” Nature 404, 365–368 (2000).
[CrossRef] [PubMed]

1999 (1)

J. Ye, D. Vernooy, H. Kimble, “Trapping of single atoms in cavity QED,” Phys. Rev. Lett. 83, 4987–4990 (1999).
[CrossRef]

Abdelrahman, A.

A. Abdelrahman, M. Vasiliev, K. Alameh, P. Hannford, “Asymmetrical two-dimensional magnetic lattices for ultracold atoms,” Phys. Rev. A 82, 012320 (2010).
[CrossRef]

Akulshin, A.

M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B At. Mol. Opt. Phys. 41, 065301 (2008).
[CrossRef]

Alameh, K.

A. Abdelrahman, M. Vasiliev, K. Alameh, P. Hannford, “Asymmetrical two-dimensional magnetic lattices for ultracold atoms,” Phys. Rev. A 82, 012320 (2010).
[CrossRef]

Awschalom, D.

D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[CrossRef]

Barb, I.

T. Fernholz, R. Gerritsma, S. Whitlock, I. Barb, R. Spreeuw, “Fully permanent magnet atom chip for Bose-Einstein condensation,” Phys. Rev. A 77, 033409 (2008).
[CrossRef]

Baur, S.

M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106, 210503 (2011).
[CrossRef] [PubMed]

Blais, A.

A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. Girvin, R. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

Bloch, I.

P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Mandel, A. Widera, T. Rom, I. Bloch, T. Hänsch, “Quantum information processing optical lattices and magnetic microtraps,” Fortschr. Phys. 54, 702 (2006).
[CrossRef]

Boca, A.

A. Boozer, A. Boca, R. Miller, T. Northup, H. Kimble, “Cooling to the ground state of axial motion for one atom strongly coupled to an optical cavity,” Phys. Rev. Lett. 97, 083602 (2006).
[CrossRef] [PubMed]

Bochmann, J.

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

M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106, 210503 (2011).
[CrossRef] [PubMed]

Böhi, P.

M. Riedel, P. Böhi, Y. Li, T. Hänsch, A. Sinatra, P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef] [PubMed]

P. Böhi, M. Riedel, J. Hoffrogge, J. Reichel, T. Haensch, P. Treutlein, “Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip,” Nat. Phys. 5, 592–597 (2009).
[CrossRef]

Boozer, A.

A. Boozer, A. Boca, R. Miller, T. Northup, H. Kimble, “Cooling to the ground state of axial motion for one atom strongly coupled to an optical cavity,” Phys. Rev. Lett. 97, 083602 (2006).
[CrossRef] [PubMed]

Bourdel, T.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Bourouina, T.

M. Malak, N. Gaber, F. Marty, N. Pavy, E. Richalot, T. Bourouina, “Analysis of Fabry-Pérot optical micro-cavities based on coating-free all-Silicon cylindrical Bragg reflectors,” Opt. Express 21, 2378–2392 (2013).
[CrossRef] [PubMed]

M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250μm-gap silicon Fabry-Perot cavities,” J. Microelectromech. Syst. 21(1), 171–180 (2012).
[CrossRef]

M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, T. Bourouina, “Micromachined Fabry-Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett. 98(21), 211113 (2011).
[CrossRef]

Brennecke, F.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Briggs, G.

D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[CrossRef]

Buckley, B.

D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[CrossRef]

Burghoff, J.

S. Nolte, M. Will, J. Burghoff, A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A 77, 109–111 (2003).
[CrossRef]

Byrnes, T.

A. Pyrkov, T. Byrnes, “Entanglement generation in quantum networks of Bose-Einstein condensates,” New J. Phys. 15093019 (2013).
[CrossRef]

T. Byrnes, “Fractality and macroscopic entanglement in two-component Bose-Einstein condensates,” Phys. Rev. A 88, 023609 (2013).
[CrossRef]

T. Byrnes, “Quantum computation using two component Bose-Einstein condensates,” World Acad. Sci. Eng. Technol. 63, 542 (2012).

T. Byrnes, K. Wen, Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012).
[CrossRef]

A. Pyrkov, T. Byrnes, “Quantum teleportation of spin coherent states,” arxiv:1305.2479.

Cirac, J.

A. Sørensen, L. Duan, J. Cirac, P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature 409, 63–66 (2000).
[CrossRef]

Colombe, Y.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Mandel, A. Widera, T. Rom, I. Bloch, T. Hänsch, “Quantum information processing optical lattices and magnetic microtraps,” Fortschr. Phys. 54, 702 (2006).
[CrossRef]

DiCarlo, L.

D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[CrossRef]

Donner, T.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Duan, L.

A. Sørensen, L. Duan, J. Cirac, P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature 409, 63–66 (2000).
[CrossRef]

Dubois, G.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

Dürr, S.

M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106, 210503 (2011).
[CrossRef] [PubMed]

Eichler, C.

C. Eichler, C. Lang, J. Fink, J. Govenius, S. Filipp, A. Wallraff, “Observation of entanglement between itinerant microwave photons and a superconducting qubit,” Phys. Rev. Lett. 109, 240501 (2012).
[CrossRef]

Esslinger, T.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Fernholz, T.

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Riedel, M.

M. Riedel, P. Böhi, Y. Li, T. Hänsch, A. Sinatra, P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
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Rom, T.

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D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[CrossRef]

A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. Girvin, R. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

Schuster, D.

D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[CrossRef]

A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. Girvin, R. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

Sears, A.

D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[CrossRef]

See, P.

G. Wilpers, P. See, P. Gill, A. Sinclair, “A monolithic array of three-dimensional ion traps fabricated with conventional semiconductor technology,” Nat. Nanotechnol. 7, 572–576 (2012).
[CrossRef] [PubMed]

Sidorov, A.

M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B At. Mol. Opt. Phys. 41, 065301 (2008).
[CrossRef]

Sinatra, A.

M. Riedel, P. Böhi, Y. Li, T. Hänsch, A. Sinatra, P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef] [PubMed]

Sinclair, A.

G. Wilpers, P. See, P. Gill, A. Sinclair, “A monolithic array of three-dimensional ion traps fabricated with conventional semiconductor technology,” Nat. Nanotechnol. 7, 572–576 (2012).
[CrossRef] [PubMed]

Singh, M.

M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B At. Mol. Opt. Phys. 41, 065301 (2008).
[CrossRef]

Smith, P.

Sørensen, A.

A. Sørensen, L. Duan, J. Cirac, P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature 409, 63–66 (2000).
[CrossRef]

Specht, H.

M. Hijlkema, B. Weber, H. Specht, S. Webster, A. Kuhn, G. Rempe, “A single-photon server with just one atom,” Nature 3, 253–255 (2007).

Spreeuw, R.

S. Whitlock, R. Gerritsma, T. Fernholz, R. Spreeuw, “Two-dimensional array of microtraps with atomic shift register on a chip,” New J. Phys. 11, 023021 (2009).
[CrossRef]

T. Fernholz, R. Gerritsma, S. Whitlock, I. Barb, R. Spreeuw, “Fully permanent magnet atom chip for Bose-Einstein condensation,” Phys. Rev. A 77, 033409 (2008).
[CrossRef]

Steck, D. A.

D. A. Steck, “Rubidium 87 D Line Data,” Los Alamos National Laboratory (2001).

Stefanov, A.

J. Matthews, A. Politi, A. Stefanov, J. OBrien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
[CrossRef]

Steinmetz, T.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Mandel, A. Widera, T. Rom, I. Bloch, T. Hänsch, “Quantum information processing optical lattices and magnetic microtraps,” Fortschr. Phys. 54, 702 (2006).
[CrossRef]

Succo, M.

M. Kohnen, M. Succo, P. Petrov, R. Nyman, M. Trupke, E. Hinds, “An array of integrated atom-photon junctions,” Nat. Photonics 5, 35–38 (2011).
[CrossRef]

Treutlein, P.

M. Riedel, P. Böhi, Y. Li, T. Hänsch, A. Sinatra, P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef] [PubMed]

P. Böhi, M. Riedel, J. Hoffrogge, J. Reichel, T. Haensch, P. Treutlein, “Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip,” Nat. Phys. 5, 592–597 (2009).
[CrossRef]

P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Mandel, A. Widera, T. Rom, I. Bloch, T. Hänsch, “Quantum information processing optical lattices and magnetic microtraps,” Fortschr. Phys. 54, 702 (2006).
[CrossRef]

Trupke, M.

J. Lepert, M. Trupke, M. Hartmann, M. Plenio, E. Hinds, “Arrays of waveguide-coupled optical cavities that interact strongly with atoms,” New J. Phys. 13, 113002 (2011).
[CrossRef]

M. Kohnen, M. Succo, P. Petrov, R. Nyman, M. Trupke, E. Hinds, “An array of integrated atom-photon junctions,” Nat. Photonics 5, 35–38 (2011).
[CrossRef]

G. Lepert, M. Trupke, E. Hinds, H. Rogers, J. Gates, P. Smith, “Demonstration of UV-written waveguides, Bragg gratings and cavities at 780 nm, and an original experimental measurement of group delay,” Opt. Express 19, 24933–24943 (2011).
[CrossRef]

Tuennermann, A.

S. Nolte, M. Will, J. Burghoff, A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A 77, 109–111 (2003).
[CrossRef]

Uphoff, M.

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

Vasiliev, M.

A. Abdelrahman, M. Vasiliev, K. Alameh, P. Hannford, “Asymmetrical two-dimensional magnetic lattices for ultracold atoms,” Phys. Rev. A 82, 012320 (2010).
[CrossRef]

Vernooy, D.

J. Ye, D. Vernooy, H. Kimble, “Trapping of single atoms in cavity QED,” Phys. Rev. Lett. 83, 4987–4990 (1999).
[CrossRef]

Vo, C.

M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106, 210503 (2011).
[CrossRef] [PubMed]

Volk, M.

M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B At. Mol. Opt. Phys. 41, 065301 (2008).
[CrossRef]

Wallquist, M.

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[CrossRef]

Wallraff, A.

C. Eichler, C. Lang, J. Fink, J. Govenius, S. Filipp, A. Wallraff, “Observation of entanglement between itinerant microwave photons and a superconducting qubit,” Phys. Rev. Lett. 109, 240501 (2012).
[CrossRef]

A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. Girvin, R. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

Waxman, A.

A. Waxman, “Coherent manipulation of the Rubidium atom ground state,” M.Sc Thesis (Ben-Gurion University of the Negev, 2007).

Weber, B.

M. Hijlkema, B. Weber, H. Specht, S. Webster, A. Kuhn, G. Rempe, “A single-photon server with just one atom,” Nature 3, 253–255 (2007).

Webster, S.

M. Hijlkema, B. Weber, H. Specht, S. Webster, A. Kuhn, G. Rempe, “A single-photon server with just one atom,” Nature 3, 253–255 (2007).

Wen, K.

T. Byrnes, K. Wen, Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012).
[CrossRef]

Whitlock, S.

S. Whitlock, R. Gerritsma, T. Fernholz, R. Spreeuw, “Two-dimensional array of microtraps with atomic shift register on a chip,” New J. Phys. 11, 023021 (2009).
[CrossRef]

T. Fernholz, R. Gerritsma, S. Whitlock, I. Barb, R. Spreeuw, “Fully permanent magnet atom chip for Bose-Einstein condensation,” Phys. Rev. A 77, 033409 (2008).
[CrossRef]

Widera, A.

P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Mandel, A. Widera, T. Rom, I. Bloch, T. Hänsch, “Quantum information processing optical lattices and magnetic microtraps,” Fortschr. Phys. 54, 702 (2006).
[CrossRef]

Will, M.

S. Nolte, M. Will, J. Burghoff, A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A 77, 109–111 (2003).
[CrossRef]

Wilpers, G.

G. Wilpers, P. See, P. Gill, A. Sinclair, “A monolithic array of three-dimensional ion traps fabricated with conventional semiconductor technology,” Nat. Nanotechnol. 7, 572–576 (2012).
[CrossRef] [PubMed]

Wu, H.

D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[CrossRef]

Yamamoto, Y.

T. Byrnes, K. Wen, Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012).
[CrossRef]

D. Press, T. Ladd, B. Zhang, Y. Yamamoto, “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature 456, 218–221 (2008).
[CrossRef] [PubMed]

Ye, J.

J. Ye, D. Vernooy, H. Kimble, “Trapping of single atoms in cavity QED,” Phys. Rev. Lett. 83, 4987–4990 (1999).
[CrossRef]

Zhang, B.

D. Press, T. Ladd, B. Zhang, Y. Yamamoto, “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature 456, 218–221 (2008).
[CrossRef] [PubMed]

Zoller, P.

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[CrossRef]

A. Sørensen, L. Duan, J. Cirac, P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature 409, 63–66 (2000).
[CrossRef]

Appl. Phys. A (1)

S. Nolte, M. Will, J. Burghoff, A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A 77, 109–111 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, T. Bourouina, “Micromachined Fabry-Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett. 98(21), 211113 (2011).
[CrossRef]

Fortschr. Phys. (1)

P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Mandel, A. Widera, T. Rom, I. Bloch, T. Hänsch, “Quantum information processing optical lattices and magnetic microtraps,” Fortschr. Phys. 54, 702 (2006).
[CrossRef]

J. Microelectromech. Syst. (1)

M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250μm-gap silicon Fabry-Perot cavities,” J. Microelectromech. Syst. 21(1), 171–180 (2012).
[CrossRef]

J. Phys. B At. Mol. Opt. Phys. (1)

M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B At. Mol. Opt. Phys. 41, 065301 (2008).
[CrossRef]

Nat. Nanotechnol. (1)

G. Wilpers, P. See, P. Gill, A. Sinclair, “A monolithic array of three-dimensional ion traps fabricated with conventional semiconductor technology,” Nat. Nanotechnol. 7, 572–576 (2012).
[CrossRef] [PubMed]

Nat. Photonics (2)

M. Kohnen, M. Succo, P. Petrov, R. Nyman, M. Trupke, E. Hinds, “An array of integrated atom-photon junctions,” Nat. Photonics 5, 35–38 (2011).
[CrossRef]

J. Matthews, A. Politi, A. Stefanov, J. OBrien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
[CrossRef]

Nat. Phys. (1)

P. Böhi, M. Riedel, J. Hoffrogge, J. Reichel, T. Haensch, P. Treutlein, “Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip,” Nat. Phys. 5, 592–597 (2009).
[CrossRef]

Nature (10)

M. Riedel, P. Böhi, Y. Li, T. Hänsch, A. Sinatra, P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef] [PubMed]

A. Sørensen, L. Duan, J. Cirac, P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature 409, 63–66 (2000).
[CrossRef]

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

P. Pinkse, T. Fischer, P. Maunz, G. Rempe, “Trapping an atom with single photons,” Nature 404, 365–368 (2000).
[CrossRef] [PubMed]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. Girvin, R. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

D. Press, T. Ladd, B. Zhang, Y. Yamamoto, “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature 456, 218–221 (2008).
[CrossRef] [PubMed]

B. Julsgaard, A. Kozhekin, E. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413, 400–403 (2001).
[CrossRef] [PubMed]

M. Hijlkema, B. Weber, H. Specht, S. Webster, A. Kuhn, G. Rempe, “A single-photon server with just one atom,” Nature 3, 253–255 (2007).

New J. Phys. (3)

A. Pyrkov, T. Byrnes, “Entanglement generation in quantum networks of Bose-Einstein condensates,” New J. Phys. 15093019 (2013).
[CrossRef]

S. Whitlock, R. Gerritsma, T. Fernholz, R. Spreeuw, “Two-dimensional array of microtraps with atomic shift register on a chip,” New J. Phys. 11, 023021 (2009).
[CrossRef]

J. Lepert, M. Trupke, M. Hartmann, M. Plenio, E. Hinds, “Arrays of waveguide-coupled optical cavities that interact strongly with atoms,” New J. Phys. 13, 113002 (2011).
[CrossRef]

Opt. Express (2)

Phys. Rev. A (1)

T. Byrnes, K. Wen, Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012).
[CrossRef]

Phys. Rev. A (3)

A. Abdelrahman, M. Vasiliev, K. Alameh, P. Hannford, “Asymmetrical two-dimensional magnetic lattices for ultracold atoms,” Phys. Rev. A 82, 012320 (2010).
[CrossRef]

T. Fernholz, R. Gerritsma, S. Whitlock, I. Barb, R. Spreeuw, “Fully permanent magnet atom chip for Bose-Einstein condensation,” Phys. Rev. A 77, 033409 (2008).
[CrossRef]

T. Byrnes, “Fractality and macroscopic entanglement in two-component Bose-Einstein condensates,” Phys. Rev. A 88, 023609 (2013).
[CrossRef]

Phys. Rev. Lett. (5)

M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106, 210503 (2011).
[CrossRef] [PubMed]

J. Ye, D. Vernooy, H. Kimble, “Trapping of single atoms in cavity QED,” Phys. Rev. Lett. 83, 4987–4990 (1999).
[CrossRef]

A. Boozer, A. Boca, R. Miller, T. Northup, H. Kimble, “Cooling to the ground state of axial motion for one atom strongly coupled to an optical cavity,” Phys. Rev. Lett. 97, 083602 (2006).
[CrossRef] [PubMed]

D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[CrossRef]

C. Eichler, C. Lang, J. Fink, J. Govenius, S. Filipp, A. Wallraff, “Observation of entanglement between itinerant microwave photons and a superconducting qubit,” Phys. Rev. Lett. 109, 240501 (2012).
[CrossRef]

Phys. Scr. (1)

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[CrossRef]

World Acad. Sci. Eng. Technol. (1)

T. Byrnes, “Quantum computation using two component Bose-Einstein condensates,” World Acad. Sci. Eng. Technol. 63, 542 (2012).

Other (3)

A. Pyrkov, T. Byrnes, “Quantum teleportation of spin coherent states,” arxiv:1305.2479.

A. Waxman, “Coherent manipulation of the Rubidium atom ground state,” M.Sc Thesis (Ben-Gurion University of the Negev, 2007).

D. A. Steck, “Rubidium 87 D Line Data,” Los Alamos National Laboratory (2001).

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Figures (5)

Fig. 1
Fig. 1

Hybrid quantum processor using permanent magnetic traps and waveguides. (a) Sketch of the proposed device (not to scale) consisting of 1⃞ a substrate of permanent magnetic material, 2⃞ reflective coating on the edges, 3⃞ silica waveguides (vertical) for delivering the control/probe photons, 4⃞ an optical microcavity etched into a 5⃞ silica transparent substrate, 6⃞ a joint silica waveguide (horizontal) for transferring photons between nodes, 7⃞ a thermal phase-shifter and 8⃞ a micropattern into the magnetic material for creating the trapping magnetic fields. (b) Density plot of the simulated magnetic field local minima combined with a cross section of the optical microcavity and the silica waveguide.

Fig. 2
Fig. 2

(a) Single BEC qubit control. Two lasers are applied to the transitions between ground states and the excited states with transition energies ga and gb, and detuned each by an amount Δ. Spontaneous emission from the excited states to the ground states with decay rate Γ is present. (b) Rabi oscillations between levels a and b in the presence of spontaneous emission. The effective decoherence rate exp(−Γefft) is shown as the dotted line. (c) Initialization of SC qubits from various initial conditions: I. |1, 0〉〉, II. | 1 2 , 1 2 . Parameters used are N = 1000, Δ/A = 1000, Γ/A = 0.1, ga/A = 100, gb/A = 100 in (b) and gb/A = 0 in (c). The timescale is t0 = h̄/A.

Fig. 3
Fig. 3

Numerically simulated magnetic field local minima of a single trap created at a working distance of dmin ≈ 13.5μm with αh = 3μm, αs = 100μm and τ = 2μm. (a) Density plot of a confining magnetic field with a displaced optical axis of a cavity (small red circule). The magnetic field local minima is created with no external magnetic bias fields applied. (b) For the alignment purpose, the trap is displaced along the positive direction of the x-axis by applying an external magnetic bias field along the x-axis, such that By-bias = Bz-bias = 0 and Bx-bias = −1G. (c) The location of the magnetic trap is below the optical axis of the cavity with no external magnetic bias fields. (d) The magnetic trap is displaced along the z-axis to overlap with the optical axis of the cavity at dmin ≈ 16.0μm with external magnetic bias fields applied along the z-axis at Bx-bias = By-bias = 0 and Bz-bias = −1G.

Fig. 4
Fig. 4

(a) Scenarios for implementing the silica microcavity where Bragg mirrors are included in the design (2). (b) Possible implementations for the hybrid quantum device with a scheme to manipulate the traveling photons using the thermal phase shifters and entangling junctions (dotted line square). The connections in (b) are not to scale where the actual physical implementation would be modified accordingly to the experiment. The red circules represent the optical micro-cavities, the solid black lines are the silica waveguides and the yellow squares represent the thermal phase shifters which are used to modify the phase of the propagating photons to exclude particular targeted SC qubit(s) from being entangled.

Fig. 5
Fig. 5

The reflected power of the composite cavity system, two micro-cavities connected via a single silica waveguide. The simulation input parameters are R 1 c = R 2 c = 0.985, R 1 w c = 0.999, R 2 w c = 0.9 with both micro-cavities at equal lengths L 1 , 2 c = 30 μ m and the silica waveguide with a length of Lw = 4mm.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

| α , β 1 N ! ( α a + β b ) N | 0
| α , β i = 1 N ( α | i + β | i ) .
S x = a b + b a
S z = a a b b .
S x = i = 1 N σ i x
S z = i = 1 N σ i z ,
| α , β | 0 , 1 .
P k = | k k |
| k = ( a ) k ( b ) N k | 0 / k ! ( N k ) ! .
| k = i = 1 N | σ i .
| F = 1 , m F = 1 = 3 4 | I = 3 / 2 , m I = 3 / 2 | J = 1 / 2 , m J = 1 / 2 + 1 2 | I = 3 / 2 , m I = 1 / 2 | J = 1 / 2 , m J = 1 / 2 | F = 2 , m F = 1 = 1 2 | I = 3 / 2 , m I = 3 / 2 | J = 1 / 2 , m J = 1 / 2 + 3 4 | I = 3 / 2 , m I = 1 / 2 | J = 1 / 2 , m J = 1 / 2 ,
H = g a ( J + + J ) + g b ( K + + K ) + A ( L + + L ) + Δ n e + Δ n f
H x = h ¯ Ω 1 S x ,
h ¯ Ω 1 = 2 g a g b A Δ 2 .
d ρ d t = i h ¯ [ ρ , H ] Γ 2 [ J + J ρ 2 J ρ J + + ρ J + J ] Γ 2 [ K + K ρ 2 K ρ K + + ρ K + K ]
Γ eff = g a g b A Γ ( N + 1 ) Δ 3 .
H z = h ¯ ω z S z ,
H QED = j G ( K j + p j + p j K j ) + h ¯ ω 0 n j e + h ¯ ω p j p j
H c-w = ν j p j p j + 1 e i ϕ j + H . c .
c k = 1 𝒩 k j sin ( π k j / 2 M ) p j
H z z = 2 h ¯ Ω 2 cos Φ i j S i z S j z + h ¯ Ω 2 [ ( S i z ) 2 + ( S j z ) 2 ]
h ¯ Ω 2 = G 2 g 2 4 Δ 3
E 1 r c E i n = r 1 c r 1 w c exp [ 2 i ϕ 1 c ] 1 r 1 c r 1 w c exp [ 2 i ϕ 1 c ]
E w E i n = r 1 w c r 2 w c exp [ 2 i ϕ w ] 1 r 1 w c r 2 w c exp [ 2 i ϕ w ]
E 2 w c E i n = r 2 w c r 2 c exp [ 2 i ϕ 2 c ] 1 r 2 w c r 2 c exp [ 2 i ϕ 2 c ]
r ˜ 1 w c = ( r 1 w c ) 2 + ( r ˜ 2 c ) 2 2 r 1 w c r ˜ 2 c η 1 + ( r 1 w c ) 2 ( r ˜ 2 c ) 2 2 r 1 w c r ˜ 2 c η
θ 1 w c = tan 1 ( ( ( r 1 w c ) 2 1 ) r ˜ 2 c ξ r 1 w c ( 1 + ( r ˜ 2 c ) 2 ) r ˜ 2 c ( ( r 1 w c ) 2 + 1 ) η )
E 1 r c = r 1 c r ˜ 1 w c exp [ i ( θ 1 w c + 2 ϕ 1 c ) ] 1 r 1 w c r ˜ 2 c exp [ i ( θ 1 w c + 2 ϕ 1 c ) ] E i n
˜ 1 c = ˜ 2 c = 2 π 2 ω 0 4 λ 2 L c + L c 2 85.6 μ m
g c w N 3 c λ 2 κ π 2 ω 0 2 ( L 1 c + L 2 c + L w )

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