Abstract

We show that a cavity optomechanical system formed by a mechanical resonator simultaneously coupled to two modes of an optical cavity can be used for the implementation of a deterministic quantum phase gate between optical qubits associated with the two intracavity modes. The scheme is realizable for sufficiently strong single-photon optomechanical coupling in the resolved sideband regime, and is robust against cavity losses.

© 2015 Optical Society of America

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2015 (1)

W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
[Crossref]

2014 (5)

C. Baker, W. Hease, D.-T. Nguyen, A. Andronico, S. Ducci, G. Leo, and I. Favero, “Photoelastic coupling in gallium arsenide optomechanical disk resonators,” Opt. Express 22, 14072–14086 (2014)
[Crossref] [PubMed]

K. C. Balram, M. Davanço, J. Y. Lim, J. D. Song, and K. Srinivasan, “Moving boundary and photoelastic coupling in GaAs optomechanical resonators,” Optica 1, 414–420 (2014).
[Crossref]

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
[Crossref]

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, “Nonlinear π phase shift for single fibre-guided photons interacting with a single resonator-enhanced atom,” Nat. Photonics. 8, 965–970 (2014).
[Crossref]

2013 (5)

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref] [PubMed]

T. P. Purdy, P-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X3, 031012 (2013).

M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013).
[Crossref] [PubMed]

O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

2012 (5)

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

T. Peyronel, O. Firstenberg, Q. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Ref. Lett. 109, 063601 (2012).
[Crossref]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

2011 (4)

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref] [PubMed]

2010 (2)

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

2006 (3)

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Cross phase modulation in a five-level atomic medium: semi-classical theory,” Eur. Phys. J. D 40, 281–296 (2006).
[Crossref]

S. Rebic, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase gate operation based on nonlinear optics,” Phys Rev A 74, 032301 (2006).
[Crossref]

2001 (2)

M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273–276 (2001).
[Crossref] [PubMed]

E. Knill, R. Laflamme, and G.J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001)
[Crossref] [PubMed]

2000 (2)

D. Vitali, M. Fortunato, and P. Tombesi, “Complete quantum teleportation with a Kerr nonlinearity,” Phys. Rev. Lett. 85, 445–448 (2000).
[Crossref] [PubMed]

M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
[Crossref] [PubMed]

1999 (1)

L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

1997 (3)

S.E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
[Crossref]

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390–393 (1997).
[Crossref]

1996 (1)

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. XXXV, 257–354 (1996).
[Crossref]

1995 (2)

S. Lloyd, “Almost any quantum logic gate is universal,” Phys. Rev. Lett. 75, 346–349 (1995).
[Crossref] [PubMed]

Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and H.J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995)
[Crossref] [PubMed]

1994 (2)

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[Crossref] [PubMed]

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Allman, M. S.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Andronico, A.

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

Arimondo, E.

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. XXXV, 257–354 (1996).
[Crossref]

Aspelmeyer, M.

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
[Crossref]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref] [PubMed]

Baker, C.

Balram, K. C.

Bawaj, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Behroozi, C.H.

L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Bennett, S. D.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

Biancofiore, C.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Bimbard, E.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Botter, T.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

Bourzeix, S.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Bouwmeester, D.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
[Crossref]

K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
[Crossref]

Brahms, N.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

Brooks, D. W. C.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

Chan, J.

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Chang, D. E.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Cheng, J.

W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Springer, 2000).

Cicak, K.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Cirac, J. I.

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390–393 (1997).
[Crossref]

Cohen, J. D.

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
[Crossref]

Davanço, M.

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

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M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013).
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[Crossref]

Regal, C. A.

T. P. Purdy, P-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X3, 031012 (2013).

Reynaud, S.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Rivière, R.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

Safavi-Naeini, A. H.

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
[Crossref]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref] [PubMed]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Ref. Lett. 109, 063601 (2012).
[Crossref]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Scheucher, M.

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, “Nonlinear π phase shift for single fibre-guided photons interacting with a single resonator-enhanced atom,” Nat. Photonics. 8, 965–970 (2014).
[Crossref]

Schilling, R.

D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement and control of a mechanical oscillator at its thermal decoherence rate,” arXiv:1410.6191 [quant-ph].

Schliesser, A.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

Schoelkopf, R. J.

M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013).
[Crossref] [PubMed]

Schreppler, S.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

Sharypov, A. V.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

Sheng, J.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

Simmonds, R. W.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Simon, C.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
[Crossref]

Sirois, A. J.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Song, J. D.

Srinivasan, K.

Stamper-Kurn, D. M.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

Stannigel, K.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

Stanojevic, J.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Sudhir, V.

D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement and control of a mechanical oscillator at its thermal decoherence rate,” arXiv:1410.6191 [quant-ph].

Teufel, J. D.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Tombesi, P.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Cross phase modulation in a five-level atomic medium: semi-classical theory,” Eur. Phys. J. D 40, 281–296 (2006).
[Crossref]

S. Rebic, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase gate operation based on nonlinear optics,” Phys Rev A 74, 032301 (2006).
[Crossref]

D. Vitali, M. Fortunato, and P. Tombesi, “Complete quantum teleportation with a Kerr nonlinearity,” Phys. Rev. Lett. 85, 445–448 (2000).
[Crossref] [PubMed]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[Crossref] [PubMed]

K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
[Crossref]

Tualle-Brouri, R.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Turchette, Q.A.

Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and H.J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995)
[Crossref] [PubMed]

Vitali, D.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Cross phase modulation in a five-level atomic medium: semi-classical theory,” Eur. Phys. J. D 40, 281–296 (2006).
[Crossref]

S. Rebic, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase gate operation based on nonlinear optics,” Phys Rev A 74, 032301 (2006).
[Crossref]

D. Vitali, M. Fortunato, and P. Tombesi, “Complete quantum teleportation with a Kerr nonlinearity,” Phys. Rev. Lett. 85, 445–448 (2000).
[Crossref] [PubMed]

K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
[Crossref]

Volz, J.

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, “Nonlinear π phase shift for single fibre-guided photons interacting with a single resonator-enhanced atom,” Nat. Photonics. 8, 965–970 (2014).
[Crossref]

Vuletic, V.

O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin1994).
[Crossref]

Weinfurter, H.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
[Crossref]

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

Whittaker, J. D.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Wilson, D. J.

D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement and control of a mechanical oscillator at its thermal decoherence rate,” arXiv:1410.6191 [quant-ph].

Winger, M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Xiao, M.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

Yu, P-L.

T. P. Purdy, P-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X3, 031012 (2013).

Zeilinger, A.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
[Crossref]

Zhang, W.-Z.

W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
[Crossref]

Zhou, L.

W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
[Crossref]

Zoller, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390–393 (1997).
[Crossref]

Eur. Phys. J. D (1)

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Cross phase modulation in a five-level atomic medium: semi-classical theory,” Eur. Phys. J. D 40, 281–296 (2006).
[Crossref]

J. Phys. B (1)

W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
[Crossref]

Nat. Photonics. (1)

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, “Nonlinear π phase shift for single fibre-guided photons interacting with a single resonator-enhanced atom,” Nat. Photonics. 8, 965–970 (2014).
[Crossref]

Nature (10)

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273–276 (2001).
[Crossref] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
[Crossref]

E. Knill, R. Laflamme, and G.J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001)
[Crossref] [PubMed]

Opt. Express (1)

Optica (1)

Phys Rev A (1)

S. Rebic, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase gate operation based on nonlinear optics,” Phys Rev A 74, 032301 (2006).
[Crossref]

Phys. Ref. Lett. (1)

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Ref. Lett. 109, 063601 (2012).
[Crossref]

Phys. Rev. (1)

T. P. Purdy, P-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X3, 031012 (2013).

Phys. Rev. A (6)

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[Crossref] [PubMed]

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
[Crossref]

Phys. Rev. Lett. (10)

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390–393 (1997).
[Crossref]

M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
[Crossref] [PubMed]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref] [PubMed]

D. Vitali, M. Fortunato, and P. Tombesi, “Complete quantum teleportation with a Kerr nonlinearity,” Phys. Rev. Lett. 85, 445–448 (2000).
[Crossref] [PubMed]

S. Lloyd, “Almost any quantum logic gate is universal,” Phys. Rev. Lett. 75, 346–349 (1995).
[Crossref] [PubMed]

Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and H.J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995)
[Crossref] [PubMed]

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

Phys. Today (1)

S.E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Prog. Opt. (1)

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. XXXV, 257–354 (1996).
[Crossref]

Science (1)

M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013).
[Crossref] [PubMed]

Science, (1)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

Other (4)

K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
[Crossref]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Springer, 2000).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin1994).
[Crossref]

D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement and control of a mechanical oscillator at its thermal decoherence rate,” arXiv:1410.6191 [quant-ph].

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Figures (1)

Fig. 1
Fig. 1 Numerical solution of the master equation Eq. (20) for the gate fidelity (t) versus the dimensionless interaction time ωmt. We compare four different cases: i) zero damping and losses γm = κ1 = κ2 = 0 and = 10 (full black line); ii) zero damping and losses and = 0 (red dashed-dotted line); iii) with damping and losses (κ1 = κ2 = 10−2ωm, Qm = 106,) and = 10 (green dashed line); iv) with damping and losses (κ1 = κ2 = 10−2 ωm, Qm = 106,) and = 0 (blue dotted line). The numerical solutions for the zero damping and loss case are indistinguishable from the analytical expression of Eq. (18) either at = 0 and at = 10. In all cases we have fixed the couplings according to the ideal strong coupling condition of Eq. (19), g1 = g2 = ωm/2.

Equations (23)

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

H ^ = h ¯ ω 1 a ^ 1 a ^ 1 + h ¯ ω 2 a ^ 2 a ^ 2 + h ¯ ω m b ^ b ^ + h ¯ ( g 1 a ^ 1 a ^ 1 + g 2 a ^ 2 a ^ 2 ) ( b ^ + b ^ ) ,
| ψ in = α 00 | 0 1 | 0 2 + α 01 | 0 1 | 1 2 + α 10 | 1 1 | 0 2 + α 11 | 1 1 | 1 2 ,
H ^ = h ¯ Δ 1 n ^ 1 + h ¯ Δ 2 n ^ 2 + h ¯ ω m b ^ b ^ + h ¯ ω m f ^ n ^ ( b ^ + b ^ ) ,
D ^ ( f ^ n ^ ) = exp [ ( b ^ b ^ ) f ^ n ^ ] ,
D ^ ( f ^ n ^ ) H ^ D ^ ( f ^ n ^ ) = H ^ opt + H ^ b ,
H ^ opt = h ¯ Δ 1 n ^ 1 + h ¯ Δ 2 n ^ 2 h ¯ ω m f ^ n ^ 2 = h ¯ Δ 1 n ^ 1 + h ¯ Δ 2 n ^ 2 h ¯ ( g 1 n ^ 1 + g 2 n ^ 2 ) 2 ω m ,
H ^ b = h ¯ ω m b ^ b ^ ,
U ^ ( t ) = U ^ opt ( t ) D ^ ( f ^ n ^ ) U ^ b ( t ) D ^ ( f ^ n ^ ) ,
| 0 1 | 0 2 | 0 1 | 0 2 ; | 0 1 | 1 2 | 0 1 | 1 2 ; | 1 1 | 0 2 | 1 1 | 0 2 ; | 1 1 | 1 2 e i 2 g 1 g 2 t ω m | 1 1 | 1 2 .
t π = π ω m 2 g 1 g 2 .
ρ ^ ( t ) = U ^ ( t ) | ψ in ψ | ρ ^ b th U ^ ( t ) = k , l = 0 3 α k α l * U ^ opt ( t ) | k l | U ^ opt ( t ) D ^ ( f k ) U ^ b ( t ) D ^ ( f k ) ρ ^ b th D ^ ( f l ) U ^ b ( t ) D ^ ( f l ) ,
D ^ ( f k ) = exp [ f k ( b ^ b ^ ) ]
ρ ^ opt ( t ) = k , l = 0 3 c k , l ( t ) α k α l * exp [ i 2 g 1 g 2 t ω m ( δ k , 3 δ l , 3 ) ] | k l | ,
c k , l ( t ) = exp [ ( f k f l ) 2 ( 1 cos ω m t ) ( 2 n ¯ + 1 ) + i ( f l 2 f k 2 ) sin ω m t ] .
| ψ t g t = α 00 | 0 1 | 0 2 + α 01 | 0 1 | 1 2 + α 10 | 1 1 | 0 2 + e i π α 11 | 1 1 | 1 2 = k = 0 3 α k e i π δ k , 3 | k .
F ( t ) ψ t g t | ρ ^ opt ( t ) | ψ t g t = k , l = 0 3 c k , l ( t ) | α k | 2 | α l | 2 exp [ i ( 2 g 1 g 2 t ω m π ) ( δ k , 3 δ l , 3 ) ] .
( t ) = ψ t g t | ρ ^ opt ( t ) | ψ t g t ¯ = 1 2 + 1 12 { exp [ g 1 2 ω m 2 ( 1 cos ω m t ) ( 2 n ¯ + 1 ) ] [ cos ( g 1 2 ω m 2 sin ω m t ) + cos ( g 1 2 + 2 g 1 g 2 ω m 2 sin ω m t + π 2 g 1 g 2 t ω m ) ] + exp [ g 2 2 ω m 2 ( 1 cos ω m t ) ( 2 n ¯ + 1 ) ] [ cos ( g 2 2 ω m 2 sin ω m t ) + cos ( g 2 2 + 2 g 1 g 2 ω m 2 sin ω m t + π 2 g 1 g 2 t ω m ) ] + exp [ ( g 1 g 2 ω m ) 2 ( 1 cos ω m t ) ( 2 n ¯ + 1 ) ] cos ( g 2 2 g 1 2 ω m 2 sin ω m t ) + exp [ ( g 1 + g 2 ω m ) 2 ( 1 cos ω m t ) ( 2 n ¯ + 1 ) ] cos [ ( g 1 + g 2 ω m ) 2 sin ω m t + π 2 g 1 g 2 t ω m ] } .
2 g 1 g 2 t π ω m = π ω m t π = 2 π g 1 g 2 = ω m 2 4 .
d d t ρ ^ ( t ) = 1 i h ¯ [ H ^ , ρ ^ ( t ) ] + κ 1 2 ( 2 a ^ 1 ρ ^ ( t ) a ^ 1 a ^ 1 a ^ 1 ρ ^ ( t ) ρ ^ ( t ) a ^ 1 a ^ 1 ) + κ 2 2 ( 2 a ^ 2 ρ ^ ( t ) a ^ 2 a ^ 2 a ^ 2 ρ ^ ( t ) ρ ^ ( t ) a ^ 2 a ^ 2 ) + γ m 2 ( n ¯ + 1 ) ( 2 b ^ ρ ^ ( t ) b ^ b ^ b ^ ρ ^ ( t ) ρ ^ ( t ) b ^ b ^ ) + γ m 2 n ¯ ( 2 b ^ ρ ^ ( t ) b ^ b ^ b ^ ρ ^ ( t ) ρ ^ ( t ) b ^ b ^ ) ,
c k , l ( t ) = Tr b [ D ^ ( f l ) U ^ b ( t ) D ^ ( f l ) D ^ ( f k ) U ^ b ( t ) D ^ ( f k ) ρ ^ b th ] ,
U ^ b ( t ) D ^ ( f l ) D ^ ( f k ) U ^ b ( t ) = exp [ ( b ^ e i ω m t b ^ e i ω m t ) ( f l f k ) ] .
c k , l ( t ) = Tr b { D ^ ( f l f k ) ( e i ω m t 1 ) ρ ^ b th } exp [ i ( f l 2 f k 2 ) sin ω m t ] .
exp [ α b ^ α * b ^ ] th = exp [ | α | 2 ( n ¯ + 1 2 ) ] ,

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