Abstract

In this paper, we propose a novel quantum approach for microwave-to-optical conversion in a multilayer graphene structure. The graphene layers are electrically connected and pumped by an optical field. The physical concept is based on using a driving microwave signal to modulate the optical input pump by controlling graphene conductivity. Consequently, upper and lower optical sidebands are generated. To achieve low noise conversion, the lower sideband is suppressed by the multilayer graphene destruction resonance. A perturbation approach is implemented to model the effective permittivity of the electrically driven multilayer graphene. Subsequently, a quantum mechanical analysis is carried out to describe the evolution of the interacting fields. It is shown that a quantum microwave-to-optical conversion is achieved for miltilayer graphene of the proper length (i.e., number of layers). The conversion rate and the number of converted photons are evaluated according to several parameters. These include the microwave signal frequency, the microwave driving voltages, the graphene intrinsic electron density, and the number of graphene layers. Owing to multilayer dispersion and to the properties of graphene, it is shown that a significant number of photons (converted from microwave to optical frequency range) is achieved for microvolt microwave driving voltages. Furthermore, a frequency-tunable operation is achieved using this technique simply by modifying the optical pump frequency.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (3)

M. Atature, D. Englund, N. Vamivakas, S. Y. Lee, and J. Wrachtrup, "Material platforms for spin-based photonic quantum technologies," Nat. Rev. Mater.,  3, 38–51 (2018).
[Crossref]

R. Igreja and C. J. Dias, "Analytical evaluation of the interdigital electrodes capacitance for a multi-layered structure," Sensor. Actuat. A,  112(2), 291–301 (2018).
[Crossref]

M. Qasymeh, "Giant Amplification of Terahertz Waves in a Nonlinear Graphene Layered Medium," IEEE Photonics Technol. Lett.,  30(1), 35–38 (2018).
[Crossref]

2017 (2)

M. Qasymeh, "Phase-Matched Coupling and Frequency Conversion of Terahertz Waves in a Nonlinear Graphene Waveguide," J. Lightwave Technol.,  35(9), 1654–1662 (2017).
[Crossref]

M. Soltani, M. Zhang, C. Ryan, G. J. Ribeill, C. Wang, and M. Loncar, "Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators," Phys. Rev. A,  96, 043808 (2017).
[Crossref]

2016 (6)

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, "On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator," Phys. Rev. A,  94, 053815 (2016).
[Crossref]

S. Pan and J. Yao, "Photonics-Based Broadband Microwave Measurement," J. Lightwave Technol.,  35(16), 3498–3513 (2016).
[Crossref]

X. Guo, X. Li, N. Liu, and Z. Y. Ou, "Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs," Sci. Rep.,  6, 30214 (2016).
[Crossref] [PubMed]

Y. Wu, L. Jiang, H. Xu, X. Dai, Y. Xiang, and D. Fan, "Hybrid nonlinear surface-phonon-plasmon-polaritons at the interface of nolinear medium and graphene-covered hexagonal boron nitride crystal," Opt. Express,  24(3), 2109–2124 (2016).
[Crossref] [PubMed]

A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, "Efficient microwave to optical photon conversion: an electro-optical realization," Optica,  3(6), 597–604 (2016).
[Crossref]

S. Awan, A. Lombardo, A. Colli, G. Privitera, T. Kulmala, J. Kivioja, and A. Ferrari, "Transport conductivity of graphene at RF and microwave frequencies," 2D Mater.,  3(1), 015010 (2016).
[Crossref]

2015 (3)

C. T. Phare, Y. D. Lee, J. Cardenas, and M. Lipson, "Graphene electro-optic modulator with 30 GHz bandwidth," Nat. Photonics,  9, 511–514 (2015).
[Crossref]

D. Ansell, I. P. Radko, Z. Han, F. J. Rodriguez, S. I. Bozhevolnyi, and A. N. Grigorenko, "Hybrid graphene plasmonic waveguide modulators," Nat. Commun.,  6, 8846 (2015).
[Crossref] [PubMed]

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Coherent frequency up-conversion of microwaves to the optical telecommunications band in an Er:YSO crystal," Phys. Rev. A,  92, 062313 (2015).
[Crossref]

2014 (5)

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, "Bidirectional and efficient conversion between microwave and optical light," Nat. Phys.,  10, 321–326 (2014).
[Crossref]

K. Fang, M. H. Matheny, X. Luan, and O. Painter, "Optical transduction and routing of microwave phonons in cavity-optomechanical circuits," Nat. Photonics.,  10, 489–496 (2014).
[Crossref]

E. A. Sete and H. Eleuch, "Strong squeezing and robust entanglement in cavity electromechanics," Phys. Rev. A 89, 013841 (2014).
[Crossref]

S. A. Eyob and H. Eleuch, "Strong squeezing and robust entanglement in cavity electromechanics," Phys. Rev. A,  89, 013841 (2014).
[Crossref]

J. H. Lee, P. E. Loya, J. Lou, and E. L. Thomas, "Dynamic mechanical behavior of multilayer graphene via supersonic projectile penetration," Science,  346(6213), 1092–1096 (2014).
[Crossref] [PubMed]

2013 (3)

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, "Transfer matrix method for optics in graphene layers," J. Phys.: Condens. Matter,  25(21), 215301 (2013).

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, "Nanomechanical coupling between microwave and optical photons," Nat. Phys.,  9, 712–716 (2013).
[Crossref]

R. A. Minasian, E. H. W. Chan, and X. Yi, "Microwave photonic signal processing," Opt. Express,  35(21), 22918–22936 (2013).
[Crossref]

2012 (1)

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Atomic interface between microwave and optical photons," Phys. Rev. A,  85, 020302 (2012).
[Crossref]

2011 (2)

M. Tsang, "Cavity quantum electro-optics. II. Input-output relations between traveling optical and microwave fields," Phys. Rev. A,  84, 043845 (2011).
[Crossref]

M. Qasymeh, W. Li, and J. Yao, "Frequency-Tunable Microwave Generation Based on Time-Delayed Optical Combs," IEEE Trans. Microw. Theory Tech.,  59(11), 2987–2993 (2011).
[Crossref]

2010 (2)

2009 (2)

J. Yao, "Microwave Photonics," J. Lightwave Technol.,  27(3), 314–335 (2009).
[Crossref]

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

2007 (1)

J. Capmany and D. Novak, "Microwave photonics combines two worlds," Nat. Photonics,  1, 319–330 (2007).
[Crossref]

2006 (1)

1991 (1)

K. K. Likharev and V. K. Semenov, “RSFQ logic/memory family: a new Josephson-junction technology for sub-terahertz-clock-frequency digital systems,"” IEEE Trans. Appl. Supercond.,  1(1), 3–28 (1991).
[Crossref]

Andrews, R. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, "Bidirectional and efficient conversion between microwave and optical light," Nat. Phys.,  10, 321–326 (2014).
[Crossref]

Ansell, D.

D. Ansell, I. P. Radko, Z. Han, F. J. Rodriguez, S. I. Bozhevolnyi, and A. N. Grigorenko, "Hybrid graphene plasmonic waveguide modulators," Nat. Commun.,  6, 8846 (2015).
[Crossref] [PubMed]

Atature, M.

M. Atature, D. Englund, N. Vamivakas, S. Y. Lee, and J. Wrachtrup, "Material platforms for spin-based photonic quantum technologies," Nat. Rev. Mater.,  3, 38–51 (2018).
[Crossref]

Awan, S.

S. Awan, A. Lombardo, A. Colli, G. Privitera, T. Kulmala, J. Kivioja, and A. Ferrari, "Transport conductivity of graphene at RF and microwave frequencies," 2D Mater.,  3(1), 015010 (2016).
[Crossref]

Awschalom, D. D.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, "Nanomechanical coupling between microwave and optical photons," Nat. Phys.,  9, 712–716 (2013).
[Crossref]

Bernier, N. R.

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, "On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator," Phys. Rev. A,  94, 053815 (2016).
[Crossref]

Bishop, L. S.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Blais, A.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Bochmann, J.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, "Nanomechanical coupling between microwave and optical photons," Nat. Phys.,  9, 712–716 (2013).
[Crossref]

Bozhevolnyi, S. I.

D. Ansell, I. P. Radko, Z. Han, F. J. Rodriguez, S. I. Bozhevolnyi, and A. N. Grigorenko, "Hybrid graphene plasmonic waveguide modulators," Nat. Commun.,  6, 8846 (2015).
[Crossref] [PubMed]

Capmany, J.

J. Capmany and D. Novak, "Microwave photonics combines two worlds," Nat. Photonics,  1, 319–330 (2007).
[Crossref]

Cardenas, J.

C. T. Phare, Y. D. Lee, J. Cardenas, and M. Lipson, "Graphene electro-optic modulator with 30 GHz bandwidth," Nat. Photonics,  9, 511–514 (2015).
[Crossref]

Chan, E. H. W.

R. A. Minasian, E. H. W. Chan, and X. Yi, "Microwave photonic signal processing," Opt. Express,  35(21), 22918–22936 (2013).
[Crossref]

Chow, J. M.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Cicak, K.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, "Bidirectional and efficient conversion between microwave and optical light," Nat. Phys.,  10, 321–326 (2014).
[Crossref]

Cleland, A. N.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, "Nanomechanical coupling between microwave and optical photons," Nat. Phys.,  9, 712–716 (2013).
[Crossref]

Colli, A.

S. Awan, A. Lombardo, A. Colli, G. Privitera, T. Kulmala, J. Kivioja, and A. Ferrari, "Transport conductivity of graphene at RF and microwave frequencies," 2D Mater.,  3(1), 015010 (2016).
[Crossref]

Collodo, M. C.

Dai, X.

Dai, Y.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, "Transfer matrix method for optics in graphene layers," J. Phys.: Condens. Matter,  25(21), 215301 (2013).

Dias, C. J.

R. Igreja and C. J. Dias, "Analytical evaluation of the interdigital electrodes capacitance for a multi-layered structure," Sensor. Actuat. A,  112(2), 291–301 (2018).
[Crossref]

DiCarlo, L.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Ding, Y.

Eleuch, H.

E. A. Sete and H. Eleuch, "Strong squeezing and robust entanglement in cavity electromechanics," Phys. Rev. A 89, 013841 (2014).
[Crossref]

S. A. Eyob and H. Eleuch, "Strong squeezing and robust entanglement in cavity electromechanics," Phys. Rev. A,  89, 013841 (2014).
[Crossref]

Englund, D.

M. Atature, D. Englund, N. Vamivakas, S. Y. Lee, and J. Wrachtrup, "Material platforms for spin-based photonic quantum technologies," Nat. Rev. Mater.,  3, 38–51 (2018).
[Crossref]

Eyob, S. A.

S. A. Eyob and H. Eleuch, "Strong squeezing and robust entanglement in cavity electromechanics," Phys. Rev. A,  89, 013841 (2014).
[Crossref]

Fan, D.

Fang, K.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, "Optical transduction and routing of microwave phonons in cavity-optomechanical circuits," Nat. Photonics.,  10, 489–496 (2014).
[Crossref]

Feofanov, A. K.

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, "On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator," Phys. Rev. A,  94, 053815 (2016).
[Crossref]

Ferrari, A.

S. Awan, A. Lombardo, A. Colli, G. Privitera, T. Kulmala, J. Kivioja, and A. Ferrari, "Transport conductivity of graphene at RF and microwave frequencies," 2D Mater.,  3(1), 015010 (2016).
[Crossref]

Fink, J. M.

Frunzio, L.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Gambetta, J. M.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Girvin, S. M.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Grigorenko, A. N.

D. Ansell, I. P. Radko, Z. Han, F. J. Rodriguez, S. I. Bozhevolnyi, and A. N. Grigorenko, "Hybrid graphene plasmonic waveguide modulators," Nat. Commun.,  6, 8846 (2015).
[Crossref] [PubMed]

Guo, X.

X. Guo, X. Li, N. Liu, and Z. Y. Ou, "Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs," Sci. Rep.,  6, 30214 (2016).
[Crossref] [PubMed]

Hafezi, M.

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Coherent frequency up-conversion of microwaves to the optical telecommunications band in an Er:YSO crystal," Phys. Rev. A,  92, 062313 (2015).
[Crossref]

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Atomic interface between microwave and optical photons," Phys. Rev. A,  85, 020302 (2012).
[Crossref]

Han, Z.

D. Ansell, I. P. Radko, Z. Han, F. J. Rodriguez, S. I. Bozhevolnyi, and A. N. Grigorenko, "Hybrid graphene plasmonic waveguide modulators," Nat. Commun.,  6, 8846 (2015).
[Crossref] [PubMed]

Igreja, R.

R. Igreja and C. J. Dias, "Analytical evaluation of the interdigital electrodes capacitance for a multi-layered structure," Sensor. Actuat. A,  112(2), 291–301 (2018).
[Crossref]

Javerzac-Galy, C.

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, "On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator," Phys. Rev. A,  94, 053815 (2016).
[Crossref]

Jiang, L.

Johnson, B. R.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Kim, Z.

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Coherent frequency up-conversion of microwaves to the optical telecommunications band in an Er:YSO crystal," Phys. Rev. A,  92, 062313 (2015).
[Crossref]

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Atomic interface between microwave and optical photons," Phys. Rev. A,  85, 020302 (2012).
[Crossref]

Kippenberg, T. J.

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, "On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator," Phys. Rev. A,  94, 053815 (2016).
[Crossref]

Kivioja, J.

S. Awan, A. Lombardo, A. Colli, G. Privitera, T. Kulmala, J. Kivioja, and A. Ferrari, "Transport conductivity of graphene at RF and microwave frequencies," 2D Mater.,  3(1), 015010 (2016).
[Crossref]

Kulmala, T.

S. Awan, A. Lombardo, A. Colli, G. Privitera, T. Kulmala, J. Kivioja, and A. Ferrari, "Transport conductivity of graphene at RF and microwave frequencies," 2D Mater.,  3(1), 015010 (2016).
[Crossref]

Lee, J. H.

J. H. Lee, P. E. Loya, J. Lou, and E. L. Thomas, "Dynamic mechanical behavior of multilayer graphene via supersonic projectile penetration," Science,  346(6213), 1092–1096 (2014).
[Crossref] [PubMed]

Lee, S. Y.

M. Atature, D. Englund, N. Vamivakas, S. Y. Lee, and J. Wrachtrup, "Material platforms for spin-based photonic quantum technologies," Nat. Rev. Mater.,  3, 38–51 (2018).
[Crossref]

Lee, Y. D.

C. T. Phare, Y. D. Lee, J. Cardenas, and M. Lipson, "Graphene electro-optic modulator with 30 GHz bandwidth," Nat. Photonics,  9, 511–514 (2015).
[Crossref]

Lehnert, K. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, "Bidirectional and efficient conversion between microwave and optical light," Nat. Phys.,  10, 321–326 (2014).
[Crossref]

Leuchs, G.

Lev, B. L.

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Coherent frequency up-conversion of microwaves to the optical telecommunications band in an Er:YSO crystal," Phys. Rev. A,  92, 062313 (2015).
[Crossref]

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Atomic interface between microwave and optical photons," Phys. Rev. A,  85, 020302 (2012).
[Crossref]

Li, W.

M. Qasymeh, W. Li, and J. Yao, "Frequency-Tunable Microwave Generation Based on Time-Delayed Optical Combs," IEEE Trans. Microw. Theory Tech.,  59(11), 2987–2993 (2011).
[Crossref]

Li, X.

X. Guo, X. Li, N. Liu, and Z. Y. Ou, "Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs," Sci. Rep.,  6, 30214 (2016).
[Crossref] [PubMed]

Likharev, K. K.

K. K. Likharev and V. K. Semenov, “RSFQ logic/memory family: a new Josephson-junction technology for sub-terahertz-clock-frequency digital systems,"” IEEE Trans. Appl. Supercond.,  1(1), 3–28 (1991).
[Crossref]

Lipson, M.

C. T. Phare, Y. D. Lee, J. Cardenas, and M. Lipson, "Graphene electro-optic modulator with 30 GHz bandwidth," Nat. Photonics,  9, 511–514 (2015).
[Crossref]

Liu, N.

X. Guo, X. Li, N. Liu, and Z. Y. Ou, "Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs," Sci. Rep.,  6, 30214 (2016).
[Crossref] [PubMed]

Liu, X.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, "Transfer matrix method for optics in graphene layers," J. Phys.: Condens. Matter,  25(21), 215301 (2013).

Lombardo, A.

S. Awan, A. Lombardo, A. Colli, G. Privitera, T. Kulmala, J. Kivioja, and A. Ferrari, "Transport conductivity of graphene at RF and microwave frequencies," 2D Mater.,  3(1), 015010 (2016).
[Crossref]

Loncar, M.

M. Soltani, M. Zhang, C. Ryan, G. J. Ribeill, C. Wang, and M. Loncar, "Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators," Phys. Rev. A,  96, 043808 (2017).
[Crossref]

Lou, J.

J. H. Lee, P. E. Loya, J. Lou, and E. L. Thomas, "Dynamic mechanical behavior of multilayer graphene via supersonic projectile penetration," Science,  346(6213), 1092–1096 (2014).
[Crossref] [PubMed]

Loya, P. E.

J. H. Lee, P. E. Loya, J. Lou, and E. L. Thomas, "Dynamic mechanical behavior of multilayer graphene via supersonic projectile penetration," Science,  346(6213), 1092–1096 (2014).
[Crossref] [PubMed]

Luan, X.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, "Optical transduction and routing of microwave phonons in cavity-optomechanical circuits," Nat. Photonics.,  10, 489–496 (2014).
[Crossref]

Majer, J.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Marquardt, C.

Matheny, M. H.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, "Optical transduction and routing of microwave phonons in cavity-optomechanical circuits," Nat. Photonics.,  10, 489–496 (2014).
[Crossref]

Minasian, R. A.

R. A. Minasian, E. H. W. Chan, and X. Yi, "Microwave photonic signal processing," Opt. Express,  35(21), 22918–22936 (2013).
[Crossref]

Novak, D.

J. Capmany and D. Novak, "Microwave photonics combines two worlds," Nat. Photonics,  1, 319–330 (2007).
[Crossref]

Orozco, L. A.

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Coherent frequency up-conversion of microwaves to the optical telecommunications band in an Er:YSO crystal," Phys. Rev. A,  92, 062313 (2015).
[Crossref]

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Atomic interface between microwave and optical photons," Phys. Rev. A,  85, 020302 (2012).
[Crossref]

Ou, Z. Y.

X. Guo, X. Li, N. Liu, and Z. Y. Ou, "Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs," Sci. Rep.,  6, 30214 (2016).
[Crossref] [PubMed]

Y. Ding and Z. Y. Ou, "Frequency downconversion for a quantum network," Opt. Lett.,  35 (15), 2591–2593 (2010).
[Crossref] [PubMed]

Painter, O.

Pan, S.

Peterson, R. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, "Bidirectional and efficient conversion between microwave and optical light," Nat. Phys.,  10, 321–326 (2014).
[Crossref]

Phare, C. T.

C. T. Phare, Y. D. Lee, J. Cardenas, and M. Lipson, "Graphene electro-optic modulator with 30 GHz bandwidth," Nat. Photonics,  9, 511–514 (2015).
[Crossref]

Plekhanov, K.

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, "On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator," Phys. Rev. A,  94, 053815 (2016).
[Crossref]

Privitera, G.

S. Awan, A. Lombardo, A. Colli, G. Privitera, T. Kulmala, J. Kivioja, and A. Ferrari, "Transport conductivity of graphene at RF and microwave frequencies," 2D Mater.,  3(1), 015010 (2016).
[Crossref]

Purdy, T. P.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, "Bidirectional and efficient conversion between microwave and optical light," Nat. Phys.,  10, 321–326 (2014).
[Crossref]

Qasymeh, M.

M. Qasymeh, "Giant Amplification of Terahertz Waves in a Nonlinear Graphene Layered Medium," IEEE Photonics Technol. Lett.,  30(1), 35–38 (2018).
[Crossref]

M. Qasymeh, "Phase-Matched Coupling and Frequency Conversion of Terahertz Waves in a Nonlinear Graphene Waveguide," J. Lightwave Technol.,  35(9), 1654–1662 (2017).
[Crossref]

M. Qasymeh, W. Li, and J. Yao, "Frequency-Tunable Microwave Generation Based on Time-Delayed Optical Combs," IEEE Trans. Microw. Theory Tech.,  59(11), 2987–2993 (2011).
[Crossref]

Radko, I. P.

D. Ansell, I. P. Radko, Z. Han, F. J. Rodriguez, S. I. Bozhevolnyi, and A. N. Grigorenko, "Hybrid graphene plasmonic waveguide modulators," Nat. Commun.,  6, 8846 (2015).
[Crossref] [PubMed]

Regal, C. A.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, "Bidirectional and efficient conversion between microwave and optical light," Nat. Phys.,  10, 321–326 (2014).
[Crossref]

Ribeill, G. J.

M. Soltani, M. Zhang, C. Ryan, G. J. Ribeill, C. Wang, and M. Loncar, "Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators," Phys. Rev. A,  96, 043808 (2017).
[Crossref]

Rodriguez, F. J.

D. Ansell, I. P. Radko, Z. Han, F. J. Rodriguez, S. I. Bozhevolnyi, and A. N. Grigorenko, "Hybrid graphene plasmonic waveguide modulators," Nat. Commun.,  6, 8846 (2015).
[Crossref] [PubMed]

Rolston, S. L.

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Coherent frequency up-conversion of microwaves to the optical telecommunications band in an Er:YSO crystal," Phys. Rev. A,  92, 062313 (2015).
[Crossref]

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Atomic interface between microwave and optical photons," Phys. Rev. A,  85, 020302 (2012).
[Crossref]

Rueda, A.

Ryan, C.

M. Soltani, M. Zhang, C. Ryan, G. J. Ribeill, C. Wang, and M. Loncar, "Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators," Phys. Rev. A,  96, 043808 (2017).
[Crossref]

Schoelkopf, R. J.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Schunk, G.

Schuster, D. I.

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Schwefel, H. G. L.

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics(Cambridge University, 1997).
[Crossref]

Sedlmeir, F.

Seeds, A. J.

Semenov, V. K.

K. K. Likharev and V. K. Semenov, “RSFQ logic/memory family: a new Josephson-junction technology for sub-terahertz-clock-frequency digital systems,"” IEEE Trans. Appl. Supercond.,  1(1), 3–28 (1991).
[Crossref]

Sete, E. A.

E. A. Sete and H. Eleuch, "Strong squeezing and robust entanglement in cavity electromechanics," Phys. Rev. A 89, 013841 (2014).
[Crossref]

Shi, X.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, "Transfer matrix method for optics in graphene layers," J. Phys.: Condens. Matter,  25(21), 215301 (2013).

Simmonds, R. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, "Bidirectional and efficient conversion between microwave and optical light," Nat. Phys.,  10, 321–326 (2014).
[Crossref]

Soltani, M.

M. Soltani, M. Zhang, C. Ryan, G. J. Ribeill, C. Wang, and M. Loncar, "Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators," Phys. Rev. A,  96, 043808 (2017).
[Crossref]

Stiller, B.

Strekalov, D. V.

Taylor, J. M.

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Coherent frequency up-conversion of microwaves to the optical telecommunications band in an Er:YSO crystal," Phys. Rev. A,  92, 062313 (2015).
[Crossref]

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Atomic interface between microwave and optical photons," Phys. Rev. A,  85, 020302 (2012).
[Crossref]

Thomas, E. L.

J. H. Lee, P. E. Loya, J. Lou, and E. L. Thomas, "Dynamic mechanical behavior of multilayer graphene via supersonic projectile penetration," Science,  346(6213), 1092–1096 (2014).
[Crossref] [PubMed]

Toth, L. D.

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, "On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator," Phys. Rev. A,  94, 053815 (2016).
[Crossref]

Tsang, M.

M. Tsang, "Cavity quantum electro-optics. II. Input-output relations between traveling optical and microwave fields," Phys. Rev. A,  84, 043845 (2011).
[Crossref]

M. Tsang, "Cavity quantum electro-optics," Phys. Rev. A,  81, 063837 (2010).
[Crossref]

Vainsencher, A.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, "Nanomechanical coupling between microwave and optical photons," Nat. Phys.,  9, 712–716 (2013).
[Crossref]

Vamivakas, N.

M. Atature, D. Englund, N. Vamivakas, S. Y. Lee, and J. Wrachtrup, "Material platforms for spin-based photonic quantum technologies," Nat. Rev. Mater.,  3, 38–51 (2018).
[Crossref]

Vogl, U.

Wang, C.

M. Soltani, M. Zhang, C. Ryan, G. J. Ribeill, C. Wang, and M. Loncar, "Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators," Phys. Rev. A,  96, 043808 (2017).
[Crossref]

Williams, K. J.

Wrachtrup, J.

M. Atature, D. Englund, N. Vamivakas, S. Y. Lee, and J. Wrachtrup, "Material platforms for spin-based photonic quantum technologies," Nat. Rev. Mater.,  3, 38–51 (2018).
[Crossref]

Wu, Y.

Xiang, Y.

Xu, H.

Yao, J.

S. Pan and J. Yao, "Photonics-Based Broadband Microwave Measurement," J. Lightwave Technol.,  35(16), 3498–3513 (2016).
[Crossref]

M. Qasymeh, W. Li, and J. Yao, "Frequency-Tunable Microwave Generation Based on Time-Delayed Optical Combs," IEEE Trans. Microw. Theory Tech.,  59(11), 2987–2993 (2011).
[Crossref]

J. Yao, "Microwave Photonics," J. Lightwave Technol.,  27(3), 314–335 (2009).
[Crossref]

Yi, X.

R. A. Minasian, E. H. W. Chan, and X. Yi, "Microwave photonic signal processing," Opt. Express,  35(21), 22918–22936 (2013).
[Crossref]

Zhan, T.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, "Transfer matrix method for optics in graphene layers," J. Phys.: Condens. Matter,  25(21), 215301 (2013).

Zhang, M.

M. Soltani, M. Zhang, C. Ryan, G. J. Ribeill, C. Wang, and M. Loncar, "Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators," Phys. Rev. A,  96, 043808 (2017).
[Crossref]

Zi, J.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, "Transfer matrix method for optics in graphene layers," J. Phys.: Condens. Matter,  25(21), 215301 (2013).

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics(Cambridge University, 1997).
[Crossref]

2D Mater. (1)

S. Awan, A. Lombardo, A. Colli, G. Privitera, T. Kulmala, J. Kivioja, and A. Ferrari, "Transport conductivity of graphene at RF and microwave frequencies," 2D Mater.,  3(1), 015010 (2016).
[Crossref]

IEEE Photonics Technol. Lett. (1)

M. Qasymeh, "Giant Amplification of Terahertz Waves in a Nonlinear Graphene Layered Medium," IEEE Photonics Technol. Lett.,  30(1), 35–38 (2018).
[Crossref]

IEEE Trans. Appl. Supercond. (1)

K. K. Likharev and V. K. Semenov, “RSFQ logic/memory family: a new Josephson-junction technology for sub-terahertz-clock-frequency digital systems,"” IEEE Trans. Appl. Supercond.,  1(1), 3–28 (1991).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

M. Qasymeh, W. Li, and J. Yao, "Frequency-Tunable Microwave Generation Based on Time-Delayed Optical Combs," IEEE Trans. Microw. Theory Tech.,  59(11), 2987–2993 (2011).
[Crossref]

J. Lightwave Technol. (4)

J. Phys.: Condens. Matter (1)

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, "Transfer matrix method for optics in graphene layers," J. Phys.: Condens. Matter,  25(21), 215301 (2013).

Nat. Commun. (1)

D. Ansell, I. P. Radko, Z. Han, F. J. Rodriguez, S. I. Bozhevolnyi, and A. N. Grigorenko, "Hybrid graphene plasmonic waveguide modulators," Nat. Commun.,  6, 8846 (2015).
[Crossref] [PubMed]

Nat. Photonics (2)

C. T. Phare, Y. D. Lee, J. Cardenas, and M. Lipson, "Graphene electro-optic modulator with 30 GHz bandwidth," Nat. Photonics,  9, 511–514 (2015).
[Crossref]

J. Capmany and D. Novak, "Microwave photonics combines two worlds," Nat. Photonics,  1, 319–330 (2007).
[Crossref]

Nat. Photonics. (1)

K. Fang, M. H. Matheny, X. Luan, and O. Painter, "Optical transduction and routing of microwave phonons in cavity-optomechanical circuits," Nat. Photonics.,  10, 489–496 (2014).
[Crossref]

Nat. Phys. (2)

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, "Nanomechanical coupling between microwave and optical photons," Nat. Phys.,  9, 712–716 (2013).
[Crossref]

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, "Bidirectional and efficient conversion between microwave and optical light," Nat. Phys.,  10, 321–326 (2014).
[Crossref]

Nat. Rev. Mater. (1)

M. Atature, D. Englund, N. Vamivakas, S. Y. Lee, and J. Wrachtrup, "Material platforms for spin-based photonic quantum technologies," Nat. Rev. Mater.,  3, 38–51 (2018).
[Crossref]

Nature (1)

L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, "Demonstration of two-qubit algorithms with a superconducting quantum processor," Nature,  460, 240–244 (2009).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (8)

M. Soltani, M. Zhang, C. Ryan, G. J. Ribeill, C. Wang, and M. Loncar, "Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators," Phys. Rev. A,  96, 043808 (2017).
[Crossref]

E. A. Sete and H. Eleuch, "Strong squeezing and robust entanglement in cavity electromechanics," Phys. Rev. A 89, 013841 (2014).
[Crossref]

S. A. Eyob and H. Eleuch, "Strong squeezing and robust entanglement in cavity electromechanics," Phys. Rev. A,  89, 013841 (2014).
[Crossref]

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Coherent frequency up-conversion of microwaves to the optical telecommunications band in an Er:YSO crystal," Phys. Rev. A,  92, 062313 (2015).
[Crossref]

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, "On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator," Phys. Rev. A,  94, 053815 (2016).
[Crossref]

M. Tsang, "Cavity quantum electro-optics," Phys. Rev. A,  81, 063837 (2010).
[Crossref]

M. Tsang, "Cavity quantum electro-optics. II. Input-output relations between traveling optical and microwave fields," Phys. Rev. A,  84, 043845 (2011).
[Crossref]

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, "Atomic interface between microwave and optical photons," Phys. Rev. A,  85, 020302 (2012).
[Crossref]

Sci. Rep. (1)

X. Guo, X. Li, N. Liu, and Z. Y. Ou, "Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs," Sci. Rep.,  6, 30214 (2016).
[Crossref] [PubMed]

Science (1)

J. H. Lee, P. E. Loya, J. Lou, and E. L. Thomas, "Dynamic mechanical behavior of multilayer graphene via supersonic projectile penetration," Science,  346(6213), 1092–1096 (2014).
[Crossref] [PubMed]

Sensor. Actuat. A (1)

R. Igreja and C. J. Dias, "Analytical evaluation of the interdigital electrodes capacitance for a multi-layered structure," Sensor. Actuat. A,  112(2), 291–301 (2018).
[Crossref]

Other (1)

M. O. Scully and M. S. Zubairy, Quantum Optics(Cambridge University, 1997).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Multilayer graphene structure connected in an interdigital fashion.
Fig. 2
Fig. 2 The equivalent electric capacitance of the multilayer graphene structure.
Fig. 3
Fig. 3 Propagation constant and the group velocity versus optical frequency. The optical pump (i.e., f1) and the upper and lower side bands (i.e., f2 and f3, respectively) are shown for f m = 50 G H z .
Fig. 4
Fig. 4 Transmittance versus optical frequency. Different numbers of layers, i.e., N, are considered.
Fig. 5
Fig. 5 The conversion rate g versus microwave signal frequency. Different intrinsic electron densities n0 are considered.
Fig. 6
Fig. 6 The conversion rate g versus the voltage of the microwave signal. Different microwave signal frequencies are considered.
Fig. 7
Fig. 7 The conversion rate g versus the medium length and the optical pump operator.
Fig. 8
Fig. 8 The optical and microwave decay coefficients versus microwave frequency.
Fig. 9
Fig. 9 The number of converted photons, i.e., A ^ 2 A ^ 2 , versus the frequency microwave signal.

Equations (48)

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

σ s = i q 2 4 π l n ( 2 μ c ( f + i τ 1 ) 2 μ c + ( f + i τ 1 ) ) + i q 2 K B T π 2 ( f + i τ 1 ) ( μ c K B T + 2 l n ( e μ c K B T + 1 ) ) ,
μ c = V f π n 0 + 2 C T q v m ,
v m = ν e i 2 π f m t + c . c . ,
μ c = μ c ' + ν μ c e i 2 π f m t + c . c . ,
μ c ' = V f π n 0 , and μ c = V f C T q π n 0 .
σ s = σ s ' + ν σ s e i 2 π f m t + c . c . ,
σ s ' = i q 2 4 π l n ( 2 μ c ' ( f + i τ 1 ) 2 μ c ' + ( f + i τ 1 ) ) + i q 2 K B T π 2 ( f + i τ 1 ) ( μ c ' K B T + 2 l n ( e μ c ' K B T + 1 ) ) ,
σ s = i q 2 π ( f + i τ 1 ) 4 ( μ c ' ) 2 ( f + i τ 1 ) 2 2 μ c + i q 2 K B T π 2 ( f + i τ 1 ) t a n h ( μ c ' 2 K B T ) μ c K B T .
cos  ( d β ) = cos  ( d ε 2 π f c ) i Z 0 2 ε sin  ( d ε 2 π f c ) σ s ,
β = β + ν β e i 2 π f m t + c . c . ,
β = i Z 0 2 d ε sin  ( d 2 π f ε c ) sin  ( d β ) σ s .
ε e f f j = ε e f f j ' + ν ε e f f j e i 2 π f m t + c . c . ,
ε e f f j ' = ( β j ' k 0 j ) 2 , and ε e f f j = 2 β j ' β j k 0 j 2 .
E j = u j ( e i 2 π f j t + i β j ' x + c . c . ) e ^ y ,
H = 1 2 V ( ε 0 ε e f f | E t |   2 + μ 0 | H t |   2 ) V ,
H = H 0 + H 1 ,
H 0 = V j = 1 2 ε e f f j ' ε 0 u j * u j + c . c . ,
H 1 = V ε e f f 2 ε 0 u 1 * ν * u 2 + c . c .
u j = ( f j ε e f f j ' ε 0 V ) 1 2 a ^ j , and ν = ( f m C T A ) 1 2 b ^ ,
H ^ = H 0 ^ + H 1 ^ ,
H 0 ^ = f m b ^ b ^ + f 1 a ^ 1 a ^ 1 + f 2 a ^ 2 a ^ 2 ,
H 1 ^ = g ( a ^ 2 b ^ a ^ 1 + h . c . ) ,
g = ε e f f 2 f 1 f 2 ε e f f 1 ' ε e f f 2 ' f m C T A .
a ^ 1 t = i f 1 a ^ 1 i g b ^ a ^ 2 ,
a ^ 2 t = i f 2 a ^ 2 i g b ^ a ^ 1 ,
b ^ t = i f m b ^ i g a ^ 1 a ^ 2 .
A ^ 2 t = i g A 1 B ^ ,
B ^ t = i g A 1 * A ^ 2 .
A ^ 2 ( t ) = A ^ 2 ( 0 ) cos  ( g | A 1 | t ) i e i ϕ 0 B ^ ( 0 ) sin  ( g | A 1 | t ) ,
g | A 1 | t = π 2 ,
| A 1 | = π v g 2 L | g | .
A ^ 2 t = Γ 2 A ^ 2 i g A 1 B ^ + Γ N 2 ,
B ^ t = Γ m 2 B ^ i g A 1 * A ^ 2 + Γ m N m ,
A ^ 2 t = Γ 2 A ^ 2 i g A 1 B ^ ,
B ^ t = Γ m 2 B ^ i g A 1 A ^ 2 .
A ^ 2 A ^ 2 = A g t e t ( Γ + Γ m ) 4 A ^ 2 B ^ | t = 0 ,
B ^ A ^ 2 | t = 0 = B ^ | t = 0 A ^ 2 | t = 0 A ^ 2 A ^ 2 | t = 0 B ^ B ^ | t = 0 .
S N R = B ^ | t = 0 A ^ 2 | t = 0 ,
A ^ 2 A ^ 2 t = Γ 2 A ^ 2 A ^ 2 + g A A ^ 2 B ^ .
B ^ B ^ t = Γ m 2 B ^ B ^ g A B ^ A ^ 2 .
A ^ 2 B ^ t = Γ m 2 A ^ 2 B ^ g A A ^ 2 A ^ 2 ,
B ^ A ^ 2 t = Γ 2 B ^ A ^ 2 + g A B ^ B ^ .
A ( t ) = C s ( t ) ,
A ^ 2 A ^ 2 = e ( Γ + Γ m ) 4 t [ A g A ^ 2 B ^ | t = 0 sin ( α t ) α + A ^ 2 A ^ 2 | t = 0 cos ( α t ) ( Γ Γ m ) 4 A ^ 2 A ^ 2 | t = 0 sin ( α t ) α ] ,
B ^ B ^ = e ( Γ + Γ m ) 4 t [ A g B A ^ 2 | t = 0 sin ( α t ) α + B ^ B ^ | t = 0 cos ( α t ) + ( Γ Γ m ) 4 B ^ B ^ | t = 0 sin ( α t ) α ] ,
α = g 2 A 2 ( Γ Γ m 4 ) 2 .
S N R = A g sin ( α t ) α cos ( α t ) Γ Γ m 4 sin ( α t ) α A ^ 2 B ^ | t = 0 A ^ 2 A ^ 2 | t = 0 ,
S N R = A ^ 2 B ^ | t = 0 A ^ 2 A ^ 2 | t = 0 .

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