Abstract

We theoretically study electro-optic light modulation based on a quantum model where the linear electro-optic effect and the externally applied microwave field result in the interaction between optical cavity modes. The model assumes that the number of interacting modes is finite, and effects of the mode overlapping coefficient on the strength of the intermode interaction can be taken into account through dependence of the coupling coefficient on the mode characteristics. We show that, under certain conditions, the model is exactly solvable and can be analyzed using the technique of the Jordan mappings for the su(2) Lie algebra. Analytical results are applied to study effects of light modulation on the frequency dependence of the photon counting rate. In contrast to the limiting case of an infinitely large number of interacting modes, when the number of interacting modes is finite, the sideband intensities reveal strongly nonmonotonic behavior supplemented with asymmetry of the intensity distribution provided the pumped mode is not central. We also analyze different regimes of two-modulator transmission and establish the conditions of validity of the semiclassical approximation by applying the methods of polynomially deformed Lie algebras for analysis of the model with quantized microwave field.

© 2017 Optical Society of America

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2016 (2)

2015 (1)

S. P. Kotova, S. A. Samagin, E. P. Pozhidaev, and A. D. Kiselev, “Light modulation in planar aligned short-pitch deformed-helix ferroelectric liquid crystals,” Phys. Rev. E 92, 062502 (2015).
[Crossref]

2014 (4)

A. D. Kiselev and V. G. Chigrinov, “Optics of short-pitch deformed-helix ferroelectric liquid crystals: symmetries, exceptional points, and polarization-resolved angular patterns,” Phys. Rev. E 90, 042504 (2014).
[Crossref]

A. Rasoloniaina, V. Huet, T. K. N. Nguyên, E. L. Cren, M. Mortier, L. Michely, Y. Dumeige, and P. Féron, “Controling the coupling properties of active ultrahigh-Q WGM microcavities from undercoupling to selective amplification,” Sci. Rep. 4, 4023 (2014).
[Crossref]

E. P. Pozhidaev, A. K. Srivastava, A. D. Kiselev, V. G. Chigrinov, V. V. Vashchenko, A. V. Krivoshey, M. V. Minchenko, and H.-S. Kwok, “Enhanced orientational Kerr effect in vertically aligned deformed helix ferroelectric liquid crystals,” Opt. Lett. 39, 2900–2903 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

2013 (1)

E. P. Pozhidaev, A. D. Kiselev, A. K. Srivastava, V. G. Chigrinov, H.-S. Kwok, and M. V. Minchenko, “Orientational Kerr effect and phase modulation of light in deformed-helix ferroelectric liquid crystals with subwavelength pitch,” Phys. Rev. E 87, 052502 (2013).
[Crossref]

2012 (2)

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

L. Olislager, I. Mbodji, E. Woodhead, J. Cussey, L. Furfaro, P. Emplit, S. Massar, K. P. Huy, and J.-M. Merolla, “Implementing two-photon interference in the frequency domain with electro-optic phase modulators,” New J. Phys. 14, 043015 (2012).
[Crossref]

2011 (2)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[Crossref]

J. Capmany and C. Fernández-Pousa, “Quantum modelling of electro-optic modulators,” Laser Photon. Rev. 5, 750–772 (2011).
[Crossref]

2010 (5)

A. Savchenkov, A. Matsko, W. Liang, V. Ilchenko, D. Seidel, and L. Maleki, “Single-sideband electro-optical modulator and tunable microwave photonic receiver,” IEEE Trans. Microw. Theory Tech. 58, 3167–3174 (2010).
[Crossref]

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

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. P. Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev. 4, 337–354 (2010).
[Crossref]

J. Capmany and C. R. Fernández-Pousa, “Quantum model for electro-optical phase modulation,” J. Opt. Soc. Am. B 27, A119–A129 (2010).
[Crossref]

2009 (4)

A. A. Savchenkov, W. Liang, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Tunable optical single-sideband modulator with complete sideband suppression,” Opt. Lett. 34, 1300–1302 (2009).
[Crossref]

S. Sensarn, G. Y. Yin, and S. E. Harris, “Observation of nonlocal modulation with entangled photons,” Phys. Rev. Lett. 103, 163601 (2009).
[Crossref]

P. Kumar and A. Prabhakar, “Evolution of quantum states in an electro-optic phase modulator,” IEEE J. Quantum Electron. 45, 149–156 (2009).
[Crossref]

M. Tsang, J. H. Shapiro, and S. Lloyd, “Quantum theory of optical temporal phase and instantaneous frequency, II: continuous-time limit and state-variable approach to phase-locked loop design,” Phys. Rev. A 79, 053843 (2009).
[Crossref]

2008 (3)

M. Tsang, J. H. Shapiro, and S. Lloyd, “Quantum theory of optical temporal phase and instantaneous frequency,” Phys. Rev. A 78, 053820 (2008).
[Crossref]

P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101, 103601 (2008).
[Crossref]

S. E. Harris, “Nonlocal modulation of entangled photons,” Phys. Rev. A 78, 021807 (2008).
[Crossref]

2007 (2)

2006 (2)

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[Crossref]

A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73, 024305 (2006).
[Crossref]

2005 (2)

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Inf. 1, 5–37 (2005).
[Crossref]

D. Wu, H. Chen, W. She, and W. Lee, “Wave coupling theory of the linear electro-optic effect in a linear absorbent medium,” J. Opt. Soc. Am. B 22, 2366–2371 (2005).
[Crossref]

2003 (2)

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Whispering-gallery-mode electro-optic modulator and photonic microwave receiver,” J. Opt. Soc. Am. B 20, 333–342 (2003).
[Crossref]

I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis-Cummings problem,” Phys. Rev. A 67, 053808 (2003).
[Crossref]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

2001 (2)

W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195, 303–311 (2001).
[Crossref]

D. A. Cohen and A. F. J. Levy, “Microphotonic components for a mm-wave receiver,” Solid State Electron. 45, 495–505 (2001).
[Crossref]

2000 (2)

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref]

G. Ribordy, J. Brendel, J.-D. Gautier, N. Gisin, and H. Zbinden, “Long-distance entanglement-based quantum key distribution,” Phys. Rev. A 63, 012309 (2000).
[Crossref]

1999 (3)

J.-M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in sidebands of phase-modulated light for quantum cryptography,” Phys. Rev. Lett. 82, 1656–1659 (1999).
[Crossref]

J.-M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60, 1899–1905 (1999).
[Crossref]

J.-M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography,” Opt. Lett. 24, 104–106 (1999).
[Crossref]

1996 (1)

A. Muller, H. Zbinden, and N. Gisin, “Quantum cryptography over 23  km in installed under-lake telecom fibre,” Europhys. Lett. 33, 335–340 (1996).
[Crossref]

1995 (1)

A. Muller, H. Zbinden, and N. Gisin, “Underwater quantum coding,” Nature 378, 449 (1995).
[Crossref]

1993 (2)

A. Muller, J. Breguet, and N. Gisin, “Experimental demonstration of quantum cryptography using polarized photons in optical fiber over more than 1  km,” Europhys. Lett. 23, 383–388 (1993).
[Crossref]

V. V. Dodonov, A. B. Klimov, and D. E. Nikonov, “Quantum phenomena in nonstationary media,” Phys. Rev. A 47, 4422–4429 (1993).
[Crossref]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref]

1979 (1)

L. S. Brown and L. J. Carson, “Quantum-mechanical parametric amplification,” Phys. Rev. A 20, 2486–2497 (1979).
[Crossref]

1963 (1)

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes: II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

1961 (1)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I,” Phys. Rev. 124, 1646–1654 (1961).
[Crossref]

1927 (1)

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. London A 114, 243–265 (1927).
[Crossref]

Andersen, U. L.

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev. 4, 337–354 (2010).
[Crossref]

Anisimov, A. A.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Bannik, O. I.

Belthangady, C.

P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101, 103601 (2008).
[Crossref]

Biedenharn, L. C.

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics: Theory and Application, Vol. 8 of Encyclopedia of Mathematics and Its Applications (Addison–Wesley, 1981).

Blinov, L. M.

L. M. Blinov and V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer-Verlag, 1994).

Bloch, M.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Breguet, J.

A. Muller, J. Breguet, and N. Gisin, “Experimental demonstration of quantum cryptography using polarized photons in optical fiber over more than 1  km,” Europhys. Lett. 23, 383–388 (1993).
[Crossref]

Brendel, J.

G. Ribordy, J. Brendel, J.-D. Gautier, N. Gisin, and H. Zbinden, “Long-distance entanglement-based quantum key distribution,” Phys. Rev. A 63, 012309 (2000).
[Crossref]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref]

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Nikonov, D. E.

V. V. Dodonov, A. B. Klimov, and D. E. Nikonov, “Quantum phenomena in nonstationary media,” Phys. Rev. A 47, 4422–4429 (1993).
[Crossref]

Ogawa, T.

M. Hayashi, S. Ishizaka, A. Kawachi, G. Kimura, and T. Ogawa, Introduction to Quantum Information Science, Graduate Texts in Physics (Springer, 2015).

Olislager, L.

L. Olislager, I. Mbodji, E. Woodhead, J. Cussey, L. Furfaro, P. Emplit, S. Massar, K. P. Huy, and J.-M. Merolla, “Implementing two-photon interference in the frequency domain with electro-optic phase modulators,” New J. Phys. 14, 043015 (2012).
[Crossref]

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. P. Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

Ortigosa-Blanch, A.

A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73, 024305 (2006).
[Crossref]

Patois, F.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Pirandola, S.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
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M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
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J.-M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography,” Opt. Lett. 24, 104–106 (1999).
[Crossref]

J.-M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60, 1899–1905 (1999).
[Crossref]

Pozhidaev, E. P.

S. P. Kotova, S. A. Samagin, E. P. Pozhidaev, and A. D. Kiselev, “Light modulation in planar aligned short-pitch deformed-helix ferroelectric liquid crystals,” Phys. Rev. E 92, 062502 (2015).
[Crossref]

E. P. Pozhidaev, A. K. Srivastava, A. D. Kiselev, V. G. Chigrinov, V. V. Vashchenko, A. V. Krivoshey, M. V. Minchenko, and H.-S. Kwok, “Enhanced orientational Kerr effect in vertically aligned deformed helix ferroelectric liquid crystals,” Opt. Lett. 39, 2900–2903 (2014).
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E. P. Pozhidaev, A. D. Kiselev, A. K. Srivastava, V. G. Chigrinov, H.-S. Kwok, and M. V. Minchenko, “Orientational Kerr effect and phase modulation of light in deformed-helix ferroelectric liquid crystals with subwavelength pitch,” Phys. Rev. E 87, 052502 (2013).
[Crossref]

Prabhakar, A.

P. Kumar and A. Prabhakar, “Evolution of quantum states in an electro-optic phase modulator,” IEEE J. Quantum Electron. 45, 149–156 (2009).
[Crossref]

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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Rasoloniaina, A.

A. Rasoloniaina, V. Huet, T. K. N. Nguyên, E. L. Cren, M. Mortier, L. Michely, Y. Dumeige, and P. Féron, “Controling the coupling properties of active ultrahigh-Q WGM microcavities from undercoupling to selective amplification,” Sci. Rep. 4, 4023 (2014).
[Crossref]

Rhodes, W. T.

J.-M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography,” Opt. Lett. 24, 104–106 (1999).
[Crossref]

J.-M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60, 1899–1905 (1999).
[Crossref]

J.-M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in sidebands of phase-modulated light for quantum cryptography,” Phys. Rev. Lett. 82, 1656–1659 (1999).
[Crossref]

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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

G. Ribordy, J. Brendel, J.-D. Gautier, N. Gisin, and H. Zbinden, “Long-distance entanglement-based quantum key distribution,” Phys. Rev. A 63, 012309 (2000).
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I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis-Cummings problem,” Phys. Rev. A 67, 053808 (2003).
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Samagin, S. A.

S. P. Kotova, S. A. Samagin, E. P. Pozhidaev, and A. D. Kiselev, “Light modulation in planar aligned short-pitch deformed-helix ferroelectric liquid crystals,” Phys. Rev. E 92, 062502 (2015).
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Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Inf. 1, 5–37 (2005).
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Schiavon, M.

M. Schiavon, G. Vallone, F. Ticozzi, and P. Villoresi, “Heralded single-photon sources for quantum-key-distribution applications,” Phys. Rev. A 93, 012331 (2016).
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S. Sensarn, G. Y. Yin, and S. E. Harris, “Observation of nonlocal modulation with entangled photons,” Phys. Rev. Lett. 103, 163601 (2009).
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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
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M. Tsang, J. H. Shapiro, and S. Lloyd, “Quantum theory of optical temporal phase and instantaneous frequency, II: continuous-time limit and state-variable approach to phase-locked loop design,” Phys. Rev. A 79, 053843 (2009).
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M. Tsang, J. H. Shapiro, and S. Lloyd, “Quantum theory of optical temporal phase and instantaneous frequency,” Phys. Rev. A 78, 053820 (2008).
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Siegman, A. E.

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U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev. 4, 337–354 (2010).
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Solomon, G.

Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Inf. 1, 5–37 (2005).
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E. P. Pozhidaev, A. K. Srivastava, A. D. Kiselev, V. G. Chigrinov, V. V. Vashchenko, A. V. Krivoshey, M. V. Minchenko, and H.-S. Kwok, “Enhanced orientational Kerr effect in vertically aligned deformed helix ferroelectric liquid crystals,” Opt. Lett. 39, 2900–2903 (2014).
[Crossref]

E. P. Pozhidaev, A. D. Kiselev, A. K. Srivastava, V. G. Chigrinov, H.-S. Kwok, and M. V. Minchenko, “Orientational Kerr effect and phase modulation of light in deformed-helix ferroelectric liquid crystals with subwavelength pitch,” Phys. Rev. E 87, 052502 (2013).
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G. Szegö, Orthogonal Polynomials, 4th ed. (American Mathematical Society, 1975).

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M. Schiavon, G. Vallone, F. Ticozzi, and P. Villoresi, “Heralded single-photon sources for quantum-key-distribution applications,” Phys. Rev. A 93, 012331 (2016).
[Crossref]

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I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis-Cummings problem,” Phys. Rev. A 67, 053808 (2003).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
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W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref]

Tsang, M.

M. Tsang, “Cavity quantum electro-optics,” Phys. Rev. A 81, 063837 (2010).
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M. Tsang, J. H. Shapiro, and S. Lloyd, “Quantum theory of optical temporal phase and instantaneous frequency, II: continuous-time limit and state-variable approach to phase-locked loop design,” Phys. Rev. A 79, 053843 (2009).
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M. Tsang, J. H. Shapiro, and S. Lloyd, “Quantum theory of optical temporal phase and instantaneous frequency,” Phys. Rev. A 78, 053820 (2008).
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I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis-Cummings problem,” Phys. Rev. A 67, 053808 (2003).
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M. Schiavon, G. Vallone, F. Ticozzi, and P. Villoresi, “Heralded single-photon sources for quantum-key-distribution applications,” Phys. Rev. A 93, 012331 (2016).
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A. Furusawa and P. van Loock, Quantum Teleportation and Entanglement. A Hybrid Approach to Optical Quantum Information Processing (Wiley-VCH, 2011).

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D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, 1988).

Vashchenko, V. V.

Villoresi, P.

M. Schiavon, G. Vallone, F. Ticozzi, and P. Villoresi, “Heralded single-photon sources for quantum-key-distribution applications,” Phys. Rev. A 93, 012331 (2016).
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Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Inf. 1, 5–37 (2005).
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Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Inf. 1, 5–37 (2005).
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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
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L. Olislager, I. Mbodji, E. Woodhead, J. Cussey, L. Furfaro, P. Emplit, S. Massar, K. P. Huy, and J.-M. Merolla, “Implementing two-photon interference in the frequency domain with electro-optic phase modulators,” New J. Phys. 14, 043015 (2012).
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Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Inf. 1, 5–37 (2005).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
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W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
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IEEE Trans. Microw. Theory Tech. (1)

A. Savchenkov, A. Matsko, W. Liang, V. Ilchenko, D. Seidel, and L. Maleki, “Single-sideband electro-optical modulator and tunable microwave photonic receiver,” IEEE Trans. Microw. Theory Tech. 58, 3167–3174 (2010).
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J. Capmany and C. Fernández-Pousa, “Quantum modelling of electro-optic modulators,” Laser Photon. Rev. 5, 750–772 (2011).
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New J. Phys. (1)

L. Olislager, I. Mbodji, E. Woodhead, J. Cussey, L. Furfaro, P. Emplit, S. Massar, K. P. Huy, and J.-M. Merolla, “Implementing two-photon interference in the frequency domain with electro-optic phase modulators,” New J. Phys. 14, 043015 (2012).
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J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes: II,” Phys. Rev. 129, 481–485 (1963).
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A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73, 024305 (2006).
[Crossref]

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

I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis-Cummings problem,” Phys. Rev. A 67, 053808 (2003).
[Crossref]

J.-M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60, 1899–1905 (1999).
[Crossref]

G. Ribordy, J. Brendel, J.-D. Gautier, N. Gisin, and H. Zbinden, “Long-distance entanglement-based quantum key distribution,” Phys. Rev. A 63, 012309 (2000).
[Crossref]

M. Schiavon, G. Vallone, F. Ticozzi, and P. Villoresi, “Heralded single-photon sources for quantum-key-distribution applications,” Phys. Rev. A 93, 012331 (2016).
[Crossref]

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. P. Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

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[Crossref]

M. Tsang, J. H. Shapiro, and S. Lloyd, “Quantum theory of optical temporal phase and instantaneous frequency,” Phys. Rev. A 78, 053820 (2008).
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M. Tsang, J. H. Shapiro, and S. Lloyd, “Quantum theory of optical temporal phase and instantaneous frequency, II: continuous-time limit and state-variable approach to phase-locked loop design,” Phys. Rev. A 79, 053843 (2009).
[Crossref]

Phys. Rev. E (3)

S. P. Kotova, S. A. Samagin, E. P. Pozhidaev, and A. D. Kiselev, “Light modulation in planar aligned short-pitch deformed-helix ferroelectric liquid crystals,” Phys. Rev. E 92, 062502 (2015).
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[Crossref]

Phys. Rev. Lett. (5)

S. Sensarn, G. Y. Yin, and S. E. Harris, “Observation of nonlocal modulation with entangled photons,” Phys. Rev. Lett. 103, 163601 (2009).
[Crossref]

P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101, 103601 (2008).
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W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref]

J.-M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, and W. T. Rhodes, “Single-photon interference in sidebands of phase-modulated light for quantum cryptography,” Phys. Rev. Lett. 82, 1656–1659 (1999).
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Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Fattal, E. Waks, and E. Diamanti, “Single photons for quantum information systems,” Prog. Inf. 1, 5–37 (2005).
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Rev. Mod. Phys. (3)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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Sci. Rep. (1)

A. Rasoloniaina, V. Huet, T. K. N. Nguyên, E. L. Cren, M. Mortier, L. Michely, Y. Dumeige, and P. Féron, “Controling the coupling properties of active ultrahigh-Q WGM microcavities from undercoupling to selective amplification,” Sci. Rep. 4, 4023 (2014).
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H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, 1993).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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G. Szegö, Orthogonal Polynomials, 4th ed. (American Mathematical Society, 1975).

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M. Hayashi, S. Ishizaka, A. Kawachi, G. Kimura, and T. Ogawa, Introduction to Quantum Information Science, Graduate Texts in Physics (Springer, 2015).

D. Bouwmeester, A. Ekert, and A. Zeilinger, eds., The Physics of Quantum Information (Springer, 2000).

P. Kok and B. W. Lovett, Introduction to Optical Quantum Information Processing (Cambridge University, 2010).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th ed. (Cambridge University, 2010).

A. Furusawa and P. van Loock, Quantum Teleportation and Entanglement. A Hybrid Approach to Optical Quantum Information Processing (Wiley-VCH, 2011).

G. Cariolaro, Quantum Communications, Signals and Communication Technology (Springer, 2015).

J.-M. Liu, Photonic Devices (Oxford University, 2005).

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University, 2007).

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Figures (7)

Fig. 1.
Fig. 1. Modulated light emerging after the modulator driven by the microwave field (MW) passes through a Fabry–Perot filter (FP) and is collected by a photodetector (D). Antireflective (AR) coating is applied to both faces of the electro-optic cavity.
Fig. 2.
Fig. 2. Photon counting rate form factor pmod as a function of the filter frequency detuning computed from Eq. (42) at the intermode coupling parameter γ/Ω=0.1 (the regime of weak intermode coupling). The other parameters are as follows: Tmax(f)=1 is the maximal transmittance of the filter; σf/Ω=0.15 is the bandwidth of the filter; T=2π/Ω is the time of intermode interaction; and ω/Ω=(ΩΩMW)/Ω=0.01.
Fig. 3.
Fig. 3. Photon counting rate form factor pmod as a function of the filter frequency detuning at the intermode coupling parameter γ/Ω=0.4 (the regime of intermediate intermode coupling). Other parameters are listed in the caption of Fig. 2.
Fig. 4.
Fig. 4. Photon counting rate form factor pmod as a function of the filter frequency detuning at the intermode coupling parameter γ/Ω=0.9 (the regime of strong intermode coupling). Other parameters are listed in the caption of Fig. 2.
Fig. 5.
Fig. 5. Photon counting rate form factor pmod as a function of the coupling parameter γ/Ω at ωf=ωopt. Other parameters are listed in the caption of Fig. 2.
Fig. 6.
Fig. 6. Photon counting rate form factor pmod as a function of the coupling parameter γ/Ω at ωf=ω2=ωopt+2Ω. Other parameters are listed in the caption of Fig. 2.
Fig. 7.
Fig. 7. Photon counting rate form factor pmod as a function of the coupling parameter γ/Ω for Nν(0)=δν,1N1(0) at (a) ωf=ω0=ωopt and (b) ωf=ω2=ωopt+2Ω.

Equations (108)

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km=2πmL,mZ.
ωopt=|kopt|vopt=Ω|mopt|,Ω=2πLvopt,
H/=ΩMWbb+ΩA0+γ0fmax{A+b+Ab},
A0=mmamam,A=mf(m)amam+1,A+=A,
H/=ΩA0+γfmax{A+exp[i(ΩMWt+ϕ)]+Aexp[i(ΩMWt+ϕ)]},
mmin<mmmax,
mopt=mmax+mmin+12,
f(m)=(mmmin)(mmaxm),
m=mopt+μ,SμS,S=mmaxmmin12,
amopt+μaμ,amopt+μaμ,
A0=moptN+J0,N=μ=SSaμaμ,J0=μ=SSμaμaμ,
AJ=μ=SS1(S+μ+1)(Sμ)aμaμ+1,
A+J+=J=μ=SS1(S+μ+1)(Sμ)aμ+1aμ,
[J0,J±]=±J±,[J+,J]=2J0.
J2=J02+(J+J+JJ+)/2=Jx2+Jy2+Jz2,
JzJ0,J±=Jx±iJy;
H(t)/=ωoptN+ΩJz+2γ2S+1{J+exp[i(ΩMWt+ϕ)]+Jexp[i(ΩMWt+ϕ)]}.
iddtU(t)=H(t)U(t),U(0)=I,
U(t)=UP(t)exp(iQt/),
U(t)=UP(t)UR(t),
UP(t)=exp[i(moptN+Jz)ΩMWt]
Q/=moptωN+ωJz+2γ2S+1{J+exp(iϕ)+Jexp(iϕ)},ω=ΩΩMW.
R(ϕ,β)=exp(iϕJz)exp(iβJy)
Qd/=R(ϕ,β)QR(ϕ,β)=moptωN+ΓJz,
ω+i4γ2S+1=Γexp[iβ],Γ=ω2+[4γ/(2S+1)]2,
E(n,mz)/=nmoptω+mzΓ,
U(t)=eiΩMWtJzR(ϕ,β)eiΓtJzR(ϕ,β)eiωopttN.
aμ(T)=U(T)aμU(T)=ν=SSMμνaν,
Mμν=ei(ωopt+μΩMW)Tei(μν)ϕUμνS(T),
UμνS(T)=μ=SSdμμS(β)dνμS(β)eiμΓT=(1)νei(μ+ν)α˜dμνS(β˜),
2α˜=π+arg{sin2β+(1+cos2β)cos(ΓT)+2icosβsin(ΓT)},
cosβ˜=cos2β+sin2βcos(ΓT),
sinβ˜=sinβ[cosα˜cosβ(1cos(ΓT))sinα˜sin(ΓT)].
ρF(T)=U(T)ρF(0)U(T),
Nμ(T)=TrF{aμaμρF(T)}=Trp{aμ(T)aμ(T)ρF(0)}.
Nμ(T)=ν,ν=SSUμνS*(T)UμνS(T)ei(νν)ϕaνaν(0),
Nμ(T)Nμ=|UμνS(T)|2Nν(0).
E(r,t)=E+(r,t)+E(r,t),E+(r,t)=E(r,t)=μ=SSEμ(+)(r)aμ(t),
E+(f)(r,t)=μ=SSE˜μ(+)(r)aμ(t),E˜μ(+)(r)=Tf(ωf,ωμ)Eμ(+)(r),
V=(d·Ef(r0)),
p(ωf)=μ=SS|Tf(ωf,ωμ)|2NμK(ωμ),
K(ωμ)=H(ωμωg)σ(ωμωg,κ)g(ωμωg,κ)×|ωμωg,κ|(d·e^)|GEμ(+)(r0)|2dκ,
p(ωf)p0(ωopt)pmod(ωf,T),p0(ωopt)=K(ωopt)Nν(0),
pmod(ωf,T)=μ=SS|Tf(ωf,ωμ)UμνS(T)|2,|UμνS(T)|2=|dμνS(β˜)|2,
|Tf(ωf,ω)|2=Tmax(f)exp[(ωfω)2/σf2],
sinβ2γ|ω|S1,
α˜Sπ+ωT2,ω=ΩΩMW,
β˜gS,g=4γ|ω|sin(|ω|T/2).
dμνS(β˜)=(S+ν)!(Sν)!(S+μ)!(Sμ)!sinνμ(β˜2)×cosν+μ(β˜2)PSν(νμ,ν+μ)(cosβ˜),
limnnαPn(α,β)(cos(z/n))=limnnαPn(α,β)(1z2n2/2)=[z2]αJα(z),
dμνS(β˜)SJμν(g),
UμνS(T)S(i)μνei(μν)ωT/2Jμν(g)eiνωT.
E+(r,t)=μ=SSEμ(+)(r)eiωμ(tT)aμ(T)0,
E+(r,t)Sμ,νEμ(+)(r)ei(μν)(Ωtψ)Jμν(g)eiωνtaν0μ,νei(μν)(Ωtψ)Jμν(g)Eν(+)(r)eiωνtaν0=eigcos(Ωtψ)E+(r,t)0,
exp[igcos(Ωt)]=μ=(i)μeiμΩtJμ(g)
M=M2M0M1,
Mμν(0)=δμνeiϕμ,Mμν(i)=eiϕμν(i)dμνS(βi),
ϕμν(i)=ϕ00(i)+μ(ΩMWTi+αi+ϕi)+ν(π+αiϕi),
μ=SSdμμS(βm)dνμS(βm)eiμϕAB=(1)νei(μ+ν)α˜dμνS(β˜),
ϕAB=ϕAϕB+Δ,Δ=ϕ0+ΩMWT+2αm,
Mμν=eiΨμνdμνS(β˜),
Ψμν=ψ0+μ(ΩMWT+αm+α˜+ϕB)+ν(π+αm+α˜ϕA),
d00S(β˜)PS(cosβ˜)=0,
JαJα=ν,μ=SSaνJνμ(α)aμaJαa,
Jνμ(±)=(Sμ)(S±μ+1)δνμ±1,Jνμ(0)=μδνμ,
[aν,aμ]=δνμ,[aν,aμ]=[aν,aμ]=0,
[Jα,Jβ]=a[Jα,Jβ]a.
exp(iβJα)aμexp(iβJα)=k=0ikβkk![Jα,aμ](k),
[Jα,aμ](k)=[Jα,[Jα,aμ](k1)],[Jα,aμ](1)=[Jα,aμ],[Jα,aμ](0)=aμ.
[Jα,aμ]=ν=SSJμν(α)aν,
exp(iβJα)aμexp(iβJα)=ν=SS[exp(iβJα)]μνaν.
eiγJzeiβJyeiαJzaμeiαJzeiβJyeiγJz=ν=SS[eiαJzeiβJyeiγJz]μνaν=ν=SSDμνS(α,β,γ)aν,
US(t)=eiβJyeiΓtJzeiβJy=eiΓt(sinβJx+cosβJz).
R(ψ,m^)=exp[iψ(m^·J)],
R(ψ,m^)=R(α˜,β˜,γ˜)=eiα˜Jzeiβ˜Jyeiγ˜Jz.
R(ψ,m^)(n·J)R(ψ,m^)=(R(ψ,m^)n·J),
R(α˜,β˜,γ˜)(n·J)R(α˜,β˜,γ˜)=(R(α˜,β˜,γ˜)n·J),
R(ψ,m^)=I3cosψ+m^m^(1cosψ)+Msinψ,
M=(0cosβ0cosβ0sinβ0sinβ0).
R(α˜,β˜,γ˜)=Rz(α˜)Ry(β˜)Rz(γ˜),
Rz(α˜)=(cosα˜sinα˜0sinα˜cosα˜0001),Ry(β˜)=(cosβ˜0sinβ˜010sinβ˜0cosβ˜).
R(ψ,m^)=R(α˜,β˜,γ˜)R.
γ˜=α˜+π,
sin(2α˜)(1+cosβ˜)=2sinβsinψ=R21,
cos(2α˜)(1+cosβ˜)=sin2β+(1+cos2β)cosψ=R11+R22,
cosβ˜=cos2β+sin2βcosψ=R33,
cosα˜sinβ˜=sinβcosβ(1cosψ)=R13,
sinα˜sinβ˜=sinβ  sinψ=R23.
β˜=0,α˜+γ˜=±ψ.
β˜=±ψ,γ˜=α˜=π/2.
H/=ΩMWNb+ωoptN+ΩJz+2γ02S+1(J+b+Jb),
|m,n,j,mz=|mmzjb|n,j,mza,
H/=nωopt+rΩ˜ωM0+2γ02S+1(M++M),
M=bJ+,M+=bJ,M0=NbJz2.
[M0,M±]=±M±,M+M=pκ(M0),
M+M=Nb(J2Jz2Jz)=p3(M0),
p3(q)=(qq1)(qq2)(qq3),
q1=jm2,q2=m3j2,q3=m+j2+1.
M0=r2S0,M+=rS0S,M=[M+]=S+rS0,
S0=Jz,S+=1Nb+1aJ+,S=Ja1Nb+1,
M0=r/2S0,M±r+1/2S,
p2(s)(M0)=(r+1/2)SS+=(r+1/2)(M0q2)(M0q3),
H0(s)/=nωopt+rΩMW+ωS0+2γ02S+1r+1/2(S++S).
M0=j2S0,M+=2jm/2S0S,M=S+2jm/2S0,
S0=m2Nb,S+=J+1jJ0a,S=a1jJ0J.
M0=S0+j2,M±(1m)/2+2jS,
p2(w)(M0)=[(1m)/2+2j](M0q1)(M0q3).
H0(w)/=nωopt+m2ΩMW+r2Ω+ωS0+2γ02S+1(1m)/2+2j(S++S).

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