Abstract

We present squeezing and anti-squeezing spectra of the output from a degenerate optical parametric oscillator (OPO) network arranged in different coherent quantum feedback configurations. One OPO serves as a quantum plant, the other as a quantum controller. The addition of coherent feedback enables shaping of the output squeezing spectrum of the plant, and is found to be capable of pushing the frequency of maximum squeezing away from the optical driving frequency and broadening the spectrum over a wider frequency band. The experimental results are in excellent agreement with the developed theory, and illustrate the use of coherent quantum feedback to engineer the quantum-optical properties of the plant OPO output.

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    [CrossRef] [PubMed]
  2. S. Lloyd, “Coherent quantum feedback,” Phys. Rev. A62(2), 022108 (2000).
    [CrossRef]
  3. M. Yanagisawa and H. Kimura, “Transfer function approach to quantum control-part I: dynamics of quantum feedback systems,” IEEE Trans. Autom. Control48(12), 2107–2120 (2003).
    [CrossRef]
  4. J. Gough and M. R. James, “The series product and its application to quantum feedforward and feedback networks,” IEEE Trans. Autom. Control54(11), 2530–2544 (2009).
    [CrossRef]
  5. V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” J. Multivariate Anal.42(2), 171–201 (1992).
    [CrossRef]
  6. L. Bouten, R. Van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM J. Control and Optim.46(6), 2199–2241 (2007).
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  7. H. Mabuchi, “Coherent-feedback control strategy to suppress spontaneous switching in ultralow power optical bistability,” Appl. Phys. Lett.98(19), 193109 (2011).
    [CrossRef]
  8. J. Kerckhoff and , “Designing quantum memories with embedded control: photonic circuits for autonomous quantum error correction,” Phys. Rev. Lett.105(19), 040502 (2010).
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  9. R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Comm. Math. Phys.93(3), 301–323 (1984).
    [CrossRef]
  10. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A31(6), 3761–3774 (1985).
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  11. C. W. Gardiner, “Driving a quantum system with the output field from another driven quantum system,” Phys. Rev. Lett.70(15), 2269–2272 (1993).
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  12. H. J. Carmichael, “Quantum trajectory theory for cascaded open systems,” Phys. Rev. Lett.70(15), 2273–2276 (1993).
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    [CrossRef]
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    [CrossRef]
  19. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett.55(22), 2409–2412 (1985).
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  23. H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D65(2), 022002 (2001).
    [CrossRef]
  24. J. Gea-Banacloche and G. Leuchs , “Squeezed states for interferometric gravitational-wave detectors,” J. Mod. Opt.34, 793–811 (1987).
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  26. N.J. Cerf, J. Clavareau, C. Macchiavello, and J. Roland, “Quantum entanglement enhances the capacity of bosonic channels with memory,” Phys. Rev. A72042330 (2005).
    [CrossRef]
  27. Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A85(4), 040305 (2012).
    [CrossRef]
  28. G. Zhang and M. R. James, “Quantum feedback networks and control: a brief survey,” Chin. Sci. Bull.57(18), 2200–2214 (2012).
    [CrossRef]
  29. J. E. Gough and S. Wildfeuer, “Enhancement of field squeezing using coherent feedback,” Phys. Rev. A80(4), 042107 (2009).
    [CrossRef]
  30. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31(2), 97–105 (1983).
    [CrossRef]
  31. H. P. Yuen and J. Shapiro, “Optical communication with two-photon coherent states–part III: quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory1(26), 78–92 (1980).
    [CrossRef]
  32. H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett.8(3), 177–179 (1983).
    [CrossRef] [PubMed]
  33. G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nat.387, 471–475 (1997).
    [CrossRef]
  34. Y. Takeno, M. Yukawa, H. Yonezawa, and A. Furusawa, “Observation of-9 dB quadrature squeezing with improvement of phase stability in homodyne measurement,” Opt. Express15(7), 4321–4327 (2007).
    [CrossRef] [PubMed]
  35. J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105(26), 263904 (2010).
    [CrossRef]
  36. A. H. Safavi-Naeini, S. Groeblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via reflection from a silicon micromechanical resonator,” arXiv:1302.6179 [quant-ph]

2012

S. Iida, M. Yukawa, H. Yonezawa, N. Yamamoto, and A. Furusawa, “Experimental demonstration of coherent feedback control on optical field squeezing,” IEEE Trans. Autom. Control57(8), 2045–2050 (2012).
[CrossRef]

J. Kerckhoff and K. W. Lehnert, “Superconducting microwave multivibrator produced by coherent feedback,” Phys. Rev. Lett.109(15), 153602 (2012).
[CrossRef] [PubMed]

Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A85(4), 040305 (2012).
[CrossRef]

G. Zhang and M. R. James, “Quantum feedback networks and control: a brief survey,” Chin. Sci. Bull.57(18), 2200–2214 (2012).
[CrossRef]

2011

H. Mabuchi, “Coherent-feedback control strategy to suppress spontaneous switching in ultralow power optical bistability,” Appl. Phys. Lett.98(19), 193109 (2011).
[CrossRef]

2010

J. Kerckhoff and , “Designing quantum memories with embedded control: photonic circuits for autonomous quantum error correction,” Phys. Rev. Lett.105(19), 040502 (2010).
[CrossRef] [PubMed]

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105(26), 263904 (2010).
[CrossRef]

2009

J. E. Gough and S. Wildfeuer, “Enhancement of field squeezing using coherent feedback,” Phys. Rev. A80(4), 042107 (2009).
[CrossRef]

J. Gough and M. R. James, “The series product and its application to quantum feedforward and feedback networks,” IEEE Trans. Autom. Control54(11), 2530–2544 (2009).
[CrossRef]

2008

H. Mabuchi, “Coherent-feedback quantum control with a dynamic compensator,” Phys. Rev. A78(3), 032323 (2008).
[CrossRef]

2007

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, “Quantum engineering of squeezed states for quantum communication and metrology,” New J. Phys.10(9), 66366 (2007).

L. Bouten, R. Van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM J. Control and Optim.46(6), 2199–2241 (2007).
[CrossRef]

Y. Takeno, M. Yukawa, H. Yonezawa, and A. Furusawa, “Observation of-9 dB quadrature squeezing with improvement of phase stability in homodyne measurement,” Opt. Express15(7), 4321–4327 (2007).
[CrossRef] [PubMed]

2006

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett.97(1) 011101 (2006).
[CrossRef] [PubMed]

2005

N.J. Cerf, J. Clavareau, C. Macchiavello, and J. Roland, “Quantum entanglement enhances the capacity of bosonic channels with memory,” Phys. Rev. A72042330 (2005).
[CrossRef]

2003

M. Yanagisawa and H. Kimura, “Transfer function approach to quantum control-part I: dynamics of quantum feedback systems,” IEEE Trans. Autom. Control48(12), 2107–2120 (2003).
[CrossRef]

2001

H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D65(2), 022002 (2001).
[CrossRef]

2000

S. Lloyd, “Coherent quantum feedback,” Phys. Rev. A62(2), 022108 (2000).
[CrossRef]

1997

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nat.387, 471–475 (1997).
[CrossRef]

1994

H. M. Wiseman and G. J. Milburn, “All-optical versus electro-optical quantum-limited feedback,” Phys. Rev. A49(5), 4110–4125 (1994).
[CrossRef] [PubMed]

1993

C. W. Gardiner, “Driving a quantum system with the output field from another driven quantum system,” Phys. Rev. Lett.70(15), 2269–2272 (1993).
[CrossRef] [PubMed]

H. J. Carmichael, “Quantum trajectory theory for cascaded open systems,” Phys. Rev. Lett.70(15), 2273–2276 (1993).
[CrossRef] [PubMed]

1992

V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” J. Multivariate Anal.42(2), 171–201 (1992).
[CrossRef]

1987

J. Gea-Banacloche and G. Leuchs , “Squeezed states for interferometric gravitational-wave detectors,” J. Mod. Opt.34, 793–811 (1987).
[CrossRef]

L.-A. Wu, M. Xiao, and H. J. Kimble, “Squeezed states of light from an optical parametric oscillator,” J. Opt. Soc. Am. B4(10), 1465–1475 (1987).
[CrossRef]

1985

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett.55(22), 2409–2412 (1985).
[CrossRef] [PubMed]

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A31(6), 3761–3774 (1985).
[CrossRef] [PubMed]

1984

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A30(3), 1386–1391 (1984).
[CrossRef]

R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Comm. Math. Phys.93(3), 301–323 (1984).
[CrossRef]

1983

H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett.8(3), 177–179 (1983).
[CrossRef] [PubMed]

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31(2), 97–105 (1983).
[CrossRef]

1981

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D23(8), 1693–1708 (1981).
[CrossRef]

1980

H. P. Yuen and J. Shapiro, “Optical communication with two-photon coherent states–part III: quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory1(26), 78–92 (1980).
[CrossRef]

Aiello, A.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105(26), 263904 (2010).
[CrossRef]

Andersen, U. L.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105(26), 263904 (2010).
[CrossRef]

Aspelmeyer, M.

A. H. Safavi-Naeini, S. Groeblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via reflection from a silicon micromechanical resonator,” arXiv:1302.6179 [quant-ph]

Belavkin, V. P.

V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” J. Multivariate Anal.42(2), 171–201 (1992).
[CrossRef]

Bouten, L.

L. Bouten, R. Van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM J. Control and Optim.46(6), 2199–2241 (2007).
[CrossRef]

Breitenbach, G.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nat.387, 471–475 (1997).
[CrossRef]

Carmichael, H. J.

H. J. Carmichael, “Quantum trajectory theory for cascaded open systems,” Phys. Rev. Lett.70(15), 2273–2276 (1993).
[CrossRef] [PubMed]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D23(8), 1693–1708 (1981).
[CrossRef]

Cerf, N.J.

N.J. Cerf, J. Clavareau, C. Macchiavello, and J. Roland, “Quantum entanglement enhances the capacity of bosonic channels with memory,” Phys. Rev. A72042330 (2005).
[CrossRef]

Chan, J.

A. H. Safavi-Naeini, S. Groeblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via reflection from a silicon micromechanical resonator,” arXiv:1302.6179 [quant-ph]

Chan, V. W. S.

Chelkowski, S.

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, “Quantum engineering of squeezed states for quantum communication and metrology,” New J. Phys.10(9), 66366 (2007).

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett.97(1) 011101 (2006).
[CrossRef] [PubMed]

Clavareau, J.

N.J. Cerf, J. Clavareau, C. Macchiavello, and J. Roland, “Quantum entanglement enhances the capacity of bosonic channels with memory,” Phys. Rev. A72042330 (2005).
[CrossRef]

Collett, M. J.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A31(6), 3761–3774 (1985).
[CrossRef] [PubMed]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A30(3), 1386–1391 (1984).
[CrossRef]

Danzmann, K.

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, “Quantum engineering of squeezed states for quantum communication and metrology,” New J. Phys.10(9), 66366 (2007).

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett.97(1) 011101 (2006).
[CrossRef] [PubMed]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31(2), 97–105 (1983).
[CrossRef]

Duan, Z.

Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A85(4), 040305 (2012).
[CrossRef]

Elser, D.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105(26), 263904 (2010).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31(2), 97–105 (1983).
[CrossRef]

Franzen, A.

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett.97(1) 011101 (2006).
[CrossRef] [PubMed]

Fürst, J. U.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105(26), 263904 (2010).
[CrossRef]

Furusawa, A.

S. Iida, M. Yukawa, H. Yonezawa, N. Yamamoto, and A. Furusawa, “Experimental demonstration of coherent feedback control on optical field squeezing,” IEEE Trans. Autom. Control57(8), 2045–2050 (2012).
[CrossRef]

Y. Takeno, M. Yukawa, H. Yonezawa, and A. Furusawa, “Observation of-9 dB quadrature squeezing with improvement of phase stability in homodyne measurement,” Opt. Express15(7), 4321–4327 (2007).
[CrossRef] [PubMed]

Gardiner, C. W.

C. W. Gardiner, “Driving a quantum system with the output field from another driven quantum system,” Phys. Rev. Lett.70(15), 2269–2272 (1993).
[CrossRef] [PubMed]

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A31(6), 3761–3774 (1985).
[CrossRef] [PubMed]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A30(3), 1386–1391 (1984).
[CrossRef]

Gea-Banacloche, J.

J. Gea-Banacloche and G. Leuchs , “Squeezed states for interferometric gravitational-wave detectors,” J. Mod. Opt.34, 793–811 (1987).
[CrossRef]

Gough, J.

J. Gough and M. R. James, “The series product and its application to quantum feedforward and feedback networks,” IEEE Trans. Autom. Control54(11), 2530–2544 (2009).
[CrossRef]

Gough, J. E.

J. E. Gough and S. Wildfeuer, “Enhancement of field squeezing using coherent feedback,” Phys. Rev. A80(4), 042107 (2009).
[CrossRef]

Groeblacher, S.

A. H. Safavi-Naeini, S. Groeblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via reflection from a silicon micromechanical resonator,” arXiv:1302.6179 [quant-ph]

Hage, B.

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett.97(1) 011101 (2006).
[CrossRef] [PubMed]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31(2), 97–105 (1983).
[CrossRef]

Hawking, S.

K. S. Thorne, S. Hawking, and W. Israel, Three Hundred Years of Gravitation (Cambridge University, 1987).

Hill, J. T.

A. H. Safavi-Naeini, S. Groeblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via reflection from a silicon micromechanical resonator,” arXiv:1302.6179 [quant-ph]

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett.55(22), 2409–2412 (1985).
[CrossRef] [PubMed]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31(2), 97–105 (1983).
[CrossRef]

Hudson, R. L.

R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Comm. Math. Phys.93(3), 301–323 (1984).
[CrossRef]

Iida, S.

S. Iida, M. Yukawa, H. Yonezawa, N. Yamamoto, and A. Furusawa, “Experimental demonstration of coherent feedback control on optical field squeezing,” IEEE Trans. Autom. Control57(8), 2045–2050 (2012).
[CrossRef]

Israel, W.

K. S. Thorne, S. Hawking, and W. Israel, Three Hundred Years of Gravitation (Cambridge University, 1987).

James, M. R.

G. Zhang and M. R. James, “Quantum feedback networks and control: a brief survey,” Chin. Sci. Bull.57(18), 2200–2214 (2012).
[CrossRef]

J. Gough and M. R. James, “The series product and its application to quantum feedforward and feedback networks,” IEEE Trans. Autom. Control54(11), 2530–2544 (2009).
[CrossRef]

L. Bouten, R. Van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM J. Control and Optim.46(6), 2199–2241 (2007).
[CrossRef]

Jia, X.

Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A85(4), 040305 (2012).
[CrossRef]

Kerckhoff, J.

J. Kerckhoff and K. W. Lehnert, “Superconducting microwave multivibrator produced by coherent feedback,” Phys. Rev. Lett.109(15), 153602 (2012).
[CrossRef] [PubMed]

J. Kerckhoff and , “Designing quantum memories with embedded control: photonic circuits for autonomous quantum error correction,” Phys. Rev. Lett.105(19), 040502 (2010).
[CrossRef] [PubMed]

Kimble, H. J.

H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D65(2), 022002 (2001).
[CrossRef]

L.-A. Wu, M. Xiao, and H. J. Kimble, “Squeezed states of light from an optical parametric oscillator,” J. Opt. Soc. Am. B4(10), 1465–1475 (1987).
[CrossRef]

Kimura, H.

M. Yanagisawa and H. Kimura, “Transfer function approach to quantum control-part I: dynamics of quantum feedback systems,” IEEE Trans. Autom. Control48(12), 2107–2120 (2003).
[CrossRef]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31(2), 97–105 (1983).
[CrossRef]

Lehnert, K. W.

J. Kerckhoff and K. W. Lehnert, “Superconducting microwave multivibrator produced by coherent feedback,” Phys. Rev. Lett.109(15), 153602 (2012).
[CrossRef] [PubMed]

Leuchs, G.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105(26), 263904 (2010).
[CrossRef]

J. Gea-Banacloche and G. Leuchs , “Squeezed states for interferometric gravitational-wave detectors,” J. Mod. Opt.34, 793–811 (1987).
[CrossRef]

Levin, Y.

H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D65(2), 022002 (2001).
[CrossRef]

Lloyd, S.

S. Lloyd, “Coherent quantum feedback,” Phys. Rev. A62(2), 022108 (2000).
[CrossRef]

Mabuchi, H.

H. Mabuchi, “Coherent-feedback control strategy to suppress spontaneous switching in ultralow power optical bistability,” Appl. Phys. Lett.98(19), 193109 (2011).
[CrossRef]

H. Mabuchi, “Coherent-feedback quantum control with a dynamic compensator,” Phys. Rev. A78(3), 032323 (2008).
[CrossRef]

Macchiavello, C.

N.J. Cerf, J. Clavareau, C. Macchiavello, and J. Roland, “Quantum entanglement enhances the capacity of bosonic channels with memory,” Phys. Rev. A72042330 (2005).
[CrossRef]

Marquardt, C.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105(26), 263904 (2010).
[CrossRef]

Matsko, A. B.

H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D65(2), 022002 (2001).
[CrossRef]

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett.55(22), 2409–2412 (1985).
[CrossRef] [PubMed]

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H. M. Wiseman and G. J. Milburn, “All-optical versus electro-optical quantum-limited feedback,” Phys. Rev. A49(5), 4110–4125 (1994).
[CrossRef] [PubMed]

Mlynek, J.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nat.387, 471–475 (1997).
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31(2), 97–105 (1983).
[CrossRef]

Painter, O.

A. H. Safavi-Naeini, S. Groeblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via reflection from a silicon micromechanical resonator,” arXiv:1302.6179 [quant-ph]

Parthasarathy, K. R.

R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Comm. Math. Phys.93(3), 301–323 (1984).
[CrossRef]

Peng, K.

Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A85(4), 040305 (2012).
[CrossRef]

Ralph, T. C.

T. C. Ralph, “Quantum key distribution with continuous variables in optics” in Quantum Information with Continuous Variables (Springer, 2003).
[CrossRef]

Roland, J.

N.J. Cerf, J. Clavareau, C. Macchiavello, and J. Roland, “Quantum entanglement enhances the capacity of bosonic channels with memory,” Phys. Rev. A72042330 (2005).
[CrossRef]

Safavi-Naeini, A. H.

A. H. Safavi-Naeini, S. Groeblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via reflection from a silicon micromechanical resonator,” arXiv:1302.6179 [quant-ph]

Schiller, S.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nat.387, 471–475 (1997).
[CrossRef]

Schnabel, R.

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, “Quantum engineering of squeezed states for quantum communication and metrology,” New J. Phys.10(9), 66366 (2007).

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett.97(1) 011101 (2006).
[CrossRef] [PubMed]

Shapiro, J.

H. P. Yuen and J. Shapiro, “Optical communication with two-photon coherent states–part III: quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory1(26), 78–92 (1980).
[CrossRef]

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett.55(22), 2409–2412 (1985).
[CrossRef] [PubMed]

Strekalov, D. V.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105(26), 263904 (2010).
[CrossRef]

Su, X.

Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A85(4), 040305 (2012).
[CrossRef]

Takeno, Y.

Thorne, K. S.

H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D65(2), 022002 (2001).
[CrossRef]

K. S. Thorne, S. Hawking, and W. Israel, Three Hundred Years of Gravitation (Cambridge University, 1987).

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H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, “Quantum engineering of squeezed states for quantum communication and metrology,” New J. Phys.10(9), 66366 (2007).

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett.97(1) 011101 (2006).
[CrossRef] [PubMed]

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R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett.55(22), 2409–2412 (1985).
[CrossRef] [PubMed]

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L. Bouten, R. Van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM J. Control and Optim.46(6), 2199–2241 (2007).
[CrossRef]

Vyatchanin, S. P.

H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D65(2), 022002 (2001).
[CrossRef]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31(2), 97–105 (1983).
[CrossRef]

Wildfeuer, S.

J. E. Gough and S. Wildfeuer, “Enhancement of field squeezing using coherent feedback,” Phys. Rev. A80(4), 042107 (2009).
[CrossRef]

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H. M. Wiseman and G. J. Milburn, “All-optical versus electro-optical quantum-limited feedback,” Phys. Rev. A49(5), 4110–4125 (1994).
[CrossRef] [PubMed]

Wu, L.-A.

Xiao, M.

Xie, C.

Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A85(4), 040305 (2012).
[CrossRef]

Yamamoto, N.

S. Iida, M. Yukawa, H. Yonezawa, N. Yamamoto, and A. Furusawa, “Experimental demonstration of coherent feedback control on optical field squeezing,” IEEE Trans. Autom. Control57(8), 2045–2050 (2012).
[CrossRef]

Yan, Z.

Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A85(4), 040305 (2012).
[CrossRef]

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M. Yanagisawa and H. Kimura, “Transfer function approach to quantum control-part I: dynamics of quantum feedback systems,” IEEE Trans. Autom. Control48(12), 2107–2120 (2003).
[CrossRef]

Yonezawa, H.

S. Iida, M. Yukawa, H. Yonezawa, N. Yamamoto, and A. Furusawa, “Experimental demonstration of coherent feedback control on optical field squeezing,” IEEE Trans. Autom. Control57(8), 2045–2050 (2012).
[CrossRef]

Y. Takeno, M. Yukawa, H. Yonezawa, and A. Furusawa, “Observation of-9 dB quadrature squeezing with improvement of phase stability in homodyne measurement,” Opt. Express15(7), 4321–4327 (2007).
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H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett.8(3), 177–179 (1983).
[CrossRef] [PubMed]

H. P. Yuen and J. Shapiro, “Optical communication with two-photon coherent states–part III: quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory1(26), 78–92 (1980).
[CrossRef]

Yukawa, M.

S. Iida, M. Yukawa, H. Yonezawa, N. Yamamoto, and A. Furusawa, “Experimental demonstration of coherent feedback control on optical field squeezing,” IEEE Trans. Autom. Control57(8), 2045–2050 (2012).
[CrossRef]

Y. Takeno, M. Yukawa, H. Yonezawa, and A. Furusawa, “Observation of-9 dB quadrature squeezing with improvement of phase stability in homodyne measurement,” Opt. Express15(7), 4321–4327 (2007).
[CrossRef] [PubMed]

Yurke, B.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett.55(22), 2409–2412 (1985).
[CrossRef] [PubMed]

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G. Zhang and M. R. James, “Quantum feedback networks and control: a brief survey,” Chin. Sci. Bull.57(18), 2200–2214 (2012).
[CrossRef]

Appl. Phys. B

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31(2), 97–105 (1983).
[CrossRef]

Appl. Phys. Lett.

H. Mabuchi, “Coherent-feedback control strategy to suppress spontaneous switching in ultralow power optical bistability,” Appl. Phys. Lett.98(19), 193109 (2011).
[CrossRef]

Chin. Sci. Bull.

G. Zhang and M. R. James, “Quantum feedback networks and control: a brief survey,” Chin. Sci. Bull.57(18), 2200–2214 (2012).
[CrossRef]

Comm. Math. Phys.

R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Comm. Math. Phys.93(3), 301–323 (1984).
[CrossRef]

IEEE Trans. Autom. Control

S. Iida, M. Yukawa, H. Yonezawa, N. Yamamoto, and A. Furusawa, “Experimental demonstration of coherent feedback control on optical field squeezing,” IEEE Trans. Autom. Control57(8), 2045–2050 (2012).
[CrossRef]

M. Yanagisawa and H. Kimura, “Transfer function approach to quantum control-part I: dynamics of quantum feedback systems,” IEEE Trans. Autom. Control48(12), 2107–2120 (2003).
[CrossRef]

J. Gough and M. R. James, “The series product and its application to quantum feedforward and feedback networks,” IEEE Trans. Autom. Control54(11), 2530–2544 (2009).
[CrossRef]

IEEE Trans. Inf. Theory

H. P. Yuen and J. Shapiro, “Optical communication with two-photon coherent states–part III: quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory1(26), 78–92 (1980).
[CrossRef]

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V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” J. Multivariate Anal.42(2), 171–201 (1992).
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Nat.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nat.387, 471–475 (1997).
[CrossRef]

New J. Phys.

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, “Quantum engineering of squeezed states for quantum communication and metrology,” New J. Phys.10(9), 66366 (2007).

Opt. Express

Opt. Lett.

Phys. Rev. A

J. E. Gough and S. Wildfeuer, “Enhancement of field squeezing using coherent feedback,” Phys. Rev. A80(4), 042107 (2009).
[CrossRef]

H. Mabuchi, “Coherent-feedback quantum control with a dynamic compensator,” Phys. Rev. A78(3), 032323 (2008).
[CrossRef]

N.J. Cerf, J. Clavareau, C. Macchiavello, and J. Roland, “Quantum entanglement enhances the capacity of bosonic channels with memory,” Phys. Rev. A72042330 (2005).
[CrossRef]

Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A85(4), 040305 (2012).
[CrossRef]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A30(3), 1386–1391 (1984).
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[CrossRef] [PubMed]

S. Lloyd, “Coherent quantum feedback,” Phys. Rev. A62(2), 022108 (2000).
[CrossRef]

Phys. Rev. D

H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D65(2), 022002 (2001).
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[CrossRef]

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H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett.97(1) 011101 (2006).
[CrossRef] [PubMed]

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105(26), 263904 (2010).
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R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett.55(22), 2409–2412 (1985).
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L. Bouten, R. Van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM J. Control and Optim.46(6), 2199–2241 (2007).
[CrossRef]

Other

T. C. Ralph, “Quantum key distribution with continuous variables in optics” in Quantum Information with Continuous Variables (Springer, 2003).
[CrossRef]

A. H. Safavi-Naeini, S. Groeblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via reflection from a silicon micromechanical resonator,” arXiv:1302.6179 [quant-ph]

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Figures (5)

Fig. 1
Fig. 1

Schematic of the OPO network system. Red: 1064 nm, green: 532 nm, orange: coherent feedback, blue: system output. The simplified functional schematic is also shown in right. (ISO: isolator, SHG: second-harmonic generation, DCM: dichroic mirror, PZT: piezoelectric transducer, C-OPO: controller OPO, P-OPO: plant OPO, MCC: mode-cleaning cavity, CVT LCK: cavity locking servo, AOM: acousto-optic modulator, EOM: electro-optic modulator, PUMP LCK: pump phase locking servo, CFB LCK: coherent feedback phase locking servo, HD: Homodyne detection)

Fig. 2
Fig. 2

Squeezing spectra of the plant OPO without feedback at two different pump powers. Blue (cyan): anti-squeezing, red (magenta): squeezing for x = 0.79 (x = 0.3). Here, x = P p P / P th P. Green: vacuum spectrum. Theoretical curves are shown in dashed lines.

Fig. 3
Fig. 3

Squeezing spectra for the empty-cavity destructive interference feedback, for different pump powers. x = P p P / P th P. Red (magenta): closed-loop (open-loop) squeezing, Green: vacuum, blue (cyan): closed-loop (open-loop) anti-squeezing. The theoretical calculations are shown in dashed lines in corresponding colors.

Fig. 4
Fig. 4

Measured squeezing spectra (red) and anti-squeezing spectra (blue) of an OPO network for different pump parities. Here, x = P p P / P th P (plant pump) and | y | = P p C / P th C (controller pump). The open loop squeezing (magenta) and the anti-squeezing spectra (cyan) are shown for comparison. Also shown are the corresponding theoretical curves in dashed lines.

Fig. 5
Fig. 5

Squeezing (red) and anti-squeezing (blue) spectra for a 16 MHz detuned controller OPO cavity (left). Here, x = 0.29 (plant pump) and y = 0 (controller pump). The open loop squeezing (magenta) and the anti-squeezing (cyan) spectra are shown for comparison. Theoretical curves for corresponding squeezing spectra are shown in dotted lines. The theoretical relative circulating power as a function of feedback phase (right).

Tables (1)

Tables Icon

Table 1 Parameters used to fit measured squeezing spectra of the empty-cavity destructive interference coherent feedback configuration.

Equations (31)

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

S P = I 5 , S C = I 2 ,
L P = ( γ 1 a γ 2 a γ 3 a γ 4 a γ L a ) , L C = ( κ b κ L b ) ,
H P = Δ a a + 1 2 i ( ε * a 2 ε a 2 ) , H C = δ b b + 1 2 i ( η * b 2 η b 2 ) .
G P 1 = ( 1 , γ 1 a , Δ a a + 1 2 i ( ε * a 2 ε a 2 ) ) , G P j = ( 1 , γ j a , 0 ) , j = 2 , 3 , 4 , L ,
G C 1 = ( 1 , κ b , δ b b + 1 2 i ( η * b 2 η b 2 ) ) , G C 2 = ( 1 , κ L b , 0 ) .
( G P 2 G ϕ G C 1 G P 1 ) G P 3 G P 4 G P L G C 2 ,
G L j = ( ( α j β j β j α j ) , 0 , 0 ) ,
S = ( e i ϕ α 1 α 2 α 3 e i ϕ β 1 α 2 α 3 β 2 α 3 β 3 0 0 0 0 e i ϕ β 3 α 1 α 2 e i ϕ β 1 β 3 α 2 β 2 β 3 α 3 0 0 0 0 e i ϕ β 2 α 1 e i ϕ β 1 β 2 α 2 0 0 0 0 0 β 1 α 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 ) ,
L = ( ( γ 1 α 1 α 2 α 3 e i ϕ + γ 2 α 3 ) a + κ α 2 α 3 e i ϕ b ( γ 1 β 3 α 1 α 2 e i ϕ + γ 2 β 3 ) a + κ β 3 α 2 e i ϕ b γ 1 β 2 α 1 e i ϕ a + κ β 2 e i ϕ b γ 1 β 1 a γ 3 a γ 4 a γ L a κ L b ) ,
H = ( Δ + γ 1 γ 2 α 1 α 2 sin ϕ ) a a + δ b b + 1 2 i ( η * b 2 η b 2 ) + 1 2 i ( ε * a 2 ε a 2 ) + κ 2 i [ ( γ 2 α 2 e i ϕ γ 1 α 1 ) a b ( γ 2 α 2 e i ϕ γ 1 α 1 ) a b ] .
d X ( t ) = ( ( X ( t ) ) i [ X ( t ) , H ] ) dt + d A ( t ) S [ X ( t ) , L ( t ) ] + [ L , X ( t ) ] SdA ( t ) ,
( X ) = 1 2 L ( t ) [ X , L ( t ) ] + 1 2 [ L ( t ) , X ] L ( t ) .
d A ( t ) = SdA ( t ) + L ( t ) dt .
d a ( t ) = A a ( t ) dt + Bd A ( t ) ,
d A ( t ) = C a ( t ) dt + Dd A ( t ) ,
a = ( a b a b ) , d A = ( dA dA * ) , d A = ( dA dA * ) .
A = Δ ( A , A + ) , B Δ ( C S , 0 ) , C = Δ ( C , 0 ) , D Δ ( S , 0 ) ,
A = ( i Δ γ T 2 Γ e i ϕ κ γ 2 α 2 e i ϕ γ 1 κ α 1 i δ κ T 2 ) , A + = ( ε 0 0 η ) ,
C = ( γ 1 α 1 α 2 α 3 e i ϕ + γ 2 α 3 κ α 2 α 3 e i ϕ γ 1 β 3 α 1 α 2 e i ϕ + γ 2 β 3 κ β 3 α 2 e i ϕ γ 1 β 2 α 1 e i ϕ κ β 2 e i ϕ γ 1 β 1 0 γ 3 0 γ 4 0 γ L 0 0 κ L )
Γ = γ 1 γ 2 α 1 α 2 ,
γ T = γ 1 + γ 2 + γ 3 + γ 4 + γ L ,
κ T = κ + κ L .
A ˜ ( ω ) = Ξ ( ω ) A ˜ ( ω ) ,
Ξ ( ω ) D + C ( i ω I 4 A ) 1 B .
Ξ ( ω ) = ( 𝒮 ( ω ) 𝒮 + ( ω ) 𝒮 + * ( ω ) 𝒮 * ( ω ) ) .
𝒫 j θ ( ω ) = 1 + 𝒩 j ( ω ) + 𝒩 j ( ω ) + e 2 i θ j ( ω ) + e 2 i θ j ( ω ) ,
𝒩 j ( ω ) j | 𝒮 j k + ( ω ) | 2 , j ( ω ) k S j k ( ω ) 𝒮 j k + ( ω ) ,
𝒫 j θ ( ω ) 𝒫 j θ ( ω ) + 4 { Re [ e i θ v j ( w ) ] } 2 δ ( ω ) ,
v j ( w ) : = ( Ξ ( 0 ) w ) j = ( 𝒮 ( 0 ) w + 𝒮 + ( 0 ) w * ) j .
P j ss = | v j ( w ) | 2 = ( C Δ N ss C ) n + j , n + j ,
0 = A Δ N ss + Δ N ss A + Q , with Q : = B d A d A dt B = ( C C 0 0 0 )

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