Abstract

The observed output of an interferometer is the result of interference among the parts of the input light beam traveling along each possible optical path. In complex systems, writing down all these possible optical paths and computing their cumulative effect can become a difficult task. We present an intuitive graph-based method for solving this problem and calculating electric fields within an interferometric setup, classical and quantum. We show how to associate a weighted directed graph with an interferometer and define rules to simplify these associated graphs. Successive application of the rules results in a final graph containing information on the desired field amplitudes. The method is applied to a number of examples in cavity optomechanics and cavity-enhanced interferometers.

© 2020 Optical Society of America

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2020 (1)

D. C. Newsom, F. Luna, V. Fedoseev, W. Löffler, and D. Bouwmeester, “Optimal optomechanical coupling strength in multimembrane systems,” Phys. Rev. A 101, 033829 (2020).
[Crossref]

2019 (4)

L. J. Maczewsky, K. Wang, A. A. Dovgiy, A. E. Miroshnichenko, A. Moroz, M. Ehrhardt, M. Heinrich, D. N. Christodoulides, A. Szameit, and A. A. Sukhorukov, “Synthesizing multi-dimensional excitation dynamics and localization transition in one-dimensional lattices,” Nat. Photonics 14, 76–81 (2019).
[Crossref]

P. C. Gibson, “Disk polynomials and the one-dimensional wave equation,” J. Approx. Theory 244, 37–56 (2019).
[Crossref]

X. Gu, M. Erhard, A. Zeilinger, and M. Krenn, “Quantum experiments and graphs ii: quantum interference, computation, and state generation,” Proc. Natl. Acad. Sci. USA 116, 4147–4155 (2019).
[Crossref]

X. Wei, J. Sheng, Y. Wu, W. Liu, and H. Wu, “Twin-beam-enhanced displacement measurement of a membrane in a cavity,” Appl. Phys. Lett. 115, 251105 (2019).
[Crossref]

2018 (4)

L. Magrini, R. A. Norte, R. Riedinger, I. Marinković, D. Grass, U. Delić, S. Gröblacher, S. Hong, and M. Aspelmeyer, “Near-field coupling of a levitated nanoparticle to a photonic crystal cavity,” Optica 5, 1597–1602 (2018).
[Crossref]

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
[Crossref]

S. Ataman, “A graphical method in quantum optics,” J. Phys. Commun. 2, 035032 (2018).
[Crossref]

P. C. Gibson, “Acoustic imaging of layered media,” J. Comput. Phys. 372, 524–545 (2018).
[Crossref]

2017 (2)

M. Krenn, X. Gu, and A. Zeilinger, “Quantum experiments and graphs: multiparty states as coherent superpositions of perfect matchings,” Phys. Rev. Lett. 119, 240403 (2017).
[Crossref]

D. V. Martynov and LSC Instrument Group, “Quantum correlation measurements in interferometric gravitational-wave detectors,” Phys. Rev. A 95, 043831 (2017).
[Crossref]

2016 (5)

2015 (3)

M. Tillmann, S.-H. Tan, S. E. Stoeckl, B. C. Sanders, H. de Guise, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Generalized multiphoton quantum interference,” Phys. Rev. X 5, 041015 (2015).
[Crossref]

S. Ataman, “The quantum optical description of three experiments involving non-linear optics using a graphical method,” Eur. Phys. J. D 69, 44 (2015).
[Crossref]

S. Ataman, “The quantum optical description of a Fabry-Perot interferometer and the prediction of an antibunching effect,” Eur. Phys. J. D 69, 187 (2015).
[Crossref]

2014 (2)

S. Ataman, “Field operator transformations in quantum optics using a novel graphical method with applications to beam splitters and interferometers,” Eur. Phys. J. D 68, 288 (2014).
[Crossref]

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

2013 (3)

M. Rossi, M. Huber, D. Bruß, and C. Macchiavello, “Quantum hypergraph states,” New J. Phys. 15, 113022 (2013).
[Crossref]

X.-W. Xu, Y.-J. Zhao, and Y.-X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
[Crossref]

N. Kiesel, F. Blaser, U. Delic, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

2012 (1)

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
[Crossref]

2011 (1)

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
[Crossref]

2009 (1)

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

2008 (3)

M. Bhattacharya and P. Meystre, “Multiple membrane cavity optomechanics,” Phys. Rev. A 78, 041801 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

2006 (1)

M. A. Nielsen, “Cluster-state quantum computation,” Rep. Math. Phys. 57, 147–161 (2006).
[Crossref]

2005 (1)

A. Thüring, H. Lück, and K. Danzmann, “Analysis of a four-mirror-cavity enhanced Michelson interferometer,” Phys. Rev. E 72, 066615 (2005).
[Crossref]

2003 (2)

2002 (2)

D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Mirror quiescence and high-sensitivity position measurements with feedback,” Phys. Rev. A 65, 063803 (2002).
[Crossref]

R. Albert and A.-L. Barabási, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47–97 (2002).
[Crossref]

2001 (1)

2000 (1)

1998 (2)

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

1992 (1)

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[Crossref]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref]

Abbott, B. P.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Abbott, R.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[Crossref]

Adhikari, R.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Ajith, P.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Albert, R.

R. Albert and A.-L. Barabási, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47–97 (2002).
[Crossref]

Allen, B.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Allen, G.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Althouse, W. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[Crossref]

Amin, R. S.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Anderson, S. B.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Anderson, W. G.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Arain, M. A.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Araya, M.

B. P. Abbott, R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and M. Araya, “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72, 076901 (2009).
[Crossref]

Aspelmeyer, M.

L. Magrini, R. A. Norte, R. Riedinger, I. Marinković, D. Grass, U. Delić, S. Gröblacher, S. Hong, and M. Aspelmeyer, “Near-field coupling of a levitated nanoparticle to a photonic crystal cavity,” Optica 5, 1597–1602 (2018).
[Crossref]

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

N. Kiesel, F. Blaser, U. Delic, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

Ataman, S.

S. Ataman, “A graphical method in quantum optics,” J. Phys. Commun. 2, 035032 (2018).
[Crossref]

S. Ataman, “The quantum optical description of three experiments involving non-linear optics using a graphical method,” Eur. Phys. J. D 69, 44 (2015).
[Crossref]

S. Ataman, “The quantum optical description of a Fabry-Perot interferometer and the prediction of an antibunching effect,” Eur. Phys. J. D 69, 187 (2015).
[Crossref]

S. Ataman, “Field operator transformations in quantum optics using a novel graphical method with applications to beam splitters and interferometers,” Eur. Phys. J. D 68, 288 (2014).
[Crossref]

Barabási, A.-L.

R. Albert and A.-L. Barabási, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47–97 (2002).
[Crossref]

Bawaj, M.

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
[Crossref]

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Bhattacharya, M.

M. Bhattacharya and P. Meystre, “Multiple membrane cavity optomechanics,” Phys. Rev. A 78, 041801 (2008).
[Crossref]

Blaser, F.

N. Kiesel, F. Blaser, U. Delic, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, “Cavity cooling of an optically levitated submicron particle,” Proc. Natl. Acad. Sci. USA 110, 14180–14185 (2013).
[Crossref]

Bork, R.

Bouwmeester, D.

D. C. Newsom, F. Luna, V. Fedoseev, W. Löffler, and D. Bouwmeester, “Optimal optomechanical coupling strength in multimembrane systems,” Phys. Rev. A 101, 033829 (2020).
[Crossref]

Bruno, N.

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
[Crossref]

Bruß, D.

M. Rossi, M. Huber, D. Bruß, and C. Macchiavello, “Quantum hypergraph states,” New J. Phys. 15, 113022 (2013).
[Crossref]

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Catalini, L.

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M. Tillmann, S.-H. Tan, S. E. Stoeckl, B. C. Sanders, H. de Guise, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Generalized multiphoton quantum interference,” Phys. Rev. X 5, 041015 (2015).
[Crossref]

Tan, S.-H.

M. Tillmann, S.-H. Tan, S. E. Stoeckl, B. C. Sanders, H. de Guise, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Generalized multiphoton quantum interference,” Phys. Rev. X 5, 041015 (2015).
[Crossref]

Tanner, D. B.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Thew, R. T.

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
[Crossref]

Thompson, J. D.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref]

Thorne, K. S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[Crossref]

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A. Thüring, H. Lück, and K. Danzmann, “Analysis of a four-mirror-cavity enhanced Michelson interferometer,” Phys. Rev. E 72, 066615 (2005).
[Crossref]

Tillmann, M.

M. Tillmann, S.-H. Tan, S. E. Stoeckl, B. C. Sanders, H. de Guise, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Generalized multiphoton quantum interference,” Phys. Rev. X 5, 041015 (2015).
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[Crossref]

Vitali, D.

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
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Walsmley, I. A.

Walther, P.

M. Tillmann, S.-H. Tan, S. E. Stoeckl, B. C. Sanders, H. de Guise, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Generalized multiphoton quantum interference,” Phys. Rev. X 5, 041015 (2015).
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L. J. Maczewsky, K. Wang, A. A. Dovgiy, A. E. Miroshnichenko, A. Moroz, M. Ehrhardt, M. Heinrich, D. N. Christodoulides, A. Szameit, and A. A. Sukhorukov, “Synthesizing multi-dimensional excitation dynamics and localization transition in one-dimensional lattices,” Nat. Photonics 14, 76–81 (2019).
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Webb, M. S.

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X. Wei, J. Sheng, Y. Wu, W. Liu, and H. Wu, “Twin-beam-enhanced displacement measurement of a membrane in a cavity,” Appl. Phys. Lett. 115, 251105 (2019).
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A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
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Whitcomb, S. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
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X. Wei, J. Sheng, Y. Wu, W. Liu, and H. Wu, “Twin-beam-enhanced displacement measurement of a membrane in a cavity,” Appl. Phys. Lett. 115, 251105 (2019).
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X. Wei, J. Sheng, Y. Wu, W. Liu, and H. Wu, “Twin-beam-enhanced displacement measurement of a membrane in a cavity,” Appl. Phys. Lett. 115, 251105 (2019).
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X.-W. Xu, Y.-J. Zhao, and Y.-X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
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J. Li, A. Xuereb, N. Malossi, and D. Vitali, “Cavity mode frequencies and strong optomechanical coupling in two-membrane cavity optomechanics,” J. Opt. 18, 084001 (2016).
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Yang, C.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
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Yu, S.

Zbinden, H.

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
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M. Krenn, M. Malik, R. Fickler, R. Lapkiewicz, and A. Zeilinger, “Automated search for new quantum experiments,” Phys. Rev. Lett. 116, 090405 (2016).
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M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
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X.-W. Xu, Y.-J. Zhao, and Y.-X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
[Crossref]

Zippilli, S.

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
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Zucker, M. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[Crossref]

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J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
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Appl. Phys. Lett. (1)

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Nature (1)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
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A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
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P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
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Phys. Rev. E (1)

A. Thüring, H. Lück, and K. Danzmann, “Analysis of a four-mirror-cavity enhanced Michelson interferometer,” Phys. Rev. E 72, 066615 (2005).
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M. Krenn, M. Malik, R. Fickler, R. Lapkiewicz, and A. Zeilinger, “Automated search for new quantum experiments,” Phys. Rev. Lett. 116, 090405 (2016).
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F. De Martini, G. Denardo, and A. Zeilinger, Quantum Interferometry (World Scientific, 1994).

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

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Figures (16)

Fig. 1.
Fig. 1. (a) Optical schematics of a Michelson interferometer. Dots and arrows define the position and direction, respectively, of states A, B, C, and D. The red lines represent light passing by the interferometer. (b) Directed graph G corresponding to the Michelson interferometer. Green and red circles indicate, respectively, the input and the output vertices.
Fig. 2.
Fig. 2. (a) Optical schematics of a Fabry–Perot cavity. Since in this example the cavity’s reflection is not of interest, only the output state C is defined. (b) Graph G corresponding to the Fabry–Perot cavity. The loop in vertex B corresponds to the situation in which the wave-front undergoes one roundtrip inside the cavity, going from state B back to state B.
Fig. 3.
Fig. 3. (a) Optical schematics of a cavity with two membranes in the middle. The red line represents light passing by the cavity, while the blue, green, and pink lines indicate different optical paths inside the cavity. (b) Graph corresponding to a cavity with two membranes in the middle. The blue, green, and pink edges correspond to the blue, green, and pink optical paths displayed in (a), respectively.
Fig. 4.
Fig. 4. Graphs G and $\hat{\rm G}$ for two different rules in which multiple edges are replaced by a single edge with equivalent weight equal to (a) the product of the weights of each individual edge, if they are consecutive edges, or (b) the sum of the weights of each individual edge, if they are in parallel.
Fig. 5.
Fig. 5. (a) Loop in vertex C, connected only to vertices B and D, is contracted by joining B and D with an equivalent edge. (b) Multiple loops at a vertex C can be replaced by a single loop with weight equal to the sum of the weights of each individual loop.
Fig. 6.
Fig. 6. Vertex detaching: $\hat{\rm G}$ is obtained by creating copies of the vertex D so that walks are conserved; each copy of D has a single incoming and outgoing edge, which allows for the application of previous rules.
Fig. 7.
Fig. 7. Successive application of simplification rules to the graph corresponding to the two membranes in the middle setup. The rules are applied until only the input and the output states are left. The weight of the edge connecting A to E in the final graph is equal to the response factor $\Gamma$.
Fig. 8.
Fig. 8. Schematics for ${\rm N}$ membranes inside a cavity. Since one state is defined before the input mirror and one state is defined after each optical element, there is a total of ${\rm N} + 3$ states.
Fig. 9.
Fig. 9. Walks arising from the detachment of the state ${{\rm A}_{{\rm N} + {2}}}$: (a) walk from vertex ${{\rm A}_{{\rm N} + {1}}}$ to vertex ${{\rm A}_k}$, for $k$ such that $2 \le k \le {\rm N} + {1}$. This walk and the edge ${\alpha _{{\rm N} + {1},k}}$ are in parallel. (b) Walk from vertex ${{\rm A}_{{\rm N} + {1}}}$ to the output vertex ${{\rm A}_{{\rm N} + {3}}}$.
Fig. 10.
Fig. 10. Graph associated with the transmission of a cavity with a membrane in the middle. Applying the simplification described in the present section to this graph yields the graph associated with a cavity with no membranes inside of it. Conversely, applying the substitutions prescribed in Eq. (14) to the response factor of a Fabry–Perot cavity gives the response factor for the displayed graph.
Fig. 11.
Fig. 11. (a) Optical schematics of a cavity-enhanced Michelson interferometer: a cavity is placed at the end of each arm of a Michelson interferometer to increase sensitivity, and a mirror is placed before the BS to increase the power stored in the system. (b) Graph corresponding to the cavity-enhanced Michelson interferometer.
Fig. 12.
Fig. 12. Successive application of simplification rules to the graph corresponding to a cavity-enhanced Michelson interferometer. As in the two membranes in the middle case, rules are applied until only the input and output vertices remain.
Fig. 13.
Fig. 13. Simplification of a graph corresponding to the cavity-enhanced Michelson interferometer when the resultant electric field in the intermediate state B is of interest. The rules are applied until only vertices A and B are left. The weight of the edge connecting A to B is equal to the response factor ${\Gamma _{\rm AB}}$ that relates the input electric field to the electric field at B.
Fig. 14.
Fig. 14. (a) Optical schematics corresponding to a Mach–Zehnder interferometer. Differences between the two paths inside the interferometer are summarized by a phase $\theta$. (b) Graph corresponding to the Mach–Zehnder interferometer, with two input states, indicated by the green circles, and two output states, indicated by the red circles.
Fig. 15.
Fig. 15. Simplification of the graph corresponding to a Mach–Zehnder interferometer. The graph is simplified until all that remains are the input and output vertices.
Fig. 16.
Fig. 16. (a) Optical schematics corresponding to a beam splitter. (b) Graph corresponding to the beam splitter. The graph is already in its simplest form, which means that the response factor between a given input and output is simply the weight of the edge connecting the corresponding vertices.

Equations (36)

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E o u t , α = Γ α E i n .
E o u t = Γ E i n ,
Γ = Φ A B Φ B D + Φ A C Φ C D ,
E o u t = i r t ( e 2 i k d 1 + e 2 i k d 2 ) E i n .
Γ = Φ A B ( n = 0 Φ B B n ) Φ B C = Φ A B Φ B C 1 Φ B B .
E o u t = t 2 e ikd 1 r 2 e 2 i k d E i n .
Φ A B = i t , Φ B C = i t 1 e i k d 1 , Φ B B = r 1 r e i k 2 d 1 , Φ C C = r 2 r 1 e i k 2 d 2 , Φ C D = i t 2 e i k d 2 , Φ C B = r 2 i t 1 r e i k ( 2 d 2 + d 1 ) , Φ D B = r i t 2 i t 1 r e i k ( 2 d 3 + d 2 + d 1 ) , Φ D C = r i t 2 r 1 e i k ( 2 d 3 + d 2 ) , Φ D D = r r 2 e i k 2 d 3 , Φ D E = i t e i k d 3 .
Γ = Φ A B Φ B C Φ C D Φ D E Δ ,
Δ 1 Φ B B Φ C C Φ D D + Φ B B Φ C C + Φ C C Φ D D + Φ D D Φ B B Φ C D Φ D C Φ B C Φ C B + Φ B C Φ C B Φ D D + Φ C D Φ D C Φ B B Φ B C Φ C D Φ D B Φ B B Φ C C Φ D D .
Γ = t 2 t 1 t 2 e i k ( d 1 + d 2 + d 3 ) Δ ,
Δ = 1 r 2 e 2 i k ( d 1 + d 2 + d 3 ) ( r 1 2 r 2 2 + r 1 2 t 2 2 + r 2 2 t 1 2 + t 1 2 t 2 2 ) + ( e 2 i k ( d 1 + d 2 ) ( r 1 2 r 2 + r 2 t 1 2 ) + e 2 i k ( d 2 + d 3 ) r 1 ( r 2 2 + t 2 2 ) ) r ( e 2 i k d 1 r 1 + e i k d 3 r 2 ) r + e 2 i k ( d 1 + d 3 ) r 1 r 2 r 2 e 2 i k d 2 r 1 r 2 ,
Φ N + 1 , k ( N ) = Φ N + 1 , k ( N ) + Φ N + 1 , N + 2 ( N ) Φ N + 2 , k ( N ) 1 Φ N + 2 , N + 2 ( N ) ,
Φ N + 1 , N + 3 ( N ) = Φ N + 1 , N + 2 ( N ) Φ N + 2 , N + 3 ( N ) 1 Φ N + 2 , N + 2 ( N ) .
Φ N + 1 , N + 2 ( N 1 ) Φ N + 1 , N + 2 ( N ) Φ N + 2 , N + 3 ( N ) 1 Φ N + 2 , N + 2 ( N ) ; Φ N + 1 , N + 2 ( N 1 ) Φ N + 1 , k ( N ) + Φ N + 1 , N + 2 ( N ) Φ N + 2 , k ( N ) 1 Φ N + 2 , N + 2 ( N ) , f o r 2 k N + 1 ; Φ i , j ( N 1 ) Φ i , j ( N ) , for the remaining weights .
Γ ( 0 ) = Φ 1 , 2 ( 0 ) Φ 2 , 3 ( 0 ) 1 Φ 22 ( 0 ) .
Γ ( 1 ) = Φ 1 , 2 [ Φ 2 , 3 Φ 3 , 4 / ( 1 Φ 3 , 3 ) ] 1 [ Φ 2 , 2 + Φ 2 , 3 Φ 3 , 2 / ( 1 Φ 3 , 3 ) ] = Φ 1 , 2 Φ 2 , 3 Φ 3 , 4 1 Φ 2 , 2 Φ 3 , 3 + Φ 2 , 2 Φ 3 , 3 Φ 2 , 3 Φ 3 , 2 ,
Φ A B = i t 1 e i k d 0 , Φ B C = r 2 , Φ B F = i t 2 , Φ C D = r 3 e i k d 1 , Φ C E = i t 3 r 3 e i k ( d 1 + d 2 ) , Φ E E = r 3 2 e i k 2 d 2 , Φ E D = i t 3 e i k d 2 , Φ D I = i t 2 e i k d 1 , Φ D B = r 1 r 2 e i k ( d 1 + 2 d 0 ) , Φ F G = r 3 e i k d 3 , Φ F H = i t 3 r 3 e i k ( d 3 + d 4 ) , Φ H H = r 3 2 e i k 2 d 4 , Φ H G = i t 3 e i k d 4 , Φ G I = r 2 e i k d 3 , Φ G B = i t 2 r 1 e i k ( d 3 + 2 d 0 ) .
Γ = Φ A B [ Φ B F ( Φ F G + Φ F G ) Φ G I + Φ B C ( Φ C D + Φ C D ) Φ D I ] 1 [ Φ B F ( Φ F G + Φ F G ) Φ G B + Φ B C ( Φ C D + Φ C D ) Φ D B ] ,
Γ = t 1 r 2 t 2 r 3 e i k d 0 [ e 2 i k d 1 a 1 + e 2 i k d 3 a 2 ] 1 r 1 r 3 e 2 i k d 0 [ r 2 2 e 2 i k d 1 a 1 t 2 2 e 2 i k d 3 a 2 ] ,
a 1 = ( 1 t 3 2 e 2 i k d 2 1 r 3 2 e 2 i k d 2 ) , a 2 = ( 1 t 3 2 e 2 i k d 4 1 r 3 2 e 2 i k d 4 ) .
d 2 = n λ 2 , d 4 = m λ 2 ,
d 1 = d 3 + ( p + 1 2 ) λ 2 ,
Γ A B = Φ A B 1 Φ B B ,
Γ A B = t 1 r 2 t 2 r 3 e i k d 0 1 r 1 r 3 e 2 i k d 0 ( r 2 2 e 2 i k d 1 a 1 t 2 2 e 2 i k d 3 a 2 ) .
d 0 + d 1 = q λ 2 , d 0 + d 3 = s λ 2 ,
E O m = n Γ nm E I n ,
Φ A C = i t , Φ A D = r , Φ B C = r , Φ B D = i t , Φ C E = r , Φ C F = i t , Φ D E = i t e i θ , Φ D F = r e i θ .
E o u t , E = ( Φ A C Φ C E + Φ A D Φ D E ) E i n , A + ( Φ B C Φ C E + Φ B D Φ D E ) E i n , B , E o u t , F = ( Φ A C Φ C F + Φ A D Φ D F ) E i n , A + ( Φ B C Φ C F + Φ B D Φ D F ) E i n , B .
E o u t , E = i r t ( 1 + e i θ ) E i n , A + ( r 2 t 2 e i θ ) E i n , B , E o u t , F = ( r 2 e i θ t 2 ) E i n , A + i r t ( 1 + e i θ ) E i n , B .
v O = U v I ,
a ^ Γ A B b ^ + Γ A C c ^ + ,
| ψ = Ψ ^ ( a ^ , b ^ , c ^ , , a ^ , b ^ , c ^ , ) | 0 ,
a Γ A C c + Γ A D d , b Γ B D d + Γ B C c ,
a b Γ A C Γ B C c 2 + Γ A D Γ B D d 2 = i 2 ( c 2 + d 2 ) .
Γ A E = i r t ( 1 + e i θ ) , Γ B E = ( r 2 t 2 e i θ ) , Γ A F = ( r 2 e i θ t 2 ) , Γ B F = i r t ( 1 + e i θ ) .
a ^ i r t ( 1 + e i θ ) e ^ + ( r 2 e i θ t 2 ) f ^ , b ^ ( r 2 t 2 e i θ ) e ^ + i r t ( 1 + e i θ ) f ^ ,