Abstract

We introduce a passively-aligned, flexure-tuned cavity optomechanical system in which a membrane is positioned microns from one end mirror of a Fabry-Perot optical cavity. By displacing the membrane through gentle flexure of its silicon supporting frame (i.e., to ∼80 m radius of curvature (ROC)), we gain access to the full range of available optomechanical couplings, finding also that the optical spectrum exhibits none of the abrupt discontinuities normally found in “membrane-in-the-middle” (MIM) systems. More aggressive flexure (3 m ROC) enables >15 μm membrane travel, milliradian tilt tuning, and a wavelength-scale (1.64 ± 0.78 μm) membrane-mirror separation. We also provide a complete set of analytical expressions for this system’s leading-order dispersive and dissipative optomechanical couplings. Notably, this system can potentially generate orders of magnitude larger linear dissipative or quadratic dispersive strong coupling parameters than is possible with a MIM system. Additionally, it can generate the same purely quadratic dispersive coupling as a MIM system, but with significantly suppressed linear dissipative back-action (and force noise).

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

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    [Crossref]

2018 (6)

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref] [PubMed]

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
[Crossref]

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref] [PubMed]

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
[Crossref]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
[Crossref] [PubMed]

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]

2017 (5)

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
[Crossref]

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
[Crossref] [PubMed]

B. Nair, A. Naesby, and A. Dantan, “Optomechanical characterization of silicon nitride membrane arrays,” Opt. Lett. 42, 1341 (2017).
[Crossref] [PubMed]

W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S. Polzik, and A. Schliesser, “Multimode optomechanical system in the quantum regime,” Proc. Natl. Acad. Sci. 114, 62–66 (2017).
[Crossref]

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano. 12, 776–783 (2017).
[Crossref]

2016 (7)

C. Reinhardt, T. Müller, A. Bourassa, and J. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref] [PubMed]

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

D. Kilda and A. Nunnenkamp, “Squeezed light and correlated photons from dissipatively coupled optomechanical systems,” J. Opt. 18, 014007 (2016).
[Crossref]

A. Barasheed, T. Müller, and J. Sankey, “Optically defined mechanical geometry,” Phys. Rev. A 93, 053811 (2016).
[Crossref]

Y. Yanay, J. Sankey, and A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).
[Crossref]

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).
[Crossref]

2015 (7)

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

T. K. Paraiso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-squared coupling in a tunable photonic crystal optomechanical cavity,” Phys. Rev. X 5, 041024 (2015).

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
[Crossref]

T. Müller, C. Reinhardt, and J. Sankey, “Enhanced optomechanical levitation of minimally supported dielectrics,” Phys. Rev. A 91, 053849 (2015).
[Crossref]

A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
[Crossref] [PubMed]

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

2014 (3)

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

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
[Crossref]

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

2013 (5)

S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
[Crossref]

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref] [PubMed]

T. P. Purdy, P.-L. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).

T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys. 15, 045017 (2013).
[Crossref]

T. Weiss and A. Nunnenkamp, “Quantum limit of laser cooling in dispersively and dissipatively coupled optomechanical systems,” Phys. Rev. A 88, 023850 (2013).
[Crossref]

2012 (4)

T. P. Purdy, R. W. Peterson, P. L. Yu, and C. A. Regal, “Cavity optomechanics with Si3N4 membranes at cryogenic temperatures,” New J. Phys. 14, 115021 (2012).
[Crossref]

K. K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
[Crossref] [PubMed]

D. E. Chang, K.-K. Ni, O. Painter, and H. J. Kimble, “Ultrahigh-Q mechanical oscillators through optical trapping,” New J. Phys. 14, 45002 (2012).
[Crossref]

N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
[Crossref]

2011 (1)

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
[Crossref]

2010 (4)

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
[Crossref]

S. R. Leslie, A. P. Fields, and A. E. Cohen, “Convex lens-induced confinement for imaging single molecules,” Anal. Chem. 82, 6224–6229 (2010).
[Crossref] [PubMed]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
[Crossref]

2009 (3)

F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
[Crossref] [PubMed]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard quantum limit for probing mechanical energy quantization,” Phys. Rev. Lett. 103, 100402 (2009).
[Crossref] [PubMed]

2008 (3)

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] [PubMed]

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (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]

2001 (1)

C. Hood, H. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001).
[Crossref]

Andrews, R. W.

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

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

Arrangoiz-Arriola, P.

P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).

Aspelmeyer, M.

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
[Crossref]

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I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
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R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
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M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
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Y. Yanay, J. Sankey, and A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).
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C. Reinhardt, T. Müller, A. Bourassa, and J. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

T. Müller, C. Reinhardt, and J. Sankey, “Enhanced optomechanical levitation of minimally supported dielectrics,” Phys. Rev. A 91, 053849 (2015).
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N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
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J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
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J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Nonlinear optomechanical couplings: Tools for dealing with solid mechanical objects in the quantum regime,” in Optics InfoBase Conference Papers (2010).

Sankey, J. 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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J. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved “position-squared” readout using degenerate cavity modes,” in ICAP Proceedings (2008), pp. 131–149.

Sawadsky, A.

A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
[Crossref] [PubMed]

Schilling, R.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref] [PubMed]

Schliesser, A.

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
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Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano. 12, 776–783 (2017).
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W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S. Polzik, and A. Schliesser, “Multimode optomechanical system in the quantum regime,” Proc. Natl. Acad. Sci. 114, 62–66 (2017).
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Schnabel, R.

A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
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S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
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Shaw, M. D.

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Shkarin, A. B.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
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D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
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T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
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R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
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Smith, G.

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
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C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
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D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
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A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schnabel, “Observation of generalized optomechanical coupling and cooling on cavity resonance,” Phys. Rev. Lett. 114, 043601 (2015).
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S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
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Taylor, J.

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
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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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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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Tombesi, P.

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
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Tsaturyan, Y.

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
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Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano. 12, 776–783 (2017).
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W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S. Polzik, and A. Schliesser, “Multimode optomechanical system in the quantum regime,” Proc. Natl. Acad. Sci. 114, 62–66 (2017).
[Crossref]

Underwood, M.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
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Urmey, M. D.

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
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Uys, H.

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
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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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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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M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
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Wallucks, A.

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical Bell test,” Phys. Rev. Lett. 121, 220404 (2018).
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R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
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S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
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Wang, Z.

P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).

Weiss, T.

T. Weiss and A. Nunnenkamp, “Quantum limit of laser cooling in dispersively and dissipatively coupled optomechanical systems,” Phys. Rev. A 88, 023850 (2013).
[Crossref]

T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys. 15, 045017 (2013).
[Crossref]

Wilson, D. J.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
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K. K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
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D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
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Witmer, J. D.

P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).

Wollack, E. A.

P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Resolving the energy levels of a nanomechanical oscillator,” arXiv:1902.04681 (2019).

Xu, H.

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
[Crossref]

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Xuereb, A.

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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Yanay, Y.

Y. Yanay, J. Sankey, and A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).
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Yang, C.

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[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]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Nonlinear optomechanical couplings: Tools for dealing with solid mechanical objects in the quantum regime,” in Optics InfoBase Conference Papers (2010).

J. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved “position-squared” readout using degenerate cavity modes,” in ICAP Proceedings (2008), pp. 131–149.

Ye, J.

C. Hood, H. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001).
[Crossref]

Yu, P. L.

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

T. P. Purdy, R. W. Peterson, P. L. Yu, and C. A. Regal, “Cavity optomechanics with Si3N4 membranes at cryogenic temperatures,” New J. Phys. 14, 115021 (2012).
[Crossref]

Yu, P.-L.

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
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Yu, P.-L. L.

T. P. Purdy, P.-L. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).

Zang, L.

T. K. Paraiso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-squared coupling in a tunable photonic crystal optomechanical cavity,” Phys. Rev. X 5, 041024 (2015).

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).
[Crossref]

Zwickl, B.

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Nonlinear optomechanical couplings: Tools for dealing with solid mechanical objects in the quantum regime,” in Optics InfoBase Conference Papers (2010).

Zwickl, B. M.

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] [PubMed]

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. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved “position-squared” readout using degenerate cavity modes,” in ICAP Proceedings (2008), pp. 131–149.

Anal. Chem. (1)

S. R. Leslie, A. P. Fields, and A. E. Cohen, “Convex lens-induced confinement for imaging single molecules,” Anal. Chem. 82, 6224–6229 (2010).
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Annalen der Physik (1)

C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, and J. Lawall, “From membrane-in-the-middle to mirror-in-the-middle with a high-reflectivity sub-wavelength grating,” Annalen der Physik 527, 81–88 (2015).
[Crossref]

Appl. Phys. Lett. (1)

N. Flowers-Jacobs, S. Hoch, J. Sankey, A. Kashkanova, A. Jayich, C. Deutsch, J. Reichel, and J. Harris, “Fiber-cavity-based optomechanical device,” Appl. Phys. Lett. 101, 221109 (2012).
[Crossref]

J. Opt. (3)

D. Kilda and A. Nunnenkamp, “Squeezed light and correlated photons from dissipatively coupled optomechanical systems,” J. Opt. 18, 014007 (2016).
[Crossref]

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).
[Crossref]

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2011).
[Crossref]

Light. Sci. & Appl. (1)

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deléglise, “High-finesse Fabry-Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light. Sci. & Appl. 6, e16190 (2017).
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Nat. Comm. (1)

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Comm. 6, 6232 (2014).
[Crossref]

Nat. Nano. (1)

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano. 12, 776–783 (2017).
[Crossref]

Nat. Phys. (3)

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

A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, “Harnessing electro-optic correlations in an efficient mechanical converter,” Nat. Phys. 14, 1038–1042 (2018).
[Crossref]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

Nature (3)

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] [PubMed]

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref] [PubMed]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473–477 (2018).
[Crossref] [PubMed]

New J. Phys. (6)

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]

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
[Crossref]

T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys. 15, 045017 (2013).
[Crossref]

T. P. Purdy, R. W. Peterson, P. L. Yu, and C. A. Regal, “Cavity optomechanics with Si3N4 membranes at cryogenic temperatures,” New J. Phys. 14, 115021 (2012).
[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]

D. E. Chang, K.-K. Ni, O. Painter, and H. J. Kimble, “Ultrahigh-Q mechanical oscillators through optical trapping,” New J. Phys. 14, 45002 (2012).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (10)

T. Weiss and A. Nunnenkamp, “Quantum limit of laser cooling in dispersively and dissipatively coupled optomechanical systems,” Phys. Rev. A 88, 023850 (2013).
[Crossref]

C. Hood, H. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001).
[Crossref]

T. Müller, C. Reinhardt, and J. Sankey, “Enhanced optomechanical levitation of minimally supported dielectrics,” Phys. Rev. A 91, 053849 (2015).
[Crossref]

A. Barasheed, T. Müller, and J. Sankey, “Optically defined mechanical geometry,” Phys. Rev. A 93, 053811 (2016).
[Crossref]

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).
[Crossref]

S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, “Anomalous dynamic backaction in interferometers,” Phys. Rev. A 88, 023809 (2013).
[Crossref]

Y. Yanay, J. Sankey, and A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93, 063809 (2016).
[Crossref]

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Phys. Rev. Lett. (8)

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref] [PubMed]

F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009).
[Crossref] [PubMed]

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Figures (3)

Fig. 1
Fig. 1 Optical resonances for a membrane in a cavity. (a) 1D model, comprising two fixed end mirrors (gray), with reflection (transmission) coefficients −rj (itj), and a thin, flexible dielectric membrane (blue) having reflection (transmission) coefficients rm (tm). Aj (Bj) indicate the right-moving (left-moving) field amplitudes just outside the membrane. (b)–(c) Dependence of the cavity resonance’s detuning (normalized by the free spectral range (FSR)) on membrane displacement Δx from (b) the cavity center (x=L/2) and (c) the first mirror (x=0), with rm values (from light to dark) −0.4, −0.6, −0.8, −0.931, and −0.9977 (rm’s phase chosen to highlight avoided crossings as in [2]). Horizontal dashed lines represent empty cavity resonant frequencies (rm=0), while other dashed lines represent the left (negatively sloped) and right (positively sloped) sub-cavity resonances when tm=0. Insets qualitatively show the field distribution of these modes. (d)–(e) Dependence of cavity’s energy decay rate κ on membrane displacement Δx from (d) the cavity center and (e) the first mirror, normalized by the empty cavity value for a single-port cavity (t2=0, r2=−1).
Fig. 2
Fig. 2 Optical response of flexural MATE system. (a) Diagram of measurement system and MATE cavity (not to scale), comprising an input mirror M1, 21 μm-thick spacers, a membrane, piezo-actuated pushers, and a “backstop” mirror M2. 10 mW of laser light (λ = 1550 nm) from a fiber collimator C passes through a 50:50 splitter, is reflected from M1, and collected with a photodiode PD. The length L = 10 cm between M1 and M2 is tuned with piezo actuators on M2; the inset shows the fractional reflected power (normalized by the off-resonance value 2.5 mW) as a function of the laser’s detuning from an L-swept cavity resonance. The violet curve is a “typical” Lorentzian fit used to extract the cavity’s energy decay rate κ. (b) Same measurement for a range of L at varied membrane displacements Δx. The white dashed line shows part of a simultaneous fit to three modes’ resonant frequencies (Eq. 2) used to eliminate piezo nonlinearities, and the arrow shows the location and direction of the sweep in (a). Higher-order transverse modes appear as faint resonances. (c) Spectrum predicted by a 1D transfer matrix model, with parameters estimated as described in the main text. (d) Dependence of cavity decay rate κ on membrane position for the central TEM00 (brightest) resonance in (b). The solid line is κ(x) obtained by numerically solving the 1D model, and the black part highlights the region used for our main fits (including the piezo corrections). The dashed line is the fit empty cavity decay rate 2π/(|t1|2 + |t2|2 + S1) (agrees with our measured value prior to incorporating the membrane). (e) Dependence of resonant reflection on membrane position for the middle TEM00 resonance in (b). The solid line shows the fit result as in (d).
Fig. 3
Fig. 3 Large flexure. Length x (top) and tilt θ (bottom) as a function of the mode index l. The starting index l0 is estimated from a fit to the transmission spectra (upper inset) described in the main text. The uncertainty on l0 is ±1, as represented by the shaded area. Below l=8, the transmission data is too broad to constrain x, and “×” symbols represent the passing of a bright fringe for further flexure. The lower inset shows the interference pattern (1310 nm light) of the mirror-membrane system while maximally flexed; dotted square represents the approximate location and size of the membrane. The dark shaded area corresponds to the uncertainty in membrane thickness and M1’s coating (included via transfer matrix calculation). The tilt was calculated assuming the measured beam radius σ = 100 μm, and the light shading corresponds to a conservative upper bound (σ = 110 μm) on the actual value at the membrane.

Equations (57)

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ω MIM c k MIM = N ω FSR + ω FSR π ( arccos [ ( 1 ) N + 1 | r m | cos ( 2 k N Δ x ) ] ϕ r )
ω MATE c k MATE = N ω FSR + ω FSR π arctan [ cos ( ϕ r ) + | r m | cos ( 2 k N Δ x ) sin ( ϕ r ) | r m | sin ( 2 k N Δ x ) ] ,
κ = ( 1 | r m | 2 ) c | t 1 | 2 + ( 1 + 2 | r m | cos ( 2 k x + ϕ r ) + | r m | 2 ) c | t 2 | 2 2 x ( 1 | r m | 2 ) + 2 ( L x ) ( 1 + 2 | r m | cos ( 2 k x + ϕ r ) + | r m | 2 ) .
P t | t 1 | 2 | t m | 2 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) { 1 k 2 θ 2 σ 2 | r 1 | | r m | ( 1 + | r 1 | 2 | r m | 2 ) cos ( 2 k x 0 + ϕ ) + | r 1 | | r m | ( cos ( 4 k x 0 + 2 ϕ ) 3 ) [ 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) ] 2 } ,
A 2 = t m A 1 + r m B 2
B 1 = t m B 2 + r m A 1 .
A 1 = B 1 e 2 i k x
B 2 = A 2 e 2 i k ( L x ) ,
( t m 2 r m 2 ) e i k L e i k L = 2 r m cos ( 2 k x k L ) .
cos ( k L + ϕ r ) = | r m | cos ( 2 k x k L ) .
L = 1 k arctan [ cos ( ϕ r ) + | r m | cos ( 2 k x ) sin ( ϕ r ) | r m | sin ( 2 k x ) ] .
k = k N + Δ k .
k Δ x k N Δ x + Δ k Δ x k N Δ x ,
k MIM N π L + 1 L ( arccos [ ( 1 ) N + 1 | r m | cos ( 2 k N Δ x ) ] ϕ r ) .
ω MIM N ω FSR + ω FSR π ( arccos [ ( 1 ) N + 1 | r m | cos ( 2 k N Δ x ) ] ϕ r ) ,
k Δ x = k N Δ x + Δ k Δ x k N Δ x
k MATE N π L + 1 L arctan [ cos ( ϕ r ) + | r m | cos ( 2 k N Δ x ) sin ( ϕ r ) | r m | sin ( 2 k N Δ x ) ] ,
ω MATE N ω FSR + ω FSR π arctan [ cos ( ϕ r ) + | r m | cos ( 2 k N Δ x ) sin ( ϕ r ) | r m | sin ( 2 k N Δ x ) ] .
E = 2 x c P 1 + 2 ( L x ) c P 2 ,
t E = P 1 | t 1 | 2 P 2 | t 2 | 2 .
| A 2 B 1 | 2 = P 2 P 1 = 1 + | r m | 2 + 2 | r m | cos ( 2 k x + ϕ r ) 1 | r m | 2 .
P 1 = c E 2 ( x + ( L x ) 1 + | r m | 2 + 2 | r m | cos ( 2 k x + ϕ r ) 1 | r m | 2 ) 1 ,
t E = κ E ,
κ = ( 1 | r m | 2 ) c | t 1 | 2 + ( 1 + 2 | r m | cos ( 2 k x + ϕ r ) + | r m | 2 ) c | t 2 | 2 2 x ( 1 | r m | 2 ) + 2 ( L x ) ( 1 + 2 | r m | cos ( 2 k x + ϕ r ) + | r m | 2 ) .
B ˜ 1 κ d κ d x x zpf ,
B ˜ MIM = 2 | r m | L | r m | + cos ( 2 k MIM x + ϕ r ) + ( k MIM + x d k MIM d x ) L sin ( 2 k MIM x + ϕ r ) 1 + | r m | cos ( 2 k MIM x + ϕ r ) x zpf .
B ˜ MIM , max | t m | 1 3 3 2 k N x zpf 1 | t m | ,
Δ x MIM , max ( B ˜ ) | t m | 1 1 2 k N ( j π + ( 1 ) j + N + 1 | t m | 3 )
B ˜ input 2 | r m | L | r m | + cos ( 2 k N Δ x + ϕ r ) + 2 k N L sin ( 2 k N Δ x + ϕ r ) 1 + | r m | 2 + 2 | r m | cos ( 2 k N Δ x + ϕ r ) x zpf .
B ˜ MATE , max | t m | 1 4 k N x zpf 1 | t m | 2
Δ x MATE , max ( B ˜ ) | t m | 1 1 4 k N ( 2 π ( 2 j + 1 ) 2 ϕ r ± | t m | 2 ) .
B ˜ MATE , max B ˜ MIM , max | t m | 1 8 3 3 1 | t m | .
G MIM ( 1 ) x ω MIM = ( 1 ) N + 1 2 ω FSR k N π | r m | sin ( 2 k N Δ x ) 1 | r m | 2 cos 2 ( 2 k N Δ x )
G MIM ( 2 ) x 2 ω MIM = ( 1 ) N + 1 4 ω FSR k N 2 π | r m | ( 1 | r m | 2 ) cos ( 2 k N Δ x ) ( 1 | r m | 2 cos 2 ( 2 k N Δ x ) ) 3 / 2 ,
G MIM , max ( 1 ) = ± 2 c k N L | r m |
G MIM , max ( 2 ) = ± 4 c k N 2 L | r m | 1 | r m | 2 | t m | 1 ± 4 c k N 2 L 1 | t m |
Δ x MIM , max ( 1 ) = ( 2 j + 1 ) λ 8
Δ x MIM , max ( 2 ) = j λ 4 ,
G MATE ( 1 ) x ω MATE = 2 k N π ω FSR | r m | ( | r m | + cos ( 2 k N Δ x + ϕ r ) ) | r m | 2 + 2 | r m | cos ( 2 k N Δ x + ϕ r ) + 1
G MATE ( 2 ) x 2 ω MATE = 4 k N 2 π ω FSR | r m | ( 1 | r m | 2 ) sin ( 2 k N Δ x + ϕ r ) ( | r m | 2 + 2 | r m | cos ( 2 k N Δ x + ϕ r ) + 1 ) 2 ,
G MATE , max ( 1 ) = 2 c k N L | r m | 1 | r m | | t m | 1 4 c k N L 1 | t m | 2
G MATE , max ( 2 ) = G MATE ( 2 ) ( Δ x MATE , max ( 2 ) ) | t m | 1 ± 18 3 c k N 2 L 1 | t m | 4
Δ x MATE , max ( 1 ) = ( 2 j + 1 ) π ϕ r 2 k N
Δ x MATE , max ( 2 ) = 1 2 k N ( 2 π j ϕ r ± 2 arctan { 6 | r m | + | r m | 4 + 34 | r m | 2 + 1 ( 1 | r m | ) 2 } )
Δ x MATE , pure ( 2 ) = ± arccos { | r m | } ϕ r + 2 π j 2 k N ,
G MATE , max ( 1 ) G MIM , max ( 1 ) = 1 1 | r m | | t m | 1 2 | t m | 2 ,
A ˜ max , MIM ( 1 ) = A ˜ max , MATE ( 1 ) | t m | 1 8 k N x zpf | t 1 | 2 | t m | 2 .
G MATE , max ( 2 ) G MIM , max ( 2 ) | t m | 1 9 2 3 1 | t m | 3 .
A ˜ max , MATE ( 2 ) A ˜ max , MIM ( 2 ) | t m | 1 4 3 3 1 | t m | .
B ˜ MIM , pure = ± 2 k N x zpf | r m | | t m |
B ˜ MATE , pure = ± 4 k N x zpf | r m | | t m |
B ˜ MATE , pure = ± 4 k N x zpf Δ x MATE , pure L | r m | | t m | ,
A 2 = E in ( t 1 t m e i k x ) j = 0 [ r 1 r m e 2 i k x ] j
= E in t 1 t m e i k x 1 r 1 r m e 2 i k x ,
p t | A 2 / E in | 2 = | t 1 | 2 | t m | 2 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + 2 k θ y + ϕ ) 2 π σ 2 e 2 y 2 σ 2 ,
p t | t 1 | 2 | t m | 2 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) 2 π σ 2 e 2 y 2 σ 2 { 1 4 k 2 θ 2 y 2 | r 1 | | r m | ( 1 + | r 1 | 2 | r m | 2 ) cos ( 2 k x 0 + ϕ ) + | r 1 | | r m | ( cos ( 4 k x 0 + 2 ϕ ) 3 ) [ 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) ] 2 } ,
P t | t 1 | 2 | t m | 2 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) { 1 k 2 θ 2 σ 2 | r 1 | | r m | ( 1 + | r 1 | 2 | r m | 2 ) cos ( 2 k x 0 + ϕ ) + | r 1 | | r m | ( cos ( 4 k x 0 + 2 ϕ ) 3 ) [ 1 + | r 1 | 2 | r m | 2 2 | r 1 | | r m | cos ( 2 k x 0 + ϕ ) ] 2 } .

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