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Characterizing stationary optomechanical entanglement in the presence of non-Markovian noise

Su Direkci1,*, Klemens Winkler2, Corentin Gut2,†, Markus Aspelmeyer2,3, and Yanbei Chen1

  • *Contact author: sdirekci@caltech.edu
  • †Present address: KEEQuant GmbH, Gebhardtstraße 28, 90762 Fürth, Germany.
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Phys. Rev. A 112, 043512 – Published 9 October, 2025

DOI: https://doi.org/10.1103/gysx-m9p2

Abstract

We study an optomechanical system, where a mechanical oscillator interacts with a Gaussian input optical field. In the linearized picture, we analytically prove that if the input light field is the vacuum state, or is frequency-independently squeezed, the stationary entanglement between the oscillator and the output optical field is independent of the coherent coupling between them, which we refer to as the universality of entanglement. Furthermore, we demonstrate that entanglement cannot be generated by performing arbitrary frequency-dependent squeezing on the input optical field. Our results hold in the presence of general, Gaussian environmental noise sources, including non-Markovian noise.

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Physics Subject Headings (PhySH)

Corrections

7 January, 2026

Correction: The address in the byline footnote for the third author contained a typographical error and has been fixed.

See Also

Universality of Stationary Entanglement in an Optomechanical System Driven by Non-Markovian Noise and Squeezed Light

Su Direkci, Klemens Winkler, Corentin Gut, Markus Aspelmeyer, and Yanbei Chen
Phys. Rev. Lett. 135, 153601 (2025)

Article Text

References (61)

  1. W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75, 715 (2003).
  2. M. Schlosshauer, Quantum decoherence, Phys. Rep. 831, 1 (2019).
  3. D. Aharonov, Quantum to classical phase transition in noisy quantum computers, Phys. Rev. A 62, 062311 (2000).
  4. C. Genes, A. Mari, D. Vitali, and P. Tombesi, Quantum effects in optomechanical systems, in Advances in Atomic Molecular and Optical Physics, Advances In Atomic, Molecular, and Optical Physics Vol. 57 (Academic, San Diego, 2009), Chap. 2, pp. 33–86.
  5. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
  6. Y. Ma, H. Miao, B. H. Pang, M. Evans, C. Zhao, J. Harms, R. Schnabel, and Y. Chen, Proposal for gravitational-wave detection beyond the standard quantum limit through EPR entanglement, Nat. Phys. 13, 776 (2017).
  7. S. G. Hofer and K. Hammerer, Entanglement-enhanced time-continuous quantum control in optomechanics, Phys. Rev. A 91, 033822 (2015).
  8. D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, Optomechanical quantum control of a nitrogen-vacancy center in diamond, Phys. Rev. Lett. 116, 143602 (2016).
  9. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, Laser cooling of a nanomechanical oscillator into its quantum ground state, Nature (London) 478, 89 (2011).
  10. H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, and Y. Chen, Probing macroscopic quantum states with a sub-heisenberg accuracy, Phys. Rev. A 81, 012114 (2010).
  11. H. Yang, H. Miao, D.-S. Lee, B. Helou, and Y. Chen, Macroscopic quantum mechanics in a classical spacetime, Phys. Rev. Lett. 110, 170401 (2013).
  12. M. Wang, X.-Y. Lü, Y.-D. Wang, J. Q. You, and Y. Wu, Macroscopic quantum entanglement in modulated optomechanics, Phys. Rev. A 94, 053807 (2016).
  13. S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, Reversible optical-to-microwave quantum interface, Phys. Rev. Lett. 109, 130503 (2012).
  14. P. K. Shandilya, D. P. Lake, M. J. Mitchell, D. D. Sukachev, and P. E. Barclay, Optomechanical interface between telecom photons and spin quantum memory, Nat. Phys. 17, 1420 (2021).
  15. H. Miao, S. Danilishin, and Y. Chen, Universal quantum entanglement between an oscillator and continuous fields, Phys. Rev. A 81, 052307 (2010).
  16. M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, Effect of phase noise on the generation of stationary entanglement in cavity optomechanics, Phys. Rev. A 84, 032325 (2011).
  17. S. Direkci, K. Winkler, C. Gut, K. Hammerer, M. Aspelmeyer, and Y. Chen, Macroscopic quantum entanglement between an optomechanical cavity and a continuous field in presence of non-markovian noise, Phys. Rev. Res. 6, 013175 (2024).
  18. T. Hong, H. Yang, E. K. Gustafson, R. X. Adhikari, and Y. Chen, Brownian thermal noise in multilayer coated mirrors, Phys. Rev. D 87, 082001 (2013).
  19. S. Kroker, J. Dickmann, C. B. Rojas Hurtado, D. Heinert, R. Nawrodt, Y. Levin, and S. P. Vyatchanin, Brownian thermal noise in functional optical surfaces, Phys. Rev. D 96, 022002 (2017).
  20. G. D. Hammond, A. V. Cumming, J. Hough, R. Kumar, K. Tokmakov, S. Reid, and S. Rowan, Reducing the suspension thermal noise of advanced gravitational wave detectors, Classical Quantum Gravity 29, 124009 (2012).
  21. J. Harms and C. M. Mow-Lowry, Suspension-thermal noise in spring–antispring systems for future gravitational-wave detectors, Classical Quantum Gravity 35, 025008 (2018).
  22. P. R. Saulson, Thermal noise in mechanical experiments, Phys. Rev. D 42, 2437 (1990).
  23. S. Direkci, K. Winkler, C. Gut, M. Aspelmeyer, and Y. Chen, companion paper, Universality of stationary entanglement in an optomechanical system driven by non-Markovian noise and squeezed light, Phys. Rev. Lett. 135, 153601 (2025).
  24. H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, Entanglement of macroscopic test masses and the standard quantum limit in laser interferometry, Phys. Rev. Lett. 100, 013601 (2008).
  25. In terms of system parameters, Ωq=Iprobeωprobemc2, where Iprobe and ωprobe are the intensity and carrier frequency of the light probing the mechanics, c is the speed of light in vacuum, and m is the mass of the oscillator.
  26. We choose to normalize the quadratures of the mechanical oscillator such that the entries of the commutator matrix, defined in Eq. (9), do not have a factor of 1/2.
  27. L. Trozzo and F. Badaracco, Seismic and newtonian noise in the gw detectors, Galaxies 10, 20 (2022).
  28. K. Janssens et al., Correlated 0.01–40 Hz seismic and newtonian noise and its impact on future gravitational-wave detectors, Phys. Rev. D 109, 102002 (2024).
  29. G. González, Suspensions thermal noise in the ligo gravitational wave detector, Classical Quantum Gravity 17, 4409 (2000).
  30. A. V. Cumming, R. Jones, G. D. Hammond, J. Hough, I. W. Martin, and S. Rowan, Large-scale monolithic fused-silica mirror suspension for third-generation gravitational-wave detectors, Phys. Rev. Appl. 17, 024044 (2022).
  31. A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, and N. Mavalvala, Structural thermal noise in gram-scale mirror oscillators, New J. Phys. 14, 115008 (2012).
  32. G. I. González and P. R. Saulson, Brownian motion of a torsion pendulum with internal friction, Phys. Lett. A 201, 12 (1995).
  33. B. M. Brubaker, J. M. Kindem, M. D. Urmey, S. Mittal, R. D. Delaney, P. S. Burns, M. R. Vissers, K. W. Lehnert, and C. A. Regal, Optomechanical ground-state cooling in a continuous and efficient electro-optic transducer, Phys. Rev. X 12, 021062 (2022).
  34. V. Jain, J. Gieseler, C. Moritz, C. Dellago, R. Quidant, and L. Novotny, Direct measurement of photon recoil from a levitated nanoparticle, Phys. Rev. Lett. 116, 243601 (2016).
  35. J. M. H. Gosling, A. Pontin, J. H. Iacoponi, P. F. Barker, and T. S. Monteiro, Sensing directional noise baths in levitated optomechanics, Phys. Rev. Res. 6, 013129 (2024).
  36. C. Gonzalez-Ballestero, M. Aspelmeyer, L. Novotny, R. Quidant, and O. Romero-Isart, Levitodynamics: Levitation and control of microscopic objects in vacuum, Science 374, eabg3027 (2021).
  37. G. M. Harry, H. Armandula, E. Black, D. R. M. Crooks, G. Cagnoli, J. Hough, P. Murray, S. Reid, S. Rowan, P. Sneddon, M. M. Fejer, R. Route, and S. D. Penn, Thermal noise from optical coatings in gravitational wave detectors, Appl. Opt. 45, 1569 (2006).
  38. S. Gras and M. Evans, Direct measurement of coating thermal noise in optical resonators, Phys. Rev. D 98, 122001 (2018).
  39. X. Yao, M. H. J. de Jong, J. Li, and S. Gröblacher, Long-range optomechanical interactions in sin membrane arrays, Phys. Rev. X 15, 011014 (2025).
  40. M. Ho, E. Oudot, J.-D. Bancal, and N. Sangouard, Witnessing optomechanical entanglement with photon counting, Phys. Rev. Lett. 121, 023602 (2018).
  41. We use the convention F{f(t)}=∫−∞∞f(t)eiωtdt.
  42. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  43. R. Simon, Peres-Horodecki separability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2726 (2000).
  44. L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Inseparability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2722 (2000).
  45. R. F. Werner and M. M. Wolf, Bound entangled gaussian states, Phys. Rev. Lett. 86, 3658 (2001).
  46. G. Adesso and F. Illuminati, Entanglement in continuous variable systems: Recent advances and current perspectives, J. Phys. A: Math. Theor. 40, 7821 (2007).
  47. A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods (CRC Press/Taylor & Francis Group, Boca Raton, FL, 2017).
  48. G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  49. M. B. Plenio, Logarithmic negativity: A full entanglement monotone that is not convex, Phys. Rev. Lett. 95, 090503 (2005).
  50. Note that in general, the operator M is a function of two time indices, in the form of M(t,t′). Here, assume M(τ)=M(t−t′) for simplicity, which is true when the input light field is the vacuum state.
  51. H. B. Callen and T. A. Welton, Irreversibility and generalized noise, Phys. Rev. 83, 34 (1951).
  52. R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966).
  53. Y. Chen, Macroscopic quantum mechanics: Theory and experimental concepts of optomechanics, J. Phys. B: At., Mol. Opt. Phys. 46, 104001 (2013).
  54. Note that the space is unchanged for frequency-independent squeezing.
  55. The squeezing cannot be reversed perfectly due to the interaction with the mechanical oscillator.
  56. https://github.com/sudirekci/universality_of_stationary_optomechanical_entanglement.
  57. A. V. Kisil, I. D. Abrahams, G. Mishuris, and S. V. Rogosin, The wiener–hopf technique, its generalizations and applications: Constructive and approximate methods, Proce. R. Soc. A 477, 20210533 (2021).
  58. G. Adesso, S. Ragy, and A. R. Lee, Continuous variable quantum information: Gaussian states and beyond, Open Syst. Inf. Dyn. 21, 1440001 (2014).
  59. 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. D 65, 022002 (2001).
  60. L. McCuller, C. Whittle, D. Ganapathy, K. Komori, M. Tse, A. Fernandez-Galiana, L. Barsotti, P. Fritschel, M. MacInnis, F. Matichard, K. Mason, N. Mavalvala, R. Mittleman, H. Yu, M. E. Zucker, and M. Evans, Frequency-dependent squeezing for advanced ligo, Phys. Rev. Lett. 124, 171102 (2020).
  61. S. L. Danilishin and F. Y. Khalili, Quantum measurement theory in gravitational-wave detectors, Living Rev. Relativ. 15, 5 (2012).

Outline

Information

Phys. Rev. A 112, 043512– Published 9 October, 2025
  • Received 10 February 2025
  • Revised 30 June 2025
  • Accepted 21 August 2025
  • Corrected 7 January 2026
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