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

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. Lett. 135, 153601 – Published 9 October, 2025

DOI: https://doi.org/10.1103/vzhk-cdc3

Abstract

Optomechanical systems subjected to environmental noise give rise to rich physical phenomena. We investigate entanglement between a mechanical oscillator and the reflected coherent optical field in a general, not necessarily Markovian environment. For the input optical field, we consider stationary Gaussian states and frequency-dependent squeezing. We demonstrate that for a coherent laser drive, either unsqueezed or squeezed in a frequency-independent manner, optomechanical entanglement is destroyed after a threshold that depends only on the environmental noises—independent of the coherent coupling between the oscillator and the optical field, or the squeeze factor. In this way, we have found a universal entangling-disentangling transition. We also show that for a configuration in which the oscillator and the reflected field are separable, entanglement cannot be generated by incorporating frequency-dependent squeezing in the optical field.

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Physics Subject Headings (PhySH)

Corrections

22 January, 2026

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

See Also

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

Su Direkci, Klemens Winkler, Corentin Gut, Markus Aspelmeyer, and Yanbei Chen
Phys. Rev. A 112, 043512 (2025)

Article Text

References (54)

  1. J. von Neumann and R. T. Beyer, Mathematical Foundations of Quantum Mechanics: New Edition (Princeton University Press, Princeton, NJ, 2018).
  2. W. H. Zurek, Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?, Phys. Rev. D 24, 1516 (1981).
  3. E. Joos and H. D. Zeh, The emergence of classical properties through interaction with the environment, Z. Phys. B 59, 223 (1985).
  4. M. Schlosshauer-Selbach, Decoherence and the Quantum-to-Classical Transition (Springer, Berlin, Heidelberg, 2008).
  5. M. Schlosshauer, Quantum decoherence, Phys. Rep. 831, 1 (2019).
  6. C. Genes, A. Mari, D. Vitali, and P. Tombesi, Quantum effects in optomechanical systems, in Advances in Atomic Molecular and Optical Physics, Vol. 57 (Academic Press, San Diego, 2009), pp. 33–86.
  7. J.-D. Tang, Q.-Z. Cai, Z.-D. Cheng, N. Xu, G.-Y. Peng, P.-Q. Chen, D.-G. Wang, Z.-W. Xia, Y. Wang, H.-Z. Song, Q. Zhou, and G.-W. Deng, A perspective on quantum entanglement in optomechanical systems, Phys. Lett. A 429, 127966 (2022).
  8. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
  9. S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, Quantum entanglement and teleportation in pulsed cavity optomechanics, Phys. Rev. A 84, 052327 (2011).
  10. T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, Entangling mechanical motion with microwave fields, Science 342, 710 (2013).
  11. H. Miao, S. Danilishin, and Y. Chen, Universal quantum entanglement between an oscillator and continuous fields, Phys. Rev. A 81, 052307 (2010).
  12. 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).
  13. C. Gut, K. Winkler, J. Hoelscher-Obermaier, S. G. Hofer, R. M. Nia, N. Walk, A. Steffens, J. Eisert, W. Wieczorek, J. A. Slater, M. Aspelmeyer, and K. Hammerer, Stationary optomechanical entanglement between a mechanical oscillator and its measurement apparatus, Phys. Rev. Res. 2, 033244 (2020).
  14. 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).
  15. M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, Creating and probing multipartite macroscopic entanglement with light, Phys. Rev. Lett. 99, 250401 (2007).
  16. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, Optomechanical entanglement between a movable mirror and a cavity field, Phys. Rev. Lett. 98, 030405 (2007).
  17. See the companion article , Characterizing stationary optomechanical entanglement in the presence of non-Markovian noise, Phys. Rev. A, 112, 043512 (2025).
  18. S. L. Danilishin and F. Y. Khalili, Quantum measurement theory in gravitational-wave detectors, Living Rev. Relativity 15, 5 (2012).
  19. Y. Chen, Macroscopic quantum mechanics: Theory and experimental concepts of optomechanics, J. Phys. B 46, 104001 (2013).
  20. LIGO Scientific Collaboration, Advanced LIGO, Classical Quantum Gravity 32, 074001 (2015).
  21. H. Yu, L. McCuller, M. Tse, N. Kijbunchoo, L. Barsotti, and N. Mavalvala, Quantum correlations between light and the kilogram-mass mirrors of LIGO, Nature (London) 583, 43 (2020).
  22. 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).
  23. M. Ho, E. Oudot, J.-D. Bancal, and N. Sangouard, Witnessing optomechanical entanglement with photon counting, Phys. Rev. Lett. 121, 023602 (2018).

  24. The factor of 2 is a particular normalization that simplifies the derivations, and it is found in several works with continuous variables; see Ref. [25], for example.

  25. G. Adesso, S. Ragy, and A. R. Lee, Continuous variable quantum information: Gaussian states and beyond, Open Syst. Inf. Dyn. 21, 1440001 (2014).
  26. 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).
  27. S. Hofer, Quantum control of optomechanical systems, Ph.D. thesis, University of Vienna, 2015.
  28. G. Adesso and F. Illuminati, Entanglement in continuous variable systems: Recent advances and current perspectives, J. Phys. A 40, 7821 (2007).

  29. In terms of system parameters, Ωq=(Iprobeωprobe/Mc2), where Iprobe and ωprobe are the intensity and the frequency of the light probing the mechanics, c is the speed of light in vacuum, and M is the mass of the oscillator.

  30. R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966).
  31. H. B. Callen and T. A. Welton, Irreversibility and generalized noise, Phys. Rev. 83, 34 (1951).
  32. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, England, 2009).
  33. C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, Springer Complexity, 3rd ed. (Springer, Berlin, 2004).

  34. Even though it is unavoidable, back action noise can be evaded through specific measurement schemes [35, 36, 37].

  35. K. S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, Quantum nondemolition measurements of harmonic oscillators, Phys. Rev. Lett. 40, 667 (1978).
  36. J. Suh, A. J. Weinstein, C. U. Lei, E. E. Wollman, S. K. Steinke, P. Meystre, A. A. Clerk, and K. C. Schwab, Mechanically detecting and avoiding the quantum fluctuations of a microwave field, Science 344, 1262 (2014).
  37. I. Shomroni, L. Qiu, D. Malz, A. Nunnenkamp, and T. J. Kippenberg, Optical backaction-evading measurement of a mechanical oscillator, Nat. Commun. 10, 2086 (2019).
  38. S. Gras and M. Evans, Direct measurement of coating thermal noise in optical resonators, Phys. Rev. D 98, 122001 (2018).
  39. R. Simon, Peres-Horodecki separability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2726 (2000).
  40. R. F. Werner and M. M. Wolf, Bound entangled Gaussian states, Phys. Rev. Lett. 86, 3658 (2001).
  41. S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005).
  42. A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods (CRC Press, Taylor & Francis Group, Boca Raton, 2017).

  43. We disregard the trivial case where Ωq=0: the joint state will be separable, as there is no optomechanical interaction.

  44. Note that αF cannot be zero due to the fluctuation-dissipation theorem.

  45. We note that this trivial entanglement in the absence of sensing noise is a potential pitfall. Indeed, models for high-frequency mechanical oscillators typically disregard sensing noise [8]. Then, studying entanglement with these models might yield misleading results where entanglement seems very robust, as is the case in Ref. [13]. This point is discussed in detail in Ref. [46].

  46. C. Gut, Stationary optomechanical entanglement detection, Ph.D. thesis, University of Vienna, 2024.
  47. P. Marian and T. A. Marian, Entanglement of formation for an arbitrary two-mode Gaussian state, Phys. Rev. Lett. 101, 220403 (2008).
  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. 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).
  51. J. Hoelscher, Generation and detection of quantum entanglement in optomechanical systems, Ph.D. thesis, University of Vienna, 2017.
  52. R. M. Nia, Multimode optomechanics in the strong cooperativity regime: Towards optomechanical entanglement with micromechanical membranes, Ph.D. thesis, University of Vienna, 2018.
  53. S. Gröblacher, A. Trubarov, N. Prigge, G. D. Cole, M. Aspelmeyer, and J. Eisert, Observation of non-Markovian micromechanical Brownian motion, Nat. Commun. 6, 7606 (2015).
  54. https://github.com/sudirekci/universality_of_stationary_optomechanical_entanglement

Outline

Information

Phys. Rev. Lett. 135, 153601– Published 9 October, 2025
  • Received 10 February 2025
  • Accepted 19 August 2025
  • Corrected 22 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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