Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Spin–orbit microlaser emitting in a four-dimensional Hilbert space

Nature volume 612, pages 246–251 (2022)Cite this article

Subjects

Abstract

A step towards the next generation of high-capacity, noise-resilient communication and computing technologies is a substantial increase in the dimensionality of information space and the synthesis of superposition states on an N-dimensional (N > 2) Hilbert space featuring exotic group symmetries. Despite the rapid development of photonic devices and systems, on-chip information technologies are mostly limited to two-level systems owing to the lack of sufficient reconfigurability to satisfy the stringent requirement for 2(N − 1) degrees of freedom, intrinsically associated with the increase of synthetic dimensionalities. Even with extensive efforts dedicated to recently emerged vector lasers and microcavities for the expansion of dimensionalities1,2,3,4,5,6,7,8,9,10, it still remains a challenge to actively tune the diversified, high-dimensional superposition states of light on demand. Here we demonstrate a hyperdimensional, spin–orbit microlaser for chip-scale flexible generation and manipulation of arbitrary four-level states. Two microcavities coupled through a non-Hermitian synthetic gauge field are designed to emit spin–orbit-coupled states of light with six degrees of freedom. The vectorial state of the emitted laser beam in free space can be mapped on a Bloch hypersphere defining an SU(4) symmetry, demonstrating dynamical generation and reconfiguration of high-dimensional superposition states with high fidelity.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: The hyperdimensional spin–orbit microlaser.
Fig. 2: Independent emission control on two distinguished HOPS.
Fig. 3: SU(4) Bloch hypersphere by delicate control of inter-ring coupling.
Fig. 4: Generation and reconfiguration of SU(4) states.

Similar content being viewed by others

Data availability

 Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The computer codes that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

References

  1. Zhang, Z. et al. Tunable topological charge vortex microlaser. Science 368, 760–763 (2020).

    Article  CAS  ADS  Google Scholar 

  2. Shao, Z., Zhu, J., Chen, Y., Zhang, Y. & Yu, S. Spin–orbit interaction of light induced by transverse spin angular momentum engineering. Nat. Commun. 9, 926 (2018).

    Article  ADS  Google Scholar 

  3. Qiao, X. et al. Higher-dimensional supersymmetric microlaser arrays. Science 372, 403–408 (2021).

    Article  MathSciNet  CAS  MATH  ADS  Google Scholar 

  4. Ma, X. et al. High-speed directly modulated cylindrical vector beam lasers. ACS Photon. 6, 3261–3270 (2019).

    Article  CAS  Google Scholar 

  5. Papič, M. et al. Topological liquid crystal superstructures as structured light lasers. Proc. Natl Acad. Sci. USA 118, e2110839118 (2021).

    Article  Google Scholar 

  6. Komisar, D., Kumar, S., Kan, Y., Wu, C. & Bozhevolnyi, S. I. Generation of radially polarized single photons with plasmonic bullseye antennas. ACS Photon. 8, 2190–2196 (2021).

    Article  CAS  Google Scholar 

  7. Lin, W., Ota, Y., Arakawa, Y. & Iwamoto, S. Microcavity-based generation of full Poincaré beams with arbitrary skyrmion numbers. Phys. Rev. Res. 3, 023055 (2021).

    Article  CAS  Google Scholar 

  8. Mohamed, S. et al. Controlling topology and polarization state of lasing photonic bound states in continuum. Laser Photon. Rev. 16, 2100574 (2022).

  9. He, C., Shen, Y. & Forbes, A. Towards higher-dimensional structured light. Light Sci. Appl. 11, 205 (2022).

    Article  CAS  ADS  Google Scholar 

  10. Miao, P. et al. Orbital angular momentum microlaser. Science 353, 464–467 (2016).

    Article  CAS  ADS  Google Scholar 

  11. Bloch, F. Nuclear induction. Phys. Rev. 70, 460–474 (1946).

    Article  CAS  ADS  Google Scholar 

  12. Padgett, M. J. & Courtial, J. Poincaré-sphere equivalent for light beams containing orbital angular momentum. Opt. Lett. 24, 430–432 (1999).

    Article  CAS  ADS  Google Scholar 

  13. Kimura, G. The Bloch vector for N-level systems. Phys. Lett. A 314, 339–349 (2003).

    Article  MathSciNet  CAS  MATH  ADS  Google Scholar 

  14. Kemp, C. J., Cooper, N. R. & Ünal, F. N. Nested-sphere description of the N-level Chern number and the generalized Bloch hypersphere. Phys. Rev. Res. 4, 023120 (2022).

    Article  CAS  Google Scholar 

  15. Barreiro, J. T., Wei, T.-C. & Kwiat, P. G. Beating the channel capacity limit for linear photonic superdense coding. Nat. Phys. 4, 282–286 (2008).

    Article  CAS  Google Scholar 

  16. Bouchard, F., Fickler, R., Boyd, R. W. & Karimi, E. High-dimensional quantum cloning and applications to quantum hacking. Sci. Adv. 3, e1601915 (2017).

    Article  ADS  Google Scholar 

  17. Wang, Y., Hu, Z., Sanders, B. C. & Kais, S. Qudits and high-dimensional quantum computing. Front. Phys. 8, 589504 (2020).

    Article  Google Scholar 

  18. Gao, X., Erhard, M., Zeilinger, A. & Krenn, M. Computer-inspired concept for high-dimensional multipartite quantum gates. Phys. Rev. Lett. 125, 050501 (2020).

    Article  MathSciNet  CAS  ADS  Google Scholar 

  19. Wang, F. et al. Generation of the complete four-dimensional Bell basis. Optica 4, 1462–1467 (2017).

    Article  CAS  ADS  Google Scholar 

  20. Tilma, T., Byrd, M. & Sudarshan, E. A parametrization of bipartite systems based on SU(4) Euler angles. J. Phys. A 35, 10445 (2002).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  21. Forbes, A. & Nape, I. Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light. AVS Quantum Sci. 1, 011701 (2019).

    Article  ADS  Google Scholar 

  22. Milione, G., Sztul, H., Nolan, D. & Alfano, R. Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light. Phys. Rev. Lett. 107, 053601 (2011).

    Article  ADS  Google Scholar 

  23. Naidoo, D. et al. Controlled generation of higher-order Poincaré sphere beams from a laser. Nat. Photon. 10, 327–332 (2016).

    Article  CAS  ADS  Google Scholar 

  24. Shen, Y., Yang, X., Naidoo, D., Fu, X. & Forbes, A. Structured ray-wave vector vortex beams in multiple degrees of freedom from a laser. Optica 7, 820–831 (2020).

    Article  CAS  ADS  Google Scholar 

  25. Shen, Y. et al. Creation and control of high-dimensional multi-partite classically entangled light. Light Sci. Appl. 10, 50 (2021).

    Article  CAS  ADS  Google Scholar 

  26. Wang, J. et al. Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat. Photon. 6, 488–496 (2012).

    Article  CAS  ADS  Google Scholar 

  27. Bahari, B. et al. Photonic quantum Hall effect and multiplexed light sources of large orbital angular momenta. Nat. Phys. 17, 700–703 (2021).

    Article  CAS  Google Scholar 

  28. Fang, X. et al. High-dimensional orbital angular momentum multiplexing nonlinear holography. Adv. Photon. 3, 015001 (2021).

    Article  ADS  Google Scholar 

  29. Hatano, N. & Nelson, D. R. Localization transitions in non-Hermitian quantum mechanics. Phys. Rev. Lett. 77, 570–573 (1996).

    Article  CAS  ADS  Google Scholar 

  30. Longhi, S. Non‐Hermitian gauged topological laser arrays. Ann. Phys. 530, 1800023 (2018).

    Article  MathSciNet  Google Scholar 

  31. Yu, Y., Jung, M. & Shvets, G. Zero-energy corner states in a non-Hermitian quadrupole insulator. Phys. Rev. B 103, L041102 (2021).

    Article  CAS  ADS  Google Scholar 

  32. McMaster, W. H. Polarization and the Stokes parameters. Am. J. Phys. 22, 351–362 (1954).

    Article  MATH  ADS  Google Scholar 

  33. Shen, Y. & Rosales‐Guzmán, C. Nonseparable states of light: from quantum to classical. Laser Photon. Rev. 16, 2100533 (2022).

  34. Lo, H.-K., Ma, X. & Chen, K. Decoy state quantum key distribution. Phys. Rev. Lett. 94, 230504 (2005).

    Article  ADS  Google Scholar 

  35. Ding, Y. et al. High-dimensional quantum key distribution based on multicore fiber using silicon photonic integrated circuits. npj Quantum Inf. 3, 25 (2017).

    Article  ADS  Google Scholar 

  36. Kagalwala, K. H., Di Giuseppe, G., Abouraddy, A. F. & Saleh, B. E. Bell’s measure in classical optical coherence. Nat. Photon. 7, 72–78 (2013).

    Article  CAS  ADS  Google Scholar 

  37. Töppel, F., Aiello, A., Marquardt, C., Giacobino, E. & Leuchs, G. Classical entanglement in polarization metrology. New J. Phys. 16, 073019 (2014).

    Article  MATH  ADS  Google Scholar 

  38. Ndagano, B. et al. Characterizing quantum channels with non-separable states of classical light. Nat. Phys. 13, 397–402 (2017).

    Article  CAS  Google Scholar 

  39. Qian, X.-F., Little, B., Howell, J. C. & Eberly, J. Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields. Optica 2, 611–615 (2015).

    Article  ADS  Google Scholar 

  40. Bogaerts, W. et al. Programmable photonic circuits. Nature 586, 207–216 (2020).

    Article  CAS  ADS  Google Scholar 

  41. Bliokh, K. Y., Rodríguez-Fortuño, F. J., Nori, F. & Zayats, A. V. Spin–orbit interactions of light. Nat. Photon. 9, 796–808 (2015).

    Article  CAS  ADS  Google Scholar 

  42. Van Mechelen, T. & Jacob, Z. Universal spin-momentum locking of evanescent waves. Optica 3, 118–126 (2016).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We acknowledge the support from the US Army Research Office (ARO) (W911NF-19-1-0249 and W911NF-21-1-0148), National Science Foundation (NSF) (ECCS-1932803, ECCS-1842612, OMA-1936276 and PHY-1847240), Defense Advanced Research Projects Agency (DARPA) (W91NF-21-1-0340), Office of Naval Research (ONR) (N00014-20-1-2558) and King Abdullah University of Science & Technology (OSR-2020-CRG9-4374.3). L.F. also acknowledges the support from Sloan Research Fellowship. This work was partially supported by NSF through the University of Pennsylvania Materials Research Science and Engineering Center (MRSEC) (DMR-1720530) and carried out in part at the Singh Center for Nanotechnology, which is supported by the NSF National Nanotechnology Coordinated Infrastructure Program under grant NNCI-1542153.

Author information

Author notes

  1. These authors contributed equally: Zhifeng Zhang, Haoqi Zhao

Authors and Affiliations

  1. Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA, USA

    Zhifeng Zhang, Shuang Wu, Tianwei Wu, Zihe Gao, Ritesh Agarwal & Liang Feng

  2. Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA

    Zhifeng Zhang, Haoqi Zhao, Xingdu Qiao & Liang Feng

  3. Dipartimento di Fisica, Politecnico di Milano, Milan, Italy

    Stefano Longhi

  4. IFISC (UIB-CSIC), Instituto de Fisica Interdisciplinary Sistemas Complejos, Palma de Mallorca, Spain

    Stefano Longhi

  5. Department of Electrical and Computer Engineering and Department of Physics, Duke University, Durham, NC, USA

    Natalia M. Litchinitser

  6. Department of Physics and Astronomy, College of Staten Island, CUNY, Staten Island, NY, USA

    Li Ge

  7. The Graduate Center, CUNY, New York, NY, USA

    Li Ge

Authors
  1. Zhifeng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haoqi Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shuang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tianwei Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xingdu Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Zihe Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Ritesh Agarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Stefano Longhi
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Natalia M. Litchinitser
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. Li Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

  11. Liang Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Z.Z., H.Z. and L.F. designed the experiment. H.Z., Z.Z. and L.F. developed the concept on the high-dimensional states. L.G., Z.Z., H.Z., S.L. and L.F. constructed the theoretical model. Z.Z., H.Z., T.W. and Z.G. conducted numerical simulations. X.Q., T.W. and H.Z. fabricated the samples. Z.Z., H.Z., S.W. and T.W. performed the measurements and data processing. All authors contributed to discussions and manuscript preparation.

Corresponding author

Correspondence to Liang Feng.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Yijie Shen and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Optical set-up and spectral characterization of the microlaser.

a, Optical set-up for the characterization of the microlaser. b, Lasing spectrum of the microlaser under a pumping intensity of 25 kW/cm2 which shows a robust single mode lasing. c, Light–light curve of the microlaser.

Extended Data Fig. 2 Chiral control and OAM characterization of spin–orbit-coupled emissions from two microrings by a cylindrical lens.

a, Characterization of OAM emissions from the left microring under different pumping and measurement conditions (\({g}_{1}\gg {g}_{2}\), \({g}_{1}={g}_{2}\), and \({g}_{2}\gg {g}_{1}\)). b, Characterization of OAM emissions from the right microring under different pumping and measurement conditions (\({g}_{4}\gg {g}_{3}\), \({g}_{3}={g}_{4}\), and \({g}_{3}\gg {g}_{4}\)). In both a and b, top, middle and bottom rows show unpolarized, left-handed polarized, and right-handed polarized components of laser emission.

Extended Data Fig. 3 Experimental demonstration of chiral control on HOPS I and II.

a, HOPS I and b, HOPS II.

Extended Data Fig. 4 Stoke polarimetry to retrieve the relative phase between two pole states on HOPS.

a, Six polarization states are recorded corresponding to \({I}_{0}(x,y)\), \({I}_{45}(x,y)\), \({I}_{90}(x,y)\), \({I}_{135}(x,y)\), \({I}_{\uparrow }\), and \({I}_{\downarrow }\) for phase retrieval using the Stokes polarimetry. White arrows denote the direction of polarizations. b, The retrieved relative phase distribution between \(|\,+\,2,\uparrow \rangle \) and \(|\,-\,2,\downarrow \rangle \) components, showing 8\(\pi \) phase winding in the azimuthal direction.

Extended Data Fig. 5 Experimental phase tuning in two individual HOPS associated with the left and right microrings.

a, Phase tuning on HOPS I associated with the left microring (\({\varphi =\phi }_{2}-{\phi }_{1}\)) under continuous-wave laser heating. Positive/negative heating power here represents heating on heater 1/2, respectively. b, Phase tuning on HOPS II associated with the right microring (\({\varphi =\phi }_{3}-{\phi }_{4}\)) under continuous-wave laser heating. Positive/negative heating power here represents heating on heater 3/4, respectively. The slight difference in slopes in both panels results from small variance in absorption efficiency associated with different heaters.

Extended Data Fig. 6 Controlled frequency detuning in the microlaser.

The frequency detuning between the two microrings under different heating power from the continuous-wave laser, showing the increase of the detuning as the increase of heater power.

Supplementary information

Supplementary Information

This Supplementary Information file includes 5 sections, 5 figures and 7 references.

Supplementary Video 1

Controlled generation on HOPS I along the latitude at different θ1 by the manipulation of pumping and heating power on waveguides 1 and 2.

Supplementary Video 2

Controlled generation on HOPS II along the latitude at different θ2 by the manipulation of pumping and heating power on waveguides 3 and 4.

Supplementary Video 3

Controlled generation on HOPS III along the latitude at different θ3 by the manipulation of pumping power on two microrings and heating power on pad 5. State vector is near the equatorial plane on HOPS I but close to the south pole on HOPS II (the case in Fig. 3c).

Supplementary Video 4

Controlled generation on HOPS III along the latitude at different θ3 by the manipulation of pumping power on two microrings and heating power on pad 5. State vector is near the equatorial planes on both HOPS I and HOPS II (the case in Fig. 3d).

Source data

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Z., Zhao, H., Wu, S. et al. Spin–orbit microlaser emitting in a four-dimensional Hilbert space. Nature 612, 246–251 (2022). https://doi.org/10.1038/s41586-022-05339-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-022-05339-z

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing