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


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.

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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.

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.


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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

Authors and Affiliations



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.

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The authors declare no competing interests.

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

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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).

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Zhang, Z., Zhao, H., Wu, S. et al. Spin–orbit microlaser emitting in a four-dimensional Hilbert space. Nature 612, 246–251 (2022).

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