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    Area of Science:

    • Quantum optics
    • Laser physics
    • Information theory

    Background:

    • Spatial entanglement is crucial for quantum information processing.
    • Laser transverse modes can exhibit complex spatial structures.
    • Quantum Lissajous states offer a unique platform for studying entanglement.

    Purpose of the Study:

    • To investigate the spatial entanglement of laser transverse modes.
    • To analyze the relationship between localization and entanglement in these modes.
    • To utilize the Schmidt decomposition for characterizing quantum Lissajous states.

    Main Methods:

    • Applying Schmidt decomposition to laser transverse modes.
    • Analytically deriving stationary Lissajous states via inverse Fourier transform.
    • Using Schmidt modes and participation number (N) for evaluation.

    Main Results:

    • Stationary Lissajous states are coherent superpositions of Hermite-Gaussian eigenmodes.
    • The participation number (N) quantifies spatial localization on the Lissajous figure.
    • A higher participation number indicates increased spatial entanglement.

    Conclusions:

    • Schmidt decomposition effectively characterizes spatial entanglement in laser modes.
    • The participation number serves as a key metric for entanglement and localization.
    • This work provides insights into the quantum nature of optical fields.