Characterizing the spatial entanglement from laser modes analogous to quantum wave functions

Abstract

The Schmidt decomposition is exploited to study the spatial entanglement of laser transverse modes analogous to quantum Lissajous states. Based on the inverse Fourier transform, the stationary Lissajous state can be analytically derived as a coherent superposition of degenerate Hermite–Gaussian eigenmodes. With the derived stationary state, the Schmidt modes and the participation number N can be employed to evaluate the spatial localization and the quantum entanglement. The larger the participation number, the more localized is the stationary coherent state on the Lissajous figure. Moreover, the larger the participation number, the higher is the spatial entanglement.

Laser Transverse Modes with Ray-Wave Duality: A Review

“…It has been demonstrated that Lissajous geometric modes can be well generated from an off-axis diode-end-pumped Nd:YVO 4 laser. The detailed description for the experiment can be obtained from previous papers [14,40,41]. When a short laser crystal with length of 1~2 mm was used in experiment, the value of M was found to be greater than 300 by far.…”

“…for the new degeneracy of ( , ) p q can be obtained from Equation (33) as with z R,x and z R,y , the beam waists are w o,x = λz R,x /π and w o,y = λz R,y /π, where λ is the laser wavelength. In terms of ∆ f T,x , ∆ f T,y , z R,x and z R,y , the parametric equations for an astigmatic cavity can be derived as [18,41] x s (z) = √…”

“…One of the most striking geometric modes is the transverse intensities exhibiting the Lissajous figures [31][32][33][34][39][40][41]. In 1964, Herriot et al [42] proved that under foldback conditions, an incident laser beam can be reflected forth and back between two spherical mirrors to exhibit an elliptical pattern.…”

“…We previously used the SU(2) quantum theory to derive the Lissajous geometric mode to be a superposition of different order HG modes combined with different longitudinalorder modes subject to the degeneracy [39,41]. Recently, we have exploited the generalized wave-packet state representation to construct the ray-wave connection for the multiple-pass Lissajous dotted patterns [52].…”

Period-multiplying pulses generation by total self-mode-locking of Nd:YVO4 lasers

“…It is not surprising that SML lasers exhibit more complex transverse dynamics since SML is inherently a nonlinear phenomenon, and 'soft aperture' is the only mechanism to limit higher-order transverse modes. For example, the Lissajous geometric mode derived by Chen et al [11,12] using SU(2) quantum theory, is the superposition of different orders of Hermite-Gaussian(HG) modes and different longitudinal modes.…”