Quantum and classical model
The theory of Talbot–Lau interference is best formulated in phase space using the Wigner–Weyl representation of quantum mechanics42. This framework can account for incoherent particle sources, phase and absorption gratings, and all laser-induced photophysical effects, as well as any relevant decoherence process. It also allows for a direct comparison between the predictions of quantum and classical mechanics within the same formalism and set of assumptions.
For a cluster with mass m and longitudinal velocity v z , the probability of being detected behind the interferometer can be written as a Fourier series in the transverse position x 3 of G 3 :
$$S({x}_{3})=\mathop{\sum }\limits_{{\ell }=-\infty }^{\infty }{S}_{{\ell }}\exp \left({\rm{i}}\frac{2{\rm{\pi }}{\ell }}{d}{x}_{3}\right).$$ (1)
In a symmetric setup with equal grating separations L and periods d, the Fourier coefficients are
$${S}_{{\ell }}={B}_{-{\ell }}^{(1)}(0){B}_{2{\ell }}^{(2)}\left({\ell }\frac{L}{{L}_{{\rm{T}}}}\right){B}_{{\ell }}^{(3)}(0),$$ (2)
where the Talbot–Lau coefficients \({B}_{{\ell }}^{(j)}\) of order ℓ for the jth grating still need to be determined as a function of the Talbot length L T = mv z d2/h.
We assume that every absorbed grating photon results in the ionization of the sodium cluster. The transmission of the particle beam through a standing wave of incident laser power P, wavelength λ L and Gaussian beam waist w y is then characterized by the mean number of ionizing photons absorbed in each grating antinode
$${n}_{0}=\frac{8{\sigma }_{{\rm{ion,266}}}P{\lambda }_{{\rm{L}}}}{\sqrt{2{\rm{\pi }}}hc{w}_{y}{v}_{z}},$$ (3)
as well as by the phase shift induced by the optical dipole potential
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