Talbot-Lau effect beyond the point-particle approximation

Recent progress in matter-wave interferometry aims to directly probe the quantum properties of matter on ever increasing scales. However, in order to perform interferometric experiments with massive mesoscopic objects, taking into account the constraints on the experimental setups, the pointlike-particle approximation needs to be cast aside. In this work, we consider near-field interferometry based on the Talbot effects with a single optical grating for large spherical particles beyond the point-particle approximation. We account for the suppression of the coherent grating effect and, at the same time, the enhancement of the decoherence effects due to scattering and absorption of grating photons.

Figure 1

Optical standing-wave grating. The figure shows the coordinate convention that we use, in which $z$ is the relevant direction in the longitudinal eikonal approximation. We consider a standing wave linearly polarized in the $x$ direction which gives rise to a grating of period $d=\lambda /2$.

Figure 2

$F_0$ [in units of $I_0/\left(c{k}^2\right)]$ as a function of the size parameter $kR$ for a silicon (Si) sphere at $\lambda =354$ nm, where the bulk refractive index [26] is $n=5.656+i2.952$. The solid green line is the result of Mie theory. The dashed red line is the prediction resulting from Rayleigh approximation.

Figure 3

$F_0$ [in units of $I_0/\left(c{k}^2\right)]$ as a function of the size parameter $kR$ for a fused silica (${\mathrm{SiO}}_2$) sphere at $\lambda =100$ nm, where the bulk refractive index has been roughly estimated as $n=1.3+i0.8$ from tables in the Supplementary Material of [5] (see also [27]). The dashed black line is $F_0$ at $n=1.3+i0.8$. The green region represents the result of a $\pm 5\%$ error in the real part of the refractive index. Inset: The red region represents the result of a $\pm 5\%$ error in the imaginary part of the refractive index.

Figure 4

Effect of the optical grating, photon scattering, and absorption on the interference pattern in the classical limit. Here we use the parameters in Table 1 with $m={10}^6$ a.m.u. and the corresponding $x=kR\sim 0.098$, where the use of the Rayleigh limit is well justified. We have assumed the same experimental arrangements and configuration as in Ref. [5], with the exception of (gravitational) acceleration and environmental decoherence effects during the free evolution times $t_1$ and $t_2$, which have been neglected to highlight the effects of the grating, scattering, and absorption mechanisms. (a) The classical fringe pattern, when varying the maximum phase modulation ${\varphi }_0$, without decoherence, i.e., what we obtain by taking the classical limit as discussed in the text. (b) The decoherence terms due to scattering and absorption are evaluated using the same integral kernels as for the quantum case. This is what has been advocated previously in the literature (cf. Ref. [17]). (c) An instance of the interference patterns for ${\varphi }_0=4$. The solid black curve corresponds to (a); the dashed blue curve, to (b); and the dotted red curve, to the quantum interference pattern including decoherence due to scattering and absorption of the grating photons. Due to different notations, the Rayleigh limit of the Lindblad superoperator in Eq. (27) differs from Eq. (2.24) in Ref. [18] by a factor of $1/4\pi$.

Figure 5

Effect of the coherent grating and photon scattering on the interference pattern in the classical limit. Here we used the parameter in Table 1 with a mass $m={10}^8$ a.m.u., which corresponds to a value of $x=kR\sim 0.46$, for which the Rayleigh approximation is not well justified. The figure shows an instance of the interference patters for ${\varphi }_0=2$, where only the decoherence due to scattering of grating photons, of which we have full control, is considered. The solid black curve corresponds to the classical limit as obtained in this work. The dashed blue curve corresponds to the classical limit obtained without modifying the incoherent kernels from the quantum case. The dotted red curve represents the quantum case. Note the striking difference between the two classical interference patters, which is larger with respect to the Rayleigh case given the greater effect of scattering decoherence in the Mie regime.

Figure 6

Sinusoidal visibility of the interference pattern for a particle of mass $m={10}^8$ a.m.u., which corresponds to a value $x=kR\sim 0.46$ for the parameters reported in Table 1. For these parameters, the use of the Rayleigh approximation is not fully justified. The dashed blue curve represents the interference pattern visibility in the case in which Rayleigh approximation is used. The solid red curve represents the visibility when Mie theory is employed. Inset: An instance of the interference patters for ${\varphi }_0=2$ showing a significant difference between the two situations.