Theory of near-field matter-wave interference beyond the eikonal approximation

A generalized description of Talbot-Lau interference with matter waves is presented, which accounts for arbitrary grating interactions and realistic beam characteristics. The dispersion interaction between the beam particles and the optical elements strongly influences the interference pattern in this near-field effect, and it is known to dominate the fringe visibility if increasingly massive and complex particles are used. We provide a general description of the grating interaction process by combining semiclassical scattering theory with a phase space formulation. It serves to systematically improve the eikonal approximation used so far, and to assess its regime of validity.

Figure 1

Schematic of a Talbot-Lau interferometer with the relevant setup parameters for a material diffraction grating. The beam emanating from the left-hand side is constrained by collimation slits to an angular spread $\alpha$ before it enters the interferometer. The first grating of period $d_1$ and size $G⪢d_1$ modulates the beam so that spatial coherence is created at the position of the second grating $(z=0)$ where the diffraction takes place. This leads to an interference pattern in the transverse beam density, ideally located at the position of the third grating. The latter can be shifted in the transverse direction by a variable distance $x_S$. It thus modulates the flux provided its longitudinal position $\eta L$ and its period $d_3$ match the interference pattern. A detector of size $W$ further downstream measures the total incoming beam intensity as a function of $x_S$. The central grating may be replaced by the phase grating created by a standing light field.

Figure 2

Talbot-Lau visibilities (14) for the equidistant setup with three material gratings [20] as a function of the de Broglie wavelength of the interfering ${\mathrm{C}}_{70}$ fullerenes. At the second grating we assume no dispersive interaction (dotted line), a van der Waals interaction (dashed line), and a retarded van der Waals interaction (solid line). The dashed-dotted line represents the classical calculation with a retarded interaction. It depends on the fictitious “wavelength” $\lambda =h∕\left(m{v}_z\right)$ via the velocity $v_z$ because the interaction time is velocity dependent.

Figure 3

Fringe visibilities (14) for a Kapitza-Dirac-Talbot-Lau interferometer [22] in the eikonal approximation. The solid line gives the quantum result for ${\mathrm{C}}_{70}$ fullerenes, while the dashed line represents the corresponding classical calculation (neglecting photon absorption). Here we assume a laser power of $P_L=6\mathrm{W}$, a vertical waist of $w_y=900\mu \mathrm{m}$, a grating distance of $L=105\mathrm{mm}$, a grating period of $d=266\mathrm{nm}$, and a dynamic polarizability ${\alpha }_{\omega }=118{\mathrm{\AA }}^3$ corresponding to a laser wavelength of $532\mathrm{nm}$.

Figure 4

Sketch of the stationary phase conditions for the semiclassical scattering factor $\Phi (\mathit{r},\mathit{p})$ defined in (52). While the momentum $\mathit{p}$ fixes the asymptotic initial momentum ${\mathit{p}}_0$ of the classical trajectory, the initial position ${\mathit{r}}_0$ is determined implicitly by the spatial point $\mathit{r}$. The latter defines the asymptotic final position ${\mathit{r}}_{2T}$ of the trajectory by means of a free evolution determined by the asymptotic final momentum ${\mathit{p}}_{2T}$. For interaction potentials that admit a scattering theory description this construction guarantees that the action (46) and the stability amplitude (53) converge as $T\to \infty$. The range of the scattering potential is indicated by the shaded area.

Figure 5

Phase difference of the semiclassical (solid line) and the eikonal approximation (dotted line) to the exact result, plotted versus the distance to one grating slit wall. We use grating and particle parameters as in Fig. 2 (corresponding to the fullerene experiment [20]).

Figure 6

Phase difference with respect to the exact scattering factor over one period $d=266\mathrm{nm}$ of the laser grating (parameters as in Fig. 3, with $\lambda =3\mathrm{pm}$, corresponding to the experiment [22]). Panel (a) shows the results for an incoming plane wave with vanishing transverse momentum, $p=0$, while (b) corresponds to $p={10}^{-3}{p}_z$. The solid line gives the error of the semiclassical approximation, which is not resolved on the scale given by the errors of the Glauber approximation (dashed line) and the elementary eikonal approximation (dotted line). The latter are indistinguishable in (a); notice the different scales of the $y$ axes.

Figure 7

(a) Phase differences with respect to the exact calculation as in Fig. 6, but for an increased de Broglie wavelength $\lambda =30\mathrm{pm}$ and $p={10}^{-3}{p}_z$. The elementary eikonal approximation (dotted line) and the Glauber eikonal approximation (dashed line) now differ markedly from the exact phase result, while the semiclassical approximation (solid line) deviates by less that $100\mathrm{mrad}$. (b) Corresponding amplitude modification as obtained from the semiclassical approximation (solid line) and the exact calculation (dashed line).

Figure 8

Interference visibility as a function of the longitudinal waist $w_z$ of the laser grating (for $\lambda =3\mathrm{pm}$, starting from the experimental value $w_z=20\mu \mathrm{m}$ [22]). The dotted line gives the result of the elementary eikonal approximation (33), while the solid curve represents the semiclassical result (66).

Figure 9

Fringe visibility as a function of the laser power $P_L$, starting from the experimental value $P_L=6\mathrm{W}$ [22]. For considerably larger powers, or, equivalently, for larger polarizabilities of the particles, the elementary eikonal approximation (dotted line) deviates markedly from the semiclassical result (66).

Figure 10

Talbot-Lau interference visibilities for the laser grating setup [22] with ${\mathrm{C}}_{70}$ fullerenes. We use increased de Broglie wavelengths corresponding to velocities around $v_z=16\mathrm{m}∕\mathrm{s}$ [51]. The semiclassical approximation (solid line) yields values which are clearly distinguishable from the Glauber eikonal approximation (dotted line). The elementary eikonal approximation differs more strongly.