# The analysis of the measured data

After the detection of the particles, we want to evaluate the data of the interference measurement. A measure of the quality of an interferogram is the interference contrast (visibility). It specifies how much the interference pattern deviates from the mean. One can express this relationship for sinusoidal signals, like in our experiment, with:

$$\mathrm{Visibility}=\frac{S_{\mathrm{max}} – S_{\mathrm{min}}}{S_{\mathrm{max}} + S_{\mathrm{min}}}=\frac{\mathrm{Amplitude}}{\mathrm{Average}}$$

 Amplitude $$A$$ - + Offset $$O$$ - + Phase $$\varphi$$ - +

# Interpretation

However, a high visibility alone does not tell us if we have really measured quantum interference. Due to the arrangement of the gratings and a lens effect at the light grating also Moiré effects can occur and, thereby, a purely classical intensity modulation can arise. To make sure that we really observe quantum effects, we need to compare the measured data with the contrasts predicted by quantum physical or classical theories.

The measurement data can be explained very well by the quantum physical model; the classical description, however, does not provide an adequate prediction of the measured data.

Extra: quantum or classical

Quantum mechanically, in a symmetrical ($$d_1 = \lambda /2 = d_3$$ and $$L_1 = L_2$$) interferometer we expect a visibility of:

$$V^{QM}=2\left| \mathrm{sinc}\left(\pi f_{1}\right) \mathrm{sinc}\left(\pi f_{3}\right)\int \mathrm{d} v_{z}\,\mu\left(v_{z}\right)J_{2}\left[\phi_{0}\sin\left(\pi\frac{L}{L_{T}}\right)\right]\right|$$

Here are:

• $$f_1$$ and $$f_3$$ the opening fractions of the mechanical gratings (G1 and G3).
• $$\mu(v_z)$$ the velocity distribution of the molecules
• $$J_2$$ a second order Bessel function
• $$\phi_0 \propto \alpha P_L / w_yv_z$$ is  the phase shift of the molecular matter wave at the maximum of the standing wave. It is dependent of the laser intensity $$P_L$$ and the beam waist $$w_z$$, the velocity $$v_z$$ and the polarizability $$\alpha$$ of the molecules.
• $$L$$ is the distance between two gratings, half of the interferometer length.
• $$L_T$$ is the Talbot length$$\frac{d^2}{\lambda}$$, with  $$d$$ the grating period and $$\lambda$$ the wave length.

The classically expected Moiré visibility differs in the argument of the Bessel function. There $$\sin\left(\pi \cdot L / L_{T}\right)$$ is exchanged with $$\left(\pi \cdot L/ L_{T}\right)$$.

• $$V^{QM} \propto \left| \int \mathrm{d} v_{z}\,\mu\left(v_{z}\right)J_{2}\left[\phi_{0}\sin\left(\pi\frac{L}{L_{T}}\right)\right]\right|$$
• $$V^{KM} \propto \left| \int \mathrm{d} v_{z}\,\mu\left(v_{z}\right)J_{2}\left[\phi_{0}\left(\pi\frac{L}{L_{T}}\right)\right]\right|$$

We see that for short grating distances the classical and the quantum mechanical predictions are very similar. But for for $$L \ge L_T$$ we can clearly distinguish the two models in our experiment.

At the experiment the phase $$\phi_0$$ is modified with the laser intensity $$P_L$$ to show the functional difference between the two predictions (see figure).