2 The strategy

The first stage of the QUPLAS experiment featured a positron beam coupled with a Talbot–Lau interferometer and a high-resolution emulsion detector (Anzi et al. Reference Anzi, Ariga, Ereditato, Ferragut, Giammarchi, Maero, Pistillo, Romé, Scampoli and Toso2020; Ariga et al. Reference Ariga, Cialdi, Costantini, Ereditato, Ferragut, Giammarchi, Leone, Maero, Miramonti and Pistillo2020). In the experimental configuration, the two gratings have different periods ($d_1=1.2\,\mathrm {\mu }{\rm m}$ and $d_2=1\,\mathrm {\mu }{\rm m}$, respectively) so that the configuration acts as a magnifier of the diffraction pattern. The distance $L_1$ between the two gratings (see figure 1) is set to $L_1=d_1 d_2 / \lambda$ (Sala et al. Reference Sala, Ariga, Ereditato, Ferragut, Giammarchi, Leone, Pistillo and Scampoli2019), where $\lambda = h/mv$ is the De Broglie wavelength, with $m$ and $v$ the positron mass and velocity, respectively, and $h$ Planck's constant. In order to obtain the maximum contrast on the detector, this is located at a distance $L_2$ from the second grating given by the relation $L_2= d_2 L_1/(d_1-d_2)$.

Figure 1. The QUPLAS configuration for this study and principle of operation. A collimated positron beam is followed by a system of gratings. The interference pattern is recorded by a high-resolution emulsion detector. The set-up is equipped with a microwave cavity to study the proposed particle–wave interaction.

The experiment is powered by a 50 mCi $^{22}$Na positron source. After a few preparation steps, an almost monochromatic (energy spread of a few eV) positron beam with energies tuneable in the range 10–20 keV is obtained. At these velocities, the particles pass through the interferometer in a fraction of $\mathrm {\mu }$s. Taking into account all the losses due to solid angle, moderation and other processes, the flux turns out to be $\simeq 10^4\,{\rm particles}\,{\rm s}^{-1}$. Taking into account the paucity of this beam and the fact that the radioactive material is an incoherent fermionic source, the experiment is definitely in the single-particle regime.

The present configuration, featuring two silicon nitride material gratings, will be supplemented by a microwave cavity, located after the second grating, as shown in figure 1, providing a standing ($\sim$10 GHz) electromagnetic field. Therefore, the positrons crossing the second grating will interact with the photons in the microwave cavity. This interaction can be described by a Hamiltonian of the form (Friedman et al. Reference Friedman, Gover, Kurizki, Ruschin and Yariv1988; Henke et al. Reference Henke, Raja, Feist, Huang, Arend, Yang, Kappert, Wang, Möller and Pan2021)

(2.1)\begin{equation} \hat{H} \simeq \hbar g (\hat{a}^{\dagger} \hat{b} + \hat{a} \hat{b}^{\dagger}) , \end{equation}

where $a$ and $a^{\dagger}$ are the boson field annihilation and creation operators, $[a,a^{\dagger} ] = 1$, $g$ is a coupling constant and $b$ the positron-energy ladder operator, reducing its energy by the energy of one microwave photon and changing its linear momentum. This interaction produces an entanglement between positrons and photons, having the form

(2.2)\begin{equation} | \psi_e , \psi_\gamma \rangle \simeq \sum_n c_n | E - n\hbar \omega \rangle | n \rangle , \end{equation}

where $n$ is the number of exchanged photons (having frequency $\omega$) and $c_n$ are amplitudes. Different effects are then obtained as a function of the power in the microwave cavity, which is related to the number of photons in the expansion. In the case of a small coupling constant, $| \psi _e, \psi _\gamma \rangle$ is dominated by the effect of one or two photons

(2.3)\begin{equation} | \psi_e, \psi_\gamma \rangle \simeq | E \rangle | 0 \rangle + g | E- \hbar \omega \rangle | 1 \rangle + g^2 | E- 2 \hbar \omega \rangle | 2 \rangle . \end{equation}

In the presence of a macroscopic, monochromatic microwave field, namely, a coherent state of the radiation field, we can substitute for $\hat {a}$ in (2.1) the complex amplitude $\alpha$ of the microwave field, that reduces to a momentum-displacement operation, in analogy with the same kind of operation in quantum optics (Olivares Reference Olivares2021).

In principle, one should solve the Schrödinger equation with the above Hamiltonian for the proposed experimental configuration. In order to simplify the problem we have developed two complementary approaches:

1. (i) the two-slit model;

2. (ii) the Gaussian–Schell model (GSM).

In the two-slit model, the grating structure is geometrically simplified down to the simplest Young-like configuration, taken to grasp the essence of the phenomenon. The idea is to model a single positron passing through the interferometer and interacting with the microwave cavity. On the other hand, the GSM is an analytical model used for the statistical treatment of partially coherent beams. Both models will be shown to give consistent results.

3 The two-slit model

The two-slit model assumes that the essence of the phenomenon is equivalent to considering two neighbouring slits belonging to the second grating, as shown in figure 2. We assume that a non-relativistic positron is propagating along the $z$ direction, while the grating slits are parallel to the $x$ direction (orthogonal to the $(y,z)$ plane shown in figure 2); under these conditions, interference will show up along the $y$ axis. The microwave cavity is modelled as a rectangular wave guide with length $w$ and transverse dimensions $a$ and $b$. Considering only the fundamental mode TE$_{10}$ of the wave guide, the stationary field is characterized by a longitudinal wavenumber

(3.1)\begin{equation} k_L= \sqrt{\left(\frac{2 {\rm \pi}\nu_0}{c}\right)^2-\left(\frac{2 {\rm \pi}\nu_c}{c}\right)^2}, \end{equation}

where $\nu _0$ is the frequency of the microwave field, and $\nu _c = c/2a$ the cutoff frequency of the TE$_{10}$ mode.

Figure 2. The two-slit model can be visualized as two independent sources located at the end of the second grating of QUPLAS, where a microwave cavity is positioned. The $x$ axis is pointing into the figure, away from the reader.

The vector potential of the confined microwave field in the cavity is given by

(3.2)\begin{equation} A_z = \frac{E_0}{\omega} \cos \left( \frac{{\rm \pi} x}{a} \right) \sin (k_Ly) \cos (\omega t + \varphi ) , \end{equation}

where $E_0$ is the amplitude of the electric field, $\omega =2{\rm \pi} \nu _0$ and $\varphi$ is a phase. The origin of the coordinates is placed at the centre of the opening for the passage of positrons into the microwave cavity. Assuming the diameter of this opening is much smaller than the linear dimensions of the microwave cavity and of the wavelength of the TE$_{10}$ mode inside the cavity, the potential (3.2) can be simplified as

(3.3)\begin{equation} A_z \simeq \frac{E_0}{\omega} k_L y \cos (\omega t+ \varphi) . \end{equation}

The amplitude of the wave electric field in the wave guide can be written in terms of the time-averaged (over a wave period) peak power $P$ by means of the relation

(3.4)\begin{equation} E_0 = \sqrt{\frac{4P\eta}{wa}}, \end{equation}

where $\eta = 1/\epsilon _0 c \sqrt {1-(\nu _c/\nu _0)^2}$ is the wave impedance.

Neglecting the term proportional to the square of the vector potential (low amplitude wave electric field), the Hamiltonian of the interaction between the particle and the field reads

(3.5)\begin{equation} H \simeq{-}\frac{e}{m} A_z p_z , \end{equation}

where $e$ is the positron charge, and $p_z \simeq m v$ (neglecting the transverse velocity components of the particle).

The effect due to two neighbouring slits is summarized by an overall shift

(3.6)\begin{align} \Delta \phi (P, \varphi; w, a, \omega, d_2) & ={-}\frac{1}{\hbar} \int_0^{\tau} H_2 \,{\rm d}t+\frac{1}{\hbar} \int_0^{\tau} H_1 \,{\rm d}t \nonumber\\ & \simeq{-} \frac{1}{\hbar} \sqrt{P\eta\frac{w}{a}} \frac{e}{{\rm \pi} v} k_L d_2 \frac{1}{\tau} \int_0^\tau \cos (\omega t + \varphi ) \nonumber\\ & ={-} \frac{1}{\hbar} \sqrt{P\eta\frac{w}{a}} \frac{e}{{\rm \pi} v} k_L d_2 \frac{\sin (\omega \tau + \varphi ) -\sin (\varphi)}{\omega \tau} , \end{align}

where $\tau \simeq w/v$ is the crossing time of the positron in the wave field, and $H_1$ and $H_2$ denote the interaction Hamiltonian relevant to the passage of the particle through slits 1 and 2, respectively, separated by a distance $d_2$ (i.e. the period of the second grating).

For a uniform random distribution of the phase $\varphi$, the degree of decoherence introduced by the stationary field can be quantified by means of

(3.7)\begin{equation} D (P; w, a, \omega, d_2) = \frac{1}{2{\rm \pi}} \int_{-{\rm \pi}}^{\rm \pi} \exp [ {\rm i} \Delta \phi (P, \varphi; w, a, \omega, d_2) ] \,{\rm d} \varphi . \end{equation}

The function $D$ is real and coincides with the visibility of the interference pattern. An example of the dependence of $D$ on microwave power is given in figure 3.

Figure 3. Visibility of the fringes as a function of the power of the microwave field in both the two-slit model and in the GSM model for positrons with 14 keV kinetic energy, $\nu _0 = 10$ GHz, $w = 4$ mm, $a=21.2$ mm and $d_2 = 1\,\mathrm {\mu }{\rm m}$. The additional data used in the GSM model are $d_1=1.2\,\mathrm {\mu }{\rm m}$, $l_0= 0.65$ nm, $w_0=2.8$ mm; the two gratings are assumed to have a transparency $f_1 = f_2= 0.5$. The visibility is computed at the $z$-coordinate of the Talbot–Lau maximum.

4 The Gaussian–Schell model

The same physical problem has been addressed by means of a simulation that makes use of an analytical description of the beam, namely the GSM (McMorran & Cronin Reference McMorran and Cronin2008).

In the GSM, the partial coherence and wavefront curvature of a beam are taken into account by using a series of parameters describing the non-ideality of both the beam and of the related grating system. This approach has a great versatility, and can describe a variety of interferometric situations in different regimes (Talbot–Lau, Fraunhofer, Mach–Zehnder$\ldots$).

The model describes a partially coherent beam as a statistical distribution of Gaussian modes, characterized by its initial width, coherence length and wavefront curvature. In the simulation, we have assumed that only the transverse part contributes to the coherence length; this is due to the good monochromaticity of the QUPLAS positron beam (good longitudinal coherence) (Ariga et al. Reference Ariga, Cialdi, Costantini, Ereditato, Ferragut, Giammarchi, Leone, Maero, Miramonti and Pistillo2020). The ingredients used in the GSM calculation are:

1. (i) The mutual intensity function.

2. (ii) Its propagation in free space.

3. (iii) Its modification due to the effect of the gratings and microwave cavity.

The mutual intensity function is used in partially coherent optics, where it is considered that a beam has some momentum distribution in the direction transverse to propagation. It is defined by

(4.1)\begin{equation} J(\boldsymbol{x}_a,\boldsymbol{x}_b,z) = \left\langle \varPhi^*(\boldsymbol{x}_a,z,t) \varPhi(\boldsymbol{x}_b,z,t)\right\rangle , \end{equation}

where $\varPhi$ is a field, $\boldsymbol {x}_a, \boldsymbol {x}_b$ are coordinates orthogonal to the $z$ propagation direction of the beam and the angular brackets denote a time average over the statistical fluctuations of the field. This quantity describes both the spatial coherence of the beam and its intensity, as given by the relation

(4.2)\begin{equation} I(\boldsymbol{x}, z) = J(\boldsymbol{x},\boldsymbol{x},z) . \end{equation}

Once this function is given at $z=0$, the beam propagation in space can be described by means of the Van Cittert–Zernike theorem, that links the mutual intensity function $J$ to the Fourier transform of the intensity of a partially incoherent source, enabling us to devise a free propagation law for $J$

(4.3)\begin{gather} J(\boldsymbol{x},\Delta \boldsymbol{x}, z) = \frac{1}{\lambda^2 z^2} \int \,{\rm d} \Delta {\boldsymbol{x}} ' \int \,{\rm d} {\boldsymbol{x}} ' \exp({-(2 {\rm \pi}{\rm i}/\lambda z) ({\boldsymbol{x}}' \boldsymbol{\cdot} \Delta {\boldsymbol{x}}' - {\boldsymbol{x}}' \boldsymbol{\cdot} \Delta {\boldsymbol{x}} -{\boldsymbol{x}}\boldsymbol{\cdot} \Delta {\boldsymbol{x}}' + {\boldsymbol{x}}\boldsymbol{\cdot} \Delta {\boldsymbol{x}} )})\nonumber\\ J({\boldsymbol{x}}',\Delta{\boldsymbol{x}}', 0) , \end{gather}

where $\boldsymbol {x} = (\boldsymbol {x}_a + \boldsymbol {x}_b)/2$ and $\Delta \boldsymbol {x} = \boldsymbol {x}_b - \boldsymbol {x}_a$. This law gives the mutual intensity function $J$ at some $z$, given its form at $z=0$ for the case of free propagation.

The structure of the initial mutual intensity function is specified as

(4.4)\begin{equation} J(\boldsymbol{x},\Delta \boldsymbol{x}, 0) = \exp(-{\rm \pi}(\boldsymbol{x}^2 / w_0^2 + \Delta \boldsymbol{x}^2 /l_0^2 + 2i \boldsymbol{x} \boldsymbol{\cdot} \Delta \boldsymbol{x}/\lambda r_0)) , \end{equation}

where $w_0$ is the beam opening (width), $l_0$ the transverse coherence length and $r_0$ the radius of curvature of the wavefront. In the free propagation of the beam from $z_0=0$ to a generic position $z$, the mutual intensity function $J$ retains the Gaussian structure, with the values of the parameters evolved as

(4.5)\begin{gather} w(z) = w(z_0) \sqrt{\left (1+\frac{z}{r_0} \right )^2 + \left ( \frac{\lambda z}{w_0 l_0} \right) ^2} , \end{gather}

(4.6)\begin{gather}l(z) = l(z_0) \sqrt{\left (1+\frac{z}{r_0} \right )^2 + \left ( \frac{\lambda z}{w_0 l_0} \right ) ^2}, \end{gather}

(4.7)\begin{gather}r(z) = z \left [ \frac{(1+z/r_0)^2+( \lambda z/w_0 l_0)^2}{(z/r_0)(1+z/r_0)+( \lambda z/w_0l_0)^2} \right ] . \end{gather}

Concerning the initial beam parameters, they were experimentally determined by a series of runs in the standard QUPLAS configuration (without the microwave cavity) as $l_0 = 0.65$ nm and $w_0 = 2.4$ mm, neglecting the effect of the curvature radius (Sala Reference Sala2018).

The action of a grating is represented by means of its transmission function $\mu$, so that the mutual intensity function at the $z$-coordinate of the grating is changed as

(4.8)\begin{equation} J'(\boldsymbol{x},\Delta \boldsymbol{x}, z) = \mu^*(\boldsymbol{x}-\Delta \boldsymbol{x} /2) \mu(\boldsymbol{x}+\Delta \boldsymbol{x} /2) J(\boldsymbol{x}, \Delta \boldsymbol{x}, z) . \end{equation}

In order to describe material gratings realized as open slits in a substrate, $\mu$ is simply assumed in each period $d$ as 1 in an interval of width $fd$ (open slit), and 0 in a following interval of width $(1-f)d$ (substrate), where $f$ is the transparency of the grating.

After an initial phase of free propagation, the operation (4.8) is performed a first time in the position of the first grating (period and transparency $d_1$ and $f_1$, respectively), followed by a new phase of free propagation for the modified $J$ up to the second grating. Then the mutual intensity function is changed again on the basis of (4.8) in the position of the second grating (period and transparency $d_2$ and $f_2$, respectively).

The insertion of the microwave cavity changes the propagation after the second grating. Its effect is modelled again using (4.8) with a transmission function $\mu = \exp (\tilde {\varPhi })$. Using the same approximations adopted in the two-slit model

(4.9)\begin{equation} \tilde{\varPhi} \simeq{-} \frac{{\rm i}}{\hbar} \int_0^\tau H \,{\rm d}t \simeq 2{\rm i} \frac{e v}{\hbar \omega^2} \sqrt{\frac{\eta P}{w a}} k_L y \left[\sin (\omega \tau + \varphi ) - \sin (\varphi) \right]. \end{equation}

The intensity at the $z$-coordinate of the detector is finally computed after a final phase of free propagation, and averaging over the phase $\varphi$. The resulting visibility of the interferometric pattern is shown for fixed values of the relevant parameters and compared with the result of the two-slit model in figure 3.