Requirements for probing chiral Casimir-Polder forces in a molecular Talbot-Lau interferometer

Chirality-dependent dispersion forces, such as the van der Waals (vdW) forces, the Casimir-Polder (CP) forces and the Casimir effects have been theoretically predicted and analyzed in many prior studies, more recently also with a growing attention towards their experimental detection Casimir1948 ; chiral ; chiral2 ; chiral3 ; chiral4 ; suzuki ; chiral5 . A better understanding of these interactions promises deeper insight into quantum field theory chiral2 . Enantiospecific interactions are also important in the understanding of Hund’s paradox hund ; hund2 ; hund3 and of homochirality, i.e., the question why many biomolecules in nature appear with only one chirality homo .

Recent studies of chirality span a diverse range of topics, including chirality-induced spin selectivity CISS ; CISS2 , proximity-induced chiral quantum light chirallight , induced chirality in clusters ref3 , circular dichroism in carbon nanotubes ref , chiral AC Stark effects ac and chiral resonance energy transfer energy . Although enantiospecific optical forces have been observed experimentally optical ; optical2 ; optical3 ; optical4 ; optical5 , the detection of chirality-dependent dispersion forces has remained a great challenge.

To enhance the sensitivity to chiral CP forces, suzuki proposed to use slow molecules in electronically excited states and separate enantiomers in a planar chiral cavity. Excited states are appealing as they can exhibit strongly enhanced interaction strengths, but the short lifetime of dipole allowed transitions also limits the interaction time and the maximal momentum transfer and separation in molecular beam experiments. While the separation and analysis of enantiomers is commonplace in chemistry, our following proposal relies on non-contact distant interactions in high vacuum, which are not commonly accessible. In our present work, we analyse in particular how the interaction of chiral molecules with chiral nanogratings influences both the molecular transmission and the molecular fringe visibility in a near-field matter-wave interferometer (see Fig. 1).

The CP potential can be understood as the vdW potential augmented by the influence of retardation Casimir1948 , and it has already been discussed in the context of matter-wave diffraction of neutral atoms Grisenti1999 ; Bruehl2002 ; Lonij2010 ; Garcion2021 ; Bouton2023 , non-polar Arndt1999 ; Brezger2002 ; Brand2015 and polar molecules Knobloch2017 . The phase accumulated by a molecule in transit through a grating depends on the molecular polarizability, anisotropy, dipole moment, conformation, the molecular velocity as well as the grating slit width, thickness and shape. The dielectric function of the grating is relevant across all frequencies and one may also need to account for charge impurities in the grating Knobloch2017 . The interaction with nanostructures is favorable as chiral CP forces decay faster with distance than their electric and magnetic counterparts in the retarded regime. We propose specifically Talbot-Lau interferometry (TLI), as shown in Fig. 1, as a suitable tool to explore proximity effects since TLI can create coherent self-images of several thousand molecular beams in parallel, all with a low velocity normal to the grating wall.

In the following, we study the conditions under which chiral molecules of sufficiently high rotatory strength may be discriminated by the interferometer transmission and fringe visibility. The paper is organized as follows: In Sec. II, we review the theoretical framework to study the influence of CP forces in Talbot-Lau interferometry with three gratings. In Sec. III, we investigate the impact of chiral CP forces when the second grating $G_{2}$ is perfectly chiral and show requirements for observing differences in interferometer transmission and fringe visibility between the enantiomers. In Sec. IV, we study a more realistic scenario where the second grating is inherently non-chiral but coated with chiral substances of high rotatory strength. In Sec. V, we then assume all gratings to be equal and find that chiral coatings of gratings $G_{1}$ and $G_{3}$ have a strong influence on the enantiomer discriminating power of the instrument, as the chiral CP forces modify the grating transmission function substantially. This can result in observable differences between enantiomers in the total signal strength. Finally, we discuss experimental considerations in Sec. VI and conclude in Sec. VII.

We begin by presenting an overview of the framework employed for studying the effects of CP forces on chiral molecules in a Talbot-Lau interferometer (Fig. 1) composed of three nanogratings, G1-G3, each with a thickness of $b$ and a periodicity of $d$. We choose a grating distance of $L$. The grating slit width $s$ may be augmented by a chiral coating of thickness $a$. It can be also reduced by the forces to a cutoff $x_{c}$, discussed below. Note that such an interferometer can also be operated with only two gratings, if G3 is replaced by a molecular binding surface, which can be imaged with 10 nm resolution and high efficiency Juffmann2009 . Even G1 could be replaced by a point source.

We start by summarizing the expressions for the expected interferometer signal and fringe visibility and will then add the effect of chiral potentials to the result. This section uses Wigner functions in phase space as described in klaus :

where $\rho(\mathbf{r},\mathbf{r^{\prime}})$ is the position density matrix and $\mathbf{\Delta}=\mathbf{r}-\mathbf{r}^{\prime}$.

In a Talbot-Lau setup, the molecular beam traverses three gratings $G_{1}-G_{3}$, spaced at equal distances $L$. The Wigner function after the first grating can be expressed as

assuming that the initial Wigner function is normalized and homogeneous, $w_{0}(\mathbf{r},\mathbf{p})=1$, i.e., the molecules enter the grating without any prior spatial coherence and evenly from all directions. We further assume a symmetric interferometer where all gratings have the same period $d$ and distance $L$. The geometrical transmission function can be written as the Fourier series $t_{1}(x)=\sum_{l}a_{l}\exp(2\pi ilx/d)$, where the coefficients $a_{l}$ depend on the slit width, slit thickness, coating and also on the interaction between the molecules and the grating wall. We assume a monochromatic molecular beam and will analyze the dependence of the interference fringes on the velocity, i.e., on the de Broglie wavelength.

A free evolution of the molecule over a distance $L$ between the gratings changes the Wigner function

where $p_{z}$ is the momentum of the molecule in $z$-direction. We neglect all external accelerations due to gravity, assuming that all gratings are aligned to better than 100 $\mu$rad with the line of gravity. We also neglect the Coriolis acceleration $a_{c}=2v\times\Omega_{E}$ which is about thousand times smaller than the gravitational acceleration for molecules travelling at about 200 m/s, when $\Omega_{E}=72$ µrad/s is the frequency of the rotation of the Earth and can be compensated by gravity under suitable interferometer alignment Fein2020 .

The passage through each second grating modulates the wavefunction according to $\psi_{1}(\mathbf{r})\rightarrow t_{2}(\mathbf{r})\psi_{1}(\mathbf{r})$ where $t_{2}(\mathbf{r})$ encodes the complex transmission function. The Wigner function after passage through a second grating is thus

where

Talbot-Lau interferometers can generate spatial coherence even from an initially incoherent molecular beam by diffracting the molecules at the slits of the first grating, if the first grating is an absorptive, spatially selective mask – such as a nanomechanical structure or a photo-depletion grating Nimmrichter2011 . Diffraction within each slit of grating $G_{1}$ then prepares the transverse coherence required to achieve spatial superposition and diffraction at $G_{2}$. While we can treat $G_{1}$ as a pure transmission filter, its CP interaction with the molecule must still be treated carefully. Molecular trajectories are deflected to undetected regions if they pass very close by the grating walls. The effective slit width is thus narrowed by a cut-off distance $2x_{c}^{(1)}$ which influences both the interference contrast and the transmitted signal. Furthermore, this effect is dispersive: slower molecules are even further deflected and do no longer contribute to the signal.

The second grating $G_{2}$ then modifies both the amplitude and the phase of the already delocalized matter-wave. In eikonal approximation the transmission function of the second grating reads:

where $m$ is the mass of the molecule. Note that this additional phase depending on $V(x)$ is canceled out for G1 and G3, leaving only the influence of a cutoff induced by the CP force, as discussed below. The quantum coherence term only exists for waves traversing through G2. Nonetheless, the contribution of G1 and G3 to the final signal and visibility is relevant as they modulate the effective transmission.

Assuming that the second grating has the Fourier coefficients $b^{\prime}_{m}$, we find

where

After free evolution over the distance $L$, passage through grating $G_{2}$ and another free evolution over the distance $L$, the Wigner function reads

We obtain the molecule density by integrating the Wigner function over all momenta and obtain

Here, the Talbot coefficients of the first grating are

while the coefficients of the second grating are

Here we have introduced the Talbot length as $L_{\lambda}=d^{2}/\lambda$ and the de Broglie wavelength as $\lambda=h/p_{z}$.

The Talbot coefficients above are modified by the CP potential,

where

This includes the chiral CP potential $V(x)$, which will be discussed below.

While the molecular interference pattern can be captured and imaged on a surface Juffmann2009 ; Juffmann2012 , for non-fluorescent molecules it has become common to use the spatial resolution of a third transmission mask $G_{3}$. The transmitted molecular flux behind $G_{3}$ is then a convolution of the density $w(x)$ before the grating and the grating intensity transmission function $|t_{3}(x)|$:

It then suffices to add a mass-selective large molecule counter Hackermuller2003 ; Strauss2023 . The expected signal is then klaus

where $A_{l}^{\prime}$ are the Talbot coefficients of the third grating and we extract the sinusoidal fringe visibility

In the following, we employ these equations to explore the differences in signal and fringe visibility between molecular enantiomers in the presence of chiral CP forces.

Note that molecular orientation is generally relevant in diffraction at nanomechanical masks, since it exposes different components of the polarizability tensor to the grating surface. We assume the chirality to be at a fixed temperature and we average over the molecular orientations.

We start by investigating a scenario in which the second grating is made from a chiral material, characterized by the reflection matrix:

It describes how the $p$- and $s$-polarized components of linearly polarized light, $\mathbf{E}_{s}$, $\mathbf{E}_{p}$ are reflected by the chiral material: $\mathbf{E}_{s}\rightarrow r_{sp}\mathbf{E}_{p}+r_{ss}\mathbf{E}_{s}$ and $\mathbf{E}_{p}\rightarrow r_{pp}\mathbf{E}_{p}+r_{ps}\mathbf{E}_{s}$. For a non-chiral surface, $r_{sp}=r_{ps}=0$ isotropic .

The CP potential between the grating and a chiral molecule in the ground state can be expressed as the sum of its electric, magnetic, and chiral components $V=V_{e}+V_{m}+V_{c}$, where chiral3 ; scheel ; stefan

and

Here $\nabla$ and $\stackrel{{\scriptstyle\leftarrow\mbox{ }}}{{\nabla^{\prime}}}$ act on the first and second position arguments of the Green’s tensor only, $\mu_{0}$ is the vacuum permeability, $\omega_{k}$ the transition frequency, $\mathbf{d}_{0k}$ is the electric dipole transition matrix element, $\mathbf{m}_{k0}$ is the magnetic dipole matrix element, and $R_{0k}=\mbox{Im}(\mathbf{d}_{0k}\cdot\mathbf{m}_{k0})$ is the optical rotatory strength. Formally, the sum is over all optical transitions (see Eq. III) but we can limit the computations to the dominant transition (i.e., $0\rightarrow k=1$). Both enantiomers have approximately the same modulus but opposite signs of the rotatory strength $R_{0k}$ and the distinct interference patterns observed between the two enantiomers can be attributed to the different signs of their chiral potential $V_{c}$.

The Green’s tensor $\mathbf{G}(\mathbf{r},\mathbf{r},\omega)$ includes the relevant information about the scattering of photons at the grating walls and it is written as green

where $k^{\parallel}$ and $k^{\perp}$ are the components of the wave vector parallel and perpendicular to the grating surface and $k^{\perp 2}=\omega^{2}/c^{2}-k^{\parallel 2}$. Substituting (III) and (III) into (III), we obtain chiral3 ; scheel ; stefan

where $\epsilon_{0}$ is the vacuum permittivity and $x$ is the distance between the molecule and the grating wall. Since we are considering the opening of the grating slit to be around 100 nm in size, we employed the non-retarded limit approximation $x\omega_{1}/c\ll 1$, which is favorable for the chiral CP potential. In the retarded regime ($x\omega_{1}/c\gg 1$) the chiral potential ($V_{c}\propto 1/x^{5}$) decays faster with distance than its electric and magnetic counterparts ($V_{e,m}\propto 1/x^{4}$) chiral3 . Since the molecule is significantly smaller than the grating thickness, we also use the proximity force approximation, wherein a molecule is assumed to interact with the grating like with an infinitely extended surface. The approach and departure of the molecule, i.e., forces acting before and after grating can modify the interference fringes. However, even after including such effects in a careful theory using a Green’s function approach nano , every grating model had to be corrected by a fudge factor in the Casimir force. This correction factor shrunk, however, from $\eta\simeq 8/5/2$ for gratings with a period of d=100 nm and a thickness of $t=12/45/87$ nm. It should be negligible in our case with $t=160$ nm. Moreover, all fringe effects and effects of defects in the grating are stationary and even in their presence one can see differences of interactions with molecules of different properties. This has also recently been exploited in nano-grating interference of molecules with and without electric dipole moments Simonovic2024 . In our case, the influence of chirality can then be probed differentially by swapping between the enantiomers.

Molecules close to the grating wall are attracted by the dispersive force. If they come within a distance shorter than a cut-off distance $x_{c}$, which depends on the materials and their chirality, they may be deflected to beyond the acceptance angle of the detector or even collide with the grating wall. In a typical Talbot Lau setup, the cut-off distance for the first and second grating $x_{c}^{(1)}$ and $x_{c}^{(2)}$ can be estimated to be

where $\theta=1$ mrad and 2 mrad are common experimental values at the first and second grating, respectively Brezger2002 . Here, the force $F(x)=-V^{\prime}(x)$ is the gradient of the CP potential. For the third grating, $x_{c}^{(3)}$ is given by cutoff ; cutoff2

where $v_{z}$ is the longitudinal velocity of the molecule. The chirality dependence of the cut-off distance $x_{c}$ results in different effective slit openings for different enantiomers. This influences both the interferometer fringe visibility and even more its transmission amplitude.

Since a large optical rotatory strength $R_{01}$ and a large value of $r_{c}$ result in a strong chiral CP force (III), we consider a beam of molecules whose optical rotatory strength is large compared to their electric and magnetic dipole moment. For the sake of the argument we start by assuming $G2$ to be made from a perfectly chiral material. In theory such a structure is conceivable even though it is hard to realize in the lab. In that case, $r_{c}\rightarrow 1$ and $r\rightarrow 0$, which implies that $V_{e}$ and $V_{m}$ vanish and that only the chirality-dependent $V_{c}$ contributes. A molecule with a positive value of $R_{01}$ is referred to as right-handed and it is attracted to the grating. Its left-handed enantiomer with $R_{01}<0$ is repelled and only molecules with $R_{01}>0$ experience a chirality dependent cut-off $x_{c}$ at a grating of positive chirality.

We therefore expect a difference between two enantiomers in their interferometer transmission and fringe contrast when we include the effect of the chiral CP potential,

where we let the grating walls be located at $\pm x_{o}$ with $x_{o}=fd/2$, and the molecules fly through them, i.e., $x\in(-x_{o},x_{o})$

Substituting $V(x)$ into (14) and performing the integration including the cut-off $x_{c}$, we obtain Fig. 2. Here we let the grating thickness be $b=160\,$nm, the periodicity $d=257\,$nm, the grating distance $L=50\,$mm, and the open fraction $f=0.45$. These parameters are chosen close to non-chiral nanostructures used in earlier experiments Juffmann2009 . The top panel shows one period of the matter-wave fringe, i.e., the transmission signal behind $G3$ as a function of the position of the third grating. This is done both for right-handed (solid red line) and left-handed (dashed blue line) molecules. In the left panel (i), the simulation is performed for hexahelicene with a monochromatic velocity of $v_{z}=180$ m/s. Their mass is $m=328$ Da, and a rotatory strength of $|R_{01}|=700\times 10^{-40}$ cgs has been reported with a transition frequency of $\omega_{1}=2\pi\times 10^{15}\,$s^-1 hexadata . From such curves we can extract the interference fringe visibility which we display in the bottom panel as a function of the molecular velocity (bottom scale) and de Broglie wavelength (top scale). The plots show that chirality of hexahelicene can be distinguished in the transmission signal. The panels of column (ii) depict the simulation for a hypothetical molecule with a mass $m=1000$ Da and a rotatory strength 10 times greater than that of hexahelicene with same transition frequency. For the evaluation of the transmission signal, we set $v_{z}=140$ m/s. This simulation demonstrates that, with such molecules, both the transmission signal and visibility as functions of velocity, could be differentiated for enantiomers due to the chiral CP forces, even though in a small velocity band.

The figure is the result of numerical computations based on Eq. (11) to Eq. (17), which is in agreement with the expectation that the final signal and fringe visibility must be periodic in the Talbot Length and thus the de Broglie wavelength. Electromagnetic properties of the molecules determine an effective open slit width of the grating, which enter the above equations. Due to chiral CP forces, enantiomers perceive different effective forces with the gratings, resulting in slight disparities in signal and visibility even for identical de Broglie wavelengths. This serves as the primary cause of the observed differences between enantiomers, as depicted in the figure.

Note that the assumption so far is strongly idealized: real-world materials generally have $r\neq 0$, and we require that the electric and magnetic anisotropy factor ani $g_{d}:=|R_{01}/c|\,/\,|\mathbf{d}_{01}|^{2}$ and $g_{m}:=|R_{01}\cdot c|\,/\,|\mathbf{m}_{01}|^{2}$ are sufficiently large to ensure the chiral CP potential to be significant in comparison to its electric and magnetic counterparts (III):

A possible solution is to use CP forces with opposite signs for the electric and magnetic components as observed over a perfectly conducting plate scheel . When the magnitudes of $\mathbf{d}_{01}$ and $\mathbf{m}_{01}/c$ are comparable, these two components may cancel, leaving only the chiral component.

Our calculations and similar ones in the literature chiral ; chiral3 ; suzuki rely on an effective-medium approximation, where the spatial variation of permittivity and permeability are neglected. This is only justified when the distance of the molecule to the chiral material exceeds the size of the chiral structure of the metamaterial diego . This requires nanoscale artefacts inside the grating bars, which is technologically very demanding. To overcome this challenge, we investigate a scenario where the second grating is coated with chiral molecules in the following section.

The electric, magnetic, and chiral components of the CP potential between two molecules can be written as stefan ; andrews ; salam

where the electric dipole transition matrix elements, the magnetic dipole moment matrix elements, the optical rotatory strengths, and the energy differences between the $|0\rangle$, $|k\rangle$ or $|k^{\prime}\rangle$-state for molecule $A$ and molecule $B$ are denoted by $\mathbf{d}_{0k}^{A}$, $\mathbf{d}_{0k^{\prime}}^{B}$, $\mathbf{m}_{0k}^{A}$, $\mathbf{m}_{0k^{\prime}}^{B}$, $R_{0k}^{A}$, $R_{0k^{\prime}}^{B}$, $E_{k}^{A}$, $E_{k^{\prime}}^{B}$, respectively. Here, $A$ refers to the molecules propagating as matter waves, while $B$ represents the molecules coating the second grating G2.

By integrating over all $B$-molecules in a layer of thickness $a$ and number density $n_{B}$, we obtain the potential exerted by the coating:

where $x$ is the distance between the molecule and the surface of the grating.

In addition, we include the CP potential arising from the bare grating itself:

where $\alpha^{A}(i\xi)$ is given by (III) and

We consider a grating made of silicon nitride, where the resonance frequency of longitudinal mode $\Omega_{L}=2.69\times 10^{16}$ rad/s, the resonance frequency of the transverse mode $\Omega_{T}=1.33\times 10^{16}$ rad/s, the longitudinal mode damping $\gamma_{L}=3.05\times 10^{16}$ rad/s, and the transverse mode damping $\gamma_{T}=6.40\times 10^{15}$ rad/s SiN .

To obtain the effect of the CP potentials on transmission signal and fringe visibility, we substitute

into (14), and compute (16) and (17). The cut-off distance $x_{c}$ can again be evaluated using (23) and (24).

Due to the limited availability of data regarding rotatory strength, electric, and magnetic anisotropy factors of molecules in chemistry, we base our following simulations on a reasonable estimate of molecular properties to see the trends. Since the CP force from the coating is generally weak compared to that from the grating itself, we proceed under the assumption that the grating is characterized by a thickness of grating $b=160\,$nm, a reduced periodicity $d=80\,$nm, a shorter grating distance $L=10\,$mm, an open fraction $f=0.45$ and coating thickness of $a=10\,$nm.

Fig. 3 (i) depicts the simulation where $|R_{01}|=1000\times 10^{-40}\,$cgs, $g_{e}=0.2$ and $g_{m}=5$ for both matter-wave molecules and molecules coating the second grating G2. For (ii), it is assumed that $|R_{01}|=5000\times 10^{-40}\,$cgs, $g_{e}=0.3$ and $g_{m}=3.3$ for both molecules. We set the transition frequency $\omega_{1}=2\pi\times 10^{15}\,$s^-1, the number density of the coating molecules $n_{B}=5\times 10^{28}$/m³ and $m=1000\,$Da for the molecules travelling as matter-waves. The second grating is always coated with right-handed molecules. As depicted in the figure, we only observe a small difference between both enantiomers in the transmission signal. This is hard to measure in an experiment and motivates us to expand our analysis to the case where chiral coatings are additionally applied to both the first and the third grating. We demonstrate that this can improve the overall chiral selectivity of the interferometer.

Here, we assume to have three identical SiN gratings, all coated with a layer of enantiomer pure, right-handed ($R_{01}^{B}>0$) molecules, while all other parameters are the same as in the previous section. As before, the molecules see a combination of electric, magnetic, and chiral CP forces from the coating, as well as electric CP forces from the grating, resulting in the interferogram and fringe visibility as illustrated in Fig. 4. The panel (i) is simulated for molecules with $|R_{01}|=1000\times 10^{-40}\,$cgs, $g_{e}=0.2$ and $g_{m}=5$, while panel (ii) shows the result for molecules with $|R_{01}|=5000\times 10^{-40}\,$cgs, $g_{e}=0.3$ and $g_{m}=3.3$. The right-handed molecules experience a strong attractive force because the electric, magnetic, and chiral components all share the same sign. This leads to a reduction in the effective grating slit width and to a loss of molecules near the coatings. Fig. 4 thus demonstrates that a coating of chiral molecules to all gratings can yield an observable effect of chiral CP forces in the transmission signal. Values close to our simulation parameters can be obtained for tailored molecules, such as [4]CC isobe ; isobe2 .

To explore a comprehensive range of potential test molecules, we plot in the upper panel of Fig. 5 the difference in mean interferometer transmission signals

as a function of the rotatory strength $R_{01}$ and the electric anisotropy factor $g_{e}$. Here, $S_{L}$ and $S_{R}$ are the transmission signals of the left-handed and right-handed molecules, respectively. The lower panel of Fig. 5 shows the corresponding maximum difference in fringe visibility $\Delta\mathcal{V}_{\rm max}.$ Our simulations capture the maximum values across a molecular velocity range from $v_{z}=100\,$m/s to $200\,$m/s, and they allow for an experimental velocity resolution of $\Delta v_{z}=10\,$m/s. This provides clear guidance for possible experiments, as soon as molecular parameters of a given species are exactly known.

Note that the magnetic component of the CP force is generally negligible compared to the electric and the chiral components of the CP force. Here we assume that $m=1000\,$Da and $\omega_{1}=2\pi\times 10^{15}\,$s^-1. The effects of the chiral CP force are more best seen in the transmission signal but also apparent in the fringe visibility. A particularly distinct difference between enantiomers becomes apparent for molecules with a rotatory strength $>10^{-37}\,$cgs and $g_{e}>0.1$. In the existing literature, it has been reported that molecules such as [4]CC can demonstrate a rotatory strength exceeding $10^{-37}\,$cgs isobe ; isobe2 ; ref5 , and molecules such as 3-methyl-cyclopentanone exhibit $g_{e}$ values exceeding 0.1 3mcp . We anticipate that data pertaining to these parameters for various molecules will become more accessible in the future.

Talbot-Lau interferometers with nanomechanical gratings have been realized with electrons McMorran2009 , positrons Ariga2019 , atoms Clauser1994 and molecules Brezger2002 ; Juffmann2009 , before. The interferometer technology is demanding but well under control. We therefore focus on the specific requirements for the observation of chiral dispersion forces. This requires the realization of chiral gratings and beams of molecules with large optical rotatory strength such as [4]CC.

Gratings could be coated by self-assembled monolayers of chiral substances. This may be suitable for narrow gratings and one can even orient all molecules within such a layer SAM ; SAM2 ; SAM3 . For larger and mechanically robust structures, dip coating is a feasible technology. For structures in the 100 nm range, however, surface tension is a challenge in all wet-chemical processes with nanostructures and needs to be addressed Robertson-Tessi2006 . We also need many mono-layers for the proposed work. Vapor deposition vapor ; vapor2 is conceivable for hexahelicene and 3-methylyclopentanone but still needs to be experimentally verified for [4]CC. We also assume the absence of local surface charges, which may dephase matter-waves of polar molecules in transit through a nanograting Knobloch2017 . While several highly chiral molecules, such as [4]CC, are not expected to have a permanent electric dipole moment in their thermal ground state, molecular modelling suggests that conformers with dipole moments up to 5 Debye may transiently develop on the time scale of nanoseconds. This can be suppressed by launching the molecules into a cryogenic buffer gas to freeze out all vibrational dynamics Piskorski2014 .

We have examined three scenarios in which chiral CP forces can influence the transmission signal and the visibility of chiral molecules behind a Talbot-Lau matter-wave interferometer with nanogratings. When the second grating consists of a perfectly chiral nanostructure ($r_{c}\rightarrow 1$, $r\rightarrow 0$), a small chiral effect should be observed even for the realistic case of hexahelicene and a very pronounced effect would be expected if we found a molecule with ten times larger rotatory strength (Fig. 2). Since such nanochiral materials are experimentally demanding, we have considered the case where the second grating $G_{2}$ is coated with highly chiral substances. We have identified examples of molecular rotatory strengths and asymmetry factors that would make an observable difference in molecular transmission but only a small difference in matter-wave fringe visibility (Fig. 3). The effect can be enhanced by applying the chiral coating to all gratings (Fig. 4) since even gratings that do not contribute to the interferometer phase shift influence the transmission and contrast by an effective slit narrowing. In this case, effects become visible already when both the coating molecules and matter-wave molecules possess a rotatory strength $>10^{-37}\,$cgs and an electric anisotropy factor $g_{e}>0.1$ (Fig. 4). Again the transmission is more affected than the fringe visibility. Finally, we have analyzed interference figures for the same experimental geometry but for molecules now varying vastly in their rotatory strength ($10^{-38}-10^{-36}$ cgs), in their electric anisotropy factor ($0.1-0.5$), and samples from an experimentally plausible velocity range ($100-200$ m/s) with finite band width (10 m/s). Figure 5 thus reveals the influence of these molecular parameters and while differences in transmission and fringe visibility of 1% may still be resolved, effects on the level of 10% would be clearly accessible. The molecular parameters are certainly challenging, but not unfeasible and the proposed experiment may allow probing for the first time the effect of static chiral Casimir Polder forces on isolated and here even delocalized molecules.