A Green's function approach to modeling molecular diffraction in the limit of ultra‐thin gratings

In recent years, matter‐wave diffraction at nanomechanical structures has been used by several research groups to explore the quantum nature of atoms and molecules, to prove the existence of weakly bound molecules or to explore atom‐surface interactions with high sensitivity. The particles' Casimir‐Polder interaction with the diffraction grating leads to significant changes in the amplitude distribution of the diffraction pattern. This becomes particularly intriguing in the thin‐grating limit, i.e. when the size of a complex molecule becomes comparable with the grating thickness and its rotation period comparable to the transit time through the mask. Here we analyze the predictive power of a Green's function scattering model and the constraints imposed by the finite control over real‐world experimental factors on the nanoscale.


Introduction
Matter-wave diffraction at a double slit or grating is a paradigmatic example of fundamental quantum physics [1], which has been demonstrated with electrons [2], neutrons [3], atoms [4,5], and small [6,7] and complex molecules [8,9]. Babinet's theorem predicts also quantum fringes in diffraction at opaque obstacles, and Poisson's spot was observed with atoms [10] and diatomic molecules [11] -with an application also in Fresnel zone plates [12]. In all these experiments attractive interactions between the grating and the matter-wave play an important role. They determine the relative population of the diffraction orders, influence the overall transmission through the grating and lead to substantial phase shifts. Hence, a detailed understanding of the interactions is of major importance for the design of any new experiment.
Atomic diffraction at nanomechanical masks has often been analyzed using the near-field potential V CP = −C 3 /r 3 as the short-range approximation of the Casimir-Polder interaction [13] -often referred to as van der Waals potential -with a constant C 3 which depends on the particle's polarizability and the dielectric boundary. However, for large composite systems the complexity of this interaction increases considerably: Molecules may vibrate in many modes, rotate at high frequencies, and change their states even while they are flying across the grating. Furthermore, some theoretical approximations that served well in earlier work with atoms [7,14,15] have to be reconsidered in the limit of ultra-thin gratings. It is then no longer justified to distinguish between the inside of a diffraction slit and a position at the slit entrance. In addition, the attraction to the wall may become timemodulated when an anisotropic molecule exposes different axes of its polarizability tensor to the grating wall over time. A full description then has to include the initial random orientation of all molecules as well as their rotation in transit through the grating. Since a single C 3 factor may then be insufficient to describe the interaction we here present a theoretical model which builds on the full Casimir-Polder potential. We explore the predictive power of the integrated phase that is accumulated by the molecular de Broglie wave during its approach towards, transit through, and exit from the grating. In hindsight this will justify the use of a coarse simplification, i.e. the parametrization of many subtle diffraction effects in an effective slit width -as has often been done in various earlier experiments with atoms and molecules.

Experiment
In Fig. 1a we show a typical molecular diffraction experiment, which has been described in more detail in Ref. [9]. It is contained in a vacuum chamber at p < 10 −7 mbar to make sure that the molecules propagate freely from the source to the detector after they were evaporated from the entrance window by a focused laser beam. The molecular beam is laterally defined by the size of the source slit S(x 1 ) of 1-2 μm as well as by one piezo-driven vertical slit S(x 2 ). It reaches a divergence of about 10 μrad for a width of 20 μm at the detector screen. This prevents the diffraction orders from overlapping. A horizontal slit S(y) is used to constrain the free-flight parabola between source and detector to select a given velocity class at a certain height of the detector. The diffraction pattern is monitored by collecting all transmitted molecules on a quartz plate at distance L 2 = 564 mm behind the grating. The laser-induced fluorescence is collected by a microscope objective and registered by an electron multiplying CCD camera. Even though the interaction with the laser in the source is limited to about 1 ms the molecules are substantially heated and reach a speed of 100 − 350 m/s. The test particle phthalocyanine (PcH 2 ) has a mass of m = 514 atomic mass units, covering de Broglie wavelengths between λ dB = h/mv 8 × 10 −12 − 2.2 × 10 −12 m in this experiments. If we assume thermalization of the internal and external degrees of freedom in the source we can assign a microcanonical internal temperature of around 1200 K to PcH 2 . Even though this is low enough to avoid decoherence caused by the emission of thermal radiation [16] it influences the interaction of the molecules with the diffracting element also through the molecular rotation. In total we compare the performance of four different silicon nitride (SiN x ) nanogratings G1-G4 to each other, which vary in their thickness by up to a factor of eight. Their geometry parameters are compiled in Fig. 1. Sending the molecules with a mean molecular velocity of 215 m/s through either one of these grat-ings we have obtained diffraction patterns as shown in Fig. 2. In all cases we observe almost maximal fringe contrast, i.e. between the diffraction peaks the fluorescence signal falls to its background level -with only a minimal contrast reduction for the thickest of all gratings. In the following we will discuss different approaches to analyze these observed molecular diffraction patterns. 3 Casimir-Polder potential in the thick grating limit If we treat the molecules as structureless point particles, defined only by their mass and momentum, the formal equivalence between the stationary Schrödinger equation and the Helmholtz equation allows us to use the Kirchhoff-Fresnel integral to compute the propagation of the de Broglie waves from the source to the detector. The situation is simplified if the detector is situated in the far-field at L 2 > (N · d) 2 /λ dB depending on the number of coherently illuminated slits N and the periodicity d. The diffraction pattern can then be described by the Fourier transform of the complex grating transmission function. From a transverse coherence of 1.5 μm at the grating (N ∼ 15) it follows that the detector is placed in the transition regime between far-and near field. Hence, the quadratic terms of the wave function have to be considered for our experiment.
In earlier experiments on atom diffraction at 'thick' gratings [17,18] it has been useful to approximate the wall inside of each grating slit as a semi-infinite plane, such that the particle's interaction with one side of the slit can be described by V CP with the Casimir-Polder coefficient C 3 is determined by the particle polarizability α(ω), the vacuum permittivity 0 and the dielectric function (ω) of the grating material, where r(ω) = ( (ω) − 1)/( (ω) + 1) is its surface reflection coefficient. All material constants depend on the frequency ω of the virtual photons that mediate the Casimir-Polder force. The phase shift that is imprinted onto the molecular matter-wave in the diffraction process results from the attraction to both walls of each slit, integrated over the grating thickness b. The interference pattern at the detector position x d in the distance L 2 behind the grating is then computed via the one-dimensional Kirchhoff-Fresnel integral over the grating aperture A, i.e. a comb of N slits of width s, separated by the period d. Pictorially, this integral sums over all elementary wavelets of wave vector |k| = 2π/λ dB emanating from the grating aperture with an interactiondependent phase: with the transmission function which includes the z-dependent slit width for a wedge angle β. In the Fresnel approximation (Eq. 2) we retain the quadratic terms in x, which still allows to describe the near-field and the far-field on an equal footing. Fur-thermore, we allow for an arbitrary initial wave function ψ(x). The molecular density observed on the detector on the right-hand side is proportional to the modulus squared of the coherent sum of all amplitudes However, because the molecular source emits particles without any mutual phase coherence we also need to integrate the molecular density pattern incoherently over all longitudinal velocity classes that may contribute with a distribution function f (v z ) and all positions x 0 in the source extending homogeneously from −D/2 to D/2.
All arguments above assume the validity of the Eikonal approximation, i.e. straight trajectories through the slits. This is justified by the fact that a molecule can only reach the detector further downstream if its interaction with the grating wall deflects it by less than 340 μrad. We introduce the cut-off distance as the closest possible separa- c) d) Figure 2 Molecular diffraction pattern recorded in the band of (215 ± 15) m/s (solid line) and fitted using the Kirchhoff-Fresnel approximation and an approximate Casimir-Polder potential as in Eq.(4) (red circles). In this fit the C 3 coefficient was the only fit parameter. The Casimir-Polder coefficients extracted from the fit are a) C 3 = 170 meV· nm 3 , for grating G1, b) C 3 = 40 meV· nm 3 for grating G2, c) C 3 = 80 meV· nm 3 for grating G3, and d) C 3 = 50 meV· nm 3 for grating G4. Within the experimental uncertainties the experiment is always very well reproduced, but with C 3 varying by 400% in this simplified model. Table 1 Characteristic fit results for all four gratings: a) When a phthalocyanine molecule is deflected by more than ±340 μrad it misses the detector. For a classical particle this will happen if its initial lateral distance to the grating is smaller than the cut-off. It depends on in the forward velocity which we take to be v z = 215 m/s, as in Fig. 2. The respective values are doubled as the deflection is present on both sides of the slit. b) the slit width reduction is the difference between the measured geometrical width and the slit width fitted as in Sec 4. c) the C 3 values are derived from a fit based on Eq. (3) and (4). d) the phase shift factor η, introduced in Eq. (21), allows one to restore a perfect fit between experiment and the refined theory of Sec. 6. All methods together provide strong evidence for the influence of surface bound charge effects that add to the Casimir-Polder potential with the same scaling law.
Slit width reduction/nm 36 38 28 28 Fitted C 3 /meV· nm 3 170 40 80 50 Phase factor η 7.2 2.0 8.2 5.0 tion between the molecule and the grating wall compatible with that requirement. We estimate it based on the deflection by the non-retarded Casimir Polder force inside each slit F x = −η · C 3 /x 4 , as described in Section 5 and listed in Table 1. Since the true C 3 value depends on many nanoscale material properties (accuracy of the geometry, patch fields, local charges and impurities etc.) which may even vary somewhat across the grating, the listed values may have larger error bars. The correction factor η accounts for this uncertainty. From a fit of Eq. (4) to the molecular diffraction patterns we find the C 3 -coefficients listed in Fig. 2. The overall reproduction of the shape and amplitude distribution is excellent for all gratings, with only small deviations for the thickest grating G2. However, the fitted C 3 coefficients vary by a factor up to four, even though they describe the same molecules and the same material. This indicates that the physics is not fully captured by this simpflied approach.

Effective slit approximation
Since the exact determination of C 3 (ω) is non-trivial, it has often proved useful in atom interferometry to encode an 'effective potential' in the transmission function. This assumes that a virtual reduction of a grating slit can mimic the transverse spreading of the molecular wave function that would otherwise be obtained at a larger slit in the presence of the attractive Casimir-Polder interaction [7,19]. An 'effective slit' description is capable of predicting the population of higher diffraction orders since the far-field interference pattern can be described as the convolution of single slit and grating diffraction, i.e. a product of their Fourier transforms.
The method of 'effective slits' has proven particularly useful when the key goal is to compare atomic or molecular polarizabilities in diffraction at one and the same grating [15]. We implement it here by replacing the trans-  Fig. 3, is surprisingly well captured by this method. We see, however, that this description also misses systematically at least one diffraction peak in either case, namely the second peak in G1, the fifth peak in G2 and G3 and the sixth and higher peaks in G4. Moreover, a best fit to the data can only be obtained when we reduce the effective slit width s eff by 37 nm for G1 and G2 and by about 28 nm for G3 and G4. This is about 50 % of the measured geometrical slit width of G1-G4 and considerably larger than the effective slit width that remains if we subtract twice the aforementioned cut-off from the geometrical slit width. While one may expect that the Casimir-Polder forces are dominant for either very thick gratings, such as G2, or very narrow slits, such as G4, we see that the effective reduction is unexpectedly high for G1. This indicates that knowledge of the grating's geometry and dielectric function as well as of the molecular polarizability alone are not sufficient to describe the real-world interaction process.

Scattering theory: point-wise evaluation of the Casimir-Polder potential
In Eq. (3) we have approximated the transmission function in the non-retarded limit of a Casimir Polder interaction between a point-like polarizable particle and an infinite half-space. This approximation has often been used in the literature for >100 nm thick gratings where effects related to the finite size of the grating can be neglected. For ultra-thin gratings this assumption is not satisfied. We here explore possible improvements in the description of the data that could be obtained by reverting explicitly to the viewpoint that the interaction potential results from spontaneous and thermal fluctuations of the molecular polarizability as well as from the electromagnetic field which induce molecular dipoles that subsequently interact with their images inside the grating.
In second-order perturbation theory the Casimir-Polder potential at zero temperature for a particle positioned at r A is given by [20] (584 of 591) The coordinates x , y , z refer to the principal axes of PcH 2 , as shown in Figure 1. The numerically calculated and experimentally measured values [24] of the polarisability agree in the static limit (ξ → 0, ω → 0) with the reduced Planck constant , the vacuum permeability μ 0 , the molecular polarisability tensor α(iξ ) and the scattering Green tensor G(r A , r A , iξ ). This relation accounts for the interaction of the fluctuating field at the (imaginary) frequency ξ with the molecule, thereby creating a dipole moment whose strength is governed by the polarisability, and whose interaction with its own image is encoded in the classical propagator function G(r A , r A , iξ ). The Green function is the fundamental solution of the vector Helmholtz equation [20] ∇ where the material properties and the grating geometry are encoded in the position-and frequency-dependent dielectric function ε(r, ω) = 1 + χ(r, ω). With the boundary condition G(r, r , ω)→0 for |r − r | → ∞, which is equivalent to the Sommerfeld radiation condition, the Green function is the unique solution to Eq. (6). In the scattering representation, the Green function is decomposed into a bulk component G (0) (r, r , ω) that governs the propagation within a homogeneous materials, and a scattering contribution G (S) (r, r , ω) that is necessary to fulfill the Maxwell boundary conditions at the interface(s) between free space and the grating. Analytical solutions are only known for simple geometries with high symmetry. Constructing a Green function therefore requires usually considerable numerical effort. However, it is often sufficient to treat the scattering perturbatively using a Born series expansion, i.e. a multiple scattering expansion. For that the dielectric body is divided into small disjoint volume elements dV , which are still large enough to be treated as a homogeneous medium. We can then write [20] The approximate scattering Green function G (S) is constructed from a reference Green function G (0) , which we choose to be the free space propagator since this is analytically known [20]. The first term in Eq. www.ann-phys.org falls rapidly with χ n (ω) we can truncate the expansion after the first term. The correction due to higher-order terms around the grating bars is estimated in analogy to the case of a molecule in front of an infinite plate. It yields in the non-retarded limit the correction factor as the ratio between the exact Casimir-Polder potential for an infinite plate [21] and the potential in first-order of the Born series The scattering Green function contains all information about the electromagnetic response and the geometry of the grating. In case of the grating material SiN x , its magnetic response can be neglected and only the electric part of the Casimir-Polder potential (5) contributes. For the dielectric susceptibility we used the model [18,22] Im with the parameters T = 3.48 · 10 15 rad/s, = 1.09 · 10 16 rad/s, f = 1.13 · 10 17 rad/s, and γ = 1.16 · 10 16 rad/s. In order to convert the imaginary part of the permittivity at real frequencies to the permittivity at imaginary frequencies, we use the Kramers-Kronig relation [22] Evaluating Eq. (5) requires in principle the complete knowledge of the polarisability at all imaginary frequencies. For atoms it is often sufficient to know the polarizability around a few strong electronic transitions which can be measured or computed with high accuracy [15]. For large molecules, on the other hand, the strongest transitions are often situated in the ultraviolet and therefore in an experimentally inaccessible range. They need to be derived from quantum chemical calculations which we have performed for PcH 2 using the software package TURBOMOLE [23]. The results are depicted in Fig. 5.
We also need to include thermal excitations and transitions in the molecules, as shown in Fig. 6. Even though we can neglect the thermal population of the electronic states the vibrational modes will be populated according to a Bose-Einstein distribution n(ω) which we use in the derivation of the Casimir-Polder potential. The poles of the Bose-Einstein distribution in the upper complex plane allows us to replace the integral over the imaginary frequencies in Eq. (5) by an infinite sum over the discrete where the primed sum denotes the relation Here k B is the Boltzmann constant. The potential (12) is an average over all possible -and therefore 'nonresonant' -intramolecular states |n weighted by their population probability p n . Since we neglect thermal population of the electronic states the polarisability is constant α n = α and the average over all excitations trivial. On the other hand, because the molecule is not in thermal equilibrium with its environment, the Casimir-Polder potential may acquire a resonant part which causes attractive and repulsive forces for transitions into higher states and lower states, respectively [25]. They are determined by the dipole matrix elements d nk The total potential is the sum of the resonant and nonresonant parts Both components are included in our numerical evaluation with the resonant part providing the dominant contribution.
When the grating thickness is reduced to a few nanometers, and since the detected molecules may approach the grating to within 10 − 20 nm we need to consider the finite extension and shape asymmetry of phthalocyanine, as well. The molecule is an oblate (flat) top (586 of 591) that measures ∼ 1.5 nm across. We employ a polynomial correction function to the Casimir-Polder potential for point dipoles V dip C P (z) which now treats the molecules as an ellipsoidally shaped electron density distribution The length of the principal axis a and the eccentricity e are defined by the distance beyond which the electron density has fallen below ρ e < 10 −3 e · a −3 0 , where a 0 denotes the Bohr radius. Both density shape parameters have been computed using the software package GAUS-SIAN (LC-BLYP/cc-pVTZ) [26]. They represent the geometry of the molecule and can be used to compute the polarizability ellipsoid [24]. The coefficients c i are analytically known. For phthalocyanine we find a = 17a 0 = 0.9 nm and e = 4 which yields The isotropic distribution of initial orientations and the rotational states that are sampled in transit through the grating are taken into account in a coherent sum over the molecular density patterns for different rotational interactions. Based on the known rotational constants of phthalocyanine [27] we compute a mean rotational quantum number of J 600 at T = 1200 K. This corresponds to a rotational period of around 10 ps, which must be compared to the 50 ps transit time of a molecule flying at 200 m/s through a 10 nm thick grating. While for most molecules in the ensemble the parameters are such that we can treat the molecule-wall interaction as effectively isotropic, the situation will change for the slowly rotating part of the ensemble, for gratings with large wedge angles and atomically thin gratings [28].
For ultra-thin gratings and fast molecules, the orientation and rotation of the molecules are no longer negligible, which means that the different rotational states enter the interference pattern incoherently. The interference pattern thus becomes orientation-dependent, where denotes the solid angle. In analogy to Eq. (4) one would therefore have to average incoherently over all interferograms for the different orientational states as Here, the integration measure d is assumed to include the proper normalisation factor and a possible (e.g. temperature-dependent) weight function. For the rela-tively thick gratings used in the present experiments, however, we used the simpler approach of an averaged phase.

Scattering analysis of the diffraction patterns
Following the results of Section 5 we compute the potential for all distances and integrate the resulting de Broglie phase shift along straight trajectories -i.e. within the Eikonal approximation. Since the potential falls with at least the third power of the distance to the grating, we can limit the integral to a region z = ±30 μm before and after the slit. The integrated phase then simplifies the transmission function of Eq. (3) which then reads with This is used in Eq. (4) to compute the diffraction patterns. When we compare our ab initio calculation with the experimental data we still see discrepancies in the representation of the higher diffraction orders. However, a near-perfect fit to the experiment can be retrieved if we introduce a multiplicative phase factor η, which adjusts the strength of the potential for each grating. The transmission function then reads The result is shown in Fig. 7. The values of η we extract for the single gratings varies between 2.0 and 8.2, e.g. by 410 %.

Discussion
All three models of Sections 3-6 indicate that each grating has its own individual character and requires its own correction to either the Casimir-Polder potential or the effective slit width. They all exert stronger forces than expected based on the potential between a neutral dielectric and polarizable particles alone. We therefore need to discuss the possible sources for these deviations and their relevance for a number of emerging experiments, also in modern atom interferometry. On the one hand, an uncertainty in the theoretical description is due to the unknown stoichiometry of the silicon nitride membranes. However, no reasonable variation in the nitrogen concentration ('x' in SiN x ) can explain η > 5, as observed above. On the other hand, an experimental accuracy in the original fabrication and measurement of the slit width may enter. Even though the grating period can be written with a precision higher than 1 nm using interferometric tables in focused ion beam writing (as done for gratings G1, G3, G4) or even with an accuracy of 0.01 nm using photolithography [29,30] (as was the case for grating G2), the local etching process will introduce variations on the scale of several nanometers. The lateral etching accuracy of focused ion beam writing is of the order of x = 5 nm and earlier atom diffraction experiments showed that changing the slit width by 1 nm could change the atomic C 3 coefficient by 25 % [31]. In order to estimate the influence of this effect we have fitted the diffraction pattern of G2 using the full Casimir-Polder treatment for a series of different slit widths, separated by 1 nm in the interval 57 − 48 nm. The resulting phase factor η = 2.0(3) at s = 57 nm decreases linearly with the slit width to η = 1 for s = 50 nm. This slit width reproduces the experimental results best with the analytically computed potential, but contradicts the width that was determined in electron microscopy as shown in Fig. 1d. Geometry effects alone therefore are insufficient to explain the observed variations between the gratings.
In the comprehensive theoretical treatment in Section 5, the temperature influences both the resonant and the non-resonant part of the Casimir-Polder potential. For the non-resonant part it is argued that the po-larizability is determined by the electronic state of the molecule. Here, it is safe to neglect contributions from electronically excited states as the internal temperature is not high enough to populate them and potential excitations would definitely have decayed by the time the molecules reach the grating, i.e after milliseconds. The resonant contributions to the Casimir-Polder based on the molecular eigenstates are roughly three orders of magnitude smaller than the non-resonant ones. This is due to the small vibrational transition dipole moments d mn . However, these vibrations may cause a time-varying dipole moment in the molecule, which could couple to the grating. Molecular modeling suggests that even at high temperatures, phthalocyanine will at most develop dipole moment fluctuations below 1 Debye. Since we don't see any indications of decoherence this dipole moment does not shift the pattern but at most contributes to the effective molecular polarizability. We may therefore ignore this effect as an explanation for the variations in C 3 for the different gratings. The effect of rotations onto the molecular diffraction patterns has recently been studied in detail by Fiedler and Scheel [32] who find that the rotational motion has a pronounced influence on the relative population and a small effect on the position of different diffraction orders.
An important clue to the interpretation of our results is given by the observation that the fabrication of silicon nitride membranes is always accompanied by processes which lead to the deposition or implantation of charges. Focused ion beams have been reported to cause surface charge densities up to 10 13 e/cm 2 in SiN x [33], whereas reactive ion etching after photolithography (used for (588 of 591) G2) is expected to generate lower charge densities. This reasoning is consistent with our Casimir-Polder fits since the value of η = 2.0 for G2 is much lower than the respective values for all FIB milled gratings, η ∈ [5.0 − 8.2]. The detailed charge variation is difficult to image with established technologies, such as Kelvin probe microscopy, since the local forces exerted by that method are sufficient to deform the very fragile gratings. Also electron microscopic methods are not ideally suited as they may add even further charges to the surface. Molecule diffraction is interestingly a non-destructive and sensitive probe of charges on freely suspended nanostrings. An efficient means to calibrate this method for a quantitative analysis, is still to be developed. Conceptually it will be interesting to revisit this problem with atoms in the future, where the internal particle dynamics is of lesser relevance. Additionally to the charging of the grating, gallium ions may be implanted into the silicon nitride membrane, changing the composition of the membrane material locally. This doping, which depend on the materials' thickness and the FIB parameters, should be pronounced most for grating G4. However, the higher values of η for G1 and G3 make it unrealistic that the gallium ions play a leading role.

Conclusions
We have presented a quantitative comparison of molecular interferograms that emerge in the diffraction at ultra-thin nanomechanical gratings. Three complementary approaches were used to describe the patterns: first, a simplified Casimir-Polder approach that includes the grating geometry. It is based on the assumption that one can approximate each grating as an infinite semi-space. Second, an approximation that reduces the entire problem to one single slit width, and finally a detailed scattering Green's function approach that includes many details of the molecular shape, excitation and dynamics as well as the details of the grating. We find that they all have their merits and drawbacks.
The scattering approach is conceptually most comprehensive and most demanding as it includes a large variety of fundamental and experiment specific details. It allows us to draw conclusions about the presence of charges in the grating, even though a quantitative comparison would still profit from a well-defined calibration standard.
The diffraction pattern can, however, also be rather well described by the much simpler approximation of non-retarded Casimir-Polder interactions in front of an extended grating. At low diffraction orders, the model of effectively narrowed slits still describes the reality surprisingly well. We attribute the high qualitative agreement between very different theoretical approaches and levels of sophistication in the description of the molecular diffraction patterns to the fact that molecules close to the vicinity of the material walls will be effectively removed from the detected beam. This justifies an effective slit description.
A quantitative fit to the data always requires the assumption of additional local forces between the molecules and the grating walls that surpass the Casimir-Polder expectation by several hundred percent. We argue that it is unlikely that these forces and their variation between the gratings can be explained by any molecular property or the grating geometry alone. Our observation rather naturally indicates the presence of charges that have most likely been deposited in the manufacturing process, may have been accumulated during electron microscopy, or might have been added in vacuum by residual contaminations. Any way, they are difficult to control or eliminate selectively on the nanometer level.
Matter-wave diffraction may therefore be developed into a sensitive tool for detecting surface charges on the nanoscale if further work is invested into a better characterization and calibration of these distributions, possibly also using optical imaging of dispersed NV centers in nanodiamonds which are very sensitive and very localized field probes [34].
Our experiments rise a flag of warning for experiments that aim at quantitative measurements of Casimir-Polder forces or studies that envisage to explore deviations from Newtons law of gravity on the micron scale. Whenever they operate with dielectric materials they will have to devise means to verify the level of surface charges.
For matter-wave interferometry it is an interesting challenge to further explore the role of these charges on the diffraction of polar molecules, as almost every biomolecule exhibits a permanent dipole moment. Hence, they are expected to interact even more strongly and more dispersively with nanomechanical masks.