Experimental methods of molecular matter-wave optics

Thermal effusive sources have served as workhorses in various interference experiments with complex molecules [84, 85]. Stable particles can be heated up to the point where they sublimate or evaporate in a free effusive regime, where their velocity distribution is determined by a Maxwell–Boltzmann law. This is strictly only the case when the collisional mean free path of the molecules is larger than the smallest dimension of the source opening [85] and for that reason, one finds a supersonic contribution in many practical realizations, also.

The volatilization temperature has to stay low enough to avoid particle fragmentation. This is a challenge for complex particles with large polarizabilities and therefore strong van der Waals interactions to neighbouring molecules—which increase the enthalpy of evaporation. Based on this insight it is now possible to alleviate this problem by chemical tailoring the molecular bonds, appropriately [84]. The attachment of perfluoroalkyl chains to organic molecules by Mayor and co-workers in Basel has proven to be an important and very successful method that paved the path to the realization of quantum interference of particles in excess of 10 000 amu [72, 73, 84, 86].

The most probable velocity in a thermal beam is $v=\sqrt {2k_{{\rm B}}T}/m$. This corresponds to a de Broglie wavelength of λ_dB ≃ 5 × 10⁻¹² m in the case of C₆₀ at 900 K [32]. Current state-of-the-art matter-wave interferometers can cope with λ_dB ≃ 200 fm [72, 87], corresponding to a particle of m = 10⁵ amu thermalized at room temperature or m ⩾ 10⁶ amu at 10 K.

The potentially detrimental effect of heating on the integrity of fragile molecules is a particular problem for biomolecules. The thermal load may, however, be reduced by limiting the heating power to a local spot, from which the molecules are rapidly laser evaporated. This method induces substantially less thermal damage than a Knudsen cell and it proved useful in far-field diffraction of tailor-made phthalocyanine derivatives [41]. A laser beam can be focused to less than a micrometre, which provides the transverse coherence that is required for molecule interferometry.

Molecule interferometry has also been performed with particles entrained in a supersonically expanding carrier gas [43, 49, 50, 58, 87, 88]. The adiabatic expansion provides translational cooling and improves the longitudinal coherence as well as the forward-directed particle flux. This process affects the temperature of all degrees of freedom including translation, rotation and vibration with a temperature ranking T_trans < T_rot < T_vib [89]. The finite energy level spacing in molecules is the reason why vibrational modes may be completely frozen in small molecules [85], while in massive particles cooling to the rotational ground state is a big challenge. Jet cooling of internal states is only effective for molecules up to a few dozen atoms. The final temperature is determined by the molecular heat capacity as well as the number of thermalizing collisions with the carrier gas.

Supersonic beams have a wide range of applications since they can be loaded with free gases [43, 49, 50, 58] or particles that are volatilized in laser desorption (LD) [90–92] or thermal sources [51].

Adiabatic expansion comes, however, at the price of an overall acceleration to velocities in the range of 300 m s⁻¹ for cryogenic nozzles and more than 1000 m s⁻¹ for helium at room temperature [51]. This high velocity can be reduced in the lab-frame when the nozzle is placed on a backward-rotating wheel [93]. At velocities below 100 m s⁻¹ [94] small molecules would be compatible with matter-wave interferometry.

Beams of large neutral clusters may be formed in aggregation sources, based on magnetron sputtering [95], laser ablation or thermal evaporation [96]. They generate neutral or singly charged particles of either sign and are prime candidates for quantum interference experiments with metal or semiconductor clusters. Aggregation sources may produce beams of internally cold particles but still require further slowing or cooling for high-mass interferometry.

Charged particles may be acceptable precursors for molecular interference experiments. Charge may provide a handle for beam preparation and cooling. It seems, however, advised to add a neutralization scheme and to perform all quantum delocalization studies with neutral particles to avoid decoherence and phase averaging in stray electromagnetic fields. Matrix-assisted laser desorption ionization [91, 97] (MALDI) and electrospray ionization [92] (ESI) could serve as promising beam sources for such experiments. This is particularly true for thermolabile biomolecules of almost any size [98].

In MALDI, molecules are initially isolated in and desorbed from an organic matrix. The matrix molecules act both as absorbers for the desorbing laser light pulse and as collision partners in the following expansion into the vacuum. They are responsible for cooling as well as for charge transfer with the analyte particles. The result is a mixed molecular beam composed of matrix molecules and isolated analyte particles travelling at 300–800 m s⁻¹. The method produces singly charged molecules and a substantial fraction of neutrals [99]. MALDI has been shown to volatilize objects with masses in excess of 10¹⁰ amu [99, 100], even functional biomaterials, which may become relevant for far-future quantum experiments.

A number of matrix-free optical methods are conceivable and currently being explored in several laboratories. This includes laser-induced acoustic desorption, LIAD [101], laser-induced forward transfer, LIFT [102], or matrix-free LD [103]. They are all compatible with a high-vacuum environment.

In contrast to that, ESI experiments start at atmospheric pressure and require the subsequent transfer of the isolated molecules through differential pumping stages into the ultra-high-vacuum science chamber. The molecules are first dissolved in a carrier liquid and filled into a thin capillary. High voltage (1–2 kV) between the capillary and the vacuum inlet leads to the formation of a liquid cone which fractions when the electrical forces overcome the surface tension (Coulomb limit). Release of small droplets and a repetitive cycle of solvent evaporation and charge instabilities finally lead to the isolation of individual solvent-free molecules. This method is again particularly useful for labile biomolecules. However, since electrosprays are intrinsically based on Coulomb fission, even a single protein of m ⩽ 10⁵ amu ends up loaded with dozens of elementary charges. Charge reduction, efficient transfer into the vacuum chamber and slowing of the beam remain the key challenges for quantum experiments based on this source [66].

A continuous molecular beam can be made compatible with TOF selection by encoding a time sequence on it. The beam can be chopped using a single rotating disc with a slit. Two discs are needed to define a travel time when both the source and the detector are operating in continuous mode. In practice several, often up to six, discs are used to suppress higher order velocity bands [85]. The transmitted velocity range can typically be reduced down to Δv_long/v₀ = 1%.

When a single disc is used in combination with a time-resolving detector, the signal can be enhanced by etching many slits into this disc. The resulting convolution of molecular packages travelling at different velocities and partially overtaking each other can complicate the velocity analysis. The signal can, however, be deconvoluted a posteriori if a known pseudo-random sequence is modulated onto the molecular beam [107]. Many different velocity classes can then be used in a single experiment.

The particle velocity can also be selected by a rotating turbine with helical grooves [42, 108]. Depending on the groove geometry and the rotational speed, only molecules within a certain velocity band can pass. This method has been widely used in neutron interferometry, but the vibrational noise of a mechanical device spinning at 100 Hz may also scramble quantum fringes: while far-field diffraction is rather inert against such perturbations [32, 51, 52] because the fringe spacing exceeds the grating period by orders of magnitude, vibration amplitudes as tiny as 10 nm may be detrimental for near-field interferometry [109]. It is, therefore, often advised to choose passive selection methods.

In the early days of molecular beam physics, Otto Stern selected the molecular beam velocity by rotating the entire vacuum apparatus [110]. Nowadays, it appears rather favourable to employ a static scheme that exploits the free fall in the Earth's gravitational field [111]. Three horizontal slits placed between the source and the detector can define a free-flight parabola and select molecules that travel at a desired velocity [41, 111]. The selectivity of this scheme depends on the height and separation of all slits. It improves with decreasing velocity and enables the selection of velocity bands with a typical width of Δv/v ∼ 5 − 30%. In practice, the first and the third slit can be realized by the source and the detection aperture, respectively.

Slowing and cooling of complex molecules is still a grand challenge, but a number of sophisticated techniques have been demonstrated in recent years: laser cooling, which is tremendously successful for atoms [112], has recently been extended to diatomic molecules, such as SrF [113] and current efforts explore the possibility of going even further. It has also been shown that cold molecules could be generated by joining cold atoms [114, 115].

Since these techniques require cycling optical transitions and the exchange of many photons it is difficult to extrapolate them to compounds with dense and open energy level systems. A number of research teams have started sympathetic cooling of molecules co-trapped with laser-cooled atomic ions. This is promising but in order to trap two species in the same Paul trap it seems that the atom and the molecule should have the same mass-to-charge ratio [116–120].

When optical cycling is no option, one may also exploit selective removal from the ensemble following the idea of Maxwell's demon: open and close a door between two chambers when you 'know' a slow particle wants to pass. An atom or molecule oscillating in any kind of trap reaches its lowest velocity at its turning point. When it is extracted from the ensemble at this very point, its motional energy is minimized. This trick has been implemented with atoms and it will be interesting to see how to extend it to larger objects [121].

It has been predicted that laser cooling of complex dielectric nanoparticles could also be realized using off-resonant optical forces in a cavity-assisted Sisyphus cooling scheme [122, 123]. Although several groups succeeded in proving the working principle for atoms [124, 125], substantial cooling of mesoscopic particles is still a challenge [126–128].

Another concept relies on non-radiative but modulated electric or magnetic fields [129]. Since neutral molecules couple only weakly to external fields, many interaction steps are required to achieve sizeable deceleration. Electric [130–132] and optical Stark deceleration [133], reverse magnetic coil guns [134, 135] and Zeeman slowers [136] have been used to manipulate molecules up to about 100 amu. Extrapolation of these methods to more complex objects is a subject for future research.

For gaseous polar molecules, Stark selection [137] rather than deceleration was shown to provide high phase-space densities and a good starting point for further storage and advanced laser cooling schemes [138–140]. This technique works best when it can select from an existing dense cold and gaseous ensemble—which makes it less compatible with massive objects such as biomolecules or metal clusters.

Finally, cryogenic buffer gas cooling [141] and the implantation of molecules into liquid helium droplets [142, 143] are techniques that allow one to cool both the internal and the translational degrees of freedom. As many other schemes, they are still waiting to be integrated into preparatory stages for future matter-wave experiments with nanoparticles.

When a molecular beam is bombarded with electrons of energy E the ionization efficiency η_ion ∝ n_e · σ_ion(E) · l depends on the electron density n_e, the molecular ionization cross section σ_ion(E) and the interaction length l. For many molecules σ_ion(E) is maximized for electron energies between 40 and 100 eV [85, 86]. The maximal electron density is limited by the energy-dependent space charge, i.e. by Coulomb repulsion between the electrons. With ionization cross sections ranging between 10⁻¹⁶ and 10⁻¹⁴ cm², a total ionization yield of 10⁻³–10⁻⁵ can typically be achieved. For complex molecules, fragmentation is an important issue, since a large manifold of internal states enables fast conversion between electronic, vibrational and rotational energies.

In spite of these difficulties, electron impact was the method of choice for a large number of quantum diffraction and interference experiments with diatomic and organic molecules up to 10⁴ amu [8, 15, 16, 39, 50, 58, 65, 69, 73].

A neutral particle may also ionize when it hits a surface whose work function is higher than its own ionization potential. If the surface is sufficiently hot, the ionized particle may re-desorb and be collected by an ion counter [151]. That is the basic idea of Langmuir–Taylor or hot-wire detectors that have been widely used in alkali atom interferometers [51, 152].

For certain atom/surface combinations this process is highly efficient [153]. It is, however, less compatible with TOF mass spectrometry, since time constants between 10⁻⁶ and 10⁻² s spoil the mass resolution [85]. On materials with a low work function and particles with a high electron affinity the reverse process, electron attachment, can also occur [154].

Some molecule/surface combinations, in particular those involving molecules, may favour molecular dissociation into products with lower ionization potentials [144]. In selected cases, up to the size of insulin, intact surface ionization has been reported in interaction with hot rhenium oxide, which exhibits a particularly high work function. Even then it was necessary to compensate for the missing potential energy by the kinetic energy of the incident hyperthermal molecular beam [155]. Since 'hyperthermal' is tantamount to saying 'very fast' this technique is not well adapted to macromolecule quantum experiments with long de Broglie wavelengths.

The typical photoionization cross-sections of clusters and molecules up to 10 000 amu vary between 10⁻¹⁸ and 10⁻¹⁶ cm² [149], i.e. about one hundred times smaller than that for electron impact ionization. In spite of that, photoionization is usually substantially more efficient. The reduced cross section can be overcompensated by the available photon flux, and photoionization is generally softer since it deposits less excess energy in the ionized particles.

Single-photon ionization (SPI) can be applied to organic molecules and clusters even above 10 000 amu [103, 150] and it has also been proposed [66, 67] and implemented [87] as the basis for a universal measurement-induced diffraction grating for matter waves.

If the energy of a single photon is not enough to overcome the ionization potential, a resonant or non-resonant two- (2PI) or multi-photon (MUPI) process may be used. Resonant MUPI can succeed if subsequent absorption events occur faster than the competing internal relaxation processes. In that case, the absorbed energy just barely exceeds the ionization potential and MUPI can also be a soft ionization scheme [156–158].

Multi-photon thermal ionization has been studied for highly stable molecules, such as the fullerenes C₆₀ and C₇₀ [159, 160]. When they are internally heated to energies between 50 and 200 eV several competing processes may be observed [161, 162]: the emission of thermal radiation, the emission of electrons and finally fragmentation.

The rotational and vibrational modes may couple energy to the electronic states and, given the thermo-statistical nature of the energy randomization, ionization may be delayed by many microseconds after the exciting laser pulse.

Delayed ionization was an enabling technique for the first fullerene diffraction experiments [32] where the total ionization efficiency was estimated to be above 10% and the spatial resolution was fixed to a few micrometres, as determined by the waist of the laser beam [163].

When a positive electric potential is applied to a tip as sharp as 10 nm, it is possible to create a local electric field of up to 10¹⁰ V m⁻¹. A molecule in the vicinity of the needle will be polarized and attracted to the tip. The internal states are then shifted to the point that an electron can tunnel between the molecule and the surface. The molecular ions can again be counted [164]. Since field ionization is a soft process it is also applicable to complex molecules [144]. However, the effective interaction region around the tip is limited to 200 nm [85], which is both an asset and a drawback: it can be used to image a molecular beam distribution with high resolution, but it also takes a long time to scan a two-dimensional (2D) distribution. When the interaction zone is reduced to the size of a single atom [165] the emitter also becomes promising as a coherent source for ion interferometry [3].

Field ionization may also be interesting for studying Rydberg molecules, since the required electric fields can be very low for such highly excited molecules [136]. The combination of molecular Rydberg physics with matter-wave interferometry is still open for future research.

For particles with masses up to about 10⁴ amu the overall detection efficiency is dominated by the ionization process. At higher masses the subsequent secondary electron yield η_SE becomes an additional factor. Depending on the particle species and the overall energy, η_SE ∝ vⁿ grows with a power law in the ion velocity v, where n = 4 is typically valid for velocities up to 10⁴ m s⁻¹ [166]. Efficient detection, therefore, requires acceleration voltages of several 10 kV for masses beyond 10⁵ amu.

While commercial mass spectrometers are equipped with secondary electron multipliers to register energetic ions, high-mass ions may also be counted using image charge detectors. Their current sensitivity of 1–10 fA sets a detection limit of about 10⁴ elementary charges per detector bin and second.

Small and cold molecules with narrow optical resonances and strong absorption lines are routinely detected via optical ionization, absorption or fluorescence in free flight. The access to specific internal states allows one to couple and even entangle internal and external degrees of freedom. This has been exploited in molecular Ramsey Bordé interferometry [48].

Free-flight optical detection is, however, hampered for complex molecules since they lack closed fluorescent transitions. Only in selected cases the short light–matter interaction time will be sufficient to capture sufficient fluorescence to detect a single particle. Also, the repetitive deposition of Stokes energy in the particle can lead to its rapid heating and damage. Such limitations are overcome when the molecule is bound to a solid substrate. Fluorescence can be collected over extended times and the substrate stabilizes the molecule by dissipating the Stokes energy.

Scanning tunnelling microscopy is known to excel in imaging single atoms and molecules [168–171]. Given its high spatial resolution and the possibility of post-processing surface-deposited interference patterns, this technique is also suitable for exploring quantum-assisted nanolithography.

In molecule interferometry it has been applied as a fullerene detector, specifically for C₆₀ molecules. Figure 2 shows an atomically resolved, flat silicon 7 × 7 (1 1 1) surface reconstruction, which provides 19 dangling bonds per unit cell and therefore plenty of possibilities for each fullerene to bind in the vicinity of where it lands on the surface, even at room temperature [172]. The surface physics involved in this process has been studied in detail by others with an interest in doped fullerenes as quantum logic elements [173]. The specific preparation of a reconstructed silicon surface requires ultra-high vacuum conditions around 10⁻¹⁰ mbar. This is a challenge in combination with molecular beams that may even be initially created at the mbar pressure level.

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STM imaging offers the highest possible spatial resolution that is physically meaningful for the purpose of macromolecule interferometry, but a single image scan may take 45 min for an area as small as 2 µm².

Fluorescence microscopy provides an excellent tool to discriminate molecules of known absorption and emission bands from their surroundings. Single-molecule fluorescence imaging was introduced about 20 years ago [174, 175] and it has found numerous applications in physics, chemistry and biology ever since. Recently, it has been used to record the build-up of a quantum interference pattern from individually arriving phthalocyanine molecules in real time (see figure 3) [41].

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Fluorescence microscopy [176, 177] can be directly interfaced to matter-wave interferometry when the molecular fringe pattern is deposited on a quartz window that closes the vacuum chamber. All optics can be mounted in air.

One may object that the spatial resolution of tunnelling microscopy exceeds that of optical microscopy by two or three orders of magnitude, because of Abbé's diffraction limit for optical imaging devices. In recent years, however, several methods have been shown to overcome these limits in optical microscopy [178–183]. They rely on the selective isolation of molecules by bleaching or on non-linear optical processes and they might eventually enable the detection of molecular near-field interference patterns with features much smaller than the wavelength of light, down to the level of a few nanometres.

With a fixed optical resolution, the position accuracy in the measurement of each individual molecule can be enormously increased when the molecular surface density is low—which is even desired in single-particle interferometry [184]. If two neighbouring molecules are further apart than the width of the point spread function (PSF) of the imaging system, one can fit the PSF to the detected signal for each molecule to determine its centre. The ultimate accuracy is determined by the signal-to-noise ratio in the image. Values as tiny as 1.5 nm [184] and 10 nm [41] have been achieved in recent experiments.

Optical microscopy hinges on molecular parameters as well as the experimental design. Dye molecules, such as for instance phthalocyanines and rhodamines are particularly well suited for that purpose, since they combine high optical absorption cross-sections and a high fluorescence quantum yield with long bleaching times. Larger molecules can be tailor-made and this enabled far-field diffraction with the most massive molecules to date.

Similarly, almost all proteins and other biomolecules can be labelled with a wide range of dye markers, if they are not already fluorescent, such as for instance the green fluorescent protein GFP [185]. Many other nanoparticles, in particular quantum dots, also show strong fluorescence.

Recent studies have shown that even non-fluorescent nanoparticles can be optically detected either by their stimulated emission [186], their absorption [187–189] or a change in the refractive index [191]. In order to reach a sufficient signal-to-noise ratio, these methods need a tightly focused and scanning probe beam, which increases the recording time when a wide area needs to be imaged.

Neutral molecules could also be detected by cryogenic bolometers. While these are not sensitive to the molecular mass, they are sensitive to energy. This can be used in experiments, where the final internal energy is correlated with the de Broglie phase accumulated throughout the interferometer. The detection principle would be based on the transfer of energy from internal excitations to the detector. When the detector is cooled to its superconducting state the transferred energy can trigger the phase transition to the normal conducting state, which is detected as an increase in the electronic resistance. Already in the early days of molecular coherence, with SF₆ molecules in a Ramsey–Bordé interferometer, the output was monitored by recording the energy increase that resulted from a resonant laser excitation of a specific vibrational state in one arm of the interferometer [48]. In future experiments, this idea may be extended to a large class of molecules when it becomes possible to refine the detector sensitivity to the level of single particles. This is expected to be achievable with superconducting single-particle detectors (SSPD) that have emerged over recent years [190].

A key challenge here is to keep the detector sufficiently clean throughout every experiment to avoid energy dissipation in a cushion of surface adsorbates [192].

Since quantum reflection is a peculiarity of small and less-polarizable particles, transmission line gratings have been regarded as more generic and essential in a large number of studies with molecules. SiN_x gratings are technologically particularly interesting since they remain pre-stressed and stretched even when they are pierced with holes of 50–100 nm width in periods of 100–300 nm [32, 42, 50, 51]. Far-field diffraction at a single grating is tolerant to small deviations from perfect periodicity and the required nanomask can readily be written using focused ion beam machines [41]. The concatenation of three gratings to a full interferometer requires additionally extraordinary reproducibility and precision between independent masks. This is especially important for wide-beam Talbot–Lau interferometers [51], where the average periodicity error needs to be smaller than 1 Å [65, 83]. Photolithography can provide large masks with a precision of Δd/d ≃ 10⁻⁵ in the grating period. This feat can actually even be achieved for gratings with a period down to 100 nm, using achromatic interference lithography [203].

Textbooks teach us that far-field interference peaks appear at sin β_n = nλ_dB/d, where β is the angle to the optical axis. Near-field physics is usually less well covered in books but equally important for a number of coherence experiments.

In order to visualize the transition between both regimes we plot the evolution of a plane wave diffracted at a grating in figure 8. The vertical axis represents the decadic logarithm of the scaled distance behind the gating, where L/L_T is measured in units of the Talbot length L_T ≡ d²/λ_dB. The horizontal axis is linear in the scaled position $x/\bar{x}$, where $\bar{x} = \lambda_{\rm dB}L/d +N\lambda_{\rm dB}L_{\rm T}/{6\rmd}$. In the far-field (L ≫ L_T) the scaled position corresponds to the diffraction angle in units of β₁, i.e. the angle to the first diffraction maximum. In the near-field, i.e. for L < L_T, $\bar{x}$ reduces to Nd/6L, which is proportional to the width of the grating.

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The scaling in figure 8 allows us to follow the wave evolution from the emergence of the Talbot carpets in the near-field (top) [204, 205] to the domain of far-field diffraction with well-resolved side peaks (bottom).

An interesting application of de Broglie interference in physical chemistry is to prove the existence of the weakly bound He₂ through a mass assignment of its molecular diffraction peaks [50].

The helium dimer usually escapes mass spectrometry as it is bound with an energy of about 1.1 mK (100 neV) [206], which is too small to survive impact ionization with typical electron energies in the range 60–100 eV. The de Broglie relation, however, allows one to deduce the mass of a particle from its diffraction angle at a given nanostructure when its velocity is known (figure 9). Supersonic expansion of helium from a small nozzle leads to a fast and well-collimated beam, which is sufficiently cold and dense to form helium dimers and clusters. Transverse coherence can be established by tight collimation, longitudinal coherence by the intrinsic velocity compression during the adiabatic expansion of the noble gas jet. It is interesting to see that diffraction at a mechanical mask is sufficiently non-invasive to leave the dimers intact during the scattering process, in spite of the fact that the kinetic energy exceeds the intra-molecular binding energy by orders of magnitude. The particles are thus first sorted by quantum interference and then detected using electron ionization quadrupole mass spectrometry.

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Far-field diffraction was also used to observe the quantum behaviour of supersonically expanded Na₂ [51] and later also of clusters up to about He₁₀₀. This also facilitated, in particular, the investigation of 'magic clusters', i.e. clusters of particular structural stability [56, 207] and bound HeH₂ halo molecules [208].

All experiments described in this report are compatible with the rules of quantum mechanics. While this is a rather satisfying fact it is still in marked discrepancy with the observation that we do not see quantum superposition in our daily lives. Many reasons can be invoked for this phenomenon: the kinematic explanation of non-interference is inherent to quantum physics and points to the fact that the de Broglie wavelength of macroscopic objects is just way too small for us to have hope to ever resolve it. Equally well rooted in quantum mechanics are the predictions of decoherence theory [20, 21, 209–212], which points to the fact that quantum observations are usually made on isolated systems. Once the individual system is coupled to a complex environment, quantum information diffuses into a large ensemble and becomes essentially unobservable in the partial subsystem. Genuine decoherence is based on quantum entanglement between systems. Quantum physics itself thus ensures that certain quantum phenomena become unobservable on a large scale. Also experiments with molecular matter waves corroborate the presence and significance of this effect [22, 70].

This also motivates a growing set of quantum interference experiments with massive and complex molecules, clusters and nanoparticles [1, 33], which was experimentally initiated with a demonstration of quantum interference with C₆₀ and C₇₀ [32] (see figure 10).

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When teaching quantum interference to students we usually discuss the deterministic fringe pattern that is formed from individually and stochastically arriving quanta on a screen at some distance behind a diffraction grating. This aspect of the wave–particle duality has recently been visualized with molecules in two experiments, where an interference pattern was first deposited onto a substrate and then imaged using either fluorescence [41] or tunnelling [42] microscopy. They both offer single-molecule sensitivity and a spatial resolution down to the molecular level in two dimensions.

Figure 11 shows the quantum fringe pattern that was recorded with phthalocyanine molecules after diffraction at a SiN_x mask with a period of 100 nm and a membrane thickness as small as 10 nm. Due to the gravitational velocity selection applied in these experiments, slower molecules are deposited further down on the detection screen. Their longer de Broglie wavelength implies larger diffraction angles and therefore larger separations between the interference orders than those observed for the fast molecules, which arrive further up on the screen. An area of 400 × 400 µm² can be imaged within a few seconds, orders of magnitude faster than using STM or AFM techniques. A sequence of images recorded at different deposition times can show how the deterministic fringe pattern is built-up from individual and randomly arriving molecules.

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Material gratings are often regarded as binary transmission masks, i.e. as purely absorptive structures. However, due to the van der Waals interaction of each polarizable molecule with the grating bars we also have to consider an additional phase contribution [41, 83]. Quantum physics causes that every polarizable particle undergoes stochastic charge fluctuations, typically most pronounced on the femtosecond time scale. They induce image charges in near-by surfaces such as the side wall of a diffraction slit. The resulting attraction to the wall has three consequences: first, it removes particles if they get too close to the wall within a cut-off distance x_c. Second, it dephases the matter wave, the stronger the closer the molecule approaches its image charge. Third, this phenomenon is also dispersive. Slow molecules acquire a stronger dephasing than fast ones. While a correct description of the van der Waals forces requires a detailed knowledge of the frequency spectrum both of the molecule and the surface, we estimate the cut-off distance based on the computationally simpler Casimir–Polder potential, V_CP = −C₄/r⁴. One finds x_c = (18C₄b²/mv²)^1/6 to be of the order of 20 nm for gratings with a thickness of b = 100 nm and molecules with a static polarizability of the order of $\alpha \simeq 100\,{\AA}^{3}\times 4\pi \varepsilon_{0}$, a typical mass of 500–700 amu and a velocity around 150 m s⁻¹ [83]. The potential V_CP generally scales with the fourth inverse power of the distance or length parameter r, while the interaction strength is defined by the constant C₄. More information about the construction of the interaction constant can be found elsewhere [213].

These effects have been investigated in great detail in diffraction [214, 215] and Mach–Zehnder interference [216] of atoms and small molecules and they turned out to reduce the effective open slit width in the diffraction of fullerenes by 50%, for grating slits with an initial width of 50 nm [32, 52].

This indicates that the Casimir–Polder approximation overestimates the cut-off x_c.

To minimize these effects, recent quantum diffraction experiments with phthalocyanines were performed with gratings of only 10 nm thickness. And yet the van der Waals forces are still substantial, as also seen in figure 11. Further experiments will, therefore, be needed to explore the extreme limit of material gratings, which shall finally even enable quantum diffraction at graphene, i.e. a membrane whose thickness measures only a single atom.

An optical absorption grating is based on the spatially periodic depletion of the molecular beam. It thus represents the idea of a measurement-induced grating [217] based on the selection of positions at which the molecules are removed from the beam—either by extracting them or by pumping them into non-detectable states [218].

Various molecules undergo SPI at photon energies beyond 7.9 eV, the energy of a VUV photon at 157 nm. One can achieve high ionization probability at the antinodes of a standing light wave and full transmission of neutral particles at its nodes. Since ions can be easily extracted by electric fields an optical ionization grating is effectively absorptive [66, 68].

Molecular absorption cross-sections between 10⁻¹⁸ and 10⁻¹⁶ cm² necessitate the use of pulsed lasers with energies in the range of 1 mJ or very high-power (cavity-enhanced) continuous lasers. Pulsed lasers are advantageous as they also enable the realization of interferometers in the time domain [67, 68]. In practice, the molecules will not only experience a spatially modulated transmission but also a position-dependent phase shift, which is determined by the optical dipole force.

The success of atom optics throughout the last decades was strongly driven by methods to manipulate the internal and external degrees of freedom with tailored laser light [219]. In contrast to atoms, with resonant absorption cross-sections as high as $\lambda_{{\rm dB}}^{2}/2\pi \simeq 10^{-9}\,{\rm cm}^{2}$ and strong optical polarizabilities close to narrow resonance lines, molecular line widths can extend to dozens of nanometres. Molecular absorption cross sections therefore typically range between 10⁻¹⁸ and 10⁻¹⁵ cm².

Even through there is no substantial resonance enhancement of dipole forces in warm and complex molecules, recent experiments have successfully utilized the coherent interaction between a non-resonant standing light field with a molecular optical polarizability α_opt: the optical field E induces an oscillating dipole moment, which interacts again with the optical field. The resulting force F = −α_opt∇E²/2 depends on the intensity gradient.

In atomic physics this force can be attractive or repulsive, depending on the relative detuning between the laser frequency and the internal resonance line. While this choice is still possible in diatomic molecules, most complex compounds studied so far were effective high-field seekers, i.e. attracted towards the field maximum.

When laser light is retro-reflected at a smooth mirror surface it generates a standing wave with a periodicity of λ_L/2 whose periodicity and contrast depends on about half the coherence length of the incident laser beam. This may extend to thousands of kilometres for narrow-band continuous wave (cw) lasers and it is limited to several millimetres for pulsed excimer lasers, in practice.

When a spatially extended molecular matter wave falls onto a dipole phase grating, the incident molecules will leave the interaction zone in a superposition of transverse momenta with p_o = p_in + Δp and ${\Delta}p=2{\rm n}\cdot \hslash k$, where n ∈ N. The relative weight of the nth momentum states is determined by the Bessel function J_n(φ), where the modulation index φ ∝ P_Lτ/w_xw_y is proportional to the laser power P_L, its beam waists w_x and w_y and the molecular transit time τ ∝ 1/v_long.

Since the molecular lines are broad and the interaction cross-sections are small, intense and well-focused laser beams are required to induce a significant dipole force. At the same time, it is important to avoid real absorption processes, which would be accompanied by three effects that are usually undesirable.

First, photon absorption can be followed by partial non-radiative internal conversion of its energy into excitations of vibrational and rotational states. A repeating sequence of many such processes may eventually heat the molecule to a degree that it fragments.

Second, photon absorption may be followed by spontaneous reemission and momentum recoil whose direction varies stochastically from event to event. Integration over many molecules and many recoils then results in a smearing of the interference pattern.

Third, it is interesting to realize that absorption from a coherent laser wave does not induce decoherence, by itself. Removal of a photon from a coherent beam does not leave any information behind and all phases are conserved. Pictorially, the interferogram is only laterally shifted by a fixed amount, which is related to the recoil of a single photon. Since each photon is in a superposition of two momentum states—being incident onto the mirror and being reflected—the absorption of a single photon is by itself a coherent beam splitting process with momentum transfer $\Delta p=\pm \hslash k$ (see [220] for single-photon emission close to a mirror).

The internal heating associated with the absorption and internal storage of a photon does not deteriorate the de Broglie interference pattern as long as it does not invoke any photo-activated or thermal emission process, which could release 'which-path' information to the environment. Even in spite of its coherent nature absorption will reduce the interference contrast, since the diffraction peaks related to the dipole phase grating are separated by $\Delta p=2n\cdot \hslash k$ and will be interlaced with those related to the single-photon absorption peaks at $\Delta p=\hslash k$ or even higher (odd) orders at high laser intensities. This relates molecular diffraction at optical gratings to studies of quantum random walks as well [221].

In [60] a green laser beam was reflected from a mirror to form a standing light wave with a period of 257 nm at which a coherent beam of fullerenes C₆₀ was successfully diffracted, as shown in figure 12.

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The interaction time of molecules with a cw optical grating is velocity-dependent (dispersive) and the phase shift reads $\Delta\Phi \propto\alpha_{{\rm opt}}E^{2}(xyz)/v\hslash$. Even though the spatial phase gradient is much weaker than that in the case of van der Waals interactions it is still necessary to pre-select a velocity class to maintain a high interference contrast.

The advantage of light gratings is particularly important in quantum interference experiments with highly polarizable molecules. Light gratings were therefore also integrated into KDTL [65, 222] and OTIMA [68] interferometry.

Mach–Zehnder interferometry with atoms and molecules was pioneering in matter-wave physics but also showed the challenges for using rare materials such as macromolecules. In the absence of coherent sources, far-field experiments require the preparation of spatial coherence, and thus in practice tight collimation. This is acceptable for intense atomic beams but it represents an almost insurmountable obstacle for macromolecular quantum optics—until new cooling schemes are developed. In the meantime, molecular quantum delocalization experiments can also be pursued in a near-field matter-wave interferometer.

The Talbot–Lau concept was already known from optics [204, 230] and it has been realized for the first time with atoms by Clauser [61] who also suggested that this idea would be a key to super-massive interferometry [62].

The implementation shown in figure 15 refers to a series of experiments with macromolecules performed in Vienna [63]. Molecules may arrive at the first transmission mask G₁ without any initial spatial coherence. The comb of narrow slits pre-selects the possible starting points for the subsequent matter waves to evolve. All slits act individually as coherence-preparing elements. The tight localization in each slit of G₁ imposes sufficient quantum uncertainty in the momentum of the particles to establish the transverse delocalization and coherence that is required to illuminate more than two neighbouring slits in the second grating. Quantum near-field interference then generates a particle density pattern at the position of G₃, which is a self-image of G₂ [62, 64, 231], as also seen in figure 8. If G₃ is scanned across the interference pattern, the totally transmitted molecular flux will give a sine-like signal.

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The idea of TLI is nowadays used for verifying the wave–particle duality of massive molecules, for quantum-enhanced lithography and quantum-enhanced molecule metrology.

The key advantage of TLI over MZI is its high multiplexing capability. Without the need of collimation one can use broader molecular beams. The signal enhancement of three to five orders of magnitude was the enabling step for a multitude of novel experiments. At the same time, the scaling behaviour of near-field optics is favourable for the very short de Broglie wavelengths of very massive particles.

The Talbot–Lau setup is based on the idea that coherent self-imaging may occur when the diffraction orders of neighbouring slits overlap at the same positions on the screen. While the diffraction pattern originating from each individual starting point in G₁ is a pure wave phenomenon, the mutual overlap between neighbouring interference patterns is a consequence of the experimental geometry.

This resonance condition is fulfilled if the gratings are positioned at the Talbot distance L_T = d²/λ_dB, with d the grating period and λ_dB the de Broglie wavelength. It is also fulfilled at multiples of the Talbot length and it even leads to (weaker) fringes with a smaller spatial period at rational fractions of L_T [204, 232].

The Talbot criterion implies that for a given experimental dimension—for instance limited by the lab size—the grating period only shrinks as $d\propto \sqrt \lambda_{{\rm dB}}$. Two molecular species that travel at the same speed and differ in mass by a factor of 100 require only a ratio of 10 in their mask periods. For high-mass interferometry this is a key advantage compared with far-field interferometry where $d\propto \sqrt \lambda_{{\rm dB}}$.

Over the years, different requirements led to a number of variations of this basic layout: first, it turns out that massive things are usually also highly polarizable. The dephasing van der Waals force that influenced earlier molecular far-field experiments is already detrimental for near-field interferometry with molecules in the range of 10 000 amu and grating openings of several dozen nanometres [65]. This is the reason why it proved necessary and successful to replace the central grating G₂ in figure 15 by an optical phase grating. While the coherence preparation and fringe selection still have to be performed by absorptive gratings, here realized by the material masks G₁ and G₃, the diffraction is then caused by the Kapitza–Dirac effect, i.e. the interaction of delocalized matter with a non-resonant standing light wave. Originally proposed by Kapitza and Dirac for electrons [233] the effect of phase gratings was first seen for atoms [234], then with electrons [235] and molecules [60] in far-field diffraction. In combination with the near-field idea of the TLI it becomes the Kapitza–Dirac–Talbot–Lau interferometer (KDTLI) [65].

While the KDTLI interferometer still integrates the position-selective nanomechanical gratings G₁ and G₃ one can replace them also by absorptive structures of light. They can, for instance, be achieved when the photon energy exceeds the molecular ionization energy [66]. An interferometer based on three (typically ultraviolet) ionizing laser gratings can be operated both in space and in the time domain [67, 87]. An optical time-domain ionizing (OTIMA) interferometer is universal in the sense that it can be applied to a large variety of different particles, from single atoms to clusters of molecules. The quantum state in such interferometers may in the future also be retrieved by using state tomography [234].

Molecular TLI as shown in figure 15 intrinsically generates a nanosized molecular density pattern at a well-defined position behind the second grating. While in many experiments this pattern is sampled and visualized by scanning the third grating G₃ across the molecular beam, it is also possible to capture the interference fringes on a screen to generate a surface-bound molecular nanostructure [42]. This idea has been pursued in a recent matter-wave experiment with C₆₀ molecules, where STM imaging was used to visualize the resulting interference pattern after its deposition on a reconstructed Si (1 1 1) 7 × 7 surface. Such images also permit us to visualize the quantum wave–particle duality. In figure 16, every single spot represents a detected C₆₀ molecule and although there is no way to predict a priori the position that any of them will have, the vertical ensemble average (bottom) shows the expected highly regular fringe pattern. And even though one would also expect a classical fringe shadow behind a two-grating arrangement [232], the expected and observed quantum fringe contrast are significantly higher—even in the presence of van der Waals forces. Quantum-enhanced molecule lithography may become relevant in nanotechnology as a positive and soft non-contact deposition technique for creating surface modifications on the nanoscale and with deposition energies as small as 0.1 eV.

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Spatially highly resolving imaging methods, such as scanning tunnelling or single-molecule fluorescence microscopy require the suppression of molecular surface diffusion to less than a few ten nanometres. This is not granted for all molecule–surface combinations. But even then, position-encoded fluorescence detection can still be implemented using a mechanical scheme that magnifies the quantum fringe pattern to the extent that even sizeable surface diffusion can be completely neglected [237].

In a three-grating interferometer of the Talbot–Lau type (figure 17), the third grating, G₃, can be regarded as part of the detector. It is scanned in the direction of the grating vector $\vec{k}=\vec{e}_{x}\cdot {2\pi}/d$, i.e. normal to the molecular beam and normal to the extension of the diffraction slits. For each position, the transmitted molecular flux is then recorded.

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In a recent experiment this method was combined with fluorescence imaging [237] of molecules that were accumulated on a quartz substrate mounted onto a translation stage behind G₃. For each position shift of the third grating Δx₃, the quartz plate was shifted by ∼4300Δx₃. This trick magnifies the molecular interference pattern mechanically to an extent that it can be easily analysed and that molecular diffusion on the detection plate can be completely neglected. When using gravitational velocity selection, a 2D image, such as shown in figure 17 allows visualizing the velocity-dependent near-field matter-wave interferogram.

A key distinguishing element in this comparison is the role of quantum information in the transition between observable quantum coherences and classical probabilities. Decoherence theory [21, 211, 212, 239–242] is applied quantum mechanics and it describes the non-observability of coherence in a system as a result of the coupling to its complex physical environment: The exchange of quantum information, the diffusion of coherence into the environment by virtue of entangling interactions or 'measurements'—even without the need for invoking any human observer—is fully described by established quantum theory. It reduces quantum coherence effectively if we neglect to include all partners of the extended many-body system.

Understanding quantum decoherence is also central to the overall enterprise that is currently being pursued under the heading of quantum information technologies. Only if we know and master the effects that destroy quantum phenomena we can also take measures to extend coherence to an extent that enables us to build quantum-based sensors or information processing devices on the mesoscopic scale.

A number of matter-wave experiments were able to corroborate the ideas of decoherence theory: when particles are excited, such that each of them spontaneously emits a photon while they are delocalized inside a MZI (see figure 14), the overall quantum fringe contrast is reduced, if the emitted photon wavelength suffices to yield which-path information, i.e. information about where the particle was [243].

The 'Copenhagen complementarity' interpretation, which states that path information and quantum interference exclude each other, can nowadays be rephrased in terms of quantum information theory: information can only be extracted by a physical particle. If this scattering partner becomes quantum entangled with the originally delocalized nano-object, quantum coherence is diluted into the more complex environment.

In pure de Broglie interference, i.e. experiments which only consider the centre-of-mass motion, this information theoretical view can finally be complemented by a third perspective, namely the analogy to 'Heisenberg's microscope': when there is no coupling to the internal states, the only way to extract position information is momentum exchange. In that case, random kicks by the interacting environment can dephase each individual delocalized molecule differently to a degree that the coherence of the ensemble is finally lost. This effect depends on the separation of the wave packets as well as on the momentum exchange with the probing radiation [244].

In TLI with C₆₀ decoherence was observed for the first time both caused by collisions with residual gases in the vacuum chamber [70, 245] and by thermal radiation emitted by the molecules themselves [22, 246]. Decoherence is also believed to be an important reason why molecules appear with a certain chirality in nature, instead of the energetically favoured superposition of left- and right-handed configurations [247].

The only way to circumvent decoherence is to isolate a particle from all external agents. In practice, this imposes rather stringent requirements on the background vacuum or temperature [35, 74–76]. Extrapolations of particle interferometry with masses around 10¹⁰ amu are conceptually still compatible with existing technologies, demanding as they may be in practice [35, 74]. This will require cooling of the entire experiment to the temperature of liquid helium in order to reduce the base pressure and the level of black body radiation to a level acceptable for high-mass interference [248].

In contrast to decoherence, phase averaging may appear trivial: any advanced quantum experiment may suffer from many external perturbations, which do not invoke quantum information arguments and yet they impose very relevant boundary conditions to high-mass interference experiments. Some of the phase averaging mechanisms may even be related to fundamental physics from outside of quantum mechanics, such as gravitational wave background noise [249] or space–time fluctuations [250].

However, most causes for phase averaging are related to a much more basic and lab-oriented level. This includes dispersive effects in interferometry with molecular beams of finite velocity spread: the Coriolis-force on the Earth, the particle's free fall in the Earth's gravitational field, the van der Waals interaction with solid surfaces or the deflection in an inhomogeneous electric or magnetic field may all influence the phase of the quantum fringe very much in the same way as they shift a classical particle beam. Also different modes of interferometer vibrations will reduce the quantum contrast, without extracting quantum information [109].

The OTIMA concept [67, 68] eliminates a large number of dispersive effects and is therefore particularly promising for this kind of macro-interference. In addition, it starts from spatially and spectrally incoherent sources, which is a tremendous benefit, given the difficulties in preparing large ensembles of ultracold and similar nanoparticles. Also single-grating experiments become feasible again, once ultracold nanoparticle sources become available.

In neither one of the two above arguments the superposition principle is ever broken. Conceptually, this leads either to an ever-growing entanglement in the universe or a many-world interpretation of quantum mechanics [251].

This is the reason why a number of proposals have been made in the attempt to establish 'reality', i.e. physical states without the superposition of classically mutually exclusive phenomena. This has for instance been done by adding non-linearities to the Schrödinger equation [17, 19, 252] or by exploring the relation of quantum mechanics in combination with gravity theory [18, 36, 37].

This summarizes to a picture where quantum mechanics, at present, is the best-confirmed theory of non-relativistic physics—and even better tested in its relativistic form of quantum electrodynamics—but where a growing number of models emerge to explore potential limits and modifications of the superposition principle for very massive bodies.