Molecular double-slit experiments

At first glance, the molecular double slit seems to be simply a microscopic version of a wavefront splitter experiment. In both cases, interference is the result of coherence between two spatially separated sites that give rise to a two-slit interference pattern. These interference fringes are the signature of phase-locked two-centre emission either in space, as in the famous double-slit experiment with single electrons¹, or more recently in time^{18, 19}. This is also the case in the molecule-frame photoelectron emission pattern of the gerade (g) state of a diatomic homonuclear molecule such as N₂, where two-centre interference causes a characteristic oscillation of the electron emission intensity along the molecular axis in momentum space. The ungerade (u) state with its phase shift of between the two emitters also has an equivalent in macroscopic double-slit experiments, either with a tilted incoming wave or, more recently, via entanglement of the slits through the polarization properties of the photons²⁰. The result is a pattern of antifringes, where the dark and the bright stripes are interchanged²¹. This corresponds to a parity change in the ungerade electron emission pattern of the molecular double-slit experiment with its phase shift of in the momentum space oscillation. We identified these oscillations by measuring the photoelectron diffraction intensities using angle-resolved high-resolution photoelectron–fragment ion coincidence spectroscopy. Furthermore, we compared the case of core-electron photoemission in the homonuclear molecule N₂ with that in a heteronuclear diatomic molecule such as CO, which may be considered as a double-slit experiment with only one of the two slits open. In the photoemission experiment, the position of the open slit is selected by the choice of the appropriate kinetic energy of the emitted electron, that is, either the CO: C(1s) or O(1s) photoemission line from the carbon or oxygen site of the CO molecule, respectively.

At this point of comparison between the different double-slit experiments, the question of the limitations on the equivalence between molecular double-slit and macroscopic Young-type double-slit experiments arises. Here, we show that there are indeed distinct differences. Macroscopic wavefront splitting by a directionally fixed-in-space system of physically separated slits causes either pure spatial interference patterns or smeared-out slit projections, but does not include feedback between the two emission sites. The molecular double-slit experiment, however, differs in this respect. Here, photoelectron scattering or diffraction, a coherence effect in momentum space widely exploited in surface science^{22, 23}, acts as an extra source of interfering waves and governs the angle-dependent photoelectron intensity in the molecule frame^{24, 25, 26}.

To explore this intriguing feedback effect of interfering waves on the scattering, we carried out photoelectron diffraction experiments on the isoelectronic hetero- and homonuclear molecules, CO and N₂, in the gas phase. Figure 1 shows a schematic diagram of the experimental set-up for CO. It is an angle-resolved high-resolution electron–fragment ion coincidence experiment²⁷ in which the electrons are detected by an array of time-of-flight spectrometers in a circle shown by the solid arrows. The mass- and angle-resolved detection of the fragment ions is carried out by a time-of-flight ion spectrometer with a position-sensitive anode, which makes it possible to determine the momentum of the fragment ions within a chosen acceptance angle of 25^° here. Measuring electrons and ions in coincidence in a gas-phase experiment is equivalent to angle-resolved photoelectron spectrometry on free, oriented molecules. The analysis of the experiment hinges on the so-called axial recoil approximation, which assumes that fragmentation of the molecule occurs on a timescale much faster than rotation. The emitter atom can be selected by the kinetic energy of the photoelectron. The direction of its emission either into the molecule (e_in), which is towards the neighbouring atom, or out of the molecule (e_out), is determined by coincidence with an ion travelling either along with or counterparallel to the electron. The two directions are also referred to as forward (e_in) and backward (e_out) scattering in the terminology of photoelectron diffraction. For both emission directions, the scattered electron wave is superposed on the directly emitted electron wave, giving rise to pronounced oscillations in the emission intensities along the molecular axis, as shown in Fig. 2. The two forward (f) and backward (b) states that define the emission direction and therefore the emitter position are in the case of CO energetically non-degenerate and hence not entangled (for further details, see the Supplementary Information).

Figure 1: Schematic diagram of the gas-phase photoelectron diffraction experiment on CO.

The experimental configuration chosen for recording the data reduces the experiment to a one-dimensional scattering experiment along the molecular axis and electric vector of the ionizing radiation. The orientation of the molecular axis and the emission direction of the electron are determined by the coincidence of the photoelectron with the ionic fragments of the subsequently dissociating CO molecule. The photoelectron angular distributions in the molecular frame are shown for both emitter sites on their corresponding screens. The shape of the patterns is asymmetric peanut-like (data from ref. 25) for both emitter sites (C and O), but exhibits an inverted pattern structure for C and O photoelectron lines, with respect to each other.

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Figure 2: Photoelectron diffraction intensities versus electron momentum k and de Broglie wavelength (in units of the internuclear distance R) for CO molecules in the gas phase.

Forward and backward scattering are differentiated by angle-resolved photoelectron–fragment ion coincidence. The red curves and data points represent the backward channel, the blue curves and data points the forward channel and the black curves and data points the sum of both. The solid lines are results of a partially RCHF calculation and the dotted lines are results of a semi-empirical one-centre interference model (see the text). Our experimental results include data points taken at BESSY and at HASYLAB/DESY (circles) as well as data from preliminary measurements at the Advanced Light Source (squares). The lozenges are reanalysed data from ref. 26. All diffraction intensities shown here are normalized to the corresponding partial cross-sections to remove the exponential decay behaviour, which would otherwise mask the oscillatory structure. The experimental data and their fitting curve shown in the inset schematic diagram are from ref. 25.

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Figure 2 shows the results of calculations in the partially relaxed-core Hartree–Fock approximation (RCHF) using the Lippmann–Schwinger equation^{28, 29} and the experimental data, as well as a fit of these data to a superposition of damped sine functions with continuous and discrete, bell-shape-like, damping curves (for details see the Methods section). The first damping curve describes the extended X-ray absorption fine structure (EXAFS)-type oscillations, whereas the latter represents the shape resonance components in the form of wave packets in momentum space representing quasibound high-l states at multiples of the de Broglie wavelength . The agreement between theory and experiment is surprisingly good in view of the complexity of the problem. Both theory and experiment exhibit very pronounced oscillations in the backward channel, a fingerprint of a high degree of interference between emitted and scattered electron waves. In contrast, the oscillation in the forward channel is less pronounced because double scattering is much less probable than single scattering. The data also clearly show virtually identical oscillation frequencies in both channels, but with a phase shift of between them because of the difference of one in the number of forward and backward scattering events (see Fig. 2, inset). This distinction between the behaviour of odd/even scattering events corresponds to the slit distinction by left- and right-handed circularly polarized light in the macroscopic double-slit experiment of Walborn et al.²⁰. The equivalence between the two experiments also holds for the case of homonuclear diatomic molecules where the two complementary superpositions, gerade and ungerade, corresponding to the fringes and antifringes in the Walborn experiment, are formed.

In analogy to Fig. 1 for the heteronuclear case, Fig. 3 schematically shows how the coherence properties in the homonuclear case were studied experimentally. The key instrumental issue in this case is the ability of the electron detector to resolve the gerade/ungerade splitting of the N₂⁺:N(1s) photoelectron lines^{30, 31}. Angle-resolved photoelectron–fragment-ion measurements with high kinetic energy resolution of the photoelectron are very demanding and are at the limit of current instrumentation for the photon beam and particle detectors²⁷. Our set of resolved data points supports the model of two spatially separated coherent emitters giving rise to distinct oscillations with a frequency inversely proportional to an apparent 'single' bond length. The effective dynamical phase shift is kr_s with r_s=0.37R. The oscillatory behaviour of the sum of the g and u diffraction intensities of N₂ is similar to the sum of the forward/backward contributions in the case of CO. For observation at 0^° with respect to the polarization vector and molecular axis, the frequency of this curve is proportional to 'double' bond length. This behaviour, which is puzzling at first glance, is the result of intramolecular scattering, that is, photoelectron diffraction in N₂. It exhibits spatial and momentum coherence corresponding to wave- and particle-like behaviour at the same time. To separate the intensity of the unscattered coherent electron emission from the scattered intensities, a sine function with a frequency proportional to the 'single' bond length was fitted to the data and subsequently subtracted as described below. This sine function is referred to as CF, because it is based on the Cohen–Fano model^{10, 32, 33} adapted to fixed-in-space molecules. In the following, we will refer to this combined one- and two-centre coherence model as the generalized Cohen–Fano (GCF) model (see the Methods section).

Figure 3: Schematic diagram of the experimental set-up for the high-resolution angle-resolved photoelectron–fragment ion coincidence experiment on N₂.

The experimental set-up was similar to that for CO (Fig. 1), but with an emphasis on the energy-resolved detection of the gerade and ungerade states rather than on the directional separation of forward and backward scattering events. The molecule-frame photoelectron angular distributions for the gerade and ungerade state shown on the corresponding screens are taken from ref. 16.

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Figure 4 shows the fitted and extended sine curves CF_g,u of the GCF model and the calculated g and u diffraction intensities RCHF_g,u along with the sum of the g and u curves as dashed, solid and dotted lines respectively. The raw data for this fit were derived from the various RCHF curves as follows. For k>7R^-1, CF_g,u is given by RCHF_g,u-EXAFS, with EXAFS taken as half of the intensity oscillation of the RCHF_g,u sum curve. Subtraction of extrapolations of the fitted curves to low kinetic energies from the diffraction intensities yields the scattered intensity within the GCF model GCF^sc_g,u=RCHF_g,u-CF_g,u,, which is shown in Fig. 5 along with representative data points for the different scattering regimes. Because of the ungerade character of the total wavefunction, the f-wave enhancement of the shape resonance essentially appears as an odd partial-wave effect only in the gerade channel and complicates the analysis of the coherence behaviour. We therefore choose the ungerade channel (shown in yellow in Fig. 5a) as a starting point of our analysis. In addition to the various fractional intensity curves, this figure shows schematic diagrams of the different two- and one-centre scattering processes of the GCF model (see the Methods section). The corresponding analysis indicates that, at the onset, the scattered photoelectron intensity oscillates predominantly with a 'single' bond length frequency (actually 0.63 R due to the same energy-dependent phase shift as for CO but with a potential-dependent dynamical phase shift of kr_s with r_s=0.37R), which changes to a 'double' bond length frequency at an energy around k(R^-1)=2. The 'double' bond length frequency is the frequency exhibited by the sum of the intensity curves for the gerade and ungerade channels, where the separate intensities have been modulated with an amplitude of half of their sum but with no relative phase shift with respect to each other. In contrast to the g–u state-specific modulation of the two-centre oscillations shown in Fig. 5a, this final state behaviour shows no hint of the character of either symmetry state. The phase shift of of the scattered intensity with respect to the emitter intensity oscillations in the two-centre case gives rise to a decreasing modulation in the ungerade channel for k values between 1 and 4 in units of the inverse bond lengths. This reflects the severe degradation of the Cohen–Fano emitter oscillation by an order of magnitude. In this regime, the de Broglie wavelength of the photoelectron is much larger than the bond length, and the spatial coherence between the two emitter sites is the dominating factor for both the scattered and unscattered electron to show wave-like behaviour. The -shifted superposition of the two waves removes the primary quantum interference over several oscillation periods, thereby generating an unmodulated source of constant electron emission at the inversion centre of the molecule. In analogy to the standing-wave regime³⁴, where pronounced partial-wave interference effects arise, the relevant wavelength scale for this behaviour is the bond length.

Figure 4: Photoelectron diffraction intensity versus electron kinetic energy for electron emission along the direction of the molecular axis in the homonuclear molecule N₂.

The horizontal axis notation is the same as in Fig. 2. The inset schematic diagram showing the superposition of the electron emitted from two separated spatial positions is shown along with data from ref. 16. The filled coloured circles are the experimental data for the gerade (green) and ungerade (yellow) state, and for the sum of both (black). Statistical and systematic errors are shown for the high-resolution data points at low kinetic energies and the data points analysed by two complementary methods. The latter error bars represent the bulk of the low-resolution data points. The non-scattered intensities shown by dashed lines are derived from the GCF model, which simulates the photoelectron diffraction intensities of the gerade and ungerade states by a superposition of two sine functions representing fractional intensities of the scattered (shown in Fig. 5) and non-scattered electrons, respectively (see discussion in text). These simulated GCF intensities are shown as purple dashed lines for both symmetry states; their sum is shown as a black dotted line. The normalization of the N₂ diffraction data to the cross-section was done in two ways, depending on the experimental resolution: for the high-resolution data below k=4.7R^-1, the separate _g- and _u-values were used, whereas for the data at all higher momentum values, (_g+_u)/2 was used for both diffraction components.

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Figure 5: Scattering behaviour of the N⁺₂(N(1s):g, u) photoelectrons.

a, Scattering behaviour for the g (green curves and data points) and u (yellow curves and data points) final ionic states. b, The decoherence curve (described in the Methods section). Here, the direct emission intensity (CF curve), shown in Fig. 4 as a dashed line, has been subtracted from the corresponding total diffraction curve, shown in Fig. 4 as the solid lines. Panel a shows the two-centre oscillation as thin yellow (u) and green (g) and thick (purple for both u and g) dashed curves for two different envelopes shown as thin and thick dotted lines. The thin dashed curves for u and g have a slope analogous to the one-centre curve in c, whereas the corresponding thick dashed curves are modified by the 'decoherence' curve in b with the area between the two curves being shaded. The error bars of these data points are shown by a single representative bar. c, EXAFS-like one-centre oscillation (red solid line) with an envelope marked by a red dotted line. The filled black circles represent the sum of g and u scattered intensities. The open black circles show the corresponding data points of CO derived from the diffraction intensities of Fig. 2. The red and blue dashed curves represent the fractional fitted intensities of the shape resonances in the standing-wave regime. The inset schematic diagrams illustrate the different intramolecular scattering processes schematically. The solid and dashed lines with arrows represent interfering pathways of the photoelectron. d, Diffraction curves for three different observation angles of the emitted photoelectron. Inset: The observation angle dependence of the oscillation frequency.

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Scattering-induced loss of coherence

As the de Broglie wavelength of the emitted electron becomes shorter than the bond length, the single-scattering process leads to a scattering-induced momentum change of the emitted electron larger than the inverse bond length and hence to particle-like localization at one atomic site, a process much faster than the tunnelling time. The resulting lack of overlap between the scattered electron wave and its neighbouring wave promotes their spatial localization and formation of a self-interference pattern, a behaviour closely related to the cancellation of geometrical phase effects in chemical reactions³⁵. This momentum-dependent transition, which transforms the coherent state into the non-coherent subtunnelling localized state, is equivalent to that seen in environment-coupled decoherence experiments with their exponential loss of coherence with time and path separation. Excellent examples sharing the same physical footing as Heisenberg's uncertainty principle for position and momentum are photon scattering on coherent matter waves⁸ and the temperature-dependent wavelength of radiation emitted by heated fullerenes⁹. In contrast to other 'Gedanken' decoherence experiments³⁶ that lead to left/right asymmetries due to the gain of 'which way' information in a spatially fixed system, our transition, which is described by the same type of transition curve as shown in Fig. 5b, is a transition from two-centre spatial coherence to one-centre self-interference that gives rise to site-resolving, but random localization with no appearance of time-integrated left/right asymmetries. Instead, wave-like behaviour between scattered and unscattered electrons emitted from two spatially separated points becomes particle-like scattering behaviour, resulting in position-resolving localization on the atomic scale in the sense of a Heisenberg microscope.

Electron entanglement

At this stage, more explicit definitions of the quantities involved may be helpful. Our system is one-dimensional and consists of two orthonormal basis sets, namely, the non-localized gerade (g) and ungerade (u) basis set of the parity states and the localized forward (f) and backward (b) basis set of the momentum states. These basis states are related to each other by

The sum of the two components must be the same for both sets and hence unresolved data, either for energy or momentum, do not provide any information on our problem of localization and coherence in molecular double-slit experiments. The spatially coherent g/u system (equation (1a)) is transformed into the spatially non-coherent f/b system (equation (1b)) on scattering. These f and b states, representing n and n-1 collision events, respectively, show different k-dependent behaviour due to the relative phase shift of between them as for the unscattered electrons, but with a different oscillation frequency. They are still superimposed forming g and u states but with a scattering-induced cancellation of their interference terms. In turn, for the case of the dominant backward amplitude, the probability of detecting a scattered electron in a certain direction is given by b²=1/2(g²+u²-2gu), but here the interference term vanishes owing to the orthogonality of the two non-degenerate basis states g and u. The angular distribution of the scattered electron in the molecular frame is therefore still symmetric, but the phase coupling between the two separated emitter points is diminished by the pure appearance of the f² and b² amplitudes without the phase-sensitive interference term 2fb. Hence, this scattering- and symmetry-induced situation is associated with a loss of coherence in both basis sets g/u and f/b, not anticipated by decoherence theory. The diminished interference term between f and bcauses localization, but the vanishing interference term between g and u still preserves the symmetry, because their orthogonality is based on non-degeneracy in terms of the g/u energy splitting E_gu compared with the linewidth _A of the subsequent Auger decay, and not on the de Broglie wavelength-dependent localization x>/(2p_x). The scattering channels f_s and b_s are equally strong in both of the corresponding symmetry channels g_s and u_s owing to the relationship b_s²+f_s²=g_s²+u_s². Therefore, there is no phase-specific element left in the two symmetry channels for the scattered electron, a situation corresponding to depolarization in angular momentum space. Briefly, this is based on the fact that the parity eigenstates of two spatial positions that are the basis of equal probability for emission either from the left or from the right site are non-local¹⁴. These g/u symmetry states become entangled to a singlet state built on the joint photoelectron–ion system with its ungerade symmetry. This non-local symmetry state is now transformed into our non-coherent randomly localized scattering state that is characterized by phase- and hence parity-insensitive behaviour. The singlet state of the complementary f/b system is the coherent localized state describing two antisymmetrically oscillating electron emission patterns at the individual sites, but with mirror symmetry regarding the inversion centre of the two emitter sites. This state, caused by the degenerate part of the overlapping g and u states, will be mentioned only briefly here. The corresponding Bell states have the form:

where l and r indicate the left and right position of the emitter site in the laboratory system, respectively. This situation may be compared most closely to the Walborn experiment^{20, 21} without selection of any component of the two 45^° polarizers for circular polarization behind the double slit. However, there is no tunnelling between the two slit sites. Hence, the rotating linear polarizer in the double-slit free path of the entangled photon beam projects the 'which way' information on the screen, showing fringes, left slit projections, antifringes and right slit projections. To reveal this inherent asymmetry in our case requires a second electron to be removed on a timescale similar to or smaller than the tunnelling time as a 'which way' marker in a reference system defined by the detection direction of the reference electron. In practice, this would be a subsequent Auger electron with _A<E_gu causing partial degeneracy of the symmetry states through lifetime broadening. This gives rise to a non-vanishing interference term 2gu and hence to f/b asymmetries. This degenerate f/bsystem is an entangled state regarding the continuous variable position (see equation (2)). In contrast to entanglement of discrete variables, position already exhibits interference behaviour on the level of the symmetry basis states. Their character as non-local spatial superpositions¹⁴ is clearly exhibited as an oscillation in momentum space (Fig. 4).

Both the two- and the one-centre coherence between scattered and unscattered electrons can be unambiguously distinguished by their characteristic frequencies in momentum space, which is a single fixed frequency for coherent two-centre superposition, but is angle dependent for localized one-centre self-interference. This can be clearly seen in Fig. 5 for a variety of angles for N₂ and CO. Asymptotically, the EXAFS oscillation disappears at shorter wavelength owing to the decrease of scattering probability with increasing photoelectron kinetic energy. Hence, the dominant spatial coherence pattern of the undisturbed emitter, which reflects the core-hole tunnelling, is the only coherence-driven oscillation surviving in photoelectron emission at high kinetic energies^{37, 38}. In this limit, the amplitudes of the g and u oscillations are the same.

In summary, we have presented evidence for a scattering-induced scale-dependent transition of quantum coherence in molecular core-electron photoionization. The observed effect is manifested as an oscillation with an angle-dependent frequency in momentum space that is caused by the self-interference of photoelectrons emitted from only one atomic site, the fingerprint of localization. However, this effect conserves the symmetry of the two emitter sites but without spatial phase coupling owing to local separation of the emitted electron. This scenario is the result of ongoing tunnelling after collapse of the wavefunction in real space. It exhibits a loss of coherence along with a subtle relationship between three time domains, that is, the scattering-induced localization, the tunnelling and the core-hole lifetime. The observed effects in the photoelectron diffraction of N₂ might not only contribute to a deeper understanding of the transition between the classical and the quantum world^{2, 3}, as decoherence theory, but also to more actual problems as the occurrence of single-particle non-locality¹⁴.

Note added in proof. The coherent localized state in the complementary f/b system predicted here has been confirmed by recent photoelectron–Auger electron coincidence experiments in the molecule frame³⁹.

Methods

The semiclassical model function f_model, used here to describe the states f_s and b_s of the scattered electron, is the sum of two damped sine functions f_2c^T,damp and f_1c^T,damp, where the subscripts 1c and 2c denote one- and two-centre scattering coherence, respectively. The transition function T_2c(k) shown in Fig. 5b is chosen in analogy to the decoherence curves of refs 8, 9, whereas T_1c(k) describes the onset of the one-centre scattering:

Here, R denotes the internuclear distance or bond length and kr_s and _1c,2c the energy-dependent and energy-independent phase shifts, respectively. ₀ is an empirical parameter describing the declining scattering intensity. In the text, we referred to this combined one- and two-centre coherence model of the scattered electron as the GCF model.

The RCHF code treats the symmetry basis states g and u as pure states as defined in equation (1a). The Cohen–Fano oscillation is the result of the interference terms 2fb. The EXAFS oscillation however, is an independent modulation of the intensity of the two symmetry channels g and u. It can be described by non-interfering f_s and b_s scattering channels, which appear with the same strength in the still non-interfering symmetry channels g and u.

Author contributions

The calculations were carried out by B.Z. and V.M., whereas the measurements and the analysis of the experimental data were done by the other authors.