Atoms Under the Magnifying Glass: Direct Observation of the Nodal Structures of Electronic States
Authors: Aneta Stodolna1, Marc J.J. Vrakking2
1FOM Institute AMOLF, Amsterdam, Netherlands,
2Max-Born-Institut, Berlin, Germany.
To describe the microscopic properties of matter and its interaction with the external world, quantum mechanics uses wave functions, whose structure and time dependence is governed by the Schrödinger equation. In atoms, electronic wave functions describe - among other things - charge distributions existing on length-scales that are many orders of magnitude removed from our daily experience. In physics laboratories, experimental observations of charge distributions are usually precluded by the fact that the process of taking a measurement changes a wave function and selects one of its many possible realizations. For this reason, physicists usually know the shape of charge distributions through calculations that are shown in textbooks. But in the last few years, this has started to change. Recent experiments have visualized the nodal structure of electronic states of hydrogen and helium on two-dimensional detectors.
The development of quantum mechanics in the early part of the last century had a profound influence on the way that scientists understand the world. Central to quantum mechanics is the concept of a wave function that satisfies the time-dependent Schrödinger equation. According to the Copenhagen interpretation, this wave function describes the probability of observing the outcome of measurements that are performed on a quantum mechanical system, such as measurements of the energy of the system or the position or momenta of its constituents. This allows reconciling the occurrence of non-classical phenomena on the micro-scale with manifestations and observations made on the macro-scale, which correspond to viewing one or more of countless realizations described by the wave function.
Despite the overwhelming impact on modern electronics and photonics, grasping quantum mechanics and the many possibilities that it describes continues to be intellectually challenging, and has motivated numerous experiments illustrating the intriguing predictions contained in the theory. For example, the 2012 Nobel Prize in Physics was awarded to Haroche and Wineland for their work on measurement and control of individual quantum systems in quantum non-demolition experiments, paving the way to more accurate optical clocks and, potentially, future quantum computers.
About thirty years ago, Russian theoreticians proposed an intriguing method for measuring properties of wave functions. They suggested studying atomic ionization in a static electric field that projects the electrons onto a two-dimensional detector and predicted interference patterns, with one of two possible origins. First of all, interference patterns result from path length differences between different trajectories that the electron can take between the atom and the detector. As clearly shown in the famous double-slit experiment on interference of single electrons (voted "the most beautiful physics experiment", in a poll conducted by Physicsworld about a decade ago) electrons exhibit both particle- and wave-like behavior.
The wave-like behavior derives from the de Broglie wavelength that quantum mechanics associates with any moving particle. The lower the kinetic energy of the electron, the larger the de Broglie wavelength is. Correspondingly, for low enough kinetic energies, the de Broglie wavelength becomes observable on macroscopic length scales. Secondly, in the case of hydrogen, the interference patterns can directly reflect the nodal structure of the electronic wave function. The fact that this is so, is due to the special status of hydrogen as nature´s only single-electron atom. Due to this circumstance, the hydrogen wave function can be written as the product of two functions that describe how the wave function changes as a function of two, so-called “parabolic coordinates”, which are linear combinations of the distance of the electron from the H+ nucleus “r”, and the displacement of the electron along the electric field axis “z”. Importantly, the shape of the two parabolic wave functions is independent of the strength of the static electric field, and therefore stays the same as the electron travels from the place where the ionization takes place to the two-dimensional detector.
Last year we published a paper, where we reported experiments for hydrogen atoms . Ground state hydrogen atoms were optically excited to electronic states of interest, using two precisely tunable laser sources, and a delicate electrostatic lens was used to magnify the imprint of the electrons on the two-dimensional detector to millimeter-scale dimensions, so the nodal patterns of the wave functions could be observed with the naked eye. The main result is shown in Figure 1. This figure shows raw camera data for four measurements, where the hydrogen atoms were excited to states with 0, 1, 2 and 3 nodes in the wave function for the ξ = r+z parabolic coordinate. The nodes can be easily recognized. The experimental arrangement served as a microscope, allowing us to look deep inside the hydrogen atom, with a magnification of approximately a factor twenty-thousand.
Figure 1: (left) two-dimensional projection of electrons resulting from excitation of hydrogen atoms to four electronic states labeled with a set of quantum numbers (n1,n2,m) and having (from top to bottom) 0, 1, 2 and 3 nodes in the wave function for the ξ = r+z parabolic coordinate; (right) comparison of the experimentally measured radial distributions (solid lines) with results from quantum mechanical calculations (dashed lines), illustrating that the experiment has measured the nodal structure of the quantum mechanical wave function (copyright: American Physical Society).
More recently, we have performed similar experiments for the helium atom . After the hydrogen atom, the helium atom is nature´s simplest atom, consisting of a doubly-charged nucleus surrounded by two electrons. The presence of two electrons in the atom introduces the concept of electron correlation. Remarkably, we saw that we could turn the electron correlation in helium on or off at will.
In the experiment, helium atoms were ionized by the absorption of an ultra-violet (UV) photon. Like in the hydrogen experiment, the photon energy of the UV light was tuned in such a manner that it was only just sufficient for ionization of the atom, thus producing very slow photoelectrons that were accelerated by an electric field towards a two-dimensional detector. At most of the UV photon energies, interference patterns were measured that could be explained by considering differences in the lengths of possible paths of the electron on the way to the detector (see Figure 2). Here, two paths differing by an integral number of de Broglie wavelengths interfere constructively, whereas two paths differing by a half-integer number of de Broglie wavelengths interfere destructively.
Figure 2: Sample images recorded for ionization of helium atoms. The four images contain interference patterns that result from path length differences along trajectories that the electron can take between the atom and the detector. The labels contained in the figures indicate the energy of the states used in the experiment defined with respect to the field-free ionization limit (copyright: American Physical Society).
However, at a number of UV photon energies the interference patterns looked markedly different, extending out to a much larger radius and containing a different number of nodes compared to measurements at slightly lower or higher photon energy (see Figure 3). A theoretical analysis revealed that at these energies the effect of electron correlation was momentarily suppressed. The suppression occurs when two electronic states, whose precise energies depend on the strength of the electric field, accidentally occur at almost identical energies. These two states then interact with each other, and for a particular value of the electric field, the energy exchange between the two parabolic coordinates is almost completely turned off. In other words, the atom becomes hydrogenic.
Figure 3: Sometimes helium behaves like a hydrogen atom, and interference patterns are measured that reveal the nodal structure of the electronic wave function that is excited (middle image). These cases stand out because the nodal pattern of these images is very different from those recorded at nearby excitation energies (left and right image), and the images extends farther radially. The labels contained in the figures indicate the energy of the states used in the experiment defined with respect to the field-free ionization limit (copyright: American Physical Society).
Correspondingly, the nodal pattern measured on the detector is once again the nodal pattern of the electronic state that is optically excited. The effect was found to be very subtle: tiny changes (<< 1%) in the strength of the electric field are sufficient to convert an atom that reveals the nodal pattern of its wave function in a hydrogen-like manner, into an atom where electron correlation removes the observability of this nodal pattern, and where the observed interference patterns are completely determined by path length differences between the atom and the detector.
In this manner, the hydrogen and helium atom constitute a wonderful nano-scale laboratory for studies of fundamental quantum mechanics, providing text-book images of nodal patterns in the case of hydrogen, and revealing the onset of electron correlation in the case of helium.
 Aneta Stodolna, Ymkje Huismans, Arnaud Rouzée, Frank Lépine, Marc J. J. Vrakking, "Photoelectron holography in strong optical and dc electric fields". Journal of Physics: Conference Series 488, 012007 (2014). Full Article.
 A. S. Stodolna, F. Lépine, T. Bergeman, F. Robicheaux, A. Gijsbertsen, J. H. Jungmann, C. Bordas, M. J. J. Vrakking, "Visualizing the Coupling between Red and Blue Stark States Using Photoionization Microscopy". Physical Review Letters, 113, 103002 (2014). Abstract.