A new protocol to image wave functions in continuous space

In recent years, physicists have been trying to better understand the behavior of individual quantum particles as they move in space. Yet directly imaging these particles with high precision has so far proved challenging, due to the limitations of existing microscopy methods.

Researchers at CNRS and École Normale Supérieure in Paris, France, have now developed a new protocol to directly image the evolution of a single-atom wave packet, a delocalized quantum state that determines the probability that an associated atom will be found in a specific location. This imaging technique, introduced in Physical Review Letters, could open exciting possibilities for the precise study of complex quantum systems in continuous space.

“Our group is interested in the study of ultracold atoms, the coldest systems in the universe, just a few billionths of degrees above absolute zero, where matter displays fascinating behaviors,” Tarik Yefsah, senior author of the paper, told Phys.org. “One of these behaviors is the so-called superfluidity, a remarkable state of matter, where particles flow without friction.

“This is a purely quantum phenomenon where the system, although composed of particles, behaves as a giant wave. When the interaction between particles is large, the precise behavior of such a state is extremely difficult to predict theoretically, especially at the microscopic, atomic, scale.”

The primary objective of the recent study by Yefsah and his colleagues was to probe these quantum systems by resolving each individual atom involved in them. They were aware that by turning on an optical lattice (i.e., a laser-made egg carton–like box), using it to pin atoms and then shining a carefully selected light onto them, they would be able to individually image them becoming fluorescent while being trapped.

“Atoms are initially in free space, and without an optical lattice for the detection phase they would move around too much and we would lose information on their initial position,” said Yefsah. “With the use of an optical lattice, the question is then, how do we ensure that the atoms are pinned in the closest lattice well (egg slot) to their initial position rather than moving to a distant one?

“The latter scenario would yield incorrect information. We realized that the answer to this question is not obvious even for a single-atom wave function living in free space—before even considering the complicated many-body wave function of an interacting ensemble.”

Based on these initial predictions and considerations, the researchers realized that it would make sense to first perform their experiment with single-atom wave functions. If this succeeded, they could then employ the same methods to study more complex quantum systems.

“This experiment is crucial: if pinning fails there, there is not much hope for it to work with more complicated systems,” said Yefsah. “Luckily, we found a regime where it worked beyond expectations, with a near 100% fidelity.”

To prepare the wave packets they imaged as part of their experiment, the researchers first turned the optical lattice on and cooled atoms inside of it. Following this cooling processes, the atoms were brought to the bottom of the lattice wells, which ultimately prompted them to behave as a trapped Gaussian quantum wave.

“This first step also allows us to identify the starting position of each atom,” explained Yefsah. “After switching the lattice off, the wave packets are free to expand in a plane until they grow larger than a few lattice sites (this is necessary because we want the atom to ‘forget’ about its original position and behave like a true continuous system). All experimental stages until this one are used purely for preparing our system of interest.”

After completing these initial preparation steps, the researchers turned the optical lattice back on to recapture each of the atoms into a given well. This then allowed them to obtain a precise measurement of the atoms’ displacement.

“By repeating this experiment with a large number of atoms, we reconstructed the time evolution of the wave function with unprecedented fidelity,” said Yefsah.

“This is only achieved when the turn-on of the lattice is well-timed: it cannot be too fast or too slow. The measurement of the expansion of the gaussian wave packet as predicted by the Schrödinger equation allowed us to find the sweet spot.”

Ultimately, Yefsah and his colleagues were able to devise an effective protocol to project an atom from continuous space onto the nearest lattice site in a controlled way, attaining a fidelity that exceeds 99%. The researchers have so far used this protocol to image a single-atom wave packet expanding in continuous space, yet it could soon be used to study other complex quantum systems that are sufficiently diluted.

“The imaging method we developed is a bit like a CCD camera for atomic wave functions,” added Yefsah. “Our single-atom imaging method for the continuum is applicable to any sufficiently dilute system that evolves in free space, including many-body systems for which theoretical results might be difficult to compute.

“For instance, in a recent publication posted on the arXiv preprint server and now accepted by PRL, the team uses the imaging technique pioneered in their wave packet article to image Fermi gases. We were able to reveal the very intricate manner in which atoms act collectively in these systems.”

Now that they have confirmed the effectiveness of their imaging protocol, the researchers are using it to tackle the research problems that they had in mind when they first developed it. Specifically, they are now employing it to study the behavior of strongly interacting superfluids, which is very difficult to predict theoretically.

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