Three teams of researchers in the US and France have independently developed a new technique to visualize the positions of atoms in real, continuous space, rather than at discrete sites on a lattice. By applying this method, the teams captured “snapshots” of weakly interacting bosons, non-interacting fermions and strongly interacting fermions and made in-situ measurements of the correlation functions that characterize these different quantum gases. Their work constitutes the first experimental measurements of these correlation functions in continuous space – a benchmark in the development of techniques for understanding fermionic and bosonic systems, as well as for studying strongly interacting systems.
Quantum many-body systems exhibit a rich and complex range of phenomena that cannot be described by the single-particle picture. Simulating such systems theoretically is thus rather difficult, as their degrees of freedom (and the corresponding size of their quantum Hilbert spaces) increase exponentially with the number of particles. Highly controllable quantum platforms like ultracold atoms in optical lattices are therefore useful tools for capturing and visualizing the physics of many-body phenomena.
The three research groups followed similar “recipes” in producing their atomic snapshots. First, they prepared a dilute quantum gas in an optical trap created by a lattice of laser beams. This lattice was configured such that the atoms experienced strong confinement in the vertical direction but moved freely in the xy-plane of the trap. Next, the researchers suddenly increased the strength of the lattice in the plane to “freeze” the atoms’ motion and project their positions onto a two-dimensional square lattice. Finally, they took snapshots of the atoms by detecting the fluorescence they produced when cooled with lasers. Importantly, the density of the gases was low enough that the separation between two atoms was larger than the spacing between the sites of the lattice, facilitating the measurement of correlations between atoms.
What does a Fermi gas look like in real space?
One of the three groups, led by Tarik Yefsah in Paris’ Kastler Brossel Laboratory (KBL), studied a non-interacting two-dimensional gas of fermionic lithium-6 (6Li) atoms. After confining a low-density cloud of these atoms in a two-dimensional optical lattice, Yefsah and colleagues registered their positions by applying a technique called Raman sideband laser cooling.
The KBL team’s experiment showed, for the first time, the shape of a parameter called the two-point correlator (g2) in continuous space. These measurements clearly demonstrated the existence of a “fermi hole”: at small interatomic distances, the value of this two-point correlator tends to zero, but as the distance increases, it tends to one. This behaviour was expected, since the Pauli exclusion principle makes it impossible for two fermions with the same quantum numbers to occupy the same position. However, the paper’s first author Tim de Jongh, who is now a postdoctoral researcher at the University of Colorado Boulder in the US, explains that being able to measure “the exact shape of the correlation function at the percent precision level” is new, and a distinguishing feature of their work.
The KBL team’s measurement also provides both two-body and three-body correlation functions for the atoms, making it possible to compare them directly. In principle, the technique could even be extended to correlations of arbitrarily high order.
What about a Bose gas?
Meanwhile, researchers directed by Wolfgang Ketterle of the Massachusetts Institute of Technology (MIT) developed and applied quantum gas microscopy to study how bosons bunch together. Unlike fermions, bosons do not obey the Pauli exclusion principle. In fact, if the temperature is low enough, they can enter a phase known as a Bose-Einstein condensate (BEC) in which their de Broglie wavelengths overlap and they occupy the same quantum state.
By confining a dilute bosonic gas of approximately 100 rubidium atoms in a sheet trap and cooling them to just above the critical temperature (Tc) for the onset of BEC, Ketterle and colleagues were able to make the first in situ measurement of the correlation length in a two-dimensional ultracold bosonic gas. In contrast to Yefsah’s group, Ketterle and colleagues employed polarization cooling to detect the atoms’ positions. They also focused on a different correlation function; specifically, the second-order correlation function of bosonic bunching at T>Tc.
When the system’s temperature is high enough (54 nK above absolute zero, in this experiment), the correlation function is nearly 1, meaning that the atoms’ thermal de-Broglie waves are too short to “notice” each other. But when the sample is cooled to a lower temperature of 6.4 nK, the thermal de-Broglie wavelength becomes commensurate with the interparticle spacing r, and the correlation function exhibits the bunching behavior expected for bosons in this regime, decreasing from its maximum value at r = 0 down to 1 as the interparticle spacing increases.
In an ideal system, the maximum value of the correlation function would be 2. However, in this experiment, the spatial resolution of the grid and the quasi-two-dimensional nature of the trapped gas reduce the maximum to 1.3. Enid Cruz Colón, a PhD student in Ketterle’s group, explains that this experiment is sensitive to parity projection, meaning that the count number of atoms per site is either even or odd. This implies that doubly occupied sites are registered as empty sites, which directly shrinks the measured value of g2
What does an interacting quantum gas look like in real space?
With Yefsah and colleagues focusing on fermionic correlations, and Ketterle’s group focusing on bosons, a third team led by MIT’s Martin Zwierlein found its niche by studying mixtures of bosons and fermions. Specifically, the team measured the pair correlation function for a mixture of a thermal Bose gas composed of sodium-23 (23Na) atoms and a degenerate Fermi gas of 6Li. As expected, they found that the probability of finding two particles together is enhanced for bosons and diminished for fermions.
In a further experiment, Zwierlein and colleagues studied a strongly interacting Fermi gas and measured its density-density correlation function. By increasing the strength of the interactions, they caused the atoms in this gas to pair up, triggering a transition into the BCS (Bardeen-Cooper-Schriefer) regime associated with paired electrons in superconductors. For atoms in a BEC, the density-density correlation function shows a strong bunching tendency at short distances; in the BCS regime, in contrast, the correlation depicts a long-range pairing where atoms form so-called Cooper pairs as the strength of their interactions increases.
By applying the new quantum gas microscopy technique to the study of strongly interacting Fermi gases, Ruixiao Yao, a PhD student in Zwierlein’s group and the paper’s first author, notes that they have opened the door to applications in quantum simulation. Such strongly correlated systems, Yao highlights, are especially difficult to simulate on classical computers.
The three teams describe their work in separate papers in Physical Review Letters.
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