Atomic arrays consist of atoms arranged in a regular “crystal” – be it a 1D line or some 2d or 3d lattice. They are held in place by “optical tweezers” or “optical lattices” – laser fields that can trap atoms in certain points in space.
The main reason for arranging the atoms into arrays is that they “feel” the presence of each other, because they can exchange photons. As an approximation, one can treat atoms (at least some kinds of them) as “two level systems” – there is one electron in each atom, which can be either in a “ground state”, or an “excited state” which has higher energy. By default, all atoms are in the ground states. However, if we illuminate the atom with a laser light with a right frequency, it can absorb a photon and move to the excited state. Conversely, it can switch back to the ground state by emitting a photon. This photon can then be re-absorbed by an another atom – especially when the array is dense and the atoms are closer together, so the photons have less chance of escaping without encountering other atoms. Such an emission and re-absorption looks as if there was a particle – an “excitation” – which “hopped” from one atom to the other. If there are no photons which come from the outside and get absorbed, and if no photons escape from the system, we can indeed treat the excitations as particles, and the atoms as lattice sites which can be occupied by these particles. Although photons don’t “feel” the presence of each other, they start to interact once they are converted into the excitations. The reason is simple: two excitations cannot occupy the same atom. The atom is either in the ground state (no excitation) or excited state (one excitation) – there is no other option.
The “hopping” of excitations is affected by interference effects. Two light waves superimposed onto each other can form interference patterns, such as the stripes visible if you shine a laser onto a diffraction grating. The interference occurs not only between two different photons, but a single photon can interfere with itself! Because we are in the realm of quantum mechanics, it is possible that the excitation does not occupy a single site, but is spread over several sites, occupying each of them with some probability (just like inside an atom, an electron does not orbit around the nucleus like Earth around the sun, but is everywhere around the nucleus at once, with probability given by the atomic orbital, see wavefunction). If such an excitation emits a photon, it is emitted from every site at once, and interferes with itself, which can influence e.g. the direction in which it propagates. In an array where the atoms are spaced evenly – just as the rows in a diffraction grating – these interference effects are more pronounced than e.g. an ensemble of atoms distributed randomly, especially if the array is sub-wavelength, i.e. the spacing between adjacent atoms is smaller than the wavelength of emitted/absorbed photons. These effects can be used to control how the array absorbs and emits light, and how the excitations hop from one site to each other.
Why would we want to study atomic arrays in this project? First, out of pure curiosity. The situation when the photons can interact with each other is pretty unusual. Interaction of electrons in crystals gives rise to many intriguing phenomena, such as magnetism, superconductivity and topological order. What would happen if we let photons interact?
The second reason is that physicists are searching for alternative platforms for realizing many-body condensed-matter phenomena (such as topological order), which can be easier to control and study (for example, have access to individual sites, which would be difficult in a “normal” crystals with 10^23 atoms really close to each other). There are already several platforms where this can be done. We believe that arrays serve as an alternative platform, with its own characteristic tools and methods. For example thing that makes the arrays special is that one can infer their properties from the absorbed/emitted light. Also, excitations hop in a different way than atoms: while the latter typically hop just to the nearest-neighboring sites, the excitation can hop long distances. Such long-range hopping is interesting for many reasons. For example, a number of things we know about topological orders relies on the assumption of short-range hopping and interaction. The atomic arrays would allow us to test this assumption.
In the QUINTO project, we focus on two kinds of arrays:
Free-space arrays. One of the systems we study is an array of atoms held in free space (as in the first picture). In particular, we are interested in the arrays of three-level atoms. That is, the atom can be either in one of “ground state” or in one of two “excited states” (instead of one). Thus, an excitation is a “particle” which can “hop” between the sites, but also can change its “internal state” between “+” and “-”. If there is a magnetic field permeating the system, these “internal states” have different energies. These two levels can be excited with two different kinds of photons: “left” and “right” circularly polarized ones. Polarization refers to how the electric field oscillates in the plane perpendicular to the field propagation direction: it can e.g. go back an forth on a certain line (“linear polarization”) or run around a circle left or right (“circular polarization”).
The band structure of such systems contain “topological bands”, which can lead to the presence of topological orders if sufficiently flat. But in the arrays, they are not flat at all! Therefore, we have to find factors responsible for the different properties of band structure and engineer the bands the way we like them. We already have an idea that a combination of the small distance between atoms and small number of atoms leads to conditions where the bands behave as flat. The challenge is to describe how the excitations interact within the band and to find a way to engineer similar situation with more atoms and more excitations. The more excitations – the more complicated and more interesting is the system.
Cavity Rydberg arrays. Instead of free space, the array can be placed in an optical cavity. That is: a pair of mirrors, between which the photons can bounce back and forth instead of escaping. In such a case, the distance between atoms becomes irrelevant. The excitations can hop with the same probability to the nearest-neighboring atom as to the atom that is further away. In some sense, such a case is boring, because there is no difference to which site the excitation would hop, making the system quite featureless. But it becomes interesting if the exciting states of our atoms are Rydberg states. These are high-energy excited states which correspond to atom increasing its size [fold]. The increased size of the atom makes Rydberg excitations on neighboring atoms will “feel” each other’s presence, which introduces new kind of interaction to our system. We have already found a way in which such a system, combining long- and short-range processes, resemble “frustrated magnets” where topological orders are expected. This opens up a range of questions: how can we access this system experimentally? What new techniques of manipulating and probing the system can we propose? Does the output light carry information about anyons? How much does the long-range hopping change? Can we still rely to what we know about topological orders, where the crucial elements of theory were assuming short-range interaction?
