A major topic of research in our theory group is the study of many-body optical phenomena. An ensemble of atoms (or more generally, quantum emitters) that share an environment can be described as an open, out-of-equilibrium many-body system with long-range interactions, where two tantalizing possibilities appear. First, “dark” and “bright” states emerge. The former have very extended lifetimes and can be exploited to store information and realize almost-coherent operations. The latter decay very rapidly and form the underpinning of new light sources, such as superradiant lasers. Second, the interplay between pumping, coherent evolution and dissipation may stabilize some form of macroscopic order that eludes the conventional paradigms of equilibrium statistical mechanics. All these ingredients make many-body quantum optics a fascinating theoretical challenge. A more detailed description of our research interests and some research highlights are listed below.

### Many-body quantum optics and open quantum systems

We are interested in understanding the many-body optical response of extended quantum systems. In contrast to (single-mode) cavity quantum electrodynamics (QED), where many problems can be solved exactly due to permutational symmetry, in extended systems atom-atom interactions depend on the distance and, therefore, the dynamics can explore a larger part of the Hilbert space as it is not constrained by symmetry. One thus expects that finding exact solutions for dynamical evolution is unfeasible for more than just a few atoms.

Instead of computing some observable in a specific system obeying some particular dynamics, we have recently followed a different path: to set bounds on its maximal value. In particular, we have derived analytical scaling laws that set fundamental limits on the decay of (a broad class of) Markovian many-body open quantum systems. To do this, we have reformulated the problem of maximal decay rate into finding the ground state energy of a spin Hamiltonian (see figure in the right), harnessing tools developed within the community of Hamiltonian complexity theory and applying them in the context of out-of-equilibrium dynamics. For many physically-relevant systems, upper and lower bounds are tight (except for a constant factor), yielding universal scaling laws with system size. While these laws are valid for a broad class of systems, we uncovered them thanks to our exploration of superradiant decay in atomic arrays in free space, which is described below.

### Quantum optics in atomic arrays

Atomic arrays in free space are the perfect playground for studying many-body quantum optics. Some of our earlier work was devoted to single-excitation physics, but we are currently interested in physics that emerges at large excitation densities. To advance in this research, we develop new analytical methods as well as computational techniques.

We have recently investigated dissipative quantum dynamics in arrays, in particular many-body superradiant decay, as shown in the top left plot. It is known that initially-inverted atoms that are in close proximity (in a volume much smaller than the atomic resonance wavelength) strongly synchronize as they decay, and light is emitted in a burst (known as “Dicke’s superradiance”). We have found that this physics also occurs in extended atomic arrays in free space, and determined that the arrays’ dimensionality is a key ingredient for superradiance to appear. Many-body superradiance occurs because the initial fluctuation that triggers emission is amplified throughout the decay process. Effectively, the system develops memory (and, depending on the dimension, this is possible or not). We have also studied how to observe bursts under realistic experimental conditions. Moving forward, we plan to investigate whether this order/memory can be stabilized in the steady state.

Using the techniques described in the previous section, we have recently found that the maximal decay rate is bounded by quantities that can easily be computed from the decoherence matrix that describes the dissipative couplings between the atoms. For two- and three-dimensional arrays, the largest possible decay rates scale superlinearly with the number of particles. The left bottom figure shows the asymptotic scaling with system size (i.e., number of atoms N) of the maximal many- body decay rate, for different lattice dimensionalities D. The scalings (obtained using a variational approach and framing the problem as a semidefinite program) agree with our analytical predictions. Beyond setting a bound on the superradiant peak, these scalings pose interesting questions in the field of quantum error correction, driven-dissipative phase transitions, and metrology.

### Waveguide quantum electrodynamics

Boundary conditions determine the spectral and spatial properties of the propagator of the electromagnetic vacuum fluctuations. In waveguide QED, qubits are coupled via one-dimensional baths, such as fibers, transmission lines or photonic crystals. These environments mediate long-range interactions between qubits, and the character of the interaction (whether it is more coherent or dissipative) is tailored by changing the particles’ relative positions. Despite its apparent simplicity, waveguide QED is a very rich system, displaying Hamiltonian evolution together with two competing decay channels (corresponding to photon emission to the left and right).

In recent work, we have shown that the avalanche-like behavior of many-body decay leads to spontaneous mirror symmetry breaking, with large shot- to-shot fluctuations in the number of photons emitted to the left and right. Of course, the symmetry is recovered when averaged over realizations, as the first photon is randomly emitted into either direction (as shown by the figure in the left). For chiral waveguides, many-body decay boost their chirality, effectively making them one-directional. This “winner takes all” dynamics in the presence of multiple competing decay channels also occurs when the competition occurs between “internal” degrees of freedom (for instance in systems with multiple ground states). We have recently put forward analytical work that shows the quenching of all competing transitions except the dominant one. This phenomenon could be used to develop efficient protocols for molecular photo-association.

On a separate note, we have recently proposed how to stabilize dark quantum dimers dissipatively, via squeezed vacuum in waveguide QED.

### Other projects and experimental collaborations

We sometimes dabble into problems of more applied nature, typically in the context of collaborations. For example, together with the Ochoa group, we have explored the optical response of moire insulators. Owing to their chiral symmetry, twisted bilayers display circular dichroism. Our calculations exemplify how subtle properties of the electronic wave functions, encoded in current correlations between the layers, control physical observables of moire materials. In other recent collaboration, with the Lipson and Gaeta groups, we have studied the encoding of information in the frequency domain of photons, and the subsequent coherent control of different logical channels by means of a N-photon beamsplitter using Bragg-scattering four-wave mixing.

We regularly partner up with experimental groups at Columbia and other universities. Some recent collaborators include Prof. Dan Stamper-Kurn at UC Berkeley, Profs. Antoine Browaeys and Igor Ferrier-Barbut at Institut d’Optique (Univ Paris-Saclay), Prof. Luis Orozco at JQI and the University of Maryland, and Prof. Jake Covey at UIUC. At Columbia, we work with Profs. Tanya Zelevinsky, Sebastian Will, Nanfang Yu, Dmitri Basov, Jim Schuck, Michal Lipson and Alex Gaeta.