## Periodic Reporting for period 3 - QUENOCOBA (Quantum Emitters in non-conventional baths)

Reporting period: 2020-09-01 to 2022-02-28

The field of Quantum Optics investigates the interaction between light and matter at the fundamental level. There, emitters (typically atoms) absorb and emit light (photons), giving rise to many phenomena that we observe in our lives. Furthermore, this simple process is at the basis of many devices, ranging from lasers to the most sophisticated medical tools. It also plays a crucial role in the development of new applications, like quantum computers, quantum communication devices, or quantum sensors.

The emission and absorption of photons triggered the discovery of Quantum Physics more than hundred years ago. Max-Planck or Einstein already introduced simple models to understand this phenomenon. But it was not until the development of Quantum Electrodynamics, in the middle of the past century, that a proper understanding was achieved in terms of that theory. A bit later, scientists like Glauber (who received the Nobel Prize fifteen years ago) introduced a quantum mechanical description of the light that passed from being a simple spectator to a central ingredient of light-matter interaction, originating what we call now Quantum Optics. The development of experimental techniques led to very precise tests of that theory, and the advent of applications like quantum computing triggered their use in those and other devices. One can fairly say that the theory of Quantum Optics is well established and the phenomena observed in the experiments are well understood.

In very recent years, technological advances have allowed us to trespass microscopic frontiers and enter a new dimension in the interaction of light and matter. Now it is possible to put atoms close to some dieletric materials (isolators), where they behave in a completely different way to what it has been observed so far. In particular, if one is able to write some microscopic structures on that material (yielding, what is called, a photonic crystal) atoms absorb and emit photons in a very different way as in free space. For instance, one atom (emitter) can emit and reabsorb a photon many times before it finally reaches out, or several of them can interchange one photon, giving rise to effective atom-atom interactions. One speaks of a “structured” bath for the emitter, in the sense that the density of states, that defines how photons move in the photonic crystal, has unconventional properties. In fact, we know neither what new phenomena can occur in this situation, nor how to describe them theoretically. The main objective of QUENCOBA is precisely to investigate this QUantum Emitters in NOn-COnventinal BAths and to extend the theory of Quantum Optics to this new and complex scenario. It also aims at developing the theoretical and computational tools to address the difficult problems that arise in the description of those experiments, as well as to harness those new phenomena to build better quantum devices. Although this project deals with basic science, as it aims at understanding fundamental processes that occur under special situations, it will develop the theoretical and computational techniques that will be required to build such devices.

The overall objectives of the project are: (i) to develop the theoretical tools required to investigate this new area of research; (ii) to explore and characterize novel phenomena and to develop the analytical and computational techniques required for that; and (iii) propose and analyze other physical setups where those phenomena can be observed and exploited (in the context of quantum information and simulation). The research involves the development of innovative techniques to describe new scenarios in quantum optics and many-body physics, as well as research on atoms interacting with photonic crystals, in optical lattices, and quantum dots interacting with surface acoustic waves.

The emission and absorption of photons triggered the discovery of Quantum Physics more than hundred years ago. Max-Planck or Einstein already introduced simple models to understand this phenomenon. But it was not until the development of Quantum Electrodynamics, in the middle of the past century, that a proper understanding was achieved in terms of that theory. A bit later, scientists like Glauber (who received the Nobel Prize fifteen years ago) introduced a quantum mechanical description of the light that passed from being a simple spectator to a central ingredient of light-matter interaction, originating what we call now Quantum Optics. The development of experimental techniques led to very precise tests of that theory, and the advent of applications like quantum computing triggered their use in those and other devices. One can fairly say that the theory of Quantum Optics is well established and the phenomena observed in the experiments are well understood.

In very recent years, technological advances have allowed us to trespass microscopic frontiers and enter a new dimension in the interaction of light and matter. Now it is possible to put atoms close to some dieletric materials (isolators), where they behave in a completely different way to what it has been observed so far. In particular, if one is able to write some microscopic structures on that material (yielding, what is called, a photonic crystal) atoms absorb and emit photons in a very different way as in free space. For instance, one atom (emitter) can emit and reabsorb a photon many times before it finally reaches out, or several of them can interchange one photon, giving rise to effective atom-atom interactions. One speaks of a “structured” bath for the emitter, in the sense that the density of states, that defines how photons move in the photonic crystal, has unconventional properties. In fact, we know neither what new phenomena can occur in this situation, nor how to describe them theoretically. The main objective of QUENCOBA is precisely to investigate this QUantum Emitters in NOn-COnventinal BAths and to extend the theory of Quantum Optics to this new and complex scenario. It also aims at developing the theoretical and computational tools to address the difficult problems that arise in the description of those experiments, as well as to harness those new phenomena to build better quantum devices. Although this project deals with basic science, as it aims at understanding fundamental processes that occur under special situations, it will develop the theoretical and computational techniques that will be required to build such devices.

The overall objectives of the project are: (i) to develop the theoretical tools required to investigate this new area of research; (ii) to explore and characterize novel phenomena and to develop the analytical and computational techniques required for that; and (iii) propose and analyze other physical setups where those phenomena can be observed and exploited (in the context of quantum information and simulation). The research involves the development of innovative techniques to describe new scenarios in quantum optics and many-body physics, as well as research on atoms interacting with photonic crystals, in optical lattices, and quantum dots interacting with surface acoustic waves.

The work is divided into three WP, each of them dealing with concrete subjects: WP1 is about few emitters, WP2 is about many, and WP3 is about to new methodologies to solve this kind of problems. The main results achieved so far are:

- The analysis of one and few quantum emitters coupled to structured baths modelled by tight-binding models in one (linear), two (square), and three (cubic and other configurations) dimensional lattices. Derivation of non-conventional long-range models.

- In a honey-comb lattice in two dimensions, the effective model has true long-range interactions decaying inversely proportional to the distance without an exponential overall correction.

- In a topological bath (defined by the so-called SSH model) in one dimension, chiral bound sates appear, and mediate a very non-conventional spin model with, eg, a double Neel phase.

- In a fermionic bath (ie. similar to the Kondo model), the dynamics can be understood as a renormalization group flow.

- In a waveguide, one can efficiently generate multi-photon Fock states which can be used in metrology applications.

- Proposals for experimental observation of the emerging phenomena with atoms close to photonic crystals, cold atoms in optical lattices, Rydberg atoms in Bose-Einstein condensates, or electrons in magnetic traps.

- A proposal for quantum simulation of chemistry compounds with cold atoms in optical lattices.

- A new technique to measure Renyi entropies in many-body systems.

- A computational variational method for many body system based on Gaussian and non-Gaussian states.

- A computational method to analyze quantum emitters interacting with two dimensional many-body reservoirs or disordered systems based on tensor networks.

- A computational method for quantum problems combining tensor networks and Monte Carlo techniques.

- A computational method for quantum problems combining machine learning and tensor network techniques.

- The analysis of one and few quantum emitters coupled to structured baths modelled by tight-binding models in one (linear), two (square), and three (cubic and other configurations) dimensional lattices. Derivation of non-conventional long-range models.

- In a honey-comb lattice in two dimensions, the effective model has true long-range interactions decaying inversely proportional to the distance without an exponential overall correction.

- In a topological bath (defined by the so-called SSH model) in one dimension, chiral bound sates appear, and mediate a very non-conventional spin model with, eg, a double Neel phase.

- In a fermionic bath (ie. similar to the Kondo model), the dynamics can be understood as a renormalization group flow.

- In a waveguide, one can efficiently generate multi-photon Fock states which can be used in metrology applications.

- Proposals for experimental observation of the emerging phenomena with atoms close to photonic crystals, cold atoms in optical lattices, Rydberg atoms in Bose-Einstein condensates, or electrons in magnetic traps.

- A proposal for quantum simulation of chemistry compounds with cold atoms in optical lattices.

- A new technique to measure Renyi entropies in many-body systems.

- A computational variational method for many body system based on Gaussian and non-Gaussian states.

- A computational method to analyze quantum emitters interacting with two dimensional many-body reservoirs or disordered systems based on tensor networks.

- A computational method for quantum problems combining tensor networks and Monte Carlo techniques.

- A computational method for quantum problems combining machine learning and tensor network techniques.

All the main results correspond to progress beyond the state of the art, and have given rise to 75 publications in international journals, including Nature, Nature Communication, Science Advances, Physical Review Letters, Physical Review X, Physical Review A, B, and D, and New Journal of Physics.

We plan to continue working in all the WPs following the time-line in the description of the action. The expected results until the end of the project are as follows:

WP1: (i) discover new phenomena when the bath is composed of arrays of atoms in optical lattices, in 1, 2 or 3 dimensions, and the emitters are other atoms that levitate around them. In particular, related to the fate of bound states and the interplay between the appearance of sub-radiant and super-radiant states in the arrays; (ii) put up and investigate new proposals for processing quantum information in the previous setup using effective interactions mediated by the atoms in the arrays; (iii) analyze a new setup for the generation of arbitrary states of light; (iv) investigate how to realize those setups experimentally.

WP2: (i) characterize the behavior of many-body models arising from quantum emitters coupled to structured baths; (ii) study and find effective descriptions of such models beyond the Markovian limit, where the bath possess some memory and acts back on the emitters; (iii) discover new many-body phenomena in the setup studied in WP1; (iv) extend the work on quantum chemistry simulation to make it more experimental friendly.

WP3: (i) Formalize the methods based on variational families (Gaussian and non-Gaussian) and extend their range of applicability to finite temperatures and open systems; (ii) Develop new tensor network techniques to address problems like thermalization, or the simulation of 2D systems; (iii) To propose new techniques to solve many-body quantum systems by combining ideas in machine learning, quantum information theory, and Monte Carlo Methods. In particular to simulate the dynamics of avoiding the sign problem that appears in Monte Carlo methods.

We plan to continue working in all the WPs following the time-line in the description of the action. The expected results until the end of the project are as follows:

WP1: (i) discover new phenomena when the bath is composed of arrays of atoms in optical lattices, in 1, 2 or 3 dimensions, and the emitters are other atoms that levitate around them. In particular, related to the fate of bound states and the interplay between the appearance of sub-radiant and super-radiant states in the arrays; (ii) put up and investigate new proposals for processing quantum information in the previous setup using effective interactions mediated by the atoms in the arrays; (iii) analyze a new setup for the generation of arbitrary states of light; (iv) investigate how to realize those setups experimentally.

WP2: (i) characterize the behavior of many-body models arising from quantum emitters coupled to structured baths; (ii) study and find effective descriptions of such models beyond the Markovian limit, where the bath possess some memory and acts back on the emitters; (iii) discover new many-body phenomena in the setup studied in WP1; (iv) extend the work on quantum chemistry simulation to make it more experimental friendly.

WP3: (i) Formalize the methods based on variational families (Gaussian and non-Gaussian) and extend their range of applicability to finite temperatures and open systems; (ii) Develop new tensor network techniques to address problems like thermalization, or the simulation of 2D systems; (iii) To propose new techniques to solve many-body quantum systems by combining ideas in machine learning, quantum information theory, and Monte Carlo Methods. In particular to simulate the dynamics of avoiding the sign problem that appears in Monte Carlo methods.