## Periodic Reporting for period 1 - SoftRyd (Soft-matter collective phenomena in Rydberg gases)

Reporting period: 2016-10-01 to 2018-09-30

In the last few decades, it has become possible to experimentally study systems of interacting atoms or molecules evolving according to the laws of quantum mechanics, which has stimulated the theoretical study of such quantum many-body systems. The field is now characterised by an active exchange between theory and experiment. In this context, the aim of this project is to gain a deeper understanding of emerging collective effects in Rydberg gases, i.e. systems of atoms excited to high-lying states. Rydberg gases are becoming ever more controllable in experiment, and have shown great potential for applications in quantum information and sensing technologies. Surprisingly, the collective effects resulting from the very strong interactions of such systems resemble those found in the physics of liquids close to the glass transition. Such compelling analogies remained unexplored before the beginning of this project, as the community of researchers that study glassy systems and those who study quantum many-body systems use very different conceptual frameworks. By combining the expertise of the Supervisor in the physics of Rydberg gases and the previous research experience of the Researcher in glassy physics, a fruitful collaboration that lives on in several ongoing projects has been established.

The main objectives of the project are:

1) The study of the quantum regime of Rydberg gases far from equilibrium. While the dynamics in the classical limit (where quantum superpositions decay very rapidly) had been studied to some extent, Rydberg gases in regimes where quantum effects strongly influence the dynamics remained unexplored.

2) Exploring the dynamics of multi-component Rydberg gases. While most of the previous work deals with the case where there is only one excited state (as excited by an electromagnetic field oscillating at a particular frequency), in the presence of more complex fields, several excited levels become involved in the dynamics. This case had not been explored despite its experimental relevance, and the fact that the engineering of multistable Rydberg gases, where applied electric fields lead to vastly different dynamics, may have implications for sensing technologies.

3) This objective deals with two aspects common to both (1) and (2).This include: (a) to study the dynamics of Rydberg gases using critical phenomena approaches, (b) to establish collaborations with experimental groups in order to validate new theoretical results, with the Researcher leading the exchange between theorists and experimentalists.

The main objectives of the project are:

1) The study of the quantum regime of Rydberg gases far from equilibrium. While the dynamics in the classical limit (where quantum superpositions decay very rapidly) had been studied to some extent, Rydberg gases in regimes where quantum effects strongly influence the dynamics remained unexplored.

2) Exploring the dynamics of multi-component Rydberg gases. While most of the previous work deals with the case where there is only one excited state (as excited by an electromagnetic field oscillating at a particular frequency), in the presence of more complex fields, several excited levels become involved in the dynamics. This case had not been explored despite its experimental relevance, and the fact that the engineering of multistable Rydberg gases, where applied electric fields lead to vastly different dynamics, may have implications for sensing technologies.

3) This objective deals with two aspects common to both (1) and (2).This include: (a) to study the dynamics of Rydberg gases using critical phenomena approaches, (b) to establish collaborations with experimental groups in order to validate new theoretical results, with the Researcher leading the exchange between theorists and experimentalists.

The following results have been achieved so far:

A) The transition between classical and quantum dynamics in Rydberg gases out of equilibrium has been clarified [1]. This study has confirmed that the behaviour in the classical limit (see Ref. [2]) remains qualitative unchanged even for relatively high levels of 'quantumness'. This addresses the major goal of Objective 1.

B) The dynamics of multi-level Rydberg gases has been thoroughly explored for the first time [3]. The time evolution of these systems, which is characterised by several typical length scales, has been related to those of mixtures of glass-forming liquids. As in such glassy systems, we find metastable behaviour (features that persist for very long times before eventually disappearing) arising from complex interaction patterns. These results address the main goal of Objective 2.

C) The existence of collective effects in a Rydberg gas of rubidium atoms has been studied in a collaboration led by the Researcher between theorists at Nottingham and experimentalists at the University of Pisa [4]. For the first time, a non-equilibrium phase transition (i.e. a situation similar to a change of state of aggregation when a system is not in thermodynamic equilibrium, but rather driven externally) has been experimentally observed in a quantum system. This fulfilled the second point of Objective 3 above.

D) A quantum version of a well-known classical spreading model based on Rydberg atoms has been proposed and analysed [5]. The model undergoes a non-equilibrium phase transition equivalent to that of old model in the classical limit, but behaves in a completely different way (showing a sequence of discontinuous transitions) in the quantum limit. This work, which originated from discussions while working on (C), addresses a new facet of Objective 1.

E) A study of the dynamics of dissipative Rydberg gases under a quench (i.e. a sudden change of parameters) [6]. The quench dynamics is described by an extension of the classical model for the description of the kinetics of phase transformations (i.e. how a liquid becomes a solid when the system is moved below the melting point). We verify numerically our predictions for the classical and the quantum regimes. This work originates from previous work by the Researcher on stable glasses [7], and falls within Objective 1.

Since the beginning of the project, these and related results have been orally presented at three conferences and scientific meetings to an overall audience of over 200 researchers.

Some of this work was actually carried out before the starting date of the project, as the Researcher had the opportunity to be funded by the host organisation before becoming a Marie Sklodowska-Curie fellow. As a result, (A) and (B) were published before the starting date of the project, while (C), (D) and (E) were carried out during the 11 months of the fellowship. Two ongoing projects will help us make further progress towards the achievement of Objective 2.

[1] Levi, E., Gutiérrez, R., & Lesanovsky, I. J. Phys. B, 49, 184003 (2016).

[2] Gutiérrez, R., Garrahan, J. P., & Lesanovsky, I. Phys. Rev. E, 92, 062144 (2015).

[3] Gutiérrez, R., Garrahan, J. P., & Lesanovsky, I. New J. Phys., 18, 093054 (2016).

[4] Gutiérrez, R. et al. Phys. Rev. A, 96, 041602(R) (2017).

[5] Pérez-Espigares, C., Marcuzzi, M., Gutiérrez, R. and Lesanovsky, I. Phys. Rev. Lett. 119, 140401 (2017).

[6] Gribben, D., Lesanovsky, I., & Gutiérrez, R. arXiv:1709.10383 (2017).

[7] Gutiérrez, R., & Garrahan, J. P. J. Stat. Mech. Theor. Exp., 2016, 074005 (2016).

A) The transition between classical and quantum dynamics in Rydberg gases out of equilibrium has been clarified [1]. This study has confirmed that the behaviour in the classical limit (see Ref. [2]) remains qualitative unchanged even for relatively high levels of 'quantumness'. This addresses the major goal of Objective 1.

B) The dynamics of multi-level Rydberg gases has been thoroughly explored for the first time [3]. The time evolution of these systems, which is characterised by several typical length scales, has been related to those of mixtures of glass-forming liquids. As in such glassy systems, we find metastable behaviour (features that persist for very long times before eventually disappearing) arising from complex interaction patterns. These results address the main goal of Objective 2.

C) The existence of collective effects in a Rydberg gas of rubidium atoms has been studied in a collaboration led by the Researcher between theorists at Nottingham and experimentalists at the University of Pisa [4]. For the first time, a non-equilibrium phase transition (i.e. a situation similar to a change of state of aggregation when a system is not in thermodynamic equilibrium, but rather driven externally) has been experimentally observed in a quantum system. This fulfilled the second point of Objective 3 above.

D) A quantum version of a well-known classical spreading model based on Rydberg atoms has been proposed and analysed [5]. The model undergoes a non-equilibrium phase transition equivalent to that of old model in the classical limit, but behaves in a completely different way (showing a sequence of discontinuous transitions) in the quantum limit. This work, which originated from discussions while working on (C), addresses a new facet of Objective 1.

E) A study of the dynamics of dissipative Rydberg gases under a quench (i.e. a sudden change of parameters) [6]. The quench dynamics is described by an extension of the classical model for the description of the kinetics of phase transformations (i.e. how a liquid becomes a solid when the system is moved below the melting point). We verify numerically our predictions for the classical and the quantum regimes. This work originates from previous work by the Researcher on stable glasses [7], and falls within Objective 1.

Since the beginning of the project, these and related results have been orally presented at three conferences and scientific meetings to an overall audience of over 200 researchers.

Some of this work was actually carried out before the starting date of the project, as the Researcher had the opportunity to be funded by the host organisation before becoming a Marie Sklodowska-Curie fellow. As a result, (A) and (B) were published before the starting date of the project, while (C), (D) and (E) were carried out during the 11 months of the fellowship. Two ongoing projects will help us make further progress towards the achievement of Objective 2.

[1] Levi, E., Gutiérrez, R., & Lesanovsky, I. J. Phys. B, 49, 184003 (2016).

[2] Gutiérrez, R., Garrahan, J. P., & Lesanovsky, I. Phys. Rev. E, 92, 062144 (2015).

[3] Gutiérrez, R., Garrahan, J. P., & Lesanovsky, I. New J. Phys., 18, 093054 (2016).

[4] Gutiérrez, R. et al. Phys. Rev. A, 96, 041602(R) (2017).

[5] Pérez-Espigares, C., Marcuzzi, M., Gutiérrez, R. and Lesanovsky, I. Phys. Rev. Lett. 119, 140401 (2017).

[6] Gribben, D., Lesanovsky, I., & Gutiérrez, R. arXiv:1709.10383 (2017).

[7] Gutiérrez, R., & Garrahan, J. P. J. Stat. Mech. Theor. Exp., 2016, 074005 (2016).

The results summarised above increase the present understanding of the dynamics of Rydberg gases. This brings us a step closer to one of the major goals of condensed-matter physics, namely, understanding the principles governing quantum matter out of equilibrium. Moreover, the fact that these results are not based on idealised models without clear experimental counterparts, but rather refer to systems that are accessible in modern settings based on cold atoms, makes the experimental observability of our theoretical predictions perfectly possible and relevant, as (C) above highlights.

As Rydberg gases are among the most versatile platforms for the study and development of quantum technologies (including sensing devices and quantum computing realisations), it is to be expected that these advances (together with many other contributions from scientists in the European Union and around the world) will help us take advantage of the possibilities that are to be found in the collective behaviour of such quantum systems. As the history of science teaches us, a greater understanding of both the fundamental and the emerging laws of nature leads to a greater controllability, which is the starting point of new technological developments.

As Rydberg gases are among the most versatile platforms for the study and development of quantum technologies (including sensing devices and quantum computing realisations), it is to be expected that these advances (together with many other contributions from scientists in the European Union and around the world) will help us take advantage of the possibilities that are to be found in the collective behaviour of such quantum systems. As the history of science teaches us, a greater understanding of both the fundamental and the emerging laws of nature leads to a greater controllability, which is the starting point of new technological developments.