CORDIS - EU research results

Quantum Dynamics of Strongly Correlated Systems and Ultra-Cold Atomic Gases

Final Report Summary - MASCARA (Quantum Dynamics of Strongly Correlated Systems and Ultra-Cold Atomic Gases)

The realization of strongly correlated quantum systems by means of ultra-cold atoms is now an achieved reality. Beyond the early studies of thermodynamic equilibrium properties, these systems offer unprecedented possibilities to study their dynamics. Crucial theoretical challenges therefore concern modeling and understanding quantum dynamics in the presence of strong correlations.

Main goal of our project was to develop new numerical approaches to tackle questions currently intractable by state-of-the-art techniques. To this purpose, we have used and suitably extended the “Time-dependent Variational Monte Carlo” method, to whose early development the Fellow (Dr Giuseppe Carleo) had tightly contributed. During the present Project, this approach has been further developed and applied to specific problems in strongly interacting ultra-cold atoms.

– General Review of Objectives and Results –

The Project has focused on two research areas (Lattice Bosons on one hand and Continous-Space Quantum Gases on the other hand). In the following we provide a general overview of the results we have achieved in both research areas.

Lattice Bosons –

The first research lines concerns the study of the dynamical properties of lattice bosons. These constitute a relevant description of ultra-cold atoms in optical lattices.
In this context, our research has focused onto the dynamics of interacting ultra-cold atoms after one of the microscopic parameters (for example the interaction strength) is abruptly changed. This protocol goes under the name of quantum quench, and it is now experimentally realized in many laboratories.
Specifically, we wanted to study how, and how fast, correlations can spread in these systems after the quantum quench is realized.

One of the critical tasks that we wanted to achieve during this Project was the study of specific regimes traditionally unaccessible by state-of-the-art numerical approaches. Thanks to the application and the extension of our Time-Dependent Variational Monte Carlo method, we substantially extendend the state-of-the-art in this field and specifically provided answers to two main problems :

1) First, we wanted to understand what is the role of the dimensionality in the spreading of information in a quantum system driven out of equilibrium. A fundamental open question in the field was to understand how geometrical effects can alter, or enhance the speed at which information can travel in a correlated quantum system.
Application of our t-VMC approach has allowed for the first time the study of information spreading in a strongly-interacting 2D system.

2) Second, we wanted to understand what is the effect of long-range interactions in the time-propagation of quantum correlations. In particular, an open fundamental question is to understand if a sufficiently long-ranged interaction can determine a breaking of locality and an instantaneous spreading of information. Extension and application of our t-VMC approach to long-range quantum systems (both bosons and spins) has allowed us to provide new answers in this field.

Quantum Gases in Continuous Space –

The second research line concerns the study of the dynamical properties of interacting quantum systems in continuous space. The main goal of the Project was to investigate interacting quantum models that constitute a relevant description of low-dimensional ultra-cold atoms, and that are now routinely realized in the laboratory.

1) First, we wanted to provide a theoretical understanding of both transport and collective dynamical properties in interacting quantum gases. These goals have been achieved thanks to the use of state-of-the-art Quantum Monte Carlo simulations (for the transport properties) and a semi-analytical treatment of collective excitations at finite temperature. In both cases, we have achieved succesful collaborations with experimental groups in ultra-cold atoms

2) Second, we aimed at specific methodological developments in the theoretical description of dynamical properties of interacting quantum gases, beyond the mean-field level. This goal has been achieved extending the time-dependent variational Monte Carlo approach, to systems in continuous space.

The main results obtained during the Project have been the object of four published papers and of two papers now in preparation. Three of these studies concern the first research line, on interacting lattice bosons :

1) In the published [1], we have studied the spreading of density-density correlations in Bose-Hubbard models after a quench of the interaction strength, using time-dependent variational Monte Carlo simulations. We have studied this phenomenon in one dimensional systems and, for the first time, also in two dimensions. We have shown that the spreading of correlations is generally supersonic. Further, we have shown that in two dimensions the correlation spreading is highly anisotropic and presents nontrivial interference effects.

2) In the published [2], we have studied the out-of-equilibrium behaviour of interacting quantum systems with long range algebraic interactions. Our study is based on a combined approach, based on one hand on the time-dependent Variational Monte Carlo method, and on the other hand on a quasi-particle microscopic picture. We have determined the conditions on the interaction potentials for a quasi-local propagation to be achieved, for specific models of lattice spins and bosons. We have also provided a microscopic justification of the different regimes observed.

3) In a paper in preparation [3], we detail the technical aspects that have led to the realization of the studies [1] and [2]. In particular, the extension of the time-dependent variational Monte Carlo approach to long-range interacting systems and the implementation of an implicit integration scheme, highly enhancing the overall numerical stability and precision of the method.

Other three studies concern the second research line, on interacting bosons in continuous space.

4) In a paper in collaboration with an experimental team at the Host Institution [4], we have studied the position- and momentum-space breathing dynamics of trapped one-dimensional Bose gases at finite temperature. A comparison with theoretical models taking temperature into account has been provided. In momentum space, we have found a frequency doubling in the quasi-condensate regime, corresponding to a self-reflection mechanism due to the repulsive interactions.

5) In a work [5] in collaboration with an experimental team in Florence, Italy, we have studied transport properties in one-dimensional quantum gases subjecteded to a weak periodic potential. We have compared a theoretical analysis based on quantum Monte Carlo simulations in continuous space and Luttinger liquid approach with experiments on ultracold atoms with tunable interactions and optical lattice depth. Experiments and theory are in excellent agreement. Our study has also provided the first quantitative determination of the critical parameters for the Mott transition in continuous space and has defined the regimes of validity of widely used approximate models, namely, the Bose-Hubbard and sine-Gordon models.

6) In a work being now finalized [6], we have provided an extension of the Time-Dependent Variational Monte Carlo to continuous-space quantum hamiltonians. This approach is used in conjunction with a systematic multi-body expansion of the many-body wave-function in terms of time-dependent Jastrow-Feenberg variational states. The resulting variational description has been demonstrated for the Lieb-Liniger model of interacting one-dimensional quantum gases, susbstantially extending the state-of-the-art in the numerical simulation of quantum gases in continous space.

--Final results and future impact--

The present Project has succesfully achieved the main scientific goals we had fixed at its beginning.
Significative advances both from the methodological point of view and from the point of view of physical insights have been obtained.
We believe that methodological advancements obtained during this Project will most probably have a lasting impact on the study of dynamical properties of strongly-interacting quantum systems. In particular, several new systems and regimes that were previously unaccessible by means of state-of-the-art methods are now accessible with good accuracy. These include : high-dimensional systems, long-time dynamical phenomena and quantum gases in continous space.

[1] Light-cone effect and supersonic correlations in one- and two-dimensional bosonic superfluids G Carleo, F Becca, L Sanchez-Palencia, S Sorella, M Fabrizio, Physical Review A 89 (3), 031602(R), (2014)

[2] Protected quasi-locality in quantum systems with long-range interactions – by L Cevolani, G Carleo, and L. Sanchez-Palencia – published on Physical Review A 92, 041603(R) (2015)

[3] G Carleo et al, in preparation

[4] Quench-induced Breathing Mode of One-Dimensional Bose Gases – by B Fang, G Carleo, A Johnson, I Bouchoule – published on Physical Review Letters 113, 035301 (2014)

[5] Mott transition for strongly interacting one-dimensional bosons in a shallow periodic potential – by G Boeris, L Gori, MD Hoogerland, A Kumar, E Lucioni, L Tanzi, M Inguscio, T Giamarchi, C D'Errico, G Carleo, G Modugno, L Sanchez-Palencia – published on Physical Review A 93, 011601(R) (2016)

[6] Time-dependent variational Monte Carlo for Many-Body Quantum Systems in Continous Space – by G Carleo, L Cevolani, L. Sanchez-Palencia, and M. Holzmann – To be submitted (2016)