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Tabanera-Bravo J, Parrondo JMR, Esposito M, Barra F. Thermalization and Dephasing in Collisional Reservoirs. PHYSICAL REVIEW LETTERS 2023; 130:200402. [PMID: 37267544 DOI: 10.1103/physrevlett.130.200402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2023] [Accepted: 04/10/2023] [Indexed: 06/04/2023]
Abstract
We introduce a wide class of quantum maps that arise in collisional reservoirs and are able to thermalize a system if they operate in conjunction with an additional dephasing mechanism. These maps describe the effect of collisions and induce transitions between populations that obey detailed balance, but also create coherences that prevent the system from thermalizing. We combine these maps with a unitary evolution acting during random Poissonian times between collisions and causing dephasing. We find that, at a low collision rate, the nontrivial combination of these two effects causes thermalization in the system. This scenario is suitable for modeling collisional reservoirs at equilibrium. We justify this claim by identifying the conditions for such maps to arise within a scattering theory approach and provide a thorough characterization of the resulting thermalization process.
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Affiliation(s)
- Jorge Tabanera-Bravo
- Departamento de Estructura de la Materia, Física Térmica y Electrónica and GISC, Universidad Complutense de Madrid, Plaza de las Ciencias 1. 28040 Madrid, Spain
| | - Juan M R Parrondo
- Departamento de Estructura de la Materia, Física Térmica y Electrónica and GISC, Universidad Complutense de Madrid, Plaza de las Ciencias 1. 28040 Madrid, Spain
| | - Massimiliano Esposito
- Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, G.D. Luxembourg
| | - Felipe Barra
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, 837.0415 Santiago, Chile
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2
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Palafox S, Román-Ancheyta R, Çakmak B, Müstecaplıoğlu ÖE. Heat transport and rectification via quantum statistical and coherence asymmetries. Phys Rev E 2022; 106:054114. [PMID: 36559439 DOI: 10.1103/physreve.106.054114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2022] [Accepted: 10/19/2022] [Indexed: 06/17/2023]
Abstract
Recent experiments at the nanoscales confirm that thermal rectifiers, the thermal equivalent of electrical diodes, can operate in the quantum regime. We present a thorough investigation of the effect of different particle exchange statistics, coherence, and collective interactions on the quantum heat transport of rectifiers with two-terminal junctions. Using a collision model approach to describe the open system dynamics, we obtain a general expression of the nonlinear heat flow that fundamentally deviates from the Landauer formula whenever quantum statistical or coherence asymmetries are present in the bath particles. Building on this, we show that heat rectification is possible even with symmetric medium-bath couplings if the two baths differ in quantum statistics or coherence. Furthermore, the associated thermal conductance vanishes exponentially at low temperatures as in the Coulomb-blockade effect. However, at high temperatures it acquires a power-law behavior depending on the quantum statistics. Our results can be significant for heat management in hybrid open quantum systems or solid-state thermal circuits.
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Affiliation(s)
- Stephania Palafox
- Instituto Nacional de Astrofísica, Óptica y Electrónica, Calle Luis Enrique Erro No.1 Santa María Tonantzintla, Puebla CP 72840, Mexico
| | - Ricardo Román-Ancheyta
- Instituto Nacional de Astrofísica, Óptica y Electrónica, Calle Luis Enrique Erro No.1 Santa María Tonantzintla, Puebla CP 72840, Mexico
| | - Barış Çakmak
- College of Engineering and Natural Sciences, Bahçeşehir University, Beşiktaş, Istanbul 34353, Türkiye
- TUBITAK Research Institute for Fundamental Sciences, 41470 Gebze, Türkiye
| | - Özgür E Müstecaplıoğlu
- TUBITAK Research Institute for Fundamental Sciences, 41470 Gebze, Türkiye
- Department of Physics, Koç University, İstanbul, Sarıyer, 34450, Türkiye
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3
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Cusumano S. Quantum Collision Models: A Beginner Guide. ENTROPY (BASEL, SWITZERLAND) 2022; 24:1258. [PMID: 36141146 PMCID: PMC9497832 DOI: 10.3390/e24091258] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/05/2022] [Revised: 08/26/2022] [Accepted: 08/28/2022] [Indexed: 06/16/2023]
Abstract
In recent years, quantum collision models, sometimes dubbed repeated interaction models, have gained much attention due to their simplicity and their capacity to convey ideas without resorting to technical complications typical of many approaches and techniques used in the field of open quantum systems. In this tutorial, we show how to use these models, highlighting their strengths and some technical subtleties often overlooked in the literature. We do this by deriving the Markovian master equation and comparing the standard collisional derivation with the standard microscopic one. We then use the collision model to derive the master equation of a two-level system interacting with either a bosonic or fermionic bath to give the reader a flavour of the real use of the model.
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Affiliation(s)
- Stefano Cusumano
- International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308 Gdańsk, Poland
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4
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Entropy Production in Non-Markovian Collision Models: Information Backflow vs. System-Environment Correlations. ENTROPY 2022; 24:e24060824. [PMID: 35741544 PMCID: PMC9222943 DOI: 10.3390/e24060824] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/26/2022] [Revised: 06/10/2022] [Accepted: 06/12/2022] [Indexed: 12/04/2022]
Abstract
We investigate the irreversible entropy production of a qubit in contact with an environment modelled by a microscopic collision model in both Markovian and non-Markovian regimes. Our main goal is to contribute to the discussions on the relationship between non-Markovian dynamics and negative entropy production rates. We employ two different types of collision models that do or do not keep the correlations established between the system and the incoming environmental particle, while both of them pertain to their non-Markovian nature through information backflow from the environment to the system. We observe that as the former model, where the correlations between the system and environment are preserved, gives rise to negative entropy production rates in the transient dynamics, the latter one always maintains positive rates, even though the convergence to the steady-state value is slower as compared to the corresponding Markovian dynamics. Our results suggest that the mechanism underpinning the negative entropy production rates is not solely non-Markovianity through information backflow, but rather the contribution to it through established system-environment correlations.
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Beyer K, Luoma K, Lenz T, Strunz WT. Measured Composite Collision Models: Quantum Trajectory Purities and Channel Divisibility. ENTROPY 2022; 24:e24050715. [PMID: 35626598 PMCID: PMC9142057 DOI: 10.3390/e24050715] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/17/2022] [Revised: 05/06/2022] [Accepted: 05/07/2022] [Indexed: 12/10/2022]
Abstract
We investigate a composite quantum collision model with measurements on the memory part, which effectively probe the system. The framework allows us to adjust the measurement strength, thereby tuning the dynamical map of the system. For a two-qubit setup with a symmetric and informationally complete measurement on the memory, we study the divisibility of the resulting dynamics in dependence of the measurement strength. The measurements give rise to quantum trajectories of the system and we show that the average asymptotic purity depends on the specific form of the measurement. With the help of numerical simulations, we demonstrate that the different performance of the measurements is generic and holds for almost all interaction gates between the system and the memory in the composite collision model. The discrete model is then extended to a time-continuous limit.
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Affiliation(s)
- Konstantin Beyer
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany; (T.L.); (W.T.S.)
- Correspondence:
| | - Kimmo Luoma
- Laboratory of Quantum Optics and Turku Center for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun Yliopisto, Finland;
| | - Tim Lenz
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany; (T.L.); (W.T.S.)
| | - Walter T. Strunz
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany; (T.L.); (W.T.S.)
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6
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Li Y, Li X, Jin J. Dissipation-Induced Information Scrambling in a Collision Model. ENTROPY (BASEL, SWITZERLAND) 2022; 24:345. [PMID: 35327856 PMCID: PMC8947106 DOI: 10.3390/e24030345] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/27/2022] [Revised: 02/25/2022] [Accepted: 02/25/2022] [Indexed: 12/04/2022]
Abstract
In this paper, we present a collision model to stroboscopically simulate the dynamics of information in dissipative systems. In particular, an all-optical scheme is proposed to investigate the information scrambling of bosonic systems with Gaussian environmental states. Varying the states of environments, in the presence of dissipation, transient tripartite mutual information of system modes may show negative value, signaling the appearance of information scrambling. We also find that dynamical indivisibility based non-Markovianity plays dual roles in affecting the dynamics of information.
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Affiliation(s)
| | | | - Jiasen Jin
- School of Physics, Dalian University of Technology, Dalian 116024, China; (Y.L.); (X.L.)
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7
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Heat Modulation on Target Thermal Bath via Coherent Auxiliary Bath. ENTROPY 2021; 23:e23091183. [PMID: 34573807 PMCID: PMC8464766 DOI: 10.3390/e23091183] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/16/2021] [Revised: 08/22/2021] [Accepted: 09/05/2021] [Indexed: 12/22/2022]
Abstract
We study a scheme of thermal management where a three-qubit system assisted with a coherent auxiliary bath (CAB) is employed to implement heat management on a target thermal bath (TTB). We consider the CAB/TTB being ensemble of coherent/thermal two-level atoms (TLAs), and within the framework of collision model investigate the characteristics of steady heat current (also called target heat current (THC)) between the system and the TTB. It demonstrates that with the help of the quantum coherence of ancillae the magnitude and direction of heat current can be controlled only by adjusting the coupling strength of system-CAB. Meanwhile, we also show that the influences of quantum coherence of ancillae on the heat current strongly depend on the coupling strength of system—CAB, and the THC becomes positively/negatively correlated with the coherence magnitude of ancillae when the coupling strength below/over some critical value. Besides, the system with the CAB could serve as a multifunctional device integrating the thermal functions of heat amplifier, suppressor, switcher and refrigerator, while with thermal auxiliary bath it can only work as a thermal suppressor. Our work provides a new perspective for the design of multifunctional thermal device utilizing the resource of quantum coherence from the CAB.
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8
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Seah S, Perarnau-Llobet M, Haack G, Brunner N, Nimmrichter S. Quantum Speed-Up in Collisional Battery Charging. PHYSICAL REVIEW LETTERS 2021; 127:100601. [PMID: 34533344 DOI: 10.1103/physrevlett.127.100601] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2021] [Revised: 07/01/2021] [Accepted: 08/06/2021] [Indexed: 06/13/2023]
Abstract
We present a collision model for the charging of a quantum battery by identical nonequilibrium qubit units. When the units are prepared in a mixture of energy eigenstates, the energy gain in the battery can be described by a classical random walk, where both average energy and variance grow linearly with time. Conversely, when the qubits contain quantum coherence, interference effects buildup in the battery and lead to a faster spreading of the energy distribution, reminiscent of a quantum random walk. This can be exploited for faster and more efficient charging of a battery initialized in the ground state. Specifically, we show that coherent protocols can yield higher charging power than any possible incoherent strategy, demonstrating a quantum speed-up at the level of a single battery. Finally, we characterize the amount of extractable work from the battery through the notion of ergotropy.
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Affiliation(s)
- Stella Seah
- Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland
| | | | - Géraldine Haack
- Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland
| | - Nicolas Brunner
- Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland
| | - Stefan Nimmrichter
- Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Siegen 57068, Germany
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9
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Tian F, Zou J, Li L, Li H, Shao B. Effect of Inter-System Coupling on Heat Transport in a Microscopic Collision Model. ENTROPY 2021; 23:e23040471. [PMID: 33923510 PMCID: PMC8073798 DOI: 10.3390/e23040471] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/15/2021] [Revised: 04/07/2021] [Accepted: 04/12/2021] [Indexed: 11/26/2022]
Abstract
In this paper we consider a bipartite system composed of two subsystems each coupled to its own thermal environment. Based on a collision model, we mainly study whether the approximation (i.e., the inter-system coupling is ignored when modeling the system–environment interaction) is valid or not. We also address the problem of heat transport unitedly for both excitation-conserving system–environment interactions and non-excitation-conserving system–environment interactions. For the former interaction, as the inter-system interaction strength increases, at first this approximation gets worse as expected, but then counter-intuitively gets better even for a stronger inter-system coupling. For the latter interaction with asymmetry, this approximation gets progressively worse. In this case we realize a perfect thermal rectification, and we cannot find an apparent rectification effect for the former interaction. Finally and more importantly, our results show that whether this approximation is valid or not is closely related to the quantum correlations between the subsystems, i.e., the weaker the quantum correlations, the more justified the approximation and vice versa.
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Affiliation(s)
- Feng Tian
- School of Physics, Beijing Institute of Technology, Beijing 100081, China; (F.T.); (B.S.)
| | - Jian Zou
- School of Physics, Beijing Institute of Technology, Beijing 100081, China; (F.T.); (B.S.)
- Correspondence:
| | - Lei Li
- School of Physical Science and Technology, Inner Mongolia University, Hohhot 010021, China;
| | - Hai Li
- School of Information and Electronic Engineering, Shandong Technology and Business University, Yantai 264005, China;
| | - Bin Shao
- School of Physics, Beijing Institute of Technology, Beijing 100081, China; (F.T.); (B.S.)
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10
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Cattaneo M, De Chiara G, Maniscalco S, Zambrini R, Giorgi GL. Collision Models Can Efficiently Simulate Any Multipartite Markovian Quantum Dynamics. PHYSICAL REVIEW LETTERS 2021; 126:130403. [PMID: 33861108 DOI: 10.1103/physrevlett.126.130403] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2020] [Accepted: 03/02/2021] [Indexed: 06/12/2023]
Abstract
We introduce the multipartite collision model, defined in terms of elementary interactions between subsystems and ancillas, and show that it can simulate the Markovian dynamics of any multipartite open quantum system. We develop a method to estimate an analytical error bound for any repeated interactions model, and we use it to prove that the error of our scheme displays an optimal scaling. Finally, we provide a simple decomposition of the multipartite collision model into elementary quantum gates, and show that it is efficiently simulable on a quantum computer according to the dissipative quantum Church-Turing theorem, i.e., it requires a polynomial number of resources.
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Affiliation(s)
- Marco Cattaneo
- Instituto de Física Interdisciplinar y Sistemas Complejos (IFISC, UIB-CSIC), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
- QTF Centre of Excellence, Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun Yliopisto, Finland
- QTF Centre of Excellence, Department of Physics, University of Helsinki, P.O. Box 43, FI-00014 Helsinki, Finland
| | - Gabriele De Chiara
- Centre for Theoretical Atomic, Molecular and Optical Physics, Queen's University Belfast, Belfast BT7 1NN, United Kingdom
| | - Sabrina Maniscalco
- QTF Centre of Excellence, Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun Yliopisto, Finland
- QTF Centre of Excellence, Department of Physics, University of Helsinki, P.O. Box 43, FI-00014 Helsinki, Finland
- QTF Centre of Excellence, Department of Applied Physics, School of Science, Aalto University, FI-00076 Aalto, Finland
| | - Roberta Zambrini
- Instituto de Física Interdisciplinar y Sistemas Complejos (IFISC, UIB-CSIC), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Gian Luca Giorgi
- Instituto de Física Interdisciplinar y Sistemas Complejos (IFISC, UIB-CSIC), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
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11
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Reis LH, Silva SHS, Pereira E. Beyond the Lindblad master equation: Heat, work, and energy currents in boundary-driven spin chains. Phys Rev E 2020; 101:062107. [PMID: 32688597 DOI: 10.1103/physreve.101.062107] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2020] [Accepted: 05/18/2020] [Indexed: 11/07/2022]
Abstract
We consider accurate investigation of the energy current and its components, heat and work, in some boundary-driven quantum spin systems. The expressions for the currents, as well as the associated Lindblad master equation, are obtained via a repeated interaction scheme. We consider small systems in order to analytically compute the steady distribution to study the current in the steady state. Asymmetrical XXZ and quantum Ising models are detailed analyzed. For the XXZ chain we present cases in which different compositions of heat and work currents, obtained via the repeated interaction protocol, lead to the same energy current, which may be obtained via the Lindblad master equation. For the quantum Ising chain, we describe a case of zero energy current and novanishing heat and work currents. Our findings make clear that to talk about heat in these boundary-driven spin quantum systems we must go beyond an investigation involving only the Lindblad master equation.
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Affiliation(s)
- Luís H Reis
- Departamento de Física-Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, CP 702, 30.161-970 Belo Horizonte MG, Brazil
| | - Saulo H S Silva
- Departamento de Física-Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, CP 702, 30.161-970 Belo Horizonte MG, Brazil
| | - Emmanuel Pereira
- Departamento de Física-Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, CP 702, 30.161-970 Belo Horizonte MG, Brazil
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12
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Joint Fluctuation Theorems for Sequential Heat Exchange. ENTROPY 2020; 22:e22070763. [PMID: 33286535 PMCID: PMC7517312 DOI: 10.3390/e22070763] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/05/2020] [Revised: 07/06/2020] [Accepted: 07/08/2020] [Indexed: 11/16/2022]
Abstract
We study the statistics of heat exchange of a quantum system that collides sequentially with an arbitrary number of ancillas. This can describe, for instance, an accelerated particle going through a bubble chamber. Unlike other approaches in the literature, our focus is on the joint probability distribution that heat Q1 is exchanged with ancilla 1, heat Q2 is exchanged with ancilla 2, and so on. This allows us to address questions concerning the correlations between the collisional events. For instance, if in a given realization a large amount of heat is exchanged with the first ancilla, then there is a natural tendency for the second exchange to be smaller. The joint distribution is found to satisfy a Fluctuation theorem of the Jarzynski–Wójcik type. Rather surprisingly, this fluctuation theorem links the statistics of multiple collisions with that of independent single collisions, even though the heat exchanges are statistically correlated.
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13
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Schaller G, Ablaßmayer J. Thermodynamics of the Coarse-Graining Master Equation. ENTROPY (BASEL, SWITZERLAND) 2020; 22:e22050525. [PMID: 33286296 PMCID: PMC7517020 DOI: 10.3390/e22050525] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/03/2020] [Revised: 04/30/2020] [Accepted: 05/01/2020] [Indexed: 06/12/2023]
Abstract
We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad-Gorini-Kossakowski-Sudarshan generator. By combining the formalism with full counting statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be defined with respect to the reservoir. We exemplify our findings with simple but pedagogical examples.
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14
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Quantum-Heat Fluctuation Relations in Three-Level Systems Under Projective Measurements. CONDENSED MATTER 2020. [DOI: 10.3390/condmat5010017] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
We study the statistics of energy fluctuations in a three-level quantum system subject to a sequence of projective quantum measurements. We check that, as expected, the quantum Jarzynski equality holds provided that the initial state is thermal. The latter condition is trivially satisfied for two-level systems, while this is generally no longer true for N-level systems, with N > 2 . Focusing on three-level systems, we discuss the occurrence of a unique energy scale factor β eff that formally plays the role of an effective inverse temperature in the Jarzynski equality. To this aim, we introduce a suitable parametrization of the initial state in terms of a thermal and a non-thermal component. We determine the value of β eff for a large number of measurements and study its dependence on the initial state. Our predictions could be checked experimentally in quantum optics.
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15
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Hewgill A, González JO, Palao JP, Alonso D, Ferraro A, De Chiara G. Three-qubit refrigerator with two-body interactions. Phys Rev E 2020; 101:012109. [PMID: 32069534 DOI: 10.1103/physreve.101.012109] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2019] [Indexed: 06/10/2023]
Abstract
We propose a three-qubit setup for the implementation of a variety of quantum thermal machines where all heat fluxes and work production can be controlled. An important configuration that can be designed is that of an absorption refrigerator, extracting heat from the coldest reservoir without the need of external work supply. Remarkably, we achieve this regime by using only two-body interactions instead of the widely employed three-body interactions. This configuration could be more easily realized in current experimental setups. We model the open-system dynamics with both a global and a local master equation thermodynamic-consistent approach. Finally, we show how this model can be employed as a heat valve, in which by varying the local field of one of the two qubits allows one to control and amplify the heat current between the other qubits.
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Affiliation(s)
- Adam Hewgill
- Centre for Theoretical Atomic, Molecular and Optical Physics, Queen's University Belfast, Belfast BT7 1NN, United Kingdom
| | - J Onam González
- Dpto. de Física and IUdEA: Instituto Universitario de Estudios Avanzados, Universidad de La Laguna, 38203 Spain
| | - José P Palao
- Dpto. de Física and IUdEA: Instituto Universitario de Estudios Avanzados, Universidad de La Laguna, 38203 Spain
| | - Daniel Alonso
- Dpto. de Física and IUdEA: Instituto Universitario de Estudios Avanzados, Universidad de La Laguna, 38203 Spain
| | - Alessandro Ferraro
- Centre for Theoretical Atomic, Molecular and Optical Physics, Queen's University Belfast, Belfast BT7 1NN, United Kingdom
| | - Gabriele De Chiara
- Centre for Theoretical Atomic, Molecular and Optical Physics, Queen's University Belfast, Belfast BT7 1NN, United Kingdom
- Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, F-34095 Montpellier, France
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