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Arranz FJ, Montes J, Borondo F. Order-chaos transition in correlation diagrams and quantization of period orbits. Phys Rev E 2023; 108:034210. [PMID: 37849198 DOI: 10.1103/physreve.108.034210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2023] [Accepted: 08/03/2023] [Indexed: 10/19/2023]
Abstract
Eigenlevel correlation diagrams has proven to be a very useful tool to understand eigenstate characteristics of classically chaotic systems. In particular, we showed in a previous publication [Phys. Rev. Lett. 80, 944 (1998)0031-900710.1103/PhysRevLett.80.944] how to unveil the scarring mechanism, a cornerstone in the theory of quantum chaos, using the Planck constant as the correlation parameter. By increasing the Planck constant, we induced a transition from order to chaos, in which scarred wave functions appeared as the interaction of pairs of eigenstates in broad avoided crossings, forming a well-defined frontier in the correlation diagram. In this paper, we demonstrate that this frontier can be obtained by means of the semiclassical quantization of the involved scarring periodic orbits. Additionally, in order to calculate the Maslov index of each scarring periodic orbit, which is necessary for the semiclassical quantization procedure, we introduce a straightforward method based on Lagrangian descriptors. We illustrate the theory using the vibrational eigenstates of the LiCN molecular system.
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Affiliation(s)
- F J Arranz
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - J Montes
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
| | - F Borondo
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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2
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Arranz FJ, Benito RM, Borondo F. Correspondence between classical and quantum resonances. Phys Rev E 2021; 103:062207. [PMID: 34271628 DOI: 10.1103/physreve.103.062207] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2020] [Accepted: 05/12/2021] [Indexed: 11/07/2022]
Abstract
Bifurcations take place in molecular Hamiltonian nonlinear systems as the excitation energy increases, leading to the appearance of different classical resonances. In this paper, we study the quantum manifestations of these classical resonances in the isomerizing system CN-Li⇆Li-CN. By using a correlation diagram of eigenenergies versus Planck constant, we show the existence of different series of avoided crossings, leading to the corresponding series of quantum resonances, which represent the quantum manifestations of the classical resonances. Moreover, the extrapolation of these series to ℏ=0 unveils the correspondence between the bifurcation energy of classical resonances and the energy of the series of quantum resonances in the semiclassical limit ℏ→0. Additionally, in order to obtain analytical expressions for our results, a semiclassical theory is developed.
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Affiliation(s)
- F J Arranz
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, Avenida Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - R M Benito
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, Avenida Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain.,Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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3
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Párraga H, Arranz FJ, Benito RM, Borondo F. Using correlation diagrams to study the vibrational spectrum of highly nonlinear floppy molecules: The K-CN case. Phys Rev E 2020; 101:062215. [PMID: 32688478 DOI: 10.1103/physreve.101.062215] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2019] [Accepted: 05/18/2020] [Indexed: 06/11/2023]
Abstract
The correlation diagrams of vibrational energy levels considering the Planck constant as a variable parameter have proven as a very useful tool to study vibrational molecular states, and more specifically in relation to the quantum manifestations of chaos in such dynamical systems. In this paper, we consider the highly nonlinear K-CN molecule, showing how the regular classical structures, i.e., Kolmogorov-Arnold-Moser tori, existing in the mixed classical phase space appear in the quantum levels correlation diagram as emerging diabatic states, something that remains hidden when only the actual value of the Planck constant is considered. Additionally, a quantum transition from order to chaos is unveiled with the aid of these correlation diagrams, where it appears as a frontier of scarred functions.
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Affiliation(s)
- H Párraga
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F J Arranz
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - R M Benito
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco-28049 Madrid, Spain
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco-28049 Madrid, Spain
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4
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Arranz FJ, Benito RM, Borondo F. Shannon entropy at avoided crossings in the quantum transition from order to chaos. Phys Rev E 2019; 99:062209. [PMID: 31330703 DOI: 10.1103/physreve.99.062209] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2019] [Indexed: 11/06/2022]
Abstract
Shannon entropy is studied for the series of avoided crossings that characterize the transition from order to chaos in quantum mechanics. In order to be able to study jointly this entropy for discrete and continuous probability, calculations have been performed on a quantized map, the kicked Harper map, resulting in a different behavior, as order-chaos transition takes place, for the discrete (position representation) and continuous (coherent state representation) cases. This different behavior is analyzed in terms of the distribution of zeros of the Husimi function.
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Affiliation(s)
- F J Arranz
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, av. Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - R M Benito
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, av. Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain.,Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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Park KW, Moon S, Shin Y, Kim J, Jeong K, An K. Shannon entropy and avoided crossings in closed and open quantum billiards. Phys Rev E 2018; 97:062205. [PMID: 30011575 DOI: 10.1103/physreve.97.062205] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2017] [Indexed: 11/07/2022]
Abstract
The relation between Shannon entropy and avoided crossings is investigated in dielectric microcavities. The Shannon entropy of the probability density for eigenfunctions in an open elliptic billiard as well as a closed quadrupole billiard increases as the center of the avoided crossing is approached. These results are opposite to those of atomic physics for electrons. It is found that the collective Lamb shift of the open quantum system and the symmetry breaking in the closed chaotic quantum system have equivalent effects on the Shannon entropy.
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Affiliation(s)
- Kyu-Won Park
- School of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
| | - Songky Moon
- School of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
| | - Younghoon Shin
- School of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
| | - Jinuk Kim
- School of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
| | - Kabgyun Jeong
- IMDARC, Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea
| | - Kyungwon An
- School of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
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Párraga H, Arranz FJ, Benito RM, Borondo F. Above Saddle-Point Regions of Order in a Sea of Chaos in the Vibrational Dynamics of KCN. J Phys Chem A 2018. [DOI: 10.1021/acs.jpca.8b00113] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- H. Párraga
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F. J. Arranz
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - R. M. Benito
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F. Borondo
- Instituto de Ciencias Matemáticas, Cantoblanco, 28049 Madrid, Spain
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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Johnson BA, Sibert EL. Assigning the low lying vibronic states of CH 3O and CD 3O. J Chem Phys 2017; 146:174112. [PMID: 28477600 DOI: 10.1063/1.4981795] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023] Open
Abstract
The assignment of lines in vibrational spectra in strongly mixing systems is considered. Several low lying vibrational states of the ground electronic X∼2E state of the CH3O and CD3O radicals are assigned. Jahn-Teller, spin-orbit, and Fermi couplings mix the normal mode states. The mixing complicates the assignment of the infrared spectra using a zero-order normal mode representation. Alternative zero-order representations, which include specific Jahn-Teller couplings, are explored. These representations allow for definitive assignments. In many instances it is possible to plot the wavefunctions on which the assignments are based. The plots, which are shown in the adiabatic representation, allow one to visualize the effects of various higher order couplings. The plots also enable one to visualize the conical seam and its effect on the wavefunctions. The first and the second order Jahn-Teller couplings in the rocking motion dominate the spectral features in CH3O, while first order and modulated first order couplings dominate the spectral features in CD3O. The methods described here are general and can be applied to other Jahn-Teller systems.
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Affiliation(s)
- Britta A Johnson
- Department of Chemistry and Theoretical Chemistry Institute, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
| | - Edwin L Sibert
- Department of Chemistry and Theoretical Chemistry Institute, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
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Revuelta F, Vergini E, Benito RM, Borondo F. Semiclassical basis sets for the computation of molecular vibrational states. J Chem Phys 2017; 146:014107. [PMID: 28063420 DOI: 10.1063/1.4973376] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022] Open
Abstract
In this paper, we extend a method recently reported [F. Revuelta et al., Phys. Rev. E 87, 042921 (2013)] for the calculation of the eigenstates of classically highly chaotic systems to cases of mixed dynamics, i.e., those presenting regular and irregular motions at the same energy. The efficiency of the method, which is based on the use of a semiclassical basis set of localized wave functions, is demonstrated by applying it to the determination of the vibrational states of a realistic molecular system, namely, the LiCN molecule.
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Affiliation(s)
- F Revuelta
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - E Vergini
- Departamento de Física, Comisión Nacional de Energía Atómica, Av. del Libertador 8250, 1429 Buenos Aires, Argentina
| | - R M Benito
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), 28049 Cantoblanco, Madrid, Spain
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Lopez-Pina A, Losada JC, Benito RM, Borondo F. Frequency analysis of the laser driven nonlinear dynamics of HCN. J Chem Phys 2016; 145:244309. [PMID: 28049318 DOI: 10.1063/1.4972260] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We study the vibrational dynamics of a model for the HCN molecule in the presence of a monochromatic laser field. The variation of the structural behavior of the system as a function of the laser frequency is analyzed in detail using the smaller alignment index, frequency maps, and diffusion coefficients. It is observed that the ergodicity of the system depends on the frequency of the excitation field, especially in its transitions from and into chaos. This provides a roadmap for the possibility of bond excitation and dissociation in this molecule.
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Affiliation(s)
- A Lopez-Pina
- Grupo de Sistemas Complejos and Unidad de Física, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, E-28040 Madrid, Spain
| | - J C Losada
- Grupo de Sistemas Complejos and Unidad de Física, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, E-28040 Madrid, Spain
| | - R M Benito
- Grupo de Sistemas Complejos and Unidad de Física, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, E-28040 Madrid, Spain
| | - F Borondo
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain
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Yi CH, Kim JH, Yu HH, Lee JW, Kim CM. Fermi resonance in dynamical tunneling in a chaotic billiard. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022916. [PMID: 26382485 DOI: 10.1103/physreve.92.022916] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/19/2014] [Indexed: 06/05/2023]
Abstract
We elucidate that Fermi resonance ever plays a decisive role in dynamical tunneling in a chaotic billiard. Interacting with each other through an avoided crossing, a pair of eigenfunctions are coupled through tunneling channels for dynamical tunneling. In this case, the tunneling channels are an islands chain and its pair unstable periodic orbit, which equals the quantum number difference of the eigenfunctions. This phenomenon of dynamical tunneling is confirmed in a quadrupole billiard in relation with Fermi resonance.
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Affiliation(s)
- Chang-Hwan Yi
- Department of Emerging Materials Science, DGIST. Hyeonpung-myeon Dalseong-gun, Daegu 711-873, Korea
| | - Ji-Hwan Kim
- Department of Physics, Sogang University, Seoul, 121-742, Korea
| | - Hyeon-Hye Yu
- Department of Physics, Sogang University, Seoul, 121-742, Korea
| | - Ji-Won Lee
- Department of Emerging Materials Science, DGIST. Hyeonpung-myeon Dalseong-gun, Daegu 711-873, Korea
| | - Chil-Min Kim
- Department of Emerging Materials Science, DGIST. Hyeonpung-myeon Dalseong-gun, Daegu 711-873, Korea
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Yi CH, Yu HH, Lee JW, Kim CM. Fermi resonance in optical microcavities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042903. [PMID: 25974558 DOI: 10.1103/physreve.91.042903] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/12/2014] [Indexed: 06/04/2023]
Abstract
Fermi resonance is a phenomenon of quantum mechanical superposition, which most often occurs between normal and overtone modes in molecular systems that are nearly coincident in energy. We find that scarred resonances in deformed dielectric microcavities are the very phenomenon of Fermi resonance, that is, a pair of quasinormal modes interact with each other due to coupling and a pair of resonances are generated through an avoided resonance crossing. Then the quantum number difference of a pair of quasinormal modes, which is a consequence of quantum mechanical superposition, equals periodic orbits, whereby the resonances are localized on the periodic orbits. We derive the relation between the quantum number difference and the periodic orbits and confirm it in an elliptic, a rectangular, and a stadium-shaped dielectric microcavity.
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Affiliation(s)
- Chang-Hwan Yi
- Department of Physics, Sogang University, Seoul 121-742, Korea
| | - Hyeon-Hye Yu
- Department of Physics, Sogang University, Seoul 121-742, Korea
| | - Ji-Won Lee
- Department of Physics, Sogang University, Seoul 121-742, Korea
| | - Chil-Min Kim
- Department of Emerging Materials Science, Daegu Gyeongbuk Institute of Science and Technology, Hyeonpung-myeon Dalseong-gun, Daegu 711-873, Korea
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Garcia-Muller PL, Hernandez R, Benito RM, Borondo F. The role of the CN vibration in the activated dynamics of LiNC⇌LiCN isomerization in an argon solvent at high temperatures. J Chem Phys 2014; 141:074312. [DOI: 10.1063/1.4892921] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022] Open
Affiliation(s)
- Pablo L. Garcia-Muller
- Centro de Investigaciones Energéticas Medioambientales y Tecnológicas, Avda. Complutense 40, 28040 Madrid, Spain
| | - Rigoberto Hernandez
- Center for Computational Molecular Sciences and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, USA
| | - R. M. Benito
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, 28040 Madrid, Spain and Departamento de Física y Mecánica, Escuela Técnica Superior de Ingenieros Agrónomos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F. Borondo
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain and Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
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Párraga H, Arranz FJ, Benito RM, Borondo F. Ab initio potential energy surface for the highly nonlinear dynamics of the KCN molecule. J Chem Phys 2013; 139:194304. [DOI: 10.1063/1.4830102] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Arranz FJ, Seidel L, Giralda CG, Benito RM, Borondo F. Onset of quantum chaos in molecular systems and the zeros of the Husimi function. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062901. [PMID: 23848742 DOI: 10.1103/physreve.87.062901] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2013] [Indexed: 06/02/2023]
Abstract
The distribution of zeros of the Husimi function has proven a useful tool in the characterization of regular and chaotic quantum states of dynamical systems. In a previous paper [Phys. Rev. E 82, 026201 (2010)], we showed how the quantum transition from order to chaos in three different molecular systems (LiCN, HCN, and HO(2)) can be understood by means of the correlation diagram of eigenenergies versus the Planck's constant. An order-chaos frontier of scars (eigenstates localized over unstable and stable complementary, in the sense of the Poincaré-Birkhoff theorem, periodic orbits) was observed. In this paper, we show how the distribution of zeros of the Husimi function can be related to the onset of chaos in these molecular systems, showing a very interesting feature at the frontier of scars; namely, some zeros of the Husimi function localize over the stable and unstable fixed points corresponding to the two complementary periodic orbits, this representing the quantum equivalent to the Poincaré-Birkhoff theorem.
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Affiliation(s)
- F J Arranz
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
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