1
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Ramesh VG, Peters KJH, Rodriguez SRK. Arcsine Laws of Light. PHYSICAL REVIEW LETTERS 2024; 132:133801. [PMID: 38613295 DOI: 10.1103/physrevlett.132.133801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2022] [Accepted: 02/06/2024] [Indexed: 04/14/2024]
Abstract
We demonstrate that the time-integrated light intensity transmitted by a coherently driven resonator obeys Lévy's arcsine laws-a cornerstone of extreme value statistics. We show that convergence to the arcsine distribution is algebraic, universal, and independent of nonequilibrium behavior due to nonconservative forces or nonadiabatic driving. We furthermore verify, numerically, that the arcsine laws hold in the presence of frequency noise and in Kerr-nonlinear resonators supporting non-Gaussian states. The arcsine laws imply a weak ergodicity breaking which can be leveraged to enhance the precision of resonant optical sensors with zero energy cost, as shown in our companion manuscript [V. G. Ramesh et al., companion paper, Phys. Rev. Res. (2024).PPRHAI2643-1564]. Finally, we discuss perspectives for probing the possible breakdown of the arcsine laws in systems with memory.
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Affiliation(s)
- V G Ramesh
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
| | - K J H Peters
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
| | - S R K Rodriguez
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
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2
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Cabot A, Carollo F, Lesanovsky I. Continuous Sensing and Parameter Estimation with the Boundary Time Crystal. PHYSICAL REVIEW LETTERS 2024; 132:050801. [PMID: 38364170 DOI: 10.1103/physrevlett.132.050801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/2023] [Revised: 11/07/2023] [Accepted: 11/28/2023] [Indexed: 02/18/2024]
Abstract
A boundary time crystal is a quantum many-body system whose dynamics is governed by the competition between coherent driving and collective dissipation. It is composed of N two-level systems and features a transition between a stationary phase and an oscillatory one. The fact that the system is open allows one to continuously monitor its quantum trajectories and to analyze their dependence on parameter changes. This enables the realization of a sensing device whose performance we investigate as a function of the monitoring time T and of the system size N. We find that the best achievable sensitivity is proportional to sqrt[T]N, i.e., it follows the standard quantum limit in time and Heisenberg scaling in the particle number. This theoretical scaling can be achieved in the oscillatory time-crystal phase and it is rooted in emergent quantum correlations. The main challenge is, however, to tap this capability in a measurement protocol that is experimentally feasible. We demonstrate that the standard quantum limit can be surpassed by cascading two time crystals, where the quantum trajectories of one time crystal are used as input for the other one.
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Affiliation(s)
- Albert Cabot
- Institut für Theoretische Physik, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
| | - Federico Carollo
- Institut für Theoretische Physik, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
| | - Igor Lesanovsky
- Institut für Theoretische Physik, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
- School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
- Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
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3
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Downing CA, Vidiella-Barranco A. Parametrically driving a quantum oscillator into exceptionality. Sci Rep 2023; 13:11004. [PMID: 37419917 PMCID: PMC10329046 DOI: 10.1038/s41598-023-37964-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2023] [Accepted: 06/30/2023] [Indexed: 07/09/2023] Open
Abstract
The mathematical objects employed in physical theories do not always behave well. Einstein's theory of space and time allows for spacetime singularities and Van Hove singularities arise in condensed matter physics, while intensity, phase and polarization singularities pervade wave physics. Within dissipative systems governed by matrices, singularities occur at the exceptional points in parameter space whereby some eigenvalues and eigenvectors coalesce simultaneously. However, the nature of exceptional points arising in quantum systems described within an open quantum systems approach has been much less studied. Here we consider a quantum oscillator driven parametrically and subject to loss. This squeezed system exhibits an exceptional point in the dynamical equations describing its first and second moments, which acts as a borderland between two phases with distinctive physical consequences. In particular, we discuss how the populations, correlations, squeezed quadratures and optical spectra crucially depend on being above or below the exceptional point. We also remark upon the presence of a dissipative phase transition at a critical point, which is associated with the closing of the Liouvillian gap. Our results invite the experimental probing of quantum resonators under two-photon driving, and perhaps a reappraisal of exceptional and critical points within dissipative quantum systems more generally.
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Affiliation(s)
- C A Downing
- Department of Physics and Astronomy, University of Exeter, Exeter, EX4 4QL, UK.
| | - A Vidiella-Barranco
- Gleb Wataghin Institute of Physics, University of Campinas - UNICAMP, Campinas, SP, 13083-859, Brazil
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4
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He X, Yousefjani R, Bayat A. Stark Localization as a Resource for Weak-Field Sensing with Super-Heisenberg Precision. PHYSICAL REVIEW LETTERS 2023; 131:010801. [PMID: 37478450 DOI: 10.1103/physrevlett.131.010801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2023] [Accepted: 06/05/2023] [Indexed: 07/23/2023]
Abstract
Gradient fields can effectively suppress particle tunneling in a lattice and localize the wave function at all energy scales, a phenomenon known as Stark localization. Here, we show that Stark systems can be used as a probe for the precise measurement of gradient fields, particularly in the weak-field regime where most sensors do not operate optimally. In the extended phase, Stark probes achieve super-Heisenberg precision, which is well beyond most of the known quantum sensing schemes. In the localized phase, the precision drops in a universal way showing fast convergence to the thermodynamic limit. For single-particle probes, we show that quantum-enhanced sensitivity, with super-Heisenberg precision, can be achieved through a simple position measurement for all the eigenstates across the entire spectrum. For such probes, we have identified several critical exponents of the Stark localization transition and established their relationship. Thermal fluctuations, whose universal behavior is identified, reduce the precision from super-Heisenberg to Heisenberg, still outperforming classical sensors. Multiparticle interacting probes also achieve super-Heisenberg scaling in their extended phase, which shows even further enhancement near the transition point. Quantum-enhanced sensitivity is still achievable even when state preparation time is included in resource analysis.
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Affiliation(s)
- Xingjian He
- Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610051, China
| | - Rozhin Yousefjani
- Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610051, China
| | - Abolfazl Bayat
- Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610051, China
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5
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Saha M, Agarwalla BK, Kulkarni M, Purkayastha A. Universal Subdiffusive Behavior at Band Edges from Transfer Matrix Exceptional Points. PHYSICAL REVIEW LETTERS 2023; 130:187101. [PMID: 37204882 DOI: 10.1103/physrevlett.130.187101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2022] [Revised: 11/11/2022] [Accepted: 03/31/2023] [Indexed: 05/21/2023]
Abstract
We discover a deep connection between parity-time symmetric optical systems and quantum transport in one-dimensional fermionic chains in a two-terminal open system setting. The spectrum of one dimensional tight-binding chain with periodic on-site potential can be obtained by casting the problem in terms of 2×2 transfer matrices. We find that these non-Hermitian matrices have a symmetry exactly analogous to the parity-time symmetry of balanced-gain-loss optical systems, and hence show analogous transitions across exceptional points. We show that the exceptional points of the transfer matrix of a unit cell correspond to the band edges of the spectrum. When connected to two zero temperature baths at two ends, this consequently leads to subdiffusive scaling of conductance with system size, with an exponent 2, if the chemical potential of the baths are equal to the band edges. We further demonstrate the existence of a dissipative quantum phase transition as the chemical potential is tuned across any band edge. Remarkably, this feature is analogous to transition across a mobility edge in quasiperiodic systems. This behavior is universal, irrespective of the details of the periodic potential and the number of bands of the underlying lattice. It, however, has no analog in absence of the baths.
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Affiliation(s)
- Madhumita Saha
- Department of Physics, Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Ward No. 8, NCL Colony, Pashan, Pune, Maharashtra 411008, India
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
| | - Bijay Kumar Agarwalla
- Department of Physics, Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Ward No. 8, NCL Colony, Pashan, Pune, Maharashtra 411008, India
| | - Manas Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
| | - Archak Purkayastha
- School of Physics, Trinity College Dublin, Dublin 2, Ireland
- Center for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
- Department of Physics, Indian Institute of Technology, Hyderabad 502284, India
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6
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Del Pino J, Zilberberg O. Dynamical Gauge Fields with Bosonic Codes. PHYSICAL REVIEW LETTERS 2023; 130:171901. [PMID: 37172225 DOI: 10.1103/physrevlett.130.171901] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2022] [Revised: 03/31/2023] [Accepted: 04/04/2023] [Indexed: 05/14/2023]
Abstract
The quantum simulation of dynamical gauge field theories offers the opportunity to study complex high-energy physics with controllable low-energy devices. For quantum computation, bosonic codes promise robust error correction that exploits multiparticle redundancy in bosons. Here, we demonstrate how bosonic codes can be used to simulate dynamical gauge fields. We encode both matter and dynamical gauge fields in a network of resonators that are coupled via three-wave mixing. The mapping to a Z_{2} dynamical lattice gauge theory is established when the gauge resonators operate as Schrödinger cat states. We explore the optimal conditions under which the system preserves the required gauge symmetries. Our findings promote realizing high-energy models using bosonic codes.
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Affiliation(s)
- Javier Del Pino
- Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland
| | - Oded Zilberberg
- Department of Physics, University of Konstanz, 78464 Konstanz, Germany
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7
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Mylnikov VY, Potashin SO, Sokolovskii GS, Averkiev NS. Dissipative Phase Transition in Systems with Two-Photon Drive and Nonlinear Dissipation near the Critical Point. NANOMATERIALS 2022; 12:nano12152543. [PMID: 35893511 PMCID: PMC9332203 DOI: 10.3390/nano12152543] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/14/2022] [Revised: 07/15/2022] [Accepted: 07/21/2022] [Indexed: 11/16/2022]
Abstract
In this paper, we examine dissipative phase transition (DPT) near the critical point for a system with two-photon driving and nonlinear dissipations. The proposed mean-field theory, which explicitly takes into account quantum fluctuations, allowed us to describe properly the evolutionary dynamics of the system and to demonstrate new effects in its steady-state. We show that the presence of quantum fluctuations leads to a power-law dependence of the anomalous average at the phase transition point, with which the critical exponent is associated. Also, we investigate the effect of the quantum fluctuations on the critical point renormalization and demonstrate the existence of a two-photon pump “threshold”. It is noteworthy that the obtained results are in a good agreement with the numerical simulations.
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Affiliation(s)
- Valentin Yu. Mylnikov
- Ioffe Institute, 194021 St. Petersburg, Russia; (S.O.P.); (N.S.A.)
- Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia
- Correspondence: (V.Y.M.); (G.S.S.)
| | | | - Grigorii S. Sokolovskii
- Ioffe Institute, 194021 St. Petersburg, Russia; (S.O.P.); (N.S.A.)
- Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia
- Correspondence: (V.Y.M.); (G.S.S.)
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Ying ZJ, Felicetti S, Liu G, Braak D. Critical Quantum Metrology in the Non-Linear Quantum Rabi Model. ENTROPY 2022; 24:e24081015. [PMID: 35892995 PMCID: PMC9330817 DOI: 10.3390/e24081015] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/16/2022] [Revised: 07/17/2022] [Accepted: 07/19/2022] [Indexed: 02/01/2023]
Abstract
The quantum Rabi model (QRM) with linear coupling between light mode and qubit exhibits the analog of a second-order phase transition for vanishing mode frequency which allows for criticality-enhanced quantum metrology in a few-body system. We show that the QRM including a nonlinear coupling term exhibits much higher measurement precisions due to its first-order-like phase transition at finite frequency, avoiding the detrimental slowing-down effect close to the critical point of the linear QRM. When a bias term is added to the Hamiltonian, the system can be used as a fluxmeter or magnetometer if implemented in circuit QED platforms.
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Affiliation(s)
- Zu-Jian Ying
- School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
- Correspondence: (Z.-J.Y.); (S.F.); (D.B.)
| | - Simone Felicetti
- Institute for Complex Systems, National Research Council (ISC-CNR), 00185 Rome, Italy
- Correspondence: (Z.-J.Y.); (S.F.); (D.B.)
| | - Gang Liu
- School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
| | - Daniel Braak
- EP VI and Center for Electronic Correlations and Magnetism, University of Augsburg, 86135 Augsburg, Germany
- Correspondence: (Z.-J.Y.); (S.F.); (D.B.)
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9
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Luo XW, Zhang C, Du S. Quantum Squeezing and Sensing with Pseudo-Anti-Parity-Time Symmetry. PHYSICAL REVIEW LETTERS 2022; 128:173602. [PMID: 35570450 DOI: 10.1103/physrevlett.128.173602] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2021] [Accepted: 03/28/2022] [Indexed: 06/15/2023]
Abstract
The emergence of parity-time (PT) symmetry has greatly enriched our study of symmetry-enabled non-Hermitian physics, but the realization of quantum PT symmetry faces an intrinsic issue of unavoidable symmetry-breaking Langevin noises. Here we construct a quantum pseudo-anti-PT (pseudo-APT) symmetry in a two-mode bosonic system without involving Langevin noises. We show that the spontaneous pseudo-APT symmetry breaking leads to an exceptional point, across which there is a transition between different types of quantum squeezing dynamics; i.e., the squeezing factor increases exponentially (oscillates periodically) with time in the pseudo-APT-symmetric (broken) region. Such dramatic changes of squeezing factors and quantum dynamics near the exceptional point are utilized for ultraprecision quantum sensing. These exotic quantum phenomena and sensing applications can be experimentally observed in two physical systems: spontaneous wave mixing nonlinear optics and atomic Bose-Einstein condensates. Our Letter offers a physical platform for investigating exciting APT symmetry physics in the quantum realm, paving the way for exploring fundamental quantum non-Hermitian effects and their quantum technological applications.
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Affiliation(s)
- Xi-Wang Luo
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA
| | - Chuanwei Zhang
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA
| | - Shengwang Du
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA
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10
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Purkayastha A, Saha M, Agarwalla BK. Subdiffusive Phases in Open Clean Long-Range Systems. PHYSICAL REVIEW LETTERS 2021; 127:240601. [PMID: 34951774 DOI: 10.1103/physrevlett.127.240601] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2021] [Revised: 09/26/2021] [Accepted: 11/03/2021] [Indexed: 06/14/2023]
Abstract
We show that a one-dimensional ordered fermionic lattice system with power-law-decaying hopping, when connected to two baths at its two ends with different chemical potentials at zero temperature, features two phases showing subdiffusive scaling of conductance with system size. These phases have no analogues in the isolated system (i.e., in absence of the baths) where the transport is perfectly ballistic. In the open system scenario, interestingly, there occurs two chemical-potential-driven subdiffusive to ballistic phase transitions at zero temperature. We discuss how these phase transitions, to our knowledge, are different from all the known nonequilibrium quantum phase transitions. We provide a clear understanding of the microscopic origin of these phases and argue that the subdiffusive phases are robust against the presence of arbitrary number-conserving many-body interactions in the system. These phases showing subdiffusive scaling of conductance with system size in a two-terminal setup are therefore universal properties of all ordered one-dimensional number-conserving fermionic systems with power-law-decaying hopping at zero temperature.
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Affiliation(s)
- Archak Purkayastha
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
| | - Madhumita Saha
- Department of Physics, Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Ward No. 8, NCL Colony, Pashan, Pune, Maharashtra 411008, India
- Department of Physics, Indian Institute of Technology Gandhinagar, Palaj, Gujarat 382355, India
| | - Bijay Kumar Agarwalla
- Department of Physics, Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Ward No. 8, NCL Colony, Pashan, Pune, Maharashtra 411008, India
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11
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Han J, Cai W, Hu L, Mu X, Ma Y, Xu Y, Wang W, Wang H, Song YP, Zou CL, Sun L. Experimental Simulation of Open Quantum System Dynamics via Trotterization. PHYSICAL REVIEW LETTERS 2021; 127:020504. [PMID: 34296922 DOI: 10.1103/physrevlett.127.020504] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2020] [Revised: 04/23/2021] [Accepted: 06/14/2021] [Indexed: 06/13/2023]
Abstract
Digital quantum simulators provide a diversified tool for solving the evolution of quantum systems with complicated Hamiltonians and hold great potential for a wide range of applications. Although much attention is paid to the unitary evolution of closed quantum systems, dissipation and noise are vital in understanding the dynamics of practical quantum systems. In this work, we experimentally demonstrate a digital simulation of an open quantum system in a controllable Markovian environment with the assistance of a single ancillary qubit. By Trotterizing the quantum Liouvillians, the continuous evolution of an open quantum system is effectively realized, and its application in error mitigation is demonstrated by adjusting the simulated noise intensities. High-order Trotter for open quantum dynamics is also experimentally investigated and shows higher accuracy. Our results represent a significant step toward hardware-efficient simulation of open quantum systems and error mitigation in quantum algorithms in noisy intermediate-scale quantum systems.
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Affiliation(s)
- J Han
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - W Cai
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - L Hu
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - X Mu
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - Y Ma
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - Y Xu
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - W Wang
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - H Wang
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - Y P Song
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - C-L Zou
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - L Sun
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
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12
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Garbe L, Bina M, Keller A, Paris MGA, Felicetti S. Critical Quantum Metrology with a Finite-Component Quantum Phase Transition. PHYSICAL REVIEW LETTERS 2020; 124:120504. [PMID: 32281838 DOI: 10.1103/physrevlett.124.120504] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2019] [Accepted: 03/05/2020] [Indexed: 06/11/2023]
Abstract
Physical systems close to a quantum phase transition exhibit a divergent susceptibility, suggesting that an arbitrarily high precision may be achieved by exploiting quantum critical systems as probes to estimate a physical parameter. However, such an improvement in sensitivity is counterbalanced by the closing of the energy gap, which implies a critical slowing down and an inevitable growth of the protocol duration. Here, we design different metrological protocols that exploit the superradiant phase transition of the quantum Rabi model, a finite-component system composed of a single two-level atom interacting with a single bosonic mode. We show that, in spite of the critical slowing down, critical quantum optical probes can achieve a quantum-enhanced time scaling of the sensitivity in frequency-estimation protocols.
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Affiliation(s)
- Louis Garbe
- Université de Paris, Laboratoire Matériaux et Phénomènes Quantiques UMR 7162, CNRS, 75013, Paris, France
| | - Matteo Bina
- Quantum Technology Lab, Dipartimento di Fisica Aldo Pontremoli, Università degli Studi di Milano, I-20133 Milano, Italy
| | - Arne Keller
- Université de Paris, Laboratoire Matériaux et Phénomènes Quantiques UMR 7162, CNRS, 75013, Paris, France
- Université Paris-Saclay, 91405 Orsay, France
| | - Matteo G A Paris
- Quantum Technology Lab, Dipartimento di Fisica Aldo Pontremoli, Università degli Studi di Milano, I-20133 Milano, Italy
| | - Simone Felicetti
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
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