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Mao YL, Chen H, Niu C, Li ZD, Yu S, Fan J. Testing Heisenberg-Type Measurement Uncertainty Relations of Three Observables. PHYSICAL REVIEW LETTERS 2023; 131:150203. [PMID: 37897772 DOI: 10.1103/physrevlett.131.150203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2022] [Revised: 04/18/2023] [Accepted: 09/15/2023] [Indexed: 10/30/2023]
Abstract
Heisenberg-type measurement uncertainty relations (MURs) of two quantum observables are essential for contemporary research in quantum foundations and quantum information science. Going beyond, here we report the first experimental study of MUR of three quantum observables. We establish rigorously MURs for triplets of unbiased qubit observables as combined approximation errors lower bounded by an incompatibility measure, inspired by the proposal of Busch et al. [Phys. Rev. A 89, 012129 (2014)PLRAAN1050-294710.1103/PhysRevA.89.012129]. We develop a convex programming protocol to numerically find the exact value of the incompatibility measure and the optimal measurements. We propose a novel implementation of the optimal joint measurements and present several experimental demonstrations with a single-photon qubit. We stress that our method is universally applicable to the study of many qubit observables. Besides, we theoretically show that MURs for joint measurement can be attained by sequential measurements in two of our explored cases. We anticipate that this work may stimulate broad interests associated with Heisenberg's uncertainty principle in the case of multiple observables, enriching our understanding of quantum mechanics and inspiring innovative applications in quantum information science.
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Affiliation(s)
- Ya-Li Mao
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China
| | - Hu Chen
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China
- Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China
| | - Chang Niu
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zheng-Da Li
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China
- Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China
| | - Sixia Yu
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jingyun Fan
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China
- Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China
- Center for Advanced Light Source, Southern University of Science and Technology, Shenzhen, 518055, China
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2
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Sahoo SN, Chakraborti S, Pati AK, Sinha U. Quantum State Interferography. PHYSICAL REVIEW LETTERS 2020; 125:123601. [PMID: 33016750 DOI: 10.1103/physrevlett.125.123601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/13/2020] [Accepted: 08/11/2020] [Indexed: 06/11/2023]
Abstract
Quantum state tomography (QST) has been the traditional method for characterization of an unknown state. Recently, many direct measurement methods have been implemented to reconstruct the state in a resource efficient way. In this Letter, we present an interferometric method, in which any qubit state, whether mixed or pure, can be inferred from the visibility, phase shift, and average intensity of an interference pattern using a single-shot measurement-hence, we call it quantum state interferography. This provides us with a "black box" approach to quantum state estimation, wherein, between the incidence of the photon and extraction of state information, we are not changing any conditions within the setup, thus giving us a true single shot estimation of the quantum state. In contrast, standard QST requires at least two measurements for pure state qubit and at least three measurements for mixed state qubit reconstruction. We then go on to show that QSI is more resource efficient than QST for quantification of entanglement in pure bipartite qubits. We experimentally implement our method with high fidelity using the polarization degree of freedom of light. An extension of the scheme to pure states involving d-1 interferograms for d-dimensional systems is also presented. Thus, the scaling gain is even more dramatic in the qudit scenario for our method, where, in contrast, standard QST, without any assumptions, scales roughly as d^{2}.
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Affiliation(s)
| | | | - Arun K Pati
- Quantum Information and Computation Group, Harish-Chandra Research Institute, HBNI, Allahabad 211019, India
| | - Urbasi Sinha
- Light and Matter Physics, Raman Research Institute, Bengaluru 560080, India
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3
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Pan WW, Xu XY, Kedem Y, Wang QQ, Chen Z, Jan M, Sun K, Xu JS, Han YJ, Li CF, Guo GC. Direct Measurement of a Nonlocal Entangled Quantum State. PHYSICAL REVIEW LETTERS 2019; 123:150402. [PMID: 31702297 DOI: 10.1103/physrevlett.123.150402] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2019] [Revised: 06/24/2019] [Indexed: 06/10/2023]
Abstract
Entanglement and the wave function description are two of the core concepts that make quantum mechanics such a unique theory. A method to directly measure the wave function, using weak values, was demonstrated by Lundeen et al. [Nature 474, 188 (2011)]. However, it is not applicable to a scenario of two disjoint systems, where nonlocal entanglement can be a crucial element, since that requires obtaining weak values of nonlocal observables. Here, for the first time, we propose a method to directly measure a nonlocal wave function of a bipartite system, using modular values. The method is experimentally implemented for a photon pair in a hyperentangled state, i.e., entangled both in polarization and momentum degrees of freedom.
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Affiliation(s)
- Wei-Wei Pan
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Xiao-Ye Xu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Yaron Kedem
- Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden
| | - Qin-Qin Wang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Zhe Chen
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Munsif Jan
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Kai Sun
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Jin-Shi Xu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Yong-Jian Han
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Chuan-Feng Li
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Guang-Can Guo
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
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4
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Calderaro L, Foletto G, Dequal D, Villoresi P, Vallone G. Direct Reconstruction of the Quantum Density Matrix by Strong Measurements. PHYSICAL REVIEW LETTERS 2018; 121:230501. [PMID: 30576212 DOI: 10.1103/physrevlett.121.230501] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2018] [Indexed: 06/09/2023]
Abstract
New techniques based on weak measurements have recently been introduced to the field of quantum state reconstruction. Some of them allow the direct measurement of each matrix element of an unknown density operator and need only O(d) different operations, compared to d^{2} linearly independent projectors in the case of standard quantum state tomography, for the reconstruction of an arbitrary mixed state. However, due to the weakness of these couplings, these protocols are approximated and prone to large statistical errors. We propose a method which is similar to the weak measurement protocols but works regardless of the coupling strength: our protocol is not approximated and thus improves the accuracy and precision of the results with respect to weak measurement schemes. We experimentally apply it to the polarization state of single photons and compare the results to those of preexisting methods for different values of the coupling strength. Our results show that our method outperforms previous proposals in terms of accuracy and statistical errors.
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Affiliation(s)
- Luca Calderaro
- Dipartimento di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6B, 35131 Padova, Italy
- Centro di Ateneo di Studi e Attività Spaziali "Giuseppe Colombo", Università di Padova, via Venezia 15, 35131 Padova, Italy
| | - Giulio Foletto
- Dipartimento di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6B, 35131 Padova, Italy
| | - Daniele Dequal
- Matera Laser Ranging Observatory, Agenzia Spaziale Italiana, Matera 75100, Italy
| | - Paolo Villoresi
- Dipartimento di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6B, 35131 Padova, Italy
- Istituto di Fotonica e Nanotecnologie, CNR, via Trasea 7, 35131 Padova, Italy
| | - Giuseppe Vallone
- Dipartimento di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6B, 35131 Padova, Italy
- Istituto di Fotonica e Nanotecnologie, CNR, via Trasea 7, 35131 Padova, Italy
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5
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Zhou F, Yan L, Gong S, Ma Z, He J, Xiong T, Chen L, Yang W, Feng M, Vedral V. Verifying Heisenberg's error-disturbance relation using a single trapped ion. SCIENCE ADVANCES 2016; 2:e1600578. [PMID: 28861461 PMCID: PMC5566201 DOI: 10.1126/sciadv.1600578] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/18/2016] [Accepted: 09/20/2016] [Indexed: 06/07/2023]
Abstract
Heisenberg's uncertainty relations have played an essential role in quantum physics since its very beginning. The uncertainty relations in the modern quantum formalism have become a fundamental limitation on the joint measurements of general quantum mechanical observables, going much beyond the original discussion of the trade-off between knowing a particle's position and momentum. Recently, the uncertainty relations have generated a considerable amount of lively debate as a result of the new inequalities proposed as extensions of the original uncertainty relations. We report an experimental test of one of the new Heisenberg's uncertainty relations using a single 40Ca+ ion trapped in a harmonic potential. By performing unitary operations under carrier transitions, we verify the uncertainty relation proposed by Busch, Lahti, and Werner (BLW) based on a general error-trade-off relation for joint measurements on two compatible observables. The positive operator-valued measure, required by the compatible observables, is constructed by single-qubit operations, and the lower bound of the uncertainty, as observed, is satisfied in a state-independent manner. Our results provide the first evidence confirming the BLW-formulated uncertainty at a single-spin level and will stimulate broad interests in various fields associated with quantum mechanics.
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Affiliation(s)
- Fei Zhou
- State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China
| | - Leilei Yan
- State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China
- University of the Chinese Academy of Sciences, Beijing
100049, China
| | - Shijie Gong
- State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China
- University of the Chinese Academy of Sciences, Beijing
100049, China
| | - Zhihao Ma
- Department of Mathematics, Shanghai Jiaotong University,
Shanghai 200240, China
| | - Jiuzhou He
- State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China
- University of the Chinese Academy of Sciences, Beijing
100049, China
| | - Taiping Xiong
- State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China
- University of the Chinese Academy of Sciences, Beijing
100049, China
| | - Liang Chen
- State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China
| | - Wanli Yang
- State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China
| | - Mang Feng
- State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, China
- Synergetic Innovation Center for Quantum Effects and
Applications, Hunan Normal University, Changsha 410081, China
| | - Vlatko Vedral
- Department of Physics, Clarendon Laboratory, University
of Oxford, Parks Road, Oxford OX1 3PU, U.K
- Centre for Quantum Technologies, National University of
Singapore, Singapore 117543, Singapore
- Department of Physics, National University of Singapore,
2 Science Drive 3, Singapore 117551, Singapore
- Center for Quantum Information, Institute for
Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084,
China
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6
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Thekkadath GS, Giner L, Chalich Y, Horton MJ, Banker J, Lundeen JS. Direct Measurement of the Density Matrix of a Quantum System. PHYSICAL REVIEW LETTERS 2016; 117:120401. [PMID: 27689255 DOI: 10.1103/physrevlett.117.120401] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2016] [Indexed: 06/06/2023]
Abstract
One drawback of conventional quantum state tomography is that it does not readily provide access to single density matrix elements since it requires a global reconstruction. Here, we experimentally demonstrate a scheme that can be used to directly measure individual density matrix elements of general quantum states. The scheme relies on measuring a sequence of three observables, each complementary to the last. The first two measurements are made weak to minimize the disturbance they cause to the state, while the final measurement is strong. We perform this joint measurement on polarized photons in pure and mixed states to directly measure their density matrix. The weak measurements are achieved using two walk-off crystals, each inducing a polarization-dependent spatial shift that couples the spatial and polarization degrees of freedom of the photons. This direct measurement method provides an operational meaning to the density matrix and promises to be especially useful for large dimensional states.
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Affiliation(s)
- G S Thekkadath
- Department of Physics and Max Planck Centre for Extreme and Quantum Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada
| | - L Giner
- Department of Physics and Max Planck Centre for Extreme and Quantum Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada
| | - Y Chalich
- Department of Physics and Max Planck Centre for Extreme and Quantum Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada
| | - M J Horton
- Department of Physics and Max Planck Centre for Extreme and Quantum Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada
| | - J Banker
- Department of Physics and Max Planck Centre for Extreme and Quantum Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada
| | - J S Lundeen
- Department of Physics and Max Planck Centre for Extreme and Quantum Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada
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7
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Mirhosseini M, Magaña-Loaiza OS, Hashemi Rafsanjani SM, Boyd RW. Compressive direct measurement of the quantum wave function. PHYSICAL REVIEW LETTERS 2014; 113:090402. [PMID: 25215964 DOI: 10.1103/physrevlett.113.090402] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2014] [Indexed: 05/20/2023]
Abstract
The direct measurement of a complex wave function has been recently realized by using weak values. In this Letter, we introduce a method that exploits sparsity for the compressive measurement of the transverse spatial wave function of photons. The procedure involves weak measurements of random projection operators in the spatial domain followed by postselection in the momentum basis. Using this method, we experimentally measure a 192-dimensional state with a fidelity of 90% using only 25 percent of the total required measurements. Furthermore, we demonstrate the measurement of a 19,200-dimensional state, a task that would require an unfeasibly large acquiring time with the standard direct measurement technique.
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Affiliation(s)
- Mohammad Mirhosseini
- The Institute of Optics, University of Rochester, Rochester, New York 14627, USA
| | - Omar S Magaña-Loaiza
- The Institute of Optics, University of Rochester, Rochester, New York 14627, USA
| | - Seyed Mohammad Hashemi Rafsanjani
- Center for Coherence and Quantum Optics and the Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - Robert W Boyd
- The Institute of Optics, University of Rochester, Rochester, New York 14627, USA and Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
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8
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Bamber C, Lundeen JS. Observing Dirac's classical phase space analog to the quantum state. PHYSICAL REVIEW LETTERS 2014; 112:070405. [PMID: 24579574 DOI: 10.1103/physrevlett.112.070405] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/05/2013] [Indexed: 06/03/2023]
Abstract
In 1945, Dirac attempted to develop a “formal probability” distribution to describe quantum operators in terms of two noncommuting variables, such as position x and momentum p [Rev. Mod. Phys. 17, 195 (1945)]. The resulting quasiprobability distribution is a complete representation of the quantum state and can be observed directly in experiments. We measure Dirac’s distribution for the quantum state of the transverse degree of freedom of a photon by weakly measuring transverse x so as to not randomize the subsequent p measurement. Furthermore, we show that the distribution has the classical-like feature that it transforms (e.g., propagates) according to Bayes’ law.
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Affiliation(s)
- Charles Bamber
- Measurement Science and Standards, National Research Council, Ottawa, Canada K1A 0R6
| | - Jeff S Lundeen
- Physics Department, University of Ottawa, 150 Louis Pasteur, Ottawa, Canada K1N 6N5
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9
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Di Lorenzo A. Correlations between detectors allow violation of the Heisenberg noise-disturbance principle for position and momentum measurements. PHYSICAL REVIEW LETTERS 2013; 110:120403. [PMID: 25166781 DOI: 10.1103/physrevlett.110.120403] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2012] [Revised: 01/30/2013] [Indexed: 06/03/2023]
Abstract
Heisenberg formulated a noise-disturbance principle stating that there is a tradeoff between noise and disturbance when a measurement of position and a measurement of momentum are performed sequentially, and another principle imposing a limitation on the product of the uncertainties in a joint measurement of position and momentum. We prove that the former, the Heisenberg sequential noise-disturbance principle, holds when the detectors are assumed to be initially uncorrelated from each other, but that it can be violated for some properly correlated initial preparations of the detectors.
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Affiliation(s)
- Antonio Di Lorenzo
- Instituto de Física, Universidade Federal de Uberlândia, 38400-902 Uberlândia, Minas Gerais, Brazil
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