1
|
Concha D, Pereira L, Zambrano L, Delgado A. Training a quantum measurement device to discriminate unknown non-orthogonal quantum states. Sci Rep 2023; 13:7460. [PMID: 37156829 PMCID: PMC10167228 DOI: 10.1038/s41598-023-34327-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2022] [Accepted: 04/27/2023] [Indexed: 05/10/2023] Open
Abstract
Here, we study the problem of decoding information transmitted through unknown quantum states. We assume that Alice encodes an alphabet into a set of orthogonal quantum states, which are then transmitted to Bob. However, the quantum channel that mediates the transmission maps the orthogonal states into non-orthogonal states, possibly mixed. If an accurate model of the channel is unavailable, then the states received by Bob are unknown. In order to decode the transmitted information we propose to train a measurement device to achieve the smallest possible error in the discrimination process. This is achieved by supplementing the quantum channel with a classical one, which allows the transmission of information required for the training, and resorting to a noise-tolerant optimization algorithm. We demonstrate the training method in the case of minimum-error discrimination strategy and show that it achieves error probabilities very close to the optimal one. In particular, in the case of two unknown pure states, our proposal approaches the Helstrom bound. A similar result holds for a larger number of states in higher dimensions. We also show that a reduction of the search space, which is used in the training process, leads to a considerable reduction in the required resources. Finally, we apply our proposal to the case of the phase flip channel reaching an accurate value of the optimal error probability.
Collapse
Affiliation(s)
- D Concha
- Instituto Milenio de Investigación en Óptica y Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
| | - L Pereira
- Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid, 28006, Spain.
| | - L Zambrano
- Instituto Milenio de Investigación en Óptica y Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860, Castelldefels, Barcelona, Spain
| | - A Delgado
- Instituto Milenio de Investigación en Óptica y Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
| |
Collapse
|
2
|
Park SI, Noh C, Lee C. Quantum loss sensing with two-mode squeezed vacuum state under noisy and lossy environment. Sci Rep 2023; 13:5936. [PMID: 37045874 PMCID: PMC10097776 DOI: 10.1038/s41598-023-32770-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2023] [Accepted: 04/02/2023] [Indexed: 04/14/2023] Open
Abstract
We investigate quantum advantages in loss sensing when the two-mode squeezed vacuum state is used as a probe. Following an experimental demonstration in PRX 4, 011049, we consider a quantum scheme in which the signal mode is passed through the target and a thermal noise is introduced to the idler mode before they are measured. We consider two detection strategies of practical relevance: coincidence-counting and intensity-difference measurement, which are widely used in quantum sensing and imaging experiments. By computing the signal-to-noise ratio, we verify that quantum advantages persist even under strong thermal background noise, in comparison with the classical scheme which uses a single-mode coherent state that directly suffers from the thermal noise. Such robustness comes from the fact that the signal mode suffers from the thermal noise in the classical scheme, while in the quantum scheme, the idler mode does. For a fairer comparison, we further investigate a different setup in which the thermal noise is introduced to the signal mode in the quantum schemes. In this new setup, we show that the quantum advantages are significantly reduced. Remarkably, however, under an optimum measurement scheme associated with the quantum Fisher information, we show that the two-mode squeezed vacuum state does exhibit a quantum advantage over the entire range of the environmental noise and loss. We expect this work to serve as a guide for experimental demonstrations of quantum advantages in loss parameter sensing, which is subject to lossy and noisy environment.
Collapse
Affiliation(s)
- Sang-Il Park
- Kyungpook National University, Daegu, 41566, Korea
| | - Changsuk Noh
- Kyungpook National University, Daegu, 41566, Korea.
| | - Changhyoup Lee
- Korea Research Institute of Standards and Science, Daejeon, 34113, Korea.
| |
Collapse
|
3
|
Nair R, Tham GY, Gu M. Optimal Gain Sensing of Quantum-Limited Phase-Insensitive Amplifiers. PHYSICAL REVIEW LETTERS 2022; 128:180506. [PMID: 35594116 DOI: 10.1103/physrevlett.128.180506] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/16/2021] [Accepted: 04/01/2022] [Indexed: 06/15/2023]
Abstract
Phase-insensitive optical amplifiers uniformly amplify each quadrature of an input field and are of both fundamental and technological importance. We find the quantum limit on the precision of estimating the gain of a quantum-limited phase-insensitive amplifier using a multimode probe that may also be entangled with an ancilla system. In stark contrast to the sensing of loss parameters, the average photon number N and number of input modes M of the probe are found to be equivalent and interchangeable resources for optimal gain sensing. All pure-state probes whose reduced state on the input modes to the amplifier is diagonal in the multimode number basis are proven to be quantum optimal under the same gain-independent measurement. We compare the best precision achievable using classical probes to the performance of an explicit photon-counting-based estimator on quantum probes and show that an advantage exists even for single-photon probes and inefficient photodetection. A closed-form expression for the energy-constrained Bures distance between two product amplifier channels is also derived.
Collapse
Affiliation(s)
- Ranjith Nair
- Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 639673
- Complexity Institute, Nanyang Technological University, 61 Nanyang Drive, Singapore 637460
| | - Guo Yao Tham
- Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 639673
| | - Mile Gu
- Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 639673
- Complexity Institute, Nanyang Technological University, 61 Nanyang Drive, Singapore 637460
- Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
| |
Collapse
|
4
|
Yin P, Zhang WH, Xu L, Liu ZG, Zhuang WF, Chen L, Gong M, Ma Y, Peng XX, Li GC, Xu JS, Zhou ZQ, Zhang L, Chen G, Li CF, Guo GC. Improving the precision of optical metrology by detecting fewer photons with biased weak measurement. LIGHT, SCIENCE & APPLICATIONS 2021; 10:103. [PMID: 34001846 PMCID: PMC8128924 DOI: 10.1038/s41377-021-00543-4] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/26/2020] [Revised: 04/15/2021] [Accepted: 04/21/2021] [Indexed: 06/12/2023]
Abstract
In optical metrological protocols to measure physical quantities, it is, in principle, always beneficial to increase photon number n to improve measurement precision. However, practical constraints prevent the arbitrary increase of n due to the imperfections of a practical detector, especially when the detector response is dominated by the saturation effect. In this work, we show that a modified weak measurement protocol, namely, biased weak measurement significantly improves the precision of optical metrology in the presence of saturation effect. This method detects an ultra-small fraction of photons while maintains a considerable amount of metrological information. The biased pre-coupling leads to an additional reduction of photons in the post-selection and generates an extinction point in the spectrum distribution, which is extremely sensitive to the estimated parameter and difficult to be saturated. Therefore, the Fisher information can be persistently enhanced by increasing the photon number. In our magnetic-sensing experiment, biased weak measurement achieves precision approximately one order of magnitude better than those of previously used methods. The proposed method can be applied in various optical measurement schemes to remarkably mitigate the detector saturation effect with low-cost apparatuses.
Collapse
Grants
- This work was supported by the National Key Research and Development Program of China (Nos. 2016YFA0302700, 2017YFA0304100), National Natural Science Foundation of China (Grant Nos. 11874344, 61835004, 61327901, 11774335, 91536219, 11821404), Key Research Program of Frontier Sciences, CAS (No. QYZDY-SSW-SLH003), Anhui Initiative in Quantum Information Technologies (AHY020100, AHY060300), the Fundamental Research Funds for the Central Universities (Grant No. WK2030020019, WK2470000026), Science Foundation of the CAS (No. ZDRW-XH-2019-1).
Collapse
Affiliation(s)
- Peng Yin
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Wen-Hao Zhang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Liang Xu
- National Laboratory of Solid State Microstructures and College of Engineering and Applied Sciences, Nanjing University, Nanjing, China
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, 210093, Nanjing, China
| | - Ze-Gang Liu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Wei-Feng Zhuang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Lei Chen
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Ming Gong
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Yu Ma
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Xing-Xiang Peng
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Gong-Chu Li
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Jin-Shi Xu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Zong-Quan Zhou
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| | - Lijian Zhang
- National Laboratory of Solid State Microstructures and College of Engineering and Applied Sciences, Nanjing University, Nanjing, China
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, 210093, Nanjing, China
| | - Geng Chen
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China.
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China.
| | - Chuan-Feng Li
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China.
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China.
| | - Guang-Can Guo
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, Anhui, China
| |
Collapse
|
5
|
Abstract
The extraordinary sensitivity of plasmonic sensors is well-known in the optics and photonics community. These sensors exploit simultaneously the enhancement and the localization of electromagnetic fields close to the interface between a metal and a dielectric. This enables, for example, the design of integrated biochemical sensors at scales far below the diffraction limit. Despite their practical realization and successful commercialization, the sensitivity and associated precision of plasmonic sensors are starting to reach their fundamental classical limit given by quantum fluctuations of light-known as the shot-noise limit. To improve the sensing performance of these sensors beyond the classical limit, quantum resources are increasingly being employed. This area of research has become known as "quantum plasmonic sensing", and it has experienced substantial activity in recent years for applications in chemical and biological sensing. This review aims to cover both plasmonic and quantum techniques for sensing, and it shows how they have been merged to enhance the performance of plasmonic sensors beyond traditional methods. We discuss the general framework developed for quantum plasmonic sensing in recent years, covering the basic theory behind the advancements made, and describe the important works that made these advancements. We also describe several key works in detail, highlighting their motivation, the working principles behind them, and their future impact. The intention of the review is to set a foundation for a burgeoning field of research that is currently being explored out of intellectual curiosity and for a wide range of practical applications in biochemistry, medicine, and pharmaceutical research.
Collapse
Affiliation(s)
- Changhyoup Lee
- Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany.,Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea
| | - Benjamin Lawrie
- Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States
| | - Raphael Pooser
- Quantum Information Science Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States
| | - Kwang-Geol Lee
- Department of Physics, Hanyang University, Seoul 04763, Republic of Korea
| | - Carsten Rockstuhl
- Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany.,Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021Karlsruhe, Germany.,Max Planck School of Photonics, 07745 Jena, Germany
| | - Mark Tame
- Department of Physics, Stellenbosch University, Stellenbosch 7602, South Africa
| |
Collapse
|
6
|
Pirandola S. On quantum reading, quantum illumination, and other notions. IOP SCINOTES 2021. [DOI: 10.1088/2633-1357/abe99e] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
|
7
|
Nair R. Quantum-Limited Loss Sensing: Multiparameter Estimation and Bures Distance between Loss Channels. PHYSICAL REVIEW LETTERS 2018; 121:230801. [PMID: 30576187 DOI: 10.1103/physrevlett.121.230801] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2018] [Indexed: 06/09/2023]
Abstract
The problem of estimating multiple loss parameters of an optical system using the most general ancilla-assisted parallel strategy is solved under energy constraints. An upper bound on the quantum Fisher information matrix is derived assuming that the environment modes involved in the loss interaction can be accessed. Any pure-state probe that is number diagonal in the modes interacting with the loss elements is shown to exactly achieve this upper bound even if the environment modes are inaccessible, as is usually the case in practice. We explain this surprising phenomenon, and show that measuring the Schmidt bases of the probe is a parameter-independent optimal measurement. Our results imply that multiple copies of two-mode squeezed vacuum probes with an arbitrarily small nonzero degree of squeezing, or probes prepared using single-photon states and linear optics, can achieve quantum-optimal performance in conjunction with on-off detection. We also calculate explicitly the energy-constrained Bures distance between any two product loss channels. Our results are relevant to standoff image sensing, biological imaging, absorption spectroscopy, and photodetector calibration.
Collapse
Affiliation(s)
- Ranjith Nair
- Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, 117583 Singapore
| |
Collapse
|
8
|
Chen G, Zhang L, Zhang WH, Peng XX, Xu L, Liu ZD, Xu XY, Tang JS, Sun YN, He DY, Xu JS, Zhou ZQ, Li CF, Guo GC. Achieving Heisenberg-Scaling Precision with Projective Measurement on Single Photons. PHYSICAL REVIEW LETTERS 2018; 121:060506. [PMID: 30141679 DOI: 10.1103/physrevlett.121.060506] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2018] [Indexed: 06/08/2023]
Abstract
It has been suggested that both quantum superpositions and nonlinear interactions are important resources for quantum metrology. However, to date the different roles that these two resources play in the precision enhancement are not well understood. Here, we experimentally demonstrate a Heisenberg-scaling metrology to measure the parameter governing the nonlinear coupling between two different optical modes. The intense mode with n (more than 10^{6} in our work) photons manifests its effect through the nonlinear interaction strength which is proportional to its average photon number. The superposition state of the weak mode, which contains only a single photon, is responsible for both the linear Hamiltonian and the scaling of the measurement precision. By properly preparing the initial state of single photon and making projective photon-counting measurements, the extracted classical Fisher information (FI) can saturate the quantum FI embedded in the combined state after coupling, which is ∼n^{2} and leads to a practical precision ≃1.2/n. Free from the utilization of entanglement, our work paves a way to realize Heisenberg-scaling precision when only a linear Hamiltonian is involved.
Collapse
Affiliation(s)
- Geng Chen
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lijian Zhang
- National Laboratory of Solid State Microstructures and College of Engineering and Applied Sciences, Nanjing University, Nanjing, China
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Wen-Hao Zhang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xing-Xiang Peng
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Liang Xu
- National Laboratory of Solid State Microstructures and College of Engineering and Applied Sciences, Nanjing University, Nanjing, China
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Zhao-Di Liu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xiao-Ye Xu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jian-Shun Tang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yong-Nan Sun
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - De-Yong He
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jin-Shi Xu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zong-Quan Zhou
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Chuan-Feng Li
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Guang-Can Guo
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| |
Collapse
|
9
|
Solís-Prosser MA, Fernandes MF, Jiménez O, Delgado A, Neves L. Experimental Minimum-Error Quantum-State Discrimination in High Dimensions. PHYSICAL REVIEW LETTERS 2017; 118:100501. [PMID: 28339223 DOI: 10.1103/physrevlett.118.100501] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2016] [Indexed: 05/25/2023]
Abstract
Quantum mechanics forbids perfect discrimination among nonorthogonal states through a single shot measurement. To optimize this task, many strategies were devised that later became fundamental tools for quantum information processing. Here, we address the pioneering minimum-error (ME) measurement and give the first experimental demonstration of its application for discriminating nonorthogonal states in high dimensions. Our scheme is designed to distinguish symmetric pure states encoded in the transverse spatial modes of an optical field; the optimal measurement is performed by a projection onto the Fourier transform basis of these modes. For dimensions ranging from D=2 to D=21 and nearly 14 000 states tested, the deviations of the experimental results from the theoretical values range from 0.3% to 3.6% (getting below 2% for the vast majority), thus showing the excellent performance of our scheme. This ME measurement is a building block for high-dimensional implementations of many quantum communication protocols, including probabilistic state discrimination, dense coding with nonmaximal entanglement, and cryptographic schemes.
Collapse
Affiliation(s)
- M A Solís-Prosser
- Center for Optics and Photonics and MSI-Nucleus on Advanced Optics, Universidad de Concepción, Casilla 4016, Concepción, Chile
- Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile
| | - M F Fernandes
- Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG 31270-901, Brazil
| | - O Jiménez
- Departamento de Física, Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile
| | - A Delgado
- Center for Optics and Photonics and MSI-Nucleus on Advanced Optics, Universidad de Concepción, Casilla 4016, Concepción, Chile
- Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile
| | - L Neves
- Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG 31270-901, Brazil
| |
Collapse
|