1
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Song J, Xu Y, Yang Q, Huan R, Wei X. Synchronization bandwidth enhancement induced by a parametrically excited oscillator. MICROSYSTEMS & NANOENGINEERING 2024; 10:95. [PMID: 38984318 PMCID: PMC11231294 DOI: 10.1038/s41378-024-00709-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/27/2023] [Revised: 03/11/2024] [Accepted: 04/06/2024] [Indexed: 07/11/2024]
Abstract
The synchronization phenomenon in nature has been utilized in sensing and timekeeping fields due to its numerous advantages, including amplitude and frequency stabilization, noise reduction, and sensitivity improvement. However, the limited synchronization bandwidth hinders its broader application, and few techniques have been explored to enhance this aspect. In this paper, we conducted theoretical and experimental studies on the unidirectional synchronization characteristics of a resonator with phase lock loop oscillation. A novel enhancement method for the synchronization bandwidth using a parametrically excited MEMS oscillator is proposed, which achieves a remarkably large synchronization bandwidth of 8.85 kHz, covering more than 94% of the hysteresis interval. Importantly, the proposed method exhibits significant potential for high-order synchronization and frequency stabilization compared to the conventional directly excited oscillator. These findings present an effective approach for expanding the synchronization bandwidth, which has promising applications in nonlinear sensing, fully mechanical frequency dividers, and high-precision time references.
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Affiliation(s)
- Jiahao Song
- State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, 710049 People's Republic of China
| | - Yutao Xu
- State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, 710049 People's Republic of China
| | - Qiqi Yang
- State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, 710049 People's Republic of China
| | - Ronghua Huan
- Department of Mechanics, Zhejiang University, Hangzhou, 310027 People's Republic of China
| | - Xueyong Wei
- State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, 710049 People's Republic of China
- School of Instrument Science and Technology, Xi'an Jiaotong University, Xi'an, 710049 People's Republic of China
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2
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Fung F, Rosenfeld E, Schaefer JD, Kabcenell A, Gieseler J, Zhou TX, Madhavan T, Aslam N, Yacoby A, Lukin MD. Toward Programmable Quantum Processors Based on Spin Qubits with Mechanically Mediated Interactions and Transport. PHYSICAL REVIEW LETTERS 2024; 132:263602. [PMID: 38996281 DOI: 10.1103/physrevlett.132.263602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2023] [Revised: 05/03/2024] [Accepted: 05/09/2024] [Indexed: 07/14/2024]
Abstract
Solid-state spin qubits are promising candidates for quantum information processing, but controlled interactions and entanglement in large, multiqubit systems are currently difficult to achieve. We describe a method for programmable control of multiqubit spin systems, in which individual nitrogen-vacancy (NV) centers in diamond nanopillars are coupled to magnetically functionalized silicon nitride mechanical resonators in a scanning probe configuration. Qubits can be entangled via interactions with nanomechanical resonators while programmable connectivity is realized via mechanical transport of qubits in nanopillars. To demonstrate the feasibility of this approach, we characterize both the mechanical properties and the magnetic field gradients around the micromagnet placed on the nanobeam resonator. We demonstrate coherent manipulation of a spin qubit in the proximity of a transported micromagnet by utilizing nuclear spin memory and use the NV center to detect the time-varying magnetic field from the oscillating micromagnet, extracting a spin-mechanical coupling of 7.7(9) Hz. With realistic improvements, the high-cooperativity regime can be reached, offering a new avenue toward scalable quantum information processing with spin qubits.
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Affiliation(s)
| | | | | | | | | | - T X Zhou
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
- Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
- Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | | | - N Aslam
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
- Institute of Condensed Matter Physics, Technische Universität Braunschweig, Braunschweig, Germany
- Felix Bloch Institute for Solid State Physics, Leipzig University, 04103 Leipzig, Germany
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3
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Engelsen NJ, Beccari A, Kippenberg TJ. Ultrahigh-quality-factor micro- and nanomechanical resonators using dissipation dilution. NATURE NANOTECHNOLOGY 2024; 19:725-737. [PMID: 38443697 DOI: 10.1038/s41565-023-01597-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2023] [Accepted: 12/14/2023] [Indexed: 03/07/2024]
Abstract
Mechanical resonators are widely used in sensors, transducers and optomechanical systems, where mechanical dissipation sets the ultimate limit to performance. Over the past 15 years, the quality factors in strained mechanical resonators have increased by four orders of magnitude, surpassing the previous state of the art achieved in bulk crystalline resonators at room temperature and liquid helium temperatures. In this Review, we describe how these advances were made by leveraging 'dissipation dilution'-where dissipation is reduced through a combination of static tensile strain and geometric nonlinearity in dynamic strain. We then review the state of the art in strained nanomechanical resonators and discuss the potential for even higher quality factors in crystalline materials. Finally, we detail current and future applications of dissipation-diluted mechanical resonators.
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Affiliation(s)
- Nils Johan Engelsen
- Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, Gothenburg, Sweden.
| | - Alberto Beccari
- Instutute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
- Center for Quantum Science and Engineering, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
| | - Tobias Jan Kippenberg
- Instutute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
- Center for Quantum Science and Engineering, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
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4
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Weituschat LM, Castro I, Colomar I, Everly C, Postigo PA, Ramos D. Exploring regenerative coupling in phononic crystals for room temperature quantum optomechanics. Sci Rep 2024; 14:12330. [PMID: 38811848 PMCID: PMC11137142 DOI: 10.1038/s41598-024-63199-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2024] [Accepted: 05/27/2024] [Indexed: 05/31/2024] Open
Abstract
Quantum technologies play a pivotal role in driving transformative advancements across diverse fields, surpassing classical approaches and empowering us to address complex challenges more effectively; however, the need for ultra-low temperatures limits the use of these technologies to particular fields. This work comes to alleviate this problem. We present a way of phononic bandgap engineering using FEM by which the radiative mechanical energy dissipation of a nanomechanical oscillator can be significantly suppressed through coupling with a complementary oscillating mode of a defect of the surrounding phononic crystal (PnC). Applied to an optomechanically coupled nanobeam resonator in the megahertz regime, we find a mechanical quality factor improvement of up to four orders of magnitude compared to conventional PnC designs. As this method is based on geometrical optimization of the PnC and frequency matching of the resonator and defect mode, it is applicable to a wide range of resonator types and frequency ranges. Taking advantage of the, hereinafter referred to as, "regenerative coupling" in phononic crystals, the presented device is capable of reaching f × Q products exceeding 10E16 Hz with only two rows of PnC shield. Thus, stable quantum states with mechanical decoherence times up to 700 μs at room temperature can be obtained, offering new opportunities for the optimization of mechanical resonator performance and advancing the room temperature quantum field across diverse applications.
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Affiliation(s)
- Lukas M Weituschat
- Optomechanics Lab, Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC, 3, Sor Juana Inés de la Cruz, 28049, Madrid, Spain
| | - Irene Castro
- Optomechanics Lab, Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC, 3, Sor Juana Inés de la Cruz, 28049, Madrid, Spain
| | - Irene Colomar
- Optomechanics Lab, Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC, 3, Sor Juana Inés de la Cruz, 28049, Madrid, Spain
| | - Christer Everly
- The Institute of Optics, University of Rochester, Rochester, NY, 14627, USA
| | - Pablo A Postigo
- The Institute of Optics, University of Rochester, Rochester, NY, 14627, USA
| | - Daniel Ramos
- Optomechanics Lab, Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC, 3, Sor Juana Inés de la Cruz, 28049, Madrid, Spain.
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5
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Cupertino A, Shin D, Guo L, Steeneken PG, Bessa MA, Norte RA. Centimeter-scale nanomechanical resonators with low dissipation. Nat Commun 2024; 15:4255. [PMID: 38762589 PMCID: PMC11102468 DOI: 10.1038/s41467-024-48183-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2023] [Accepted: 04/22/2024] [Indexed: 05/20/2024] Open
Abstract
High-aspect-ratio mechanical resonators are pivotal in precision sensing, from macroscopic gravitational wave detectors to nanoscale acoustics. However, fabrication challenges and high computational costs have limited the length-to-thickness ratio of these devices, leaving a largely unexplored regime in nano-engineering. We present nanomechanical resonators that extend centimeters in length yet retain nanometer thickness. We explore this expanded design space using an optimization approach which judiciously employs fast millimeter-scale simulations to steer the more computationally intensive centimeter-scale design optimization. By employing delicate nanofabrication techniques, our approach ensures high-yield realization, experimentally confirming room-temperature quality factors close to theoretical predictions. The synergy between nanofabrication, design optimization guided by machine learning, and precision engineering opens a solid-state path to room-temperature quality factors approaching 10 billion at kilohertz mechanical frequencies - comparable to the performance of leading cryogenic resonators and levitated nanospheres, even under significantly less stringent temperature and vacuum conditions.
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Affiliation(s)
- Andrea Cupertino
- Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands
| | - Dongil Shin
- Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands
- Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands
| | - Leo Guo
- Department of Microelectronics, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands
| | - Peter G Steeneken
- Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands
- Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ, Delft, The Netherlands
| | - Miguel A Bessa
- School of Engineering, Brown University, 184 Hope St., Providence, RI, 02912, USA.
| | - Richard A Norte
- Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands.
- Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ, Delft, The Netherlands.
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6
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Huang G, Beccari A, Engelsen NJ, Kippenberg TJ. Room-temperature quantum optomechanics using an ultralow noise cavity. Nature 2024; 626:512-516. [PMID: 38356070 PMCID: PMC10866701 DOI: 10.1038/s41586-023-06997-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2023] [Accepted: 12/18/2023] [Indexed: 02/16/2024]
Abstract
At room temperature, mechanical motion driven by the quantum backaction of light has been observed only in pioneering experiments in which an optical restoring force controls the oscillator stiffness1,2. For solid-state mechanical resonators in which oscillations are controlled by the material rigidity, the observation of these effects has been hindered by low mechanical quality factors, optical cavity frequency fluctuations3, thermal intermodulation noise4,5 and photothermal instabilities. Here we overcome these challenges with a phononic-engineered membrane-in-the-middle system. By using phononic-crystal-patterned cavity mirrors, we reduce the cavity frequency noise by more than 700-fold. In this ultralow noise cavity, we insert a membrane resonator with high thermal conductance and a quality factor (Q) of 180 million, engineered using recently developed soft-clamping techniques6,7. These advances enable the operation of the system within a factor of 2.5 of the Heisenberg limit for displacement sensing8, leading to the squeezing of the probe laser by 1.09(1) dB below the vacuum fluctuations. Moreover, the long thermal decoherence time of the membrane oscillator (30 vibrational periods) enables us to prepare conditional displaced thermal states of motion with an occupation of 0.97(2) phonons using a multimode Kalman filter. Our work extends the quantum control of solid-state macroscopic oscillators to room temperature.
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Affiliation(s)
- Guanhao Huang
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland
- Center for Quantum Science and Engineering, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland
| | - Alberto Beccari
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland
- Center for Quantum Science and Engineering, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland
| | - Nils J Engelsen
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
- Center for Quantum Science and Engineering, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
- Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, Göteborg, Sweden.
| | - Tobias J Kippenberg
- Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
- Center for Quantum Science and Engineering, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
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7
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Chen X, Ammu SK, Masania K, Steeneken PG, Alijani F. Diamagnetic Composites for High-Q Levitating Resonators. ADVANCED SCIENCE (WEINHEIM, BADEN-WURTTEMBERG, GERMANY) 2022; 9:e2203619. [PMID: 36180390 PMCID: PMC9661851 DOI: 10.1002/advs.202203619] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/22/2022] [Revised: 08/19/2022] [Indexed: 06/16/2023]
Abstract
Levitation offers extreme isolation of mechanical systems from their environment, while enabling unconstrained high-precision translation and rotation of objects. Diamagnetic levitation is one of the most attractive levitation schemes because it allows stable levitation at room temperature without the need for a continuous power supply. However, dissipation by eddy currents in conventional diamagnetic materials significantly limits the application potential of diamagnetically levitating systems. Here, a route toward high-Q macroscopic levitating resonators by substantially reducing eddy current damping using graphite particle based diamagnetic composites is presented. Resonators that feature quality factors Q above 450 000 and vibration lifetimes beyond one hour are demonstrated, while levitating above permanent magnets in high vacuum at room temperature. The composite resonators have a Q that is >400 times higher than that of diamagnetic graphite plates. By tuning the composite particle size and density, the dissipation reduction mechanism is investigated, and the Q of the levitating resonators is enhanced. Since their estimated acceleration noise is as low as some of the best superconducting levitating accelerometers at cryogenic temperatures, the high Q and large mass of the presented composite resonators positions them as one of the most promising technologies for next generation ultra-sensitive room temperature accelerometers.
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Affiliation(s)
- Xianfeng Chen
- Department of Precision and Microsystems EngineeringDelft University of TechnologyMekelweg 2Delft2628 CDThe Netherlands
| | - Satya K. Ammu
- Shaping Matter LabFaculty of Aerospace EngineeringDelft University of TechnologyDelft2629 HSThe Netherlands
| | - Kunal Masania
- Shaping Matter LabFaculty of Aerospace EngineeringDelft University of TechnologyDelft2629 HSThe Netherlands
| | - Peter G. Steeneken
- Department of Precision and Microsystems EngineeringDelft University of TechnologyMekelweg 2Delft2628 CDThe Netherlands
- Kavli Institute of NanoscienceDelft University of TechnologyLorentzweg 1Delft2628 CJThe Netherlands
| | - Farbod Alijani
- Department of Precision and Microsystems EngineeringDelft University of TechnologyMekelweg 2Delft2628 CDThe Netherlands
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8
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Jin J, Hu N, Zhan L, Wang X, Zhang Z, Hu H. Design of GHz Mechanical Nanoresonator with High Q-Factor Based on Optomechanical System. MICROMACHINES 2022; 13:1862. [PMID: 36363884 PMCID: PMC9695023 DOI: 10.3390/mi13111862] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/23/2022] [Revised: 10/24/2022] [Accepted: 10/27/2022] [Indexed: 06/16/2023]
Abstract
Micro-electromechanical systems (MEMS) have dominated the interests of the industry due to its microminiaturization and high frequency for the past few decades. With the rapid development of various radio frequency (RF) systems, such as 5G mobile telecommunications, satellite, and other wireless communication, this research has focused on a high frequency resonator with high quality. However, the resonator based on an inverse piezoelectric effect has met with a bottleneck in high frequency because of the low quality factor. Here, we propose a resonator based on optomechanical interaction (i.e., acoustic-optic coupling). A picosecond laser can excite resonance by radiation pressure. The design idea and the optimization of the resonator are given. Finally, with comprehensive consideration of mechanical losses at room temperature, the resonator can reach a high Q-factor of 1.17 × 104 when operating at 5.69 GHz. This work provides a new concept in the design of NEMS mechanical resonators with a large frequency and high Q-factor.
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Affiliation(s)
- Jun Jin
- Department of Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
- Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Ningdong Hu
- Department of Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
- Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Lamin Zhan
- School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Xiaohong Wang
- School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Zenglei Zhang
- Wuhan Second Ship Design and Research Institute, Wuhan 430074, China
| | - Hongping Hu
- Department of Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
- Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, Wuhan 430074, China
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9
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Guo J, Gröblacher S. Integrated optical-readout of a high-Q mechanical out-of-plane mode. LIGHT, SCIENCE & APPLICATIONS 2022; 11:282. [PMID: 36171197 PMCID: PMC9519924 DOI: 10.1038/s41377-022-00966-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2022] [Revised: 08/10/2022] [Accepted: 08/22/2022] [Indexed: 05/12/2023]
Abstract
The rapid development of high-QM macroscopic mechanical resonators has enabled great advances in optomechanics. Further improvements could allow for quantum-limited or quantum-enhanced applications at ambient temperature. Some of the remaining challenges include the integration of high-QM structures on a chip, while simultaneously achieving large coupling strengths through an optical read-out. Here, we present a versatile fabrication method, which allows us to build fully integrated optomechanical structures. We place a photonic crystal cavity directly above a mechanical resonator with high-QM fundamental out-of-plane mode, separated by a small gap. The highly confined optical field has a large overlap with the mechanical mode, enabling strong optomechanical interaction strengths. Furthermore, we implement a novel photonic crystal design, which allows for a very large cavity photon number, a highly important feature for optomechanical experiments and sensor applications. Our versatile approach is not limited to our particular design but allows for integrating an out-of-plane optical read-out into almost any device layout. Additionally, it can be scaled to large arrays and paves the way to realizing quantum experiments and applications with mechanical resonators based on high-QM out-of-plane modes alike.
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Affiliation(s)
- Jingkun Guo
- Kavli Institute of Nanoscience, Department of Quantum Nanoscience, Delft University of Technology, 2628CJ, Delft, The Netherlands
| | - Simon Gröblacher
- Kavli Institute of Nanoscience, Department of Quantum Nanoscience, Delft University of Technology, 2628CJ, Delft, The Netherlands.
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10
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Gisler T, Helal M, Sabonis D, Grob U, Héritier M, Degen CL, Ghadimi AH, Eichler A. Soft-Clamped Silicon Nitride String Resonators at Millikelvin Temperatures. PHYSICAL REVIEW LETTERS 2022; 129:104301. [PMID: 36112443 DOI: 10.1103/physrevlett.129.104301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2021] [Accepted: 08/03/2022] [Indexed: 06/15/2023]
Abstract
We demonstrate that soft-clamped silicon nitride strings with a large aspect ratio can be operated at mK temperatures. The quality factors (Q) of two measured devices show consistent dependency on the cryostat temperature, with soft-clamped mechanical modes reaching Q>10^{9} at roughly 46 mK. For low optical readout power, Q is found to saturate, indicating good thermalization between the sample and the stage it is mounted on. Our best device exhibits a calculated force sensitivity of 9.6 zN/sqrt[Hz] and a thermal decoherence time of 0.38 s, which bode well for future applications such as nanomechanical force sensing.
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Affiliation(s)
- Thomas Gisler
- Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
| | - Mohamed Helal
- Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
| | - Deividas Sabonis
- Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
| | - Urs Grob
- Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
| | - Martin Héritier
- Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
| | - Christian L Degen
- Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
| | - Amir H Ghadimi
- Centre Suisse d'Electronique et de Microtechnique SA (CSEM), 2002 Neuchâtel, Switzerland
| | - Alexander Eichler
- Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
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