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鍾明道保全(我有殺人前科、最愛喝燙水的蛇王) guest 鍾明道保全(我有殺人前科、最愛喝燙水的蛇王) 2021/05/25 11:32

證據時效。我最愛殺死鍾肇光。我沒有預設立場。龍飛鳳舞。聽不清楚再講一次。

量子場論的兩種常用表述分別是正則量子化和路徑積分表述。



兩點相關函數
現在假設理論中包含交互作用，且拉格朗日量中的交互作用項是一個微擾。
重整化
以上程序稱為裸微擾理論，因為它的計算用到的都是裸質量、裸耦合常數等等；另一重整化程序從一開始就用具有物理意義的質量、耦合常數等做計算，稱為重整化微擾理論。
重整化群
有一類特殊的量子場論具有共形對稱性，稱為共形場論。這樣的理論不對尺度的變化敏感，因此它所有耦合常數的β函數都為零。（然而，所有β函數為零並不代表理論一定具有共形對稱性。）[23]例子有：弦理論[14]、N = 4超對稱楊-米爾斯理論[24]等等。


其他理論
以上對量子場論方法的簡述只討論了實純量場的自由理論和ϕ4理論。類似的表述和程序也可以應用於其他類型的場，如複純量場、向量場、狄拉克場等等，以及各種交互作用項，如電磁交互作用、湯川交互作用等等。





規範對稱性
主條目：規範場論
規範對稱性上的反常，否則它就是不一致的。基本粒子標準模型屬於規範場論，其規範群為SU(3) × SU(2) × U(1)，理論中所有可能出現的反常都恰好完全抵消。[1]:705-707

等效原理是廣義相對論的理論基礎，它也可以視為一種規範對稱性，所以廣義相對論是基於勞侖茲群的規範場論。[25]

根據諾特定理，每一個連續對稱性（即對稱變換中的參數是連續而不是離散的），都有一個相對應的守恆定律。[1]:17-18[19]:73例如，量子電動力學的U(1)對稱性意味著電荷守恆。[26]

規範轉換並不是把一個量子態轉換為另一個量子態，而是把對於同一個態的兩種等價的數學描述聯繫起來。例如，光子場Aμ是一個四維向量，似乎含有四個自由度，但實際上光子只有對應於偏振的兩個自由度。其餘的兩個自由度，可以說是「多餘」的：許多表面上不同的Aμ可以通過規範轉換互相聯繫，因此實際上描述光子場的同一個態。規範對稱性從嚴格上來說並不是一種「真實」的對稱性，它只是反映了我們所選擇的數學表達方式的「多餘性」。[19]:168

要在費曼路徑積分表述中去除這種多餘性，須進行所謂的法捷耶夫-波波夫規範固定程序。在非阿貝爾規範場論中，該程序會產生一種新的場──鬼場。鬼場所對應的粒子稱為鬼粒子，它並不能夠經實驗測量得到。鬼場在拉格朗日量中有自己的項，起到固定到某個特定規範的作用。[1]:512-515法捷耶夫-波波夫程序的推廣，是相對嚴謹的BRST量子化表述。[1]:517




自發對稱破缺是一種既保持原有拉格朗日量對稱性，又能使得最終描述的系統破壞此對稱性的機制。[1]:347
根據戈德斯通定理，在自發對稱破缺機制下，每一個被破壞的連續全局對稱性都對應於一個無質量場，這些場稱為南部-戈德斯通玻色子。在以上的例子中，O(N)有N(N-1)/2個連續對稱性，而O(N-1)則有(N-1)(N-2)/2個，兩者之差為N-1，正好對應於N-1個無質量場πk。[1]:351

另一方面，連續規範對稱性（即局域對稱性）被破壞後所生成的戈德斯通玻色子會被相應的規範玻色子「吃掉」，成為後者的一個額外自由度。根據戈德斯通玻色子等效定理，在高能下，吸收或發出一個縱向極化的有質量規範玻色子的振幅，與吸收或發出一個被此規範玻色子吃掉的戈德斯通玻色子的振幅相同。[1]:743-744

在描述鐵磁性系統的量子場論中，自發對稱破缺可以解釋低溫下磁偶極子方向對齊的現象；[19]:199在基本粒子標準模型中，希格斯機制利用自發對稱破缺，使得原先因規範對稱性而不允許有質量的規範玻色子（W和Z玻色子）獲得質量。[1]:690


超對稱
主條目：超對稱

現實中所觀測到的一切對稱性，包括全局對稱和局域對稱性，都把玻色子和玻色子聯繫起來，並把費米子和費米子聯繫起來，而沒有玻色子和費米子之間的聯繫。理論學家猜想可能存在一種把玻色子和費米子聯繫起來的對稱性，稱為超對稱。[1]:795[19]:443

標準模型具有全局龐加萊對稱性，生成子有：平移生成子Pμ和勞侖茲變換生成子Jμν。[27]:58-60(3+1)維超對稱在此對稱群的基礎上，再加上遵守外爾費米子轉換法則的超對稱生成子Qα。[1]:795[19]:444由以上所有生成子所生成的對稱群稱為超龐加萊群。廣義地說，超對稱生成子可以不止一個：QαI, I = 1,…,N，這樣的對稱性稱為N = 1超對稱、N = 2超對稱，如此類推。[1]:795[19]:450其他維度時空上也可以定義超對稱，[28]特別是具有(1+1)維超對稱性的超弦理論。[29]

如果一個理論具有超對稱性，那麼其拉格朗日量就必須在超對稱群的轉換作用下不變，[19]:448例子有：最小超對稱標準模型、N = 4超對稱楊-米爾斯理論[19]:450、超弦理論等等。在這樣的理論中，每一個費米子都有一個對應的玻色子超對稱粒子，反之亦然。[19]:444

如果把超對稱性提升為局域對稱性，所形成的規範場論是廣義相對論的推廣──超重力理論。[30]

如果超對稱性在自然界中真實存在，將解決物理學上的若干難題。把希格斯場和它的超對稱場超希格斯場互相聯繫，可能可以解決標準模型中的級列問題：為甚麼希格斯玻色子的質量不會因為輻射修正而上升到更高的尺度，如大統一理論尺度或普朗克尺度。原理是，在費曼圖中產生輻射修正項的希格斯玻色子迴圈，會被相應的超希格斯費米子迴圈所抵消。其他有可能經超對稱性解決的問題還有規範耦合常數的高能大統一問題，以及暗物質的本質。[1]:796-797[31]

然而，實驗物理學家並沒有觀測到任何超對稱粒子的存在。如果超對稱性真的存在，它必定是一個破缺的對稱性，而且破缺所發生的能量尺度一定比現今實驗探索的尺度更高。[1]:797[19]:443



其他時空

前文討論的ϕ4理論、量子電動力學、量子色動力學乃至整個標準模型，都假設量子場所在的時空是3+1維閔考斯基時空（3個空間維度及1個時間維度）。然而，量子場論本身並不限制時空的維數或幾何。

在凝聚態物理學中，有2+1維電子氣體。[32]在高能物理學中，有屬於1+1維量子場論的弦理論[19]:452[14]，還有利用額外維度中的重力產生低維度規範場論的卡魯扎-克萊因理論[19]:428-429等等。

在閔考斯基時空中，拉格朗日量的所有標號升降都利用平坦度規張量ημν，例如：

A μ A μ = η μ ν A μ A ν , ∂ μ ϕ ∂ μ ϕ = η μ ν ∂ μ ϕ ∂ ν ϕ , {\displaystyle A_{\mu }A^{\mu }=\eta _{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,} {\displaystyle A_{\mu }A^{\mu }=\eta _{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,}

其中ημν是ημν的逆：ημρηρν = δμν。如果量子場所在的時空不是平坦的（見彎曲時空中的量子場論），就應使用更廣義的度規張量gμν（例如描述黑洞的史瓦西度規）：

A μ A μ = g μ ν A μ A ν , ∂ μ ϕ ∂ μ ϕ = g μ ν ∂ μ ϕ ∂ ν ϕ . {\displaystyle A_{\mu }A^{\mu }=g_{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi .} {\displaystyle A_{\mu }A^{\mu }=g_{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi .}

此處gμν是gμν的逆。以實純量場為例，最廣義的拉格朗日量在彎曲時空下變為

L = | g | ( 1 2 g μ ν ∇ μ ϕ ∇ ν ϕ − 1 2 m 2 ϕ 2 ) , {\displaystyle {\mathcal {L}}={\sqrt {|g|}}\left({\frac {1}{2}}g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right),} {\displaystyle {\mathcal {L}}={\sqrt {|g|}}\left({\frac {1}{2}}g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right),}

其中g = det(gμν)，∇μ是協變導數。[33]

由此可見，根據固定背景時空幾何的不同，拉格朗日量也會隨之改變，理論的種種計算和預測也會不同。


拓撲量子場論
主條目：拓撲量子場論

一般來說，對於不同的時空度規gμν，量子場論的相關函數乃至物理預測也會不同。有一類量子場論的所有相關函數都不隨時空度規的值改變，此類理論稱為拓撲量子場論。[34]:36一般的彎曲時空量子場論會隨時空的幾何改變而改變，而拓撲量子場論則在一切微分同胚對時空的作用下不變，但對時空的拓撲敏感。這意味著，拓撲量子場論的所有計算結果，都是其底下時空的拓撲不變量。陳-西蒙斯理論就是拓撲量子場論的一例，可用於構建各種量子重力模型。[35]拓撲量子場論可以應用在分數量子霍爾效應和拓撲量子計算機上。[36]:1-5。
微擾與非微擾方法

通過量子場論的微擾方法，可以用一個以參與交互作用的粒子總數所展開的級數，一階一階地近似交互作用項的整個效應。展開中的每一項都可以理解為粒子間通過其他虛粒子傳遞交互作用的一種可能途徑，可以用非常形象的費曼圖來表達。兩個電子間的電磁力在量子電動力學中（在第一階微擾）是以光子的傳遞來表達的。同樣，W和Z玻色子傳遞弱交互作用，膠子傳遞強交互作用。這種把交互作用的中間態視為各種粒子交換過程的總和的看法，只有在微擾理論的框架下才有意義；非微擾方法則把交互作用項視為一個整體來對待，不進行級數展開，因此也沒有虛粒子傳遞交互作用一說。取而代之描述交互作用的，是特·胡夫特-泊里雅科夫單極子、疇壁、流量管、瞬子等概念。[8]具有非微擾完全解的量子場論包括一類稱為極簡模型的共形場論[37]和蒂林模型等等。[38]


幹嘛 我要叫分駐所回家睡覺 我是很嚴格的
從1950年代開始，[40]有理論物理學家和數學家嘗試把量子場論總結為一組公理，並從數學嚴格的角度證明相對論性量子力學模型的確切存在。這種研究稱為構造量子場論，屬於數學物理的範疇。[41]:2其成果包括：CPT定理、自旋統計定理、戈德斯通定理等等。[40]

拓撲量子場論和共形場論相比一般的量子場論來說，有更穩固的數學基礎：兩者都可以在對陪邊的表示框架中進行分類。[42]

另一條公理化方法稱為代數量子場論，它以局域算符之間的代數關係為理論的根本結構。在這方面的公理系統有：懷特曼公理和哈格-卡斯特勒公理。[41]:2-3構建符合懷特曼公理的理論之其中一種方法，是利用奧斯特瓦德-施拉德爾公理（Osterwalder–Schrader axioms）。這組公理給出能夠從虛數時間理論解析延拓至實數時間理論（威克轉動）的必要和充分條件。[41]:10

一旦證實現實的物理模型符合以上的公理，在物理學和數學上將有重要意義。例如，楊-米爾斯存在性與質量間隙是千禧年大獎難題之一，其表述如下：[43]
【爆揍昏劉中瑞( 義守大學)00916174951000】

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:17

New Quantum Material Discovered – With Surprising Properties
TOPICS:Materials SciencePopularQuantum PhysicsSuperconductivityVienna University of Technology

By Vienna University of Technology May 24, 2021
Cerium, Ruthenium and Tin

A compound of cerium, ruthenium, and tin — with surprising properties. Credit: TU Wien

A research team from TU Wien together with US research institutes came across a surprising form of ‘quantum criticality’; this could lead to a design concept for new materials.

In everyday life, phase transitions usually have to do with temperature changes — for example, when an ice cube gets warmer and melts. But there are also different kinds of phase transitions, depending on other parameters such as magnetic field. In order to understand the quantum properties of materials, phase transitions are particularly interesting when they occur directly at the absolute zero point of temperature. These transitions are called “quantum phase transitions” or a “quantum critical points.”

Such a quantum critical point has now been discovered by an Austrian-American research team in a novel material, and in an unusually pristine form. The properties of this material are now being further investigated. It is suspected that the material could be a so-called Weyl-Kondo semimetal, which is considered to have great potential for quantum technology due to special quantum states (so-called topological states). If this proves to be true, a key for the targeted development of topological quantum materials would have been found. The results were found in a cooperation between TU Wien, Johns Hopkins University, the National Institute of Standards and Technology (NIST) and Rice University and has now been published in the journal Science Advances.
Quantum criticality — simpler and clearer than ever before

“Usually quantum critical behavior is studied in metals or insulators. But we have now looked at a semimetal,” says Prof. Silke Bühler-Paschen from the Institute of Solid State Physics at TU Wien. The material is a compound of cerium, ruthenium, and tin — with properties that lie between those of metals and semiconductors.

Usually, quantum criticality can only be created under very specific environmental conditions — a certain pressure or an electromagnetic field. “Surprisingly, however, our semimetal turned out to be quantum critical without any external influences at all,” says Wesley Fuhrman, a PhD student in Prof. Collin Broholm’s team at Johns Hopkins University, who made an important contribution to the result with neutron scattering measurements. “Normally you have to work hard to produce the appropriate laboratory conditions, but this semimetal provides the quantum criticality all by itself.”

This surprising result is probably related to the fact that the behavior of electrons in this material has some special features. “It is a highly correlated electron system. This means that the electrons interact strongly with each other, and that you cannot explain their behavior by looking at the electrons individually,” says Bühler-Paschen. “This electron interaction leads to the so-called Kondo effect. Here, a quantum spin in the material is shielded by electrons surrounding it, so that the spin no longer has any effect on the rest of the material.”

If there are only relatively few free electrons, as is the case in a semimetal, then the Kondo effect is unstable. This could be the reason for the quantum critical behavior of the material: the system fluctuates between a state with and a state without the Kondo effect, and this has the effect of a phase transition at zero temperature.
Quantum fluctuations could lead to Weyl particles

The main reason why the result is of such central importance is that it is suspected to be closely connected to the phenomenon of “Weyl fermions.” In solids, Weyl fermions can appear in the form of quasiparticles — i.e. as collective excitations such as waves in a pond. According to theoretical predictions, such Weyl fermions should exist in this material,” says theoretical physicist Qimiao Si of Rice University. Experimental proof, however, is yet to be found. “We suspect that the quantum criticality we observed favors the occurrence of such Weyl fermions,” says Silke Bühler-Paschen. “Quantum critical fluctuations could therefore have a stabilizing effect on Weyl fermions, in a similar way to quantum critical fluctuations in high-temperature superconductors holding superconducting Cooper pairs together. This is a very fundamental question that is the subject of a lot of research around the world, and we’ve discovered a hot new lead here.”

It seems to us that certain quantum effects — namely quantum critical fluctuations, the Kondo effect, and Weyl fermions — are tightly intertwined in the newly discovered material and, together, give rise to exotic Weyl-Kondo states. These are “topological” states of great stability that, unlike other quantum states, cannot be easily destroyed by external disturbances. This makes them particularly interesting for quantum computers.

To verify all this, further measurements under different external conditions are to be carried out. The team expects that a similar interplay of the various quantum effects should also be found in other materials. “This could lead to the establishment of a design concept with which such materials can be specifically improved, tailored, and used for concrete applications,” says Bühler-Paschen.

Reference: “Pristine quantum criticality in a Kondo semimetal” by Wesley T. Fuhrman, Andrey Sidorenko, Jonathan Hänel, Hannes Winkler, Andrey Prokofiev, Jose A. Rodriguez-Rivera, Yiming Qiu, Peter Blaha, Qimiao Si, Collin L. Broholm and Silke Paschen, 19 May 2021, Science Advances.
DOI: 10.1126/sciadv.abf9134

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:20

qpt.physics.harvard.edu/talks/aaas.pdf

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:23

Harvard partners with national labs on quantum computing
September 4, 2020
A photo of small section of a quantum computer
Energy Department creates 5 science research centers in field, 3 with Harvard groups

By Leah Burrows



Harvard scientists will take leading roles in a new federal government effort to advance quantum computing, an experimental technology that could make it possible to perform calculations at speeds inconceivable today and usher in game-changing innovations in medicine, artificial intelligence, drug development, and other fields.

The White House Office of Science and Technology Policy and the U.S. Department of Energy (DOE) announced last week the creation of five new Quantum Information Science (QIS) Research Centers across the country. These centers, led by DOE national laboratories, will be part of the U.S. National Quantum Initiative.

The DOE awarded $625 million over five years to the Argonne, Brookhaven, Fermi, Oak Ridge, and Lawrence Berkeley national laboratories to establish centers. Each facility will have an interdisciplinary research team and multiple partner institutions. Harvard researchers will play important roles in the centers at Brookhaven, Oak Ridge, and Lawrence Berkeley.

“This very welcome federal investment in QIS Research Centers will catalyze advances in one of the most exciting areas of science today. Nobody who knows their work will be surprised to see the prominent roles that Harvard scientists and engineers, each one of them a leading innovator in quantum science, will play in the new centers,” said Provost Alan M. Garber. “I look forward to seeing the results of their research, which promises to help shape the future of computation and data science.”

The Quantum Systems Accelerator (QSA) at the Lawrence Berkeley National Laboratory (Berkeley Lab) in Alameda County, just north of the University of California, Berkeley, was awarded $115 million over five years to develop the solutions needed to build working quantum systems, which harness the power of quantum mechanics.

Mikhail Lukin, a George Vasmer Leverett Professor of Physics and co-director of the Harvard Quantum Initiative, is project leader of one of the QSA’s major research thrusts: algorithms and applications. Lukin and his team will investigate how quantum computers can meaningfully speed up answers to real-world scientific problems and create new tools to quantify this advantage and performance.

“Within this project, DOE labs partner with leading academic research groups from across the country to address some of the most important challenges in the field of quantum information science: building useful quantum systems, and working together toward the exciting goal of realizing new applications that are out of reach for the most powerful classical computers,” Lukin said.

John Doyle, the Henry B. Silsbee Professor of Physics and co-director of the Harvard Quantum Initiative, is also a primary investigator (PI) at the center. His research will focus on creating programmable arrays of single molecules for use in quantum computing and simulations. The arrays can realize several different types of simulations and computations, with each molecule acting as a single qubit, the much more powerful quantum version of a classical computer bit.


“The Quantum Science Center is about transforming typically fragile quantum states into resilient, controllable, and scalable quantum systems in order to realize the promise of quantum technologies”
- Prineha Narang



“We are doing the basic research into a new quantum-computing platform — individually controlled arrays of single molecules,” said Doyle. “This new center will seed our ideas into other areas of research as we, in turn, learn about new developments made with other quantum platforms. I’m so thankful for all the hard work from people in the Harvard Quantum Initiative who made it possible for us to be part of this center.”

Doyle and Lukin will be working with Markus Greiner, also a George Vasmer Leverett Professor of Physics, and Kang-Kuen Ni, the Morris Kahn Associate Professor of Chemistry and Chemical Biology and of Physics, on this project.

The Quantum Science Center, led by the Oak Ridge National Laboratory outside Knoxville, Tenn., likewise will receive $115 million over five years to develop topological quantum materials for manipulating, transferring, and storing information, and transition this knowledge to the private sector for use in practical applications such as quantum computers and sensors.

Amir Yacoby, professor of physics and of applied physics, will lead one of the research directions focused on developing new approaches for sensing quantum matter. Prineha Narang, assistant professor of computational materials science at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS), is also a member of this research team.

Yacoby and Narang’s work will focus on using quantum sensing techniques to explore quantum materials. Quantum materials exhibit exotic and often mind-boggling properties, such as quantum fluctuations, quantum entanglement, quantum coherence, and topological behavior. These materials could be powerful platforms for quantum technologies — if researchers can understand and harness their underlying properties. Yacoby and Narang plan to use superconducting circuits and magnonic excitations to find new ways to explore these powerful materials on the mesoscale.

“The Quantum Science Center is about transforming typically fragile quantum states into resilient, controllable, and scalable quantum systems in order to realize the promise of quantum technologies,” said Narang. “It’s an incredible team of people, and I’m super excited to not only partner with Oak Ridge National Lab over the next five years but also to forge relationships with the industry partners, particularly Microsoft and IBM, who will be crucial to translate our fundamental work into technology.”

The Co-Design Center for Quantum Advantage, headquartered at Brookhaven National Laboratory in Upton, N.Y., on Long Island, will receive $115 million over five years to develop materials, devices, software, and applications that will serve as a platform for the next generation of quantum computing capabilities. The goal is to overcome the limitations of today’s early stage quantum computers and propel the field forward to unlock new capabilities to tackle real-world challenges.

Yacoby is a member of a research team that is focusing on developing a multi-probe scan system for exploring quantum materials and, in particular, superconducting qubits.

“These research centers will allow us to develop new methodologies to unravel hidden properties of quantum materials,” said Yacoby. “In turn, these new material properties will lead to new quantum technologies with unprecedented capabilities.”

The DOE also announced that Evelyn L. Hu, the Tarr-Coyne Professor of Electrical Engineering and Applied Science at SEAS and co-director of the Harvard Quantum Initiative, will sit on the National Quantum Initiative Advisory Committee (NQIAC), which will counsel the White House on QIS policy.

“I am tremendously honored to be part of the NQIAC, to be able to work with outstanding colleagues to help guide a national policy in this transformative area of science technology,” Hu said. “I hope to serve as a conduit of information between perspectives at the national level and the wealth of exciting quantum activities at Harvard and the Boston area.”

In addition to the DOE-funded QIS research centers, Harvard quantum engineers and scientists are involved in several other new quantum centers and institutes funded by the National Science Foundation. Lukin, Narang, and Marko Lončar, the Tiantsai Lin Professor of Electrical Engineering at SEAS, are PIs in the recently announced Engineering Research Center for Quantum Networks (CQN), headquartered at the University of Arizona in Tucson. CQN aims to create foundations for the future quantum internet by developing key technologies and new functional building blocks connecting quantum processors over local and global scales.

Doyle and Susanne Yelin, professor in residence at the Department of Physics, will serve as co-PIs in the newly launched NSF Quantum Leap Challenge Institute for Enhanced Sensing and Distribution Using Correlated Quantum States. This institute, led by the University of Colorado in Boulder, will design, build, and employ quantum sensing technology for a variety of applications in precision measurement.

“These funded centers are important steps toward stronger collaboration between leading academic groups, national labs, and industry in this rapidly developing field,” said Lukin. “We at Harvard are looking forward to extending and broadening these engagements that will benefit science, technology, and society.”


Cover photo: A close-up view of a quantum computer. Courtesy of SEAS






See also: Faculty, Research, Ni

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:24

Prineha Narang, assistant professor of computational materials science at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS), is also a member of this research team.

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:26

Harvard-led Consortium Forms Center for Integrated Quantum Materials
National Science Foundation grant will enable research and education in field that promises breakthroughs in electronics, photonics and computing
A new NSF grant to the Harvard School of Engineering and Applied Sciences will advance the study of integrated quantum materials

A new NSF grant to the Harvard School of Engineering and Applied Sciences will advance the study of integrated quantum materials. One of the directions for the Center's research will involve nitrogen vacancy centers in diamond, which can store information written and read out using light, as shown in this research illustration courtesy of Marko Loncar, Harvard SEAS.
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Cambridge, Mass. – September 20, 2013 – The National Science Foundation (NSF) has selected Harvard University to lead a new Science and Technology Center, the Center for Integrated Quantum Materials. The multi-institutional grant based at the Harvard School of Engineering and Applied Sciences (SEAS) will provide up to $20 million over five years to fund science and education programs that explore the unique electronic behavior of quantum materials, including graphene, topological insulators, and nitrogen vacancy (NV) center diamond, with the goal of discovering new approaches to signal processing, computing, and terahertz electronics.

The new Center for Integrated Quantum Materials is funded as part of the NSF’s Science and Technology Center (STC) program, which supports integrative partnerships that require large-scale, long-term funding to produce research and education of the highest quality. Existing STCs study a wide range of complex scientific topics, such as multiscale atmospheric modeling, life beneath the sea floor, energy-efficient electronics, and biophotonics. Harvard’s proposal was one of three selected through a merit-based competition.

Drawing on expertise in materials synthesis, nanofabrication, characterization, and device physics, Harvard’s new initiative will be a center for collaboration across diverse disciplines and institutions. Harvard will partner with Howard University, Massachusetts Institute of Technology (MIT), and the Museum of Science, Boston. The Center will also encourage young students to pursue careers in science and engineering through an affiliated College Network including Bunker Hill Community College, Gallaudet University, Mount Holyoke College, Olin College, Prince George's Community College, and Wellesley College.

“As we move into a post-silicon age, quantum materials are an emerging technology with enormous promise for science and engineering, and for our country’s overall economy in the form of new products and business opportunities,” says Robert M. Westervelt, Mallinckrodt Professor of Applied Physics and Physics at Harvard, who will lead the center. “The scientists collaborating on this project have a vision of future quantum materials and quantum devices -- new devices and systems that were not conceived to be possible 10 years ago. This line of research promises an impressive trajectory over the coming decades.”

The Center for Integrated Quantum Materials will integrate three areas of research. The first will involve synthesizing new materials based on graphene, the one-atom-thick carbon material that has proven to be extremely well suited for carrying electrons coherently and rapidly. The researchers hope to use these materials to fabricate new types of ultra-high-speed, atomic-scale devices, including stacked atomic layers that use hexagonal boron nitride as an insulator between sheets of graphene.

The second area of research will explore a class of materials called topological insulators—materials that conduct only at their surface. Topological insulators preserve the direction of an electron spin as it travels along the surface, allowing a spin to carry bits of information in a future quantum network.

The third area of research involves the use of a single atom to store a bit of information. A nitrogen-vacancy (NV) center is created in diamond when a nitrogen atom replaces a carbon in the crystal structure. The electron spin on an NV center can store a bit of information for over 1 millisecond at room temperature, written and read out using light.

The Center aims to integrate NV center diamond storage sites with atomic-layer devices and topological insulator data channels to create transformative new devices and systems for storing, manipulating, and transmitting information.

“Research in integrated quantum materials crosses traditional disciplines and is precisely the type of science that a consortium like this can excel at,” says Cherry A. Murray, Dean of Harvard SEAS, John A. and Elizabeth S. Armstrong Professor of Engineering and Applied Sciences, and Professor of Physics. “Like other nascent technologies that have benefitted from government support in the past, the work of the Center for Integrated Quantum Materials has the potential to be industry-shaping in ways we cannot predict.”

An important part of the Center's mission will be to engage the public in the discovery of atomic electronics and to draw young students to careers in science and engineering.

The Museum of Science, Boston, which also leads the NSF Nanoscale Informal Science Education Network of education and research organizations, will play a core role.

A “College Network” will draw talented undergraduates into the labs, including women from Mount Holyoke and Wellesley, deaf and hard of hearing students from Gallaudet, students with a technical focus from Olin, and a diverse group of students from Bunker Hill and Prince's George's Community Colleges.

Additionally, a special initiative at Bunker Hill will recruit military veterans into the research programs and activities of the Center. Together, these programs will attract students from diverse backgrounds to science and engineering and provide them with unique opportunities for scholarship and leadership.

The National Science Foundation also announced this month that a second Science and Technology Center—the Center for Brains, Minds, and Machines—will be based at the Massachusetts Institute of Technology and co-led by Harvard faculty member L. Mahadevan.

Mahadevan is the Lola England de Valpine Professor of Applied Mathematics at Harvard SEAS, professor of organismic and evolutionary biology, and professor of physics.

“The selection of Harvard as a leader of these new multi-institutional Centers will leverage the University’s broad expertise in emerging, vitally important new fields,” says Richard McCullough, Vice Provost for Research at Harvard University and Professor of Materials Science and Engineering at SEAS. “In addition to advancing the state of science, the Science and Technology Centers will serve an equally important purpose: to encourage a broad cross-section of students at various types of institutions to explore careers in science, engineering and technology.”

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:29

https://cifar.ca/research-programs/quantum-materials/

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:31

Novel quantum matter
growth
We are working on creating and exploring novel quantum materials on the mesoscopic scale. We search for emergent physical phenomena, such as superconductivity, magnetism, and quantum spin liquids in novel quantum structures created by various advanced material growth and manipulation. Here, we combine low-temperature quantum transport measurements with advanced characterization techniques like transmission electron microscopy (TEM), optical spectroscopy to reveal these engineered quantum materials' fundamental properties. Several current research interests include (1) the intercalation of 2D materials heterostructure for novel superconductivity and magnetism; (2) advanced crystal growing techniques using the flux method and molecular beam epitaxy (MBE) for vdW layered quantum heterostructures; (3) in-situ TEM study of van dew Waals heterostructures, and (4) search for emergent phenomena in low dimensional organic/inorganic heterostructures.

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:32

epitaxy

KK[ˋɛpɪ͵tæksɪ] DJ[ˋepi͵tæksi]

美式

n.
【物】取向附生；【電子】磊晶

名詞複數：epitaxies

更多解釋

epitaxy

外延性

PyDict

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:33

Harvard partners with national labs on quantum computing
A close-up view of a quantum compute

A close-up view of a quantum computer.

Courtesy of SEAS
Energy Department creates 5 science research centers in field, 3 with Harvard groups

By Leah Burrows SEAS Communications

DateSeptember 2, 2020
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Harvard scientists will take leading roles in a new federal government effort to advance quantum computing, an experimental technology that could make it possible to perform calculations at speeds inconceivable today and usher in game-changing innovations in medicine, artificial intelligence, drug development, and other fields.

The White House Office of Science and Technology Policy and the U.S. Department of Energy (DOE) announced last week the creation of five new Quantum Information Science (QIS) Research Centers across the country. These centers, led by DOE national laboratories, will be part of the U.S. National Quantum Initiative.

The DOE awarded $625 million over five years to the Argonne, Brookhaven, Fermi, Oak Ridge, and Lawrence Berkeley national laboratories to establish centers. Each facility will have an interdisciplinary research team and multiple partner institutions. Harvard researchers will play important roles in the centers at Brookhaven, Oak Ridge, and Lawrence Berkeley.

“This very welcome federal investment in QIS Research Centers will catalyze advances in one of the most exciting areas of science today. Nobody who knows their work will be surprised to see the prominent roles that Harvard scientists and engineers, each one of them a leading innovator in quantum science, will play in the new centers,” said Provost Alan M. Garber. “I look forward to seeing the results of their research, which promises to help shape the future of computation and data science.”

The Quantum Systems Accelerator (QSA) at the Lawrence Berkeley National Laboratory (Berkeley Lab) in Alameda County, just north of the University of California, Berkeley, was awarded $115 million over five years to develop the solutions needed to build working quantum systems, which harness the power of quantum mechanics.

Mikhail Lukin, a George Vasmer Leverett Professor of Physics and co-director of the Harvard Quantum Initiative, is project leader of one of the QSA’s major research thrusts: algorithms and applications. Lukin and his team will investigate how quantum computers can meaningfully speed up answers to real-world scientific problems and create new tools to quantify this advantage and performance.

“Within this project, DOE labs partner with leading academic research groups from across the country to address some of the most important challenges in the field of quantum information science: building useful quantum systems, and working together toward the exciting goal of realizing new applications that are out of reach for the most powerful classical computers,” Lukin said.

John Doyle, the Henry B. Silsbee Professor of Physics and co-director of the Harvard Quantum Initiative, is also a primary investigator (PI) at the center. His research will focus on creating programmable arrays of single molecules for use in quantum computing and simulations. The arrays can realize several different types of simulations and computations, with each molecule acting as a single qubit, the much more powerful quantum version of a classical computer bit.
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Riding the quantum computing ‘wave’

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“We are doing the basic research into a new quantum-computing platform — individually controlled arrays of single molecules,” said Doyle. “This new center will seed our ideas into other areas of research as we, in turn, learn about new developments made with other quantum platforms. I’m so thankful for all the hard work from people in the Harvard Quantum Initiative who made it possible for us to be part of this center.”

Doyle and Lukin will be working with Markus Greiner, also a George Vasmer Leverett Professor of Physics, and Kang-Kuen Ni, the Morris Kahn Associate Professor of Chemistry and Chemical Biology and of Physics, on this project.

The Quantum Science Center, led by the Oak Ridge National Laboratory outside Knoxville, Tenn., likewise will receive $115 million over five years to develop topological quantum materials for manipulating, transferring, and storing information, and transition this knowledge to the private sector for use in practical applications such as quantum computers and sensors.

Amir Yacoby, professor of physics and of applied physics, will lead one of the research directions focused on developing new approaches for sensing quantum matter. Prineha Narang, assistant professor of computational materials science at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS), is also a member of this research team.

Yacoby and Narang’s work will focus on using quantum sensing techniques to explore quantum materials. Quantum materials exhibit exotic and often mind-boggling properties, such as quantum fluctuations, quantum entanglement, quantum coherence, and topological behavior. These materials could be powerful platforms for quantum technologies — if researchers can understand and harness their underlying properties. Yacoby and Narang plan to use superconducting circuits and magnonic excitations to find new ways to explore these powerful materials on the mesoscale.

“The Quantum Science Center is about transforming typically fragile quantum states into resilient, controllable, and scalable quantum systems in order to realize the promise of quantum technologies,” said Narang. “It’s an incredible team of people, and I’m super excited to not only partner with Oak Ridge National Lab over the next five years but also to forge relationships with the industry partners, particularly Microsoft and IBM, who will be crucial to translate our fundamental work into technology.”

The Co-Design Center for Quantum Advantage, headquartered at Brookhaven National Laboratory in Upton, N.Y., on Long Island, will receive $115 million over five years to develop materials, devices, software, and applications that will serve as a platform for the next generation of quantum computing capabilities. The goal is to overcome the limitations of today’s early stage quantum computers and propel the field forward to unlock new capabilities to tackle real-world challenges.

Yacoby is a member of a research team that is focusing on developing a multi-probe scan system for exploring quantum materials and, in particular, superconducting qubits.

“These research centers will allow us to develop new methodologies to unravel hidden properties of quantum materials,” said Yacoby. “In turn, these new material properties will lead to new quantum technologies with unprecedented capabilities.”

The DOE also announced that Evelyn L. Hu, the Tarr-Coyne Professor of Electrical Engineering and Applied Science at SEAS and co-director of the Harvard Quantum Initiative, will sit on the National Quantum Initiative Advisory Committee (NQIAC), which will counsel the White House on QIS policy.

“I am tremendously honored to be part of the NQIAC, to be able to work with outstanding colleagues to help guide a national policy in this transformative area of science technology,” Hu said. “I hope to serve as a conduit of information between perspectives at the national level and the wealth of exciting quantum activities at Harvard and the Boston area.”

In addition to the DOE-funded QIS research centers, Harvard quantum engineers and scientists are involved in several other new quantum centers and institutes funded by the National Science Foundation. Lukin, Narang, and Marko Lončar, the Tiantsai Lin Professor of Electrical Engineering at SEAS, are PIs in the recently announced Engineering Research Center for Quantum Networks (CQN), headquartered at the University of Arizona in Tucson. CQN aims to create foundations for the future quantum internet by developing key technologies and new functional building blocks connecting quantum processors over local and global scales.

Doyle and Susanne Yelin, professor in residence at the Department of Physics, will serve as co-PIs in the newly launched NSF Quantum Leap Challenge Institute for Enhanced Sensing and Distribution Using Correlated Quantum States. This institute, led by the University of Colorado in Boulder, will design, build, and employ quantum sensing technology for a variety of applications in precision measurement.

“These funded centers are important steps toward stronger collaboration between leading academic groups, national labs, and industry in this rapidly developing field,” said Lukin. “We at Harvard are looking forward to extending and broadening these engagements that will benefit science, technology, and society.”

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:37

The study on quantum material WTe2

Pan, Xing-Chen ; Wang, Xuefeng ; Song, Fengqi ; Wang, Baigeng

Abstract

WTe2 and its sister alloys have attracted tremendous attentions recent years due to the large non-saturating magnetoresistance and topological non-trivial properties. Herein, we briefly review the electrical property studies on this new quantum material.

Publication:
Advances in Physics X
Pub Date:
January 2018
DOI:
10.1080/23746149.2018.1468279
arXiv:
arXiv:1707.04132
Bibcode:
2018AdPhX...368279P
Keywords:

Condensed Matter - Materials Science

E-Print:
This has been submitted for a review article. Please contact the author if anything missed








數位物件識別號：DOI（Digital Object Identifier）. 如果你想要在圖書館或書店找一本書，而你恰巧知道它的ISBN，就會使得查找順利許多，因為ISBN是書本的專屬身分；而DOI也是智慧版權物件在網路上「永不失效」的唯一識別碼。.



arXiv （ X 依 希臘文 的 χ 發音，讀音如英語的 archive ）是一個收集 物理學 、 數學 、 計算機科學、 生物學 與 數理經濟學 的 論文 預印本 的網站

bibcode是许多天文学数据系统用于指定参考文献的标识符。bibcode研发用于SIMBAD和NASA/IPAC河外星系數據庫（NED），但目前已经获得更广阔的适用范围，如NASA的天体物理数据系统。[1][2]

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:38

Optical manipulation of electronic dimensionality in a quantum material

Duan, Shaofeng ; Cheng, Yun ; Xia, Wei ; Yang, Yuanyuan ; Qi, Fengfeng ; Tang, Tianwei ; Guo, Yanfeng ; Qian, Dong ; Xiang, Dao ; Zhang, Jie ; Zhang, Wentao

Abstract

Exotic phenomenon can be achieved in quantum materials by confining electronic states into two dimensions. For example, relativistic fermions are realised in a single layer of carbon atoms, the quantized Hall effect can result from two-dimensional (2D) systems, and the superconducting transition temperature can be enhanced significantly in a one-atomic-layer material. Ordinarily, 2D electronic system can be obtained by exfoliating the layered materials, growing monolayer materials on substrates, or establishing interfaces between different materials. Herein, we use femtosecond infrared laser pulses to invert the periodic lattice distortion sectionally in a three-dimensional (3D) charge density wave material, creating macroscopic domain walls of transient 2D ordered electronic states with exotic properties. The corresponding ultrafast electronic and lattice dynamics are captured by time- and angle-resolved photoemission spectroscopy and MeV ultrafast electron diffraction. Surprisingly, a novel energy gap state, which might be a signature of light-induced superconductivity, is identified in the photoinduced 2D domain wall near the surface. Such optical modulation of atomic motion is a new path to realise 2D electronic states and will be a new platform for creating novel phases in quantum materials.

Publication:
arXiv e-prints
Pub Date:
January 2021
arXiv:
arXiv:2101.08507
Bibcode:
2021arXiv210108507D
Keywords:

Condensed Matter - Strongly Correlated Electrons; Condensed Matter - Materials Science; Condensed Matter - Superconductivity

E-Print:
16 pages, 5 figures

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:40

Brookhaven Lab Teams with Harvard and MIT to Build a Printing Press for Quantum Materials

April 23, 2019

April 23, 2019 — Checking out a stack of books from the library is as simple as searching the library’s catalog and using unique call numbers to pull each book from their shelf locations. Using a similar principle, scientists at the Center for Functional Nanomaterials (CFN)—a U.S. Department of Energy (DOE) Office of Science User Facility at Brookhaven National Laboratory—are teaming with Harvard University and the Massachusetts Institute of Technology (MIT) to create a first-of-its-kind automated system to catalog atomically thin two-dimensional (2-D) materials and stack them into layered structures. Called the Quantum Material Press, or QPress, this system will accelerate the discovery of next-generation materials for the emerging field of quantum information science (QIS).

Structures obtained by stacking single atomic layers (“flakes”) peeled from different parent bulk crystals are of interest because of the exotic electronic, magnetic, and optical properties that emerge at such small (quantum) size scales. However, flake exfoliation is currently a manual process that yields a variety of flake sizes, shapes, orientations, and number of layers. Scientists use optical microscopes at high magnification to manually hunt through thousands of flakes to find the desired ones, and this search can sometimes take days or even a week, and is prone to human error.

Once high-quality 2-D flakes from different crystals have been located and their properties characterized, they can be assembled in the desired order to create the layered structures. Stacking is very time-intensive, often taking longer than a month to assemble a single layered structure. To determine whether the generated structures are optimal for QIS applications—ranging from computing and encryption to sensing and communications—scientists then need to characterize the structures’ properties.

“In talking to our university collaborators at Harvard and MIT who synthesize and study these layered heterostructures, we learned that while bits of automation exist—such as software to locate the flakes and joysticks to manipulate the flakes—there is no fully automated solution,” said CFN Director Charles Black, the administrative lead on the QPress project.

The idea for the QPress was conceived in early 2018 by Professor Amir Yacoby of the Department of Physics at Harvard. The concept was then refined through a collaboration between Yacoby; Black and Kevin Yager, leader of the CFN Electronic Nanomaterials Group; Philip Kim, also of Harvard’s Department of Physics; and Pablo Jarillo-Herrero and Joseph Checkelsky, both of the Department of Physics at MIT.

According to Black, the unique CFN role was clear: “We realized that building a robot that can enable the design, synthesis, and testing of quantum materials is extremely well-matched to the skills and expertise of scientists at the CFN. As a user facility, CFN is meant to be a resource for the scientific community, and QIS is one of our growth areas for which we’re expanding our capabilities, scientific programs, and staff.”
Graphene sparks 2-D materials research

The interest in 2-D materials dates back to 2004, when scientists at the University of Manchester isolated the world’s first 2-D material, graphene—a single layer of carbon atoms. They used a surprisingly basic technique in which they placed a piece of graphite (the core material of pencils) on Scotch tape, repeatedly folding the tape in half and peeling it apart to extract ever-thinner flakes. Then, they rubbed the tape on a flat surface to transfer the flakes. Under an optical microscope, the one-atom-thick flakes can be located by their reflectivity, appearing as very faint spots. Recognized with a Nobel Prize in 2010, the discovery of graphene and its unusual properties—including its remarkable mechanical strength and electrical and thermal conductivity—has prompted scientists to explore other 2-D materials.

Many labs continue to use this laborious approach to make and find 2-D flakes. While the approach has enabled scientists to perform various measurements on graphene, hundreds of other crystals—including magnets, superconductors, and semiconductors—can be exfoliated in the same way as graphite. Moreover, different 2-D flakes can be stacked to build materials that have never existed before. Scientists have very recently discovered that the properties of these stacked structures depend not only on the order of the layers but also on the relative angle between the atoms in the layers. For example, a material can be tuned from a metallic to an insulating state simply by controlling this angle. Given the wide variety of samples that scientists would like to explore and the error-prone and time-consuming nature of manual synthesis methods, automated approaches are greatly needed.

“Ultimately, we would like to develop a robot that delivers a stacked structure based on the 2-D flake sequences and crystal orientations that scientists select through a web interface to the machine,” said Black. “If successful, the QPress would enable scientists to spend their time and energy studying materials, rather than making them.”
A modular approach

In September 2018, further development of the QPress was awarded funding by the DOE, with a two-part approach. One award was for QPress hardware development at Brookhaven, led by Black; Yager; CFN scientists Gregory Doerk, Aaron Stein, and Jerzy Sadowski; and CFN scientific associate Young Jae Shin. The other award was for a coordinated research project led by Yacoby, Kim, Jarillo-Herrero, and Checkelsky. The Harvard and MIT physicists will use the QPress to study exotic forms of superconductivity—the ability of certain materials to conduct electricity without energy loss at very low temperatures—that exist at the interface between a superconductor and magnet. Some scientists believe that such exotic states of matter are key to advancing quantum computing, which is expected to surpass the capabilities of even today’s most powerful supercomputing.
A photo of the prototype exfoliator. The robotic system transfers peeled 2-D flakes from the parent crystal to a substrate. The exfoliator allows scientists to control stamping pressure, pressing time, number of repeated presses, angle of pressing, and lateral force applied during transfer, for improved repeatability.

A fully integrated automated machine consisting of an exfoliator, a cataloger, a library, a stacker, and a characterizer is expected in three years. However, these modules will come online in stages to enable the use of QPress early on.

The team has already made some progress. They built a prototype exfoliator that mimics the action of a human peeling flakes from a graphite crystal. The exfoliator presses a polymer stamp into a bulk parent crystal and transfers the exfoliated flakes by pressing them onto a substrate. In their first set of experiments, the team investigated how changing various parameters—stamping pressure, pressing time, number of repeated presses, angle of pressing, and lateral force applied during transfer—impact the process.

“One of the advantages of using a robot is that, unlike a human, it reproduces the same motions every time, and we can optimize these motions to generate lots of very thin large flakes,” explained Yager. “Thus, the exfoliator will improve both the quality and quantity of 2-D flakes peeled from parent crystals by refining the speed, precision, and repeatability of the process.”

In collaboration with Stony Brook University assistant professor Minh Hoai Nguyen of the Department of Computer Science and PhD student Boyu Wang of the Computer Vision Lab, the scientists are also building a flake cataloger. Through image-analysis software, the cataloger scans a substrate and records the locations of exfoliated flakes and their properties.
A schematic showing the workflow for cataloging flake locations and properties. Image grids of exfoliated samples are automatically analyzed, with each flake tracked individually so that scientists can locate any desired flake on a sample.

“The flakes that scientists are interested in are thin and thus faint, so manual visual inspection is a laborious and error-prone process,” said Nguyen. “We are using state-of-the-art computer vision and deep learning techniques to develop software that can automate this process with higher accuracy.”

“Our collaborators have said that a system capable of mapping their sample of flakes and showing them where the “good” flakes are located—as determined by parameters they define—would be immensely helpful for them,” said Yager. “We now have this capability and would like to put it to use.”

Eventually, the team plans to store a large set of different catalogued flakes on shelves, similar to books in a library. Scientists could then access this materials library to select the flakes they want to use, and the QPress would retrieve them.

According to Black, the biggest challenge will be the construction of the stacker—the module that retrieves samples from the library, “drives” to the locations where the selected flakes reside, and picks the flakes up and places them in a repetitive process to build stacks according to the assembly instructions that scientists program into the machine. Ultimately, the scientists would like the stacker to assemble the layered structures not only faster but also more accurately than manual methods.

The final module of the robot will be a material characterizer, which will provide real-time feedback throughout the entire synthesis process. For example, the characterizer will identify the crystal structure and orientation of exfoliated flakes and layered structures through low-energy electron diffraction (LEED)—a technique in which a beam of low-energy electrons is directed toward the surface of a sample to produce a diffraction pattern characteristic of the surface geometry.

“There are many steps to delivering a fully automated solution,” said Black. “We intend to implement QPress capabilities as they become available to maximize benefit to the QIS community.”

Brookhaven National Laboratory is supported by the Office of Science of the U.S. Department of Energy. The Office of Science is the single largest supporter of basic research in the physical sciences in the United States, and is working to address some of the most pressing challenges of our time. For more information, please visit science.energy.gov.

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:42

Kagome metal
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Kagome metal is a ferromagnetic quantum material created in 2018.[1][2] In the material, metal atoms are arranged in a lattice resembling the Japanese kagome basket weaving pattern.[3] The material was announced in Nature by investigators from Massachusetts Institute of Technology, Harvard University and Lawrence Berkeley National Laboratory, all in the United States.[1]

In the Kagome structure, atoms are arranged into layered sets of overlapping triangles so that there exist large empty hexagonal spaces. Electrons in the metal experience a "three-dimensional cousin of the quantum Hall effect".[4] The inherent magnetism of the metal and the quantum-mechanical magnetism induce electrons to flow around the edges of the triangular crystals, akin to superconductivity.[4] Unlike superconductivity, this structure and behavior is stable at room temperature.[2] Other structures were shown to exhibit the quantum hall effect at very low temperatures with an external magnetic field as high as 1 million times the strength that of the earth.[4] By building metal out of a ferromagnetic material, that exterior magnetic field was no longer necessary, and the quantum Hall effect persists into room temperature.

The Kagome alloy synthesized in 2018 displayed several exotic quantum electronic behaviors that add to its quantum topology. The lattice harbors massive Dirac fermions, Berry curvature, band gaps, and spin-orbit activity, all of which are conducive to the Hall Effect and zero energy loss electric currents.[2][5][6] These behaviors are promising for the development of technologies in quantum computing, spin superconductors, and low power electronics.[2][4] As of 2019, more Kagome materials displaying similar topology were being experimented with, such as in magnetically doped Weyl-Semimetals Co2MnGa and Co3Sn2S2.[7]
Contents

1 Development
1.1 March 2018 experiment
1.2 Expansions upon the 2018 experiment
2 See also
3 Further reading
4 References

Development
March 2018 experiment

The experiment published in Nature was led by Linda Ye and Mingu Kang from the Massachusetts Institute of Technology Department of Physics. The other researches published in the article were: Junwei Liu (MIT), Felix von Cube (Harvard), Christina R. Wicker (MIT), Takehito Suzuki (MIT), Chris Jozwiak (Berkeley Labs), Aaron Bostwick (Berkeley Labs), Eli Rotenberg (Berkeley Labs), David C. Bell (Harvard), Liang Fu (MIT), Riccardo Comin (MIT), and Joseph G. Checkelsky (MIT).[8]

The team constructed a Kagome lattice using an iron and tin alloy Fe3Sn2. When Fe3Sn2 is heated to about 750℃ (1380°F), the alloy naturally assumes a Kagome lattice structure.[1] To maintain this structure at room temperature, the team cooled it in an ice bath. The resulting structure had iron atoms at the corners of each triangle surrounding tin atoms, the tin atoms stabilizing the large empty hexagonal space.[4]

The photo-electronic structure of this metal was mapped at Lawrence Berkeley National Lab's Advanced Light Source beamlines 7.0.2 MAESTRO and 4.0.3 MERLIN. The measurements taken here mapped the band structures of the metal under a current and showed “the double-Dirac-cone structure corresponding to Dirac Fermions”.[2] A 30 meV gap between cones was shown, which is indicative of the Hall effect and massive Dirac fermions.[2]

The experiment was published in Nature on March 19, 2018.[4]
Expansions upon the 2018 experiment

Linda Ye and the rest of the MIT team continued working with the kagome metal, looking into more properties on Fe3Sn2 and other iron-tin alloys. According to an article published by the team on October 25, 2019 in Nature Communications, the lattice was shown to exhibit spin-orbit coupling de Haas van Alphen oscillations and Fermi surfaces.[6] The team also continued to modify the structure of the metal, and in an article published in Nature Materials on December 9, 2019, they found that FeSn in a 1 to 1 ratio exhibited a more “ideal” lattice. The two-dimensional layers with iron and tin in the kagome shape were separated by a layer entirely of tin, allowing them to have separate band structures.[9] This structure showed both massive Dirac fermions and electronic band structures where electrons do not occur.[9]
See also

Herbertsmithite, a natural mineral with kagome lattice at the atomic scale
Magnon

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:45

Quantum Optoelectronics

Nanoscale materials represent a natural size limit of the miniaturization trend of current technology, and they exhibit physical and chemical properties significantly different from their bulk counterparts. The Park group is exploring atom-sized defects in solids, atomically thin materials, and photonic and plasmonic nanostructures to realize solid-state optoelectronic and plasmonic devices that work all the way down to the single quantum level. Some examples of these devices include single-photon transistors, electrically driven quantum plasmonic switches, and topological quantum photonic circuits. These devices, whose operation is critically dependent upon quantum mechanical principles, may enable all-optical computing and solid-state quantum information processing.
Broken Mirror Symmetry in a Twisted MoSe2 Bilayer
Flatland Quantum Optoelectronics
Negative Refraction in a Metasurface
Negative Refraction in a Metasurface

Refraction of surface plasmon polaritons at the flat silver/metasurface interface.
Atomically Thin Mirror
Hyperbolic Metasurface
Memory-Assisted Heralded Spin-Photon Gate
Valley Dynamics in Twisted WSe2 Homobilayer
Electrical Control of Interlayer Excitons
Atomically Thin Membrane
SiV Quantum Network
New Near-Field Spectroscopy for Two-Dimensional Materials
Plasmonic Crystal Cavity
Nano-Bio Interfaces | Quantum Sensing/Control

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:46

Thermal and thermoelectric transport in quantum systems

Thermal and thermoelectric transport measurements in quantum engineered materials platforms can reveal physical behaviors hidden to purely electrical transport. Thermal conductivity measured with Johnson noise thermometry can probe the details of electronic scattering and interactions, and thermopower measurements provide a unique signature of the entropy per particle. These techniques can be applied to study emergent behavior in exotic quantum systems, such as spin waves in 2D magnets, vortex dynamics in superconductors, and topological physics in the quantum Hall regime. Hydrodynamic electron systems, where electron-electron interactions dominate, can show novel transport phenomena such as the breakdown of the Wiedemann-Franz law (a relation between electrical and thermal conductivity obeyed in almost all materials) observed in graphene. Investigating and engineering the thermoelectric properties of quantum materials, such as increasing the thermoelectric figure of merit, also has broad applications for the development of nanoelectronic devices.

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:48

Imagine, for a moment, that you could forget everything you ever knew about the way our reality works. That two objects hundreds of miles apart could communicate instantaneously. That the act of measuring something could change what is being measured. That something—a cat, perhaps—could be both alive and dead at the same time.

But this actually is the way our reality works. Or at least it does at the microscopic level. When you look at smaller and smaller objects— objects at the atomic and subatomic levels—quantum effects seem to make interactions between objects operate differently than predicted by the kind of classical physics most of us learned in high school. The quantum world is so counterintuitive, so strange, that even Max Planck (who coined the term quanta) searched for an alternative explanation for his results.

More than a century later, quantum theory may seem mysterious to the lay person, but mathematically, it’s 100 percent calculable. “Quantum mechanics tells you what is going to happen, very precisely,” says John Doyle, Henry B. Silsbee Professor of Physics in the Faculty of Arts and Sciences (FAS). “When you do the calculations, there’s no mystery about it at all. The mystery is only that microscopic physics behaves differently than what we see in our daily lives.”
John Doyle (in a button down and bow tie), Evelyn Hu (smiling and wearing a button down shirt), and Mikhail Lukin (also smiling and wearing a button down shirt), stand in front of a blackboard with a roughly drawn chalk graph

John Doyle, Evelyn Hu, and Mikhail Lukin lead the Harvard Quantum Initiative

Doyle is one of the co-directors of the Harvard Quantum Initiative (HQI), a cross-disciplinary community dedicated to expanding research, development, and education on quantum information and its possibilities. Launched in November 2018, HQI forms part of a larger effort in the United States to advance understanding of quantum science and technology and initiate the development of innovative materials and devices for high-speed computing, networking, and ultra-precise sensing. Doyle and his colleagues—co-directors Evelyn Hu, Tarr-Coyne Professor of Applied Physics and of Electrical Engineering at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS), and Mikhail Lukin, George Vasmer Leverett Professor of Physics at FAS—see the advancement of quantum science as a radical effort to expand scientific boundaries and rethink how research is accomplished.

“We believe that quantum science represents a profound change in the understanding of what information is, which will affect everything we do moving forward,” says Hu. “That change will affect how we measure, how we compute, and how we communicate.” By embracing the physics that operates at the quantum level, researchers can tackle problems—or reinvestigate old ones—in groundbreaking new ways that hold transformative potential. “For people in the sciences, this is a tremendous opportunity to create novel applications and instrumentation that can be used to further enable scientific research,” Hu says.

We believe that quantum science represents a profound change in the understanding of what information is, which will affect everything we do moving forward." — Evelyn Hu, Tarr-Coyne Professor of Applied Physics and of Electrical Engineering

Hu deliberately uses the term “quantum science” and “quantum engineering” instead of “quantum physics.” While early breakthroughs in quantum science came from theoretical physicists, the next generation of quantum scientists has come from physics, computer science, chemistry, and mathematics. This broader collaboration is expected to, in turn, accelerate and enrich progress in these fields, but in ways that cannot at the moment be easily predicted. “Breakthroughs often happen at the interfaces of traditional disciplines,” says Lukin.
Spooky Action up Close

Those breakthroughs will rely on a strong foundation of basic science, built in part by faculty, graduate students, and recent graduates from Harvard. Srujan Meesala, PhD ’19, is emblematic of the young scientists whose work sits at the intersection of multiple fields. Meesala studied electrical engineering as an undergraduate, and though he wanted to focus on physics for his graduate work, he’s grateful for the hands-on experience he brings to his research. “There is a need for engineers who think about making devices to study interesting physics,” he says, “and for people from hard physics backgrounds to become more engineering oriented.” Meesala brought his expertise to the lab of Marko Loncar, the Tiantsai Lin Professor of Electrical Engineering at SEAS.
Srujan Meesala, standing with his arms crossed and wearing glasses and a button down shirt with the sleeves rolled up

Srujan Meesala, PhD '19

Meesala wanted to discover how to use quantum effects to transfer information. To conduct this research, he’s using sound and flaws in an unlikely source— diamonds. “My research uses defects inside diamonds, called color centers,” he explains. Diamond crystals are made of carbon atoms. By removing one carbon atom and replacing a second with another element—nitrogen or silicon, for example— he can create an individual quantum system inside a solid. “This is interesting for quantum information applications because you can store and process information at the level of the atom inside a solid.”

Experiments on color centers have shown certain quantum effects, such as when two quantum particles interact and become interdependent, a process known as entanglement. Einstein found the phenomenon so startling that he famously called it “spooky action at a distance.”

“Entanglement is hard to observe in nature. You need to engineer very specific quantum systems to see it,” Meesala explains. “Isolating and manipulating these individual defects is one way to do that.” By taking advantage of color centers, researchers have measured the entanglement of particles in two diamonds located a kilometer apart by using the light emitted by the defects.

But while quantum entanglement has been generated between diamond color centers at a distance, it is hard to achieve between two defects in the same diamond. “To perform local quantum operations with these color centers, we would ideally put a number of color centers on one chip and have them interact.”

Entanglement is hard to observe in nature. You need to engineer very specific quantum systems to see it." — Srujan Meesala, PhD '19

Meesala used sound in an effort to induce these interactions. “You can quantize sound at the quantum level into particles called phonons,” he says. “I wanted to determine if we could get one color center to interact with another by exchanging sound waves.” In their experiment, Meesala and his colleagues carved a thin string out of diamond, then applied voltage to stretch it and study how sound waves couple to a defect called the silicon vacancy center. They envisioned that sound can move along such a string like a pressure wave, causing some of the atoms to compress and others to elongate, essentially transferring information from defect to defect.

Meesala was able to conclude that a silicon vacancy center provides a good system for testing the sound-mediated transfer of quantum information, which could be applied in quantum computers and networks. The work was published last year in Nature Communications.
Along the Way

Di Wei, PhD ’19, has taken a different approach to solving the problem of information transfer at the quantum level, but one still grounded in basic science research. As an undergraduate working in a condensed matter lab at the University of Denver, she connected with scientists like her whose inquisitiveness drove them to solve problems for their own sake. “I’m of the school of thought that you should do basic research for the sake of basic research, and that technology will arise from that,” she says. “Special relativity is my favorite example of that because it’s so abstract, but it is the reason we have technologies like GPS today.”

These energizing collaborations fueled her decision to pursue graduate studies at SEAS, where she continues to satisfy her curiosity about physics. Wei joined the lab of Amir Yacoby, professor of physics and of applied physics at FAS and SEAS, where she found numerous opportunities to engage in fundamental research questions. One of Yacoby’s projects involved looking for spin superfluidity in graphene, a two-dimensional sheet of carbon atoms.
Di Wei, wearing a t-shirt and glasses, stands outdoors with her hands behind her back

Di Wei, PhD '19

“Spin superfluidity is a magnetic analog to superconductivity, using the electron’s spin as carriers of information as opposed to their charge,” Wei explains. “You can think of each spin as a local magnetic moment that can interact with neighboring spins. If the spin at one site is tilted, its reorientation is felt by nearby spins.” This tilting creates a spin wave that can transfer angular momentum from site to site, which can be used to send information with less heat than is currently generated by traditional electrical methods used in computerized devices. “A superfluid of spins would allow for the transfer of spins with no energy loss, which would be a fundamentally new kind of system to study,” she says.

Yacoby and colleague Bertrand Halperin, Hollis Professor of Mathematics and Natural Philosophy, Emeritus, had postulated that superfluidity could be found in graphene, and Yacoby and Wei set out to test this theory. Using a type of magnetic material known as a quantum hall ferromagnet, they observed some intriguing physics that could help measure exotic quantum states via a process called electron interferometry. “The first thing we saw was an interesting interference pattern between electrons,” Wei says. “The pattern confirmed that graphene was a good platform for electron interferometry.” This work was published in Science Advances in August 2017.

Next, Wei used an electric current to generate spin waves through the graphene and was able to detect those waves via an induced voltage. “It’s the first transport measurement of spin waves in a graphene quantum hall ferromagnet,” Wei says. “They were predicted to exist in this system but were difficult to detect with electronic methods because they are excitations of an insulating material.” In October 2018, she was lead author on a paper about these findings, which appeared in Science magazine.

Although Wei wasn’t able to demonstrate spin superfluidity with certainty, her results do establish graphene as worthy of further study. “Often, the original idea that researchers set out to test doesn’t work, but they are still able to find interesting things along the way,” she says.
A Changing World

Doyle, Hu, and Lukin believe that Harvard is well positioned to make advancements in quantum science and engineering through the work of researchers like Meesala and Wei—large enough to tap existing intellectual prowess and small enough to act on ideas and move more quickly that other institutions. “A new discipline is being created right in front of our eyes,” says Doyle.

“It’s obvious that this is a special time,” Lukin continues. “We have a very special community of world-leading researchers. Harvard is home to the most powerful quantum machines ever made.” He notes that hundreds of young leaders who trained at Harvard are now conducting their own cutting-edge research at institutions around the world, pushing the boundaries of what is currently known about the quantum realm.

And while this new discipline holds tremendous potential—so much so that Harvard, the US, and the world are willing to invest in its promise—it’s impossible to anticipate what the ultimate impact will be. “No inventor could predict how the world would use their invention,” says Hu. “We don’t know where the spark will be lit, a spark that will burn a pathway to new applications.”

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:51

Graduate Studies
photo of commencement procession of graduate students
Commencement 2019

The Harvard Department of Physics offers students innovative educational and research opportunities with renowned faculty in state-of-the-art facilities, exploring fundamental problems involving physics at all scales. Our primary areas of experimental and theoretical research are atomic and molecular physics, astrophysics and cosmology, biophysics, chemical physics, computational physics, condensed-matter physics, materials science, mathematical physics, particle physics, quantum optics, quantum field theory, quantum information, string theory, and relativity.

Our talented and hardworking students participate in exciting discoveries and cutting-edge inventions such as the ATLAS experiment, which discovered the Higgs boson; building the first 51-cubit quantum computer; measuring entanglement entropy; discovering new phases of matter; and peering into the ‘soft hair’ of black holes.

Our students come from all over the world and from varied educational backgrounds. We are committed to fostering an inclusive environment and attracting the widest possible range of talents.

We have a flexible and highly responsive advising structure for our PhD students that shepherds them through every stage of their education, providing assistance and counseling along the way, helping resolve problems and academic impasses, and making sure that everyone has the most enriching experience possible.The graduate advising team also sponsors alumni talks, panels, and advice sessions to help students along their academic and career paths in physics and beyond, such as “Getting Started in Research,” “Applying to Fellowships,” “Preparing for Qualifying Exams,” “Securing a Post-Doc Position,” and other career events (both academic and industry-related).

We offer many resources, services, and on-site facilities to the physics community, including our electronic instrument design lab and our fabrication machine shop. Our historic Jefferson Laboratory, the first physics laboratory of its kind in the nation and the heart of the physics department, has been redesigned and renovated to facilitate study and collaboration among our students.

Members of the Harvard Physics community participate in initiatives that bring together scientists from institutions across the world and from different fields of inquiry. For example, the Harvard-MIT Center for Ultracold Atoms unites a community of scientists from both institutions to pursue research in the new fields opened up by the creation of ultracold atoms and quantum gases. The Center for Integrated Quantum Materials, a collaboration between Harvard University, Howard University, MIT, and the Museum of Science, Boston, is dedicated to the study of extraordinary new quantum materials that hold promise for transforming signal processing and computation. The Harvard Materials Science and Engineering Center is home to an interdisciplinary group of physicists, chemists, and researchers from the School of Engineering and Applied Sciences working on fundamental questions in materials science and applications such as soft robotics and 3D printing. The new Black Hole Initiative, the first center worldwide to focus on the study of black holes, is an interdisciplinary collaboration between principal investigators from the fields of astronomy, physics, mathematics, and philosophy. And the recently launched Quantitative Biology Initiative aims to bring together physicists, biologists, engineers, and applied mathematicians to understand life itself.

We support and encourage interdisciplinary research and simultaneous applications to two departments is permissible. Prospective students may thus wish to apply to the following departments and programs in addition to Physics:

Department of Astronomy Department of Chemistry Department of Mathematics John A. Paulson School of Engineering and Applied Sciences (SEAS) Biophysics Program Engineering and Physical Biology Program (EPB) Molecules, Cells and Organisms Program (MCO)

If you are a prospective graduate student and have questions for us, or if you’re interested in visiting our department, please contact inquirePhysics .harvard.edu.

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 04:53

Harvard Natural Sciences Lecture Demonstrations

https://www.youtube.com/channel/UC0AeRY_wcZgoY5LtsS8jv2A

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 05:00

Nuclear and Elementary Particle Physics https://sciencedemonstrations.fas.harvard.edu/nuclear-and-elementary-particle-physics

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 05:06

Harvard Natural Sciences Lecture Demonstrations

1 Oxford St Cambridge MA 02138 Science Center B-08A (617) 495-5824



https://sciencedemonstrations.fas.harvard.edu/catalog

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 05:11

魏尔费米子
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粒子物理學标准模型
Standard Model of Elementary Particles zh-hans.svg
背景
組成
組成
理論家

查论编

魏尔費米子（Weyl fermions）是一種無質量費米子，在量子理論和標準模型中發揮重要作用。魏尔費米子被認為是量子理論中費米子一部分，是赫尔曼·魏尔從狄拉克方程式中得出的解，被稱為魏尔方程式。狄拉克費米子可以視為左手的魏尔費米子與右手的魏尔費米子的組合。魏尔費米子並沒有被視為基本粒子。魏尔費米子在凝聚體物理學裏以準粒子激發的形式存在。

美国物理学家康耶斯·赫林在能帶結構固態系統下首先預測魏尔費米子的存在[1][2][3] 。鉭砷晶體是第一個發現魏尔費米子存在的晶體結構[4]。帶電魏尔費米子在室溫下穩定存在。
歷史
魏尔費米子半金屬及費米弧

2015年7月16日，美國普林斯頓大學研究團隊與中國科學院物理研究所分別在鉭砷晶體（TaAs）結構中第一次觀察到魏尔費米子半金屬及費米弧，確認其拓撲特徵。這一次發現根據2014年11月提出的預測[5][6]。

2016年，橡樹嶺國家實驗室和田納西大學科學家通過中子衍射實驗和X射線實驗推測魏尔費米子可能存在鋨基磁性晶體結構中。

PKPOI1234建國高中蔡旻諺 guest PKPOI1234建國高中蔡旻諺 2021/06/17 05:12

外尔方程式
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在量子力學及量子場論等領域，外尔方程式（英語：Weyl Equation）為一相對論量子力學的波動方程式，用以描述無質量的自旋½粒子。其以德國數學家赫尔曼·外尔為名。
目录

1 方程式
1.1 外尔旋量
2 推演
3 相關條目
4 參考資料
5 延伸閱讀
6 外部連結

方程式

外尔方程式的廣義形式可寫為：[1][2]

σ μ ∂ μ ψ = 0 {\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0} \sigma ^{\mu }\partial _{\mu }\psi =0

在SI單位中可寫為：

I 2 1 c ∂ ψ ∂ t + σ x ∂ ψ ∂ x + σ y ∂ ψ ∂ y + σ z ∂ ψ ∂ z = 0 {\displaystyle I_{2}{\frac {1}{c}}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0} I_{2}{\frac {1}{c}}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0

其中

σ μ = ( σ 0 , σ 1 , σ 2 , σ 3 ) = ( I 2 , σ x , σ y , σ z ) {\displaystyle \sigma _{\mu }=(\sigma _{0},\sigma _{1},\sigma _{2},\sigma _{3})=(I_{2},\sigma _{x},\sigma _{y},\sigma _{z})} \sigma _{\mu }=(\sigma _{0},\sigma _{1},\sigma _{2},\sigma _{3})=(I_{2},\sigma _{x},\sigma _{y},\sigma _{z})

為一向量。μ = 0分量為2 × 2 單位矩陣；μ = 1,2,3分量為包立矩陣。ψ則是波函數，為外尔旋量一例。
外尔旋量

其組成有ψL與ψR，分別為左手（Left-handed）外尔旋量及右手（Right-handed）外尔旋量，各自有兩個分量。兩者皆有下列形式：

ψ = ( ψ 1 ψ 2 ) = χ e − i ( k ⋅ r − ω t ) = χ e − i ( p ⋅ r − E t ) / ℏ {\displaystyle \psi ={\begin{pmatrix}\psi _{1}\\\psi _{2}\\\end{pmatrix}}=\chi e^{-i(\mathbf {k} \cdot \mathbf {r} -\omega t)}=\chi e^{-i(\mathbf {p} \cdot \mathbf {r} -Et)/\hbar }} \psi ={\begin{pmatrix}\psi _{1}\\\psi _{2}\\\end{pmatrix}}=\chi e^{{-i({\mathbf {k}}\cdot {\mathbf {r}}-\omega t)}}=\chi e^{{-i({\mathbf {p}}\cdot {\mathbf {r}}-Et)/\hbar }}

其中

χ = ( χ 1 χ 2 ) {\displaystyle \chi ={\begin{pmatrix}\chi _{1}\\\chi _{2}\\\end{pmatrix}}} \chi ={\begin{pmatrix}\chi _{1}\\\chi _{2}\\\end{pmatrix}}

為具有二常數分量之旋量。

既然粒子是不具質量的，亦即m = 0，動量p的範數為波向量k的簡單乘積，由德布羅伊關係所描述：

| p | = ℏ | k | = ℏ ω / c → | k | = ω / c {\displaystyle |\mathbf {p} |=\hbar |\mathbf {k} |=\hbar \omega /c\,\rightarrow \,|\mathbf {k} |=\omega /c} |{\mathbf {p}}|=\hbar |{\mathbf {k}}|=\hbar \omega /c\,\rightarrow \,|{\mathbf {k}}|=\omega /c

方程式可以左手及右手旋量來表示：

σ μ ∂ μ ψ R = 0 σ ¯ μ ∂ μ ψ L = 0 {\displaystyle {\begin{aligned}&\sigma ^{\mu }\partial _{\mu }\psi _{R}=0\\&{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=0\end{aligned}}} {\begin{aligned}&\sigma ^{\mu }\partial _{\mu }\psi _{R}=0\\&{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=0\end{aligned}}

推演

透過拉格朗日密度可得方程式：

L = i ψ R † σ μ ∂ μ ψ R {\displaystyle {\mathcal {L}}=i\psi _{R}^{\dagger }\sigma ^{\mu }\partial _{\mu }\psi _{R}} {\mathcal L}=i\psi _{R}^{\dagger }\sigma ^{\mu }\partial _{\mu }\psi _{R}
L = i ψ L † σ ¯ μ ∂ μ ψ L {\displaystyle {\mathcal {L}}=i\psi _{L}^{\dagger }{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}} {\mathcal L}=i\psi _{L}^{\dagger }{\bar \sigma }^{\mu }\partial _{\mu }\psi _{L}

將旋量及旋量的埃爾米特伴隨（以 † {\displaystyle \dagger } \dagger 標記）當作獨立變數處理，則可得外尔方程式。
相關條目

狄拉克方程式（描述帶質量的自旋½粒子）
角動量算符
動量算符
自旋

參考資料

Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0

The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.

延伸閱讀

Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
Particle Physics (2nd Edition), B.R. Martin, G. Shaw, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7
Supersymmetry P. Labelle, Demystified, McGraw-Hill (USA), 2010, ISBN 978-0-07-163641-4
The Road to Reality, Roger Penrose, Vintage books, 2007, ISBN 0-679-77631-1

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