時間晶體：神奇新物種 | 諾獎得主Wilczek專欄_風聞

撰文 | Frank Wilczek (麻省理工學院教授、2004年諾貝爾獎得主）

翻譯 | 吳飆**（北京大學量子材料科學中心教授）******

來源：蔻享學術

Frank Wilczek

弗蘭克·維爾切克是麻省理工學院物理學教授、量子色動力學的奠基人之一。因在夸克粒子理論（強作用）方面所取得的成就，他在2004年獲得了諾貝爾物理學獎。

序

晶體是自然界最有序的物質。在它們的內部，原子和分子按照重複的結構規則排列，如此形成的固體不但穩定、有剛性，而且看起來非常漂亮。

Crystals are nature’s most orderly substance. Inside them, atoms and molecules are arranged in regular, repeating structures, giving rise to solids that are stable and rigid-and often beautiful to behold. 

在現代科學到來之前，人們已經發現晶體非常炫目和迷人，因此經常把它們當作珠寶珍藏。十九世紀的科學家對晶體形式進行了分類，並理解了它們對光的作用，這些努力促進了數學和物理的發展。後來在二十世紀，科學家對電子在晶體中基本量子行為的研究，直接催生了現代半導體電子學，最後給我們帶來了智能手機和互聯網。

People have found crystals fascinating and attractive since before the dawn of modern science, often prizing them as jewels. In the 19th century scientists’ quest to classify forms of crystals and understand their effect on light catalyzed important progress in mathematics and physics. Then, in the 20th century, study of the fundamental quantum mechanics of electrons in crystals led directly to modern semiconductor electronics and eventually to smartphones and the internet. 

我們對晶體的理解正在更上一層樓，這要歸功於愛因斯坦的相對論：空間和時間是緊密相連的，它們在本質上屬於同一個框架。所以，一個很自然的問題就是物體在時間上可不可以具有普通晶體在空間擁有的性質。為了回答這個問題，我們發現了“時間晶體”。這個新概念和日益增多的相關新材料，不但導致了令人激動的物理認識，而且帶來了新的應用希望，比如實現比現在所有時鐘更精確的計時技術。

The next step in our understanding of crystals is occurring now, thanks to a principle that arose from Albert Einstein’s relativity theory: space and time are intimately connected and ultimately on the same footing. Thus, it is natural to wonder whether any objects display properties in time that are analogous to the properties of ordinary crystals in space. In exploring that question, we discovered “time crystals.” This new concept, along with the growing class of novel materials that fit within it, has led to exciting insights about physics, as well as the potential for novel applications, including clocks more accurate than any that exist now.

對稱 SYMMETRY

在完整介紹這個新想法之前，我必須先闡明晶體到底是什麼。這個問題的答案充滿了科學的內涵，涉及兩個深刻的概念：對稱和對稱性自發破缺。Before I fully explain  this new idea, I must clarify what, exactly, a crystal is. The most fruitful answer for scientific purposes brings in two profound concepts: symmetry and spontaneous symmetry breaking. 

在日常生活中，“對稱”的意義非常寬泛，有平衡、和諧，甚至公正的意思。在物理和數學裏，它的意義更加準確。如果對一個物體做一些變換（比如旋轉），它的狀態卻沒有發生改變，那麼我們説這個物體是對稱的或具有某種對稱性。

In common usage, “symmetry” very broadly indicates balance, harmony or even justice. In physics and mathematics, the meaning is more precise. We say that an object is symmetric or has symmetry if there are transformations that could change it but do not. 

這個定義乍聽起來可能有些玄妙和抽象，讓我們看一個簡單的例子：一個圓。當我們繞着圓心旋轉這個圓，無論轉多大的角度，儘管圓上的每個點可能都移動了，但這個圓看上去卻沒有任何變化——它有完美的對稱性。一個正方形也有對稱性，但不如圓，因為你必須旋轉90°的整數倍，正方形才能恢復原貌。這些例子説明，對稱的數學定義抓住了對稱的實質，並且讓它的意義更精確。

In common usage, “symmetry” very broadly indicates balance, harmony or even justice. In physics and mathematics, the meaning is more precise. We say that an object is symmetric or has symmetry if there are transformations that could change it but do not. These examples show that the mathematical concept of symmetry captures an essential aspect of its common meaning while adding the virtue of precision. 

這樣定義對稱還有一個好處，它可以被推廣。我們可以改進這個想法讓它不只適用於形狀，而是廣泛地適用於物理規律。對於一條物理定律，如果我們改變它的應用環境而定律不變，我們就説這條物理定律具有對稱性。比如，狹義相對論的基本原則是：當我們在不同的相對勻速運動的參考系裏看世界時，物理定律是一樣的。所以，相對論要求物理定律具有某種對稱性——即，在改變參照系的情況下，物理定律不會發生改變。

A second virtue of this concept of symmetry is that it can be generalized. We can adapt the idea so that it applies not just to shapes but more widely to physical laws. We say a law has symmetry if we can change the context in which the law is applied without changing the law itself. For example, the basic axiom of special relativity is that the same physical laws apply when we view the world from different platforms that move at constant velocities relative to one another. Thus, relativity demands that physical laws display a kind of symmetry-namely, symmetry under the platformchanging transformations that physicists call “boosts.”

對於晶體（包括時間晶體）來説，重要的是另一類變換，這些變換很簡單，但有很重要的意義——這就是“平移”（translation）。相對論提出，對於移動速度不同的參考系，物理定律是相同的；類似地，在空間平移對稱中，對於不同地點的觀察者，物理定律是相同的。如果你將實驗室從一個地點移到另一個地點，即“平移”，你會發現在新的地方物理定律是一樣的。換句話説，空間平移對稱是指，我們在任何地點發現的物理定律適用於所有地點。

A different class of transformations is important for crystals, including time crystals. They are the very simple yet profoundly important transformations known as “translations”. Whereas relativity says the same laws apply for observers on moving platforms, spatial translation symmetry says the same laws apply for observers on platforms in different places. If you move-or “translate”-your laboratory from one place to another, you will find that the same laws hold in the new place. Spatial translation symmetry, in other words, asserts that the laws we discover  anywhere  apply  everywhere. 

時間平移對稱表達的是一個類似的想法，不過它是針對時間而不是空間：現在運行的物理定律同樣適用於過去或將來的觀測者。也就是説，我們在任何時間發現的物理定律適用於所有時間。由於它基礎性和重要性，時間平移對稱應該有一個更簡單的名字。在這裏我把它叫做tau，用希臘字母τ表示。

Time translation symmetry expresses a similar idea but for time instead of space. It says the same laws we operate under now also apply for observers in the past or in the future. In other words, the laws we discover at any time apply at every time. In view of its basic importance, time translation symmetry deserves to have a less forbidding name, with fewer than seven syllables. Here I will call it tau, denoted by the Greek symbol τ. 

如果沒有空間和時間平移對稱，不同地方不同時間做的實驗將無法重複。科學家在日常工作中把這些對稱看作是理所當然的。確實，如果沒有這些對稱性，那麼我們所知道的科學是不可能存在的。但重要的是，我們可以在實驗中測試空間和時間的平移對稱。我們以遙遠的天體來具體説明。這些天體顯然處於不同的地方，同時由於光速有限，我們現在觀測到的其實是天體在過去的運動。通過非常詳細、準確的觀測，天文學家已經確認，遙遠天體所遵循的物理定律和此時此地在地球上的物理規律完全相同。 

Without space and time translation symmetry, experiments carried out in different places and at different times would not be reproducible. In their everyday work, scientists take those symmetries for granted. Indeed, science as we know it would be impossible without them. But it is important to emphasize that we can test space and time translation symmetry empirically. Specifically, we can observe behavior in distant astronomical objects. Such objects are situated, obviously, in different places, and thanks to the finite speed of light we can observe in the present how they behaved in the past. Astronomers have determined, in great detail and with high accuracy, that the same laws do in fact apply. 

對稱性破缺 SYMMETRY BREAKING 

晶體因對稱而美，但對於物理學家來説，晶體最顯著的特徵卻是它們缺失了對稱。

For all their aesthetic symmetry,  it is actually the way crystals lack symmetry that is, for physicists, their defining characteristic. 

考慮一個特別簡單的晶體。它是一維的，它的原子核規則地排列在一條直線上，相鄰間距是d【因此，每個原子核的座標是nd，其中n是整數】。如果我們將這個晶體往右平移一丁點兒，那麼它和移動前是不一樣的。只有平移了特定的距離d，我們才會得到相同的晶體。所以，我們的理想化晶體只具有部分平移對稱性，這與前面介紹的正方形只具有部分旋轉對稱性是一樣的道理。

Consider a drastically idealized crystal. It will be one-dimensional, and its atomic nuclei will be located at regular intervals along a line, separated by the distance  d.  (Their coordinates therefore will be nd,  where n is a whole number.) If we translate this crystal to the right by a tiny distance, it will not look like the same object. Only after we translate through the specific distance d will we see the same crystal. Thus, our idealized crystal has a reduced degree of spatial translation symmetry, similarly to how a square has a reduced degree of rotation symmetry. 

物理學家認為，在晶體中，物理基本定律的平移對稱性“破缺”了，只剩下部分平移對稱性。這些遺留的對稱性卻描述了晶體的本質特徵。事實上，一旦我們知道晶體的對稱是平移距離d的整數倍，我們就知道晶體中原子的相對位置。

Physicists say that in a crystal the translation symmetry of the fundamental laws is “broken,” leading to a lesser translation symmetry. That remaining symmetry conveys the essence of our crystal. Indeed, if we know that a crystal’s symmetry involves translations through multiples of the distance  d,  then we know where to place its atoms relative to one another.

二維和三維的晶體會更復雜，它們的種類非常多，可以同時具有部分旋轉和平移對稱性。十四世紀的藝術家在裝飾西班牙格拉納達的阿爾罕布拉宮時，利用想象和經驗發現了很多可能的二維晶體。而十九世紀的數學家則對三維晶體進行了分類。

Crystalline patterns in two and three dimensions can be more complicated, and they come in many varieties. They can display partial rotational and partial translational symmetry. The 14th-century artists who decorated the Alhambra palace in Granada, Spain, discovered many possible forms of two-dimensional crystals by intuition and experimentation, and mathematicians in the 19th century classified the possible forms of three-dimensional crystals. 

2011年的夏天，我開了一門課，主要講授物理中的對稱。我在準備晶體分類那章時覺得相關的數學非常優雅。在備課過程中，我總是嘗試從一個新的角度來審視我的課程，儘可能增加一些新的內容。我突然意識到，三維空間晶體的分類可以推廣到四維時空晶體。

In the summer of 2011 I was preparing to teach this elegant chapter of mathematics as part of a course on the uses of symmetry in physics. I always try to take a fresh look at material I will be teaching and, if possible, add something new. It occurred to me then that one could extend the classification of possible crystalline patterns in three-dimensional space to crystalline patterns in four-dimensional spacetime. 

我把相關的數學研究告訴了阿爾弗雷德·薩皮爾（Alfred Shapere），他曾是我的學生，現在是我親密的合作者。他目前在肯塔基大學工作。他希望我先回答兩個基本的物理問題：**時空晶體能描述什麼實際的物理體系？**這些晶體會引導我們發現不同的物質狀態嗎？

When I mentioned this mathematical line of investigation to Alfred Shapere, my former student turned valued colleague, who is now at the University of Kentucky, he urged me to consider two very basic physical questions. They launched me on a surprising scientific adventure:**What real-world systems could crystals in spacetime describe?**Might these patterns lead us to identify distinctive states of matter?

這兩個問題帶我走上一個充滿驚喜的科學歷程。

They launched me on a surprising scientific adventure.

第一個問題的答案相當直接。既然普通晶體是物體在空間的有序排列，那麼時空晶體應該是事件在時空中的有序排列。

The answer to the first question is fairly straight-forward. Whereas ordinary crystals are orderly arrangements of objects in space, spacetime crystals are orderly arrangements of events in spacetime. 

我們效仿上面對普通晶體的討論，先考慮一維時空晶體來找找感覺。這個特殊情況下，時空晶體就成了純粹的時間晶體。我們這時需要尋找的系統應該這樣：它的狀態每隔一段時間就會重複。令人尷尬的是，這樣的系統早已為人熟知。比如，地球在空間中的姿態每隔一天就重複一 遍，地球與太陽的相對位置每隔一年也重複一次。

As we did for ordinary crystals, we can get our bearings by considering the one-dimensional case, in which spacetime crystals simplify to purely time crystals. We are looking, then, for systems whose overall state repeats itself at regular intervals. Such systems are almost embarrassingly familiar. For example, Earth repeats its orientation in space at daily intervals, and the Earth-sun system repeats its configuration at yearly intervals. 

發明家和科學家在過去幾十年裏發展了很多時鐘系統，這些時鐘每重複一次的時間間隔的精度越來越高。單擺和彈簧鍾已經被基於（傳統）晶體振動的晶鍾超越，後者又被基於原子振動的原子鐘超越了。原子鐘已經取得了令人驚歎的精度，但我們有很多理由去繼續提高精度——我們後面將會看到，在這個問題上，時間晶體極有可能會幫上忙。

Inventors and scientists have, over many decades, developed systems that repeat their arrangements at increasingly accurate intervals for use as clocks. Pendulum and spring clocks were superseded by clocks based on vibrating (traditional) crystals, and those were eventually superseded by clocks based on vibrating atoms. Atomic clocks have achieved extraordinary accuracy, but there are important reasons to improve them further-and time crystals might help, as we will see later. 

一些大家熟知的真實體系則是高維時空晶體。比如下圖中的平面聲波，其曲面的高度表示隨空間和時間變化的密度。更復雜的時空晶體可能很難在自然界找到，但它們可能成為藝術家和工程師追求的目標——想象一下，一個會動的增強版阿爾罕布拉宮也是一個時空晶體。

Some familiar real-world systems also embody higher-dimensional spacetime crystal patterns. For example, the pattern shown here can represent a planar sound wave, where the height of the surface indicates compression as a function of position and time. More elaborate spacetime crystal patterns might be difficult to come by in nature, but they could be interesting targets for artists and engineers-imagine a dynamic Alhambra on steroids. 

對於這類時空晶體，我們只是新瓶裝舊酒，換了一個不同的標籤。而回答薩皮爾的第二個問題則會將我們帶入一個真正創新的物理領域。為此，我們現在必須介紹一個概念：對稱性自發破缺。 

These types of spacetime crystals, though, simply repackage known phenomena under a different label. We can move into genuinely new territory in physics by considering Shapere’s second question. To do that, we must now bring in the idea of  spontaneous  symmetry breaking.

對稱性自發破缺 SPONTANEOUS SYMMETRY BREAKING

當液體或氣體冷卻成晶體時，一件非常基本且神奇的事發生了：晶體——這個物理定律的解——具有的對稱性少於物理定律本身的對稱性。由於這個對稱性的減少只是通過降温而獲得的，在這個過程中並沒有其他外界因素的干預，於是我們認為在晶體形成過程中，物質“自發”破壞了空間平移對稱性。

When a lIquId or gas cools into a crystal, something fundamentally remarkable occurs: the emergent solution of the laws of physics-the crystal-displays less symmetry than the laws themselves. As this reduction of symmetry is brought on just by a decrease in temperature, without any special outside intervention, we can say that in forming a crystal the material breaks spatial translation symmetry “spontaneously.” 

晶體形成的一個重要特徵是物質系統的行為有一個急劇的變化，或者按專業説法，一個急劇的相變。在臨界温度上（這個温度的高低取決於系統的化學成分和環境壓強），系統是液體；臨界温度下，系統則變成了晶體——晶體的各種性質都和液體非常不同。這個相變可以預測，並伴有能量的釋放（一般是以熱的形式）。環境條件的微小變化會讓物質重組，成為非常不同的材料，比如水變冰。人們雖然很熟悉這個現象但依然會覺得神奇。

An important feature of crystallization is a sharp change in the system’s behavior or, in technical language, a sharp phase transition. Above a certain critical temperature (which depends on the system’s chemical composition and the ambient pressure), we have a lIquId; below it we have a crystal-objects with quite different properties. The transition occurs predictably and is accompanied by the emission of energy (in the form of heat). The fact that a small change in ambient conditions causes a substance to reorganize into a qualitatively distinct material is no less remarkable for being, in the case of water and ice, very familiar. 

晶體的剛性是另一個不同於液體和氣體的性質。從微觀上看，晶體之所以有剛性是因為晶體中的原子在很大範圍內的有序排列，任何試圖破壞這種有序性的行為都會遭到晶體的抵抗。

The rigidity of crystals is another emergent property that distinguishes them from lIquIds and gases. From a microscopic perspective, rigidity arises because the organized pattern of atoms in a crystal persists over long distances and the crystal resists attempts to disrupt that pattern.

我們剛剛討論了晶體形成的三個特徵——減少的對稱性、急劇的相變和剛性——它們是緊密相關的。這三個特徵都源於一個基本原則，原子“希望”按照一個能量儘可能小的方式排列。在不同的外部條件（比如不同的壓強和温度）下，原子會按不同的方式排列——這些就是不同的 “相”。當外界條件改變時，我們經常會看到急劇的相變。有序排列的形成要求原子們集體行動，整個材料中的原子都會被要求按照同樣的方式排列。這種排列即使受到小的擾動，也會自動恢復。

The three features of crystallization that we have just discussed-reduced symmetry, sharp phase transition and rigidity-are deeply related. The basic principle underlying all three is that atoms “want” to form patterns with favorable energy. Different choices of pattern-in the jargon, different phases-can win out under different conditions (for instance, various pressures and temperatures). When conditions change, we often see sharp phase transitions. And because pattern formation requires collective action on the part of the atoms, the winning choice will be enforced over the entire material, which will snap back into its previous state if the chosen pattern is disturbed. 

由於對稱性自發破缺能將不同的想法連接起來解釋很多物理現象，我感到探索τ被自發破缺的可能性是非常重要的。當我把這個想法具體寫下來時，我向我的妻子貝茜·迪瓦恩（Betsy Devine）解釋了這個想法：

Because spontaneous symmetry breaking unites such a nice package of ideas and powerful implications, I felt it was important to explore the possibility that τ can be broken spontaneously. As I was writing up this idea, I explained it to my wife, Betsy Devine: 

“它看起來是一個晶體，但它是關於時間的晶體。”受到我激情的感染，她好奇地問道：“你準備叫它什麼？” 我回答道：“時間平移對稱性的自發破缺。”她立即反對説：“不會吧。應該叫它時間晶體。”我選擇了她的叫法。2012年，我發表了兩篇論文，介紹了這個想法，其中一篇是與薩皮爾合作的。時間晶體是這樣的系統，它的τ自發破缺了。

“It’s like a crystal but in time.” Drawn in by my excitement, she was curious: “What are you calling it?” “Spontaneous breaking of time translation symmetry,” I said. “No way,” she countered. “Call it time crystals.” Which, naturally, I did. In 2012 I published two papers, one co-authored by Shapere, introducing the concept. A time crystal, then, is a system in which τ is spontaneously broken.

有人可能會問，既然τ和自發破缺都早已為人熟知，為什麼在更早的時候科學家沒有想到把二者結合起來？這是因為τ和其他對稱有一個重大的區別，使得它的自發破缺變得更加微妙。這個區別來自數學家埃米·諾特（Emmy Noether）在1915年證明的一個深刻的物理定理。諾特的定理建立了對稱和守恆量之間的聯繫——每一種對稱對應一種守恆量。

One might wonder why it took so long for the concepts of τ and spontaneous symmetry breaking to come together, given that separately they have been understood for many years. It is because τ differs from other symmetries in a crucial way that makes the question of its possible spontaneous breaking much subtler. The difference arises because of a profound theorem proved by mathematician Emmy Noether in 1915. Noether’s theorem makes a connection between symmetry principles and conservation laws-it shows that for every form of symmetry, there is a corresponding quantity that is conserved.

應用到這裏我們可以發現，諾特定理表明τ其實等價於能量守恆。反過來説，當一個系統的τ破缺時，能量就不再守恆，能量這個概念不再能有效地刻畫這個系統。（更精確地講，沒有τ，你不再能夠將系統不同部分的貢獻加起來得到一個類似能量的物理量，更不能保證這個物理量不隨時間變化。） 

物理學家通常這樣解釋，對稱自發破缺之所以能發生，是因為它能降低能量。如果能量最低的狀態打破了空間對稱而系統的能量又同時保持守恆，那麼一旦進入對稱破缺的狀態，系統就會持續保持這個狀態。這就是普通晶體能夠存在的物理原因。

The usual explanation for why spontaneous symmetry breaking occurs is that it can be favorable energetically. If the lowest-energy state breaks  spatial  symmetry and the energy of the system is conserved, then the broken symmetry state, once entered, will persist. That is how scientists account for ordinary crystallization, for example.

但是，這種基於能量的解釋不適用於τ的破缺，因為τ一旦破缺，能量就不再守恆，能量不再能度量這個系統。由於這個困難，大多數物理學家從來沒有考慮過τ自發破缺的可能性，當然也就沒考慮過時間晶體這種奇怪的東西。

But that energy-based explanation will not work for τ breaking, because τ breaking removes the applicable measure of energy. This apparent difficulty put the possibility of spontaneous τ breaking, and the associated concept of time crystals, beyond the conceptual horizon of most physicists. 

但是，對稱性自發破缺還有一種更廣泛的理解方式，適用於τ的破缺。除了自發重組成一個能量更低的狀態，材料可以重組進入一個更穩定的狀態。比如，許多粒子可以在一個大的空間或時間範圍形成一個有序排列，如果破壞有序的力是小尺度和局域的，那麼這個有序排列就很難被打破。這樣，如果相比以前的狀態，材料新的有序排列發生在一個更大的尺度上，那麼它就有可能獲得更高的穩定性。

There is, however, a more general road to spontaneous symmetry breaking, which also applies to τ breaking. Rather than spontaneously reorganizing to a lower-energy state, a material might reorganize to a state that is more stable for other reasons. For instance, ordered patterns that extend over large stretches of space or time and involve many particles are difficult to unravel because most disrupting forces act on small, local scales. Thus, a material might achieve greater stability by taking on a new pattern that occurs over a larger scale than in its previous state. 

當然，最終沒有哪種物質的狀態可以面對所有的擾動都保持穩定。比如，鑽石就是這樣。“鑽石恆久遠”這句傳奇的廣告詞已經婦孺皆知。但如果温度足夠高，鑽石在合適的空氣中會燒成灰塵，不再光彩奪目。鑽石在普通温度和氣壓下不是碳最穩定的狀態，它們是在非常高的壓強下產生的。鑽石一旦形成，它們可以在普通壓強下存在很長時間。按照物理學家的計算，如果等待足夠長的時間，你的鑽石也會變成石墨。甚至還有一個非常小但不是零的可能性，那就是量子漲落會讓鑽石變成黑洞。鑽石還可能由於質子的衰變而轉化為別的物質。在實踐中，當我們説一個物質狀態（比如鑽石）時，我們説的是它形成了一種組織，對於很多外界擾動，這個組織具有穩定性。

Ultimately, of course, no ordinary state of matter can maintain itself against all disruptions. Consider, for example, diamonds. A legendary ad campaign popularized the slogan “a diamond is forever.” But in the right atmosphere, if the temperature is hot enough, a diamond will burn into inglorious ash. More basically, diamonds are not a stable state of carbon at ordinary temperatures and atmospheric pressure. They are created at much higher pressures and, once formed, will survive for a very long time at ordinary pressures. But physicists calculate that if you wait long enough, your diamond will turn into graphite. Even less likely, but still possible, a quantum fluctuation can turn your diamond into a tiny black hole. It is also possible that the decay of a diamond’s protons will slowly erode it. In practice, what we mean by a “state of matter” (such as diamond) is an organization of a substance that has a useful degree of stability against a significant range of external changes.

新舊時間晶體 OLD AND NEW TIME CRYSTALS 

交流約瑟夫森效應是物理中的一塊寶石，它為一大類時間晶體提供了原型。以1973年諾貝爾物理學獎得主、英國物理學家布賴恩·戴維·約瑟夫森（Brian David Josephson）命名的“約瑟夫森結”是夾在兩個超導體中間的絕緣層。當在結的兩端加上一個常電壓後，我們就能觀察到約瑟夫森效應——這時會觀測到一個頻率為2eV/ℏ的交變電流流過，在這裏e是電子電荷，ℏ是約化普朗克常數。儘管整個物理設置不隨時間改變（也就是説，它遵守τ），但系統最後的行為卻隨時間變化。完整的時間平移對稱變成周期為ℏ/2eV的整數倍的對稱。所以，交變約瑟夫森效應體現了時間晶體的最基本特徵。但是，從某些角度來説，它不是完全符合期望。為了維持電流，我們必須讓電路閉合並連上一節電池。交變電流會釋放熱，而電池會衰竭。另外，變化的電流還會輻射電磁波。由於這些原因，約瑟夫森結的穩定性還不夠理想。

The ac Josephson effect  is one of the gems of physics, and it supplies the prototype for one large family of time crystals. It occurs when we apply a constant voltage  V  (a difference in potential energy) across an insulating junction separating two superconducting materials (a so-called Josephson junction, named after physicist Brian Josephson). In this situation, one observes that an alternating current at frequency 2 eV/ℏ flows across the junction, where e is the charge of an electron and ℏ is the reduced Planck’s constant. Here, although the physical setup does not vary in time (in other words, it respects τ), the resulting behavior does vary in time. Full time translation symmetry has been reduced to symmetry under time translation by multiples of the period  ℏ/2eV.  Thus, the AC Josephson effect embodies the most basic concept of a time crystal. In some respects, however, it is not ideal. To maintain the voltage, one must somehow close the circuit and supply a battery. But AC circuits tend to dissipate heat, and batteries run down. Moreover, oscillating currents tend to radiate electromagnetic waves. For all these reasons, Josephson junctions are not ideally stable. 

通過各種改進（比如改用完全超導的線路，用高品質電容代替通常的電池，用閉合的籠罩防止輻射外泄），我們可以大幅降低這些效應。另外，通過用超流體或磁鐵取代超導體，我們可以觀察到類似的效應，同時將各種耗散降到最低。

By using various refinements (such as fully superconducting circuits, excellent capacitors in place of ordinary batteries and enclosures to trap radiation), it is possible to substantially reduce the levels of those effects. And other systems that involve superfluids or magnets in place of superconductors exhibit analogous effects while minimizing those problems. In very recent work, Nikolay Prokof’ev and Boris Svistunov have proposed extremely clean examples involving two interpenetrating superfluids. 

對τ破缺的大膽思考讓這些問題受到了很多關注，物理學家因此發現了新的物理系統並做了許多富有成果的實驗。但是，由於核心思想已經隱含在了約瑟夫森1962年的工作裏，我們不妨稱這些物理系統為“舊”時間晶體。

Thinking explicitly about τ breaking has focused attention on these issues and led to the discovery of new examples and fruitful experiments. Still, because the central physical idea is already implicit in Josephson’s work of 1962, it seems appropriate to refer to all these as “old” time crystals. 

2017年3月9日，《自然》雜誌宣佈了“新”時間晶體的到來。這期雜誌的封面是漂亮的、象徵性的時間晶體，上面還有一句宣言：“時間晶體：神奇新物態的首次觀測。”雜誌裏面是兩篇獨立的開拓性論文。這兩篇論文顯示，在一個實驗裏，美國馬里蘭大學的克里斯托弗·門羅（Christopher Monroe）領導的小組用精心設計的鐿離子鍊形成了時間晶體。在另一個實驗裏，哈佛大學米哈伊爾·盧金 （Mikhail Lukin）的小組利用鑽石裏的幾千個氮空位缺陷實現了時間晶體。

“New” time crystals arrived with the March 9, 2017, issue of  Nature, which featured gorgeous (metaphorical) time crystals on the cover and announced “Time crystals: First observations of exotic new state of matter.” Inside were two independent discovery papers. In one experiment, a group led by Christopher Monroe of the University of Maryland, College Park, created a time crystal in an engineered system of a chain of ytterbium ions. In the other, Mikhail Lukin’s group at Harvard University realized a time crystal in a system of many thousands of defects, called nitrogen vacancy centers, in a diamond. 

在這兩個實驗中，原子（鐿離子或鑽石缺陷）的自旋方向會規則變化，每隔一定週期，原子們會回到初始的形態。在門羅的實驗裏，研究人員用激光翻轉離子的自旋，並將這些自旋關聯起來形成“糾纏”態。最後，離子的自旋開始以兩倍於激光脈衝速率的頻率振盪。在盧金的實驗裏，科學家用微波脈衝翻轉鑽石缺陷的自旋，發現在微波脈衝之間系統會重複變化好幾次。在這兩個實驗中，系統都需要外界的激發——激光或微波脈衝——但系統最終的振盪週期卻和激發頻率不同。換句話説，它們都自發地破壞了時間對稱。

In both systems, the spin direction of the atoms (either the ytterbium ions or the diamond defects) changes with regularity, and the atoms periodically come back into their original configurations. In Monroe’s experiment, researchers used lasers to flip the ions’ spins and to correlate the spins into connected, “entangled” states. As a result, though, the ions’ spins began to oscillate at only half the rate of the laser pulses. In Lukin’s project, the scientists used microwave pulses to flip the diamond defects’ spins. They observed time crystals with twice and three times the pulse spacing. In all these experiments, the materials received external stimulation-lasers or microwave pulses-but they displayed a different period than that of their stimuli. In other words, they broke time symmetry spontaneously. 

這兩個實驗在材料物理領域開拓了一個新方向。基於相同的一般原則，人們發現了更多類似的材料體系。它們現在被稱為弗洛克（Floquet）時間晶體。

These experiments inaugurated a direction in materials physics that has grown into a minor industry. More materials utilizing the same general principles-which have come to be called Floquet time crystals-have come on the scene since then, and many more are being investigated.

弗洛克時間晶體和一些早期發現的現象有些類似，但有本質的不同。1831年，邁克爾·法拉第（Michael Faraday）發現，當他以週期T垂直晃動一水槽的汞時，汞的流動週期通常是2T。但是，在對稱破缺的法拉第系統以及很多2017年以前研究過的其他體系中，材料和驅動力（這裏指晃動）沒有清晰的分離，它們沒有展示對稱性自發破缺的核心特徵。驅動一直在不停地向系統注入能量（或更準確地説是熵），它們最後以熱輻射的形式散去。

Floquet time crystals are distinct in important ways from related phenomena discovered much earlier. Notably, in 1831 Michael Faraday found that when he shook a pool of mercury vertically with period  T, the resulting flow often displayed period 2 T. But the symmetry breaking in Faraday’s system-and in many other systems studied in the intervening years prior to 2017-does not allow a clean separation between the material and the drive (in this case, the act of shaking), and it does not display the hallmarks of spontaneous symmetry breaking. The drive never ceases to pump energy (or, more accurately, entropy), which is radiated as heat, into the material. 

實際效果是，由材料加驅動組成的整個系統的對稱性要少於驅動或材料各自的對稱性。2017年的兩個實驗所用的物理系統顯著不同：這兩個系統在經過短暫的遲滯後會進入一個穩態，材料不再和驅動力交換能量或熵。這個區別很微妙，但在物理上非常關鍵。這些新的弗洛克時間晶體代表了一種新的物質狀態，展示了對稱性自發破缺的核心特徵。

In effect, the entire system consisting of material plus drive-whose behavior, as noted, cannot be cleanly separated-simply has less symmetry than the drive considered separately. In the 2017 systems, in contrast, after a brief settling-down period, the material falls into a steady state in which it no longer exchanges energy or entropy with the drive. The difference is subtle but physically crucial. The new Floquet time crystals represent distinct phases of matter, and they display the hallmarks of spontaneous symmetry breaking, whereas the earlier examples, though extremely interesting in their own right, do not. 

在這個意義上，地球的自轉和繞太陽的公轉也不是時間晶體。它們令人深刻的穩定性是由能量和角動量的大致守恆來保證的。它們的能量和角動量都不是最小的，因此前面關於穩定性的能量分析在這裏不適用。因為這兩個量都特別大，所以要顯著改變它們，你需要特別大的擾動或者長時間積累的小擾動。潮汐、其他行星的引力，甚至太陽自己的轉動都會影響這些天文體系。因此，與此相關的時間，比如“日”和“年”的測量，都非常麻煩，需要不斷修正。

Likewise, Earth’s rotation and its revolution around the sun are not time crystals in this sense. Their impressive degree of stability is enforced by the approximate conservation of energy and angular momentum. They do not have the lowest possible values of those quantities, so the preceding energetic argument for stability does not apply; they also do not involve longrange patterns. But precisely because of the enormous value of energy and angular momentum in these systems, it takes either a big disturbance or small disturbances acting over a long time to significantly change them. Indeed, effects that include the tides, the gravitational influence of other planets and even the evolution of the sun do slightly alter those astronomical systems. The associated measures of time such as “day” and “year” are, notoriously, subject to occasional correction. 

與之形成鮮明對比的是這些新的時間晶體，它們的重複行為有很強的剛性和穩定性。這個特徵給出了一種精確分割時間的方式，這是製造先進時鐘的關鍵。現代原子鐘有令人驚歎的精度，但是它們沒有時間晶體擁有的長時間穩定性。基於這些時間晶體，我們可能製造更精確、更簡潔的時鐘，從而能以極高精度測量距離和時間，而這樣的時鐘可以用來改進GPS，也可以根據地下洞穴和礦產對重力或引力波的影響來探測這些洞穴和礦產。由於這些可能的應用，美國國防高級研究計劃局（DARPA）正在資助時間晶體的研究。

In contrast, these new time crystals display strong rigidity and stability in their patterns-a feature that offers a way of dividing up time very accurately, which could be the key to advanced clocks. Modern atomic clocks are marvels of accuracy, but they lack the guaranteed long-term stability of time crystals. More accurate, less cumbersome clocks based on these emerging states of matter could empower exquisite measurements of distances and times, with applications from improved GPS to new ways of detecting underground caves and mineral deposits through their influence on gravity or even gravitational waves. darpa-the De  fense Advanced Research Projects Agency-is funding research on time crystals with such possibilities in mind. 

宇宙學和黑洞中的τ THE TAO OF τ 

圍繞時間晶體和自發τ破缺的想法和實驗還處於“嬰兒期”。在這方面還有很多沒有回答的問題。一個正在努力的方向是通過設計和發現新的時間晶體材料，從而擴展時間晶體家族，讓它們包括更大更方便的系統，展示更多變的時空排列。物理學家對研究這些狀態的相變也非常感興趣。

The circle of ideas  and experiments around time crystals and spontaneous τ breaking represents a subject in its infancy. There are many open questions and fronts for growth. One ongoing task is to expand the census of physical time crystals to include larger and more convenient examples and to embody a wider variety of spacetime patterns, by both designing new time crystal materials and discovering them in nature. Physicists are also interested in studying and understanding the phase transitions that bring matter into and out of these states. 

另一個方向是細緻研究時間晶體的物理性質。前面提到過的半導體晶體提供了令人鼓舞的榜樣。我們可以研究電子和光在時間晶體中會受到怎樣的影響，以及什麼新發現會從中湧現。

Another task is to examine in detail the physical properties of time crystals (and spacetime crystals, in which space symmetry and τ are both spontaneously broken). Here the example of semiconductor crystals, mentioned earlier, is inspiring. What discoveries will emerge as we study how time crystals modify the behavior of electrons and light moving within them? 

我們已經在思考和時間相關的物質狀態的各種可能性。我們不但可以考慮時間晶體，還可以考慮時間準晶體（規則但是沒有重複排列的材料）、時間液體（時間軸上的事件密度是常數但不是週期的）和時間玻璃（具有剛性的結構但是和規則排列有偏離）。研究人員正在探索這些材料以及其他可能的相關材料。事實上，某些形式的時間準晶體和時間液體已經找到了。

Having opened our minds to the possibility of states of matter that involve time, we can consider not only time crystals but also time quasicrystals (materials that are very ordered yet lack repeating patterns), time liquids (materials in which the density of events in time is constant but the period is not) and time glasses (which have a pattern that looks perfectly rigid but actually shows small deviations). Researchers are actively exploring these and other possibilities. Indeed, some forms of time quasicrystals and a kind of time lIquId have been identified already. 

迄今，我們已經考慮了各種基於τ的物質相。我最後簡短評論一下宇宙學和黑洞中的τ。

So far we have considered phases of matter that put τ into play. Let me conclude with two brief comments about τ in cosmology and in black holes. 

穩恆態宇宙模型試圖從原則上在宇宙學中維持τ。在這個二十世紀中期非常流行的模型中，天文學家假設宇宙的狀態或形貌在大尺度上是不依賴於時間的——也就是説，它是時間對稱的。宇宙一直在擴張，穩恆態宇宙模型則假設物質不斷產生，從而保持宇宙的平均密度不變。但是穩恆態宇宙模型沒能經受住時間的考驗。天文學家已經掌握的證據表明，137億年以前宇宙大爆炸時刻的宇宙和現在的宇宙非常不同，儘管物理規律是一樣的。在這個意義上，τ對稱在宇宙中自發破缺了。有些宇宙學家還認為， 我們的宇宙是循環的或者宇宙曾經歷過一段快速振盪期。這些猜想很接近時間晶體的相關思想。

The steady-state-universe model was a principled attempt to maintain τ in cosmology. In that model, popular in the mid-20th century, astronomers postulated that the state, or appearance, of the universe on large scales is independent of time-in other words, it upholds time symmetry. Although the universe is always expanding, the steady-state model postulated that matter is continuously being created, allowing the average density of the cosmos to stay constant. But the steady-state model did not survive the test of time. Instead astronomers have accumulated overwhelming evidence that the universe was a very different place 13.7 billion years ago, in the immediate aftermath of the big bang, even though the same physical laws applied. In that sense, τ is (perhaps spontaneously) broken by the universe as a whole. Some cosmologists have also suggested that ours is a cyclic universe or that the universe went through a phase of rapid oscillation. These speculations-which, to date, remain just that-bring us close to the circle of ideas around time crystals. 

最後談一下廣義相對論。它是至今關於時空結構的最好理論。廣義相對論建立了這樣的概念：我們能夠確定空間內任意鄰近兩點間的距離。可惜，這個簡單的想法至少在兩種極端條件下不成立：當我們將宇宙大爆炸外推到它的初始時刻或者在黑洞中心的時候。在其他物理領域，如果一個方程不再能描述某個物體的行為時，它通常意味着系統將經歷一個相變。難道是時空在高壓、高温或極速變化下自己放棄了τ對稱？

Finally, the equations of general relativity, which embody our best present understanding of spacetime structure, are based on the concept that we can specify a definite distance between any two nearby points. This simple idea, though, is known to break down in at least two extreme conditions: when we extrapolate big bang cosmology to its initial moments and in the central interior of black holes. Elsewhere in physics, breakdown of the equations that describe behavior in a given state of matter is often a signal that the system will undergo a phase transition. Could it be that spacetime itself, under extreme conditions of high pressure, high temperature or rapid change, abandons τ? 

最終，時間晶體這個概念可能會同時在理論和實踐上，從另一個角度推動物理學家對宇宙學和黑洞的理解。在可見的將來，我們非常可能會發現新型時間晶體，它們會讓我們造出更完美的時鐘，展示更多有用的性質。簡而言之，時間晶體非常有趣，為我們打開了更多的窗户，擴展了我們對物質組織結構的理解。

Ultimately the concept of time crystals offers a chance for progress both theoretically-in terms of understanding cosmology and black holes from another perspective-and practically. The novel forms of time crystals most likely to be revealed in the coming years should move us closer to more perfect clocks, and they may turn out to have other useful properties. In any case, they are simply interesting, and offer us opportunities to expand our ideas about how matter can be organized.

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