撰文 | Frank Wilczek (麻省理工学院教授、2004年诺贝尔奖得主）

 

翻译 | 吴飙（北京大学量子材料科学中心教授）

 

序

 

晶体是自然界最有序的物质。在它们的内部，原子和分子按照重复的结构规则排列，如此形成的固体不但稳定、有刚性，而且看起来非常漂亮。

 

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.

 

在现代科学到来之前，人们已经发现晶体非常炫目和迷人，因此经常把它们当作珠宝珍藏。十九世纪的科学家对晶体形式进行了分类，并理解了它们对光的作用，这些努力促进了数学和物理的发展。后来在二十世纪，科学家对电子在晶体中基本量子行为的研究，直接催生了现代半导体电子学，最后给我们带来了智能手机和互联网。

 

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.

 

我们对晶体的理解正在更上一层楼，这要归功于爱因斯坦的相对论：空间和时间是紧密相连的，它们在本质上属于同一个框架。所以，一个很自然的问题就是物体在时间上可不可以具有普通晶体在空间拥有的性质。为了回答这个问题，我们发现了“时间晶体”。这个新概念和日益增多的相关新材料，不但导致了令人激动的物理认识，而且带来了新的应用希望，比如实现比现在所有时钟更精确的计时技术。

 

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.

 

对称 SYMMETRY

 

在完整介绍这个新想法之前，我必须先阐明晶体到底是什么。这个问题的答案充满了科学的内涵，涉及两个深刻的概念：对称和对称性自发破缺。

 

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，“对称”的意义非常宽泛，有平衡、和谐，甚至公正的意思。在物理和数学里，它的意义更加准确。如果对一个物体做一些变换（比如旋转），它的状态却没有发生改变，那么我们说这个物体是对称的或具有某种对称性。

 

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.

 

这个定义乍听起来可能有些玄妙和抽象，让我们看一个简单的例子：一个圆。当我们绕着圆心旋转这个圆，无论转多大的角度，尽管圆上的每个点可能都移动了，但这个圆看上去却没有任何变化——它有完美的对称性。一个正方形也有对称性，但不如圆，因为你必须旋转90°的整数倍，正方形才能恢复原貌。这些例子说明，对称的数学定义抓住了对称的实质，并且让它的意义更精确。

 

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.

这样定义对称还有一个好处，它可以被推广。我们可以改进这个想法让它不只适用于形状，而是广泛地适用于物理规律。对于一条物理定律，如果我们改变它的应用环境而定律不变，我们就说这条物理定律具有对称性。比如，狭义相对论的基本原则是：当我们在不同的相对匀速运动的参考系里看世界时，物理定律是一样的。所以，相对论要求物理定律具有某种对称性——即，在改变参照系的情况下，物理定律不会发生改变。

 

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."

 

对于晶体（包括时间晶体）来说，重要的是另一类变换，这些变换很简单，但有很重要的意义——这就是“平移”（translation）。相对论提出，对于移动速度不同的参考系，物理定律是相同的；类似地，在空间平移对称中，对于不同地点的观察者，物理定律是相同的。如果你将实验室从一个地点移到另一个地点，即“平移”，你会发现在新的地方物理定律是一样的。换句话说，空间平移对称是指，我们在任何地点发现的物理定律适用于所有地点。

 

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.

 

时间平移对称表达的是一个类似的想法，不过它是针对时间而不是空间：现在运行的物理定律同样适用于过去或将来的观测者。也就是说，我们在任何时间发现的物理定律适用于所有时间。由于它基础性和重要性，时间平移对称应该有一个更简单的名字。在这里我把它叫做tau，用希腊字母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.

 

如果没有空间和时间平移对称，不同地方不同时间做的实验将无法重复。科学家在日常工作中把这些对称看作是理所当然的。确实，如果没有这些对称性，那么我们所知道的科学是不可能存在的。但重要的是，我们可以在实验中测试空间和时间的平移对称。我们以遥远的天体来具体说明。这些天体显然处于不同的地方，同时由于光速有限，我们现在观测到的其实是天体在过去的运动。通过非常详细、准确的观测，天文学家已经确认，遥远天体所遵循的物理定律和此时此地在地球上的物理规律完全相同。

 

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.

 

对称性破缺 SYMMETRY BREAKING

 

晶体因对称而美，但对于物理学家来说，晶体最显著的特征却是它们缺失了对称。

 

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

 

考虑一个特别简单的晶体。它是一维的，它的原子核规则地排列在一条直线上，相邻间距是d【因此，每个原子核的坐标是nd，其中n是整数】。如果我们将这个晶体往右平移一丁点儿，那么它和移动前是不一样的。只有平移了特定的距离d，我们才会得到相同的晶体。所以，我们的理想化晶体只具有部分平移对称性，这与前面介绍的正方形只具有部分旋转对称性是一样的道理。

 

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.

物理学家认为，在晶体中，物理基本定律的平移对称性“破缺”了，只剩下部分平移对称性。这些遗留的对称性却描述了晶体的本质特征。事实上，一旦我们知道晶体的对称是平移距离d的整数倍，我们就知道晶体中原子的相对位置。

 

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.

 

二维和三维的晶体会更复杂，它们的种类非常多，可以同时具有部分旋转和平移对称性。十四世纪的艺术家在装饰西班牙格拉纳达的阿尔罕布拉宫时，利用想象和经验发现了很多可能的二维晶体。而十九世纪的数学家则对三维晶体进行了分类。

 

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，我开了一门课，主要讲授物理中的对称。我在准备晶体分类那章时觉得相关的数学非常优雅。在备课过程中，我总是尝试从一个新的角度来审视我的课程，尽可能增加一些新的内容。我突然意识到，三维空间晶体的分类可以推广到四维时空晶体。

 

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.

 

我把相关的数学研究告诉了阿尔弗雷德·萨皮尔（Alfred Shapere），他曾是我的学生，现在是我亲密的合作者。他目前在肯塔基大学工作。他希望我先回答两个基本的物理问题：

 

时空晶体能描述什么实际的物理体系？

 

这些晶体会引导我们发现不同的物质状态吗？

 

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?

 

这两个问题带我走上一个充满惊喜的科学历程。

 

They launched me on a surprising scientific adventure.

 

第一个问题的答案相当直接。既然普通晶体是物体在空间的有序排列，那么时空晶体应该是事件在时空中的有序排列。

 

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.

 

我们效仿上面对普通晶体的讨论，先考虑一维时空晶体来找找感觉。这个特殊情况下，时空晶体就成了纯粹的时间晶体。我们这时需要寻找的系统应该这样：它的状态每隔一段时间就会重复。令人尴尬的是，这样的系统早已为人熟知。比如，地球在空间中的姿态每隔一天就重复一 遍，地球与太阳的相对位置每隔一年也重复一次。

 

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.

 

发明家和科学家在过去几十年里发展了很多时钟系统，这些时钟每重复一次的时间间隔的精度越来越高。单摆和弹簧钟已经被基于（传统）晶体振动的晶钟超越，后者又被基于原子振动的原子钟超越了。原子钟已经取得了令人惊叹的精度，但我们有很多理由去继续提高精度——我们后面将会看到，在这个问题上，时间晶体极有可能会帮上忙。

 

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.

 

一些大家熟知的真实体系则是高维时空晶体。比如下图中的平面声波，其曲面的高度表示随空间和时间变化的密度。更复杂的时空晶体可能很难在自然界找到，但它们可能成为艺术家和工程师追求的目标——想象一下，一个会动的增强版阿尔罕布拉宫也是一个时空晶体。

 

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.

对于这类时空晶体，我们只是新瓶装旧酒，换了一个不同的标签。而回答萨皮尔的第二个问题则会将我们带入一个真正创新的物理领域。为此，我们现在必须介绍一个概念：对称性自发破缺。

 

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.

 

对称性自发破缺 SPONTANEOUS SYMMETRY BREAKING

 

当液体或气体冷却成晶体时，一件非常基本且神奇的事发生了：晶体——这个物理定律的解——具有的对称性少于物理定律本身的对称性。由于这个对称性的减少只是通过降温而获得的，在这个过程中并没有其他外界因素的干预，于是我们认为在晶体形成过程中，物质“自发”破坏了空间平移对称性。

 

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."

 

晶体形成的一个重要特征是物质系统的行为有一个急剧的变化，或者按专业说法，一个急剧的相变。在临界温度上（这个温度的高低取决于系统的化学成分和环境压强），系统是液体；临界温度下，系统则变成了晶体——晶体的各种性质都和液体非常不同。这个相变可以预测，并伴有能量的释放（一般是以热的形式）。环境条件的微小变化会让物质重组，成为非常不同的材料，比如水变冰。人们虽然很熟悉这个现象但依然会觉得神奇。

 

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.

 

晶体的刚性是另一个不同于液体和气体的性质。从微观上看，晶体之所以有刚性是因为晶体中的原子在很大范围内的有序排列，任何试图破坏这种有序性的行为都会遭到晶体的抵抗。

 

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.

 

我们刚刚讨论了晶体形成的三个特征——减少的对称性、急剧的相变和刚性——它们是紧密相关的。这三个特征都源于一个基本原则，原子“希望”按照一个能量尽可能小的方式排列。在不同的外部条件（比如不同的压强和温度）下，原子会按不同的方式排列——这些就是不同的 “相”。当外界条件改变时，我们经常会看到急剧的相变。有序排列的形成要求原子们集体行动，整个材料中的原子都会被要求按照同样的方式排列。这种排列即使受到小的扰动，也会自动恢复。

 

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.

 

由于对称性自发破缺能将不同的想法连接起来解释很多物理现象，我感到探索τ被自发破缺的可能性是非常重要的。当我把这个想法具体写下来时，我向我的妻子贝茜·迪瓦恩（Betsy Devine）解释了这个想法：

 

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:

 

“它看起来是一个晶体，但它是关于时间的晶体。”受到我激情的感染，她好奇地问道：“你准备叫它什么？” 我回答道：“时间平移对称性的自发破缺。”她立即反对说：“不会吧。应该叫它时间晶体。”我选择了她的叫法。2012年，我发表了两篇论文，介绍了这个想法，其中一篇是与萨皮尔合作的。时间晶体是这样的系统，它的τ自发破缺了。

 

"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.

 

有人可能会问，既然τ和自发破缺都早已为人熟知，为什么在更早的时候科学家没有想到把二者结合起来？这是因为τ和其他对称有一个重大的区别，使得它的自发破缺变得更加微妙。这个区别来自数学家埃米·诺特（Emmy Noether）在1915年证明的一个深刻的物理定理。诺特的定理建立了对称和守恒量之间的联系——每一种对称对应一种守恒量。

 

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.

 

应用到这里我们可以发现，诺特定理表明τ其实等价于能量守恒。反过来说，当一个系统的τ破缺时，能量就不再守恒，能量这个概念不再能有效地刻画这个系统。（更精确地讲，没有τ，你不再能够将系统不同部分的贡献加起来得到一个类似能量的物理量，更不能保证这个物理量不随时间变化。）

 

In the application relevant here, Noether's theorem states that τ is basically equivalent to the conservation of energy. Conversely, when a system breaks τ, energy is not conserved, and it ceases to be a useful characteristic of that system. (More precisely: without τ, you can no longer obtain an energylike, time-independent quantity by summing up contributions from the system's parts.)

 

物理学家通常这样解释，对称自发破缺之所以能发生，是因为它能降低能量。如果能量最低的状态打破了空间对称而系统的能量又同时保持守恒，那么一旦进入对称破缺的状态，系统就会持续保持这个状态。这就是普通晶体能够存在的物理原因。

 

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，这种基于能量的解释不适用于τ的破缺，因为τ一旦破缺，能量就不再守恒，能量不再能度量这个系统。由于这个困难，大多数物理学家从来没有考虑过τ自发破缺的可能性，当然也就没考虑过时间晶体这种奇怪的东西。

 

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，对称性自发破缺还有一种更广泛的理解方式，适用于τ的破缺。除了自发重组成一个能量更低的状态，材料可以重组进入一个更稳定的状态。比如，许多粒子可以在一个大的空间或时间范围形成一个有序排列，如果破坏有序的力是小尺度和局域的，那么这个有序排列就很难被打破。这样，如果相比以前的状态，材料新的有序排列发生在一个更大的尺度上，那么它就有可能获得更高的稳定性。

 

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.

 

当然，最终没有哪种物质的状态可以面对所有的扰动都保持稳定。比如，钻石就是这样。“钻石恒久远”这句传奇的广告词已经妇孺皆知。但如果温度足够高，钻石在合适的空气中会烧成灰尘，不再光彩夺目。钻石在普通温度和气压下不是碳最稳定的状态，它们是在非常高的压强下产生的。钻石一旦形成，它们可以在普通压强下存在很长时间。按照物理学家的计算，如果等待足够长的时间，你的钻石也会变成石墨。甚至还有一个非常小但不是零的可能性，那就是量子涨落会让钻石变成黑洞。钻石还可能由于质子的衰变而转化为别的物质。在实践中，当我们说一个物质状态（比如钻石）时，我们说的是它形成了一种组织，对于很多外界扰动，这个组织具有稳定性。

 

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.

 

新旧时间晶体 OLD AND NEW TIME CRYSTALS

 

交流约瑟夫森效应是物理中的一块宝石，它为一大类时间晶体提供了原型。以1973年诺贝尔物理学奖得主、英国物理学家布赖恩·戴维·约瑟夫森（Brian David Josephson）命名的“约瑟夫森结”是夹在两个超导体中间的绝缘层。当在结的两端加上一个常电压后，我们就能观察到约瑟夫森效应——这时会观测到一个频率为2eV/ℏ的交变电流流过，在这里e是电子电荷，ℏ是约化普朗克常数。尽管整个物理设置不随时间改变（也就是说，它遵守τ），但系统最后的行为却随时间变化。完整的时间平移对称变成周期为ℏ/2eV的整数倍的对称。所以，交变约瑟夫森效应体现了时间晶体的最基本特征。但是，从某些角度来说，它不是完全符合期望。为了维持电流，我们必须让电路闭合并连上一节电池。交变电流会释放热，而电池会衰竭。另外，变化的电流还会辐射电磁波。由于这些原因，约瑟夫森结的稳定性还不够理想。

 

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.

通过各种改进（比如改用完全超导的线路，用高品质电容代替通常的电池，用闭合的笼罩防止辐射外泄），我们可以大幅降低这些效应。另外，通过用超流体或磁铁取代超导体，我们可以观察到类似的效应，同时将各种耗散降到最低。

 

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.

 

对τ破缺的大胆思考让这些问题受到了很多关注，物理学家因此发现了新的物理系统并做了许多富有成果的实验。但是，由于核心思想已经隐含在了约瑟夫森1962年的工作里，我们不妨称这些物理系统为“旧”时间晶体。

 

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》杂志宣布了“新”时间晶体的到来。这期杂志的封面是漂亮的、象征性的时间晶体，上面还有一句宣言：“时间晶体：神奇新物态的首次观测。”杂志里面是两篇独立的开拓性论文。这两篇论文显示，在一个实验里，美国马里兰大学的克里斯托弗·门罗（Christopher Monroe）领导的小组用精心设计的镱离子链形成了时间晶体。在另一个实验里，哈佛大学米哈伊尔·卢金 （Mikhail Lukin）的小组利用钻石里的几千个氮空位缺陷实现了时间晶体。

 

"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.

 

在这两个实验中，原子（镱离子或钻石缺陷）的自旋方向会规则变化，每隔一定周期，原子们会回到初始的形态。在门罗的实验里，研究人员用激光翻转离子的自旋，并将这些自旋关联起来形成“纠缠”态。最后，离子的自旋开始以两倍于激光脉冲速率的频率振荡。在卢金的实验里，科学家用微波脉冲翻转钻石缺陷的自旋，发现在微波脉冲之间系统会重复变化好几次。在这两个实验中，系统都需要外界的激发——激光或微波脉冲——但系统最终的振荡周期却和激发频率不同。换句话说，它们都自发地破坏了时间对称。

 

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.

这两个实验在材料物理领域开拓了一个新方向。基于相同的一般原则，人们发现了更多类似的材料体系。它们现在被称为弗洛克（Floquet）时间晶体。

 

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.

 

弗洛克时间晶体和一些早期发现的现象有些类似，但有本质的不同。1831年，迈克尔·法拉第（Michael Faraday）发现，当他以周期T垂直晃动一水槽的汞时，汞的流动周期通常是2T。但是，在对称破缺的法拉第系统以及很多2017年以前研究过的其他体系中，材料和驱动力（这里指晃动）没有清晰的分离，它们没有展示对称性自发破缺的核心特征。驱动一直在不停地向系统注入能量（或更准确地说是熵），它们最后以热辐射的形式散去。

 

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.

 

实际效果是，由材料加驱动组成的整个系统的对称性要少于驱动或材料各自的对称性。2017年的两个实验所用的物理系统显著不同：这两个系统在经过短暂的迟滞后会进入一个稳态，材料不再和驱动力交换能量或熵。这个区别很微妙，但在物理上非常关键。这些新的弗洛克时间晶体代表了一种新的物质状态，展示了对称性自发破缺的核心特征。

 

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.

 

在这个意义上，地球的自转和绕太阳的公转也不是时间晶体。它们令人深刻的稳定性是由能量和角动量的大致守恒来保证的。它们的能量和角动量都不是最小的，因此前面关于稳定性的能量分析在这里不适用。因为这两个量都特别大，所以要显著改变它们，你需要特别大的扰动或者长时间积累的小扰动。潮汐、其他行星的引力，甚至太阳自己的转动都会影响这些天文体系。因此，与此相关的时间，比如“日”和“年”的测量，都非常麻烦，需要不断修正。

 

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.

 

与之形成鲜明对比的是这些新的时间晶体，它们的重复行为有很强的刚性和稳定性。这个特征给出了一种精确分割时间的方式，这是制造先进时钟的关键。现代原子钟有令人惊叹的精度，但是它们没有时间晶体拥有的长时间稳定性。基于这些时间晶体，我们可能制造更精确、更简洁的时钟，从而能以极高精度测量距离和时间，而这样的时钟可以用来改进GPS，也可以根据地下洞穴和矿产对重力或引力波的影响来探测这些洞穴和矿产。由于这些可能的应用，美国国防高级研究计划局（DARPA）正在资助时间晶体的研究。

 

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.

 

宇宙学和黑洞中的τ THE TAO OF τ

 

围绕时间晶体和自发τ破缺的想法和实验还处于“婴儿期”。在这方面还有很多没有回答的问题。一个正在努力的方向是通过设计和发现新的时间晶体材料，从而扩展时间晶体家族，让它们包括更大更方便的系统，展示更多变的时空排列。物理学家对研究这些状态的相变也非常感兴趣。

 

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.

 

另一个方向是细致研究时间晶体的物理性质。前面提到过的半导体晶体提供了令人鼓舞的榜样。我们可以研究电子和光在时间晶体中会受到怎样的影响，以及什么新发现会从中涌现。

 

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?

 

我们已经在思考和时间相关的物质状态的各种可能性。我们不但可以考虑时间晶体，还可以考虑时间准晶体（规则但是没有重复排列的材料）、时间液体（时间轴上的事件密度是常数但不是周期的）和时间玻璃（具有刚性的结构但是和规则排列有偏离）。研究人员正在探索这些材料以及其他可能的相关材料。事实上，某些形式的时间准晶体和时间液体已经找到了。

 

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，我们已经考虑了各种基于τ的物质相。我最后简短评论一下宇宙学和黑洞中的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.

 

稳恒态宇宙模型试图从原则上在宇宙学中维持τ。在这个二十世纪中期非常流行的模型中，天文学家假设宇宙的状态或形貌在大尺度上是不依赖于时间的——也就是说，它是时间对称的。宇宙一直在扩张，稳恒态宇宙模型则假设物质不断产生，从而保持宇宙的平均密度不变。但是稳恒态宇宙模型没能经受住时间的考验。天文学家已经掌握的证据表明，137亿年以前宇宙大爆炸时刻的宇宙和现在的宇宙非常不同，尽管物理规律是一样的。在这个意义上，τ对称在宇宙中自发破缺了。有些宇宙学家还认为， 我们的宇宙是循环的或者宇宙曾经历过一段快速振荡期。这些猜想很接近时间晶体的相关思想。

 

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.

 

最后谈一下广义相对论。它是至今关于时空结构的最好理论。广义相对论建立了这样的概念：我们能够确定空间内任意邻近两点间的距离。可惜，这个简单的想法至少在两种极端条件下不成立：当我们将宇宙大爆炸外推到它的初始时刻或者在黑洞中心的时候。在其他物理领域，如果一个方程不再能描述某个物体的行为时，它通常意味着系统将经历一个相变。难道是时空在高压、高温或极速变化下自己放弃了τ对称？

 

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?

 

最终，时间晶体这个概念可能会同时在理论和实践上，从另一个角度推动物理学家对宇宙学和黑洞的理解。在可见的将来，我们非常可能会发现新型时间晶体，它们会让我们造出更完美的时钟，展示更多有用的性质。简而言之，时间晶体非常有趣，为我们打开了更多的窗户，扩展了我们对物质组织结构的理解。

 

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.

 

本文经授权转载自微信公众号「蔻享学术」。