On November 30, 2021, Nature published a study by Google's quantum computer team to implement time crystal experiments, making another splash in this new field. The time crystal is a new form of matter proposed in 2012 by theoretical physicist and Nobel laureate in physics Wilczek. With the continuous exploration of researchers, the theory and experiment of time crystals have undergone tremendous changes. In terms of theory, physicists "besieged" the originally proposed concept of quantum time crystals, and later proposed a new concept of discrete time crystals. In terms of experiments, a number of indications have been controversial to achieve discrete-time crystal experiments, and the experimental presentation is not consistent with theoretical concepts. In less than 10 years, research on time crystals has developed rapidly in the face of skepticism from teams on all sides. In this article, we will witness how physicists can push their limits to explore the boundaries of human knowledge.
Written by | Guo Qitao (Southern University of Science and Technology), Yin Zhangqi (Beijing Institute of Technology)
Foreword: In recent years, the development of experimental technology in quantum computing has brought unlimited possibilities for human beings to use quantum superiority to accelerate information processing and study complex quantum physical systems. Based on superconducting circuits and optical systems, people have demonstrated the superiority of quantum computing systems over classical computers on different problems. At the same time, quantum simulations performed on quantum computers have opened up new avenues for physicists to study novel quantum states and topological materials. Condensed matter physicists have been able to break through the limitations of paper, numerical simulations, and even existing materials to explore more imaginative concepts of physics. Time crystals are a recent example of a huge impact. Starting from the origin of time crystals, this article will focus on the development of discrete time crystals and experimental developments of multibody localized protection, and the fierce debates related to it, hoping that readers will understand the theory of discrete time crystals and quantum simulations based on quantum computers.
2012-2015: Negation of Negation
The peculiar concept of time crystals stems from a bold 2012 question proposed by Nobel laureate physics laureate Frank Wilczek: whether there is a substance that, when near the ground state, spontaneously undergoes periodic changes in the temporal dimension, just as space crystals spontaneously repeat periodically in the spatial dimension. More precisely, the crystals common in life originate from the spontaneous breaking of the spatial continuous translation symmetry of many atoms, thus forming a self-organizing structure of spatial discrete translation symmetry. Similarly, Verczek's original definition of a time crystal is a timeless system in which the time-continuous translational symmetry of the ground state occurs is broken, thus allowing its state and observable measurement of periodic changes in the time self-organization structure [1, 2].
Fig. 1 Schematic diagram of the concept of time crystals: The Wegener ring crystal formed by ions, when it is in the ground state, the ions still rotate, and this space-time translational symmetry system is called space-time crystals. 丨Source: Dr. Li Tongzang, Professor Zhang Xiang Experimental Group
Along these lines, Wilczek and his collaborators proposed models of classical time crystals and quantum time crystals,[1, 2] respectively, while Li Tongzang and Zhang Xiang of the University of California, Berkeley, also proposed the theory of quantum space-time crystals based on ion traps[3]. Unlike the classical time crystal model, the quantum time crystal was besieged by various physics experts as soon as it came out, before the French Leibniz Prize winner Patrick Bruno, who pointed out that Wilcek's quantum time crystal model and Li Tongzang and others' theory of quantum space-time crystals could not be established under the condition of limited temperature[4], which he called the "non-existence theorem" of time crystals; later, There was Haruki Watanabe, a Japanese expert in condensed matter physics theory, and so on. Proceeding from a long program in the temporal dimension, they demonstrated that in the case of a finite temperature equilibrium state, a multibody physical system with only short-range interactions does not exist at the thermodynamic limits of quantum time crystals [5]. Only a few years after its birth, the beautiful physical model of the time crystal seems to be completely rejected by the rigorous analysis and arguments of various families, but the beautiful model is always loved in the dark. Although Yuki Watanabe et al. deny the possibility of finding quantum time crystals in equilibrium physical systems, their arguments cannot deny the possibility of quantum time crystals in non-equilibrium systems subject to periodic modulation [5]. Based on this, a model known as discrete-time crystals was invented and flourished at an astonishing rate in the years that followed.
2015-2018: Towards Reality
Although quantum-time crystal models that break continuous-time translational symmetry have encountered numerous difficulties, the development of discrete-time crystal theory has been thriving since Krzysztof Sacha first explicitly introduced the concept of spontaneous breakage of discrete-time symmetry in 2015.[6] Condensed matter scientists, represented by Norman Y. Yao (2019 American Physical Society Wally Prize winner), Vedika Khemani (2020 American Physical Society Wally Prize winner), and Dominic V. Else, from different perspectives, finally completed the construction of discrete quantum time crystal models in spin systems.
Before introducing their theory, let's talk about the spontaneous breaking of symmetry, which means that nature deliberately selects some solutions of special properties from all the solutions of the equations of motion allowed by a physical system. At this time, although the physical system itself has some symmetry of the Larry (Hamiltonian quantity), the state of motion and observable behavior of the system have a smaller symmetry. Specific to discrete-time crystals, in a periodic drive (Flokai) system, the system Hamiltonian quantity has a discrete-time translational symmetry of a time period, however, a considerable measurement of a discrete-time crystal is presented as a periodic translational symmetry structure (a positive integer greater than 1), which we call the "multiplier period" behavior. Not only that, but this "times-period" behavior is stable, and discrete-time crystals can always maintain their maverick dynamics in the face of subsystem heating, perturbations of drive cycles, and disturbances of interaction intensity in multibody systems [7, 8].
Fig. 2 Schematic diagram of the concept of discrete-time crystals: Time crystals composed of one-dimensional 1/2 spin chains. The Hamiltonian quantity of the system has a discrete-time translational symmetry with time T as the period, so that the T-time spin chain is completely flipped. Therefore the measurable measurement has a discrete-time translational symmetry of 2T. 丨Source: Physics World
Although the conjecture that periodic drive systems may have quantum-time crystals was grounded as early as 2015,[5] a thorny problem has been standing in the face of scientists in this direction: interacting multibody spin systems with drives, mostly subject to the thermal eigenstat hypothesis, rapidly thermally heat up subsystems during evolution and are heated to infinite temperatures, so that stable discrete-time translational symmetry-breaking phenomena cannot be observed [8, 9]. Thus, Norman Yao,[7] who has long been concerned with the theory of multibody localization that can circumvent quantum heating,[7] Vidica Kemani, who studies the phase structure of the spin-driven spin system,[10] and Dominique Els[11] who studies symmetry-protected topological phases, naturally become leaders in this field. They all turned their attention to a class of drive Isin spin chains with multibody localization properties, and the drive process within a single cycle T is divided into the application of a spin 1/2 longitudinal field Ising model with disorder and the opposing Pauli X flip operator. Because of its correspondence with a topology with a Mayonara π model, this model is also known as π-spin glass. For spin chains prepared on a direct product state, we can find that although the Frokai evolutionary operator has a time translation symmetry of T as a period, the susceptibility of each lattice point of the system always returns to the initial state after 2T, which also means that the system does have discrete time translation symmetry breaking. In addition, the model has good properties: in addition to the disordered longitudinal field Isin model itself tolerates interaction perturbations, the model also tolerates some deviation in the strength of the flip operator X due to the existence of emerging Z2 symmetry. Many of the work analyzing the long-term kinetic and thermodynamic properties of the system [7-13] shows that a achievable discrete-time crystal model appears to have been found.
Fig. 3 Experimental diagram of discrete-time crystals, ion traps, diamond color center systems, and nuclear magnetic resonance systems. Upper layer: Laser-trapped ytterbium ions, random impurities in diamond, and crystals of ammonium dihydrogen phosphate. Mesosphere: Dynamics of the magnetic susceptibility of discrete-time crystals. Lower: The secondary frequency response characteristics of discrete-time crystals (i.e., the "period-times" characteristic). 丨Source: A Brief History of Time Crystal
Testing this theory on the experimental platform is naturally a matter of course. As shown in Figure 3, during the two years from 2017 to 2018, the ion trap experimental team from the University of Maryland[14], the Diamond Color Center Experimental Team at Harvard University [15] and the Yale University NUCLEAR Resonance Quantum Platform team[16] reported on their work to achieve discrete-time crystals and observe their "multi-period" behavior in top publications such as Nature and Physical Review Letters. This news not only excited the academic community, but also the mass media rushed to report it. However, in a lively atmosphere, the controversy in the academic circles did not stop, and even caused greater waves in the following time.
2018-2021: Storms start again
After the emergence of discrete-time crystal experiments, the main question was whether these discrete-time crystals that were manufactured were really multibody localized time crystals with good properties and never heated, or whether they were doped with other dynamic mechanisms.[8]
The problems of diamond color center experiments and nuclear magnetic resonance experiments are the most prominent. In the diamond color center experiment, despite the strong disorder, its interaction form—three-dimensional long-range dipole interaction—is incompatible with multibody localization. The experimental team also admitted that this is a "critical time crystal" that will eventually heat up [17]. MrI experiments are completely out of order and therefore do not belong to the discrete-time crystals protected by multibody localization. Vidika Kemani and her phD supervisor Shivaji Sandy proposed that the NMR version of the time crystal is based on a mechanism known as "preheating" [8, 9]. In preheating systems, due to some symmetry and quasi-conservation observable measurements [9, 18-20], the heating process will be suppressed, thus providing a window time for observing the behavior of discrete-time crystals. However, there are various limitations in the preheating system, such as the preheating mechanism can only protect the dynamics of some low-temperature states close to the ground state of the system, and can only protect the system from being completely heated in exponential time, which is still mediocre compared to the multi-body localized system that is almost never heated. For example, if the multi-body localized system is a golden immortal that survives the catastrophe, then the preheating system can only be regarded as just having a baby, which is longer than the average quantum system. Vidika Kemani and Shivaji Sandy also proposed a solution to examine the properties of preheated discrete-time crystals. Many of these assertions were confirmed by the ion trap experimental team at the University of Maryland in experiments with preheating discrete-time crystals, which were subsequently published in the journal Science. The author also completed an experiment on a similar Froger preheating mechanism on a superconducting system by Zhu Xiaobo's team at the University of Science and Technology of China [21].
Subsequently, the focus of the debate shifted to the ion trap version of the discrete-time crystal, which was once considered to be the first multibody localized discrete-time crystal to work due to its disordered and multibody localized compatible interaction. However, due to the initial dependence of the model of this experiment in numerical experiments, and the short length of spin chains in the experiment (only 10), the evidence to support multibody localization is not sufficient. On September 1, 2021, Vidika Kemani and others formally challenged Norman Yao et al.,[22] questioning Yao et al.'s theoretical model on several issues such as effective Hamiltonian quantifier forms, size effects, and boundary conditions. Vidika Kemani et al. pointed out that although previous experiments on ion traps have the disorder necessary for multibody localization, the disorder at its single lattice point does not play a role in Frokai dynamics to prevent its heating. Through the analysis of the two-cycle evolutionary operator, they found that the disorder in the effective Hamiltonian quantity can be precisely eliminated by opposing the easy relation, so that the system does not produce the effect of Floquet's multibody localization during evolution [8, 22]. Therefore, experiments on ion traps can also be thought of as discrete-time crystals protected by a preheating mechanism. Vidika Kemani et al. also listed their numerical results, according to the experimental design on the ion trap, the discrete-time crystal does not traverse all the direct product states, for some high temperature states, the system will heat up rapidly in dozens of Flokai cycles, making it impossible to observe stable discrete-time crystal dynamics. In addition, the experiment has some loopholes due to the limited size and boundary effects. Vidika Kemani et al. also pointed out the conditions for achieving multibody localized discrete-time crystals based on the Norman Yao et al. model, that is, the interaction intensity must also have a large disorder, so that there is no disorder in the effective Hamiltonian quantity.
Two weeks later, on September 15, Norman Yao et al. responded,[23] calling the model of Vidica Kemani et al. the KMS (Khemani-Moessner-Sondhi) model and their own model YPPV (
Yao-Potter-Potirniche-Vishwanath) model. Norman Yao et al. showed through numerical experiments that for a sufficiently small integrability destruction horizontal field, their system can indeed keep the arbitrary direct product state oscillating. But Norman Yao and others also admit that for some "mischievous" initial states, the periodic oscillation amplitude of their model will be relatively small. Further, through extrapolation of finite sizes and analysis of boundary conditions, Yao et al. insist that their discrete-time crystal experiments were not solely due to the preheating effect, but were indeed involved in quantum multibody effects. In early 2021, Norman Yao et al. designed another experiment on the diamond color center system with a multibody localized discrete-time crystal,[24] which showed no initial dependence and was able to sustain more than 800 Flokai cycles, which was more convincing than the ion trap experiment.
Based on the above questions and answers, the author believes that the model of Norman Yao et al. does not fully demonstrate the true multibody localized discrete time crystal behavior, but many phenomena also show that this system does have a part of the quantum multibody effect involved. In this academic battle, it is not yet time to make up our minds.
Multibody localized time crystals are fabricated using digital quantum simulations
In fact, in June 2020, Vidika Kemani and Shivaji Sandy and others have been hung on the preprint website arXiv, and they intend to implement a multibody localized discrete-time crystal on Google's superconducting quantum processor sycamore. A year later, the Google team implemented discrete-time crystals based on the KMS model, which was published online in the journal Nature on November 30, 2021. This experiment is a great improvement over all previous discrete-time crystallization protocols.
The first is the length of the spin chain: the Google team used a chain of 20 superconducting bits to implement the time crystal, which really enters the region of the quantum multibody system so that the effects of size and boundary effects are small enough. Second, the hamiltonian amount of the multibody localized time crystal is completely used to achieve digital quantum simulation, that is, the Hamiltonian amount that is "spelled out" completely using the quantum gate circuit. This is more maneuverable than previous analog quantum simulation schemes that use native interactions between qubits, but are more difficult to implement. In Google's experiments, they used a single-bit gate and a two-bit fermion simulation gate to achieve a hamiltonic amount of a multibody localized time crystal. To avoid interference with the preheating mechanism, the Google team experimented with the fully polarized initial state, the Neil state, and the random binary string state. The experimental results show that their system not only demonstrates the "multi-period" dynamics of discrete-time crystals, but also does meet the characteristics of Flokai's multibody localization that does not depend on the initial state. In addition, they measured a new observable measurement, the spin glass sequence parameter [25, 26], which showed that the behavior of this sequence parameter with the size of the system showed that it did meet the spin glass characteristics. Therefore, the design of this experiment is indeed the most suitable for the requirements of multibody localized discrete-time crystals. The method of measuring the spin time correlation function using auxiliary bits is also more rigorous than previous experiments.
Figure 4 Schematic diagram of the Google time crystal experiment. (a): The process of making discrete-time crystals, which prepare the initial state of the system to the binary string state. Using the digital quantum simulation method, the Hamiltonian quantity of a discrete-time crystal is simulated with a quantum gate and reads the expectation of its Pauli-z operator at the end. (b): Discrete-time crystal dynamics averaged for different initial states and disorders. (c): Comparison of the behavior of the thermal system and the behavior of the multibody localized discrete-time crystals. (d): Discrete-time crystal dynamics obtained after noise reduction by echo lines without decoherence effects. 丨Source: Google Discrete Time Crystal Preprint Of Time-Crystalline Eigenstate Order on a Quantum Processor
The experiment also demonstrates the excellent technical capabilities of Google's superconducting quantum computing team. Prior to the experiment, the Google team performed a fine cross-entropy benchmark and Flokai proofreading of the fermion simulation gate. And after the experiment, the noise reduction process is carried out through the echo line, which greatly reduces the influence of superconducting qubit decoherence in the experiment, making the experimental data more beautiful. It should be pointed out that the mainland quantum computing experimental team has also made its own contributions - Yu Haifeng's team at the Beijing Institute of Quantum Information Science successfully realized a discrete-time crystal based on the YPPV model through analog quantum simulation on the superconducting bit system[27]; Wang Haohua's team at Zhejiang University realized a class of boundary time crystals based on symmetry-protected topological states[28]. In addition, there are a number of discrete-time crystal models waiting to be further explored, such as classical preheated discrete-time crystals[29, 30], discrete-time crystals based on cellular automata[31], and so on. Discrete-time crystals have evolved from ingenious to rigorous and thorough physical models, which have had a huge impact on quantum simulation and novel quantum matter, and prompted condensed matter theory physicists and quantum physics experimental scientists to join hands to explore more interesting man-made states of matter.
The future can be expected
In September 2021, four theoretical scientists, Norman Yao, Vidika Kemani, Dominique Els, and Yuki Watanabe, received the Scientific Breakthrough Award, marking a broader recognition of the new field of discrete time crystals. At the end of 2021, the discrete-time crystal experiment implemented by Google's quantum computing team was named one of the annual physics breakthroughs by the American Physical Society (APS) Physics and the British Physical Society (IOP) Physics World. The study of discrete-time crystals has refreshed people's understanding of periodic drive systems, multibody localization, preheating, and quantum heating processes, and has prompted more researchers in different fields to participate in them. From the development process of discrete-time crystals, it can be seen that scientific exploration is not smooth most of the time, and it needs to be negated by negation, as well as academic tit-for-tat controversy. In scientific inquiry, creative errors are more valuable than mediocre correctness, because new ideas may be born in mistakes. The time crystal has caught up with the rapid development of quantum computing technology, which has been able to develop rapidly in a short period of time rather than be buried. This leads us to ponder: how should theories and experiments look at each other, and how should they cooperate to advance a scientific field?
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