Time crystal
In condensed matter physics, a time crystal is a quantum system of particles whose lowest-energy state is one in which the particles are in repetitive motion. The system cannot lose energy to the environment and come to rest because it is already in its quantum ground state. Time crystals were first proposed theoretically by Alfred Shapere and Frank Wilczek in 2012 as a time-based analogue to common crystals – whereas the atoms in crystals are arranged periodically in space, the atoms in a time crystal are arranged periodically in both space and time. Several different groups have demonstrated matter with stable periodic evolution in systems that are periodically driven. In terms of practical use, time crystals may one day be used as quantum computer memory.
The existence of crystals in nature is a manifestation of spontaneous symmetry breaking, which occurs when the lowest-energy state of a system is less symmetrical than the equations governing the system. In the crystal ground state, the continuous translational symmetry in space is broken and replaced by the lower discrete symmetry of the periodic crystal. As the laws of physics are symmetrical under continuous translations in time as well as space, the question arose in 2012 as to whether it is possible to break symmetry temporally, and thus create a "time crystal"
If a discrete time-translation symmetry is broken, then the system is referred to as a discrete time crystal. A discrete time crystal never reaches thermal equilibrium, as it is a type of non-equilibrium matter. Breaking of time symmetry can occur only in non-equilibrium systems. Discrete time crystals have in fact been observed in physics laboratories as early as 2016. One example of a time crystal, which demonstrates non-equilibrium, broken time symmetry is a constantly rotating ring of charged ions in an otherwise lowest-energy state.
Concept
Ordinary crystals form through spontaneous symmetry breaking related to spatial symmetry. Such processes can produce materials with interesting properties, such as diamonds, salt crystals, and ferromagnetic metals. By analogy, a time crystal arises through the spontaneous breaking of a time-translation symmetry. A time crystal can be informally defined as a time-periodic self-organizing structure. While an ordinary crystal is periodic in space, a time crystal has a repeating structure in time. A time crystal is periodic in time in the same sense that the pendulum in a pendulum-driven clock is periodic in time. Unlike a pendulum, a time crystal "spontaneously" self-organizes into robust periodic motion.Time-translation symmetry
Symmetries in nature lead directly to conservation laws, something which is precisely formulated by Noether's theorem.The basic idea of time-translation symmetry is that a translation in time has no effect on physical laws, i.e. that the laws of nature that apply today were the same in the past and will be the same in the future. This symmetry implies the conservation of energy.
Broken symmetry in normal crystals
Common crystals exhibit broken translation symmetry: they have repeated patterns in space and are not invariant under arbitrary translations or rotations. The laws of physics are unchanged by arbitrary translations and rotations. However, if we hold fixed the atoms of a crystal, the dynamics of an electron or other particle in the crystal depend on how it moves relative to the crystal, and particle momentum can change by interacting with the atoms of a crystal—for example in Umklapp processes. Quasimomentum, however, is conserved in a perfect crystal.Time crystals show a broken symmetry analogous to a discrete space-translation symmetry breaking. For example, the molecules of a liquid freezing on the surface of a crystal can align with the molecules of the crystal, but with a pattern less symmetric than the crystal: it breaks the initial symmetry. This broken symmetry exhibits three important characteristics:
- the system has a lower symmetry than the underlying arrangement of the crystal,
- the system exhibits spatial and temporal long-range order,
- it is the result of interactions between the constituents of the system, which align themselves relative to each other.
Broken symmetry in discrete time crystals (DTC)
The initial symmetry, which is the discrete time-translation symmetry with, is spontaneously broken to the lower discrete time-translation symmetry with, where is time, the driving period, an integer.
Many systems can show behaviors of spontaneous time-translation symmetry breaking but may not be discrete time crystals: convection cells, oscillating chemical reactions, aerodynamic flutter, and subharmonic response to a periodic driving force such as the Faraday instability, NMR spin echos, parametric down-conversion, and period-doubled nonlinear dynamical systems.
However, discrete time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking:
- it is a broken symmetry the system shows oscillations with a period longer than the driving force,
- the system is in crypto-equilibrium these oscillations generate no entropy, and a time-dependent frame can be found in which the system is indistinguishable from an equilibrium when measured stroboscopically,
- the system exhibits long-range order the oscillations are in phase over arbitrarily long distances and time.
These characteristics makes discrete time crystals analogous to spatial crystals as described above and may be considered a novel type or phase of nonequilibrium matter.
Thermodynamics
Time crystals do not violate the laws of thermodynamics: energy in the overall system is conserved, such a crystal does not spontaneously convert thermal energy into mechanical work, and it cannot serve as a perpetual store of work. But it may change perpetually in a fixed pattern in time for as long as the system can be maintained. They possess "motion without energy"—their apparent motion does not represent conventional kinetic energy. Recent experimental advances in probing discrete time crystals in their periodically driven nonequilibrium states have led to the beginning exploration of novel phases of nonequilibrium matter.Time crystals do not evade the second law of thermodynamics, although they spontaneously break "time-translation symmetry", the usual rule that a stable object will remain the same throughout time. In thermodynamics, a time crystal's entropy, understood as a measure of disorder in the system, remains stationary over time, marginally satisfying the second law of thermodynamics by not increasing.
History
The idea of a quantized time crystal was theorized in 2012 by Alfred Shapere and Frank Wilczek, a Nobel laureate and professor at MIT. In 2013, Xiang Zhang, a nanoengineer at University of California, Berkeley, and his team proposed creating a time crystal in the form of a constantly rotating ring of charged ions.In response to Wilczek and Zhang, Patrick Bruno and Masaki Oshikawa published several articles stating that space–time crystals were impossible.
Subsequent work developed more precise definitions of time-translation symmetry-breaking, which ultimately led to the Watanabe–Oshikawa "no-go" statement that quantum space–time crystals in equilibrium are not possible. Later work restricted the scope of Watanabe and Oshikawa: strictly speaking, they showed that long-range order in both space and time is not possible in equilibrium, but breaking of time-translation symmetry alone is still possible.
Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed. In 2014 at Jagiellonian University in Kraków predicted the behaviour of discrete time crystals in a periodically driven system with "an ultracold atomic cloud bouncing on an oscillating mirror".
In 2016, research groups at Princeton and at Santa Barbara independently suggested that periodically driven quantum spin systems could show similar behaviour. Also in 2016, Norman Yao at Berkeley and colleagues proposed a different way to create discrete time crystals in spin systems. These ideas were successful and independently realized by two experimental teams: a group led by Harvard's Mikhail Lukin and a group led by Christopher Monroe at University of Maryland. Both experiments were published in the same issue of Nature in March 2017.
Later, time crystals in open systems, so-called "dissipative time crystals," were proposed in several platforms breaking a discrete and a continuous time-translation symmetry. A dissipative time crystal was experimentally realized for the first time in 2021 by the group of Andreas Hemmerich at the Institute of Laser Physics at the University of Hamburg. The researchers used a Bose–Einstein condensate strongly coupled to a dissipative optical cavity and the time crystal was demonstrated to spontaneously break discrete time-translation symmetry by periodically switching between two atomic density patterns. In an earlier experiment in the group of Tilman Esslinger at ETH Zurich, limit cycle dynamics was observed in 2019, but evidence of robustness against perturbations and the spontaneous character of the time-translation symmetry breaking were not addressed.
In 2019, physicists Valerii Kozin and Oleksandr Kyriienko proved that, in theory, a permanent quantum time crystal can exist as an isolated system if the system contains unusual long-range multiparticle interactions. The original "no-go" argument only holds in the presence of typical short-range fields that decay as quickly as for some. Kozin and Kyriienko instead analyzed a spin-1/2 many-body Hamiltonian with long-range multispin interactions, and showed it broke continuous time-translational symmetry. Certain spin correlations in the system oscillate in time, despite the system being closed and in a ground energy state. However, demonstrating such a system in practice might be prohibitively difficult, and concerns about the physicality of the long-range nature of the model have been raised.