In this article, I will explain the time crystal. This is a system in which the entire system is filled with a fixed unit structure by discrete translational and rotational operations in the time direction rather than the space direction. A crystal is a system that can transition to any unit structure in the whole system without duplication by repeating translation and rotation operations on the unit structure a finite or infinite number of times. For example, consider a two-dimensional equilateral triangular lattice as shown in Figure 1. The unit structure in this lattice is the red part of a rhombus-shaped equilateral triangle connected vertically. This unit structure can be moved to another unit structure with a different position by a 60-degree rotation about the axis of the atom and a movement along the length of one side.

Fig. 1 Equilateral triangular two-dimensional lattice and its unit structure.

A time crystal has a unit structure in the time direction. In other words, periodic structural changes occur. However, this is a spontaneous phenomenon, not like the Rabi oscillation, which occurs only when an external laser is applied. This phenomenon can be realized by alternately multiplying two or more Hamiltonians with a constant period $T=T_1+T_2$ .

$H_1=\hbar g(1-\epsilon)\sum_l\sigma_l^x (0<t<T_1)$

$H_2=\hbar\sum_{m}J_{lm}^z\sigma_l^z\sigma_m^z+\hbar\sum_lB_l^z\sigma_l^z(T_1<t<T_2)$

The first and second terms in H₂ are sometimes multiplied separately, and in some systems only the first term is used. In particular, the first term of H₂ determines the duration of the time crystal. This Hamiltonian is called the Floquet lattice, which can be multiplied to produce a time crystal oscillation with a period of T/2. Time crystals are a quantum mechanical phenomenon, but in fact they are no different from the conserved quantity systems found in mechanics and electromagnetism textbooks. In a time crystal, it is itself a resonant circuit, the transition between the states of the time crystal oscillation is a conserved quantity corresponding to the electromagnetic field in the resonant circuit, the first term of H₂ is the power generation, and H₁ is the transformer adjacent to the resonant circuit. This phenomenon can occur in any quantum system, and has been experimentally realized in a one-dimensional system using ytterbium ions [1] and nitrogen defects in diamond [2]. Furthermore, research has been done to grow entanglement structures by using the time variation of the time crystal itself [3]. The development of this field is still in its infancy. I am looking forward to the future development of this field.

[1][1609.08684] Observation of a Discrete Time Crystal (arxiv.org)

[2][1610.08057v1] Observation of discrete time-crystalline order in a disordered dipolar many-body system (arxiv.org)

[3][1907.13146] Simulating complex quantum networks with time crystals (arxiv.org)

The world's simplest theory of time crystals.

Hikaru Wakaura