This book provides the first comprehensive description of time crystals which have a repeating structure in time. It introduces the fundamental concepts behind time crystals and explores the many different branches of this new research area. The book starts with the original idea of the time crystallization in quantum systems as introduced by Wilczek and follows the development of the field up to the present day. Both spontaneous formation of crystalline structures in time and concepts of the condensed matter physics in the time domain, ranging from Anderson localization in time to many-body systems with exotic interactions, are described. The prospect of creation of novel objects by means of time engineering is also presented.

The book assumes knowledge of quantum mechanics to the graduate level. It serves as a valuable reference with pointers to future research directions for graduate students and senior scientists alike.

Autorentext
Krzysztof Sacha is a world-renowned scientist, one of the pioneers of time crystal research and author of many prominent papers related to this topic. Among his scientific achievements belongs the prediction of the behavior of discrete time crystals in a periodically-driven many-body system. His other research themes include cold quantum gases, ionization in strong laser fields and quantum chaos. He is a Professor at Jagiellonian University in Krakow, Poland. Besides his academic work he is engaged in the popularization of science and regularly runs marathons.

Inhalt
I. Spontaneous formation of space crystals 

II. Spontaneous breaking of continuous time translation symmetry

II.1 Wilczek model

II.2 Lack of time crystal formation by systems in the ground states

II.3 Excited eigenstates and time crystals formation

II.4 Superfluid phase and Josephson oscillations and their relation to time crystals

III.Discrete time crystals

III.1 Proposals of discrete time crystals

III.1.1 Formation of discrete time crystals by atoms bouncing on an oscillating mirror

III.1.2 Formation of discrete time crystals by periodically driven spin systems III.1.3 Different proposals of discrete time crystals

III.2 Experimental realizations of discrete time crystals

IV. Spontaneous formation of quasi-crystals structures in time

V. Condensed matter physics in the time domain

V.1 Emergence of crystalline structures in time in periodically driven systems

V.2 Anderson localization in time

V.3 Topological time crystals

V.4 Quasi-crystals in time

V.5 Many-body time crystals with exotic interactions

V.6 Many-body localization induced by temporal disorder

V.7 Time crystals with properties of 2D or 3D space crystals

VI. Time engineering VII. Time crystals in photonic materials