Condensed Matter Physics in Time Crystals K Giergiel1 and K Sacha1,2 1M. Smoluchowski Institute of Physics, Jagiellonian University, 11, ul. prof. Stanisława Łojasiewicza, 30-348, Kraków, Poland. Contact Phone: +48126644890 2Mark Kac Complex Systems Research Center, Jagiellonian University, 11, ul. prof. Stanisława Łojasiewicza, 30-348, Kraków, Poland. Contact Phone: +48126644890 Contact Email: [email protected] Periodically driven systems can display new many-body phases. One kind of such phases are time crystals – quantum many-body systems that, due to interactions between particles, are able to spontaneously self-organise their motion in a peri- odic way in time by analogy with the formation of crystalline structures in space in condensed matter Figure 1: Example of a system with exotic long-range physics. interactions: ultra-cold atoms bouncing resonantly on In solid-state physics properties of space crys- a harmonically oscillating mirror in a 1D model. The tals are often investigated with the help of exter- 20:1 resonance condition between the mirror oscillations and atomic motion is considered and the system is nal potentials that are spatially periodic and reflect described by the Bose-Hubbard Hamiltonian: H^eff = various crystalline structures. A similar approach J P y 1 P y y − 2 hi;ji a^i a^j + 2 i;j Uija^i a^ja^ja^i. Center panel shows can be applied for time crystals, as periodically the interaction coefficients Uij, which depend on the dis- driven systems constitute counterparts of spatially tance between sites i − j of the Bose-Hubbard Hamilto- periodic systems, but in the time domain. nian. The long-range interactions presented in the centre Employing periodically driven systems we pro- panel can be realised if the s-wave scattering length of pose quantum simulators, which promise access atoms g0(t) is modulated in time as presented in the right to programmable exotic long-range interactions in panel solid state-like many-body problems, systems with topological protection of time correlations, methods to investigate Anderson localisation physics in the time domain and quasi-crystalline structures in time. These propositions can be realised experimentally, for example, in ultra-cold atoms bouncing resonantly on an oscillating atom mirror..