Discrete Time Dynamics

Andrei Yu. Khrennikov

We analyze dynamical consequences of a conjecture that there exists a fundamental (indivisible) quant of time. In particular we study the problem of discrete energy levels of hydrogen atom. We are able to reconstruct potential which in discrete time formalism leads to energy levels of unperturbed hydrogen atom. We also consider linear energy levels of quantum harmonic oscillator and show how they are produced in the discrete time formalism. More generally, we show that in discrete time formalism finite motion in central potential leads to discrete energy spectrum, the property which is common for quantum mechanical theory. Thus deterministic (but discrete time!) dynamics is compatible with discrete energy levels.

Viktor Holubec

Andrei Yu. Khrennikov, Yaroslav Volovich

We analyze dynamical consequences of a conjecture that there exists a fundamental (indivisible) quant of time. In particular we study the problem of discrete energy levels of hydrogen atom. We are able to reconstruct potential which in discrete time formalism leads to energy levels of unperturbed hydrogen atom. We also consider linear energy levels of quantum harmonic oscillator and show how they are produced in the discrete time formalism. More generally, we show that in discrete time formalism finite motion in central potential leads to discrete energy spectrum, the property which is common for quantum mechanical theory. Thus deterministic (but discrete time!) dynamics is compatible with discrete energy levels.

D. Galetti, M. Ruzzi

Koki Chinzei

A discrete time crystal (DTC) is an essentially nonequilibrium phase that breaks discrete time-translation symmetry in periodically driven (Floquet) systems.In ideal isolated quantum systems without dissipation, many kinds of DTCs have been theoretically proposed and experimentally verified in various artificial quantum systems, such as trapped ions, NV centers in diamond, and quantum computers.However, in real physical systems, we cannot avoid dissipation stemming from coupling to an external environment.Nevertheless, the effects of dissipation on DTCs have not been fully explored yet.This thesis reports the author's two original studies on dissipative DTCs.First, we formulate a symmetry-based theory of dissipative DTCs.To this end, we introduce Floquet dynamical symmetry (FDS) governing the long-time dynamics of dissipative Floquet systems.The FDS protects the quantum coherence of the system from dissipation and dephasing and leads to two types of time crystals: the discrete time crystal (DTC) and discrete time quasicrystal (DTQC), which are different in periodicity.We show that these time crystals appear in spin-S Bose-and Fermi-Hubbard models driven by an ac field and that their periodicity can be tuned by adjusting the field strength.Since these time crystals rely on the FDS alone, they appear even in the presence of perturbations as long as the FDS exists.We also elucidate that the FDS explains the symmetry structure behind the conventional DTCs proposed in the previous studies.Second, we investigate dissipative DTCs in solid-state materials by considering a dissipative quantum Ising model periodically driven by π-pulses.While the previous studies on dissipative DTCs have focused on well-controllable artificial quantum systems, we show that realistic dissipation in solids does not destroy but can stabilize the DTC order.Using the time-dependent mean-field theory, we analyze the DTC phase transition and determine critical exponents, including a dynamical one.We also find a novel reentrant phase transition in varying the pulse interval.This transition is due to a nontrivial interplay of the periodic drive and dissipation and has no equilibrium counterparts in the corresponding undriven system.Furthermore, to demonstrate the rigidity (stability) of the DTC, we consider imperfect πpulses, showing that the DTC is robust against minor imperfections and finding that DTQC appears for major imperfections.These results uncover a general symmetry aspect of dissipative DTCs and lay the foundation for realizing DTCs in material science.

Krzysztof Sacha