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arXiv:2505.23021

Generating discrete time crystals through optimal control

Sahoo, Mrutyunjaya, Rahul Kumar Ghosh, Das, Bandita, Shishira Mahunta, Bodhaditya Santra, Victor Mukherjee

Published May 29, 2025• 0 citations
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Abstract

In this work we use optimal control to generate Discrete Time Crystals (DTC) in generic many-body quantum systems. We define appropriate cost functions, which, when optimized, result in the formation of DTCs. This hitherto unexplored method represents DTCs as an optimization problem, and allows us to find non-trivial realistic periodic control pulses and parameter regimes which result in spontaneous breaking of time-translational symmetry in quantum systems. We exemplify our approach using many-body quantum systems in the presence, as well as absence of dissipation. We also discuss possible experimental realization of the control protocol for generating DTCs.

DOI
10.48550/arxiv.2505.23021

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In this work we use optimal control to generate Discrete Time Crystals (DTC) in generic many-body quantum systems. We define appropriate cost functions, which, when optimized, result in the formation of DTCs. This hitherto unexplored method represents DTCs as an optimization problem, and allows us to find non-trivial realistic periodic control pulses and parameter regimes which result in spontaneous breaking of time-translational symmetry in quantum systems. We exemplify our approach using many-body quantum systems in the presence, as well as absence of dissipation. We also discuss possible experimental realization of the control protocol for generating DTCs.
Abstract A remarkable consequence of spontaneously breaking the time translational symmetry in a system, is the emergence of time crystals. In periodically driven systems, discrete time crystals (DTC) can be realized which have a periodicity that is n times the driving period. However, all of the experimental observations have been performed for period-doubling and period-tripling DTC. Novel physics can arise by simulating many-body physics in the time domain, which would require a genuine realisation of the n-tupling DTC. A system of ultra-cold bosonic atoms bouncing resonantly on an oscillating mirror is one of the models that can realise large period DTC. The preparation of DTC demands control in creating the initial distribution of the ultra-cold bosonic atoms along with the mirror frequency. In this work, we demonstrate that such DTC is robust against perturbations to the initial distribution of atoms. We show how Bayesian methods can be used to enhance control in the preparation of the initial state as well as to efficiently calculate the phase diagram for such a model. Moreover, we examine the stability of DTCs by analyzing quantum many-body fluctuations and show that they do not reveal signatures of heating.
Discrete time crystals (DTCs) exhibit a special nonequilibrium phase of matter in periodically driven many-body systems with spontaneous breaking of time translational symmetry. The presence of decoherence generally enhances thermalization and destroys the coherence required for the existence of DTCs. In this paper, we devise a mechanism for establishing a stable DTC with period-doubling oscillations in an open quantum system that is governed by a properly tailored non-Hermitian Hamiltonian. We find a specific class of nonreciprocal couplings in our non-Hermitian dynamics which prevents thermalization through eigenstate ordering. This choice of non-Hermitian dynamics significantly enhances the stability of the DTC against imperfect pulses. Through a comprehensive analysis, we determine the phase diagram of the system in terms of pulse imperfection.
Discrete time crystals (DTCs) are phases of matter characterized by the presence of an observable evolving with nTnTnT periodicity under a TTT-periodic Hamiltonian, where n>1n>1n>1 is an integer insensitive to small parameter variations. In particular, DTCs with n=2n=2n=2 have been extensively studied in periodically quenched and kicked spin systems in recent years. In this paper, we study the emergence of DTCs in a many-body system whose semiclassical mean-field dynamics is nonintegrable, using a rather simple model depicting a harmonically driven spin chain. We advocate to first employ a semiclassical approximation to arrive at a mean-field Hamiltonian and then identify the parameter regime at which DTCs exist, with standard tools borrowed from studies of classical chaos. Specifically, we seek symmetric-breaking solutions by examining the stable islands on the Poincar\'e surface of section of the mean-field Hamiltonian. We then turn to the actual many-body quantum system, evaluate the stroboscopic dynamics of the total magnetization in the full quantum limit, and verify the existence of DTCs. Our effective and straightforward approach indicates that in general DTCs are one natural aspect of many-body quantum chaos with mixed classical phase space structure. Our approach can also be applied to general time-periodic systems, which is thus promising for finding DTCs with n>2n>2n>2 and opening possibilities for exploring DTCs properties beyond their time-translational breaking features.
The engineering of new states of matter through Floquet driving has revolutionized the field of condensed matter physics. This technique enables the creation of hybrid topological states and ordered phases that are absent in normal systems. Crystalline structures, exemplifying spatially ordered systems under periodic driving, have been extensively studied. However, recent focus has shifted towards discrete time crystals (DTCs), periodically driven quantum many-body systems that break time translation symmetry under specific conditions. In this paper, the model of discrete time crystals is extended to allow for the formation of time-varying tesseracts, allowing for the investigation of time translational symmetry in pseudo-higher-dimensional lattice systems.
Many-body quantum systems, under suitable conditions, exhibit time-translation symmetry breaking and settle in a discrete time crystalline (DTC) phase -- an out-of-equilibrium quantum phase of matter. The defining feature of DTC is a robust subharmonic response. However, the DTC phase is fragile in the presence of environmental dissipation. Here, we propose and exemplify a DTC phase in a noninteracting system that owes its stability to environmental dissipation. The lifetime of this DTC is independent of initial conditions and the size of the system, though it depends on the frequency of the external driver. We experimentally demonstrate this realization of DTC using Nuclear Magnetic Resonance spectroscopy.
The discrete time crystal (DTC) is a nonequilibrium phase of matter that spontaneously breaks time-translation symmetry. Disorder-induced many-body localization can stabilize the DTC phase by breaking ergodicity and preventing thermalization. Here, we observe the hallmark signatures of the many-body–localized DTC using a quantum simulation platform based on individually controllable carbon-13 nuclear spins in diamond. We demonstrate long-lived period-doubled oscillations and confirm that they are robust for generic initial states, thus showing the characteristic time-crystalline order across the many-body spectrum. Our results are consistent with the realization of an out-of-equilibrium Floquet phase of matter and introduce a programmable quantum simulator based on solid-state spins for exploring many-body physics.