Quantum Time Crystal By Decoherence in Incommensurate Charge Density Wave Ring

A quantum time crystal (QTC)\footnote{F. Wilczek. PRL {\bf109}:160401, (2012)}$^,$\footnote{ T. Li et al. PRL, {\bf109}:163001, (2012)} is a novel quantum mechanical ground state which breaks time-translation symmetry $\mathcal T$. $\mathcal T$ of a system with thermodynamic (infinite-volume) limit is expected to become discrete by spontaneous $\mathcal T$ breaking$^1$, but the possibility to realize such system is still under debate \footnote{P. Bruno. PRL, {\bf111}:070402, (2013)}$^,$\footnote{H. Watanabe and M. Oshikawa. PRL, {\bf114}:251603, (2015)}. Meanwhile, Volovik\footnote{G. E. Volovik. JETP Lett., {\bf98}:491, (2013)} proposes the possibility of an effective QTC, that is, a QTC with a finite size in time. In this presentation we show that $\mathcal T$ of a finite-size system becomes discrete by a ``decoherence measurement", which is best described using an incommensurate charge density wave (ICDW). We use a ring-shaped ICDW threaded by a fluctuating magnetic flux to model an effective QTC with a finite size in space and time: $\mathcal T$ becomes discrete by measuring the ground state of an ICDW ring and the lifetime of the QTC is extended using decoherence\footnote{A. Caldeira and A. Leggett. Physica A, {\bf121}:587, (1983)}.