Quantum time crystal by decoherence: Proposal with an incommensurate charge density wave ring

We show that time translation symmetry of a ring system with a macroscopic quantum ground state is broken by decoherence. In particular, we consider a ring-shaped incommensurate charge density wave (ICDW ring) threaded by a fluctuating magnetic flux: the Caldeira-Leggett model is used to model the fluctuating flux as a bath of harmonic oscillators. We show that the charge density expectation value of a quantized ICDW ring coupled to its environment oscillates periodically. The Hamiltonians considered in this model are time independent unlike “Floquet time crystals” considered recently. Our model forms a metastable quantum time crystal with a finite length in space and in time.

Figure 1

The concept of translation symmetry breaking by decoherence is illustrated. (a) A simplified version of the two-state system considered by Leggett et al. [4]. A particle can tunnel through a potential barrier and exist at two states simultaneously. However, if this system starts to interact with its surrounding environment, then the particle will localize at one of the states. (b) Similarly, the ground state of a free particle confined on a ring is a plane wave state. Coupling to environment will localize the particle and break rotational symmetry. This “particle” corresponds to the phase of an incommensurate charge density wave ring (ICDW ring) in our model (Fig. 2).

Figure 2

(a) We consider an incommensurate CDW ring threaded by a fluctuating magnetic flux. This figure shows a commensurate CDW with $\lambda /a=2$ because it is easier to visualize. The wave (typically $\sim {10}^5$ wavelengths) represents the charge density and the dots represent the atoms of a quasi-one-dimensional crystal. (b) ICDW ring crystals such as monoclinic ${\mathrm{TaS}}_3$ ring crystals and ${\mathrm{NbSe}}_3$ ring crystals can be produced experimentally [7, 8, 9]. Our model can be tested provided clean ring crystals with almost no defects and impurities can be produced. (c) The ground state of an isolated quantized ICDW ring is a superposition of periodically oscillating ICDWs; hence the oscillation is unobservable. Coupling to environment (fluctuating magnetic flux) will break the superposition and the oscillation becomes apparent.

Figure 3

These plots are shown with $g_s=1{\mathrm{Hz}}^{2-s}$ and $\mu ={10}^{-8}$ sec. (a) The fundamental solution $G\left(t\right)$ of a classical ICDW ring coupled to its environment is shown for sub-Ohmic ($s<1$), Ohmic $(s=1),$ and super-Ohmic ($s>1$) damping. Note that $G\left(t\right)\approx t$ for $t$ less than some time scale ${\tau }_{\text{damp},s}$. The time $t$ is normalized by $4\pi \mu$, so the horizontal axis gives the number of “lattice points.” (b) The derivative of the fundamental solution, $\dot{G}\left(t\right)$, heuristically describes the velocity of the ICDW ring. Note that $\dot{G}\left(t\right)\approx 1$ for $t\ll {\tau }_{\text{damp},s}$.

Figure 4

(a) The amplitude of the charge density oscillation is shown for $g_s=1{\mathrm{Hz}}^{2-s}, \mu ={10}^{-8}\sec$, and $\mathrm{\Omega }=1/\mu$. This oscillation is an effective QTC. In general, the oscillation amplitude is larger for super-Ohmic ($s>1$) damping but has a small lifetime ${\tau }_Q$. On the other hand, the oscillation amplitude is small for sub-Ohmic ($s<1$) damping but has a long ${\tau }_Q$. (b) The charge density oscillation is shown for super-Ohmic damping with $s=1.2, g_{1.2}=1{\mathrm{Hz}}^{0.8}, \mu ={10}^{-8}\sec$, and $\mathrm{\Omega }=1/\mu$. The time $t$ is normalized such that the horizontal axis gives the number of “lattice points.”