We propose the Caldeira-Leggett quantum time crystal in which the interaction between a quantum system and a viscoelastic sub-Ohmic environment leads to periodic revivals of quantum coherence. Using the Caldeira-Leggett model of quantum decoherence with a sub-Ohmic environment characterized by the spectral function $J\left(\omega \right)\propto {\omega }^s$ ($0\le s<1$), we discover that decreasing the sub-Ohmic parameter $s$ enhances the coherence lifetime of Gaussian wave packets, with the purity (i.e. coherence) decaying as $t^{-s/2}$ throughout the regime $0\le s<1$. A revival oscillation of the purity with lifetime $e^{2/s}$ occurs in the deep sub-Ohmic regime ($s<0.1$), which we call the Caldeira-Leggett quantum time crystal. We reveal that such oscillations originate from the fact that an environment in the deep sub-Ohmic regime behaves as a gel-like fractal medium with viscosity and elasticity, suggesting the significance of fractal gel-like environments in vivo and in information processing.

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• Fractals
• Open quantum systems & decoherence
• Quantum foundations
• Time crystals