Temperature Measurement via Time Crystal Frequencies in One-Dimensional Quantum Droplets

We propose a method for temperature measurement by analyzing the frequency of generated time crystals in one-dimensional (1D) quantum droplets. The system consists of a binary Bose-Einstein condensate mixture confined in a driven quasi-periodic optical lattice (QOL) with repulsive cubic effective mean-field and attractive quadratic beyond-mean-field interactions. By solving the 1D extended Gross-Pitaevskii equation, we derive the exact analytical wavefunction and investigate the droplet dynamics under different driving conditions. Specifically, we examine three cases: (i) constant driving frequency with linearly increasing QOL depth, (ii) constant QOL depth with linearly varying driving frequency, and (iii) constant driving frequency with sinusoidally modulated QOL depth. Fast Fourier Transform analysis reveals harmonic density oscillations, confirming time crystal formation. Additionally, we establish a non-trivial correlation between time crystal frequency and system temperature, demonstrating that an increase in time crystal frequency leads to oscillatory variations in the magnitude of the droplet's negative temperature. Finally, numerical stability analysis confirms that the obtained solutions remain robust, ensuring their feasibility for experimental realization.