Fig. 4: discrete time crystal (DTC) and discrete time quasi-crystal (DTQC) oscillations.
From: Signatures of discrete time-crystallinity in transport through an open Fermionic chain
Time evolution of \(\langle {S}_{j}^{y}(t)\rangle\) [j = 1 (blue solid lines), j = 2 (orange dot-dashed lines), j = 3 (green dashed lines)] as a function of the dimensionless time t/T showing the appearance of DTC in the long-time limit for three values of system-lead couplings, (a) γ = 10−5ω, (b) γ = 10−3ω, and (c) γ = ω/50. Black dotted line represents \(\cos (\frac{2\pi t}{T})\), the driving period. d–f Discrete Fourier transform (DTF) \({{{{{{{\mathcal{F}}}}}}}}(\langle {S}_{3}^{y}\rangle )\) of \(\langle {S}_{3}^{y}(t)\rangle\) as a function of the frequency domain variable f/ω corresponding to (a–c), respectively, showing the λ = 2ω/3. Time evolution of \(\langle {S}_{j}^{y}(t)\rangle\) (j = 1, 2, 3) as a function of the dimensionless time t/T showing the appearance of DTQC in the long-time limit for three values of system-lead couplings, g γ = 10−5ω, (h) γ = 10−3ω, and i γ = ω/50. j–l DFT of \(\langle {S}_{3}^{y}(t)\rangle\) as a function of the frequency domain variable f/ω corresponding to (g–i), respectively, showing the \(\lambda =(\sqrt{2}-1)\omega\). There exists another frequency at an integer multiple of the DTC (and DTQC) frequency in the form of a tiny peak in DTF above ω.
