Cold atoms trapped in one-dimensional optical lattices and driven to the four-level N configuration are exploited for achieving an electromagnetically induced grating with parity-time-symmetry. This nontrivial grating exhibits unidirectional diffraction patterns, e.g., with incident probe photons diffracted into either negative or positive angles, depending on the sign relation between spatially modulated absorption and dispersion coefficients. Such asymmetric light diffraction is a result of the out-of-phase interplay of amplitude and phase modulations of transmission function and can be easily tuned via optical depth, probe detuning, pump Rabi frequencies, etc.

We study the synchronization behaviors of two indirectly coupled mechanical oscillators of different frequencies in a doublecavity optomechanical system. It is found that quantum synchronization is roughly vanishing though classical synchronization seems rather good when each cavity mode is driven by an external field in the absence of temporal modulations. By periodically modulating cavity detunings or driving amplitudes, however, it is possible to observe greatly enhanced quantum synchronization accompanied with nearly perfect classical synchronization. The level of quantum synchronization observed here is, in particular, much higher than that for two directly coupled mechanical oscillators. Note also that the modulation on cavity detunings is more appealing than that on driving amplitudes when the robustness of quantum synchronization is examined against the bath’s mean temperature or the oscillators’ frequency difference.

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We study the collective dynamics of a clean Floquet system of cold atoms, numerically simulating two realistic setups based on a regular chain of interacting Rydberg atoms driven by laser fields. In both cases, the population evolution and its Fourier spectrum display clear signatures of a discrete time crystal (DTC), exhibiting the appearance of a robust subharmonic oscillation which persists on a timescale increasing with the chain size, within a certain range of control parameters. We also characterize how the DTC stability is affected by dissipative processes, typically present in this atomic system even though the Rydberg state is very long lived.

By using the concept of the quantum discord (QD), we study the spin-1/2 antiferromagnetic Heisenberg chain with next-nearest-neighbor interaction. Due to the $SU(2)$ symmetry and $Z_{2}$ symmetry in this system, we obtain the analytical result of the QD and its geometric measure (GMQD), which is determined by the two-site correlators. For the 4-site and 6-site cases, the connection between GMQD (QD) and the eigenenergies was revealed. From the analytical and numerical results, we find GMQD (QD) is an effective tool in detecting the both the first-order and the infinite-order quantum-phase-transition points for the finite-size systems. Moreover, by using the entanglement excitation energy and a universal frustration measure we consider the frustration properties of the system and find a nonlinear dependence of the GMQD on the frustration.