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We systematically study the localization effect in discrete-time quantum walks on a honeycomb network and establish the mathematical framework. We focus on the Grover walk first and rigorously derive the limit form of the walker's state, showing it has a certain probability to be localized at the starting position. The relationship between localization and the initial coin state is concisely represented by a linear map. We also define and calculate the average probability of localization by generating random initial states. Further, coin operators varying with positions are considered and the sufficient condition for localization is discussed. We also similarly analyze another four-state Grover walk. Theoretical predictions are all in accord with numerical simulation results. Finally, our results are compared with previous works to demonstrate the unusual trapping effect of quantum walks on a honeycomb network, as well as the advantages of our method.

A discrete time crystal (DTC) repeats itself with a rigid rhythm, mimicking a ticking clock set by the interplay between its internal structures and an external force. Discrete time crystals promise profound applications in precision timekeeping and other quantum techniques. However, it has been facing a grand challenge of thermalization. The periodic driving supplying the power may ultimately bring DTCs to thermal equilibrium and destroy their coherence. Here we show that an all-to-all interaction delivers a DTC that evades thermalization and maintains quantum coherence and quantum synchronization regardless of spatial inhomogeneities in the driving field and the environment. Moreover, the sensitivity of this DTC scales with the total particle number to the power of 3/2, realizing a quantum device of measuring the driving frequency or the interaction strength beyond the Heisenberg limit. Our work paves the way for designing nonequilibrium phases with long coherence time to advance quantum metrology.

Particle-wave duality has allowed physicists to establish atomic interferometers as celebrated complements to their optical counterparts in a broad range of quantum devices. However, interactions naturally lead to decoherence and have been considered as a longstanding obstacle in implementing atomic interferometers in precision measurements. Here, we show that interactions lead to dynamical quantum phase transitions between Schrödinger's cats in an atomic interferometer. These transition points result from zeros of Loschmidt echo, which approach the real axis of the complex time plane in the large particle number limit, and signify pair condensates, another type of exotic quantum states featured with prevailing two-body correlations. Our work suggests interacting atomic interferometers as a new tool for exploring dynamical quantum phase transitions and creating highly entangled states to beat the standard quantum limit.

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