Discrete time crystals (DTCs) refer to a novel many-body steady state that spontaneously breaks the discrete time-translational symmetry in a periodically driven quantum system. Here, we study DTCs in a Bose-Einstein condensate bouncing resonantly on an oscillating mirror, using a two-mode model derived from a standard quantum field theory. We investigate the validity of this model and apply it to study the long-time behavior of our system. A wide variety of initial states based on two Wannier modes are considered. We find that in previous studies the investigated phenomena in the evolution time window ($⪅2000$ driving periods) are actually “short-time” transient behavior though DTC formation signaled by the subharmonic responses is still shown if the interboson interaction is strong enough. After a much longer (about 20 times) evolution time, initial states with no “long-range” correlations relax to a steady state, where time-symmetry breaking can be unambiguously defined. Quantum revivals also eventually occur. This long-time behavior can be understood via the many-body Floquet quasieigenenergy spectrum of the two-mode model. A symmetry-breaking edge for DTC formation appears in the spectrum for strong enough interaction, where all quasieigenstates below the edge are symmetry breaking while those above the edge are symmetric. The late-time steady state's time-translational symmetry depends solely on whether the initial energy is above or below the symmetry-breaking edge. A phase diagram showing regions of symmetry-broken and symmetric phases for differing initial energies and interaction strengths is presented. We find that, according to this two-mode model, the discrete time crystal survives for times out to at least $250000$ driving periods.

中文翻译：

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玻色-爱因斯坦凝聚中的离散时间晶体和简单双模理论中的对称破缺边

离散时间晶体 (DTC) 是指一种新颖的多体稳态，它自发地打破了周期性驱动的量子系统中的离散时间平移对称性。在这里，我们使用源自标准量子场论的双模式模型研究了 Bose-Einstein 凝聚体中的 DTC，该凝聚体在振荡镜上共振反弹。我们调查了该模型的有效性并将其应用于研究我们系统的长期行为。考虑了基于两种 Wannier 模式的多种初始状态。我们发现在先前的研究中，进化时间窗口中的调查现象（$⪅\mathrm{2000年}$驱动周期）实际上是“短时间”瞬态行为，尽管如果玻色子间相互作用足够强，仍会显示由次谐波响应发出的 DTC 形成信号。经过更长（大约 20 倍）的演化时间后，没有“长程”相关性的初始状态会松弛到稳定状态，此时可以明确定义时间对称性破缺。量子复兴也最终会发生。这种长时间的行为可以通过双模式模型的多体 Floquet 准特征能谱来理解。DTC 形成的对称破坏边缘出现在足够强相互作用的光谱中，其中边缘下方的所有准本征态都是对称破坏，而边缘上方的所有准本征态都是对称的。后期稳定状态' s 时间平移对称性仅取决于初始能量是高于还是低于对称破坏边缘。显示了不同初始能量和相互作用强度的对称破坏和对称相区域的相图。我们发现，根据这个双模模型，离散时间晶体至少可以存活$250000$ 驾驶期间。