Jaynes-Cummings模型的半经典研究
A semiclassical study of the Jaynes-Cummings model
摘要：
我们考虑单量子自旋$s$与谐振子耦合的Jaynes-Cummings模型，在该模型中，经典动力学表现出不稳定的平衡点。该模型的这种状态与冷原子系统的物理学有关，在非平衡情况下，通过快速扫描通过费什巴赫共振获得。我们证明了在这个具有两个自由度的可积系统中，对于任何接近不稳定点的初始条件，经典动力学都是由焦点-焦点型奇点控制的。特别是，它显示了预期的单值性，这禁止了全局作用角坐标的存在。守恒量的联合谱的显式计算揭示了量子水平上的单值性，作为本征值晶格中的位错。我们进行了详细的半经典分析相关的本征态。而大多数的水平是很好地描述了通常的玻尔-索末菲量子化规则，适当地适应极坐标，我们展示了这些规则是如何修改的临界水平附近。计算了经典不稳定态的谱分解，发现临界WKB态占主导地位。这提供了一个有用的工具来分析量子动力学从这个特定的状态，它表现出一个非周期性的孤子脉冲序列与一个相当明确的特征频率。
英文摘要：
We consider the Jaynes-Cummings model of a single quantum spin $s$ coupled to a harmonic oscillator in a parameter regime where the underlying classical dynamics exhibits an unstable equilibrium point. This state of the model is relevant to the physics of cold atom systems, in non-equilibrium situations obtained by fast sweeping through a Feshbach resonance. We show that in this integrable system with two degrees of freedom, for any initial condition close to the unstable point, the classical dynamics is controlled by a singularity of the focus-focus type. In particular, it displays the expected monodromy, which forbids the existence of global action-angle coordinates. Explicit calculations of the joint spectrum of conserved quantities reveal the monodromy at the quantum level, as a dislocation in the lattice of eigenvalues. We perform a detailed semi-classical analysis of the associated eigenstates. Whereas most of the levels are well described by the usual Bohr-Sommerfeld quantization rules, properly adapted to polar coordinates, we show how these rules are modified in the vicinity of the critical level. The spectral decomposition of the classically unstable state is computed, and is found to be dominated by the critical WKB states. This provides a useful tool to analyze the quantum dynamics starting from this particular state, which exhibits an aperiodic sequence of solitonic pulses with a rather well defined characteristic frequency.
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