Dynamics of interacting bosons using the Herman–Kluk semiclassical initial value representation
Dynamics of interacting bosons using the Herman–Kluk semiclassical initial value representation
复制标题
使用 Herman-Kluk 半经典初始值表示的相互作用玻色子动力学
DOI：
10.1088/1751-8113/49/16/165303
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
W. Strunz
中科院分区：
文献类型：
--
作者：
Shouryya Ray;Paula Ostmann;L. Simon;F. Grossmann;W. Strunz
最近在光学晶格中监测超冷气体动力学的实验进展需要对大量玻色子进行定量理论描述。在本文中，我们研究了将传播子表示为相空间上的积分并使用经典轨迹的时变半经典初值方法是否适合描述集中于单模的相互作用玻色子。尽管自相互作用有非线性贡献，但相应的经典动力学允许对半经典传播子进行很大程度的解析处理。我们发现Herman-Kluk （HK）传播子的应用在半经典极限下保持了一致性，但是对于低粒子数n，范数出现了衰减。在当前系统中，即使在半经典极限下，对于不消失的相互作用强度，冻结高斯近似（FGA）（具有单位前因子的HK）也明确地表明了违反一致性。此外，我们通过在最陡下降近似下计算相空间积分，证明了HK传播子在半经典极限(n→∞？>)。但是，在倒数第二到倒数第一的顺序中会产生错误(小参数1 / n ？>)，从积分的数值计算中可以看出，并通过考虑对最陡下降计算的有限n个修正来解析地证实。相比之下，FGA仅准确到最低阶，并且在分析中发现能谱中错误的次阶项。最后，作为一个应用实例，我们通过计算Wigner函数的时间演化来研究波包的动力学。虽然经常使用的截断维格纳近似不能捕获精确量子力学解（解析已知）中存在的任何干扰，但我们发现HK方法尽管也只使用经典信息，但正确地再现了精确解的显著特征。
Recent experimental progress in monitoring the dynamics of ultracold gases in optical lattices necessitates a quantitative theoretical description for a significant number of bosons. In the present paper, we investigate if time-dependent semiclassical initial value methodology, with propagators expressed as integrals over phase space and using classical trajectories, is suitable to describe interacting bosons, concentrating on a single mode. Despite the nonlinear contribution from the self-interaction, the corresponding classical dynamics allows for a largely analytical treatment of the semiclassical propagator. We find that application of the Herman–Kluk (HK) propagator conserves unitarity in the semiclassical limit, but a decay of the norm is seen for low particle numbers n. The frozen Gaussian approximation (FGA) (HK with unit prefactor) is explicitly shown to violate unitarity in the present system for non-vanishing interaction strength, even in the semiclassical limit. Furthermore, we show by evaluating the phase space integral in steepest descent approximation, that the HK propagator reproduces the exact spectrum correctly in the semiclassical limit ( n → ∞ ?> ). An error is, however, incurred in next-to-next-to-leading order (small parameter 1 / n ?> ), as seen upon numerical evaluation of the integral and confirmed analytically by considering finite n corrections to the steepest descent calculations. The FGA, in contrast, is only accurate to lowest order, and an erroneous next-to-leading order term in the energy spectrum was found analytically. Finally, as an example application, we study the dynamics of wave packets by computing the time evolution of the Wigner function. While the often-used truncated Wigner approximation cannot capture any interferences present in the exact quantum mechanical solution (known analytically), we find that the HK approach, despite also using classical information only, reproduces the salient features of the exact solution correctly.