Rosen-Morse势作用下超冷原子势垒上反射的透射共振
Transmission resonances in above-barrier reflection of ultra-cold atoms by the Rosen-Morse potential
摘要：
在平均场Gross-Pitaevskii近似下，解析地研究了超冷原子在Rosen-Morse势作用下的势垒上量子反射。将无反射传输问题转化为势深度的准线性本征值问题，通过对瑞利-薛定谔含时微扰理论的修正，得到了支持入射物质波无反射传输的势高度的近似表达式.如果非线性参数与入射粒子的化学势相比很小，该近似提供了对所有共振阶的共振位置的高度精确的描述。值得注意的是，第一传输谐振的结果是精确的，即，推导出的共振势高度公式给出了在所涉及的参数、非线性参数和化学势的所有允许的变化范围内第一非线性共振位置的精确值。这已经通过构造第一共振问题的精确解来证明。此外，所提出的近似表明，在线性情况下，在非线性情况下，无反射传输不仅可能发生的势阱，但也可能发生的势垒与正的电位高度。它还表明，从相应的线性谐振的位置的谐振位置的非线性移位近似描述为谐振阶数的线性函数。最后，一个紧凑的（但，高精度）的解析公式的第n阶谐振位置构造通过解析和数值方法相结合。
英文摘要：
Quantum above-barrier reflection of ultra-cold atoms by the Rosen-Morse potential is analytically considered within the mean field Gross-Pitaevskii approximation. Reformulating the problem of reflectionless transmission as a quasi-linear eigenvalue problem for the potential depth, an approximation for the specific height of the potential that supports reflectionless transmission of the incoming matter wave is derived via modification of the Rayleigh-Schroedinger time-independent perturbation theory. The approximation provides highly accurate description of the resonance position for all the resonance orders if the nonlinearity parameter is small compared with the incoming particles chemical potential. Notably, the result for the first transmission resonance turns out to be exact, i.e., the derived formula for the resonant potential height gives the exact value of the first nonlinear resonances position for all the allowed variation range of the involved parameters, the nonlinearity parameter and chemical potential. This has been shown by constructing the exact solution of the problem for the first resonance. Furthermore, the presented approximation reveals that, in contrast to the linear case, in the nonlinear case reflectionless transmission may occur not only for potential wells but also for potential barriers with positive potential height. It also shows that the nonlinear shift of the resonance position from the position of the corresponding linear resonance is approximately described as a linear function of the resonance order. Finally, a compact (yet, highly accurate) analytic formula for the n-th order resonance position is constructed via combination of analytical and numerical methods.
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