Pound-Overhauser控制非门的有效哈密顿量
The effective Hamiltonian of the Pound-Overhauser controlled-NOT gate
摘要：
在基于NMR的量子计算中，已知受控非门可以通过将低功率单色射频场施加到弱耦合双自旋系统中的双峰的一个峰来实现。这在NMR光谱学中被称为Pound-Overhauser双共振。然而，与该过程相关联的“过渡”哈密顿量只是一种近似，它忽略了非共振效应，并且不能正确地预测相关联的相位因子。本文导出了旋转坐标系中自旋态演化的精确有效哈密顿量，包括单峰辐照（非跃迁）和双峰间辐照（非共振）两种情况。这些有效的哈密尔顿算子的准确性进行了验证，通过比较的密度矩阵的可观测的产品运营商组件通过模拟得到的那些通过拟合相应的实验。它进一步示出了如何在过渡和谐振领域可以用来实现受控非门完全有条件的阶段，并推导出这些阶段的解析表达式。在附录中，解析对角化的共振哈密顿量，并给出证明，在弱耦合近似下，当射频场功率与两个自旋的共振频率之差相比很小时，非共振效应可以忽略不计。
英文摘要：
In NMR-based quantum computing, it is known that the controlled-NOT gate can be implemented by applying a low-power, monochromatic radio-frequency field to one peak of a doublet in a weakly-coupled two-spin system. This is known in NMR spectroscopy as Pound-Overhauser double resonance. The ``transition'' Hamiltonian that has been associated with this procedure is however only an approximation, which ignores off-resonance effects and does not correctly predict the associated phase factors. In this paper, the exact effective Hamiltonian for evolution of the spins' state in a rotating frame is derived, both under irradiation of a single peak (on-transition) as well as between the peaks of the doublet (on-resonance). The accuracy of these effective Hamiltonians is validated by comparing the observable product operator components of the density matrix obtained by simulation to those obtained by fitting the corresponding experiments. It is further shown how both the on-transition and on-resonance fields can be used to implement the controlled-NOT gate exactly up to conditional phases, and analytic expressions for these phases are derived. In Appendices, the on-resonance Hamiltonian is analytically diagonalized, and proofs are given that, in the weak-coupling approximation, off-resonance effects can be neglected whenever the radio-frequency field power is small compared to the difference in resonance frequencies of the two spins.
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