基本极限的多态灾变悖论与三态猜想
The paradox of the many-state catastrophe of fundamental limits and the three-state conjecture
摘要：
非线性极化率基本极限的计算假定，当量子系统在基本极限处具有非线性响应时，只有三个能量本征态对第一和第二超极化率有贡献。这就是所谓的三能级假设，也是基本极限理论中唯一未被证明的假设。所有的计算都是基于薛定谔方程的直接解，得到小于0.709的内禀超极化率和小于0.6的内禀二阶超极化率。在这项工作中，我们表明，放宽三能级的animals，并允许任意数量的国家作出贡献，导致发散的优化的内在超极化率在无限数量的国家的限制-我们称之为多状态灾难。这并不奇怪，因为发散系统很可能无法从薛定谔方程导出，但遵守求和规则。求和规则是极限理论的第二个组成部分，也适用于具有更一般哈密顿量的系统。这些奇异的哈密顿量可能无法模拟自然界中的任何真实系统。事实上，一类来自求和规则的跃迁矩和能量没有相应的可以用微分形式表示的哈密顿量。在这项工作中，我们表明，三个层次的antennas作为一个约束，排除了许多非物理的哈密顿量，并防止从发散的内在超极化率。我们认为，这意味着真正的基本面极限比之前计算的要小。由于三能级不等式并不导致最大可能的非线性响应，与其断言相反，我们提出了一种有趣的可能性，即三能级不等式对于任何服从薛定谔方程的系统都是正确的，但这种断言可能是不可证明的。
英文摘要：
The calculation of the fundamental limits of nonlinear susceptibilities posits that when a quantum system has a nonlinear response at the fundamental limit, only three energy eigenstates contribute to the first and second hyperpolarizability. This is called the three-level ansatz and is the only unproven assumption in the theory of fundamental limits. All calculations that are based on direct solution of the Schrodinger equation yield intrinsic hyperpolarizabilities less than 0.709 and intrinsic second hyperpolarizabilities less than 0.6. In this work, we show that relaxing the three-level ansatz and allowing an arbitrary number of states to contribute leads to divergence of the optimized intrinsic hyperpolarizability in the limit of an infinite number of states - what we call the many-state catastrophe. This is not surprising given that the divergent systems are most likely not derivable from the Schrodinger equation, yet obey the sum rules. The sums rules are the second ingredient in limit theory, and apply also to systems with more general Hamiltonians. These exotic Hamiltonians may not model any real systems found in nature. Indeed, a class of transition moments and energies that come form the sum rules do not have a corresponding Hamiltonian that is expressible in differential form. In this work, we show that the three-level ansatz acts as a constraint that excludes many of the nonphysical Hamiltonians and prevents the intrinsic hyperpolarizability from diverging. We argue that this implies that the true fundamental limit is smaller than previously calculated. Since the three-level ansatz does not lead to the largest possible nonlinear response, contrary to its assertion, we propose the intriguing possibility that the three-level ansatz is true for any system that obeys the Schrodinger equation, yet this assertion may be unprovable.
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