## Monte Carlo studies of the intrinsic second hyperpolarizabilityPublished in JOSA B, Vol. 27 Issue 9, pp.1849-1856 (2010) |

**Spotlight summary:**The paper of Shafei et al. belongs to a series of papers initiated a decade ago by M.G. Kuzyk, oriented toward a theoretical analysis of fundamental limits and the optimization of attainable first- and second-order off-resonant electronic nonlinear hyperpolarizabilities, which describe a molecule’s response—its nonlinear polarization—to optical fields. Nonlinear hyperpolarizabilities are expressed in terms of infinite series. In the absence of a small parameter, the optimization of hyperpolarizabilities and the search for universal features constitute a challenge, which requires, first of all, a deep physical intuition.

In earlier papers, Kuzyk et al. calculated the fundamental limits for nonlinear hyperpolarizabilities using the three-state ansatz, which states that when a hyperpolarizability is near the fundamental limit, the system can be described by a three-level model. To this end they have used sum rules—one of standard approaches in quantum theories—that put constraints on infinite sets of energy levels and transition moments.

Surprisingly, experimental studies and earlier theoretical approaches have revealed a large gap between attainable values of the first and second hyperpolarizabilities and predicted fundamental limits. This casted doubt on the validity of the three-state ansatz.

To explain the origin of this discrepancy, Kuzyk et al. proposed recently an unusual approach reminiscent of the cutting the Gordian knot by Alexander the Great. Namely, instead of solving quantum mechanical problems with some à priori postulated Hamiltonians, they sample the multidimensional space of parameters (finite number of energy states and corresponding transition moments) taking into account constraints put on them by sum rules (whether the sampled parameters are attainable with standard Hamiltonians is an open and challenging question in quantum mechanics). To this end, they apply Monte Carlo method which, in general, uses sequences of random numbers to sample a representative probe of different copies of the system, aimed at the calculation of empiric distributions of random variables (hyperpolarizabilities) of interest and their moments.

The method was recently successfully applied to the study of the first hyperpolarizability and has led to the conclusion that its values can approach fundamental limits, indicating that the three-state ansatz works nicely in this case. Shafei et al. use the same strategy to the second hyperpolarizability and come to the following conclusions.

First, there exist specific sets of values of energy levels and corresponding transition moments for which the second hyperpolarizability attains values very close to the fundamental limit, thus breaking apparent experimental limits.

Second, they prove, using Monte Carlo sampling, that close to the fundamental limit of the second hyperpolarizability, the first three states dominate its value, in accordance with the three-state ansatz.

Third and last, the authors reveal some universal features associated with the second hyperpolarizability when it is close to the fundamental limit. The necessary condition for this event is formulated in a short and elegant way; it states that the first excited state must be much closer to the ground state than the second excited state. Surprisingly, this constitutes a far analogy to the effect of repulsion of levels in complex systems. The stretched exponential distributions reported in the paper indicate that the system actually belongs to the class of complex systems.

To summarize, Monte Carlo studies lead to conclusions that may stimulate inspiration for guiding the design of molecules with nonlinear hyperpolarizabilities close to the fundamental limits.

-- Antoni C. Mitus

**Technical Division:**Light–Matter Interactions

**ToC Category:**Atomic and Molecular Physics

OCIS Codes:
| (020.0020) Atomic and molecular physics : Atomic and molecular physics |

(020.4900) Atomic and molecular physics : Oscillator strengths | |

(190.0190) Nonlinear optics : Nonlinear optics |

Posted on August 20, 2010

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