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arXiv:quant-ph/0702122 · Physical Review A

Chirping a two-photon transition in a multistate ladder

Wolfgang Merkel, Holger Mack, Wolfgang P. Schleich, Eric Lutz, G. G. Paulus, B. Girard

Published Aug 20, 2007• 12 citations
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Abstract

We consider a two-photon transition in a specific ladder system driven by a chirped laser pulse. In the weak field limit, we find that the excited state probability amplitude arises due to interference of multiple quantum paths which are weighted by quadratic phase factors. The excited state population has the form of a Gauss sum which plays a prominent role in number theory.

DOI
10.1103/physreva.76.023417

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Acta Physica Sinica•Jan 1, 2025•0 citations
We consider a two-photon transition in a specific ladder system driven by a chirped laser pulse. In the weak field limit, we find that the excited state probability amplitude arises due to interference of multiple quantum paths which are weighted by quadratic phase factors. The excited state population has the form of a Gauss sum which plays a prominent role in number theory.
We propose three implementations of the Gauss sum factorization schemes discussed in part I of this series (Wölk et al 2011 New J. Phys. 13 103007): (i) a two-photon transition in a multi-level ladder system induced by a chirped laser pulse, (ii) a chirped one-photon transition in a two-level atom with a periodically modulated excited state and (iii) a linearly chirped one-photon transition driven by a sequence of ultrashort pulses. For each of these quantum systems, we show that the excitation probability amplitude is given by an appropriate Gauss sum. We provide rules on how to encode the number N to be factored in our system and how to identify the factors of N in the fluorescence signal of the excited state.
Two-photon excitation of a quantum ladder system by a chirped pulse leads to interferences in the excited-state population, due to the phase differences between direct and sequential paths. First, based on an analytical approach, we separate and discuss the features of the contributions arising from these two paths. Then interferences with two intermediate levels are observed experimentally in atomic sodium vapor. Interplay between the two sequential paths leads to strongly contrasted oscillations and a great enhancement of the transition probability for some values of the chirp.
Two-photon excitation of a quantum ladder system by an ultrashort chirped pulse leads to interferences in the excited-state population, between direct and sequential paths. Experimental results have been obtained in atomic sodium vapor. The presence of several intermediate and final states leads to new phenomena. The interplay between the two sequential paths leads to strongly contrasted oscillations and a large enhancement of the transition probability for precise values of the chirp. A cubic spectral phase modifies significantly the behavior of the quantum interferences and can be used to enhance the two-photon absorption rate.
Excitation of a quantum ladder system by frequency-swept (-chirped) laser pulses has been investigated by studying an atomic model system, the 5s-5p-5d ladder of rubidium. The population transfer is significantly enhanced when the frequency is swept such that it follows the spacing of the ladder. At intermediate intensities we observe oscillations in the population transfer to the upper level (5d), which can be attributed to interference between two routes of excitation: sequential transfer and direct two-photon excitation via a virtual intermediate state. We determine simultaneously the population transfer to the upper level (5d) and the direct three-photon ionization, the latter corresponding to a three-step ladder with the final state being in the continuum: 5s-5p-5d-\ensuremath{\varepsilon}p/\ensuremath{\varepsilon}f. Experiments were performed, using 100 fs pulses at \ensuremath{\sim}780 nm from a self-mode-locked Ti:Al2{\mathrm{Al}}_{2}Al2​O3{\mathrm{O}}_{3}O3​ oscillator amplified at 10 Hz.
Abstract The periodicity properties of Gauss sums allow us to factor integer numbers. We show that the excitation probability amplitudes of appropriate quantum systems interacting with specific laser fields are determined by Gauss sums. The resulting probabilities are experimentally accessible by measuring the fluorescence from this level. In particular, we discuss a two‐photon transition in a ladder system driven by a chirped laser pulse. In addition, we consider two realizations of laser driven one‐photon transitions. For each quantum system we demonstrate the power of this factorization scheme using numerical examples.
In general cases of strong field excitation, the Stark effect has a significant influence on transient two-photon transitions, and the analytic description of this process is quite challenging. By combining analytical solutions and numerical simulations, the transient two-photon transition processes excited by weak and strong chirped pulses are systematically investigated, showing the important influences of parameters such as light field intensity, chirp factor, and detuning on the time-domain evolution of two-photon transition probabilities. Firstly, an approximate analytical expression is derived for the amplitude of the time-domain two-photon transition probability by using the second-order perturbation theory. This analytical solution indicates that the transient two-photon transition process under weak field excitation is similar to the Fresnel rectangular edge diffraction effect. As the light field intensity increases, the influence of the Stark effect on two-photon transitions also intensifies. Secondly, through a series of approximations, the approximate analytical solutions of the Schrödinger equation under strong field interactions are obtained. The analytical solutions show that the strong field Stark effect induces energy level to split, which disrupts the symmetry of the time-domain two-photon transition probability distribution, and its frequency domain process is similar to the “double-slit interference” effect. The research results indicate that the efficiency of population transfer during strong field excitation is closely related to the light field intensity, while the chirp factor can not only regulate the efficiency and time position of population transfer but also change the oscillation frequency of the population probability in the time domain. This work offers new insights into describing the time-domain evolution of the population probability under strong field excitation and lays a scientific basis for research on two-photon microscopy imaging.