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arXiv:2503.16772 · Physical Review A

Two-photon resonance fluorescence in a three-level ladder-type atom

Jacob Ngaha, A. S. Parkins, H. J. Carmichael

Published Sep 9, 2025• 1 citation
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Abstract

Two-photon transitions in ladder systems have been well-studied for decades, but the field is experiencing a resurgence due to interest in quantum dots and artificial atoms. Here, the authors study the experimentally relevant case of atoms with a three-level ladder structure driven by a coherent field close to two-photon resonance.
DOI
10.1103/3lrt-ygwp

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We theoretically investigate resonance fluorescence of a ladder- and a triangular-type three-level atomic system in the full quantization regime. By solving Heisenberg’s quantum kinetic equations of motion of an electron interacting with photons and two coherent laser fields, we obtain the firstorder electric field correlation function, which corresponds to photoluminescence spectrum. When the single electrons in both systems are resonantly excited by two strong cw-laser fields, in contrast to the ladder-type atomic systems emitting mainly incoherent photons, the triangular-type system shows that continuous coherent photons can be selectively generated from three optical transitions by trapping electronic population inversions and manipulated by changing the transition rates.
Multiphoton emitters are a sought-after resource in quantum photonics. Nonlinear interactions between a multilevel atomic system and a coherent drive can lead to resonant two-photon emission, but harvesting light from this process has remained a challenge due to the small oscillator strengths involved. Here we present a study of two-photon resonance fluorescence at microwave frequencies, using a superconducting, ladder-type artificial atom, a transmon, strongly coupled to a waveguide. We drive the two-photon transition between the ground and second-excited states at increasingly high powers and observe a resonance fluorescence peak whose intensity becomes comparable to single-photon emission until it splits into a Mollow-like triplet. We measure photon correlations of frequency-filtered spectral lines and find that while emission at the fundamental frequency stays antibunched, the resonance fluorescence peak at the two-photon transition is superbunched. Our results provide a route towards the realization of multiphoton sources in the microwave domain.
We consider a three-level atomic system interacting with two coherent fields in the ladder configuration. Namely, the pumping field couples the ground and first excited state while the driving field interacts with the two excited states. The intensity spectrum of resonant optical fluorescence at the two-photon transition is studied. It is shown that coherent driving in this scheme allows for efficient control of fluorescence distribution between the one-photon (elastic with respect to pumping field) and two-photon (anti-Stoke Raman with respect to the driving field) channels.
A theoretical study is made of the steady-state intensity and squeezing properties of the fluorescent light from a three-level-atom system in a ladder configuration, which is subject to spontaneous-emission decay to the electromagnetic-field vacuum. Two cases are examined: in the equispaced (ES) level case, the two atomic transition frequencies are nearly equal and two photon absorption processes occur from a single coherent laser field that couples both transitions and has a small detuning from each; in the non- equispaced (NES) level case, the atomic transition frequencies are rather different and two photon absorption processes occur from two coherent laser fields, each coupled to a single transition and in near resonance with it. In both cases, a situation of small two-photon detuning occurs. Optical Bloch equations for the atomic density matrix in rotating frames are given and matrix expressions for determining the steady-state populations and coherences are obtained. Analytic expressions are also given for special cases. The fluorescent intensity is obtained from the populations of the intermediate and upper states, with the normally ordered variance (NOV) for quadrature components of the fluorescent field involving, in addition, the atomic coherences. The fluorescent intensities are shown graphically as a function of two-photon detuning for a variety of one-photon detunings and Rabi frequencies for the ES and NES cases.In both cases, the fluorescent intensities show the well known resonances from the upper state at zero two-photon detuning and from the intermediate state at zero one-photon detuning for the lower transition. However, further resonances are also found. The intermediate-state intensity displays a resonance at zero two-photon detuning. Also, the upper-state intensity shows a resonance at zero one-photon detuning for the upper transition, but only for the NES case; evidently, the transfer rate from upper to lower transition coherences destroys this resonance for the ES case. The time-averaged NOV is also shown graphically as a function of two-photon detuning for a variety of one-photon detunings and Rabi frequencies for the ES and NES cases. For the ES case, the quadrature frequency is chosen as the single laser frequency and for the NES case, it is chosen as either the average of the two laser frequencies or the lower transition laser frequency. Squeezing occurs near zero two-photon detuning for both ES and NES cases, though only for the quadrature frequency equal to the average laser frequency in the latter case. It also occurs near zero detuning for the lower transition for both cases, though only for the quadrature frequency equal to the lower transition laser frequency in the NES case. The squeezing minima show a splitting effect for large Rabi frequencies corresponding to two-photon or one-photon Rabi splitting of the dressed atom levels for the situation near two-photon or one-photon resonance, respectively. The large squeezing near two-photon resonance for moderate Rabi frequencies and large one-photon detuning corresponds to essentially pure three-level squeezing, while that near one-photon resonance for weak Rabi frequencies involves two-level squeezing.
Resonance fluorescence and photon correlations are areas that have long been studied, and are integral to the field of quantum optics. Recently frequency filtered photon correlations and the method of frequency filtering have had a surge of interest [1]–[4]. In this work we develop a novel method of calculating frequency filtered photon correlations by coupling the atomic fluorescence - in this case, from a three-level ladder type atom [5], [6] - into an array of single-mode cavity filters. This coupling, modelled with an open cascaded systems approach [7], [8] is entirely unidirectional, allowing us to derive a closed set of coupled moment equations for the filter source operators. This provides an extremely efficient method for calculating frequency filtered second-order correlation functions.