Properties of a Three-Level $Λ$-Type Atom Driven by Coherent and Stochastic Fields

TL;DR

This work develops a theoretical framework for a three-level Λ-type atom driven by a strong coherent field and a finite-bandwidth stochastic field modeled as a complex Gaussian–Markovian process. By deriving a Lindblad master equation that incorporates noise via Z_± operators, the authors analyze steady-state populations in both bare and dressed bases and compute the incoherent resonance fluorescence spectrum using the quantum regression theorem. The results show that pump noise is not merely decoherence but a controllable parameter that reshapes the dressed-state structure, broadens spectral features, and can selectively enhance or suppress transitions when the stochastic central frequency is detuned to resonance with the generalized Rabi frequency. This noise-assisted control has potential applications in robust state preparation, spectroscopy, and quantum technologies where realistic laser fluctuations are unavoidable.

Abstract

We present a theoretical investigation of a three-level $Λ$-type atom driven by a strong coherent laser and a weak stochastic field exhibiting amplitude and phase fluctuations. The stochastic field is modeled as a complex Gaussian-Markovian random process with finite bandwidth to describe realistic laser noise. Using the Born-Markov and rotating-wave approximations, we derive a Lindblad-form master equation that incorporates spontaneous emission and noise-induced terms, and we solve for the steady-state regime. We examine level populations in both the bare and dressed bases and compute the incoherent resonance-fluorescence spectrum. Our analysis shows that the stochastic drive is not merely a source of decoherence but a versatile control parameter. By detuning the stochastic-field central frequency relative to the coherent drive (especially for narrow bandwidths), we observe pronounced changes in emission characteristics, including selective enhancement or suppression, and reshaping of the multi-peaked fluorescence spectrum when the detuning matches the generalized Rabi frequency. Numerical results reveal nontrivial steady-state modifications distinct from purely coherent driving, enabling precise control of populations and suggesting applications in quantum control, quantum technologies, spectroscopy, and noise-assisted manipulation of atomic systems.

Figures (11)

• Figure 1: Schematic of the $\Lambda$-type atomic system. The transitions $\ket{g}\leftrightarrow\ket{e}$ and $\ket{s}\leftrightarrow\ket{e}$ are driven by fields with frequencies $\omega_P$ and $\omega_C$, respectively. The system is characterized by one-photon detunings $\Delta$ and $\Delta'$, two-photon detuning $\delta$, and decay/decoherence rates $\gamma_{eg}$, $\gamma_{es}$, and $\gamma_{sg}$. Transition frequencies are defined as $\omega_{ij} = \omega_i - \omega_j$ for $i,j \in \{e,g,s\}$.
• Figure 2: Panels (a) and (b) show the steady-state populations of states $\ket{g}$ and $\ket{s}$ as functions of the single-photon detuning $\eta=\omega_s-\omega_L$ for various stochastic-field strengths $D$. The Rabi frequency is $\Omega=100$. Solid lines denote $\rho_{gg}^{(st)}$, and dashed lines denote $\rho_{ss}^{(st)}$.
• Figure 3: Panels (a) and (b) show the steady-state populations of the states $\ket{g}$ and $\ket{s}$ for several values of the Rabi frequency $\Omega$ and stochastic-field strength $D$, plotted versus the frequency difference $\eta=\omega_s -\omega_L$ between the coherent and stochastic fields. As in the previous figure, $\rho_{gg}^{(st)}$ is shown with solid lines and $\rho_{ss}^{(st)}$ with dashed lines. The single-photon detuning is $\Delta=20$. Other parameters are given in the text.
• Figure 4: Panels (a) and (b) show the steady-state population of the excited state $\ket{e}$ as a function of the frequency difference $\eta=\omega_s-\omega_L$ between the coherent and stochastic fields, for several stochastic-field strengths $D$ and detunings. The Rabi frequency is $\Omega = 100$.
• Figure 5: Panels (a) and (b) show the steady-state population of the excited state $\ket{e}$ as a function of the frequency difference $\eta=\omega_s-\omega_L$ between the coherent and stochastic fields, for several stochastic-field strengths $D$ and Rabi frequencies. The single-photon detuning is $\Delta = 20$.