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THREE-LEVEL ATOM INTERACTING WITH THERMAL LIGHT

RABUMA GAMACHU INJIFETA

Published Jun 29, 2026• 0 citations
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Abstract

ABSTRACT We investigate the steady-state properties and dynamics of a three-level cascade atom interacting with a thermal light field, focusing on how ther mal photon occupation affects population distributions, coherence decay, and fluorescence spectra. Using a rate-equation model and analytical ex pressions for the steady-state populations, we examine the time evolution of the atomic populations under low, intermediate, and high thermal pho ton numbers ¯n. At intermediate ¯n, a balance between spontaneous emis sion and thermal absorption emerges, yielding nonzero stable populations at all levels; conversely, for low ¯n, spontaneous emission dominates, and the system rapidly decays to the ground state. We also investigate the degradation of coherence |ρfe(t)| between the upper and intermediate lev els. At high thermal photon numbers, thermalization drives the system toward equipartition, such that the populations of all levels become nearly equal. We show that increasing thermal occupation shortens the life time of quantum superpositions and accelerates decoherence, as evidenced by the loss of |ρfe(t)|. In the steady-state fluorescence spectrum of the |e〉 → |g〉 transition, Lorentzian broadening, proportional to the effective decoherence rate, highlights the influence of thermal photons on phase co herence and spectral features. These findings have direct implications for quantum information processing: they demonstrate how thermal noise markedly degrades coherence, gate fidelity, and photon emission quality. Keywords: population distribution; three-level cascade atom; thermal light field; steady state population; rate-equation model; coherence decay. vii

DOI
10.20372/nadre:25510

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THREE-LEVEL ATOM INTERACTING WITH THERMAL LIGHT

RABUMA GAMACHU INJIFETA

ABSTRACT We investigate the steady-state properties and dynamics of a three-level cascade atom interacting with a thermal light field, focusing on how ther mal photon occupation affects population distributions, coherence decay, and fluorescence spectra. Using a rate-equation model and analytical ex pressions for the steady-state populations, we examine the time evolution of the atomic populations under low, intermediate, and high thermal pho ton numbers ¯n. At intermediate ¯n, a balance between spontaneous emis sion and thermal absorption emerges, yielding nonzero stable populations at all levels; conversely, for low ¯n, spontaneous emission dominates, and the system rapidly decays to the ground state. We also investigate the degradation of coherence |ρfe(t)| between the upper and intermediate lev els. At high thermal photon numbers, thermalization drives the system toward equipartition, such that the populations of all levels become nearly equal. We show that increasing thermal occupation shortens the life time of quantum superpositions and accelerates decoherence, as evidenced by the loss of |ρfe(t)|. In the steady-state fluorescence spectrum of the |e〉 → |g〉 transition, Lorentzian broadening, proportional to the effective decoherence rate, highlights the influence of thermal photons on phase co herence and spectral features. These findings have direct implications for quantum information processing: they demonstrate how thermal noise markedly degrades coherence, gate fidelity, and photon emission quality. Keywords: population distribution; three-level cascade atom; thermal light field; steady state population; rate-equation model; coherence decay. vii

•Jun 29, 2026•0 citations

Fluorescence intensity and squeezing in a driven three-level atom: Ladder case

Z. Ficek, Bryan J. Dalton, P. L. Knight

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.

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We analyze the efficiency of coherent population trapping (CPT) in a superposition of the ground states of three-level atoms under the influence of the decoherence process induced by a broadband thermal field. We show that in a single atom there is no perfect CPT when the atomic transitions are affected by the thermal field. The perfect CPT may occur when only one of the two atomic transitions is affected by the thermal field. In the case when both atomic transitions are affected by the thermal field, we demonstrate that regardless of the intensity of the thermal field the destructive effect on the CPT can be circumvented by the collective behavior of the atoms. An analytic expression was obtained for the populations of the upper atomic levels which can be considered as a measure of the level of thermal decoherence. The results show that the collective interaction between the atoms can significantly enhance the population trapping in that the population of the upper state decreases with an increased number of atoms. The physical origin of this feature is explained by the semiclassical dressed-atom model of the system. We introduce the concept of multiatom collective coherent population trapping by demonstrating the existence of collective (entangled) states whose storage capacity is larger than that of the equivalent states of independent atoms.

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Squeezing and trapping in three-level atoms

F. A. M. de Oliveira, Bryan J. Dalton, P. L. Knight

We discuss the phase-sensitive noise properties of fluorescence emitted by three-level atoms coherently excited by two laser fields, using both undressed and dressed uncoupled-state representations. The occurrence of steady-state squeezing is found to be highly sensitive to relaxation processes between the two-photon resonant levels. For purely radiative decay from the upper level, and no lower-state relaxation, the population is susceptible to coherent population trapping. Near two-photon resonance, the atom is optically pumped into a nonabsorbing (trapped) superposition state, which, although it reduces upper-level population, also leads to an interference cancellation of dipole moments and to suppression of squeezing. Lower-level relaxation unbalances this interference and restores the squeezing to some extent. Transient squeezing is also predicted in the time-dependent fluorescence.

Journal of the Optical Society of America B•Oct 1, 1987•31 citations
ABSTRACT We investigate the steady-state properties and dynamics of a three-level cascade atom interacting with a thermal light field, focusing on how ther mal photon occupation affects population distributions, coherence decay, and fluorescence spectra. Using a rate-equation model and analytical ex pressions for the steady-state populations, we examine the time evolution of the atomic populations under low, intermediate, and high thermal pho ton numbers ¯n. At intermediate ¯n, a balance between spontaneous emis sion and thermal absorption emerges, yielding nonzero stable populations at all levels; conversely, for low ¯n, spontaneous emission dominates, and the system rapidly decays to the ground state. We also investigate the degradation of coherence |ρfe(t)| between the upper and intermediate lev els. At high thermal photon numbers, thermalization drives the system toward equipartition, such that the populations of all levels become nearly equal. We show that increasing thermal occupation shortens the life time of quantum superpositions and accelerates decoherence, as evidenced by the loss of |ρfe(t)|. In the steady-state fluorescence spectrum of the |e〉 → |g〉 transition, Lorentzian broadening, proportional to the effective decoherence rate, highlights the influence of thermal photons on phase co herence and spectral features. These findings have direct implications for quantum information processing: they demonstrate how thermal noise markedly degrades coherence, gate fidelity, and photon emission quality. Keywords: population distribution; three-level cascade atom; thermal light field; steady state population; rate-equation model; coherence decay. vii
ABSTRACT We investigate the steady-state properties and dynamics of a three-level cascade atom interacting with a thermal light field, focusing on how ther mal photon occupation affects population distributions, coherence decay, and fluorescence spectra. Using a rate-equation model and analytical ex pressions for the steady-state populations, we examine the time evolution of the atomic populations under low, intermediate, and high thermal pho ton numbers ¯n. At intermediate ¯n, a balance between spontaneous emis sion and thermal absorption emerges, yielding nonzero stable populations at all levels; conversely, for low ¯n, spontaneous emission dominates, and the system rapidly decays to the ground state. We also investigate the degradation of coherence |ρfe(t)| between the upper and intermediate lev els. At high thermal photon numbers, thermalization drives the system toward equipartition, such that the populations of all levels become nearly equal. We show that increasing thermal occupation shortens the life time of quantum superpositions and accelerates decoherence, as evidenced by the loss of |ρfe(t)|. In the steady-state fluorescence spectrum of the |e〉 → |g〉 transition, Lorentzian broadening, proportional to the effective decoherence rate, highlights the influence of thermal photons on phase co herence and spectral features. These findings have direct implications for quantum information processing: they demonstrate how thermal noise markedly degrades coherence, gate fidelity, and photon emission quality. Keywords: population distribution; three-level cascade atom; thermal light field; steady state population; rate-equation model; coherence decay. vii
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.
We analyze the efficiency of coherent population trapping (CPT) in a superposition of the ground states of three-level atoms under the influence of the decoherence process induced by a broadband thermal field. We show that in a single atom there is no perfect CPT when the atomic transitions are affected by the thermal field. The perfect CPT may occur when only one of the two atomic transitions is affected by the thermal field. In the case when both atomic transitions are affected by the thermal field, we demonstrate that regardless of the intensity of the thermal field the destructive effect on the CPT can be circumvented by the collective behavior of the atoms. An analytic expression was obtained for the populations of the upper atomic levels which can be considered as a measure of the level of thermal decoherence. The results show that the collective interaction between the atoms can significantly enhance the population trapping in that the population of the upper state decreases with an increased number of atoms. The physical origin of this feature is explained by the semiclassical dressed-atom model of the system. We introduce the concept of multiatom collective coherent population trapping by demonstrating the existence of collective (entangled) states whose storage capacity is larger than that of the equivalent states of independent atoms.
We investigate the steady-state spontaneous emission spectrum of a three-level atom driven by two coherent fields and the absorption spectrum of a weak probe passing through a collection of such driven atoms. We find that the fluorescence spectrum is strongly affected by the decay rates of all the levels involved in the atomic evolution and not just by the decay parameters of the specific transition whose emission spectrum is being monitored. In particular, the spectral components can acquire very different widths and peak heights relative to the case of the standard resonance fluorescence in which a two-level system is driven by a single near-resonant field. An external probe signal passing through the gas of three-level atoms may be absorbed or amplified, as in the standard two-level case, but under specific operating conditions, amplification (or absorption) occurs over the entire range where the atomic response is appreciable. In this case the existence of amplification or absorption is controlled solely by the population difference between two dressed states of the system. We provide numerical results in support of our arguments for arbitrary values of the atomic and field parameters and also develop an analytic description in the limit of strong driving fields that leads to explicit line-shape and linewidth formulas.
We derive equations that describe the dynamics of three-level atoms with a cascade system of levels interacting with two resonant coherent fields in conditions where cooperative relaxation dominates over incoherent spontaneous emission. We calculate the temporal dynamics of the values of the atomic populations. The possibility of coherent population trapping in the presence of the cooperative decay is established. Finally, we calculate the averaged population of the intermediate level as a function of detuning for different values of the coupling constant.
We discuss the phase-sensitive noise properties of fluorescence emitted by three-level atoms coherently excited by two laser fields, using both undressed and dressed uncoupled-state representations. The occurrence of steady-state squeezing is found to be highly sensitive to relaxation processes between the two-photon resonant levels. For purely radiative decay from the upper level, and no lower-state relaxation, the population is susceptible to coherent population trapping. Near two-photon resonance, the atom is optically pumped into a nonabsorbing (trapped) superposition state, which, although it reduces upper-level population, also leads to an interference cancellation of dipole moments and to suppression of squeezing. Lower-level relaxation unbalances this interference and restores the squeezing to some extent. Transient squeezing is also predicted in the time-dependent fluorescence.