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Journal of Optics B Quantum and Semiclassical Optics

The damped and coherently-driven Jaynes$ndash$Cummings model

S. Swain, Z. Ficek

Published Aug 1, 2002• 5 citations
Publisher

Abstract

We investigate the influence of a single-mode cavity on the Autler–Townes doublet that arises when a three-level atom is strongly driven by a laser field tuned to one of the atomic transitions and probed by a tunable, weak field coupled to the other transition. We assume that the cavity mode is coupled to the driven transition and the cavity and laser frequencies are equal to the atomic transition frequency. We find that the Autler–Townes spectrum can have one, two or three peaks depending on the relative magnitudes of the Rabi frequencies of the cavity and driving fields. We show that, in order to understand the three-peaked spectrum, it is necessary to go beyond the secular approximation, leading to interesting quantum interference effects. We find that the positions and relative intensities of the three spectral components are affected strongly by the atom–cavity coupling strength g and the cavity damping κ. For an increasing g and/or decreasing κ the triplet evolves into a single peak. This results in 'undressing' of the system such that the atom collapses into its ground state. We interpret the spectral features in terms of the semiclassical dressed-atom model, and also provide complementary views of the cavity effects in terms of quantum Langevin equations and the fully quantized, 'double-dressing' model.

DOI
10.1088/1464-4266/4/4/316

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The damped and coherently-driven JaynesndashndashndashCummings model
We investigate the influence of a single-mode cavity on the Autler–Townes doublet that arises when a three-level atom is strongly driven by a laser field tuned to one of the atomic transitions and probed by a tunable, weak field coupled to the other transition. We assume that the cavity mode is coupled to the driven transition and the cavity and laser frequencies are equal to the atomic transition frequency. We find that the Autler–Townes spectrum can have one, two or three peaks depending on the relative magnitudes of the Rabi frequencies of the cavity and driving fields. We show that, in order to understand the three-peaked spectrum, it is necessary to go beyond the secular approximation, leading to interesting quantum interference effects. We find that the positions and relative intensities of the three spectral components are affected strongly by the atom–cavity coupling strength g and the cavity damping κ. For an increasing g and/or decreasing κ the triplet evolves into a single peak. This results in 'undressing' of the system such that the atom collapses into its ground state. We interpret the spectral features in terms of the semiclassical dressed-atom model, and also provide complementary views of the cavity effects in terms of quantum Langevin equations and the fully quantized, 'double-dressing' model.
We study the Autler-Townes absorption spectrum of the upper and intermediate levels of a three-level cascade atom in which the intermediate and ground levels are strongly driven by a laser field as well as weakly coupled by a cavity mode [Phys. Rev. 100, 703 (1955)]. Surprisingly, one of the Autler-Townes peaks may be greatly suppressed or enhanced by tuning the cavity resonant frequency. The larger the Rabi frequency, the more pronounced the cavity-induced suppression or enhancement. The Autler-Townes spectrum carries full information on the cavity-induced modification of the strongly driven levels. A physical interpretation is presented in the dressed-state basis.
We calculate the probe spectrum for multilevel atoms in a damped, weakly driven cavity supporting two degenerate orthogonally polarized modes. One mode is weakly driven by a linearly polarized external field. The atoms, initially prepared in a mF=0{m}_{F}=0mF​=0 hyperfine ground state, couple to the driven mode by making a $\ensuremath{\Delta}{m}_{F}=0$ transition and to the orthogonally polarized mode by making $\ensuremath{\Delta}{m}_{F}=\ifmmode\pm\else\textpm\fi{}1$ transitions to other hyperfine sublevels. We compare probe spectra, the intensity of the driven mode, the undriven mode, and the atomic excitation probability as a function of the detuning of the driving laser from the cavity resonance for three- and four-level atomic models. In both cases the spectrum of the driven mode is a vacuum Rabi doublet familiar from the driven, damped Jaynes-Cummings model. The undriven mode spectrum is a triplet with the maximum on resonance for the three-level model while the four-level model has a four-peaked spectrum. We discuss the role of strong coupling to the undriven mode in accounting for the qualitative difference between the models.
A cavity-modified master equation is derived for a coherently driven two-level atom coupled to a single-mode cavity in the bad cavity limit, in which the cavity frequency is tuned to either the center or one of the sidebands of the Mollow triplet. The atomic populations in both the bare- and dressed-state representations are analyzed in terms of the cavity-modified transition rates. In the bare-state basis, the role of the cavity may be interpreted as enhancing the stimulated absorption of the atom while suppressing the stimulated emission. The bare-state population may thus be inverted under appropriate conditions. The dressed-state inversion, however, originates from the enhancement of the atom-cavity interaction when the cavity is resonant with the atomic dressed-state transition. We show that two-phase quadratures of the atomic polarization decay at different rates. The decay of the in-phase (or out-of-phase) quadrature may be greatly inhibited as the driving intensity increases, depending on the cavity resonant frequency. The spectrum of the atomic fluorescence emitted out the side of the cavity is also studied. The spectral profiles are sensitive to the cavity frequency. When the cavity frequency is tuned to the center of the Mollow resonances, the fluorescence spectrum is symmetrical with three peaks whose linewidths and heights are intensity dependent. When the cavity frequency is tuned to one of the Mollow sidebands, however, it is asymmetric, and the central peak and the sideband on resonance with the cavity can be significantly suppressed for strong driving fields. All three spectral lines can be narrowed by increasing the Rabi frequency. The physics of these striking spectral features is explored in the dressed-state basis. We also investigate the probe absorption spectrum. When the cavity frequency is tuned to the center of the Mollow fluorescence triplet, the central component exhibits a Lorentzian line shape, while the side bands show the Rayleigh-wing line shape. Probe gain may occur at line center due to cavity-induced, bare-state population inversion. When the cavity is tuned to resonance with one sideband, only two sidebands, with Lorentzian profiles, dominate. Gain can occur at the sideband far off resonant with the cavity. The sideband gain is a consequence of an unbalanced dressed-state population distribution.
We establish a connection between the simple harmonic oscillator and a two-level atom interacting with resonant, quantized cavity and strong driving fields, which suggests an experiment to measure the harmonic-oscillator's probability distribution function. To achieve this, we calculate the Autler-Townes spectrum by coupling the system to a third level. We find that there are two different regions of the atomic dynamics depending on the ratio of the Rabi frequency ${\ensuremath{\Omega}}_{c}$ of the cavity field to that of the Rabi frequency $\ensuremath{\Omega}$ of the driving field. For ${\ensuremath{\Omega}}_{c}<\ensuremath{\Omega}$ and moderate coupling of the transition to the cavity mode the spectral peaks are composed of multiplets. A quantized dressed-atom approach provides a simple explanation of the spectral features and shows that the oscillations in the spectral components arise from the oscillations of the population distribution in the dressed states. The observation of these features would provide evidence for the quantum nature of the cavity field. The distribution is an analog of the harmonic-oscillator probability distribution function, and should be experimentally observable. For ${\ensuremath{\Omega}}_{c}>~\ensuremath{\Omega}$ there is no Autler-Townes splitting and the spectrum is composed of a single peak located at the frequency of the probe transition. We show that this effect results from the collapse of the atom to the ground state, which has been predicted by Alsing, Cardimona, and Carmichael [Phys. Rev. A 45, 1793 (1992)] for a two-level atom in a lossless cavity.
The eigenstates of the Jaynes-Cummings Hamiltonian (the dressed atomic states) are widely used to describe the interaction between a single two-level atom and a single mode of the electromagnetic field. They provide a particularly useful basis when the secular approximation is justified. For an atom interacting fog with n photons in a cavity, the secular approximation requires that the n-photon Rabi frequency 2 n g be much larger than the atomic and cavity linewidths. Normally this condition is met when n is large and the coupling constant g is much smaller than the atomic and cavity linewidths. In this paper we allow g to be larger than the linewidths, and we allow the photon number to be small. We show that when the cavity is driven by a coherent field, the usual secular approximation cannot be used. The natural basis states become are now the eigenstates of the Jaynes-Cummings Hamiltonian plus the external field interaction. We solve this eigenvalue problem by separating the field and atomic parts of the Hamiltonian using a displacement and squeeze in the Hilbert-space of the field. We find that the Rabi frequencies are renormalized downwards as the strength of the driving field is increased. They eventually vanish at a critical field strength. At this critical field strength we observe a spontaneous symmetry breaking in the steady-state solution of the master equation for a single atom interacting with a driven cavity mode.
A dressed-state rate-equation model is used to describe the Autler-Townes effect that is produced by a resonant 100% amplitude-modulated field driving a two-level resonance. The purpose of using this model is to give an intuitive interpretation of the rather complex absorption spectra that are produced when a weak probe field couples this strongly driven transition to a third atomic state. We obtain analytic expressions for the positions, strengths, and widths of the components of the absorption spectra, which are in excellent agreement with experimentally obtained spectra presented in the accompanying paper [S. Papademetriou, following paper, Phys. Rev. A 53, 997 (1996)]. \textcopyright{} 1996 The American Physical Society.