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Physical Review A

Electromagnetically induced transparency and evolution of a two-level system under a strong chaotic field

Abraham G. Kofman

Published Feb 13, 2001• 4 citations
Publisher

Abstract

The present paper treats the electromagnetically induced transparency (EIT), i.e., the absorption spectrum of a weak probe field, near-resonant to a material transition, in the presence of a strong amplitude-phase fluctuating field which couples the upper level of the transition to another excited level. The coupling field is assumed to be chaotic (i.e., a complex Gaussian process with a Lorentzian spectrum) and so strong that the material relaxation and the detuning can be neglected. In addition, we consider the Fourier transform of the spectrum ${U}_{\mathrm{aa}}(t),$ which determines absorption of a probe field with a time-dependent intensity. The quantity ${U}_{\mathrm{aa}}(t)$ equals a diagonal element of the average reduced evolution operator of the two-level system (TLS), coupled by the strong field. As a starting point, we use the stationary, partially averaged stochastic Schr\"odinger equation, which has the form of a set of two second-order differential equations. The theory is developed by combining analytical and numerical approaches. For the most important regime, when the rms Rabi frequency ${V}_{0}$ is much greater than the coupling-field bandwidth $\ensuremath{\nu},$ we obtain scaling relations for the EIT line shape and the TLS evolution. In particular, the absorption at the minimum (i.e., at the resonance frequency) scales as ${\ensuremath{\nu}}^{1/3}{V}_{0}^{\ensuremath{-}4/3},$ whereas the evolution function ${U}_{\mathrm{aa}}(t)$ performs damped oscillations with the decay rate on the order of ${(V}_{0}^{2}\ensuremath{\nu}{)}^{1/3}.$ These results differ significantly from those following from the uncorrelated-jump model. The results of this paper are shown to be applicable to more general models of amplitude-phase fluctuations than the Gaussian-Markovian chaotic field.

DOI
10.1103/physreva.63.033810

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The present paper treats the electromagnetically induced transparency (EIT), i.e., the absorption spectrum of a weak probe field, near-resonant to a material transition, in the presence of a strong amplitude-phase fluctuating field which couples the upper level of the transition to another excited level. The coupling field is assumed to be chaotic (i.e., a complex Gaussian process with a Lorentzian spectrum) and so strong that the material relaxation and the detuning can be neglected. In addition, we consider the Fourier transform of the spectrum Uaa(t),{U}_{\mathrm{aa}}(t),Uaa​(t), which determines absorption of a probe field with a time-dependent intensity. The quantity Uaa(t){U}_{\mathrm{aa}}(t)Uaa​(t) equals a diagonal element of the average reduced evolution operator of the two-level system (TLS), coupled by the strong field. As a starting point, we use the stationary, partially averaged stochastic Schr"odinger equation, which has the form of a set of two second-order differential equations. The theory is developed by combining analytical and numerical approaches. For the most important regime, when the rms Rabi frequency V0{V}_{0}V0​ is much greater than the coupling-field bandwidth \ensuremathν,\ensuremath{\nu},\ensuremathν, we obtain scaling relations for the EIT line shape and the TLS evolution. In particular, the absorption at the minimum (i.e., at the resonance frequency) scales as \ensuremathν1/3V0\ensuremath−4/3,{\ensuremath{\nu}}^{1/3}{V}_{0}^{\ensuremath{-}4/3},\ensuremathν1/3V0\ensuremath−4/3​, whereas the evolution function Uaa(t){U}_{\mathrm{aa}}(t)Uaa​(t) performs damped oscillations with the decay rate on the order of (V02\ensuremathν)1/3.{(V}_{0}^{2}\ensuremath{\nu}{)}^{1/3}.(V02​\ensuremathν)1/3. These results differ significantly from those following from the uncorrelated-jump model. The results of this paper are shown to be applicable to more general models of amplitude-phase fluctuations than the Gaussian-Markovian chaotic field.
This paper deals with electromagnetically induced transparency (EIT), i.e., a significant suppression of absorption at a material transition (in the form of a dip in an absorption spectrum), due to a strong laser field, coupling the excited level with an auxiliary excited (unpopulated) level. A comprehensive theory of EIT is developed for the cases of coherent and Markovian phase- and amplitude-phase-fluctuating coupling fields. For a coherent coupling field, a shift of the absorption minimum from the two-photon resonance is revealed in the off-resonance case. Two models of amplitude-phase fluctuating fields are considered: the chaotic field and the uncorrelated-jump field. Closed analytical expressions for EIT line shape are derived and exact limits of different regimes of EIT are obtained, the emphasis being on the near-resonance case. The main conclusion is that an amplitude-phase-fluctuating field can induce significant transparency, though reduced in comparison to a phase-fluctuating field of the same average intensity and bandwidth. EIT decreases with the increase of the bandwidth for all stochastic models considered. EIT with a chaotic field is generally less pronounced than EIT with an uncorrelated-jump field of the same intensity and bandwidth, the difference increasing with the field intensity. The possibility of experimental verification of the results obtained is discussed.
We study the absorption spectrum of a weak probe field near resonant to an atomic transition, the upper level of which is strongly coupled to a third level by the interaction with a Lorentzian bosonic reservoir, such as, e.g., a mode of a high-QQQ cavity or a local vibration in a solid. The reservoir coupling is approximated by the interaction with a classical complex Gaussian-Markovian random process (control field), which is justified when the reservoir temperature exceeds significantly the mode frequency or when the high-QQQ cavity is pumped by broadband incoherent radiation. The present theory is applicable also when the control field is chaotic laser light. We assume that the rms control-field Rabi frequency V0{V}_{0}V0​ is much greater than the field detuning ${\ensuremath{\Delta}}_{c}$, which, in turn, is much greater than the material relaxation constants. We reveal and describe analytically all qualitatively different regimes of the spectrum modification and obtain their validity conditions. The analytical results are verified by numerical calculations using the exact continued-fraction solution. The analytical formulas obtained allow one to perform fast computer calculations for arbitrarily small values of the reservoir (control-field) bandwidth $\ensuremath{\nu}$, in contrast to the known numerical methods, which require sharply increasing computational resources with a decrease of $\ensuremath{\nu}$. In the most interesting case $\ensuremath{\nu}⪡{V}_{0}$, the spectrum consists of two peaks, the nonvanishing bandwidth and material relaxation affecting mainly the dip between the peaks. The results obtained in the static limit (i.e., a very narrow reservoir) are independent of the reservoir band shape. We reveal reservoir-induced transparency (RIT)---i.e., absorption reduction due to the reservoir coupling. Moreover, two unexpected, remarkable features are uncovered in a range of intermediate values of $\ensuremath{\nu}$ and V0{V}_{0}V0​, ${\ensuremath{\Gamma}}^{2}\ensuremath{\mid}{\ensuremath{\Delta}}_{c}\ensuremath{\mid}⪡{V}_{0}^{2}\ensuremath{\nu}⪡{\ensuremath{\mid}{\ensuremath{\Delta}}_{c}\ensuremath{\mid}}^{3}$ ($\ensuremath{\Gamma}$ is the spectral width in the absence of the control field): an extra peak in the dip and a resonance-absorption decrease by a half (i.e., an increase of RIT) relative to the static limit. The existence of the above spectral features depends on the statistical details of the control-field temporal behavior.
Summary form only given. The phenomenon of electromagnetically induced transparency (EIT) has been the subject of extensive experimental and theoretical studies in recent years, since it has a fundamental significance and a host of important applications. To produce EIT one requires a strong laser field, which often undergoes random fluctuations. The case of amplitude-phase fluctuations appears to be much more complicated and richer than the case of Markovian phase fluctuations. Here we consider transparency induced by a Gaussian-Markovian amplitude-phase fluctuating (chaotic) field. The problem involves five parameters: the coupling field rms Rabi frequency, bandwidth, detuning, and the widths of the material and coupled transitions.
The propagation of a probe field through a four-level Y-type atomic system is described in the presence of two additional coherent radiation fields, namely, the control field and the coupling field. An expression for the probe response is derived analytically from the optical Bloch equations under steady state condition to study the absorptive properties of the system under probe field propagation through an ensemble of stationary atoms as well as in a Doppler broadened atomic vapor medium. The most striking result is the conversion of electromagnetically induced transparency (EIT) into electromagnetically induced absorption (EIA) as we start switching from weak probe regime to strong probe regime. The dependence of this conversion on residual Doppler averaging due to wavelength mismatch is also shown by choosing the coupling transition as a Rydberg transition.
Relaxation of a two-level system (TLS) into a resonant infinite-temperature reservoir with a Lorentzian spectrum is studied. The reservoir is described by a complex Gaussian-Markovian field coupled to the nondiagonal elements of the TLS Hamiltonian. The theory can be relevant for electromagnetic interactions in microwave high-Q cavities and muon spin depolarization. Analytical results are obtained for the strong-coupling regime, ${\ensuremath{\Omega}}_{0}\ensuremath{\gg}\ensuremath{\nu},$ where ${\ensuremath{\Omega}}_{0}$ is the rms coupling amplitude (Rabi frequency) and $\ensuremath{\nu}$ is the width of the reservoir spectrum. In this regime, the population difference and half of the initial coherence decay with two characteristic rates: most of the decay occurs at $t\ensuremath{\sim}{\ensuremath{\Omega}}_{0}^{\ensuremath{-}1},$ the relaxation being reversible for $t\ensuremath{\ll}({\ensuremath{\Omega}}_{0}^{2}\ensuremath{\nu}{)}^{\ensuremath{-}1/3},$ whereas for $t\ensuremath{\gg}({\ensuremath{\Omega}}_{0}^{2}\ensuremath{\nu}{)}^{\ensuremath{-}1/3}$ the relaxation becomes irreversible and is practically over. The other half of the coherence decays with a rate on the order of $\ensuremath{\nu},$ which may be slower by orders of magnitude than the time scale of the population relaxation. The above features are explained by the fact that at $t\ensuremath{\ll}{\ensuremath{\nu}}^{\ensuremath{-}1},$ the reservoir temporal fluctuations are effectively one-dimensional (adiabatic). Moreover, we identify the pointer basis, in which the reduction of the state vector occurs. The pointer states depend on the initial phase of the reservoir.
A perturbation method was used to solve optical Bloch equations (OBEs) for the transition F g =1→ F e =2, in order to describe the role of ground-level Zeeman coherences in the formation of electromagnetically induced absorption (EIA). A narrow Lorentzian peak, centered at zero value of the scanning magnetic field, appears in the analytical expression of the second-order correction of a density-matrix element for ground-level Zeeman coherences, (ρ g −1 , g +1 ) x 2 . Through analytical expressions for lower-order corrections of density-matrix elements, we were able to establish clear relations between the narrow Lorentzian in (ρ g − 1 , g +1 ) x 2 and higher-order corrections of optical coherences, i.e. EIA. We see from analytical expressions that these two resonances have opposite signs and that EIA becomes electromagnetically induced transparency (EIT) in the limit of low efficiency of spontaneous transfer of coherences from excited-level to ground-level Zeeman sublevels. The transient behavior of EIA follows the time evolution of (ρ g − 1 , g +1 ) x 2 . After the coupling laser is turned on, both the Lorentzian peak in (ρ g − 1 , g +1 ) x 2 and EIA reach steady state via over-damped oscillations.