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Physical Review A

Four-level atom dynamics and emission statistics using a quantum jump approach

S. N. Sandhya

Published Jan 22, 2007• 12 citations
Publisher

Abstract

Four-level atom dynamics is studied in a ladder system in the nine parameter space consisting of driving field strengths, detunings and decay constants, ${{\ensuremath{\Omega}}_{1},{\ensuremath{\Omega}}_{2},{\ensuremath{\Omega}}_{3},{\ensuremath{\Delta}}_{1},{\ensuremath{\Delta}}_{2},{\ensuremath{\Delta}}_{3},{\ensuremath{\Gamma}}_{2},{\ensuremath{\Gamma}}_{3},{\ensuremath{\Gamma}}_{4}}$. One can selectively excite or induce two-level behavior between particular levels of ones choice by appropriately tuning the driving field strengths at three-photon resonance. The dynamics may be classified into two main regions of interest (i) small ${\ensuremath{\Omega}}_{2}$ coupling the \ensuremath{\mid}2⟩-\ensuremath{\mid}3⟩ transition and (ii) large ${\ensuremath{\Omega}}_{2}$. In case (i) one sees two-level behavior consisting of adjacent levels and in a particular region in the parameter space, there is an intermittent shelving of the electrons in one of the two subsystems. In case (ii) the levels consist of the ground state and the upper most level. Emission statistics is studied using the delay function approach in both the cases. In case (i), the behavior of the second order correlation function ${g}^{2}(t)$, is similar to that of two-level emission for low ${\ensuremath{\Omega}}_{1}$ coupling the \ensuremath{\mid}1⟩-\ensuremath{\mid}2⟩ transition, and the correlation increases with ${\ensuremath{\Omega}}_{1}$ for smaller time delays. While, in case (ii) when, in addition, ${\ensuremath{\Omega}}_{3}$ coupling the \ensuremath{\mid}3⟩-\ensuremath{\mid}4⟩ transitionis kept low, ${g}^{2}(t)$ shows superpoissonian distribution, which may be attributed to three-photon processes.

DOI
10.1103/physreva.75.013809

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•Jan 1, 2002•1 citation
Four-level atom dynamics is studied in a ladder system in the nine parameter space consisting of driving field strengths, detunings and decay constants, \ensuremathΩ1,\ensuremathΩ2,\ensuremathΩ3,\ensuremathΔ1,\ensuremathΔ2,\ensuremathΔ3,\ensuremathΓ2,\ensuremathΓ3,\ensuremathΓ4{{\ensuremath{\Omega}}_{1},{\ensuremath{\Omega}}_{2},{\ensuremath{\Omega}}_{3},{\ensuremath{\Delta}}_{1},{\ensuremath{\Delta}}_{2},{\ensuremath{\Delta}}_{3},{\ensuremath{\Gamma}}_{2},{\ensuremath{\Gamma}}_{3},{\ensuremath{\Gamma}}_{4}}\ensuremathΩ1​,\ensuremathΩ2​,\ensuremathΩ3​,\ensuremathΔ1​,\ensuremathΔ2​,\ensuremathΔ3​,\ensuremathΓ2​,\ensuremathΓ3​,\ensuremathΓ4​. One can selectively excite or induce two-level behavior between particular levels of ones choice by appropriately tuning the driving field strengths at three-photon resonance. The dynamics may be classified into two main regions of interest (i) small \ensuremathΩ2{\ensuremath{\Omega}}_{2}\ensuremathΩ2​ coupling the \ensuremath{\mid}2⟩-\ensuremath{\mid}3⟩ transition and (ii) large \ensuremathΩ2{\ensuremath{\Omega}}_{2}\ensuremathΩ2​. In case (i) one sees two-level behavior consisting of adjacent levels and in a particular region in the parameter space, there is an intermittent shelving of the electrons in one of the two subsystems. In case (ii) the levels consist of the ground state and the upper most level. Emission statistics is studied using the delay function approach in both the cases. In case (i), the behavior of the second order correlation function g2(t){g}^{2}(t)g2(t), is similar to that of two-level emission for low \ensuremathΩ1{\ensuremath{\Omega}}_{1}\ensuremathΩ1​ coupling the \ensuremath{\mid}1⟩-\ensuremath{\mid}2⟩ transition, and the correlation increases with \ensuremathΩ1{\ensuremath{\Omega}}_{1}\ensuremathΩ1​ for smaller time delays. While, in case (ii) when, in addition, \ensuremathΩ3{\ensuremath{\Omega}}_{3}\ensuremathΩ3​ coupling the \ensuremath{\mid}3⟩-\ensuremath{\mid}4⟩ transitionis kept low, g2(t){g}^{2}(t)g2(t) shows superpoissonian distribution, which may be attributed to three-photon processes.
The absorption profile of a four-level ladder atomic system interacting with three driving fields is studied perturbatively and analytical results are presented. Numerical results where the driving field strengths are treated up to all orders are presented. The absorption features are studied in two regimes—(i) the weak middle transition coupling, i.e. Ω 2 ≪ Ω 1,3 and (ii) the strong middle transition coupling Ω 2 ≫ Ω 1,3 . In case (i), it is shown that the ground-state absorption and the saturation characteristics of the population of level |2⟩ reveal deviation due to the presence of upper level couplings. In particular, the saturation curve for the population of level |2⟩ shows a dip for Ω 1 = Ω 3 . While the populations of levels |3⟩ and |4⟩ show a maxima when this resonance condition is satisfied. Thus the resonance condition provides a criterion for maximally populating the upper levels. A second-order perturbation calculation reveals the nature of these minima (maxima). In the second case, I report two important features: (a) filtering of the Aulter–Townes doublet in the three-peak absorption profile of the ground state, which is achieved by detuning only the uppermost coupling field and (b) control of line-width by controlling the strength of the upper coupling fields. This filtering technique coupled with the control of linewidth could prove to be very useful for high resolution studies.
The statistics of spontaneous photon emission times in single atom resonance fluorescence are investigated through the radiative cascade of the dressed atom. The delay function which gives the distribution of the delays between two successive emissions is calculated for a coherent as well as an incoherent laser excitation. For a two-level atom, various signals that reflect the fluorescence intensity (average value, photon counting, fluctuation spectrum, etc.) are reviewed by this method. For a three-level atom, this approach is applied to the analysis of the recently observed phenomenon of intermittent fluorescence and quantum jumps.>
A single-atom optical clock proposed over ten years ago utilizes an amplification scheme which only recently was recognized as a novel problem in quantum statistics. The atom, a three-state system, has two coupled transitions that are driven continuously by two external fields, one being an allowed transition (1\ensuremath{\leftrightarrows}3) and the other a forbidden transition (2\ensuremath{\leftrightarrows}3) where \ensuremath{\Vert}3〉 is the lowest state. It has been argued that the weak transition, which is difficult to detect, could be monitored by the presence or absence of spontaneous emission of the strong transition. Thus, when the atom is shelved in the metastable state \ensuremath{\Vert}2〉, the strong transition is extinguished, but when the atom executes a single quantum jump (2\ensuremath{\rightarrow}3), it triggers a succession of perhaps a million quantum jumps (macroscopic quantum jumps) in the strong transition, an amplification that can be detected easily. This intuitive argument for alternating bright and dark intervals assumes, however, that the atom is always in an eigenstate. Should the atom be in a superposition state, because of coherent excitation, one could imagine that the weak transition would merely reduce the intensity of the strong transition slightly. This issue is resolved, in favor of the first intuitive argument, by calculating the photon-counting statistics, the probability W(n,T) of observing precisely n photon-counting events in a collection T in a quantum-mechanically consistent way. The results cannot be described by classical statistics. A compact analytic form is obtained for W(n,T) by considering the entire hierarchy of correlation functions where the emission interval peaks sharply about a particular n with a Poisson distribution and can be comparable in length to the darkness interval (n=0) while the other values of n display vanishingly small probabilities.
A quantum-statistical theory is developed for a single-atom, three-state quantum system where a strong (1-3) and a weak (2-3) transition are driven resonantly by an incoherent field. From a second-order correlation function of the fluorescence intensity, it is concluded that a quantum jump can occur, in the sense that the single atom can undergo large fluctuations between a state of full emission in the strong transition to a state of no emission when the electron is shelved in the metastable state (2).
This thesis investigates the properties of an atomic system exhibiting a giant Kerr nonlinearity. The atomic energy level scheme involves four energy levels. A three level A subsystem in the atom exhibits the effect of electromagnetically induced transparency (EIT), reducing the spontaneous emission noise. The fourth level leads to an ac-Stark shift of the ground state, which in turn leads to a giant, noiseless Kerr nonlinearity. Two different environments are explored. First, a system comprising of large number of atoms in an optical cavity is analysed. Detailed aspects of noise reduction in this system are investigated. In particular, strong squeezing in the quadrature in phase with the field driving the cavity mode is found, if the effective coupling of light to the atoms is strong. However, the linewidth of the predicted squeezing is found to be very narrow. This is attributed to a very steep linear susceptibility of the atomic medium. Since the widening of the squeezing window is possible only for weaker effective coupling, in turn reducing the squeezing level, a different environment is proposed. This involves a single four level atom, strongly coupled to the cavity mode. In such a strongly coupled system, the most appropriate approach is found to be that formulated in terms of polaritons – composite excitations of the 'atom-cavity molecule'. Adopting the polariton approach, nonclassical correlations in the field leaving the cavity are investigated. Strong photon antibunching is found and the effect of photon blockade predicted and described. The photon blockade effect can also be found in a system comprised of a two level atom coupled to the cavity mode, if the external driving is tuned to one of the vacuum Rabi resonances. A comparison between the two schemes is performed, and it is found that the four level scheme exhibits much better photon blockade. The reason for this is quantum interference between secondary transitions in the dressed states picture. Destructive interference cancels the transitions that would otherwise introduce a second photon into the system, hence producing a more robust photon blockade. All of these results are valid in the regime where external driving is weak. If the external driving strength is increased, the photon statistics (as measured by the zero-delay second order correlation function) changes from strong antibunching to strong bunching, over a relatively narrow range of driving strengths. The occurrence of this change can again be attributed to quantum interference. It is shown that the interference effect prevents the excitation of the composite system by a second photon, but not excitation by a two-photon transition (following the first excitation). Therefore, the third excitation manifold is excited, which then decays back to the first manifold in a two photon cascade. This two photon cascade is the source of correlated photon pairs causing an increase in the second order correlation function. The dynamics of forward scattering of light is presented, and nonclassical behaviour of the delay dependence of correlation function ('overshoots' and 'undershoots') is discussed. For the analytical treatment of this system, a method based on the polariton approach is devised, which includes the treatment of driving and damping. It is shown that this method is ideally suited to the analysis of strongly coupled systems, where only a few photons contribute to the dynamics.