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Physica Scripta

Comparison of a double-Λ atomic scheme with single- and two-fold coupled transitions

D. Arsenović, Jelena Dimitrijević

Published Apr 27, 2012• 3 citations
Publisher

Abstract

A four-level double-Λ atomic scheme, i.e. two Λ systems sharing the same two ground levels, that interacts with four laser light fields is studied theoretically. The peculiarity is that each of the two ground states can be coupled to each excited state by two laser light fields. A certain energy difference exists between excited-state levels. We consider this energy small enough so that a laser resonant to either transition can also couple the other transition. We test whether coupling of the more detuned laser is not non-negligible. Multiply connected states were also recently analyzed (and a comparison with experiment was presented), but for the simpler, two- and three-level atomic schemes (Stacey et al 2008 J. Phys. B: At. Mol. Opt. Phys. 41 085502). Theoretical treatment of the double-Λ atomic scheme is commonly done by solving optical Bloch equations (OBEs). When the rotating-wave approximation (RWA) is applied, OBEs become, by their form, a set of linear differential equations with constant coefficients. Theoretical treatment of the interaction scheme treated here leads to OBEs with coefficients that are not constant, but oscillate with time, even after RWA is applied. Under certain assumptions, the approximation can be used where the time-dependent coefficients are averaged over their periods. The method yields a new system of equations (similar to standard OBEs), but with more independent variables, and can also be solved in the usual way. The results presented here analyze the validity of this approximation by comparing the results for the double-Λ atomic scheme with single- and two-fold coupled transitions. We test whether in the limit of large-energy splitting between excited-state levels both approaches lead to similar results.

DOI
10.1088/0031-8949/2012/t149/014008

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A four-level double-Λ atomic scheme, i.e. two Λ systems sharing the same two ground levels, that interacts with four laser light fields is studied theoretically. The peculiarity is that each of the two ground states can be coupled to each excited state by two laser light fields. A certain energy difference exists between excited-state levels. We consider this energy small enough so that a laser resonant to either transition can also couple the other transition. We test whether coupling of the more detuned laser is not non-negligible. Multiply connected states were also recently analyzed (and a comparison with experiment was presented), but for the simpler, two- and three-level atomic schemes (Stacey et al 2008 J. Phys. B: At. Mol. Opt. Phys. 41 085502). Theoretical treatment of the double-Λ atomic scheme is commonly done by solving optical Bloch equations (OBEs). When the rotating-wave approximation (RWA) is applied, OBEs become, by their form, a set of linear differential equations with constant coefficients. Theoretical treatment of the interaction scheme treated here leads to OBEs with coefficients that are not constant, but oscillate with time, even after RWA is applied. Under certain assumptions, the approximation can be used where the time-dependent coefficients are averaged over their periods. The method yields a new system of equations (similar to standard OBEs), but with more independent variables, and can also be solved in the usual way. The results presented here analyze the validity of this approximation by comparing the results for the double-Λ atomic scheme with single- and two-fold coupled transitions. We test whether in the limit of large-energy splitting between excited-state levels both approaches lead to similar results.
We study a double-Λ atomic scheme that interacts with four laser light beams so that a closed loop of radiation-induced transitions is formed. When specific relations for field phases, frequencies and amplitudes are satisfied, coherent superpositions (the so-called ‘dark states’) can be formed in a double-Λ, which leads to the well-known effect of electromagnetically induced transparency (EIT). If the interaction scheme in a double-Λ system is such that a closed loop is formed, the relative phase of the laser light fields becomes very important. We analyze here the effect of the lasers' relative phase on the EIT in double-Λ configuration of levels. The theoretical study of interactions of lasers with a double-Λ atomic scheme is commonly conducted by solving the optical Bloch equations (OBEs). We use here a perturbative method for solving OBEs, where the interaction of lasers with double-Λ is considered a perturbation. An advantage of the perturbative method is that it generally produces simpler solutions, and analytical expressions can be obtained. We present analytical expressions for the lower-order corrections of the EIT signal. Our results show that the EIT by the perturbative method can be approximated by the sum of products of complex Lorentzians. Through these expressions, we see in what way the relative phase affects the overall EIT profile.
Two electromagnetic fields (the stronger called the pump and the weaker called the signal) are allowed to interact with a four-level double λ-type system with two pairs of closely spaced levels. The present four-level system could also be viewed as an example of a double resonant system. Under the Doppler-free condition and using a semiclassical formulation of atom–field interaction, we derive the optical Bloch equations involving the said four-level system coupled to two external fields. These coupled optical Bloch equations are unsolvable in closed analytical form. In order to obtain the absorptive lineshapes, we use the usual perturbation method for getting the approximate analytical solution to the coupled optical Bloch equations for the density matrix elements. Through the off-diagonal complex density matrix elements, the introduction of phase angles between the levels participating in dipole-allowed transitions is automatic. It is shown that these phases are appreciable if the applied electromagnetic field is strong. For on-resonance pump conditions, we obtain the usual Rabi splitting. The effects of two-photon resonances are clearly shown for off-resonance pump conditions.
The optical Bloch equations, which give the time evolution of the elements of the density matrix of an atomic system subject to radiation, are generalized so that they can be applied when transitions between pairs of states can proceed by more than one stimulated route. The case considered is that for which the time scale of interest in the problem is long compared with that set by the differences in detuning of the radiation fields stimulating via the different routes. It is shown that the Bloch equations then reduce to the standard form of linear differential equations with constant coefficients. The theory is applied to a two-state system driven by two lasers with different intensities and frequencies and to a three-state Λ-system with one laser driving one transition and two driving the second. It is also shown that the theory reproduces well the observed response of a cold 40Ca+ ion when subject to a single laser frequency driving the 4S1/2–4P1/2 transition and a laser with two strong sidebands driving 3D3/2–4P1/2.
The effects of counter-rotating terms on ground states (GS) of a lambda-type three-level atomic system coupled with two fields are examined. The GS, which are dark states in rotating wave approximation (RWA), can be expressed by a simple formula including the excited states. When the coupling strength is less than 0.2 of the maximum energy splitting in the atomic system, the component coefficients of excited states in the GS can be described by the formula similar as that in a two-level system and have linear relationship with their coupling constant. Further increasing the coupling strength will increase the excited components in the GS more rapidly, very different from the two-level system. OCIS codes: 270.1670. doi: 10.3788/COL201210.S12701.
We have investigated the interaction between bichromatic optical fields and atomic transitions. We have considered both the interaction of a saturating bichromatic field in the form of a 100% amplitude-modulated field resonantly interacting with a single atomic transition as well as the interaction of a pair of 100% amplitude-modulated fields with two coupled atomic transitions. In our studies of a single atomic transition resonantly excited by a 100% amplitude-modulated field we have used a fully quantum-electrodynamical analysis to model the Autler-Townes effect induced by a bichromatic optical field. We find that the effect induced by a bichromatic field is markedly different from the effect produced by a monochromatic exciting field. We have derived analytic expressions for the bichromatic Autler-Townes absorption spectrum. The absorption spectrum consists of equally spaced lines spaced by the modulation frequency. The linewidths of neighboring lines alternate as functions of the field strength. Experimentally, we observed this spectrum for both resonant and detuned 100% amplitude-modulated driving fields by monitoring the absorption of a weak monochromatic field as it was tuned about the resonance frequency of the 3²P₃/₂(F = 3, mF = 3) ↔ 4²D₅/₂(F = 4, mF = 4) transition while driving the 3²S₁/₂(F = 2, mF = 2) ↔ 3²P₃/₂(F = 3, mF = 3) transition with a saturating 100% amplitude-modulated field. We found excellent agreement between our predicted and measured spectra. We then investigated the resonant interaction between a pair of 100% amplitude-modulated fields and two atomic transitions sharing a common atomic level. We considered both a three-level atomic cascade system and a three-level atomic lambda system. We examined the effect that the relative phase between the two modulations has on the rate at which the modulated fields are absorbed. We find that the saturated time-averaged field absorption exhibits parametric resonances in both the cascade and lambda systems. The time-averaged absorption is strongly dependent upon the relative phase of modulation between the two fields and for certain phase differences the time-averaged absorption can be enhanced over the maximum saturated absorption experienced by cw fields of the same time-averaged intensities. We also find that the stationary atomic response exhibits periodic oscillations and for the cascade system periodic two-photon inversions of up to 40% can be achieved. We further find dark absorption resonances as a function of phase in the lambda system. We analyze the results from the point of view of semi-classical adiabatic dressed states.
I study the coherent interaction of a three-level atom in the lambda configuration with pump, Stoke, and anti-Stokes laser fields. The pump laser field drives transitions from the two non-degenerate lower levels to the excited state which is highly detuned for either transition. This excited state is also coupled to each of the lower states by Stokes or anti-Stokes transitions so that this system may represent a model of four-wave mixing. Assuming large detunings, the excited state is adiabatically eliminated from obtaining an effective two-level system with Raman coupling between the levels. Optical Bloch equations are obtained for this effective two-level system, and phenomenological damping terms are added for collisional dephasing of the Raman transitions. The equations are solved for the steady-state as well as by the method of adiabatic following. From these solutions we obtain the polarization.