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 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.
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.
In a recent experiment [ DevoeR. G.BrewerR. G., Phys. Rev. Lett.50, 1269 ( 1983)], it was found that the optical Bloch equations could not satisfactorily explain the signal that was observed for free-induction decay in the impurity ion cyrstal Pr3+:LaF3. Several theories have been proposed to explain this failure of the Bloch equations. In this paper, the general validity conditions for the optical Bloch equations are examined within the limits of a Markovian relaxation model. The specific problem to be considered is the interaction of an optical field with two-level atoms. The atoms undergo relaxation as a result of coupling to a perturber bath that, itself, is negligibly affected by the relaxation process. First, a simple decay-parameter model is assumed for the relaxation of atomic density-matrix elements. Such a model is found to lead to a set of generalized Bloch equations of which the conventional Bloch equations form a subset. Subsequently, more-realistic models for relaxation in both vapors and solids are considered within the limits of the impact approximation (i.e., the duration of a fluctuation can be viewed as instantaneous with respect to all relevant time scales in the problem). It is found that, even in the impact (Markovian) approximation, the generalized Bloch equations cannot be expected to provide an adequate description of relaxation, owing to effects in which a fluctuation-induced change in the atomic transition frequency persists between fluctuations. In vapors this frequency shift is produced by velocity-changing collisions that change the atomic resonance frequency (as seen in the laboratory frame), whereas in solids it is produced by local-field fluctuations. The limiting conditions under which one can expect both the generalized and the conventional Bloch equations to retain their validity are explored.
We study phenomenon of electromagnetically induced absorption in the Hanle configuration by solving time-dependent optical Bloch equations for the case of the closed multilevel F g = 1 → F e = 2 transition.Our model gives optical Bloch equations as a non-homogeneous system of ordinary linear differential equations.For weak laser fields (Ω Γ i.e.Rabi frequency small compared to spontaneous emission rate), a perturbative method to solve linear differential equations can be applied.Perturbative method is realized by solving (in the time-domain) higher-order corrections to the density matrix which in the sum converge to the exact solution of optical Bloch equations.By its form, each successive correction is also system of ordinary linear differential equations which depends on the solution of previous ones.Corrections are partitioned such that odd give corrections to optical coherences, while even give corrections to populations and Zeeman coherences.We present numerical results for the behavior of density matrix elements with successive corrections, and compare them with exact solution of optical Bloch equations.Electromagnetically induced absorption is observed as a 4th and higher (even) correction to populations, when behavior in respect to both time and magnetic field is viewed.Since in our method each correction depends on the solution of previous ones, we can analyze how (through mechanism of transfer of coherences and transfer of populations between Zeeman sublevels) electromagnetically induced absorption is formed.We also discuss qualitative differences in the behavior (with respect to time) of certain density matrix elements for magnetic fields "inside" and "outside" electromagnetically induced absorption resonance.
Recent experiments with single atomic ions isolated in electromagnetic traps have demonstrated abrupt changes in observed fluorescence intensities, which have been interpreted as evidence for quantum jumps as the atom changes energy levels. This presents the seeming paradox of discontinuously changing observables from a system whose dynamical evolution is governed by the continuous optical Bloch equations. In this paper I present an analysis of the dynamics of a model three-level atom interacting with two laser fields in terms of the dressed states of the atom-laser system. These dressed states are eigenstates of a Hamiltonian which includes laser-atom interactions, and thus they simplify the discussion of the time evolution of the combined laser-atom system. This dressed-state description is particularly useful in the regime in which both laser intensities are strong enough to saturate their respective transitions. I calculate the radiative lifetimes of the different species of dressed states and the branching ratios for decay to other dressed states, and demonstrate the consequences of these quantities for the observed fluorescence pattern. The calculated duration of bright and dark periods shows a pronounced dependence on the detunings of the lasers.
In this paper we report on the study of two and multi-level atoms interacting with multiple laser beams. The semi-classical approach is used to describe the system in which the atoms are treated quantum mechanically via the density matrix operator, while the laser beams are treated classically using Maxwells equations. We present results of a two level atom interacting with single and multiple laser beams and demonstrate Rabi oscillations between the levels. The effects of laser modulation on the dynamics of the atom (atomic populations and coherences) are examined by solving the optical Bloch equations. Plots of the density matrix elements as a function of time are presented for various parameters such as laser intensity, detuning, modulation etc. In addition, phase-space plots and Fourier analysis of the density matrix elements are provided. The atomic polarization, estimated from the coherence terms of the density matrix elements, is used in the numerical solution of Maxwells equations to determine the behaviour of the laser beams as they propagate through the atomic ensemble. The effects of saturation and hole-burning are demonstrated in the case of two counter propagating beams with one being a strong beam and the other being very weak. The above work is extended to include four-wave mixing in four level atoms in a diamond configuration. Two co-propagating beams of different wavelengths drive the atoms from a ground state |1〉 to an excited state |3〉 via an intermediate state |2〉. The atoms then move back to the ground state via another intermediate state |4〉, resulting in the generation of two additional correlated photon beams. The characteristics of these additional photons are studied.
The coupled ion pairs were modeled by two coupled two-level systems. The resulting Bloch equations were derived by the density-matrix formalism. They were solved numerically for identical ions and for slightly detuned ions with identical wave functions as a first approximation to the problem of inhomogeneous line broadening in solids. The following results were found: (1) intrinsic optical bistability as a function of the applied laser field; (2) regions of stability, instability, and multistability for different detunings and couplings of the ions at different laser fields; (3) asymmetric shapes of the absorption lines with unstable regions; and (4) suppression of the photon echo due to the nonlinear chaotic dynamics of the coupled ions. Experimental evidence is given which supports the computer calculations. \textcopyright{} 1996 The American Physical Society.