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Physical Review A

Constraints on dephasing widths and shifts in three-level quantum systems

P. R. Berman, Ross C. O’Connell

Published Feb 2, 2005• 9 citations
Publisher

Abstract

It is shown that the density matrix equations for a three-level quantum system interacting with external radiation fields can lead to negative populations if arbitrary dephasing rates and shifts are included in these equations. To guarantee non-negative populations, the equations themselves impose certain restrictions on the dephasing widths and shifts. The constraints on the widths are shown to be identical to those that can be derived from a model of Markovian dephasing events, independent of any atom-field interaction.

DOI
10.1103/physreva.71.022501

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It is shown that the density matrix equations for a three-level quantum system interacting with external radiation fields can lead to negative populations if arbitrary dephasing rates and shifts are included in these equations. To guarantee non-negative populations, the equations themselves impose certain restrictions on the dephasing widths and shifts. The constraints on the widths are shown to be identical to those that can be derived from a model of Markovian dephasing events, independent of any atom-field interaction.
We study the constraints imposed on the population and phase relaxation rates by the physical requirement of completely positive evolution for open NNN-level systems. The Lindblad operators that govern the evolution of the system are expressed in terms of observable relaxation rates, explicit formulas for the decoherence rates due to population relaxation are derived, and it is shown that there are additional, nontrivial constraints on the pure dephasing rates for N>2N>2N>2. Explicit, experimentally testable inequality constraints for the decoherence rates are derived for three- and four-level systems, and the implications of the results are discussed for generic ladder, $\ensuremath{\Lambda}$, and V systems and transitions between degenerate energy levels.
Exact expressions are obtained for the density matrix in the operator form and for the quasiprobabilities for the system of an atom undergoing Raman transitions between two degenerate levels on interaction with a quantized field in a lossy cavity driven by an external field. The model is generalized to include the effects of atomic dephasing collisions. The spectral properties of the radiation are discussed, and the equation of motion of the density matrix is derived in the presence of collisions and an external drive. The equation is found to be of second order with respect to time in the absence of the driving field and also analytically solvable.
Relaxation of a three-level atom with non-equidistant spectrum interacting with a thermostat and an external stochastic field is studied. The master equation for the reduced density matrix and its solution in the first order of the perturbation theory are obtained. Using quantum regression theorem, explicit formulas for two-time correlation functions and shapes of radiation lines are derived for all types of three-level atoms and the following stochastic processes: a delta-correlated process and a Kubo-Anderson process. The influence of stochastic perturbations of atom’s energy levels and damping constants of adjacent transitions on width of radiation lines is shown.
Density matrix equations for the relaxation of a two-level system are studied. Standard dogma, Γ ik ≥ (γi + γk)/2, about the relaxation constant for the off-diagonal Γ ik ) and diagonal (γi) matrix elements is verified for the case of up- and down-transitions with high rates w 21 and w 12. Non-negativity of the density matrix imposes much weaker limitation, Γ21 ≥ (½)[(w 21)½ - (w 12)½]2. An almost ‘physical’ system is suggested, which describes to a certain approximation such decrease of transverse relaxation constant Γ.
Mathematical modeling of the population dynamics is performed for states of a three-level system (atom) with a V-type configuration transforming a light pulse. It is assumed that the excited eigenstates of the atom are degenerate and coupled by coherent interaction, one of the states being radiating (radiative), while the other state is nonradiating (“dark”). The population dynamics of atomic states is described on the basis of numerical solutions of equations for the matrix elements of the density operator. The dependence of the efficiency of population of the atomic dark state from the values of the parameters of an irradiation pulse and from the ratio of the period of population oscillations of excited atomic states (caused by their coherent interaction) to the lifetime of the atomic radiating state is determined. Typical examples of the time dependence of the population of states of the atom considered are presented for the cases of irradiation by a short (as compared to the lifetime of the radiating state) sinusoidal light pulse and by a long rectangular light pulse with the resonance carrier frequency.