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Physical Review A

Dark-state evolution and self-phase modulation in a Λ medium

V. O. Chaltykyan, G. G. Grigoryan, G. Nikogosyan

Published Jul 28, 2003• 14 citations
Publisher

Abstract

We consider the propagation of two laser pulses in a medium of three-level $\ensuremath{\Lambda}$ atoms with nonvanishing two-photon detuning and different oscillator strengths of the transitions. We show that the two-photon detuning and the derivative of the mixing angle are described by one and the same equation. We analyze the solution of this equation and its physical consequences. In particular, we obtain the conditions and the distance where the nonadiabaticity and the two-photon detuning may be reduced. We derive a simple expression for the maximum length to which excitation in the medium may reach.

DOI
10.1103/physreva.68.013819

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We consider the propagation of two laser pulses in a medium of three-level \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ atoms with nonvanishing two-photon detuning and different oscillator strengths of the transitions. We show that the two-photon detuning and the derivative of the mixing angle are described by one and the same equation. We analyze the solution of this equation and its physical consequences. In particular, we obtain the conditions and the distance where the nonadiabaticity and the two-photon detuning may be reduced. We derive a simple expression for the maximum length to which excitation in the medium may reach.
The spatial and temporal evolution is studied of two powerful short laser pulses having different wavelengths and interacting with a dense three-level \ensuremath{\Lambda}-type optical medium under coherent population trapping. The general case of unequal oscillator strengths of the transitions is considered. The durations of the probe pulse and the coupling pulse T1,2(T2>T1){T}_{1,2} {(T}_{2}>{T}_{1})T1,2​(T2​>T1​) are assumed to be shorter than any of the relevant atomic relaxation times. We propose analytical and numerical solutions of a self-consistent set of coupled Schr\"odinger equations and reduced wave equations in the adiabatic limit taking account of the first nonadiabatic correction. The adiabaticity criterion is also discussed taking account of pulse propagation. The dynamics of propagation is found to be strongly dependent on the ratio of the transition oscillator strengths. It is shown that the envelopes of the pulses slightly change throughout the medium length in the initial stage of propagation. This distance can be large compared to the one-photon resonant absorption length. Eventually, the probe pulse is completely reemitted into the coupling pulse during propagation. An effect of localization of the atomic coherence was observed similar to the one predicted by Fleischhauer and Lukin [Phys. Rev. Lett. 84, 5094 (2000)].
The possibility of formation of a coherent dark state in the five-level system in the entire volume of the medium in an adiabatic pulse propagation is investigated. It is shown that the dark state formation is independent on the value of the first two-photon detuning from the resonance, which may be changed during the propagation in the medium. In the case of the M-type system with the equal strengths of oscillators at the extreme transitions it is shown that the self-phase modulation does not influence the coherent effects while in the ladder-type system it may lead to the destruction of the dark state. The estimation of the length of the medium in which the propagation effects are negligible is obtained.
We present a theory of a continuous-wave light propagation in a medium of atoms with a double-$\ensuremath{\Lambda}$ configuration of levels. This is a configuration with a closed cycle of radiation-induced transitions. An interference of excitation channels in such closed-loop systems leads to a strong dependence of the atomic state on the relative phase and the relative amplitudes of applied electromagnetic waves. Therefore, the medium response may be controlled by the phases. On the other hand, the phases themselves change during the propagation. Thus the state of the medium and all the field parameters are tightly coupled to each other in the present problem. We consider the propagation of four-frequency laser radiation through the double-$\ensuremath{\Lambda}$ medium for two situations. At resonant or near-resonant excitation of atoms, both the medium and the field evolve into a nonabsorbing state. This state implies specific coherent superposition for atoms (``dark state''), and particular relations for the field phases, amplitudes, and frequencies. In this way, the propagation results in the phase, amplitude, and frequency matching of the laser waves. In the second case, one $\ensuremath{\Lambda}$ system in double-$\ensuremath{\Lambda}$ atoms is excited resonantly, while the second $\ensuremath{\Lambda}$ system is far off resonance. Such an excitation scheme ensures the preparation of atoms in the nearly dark state throughout the medium. Therefore, the total light energy is dissipated very weakly, whereas each individual laser wave can vary considerably along the propagation path. We have found that the resonant fields change as much as the far-detuned ones. The intensities oscillate sinusoidally with the optical length, with the energy being transferred back and forth between two waves in each frequency pair, resonant and far detuned. This gives the possibility for an almost lossless amplification of two of the laser waves, or an even generation of one of them.
We study the features of spatial and temporal evolution of two powerful short different-wavelength laser pulses interacting with optical dense three-level Lambda-type medium under condition of coherent population trapping for the general case of unequal transition oscillator strengths. The pulses are simultaneously sent into an atom. The durations of probe and coupling pulses T1,2T_{1,2}T1,2​ (T2>T1T_2>T_1T2​>T1​) are supposed to be shorter than all relaxation times in the medium. We present analytical and numerical solutions of the self-consistent set of coupled Schr\\"{o}dinger equations and reduced wave equations in adiabatic limit with taking into account the first non-adiabatic correction. The adiabaticity criterion with account of pulse propagation is studied. We find that propagation dynamics depends on the relationship between the transition oscillator strengths. It is shown that at the initial stage of propagation the envelopes of the pulses slightly change along the length of the medium. This distance can be large compared to one-photon resonant absorption length of the medium. Also it is possible a regime of complete reemitting of the probe pulse into the coupling one during propagation. A spatial evolution of the atomic Raman coherence is investigated in detail too.
Propagation of a pulse pair in a nondissipative three-level medium is investigated in the adiabatic-following approximation to a trapped state. The general case of unequal oscillator strengths of two electric dipole transitions in an atom is studied analytically, and the adiabaticity criterion for the matter-field interaction is derived. It is shown that the interaction adiabaticity strongly depends on the relationship between oscillator strengths. A simple expression specifying the critical propagation length at which the stimulated Raman adiabatic passage process is still effective is derived. An estimate of the propagation distance at which a complete energy transfer from the pump pulse into the Stokes pulse occurs is made.
We consider adiabatic interaction of five-level atomic systems and their media with four short laser pulses under the condition of all two-photon detunings being zero. We derive analytical expressions for eigenvalues of the system's Hamiltonian and determine conditions of adiabaticity for both the atom and the medium. We analyze, in detail, the system's behavior when the eigenvalue with nonvanishing energy is realized. As distinct from the usual dark state of a five-level system (corresponding to zero eigenvalue), which is a superposition of three states, in our case the superposition of four states does work. We demonstrate that this seemingly unfavorable case nevertheless completely imitates a three-level system not only for a single atom but also in the medium, since the propagation equations are also split into those for three- and two-level media separately. We show that, under certain conditions, all the coherent effects observed in three-level media, such as population transfer, light slowing, light storage, and so on, may efficiently be realized in five-level media. This has an important advantage that the light storage can be performed twice in the same medium; i.e., the second pulse can be stored without retrieving the first one, and then the two pulses can be retrieved in any desired sequence.