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Annalen der Physik

Matrix perturbation theory for driven three‐level systems with damping

B.N. Sanchez, Tobias Brandes

Published Jul 20, 2004• 1 citation
Publisher

Abstract

Abstract We investigate the dynamics of the Λ system driven by two resonant laser fields in presence of dissipation for coupling strengths where the rotating‐wave approximation starts to break down. This regime is characterised by Rabi frequencies being approximately equal or smaller than the field frequencies. A systematic procedure to obtain an expansion for the solution of the Bloch evolution equations of the system is presented. The lowest contribution results to be the well‐known rotating‐wave approximation. The method is based on a semi‐classical treatment of the problem, and its predictions are interpreted fully quantum mechanically. The theory is illustrated by a detailed study of the disappearance of coherent population trapping as the intensities of the fields increase.

DOI
10.1002/andp.20045161002

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Abstract We investigate the dynamics of the Λ system driven by two resonant laser fields in presence of dissipation for coupling strengths where the rotating‐wave approximation starts to break down. This regime is characterised by Rabi frequencies being approximately equal or smaller than the field frequencies. A systematic procedure to obtain an expansion for the solution of the Bloch evolution equations of the system is presented. The lowest contribution results to be the well‐known rotating‐wave approximation. The method is based on a semi‐classical treatment of the problem, and its predictions are interpreted fully quantum mechanically. The theory is illustrated by a detailed study of the disappearance of coherent population trapping as the intensities of the fields increase.
Abstract The time evolution of the ground‐state population and coherence of a Λ‐type three‐level emitter strongly coupled to a radiation reservoir is studied by using a transformed rotating‐wave approximation (TRWA) and time‐dependent variational approach with the multiple Davydov ansatz. The study aims at illustrating the role of counter‐rotating couplings in the long‐time evolution of the Λ system. The TRWA and variational approach are found to be consistent with each other and predict that there exist long‐lived oscillatory behaviors of ground‐state populations and coherence, which are robust in the presence of strong dissipation and arise from the counter‐rotating‐coupling‐induced virtual photon processes. The oscillation amplitude and frequency depend on the system‐reservoir coupling strength and the frequency difference between the two ground states, which can be qualitatively and quantitatively captured by the TRWA analytical results. Present results offer insights into the impact of the counter‐rotating couplings on the long‐time evolution of the strongly dissipative multilevel system as well as the protocol of generation of coherent oscillations by relaxation in a strong‐coupling regime.
We explore extended $\ensuremath{\Lambda}$ systems driven by two resonant laser fields in a strong regime characterized by the fact that the Rabi frequencies are much bigger than the difference between the two field frequencies. We show that a specific coherent superposition of the lower states is trapped during the dynamics. This extended trapped state is different from the trapped state usually encountered in the rotating-wave-approximation regime. We show the robustness of this trapping in the sense that it is stable with respect to significant deviations of the field parameters (frequencies and amplitudes).
Here we use perturbation techniques based on the averaging method to investigate Rabi oscillations in cw and pulse-driven two-level systems (TLS's). By going beyond the rotating-wave approximation, especifically to second-order in perturbation, we obtain the Bloch-Siegert shift of the TLS resonant frequency, in which the resonant frequency increases with the driving field amplitude. This frequency shift implies that short resonant πππ-pulses in which the Rabi frequency is approximately 40\% or higher of the transition frequency do not achieve complete inversion in TLS's. Hence, guided by analytical results based on the averaging technique, we propose two methods for obtaining population inversions in the TLS driven by short πππ-pulses: one with chirping and the other with pulse shaping and near resonance blue-shifted detuning. Both methods minimize dephasing due to the Bloch-Siegert shift, reduce the dependance of the excitation of the TLS on the pulse phase, and are very effective in achieving complete population inversions.
In the framework of the nonsecular perturbation theory based on the Bogoliubov averaging method, the coherent dynamics of multiphoton Raman transitions in a two-level spin system driven by an amplitude-modulated microwave field is studied. Closed-form expressions for the Rabi frequencies of these transitions are obtained beyond the rotating-wave approximation for the low-frequency driving component. It is shown that spin states dressed by the high-frequency component of the driving field are shifted due to the Bloch-Siegert-like effect caused by antiresonant interactions with the strong low-frequency driving. We predict that with increasing the order of the Raman transition, the Rabi frequency decreases and the contribution of the Bloch-Siegert shift to this frequency becomes dominant. It is found that the amplitude and phase of the Rabi oscillations strongly depend on the initial phase of the low-frequency field as well as on detuning from multiphoton resonance. The recent experimental data for the second- and third-order Raman transitions observed for the nitrogen-vacancy center in diamond (Z. Shu et al., arXiv:1804.10492) are well described in the frame of our approach. Our results may provide new possibilities for coherent control of quantum systems.
We derive dynamical equations for a driven, dissipative quantum system in which the environment-induced relaxation rate is comparable to the Rabi frequency, avoiding assumptions on the frequency dependence of the environmental coupling. When the environmental coupling varies significantly on the scale of the Rabi frequency, secular or rotating wave approximations break down. We avoid these approximations, yielding dynamical steady states which account for the interaction between driven quantum dots and their phonon environment. The theory, which is motivated by recent experimental observations, qualitatively and quantitatively describes the transition from asymmetric unsaturated resonances at weak driving to population inversion at strong driving.
It is the aim of the present work to show some departures from the well-known optical Bloch equations usually obtained when the damping of a two-level system driven by a Glauber coherent mode of the radiation field is treated in the Markovian approximation. In particular, we show that for an arbitrary intensity we have a set of 4 n optical Bloch equations which can be reduced to the usual optical Bloch equations when the population of the mode becomes important. A criterion for the quasi-classical evolution can be deduced. Obviously, in both limits of an arbitrarily weak or strong excitation intensity the Mollow's results are obtained. In addition, we do not need here the introduction of a particular form like the plane wave used by Mollow for the corresponding classical field--as the problem is solved explicitly.