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Journal of Optics B Quantum and Semiclassical Optics

Two-level dynamics: Rabi flopping in the strong coupling regime

Marco Frasca

Published Jan 25, 2001• 5 citations
Publisher

Abstract

We discuss the dynamics of a two-level model in a strong coupling regime through the analysis of the probability amplitudes. It is seen that the theory recovers Rabi flopping also for strong coupling. Population trapping can occur and a decay constant for the excited state is also computed. Then, spontaneous emission can be made to disappear or, otherwise, the emitted photon turns out to have a frequency being an odd harmonic of the frequency of the perturbation.

DOI
10.1088/1464-4266/3/1/353

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We discuss the dynamics of a two-level model in a strong coupling regime through the analysis of the probability amplitudes. It is seen that the theory recovers Rabi flopping also for strong coupling. Population trapping can occur and a decay constant for the excited state is also computed. Then, spontaneous emission can be made to disappear or, otherwise, the emitted photon turns out to have a frequency being an odd harmonic of the frequency of the perturbation.
A three-level atom interacting with laser fields of different frequencies out of resonance is studied. The corresponding theory of dressed states is applied and extended beyond the leading order to obtain the higher-order corrections. These higher-order corrections can be computed by a dual Dyson series. Rather than to apply perturbation theory, the equations for the probability amplitudes of the transformed Hamiltonian in the dual interaction picture are considered. This gives us the opportunity to study the resonance properties of the quantum dynamics of the three-level atom being driven out of resonance by the external laser fields. In this way, by different resonance conditions due both to the detunings and the couplings, one gets Rabi flopping without resonance of the external fields and the atom levels, with also the Rabi frequency originating both from the detunings and the couplings.
The effects of counter-rotating terms (CRTs) on Rabi splitting and the dynamic evolution of atomic population in the Jaynes—Cummings model are studied with a coherent-state approach. When the coupling strength increases, the Rabi splitting becomes of multi-Rabi frequencies for the initial state of an excited atom in a vacuum field, and the collapses and revivals gradually disappear, and then reappear with quite good periodicity. Without the rotating-wave approximation (RWA), the initial excited state contains many eigenstates rather than two eigenstates under the RWA, which results in the multi-peak emission spectrum. An analytical approximate solution for the strong coupling regime is obtained, which gives a new oscillation frequency and explains the recovery of collapses and revivals due to the equal energy spacing.
It has been shown recently [see, for example, S.-Y. Zhu and M. O. Scully, Phys. Rev. Lett. 76, 388 (1996)] that a dynamical suppression of spontaneous emission can occur in a three-level system when an external field drives transitions between a metastable state and two decaying states. What is unusual in the decay scheme is that the decaying states are coupled directly by the vacuum radiation field. It is shown that the decay dynamics required for total suppression of spontaneous emission necessarily implies that the level scheme is isomorphic to a three-level \ensuremath{\Lambda} system, in which the lower two levels are both metastable and each is coupled to the decaying state. As such, the total suppression of spontaneous emission can be explained in terms of conventional dark states and coherent population trapping.
A two-level atom interacting with a single radiation mode is considered, without the rotating-wave approximation, in the strong coupling regime. It is shown that, in agreement with the recent results on Rabi flopping in a Josephson junction (Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, to appear in Phys. Rev. Lett.), the Rabi frequency is indeed proportional to first kind integer order Bessel functions in the limit of a large number of photons and the dressed states are Schr\\"odinger cat states. To approach this problem analytically use is made of the dual Dyson series and the rotating-wave approximation.
For a three-level atom in the ssV configuration (i.e., having two excited states each dipole-coupled to a common ground state), we have found a particular linear combination of bare-atom states in which Rabi oscillations and their associated collapses and revivals do not occur. Moving to a dressed-state picture, we discover that this particular linear combination state is just that dressed state which is decoupled from all the field modes. It is a dressed state for which the transition dipole moments with the other dressed states are zero. The existence of this decoupled dressed state depends on the tuning of the dressing laser field, which in turn depends on the bare-atom excited-state dipole moments and energy-level separation. When we include spontaneous emission, the population decays from the other dressed states into this decoupled state and remains coherently trapped there, producing a system that experiences no dynamical behavior. This is exact for \ensuremath{\delta}-function photon statistics (i.e., if there is no intensity uncertainty). The trapping becomes less perfect as the photon statistics are allowed to have a greater bandwidth. Also, if the applied field is tuned incorrectly, the spontaneous realignment of the atomic state amplitudes does not result in a totally decoupled dressed state, and the dynamics proceed normally.