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Physical Review A

Decoherence and coherent population transfer between two coupled systems

Uzma Akram, Z. Ficek, S. Swain

Published Jun 15, 2000• 76 citations
Publisher

Abstract

We show that an arbitrary system described by two dipole moments exhibits coherent superpositions of internal states that can be completely decoupled from the dissipative interactions (responsible for decoherence) and an external driving laser field. These superpositions, known as dark or trapping states, can be completely stable or can coherently interact with the remaining states. We examine the master equation describing the dissipative evolution of the system and identify conditions for population trapping and also classify processes that can transfer the population to these undriven and nondecaying states. It is shown that coherent transfers are possible only if the two systems are nonidentical, that is the transitions have different frequencies and/or decay rates. In particular, we find that the trapping conditions can involve both coherent and dissipative interactions, and depending on the energy level structure of the system, the population can be trapped in a linear superposition of two or more bare states, a dressed state corresponding to an eigenstate of the system plus external fields or, in some cases, in one of the excited states of the system. A comprehensive analysis is presented of the different processes that are responsible for population trapping, and we illustrate these ideas with three examples of two coupled systems: single V- and $\ensuremath{\Lambda}$-type three-level atoms and two nonidentical two-level atoms, which are known to exhibit dark states. We show that the effect of population trapping does not necessarily require decoupling of the antisymmetric superposition from the dissipative interactions. We also find that the vacuum-induced coherent coupling between the systems could be easily observed in $\ensuremath{\Lambda}$-type atoms. Our analysis of the population trapping in two nonidentical atoms shows that the atoms can be driven into a maximally entangled state which is completely decoupled from the dissipative interaction.

DOI
10.1103/physreva.62.013413

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We show that an arbitrary system described by two dipole moments exhibits coherent superpositions of internal states that can be completely decoupled from the dissipative interactions (responsible for decoherence) and an external driving laser field. These superpositions, known as dark or trapping states, can be completely stable or can coherently interact with the remaining states. We examine the master equation describing the dissipative evolution of the system and identify conditions for population trapping and also classify processes that can transfer the population to these undriven and nondecaying states. It is shown that coherent transfers are possible only if the two systems are nonidentical, that is the transitions have different frequencies and/or decay rates. In particular, we find that the trapping conditions can involve both coherent and dissipative interactions, and depending on the energy level structure of the system, the population can be trapped in a linear superposition of two or more bare states, a dressed state corresponding to an eigenstate of the system plus external fields or, in some cases, in one of the excited states of the system. A comprehensive analysis is presented of the different processes that are responsible for population trapping, and we illustrate these ideas with three examples of two coupled systems: single V- and \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-type three-level atoms and two nonidentical two-level atoms, which are known to exhibit dark states. We show that the effect of population trapping does not necessarily require decoupling of the antisymmetric superposition from the dissipative interactions. We also find that the vacuum-induced coherent coupling between the systems could be easily observed in \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-type atoms. Our analysis of the population trapping in two nonidentical atoms shows that the atoms can be driven into a maximally entangled state which is completely decoupled from the dissipative interaction.
The interaction of a pair of copropagating pulses with three-level \ensuremath{\Lambda}-type atoms is discussed in terms of time-dependent coupled and decoupled superpositions \ensuremath{\Vert}\ifmmode\pm\else\textpm\fi{}〉 of the lower levels. Due to the explicit time dependence of these states there is a nonadiabatic coupling between the ``bright'' state \ensuremath{\Vert}+〉 and the ``dark'' state \ensuremath{\Vert}-〉 in addition to the strong coupling between \ensuremath{\Vert}+〉 and the upper level \ensuremath{\Vert}a〉 . We show that under quasiadiabatic conditions and in the presence of decay from the upper level this coupling can be treated perturbatively and the Maxwell-Bloch equations can be solved analytically. With the help of such a perturbation approach, coherent population transfer and formstable laser pulse propagation are studied. \textcopyright{} 1996 The American Physical Society.
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.
The interaction between a three-level atom and a polychromatic field with an equidistant spectrum (Λ-scheme of the atom--field interaction) has been studied theoretically. It is shown that the interaction of an atom with such a field can be reduced to its interaction with a bichromatic field with additional light shifts of transition frequencies and an additional coupling of the lower atomic levels, which is proportional to the field intensity. Owing to this coupling, the idea of the coherent population trapping can be considered only as an approximation, because the dark state is not an eigenstate of the effective Hamiltonian in the general case of arbitrary dipole moments. The analyzed model gives a simple theoretical interpretation for the formation of the atomic state, which is close to the coherent population trapping, in the radiation field of a femtosecond laser.
We explore the quantum entanglement between two collective fields via atomic coherence effects. For three-level atoms in V\mathsf{V}V configuration driven by two applied fields on two-photon resonance, one coherent superposition of the excited states is not excited, which is the counterpart of coherent population trapping. The coherence-induced depopulation makes two cavity fields in each collection combine into a quantum-beat, i.e., equivalently, the difference mode of the two components decouples from the driven atoms. The two sum modes, when they are arranged in the four-wave mixinglike interactions, can be prepared in Einstein-Podolsky-Rosen entangled state. Correspondingly, any two individual fields from different collective modes are entangled with each other. Furthermore, the effects of thermal reservoir and laser linewidths are discussed, and a generalization is given to the case in which each quantum beat involves more than two modes.
Master equation methods are used to investigate the effects of a broad-band squeezed vacuum on a three-level atom of the lambda configuration. The two-mode squeezed vacuum is treated as a Markovian reservoir in a non-stationary phase-dependent state. In addition to the squeezed vacuum the atom is driven by two coherent laser fields each of which, depending on the polarization, can couple to one or both of the atomic transitions. We show that in general the optical Bloch equations for the atomic density matrix elements have oscillatory coefficients, thereby necessitating the use of Floquet methods. For the case of equal laser frequencies, which are also equal to the carrier frequency of the squeezed vacuum, the coefficients of the Bloch equations become time independent and stationary solutions for the populations and coherences are obtained by standard matrix methods. For the ordinary vacuum the usual coherent population trapping effect at two-photon resonance is obtained, with the upper state population being zero. An unsqueezed thermal field partially destroys the trapping effect as the upper state population is no longer zero at two-photon resonance. The squeezed vacuum has the effect of improving the trapping in that the coherence hole becomes more pronounced for some values of the relative phase between the squeezed vacuum and the driving fields. The additional effects of a coherence transfer rate between the two optical coherences, which occurs for special choices of angular momentum quantum numbers are also studied. For the case of equal laser frequencies, the inclusion of this coherence transfer process destroys population trapping and reduces the lambda system to a two-level system. However, for the case of unequal laser frequencies, the coherence transfer process in combination with the squeezed vacuum can restore to some extent the population trapping. We show that other features that do not occur for two-level atoms, such as stationary population inversions between pairs of the atomic levels, also depend on the relative phase and can be enhanced in the squeezed vacuum. In the case of unequal frequencies of the driving fields the population in the upper state depends on the relative phase only when the carrier frequency of the squeezed vacuum is equal to one of the two frequencies of the driving fields. When the carrier frequency of the squeezed vacuum is slightly detuned from both frequencies of the driving fields, the population in the upper state is insensitive to the relative phase but is dependent on the degree of squeezing. For large detunings, the population does not show any dependence on the degree of squeezing and its distribution in function of the two-photon detuning is similar to that in the thermal vacuum field.
Coherent population transfer and superposition of atomic states via a technique of stimulated Raman adiabatic passage in an excited-doublet four-level atomic system have been analyzed. It is shown that the behavior of adiabatic passage in this system depends crucially on the detunings between the laser frequencies and the corresponding atomic transition frequencies. Particularly, if both the fields are tuned to the center of the two upper levels, the four-level system has two degenerate dark states, although one of them contains the contribution from the excited atomic states. The nonadiabatic coupling of the two degenerate dark states is intrinsic, it originates from the energy difference of the two upper levels. An arbitrary superposition of atomic states can be prepared due to such nonadiabatic coupling effect.