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Physical Review A

Amplification without inversion:mUnderstanding probability amplitudes, quantum interference, and Feynman rules in a strongly driven system

J. L. Cohen, P. R. Berman

Published May 1, 1997• 38 citations
Publisher

Abstract

Density-matrix calculations provide the steady-state conditions for probe amplification or lasing between atomic levels with an uninverted population, but additional insight into the underlying physics is given by a probability amplitude approach. In this paper we derive the gain coefficient from the Feynman diagrams for a probe-laser incident on a resonantly pumped, V-type system using time-dependent perturbation theory in a dressed basis. The connection is made to density-matrix calculations for this model, which have been used recently to describe experiments in Rb [A. S. Zibrov et al., Laser Phys. 5, 553 (1995); Phys. Rev. Lett. 75, 1499 (1995)]. In the density-matrix calculation the overall gain is possible because the pump-induced coherence of a strongly driven transition leads to probe amplification, despite the lack of inversion on the probe transition. In our amplitude approach we associate a specific physical process with each of the scattering channels for the probe and show how amplification without inversion can be achieved. The amplitude calculation reveals a distinction between stepwise and two-quantum processes. Interference is shown to result from the two-quantum processes, constructive for the amplification channels and destructive for the absorption. Terms appearing in the gain coefficient are traced to different sources in the amplitude and density-matrix approaches. The physical origin of each term is discussed and compared for both approaches. Terms that arise from coherences in the density-matrix approach are shown to correspond to noninterfering stepwise contributions in the amplitude approach. In deriving these results, we find that the Feynman rules that we construct for forming the probability amplitude for an arbitrary scattering process of the electromagnetic field from the coupled atom-strong pump system are consistent with Rayleigh-Schr\"odinger perturbation theory in the quantum dressed basis. In addition, the spontaneously emitted photons become entangled with the probe field, correlating the emission spectrum with specific scattering channels.

DOI
10.1103/physreva.55.3900

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arXiv:quant-ph/9807014•Physical Review A•Oct 12, 2001•31 citations
Density-matrix calculations provide the steady-state conditions for probe amplification or lasing between atomic levels with an uninverted population, but additional insight into the underlying physics is given by a probability amplitude approach. In this paper we derive the gain coefficient from the Feynman diagrams for a probe-laser incident on a resonantly pumped, V-type system using time-dependent perturbation theory in a dressed basis. The connection is made to density-matrix calculations for this model, which have been used recently to describe experiments in Rb [A. S. Zibrov et al., Laser Phys. 5, 553 (1995); Phys. Rev. Lett. 75, 1499 (1995)]. In the density-matrix calculation the overall gain is possible because the pump-induced coherence of a strongly driven transition leads to probe amplification, despite the lack of inversion on the probe transition. In our amplitude approach we associate a specific physical process with each of the scattering channels for the probe and show how amplification without inversion can be achieved. The amplitude calculation reveals a distinction between stepwise and two-quantum processes. Interference is shown to result from the two-quantum processes, constructive for the amplification channels and destructive for the absorption. Terms appearing in the gain coefficient are traced to different sources in the amplitude and density-matrix approaches. The physical origin of each term is discussed and compared for both approaches. Terms that arise from coherences in the density-matrix approach are shown to correspond to noninterfering stepwise contributions in the amplitude approach. In deriving these results, we find that the Feynman rules that we construct for forming the probability amplitude for an arbitrary scattering process of the electromagnetic field from the coupled atom-strong pump system are consistent with Rayleigh-Schr"odinger perturbation theory in the quantum dressed basis. In addition, the spontaneously emitted photons become entangled with the probe field, correlating the emission spectrum with specific scattering channels.
A four-level system driven by two coherent fields is considered. It is shown that in the presence of an incoherent pump, the probe gain at a short wavelength can be achieved due to the quantum interference. Our density matrix calculation provides the conditions for probe amplification from different origins, including gain without population inversion on any state basis, gain with population inversion on the dressed-state basis and gain with population inversion on the bare-state basis. Also, by controlling the Rabi frequency of the coupling field a total change from non-inversion to inversion can be achieved which does not depend on the intensity of the incoherent pump.
A four-level system driven by two coherent fields is considered. the probe gain at a short wavelength can be achieved due to the quantum interference in the presence of an incoherent pump. The density matrix calculation provides the conditions for probe amplification from different origin, including gain without population inversion in any state basis, gain with population inversion in the dressed-state basis, and gain with population inversion in the bare-state basis. Also, by controlling Rabi frequency of the coupling field a totally changing from non-inversion to inversion can be achieved which does not depend on the intensity of the incoherent pump.
Amplification without population inversion in a resonant V-type atomic medium is analyzed using the theory of quantum trajectories. A global view of the dynamics underlying the amplification is provided by a quantum stochastic process governing an interplay between coherence and decoherence. The quantum trajectories decompose into distinct ``gain cycles'' and ``loss cycles'' which determine, respectively, the emission and absorption spectra that might be calculated from perturbation theory. Two methods for calculating net gain are developed, motivated by complementary views of the exchange of energy between an atom and a probe field. One time averages the energy radiated continuously by the induced dipole, while the other determines probabilities for discontinuous energy exchange through the emission and absorption of individual quanta. In the latter case, the emission and absorption probabilities are evaluated as sums over probabilities for classical records that define the unobservable exchange of a quantum with the probe field in terms of observable scattering events. Quantum trajectories for a V-type medium driven by a coherent field are compared with those for a medium driven incoherently. Two relationships which connect amplification to population inversion in the latter case are shown to be lacking in the former; hence the possibility for amplification without population inversion arises from the following: (1) a decoupling of the rate of gain-cycle (loss-cycle) initiation from the time-averaged population in the initial state for gain (loss), and (2) loss of the symmetry that the final state for emission be the initial state for absorption and vice versa. The specific influences of these general observations vary from model to model. The details are worked out for the resonant V-type medium, where the quantum trajectory analysis sheds light on the meaning of ``without population inversion'' and ``cancellation of absorption by quantum interference.''
Summary form only given.A three-level atom in the V-configuration is treated under two coherent laser fields at resonance with the two allowed transitions. The master equation for the system and full quantum mechanical Hamiltonian are derived, and equations of motion for the elements of the density matrix are obtained by projecting into the bare- and dressed-states schemes. Analysis of light amplification without population inversion in this V-type system is accomplished within the framework of the dressed-state basis. The physical content of the equations of motion for the density matrix elements, namely the coherences and the populations is provided by identifying effective decay and relaxation rates and gain/absorption constants with the appropriate composite coefficients. These depend on the Rabi frequencies of both the drive and probe coherent fields. Time-dependent solutions for both the coherences and the populations are obtained both numerically and analytically under the secular approximation. We have shown how to construct these analytical approximate time-dependent solutions for dressed-state populations and coherences. These approximate solutions, as well as the full numerical calculation quantities, exhibit oscillations at the generalized Rabi frequency.
Amplification in a V system is studied both in the framework of the density matrix formalism and using the S matrix in the usual Hilbert space of the dressed-state vector. We show that it is possible to achieve amplification without inversion both in the bare- and in the dressed-state basis. In this case there is an asymmetry between amplification and absorption because of the occurrence of interferences between Feynman diagrams that can reduce the absorption. The comparison with the two-level atom shows a close analogy with the gain observed about the central resonance of the Mollow transmission spectrum. \textcopyright{} 1996 The American Physical Society.
Light-induced absorption with population inversion and amplification without population inversion in a coherently prepared closed three level V-type system are investigated. This study is performed from the point of view of a two color dressed-state basis. Both of these processes are possible due to atomic coherence and quantum interference contrary to simple intuitive predictions. Merely on a physical basis, one would expect a complementary process to the amplification without inversion. We believe that absorption in the presence of population inversion found in the dressed-state picture utilized in this study, constitutes such a process. We derive approximate analytic time-dependent solutions, for coherences and populations that are compared with full numerical solutions exhibiting good agreement. Steady-state quantities are also calculated, and the conditions under which the system exhibits absorption and gain with and without inversion, in the dressed-state representation are derived. It is found that for weak input probe-laser field absorption with inversion and amplification without inversion may occur, for a range of system parameters. These take place at resonance and the generalized Rabi ac Stark-shifted frequencies, in agreement with exprimental investigations, and at beat frequencies, depending on the relevant parameters.