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arXiv:quant-ph/0612082 · Physical Review A

Photon storage in Λ -type optically dense atomic media. I. Cavity model

Alexey Vyacheslavovich Gorshkov, Axel André, Mikhail D. Lukin, Anders Søndberg Sørensen

Published Sep 7, 2007• 233 citations
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Abstract

In a recent paper [Gorshkov et al., Phys. Rev. Lett. 98, 123601 (2007)], we used a universal physical picture to optimize and demonstrate equivalence between a wide range of techniques for storage and retrieval of photon wave packets in $\ensuremath{\Lambda}$-type atomic media in free space, including the adiabatic reduction of the photon group velocity, pulse-propagation control via off-resonant Raman techniques, and photon-echo-based techniques. In the present paper, we perform the same analysis for the cavity model. In particular, we show that the retrieval efficiency is equal to $C∕(1+C)$ independent of the retrieval technique, where $C$ is the cooperativity parameter. We also derive the optimal strategy for storage and, in particular, demonstrate that at any detuning one can store, with the optimal efficiency of $C∕(1+C)$, any smooth input mode satisfying $TC\ensuremath{\gamma}⪢1$ and a certain class of resonant input modes satisfying $TC\ensuremath{\gamma}\ensuremath{\sim}1$, where $T$ is the duration of the input mode and $2\ensuremath{\gamma}$ is the transition linewidth. In the two subsequent papers of the series, we present the full analysis of the free-space model and discuss the effects of inhomogeneous broadening on photon storage.

DOI
10.1103/physreva.76.033804

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In a recent paper [Gorshkov et al., Phys. Rev. Lett. 98, 123601 (2007)], we used a universal physical picture to optimize and demonstrate equivalence between a wide range of techniques for storage and retrieval of photon wave packets in \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-type atomic media in free space, including the adiabatic reduction of the photon group velocity, pulse-propagation control via off-resonant Raman techniques, and photon-echo-based techniques. In the present paper, we perform the same analysis for the cavity model. In particular, we show that the retrieval efficiency is equal to C∕(1+C)C∕(1+C)C∕(1+C) independent of the retrieval technique, where CCC is the cooperativity parameter. We also derive the optimal strategy for storage and, in particular, demonstrate that at any detuning one can store, with the optimal efficiency of C∕(1+C)C∕(1+C)C∕(1+C), any smooth input mode satisfying TC\ensuremathγ⪢1TC\ensuremath{\gamma}⪢1TC\ensuremathγ⪢1 and a certain class of resonant input modes satisfying TC\ensuremathγ\ensuremath∼1TC\ensuremath{\gamma}\ensuremath{\sim}1TC\ensuremathγ\ensuremath∼1, where TTT is the duration of the input mode and 2\ensuremathγ2\ensuremath{\gamma}2\ensuremathγ is the transition linewidth. In the two subsequent papers of the series, we present the full analysis of the free-space model and discuss the effects of inhomogeneous broadening on photon storage.
In a recent paper [Gorshkov et al., Phys. Rev. Lett. 98, 123601 (2007)], we presented a universal physical picture for describing a wide range of techniques for storage and retrieval of photon wave packets in $\ensuremath{\Lambda}$-type atomic media in free space, including the adiabatic reduction of the photon group velocity, pulse-propagation control via off-resonant Raman techniques, and photon-echo based techniques. This universal picture produced an optimal control strategy for photon storage and retrieval applicable to all approaches and yielded identical maximum efficiencies for all of them. In the present paper, we present the full details of this analysis as well some of its extensions, including the discussion of the effects of non-degeneracy of the two lower levels of the $\ensuremath{\Lambda}$ system. The analysis in the present paper is based on the intuition obtained from the study of photon storage in the cavity model in the preceding paper [Gorshkov et al., previous paper, Phys. Rev. A. 76, 033804 (2007)].
In a recent paper [Gorshkov et al., Phys. Rev. Lett. 98, 123601 (2007)] and in the two preceding papers [Gorshkov et al., this issue, Phys. Rev. A 76, 033804 (2006); 76, 033805 (2006)], we used a universal physical picture to optimize and demonstrate equivalence between a wide range of techniques for storage and retrieval of photon wave packets in homogeneously broadened $\ensuremath{\Lambda}$-type atomic media, including the adiabatic reduction of the photon group velocity, pulse-propagation control via off-resonant Raman techniques, and photon-echo-based techniques. In the present paper, we generalize this treatment to include inhomogeneous broadening. In particular, we consider the case of Doppler-broadened atoms and assume that there is a negligible difference between the Doppler shifts of the two optical transitions. In this situation, we show that, at high enough optical depth, all atoms contribute coherently to the storage process as if the medium were homogeneously broadened. We also discuss the effects of inhomogeneous broadening in solid state samples. In this context, we discuss the advantages and limitations of reversing the inhomogeneous broadening during the storage time, as well as suggest a way for achieving high efficiencies with a nonreversible inhomogeneous profile.
We present a universal physical picture for describing storage and retrieval of photon wave packets in a Lambda-type atomic medium. This physical picture encompasses a variety of different approaches to pulse storage ranging from adiabatic reduction of the photon group velocity and pulse-propagation control via off-resonant Raman fields to photon-echo-based techniques. Furthermore, we derive an optimal control strategy for storage and retrieval of a photon wave packet of any given shape. All these approaches, when optimized, yield identical maximum efficiencies, which only depend on the optical depth of the medium.
We derive an optimal control strategy for storage and retrieval of a photon wavepacket of any given shape in a Lambda-type atomic medium with a limited optical depth. The control is provided by an appropriately shaped classical laser field. We present a universal physical picture encompassing different pulse-storage techniques ranging from adiabatic reduction of the photon group velocity and pulse-propagation control via off-resonant Raman fields to photon-echo based approaches. When properly optimized, all three approaches yield identical maximum efficiencies, which only depend on the optical depth of the medium. Extending the model to include Doppler broadening, we find the surprising result that at high enough optical depth all atoms contribute coherently as if the medium were homogeneously broadened.
We have performed a theoretical analysis of the photon-echo-based quantum memory realized recently in A. L. Alexander et al., Phys. Rev. Lett. 96, 043602 (2006) in the medium with linear correlation between atomic spatial coordinates and frequency detunings of the inhomogeneously broadened transition. A more general vision on the physical picture of temporal and spatial light-atoms dynamics has been obtained in the analytical solutions, taking into account the atomic phase relaxation, inhomogeneous broadening, and arbitrary parameters of the probe light pulses. We have evaluated preferable experimental conditions for the storage of incoming data light field modes where quantum memory efficiency and fidelity have been found. Physics of high efficiency in the forward geometry of the quantum memory and advantages for the realization of more universal long-lived storage on this basis are clarified.
We use the numerical gradient ascent method from optimal control theory to extend efficient photon storage in $\ensuremath{\Lambda}$-type media to previously inaccessible regimes and to provide simple intuitive explanations for our optimization techniques. In particular, by using gradient ascent to shape classical control pulses used to mediate photon storage, we open up the possibility of high efficiency photon storage in the nonadiabatic limit, in which analytical solutions to the equations of motion do not exist. This control shaping technique enables an order-of-magnitude increase in the bandwidth of the memory. We also demonstrate that the often discussed connection between time reversal and optimality in photon storage follows naturally from gradient ascent. Finally, we discuss the optimization of controlled reversible inhomogeneous broadening.