Photon storage in $\Lambda$-type optically dense atomic media. III. Effects of inhomogeneous broadening

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 $\Lambda$-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.

Figure 1

(Color online) $\Lambda$-type medium coupled to a classical field (solid) with Rabi frequency $\Omega \left(t\right)$ and a quantum field (dashed). Collective enhancement [9] effectively increases the atom-field coupling constant $g$ up to $g\sqrt{N}$, where $N$ is the number of atoms in the medium. ${\Delta }_j$ is a frequency shift due to inhomogeneous broadening.

Figure 2

Optimal spin-wave modes to retrieve from (in the forward direction) at the indicated values of $d^{\prime }$ in the presence of (Gaussian) Doppler broadening with HWHM ${\Delta }_{\mathrm{I}}=88\gamma$.

Figure 3

Error $1-\eta$ as a function of unbroadened optical depth $d$ for different efficiencies $\eta$. ${\eta }_{\mathrm{back}}^{\max }\left(d\right)$ (solid) is the Doppler-free maximum total efficiency for any storage followed by backward retrieval. The maximum total efficiency for storage followed by backward retrieval with Doppler broadening of HWHM ${\Delta }_{\mathrm{I}}=88\gamma$ (circles) does not follow ${\eta }_{\mathrm{back}}^{\max }\left(d^{\prime }\right)$ (dotted), which is the efficiency one would expect from naive rescaling of resonant optical depth, but follows more closely the heuristic model $5.8[\pi ∕\left(4d^{\prime 2}\right)+1∕d]$ (dashed) described in the text.

Figure 4

The solid lines show the optimal input modes for $d^{\prime }=20$ and $d=20$, 60, and 120 (the value of $d$ is indicated on the plot) for storage followed by backward retrieval in a medium with a Gaussian inhomogeneous profile, without the reversal of the inhomogeneous broadening, and without averaging over frequency classes during the storage time. The dashed lines show the corresponding optimal modes for $d^{\prime }=d$, i.e., for the case of no inhomogeneous broadening.

Figure 5

(Color online) (a) As a function of $d^{\prime }$ and $d∕d^{\prime }$, the optimal (smallest) error of fast storage followed by fast backward retrieval (surface). $5.8∕d^{\prime }$ (dash-dotted line) is the limiting behavior, found in paper II, of the homogeneous error $(d∕d^{\prime }=1)$ for large enough $d(=d^{\prime })$. $14.2∕d^{\prime 2}$ (dotted) approximates the $d^{\prime }$-limited error $(d∕d^{\prime }\to \infty )$ near $d^{\prime }\sim 10–20$ and, probably, above. (b) As a function of $d^{\prime }$, dash-dotted and dotted lines are the same as in (a). The thick and thin solid lines are the $d∕d^{\prime }=1$ and $d∕d^{\prime }\to \infty$ error curves, respectively, which can be read out from the surface plot in (a). $1∕(1+14.2∕d^{\prime 2})$ (dashed line) is a heuristic curve that matches the $d^{\prime }$-limited error (thin solid line) very well.

Figure 6

Total efficiency of fast storage followed by fast backward retrieval as a function of $Td\gamma$ in the limit when $T\gamma \to 0$, $d\to \infty$, but $Td\gamma$ is finite. The curves show the efficiency without inhomogeneous broadening (dash-dotted line) and with (solid lines) Gaussian (G) or Lorentzian (L) inhomogeneous profiles optimized with respect to the inhomogeneous width. The horizontal dashed line is 1.

Figure 7

${\Delta }_{I}T$ as a function of $Td\gamma$, where ${\Delta }_I$ is the optimal HWHM of the Gaussian (G) or Lorentzian (L) inhomogeneous profile. The points are connected with straight lines to guide the eye. The dashed and dotted lines are $1.4{(Td\gamma -2)}^{1∕2}$ and $2.25{(Td\gamma -2)}^{1∕2}$, respectively.

Figure 8

Storage and backward retrieval with and without reversible inhomogeneous broadening at finite optical depth $d=100$ (a), and (b) 1000. The curves show the total efficiency of storage followed by backward retrieval of the resonant Gaussian-like pulse of duration $T$ of Eq. (43) (shown in Fig. 3 of paper II) with “optimal” resonant adiabatic storage (dotted) or with fast storage without inhomogeneous broadening (dash dotted) or with reversible optimal-width Lorentzian (L) or Gaussian (G) broadening (solid). The horizontal dashed line is the optimal adiabatic efficiency while the second dashed line is $\exp (-2\gamma T)$, by which fast efficiencies are rescaled relative to the $T\gamma \to 0$ limit of Fig. 6.