Creating single collective atomic excitations via spontaneous Raman emission in inhomogeneously broadened systems: Beyond the adiabatic approximation

The creation of single collective excitations in atomic ensembles via spontaneous Raman emission plays a key role in several quantum communication protocols, starting with the seminal Duan-Lukin-Cirac-Zoller protocol [L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature (London) 414, 413 (2001)]. This process is usually analyzed theoretically under the assumptions that the write laser pulse inducing the Raman transition is far off resonance and that the atomic ensemble is only homogeneously broadened. Here we study the impact of near-resonance excitation for inhomogeneously broadened ensembles on the collective character of the created atomic excitation. Our results are particularly relevant for experiments with hot atomic gases and for potential future solid-state implementations.

Figure 1

Basic level scheme for the creation of collective atomic excitations in atomic ensembles via spontaneous Raman emission. All atoms start out in $g$. A laser pulse off-resonantly excites the $g\text{−}e$ transition, making it possible for a photon to be emitted on the $e\text{−}s$ transition (with small probability).

Figure 2

Evolution of the absolute value of the excited state amplitude for a single atom (that starts out in the ground state) under pulsed excitation for different values of detuning $\Delta$. The fine line shows the exciting laser pulse $\Omega \left(t\right)$, which is a Gaussian centered at $0.2\mu \text{s}$ with FWHM pulse duration of $0.1\mu \text{s}$ and a maximum Rabi frequency of 1 MHz. The same pulse is used in the examples throughout the paper.

Figure 3

(Color online) We consider the situation where the energy of the excited state $e$ is not the same for all the atoms in the ensemble, but it has a Gaussian distribution with FWHM $\sigma$. We have chosen $\sigma =500\text{MHz}$ in all our examples. We denote the detuning of the write laser with respect to the center of this distribution by ${\Delta }_0$, which varies between 0 and 1.25 GHz in our examples.

Figure 4

(Color online) The average excited state population $p_e$ as a function of time for different values of ${\Delta }_0$. The probability of emitting a Stokes photon is proportional to this quantity. The detuning ${\Delta }_0$ of the laser from the center of the atomic distribution takes the values of 0, 0.5, 0.75, 1.0, and 1.25 GHz from top to bottom. The solid lines correspond to a situation without spontaneous decay; the dashed lines correspond to a decay rate $\Gamma =5\text{MHz}$. The write pulse is the same as in Fig. 2. The blue bottommost line shows the profile of the square of the Rabi frequency, ${\Omega }^2\left(t\right)$. One can see that $p_e$ follows ${\Omega }^2$ for large ${\Delta }_0$, in analogy with the single-atom situation of Fig. 2.

Figure 5

(Color online) Collectivity of the created atomic excitation in $s$ as a function of the time of emission of the Stokes photon $t_S$ for different values of ${\Delta }_0$ for $\Gamma =0$ (full lines) and $\Gamma =5\text{MHz}$ (dashed lines). The pulse and inhomogeneous distribution are the same as in Fig. 4. The square of the Rabi frequency ${\Omega }^2\left(t\right)$ is shown as a fine blue line. One can see that a detuning ${\Delta }_0=1.25\text{GHz}$ is sufficient to have excellent collectivity for all relevant values of $t_S$.

Figure 6

The quantity $n\left(\Delta \right){\left|\beta \left(t_0,\Delta \right)\right|}^2$ at $t_0=0.2\mu \text{s}$ (at the center of the pulse) as a function of $\Delta$, showing which atomic frequencies contribute significantly to the emission of the Stokes photon and thus to the atomic excitation created in $s$. The laser frequency is defined to be $\Delta =0$. The center frequency of the atomic distribution takes the same values as in Figs. 4, 5. The contribution from the resonant atoms (fluorescence) dominates for small values of ${\Delta }_0$, but it becomes negligible compared to the contribution from the bulk of the atomic distribution (Raman scattering) for large ${\Delta }_0$.