Master equation for collective spontaneous emission with quantized atomic motion

We derive a Markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion. Our equation provides a unifying picture of the consequences of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and applies equally well to distinguishable and indistinguishable atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find closed-form formulas for a number of relevant states (Gaussian states, Fock states, and thermal states). In particular, we show that dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion.

Figure 1

Decomposition of the global system into bath and system of interest. The system of interest is the internal part of the atoms described by the state ${\rho }_A^{\mathrm{in}}$. The bath corresponds to the atomic external degrees of freedom, described by the state ${\rho }_A^{\mathrm{ex}}$, and the electromagnetic field degrees of freedom, initially in the vacuum state $|0\rangle \langle 0|$.

Figure 2

Characteristic time scales corresponding to the evolution of the external dynamics (${\tau }_M$), the internal dynamics (${\tau }_R\sim 1/{\gamma }_0$ with ${\gamma }_0$ the free spontaneous emission rate), the isolated system dynamics (${\tau }_S\sim 1/{\omega }_0$ with ${\omega }_0$ the atomic transition frequency), and the bath (${\tau }_B<{\tau }_S$).

Figure 3

Coordinate system $Oxyz$ where the $z$ direction corresponds to the quantization axis. The dipole moment for a $\pi$ transition is ${\mathbf{d}}_{\pi }=d_0{\mathbf{e}}_z$ with $d_0\in \mathbb{R}$, and for a ${\sigma }^{\pm }$ transition is ${\mathbf{d}}_{{\sigma }^{\pm }}=d_{\mp }{\mathbf{\varepsilon }}_{\mp }$ with $d_{\mp }\in \mathbb{C}$. The vector ${\mathbf{r}}_{ij}$ connecting the atoms $i$ and $j$ lies in the plane $y=0$ and forms an angle ${\alpha }_{ij}$ with the $z$ axis. The primed coordinate system $O{x}^{\prime }{y}^{\prime }{z}^{\prime }$, equipped with spherical coordinates $(k^{\prime },{\theta }^{\prime },{\phi }^{\prime })$, is chosen so that ${\mathbf{r}}_{ij}^{\prime }\equiv {\mathbf{r}}_{ij}$ lies along the $z^{\prime }$ axis in order to facilitate the calculation of the correlation functions.

Figure 4

Schematic view of two atoms separated by a distance $r_{ij}$ and whose positions are described by Gaussian wave packets of width ${\ell }_0$. The dipole moments and dipole radiation patterns are illustrated in red and correspond to a $\pi$ transition with ${\alpha }_{ij}=\pi /2$.

Figure 5

Regularized dipole-dipole shifts ${\mathrm{\Delta }}_{ij}^{\mathrm{sep}}$ as a function of ${\xi }_{ij}=k_0{r}_{ij}$ for atoms in Gaussian states (70) with a Lamb-Dicke parameter ${\eta }_0=1$ and different cutoffs (solid lines from left to right): $k_0\epsilon ={10}^{-1}$ (blue curve), $k_0\epsilon ={10}^{-2}$ (green curve), $k_0\epsilon ={10}^{-3}$ (orange curve), and $k_0\epsilon ={10}^{-4}$ (dark red curve). The red dashed curve corresponds to the classical dipole-dipole shift ${\mathrm{\Delta }}^{\mathrm{cl}}$ given by Eq. (51). The plots shown are for a $\pi$ transition with ${\alpha }_{ij}=\pi /2$.

Figure 6

Off-diagonal decay rates (top) and regularized dipole-dipole shifts (bottom) for the configuration illustrated in Fig. 4 as a function of ${\xi }_{ij}=k_0{r}_{ij}$ and ${\eta }_0=k_0{\ell }_0$ for (a) distinguishable atoms [Eq. (76) and (84)] and (b), (c) indistinguishable atoms [Eq. (85) and (86) for $N=2]$. The plots shown are for a $\pi$ transition with ${\alpha }_{ij}=\pi /2$ and a cutoff $k_0\epsilon =0.01$. The black solid curves at the front of each plot (for ${\eta }_0=0$) are the classical decay rate (32) and dipole-dipole shift (51). The black solid curves on the left of each plot of the decay rates (for ${\xi }_{ij}=0$) correspond, in the cases (a) and (b), to Eq. (79).

Figure 7

Decay rates in the superradiant regime in units of ${\gamma }_0$ as a function of the Lamb-Dicke parameter ${\eta }_0$ when atoms $i$ and $j$ are initially in the same motional Fock state $|{\varphi }_{(n,\mathbf{0})}\rangle$ centered around the origin with (from right to left) $n=0$ (ground state, red curve), $n=1$ (green curve), and $n=10$ (blue curve). Inset: same figure in log-log scale showing the power-law decrease of ${\gamma }_{ij}^{\mathrm{sep}}$ as $1/{\eta }_0$. The plots shown are for a $\pi$ transition with ${\alpha }_{ij}=\pi /2$.