Quantum-trajectory analysis of an optical Stern-Gerlach experiment

We present a Monte Carlo simulation of a typical cavity-QED experiment on the optical Stern-Gerlach effect in terms of the quantum trajectory method. We show that the phenomenon of wave packet splitting may be experimentally observed even in a moderate coupling regime where the atomic and cavity damping rates are comparable to the atom-field coupling strength.

Figure 1

The model. The Gaussian wave packet of a two-state atom initially in its ground state is placed on a node of a standing-wave cavity, on the cavity axis.

Figure 2

The evolution of an atomic wave packet placed on a node of the cavity field. The probability distributions in momentum space (a) and in coordinate space (b). $q=p∕\hslash k$, $\xi =x∕\lambda$, and $\tau =g_{0}t$. For a typical atom with $\mu =\hslash k^2∕2M{g}_0\sim 1.7\times {10}^{-4}$. The atom is initially in the ground state having a momentum spread $\mathrm{\Delta }p=10\hslash k$ while the field is prepared in a single-quantum Fock state. The inset is a lateral view of (b) into the time axis along with the $\cos kx$ curve.

Figure 3

Spatial modulation of the dressed energies $(\hslash =1)$ and motion of the dressed states starting from a node (a) and from an antinode (b). $A$ and $N$, respectively, denote antinode and node.

Figure 4

Spatial modulation of the dressed energies $(\hslash =1)$ in the presence of nonzero atom-field detuning. ${\omega }^{\prime }={\omega }_0+\delta ∕2$ and $g^{\prime }=\sqrt{g_0^2+{(\delta ∕2)}^2}$.

Figure 5

Wave packet evolution in coordinate space in the presence of nonzero atom-field detuning, $\delta ∕g_0=0.1$, while other parameters are the same as in Fig. 2. $\xi =x∕\lambda$.

Figure 6

The shapes of the wave packet sitting on a node right after an atomic jump arranged to occur at $\tau =1,\dots ,5,10$, and 20 (a) in momentum space and (b) in coordinate space. $q=p∕\hslash k$, $\xi =x∕\lambda$.

Figure 7

The time evolution of the atomic wave packet of an infinitely heavy atom free of damping located at a node (a) in momentum space and (b) in coordinate space. The insets are ground-level views into the time axis.

Figure 8

The dynamics of a wave packet in the presence of damping in which an atomic jump has occurred (a) in momentum space and (b) in coordinate space. $\gamma =0.1$ and $\mathrm{\Delta }q=10$, for an infinitely heavy atom initially in the ground state placed at a node, with the cavity field in a one-quantum coherent state. The insets are ground-level views along the time axis.

Figure 9

The evolution of ${\mathcal{P}}_e\left(q\right)={\sum }_n{\mid C_e(n,q)\mid }^2$. The inset shows the temporal behavior of $P_e=\int dq\mathcal{P}\left(q\right)$.

Figure 10

The “far-field” momentum distributions averaged over 300 atoms at a location approximately $40w$ away from the cavity axis. The thinner line is for the case of nonradiative atoms, while the thicker line is for the case of radiative atoms with $\gamma ∕g_0=0.5$.