Effects of detuning on tunneling and traversal of ultracold atoms through vacuum-induced potentials

In this paper, we study the tunneling and traversal of ultracold two-level atoms through the potential induced by the vacuum cavity mode. In particular, we discuss the effects of off-resonant interaction between the cavity mode and atomic transition on tunneling time of the ultracold atoms through a high-$Q$ mazer cavity. The phase time which may be considered as an appropriate measure of the time required for the atom to cross the cavity, exhibits some interesting features in the presence of off-resonant interaction. For example, switching between the sub and superclassical behaviors in phase time occurs for proper choice of detuning. Similarly, negative phase time appears for the transmission of atoms in both excited and ground states in the presence of off-resonant interaction.

Figure 1

Schematic energy-level diagram of a two-level atom having angular frequency ${\omega }_0$ with atom-field detuning $\Delta$.

Figure 2

Dimensionless phase time (solid curve) for transmission of a two-level atom in excited state vs the mean momentum for the parameters $k_{0}L=10\pi$ (a) $\tilde{\Delta }=0$ and (b) $\tilde{\Delta }=+10$. The transmission probability $T_{e,0}$ of the atom in excited state is shown in dashed curves.

Figure 3

Dimensionless phase time (solid curve) for transmission of a two-level atom in excited state vs the mean momentum for the parameters $k_{0}L=\pi /2$ (a) $\tilde{\Delta }=+3$ and (b) $\tilde{\Delta }=-3$. The transmission probability $T_{e,0}$ of the atom in excited state is shown in dashed curves.

Figure 4

Dimensionless phase time (solid curve) for transmission of a two-level atom in ground state vs the mean momentum for the parameters $k_{0}L=\pi /2$, (a) $\tilde{\Delta }=+3$ and (b) $\tilde{\Delta }=-3$. The transmission probability in the ground state $T_{g,1}$ is shown in the inset of (a) and with a dashed curve in (b).

Figure 5

Dimensionless phase time (solid curve) and transmission probability (dashed curve) for transmission of a two-level atom in ground state vs the mean momentum for the parameters $k_{0}L=3\pi /2$ and $\tilde{\Delta }=+2$.

Figure 6

Dimensionless phase time (solid curve) for transmission of a two-level atom in ground state vs the mean momentum for the parameters $k_{0}L=3\pi /2$ and $\tilde{\Delta }=-2$. The dashed curve represents the transmission probability $T_{g,1}$ of the atom in ground state.

Figure 7

Normalized probability density $P\equiv |\langle g,1|{\psi }_T(z,t)\rangle {|}^2/\sigma$ at $Z=L$ as a function of dimensionless time $t/t_{cl}$. The solid (dashed) curve represents $P$ after transmission through the cavity (in free space). The parameters used for the calculations are $k_{0}L=3\pi /2$, $\tilde{\Delta }=2$, $\sigma /k_0=0.01$, $\overline{k}/k_0=1.6$. Both the solid and dashed curves are normalized to unity.

Figure 8

Normalized probability density $P\equiv |\langle g,1|{\psi }_T(z,t)\rangle {|}^2/\sigma$ at $Z=L$ as a function of dimensionless time $t/t_{cl}$. The solid (dashed) curve represents $P$ after transmission through the cavity (in free space). The parameters used for the calculations are $k_{0}L=10\pi$, $\tilde{\Delta }=-0.55$, $\sigma /k_0=0.55$, $\overline{k}/k_0=2.055$. Both the solid and dashed curves are normalized to unity.