Ground-state cooling of a trapped ion by quantum interference pathways

We investigate the possibility of enhancement of cooling a trapped ion by combining the electromagnetically induced transparency (EIT) effect with the standing-wave coupling. Our study shows that the quantum destructive interference which is caused by the EIT effect and the standing-wave coupling can cancel all the dominant heating effects if appropriate parameters are chosen. The analytical predictions and numerical simulations show that the final temperature can be much lower than the recoil energy. In addition, this fast-cooling scheme is robust against fluctuations of the strength of the laser beams, which makes it more feasible for experimental realization.

Figure 1

Level diagram of the ion. The ion has an excited state, $|e\rangle$, and two ground states, $|g\rangle$ and $|r\rangle$. $|e\rangle \leftrightarrow |g\rangle$ and $|e\rangle \leftrightarrow |r\rangle$ are driven by two EIT laser beams with the same detuning, while $|g\rangle \leftrightarrow |r\rangle$ are driven by a resonant standing-wave laser.

Figure 2

Cooling and heating transitions. Starting with the $n\text{th}$-vibration steady state $|D,n\rangle$, there are two coherent heating and cooling processes: (1) $|D,n\rangle |e,n\pm 1\rangle$ via the EIT sideband transition, then dissipate back to $|D,n\pm 1\rangle$; (2) $|D,n\rangle |B,n\pm 1\rangle$ via standing-wave sideband transition, followed by an internal pump $|B,n\pm 1\rangle |e,n\pm 1\rangle ,$ then dissipate back to $|D,n\pm 1\rangle$.

Figure 3

Numerical simulation of the mean phonon number vs cooling time. The parameters are ${\eta }_1=-{\eta }_2=0.05$, ${\eta }_s=0.05$, $2\gamma =10\nu$, ${\Omega }_g=\nu$, ${\Omega }_r=5\nu$, ${\Omega }_s=5/13\nu$ $\Delta =25\nu$. The numerical results are shown as solid lines. The dashed lines correspond to analytical prediction.

Figure 4

Numerical simulation of the final phonon number as a function of ${\Omega }_r$. The other parameters are ${\eta }_g=-{\eta }_r=0.05$, ${\eta }_s=0.05$, ${\Omega }_g=\nu$, $2\gamma =10\nu$. ${\Omega }_s$ and $\Delta$ are chosen under the optimal conditions (17) and (31). The solid line is obtained with numerical simulations. The dashed line corresponds to analytical prediction.

Figure 5

The cooling rate as a function of ${\Omega }_g$. The other parameters are ${\eta }_1=-{\eta }_2=0.05$, ${\eta }_s=0.05$, ${\Omega }_r=5{\Omega }_g$, ${\Omega }_s=5/13\nu$, $2{\gamma }_g=2{\gamma }_r=5\nu$; $\Delta$ are chosen under the optimal conditions (31). The solid line is obtained with numerical simulations. The dashed line is the corresponding the analytical prediction.

Figure 6

The final mean phonon number as a function of ${\Omega }_g$. The other parameters are ${\eta }_1=-{\eta }_2=0.05$, ${\eta }_s=0.05$, ${\Omega }_r=5\nu$, ${\Omega }_s=5/13\nu$, $2{\gamma }_g=2{\gamma }_r=5\nu$, and $\Delta =25\nu$. The corresponding optimal value of ${\Omega }_g$ from analytical prediction is at ${\Omega }_{g0}=\nu$.

Figure 7

The final mean phonon number as a function of the positional deviation from the node of the standing wave. The other parameters are ${\eta }_1=-{\eta }_2=0.05$, ${\eta }_s=0.05$, ${\Omega }_r=5\nu$, ${\Omega }_s=5/13\nu$, $2{\gamma }_g=2{\gamma }_r=5\nu$, and $\Delta =25\nu$.

Figure 8

Implementation of the cooling scheme in a $\Lambda$-configuration three-level ion using four laser fields. The EIT coupling is achieved by two running-wave laser fields with coupling strengths ${\Omega }_g$ and ${\Omega }_r$ and Lamb-Dicke parameters ${\eta }_1$ and ${\eta }_2.$ The effective standing-wave coupling is achieved with a standing-wave laser with Lamb-Dicke parameter ${\eta }^{\prime }$ and Rabi frequency ${\Omega }^{\prime }$, and a running-wave laser coupling is achieved with Lamb-Dicke parameter ${\eta }^{\prime \prime }$ and Rabi frequency ${\Omega }^{\prime \prime }$. The two laser beams are sufficiently detuned from excited state $|e\rangle$.