Quantum-limited velocity readout and quantum feedback cooling of a trapped ion via electromagnetically induced transparency

We discuss continuous observation of the momentum of a single atom by employing the high velocity sensitivity of the index of refraction in a driven $\Lambda$-system based on electromagnetically induced transparency. In the ideal limit of unit collection efficiency this provides a quantum-limited measurement with minimal backaction on the atomic motion. A feedback loop, which drives the atom with a force proportional to measured signal, provides a cooling mechanism for the atomic motion. We derive the master equation which describes the feedback cooling and show that in the Lamb-Dicke limit the steady state energies are close to the ground state, limited only by the photon collection efficiency. Outside of the Lamb-Dicke regime the predicted temperatures are well below the Doppler limit.

Figure 1

(a) Atomic $\Lambda$ system with two ground states $\mid g⟩$ and $\mid r⟩$. The transition $\mid r⟩\to \mid e⟩$ is driven by a strong resonant light field with Rabi frequency ${\Omega }_L$, while the probe field $E_p$ couples $\mid g⟩\to \mid e⟩$ with detuning ${\Delta }_p$. (b) Susceptibility $\chi \left({\Delta }_p\right)$ of a $\Lambda$ system as a function of the detuning of the probe field, $E_p$. At the center of the transparency window the real part of the susceptibility (solid line) representing the index of refraction of the medium exhibits a steep slope, while the imaginary part (dotted line) vanishes.

Figure 2

Schematic setup for the continuous observation of the atomic momentum. The trapped atom is driven by a strong laser to create the transparency effect for the probe field $E_p$. Modulations of the probe light are detected in a homodyne measurement, i.e., by mixing $E_p$ with the strong field of the local oscillator (LO). For feedback cooling, a force proportional to the measured signal $I_c\left(t\right)$ is applied on the atom. Alternatively to the lens system, the atom can be placed inside a running wave cavity, e.g., to enhance the photon collection efficiency $ϵ$.

Figure 3

The figure shows the dependence of the steady state energy as a function of the feedback gain, $G$. The results are calculated in the Lamb-Dicke limit and for ${\Gamma }_0=0.01\nu$. The three curves are plotted for the parameters, $ϵ=0.1$ (solid line), $ϵ=0.05$ (dashed line), and $ϵ=0.01$ (dotted line).

Figure 4

Linear susceptibility $\chi \left({\Delta }_p\right)$ in arbitrary units for ${\Delta }_L∕\Gamma ={\Omega }_L∕\Gamma =1$.

Figure 5

Steady state occupation number, $\overline{n}=⟨{\widehat{a}}^{†}\widehat{a}⟩$, as a function of the detuning ${\Delta }_L$. The solid line shows the result for optimized $\varphi$ and optimized gain, $G$, while the dashed line shows the result without the feedback loop. The curves are plotted for the parameters: $ϵ=0.05,{\Omega }_L∕\Gamma =0.8$, and $\nu ∕\Gamma =0.1$.

Figure 6

(a) Laser heating and cooling rates, $A_{\pm }$, as a function of the laser detuning $\Delta$. (b) Feedback heating and cooling rates, $A_{\pm }^{fb}$, for $G=1$ and an optimized local oscillator phase $\varphi$. The parameters for the two plots are $ϵ=0.05,\Omega ∕\Gamma =0.8$, and $\nu ∕\Gamma =0.1$.