Atom microscopy via two-photon spontaneous emission spectroscopy

We study subwavelength position measurement via spontaneous emission spectroscopy with two photons. Our model systems are a single $\Lambda$-type three-level atom, in which a dual interaction generates two independent photons, and an $M$-type five-level atom, which emits two photons in a double-Raman process. The difference is that the $M$-type system emits correlated pairs of photons, while the photon pairs from the $\Lambda$ system are uncorrelated. In order to evaluate the performance of the position measurement, we introduce an analysis scheme motivated by a possible experimental measurement procedure. We find characteristic differences in the spontaneous emission spectra of the two systems arising from the photon correlation in the $M$-type system. But interestingly, the effect of this photon pair correlation on the conditional position probability distribution does not lead to an increase in the position information of the single atom. Nevertheless, both schemes allow for an efficient measurement of the particle position with unambiguous position determination over a wide range of parameters.

Figure 1

(Color online) Level scheme of the $M$-type configuration. (a) Bare-state representation. (b) Dressed-state representation.

Figure 2

(Color online) Level scheme of the $\Lambda$-type configuration. The two subpanels indicate the two steps in the dual interaction. (a) First interaction. The initial atomic state is $|c⟩$, the final state is $|b⟩$. (b) Second interaction. The initial atomic state is $|b⟩$, the final state is $|c⟩$.

Figure 3

(Color online) Single-photon spontaneous emission spectra. Dashed lines show $S\left({\nu }_{\mathbf{k}}\right)$, and the solid lines stand for $S\left({\nu }_{\mathbf{q}}\right)$. (a) depicts results for the $\Lambda$-type system with dual interactions, (b) for the $M$-level system. The parameters are ${\gamma }_1={\gamma }_2=0.1\Gamma$, $\left|{\Omega }_1\right|=\Gamma$, and $\left|{\Omega }_2\right|=0.5\Gamma$.

Figure 4

(Color online) Filter function or steady-state population of the final state as a function of the photon frequencies. (a) Equation (19) for the $\Lambda$ system with dual interaction. (b) Equation (10) for the $M$-level system. The parameters are as in Fig. 3.

Figure 5

(Color online) Conditional position probability distribution for the $M$-type configuration. The different subpanels show results for various positions $x_0$ of the particle within the standing wave as discussed in Sec. 3B. The standing-wave phases are chosen as (a) ${\varphi }_1=0$, ${\varphi }_2=\pi /4$ and (b) ${\varphi }_1=\pi /4$, ${\varphi }_2=\pi /4$. The other parameters are $\left|{\Omega }_1\right|=\left|{\Omega }_2\right|=\Gamma$, ${\gamma }_1={\gamma }_2=0.1\Gamma$, and $\eta =1.2$.

Figure 6

(Color online) Conditional position probability distribution for the $M$-type configuration. The parameters are as in Fig. 5b. The figure shows results for particle positions that are not accessible with the standing-wave configuration in Fig. 5a.