Nonclassical imaging for a quantum search of trapped ions

We discuss a simple search problem which can be pursued with different methods, either on a classical or on a quantum basis. The system is represented by a chain of trapped ions. The ion to be searched for is a member of that chain, consisting, however, of an isotopic species different from the others. It is shown that classical imaging may lead to the final result as fast as quantum imaging. However, for the discussed case the quantum method gives more flexibility and higher precision when the number of ions considered in the chain increases. In addition, interferences are observable even when the distances between the ions are smaller than half a wavelength of the incident light.

Figure 1

(a) Level scheme considered for the trapped ions: the ions are excited on the $\mid g⟩\to \mid e⟩$ transition, whereas the $\mid e⟩\to \mid f⟩$ transition is used for fluorescence detection; (b) an incident $\pi$ pulse of a wave with wave vector ${\vec{k}}_L$ excites the linearly trapped ions into their upper level $\mid e⟩$. The fluorescence is then registered in the far field by two detectors at ${\vec{r}}_i,i=1,2(\mid {\vec{r}}_1\mid =\mid {\vec{r}}_2\mid =r)$.

Figure 2

(Color) Three-dimensional plot of the second-order correlation function $G_{p,4}^{\left(2\right)}({\vec{r}}_1,{\vec{r}}_2)$ versus the two detector positions ${\varphi }_i\left({\vec{r}}_i\right),i=1$,2, for $N=4$ ions, with the off-resonant, nonradiating isotope placed in (a) the first (or fourth) position and (b) the second (or third) position of the chain. The distance between neighboring ions is $d=5.75\lambda$, the wavelength of the incident light is $\lambda =194\mathrm{nm}$, and a $\pi$ pulse is used for the excitation.

Figure 3

(Color) Interference patterns for $N=4$ ions. Blue and green curves correspond to the first (or fourth) and second (or third) ion being the isotope, respectively. Parameters are the same as in Fig. 2. (a) Classical interference pattern $G_{p,4}^{\left(1\right)}\left({\vec{r}}_1\right)$ [or second-order correlation function $G_{p,4}^{\left(2\right)}({\vec{r}}_1,{\vec{r}}_2)$ where the second detector is at ${\varphi }_2\left({\vec{r}}_2\right)=0$; see horizontal lines in Figs. 2a, 2b]. (b) Nonclassical interference pattern $G_{p,4}^{\left(2\right)}({\vec{r}}_1,{\vec{r}}_2)$; the two detectors are slightly separated with $\mathbf{\mid }\mid {\varphi }_1\mid -\mid {\varphi }_2\mid \mathbf{\mid }=1∕\pi$ [see tilted lines in Figs. 2a, 2b].

Figure 4

(Color) Interference patterns for $N=4$ ions with ion distance $\lambda ∕2$. (a) Red curve, nonclassical interference pattern $G_{1,4}^{\left(2\right)}({\vec{r}}_1,{\vec{r}}_2)$ with ${\varphi }_1\left({\vec{r}}_1\right)={\varphi }_2\left({\vec{r}}_2\right)$ [see red line in (b)]; black curve, classical interference pattern $G_{1,4}^{\left(1\right)}\left({\vec{r}}_1\right)$ [$G_{1,4}^{\left(2\right)}({\vec{r}}_1,{\vec{r}}_2)$ with ${\varphi }_2\left({\vec{r}}_2\right)=0$; see black line in (b)].

Figure 5

(Color) Interference patterns for $N=9$ ions. Black, yellow, green, blue, and red curves correspond to first (or ninth), second (or eighth), third (or seventh), fourth (or sixth) and fifth ion being the isotope, respectively. Parameters are the same as in Fig. 2. (a) Nonclassical interference pattern $G_{p,9}^{\left(2\right)}({\vec{r}}_1,{\vec{r}}_2)$; the two detectors have a constant separation $\mathbf{\mid }\mid {\varphi }_1\mid -\mid {\varphi }_2\mid \mathbf{\mid }=1∕\pi$. Dashed lines correspond to detectors’ positions being chosen in such a way that $\mathbf{\mid }\sin \mid {\varphi }_1\mid -\sin \mid {\varphi }_2\mid \mathbf{\mid }=0.378$. (b) Classical interference pattern $G_{p,9}^{\left(1\right)}\left({\vec{r}}_1\right)$.