Atom localization and center-of-mass wave-function determination via multiple simultaneous quadrature measurements

We discuss localization and center-of-mass wave-function measurement of a quantum particle using multiple simultaneous dispersive interactions of the particle with different standing-wave fields. In particular, we consider objects with an internal structure consisting of a single ground state and several excited states. The transitions between ground and the corresponding excited states are coupled to the light fields in the dispersive limit, thus giving rise to a phase shift of the light field during the interaction. We show that multiple simultaneous measurements allow both an increase in the measurement or localization precision in a single direction and the performance of multidimensional measurements or localization. Further, we show that multiple measurements may relax the experimental requirements for each individual measurement.

Figure 1

(Color online) Internal structure of the quantum particle considered. ${\lambda }_j$ $(j∊\{1,2,3\})$ are the coupling constants between transition and laser field; ${\Delta }_j$ are the detunings, which are considered large such that the interaction is dispersive.

Figure 2

(Color online) One of the setups discussed in this paper. Here, a quantum object flies through the intersection area of two cavity field modes. The cavity fields couple off-resonantly to different transitions between internal states of the quantum object and thus acquire a phase shift during the interaction due to its dispersive nature. Quadrature measurements of the cavity fields then allow a two-dimensional localization of the particle.

Figure 3

(Color online) Density plot of the conditional probability $P_{\alpha }({\chi }_{\theta }\mid x)$ for $\theta =0$, $\alpha =3$, $\xi =0$. Bright colors indicate low probability, dark colors high probability. The solid (red) line at the center of the dark maximum region indicates the solution to the implicit conditions Eq. (30a) or (32) for the maximum of the probability.

Figure 4

(Color online) Example filter functions for different results of a single measurement. This corresponds to the probability that a certain value for ${\chi }_{\theta }$ is measured for an actual position $x$ of the quantum particle. The parameters are $\theta =0$, $\alpha =3$, $\xi =0$. The solid (red) line shows the case ${\chi }_{\theta }=0$, the dashed (blue) line is for ${\chi }_{\theta }=2\alpha$, and the dash-dotted (green) line shows ${\chi }_{\theta }=-2\alpha$.

Figure 5

(Color online) Example filter function for two simultaneous measurements. The dashed (blue) curve shows $P_{\alpha }({\chi }_{{\theta }_1}\mid x)$, the dash-dotted (green) line shows $P_{\beta }({\chi }_{{\theta }_2}\mid x)$, and the solid (red) line the product $P_{\alpha }({\chi }_{{\theta }_1}\mid x)P_{\beta }({\chi }_{{\theta }_2}\mid x)$. This product corresponds to the filter function in the dual-measurement case. The parameters for the first field mode are ${\theta }_1=0$, $\alpha =3$, ${\xi }_1=\pi ∕6$, and ${\chi }_{{\theta }_1}=0$. The parameters for the second mode are ${\theta }_2=\pi ∕2$, $\beta =3$, and ${\xi }_2=\pi ∕2$. The wavelength of the second field is chosen smaller than the wavelength $\lambda$ of the first field by a factor of 1.2. In (a), the measurement outcome ${\chi }_{{\theta }_2}$ for the second mode is chosen consistent with an actual position of the particle at about $-0.2\lambda$ $({\chi }_{{\theta }_2}\approx 0)$. For (b), the position is about $0.05\lambda$ $({\chi }_{{\theta }_2}\approx 0.56\pi )$, and for (c) it is about $0.3\lambda$ $({\chi }_{{\theta }_2}\approx 1.69\pi )$.