Subwavelength position measurements with running-wave driving fields

Subwavelength position measurement of quantum particles is discussed. Our setup is based on a closed-loop driving-field configuration, which enforces a sensitivity of the particle dynamics to the phases of the applied fields. Thus, running wave fields are sufficient, avoiding limitations associated with standing-wave-based localization schemes. Reversing the directions of the driving laser fields switches between different magnification levels for the position determination. This allows us to optimize the localization, and at the same time eliminates the need for additional classical measurements common to all previous localization schemes based on spatial periodicity.

Figure 1

(a) The closed-loop diamond scheme considered as example. (b) Generic closed-loop system with $2N$ states. Red arrows indicate driving laser fields.

Figure 2

Ratio $R$ of the fluorescence intensities on transitions $|3\rangle \to |2\rangle$ and $|2\rangle \to |1\rangle$ versus closed-loop phase $\Phi$. (i) $\Omega =\gamma$, (ii) $\Omega =5\gamma$, (iii) $\Omega =10\gamma$.

Figure 3

Potential positions $z$ depending on the measured ratio $R$. Solid blue lines show $\xi =4$, dashed red lines indicate $\xi =2$, and the dash-dotted magenta line shows $\xi =0.25$. The horizontal green line indicates the assumed particle position $z=0.15\overline{\lambda }$.

Figure 4

Particle position $z$ depending on the measured ratio $R$, with parameters as in Fig. 3. The shaded area indicates uncertainties due to an imperfect measurement of $R$. The horizontal green line indicates the assumed particle position $z=0.15\overline{\lambda }$. (a) Laser configuration $\xi =2$, relative phase ${\phi }_0=0$; (b) $\xi =4$ and ${\phi }_0=0$; (c) $\xi =2$ and ${\phi }_0=\pi /4$. Note the different position axis scale in (a).

Figure 5

Estimate of the position determination uncertainty. The solid red line shows the slope $\partial \Phi /\partial R$ at the ratio $R$ equal to half the maximum ratio $R_{\text{max}}$ against the laser intensity parameter $x$. The dashed black line shows a fit of a function $y=c_0+c_1/x$ to the red curve for ratios $0<x<10$ with parameters $c_0=0.075$ and $c_1=1.402$.