Heisenberg scaling of imaging resolution by coherent enhancement

Classical imaging works by scattering photons from an object to be imaged, and achieves resolution scaling as $1/\sqrt{t}$, with $t$ the imaging time. By contrast, the laws of quantum mechanics allow one to utilize quantum coherence to obtain imaging resolution that can scale as quickly as $1/t$ – the so-called “Heisenberg limit.” However, ambiguities in the obtained signal often preclude taking full advantage of this quantum enhancement, while imaging techniques designed to be unambiguous often lose this optimal Heisenberg scaling. Here we demonstrate an imaging technique which combines unambiguous detection of the target with Heisenberg scaling of the resolution. We also demonstrate a binary search algorithm which can efficiently locate a coherent target using the technique, resolving a target trapped ion to within 0.3% of the $1/e^2$ diameter of the excitation beam.

Figure 1

(a) Coherent oscillations in a driven two-level system produce very narrow features which in principle can be used to achieve Heisenberg-limited resolution, but signal ambiguities often preclude achieving such resolution in practice. (b) A sequence of $L$ pulses of the same overall duration as the drive in (a) can produce an unambiguous single excitation peak whose width nevertheless preserves optimal Heisenberg scaling as $1/L$, by varying the phases ${\varphi }_i$ of each pulse. Here $L=13$.

Figure 2

(a) Schematic of the apparatus used to perform quantum-enhanced imaging. The 674 nm control beam passes through an intensity stabilization circuit (described in text) and then an AOM used to control its intensity. The beam passes through a fiber and then enters the apparatus, where it is focused onto the ion. PID = proportional-integral-differential feedback controller. (b) Relevant level structure in ${}^{88}\mathrm{Sr}^{+}$.

Figure 3

Examples of quantum-enhanced imaging for (a) a single pulse, (b) a seven-pulse sequence, (c) a 25-pulse sequence, (d) a 153-pulse sequence. The orange dashed curves show model-free theory predictions for the excitation as a function of drive strength. The inset in subplot (d) shows the excitation width and agreement with the model near $\theta =\pi$ for the narrow $L=153$ sequence; the inset $x$ axis spans 0.2 rad.

Figure 4

Scaling of the FWHM of the excitation region as a function of pulse sequence length $L$. The solid green curve shows the theoretical width that should be achieved (asymptotically scaling as $1/L$),while the black dot-dashed curve shows classical scaling as $L^{-0.5}$.

Figure 5

Demonstration of the logarithmic search technique. The beam is positioned so that its maximum intensity slope is located at the ion position. (a)–(f) Successive pulse sequences of length $L_1=3$, $L_2=7$, $L_3=13$, $L_4=25$, $L_5=53$, $L_6=99$ pulses are used to excite the ion. The amplitude is varied to determine the regions where the ion is excited. For iterations $i=1,...,6$, thresholds $T_i$ ranging from 0.32 to 0.36 (dashed horizontal lines) are used to maximize the probability of success (Supplemental Material [24]). Black dots indicate the average excitation probability observed when searching each of the two amplitude subspaces. Only $n=5$ repetitions need be used for each scan due to the unambiguous, narrow feature inherent to the excitation sequences. With each iteration, searched locations where the ion was not found can be excluded (gray regions), the ion position is further localized, and subsequent pulse sequences search only in the remaining target area. (g) RMS error of the quantum-enhanced technique (filled circles) as a function of total coherent drive time clearly demonstrates the asymptotic $1/\tau$ scaling with total coherent drive time $\tau$ (solid curve). The dot-dashed curve is a fit showing scaling as ${\tau }^{-0.94\pm 0.01}$, improving much more quickly than the classical $t^{-0.5}$ scaling (dashed curve).