Theory of coherent interaction-free detection of pulses

TL;DR

This work investigates interaction-free measurements implemented coherently on a three-level (qutrit) system to detect resonant $B$ pulses without absorbing them. By combining repeated Ramsey interrogations with beam-splitter and pulse unitaries $S(\phi)$ and $B(\theta_j,\varphi_j)$, the authors derive large-$N$ analytical results and show that the coherent protocol achieves near-unity efficiency and can reach the Heisenberg limit in Fisher information for small pulse strengths, outperforming the conventional projective approach. The study also demonstrates robustness to realistic imperfections, including phase noise, variable pulse strengths, detuning, decoherence, and finite temperature, and identifies practical limits such as a minimum pulse strength $\theta \approx 4\phi_N$. The results suggest a quantum-resource advantage from coherence for interaction-free detection and open avenues for detecting quantized pulses in cavities and for enhanced quantum sensing tasks.

Abstract

Quantum physics allows an object to be detected even in the absence of photon absorption by the use of so-called interaction-free measurements. We provide a formulation of this protocol using a three-level system, where the object to be detected is a pulse coupled resonantly to the second transition. In the original formulation of interaction-free measurements, the absorption is associated with a projection operator onto the third state. We perform an in-depth analytical and numerical analysis of the coherent protocol, where coherent interaction between the object and the detector replaces the projective operators, resulting in higher detection efficiencies. We provide approximate asymptotic analytical results to support this finding. We find that our protocol reaches the Heisenberg limit when evaluating the Fisher information at small strengths of the pulses we aim to detect -- in contrast to the projective protocol that can only reach the standard quantum limit. We also demonstrate that the coherent protocol remains remarkably robust under errors such as pulse-rotation phases and strengths, the effects of relaxation rates and detunings, as well as different thermalized initial states.

Figures (13)

• Figure 1: Schematic of (a) the coherent protocol and (b) the projective protocol. Here, $\phi_N = \pi/(N+1)$.
• Figure 2: The probabilities and efficiency of our protocol vs $\theta/\phi_N$ for $N = 25$. The exact numerical values are represented by solid lines, whereas the approximations are represented by dashed lines. We see that for $\theta/\phi_N \ge 4$, the ground state probability $p_0$ and efficiency $\eta_{\rm c}$ approach unity.
• Figure 3: The surface map for (a) $p_0$ and (b) $p_{\rm det}$ as functions of $\theta$ and $N$ in an ideal case of identical $B$ pulses. Red 'x' markers correspond to the threshold values of $\theta$ at $p_{0} \ge 0.85$ and the dashed black line in (a) denotes the values $\theta = 4\phi_{N}$. The inset in (a) represents $\ln(\theta/\pi)$ with circle symbols taken at four threshold values $p_{0} \ge 0.25$ (blue), $p_{0} \ge 0.5$ (magenta), $p_{0} \ge 0.85$ (red), and $p_{0} \ge 0.95$ (green). For this analysis, the range of $N$ is extended to $N \in [25,100]$. The solid lines of the same color are the lines of best fit after taking the natural logarithm of the power law $aN^{-1}$, i.e., $\ln(a) - \ln(N)$.
• Figure 4: Probabilities of success for successive $B$-pulse implementations: (a) $p_{j,0}$ for the coherent protocol and (b) $p_{j, {\rm det}}$ for the projective protocol, with values in the range $[0,1]$ indicated by the colorbar. Plots (c) and (d) show the probabilities of absorption in the coherent and projective protocols, respectively. All $B$ pulses are of strength $\theta = \pi$.
• Figure 5: The quantum Fisher informations for $N$ = 2 (a), 5 (b), and 25 (c) for both coherent and projective cases. Panel (c) also contains an inset showing a different resolution of the QFI. Each case is evaluated using the Fisher informations $\rm QFI_{\rm c}$, $\rm QFI_{\eta_{\rm c}}$, $\rm QFI_{\rm proj}$, and $\rm QFI_{\eta}$. (d) Log-log plot of $\rm{QFI_c}$ at $\theta = 0$ (blue circles) along with the exact solution of $\rm{QFI_c}$ at $\theta = 4\phi_N$ (red circles) and the corresponding approximate solution (yellow circles) using the probability amplitudes derived in Sec. \ref{['sec_3']}. Each case is evaluated in the range $N \in [25, 100]$. The blue, red, and yellow solid lines are the corresponding best-fit lines.