Quantum-limited detection of arrival time and carrier frequency of time-dependent signals

Abstract

Precise measurements of both the arrival time and carrier frequency of light pulses are essential for time-frequency-encoded quantum technologies. Quantum mechanics, however, imposes fundamental limits on the simultaneous determination of these quantities. In this work, we derive and experimentally verify the quantum uncertainty bounds governing joint time-frequency measurements. We show that when detection is restricted to finite time windows, the problem is naturally described by a quantum rotor, rendering the commonly used Heisenberg uncertainty relation inapplicable. We further propose an optimal detection scheme that saturates these fundamental limits. By sampling the Q-function, we demonstrate the reconstruction of the Wigner function beyond the harmonic oscillator. Using an experimental implementation based on a quantum pulse gate, we confirm that the proposed scheme approaches the ultimate quantum limit for simultaneous time-frequency measurements. These results provide a new framework for joint time-frequency detection with direct implications for precision measurements and quantum information processing.

Figures (7)

• Figure 1: (a) We consider a time-dependent quantum signal with an arrival time $\tau_0$ and a carrier frequency $\omega_0$ on a finite time interval $T$. (b) This signal can be periodically extended outside the time-interval, which motivates the mapping to a unit circle. (c) By mapping to the unit circle, we introduce an angular variable $\phi$ that describes the signal. The spread of the signal is characterized by the standard deviation $\Delta S$ of the rotated sine operator (\ref{['S']}). For more information, see the text.
• Figure 2: Schematic of the realization of the required POVM. An ancillary field in the von Mises fiducial state $|0,0\rangle$ sets the POVM element which is applied to the signal. Measuring the output of each POVM element then allows for the simultaneous estimation of arrival time $\tau_0$ and carrier frequency $\omega_0$ of the signal with optimal precision.
• Figure 3: The signal is projected onto temporally and spectrally shifted von Mises states, sketched in the rows and columns of the matrix. For each combination of shifts, the complex field overlap between the signal and the projection mode is measured, which yields the signal $Q$-function in the basis of von Mises states.
• Figure 4: a) Schematic depiction of the experimental setup. The measured signal pulse is generated by shaping it's complex spectrum via a commercial waveshaper. This signal is projected into varying von-Mises states (different central frequencies and times) via the QPG to implement the complete POVM. The converted SFG light from the QPG is detected and filtered via a spectrograph. b) Time and frequency depiction of von Mises states as they were used in the experiment.
• Figure 5: Quantum state tomography of the fiducial signal von Mises state. In the experiment state with $m=10000$ was used as fiducial state. Since just the relative difference in $m$ matters as well as angular variable matters, we will keep the notation as $|0,0\rangle$ with $\kappa=1$ centered at the origin of the time interval. From the measured $Q$-function (lower left), the time-frequency sector of the components $\rho_{mm'}$ is reconstructed via an iterative MaxLik algorithm. It's real and imaginary part are plotted in the central columns. For the reconstructed state $\rho_{mm'}$ we obtain the Wigner function (lower right). The upper row shows the theory for the respective quantities. See text for details.