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Time of arrival on a ring and relativistic quantum clocks

Iason Vakondios, Charis Anastopoulos

Abstract

We study the time-of-arrival problem for relativistic particles constrained to move on a ring, formulating the problem entirely within Quantum Field Theory (QFT). In contrast to its counterpart for motion in a line, the circle topology implies that particles may encounter the detector multiple times before detection, making a field-theoretic treatment of the measurement interaction essential. We employ the Quantum Temporal Probabilities (QTP) method to derive a class of Positive-Operator-Valued Measures (POVMs) for time-of-arrival observables directly from QFT. We analyze the resulting detection probabilities in both semiclassical and fully quantum regimes, identifying the relevant timescales and their dependence on the field-theoretic parameters. For ensembles of particles, the detection signal is a periodic function, providing a realization of a quantum clock whose operation reflects the local spacetime structure. We also extend the formalism to rotating rings and show that rotation induces additional noise in detection probabilities, interpretable as a manifestation of the rotational Unruh effect. Finally, we investigate multi-time measurements and demonstrate the emergence of non-classical temporal correlations due to entanglement.

Time of arrival on a ring and relativistic quantum clocks

Abstract

We study the time-of-arrival problem for relativistic particles constrained to move on a ring, formulating the problem entirely within Quantum Field Theory (QFT). In contrast to its counterpart for motion in a line, the circle topology implies that particles may encounter the detector multiple times before detection, making a field-theoretic treatment of the measurement interaction essential. We employ the Quantum Temporal Probabilities (QTP) method to derive a class of Positive-Operator-Valued Measures (POVMs) for time-of-arrival observables directly from QFT. We analyze the resulting detection probabilities in both semiclassical and fully quantum regimes, identifying the relevant timescales and their dependence on the field-theoretic parameters. For ensembles of particles, the detection signal is a periodic function, providing a realization of a quantum clock whose operation reflects the local spacetime structure. We also extend the formalism to rotating rings and show that rotation induces additional noise in detection probabilities, interpretable as a manifestation of the rotational Unruh effect. Finally, we investigate multi-time measurements and demonstrate the emergence of non-classical temporal correlations due to entanglement.

Paper Structure

This paper contains 15 sections, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. QTP measurement theory
  4. Time-of-arrival measurements in the real line
  5. Time-of-Arrival Probabilities in a ring
  6. Scalar Field Theory on a (1+1) Dimensional Cylinder
  7. Time-of-Arrival Probability Density
  8. Normalization and Post-Selection of Detection Events
  9. Localization operator
  10. The POVM of maximum localization
  11. Generalizations
  12. Particles in a Ring as Quantum Clocks
  13. Time-of-arrival probabilities in a rotating ring
  14. Multi-time measurements
  15. Conclusions

Figures (4)

  • Figure 1: The $Q$-symbol (\ref{['qsymbol']}) for $\varphi = \pi$, $\xi=1000, \mu=1000$, $r = 1$, $\alpha = 10$ at different time-scales where $T_q\approx282$ and $T_{rev}\approx35500$.
  • Figure 2: The cumulative probability density $W(t)$ as a function of the dimensionless time $t/(2\pi r)$.
  • Figure 3: The normalized rotation induced noise $\eta = \frac{P_0(\Omega_D)}{P_0(0)}$ of Eq. (\ref{['ffg']}) as a function of $\Omega_Dr$ and for different values of the parameter $a$.
  • Figure 4: Violation of measurement independence for massive particles. The initil state is a superposition of coherent Gaussian states with different mean momenta $\$z_1=1005$, $z_2=995$, $M=1000$, $a=10$, $r=1$, $t_1=50$.