Quantum Clocks Observe Classical and Quantum Time Dilation

Einstein's revolutionary insight, defining time as what a clock measures, is extended to the quantum realm. This operational view of time leads to defining time observables as positive-operator valued measures (POVMs) covariant under time translations, ensuring optimal time estimates and a rigorous formulation of the time-energy uncertainty relation. The paper addresses the possibility of quantum clocks experiencing a superposition of proper times, a scenario explored in relativistic clock interferometry, where time dilation leads to reduced interferometric visibility. This work builds upon existing research into quantum variants of the twin paradox and other non-classical effects in relativistic settings by introducing a proper time observable defined as a covariant POVM on the internal degrees of freedom of a relativistic particle moving through curved spacetime. This framework enables the analysis of two relativistic quantum clocks, allowing for calculation of the conditional probability distribution, which is a central focus of this study.

The paper reviews existing literature on quantum clocks and their applications in quantum metrology. It highlights the use of covariant POVMs to define time observables, ensuring optimal time estimation and addressing Pauli's objection to a time operator. The authors discuss the significance of the covariance property in ensuring unbiased estimation and variance independent of time. The concept of quantum clocks exhibiting superposition of proper times, examined in studies of relativistic clock interferometry and variants of the twin paradox, is critically reviewed. The previous works have mainly concentrated on special or general relativistic time dilation in matter-wave interferometers which decrease the interferometric visibility as a signature of matter experiencing a superposition of proper times, however the approach taken in this study is quite different. The concept of superpositions of different proper times and existing attempts to explore it via relativistic clock interferometry and related studies are critically analyzed as context for the authors' approach.

The authors employ the Page-Wootters approach to relational quantum dynamics, extended to relativistic particles with internal degrees of freedom. They define a relativistic particle with an internal clock degree of freedom, described by the Hilbert space H_t ⊗ H_cm ⊗ H_clock, where H_t, H_cm, and H_clock represent the temporal, center-of-mass, and internal clock degrees of freedom, respectively. The physical state satisfies the constraint equation C|Ψ⟩ = (H_C + H_S)|Ψ⟩ = 0, where H_C and H_S are the Hamiltonians of the clock and the system, respectively. A proper time observable is defined as a covariant POVM on the internal degrees of freedom, satisfying the covariance condition E_C(t + t') = U_C(t')E_C(t)U_C(t')†. The authors prove that a covariant time observable is an unbiased estimator of the parameter τ and has a variance independent of τ. This formalism is generalized to curved spacetime and shown to recover the Klein-Gordon equation (detailed in supplementary material). The conditional probability that clock A reads proper time T_A given clock B reads T_B is calculated using the Born rule and the physical state. The analysis then focuses on clocks prepared in localized momentum wave packets, evaluating this conditional probability to leading relativistic order, leading to the derivation of classical and quantum time dilation effects. The calculation involves expanding the Hamiltonian to leading relativistic order and employing Gaussian wave packets for the clock states to simplify computations. A specific idealized clock model with orthogonal clock states is used initially for simplicity. The Helstrom-Holevo lower bound is used to derive the time-energy/mass uncertainty relation. The analysis concludes by exploring specific clock models such as those based on the width of an atomic emission line.

The paper's key findings include: (1) The derivation of a conditional probability distribution for two relativistic quantum clocks, showing how the probability that one clock reads a specific proper time depends on the other clock's reading. (2) The demonstration that quantum clocks with center-of-mass states localized in momentum space exhibit classical time dilation on average. (3) The identification of a quantum correction to time dilation when one clock's center-of-mass is in a superposition of localized momentum wave packets. This quantum correction, denoted γ₀¹, is shown to depend on the parameters of the momentum superposition, and it can be either positive or negative. (4) The derivation of a proper time-energy/mass uncertainty relation using the Helstrom-Holevo lower bound, providing a fundamental limit on the precision of proper time measurements. This relation is shown to be saturated by optimal covariant proper time observables. (5) A detailed mathematical analysis illustrating how the magnitude of quantum time dilation varies with the difference and total average momentum of the momentum wave packets involved in the superposition, and how it changes with the weighting of these packets in the superposition. (6) An order of magnitude estimation indicating that observing the quantum time dilation effect may be achievable with current technology by utilizing ⁸⁷Rb atoms as clocks in a superposition of momentum wave packets, potentially through techniques like momentum beam splitters.

The findings demonstrate that relativistic quantum clocks show classical time dilation on average when their center-of-mass is localized in momentum space, thus validating special relativity in this quantum framework. However, the emergence of a quantum correction to time dilation when a clock is in a momentum superposition highlights the unique behavior of quantum systems in relativistic scenarios. This quantum effect arises from the non-classical nature of the superposition, causing a deviation from the time dilation observed in a corresponding classical mixture of states. The authors provide an order-of-magnitude estimate for the quantum correction, suggesting potential experimental verification using existing technologies such as atomic clocks and spectroscopic techniques. The time-energy/mass uncertainty relation derived emphasizes that a complete quantum description of relativistic systems requires considering both mass and proper time as dynamical quantum observables, which has been a topic of debate in the literature. The specific clock models employed in the calculations were chosen for simplicity but the fundamental principles indicate that quantum time dilation should be a universal phenomenon, affecting all types of clocks. The results open up new avenues for exploring the interplay between quantum mechanics and relativity, including the construction of relativistic quantum reference frames and the investigation of different relational perspectives. Future research may also consider entanglement effects and the influence of spatial superpositions, allowing for a broader understanding of quantum time dilation effects and their relation to gravity and quantum reference frames.

This paper presents a comprehensive theoretical framework for analyzing quantum clocks in relativistic settings, demonstrating the existence of a quantum correction to classical time dilation in momentum superposition states. The derived proper time-energy/mass uncertainty relation provides a fundamental constraint on time measurements. The feasibility of experimental verification is discussed, suggesting potential avenues for future research using existing technologies. This study contributes significantly to the field by providing a rigorous theoretical foundation for investigating the quantum nature of time and its interaction with relativistic effects.

The analysis uses idealized clock models, and the impact of more realistic clock imperfections on the quantum time dilation effect warrants further investigation. The calculation of the conditional probability distribution relies on perturbative approximations, which might limit its accuracy for systems far from the classical regime. Experimental realization of the suggested experiment will require precise control over the momentum superposition and maintaining coherence over long timescales, which present substantial experimental challenges. The extension of this framework to explore the effect of entanglement between clocks or other non-classical features is required for a more complete understanding. Further research is also needed to determine if the effect is universal for all types of clocks.