Quantum Clocks Observe Classical and Quantum Time Dilation

The study addresses how time, operationally defined by clocks, should be modeled when clocks are inherently quantum systems subject to superposition. Building on Einstein’s operational view of time and subsequent developments in quantum metrology, time observables are treated as covariant POVMs associated with quantum clocks. This framework avoids Pauli’s objection to a self-adjoint time operator and connects to relational quantum dynamics. In relativistic contexts, quantum systems (clocks) can experience superpositions of proper times, as indicated by prior work in relativistic clock interferometry and quantum variants of the twin paradox. The paper introduces a proper time observable for internal degrees of freedom of relativistic particles moving in curved spacetime and constructs the conditional probability that one clock reads a proper time given another clock’s proper time. The central questions are: (i) Do quantum clocks recover classical special-relativistic time dilation on average when their motion is classical-like? (ii) Are there quantum corrections to time dilation when clocks move in nonclassical (e.g., momentum-superposed) states? (iii) What are the ultimate precision limits relating proper time to energy/mass in this framework?

The work situates itself within quantum metrology approaches to time estimation using covariant POVMs that saturate the Cramér–Rao bound and formalize time–energy uncertainty relations. It references foundational issues around time operators (Pauli) and the role of time observables in relational quantum dynamics. Prior studies explored superpositions of proper times in matter-wave interferometry, quantum twin paradox scenarios, and nonclassical relativistic effects, including gravitationally induced time dilation and decoherence. Related models of quantum clocks (continuous and discrete spectra) and proposals for experimental observation (e.g., Penning traps, ion/atomic clocks) are discussed. The paper extends these lines by defining a proper time observable on internal degrees of freedom consistent with relativistic dynamics and by deriving a conditional time distribution between clocks.

- Formalism: Employs the Page–Wootters relational framework, where joint clock–system states |Ψ⟩ in a physical Hilbert space satisfy a Hamiltonian constraint C|Ψ⟩ = (H_C + H_S)|Ψ⟩ = 0, enabling time as a quantum observable via conditioning. For relativistic particles with internal clocks, the Hilbert space factors as H_t ⊗ H_cm ⊗ H_clock. The induced conditional Schrödinger evolution describes center-of-mass and internal clock degrees relative to an inertial time t. - Proper time observable: Defines a clock as {H_clock, ρ, H_clock, T_clock}, with T_clock a covariant POVM under U_clock(τ) = e^{-i H_clock τ}. Covariance ensures unbiasedness and τ-independent variance for proper time estimates. Proper time is measured by a POVM covariant under the internal Hamiltonian that generates evolution with respect to the particle’s proper time. - Precision bounds: Uses Helstrom–Holevo bound for unbiased estimators to derive a proper time–energy uncertainty relation and, via M_clock := m + H_clock/c^2, a proper time–mass uncertainty relation. Identifies optimal covariant observables (E_clock(τ) = |τ⟩⟨τ|) that saturate the bound and the Mandelstam–Tamm inequality. - Conditional probability between clocks: Considers two relativistic particles (A, B) with internal clocks, constructs the joint physical state, and derives the conditional probability prob[T_A = τ_A | T_B = τ_B] using the Born rule on covariant time POVMs. Expands to leading relativistic order in P/mc and H_clock/mc^2, assuming Gaussian fiducial clock states with width σ (ideal sharp time POVM for simplicity). Assumes initially unentangled center-of-mass and internal states, and localized momentum-space Gaussian wave packets. - Leading-order evaluation: Performs perturbative evaluation of traces using covariance properties of the POVM and Gaussian states to get closed-form expressions for the conditional mean and variance of T_A given T_B. Extends analysis to the case where clock A’s center-of-mass is in a coherent superposition of two Gaussian momentum wave packets (weights set by θ and relative phase), extracting classical and quantum contributions to time dilation. - Methods (curved spacetime and constraints): Provides a Hamiltonian constraint formulation for N relativistic particles with internal degrees in curved spacetimes, introduces mass function M_n = m_n + H_n^clock/c^2, factorizes constraints, and applies Dirac quantization. Shows recovery of the relativistic Schrödinger evolution via Page–Wootters conditioning and derives leading relativistic expansion of the conditional time probability distribution.

- Conditional probability distribution: Derived a nonperturbative expression (Eq. 14) for the joint/conditional probability that one clock reads a proper time given another clock’s reading, applicable to arbitrary nonclassical states in curved spacetime. A leading-order perturbative evaluation yields a Gaussian-weighted form with relativistic corrections (Eq. 16). - Classical time dilation recovered: For both clocks’ center-of-mass prepared in localized Gaussian momentum wave packets, the conditional mean proper time satisfies ⟨T_A⟩ ≈ (1 − [(⟨H_cm,A⟩ − ⟨H_cm,B⟩)/mc^2]) T_B to leading order, consistent with special relativistic time dilation for classical momenta (Eqs. 17–21). Variance scales with the clock fiducial width σ (Eq. 18). - Quantum correction for momentum superpositions: When clock A’s center-of-mass is in a coherent superposition of two Gaussian momentum wave packets with average momenta p_A and p'_A (weights cos^2θ and sin^2θ), the average time dilation decomposes into a classical mixture term γ̄ and a purely quantum interference term γ_0^1 (Eqs. 23–25). The quantum term can increase or decrease total dilation and vanishes if the state reduces to a mixture (θ ∈ {0, π/2}) or p_A = p'_A. - Behavior of the quantum contribution: γ_0^1 depends on momentum difference (p_A − p'_A), total momentum (p_A + p'_A), and θ. There exists an optimal momentum difference maximizing |γ_0^1| for given total momentum (Fig. 2). For zero total momentum, γ_0^1 is positive for all θ and maximal at θ = π/4; with increasing total momentum, γ_0^1 shifts, allowing both positive and negative values depending on θ. - Precision bounds and speed limits: Established the Helstrom–Holevo-based proper time–energy uncertainty ⟨(ΔT_clock)^2⟩ ≥ 4/⟨(ΔH_clock)^2⟩ and the proper time–mass bound ΔM_clock ΔT_clock ≥ 1/(2 c^2), providing ultimate precision limits. The optimal covariant POVM saturates the Mandelstam–Tamm bound, yielding τ_1 = π/(2 ΔM_clock). - Experimental estimate: For 87Rb atoms with momenta corresponding to ≈5 m/s and ≈15 m/s in a momentum superposition and appropriate θ (e.g., 3π/4), the predicted quantum time dilation magnitude is γ_0^1 ≈ 10^{-15}. With internal-clock resolution ~10^{-14} s, coherence times on the order of ~10 s would be required, comparable to demonstrated atomic interferometry coherence times.

The findings demonstrate that quantum clocks defined via covariant time observables recover classical special-relativistic time dilation on average when their motion is classical-like, while exhibiting distinct quantum corrections when their center-of-mass motion is nonclassical (momentum superpositions). The conditional probability framework provides a unified, operational way to compare proper times between clocks and is compatible with curved spacetime scenarios. The proper time–energy/mass uncertainty relations position proper time and mass as genuine quantum observables with fundamental precision limits and connect to quantum speed limits. The observed quantum time dilation is not a gravitational effect but arises from coherence in momentum space, offering a new, potentially observable relativistic quantum phenomenon. The approach suggests universality across clock implementations due to covariance, though detailed modifications will depend on clock spectra and measurement models. The framework opens avenues to study effects of entanglement between clocks, spatial superpositions (including gravitational time dilation), and to develop relativistic quantum reference frames and transformations between relational clocks.

The paper introduces an optimal covariant proper time observable for relativistic quantum clocks and derives a conditional time distribution enabling direct comparisons between clocks. It shows that classical special-relativistic time dilation emerges on average for localized momentum states and identifies a quantum correction for clocks in coherent momentum superpositions. It further establishes proper time–energy and proper time–mass uncertainty relations, linking to quantum speed limits and treating mass and proper time as quantum observables. Order-of-magnitude estimates indicate potential experimental observability with current or near-future technology, though feasibility requires further investigation. Future research directions include exploring universality across clock models, entanglement-assisted or suppressed time dilation, spatial superpositions and gravitational time dilation within this framework, connections to the quantum equivalence principle, and the construction of relativistic quantum reference frames and clock changes.

- Many results are evaluated to leading order in P/mc and H_clock/mc^2; higher-order relativistic effects are not included in explicit formulas. - Idealized clock models (e.g., sharp covariant time POVMs, Gaussian fiducial states) are used for analytical tractability; real clocks with discrete spectra will introduce modifications. - Primary demonstrations are in Minkowski spacetime; while a general curved-spacetime formalism is outlined, detailed gravitational scenarios are deferred. - Assumptions of initially unentangled center-of-mass and internal degrees and specific Gaussian states limit generality; effects of entanglement and different state preparations remain to be quantified. - Experimental observation requires long coherence times (~10 s in the given estimate) and precise control of momentum superpositions, posing significant practical challenges. - The estimate of γ_0^1 is order-of-magnitude; rigorous error analysis and noise/decoherence modeling are not provided.