Entanglement, Correlations, and Time Dilation in Two-Mode Quantum Systems

In a quantum clock, proper time accumulates differently across its wave packet, producing interference effects with no classical analogue. This quantum noise sets a fundamental limit to precision time-keeping. Within a covariant geometric framework valid in arbitrary background spacetimes, we derive an explicit expression for the effective time dilation experienced by a bosonic two-mode quantum system. In flat spacetime, the result takes a compact form in terms of the symplectic eigenvalues of the covariance matrix, which act as effective "quantum velocities" describing the clock's internal motion. This time-dilation formula predicts quantitatively how entanglement-assisted timekeeping with two-mode clocks can improve precision by exploiting inter-mode correlations to suppress intrinsic energy (frequency) fluctuations, analogous to the way entangled probes surpass the standard quantum limit in phase estimation.