Quantum clock and Newtonian time

Abstract

An extension of standard quantum mechanics is proposed in which the Newtonian time parameter appearing in the unitary evolution operator is replaced with the time shown by a `quantum clock'. A quantum clock is defined by the following properties: (a) the time that the clock shows is non-decreasing, (b) the clock ticks at random with random tick sizes, and (c) on average the clock shows the Newtonian time. We show that the leading term in the evolution equation for the density matrix associated with any quantum clock model gives the von Neumann equation. Modifications to the von Neumann equation are worked out in detail in a parametric family of examples for which the tick sizes have a gamma distribution. The leading correction to the von Neumann equation is given by the Lindblad equation generated by the Hamiltonian, but there are higher-order terms that generalize the von Neumann equation and the Lindblad equation. Lower bounds on the parameters of these quantum clock models are derived by use of the precision limit of an atomic clock.

Figures (3)

• Figure 1: Gamma clock. The typical behaviour of a gamma clock for different values of the rate parameter $\kappa$ is simulated for $\kappa=100$, $\kappa=10$, $\kappa=1$ and $\kappa=0.01$. For larger $\kappa$ the clock time approaches the Newtonian time. For small $\kappa$, visible ticks become rare, but the jump size can be significant.
• Figure 2: Quantum dynamics under a quantum clock. Dynamical trajectories associated with different jump rates are simulated here, when the initial state lies on the equator of the Bloch sphere. Since the quantum state jumps from one point of the equator to another, to help visualization they are joined by straight lines, although lines are not part of the dynamics.
• Figure 3: Dynamics of the density matrix. In the case of a spin-$\frac{1}{2}$ system with the Hamiltonian ${\hat{H}}= \omega{\hat{\sigma}}_z$, the dynamical trajectory ${\hat{\rho}}(t)$ for the density matrix is confined to a latitudinal disk inside the Bloch sphere. The resulting trajectories of order up to $\lambda^0$, $\lambda^1$, $\lambda^2$, and $\lambda^3$ are plotted here with the initial state given by the eigenstate of ${\hat{\sigma}}_x$ with the eigenvalue $+1$ so that the orbits lie on the equatorial disk. Here we set $\omega=0.8$ for the angular frequency. (a) For a small $\lambda$, set at $\lambda=0.005$ here, the deviation away from the leading Lindblad (of order $\lambda$) term is negligible. (b) For a larger $\lambda$, which is set at $\lambda=0.5$ here, the trajectories become more distinguishable. The overall exponential damping is slightly suppressed by including the $\lambda^3$ term, as compared to the trajectory truncated at $\lambda^2$. Here, the unitary ($\lambda=0$) orbit is omitted.