Clocks That Tick Quantumly express
A regular wall clock runs smoothly, but quantum clocks are a different story. They tick at random moments, and each tick can have a different duration. On average, if you watch such a clock for a long time, it will show the correct time. But at any given moment, its hand might jump forward by an unexpected amount.
The authors replaced the usual time in quantum mechanics equations with the readings of such 'quantum clocks.' It turns out that the leading term in the equation for the density matrix (which describes the state of a quantum system) yields the familiar von Neumann equation. The next correction, however, is the Lindblad equation, which typically describes losses and noise. Further corrections are even more subtle and were previously unknown.
For a free particle, the corrections lead to an anomaly in the Ehrenfest theorem — the rule connecting the average values of position and momentum. If such an anomaly is detected in an experiment, it would signal that time in the quantum world is indeed not as smooth as we tend to think.
Original abstract
as written by the authors arXiv:2602.02077
An extension of standard quantum mechanics is proposed in which the Newtonian time appearing as a parameter in the unitary evolution operator is replaced with the time shown by a `quantum clock'. Such a clock is defined by the following properties: (a) the time that the clock shows is nondecreasing, (b) the clock ticks at random Newtonian times 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 gives the von Neumann equation. 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 terms. Modifications to the von Neumann equation are worked out in detail in a parametric family of models for which the tick sizes are gamma distributed. Lower bounds on the parameters of these quantum clock models are derived using the precision limit of an atomic clock. An anomalous term in the Ehrenfest theorem for a free particle is derived, which in principle can be used as a basis for testing such models.
Atomic clocks are so accurate that over the age of the Universe they would have drifted by less than a second. It is precisely their accuracy that sets the lower bound on the parameters of quantum clocks in this model.