Time-of-flight measurements in quantum mechanics

A widely used method in metrology is the time-of-flight measurement. The momentum can be determined via the measurement of a time difference if fixed positions are given. Figure 1 shows such a concept.

In this work, we examine quantum mechanical time-of-flight momentum measurements, which are motivated by the classical time-of-flight concept. Our setup is modeled by a Hamiltonian for a free particle interacting with two quantum pointers, see Ref. [1-4], which are able to record information about the particle.

For this model we can solve the corresponding three-particle dynamics in the Heisenberg picture and use this solutions in order to define an operational momentum operator and an operational momentum eigenvalue, both in the spirit of classical time-of-flight measurements. This allows us to examine two fundamental aspects: First, we verify that the expectation value of the operational momentum operator equals the one of the original momentum operator describing the single-particle system. Moreover, we examine the momentum variance and characterize additional noise caused by the interaction of the particle with the pointers. This additional noise is then minimized by preparing appropriate pointer states, shown in Fig. 2.
Second, we analyze in what sense position measurements on the pointers reduce the conditional quantum state of the particle and hence define its momentum. All results can be found in Ref. [5].