Physically significant phase shifts in matter-wave interferometry

The action S of a trajectory x(t) in the time interval ${t0,tf}$ is defined to be $S=∫t0tfL(x,ẋ,t)+dt,$ where $L$ is the Lagrangian of the system. Physically, the action is proportional to the proper time evolved along the trajectory (Ref. 21).

Unlike the classical accelerometer, the interferometer does not provide information about the three position measurements individually.

It is worth emphasizing that there is no fundamental physical difference between “open” interferometers and “closed” interferometers (in which the central trajectories of the two arms overlap at the time of the final beamsplitter). In order to observe first-order coherence in a particular region, the probability density $|ψ|2$ must have the form $|ψ|2∼|ψ1+ψ2|2=|ψ1|2+|ψ2|2+ψ1*ψ2+ψ1ψ2*.$ Here, ψ₁ and ψ₂ represent the amplitudes of the wavefunction on two different trajectories, and the phenomenon of interference is represented by the terms $ψ1*ψ2$ and $ψ1ψ2*$⁠. The interference terms are zero unless the region contains nonzero amplitude from both trajectories. Physically speaking, every interferometer is closed. The notion of an “open” interferometer is useful only insofar as it indicates the parametric dependence of physical observables, e.g., the dependence of the interferometer phase on initial conditions.

To be precise, $ϕ¯sep$ cancels the boundary term that appears when $ϕprop$ is integrated by parts. See Sec. V for further details.

This approximation, which is used to simplify the calculation, is not necessary for the result. The same result is obtained if the light pulses occur over finite intervals (Ref. 15) as long as the motion of the arms during the light pulses is taken into account.