Transient Quantum Beat Oscillations in Extreme-Relativistic Diffraction in Time

In the solution of the Klein-Gordon equation for the shutter problem, we prove that, at internuclear distances, a relativistic beam of Pi-mesons has a probability density which oscillates in time in a similar way to the spatial dependence in optical Fresnel diffraction from a straight edge. However, for an extreme-relativistic beam, the Fresnel oscillations turn into quantum damped beat oscillations. We prove that quantum beat oscillations are the consequence, at extreme-relativistic velocities, of the interference between the initial incident wave function, and the Green’s function in the relativistic shutter problem. This is a pure quantum relativistic phenomenon.


Introduction
Quantum beat oscillations are a common subject in Atomic and Molecular Spectroscopy. In Atomic physics, the term quantum beat refers to a superposed oscillatory behavior in the light intensity emitted by some suddenly excited atomic systems in their subsequent decay [1]. In Molecular Spectroscopy, quantum beat spectroscopy is a Doppler-free time domain method based on the creation of molecular coherences with a laser pulse and the measurement of their subsequent time evolution [2].
In this same context is the work of Villavicencio et al. [3], where transient phenomena of phase-modulated cutoff wave packets were explored by deriving an exact general solution to Schrödinger's equation for finite-range potentials involving arbitrary initial quantum states. They show that the dynamical features of the probability density are governed by a virtual two-level system. They also found that for a system with a bound state the interplay between the virtual levels with the latter causes a quantum beat. They also find a regime characterized by a time-diffraction oscillation. This last result came as a big surprise; it happens that in the exact Schrödinger's solution of the shutter problem [4], where we find diffraction in time as a consequence of the free time-evolution of an initial space-discontinuous beam of particles, the exact analytic expression for the probability density has a structure in which quantum beats do not exist.
Diffraction in time oscillations is a pure quantum phenomenon, and similar oscillations arise at the moment of closing and opening gates in nanoscopic circuits [5]. With adequate potentials added to the model, it has been used to study transient dynamics of tunneling matter waves [6], and the transient response to abrupt changes of the interaction potential in semiconductor structures and quantum dots [7]. For a review on the subject see [8] [9]. There is, in summary,

The Klein-Gordon Shutter Problem
In the relativistic shutter problem for spin-0 particles, we assume for all 0 t ≤ , that we have a discontinuous right-moving plane wave in the left side of a perfectly absorbing shutter, and zero to the right: here ( ) which we use to solve the Klein-Gordon (K-G) equation, The exact solution of this initial-value problem is given in the appendix A, where we use dimensionless variables for "position" χ and "time" τ defined For arbitrary ( 0 τ > , 0 χ > ), on the right-hand side of the shutter, we have the exact K-G wave function solution: , ; e d e , , e .
Here, we have defined both dimensionless "wave-number" κ and "angular-frequency" Ω as: We have also defined, the real functions: , ; e d . ,

Time-Dependent Density
Given the K-G wave function ( ) , x t ψ , the probability density is calculated by: which, for dimensionless variables, turn into ( ) , ρ χ τ , we have: Notice that the initial relativistic plane-wave, This means, as we will show, that the time-evolved density ( ) will oscillate in time around the initial value: ρ = Ω . Using Equation (5) we get for arbitrary ( 0 τ > , 0 χ > ) the exact K-G probability density: .

Relativistic Diffraction in Time for Pi-Mesons
At this point, we apply our present results to a relativistic beam of neutral Pi-mesons ( 0 π ) which are 0-spin particles. We know that Pi-mesons are the carriers of nuclear forces and for this, and no other reason, we assume a particle detector fixed at a distance x from the shutter of about,

Damped Beats in Extreme-Relativistic Diffraction in Time
The next obvious question is: how does diffraction in time look like for extreme-

Conclusions
Two simultaneous properties are needed for the existence of quantum beats. The   first is that in the probability density, which is a quadratic expression of the wave function, we have products of two different oscillatory functions. In our case in Equation (12) we have the most important contribution of these products: where the agular frequency Ω is given by The second property is that only for extreme relativistic velocities the two os- In conclusion, quantum beat oscillations are the consequence, at extreme relativistic velocities, of the interference between the initial incident wave function: e i τ Ω , and the Green's function of the relativistic shutter problem: ( ) This a pure quantum relativistic phenomenon. We claim that our result is original and has never been reported before.

A. The Klein-Gordon Solution for the 1D Shutter Problem
Let us consider the Klein-Gordon (K-G) equation for  Here both functions φ < and φ > must be bounded: ( φ < at χ → −∞ ) and ( φ > at χ → +∞ ). The important boundary condition is that the two functions and their corresponding first derivatives must be continuous at the interface, 0 χ = .
Equations (17)  The constants A and B are fixed from the continuity conditions at the interface 0 χ = : We have a set of coupled algebraic equations for the constants A and B: