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arXiv:1808.02646 · Physical Review A

Quantum free-fall motion and quantum violation of the weak equivalence principle

Philip Caesar Flores, Eric A. Galapon

Published Apr 17, 2019• 16 citations
arXivHTMLPDF

Abstract

The weak equivalence principle (WEP) in the quantum regime has been the subject of many studies with a broad range of approaches to the problem. Here, we tackle the problem anew through the time of arrival (TOA) operator approach. This is done by constructing the TOA operator for a nonrelativistic and structureless particle that is projected upward in a uniform gravitational field with an intended arrival point below the classical turning point. The TOA operator is constructed under the constraint that the inertial and gravitational masses are equivalent and that Galilean invariance is preserved. These constraints are implemented by Weyl quantization of the corresponding classical TOA function for the projectile. The expectation value of the TOA operator is explicitly shown to be equal to the classical time of arrival plus mass-dependent quantum correction terms, implying incompatibility of the WEP with quantum mechanics. The full extent of the violation of the WEP is shown through the mass dependence of TOA distribution for the projectile.

DOI
10.1103/physreva.99.042113

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The weak equivalence principle (WEP) in the quantum regime has been the subject of many studies with a broad range of approaches to the problem. Here, we tackle the problem anew through the time of arrival (TOA) operator approach. This is done by constructing the TOA operator for a nonrelativistic and structureless particle that is projected upward in a uniform gravitational field with an intended arrival point below the classical turning point. The TOA operator is constructed under the constraint that the inertial and gravitational masses are equivalent and that Galilean invariance is preserved. These constraints are implemented by Weyl quantization of the corresponding classical TOA function for the projectile. The expectation value of the TOA operator is explicitly shown to be equal to the classical time of arrival plus mass-dependent quantum correction terms, implying incompatibility of the WEP with quantum mechanics. The full extent of the violation of the WEP is shown through the mass dependence of TOA distribution for the projectile.
The weak equivalence principle of gravity is examined at the quantum level in two ways.First, the position detection probabilities of particles described by a non-Gaussian wave packet projected upwards against gravity around the classical turning point and also around the point of initial projection are calculated.These probabilities exhibit mass dependence at both these points, thereby reflecting the quantum violation of the weak equivalence principle.Second, the mean arrival time of freely falling particles is calculated using the quantum probability current, which also turns out to be mass dependent.Such a mass dependence is shown to be enhanced by increasing the non-Gaussianity parameter of the wave packet, thus signifying a stronger violation of the weak equivalence principle through a greater departure from Gaussianity of the initial wave packet.The mass dependence of both the position detection probabilities and the mean arrival time vanishes in the limit of large mass.Thus, compatibility between the weak equivalence principle and quantum mechanics is recovered in the macroscopic limit of the latter.A selection of Bohm trajectories is exhibited to illustrate these features in the free fall case.
We use the Feynman path integral approach to nonrelativistic quantum mechanics twofold. First, we derive the lagrangian for a spinless particle moving in a uniformly but not necessarily constantly accelerated reference frame; then, applying the strong equivalence principle (SEP) we obtain the Schroedinger equation for a particle in an inertial frame and in the presence of a uniform and constant gravity field. Second, using the associated Feynman propagator, we propagate an initial gaussian wave packet, with the final wave function and probability density depending on the ratio m/hbar, where m is the inertial mass of the particle, thus exhibiting the fact that the weak equivalence principle (WEP) is violated by quantum mechanics. Although due to rapid oscillations the wave function does not exist in the classical limit, the probability density is well defined and mass independent when hbar goes to 0, showing the recovery of the WEP. Finally, at the quantum level, a heavier particle does not necessarily falls faster than a lighter one; this depends on the relations between the initial and final common positions and times of the particles.
We propose tests of the weak equivalence principle (WEP) using a torsion balance, in which superposition of energy eigenstates are created in a controllable way for the test masses. After general considerations on the significance of tests of the WEP using quantum states and the need for considering inertial and gravitational masses as operators, we develop a model to derive the matrix elements of the free-fall operator, showing that the variance of the acceleration operator, in addition to its mean, enables estimation of violations of the WEP due to quantum coherence in a way that is robust with respect to shot-to-shot fluctuations. Building on this analysis, we demonstrate how the validity of the WEP may be tested in a torsion balance setup, by accessing the mean and variance of a torque operator we introduce and quantize. Due to the long acquisition times of the signal as compared to the timescale on which coherent superposition states may survive, we further propose a dynamical setting, where the torsion balance is subject to a time-dependent gravitational field, and measurements of angular acceleration encode possible violations of the WEP.
We introduce a new non-relativistic quantum operator for the distance traveled by a particle in a given interval of time. The purpose of this operator is to investigate the Weak Equivalence Principle (WEP) in the quantum regime. In particular, our operator measures the integrated expected trajectory of the particle. If its expectation value depends on the particle's mass we can infer that the particle's motion is also dependent on its mass and thus violates the WEP. In this article the new operator is derived and some of its elementary properties are explored. As a proof of concept we use it to analyze the expected distance traveled by a free Gaussian wavepacket with some initial momentum. It is shown in this case that the distance such a particle travels becomes close to light-like as its mass vanishes and agrees with the classical result for macroscopic masses. This calculation begs a number of questions which are presented as topics for future research.
The weak equivalence principle (WEP), that is, the universality of free fall, states that all point-like neutral particles in a gravitational field fall in the same way. This is the basis of the geometrization of the gravitational interaction. Together with further requirements on the behavior of point particles, light propagation and clocks one can show that gravity is modeled by a Riemannian geometry. Since in the quantum domain all objects are extended, it is not clear whether the notion of a WEP in the quantum domain makes sense at all. We show that for matter wave interferometry the notion of WEP still can be given a meaning. We give a short overview over schemes which allows a violation of the WEP and emphasize that there are also schemes which show that there might be violations of the WEP in the quantum regime which are not present classically. This makes a test of the WEP with quantum matter necessary. We also give a brief outline of the efforts made for testing the WEP with interferometry with cold atoms in the Bremen drop tower carried out by the QUANTUS and PRIMUS collaboration.
Abstract We study the effect of quantum statistics on the arrival time distribution of quantum particles computed through the probability current density. It is shown that symmetrization or asymmetrization of the wave function affects the arrival time distribution for even freely propagating particles. In order to investigate the effect of statistics on the weak equivalence principle in quantum mechanics (WEQ), we then compute the mean arrival time for wavepackets in free fall. The violation of WEQ through the effect of statistics on the mass dependence of the mean arrival time is clearly exhibited. We finally evaluate the effect of spin on the violation of WEQ using a different approach by including an explicit spin-dependence in the probability current distribution, and compare it with the approach using particle statistics. Our results show WEQ re-emerges smoothly in the limit of large mass.