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Physical Review A

Generalized crossing states in the interacting case: The uniform gravitational field

Anthony Villanueva, Eric A. Galapon

Published Nov 29, 2010• 5 citations
Publisher

Abstract

We reconsider Baute et al.'s free crossing states [Phys. Rev. A 61, 022118 (2000)] and show that if we require a generalization in the interacting case that goes in complete parallel with the free-particle case, then this generalized crossing state cannot be arbitrary but is determined by the null space of the particle's quantum time-of-arrival operator. Nonetheless, the free crossing states appear as the leading term in the asymptotic expansion of our generalized crossing state in the limit of large momentum. We then examine the quantum time-of-arrival problem of a spinless particle in a uniform gravitational field. Mass-dependent time-of-arrival probability distributions emerge, signifying quantum departures from the weak equivalence principle. However, in the classical limit of large mass and vanishing uncertainty in position, the mass dependence of the quantum time-of-arrival distribution becomes exponentially small and the mean quantum time of arrival reduces to the classical time of arrival.

DOI
10.1103/physreva.82.052117

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We reconsider Baute et al.'s free crossing states [Phys. Rev. A 61, 022118 (2000)] and show that if we require a generalization in the interacting case that goes in complete parallel with the free-particle case, then this generalized crossing state cannot be arbitrary but is determined by the null space of the particle's quantum time-of-arrival operator. Nonetheless, the free crossing states appear as the leading term in the asymptotic expansion of our generalized crossing state in the limit of large momentum. We then examine the quantum time-of-arrival problem of a spinless particle in a uniform gravitational field. Mass-dependent time-of-arrival probability distributions emerge, signifying quantum departures from the weak equivalence principle. However, in the classical limit of large mass and vanishing uncertainty in position, the mass dependence of the quantum time-of-arrival distribution becomes exponentially small and the mean quantum time of arrival reduces to the classical time of arrival.
Abstract The probability distribution of a time measurement Tx at position x can be inferred from the probability distribution of a position measurement Xt at time t as given by the Born rule [1, 2]. In an application to free-fall, this finding has been used to predict the existence of a mass-dependent positive relative shift with respect to the classical time-of-arrival in the long time-of-flight regime for dropped quantum particles [1]. The present paper extends these results in two important directions. We first show that for a Gaussian quantum particle of mass m dropped in a uniform gravitational field g, the uncertainties about time and position measurements are related by the relation Δ T x Δ X t ≥ ℏ 2 m g This novel form of uncertainty relation suggests that choosing the initial state so as to obtain a lower uncertainty in the measured position leads to a higher uncertainty in the measured arrival time. Secondly, we examine the case of a free-falling particle starting from a non-Gaussian initial superposed state, for which we predict the presence of gravitationally induced interferences and oscillations in the mean time-of-arrival as a function of the detector’s position that can be interpreted as the signature of a Zitterbewegung-like effect.
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.
A realization of the concept of ``crossing state'' invoked, but not implemented, by Wigner, allows advancement in two important aspects of the formalization of the time of arrival in quantum mechanics: (i) For free motion, we find that the limitations described by Aharonov et al. in Phys. Rev. A 57, 4130 (1998) for the time-of-arrival uncertainty at low energies for certain measurement models are in fact already present in the intrinsic time-of-arrival distribution of Kijowski ${\ensuremath{\Pi}}_{K};$ (ii) we have also found a covariant generalization of this distribution for arbitrary potentials and positions.
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.
Using standard results from statistics, we show that for Gaussian quantum systems the distribution of a time measurement at a fixed position can be directly inferred from the distribution of a position measurement at a fixed time as given by the Born rule. In an application to a quantum particle of mass mmm falling in a uniform gravitational field ggg, we use this approach to obtain an exact explicit expression for the probability density of the time of arrival (TOA). In the long time-of-flight approximation, we predict that the average positive relative shift with respect to the classical TOA in case of a zero initial mean velocity is asymptotically given by $\ensuremath{\delta}=\frac{{q}^{2}}{2}$ when the factor $q\ensuremath{\equiv}\frac{\ensuremath{\hbar}}{2m\ensuremath{\sigma}\sqrt{2gx}}\ensuremath{\ll}1$ (semiclassical regime) and by $\ensuremath{\delta}=\sqrt{{\frac{2}{\ensuremath{\pi}}}^{\phantom{\ensuremath{\int}}}}q$ when $q\ensuremath{\gg}1$ (quantum regime), where $\ensuremath{\sigma}$ is the width of the initial Gaussian wave packet and xxx is the mean distance to the detector. We also discuss experimental conditions under which these predictions can be tested.
Abstract We ask the question of how the (weak) equivalence principle established in classical gravitational physics should be reformulated and interpreted for massive quantum objects that may also have internal degrees of freedom (dof). This inquiry is necessary because even elementary concepts like a classical trajectory are not well defined in quantum physics—trajectories originating from quantum histories become viable entities only under stringent decoherence conditions. From this investigation we posit two logically and operationally distinct statements of the equivalence principle for quantum systems. Version A: the probability distribution of position for a free-falling particle is the same as the probability distribution of a free particle, modulo a mass-independent shift of its mean. Version B: any two particles with the same velocity wave-function behave identically in free fall, irrespective of their masses. Both statements apply to all quantum states, including those without a classical correspondence, and also for composite particles with quantum internal dof. We also investigate the consequences of the interaction between internal and external dof induced by free fall. For a class of initial states, we find dephasing occurs for the translational dof, namely, the suppression of the off-diagonal terms of the density matrix, in the position basis. We also find a gravitational phase shift in the reduced density matrix of the internal dof that does not depend on the particle’s mass. For classical states, the phase shift has a natural classical interpretation in terms of gravitational red-shift and special relativistic time-dilation.