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Physical Review A

Quantum wave-packet size effects on neutron time-of-flight spectroscopy

Eric A. Galapon

Published Sep 30, 2009• 15 citations
Publisher

Abstract

Neutron time-of-flight interferometry assumes the validity of the classical time-of-flight expression in the interpretation of time-of-flight data and simultaneously assumes that the incident neutrons are de Broglie wave packets obeying the laws of quantum mechanics. A quantum-mechanical treatment of time of flight as an expected value of an appropriate time-of-arrival operator for a given initial wave-packet state shows that the size of the initial wave packet introduces measurable quantum corrections to the expected classical time of flight. Our results provide concrete testable predictions that may clarify the status of time as a quantum-dynamical observable.

DOI
10.1103/physreva.80.030102

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Neutron time-of-flight interferometry assumes the validity of the classical time-of-flight expression in the interpretation of time-of-flight data and simultaneously assumes that the incident neutrons are de Broglie wave packets obeying the laws of quantum mechanics. A quantum-mechanical treatment of time of flight as an expected value of an appropriate time-of-arrival operator for a given initial wave-packet state shows that the size of the initial wave packet introduces measurable quantum corrections to the expected classical time of flight. Our results provide concrete testable predictions that may clarify the status of time as a quantum-dynamical observable.
We propose a scheme to experimentally observe matter-wave interference in the time domain, specifically in the arrival time or time-of-flight (TOF) distribution for an atomic Bose–Einstein condensate (BEC) Schrödinger-cat state represented by superposition of macroscopically separated wave packets in space. This is in contrast to interference in space at a fixed time observed in the reported BEC experiments. We predict and quantify the quantum interference in the TOF distribution calculated from the modulus of the quantum probability current density (rather than the TOF distributions obtained from a purely classical or semi-classical treatment in many reported experiments). The interference and hence the coherence in the quantum TOF signal disappears in the large-mass limit. Our scheme has the potential to probe the validity of various other theoretical approaches (Muga and Leavens 2000 Phys. Rep. 338 353) of calculating the quantum arrival time distribution.
The time-of-flight measurement approach of Peres based on the Salecker-Wigner quantum clock is applied to the one-dimensional scattering of a wave packet from a rectangular barrier. By directly evaluating the expectation value of the clock-time operator in the asymptotic states of wave packet long after the scattering process, we derive an average wave-packet clock time for the barrier interaction, which is expressed as an average of the stationary-state clock time over all possible initial scattering states of the wave packet. We show that the average wave-packet clock time is identical to the average dwell time of a wave packet.
In this paper we study the construction of probability densities for time-of-flight in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time-of-flight appear naturally when one considers histories. The definition of time-of-flight probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of Schr\\"odinger's equation cannot naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in history based schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit--there are however problems related to the quantum Zeno effect and decoherence. We finally employ the histories-based description to construct Positive-Operator-Valued-Measures (POVMs) for the time-of-arrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski.
There is an ongoing tension between two neutron lifetime measurements. The so-called beam method gives a neutron lifetime of 888 seconds, while the trap (or bottle) method reports a shorter lifetime of 879 seconds. This discrepancy has sparked debates across experimental and theoretical fronts. While the intriguing possibility of a dark decay mode was initially considered, it was promptly ruled out by experimental evidence, leaving the discussion unresolved. In our work, we explore a novel approach to addressing the neutron lifetime anomaly using the quantum time operator formalism. We introduce a time operator that projects the initial system into a state evolving unitarily into a singular Dirac delta-like function, enabling localized interactions or decay. Our analysis reveals that for low-energy systems, the expectation value of this operator deviates from the classical time value due to quantum corrections. Interestingly, these corrections vanish in the classical regime of small de Broglie wavelengths. Moreover, we found that the quantum corrections are sensitive to the initial state preparation, offering a potential explanation for the observed discrepancy between the two measurement methods.
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.