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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

The interaction of propagating space-time fluctuations with wave packets

W. L. Power

Published Mar 8, 1999• 6 citations
Publisher

Abstract

It has been suggested by Percival and Strunz that fluctuations in spacetime on the Planck scale might be detected by atom interferometers. Thus far, the models used to predict the impact of these fluctuations have been limited to a description of the atoms in terms of two states corresponding to the two arms of an interferometer. In this paper we generalise this model to include the effects of spacetime fluctuations on atomic wavepackets. When the fluctuations are assumed to have simple non–commutative properties, the density matrix of a wave packet is shown to evolve according to an equation of the same general form as that hypothesised by Ghirardi, Rimini and Weber, as a way of dealing with the measurement problem in quantum mechanics.

DOI
10.1098/rspa.1999.0346

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It has been suggested by Percival and Strunz that fluctuations in spacetime on the Planck scale might be detected by atom interferometers. Thus far, the models used to predict the impact of these fluctuations have been limited to a description of the atoms in terms of two states corresponding to the two arms of an interferometer. In this paper we generalise this model to include the effects of spacetime fluctuations on atomic wavepackets. When the fluctuations are assumed to have simple non–commutative properties, the density matrix of a wave packet is shown to evolve according to an equation of the same general form as that hypothesised by Ghirardi, Rimini and Weber, as a way of dealing with the measurement problem in quantum mechanics.
In papers on primary state diffusion (Percival 1994, 1995), numerical estimates suggested that fluctuations in the spacetime metric on the scale of the Planck time (ca.10–44s) could be detected using atom interferometers. In this paper we first specify a stochastic metric obtained from fluctuations that propagate with the velocity of light, and then develop the non–Markovian quantum state diffusion theory required to estimate the resulting decoherence effects on a model matter interferometer. Both commuting and non–commuting fluctuations are considered. The effects of the latter are so large that if they applied to some real atom interferometry experiments they would have suppressed the observed interference. The model is too crude to conclude that such fluctuations do not exist, but it does demonstrate that the small numerical value of the Planck time does not alone prevent experimental access to Planck–scale phenomena in the laboratory.
One of the biggest problems faced by those attempting to combine quantum theory and general relativity is the experimental inaccessibility of the unification scale. In this paper, we show how incoherent conformal waves in the gravitational field, which may be produced by quantum mechanical zero–point fluctuations, interact with the wave packets of massive particles. The result of this interaction is to produce decoherence within the wave packets which could be accessible in experiments at the atomic scale.Using a simple nonlinear model for the coherence properties of the gravitational field, we derive an equation for the evolution of the density matrix of such a wave packet. Following the primary state diffusion programme, the most promising source of space–time fluctuations for experimental detection is these zero–point energy fluctuations. According to our model, the absence of intrinsic irremovable decoherence in matter interferometry experiments puts bounds on some of the parameters of quantum gravity theories. Current experiments give λ >18, where λtPlanck is an effective cut–off for the validity of low–energy quantum gravity theories.
We present a generalized model for the evolution of atomic wave-packets in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers using atom clouds prepared in a trap as a Bose-Einstein condensate (BEC) or as a thermal ensemble. Predictions of the model for dynamic properties such as wave-packet size and phase are in excellent agreement with explicit numerical solutions of the non-linear Gross-Pitaevskii equations and enable fast calculations of interferometric performance in regimes where full numerical solutions become impractical. As a starting point, the static Thomas-Fermi (TF) approximation for a BEC in a harmonic trap is generalized to the whole range of atom-atom interaction strengths: from non-interacting atoms (low densities) to the standard TF limit (high atomic densities, as long as the condensate approximation still holds). In particular, this generalization allows a good estimation of atomic cloud properties along the transition from a three-dimensional to a quasi-one-dimensional BEC in an elongated trap. We then develop a theoretical model of wave-packet evolution in time-dependent conditions. The model is applicable for a wide range of dynamical problems involving evolution in time-dependent potentials and in a changing mean-field atomic repulsion due to splitting and separation of wave-packets. We use the model for studying two effects that influence interferometric coherence: imperfect spatial recombination in a two-state interferometer (the so-called "Humpty-Dumpty effect") and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential).
We present a general method for calculating wave-packet evolution in the context of ultracold atoms in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers based on atom clouds initially prepared in a trap in a Bose-Einstein condensate (BEC) state or in a thermal distribution. Without having to solve explicitely the Gross-Pitaevskii or the Schrodinger equation, this provides a good estimation of dynamic wave-packet properties such as size and phase across the wave-packet, and slows to predict the coherence of the interferometer. We develop a generalized Thomas-Fermi approximation for a BEC in a harmonic trap, which is valid for any atom-atom interaction strength. In particular, this approximation also provides a good estimation of the transition from a three-dimensional to a one-dimensional BEC in an elongated trap. The approximation is followed by a theory of dynamic wave-packet evolution when the trap is fully or partially switched off and the atoms are split and propagate, for example, in free space or in a matter waveguide. The method is applied for studying two examples of interferometric effects: reduction of coherence of two-state interferometers due to imperfect spatial recombination of the two interferometer arms (the Humpty-Dumpty effect) and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential). For both effects we extend the applicability of the theory to a wide range of interferometric scenarios that were not included in previous theories.
Assuming that “quantum spacetime” is fundamentally discrete, how might this discreteness show itself? Some of its potential effects are more evident, others less so. The atomic and molecular structure of ordinary matter influences the propagation of both waves and particles in a material medium. Classically, particles can be deflected by collisions and also retarded in their motion, giving rise in particular to viscosity and Brownian motion. In the case of spatio-temporal discreteness, viscosity is excluded by Lorentz symmetry, but fluctuating deviations from rectilinear motion are still possible. Such “swerves” have been described in and. They depend (for a massive particle) on a single phenomenological parameter, essentially a diffusion constant in velocity space. As far as I know, the corresponding analysis for a quantal particle with mass has not been carried out yet, but for massless quanta such as photons the diffusion equation of can be adapted to say something, and it then describes fluctuations of both energy and polarization (but not of direction), as well as a secular “reddening” (or its opposite). A more complete quantal story, however, would require that particles be treated as wave packets, raising the general question of how spatiotemporal discreteness affects the propagation of waves . Here, the analogy with a material medium suggests effects such as scattering and extinction, as well as possible nonlinear effects. Further generalization to a “second-quantized field” might have more dramatic, if less obvious, consequences.
We establish the fundamental properties of two classes of interferometers: those forming a pattern in momentum space, based on the superposition of two different momentum states and described by their relative propagation, and those forming a pattern in real space, based on two or more interferometer arms and described by the propagation between their local ends. These two classes are characterized by different sensitivity functions and different noise sources; we will compare the sensitivity functions of two possible configurations of atom interferometer and identify the most promising realization, with a view to applications to tests of general relativity and investigations of alterations of the spacetime curvature.