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arXiv:2410.08040 · Physical Review A

Beyond the semiclassical approximation in atom interferometry

W. LaRow, Mark Edwards, Charles A. Sackett

Published Mar 19, 2025• 1 citation
arXivHTMLPDF

Abstract

We describe a quantum perturbative approach to evaluating the phase shift of an atom interferometer in a weakly anharmonic trap. This provides a simple way to evaluate quantum corrections to the standard semiclassical approximation. The calculation benefits from the use of generalized coherent states for a basis. We find that the form of the semiclassical approximation remains valid to first order in the anharmonic perturbation, but that phase differences arise because the trajectory of a quantum wave packet will generally deviate from that of a classical particle. The quantum correction to the phase is a factor ${\ensuremath{\ell}}^{2}/{A}^{2}$ smaller than the semiclassical perturbation itself, where $\ensuremath{\ell}$ is the quantum harmonic-oscillator lengthscale, and $A$ is the classical amplitude of the motion. We provide analytical results for one-dimensional perturbations of power 3 through 6 in the position coordinate.

DOI
10.1103/physreva.111.033315

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We describe a quantum perturbative approach to evaluating the phase shift of an atom interferometer in a weakly anharmonic trap. This provides a simple way to evaluate quantum corrections to the standard semiclassical approximation. The calculation benefits from the use of generalized coherent states for a basis. We find that the form of the semiclassical approximation remains valid to first order in the anharmonic perturbation, but that phase differences arise because the trajectory of a quantum wave packet will generally deviate from that of a classical particle. The quantum correction to the phase is a factor \ensuremathℓ2/A2{\ensuremath{\ell}}^{2}/{A}^{2}\ensuremathℓ2/A2 smaller than the semiclassical perturbation itself, where \ensuremathℓ\ensuremath{\ell}\ensuremathℓ is the quantum harmonic-oscillator lengthscale, and AAA is the classical amplitude of the motion. We provide analytical results for one-dimensional perturbations of power 3 through 6 in the position coordinate.
A free-oscillation interferometer uses atoms confined in a harmonic trap. Bragg scattering from an off-resonant laser is used to split an atomic wave function into two separated packets. After one or more oscillations in the trap, the wave packets are recombined by a second application of the Bragg laser to close the interferometer. Anharmonicity in the trap potential can lead to a phase shift in the interferometer output. In this paper, analytical expressions for the anharmonic phase are derived at leading order for perturbations of arbitrary power in the position coordinate. The phase generally depends on the initial position and velocity of the atom, which are themselves typically uncertain. This leads to degradation in the interferometer performance, and can be expected to limit the use of a centimeter-scale device to interaction times of about 0.1 s. Methods to improve performance are discussed.
A Sagnac atom interferometer can be constructed using a Bose–Einstein condensate trapped in a cylindrically symmetric harmonic potential. Using the Bragg interaction with a set of laser beams, the atoms can be launched into circular orbits, with two counterpropagating interferometers allowing many sources of common-mode noise to be excluded. In a perfectly symmetric and harmonic potential, the interferometer output would depend only on the rotation rate of the apparatus. However, deviations from the ideal case can lead to spurious phase shifts. These phase shifts have been theoretically analyzed for anharmonic perturbations up to quartic in the confining potential, as well as angular deviations of the laser beams, timing deviations of the laser pulses, and motional excitations of the initial condensate. Analytical and numerical results show the leading effects of the perturbations to be second order. The scaling of the phase shifts with the number of orbits and the trap axial frequency ratio are determined. The results indicate that sensitive parameters should be controlled at the 10−5 level to accommodate a rotation sensing accuracy of 10−9 rad/s. The leading-order perturbations are suppressed in the case of perfect cylindrical symmetry, even in the presence of anharmonicity and other errors. An experimental measurement of one of the perturbation terms is presented.
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In this Master's thesis, we give a pedagogical introduction to the semiclassical analysis of wave dynamics in single-band and multi-band systems. We show how the semiclassical evolutions of Wigner functions can approximate the dynamics of wave packet centers and wave packet deformations. Three systems of increasing complexity are considered: the quantum harmonic oscillator and the quantum anharmonic oscillator as single-band systems, and the Dirac oscillator as a two-band system. We explain the origins of the first-order semiclassical corrections (Moyal, Littlejohn-Flynn, and Berry) in the equations of motion and examine their effects on predictive accuracy. Another focus is set on the conditions for validity of the semiclassical analysis. For both the quantum anharmonic and the Dirac oscillators, a dimensionless parameter controlling the accuracy of the approximation is identified. In multi-band systems, the formalism applies only to states within a single adiabatic subspace, which, for the Dirac oscillator, corresponds to an energy branch. This motivates the construction of single-branch almost coherent states: approximately Gaussian wave packets formed by superpositions of states from a single energy branch. Their centers undergo oscillations with an aperiodic envelope. These states enable comparison between semiclassical Wigner function evolution and the commonly used Sundaram-Niu formalism. We show that semiclassical Wigner function evolution captures deformation-related dynamics, including the initial decay of oscillations, which the Sundaram-Niu formalism cannot describe by construction. Additionally, exact quantum evolution reveals partial relocalization of these single-branch almost coherent states, a phenomenon that cannot be captured by the semiclassical evolution.
The time-dependent WKB approximation for a coherent state is expanded to third order around a guiding real trajectory, allowing for the novel treatment of nonlinearity in its semiclassical dynamics, which is generally present in all physical systems far from the classical regime. The result is a closed-form solution consisting of a linear combination of Airy functions and their derivatives multiplied by an exponential. The expression's ability to capture nonlinearity is demonstrated by examining the autocorrelation of initial coherent states in anharmonic systems with few to many contributing periodic orbits. Its accuracy is compared to the quadratic expansion and found to be superior in regimes of ℏ where the curvature begins to be significant, as expected. Moreover, the expression is shown to be a real-trajectory uniformization over two coalescing saddle points that are emblematic of significant curvature. This extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time and establishes their regime of validity.