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arXiv:2002.04588 · Physical Review A

Analytic theory for Bragg atom interferometry based on the adiabatic theorem

Jan-Niclas Kirsten-Siemß, Florian Fitzek, Sven Abend, Ernst Maria Rasel, Naceur Gaaloul, Klemens Hammerer

Published Sep 10, 2020• 23 citations
arXivHTMLPDF

Abstract

High-fidelity Bragg pulses are an indispensable tool for state-of-the-art atom interferometry experiments. In this paper, we introduce an analytic theory for such pulses. Our theory is based on the pivotal insight that the physics of Bragg pulses can be accurately described by the adiabatic theorem. We show that efficient Bragg diffraction is possible with any smooth and adiabatic pulse shape and that high-fidelity Gaussian pulses are exclusively adiabatic. Our results give strong evidence that adiabaticity according to the adiabatic theorem is a necessary requirement for high-performance Bragg pulses. Our model provides an intuitive understanding of the Bragg condition, also referred to as the condition on the ``pulse area.'' It includes corrections to the adiabatic evolution due to Landau-Zener processes as well as the effects of a finite atomic velocity distribution. We verify our model by comparing it to an exact numerical integration of the Schr\"odinger equation for Gaussian pulses diffracting four, six, eight, and ten photon recoils. Our formalism provides an analytic framework to study systematic effects as well as limitations to the accuracy of atom interferometers employing Bragg optics that arise due to the diffraction process.

DOI
10.1103/physreva.102.033709

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High-fidelity Bragg pulses are an indispensable tool for state-of-the-art atom interferometry experiments. In this paper, we introduce an analytic theory for such pulses. Our theory is based on the pivotal insight that the physics of Bragg pulses can be accurately described by the adiabatic theorem. We show that efficient Bragg diffraction is possible with any smooth and adiabatic pulse shape and that high-fidelity Gaussian pulses are exclusively adiabatic. Our results give strong evidence that adiabaticity according to the adiabatic theorem is a necessary requirement for high-performance Bragg pulses. Our model provides an intuitive understanding of the Bragg condition, also referred to as the condition on the ``pulse area.'' It includes corrections to the adiabatic evolution due to Landau-Zener processes as well as the effects of a finite atomic velocity distribution. We verify our model by comparing it to an exact numerical integration of the Schr"odinger equation for Gaussian pulses diffracting four, six, eight, and ten photon recoils. Our formalism provides an analytic framework to study systematic effects as well as limitations to the accuracy of atom interferometers employing Bragg optics that arise due to the diffraction process.
We present an analytic theory of the diffraction of (matter) waves by a lattice in the ``quasi-Bragg" regime, by which we mean the transition region between the long-interaction Bragg and ``channeling'' regimes and the short-interaction Raman-Nath regime. The Schr\"odinger equation is solved by adiabatic expansion, using the conventional adiabatic approximation as a starting point, and reinserting the result into the Schr\"odinger equation to yield a second-order correction. Closed expressions for arbitrary pulse shapes and diffraction orders are obtained and the losses of the population to output states otherwise forbidden by the Bragg condition are derived. We consider the phase shift due to couplings of the desired output to these states that depends on the interaction strength and duration and show how these can be kept negligible by a choice of smooth (e.g., Gaussian) envelope functions even in situations that substantially violate the adiabaticity condition. We also give an efficient method for calculating the effective Rabi frequency (which is related to the eigenvalues of Mathieu functions) in the quasi-Bragg regime.
The use of retroreflection in light-pulse atom interferometry under microgravity conditions naturally leads to a double-diffraction scheme. The two pairs of counterpropagating beams induce simultaneously transitions with opposite momentum transfer that, when acting on atoms initially at rest, give rise to symmetric interferometer configurations where the total momentum transfer is automatically doubled and where a number of noise sources and systematic effects cancel out. Here we extend earlier implementations for Raman transitions to the case of Bragg diffraction. In contrast with the single-diffraction case, the existence of additional off-resonant transitions between resonantly connected states precludes the use of the adiabatic elimination technique. Nevertheless, we have been able to obtain analytic results even beyond the deep Bragg regime by employing the so-called ``method of averaging,'' which can be applied to more general situations of this kind. Our results have been validated by comparison to numerical solutions of the basic equations describing the double-diffraction process.
Abstract Multi-photon Bragg diffraction is a powerful method for fast, coherent momentum transfer of atom waves. However, laser noise, Doppler detunings, and cloud expansion limit its efficiency in large momentum transfer (LMT) pulse sequences. We present simulation studies of robust Bragg pulses developed through numerical quantum optimal control. Optimized pulse performance under noise and cloud inhomogeneities is analyzed and compared to analogous Gaussian and adiabatic rapid passage pulses in simulated LMT Mach–Zehnder interferometry sequences. The optimized pulses maintain robust population transfer and phase response over a broader range of noise, resulting in superior contrast in LMT sequences with thermal atom clouds and intensity inhomogeneities. Large optimized LMT sequences use lower pulse area than Gaussian pulses, making them less susceptible to spontaneous emission loss. The optimized sequences maintain over five times better contrast with tens of ℏ k momentum separation and offer more improvement with greater LMT. Such pulses could allow operation of Bragg atom interferometers with unprecedented sensitivity, improved contrast, and hotter atom sources.
Bragg diffraction of matter waves is an established technique used in the most accurate quantum sensors. It is also the method of choice to operate large-momentum-transfer, high-sensitivity atom interferometers. It suffers, however, from an intrinsic multi-path character. Optimal control theory (OCT) has recently led to an improved robustness of atom interferometers to a range of challenging environmental effects such as vibrations or platform accelerations. In this theoretical work, we apply OCT protocols to control the Bragg diffraction phase shifts, thereby enhancing the metrological accuracy of the interferometer. We show a minimization of the diffraction phase for realistic conditions of finite temperature of the incoming wavepacket in a multi-path, high-order Bragg interferometer in a Mach–Zehnder configuration. We study input states with different momentum widths and find that our approach mitigates diffraction phases below the microradian level in the case of 1% of the photon recoil, thereby eliminating one of the leading systematic effects in atom interferometry.
This thesis develops a general theoretical and numerical framework for achieving high-contrast atom interferometry based on double Bragg diffraction (DBD). While DBD offers intrinsic symmetry, reduced sensitivity to internal-state systematics, and suitability for microgravity experiments, its performance has long been limited by imperfect diffraction and contrast loss. This work overcomes these limitations by constructing an analytic Hamiltonian description of DBD -- including Doppler effects and polarization imperfections -- and by deriving reduced two- and five-level models via a truncated Magnus-expansion approach. These models clarify the origin of AC-Stark shifts, polarization-induced errors, and Doppler selectivity, and they provide accurate predictions for realistic input momentum distributions. Building on this theoretical foundation, the thesis introduces a tri-frequency laser scheme with dynamically tunable detuning and evaluates different detuning-control strategies using a five-level S-matrix formalism. Linear detuning sweeps and optimal-control pulses are shown to provide near-ideal beam-splitter and mirror performance, respectively, ensuring robust contrast across a wide range of experimental imperfections. Complementary full three-dimensional simulations using the GPU-accelerated Universal Atom Interferometer Simulator (UATIS) incorporate interacting Bose-Einstein condensates and realistic optical potentials, revealing transverse effects and polarization-induced distortions that extend the predictions of the one-dimensional non-interacting models. Taken together, this thesis establishes a coherent theoretical and numerical framework demonstrating that, with appropriate detuning control, double-Bragg atom interferometers can achieve the robustness required for precision inertial sensing and future space-based quantum tests of fundamental physics.
The transition efficiency of atomic Bragg diffraction as mirrors and beam splitters in Bragg atom interferometers plays an essential role in impacting the fringe contrast and measurement sensitivity. This can be attributed to the properties of atomic sources, Bragg pulse shapes, the pulse duration, and the relative position deviation of the atoms and Bragg pulses. Here, we investigate the effect of the atomic source's diffusion and velocity width on the efficiency of Bragg diffraction of the moving cold atomic cloud. The transfer efficiency of Bragg mirrors and beam splitters are numerically simulated and experimentally measured, which are well consistent in comparison. We quantify these effects of atomic diffusion and velocity width and precisely compute how Bragg pulses' efficiencies vary as functions of these parameters. Our results and methodology allow us to optimize the Bragg pulses at different atomic sources and will help in the design of large momentum transfer mirrors and beam splitters in atom interferometry experiments.