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Linking the rotation of a rigid body to the Schrodinger equation: The quantum tennis racket effect and beyond.

ABSTRACT: The design of efficient and robust pulse sequences is a fundamental requirement in quantum control. Numerical methods can be used for this purpose, but with relatively little insight into the control mechanism. Here, we show that the free rotation of a classical rigid body plays a fundamental role in the control of two-level quantum systems by means of external electromagnetic pulses. For a state to state transfer, we derive a family of control fields depending upon two free parameters, which allow us to adjust the efficiency, the time and the robustness of the control process. As an illustrative example, we consider the quantum analog of the tennis racket effect, which is a geometric property of any classical rigid body. This effect is demonstrated experimentally for the control of a spin 1/2 particle by using techniques of Nuclear Magnetic Resonance. We also show that the dynamics of a rigid body can be used to implement one-qubit quantum gates. In particular, non-adiabatic geometric quantum phase gates can be realized based on the Montgomery phase of a rigid body. The robustness issue of the gates is discussed.

SUBMITTER: Damme LV

PROVIDER: S-EPMC5479806 | biostudies-other | 2017 Jun

REPOSITORIES: biostudies-other

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Linking the rotation of a rigid body to the Schrödinger equation: The quantum tennis racket effect and beyond.

Damme L Van LV Leiner D D Mardešić P P Glaser S J SJ Sugny D D

Scientific reports 20170621 1

The design of efficient and robust pulse sequences is a fundamental requirement in quantum control. Numerical methods can be used for this purpose, but with relatively little insight into the control mechanism. Here, we show that the free rotation of a classical rigid body plays a fundamental role in the control of two-level quantum systems by means of external electromagnetic pulses. For a state to state transfer, we derive a family of control fields depending upon two free parameters, which al ...[more]

PMID: 28638097

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Project description:We present a simple, one-dimensional model of an atom exposed to a time-dependent intense, short-pulse EM field with the objective of teaching undergraduates how to apply various numerical methods to study the behavior of this system as it evolves in time using several time propagation schemes.In this model, the exact Coulomb potential is replaced by a soft-core interaction to avoid the singularity at the origin. While the model has some drawbacks, it has been shown to be a reasonable representation of what occurs in the fully three-dimensional hydrogen atom.The model can be used as a tool to train undergraduate physics majors in the art of computation and software development.Program summaryProgram Title:: 1d hydrogen light interactionProgram Files doi:: http://dx.doi.org/10.17632/2275fmvdzc.1Code Ocean Capsule:: https://doi.org/10.24433/CO.1476487.v1Licensing provisions:: MIT licenseProgramming language:: FORTRAN90Nature of problem:: The one dimensional time dependent Schrödinger equation has been shown to be quite useful as a model to study the Hydrogen atom exposed to an intense, short pulse, electromagnetic field. We use a model potential that is cut-off near x = 0 and avoids the singularity of the true 1-D potential, but retains the characteristic Rydberg series and continuum to study excitation and ionization of the true H atom. The code employs a number of numerical methods to understand and compare the efficacy and accuracy when applied to this model problem.Solution method:: The program uses and contrasts a number of approaches; the Crank-Nicolson, Short Iterative Lanczos, various incarnations of the split-operator and the Chebychev method have been programmed. These methods have been compared using a 3-point finite difference (FD) discretization of the space coordinate. For completeness, some attention has also been given to using 5-9 FD formulas in order to show how higher order discretization affects the accuracy and efficiency of the methods but the primary focus of the method is the time propagation.Additional comments including restrictions and unusual features:: The main purpose of this code is as a teaching tool for undergraduates interested in acquiring knowledge of numerical methods and programming skills useful to a practicing computational physicist.

Project description:BackgroundBased on Euler/Cardan analysis, prior investigations have reported up to 80° of glenohumeral (GH) external rotation during arm elevation, dependent on the plane of elevation (PoE). However, the subtraction of Euler/Cardan angles does not compute the rotation around the humerus' longitudinal axis (i.e. axial rotation). Clinicians want to understand the true rotation around the humerus' longitudinal axis and rely on laboratories to inform their understanding of underlying shoulder biomechanics, especially for the GH joint since its motion cannot be visually ascertained. True GH axial rotation has not been previously measured in vivo, and its difference from Euler/Cardan (apparent) axial rotation is unknown.Research questionWhat is the true GH axial rotation during arm elevation and external rotation, and does it vary from apparent axial rotation and by PoE?MethodsTwenty healthy subjects (10 M/10 F, ages 22-66) were recorded using biplane fluoroscopy while performing arm elevation in the coronal, scapular and sagittal planes, and external rotation in 0° and 90° of abduction. Apparent GH axial rotation was computed using the xz'y'' and yx'y'' sequences. True GH axial rotation was computed by integrating the projection of GH angular velocity onto the humerus' longitudinal axis. One-dimensional statistical parametric mapping was utilized to compare apparent versus true axial rotation, axial rotation versus 0°, and detect differences in axial rotation by PoE.ResultsIn contrast to apparent axial rotation, true GH axial rotation does not differ by PoE and is not different than 0° during arm elevation at higher elevation angles. The spherical area between the sequence-specific and actual humeral trajectory explains the difference between apparent and true axial rotation.SignificanceProper quantification of axial rotation is important because biomechanics literature informs clinical understanding of shoulder biomechanics. Clinicians care about true axial rotation, which should be reported in future studies of shoulder kinematics.

Dataset Information

Linking the rotation of a rigid body to the Schrodinger equation: The quantum tennis racket effect and beyond.

Publications

Linking the rotation of a rigid body to the Schrödinger equation: The quantum tennis racket effect and beyond.

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