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Angle of Twist: Problem Solving

Angle of Twist: Problem Solving

An electric motor applies a torque of 700 N·m to an aluminum shaft, triggering a stable rotation. Two pulleys, B and C, are subjected to torques of 300 N·m and 400 N·m, respectively. The modulus of rigidity is provided as 25 GPa. With the knowledge of the length and diameter of each segment, the twist angle between the two pulleys can be computed. First, a section cut is made between pulleys B and C, and the cut cross-section is analyzed using a free-body diagram. Given that the...

Spin–Spin Coupling: One-Bond Coupling

Spin–Spin Coupling: One-Bond Coupling

Coupling interactions are strongest between NMR-active nuclei bonded to each other, where spin information can be transmitted directly through the pair of bonding electrons. While nuclei polarize their electrons to the opposite spins, the bonding electron pair has opposite spins. Configurations with antiparallel nuclear spins are expected to be lower in energy. When coupling makes antiparallel states more favorable, J is considered to have a positive value. The one-bond coupling constant, 1J,...

Angle of Twist - Elastic Range

Angle of Twist - Elastic Range

Consider a cylindrical shaft with a length denoted by L and a consistent cross-sectional radius referred to as r. This shaft undergoes a torque at the free end. The highest shearing strain within the shaft is directly proportional to the twist angle and the radial distance from the shaft axis. When the shaft behaves elastically, this shearing strain can be articulated using variables such as the applied torque, radial distance, the polar moment of inertia, and the modulus of rigidity. By...

NMR Spectroscopy: Spin–Spin Coupling

NMR Spectroscopy: Spin–Spin Coupling

The spin state of an NMR-active nucleus can have a slight effect on its immediate electronic environment. This effect propagates through the intervening bonds and affects the electronic environments of NMR-active nuclei up to three bonds away; occasionally, even farther. This phenomenon is called spin–spin coupling or J-coupling. Coupling interactions are mutual and result in small changes in the absorption frequencies of both nuclei involved. While nuclei of the same element are involved...

Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)

Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)

Two NMR-active nuclei bonded to a central atom can be involved in geminal or two-bond coupling. Geminal coupling is commonly seen between diastereotopic protons in chiral molecules and unsymmetrical alkenes, among others.
The central atom need not be NMR-active because its electrons are affected by the electron polarization of the spin-active atoms. However, spin information is transmitted less effectively than in one-bond coupling, and 2J values are usually weaker than 1J values. The energy of...

Spin–Spin Coupling Constant: Overview

Spin–Spin Coupling Constant: Overview

In bromoethane, the three methyl protons are coupled to the two methylene protons that are three bonds away. In accordance with the n+1 rule, the signal from the methyl protons is split into three peaks with 1:2:1 relative intensities. The methylene protons appear as a quartet, with the relative intensities of 1:3:3:1.
Qualitatively, any spin plus-half nucleus polarizes the spins of its electrons to the minus-half state. Consequently, the paired electron in the hydrogen–carbon bond must...

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Related Experiment Video

Updated: Jun 13, 2025

Magnetic Tweezers for the Measurement of Twist and Torque

Magnetic Tweezers for the Measurement of Twist and Torque

Published on: May 19, 2014

Two-axis twisting using Floquet-engineered XYZ spin models with polar molecules.

Calder Miller¹, Annette N Carroll², Junyu Lin²

¹JILA, National Institute of Standards and Technology and Department of Physics, University of Colorado, Boulder, CO, USA. calder.miller@colorado.edu.

|September 11, 2024

Summary

Researchers engineered new quantum many-body systems using polar molecules in optical lattices. Floquet engineering with microwave pulses validated spin models, enabling complex quantum dynamics and future precision measurements.

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Area of Science:

Quantum physics
Atomic and molecular physics
Condensed matter physics

Background:

Polar molecules in optical lattices offer a tunable platform for studying spin-motion dynamics via dipolar interactions.
Precise control over Ising and spin-exchange interactions using electric fields enables engineering of complex many-body dynamics.

Purpose of the Study:

To realize novel quantum many-body systems of polar molecules using Floquet engineering.
To validate spin models tuned by Floquet microwave pulses against those tuned by static electric fields.

Main Methods:

Utilized ultracold potassium-87 rubidium (40K87Rb) molecules with spin encoded in rotational states.
Employed Floquet engineering with microwave pulse sequences.
Observed Ramsey contrast dynamics for model validation.

Main Results:

Mutually validated XXZ spin models tuned by Floquet pulses and d.c. electric fields.
Observed two-axis twisting mean-field dynamics generated by a Floquet-engineered XYZ model.
Demonstrated the capability to realize Hamiltonians inaccessible with static fields.

Conclusions:

Floquet engineering provides a powerful method to create new quantum many-body systems with polar molecules.
This approach enables the simulation of complex Hamiltonians and the generation of entangled states for precision measurements.
Future applications include quantum simulation of multi-level systems leveraging the rich molecular structure.