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NMR-active nuclei have energy levels called 'spin states' that are associated with the orientations of their nuclear magnetic moments. In the absence of a magnetic field, the nuclear magnetic moments are randomly oriented, and the spin states are degenerate. When an external magnetic field is applied, the spin states have only 2 + 1 orientations available to them. A proton with = ½ has two available orientations. Similarly, for a quadrupolar nucleus with a nuclear spin value of one, the...
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In the absence of an external magnetic field, nuclear spin states are degenerate and randomly oriented. When a magnetic field is applied, the spins begin to precess and orient themselves along (lower energy) or against (higher energy) the direction of the field. At equilibrium, a slight excess population of spins exists in the lower energy state. Because the direction of the magnetic field is fixed as the z-axis,  the precessing magnetic moments are randomly oriented around the z-axis.
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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,...
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Spin–Spin Coupling Constant: Overview01:08

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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.
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Near absolute zero temperatures, in the presence of a magnetic field, the majority of nuclei prefer the lower energy spin-up state to the higher energy spin-down state. As temperatures increase, the energy from thermal collisions distributes the spins more equally between the two states. The Boltzmann distribution equation gives the ratio of the number of spins predicted in the spin −½ (N−) and spin +½ (N+) states.
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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.
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Localization-delocalization transition in spin-orbit-coupled Bose-Einstein condensate.

Chunyan Li1, Fangwei Ye1, Yaroslav V Kartashov2,3

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Spin-orbit coupling significantly impacts the localization-delocalization transition in Bose-Einstein condensates. It can reduce the threshold for state localization, especially with band flattening effects.

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Area of Science:

  • Quantum physics
  • Atomic, molecular, and optical physics
  • Condensed matter physics

Background:

  • Bose-Einstein condensates (BECs) are quantum states of matter with unique properties.
  • Spin-orbit (SO) coupling introduces momentum-dependent forces, crucial for quantum simulations.
  • The localization-delocalization transition (LDT) is a key phenomenon in disordered or patterned quantum systems.

Purpose of the Study:

  • To investigate the influence of spin-orbit coupling on the LDT in a BEC.
  • To understand how SO coupling modifies the critical parameters for localization.
  • To explore the role of nonlinearity and simultaneous couplings in controlling the LDT.

Main Methods:

  • Theoretical analysis of a spin-orbit coupled BEC in a bichromatic potential.
  • Examination of the lowest eigenstates and their localization properties.
  • Investigation of the effects of varying SO coupling strength, potential depth, and nonlinearity.

Main Results:

  • SO coupling significantly alters the threshold potential depth for the LDT.
  • Band flattening induced by SO coupling can strongly reduce the localization threshold.
  • Simultaneous Rabi and SO coupling are essential for LDT threshold cancellation.
  • Strong SO coupling leads to behavior resembling a one-component system.
  • Defocusing nonlinearity can induce localization in states that are delocalized in the linear limit.

Conclusions:

  • Spin-orbit coupling is a critical factor in controlling quantum phase transitions in BECs.
  • The interplay between SO coupling, potential landscape, and nonlinearity offers pathways to engineer quantum states.
  • Understanding these effects is vital for designing novel quantum simulators and devices.