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Integrable Time-Dependent Quantum Hamiltonians.

Nikolai A Sinitsyn1, Emil A Yuzbashyan2, Vladimir Y Chernyak3

  • 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.

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|May 26, 2018
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Summary
This summary is machine-generated.

We found conditions for exactly solving the time-dependent Schrödinger equation. This method allows incorporating time dependence into quantum integrable models, validating solutions for models like the Tavis-Cummings system.

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Area of Science:

  • Quantum mechanics
  • Mathematical physics

Background:

  • The time-dependent Schrödinger equation governs quantum systems evolving over time.
  • Finding analytical solutions for these equations, especially with time-dependent Hamiltonians, is a significant challenge in quantum physics.

Purpose of the Study:

  • To establish conditions for the exact analytical solvability of the nonstationary Schrödinger equation with a time-dependent Hamiltonian.
  • To develop a general strategy for introducing time dependence into quantum integrable models while preserving their integrability.

Main Methods:

  • Formulating conditions based on the existence of a non-Abelian gauge field with zero curvature in the parameter space.
  • Applying these conditions to known solvable models and extending the methodology to new systems.

Main Results:

  • Identified a key requirement: the presence of a zero-curvature non-Abelian gauge field.
  • Demonstrated that established models like the multistate Landau-Zener models meet these criteria.
  • Successfully incorporated time dependence into quantum integrable models, maintaining their solvability.
  • Validated prior conjectures, including the analytical solution for the driven generalized Tavis-Cummings model.

Conclusions:

  • The established conditions provide a pathway to analytically solve complex quantum systems with time-dependent Hamiltonians.
  • This framework offers a powerful tool for exploring time-dependent phenomena in various quantum integrable models.
  • The findings open new avenues for research in quantum dynamics and control.