Dephasing by a continuous-time random walk process

Area of Science:

• Quantum Information Science
• Spectroscopy
• Statistical Physics

Background:

• Stochastic processes are crucial in understanding complex systems in magnetic resonance and optical spectroscopy.
• Evaluating ensemble-averaged functions, like exp(i ∫ Q(s)ds), is essential for theoretical treatments.
• Existing methods may not fully capture the dynamics of systems with slow, steplike environmental changes.

Purpose of the Study:

• To derive an exact analytical solution for ensemble-averaged exponential functions involving stochastic processes.
• To develop a unified framework for analyzing both Gaussian and non-Gaussian stochastic dynamics.
• To apply the derived method to practical problems in quantum systems and spectroscopy.

Main Methods:

• Utilized a continuous-time random walk (CTRW) model to represent the stochastic process Q(s).
• Performed exact mathematical evaluation of the ensemble-averaged exponential function for the CTRW process.
• Developed a theoretical framework applicable to both Gaussian and non-Gaussian limits of the CTRW.

Main Results:

• An exact evaluation of the function (exp(i ∫ Q(s)ds)) was obtained for CTRW processes.
• The CTRW framework successfully unifies the study of Gaussian and non-Gaussian stochastic dynamics.
• Demonstrated applicability by extracting qubit-lattice interaction parameters and calculating a two-dimensional spectrum.

Conclusions:

• The developed method offers a powerful tool for analyzing stochastic dynamics in spectroscopy.
• The CTRW approach provides a versatile framework for studying diverse physical phenomena.
• This work facilitates a deeper understanding of dephasing and spectral properties in quantum systems.