Abstract
Spin precession in magnetic materials is commonly modelled with the classical phenomenological Landau–Lifshitz–Gilbert (LLG) equation. Based on a quantized three-dimensional spin + environment Hamiltonian, we here derive a spin operator equation of motion that describes precession and includes a general form of damping that consistently accounts for memory, coloured noise and quantum statistics. The LLG equation is recovered as its classical, Ohmic approximation. We further introduce resonant Lorentzian system–reservoir couplings that allow a systematic comparison of dynamics between Ohmic and non-Ohmic regimes. Finally, we simulate the full non-Markovian dynamics of a spin in the semi-classical limit. At low temperatures, our numerical results demonstrate a characteristic reduction and flattening of the steady state spin alignment with an external field, caused by the quantum statistics of the environment. The results provide a powerful framework to explore general three-dimensional dissipation in quantum thermodynamics.
Highlights
The continued miniaturisation of critical components in consumer electronics and neighbouring technologies will require a deeper understanding of thermal noise and general thermodynamic principles beyond the classical macroscopic world
Attempts have pursued a path integral derivation of a quantum spin dynamics equation [51], as well as other conceptually related classical and quantum derivations [52, 53]. These derivations were not directly applied to the calculation of magnetization dynamics or steady states, nor have they been connected to recent generalizations of Gilbert damping that include inertial terms [49, 50, 54,55,56,57,58,59] or provided an assessment of quantum effects
Here we have demonstrated that Lorentzian coupling functions, kernels and power spectra provide a systematic framework to explore system dynamics that arises from inertial terms and other memory effects, while recovering the standard Ohmic limit whenever the Lorentzian resonance frequency ω0 is much larger than the typical system frequencies
Summary
The three-dimensional spin dynamics equation (11) describes the evolution of spin operators and explicitly includes memory of the past dynamics (nonMarkovianity) This contrasts with previous derivations of quantum spin dynamics in the form of a master equation [69] which can be solved numerically. Similar to (8), this model assumes that the equation of motion of the spins is coupled to a stochastic equation of motion for the lattice enabling transfer of energy and angular momentum between the lattice and the spin systems. Different coupling potentials, such as harmonic and Morse potentials, have been considered and the spin-lattice coupling impact on the magnetisation has been characterised [71]. The spin-boson model is recovered as a special case for spin 1/2 operators and rank-1 coupling tensors, see Appendix A1
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