A nonlinear dynamics semi-classical model is used to show that standard quantum spin analysis can be obtained. The model includes a classically driven nonlinear differential equation with dissipation and a semi-classical interpretation of the torque on a spin magnetic moment in the presence of a realistic magnetic field, which will represent two equilibrium positions. The highly complicated driven nonlinear dissipative semi-classical model is used to introduce chaos, which is necessary to produce the correct statistical quantum results. The resemblance between this semi-classical spin model and the thoroughly studied classical driven-damped nonlinear pendulum are shown and discussed.
In an effort to provide an alternative method to represent a quantum spin, a precise 3D nonlinear dynamics method is used. A two-sided torque function is created to mimic the unique behavior of the quantum spin. A full 3D representation of the magnetic field of a Stern–Gerlach device was used as in the original experiment. Furthermore, the temporarily driven nonlinear damped model exhibits chaos, but struggles to be consistent through azimuthal angles in reproducing the well known quantum spin statistics.
Classical nonlinear theories are highly successful in describing far-from-equilibrium dynamics of magnets, encompassing phenomena such as parametric resonance, ultrafast switching, and even chaos. However, at ultrashort length and time scales, where quantum correlations become significant, these models inevitably break down. While numerous methods exist to simulate quantum many-body spin systems, they are often limited to near-equilibrium conditions, capture only short-time dynamics, or obscure the intuitive connection between nonlinear behavior and its geometric origin in the su(2) spin algebra. To advance nonlinear magnetism into the quantum regime, we develop a theory in which semiclassical spin correlations, rather than individual spins, serve as the fundamental dynamical variables. Defined on the bonds of a bipartite lattice, these correlations are inherently nonlocal, with dynamics following through a semiclassical mapping that preserves the original spin algebra. The resulting semiclassical theory captures nonlinear dynamics that are entirely nonclassical and naturally accommodates phenomenological damping at the level of correlations, which is typically challenging to include in quantum methods. As an application, we focus on Heisenberg antiferromagnets, which feature significant quantum effects. We predict nonlinear scaling of the mean frequency of quantum oscillations in the Néel state with the spin quantum number S. These have no classical analog and exhibit features reminiscent of nonlinear parametric resonance, fully confirmed by exact diagonalization. The predicted dynamical features are embedded in the geometric structure of the semiclassical phase space of spin correlations, making their physical origin much more transparent than in full quantum methods. With this, semiclassical spin-correlation dynamics provide a foundation for exploring nonlinear quantum magnetism.
We develop a quantum model for the rotational dynamics of a freely floating levitated ferromagnetic gyroscope (LFG), emphasizing the interplay between intrinsic spin
S, mechanical angular momentum
L, and magnetic torque. The conserved total angular momentum projection along the
z-directed magnetic field
B,
Jz=Sz+Lz, is quantized, leading in the small-libration-amplitude limit to discrete precessional states
∣m⟩ (eigenstates of
Jz with eigenvalues
Jz=mℏ) and librational harmonic oscillator states
∣n⟩ (
n=0,1,2,…). The discreteness of the energies and dynamical variables is governed by the quantum precession scale
ΩQ=ℏ/I, where
I is the moment of inertia of the LFG. We find that the phenomenon of LFG precession persists into high-field regimes where the magnitude of the rotational angular momentum associated with precession exceeds the total intrinsic spin. We analyze the complementary quantum limits of localized semiclassical LFG orientation wave packets and exact
Jz-eigenstates
∣m⟩, clarifying the relation between classical precession signals and the underlying quantized spin-rotor dynamics. We further show that radio-frequency fields can drive
Δm=±1 and
Δn=±1 transitions, enabling ladder spectroscopy, tilt-angle control, and sideband-like coupling between precession and libration. The coupled dynamics also exhibit branch-point magnetic resonances where precession and librational motion become strongly coupled. These results establish a framework for using LFGs not only as ultrasensitive torque and magnetic-field sensors, but also as controllable mesoscopic quantum systems. The techniques developed here may be applied to searches for exotic, beyond-the-standard model spin-dependent interactions, ultralight dark matter, and spin-gravity couplings.
Motivated by the progress of a multi-scale approach in magnetic materials the dynamics of the Ising model in a transverse field introduced by de Gennes (1963) as a basic model for a ferroelectric order-disorder phase transition is reformulated in terms of a mesoscopic model and inherent microscopic parameters. The statical and dynamical behavior of the Ising model in a transverse field is considered as classical field theory with fields obeying Poisson bracket relations. The related classical Hamiltonian is formulated in such a manner that the quantum equations of motion are reproduced. In contrast to the isotropic magnetic system, see Tserkovnyak et al. (2005), the ferroelectric one reveals no rotational invariance in the spin space and consequently, the driving field becomes anisotropic. A further conclusion is that the resulting excitation spectrum is characterized by a soft-mode behavior, studied by Blinc & Zeks (1974) instead of a Goldstone mode which appears when a continuous symmetry is broken, compare Tserkovnyak et al. (2005). Otherwise the underlying spin operators are characterized by a Lie algebra where the total antisymmetric tensor plays the role of the structural constants. Using symmetry arguments of the underlying spin fields and expanding the driving field in terms of spin operators and including terms which break the time reversal symmetry we are able to derive a generalized evolution equation for the moments. This equation is similar to the Landau Lifshitz equation suggested by Landau& Lifshitz (1935) with Gilbert damping, see Gilbert (2004). Alternatively, such dissipative effects can be included also in the Lagrangian written in terms of the spin moments and bath variables Bose & Trimper (2011). Due to the time reversal symmetry breaking coupling the resulting equation includes under these circumstances a dissipative equation of motion relevant for ferroelectric material. The deterministic equation is extended by stochastic fields analyzed by Trimper et al. (2007). The averaged time dependent polarization offers three modes below the phase transition temperature. The two transverse excitation energies are complex, where the real part corresponds to a propagating soft mode and the imaginary part is interpreted as the wave vector and temperature dependent damping. Further there exists a longitudinal diffusive mode. All modes are influenced by the noise strength. The solution offers scaling properties below and above the phase transition. The results are preferable and applicable for ferroelectric order-disorder systems. A further extension of the approach is achieved by a symmetry allowed coupling of the polarization to themagnetization. The coupling is related to a combined space-time symmetry due to the fact that the magnetization is an axial vector with m( x,−t) = − m( x, t) whereas 23