In this paper, we derive analytically the dynamical laws of the coupled Gross-Pitaevskii equations (CGPEs) without/with an angular momentum rotation term and an external magnetic field for modelling nonrotating/rotating spin-1 Bose-Eintein condensates.We prove the conservation of the angular momentum expectation when the external trapping potential is radially symmetric in two dimensions and cylindrically symmetric in three dimensions; obtain a system of first order ordinary differential equations (ODEs) governing the dynamics of the density of each component and solve the ODEs analytically in a few cases; derive a second order ODE for the dynamics of the condensate width and show that it is a periodic function without/with a perturbation; construct the analytical solution of the CGPEs when the initial data is chosen as a stationary state with its centerof-mass shifted away from the external trap center.Finally, these dynamical laws are confirmed by the direct numerical simulation results of the CGPEs.
We investigate the generation of optical torque in coherently prepared multi-level atomic media driven by a vector vortex beam composed of two counter-rotating components carrying opposite orbital angular momenta,
+lℏ and
−lℏ. We consider a three-level
Λ configuration and a four-level tripod configuration. Using a perturbative steady-state solution of the optical Bloch equations, we obtain analytical expressions for both linear and nonlinear contributions to the optical torque. The results show that the torque is strongly controlled by atomic coherence, including the initial population imbalance and the relative phase between the vortex components. Nonvanishing torque can arise even when the two components have equal amplitudes, due to coherence-induced asymmetry in the atomic response. In the tripod configuration, the presence of a strong control field leads to electromagnetically induced transparency, which suppresses the torque near resonance and shifts the dominant response to finite detunings. These results establish a route for controlling light-induced rotational dynamics in atomic media using vector vortex fields, with potential applications in coherent optical manipulation and angular-momentum-based control in quantum systems.
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.