Quantifying Transient Chirality in Non-Equilibrium Quantum Fluid Vortices via Localized Spin-Orbit Coupling Mapping

Introduction

The description of superfluid vortices in quantum fluids often relies on idealized models assuming perfect rotational symmetry. However, experimental observations suggest the presence of complex, time-dependent topological structures, particularly in multi-component systems like $\text{He}^3$, where spin degrees of freedom couple strongly to orbital motion [1]. These vortices are not merely quantized circulation paths; they host excitations whose collective behavior can transiently possess chirality. We hypothesize that inducing controlled spin-orbit coupling ($\lambda_{\text{SO}}$) allows us to drive these vortices into a non-equilibrium state where local chiral symmetry breaking occurs, albeit transiently.

Our primary research question is to establish a robust, quantifiable metric for the degree of chirality present in these macroscopic quantum fluid vortices as they transition from equilibrium to a highly excited, non-equilibrium steady state. This requires moving beyond simple circulation measurements to probe the spin texture coupled to the orbital flow.

Theoretical Framework: Chiral Induction and Stability

We model the system using a Ginzburg-Landau approach generalized to include spinor components influenced by an external magnetic field inducing spin-orbit interaction. The order parameter $\Psi$ is a complex spinor field whose dynamics govern both superfluid velocity and spin orientation. The presence of chirality is encoded in the winding of the spin texture around the vortex core.

When the spin-orbit coupling term, $\hat{H}_{\text{SO}}$, is non-zero, it introduces an effective momentum dependence on the spin dynamics, coupling the orbital superfluid velocity $\mathbf{v}_s$ to the spin density $\mathbf{S}$. The induction of a chiral state can be mathematically described by requiring a non-zero expectation value for a chirality operator $\hat{\mathcal{C}}$:

$$\langle \hat{\mathcal{C}} \rangle = \frac{1}{V} \int_{\text{Volume}} d^3r \left( \mathbf{v}_s(\mathbf{r}) \cdot \nabla \times \mathbf{S}(\mathbf{r}) - \mathbf{S}(\mathbf{r}) \cdot \nabla \times \mathbf{v}_s(\mathbf{r}) \right) \neq 0$$

This expression quantifies the rotational imbalance between the velocity field and the spin texture, which is the microscopic signature of transient chirality in this setup [2]. The stability of this chiral state against thermalization depends critically on the ratio of the coupling energy scale to the quasiparticle scattering rate.

Experimental Protocol and Measurement

Our experimental setup involves trapping a sample of $\text{He}^3$ in a toroidal geometry cooled to $m\text{K}$ temperatures. The vortices are excited via controlled ramping of a tilted magnetic field, which drives the spin-orbit coupling. The crucial measurement involves time-resolved small-angle neutron scattering (SANS). The scattering pattern asymmetry ($\mathcal{A}$) observed at specific momentum transfers is directly proportional to the local chiral density fluctuations ($\rho_{\chi}$):

$$\mathcal{A}(t) = k_{\text{SANS}} \cdot \rho_{\chi}(t) \propto \frac{d\langle N_{\uparrow} \rangle}{dt} - \frac{d\langle N_{\downarrow} \rangle}{dt}$$

We monitor $\mathcal{A}(t)$ following a sudden perturbation pulse tuned to match the characteristic frequency of the vortex excitation modes. The data collected are presented below:

Table 1. Decay of Chirality Signal Following Perturbation Pulse

This decay profile exhibits an exponential relaxation dictated by the thermal bath coupling, but the initial rise time is sensitive to the initial $\lambda_{\text{SO}}$ setting [1].

Results and Kinetic Analysis

The fits to the time-decay data reveal a biexponential relaxation profile, suggesting two distinct loss mechanisms: rapid thermalization into the bulk liquid phase, and slower annihilation into stable low-energy vortex configurations. The initial decay constant, $\tau_{\text{decay}}$, was found to be inversely proportional to the applied $\lambda_{\text{SO}}$ when $\lambda_{\text{SO}}$ is sufficiently strong to overcome viscous damping. Specifically, we found $\tau_{\text{decay}} = \tau_0 \left( 1 + \beta \lambda_{\text{SO}} \right)^{-1}$, where $\tau_0$ is the intrinsic timescale of the bulk fluid. This confirms a direct, non-linear coupling between the spin dynamics and the orbital vortex topology.

Conversely, when $\lambda_{\text{SO}}$ is very small, the system remains locked in a quasi-chiral metastable state, exhibiting coherence times exceeding several millisecond timescales, far longer than the thermal bath timescale [2]. This metastable state hints at a topological protection mechanism afforded by the macroscopic scale of the superfluid condensate.

Discussion and Conclusion

The successful mapping of a time-dependent chirality observable onto fundamental parameters of the quantum fluid ($\lambda_{\text{SO}}$ and $\tau_{\text{decay}}$) provides a powerful analogy between macroscopic quantum hydrodynamics and microscopic spin dynamics. The existence of a measurable, transient chiral phase in this non-equilibrium vortex state suggests that macroscopic quantum fluids can exhibit behavior previously restricted to microscopic chiral molecules.

Future work will focus on engineering the annihilation pathway itself, attempting to drive the system into a controlled, high-fidelity chiral burst rather than allowing uncontrolled decay. The current success hinges on the ability to maintain the coherence of the spin texture against thermal noise during the measurement window.