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A video of the phase and density profiles of a vortex lattice formation obtained by solving the GP equation for rotating BECs.

This study proposed a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit real-time integration scheme. Specifically, the Gross–Pitaevskii equation was described as a complex representation to obtain a pair of time-dependent equations, which were then solved simultaneously following discretization based on SPH particle approximation. We adopt the fourth-order Runge–Kutta method for time evolution. We performed simulations of a rotating Bose gas trapped in a harmonic potential, showing results that qualitatively agreed with previously reported experiments and simulations. The geometric patterns of formed lattices were successfully reproduced for several cases, for example, the hexagonal lattice observed in the experiments of rotating BECs. Consequently, it was confirmed that the simulation began with the periodic oscillation of the condensate, which attenuated and maintained a stable rotation with slanted elliptical shapes; however, the surface was excited to be unstable and generated ripples, which grew into vortices and then penetrated inside the condensate, forming a lattice. We confirmed that each branch point of the phase of wavefunctions corresponds to each vortex. These results demonstrate our approach at a certain degree of accuracy. In conclusion, we successfully developed a new SPH scheme for the simulations of vortex lattice formation in rotating BECs.


Schematic of the two-fluid model with angular momentum conservation.

Our recent study suggested that a fully classical mechanical approximation of the two-fluid model of superfluid helium-4 based on smoothed-particle hydrodynamics (SPH) is equivalent to solving a many-body quantum mechanical equation under specific conditions. This study further verifies the existence of this equivalence. First, we derived the SPH form of the motion equation for the superfluid component of the two-fluid model, i.e., the motion equation driven by the chemical potential gradient obtained using the Gibbs–Duhem equation. We then derived the SPH form of the motion equation for condensates based on the Gross–Pitaevskii theory, i.e., the motion equation driven by the chemical potential gradient obtained from the Schrödinger equation of interacting bosons. Following this, we compared the two discretized equations. Consequently, we discovered that a condition maintaining zero internal energy for each fluid particle ensures the equivalence of the equations when the quantum pressure is negligible. Moreover, their equivalence holds even when the quantum pressure is non-negligible if the quantum pressure gradient force equals the mutual friction force. A zero internal energy indicates the thermodynamic ground state, which includes an elementary excitation state. Therefore, the condition can be sufficiently satisfied when the velocities of fluid particles do not exceed the Landau critical velocity, which is not a stringent condition for simulations with a characteristic velocity of a few cm·s−1cm·s−1 in a laboratory system. Based on the above, we performed a simulation of rotating liquid helium-4 and succeeded in generating a vortex lattice with quantized circulation, known as a quantum lattice.


(a) Schematic diagram showing the geometric correspondence between the characteristic particle arrangement observed in the SPH simulation, and (b) the Cauchy–Goursat theorem in the complex function theory.

Our recent study has shown that the representative phenomena of liquid helium-4 rotating in a cylinder could be simulated by solving the two-fluid model using smoothed-particle hydrodynamics (SPH) after reformulating the viscosity to conserve the rotational angular momentum. Specifically, the emergence of multiple parallel vortices and their rigid-body rotations were observed in our previous SPH simulations. The reported scheme is based on a classical approximation that assumes the fluid forces of both components and their interactions, with the expectation of functioning as a coarse-grained model of existing approximations that couple a microscopic model and the Navier–Stokes equation. Based on previous studies, this paper proposes an improved SPH scheme that explicitly incorporates vortex dynamics into SPH to reproduce vortex lattices, which was not possible in previous studies. Consequently, our improved scheme was observed to reproduce vortex lattices by introducing the Magnus force and interaction forces among vortices into the reformulated two-fluid model. The spinning of the vortices and rigid-body rotations were also observed. The number of vortices showed a certain agreement with Feynman's rule after the model parameter was optimized. Notably, from a scientific point of view, such vortex lattices are reproduced by the classical-mechanical approximation. We hope that our model will help physicists studying low-temperature physics find a new way of approaching this bizarre phenomenon that has attracted attention for more than 80 years.

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