東京大学先端科学技術研究センター

都築研究室

多粒子系コアダイナミクス研究室

Satori Tsuzuki's Lab.

Laboratory for core dynamics of many-particle interacting systems

Lecturer (Senior Lecturer), Principal Investigator

Research Center for Advanced Science and Technology (RCAST) ,

The University of Tokyo

Address: 4-6-1, Komaba, Meguro-Ku, Tokyo, 153-8904, JAPAN

E-mail : contact (at) satoritsuzuki.org

都築怜理, 東京大学先端科学技術研究センター講師（専任）

〒153-8904 東京都目黒区駒場4-6-1 東京大学先端科学技術研究センター

# Highlighted articles

# A video of the rotation simulation with the lower resolution using 2 454 000 particles from the beginning to 0.79 s in physical time.

This paper reports a three-dimensional (3D) simulation of a rotating liquid helium-4, using a two-fluid model with spin-angular momentum conservation. Our model was derived from the particle approximation of an inviscid fluid with residual viscosity. Despite the fully classical mechanical picture, the resulting system equations were consistent with those of the conventional two-fluid model. We consider bulk liquid helium-4 to be an inviscid fluid, assuming that the viscous fluid component remains at finite temperatures. As the temperature decreased, the amount of the viscous fluid component decreased, ultimately becoming a fully inviscid fluid at absolute zero. Weak compressibility is assumed to express the volume change because some helium atoms do not render fluid owing to Bose–Einstein condensations or change states because of local thermal excitation. One can solve the governing equations for an incompressible fluid using explicit smoothed-particle hydrodynamics, simultaneously reproducing density fluctuations and describing the fluid in a many-particle system. We assume the following fluid–particle duality: a hydrodynamic interfacial tension between the inviscid and viscous components or a local interaction force between two types of fluid particles. The former can be induced in the horizontal direction when non-negligible non-uniformity of the particles occurs during forced two-dimensional rotation, and the latter is non-negligible when the former is negligible. We performed a large-scale simulation of 3D liquid helium forced to rotate horizontally using 32 graphics processing units. Compared with the low-resolution calculation using 2.4 × 106 particles, the high-resolution calculation using 19.6 × 106 particles showed spinning vortices close to those of the theoretical solution. We obtained a promising venue to establish a practical simulation method for bulk liquid helium-4.

# A video of the phase and density profiles of a vortex lattice formation obtained by solving the GP equation for rotating 3D BECs

Recently, we presented a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit time-integrating scheme; our SPH scheme could reproduce the vortex lattices and their formation processes in rotating quasi-two-dimensional (2D) BECs trapped in a 2D harmonic potential. In this study, we have successfully demonstrated a simulation of rotating 3D BECs trapped in a 3D harmonic potential forming 'cigar-shaped' condensates. We have found that our scheme can reproduce the following typical behaviors of rotating 3D BECs observed in the literature: (i) the characteristic shape of the lattice formed in the cross-section at the origin and its formation process, (ii) the stable existence of vortex lines along the vertical axis after reaching the steady state, (iii) a 'cookie-cutter' shape, with a similar lattice shape observed wherever we cut the condensate in a certain range in the vertical direction, (iv) the bending of vortex lines when approaching the inner edges of the condensate, and (v) the formation of vortex lattices by vortices entering from outside the condensate. Therefore, we further validated our scheme by simulating rotating 3D BECs.

# A video of the phase and density profiles of a vortex lattice formation obtained by solving the GP equation for rotating 2D 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.