We establish self-contained hydrodynamic equations for a spin-1 spinor Bose-Einstein condensate (BEC) for an arbitrary state including nonequilibrium one. The obtained hydrodynamic equations, which yield a description equivalent to the time-dependent multi-component Gross-Pitaevskii (GP) equation, involve the continuity equations for the density, the spin, and the nematic tensor (or the quadrupolar tensor) and the equation of motion for the density. These equations are written only in terms of observable physical quantities, i.e., the mass current in addition to the density, the spin, and the nematic tensor.

We derive the low-lying collective modes in the ferromagnetic (FM) state and the polar (P) state from our hydrodynamic equations: we apply linearization to the hydrodynamic equations with respect to the fluctuations of the density, the spin, and the nematic tensor. Then, the differential equations for them are derived, which yield the dispersion relations. The obtained dispersion relations are consistent with those calculated from the time-dependent multi-component GP equation or the Bogoliubov approximation. In addition, it is found that one of the spin modes in each state originates from the fluctuations of the nematic tensor.

Since the hydrodynamic equations are described by observable quantities, they help us to understand the physical properties of a spinor BEC intuitively. Moreover, in a spinor BEC system, the in-situ and high-resolution observation of the magnetization profile is possible, which augments the significance of the spinor hydrodynamics. We can elucidate and predict features of a spinor BECs, such as spin textures and their dynamics, in a physically more transparent manner in comparison to the time-dependent multi-component GP equation written in terms of spinor order parameters (or a spinor wave function), which cannot be detected directly.

A spinor BEC, which is a BEC with spin-internal degrees of freedom, shows a rich variety of phenomena originating from its spin-degrees of freedom, such as magnon collective modes, spin texture formation, and topological excitations. Many of these properties are well understood by several variations of the mean-field approximations, i.e., Bogoliubov-de-Gennes approximation and the multi-component GP equation developed by Ho and Ohmi & Macida.

Experimentally, a spinor BEC was first realized in a sodium-23 vapor confined in an optical trap by Davis et. al. in 1998. Prompted by the realization of spinor BECs, various experiments have been performed, for example, a BEC prepared in a metastable state, quantum tunneling of a BEC, and spin-exchange dynamics. Furthermore, Higbie et. al. developed an in situ and high-resolution imaging of magnetization profiles for spinor BECs, which prompted the theoretical researches of the hydrodynamic description for spnor BECs.

The pioneering works of the spinor hydrodynamics were done by Lamacraft and Kudo & Kawaguchi, which mainly focus on the experiment by Sadler et. al., i,e., spin domain formation in a quenched ferromagnetic BEC of rubidium-87. According to their works, the obtained hydrodynamic equations for a ferromagnetic BEC are written in terms of observable quantities, that is to say, the density ρ, the mass current v_mass, and the spin density f. The hydrodynamic equation reduces to a modified Landau-Lifshitz equation with the assumption of the spatially uniform density in the work of Lamacraft, whereas the hydrodynamic equation is expressed in the form of a Landau-Lifshitz-Gilbert equation in the work of Kudo and Kawaguchi. The hydrodynamic equation for the polar BEC is also derived by Kawaguchi, and there are a couple of formalisms that can be applied to general spins and states by means of the Majorana representation of spin states; however, the hydrodynamic equations that are written in terms of observable physical quantities and enable us to treat a spinor BEC in an arbitrary state have not been established yet.

Thus, in this thesis, we aim to develop such spinor hydrodynamic equations involving only physical quantities, which can apply to a BEC in an arbitrary state in the mean-field regime.

The main results of this thesis are comprised of three parts as follows.

Firstly, the appropriate variables that describe the hydrodynamic equations are determined on the basis of the degrees of freedom of the multi-component GP equation. Corresponding to the six variables of the multi-component GP equation, the hydrodynamic equations require the thirteen variables, that is to say, the density ρ, the mass current v_mass, the spin density f, and the nematic tensor N_μν, where the nematic tensor indicates anisotropy and axes of a spinor wave function in the multi-component GP description. By introducing the nematic tensor, the hydrodynamic equations for a spin-1 BEC are generalized to arbitrary states, i.e., the FM state, the P state, and the broken-axisymmetry (BA) state, which has a broken axisymmetry about an external magnetic field.

Secondly, we derive the hydrodynamic equations in terms of the variables mentioned above. The hydrodynamic equations are made up by the continuity equations and the equation of motion.

The continuity equation for the density, which involves the time derivative of the density and the divergence of the mass current, is expressed by the same form as that for the FM state except for the explicit form of the mass current:

where M indicates the mass of a particle and α, β, and γ denote the angles of an Euler rotation of the spinor wave function, and 〓 expresses the state of the BEC. Here, as a consequence of the generalization of the mass current, the Mermin-Ho relation of the mass current for the FM state is also generalized so that the Berry phase changes by a factor of the magnitude of the spin |f|, and the difference between the circulation of the mass current and the Berry phase is given by ((h)/(M))(n-2|f|), where n is an integer.

The continuity equation for the spin is also written in the same expression as that for the FM state except for the spin current v_μ:

where the terms on the right-hand side originate from the linear and quadratic Zeeman terms. For an arbitrary state, the spin current involves not only terms originating from mass transport and a spatial profile of f but a term arising from spatial derivatives of the nematic tensor. For the particular case of the FM state, the expression of the spin current reduces to that for the FM state.

The continuity equation for the nematic tensor is obtained as

where v_μν represents the nematic current. The term in the second line on the right-hand side of the continuity equation implies that the outer product of the spin and the column vector of the nematic tensor acts like a torque on the nematic tensor, which causes the nematic current as well as mass transport.

The last equation of the hydrodynamic description is the equation of motion for the density:

which is analogous to the Euler equation. As we see, one of the features of a spinor BEC appears on the right-hand side of the equation of motion: there are the quantum pressure terms caused by not only the density but the spin and the nematic tensor.

Similarly, we also derive the equations of motion for the spin and the nematic tensor. Our hydrodynamic equations reduce to the hydrodynamic equations for the ferromagnetic BEC under the condition of the FM state.

Finally, we linearize the equations of motion for the density, spin, and the nematic tensor with respect to their fluctuation, and derive the dispersion relations for the low-lying collective modes in the FM state and the P state. The obtained results, which reproduce the elementary excitations calculated from the multi-component GP equation, imply that one of the magnons in each state is caused by the fluctuation of the nematic tensor.

In conclusion, we construct the hydrodynamic equations that are equivalent to the time-dependent multi-component GP equation. Our hydrodynamic equations are totally written in terms of the observable quantities and self-contained. We can obtain not only the low-lying collective excitations but also elucidate their physical origins in a physically transparent manner by linearizing the hydrodynamic equations.

内部自由度としてスピン自由度を持つボースアインシュタイン凝縮体(BEC)はスピノルBECと呼ばれ、1998年に実験的に初めて実現した。スピノルBECの特長の一つは内部自由度の存在によってもたらされる非自明な動力学的性質である。本論文は、スピン1の内部自由度を持つスピノルBECの動力学的性質に関する新たな定式化と知見を引き出した理論研究をまとめたものである。

本論文は5章からなる。第1章では研究の背景としてスピノルBECの概説が与えられている。第2章では本研究に密接に関連した先行研究であるスピノルBECの強磁性状態の流体力学的定式化について説明が与えられている。第3章、第4章では本研究の主要な結果について記述されている。第3章ではスピン1スピノルBECの任意の状態に対する流体力学的定式化の導出が与えられ、第4章では、その流体力学的定式化を通して得られる物理的描像と、それに基づいて見出された新たな回転モードについて説明が与えられ、また超流動速度の循環の量子化の破れを表す表式(一般化されたMermin-Ho関係式)の導出と説明が与えられている。さらにこの章では基底状態まわりの線形化励起の分散関係を求め、Gross-Pitaevskii方程式に基づく先行研究の結果との比較を行う、先行研究との整合性を確認している。第5章にはまとめと結論が述べられている。

スピノルBEC実現当初から、実験結果が多成分Gross-Pitaevskii方程式によってよく記述されることが知られていた。一方で、スピン自由度に関連した非自明な動力学的性質が期待され、内部自由度の動力学をより見通しよく記述し、新たな物理的描像を与え得る別の定式化も試みられている。そのようなものとして流体力学的方程式による動力学の定式化が知られている。スピノルBECにおける強磁性状態(空間の各点において、磁気モーメントの大きさが最大値をとり、かつ向きは空間的に変化しうる状態)における流体力学的方程式はすでに先行研究で与えられ、強磁性体におけるランダウ・リフシッツ方程式と類似の動力学方程式が得られている。しかし、スピノルBECの任意の状態に適用可能な流体力学方程式はこれまで導かれていなかった。

本研究の主な成果はスピン1のBECの任意の状態で成り立つ流体力学方程式を導出したことにある。それらの方程式系は、質量密度、超流動速度、局所磁化ベクトル、そしてスピン演算子の2階対称テンソルで与えられるネマティックテンソルを流体力学的変数とし、4種の方程式(質量密度、運動量密度、局所磁化ベクトル、ネマティックテンソルに関する「連続の方程式」タイプの方程式)からなる閉じた方程式系をなす。この方程式系は多成分Gross-Pitaevskii方程式と等価であり、線形化励起について同等な結果が得られている。

この成果の意義は、第一にBECでネマティックテンソルが本質的役割を果たす初めての流体力学方程式である点にある。先行研究で対象とされた強磁性状態のようにスピン空間で一軸対称性を持つ場合には、ネマティックテンソルは磁化ベクトルと独立な情報を与えないので、質量密度、運動量密度、局所磁化ベクトルだけで閉じた流体力学方程式で系の動力学が記述できてしまう。一方スピン空間で一軸対称性を持たない状態ではスピンダイナミックスにネマティックテンソルが関与し、スピンダイナミックスも非自明なものになる。その意味でスピン空間で一軸対称性を持たない状態でも成立する本研究成果は重要な意義を持つ。

本研究の成果の第二の意義は、質量密度、超流動速度、局所磁化ベクトル、ネマティックテンソルで閉じた方程式をなすことを示した点にある。これ以上の流体力学変数が現れない理由の一つとして局所磁化ベクトルとネマティックテンソルが閉じた代数をなしていることが挙げられる。

本研究の成果の第三の意義は、導出した流体力学方程式からネマティックテンソルの新たな回転モードを発見し、実験による検出方法を提案したことにある。流体力学的方程式から、磁化がネマティックテンソルに対してトルクを与えることが見て取ることができ、無磁場の(したがってスピンが保存量になる)状態で、磁化が分極した状態におけるネマティックテンソルが空間的に一様に歳差運動する回転モードの存在がただちに示唆される。実際にそのような運動が流体力学方程式から解析的に導かれた。

以上のように本研究成果はスピン1BECが持つ非自明な動力学的性質を記述する枠組みを与え、対称性や物理量の代数的性質から、非自明な動力学的性質を見通し良く整理することを可能したという意味で学位論文に値するものである。

本論文の研究成果は原著論文して投稿準備中である。本論文で述べられている結果は、上田正仁氏との共同研究の成果であるが論文提出者が主体となって分析および検証を行ったもので、論文提出者の寄与が十分であると判断する。したがって博士(理学)の学位を授与できると認める。