玻色-愛因斯坦凝聚中的量化渦旋及其動力學(xué)

Preface

Chapter 1 Introduction

1.1 Brief history of Bose-Einstein condensation

1.2 Quantized vortex states in BEC

1.3 Review on numerical methods for stationary states

1.4 Review on numerical methods for the time-dependent GPE

1.5 Scope of this book

Chapter 2 Stationary states for rotating BEC

2.1 GPE in a rotational frame

2.1.1 Dimensionless GPE

2.1.2 Reduction to two dimensions

2.2 Stationary states

2.2.1 Semiclassical scaling and geometrical optics

2.2.2 Ground state

2.2.3 Approximate ground state

2.2.4 Excited states

2.2.5 Critical angular velocity in symmetric trap

2.3 Numerical methods for stationary states

2.3.1 Gradient flow with discrete normalization

2.3.2 Energy diminishing

2.3.3 Continuous normalized gradient flow

2.3.4 Fully numerical discretization

2.4 Numerical results

2.4.1 Initial data for computing ground state

2.4.2 Results in 2D

2.4.3 Results in 3D

2.4.4 Critical angular velocity

2.4.5 Numerical verification for dimension reduction

2.4.6 Errors of the TF approximation

2.4.7 Spurious numerical ground states when |Ω|≥γxy=1

2.5 Conclusion

Chapter 3 Dynamics of rotating BEC

3.1 Some properties of the GPE

3.2 A TSSP method for the GPE

3.2.1 Time-splitting

3.2.2 Discretization in 2D

3.2.3 Discretization in 3D

3.2.4 Stability

3.3 Numerical results

3.3.1 Accuracy test

3.3.2 Dynamics of a vortex lattice in rotating BEC

3.3.3 Generation of giant vortex in rotating BEC

3.4 Conclusion

Chapter 4 Applications to stationary states of rotating two-component BEC

4.1 The time-dependent coupled GPEs

4.1.1 Dimensionless coupled GPEs

4.1.2 Reduction to two dimensions

4.1.3 Semiclassical scaling

4.2 Stationary states

4.2.1 Ground state

4.2.2 Symmetric and central vortex states

4.2.3 Numerical methods for the stationary states

4.2.4 Numerical results for the stationary states

4.3 Conclusion

Chapter 5 Applications to dynamics of rotating two-component BEC

5.1 Some properties of the coupled GPEs

5.2 A TSSP method for the coupled GPEs

5.2.1 Time-splitting

5.2.2 Discretization in 2D

5.2.3 Stability

5.3 Numerical results for the dynamics

5.4 Conclusion

Chapter 6 Application into ground state of spinor BEC

6.1 A continuous normalized gradient flow

6.1.1 Euler-Lagrange equations

6.1.2 A continuous normalized gradient flow

6.2 Normalization and magnetization conservative and energy diminishing numerical scheme

6.2.1 Semi-discretization in time

6.2.2 A fully discretized method

6.3 Numerical results

6.3.1 Choice of initial data and energy diminishing

6.3.2 Accuracy test

6.3.3 Applications

6.4 Conclusion

Chapter 7 Applications to dynamics of spinor F=1 BEC

7.1 The generalized Gross-Pitaevskii equations

7.2 A TSSP method for the generalized GPEs

7.2.1 Time-splitting

7.2.2 Discretization in 2D

7.2.3 Stability

7.3 Numerical results

7.3.1 Accuracy tests

7.3.2 Generation of vortices

7.3.3 Dynamics of a vortex lattice

7.4 Conclusion

Chapter 8 Concluding remarks and future work

Bibliography 2100433B

《玻色:愛因斯坦凝聚中的量化渦旋及其動力學(xué)》的特點與獨到之處是我們設(shè)計了一種模守恒且能量遞減的數(shù)值方法來求得靜態(tài)的Gross-Pitaevskii方程（組）的數(shù)值解；我們也設(shè)計了一種高精度且快速的方法-時間分裂譜方法來求解動態(tài)的Gross-Pitaevskii方程（組）的數(shù)值解；并用所求得的數(shù)值解來分別模擬玻色-愛因斯坦凝聚體的基態(tài)與動力學(xué)，特別是揭示了基態(tài)中的渦旋現(xiàn)象及渦旋運動規(guī)律。《玻色:愛因斯坦凝聚中的量化渦旋及其動力學(xué)》提出的高效數(shù)值方法可以為人們利用計算機(jī)研究玻色-愛因斯坦凝聚現(xiàn)象提供理論方法，加深人們對第五種物質(zhì)-玻色-愛因斯坦凝聚體的理解，最終使人們更方便掌握這種物質(zhì)現(xiàn)象的各種規(guī)律，以便更好地使之在國民經(jīng)濟(jì)建設(shè)中發(fā)揮作用。《玻色:愛因斯坦凝聚中的量化渦旋及其動力學(xué)》提出的高效數(shù)值方法使用方便，不僅僅只可以用在研究玻色-愛因斯坦凝聚現(xiàn)象，還可以推廣應(yīng)用到其它科學(xué)問題之中：例如一般能量泛函在有限制性條件下的求極值計算問題、具有守恒率的偏微分方程（組）的數(shù)值求解問題等?！恫Ｉ?愛因斯坦凝聚中的量化渦旋及其動力學(xué)》在在描述理論和數(shù)值方法過程中深入淺出，從簡單到復(fù)雜，循序漸進(jìn)。既有深奧的理論說明，又有詳細(xì)的算法推導(dǎo)過程；既有原始的物理模型，又有數(shù)學(xué)的簡化過程；這些讓讀者既領(lǐng)悟到了數(shù)值模擬的具體過程，又了解了玻色-愛因斯坦凝聚這一極低溫度的物理現(xiàn)象。

玻色氣體（英語：Bose gas）是一個經(jīng)典的理想氣體的量子力學(xué)模型。其概念相似于費米氣體。

結(jié)合薩特延德拉·玻色和愛因斯坦共同提出的理想的玻色氣體，指的是在足夠低的溫度下〈接近0K〉一群玻色子會形成所謂的固化物。但這樣的行為和經(jīng)典的理想氣體不同。而固化物的形成即所認(rèn)知的玻色–愛因斯坦凝聚。

玻色子具有整數(shù)自旋，并遵守玻色-愛因斯坦統(tǒng)計。薩特延德拉·納特·玻色闡明了光子的表現(xiàn)，并為統(tǒng)計力學(xué)遵從量子規(guī)則的微系統(tǒng)提供了機(jī)會，1924年寫了一篇推導(dǎo)普朗克量子輻射定律的論文寄給當(dāng)時在德國的愛因斯坦，愛因斯坦意識到這篇論文的重要性，并將其擴(kuò)展到不同的經(jīng)典理想氣體的宏觀粒子，不但親自把它翻譯成德語，還以玻色的名義把論文遞予名望頗高的《德國物理學(xué)刊》（"Zeitschrift für Physik"）發(fā)表。