非共振激發之雷德堡量子氣體為新穎且具有前景的極冷原子系統，在此系統中，原子間的交互作用力具有高度的可調控性以及長的同調時間。在這篇論文中，我們研究了一個在三維自由空間下具有排斥交互作用力的非共振激發之雷德堡費米氣體。雷德堡交互作用力因為阻塞效應 (blockade effect) 而帶有一個特徵長度R_c (稱為阻塞長度)，以至於在動量空間下，存在負的極小值。因為這個特徵，在隨機相位近似 (random phase approximation) 中的計算指出集體激發會在強交互作用力或長阻塞長度時軟化 (soften)，暗示了一個由費米液體到一個未知態的量子相變的存在。藉由猜測此未知態為一個密度波態，我們使用了平均場理論來研究系統的基態性質 (包括相圖、臨界溫度、密度波的型態)。結果指出系統存在一個由費米液體到體心立方堆積之密度波的一階相變。更進一步地，我們探討了Pomeranchuk不穩定 (Pomeranchuk instability) 是否存在於我們所研究的系統中，使用了以下三種方法：數值計算系統總能量 (有限大小之費米面形變)、金茲堡－朗道展開 (Ginzburg-Landau expansion) 以及 Pomeranchuk不穩定條件 (後兩者為無限小之費米面形變)。三種方法皆指出在此系統中不會有Pomeranchuk不穩定發生。我們發現了一種新的密度波態，此種密度波態是由阻塞效應引入的特徵長度所造成，不同於維格納晶體 (特徵長度為粒子密度) 或是固態中的密度波態 (特徵長度由晶格結構給出)。

Rydberg-dressed quantum gas is a promising ultracold atom system, which provides highly controllable interaction and long coherence time. Here we study a Rydberg-Dressed Fermi Gas with repulsive interaction at three-dimensional free space. The interaction between Rydberg-dressed atoms has a length scale Rc introduced by the blockade effect, which features a notable negative minimum in momentum space. As such, calculation within random phase approximation shows
that a collective mode softens in strong interaction or long blockade radius regime, indicating a quantum phase transition from a Fermi liquid to an unknown phase. By the guess that this phase is a density wave phase, we develop a mean field theory to study the ground state properties (the phase diagram, critical temperature and the form of density modulation) of the system. The result shows there is a first order phase transition from the Fermi liquid phase to a rippled density wave phase with body centered cubic (BCC) structure. Furthermore, we investigate Pomeranchuk instability (PI) of our system with three different methods: numerical computation of energy (finite distortion of Fermi surface), Ginzberg-Landau expansion and the PI condition (both are small distortion of Fermi surface). These three results consistently show that there is no PI in the system. Our finding adds an new member to the density wave phase. This new phase has lattice constant introduced by the blockade effect, which is different from the Wigner crystal (length scale given by particle density) or charge density wave in solid states (length scale given by the underlying crystal structure).

1 Introduction 3
2 Preliminary analysis of the system 5
2.1 Model for Rydberg-dressed Fermi gas . . . . . . . . . . . . . . . . . 5
2.2 Softening of the collective mode . . . . . . . . . . . . . . . . . . . . 9
3 Mean eld theory: ansatz with lattice order 14
3.1 The mean eld method . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 1D case: illustration of the concept and the numerical method . . . 17
3.3 3D case: rippled density wave . . . . . . . . . . . . . . . . . . . . . 18
4 Pomeranchuk Instability 24
4.1 Introduction to Pomeranchuk instability . . . . . . . . . . . . . . . 24
4.2 Microscopic description: a mean eld approach for PI . . . . . . . . 26
5 Summary and discussion 31

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