此文章研究雷德保裝束(Rydberg-dressed)費米氣體在三維自由空間中的量子相轉變現象。由隨機相位近(random phase approximation)計算得知，若原子的雷德堡阻絕長度(Rydberg blockade length)夠長，或者系統密度夠高，微弱的交互作用即可使均勻的費米液體系統中出現有序密度波。我們的平均場理論(mean-field theory)結果顯示，系統經由一級相轉變過程形成具有體心立方晶格結構的有序密度波，並且其相轉變溫度約為費米能階的十分之一。再者，此研究涉及的理論徹底不同於已知的藍道費米液體理論，意即此有序密度結構非藍道費米液體理論所能預期。

We study quantum phase transitions of a repulsive Rydberg-dressed Fermi gas
in 3D free space. Under random phase approximation(RPA), it is shown that a
density wave order emerges from a Fermi liquid within week interaction and long
blockade length (or high density) regime. Our mean-field theory results show that,
this density wave order forms BCC structure through a first order phase transition,
and the transition temperature is about one-tenth of the Fermi energy. Further-
more, we argue that the theory associated with this novel phenomenon of repulsive
Fermionic system differs completely from the well known Landau Fermi liquid
theory.

Abstract i
1 Introduction 1
2 The Effective Interaction 4
3 The Collective Mode 7
3.1 Random phase approximation and linear response theory 7
3.2 The density wave 10
4 The Mean-Field Theory 13
4.1 The density wave 13
4.2 Local density approximation (LDA) 18
5 Fermi Surface Distortion 22
6 Summery 28
Appendices 29
Appendix A Zero order polarization 29
Appendix B Collective mode with Hubbard approximation and in lower dimension
system 31
Appendix C Fock term of the effective interaction for density wave ansatz 33
Appendix D Mean field matrixes of BCC density wave 35
Appendix E Local density approximation (LDA) for a trapped Rydberg-dressed
cloud 38
Reference 43

[1] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote and M. D. Lukin Fast
quantum gates for neutral atoms. Phys. Rev. Lett. 85, 2208-2211 (2000)
[2] M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and
P. Zoller. Dipole blockade and quantum information processing in mesoscopic
atomic ensembles. Phys. Rev. Lett. 87, 037901 (2001)
[3] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker
and M. Saffman, Observation of Rydberg blockade between two atoms, Nature
Physics 5, 110 (2009)
[4] A. Ga¨etan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P.
Pillet, A. Browaeys and P. Grangier, Observation of collective excitation of two
individual atoms in the Rydberg blockade regime, Nature Physics 5, 115 - 118
(2009)
[5] T. A. Fulton, and G. J. Dolan, Observation of single-electron charging effects in
small tunnel junctions. Phys. Rev. Lett. 59, 190112 (1987)
[6] D. V. Averin, and K. K. Likharev, Coulomb blockade of single-electron tunneling,
and coherent oscillations in small tunnel-junctions. J. Low. Temp. Phys. 62,
345373 (1986)
[7] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Current rectification by Pauli
exclusion in a weakly coupled double quantum dot system. Science 297, 1313-
1317 (2002)
[8] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup and H. J. Kimble,
Photon blockade in an optical cavity with one trapped atom. Nature 436, 87-90
(2005)
[9] N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, Sub-poissonian loading
of single atoms in a microscopic dipole trap. Nature 411, 1024-1027(2001)
[10] M. Saffman, T. G. Walker, and K. MØmer, Quantum information with Rydberg
atoms. Rev. Mod. Phys. 82, 2313-2363 (2010)
[11] E. Brion, A. S. Mouritzen, and K. Mølmer. Conditional dynamics induced by new
configurations for Rydberg dipole-dipole interactions. Phys. Rev. A 76, 022334
(2007)
[12] E. Brion, K. Mølmer, and M. Saffman. Quantum computing with collective ensembles
of multilevel systems. Phys. Rev. Lett. 99, 260501 (2007)
[13] D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y.P. Zhang, R. C.ot’e, E. E.
Eyler, and P. L. Gould, Local blockade of Rydberg excitation in an ultracold gas.
Phys. Rev. Lett. 93, 063001 (2004)
[14] K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidem¨uller,
Suppression of excitation and spectral broadening induced by interactions in a
cold gas of Rydberg atoms. Phys. Rev. Lett. 93, 163001 (2004)
[15] , T. C. Liebisch, A. Reinhard, P. R. Berman, and G. Raithel, Atom counting statistics
in ensembles of interacting Rydberg atoms. Phys. Rev. Lett. 95 253002
(2005); erratum 98, 109903 (2007)
[16] T. Vogt, M. Viteau, J. Zhao, A. Chotia, D. Comparat, and P. Pillet, Dipole blockade
at F¨orster resonances in high resolution laser excitation of Rydberg states of
cesium atoms. Phys. Rev. Lett. 97, 083003 (2006)
[17] P. Bohlouli-Zanjani, J. A. Petrus, and J. D. D. Martin, Enhancement of Rydberg
atom interactions using ac stark shifts. Phys. Rev. Lett. 98, 203005 (2007)
[18] R. Heidemann, U. Raitzsch, V. Bendkowsky, B. Butscher, R. L¨ow, L. Santos,
and T. Pfau, Evidence for coherent collective Rydberg excitation in the strong
blockade regime. Phys. Rev. Lett. 99, 163601 (2007)
[19] T. Peyronel, O. Firstenberg, Q. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M.
D. Lukin, and V. Vuleti, Quantum nonlinear optics with single photons enabled by
strongly interacting atoms. Nature 488, 57-60 (2012)
[20] Y. O. Dudin and A. Kuzmich, Strongly Interacting Rydberg Excitations of a Cold
Atomic Gas. Science 336, 887 (2012)
[21] U. Raitzsch, V. Bendkowsky, R. Heidemann, B. Butscher, R. L¨ow, and T. Pfau,
Echo Experiments in a Strongly Interacting Rydberg Gas. Phys. Rev. Lett. 100,
013002 (2008)
[22] J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, C. S.
Adams, Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble.
Phys. Rev. Lett. 105, 193603 (2010)
[23] J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Low, and T. Pfau, Highly
Resolved Measurements of Stark-Tuned F¨orster Resonances between Rydberg
Atoms. Phys. Rev. Lett. 108, 113001 (2012)
[24] T. E. Lee, H. Haffner, and M. C. Cross, Antiferromagnetic phase transition in a
nonequilibrium lattice of Rydberg atoms. Phys. Rev. A 84, 031402(R) (2011)
[25] Jing Qian, Guangjiong Dong, Lu Zhou, and Weiping Zhang, Phase diagram of
Rydberg atoms in a nonequilibrium optical lattice. Phys. Rev. A 85, 065401 (2012)
[26] Xue-Feng Zhang, Qing Sun, Yu-Chuan Wen, Wu-Ming Liu, Sebastian Eggert,
and An-Chun Ji, Rydberg Polaritons in a Cavity: A Superradiant Solid. Phys.
Rev. Lett. 110, 090402 (2013)
[27] K. Saha, S. Sinha, and K. Sengupta, Phases and collective modes of Rydberg
atoms in an optical lattice. Phys. Rev. A 89, 023618 (2014)
[28] M. Hoening, W. Abdussalam, M. Fleischhauer, and T. Pohl, Antiferromagnetic
long-range order in dissipative Rydberg lattices. Phys. Rev. A 90, 021603(R)
(2014)
[29] H. Weimer, Robert L¨ow, Tilman Pfau, and Hans Peter B¨uchler, Quantum Critical
Behavior in Strongly Interacting Rydberg Gases. Phys. Rev. Lett. 101, 250601
(2008)
[30] N. Henkel, R. Nath, and T. Pohl, Three-Dimensional Roton Excitations and Supersolid
Formation in Rydberg-Excited Bose-Einstein Condensates. Phys. Rev. Lett.
104,195302 (2010)
[31] N. Henkel, F. Cinti, P. Jain, G. Pupillo, and T. Pohl, Supersolid Vortex Crystals
in Rydberg-Dressed Bose-Einstein Condensates. Phys. Rev. Lett. 108, 265301
(2012)
[32] G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, Strongly
Correlated Gases of Rydberg-Dressed Atoms: Quantum and Classical Dynamics.
Phys. Rev. Lett. 104, 223002 (2010)
[33] F. Cinti, P. Jain, M. Boninsegni, A. Micheli, P. Zoller, and G. Pupillo, Supersolid
Droplet Crystal in a Dipole-Blockaded Gas. Phys. Rev. Lett. 105, 135301 (2010)
[34] Bo Xiong, H. H Jen and Daw-Wei Wang, Topological superfluid by blockade
effects in a Rydberg-dressed Fermi gas. Phys. Rev. A 90, 013631 (2014)
[35] G.Gr¨uner, The dynamics of charge-density waves. Rev. Mod. Phys. 60, 1129
(1988)
[36] T. G. Walker and M. Saffman, Consequences of Zeeman degeneracy for the van
der Waals blockade between Rydberg atoms. Phys. Rev. A 77, 032723(2008)
[37] K. Singer, J. Stanojevic, M. Weidem¨uller and R. Cˆot´e, Long-range interactions
between alkali Rydberg atom pairs correlated to the ns-ns, np-np and nd-nd
asymptotes. J. Phys. B: At. Mol. Opt. Phys. 38 S295(2005)
[38] F. Maucher, N. Henkel, M. Saffman, W. Kr’Olikowski, S. Skupin and T. Pohl,
Rydberg-Induced Solitons: Three-Dimensional Self-Trapping of Matter Waves.
Phys. Rev. Lett. 106, 170401(2011)
[39] J. Honer, H.Weimer, T. Pfau and H. P. B¨uchler , Collective Many-Body Interaction
in Rydberg Dressed Atoms. Phys. Rev. Lett. 105, 160404 (2010)
[40] L. D. Landau: On the theory of superfluidity of helium II. USSR J. Phys. 11, 91-92
(1947)
[41] L. D. Landau: The theory of a Fermi liquid. Sov. Phys. JETP 3, 920-925 (1957)
[42] D. Pines, P. Nozi`eres: The Theory of Quantum Liquids (Benjamin, 1966)
[43] A. L. Fetter and J. D. Walecka, Quantum Theory Of Many-Particle Systems, Sec
9, 12, 13, 15, 17
[44] D. Pines, Elementary excitations in quantum liquids. Phys. Today 34, 106V131
(1981)
[45] P. Nozi`eres, Is the roton in superfluid 4He the ghost of a Bragg spot? J. Low Temp.
Phys. 137, 45V67 (2004)
[46] L. D. Landau, Phys. Z. Sowiet 11, 26, 545 (1937)[The Collected Papers of L. D.
Landau, edited by D. ter Haar’(Gordon and Breach-Pergamon, New York, 1965),
p. 198].
[47] J. Quintanilla, M. Haque, and A. J. Schofield , Symmetry-breaking Fermi surface
deformations from central interactions in two dimensions. Phys. Rev. B 78,
035131 (2008)
[48] J. Quintanilla, and A. J. Schofield , Pomeranchuk and topological Fermi surface
instabilities from central interactions. Phys. Rev. B 74, 115126 (2006)
[49] P. W. Anderson and P. Morel, Generalized Bardeen-Cooyer-Schrieffer States and
the Proposed Low-Temperature Phase of Liquid He III. Phys. Rev. 123, 6 (1961)
[50] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases