有限温度超流—莫特绝缘体相变
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摘要
光晶格为量子隧穿模型和量子相变等物理问题的实现提供了可能。光晶格中超冷原子从超流相(SF)到莫特绝缘相(MI)的跃迁,翻开了可控条件下强关联机制领域研究的新篇章。目前,荧光成像技术可精确测量到单原子单格点的原子随空间和时间变化的量子相变。本文首先介绍了光晶格中玻色-哈伯德模型,以及将玻色-哈伯德模型推广到包含三体相互作用得到的扩展的玻色-哈伯德模型,再结合平均场近似和微扰理论,计算了超流相与莫特绝缘相边界,同时给出了相互作用U和隧穿矩阵t的详细计算。接着利用紧束缚近似讨论了光晶格中自旋为1的玻色一哈伯德模型,研究了极限条件下的莫特绝缘相,进而理论计算得到了在奇偶占据数下超流相到莫特绝缘相的边界,并画出了相图。最后在前面的基础上,结合热力学相关理论计算有限温度下位于二维(三维)光晶格中的冷原子的数密度、超流密度和熵随空间变化的图,结合超流相与莫特绝缘相的知识分析其物理意义。该研究方案在目前冷原子实验中有望被实现和直接观察。
Optical lattices provided possibility for the realization of many physical problems, such as quantum tunneling model and quantum phase transitions. The phase transition of ultra cold atoms in the optical lattices from superfluid phase to Mott insulator phase opened a new door for the investigation of strong correlation mechanism under well controlled conditions. Currently the fluorescence imaging technique allows us to observe space- and time-resolved characterization of single-atom and single-site across the quantum phase transition. In this thesis, we first give a brief introduction of Bose-Hubbard model, and extended-Bose-Hubbard model with two-or three-body on-site interactions, and calculate the phase boundary of the superfluid state and the Mott-insulator state combining the mean-field approximation with the perturbation theory. Meanwhile we give details on how to evaluate the interaction parameter U and the hopping matrix t. Then starting from the spin-1 Bose-Hubbard model under the tight-binding approximation, we study the MI phase in the limit of t=0, calculate the phase boundary of the SF-MI transition, and discuss the phase diagram with the even-odd dependence of the MI phase. Finally, based on the above knowledge and the theory of thermodynamics, we study the number density, the superfluid density and the entropy of the atoms for the SF and MI states in two and three- dimensional optical lattices at finite temperature. Our results can be readily verified in the current experiments.
Optical lattices provided possibility for the realization of many physical problems, such as quantum tunneling model and quantum phase transitions. The phase transition of ultra cold atoms in the optical lattices from superfluid phase to Mott insulator phase opened a new door for the investigation of strong correlation mechanism under well controlled conditions. Currently the fluorescence imaging technique allows us to observe space- and time-resolved characterization of single-atom and single-site across the quantum phase transition. In this thesis, we first give a brief introduction of Bose-Hubbard model, and extended-Bose-Hubbard model with two-or three-body on-site interactions, and calculate the phase boundary of the superfluid state and the Mott-insulator state combining the mean-field approximation with the perturbation theory. Meanwhile we give details on how to evaluate the interaction parameter U and the hopping matrix t. Then starting from the spin-1 Bose-Hubbard model under the tight-binding approximation, we study the MI phase in the limit of t=0, calculate the phase boundary of the SF-MI transition, and discuss the phase diagram with the even-odd dependence of the MI phase. Finally, based on the above knowledge and the theory of thermodynamics, we study the number density, the superfluid density and the entropy of the atoms for the SF and MI states in two and three- dimensional optical lattices at finite temperature. Our results can be readily verified in the current experiments.
引文
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[18]T.-L. Ho, Spinor Bose Condensates in Optical Traps, Phys. Rev. Lett.,1998,81,742.
[19]S. Tsuchiya, S. Kurihara, and T. Kimura, Superfluid-Mott insulator transition of spin-1 bosons in an optical lattice, Phys. Rev. A,2004,70,043628.
[20]T. Ohmi and K. Machida, Bose-Einstein Condensation with Internal Degrees of Freedom in Alkali Atom Gases, J. Phys. Soc. Jpn.,1998,67,1822.
[21]A. A. Svidzinsky and S. T. Chui, Insulator-superfluid transition of spin-1 bosons in an optical lattice in magnetic field, Phys. Rev. A,2003,68,043612.
[22]Y. Wu, Simple algebraic method to solve a coupled-channel cavity QED model. Phys. Rev. A,1996,54,4534.
[23]T.-L. Ho, and S. K. Yip, Fragmented and Single Condensate Ground States of Spin-1 Bose Gas, Phys. Rev. Lett.,2000,84,4031. C.K. Law, H. Pu, and N.P. Bigelow, Quantum Spins Mixing in Spinor Bose-Einstein Condensates, Phys. Rev. Lett.,1998, 81,5257.
[24]J. K. Freericks and H. Monien, Phase diagram of the Bose-Hubbard Model, Europhys. Lett.1994,26,545.
[25]G. G. Batrouni, R.T. Scalettar, and G.T. Zimanyi, Quantum critical phenomena in one-dimensional Bose systems, Phys. Rev.Lett.,1990,65,1765.
[26]G. Birkl, M. Gatzke, l.H. Deutsch, S.L. Rolston, and W.D. Phillips, Bragg Scattering from Atoms in Optical LatticesPhys, Rev. Lett.,1995,75,2823.
[27]M. Weidemuller, A. Hemmerich. A.Gorlitz, T. Esslinger, and T.W. Hansch, Bragg Diffraction in an Atomic Lattice Bound by Light, Phys. Rev. Lett.,1995,75,4583.
[28]K. W, Mahmud, G. G. Batrouni, and R. T. Scalettar, Isentropes of spin-l bosons in an optical lattice, Phys. Rev. A,2010,81,033609.
[29]G. G. Batrouni, V. G. Rousseau, and R. T. Scalettar, Magnetic and Superfluid Transitions in the One-Dimensional Spin-1 Boson Hubbard Model, Phys. Rev. Lett., 2009,102,140402.
[30]V. Apaja, O. F. Syljuasen, Dimerized ground state in the one-dimensional spin-1 boson Hubbard model, Phys. Rev. A,2006,74,035601.
[31]S. Bergkvist, I. P. McCulloch, and A. Rosengren, Spinful bosons in an optical lattice, Phys. Rev. A,2006,74,053419.
[32]M. Rizzi, D. Rossini, G. D. Chiara, S. Montangero, and R. Fazio, Phase Diagram of Spin-1 Bosons on One-Dimensional Lattices, Phys. Rev. Lett.,2005,95,240404.
[33]K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, Superfluid and Insulating Phases in an Interacting-Boson Model:Mean-Field Theory and the RPA, Europhys. Lett.,1993,22,257
[34]R. V. Pai, K. Sheshadri, and R. Pandit, Phases and transitions in the spin-1 Bose-Hubbard model:Systematics of a mean-field theory, Phys. Rev. B,2008,77, 014503.
[35]T. Kimura, S. Tsuchiya, and S. Kurihara, Possibility of a First-Order Superfluid-Mott-Insulator Transition of Spinor Bosons in an Optical Lattice, Phys. Rev. Lett., 2005.94,11043.
[36]K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, Percolation -Enhanced Localization in the Disorderd Bosonic. Hubbard Model, Phys. Rev. Lett., 1995,75,4075.
[37]J.-N. Zhang and S. Yi, Thermodynamic properties of a dipolar Fermi gas, Phys. Rev. A.2010,81,033617.
[38]P. B. Blakie, A. Bezett, and P. Buonsante, Degenerate Fermi gas in a combined harmonic-lattice potential, Phys. Rev. A,2007,75,063609.
[39]L. Hackermuller, U. Schneider, M. M.-Cardoner, T. Kitagawa, T. Best, S. Will, E. Demler, E. Altman, I. Bloch, and B. Paredes, Anomalous Expansion of Attractively Interacting Fermionic Atoms in an Optical Lattice, Science,2010,327,1621.
[40]S. Folling, A. Widera, T. Miiller, F. Gerbier, and I. Bloch, Formation of Spatial Shell Structure in the Superfluid to Mott Insulator Transition, Phys. Rev. Lett.,2006,97, 060403.
[41]S. Wessel, F. Alet, M. Troyer, and G. George Batrouni, Quantum Monte Carlo simulations of confined bosonic atoms in optical lattices, Phys. Rev.A,2004,70, 053615.
[42]M. Rigol, G. G. Batrouni, and V. G. Rousseau, State diagrams for harmonically trapped bosons in optical lattices, Phys. Rev. A,2009,79,053605.
[2]C. Cohen--Tannoudji and W.D. Phillips, New Mechanisms for Laser Cooling, Phys. Today,1990,33.
[3]R. Grimm, M. Weidemuller, Y. B. Ovchinnikov, Optical Dipole Traps for Neutral Atoms, Advances In Atomic, Molecular, and Optical Physics,2000,42,95-170.
[4]M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature, 2002,415,39.
[5]J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature,2010,467, 68.
[6]W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Folling, L. Pollet, M. Greiner, Probing the Superfluid-to-Mott Insulator Transition at the Single-Atom Level, Science,2010,329,547.
[7]D. van Oosten, P. van der Straten, and H. T. C. Stoof, Quantum phases in an optical lattice, Phys. Rev. A,2001,63,053601.
[8]D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold Bosonic Atoms in Optical Lattices, Phys. Rev. Lett.,1998,81,3108.
[9]B.-l. Chen, X.-b. Huang, and S.-p. Kou, Mott-Hubbard transition of bosons in optical lattices with three-body interactions, Phys. Rev. A,2008,78,043603.
[10]张礼,近代物理学进展,北京,清华大学出版社,2009,276-295.
[11]W. Zwerger, Mott-Hubbard transition of cold atoms in optical lattices, J. Opt. B: Quantum Semiclass. Opt.,2003,5, S9.
[12]A. Posazhennikova, Colloquium:Weakly interacting, dilute Bose gases in 2D, REVIEWS OF MODERN PHYSICS,2006,78,1111.
[13]M. Holthaus, Bloch oscillations and Zener breakdown in an optical lattice, J. Opt. B: Quantum Semiclass. Opt.,2000,2,589.
[14]D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov, Bose-Einstein Condensation in Quasi-2D Trapped Gases, Phys. Rev. Lett.,2000,84,2551.
[15]K. Drese, M. Holthaus, Ultracold atoms in modulated standing light waves, Chemical Physics,1997,217,201.
[16]M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover Publications,1970,721.
[17]J. C. Slater, A Soluble Problem in Energy Band, Phys.Rev.,1952,87,807.
[18]T.-L. Ho, Spinor Bose Condensates in Optical Traps, Phys. Rev. Lett.,1998,81,742.
[19]S. Tsuchiya, S. Kurihara, and T. Kimura, Superfluid-Mott insulator transition of spin-1 bosons in an optical lattice, Phys. Rev. A,2004,70,043628.
[20]T. Ohmi and K. Machida, Bose-Einstein Condensation with Internal Degrees of Freedom in Alkali Atom Gases, J. Phys. Soc. Jpn.,1998,67,1822.
[21]A. A. Svidzinsky and S. T. Chui, Insulator-superfluid transition of spin-1 bosons in an optical lattice in magnetic field, Phys. Rev. A,2003,68,043612.
[22]Y. Wu, Simple algebraic method to solve a coupled-channel cavity QED model. Phys. Rev. A,1996,54,4534.
[23]T.-L. Ho, and S. K. Yip, Fragmented and Single Condensate Ground States of Spin-1 Bose Gas, Phys. Rev. Lett.,2000,84,4031. C.K. Law, H. Pu, and N.P. Bigelow, Quantum Spins Mixing in Spinor Bose-Einstein Condensates, Phys. Rev. Lett.,1998, 81,5257.
[24]J. K. Freericks and H. Monien, Phase diagram of the Bose-Hubbard Model, Europhys. Lett.1994,26,545.
[25]G. G. Batrouni, R.T. Scalettar, and G.T. Zimanyi, Quantum critical phenomena in one-dimensional Bose systems, Phys. Rev.Lett.,1990,65,1765.
[26]G. Birkl, M. Gatzke, l.H. Deutsch, S.L. Rolston, and W.D. Phillips, Bragg Scattering from Atoms in Optical LatticesPhys, Rev. Lett.,1995,75,2823.
[27]M. Weidemuller, A. Hemmerich. A.Gorlitz, T. Esslinger, and T.W. Hansch, Bragg Diffraction in an Atomic Lattice Bound by Light, Phys. Rev. Lett.,1995,75,4583.
[28]K. W, Mahmud, G. G. Batrouni, and R. T. Scalettar, Isentropes of spin-l bosons in an optical lattice, Phys. Rev. A,2010,81,033609.
[29]G. G. Batrouni, V. G. Rousseau, and R. T. Scalettar, Magnetic and Superfluid Transitions in the One-Dimensional Spin-1 Boson Hubbard Model, Phys. Rev. Lett., 2009,102,140402.
[30]V. Apaja, O. F. Syljuasen, Dimerized ground state in the one-dimensional spin-1 boson Hubbard model, Phys. Rev. A,2006,74,035601.
[31]S. Bergkvist, I. P. McCulloch, and A. Rosengren, Spinful bosons in an optical lattice, Phys. Rev. A,2006,74,053419.
[32]M. Rizzi, D. Rossini, G. D. Chiara, S. Montangero, and R. Fazio, Phase Diagram of Spin-1 Bosons on One-Dimensional Lattices, Phys. Rev. Lett.,2005,95,240404.
[33]K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, Superfluid and Insulating Phases in an Interacting-Boson Model:Mean-Field Theory and the RPA, Europhys. Lett.,1993,22,257
[34]R. V. Pai, K. Sheshadri, and R. Pandit, Phases and transitions in the spin-1 Bose-Hubbard model:Systematics of a mean-field theory, Phys. Rev. B,2008,77, 014503.
[35]T. Kimura, S. Tsuchiya, and S. Kurihara, Possibility of a First-Order Superfluid-Mott-Insulator Transition of Spinor Bosons in an Optical Lattice, Phys. Rev. Lett., 2005.94,11043.
[36]K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, Percolation -Enhanced Localization in the Disorderd Bosonic. Hubbard Model, Phys. Rev. Lett., 1995,75,4075.
[37]J.-N. Zhang and S. Yi, Thermodynamic properties of a dipolar Fermi gas, Phys. Rev. A.2010,81,033617.
[38]P. B. Blakie, A. Bezett, and P. Buonsante, Degenerate Fermi gas in a combined harmonic-lattice potential, Phys. Rev. A,2007,75,063609.
[39]L. Hackermuller, U. Schneider, M. M.-Cardoner, T. Kitagawa, T. Best, S. Will, E. Demler, E. Altman, I. Bloch, and B. Paredes, Anomalous Expansion of Attractively Interacting Fermionic Atoms in an Optical Lattice, Science,2010,327,1621.
[40]S. Folling, A. Widera, T. Miiller, F. Gerbier, and I. Bloch, Formation of Spatial Shell Structure in the Superfluid to Mott Insulator Transition, Phys. Rev. Lett.,2006,97, 060403.
[41]S. Wessel, F. Alet, M. Troyer, and G. George Batrouni, Quantum Monte Carlo simulations of confined bosonic atoms in optical lattices, Phys. Rev.A,2004,70, 053615.
[42]M. Rigol, G. G. Batrouni, and V. G. Rousseau, State diagrams for harmonically trapped bosons in optical lattices, Phys. Rev. A,2009,79,053605.