低维玻色—爱因斯坦凝聚的转变温度和基态性质
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摘要
玻色-爱因斯坦凝聚(BEC)是近年来物理学界的研究热点之一.它揭示了一类新的物质状态,高度密集的大量原子以相干的方式演变,将微观的量子现象带到了宏观尺度.激子BEC是在半导体材料中实现的低维玻色-爱因斯坦凝聚,和原子BEC相比,有着更为广阔的前景:如最有希望在大功率、低功耗发光器件和可能在超快逻辑器件及量子计算中得到应用,因而越来越受到人们的重视.凝聚温度是普通玻色气体转变为玻色-爱因斯坦凝聚体的相变温度.本文考虑体系中粒子第一激发态能量不为零的条件下,对低维体系的凝聚温度进行相关计算,考查体系中粒子的第一激发态能量的大小对体系的凝聚温度的影响;以及在第一激发态能量大小一定的前提下,不同维度的空间中体系的凝聚温度、基态占据数随总粒子数变化的关系,得到了与实验较为符合的结果.研究发现:随着粒子数密度的增加,低维体系的凝聚温度比高维情形增长的快;体系的凝聚温度与第一激发态能量密切相关,解释了对于囚禁在GaAs量子阱中的激子气体,即使在无谐振势或幂指数等特殊外势的条件下,也可以在几个K实现玻色-爱因斯坦凝聚的实验现象.玻色-爱因斯坦凝聚体波函数藉由Gross-Pitaevskii方程(G-P方程)描述.最后,我们使用傅立叶-格雷德-哈密顿方法数值求解G-P方程,研究了总粒子数、粒子间相互作用、谐振频率和一般幂指数外势对玻色凝聚体粒子数密度分布、基态能量的影响.研究结果表明:增大幂指数外势、谐振频率,降低粒子间的排斥作用都会增加凝聚体中心的粒子数密度、缩小凝聚体半径;而增大总粒子数、谐振频率、粒子间的排斥作用及幂指数外势的指数都会增大体系的基态能量;随着总粒子数增大,数值结果与托马斯-费米近似结果渐趋一致,托马斯-费米近似在大粒子数条件下是一种较好的近似方法,在粒子数有限时,托马斯-费米结果与真实情形偏差较大,应采用数值解法.
Bose-Einstein Condensation (BEC) is one of the hotspot in the physics research recently. It reveals a new form of matter in the universe, a large number of dense atoms evolves coherently, bringing the quantum phenomenon from the microscopic world to the macroscopic world. Excitons BEC is the Bose-Einstein Condensation realized in the semi-conductor materials. The physists lay more and more emphasis on it due to it’s speciality and wide prospect, such as the application on high-powered and low-consumption light-emitting devices, ultra-fast logic devices, and quantum computing. Critical temperature is the phase transition temperature at which the common bose gas transforms into the Bose-Einstein Condensation. This thesis calculates the critical temperature in the low dimensional spaces according to the first-excited-state energy of particles in the traped condensate being different from zero, investigates the effect of first-excited-state energy on the critical temperature as well as the relationship of the critical temperature and the ground-state faction versus the total number of particles in different dimensional spaces. The results we get are consistent with the experiment data. It is shown that the critical temperature in low-dimensional spaces increases more quickly than that in high-dimensional space with density of particle number increasing, and the critical temperature of system has close relation to the first-excited-state energy. Our theoretical method can explain the phenomena that even under the condition of absent special external potential, such as harmonic potential and power-law potential, the Bose-Einstein Condensation can be realized at several K for excitons gas trapped in GaAs quantum well. The wave function of BEC can be described by the Gross-Pitaevskii equation. In the last part of the thesis, we study the effect of total particle number, the interaction between particles, the frequency of harmonic potential and the power-law potential on the distribution of the particles in BEC and the ground state energy of the condensate by solving the G-P equation using the Fourier-Grid-Hamiltonian method numerically. The results we get show that with the intensity of power-law potential or the frequency of harmonic potential increases or the repulsive interaction between particles decreases, the particle density in the condensate center will increase and the radius of the condensate will decrease. The ground state energy of the BEC will rise with the increase of the total particle number, the repulsive interaction between particles, the frequency of harmonic potential and the intensity of power-law potential. Thomas-Fermi approximation grows more and more resembling with the numeric result when the particle number increases, which shows Thomas-Fermi limit is a good method in the condition of large particle number, if the particle number in the system is less, the numeric way should be use when study the BEC problem.
Bose-Einstein Condensation (BEC) is one of the hotspot in the physics research recently. It reveals a new form of matter in the universe, a large number of dense atoms evolves coherently, bringing the quantum phenomenon from the microscopic world to the macroscopic world. Excitons BEC is the Bose-Einstein Condensation realized in the semi-conductor materials. The physists lay more and more emphasis on it due to it’s speciality and wide prospect, such as the application on high-powered and low-consumption light-emitting devices, ultra-fast logic devices, and quantum computing. Critical temperature is the phase transition temperature at which the common bose gas transforms into the Bose-Einstein Condensation. This thesis calculates the critical temperature in the low dimensional spaces according to the first-excited-state energy of particles in the traped condensate being different from zero, investigates the effect of first-excited-state energy on the critical temperature as well as the relationship of the critical temperature and the ground-state faction versus the total number of particles in different dimensional spaces. The results we get are consistent with the experiment data. It is shown that the critical temperature in low-dimensional spaces increases more quickly than that in high-dimensional space with density of particle number increasing, and the critical temperature of system has close relation to the first-excited-state energy. Our theoretical method can explain the phenomena that even under the condition of absent special external potential, such as harmonic potential and power-law potential, the Bose-Einstein Condensation can be realized at several K for excitons gas trapped in GaAs quantum well. The wave function of BEC can be described by the Gross-Pitaevskii equation. In the last part of the thesis, we study the effect of total particle number, the interaction between particles, the frequency of harmonic potential and the power-law potential on the distribution of the particles in BEC and the ground state energy of the condensate by solving the G-P equation using the Fourier-Grid-Hamiltonian method numerically. The results we get show that with the intensity of power-law potential or the frequency of harmonic potential increases or the repulsive interaction between particles decreases, the particle density in the condensate center will increase and the radius of the condensate will decrease. The ground state energy of the BEC will rise with the increase of the total particle number, the repulsive interaction between particles, the frequency of harmonic potential and the intensity of power-law potential. Thomas-Fermi approximation grows more and more resembling with the numeric result when the particle number increases, which shows Thomas-Fermi limit is a good method in the condition of large particle number, if the particle number in the system is less, the numeric way should be use when study the BEC problem.
引文
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[38] Zhu X., Littlewood P. B.,Hybertsen M. Exciton condensate in semiconductorquantum well structures[J]. Phys. Rev. Lett., 1995, 74(9):1633-1636
[39] Butov L. V., Filin A. I., Anomalous transport and luminescence of indirect excitons in A1As/GaAs coupled quantum wells as evidence for exciton condensation[J]. Phys. Rev. B, 1998, 58(4):1980-2000
[40] Reppy J. D., Crooker B. C., Hebral B., Corwin A.D., He J., Zassenhaus G. M. Density Dependence of the Transition Temperature in a Homogeneous Bose-Einstein Condensate[J]. Phys. Rev. Lett., 2000, 84(10):2060-2063
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[42] Butov L. V. Condensation and pattern formation in cold exciton gases in coupled quantum wells[J]. J. Phys. Condens. Matter, 2004, 16:1577-1613
[43] Bogoliubov N. On the theory of superfluidity[J]. J. Phys. (U.S.S.R) 1947, 11:23-32
[44] Fetter A. L. Nonuniform states of an imperfect Bose gas[J]. Ann. Phys. (N. Y.), 1972, 70: 67-101
[45] Pitaevskii L. P. Vortex lines in an imperfect Bose gas[J]. Eksp. Teor. Fiz., Sov. Phys. JETP, 1961, 40:646-651
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[51]王□,闫珂柱.简谐势阱中非理想气体玻色-爱因斯坦转变温度的数值研究.物理学报, 2004; 53(5): 1284-1288
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[2] Bose S. N., The Planck law and the hypothesis[J]. Z. Phys., 1924, 26(2):178-185
[3] Einstein A., Quantentheorie des einatomigen idealen Gases[J]. Sitzungsber. Kgl. Preuss. Akad. Wiss., 1924, 261-267
[4] Hecht C.E. The possible superfluid behaviour of hydrogen atom gased and liquids[J]. Physica, 1959, 25(12):1159-1161
[5] Hulin D., Mysyriwucz A., Guillaume C. B. Evidence for Bose-Einstein statistics in an exciton gas[J]. Phys. Rev. Lett., 1980, 45(24):1970-1973
[6] The Nobel Foundation. The Nobel Prize in Physics 1997 "for development of methods to cool and trap atoms with laser light"[EB/OL], http://nobelprize.org/physics/laureates/1997/index.html, 2008-01-23
[7] Hess H. F., Evaporative cooling of magnetically trapped and compressed spin-polarized hydrogen[J]. Phys. Rev. B, 1986, 34(2):3476-3479
[8] Anderson M. H., Ensher J. R., Matthews M. R., Wieman C. E., Cornell E. A., Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor[J]. Science, 1995, 269:198-201
[9] Davis K. B., Mewes M.-O., Anderson M. R., Van Druten N. J., Durfee D. S., Kurn D. M., Ketterle W. Bose-Einstein Condensation in a Gas of Sodium Atoms[J]. Phys. Rev. Lett., 1995, 75(22):3969-3973
[10] Fried D. G., Killian T. C., Willmann L. et al. Bose-Einstein condensation of atomic hydrogen[J]. Phys. Rev. Lett., 1998, 81(18):3811-3814
[11]郝柏林.中性原子的玻色-爱因斯坦凝聚[J].物理学进展, 1997, 17(3):223-232
[12] http://bec01.phy.georgiasouthern.edu/bec.html/
[13] Grlitz A., Vogels J. M., et al. Realization of Bose-Einstein Condensates in Lower Dimensions[J]. Phys. Rev. Lett., 2001, 87(13):130402-130405
[14] Butov L. V., Lai C. W., Ivanov A. L., Gossard A. C., Chemla D. S. Towards Bose-Einstein condensation of excitons in potential traps[J]. Nature, 2002,417:47-52
[15] Keldysh L. V., Kozlov A. N. Collective properties of excitons in semiconductors[J]. Sov. Phys. JETP, 1968, 27:521-525
[16] Lai C. W., Zoch J., Gossaxd A. C., et a1. Phase diagram of degenerate exciton systems[J]. Science, 2004, 303:503-506
[17] Bgnyai L. A., et al. Quasiclassical approach to Bose condensation in a finite potential well[J]. Phys. Rev. B, 2004, 70(4):045201-045208
[18] Bradley C. C., Sackett C. A., Tollett J. J., et al. Hulet evidence of bose-einstein condensation in an atomic gas with attractive interaction[J]. Phys. Rev. Lett., 1995, 75(9):1687-1690
[19] Zwierlein M. W. et al. Observation of Bose-Einstein Condensation of Molecules[J]. Phys. Rev. Lett., 2003, 91(25):250401-250404
[20] Jochim S. et al. Bose-Einstein Condensation of Molecules[J]. Science. 2003, 302:2101-2103
[21] Zhang Yue-qing, Wang Yi-qiu. Shift of critical temperature in Bose-Einstein condensate [J]. J.At. Mol.phys., 2007, 24(2):397-402(in Chinese) [张月清,王义通.玻色-爱因斯坦凝聚态的临界温度移动[J].原子与分子物理学报, 2007, 24(2):397-402]
[22] Chen Ai-xi, Deng Li. Quantum coherent oscillations between two coupled bose-einstein condensates[J]. J.At. Mol.phys., 2006, 23(5):809-814(in Chinese) [陈爱喜,邓黎.两耦合玻色-爱因斯坦凝聚体间的量子相干振荡[J].原子与分子物理学报, 2006, 23(5):809-814]
[23] Luo Xiang-yi, Liu Xue-shen. Ding Pei-zhu. Numerical study of interaction of two Bose-Einstein condensates[J]. J.At.Mol.phys., 2007, 24(2):418-420(in Chinese) [罗香怡,刘学深,丁培柱.两个玻色-爱因斯坦凝聚体间相互作用的数值研究[J].原子与分子物理学报, 2007, 24(2):418-420]
[24] Bagnato V., Kleppner D. Bose-Einstein condensation in low-dimensional traps[J]. Phys. Rev. A, 1991, 44(11):7439-7441
[25] Ketterle W., Van Druten N. J. Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions[J]. Phys. Rev. A, 1996, 54(1):656-660
[26] Menotti C., Stringari S. Collective oscillations of a one-dimensional trapped Bose-Einstein gas[J]. Phys. Rev. A, 2002, 66(4):043610-043615
[27] Cui Hai-tao, Wang Lin-cheng, Yi Xue-xi. Low-dimensional Bose-Einstein condensation in finite-number trapped atoms[J]. Acta Phys.Sin., 2004, 53(4):991-995(in Chinese) [崔海涛,王林成,衣学喜.低维俘获原子的玻色-爱因斯坦凝聚中的有限粒子数效应[J].物理学报, 2004, 53(4):991-995]
[28] Jin Li-xia, Wu Xu-dong, Zhan Zhi-ming, Chen Ai-xi. The finite-size interacting Bose gas trapped in a quasi two-dimensional harmonic trap[J]. J. Huazhong Univ. of Sci. & Tech.(Nature Science Edition), 2002, 30(11):102-104 (in Chinese)[金丽霞,吴曙东,詹志明,陈爱喜.准二维谐振势阱中有限尺度的非理想玻色气体[J].华中科技大学学报(自然科学版), 2002, 30(11):102-104]
[29] Su Xiaoqin, Guo Guangcan. Quantum communication and quantum computation[J]. Chinese Journal of Quantum Electronics(量子电子学报), 2004, 21(6):706-718(in Chinese)
[30] Robert Eisberg, Robert Resnick Quantum Physics[M]. John Wiley& Sons, 1985
[31] Ziff R. M., Uhlenbeck G. E., Aac M. The ideal Bose-Einstein gas, revisited[J]. Phys. Rep., 1977, 32(4):169-248
[32] Kerson Huang Statistical Mechanics[M]. John Wiley&Sons, 1987
[33] Hohenberg P. C. Existence of Long-Range Order in One and Two Dimensions[J]. Phys. Rev., 1967, 158(2):383-386
[34] Pethick C. J., Smith H. Bose-Einstein Condensation in Dilute Gases[M]. Cambridge University Press, 2002, 4
[35] Snoke D. W., Wolfe J. P., Mysyrowicz A. Quantum saturation of a Bose gas: excitons in Cu2O[J]. Phys. Rev. Lett., 1987, 59(7):827-830
[36] Lin J. L., Wolfe J. P. Bose-Einstein condensation of paraexcitons in stressed Cu2O[J]. Phys. Rev. Lett., 1993, 71(8):1222-1225
[37] O’Hara K. E., Suilleabhain L. O., Wolfe J. P. Strong non-radiative recombination of excitons in Cu2O and its impact on Bose-Einstein statistics[J]. Phys. Rev. B, 1999, 60(15): 10565-10568
[38] Zhu X., Littlewood P. B.,Hybertsen M. Exciton condensate in semiconductorquantum well structures[J]. Phys. Rev. Lett., 1995, 74(9):1633-1636
[39] Butov L. V., Filin A. I., Anomalous transport and luminescence of indirect excitons in A1As/GaAs coupled quantum wells as evidence for exciton condensation[J]. Phys. Rev. B, 1998, 58(4):1980-2000
[40] Reppy J. D., Crooker B. C., Hebral B., Corwin A.D., He J., Zassenhaus G. M. Density Dependence of the Transition Temperature in a Homogeneous Bose-Einstein Condensate[J]. Phys. Rev. Lett., 2000, 84(10):2060-2063
[41] Butov L. V. Exciton condensation in coupled quantum wells[J]. Solid State Communications, 2003, 127:89-98
[42] Butov L. V. Condensation and pattern formation in cold exciton gases in coupled quantum wells[J]. J. Phys. Condens. Matter, 2004, 16:1577-1613
[43] Bogoliubov N. On the theory of superfluidity[J]. J. Phys. (U.S.S.R) 1947, 11:23-32
[44] Fetter A. L. Nonuniform states of an imperfect Bose gas[J]. Ann. Phys. (N. Y.), 1972, 70: 67-101
[45] Pitaevskii L. P. Vortex lines in an imperfect Bose gas[J]. Eksp. Teor. Fiz., Sov. Phys. JETP, 1961, 40:646-651
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