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Acta Physica Polonica A

Statistical Physics of Bose-Einstein Condensation

Mariusz Gajda, Kazimierz Rza̧żewski

Published Dec 1, 2001• 2 citations

Abstract

cs, Po l i sh Ac ademy of Sciences al .Lo tni k §w 32/ 46, 02-668 W arsaw, Po l and b Center for Theo reti cal Physi cs, Po l ish Aca demy of Sciences al .Lo tni k §w 32/ 46, 02-668 W arsaw, Po l and ( Recei ved Novem ber 15, 2001)W e rev iew a pr ogress i n un derstandi ng of statisti cal prop erties of a quantu m degenerate Bose gas.W e show how the Ma xw ell demon ensem ble helps to compute Ûuctuations of the Bose{ Eins tein condensate of an ideal Bose gas according to the micro canoni cal ensemble.T hen, we review a metho d of measuring these Ûuctuations.U sing a solubl e mo del of interacting Bose gas w e also stress the imp ortance of higher-order correlation functions.Finall y, w e review our no vel computational metho d of studying an interacting Bose gas near its critical temp erature.

DOI: 10.12693/aphyspola.100.7

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cs, Po l i sh Ac ademy of Sciences al .Lo tni k §w 32/ 46, 02-668 W arsaw, Po l and b Center for Theo reti cal Physi cs, Po l ish Aca demy of Sciences al .Lo tni k §w 32/ 46, 02-668 W arsaw, Po l and ( Recei ved Novem ber 15, 2001)W e rev iew a pr ogress i n un derstandi ng of statisti cal prop erties of a quantu m degenerate Bose gas.W e show how the Ma xw ell demon ensem ble helps to compute Ûuctuations of the Bose{ Eins tein condensate of an ideal Bose gas according to the micro canoni cal ensemble.T hen, we review a metho d of measuring these Ûuctuations.U sing a solubl e mo del of interacting Bose gas w e also stress the imp ortance of higher-order correlation functions.Finall y, w e review our no vel computational metho d of studying an interacting Bose gas near its critical temp erature.

Sch oo l o f Sem icond uct i ng Co m p ou n ds, Ja szo wi ec 200 2 B ose{ E i n ste in C on d en sation in A to m i c G ases J. Z ac horo w sk i and W. Ga wl ik M. Sm ol ucho wski I nsti tut e of Ph ysi cs, Jagell oni an Uni versity Reym on ta 4, 30-059 Kra k §w, Po l an dT hi s pap er presen ts fun damenta l prin ciple s, characteristics, and limitati ons of v arious exp erimental metho ds of coolin g and trapping of neutral atoms by laser light and magnetic Ùelds.I n additio n to surveying the exp erimental techni ques, basic prop erties of quantum degenerate gases are discussed w ith particular emphasis on the Bose{ Einstei n condensate.W e also present main parameters and exp ected characteristics of the Ùrst Polish Bose{Einstei n condensate apparatus built in the N ational Lab oratory of A tomic, Molecula r, and O ptical Physics in T oru ¥, Poland.

This work is an up-to-date and partially improved analysis of the basics of the mathe- matical description of Bose-Einstein condensation in terms of first principles of Quantum Mechanics. The main aims are: 1) to provide a compact, yet coherent overview: - of the basic mathematical tools used to formalise Bose-Einstein condensation, - of the mathematical techniques for studying several features of this physical phenomenon, - and of how such means emerge as the natural ones in connection with their physical interpretation; 2) to discuss and to place in the above perspective some new contributions and improvements: - on equivalent characterizations of Bose-Einstein condensation, - on the strength of the convergence for some currently available asymptotic results, - and on the effects of interparticle correlations on the energy and the dynamics of the many-body condensate.

c h a and K. R z¨âew ski b a I nst i t ut e of E xp eri m ental P hysi cs, Opti cs D i vi sion W arsaw Uni versi ty , Hoâa 69, 00-681 W arsaw, Pol an d b Center for Theo reti cal Physi cs, Pol ish Aca dem y of Sciences al .Lo tni k §w 32/ 46, 02-668 W arsaw, Po l and W e st udy an ult ra cold dilute gas of bosonic ato ms in an optical lattice induced by intersecting laser b eams.A s a Ùrst appro ximation w e neglect conÙning p otential and atom {atom interactions.In t his case the Gross{ Pitae vs ki j equation reduces to simple Mathieu equation.U pon cho osing p erio dic boundary conditions this problem has well know n perio dic solution.T his simple picture allow s to demonstrate lo cali zation of the w ave packet and formation of the band structure.W e calcula te spectrum of the excited states as a function of the strength of mo dulati ng p otential and using a standard adiabatici ty criterion w e predict the most e£cient w ay to ramp up optical lattice , w ithout higher state excitation.Finall y, we discuss the in Ûuence of the atom{atom interaction (nonlinea rity ) on the adiabatic ity of the pro cess.