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Unformatted text preview:  HH ¨7 ƒ½ë Sae Mulli (The Korean Physical Society), Volume 50, Number 5, 2005¸ 5, pp. 323∼328 4 Z ELTB UXÅù T“A 4” BEC V?8 øR ;U sËħ Þ êNkp Ó0 ˜X m¢ s ê0ý mê c. És »  »' c£å c¢Ú cä× ×> ¨* · ‡ · +* · *.H∗ »< <  /<§ ü<  “ @†“ Óo†õ, “… 402-751 Æ tÆ ; (2004¸ 12 9{ ~6, 2005¸ 4 26{ þ7‘ ~6)  4  Χ  Z 9 ã  4  xr Χ  Z 9 já: ã 1995¸ JILA ªÒ\ _K   Ձ  ¨ 87 Rb "\" BEC I ½‰H\   ·ºo l^\" BEC ¶ éf œ © ¨&†  É úú ‰f ³d r ˜˜ ¦³  ¨& 9 ¸ ½ ª ¨& Af  :¸ úÒ ¯ 9ºh¸  \ ½‰ ”4s ô‚s. Õ ½‰` 0K" Òe_ “•\ ±Æ s €Ã&sl• t H § ÇÓ ³¦ H r¦ r ¦  H   –¶ ß é hº Q¼ ñ¸  ú Ô ¯¸ 9¹ ë, "_ >à #Ö &• °\ s؍ • €¯ 9, sô &˜\ %l 0K" BEC\ H¯ H  Q ñÐ 3 Af Ç ¦ H ¦  ¸ Õ½ º e î1 ~&ds €¯ . ‘ ë\" BEC I\ ¸ lÕ  ܖ ·” ˜ tÉ ú lü+ à ” l ½ñ” 9¹ : 7f H rx Ó   r HH H œ ¦˜ t H © ú ü ¯¼Ð ú9 ˜ Gross-Pitaevskii (GP) ~&dܖÂ' ĕ ‚+s^(ELTB) ~&d` 7 # ˜€. GP ~ Ó  ½ñ”¼ÐÒ »¸) þ‰ a A Ó ¦ x ½ñ” Ž£Œ Ќ ¤ Ó ½  ñ”\f ú xÐß î©  é ŒŒ 9Ðß ª¦   þ &d" 8 Kžmîs ¨ç{\ _ô " yy_ Kxžmî, Õo“ Ø[\ _ô q‚+ Hx 9 – H œ Ç ¶ ••  – æt Ç A KxÐmî_ ½¼– ½$÷X, #l\ hypercoordinate` •{ € q‚+ †\ _K "&\" 9žß +ÜÐ ¨í&< Œ  –Ë H ¦   A Ó  ¸9 þ ½  éhf ¶ ϖ H 1ß  > h Ó ú þ‰ ½ñ” 3 )  ½ñ” ª i<f ú ú9  µí Y >_ ½` ° ‚+s^ ~&d` %> . s ~&d\ € %†\" ¸ ·”  †¦ H A Ó ¦  a Ó  œ Æ ˜˜  r¦ x O h Œ >ß 9© -ú ìZ` &6# íô {{ \t°` GP ~&d` Ãu&ܖ Û# %“ °õ q“ # ˜€ –Ç  œ ¯¦ Ó ¦ ½ñ” ºh¼Ð Q 3É ú §Œ Ќ  ¦ r ¯ ¤ H < hÉ hº éf ú þQ ú¯ X+ º ”3¼ X, &“ >Ã_ "\" ¸ [# ´` S“½ à e%Ü9, >à &f\  °s &t r ¶ H ˜ t H ¦ ‰É  hº ”  ú   t ú“ š  12 %\ ç  ` · à e%. s–+ ‚+ s^ ~&ds &“ >Ã_ "[  §¦ ¸ ·  º4 ¯ ú º ”3 Ћ þ ‰ ½ñ” hÉ hº éþ  H ¦ ˜  A Ó  r ¶t – À# BEC\" I\ ¸ lÕô“ t` à e. Ð sÒQ”  f © ú ü¦ : º ” H œ ¦ ˜ tÇ r ¦  PACS numbers: 30 Keywords: ˜Ý-“à “ 6|I м » £9©   xœ I. "  e] Ø 1920¸@ ×ì\ Bose [1]ü Einstein [2, 3]\ _K {  æÍ / ø <   9  þ :¸ ú ¸ 9þ úÉ ª © [_ “• ±t€ —Ž {[s °“ €  tr   H  t r œ œ  9 ” & м » £9© I\ ] e> ÷ ˜Ý-“à “ 6|I(Boset H   xœ Einstein Condensation, BEC)\ sµ s&ܖ µ ¦ §` r  ¡¦ :h¼Ð 1 ß €· êÐ úÉ ü<þ ª © ¨& / ) sʖ ´“ Óo†[s Õ I\ ½‰K ?l p §r t Æ t œ¦ ³ A 9¹ ¸4 Œ ® 0K €¯ô ”§` lÖ# M. 1939¸ Fritz London\  Ç ¦ ¦ o     ‰ ¡¢ í»‰ &© _K Ó^ óµ_ œÄ^ ‰s BEC–Â' l“Hs o š§ ³œ ÐÒ † d ˜ ú9’¼ ·&Ü [4], Ó^ óµ "[ s_ © “§õ o š§ ¶ t ‰ ¡¢ éþ  œÇ 4 yô  œ ©/h¼Ð úÉ ºî @&ܖ “ Ã"(10−9 sec)“ BEC I\ @ô  ªr r É œ © /  ǃ ¨ jh¼Ð Q> ßþQ ® f ½\ z]&ܖ #§> ë[# M. " BEC–_ ¦´   –t o Ð   ñ ª üh 9 ú/ Af „s õ&s~ Õ Óo& í|` ·?l 0K"  t  $¦ ˜ H  t –  •x  9þ ß œñŒ  §h h¦ Å ß ©  {[ ç_ © 6s q“& &“, t5 rçs   q–œ /h¼Ð  9þÐ ÀQ ¼7 9¹  @&ܖ | {[– sÒ#” rÛ%s €¯ 9 s   t  › ¦  AŒ ŒQ ªf úú ‰ ‰Œ \ 0 # # ÕÒ\" ·ºo l^\ Íyr& BEC ¨ ˜˜ ¦ t• ¦ –t  ßþQ /9 ¸4 âÒ÷3 ¼Q \ ë[# ? ”§s Å&%. ×n# 1995¸ c H  t + и /< –• @†_ JILA ÕÒ\ _K Rb "– sÀ#” Æ ª  ¨ ¶ éÐ ÒQ  BEC þœ– ½‰÷%“ [5], +\ s# àµ, o½, jíÐ ¨&&3¦ ³ ' Q Ô¢ ¢ ¦ § § ∗ E-mail: [email protected] ºè éþÐ ÒQ Ù "[– sÀ#” BEC z+z\" ƒs# $ ¶t  ´>´ «f Q í  N B&3¼ /÷%Ü9 [6–8], s D–î Óo†_ I1` _p  hÐ ü< l  H rt Æ x¦ H    &3 ª êÐ  |s ÷%. Õ sʖ BEC I_ :f ×_  œ © £ç   ¤ æ  í»‰ è < úÉ &©þ úú  “ œÄ^~ ™6[sü °“ ‰[s ·ºo l   xt r ³œt ˜˜ ‰Ð Ò# ^– sÀQ” BEC\"•  è s z+&ܖ  f¸ ß ¯ «h¼Ð – H ´> ‰a X)  ” S“  e [9,10].  Ð>¼Ð ÒQ  9¸  Ð>þ ©ñß ˜”Ü– sÀ#” >_ x• &t€ ˜”[  ç r    rt œ –   ú _ o Ât9, “• ?€ ˜”[_ 5• ª r :¸ /9  Ð>þ Ÿ  rt q ¦t QþQ Ð> ü9 © Q Q ¦9 ×#[# ˜”_ Ó| s U#”. sô “x r t œ ´ Ç ¸ G: ©f ü9 © Ð>þ ß •, F$“_ I\" Ó| _ s ˜”[ ç_ rœ H t œ rt – Ð Q : l<ºþß o 9Q o˜ U#tl Më\ 1†Ã[ç_ ×^s {# ´ H xÊ t– æ?  ¦ “, BEC–_ „s ”'. Ð  Ÿ)  a ˜˜ úú ‰ /©¼Ð  «f ¦9¸ Ð ·ºo l^\ @ܖ  z+\" “x• ˜ ¦œ H ´> H  G: Gf  F$“_ Fô\"_ BEC–_ „s\ œ&` ´ H r Ç Ð  íh ú  ¦  Ò ) º úÉ :¸f 9þ ß œñŒ Æ> . BÄ ±“ “•\" {[ ç_ © 6 a r r H  t –  •x r ¤ É Xú  ÒQ¦ QË+ º ”¼ “ ]8\ _K sÀ#”“ #a½ à eÜ9, s   >É  â º 9þ l¦ ü rl ½ñ”É @ҁf Ä {[_ î1 lÕ  1 ~&d“ üÂ\"  t rx` t H îx Ó r K” îç (J[õ Ø[– “ô (J[_ +ܖ ³   $> Ð  $> ˼Рð  ¨H ™ æt Ç ™ ½ ³H &&<  ½ñ” ‰÷X, s ~&d` Gross-Pitaevskii(GP) ~&ds Ó ¦ Ó  ½ñ”    ô [11]. GP ~&d“ “x•, F$“_ BEC  Ç Ó r  ½ñ”É ¦9¸ G: r œ ©  ú ü¦ ú9R ” Ð  $> I\ ¸ lÕô“ ·4 e. Ø[– “ô (J[ ¦ ˜ tÇ ˜  æt Ç ™ -323- -324- ÇGt Æ Dü<r hü ô²Óo†t “DÓo”, Volume 50, Number 5, 2005¸ 5 t 4 Z r A Ó É þh ½¼Ð /Q<  “ q‚+&“ †Ü–  ?#tX, s GP ~& H H Ó ½ñ ¦  ” 3h¼Ð Ò ¯ Q> ß d` K$&ܖ ɍ ` #§> ëŽ. 1999¸ p² H ¦  –H G  D Purdue ÕÒ\" ÙÓo_ ^ ~&d` ɍ X &6 ªf þü ‰ ½ñ” Ò < h ¨ ˜t Ó ¦ H x Ç  ô hypercoordinate` s6 # GP ~&dõ 11ô  ¦x   Œ Ó  xxÇ ‚ ½ñ” lp  A þ ‰ ½ñ” ßþQ ·  ” + s^ ~&d` ë[#   e [12]. s ÕÒ\" Ó ¦ –t p   ªf ¨ H   ½ñ” é  s ~&d` 1 " N-body >\ &6 #, Ns  Ó ¦ ¶  h Œ   â x H º ¸ 3A Ä š #0 4.5 % ?\" z+&“ õ\ ¸ ["½  /f «h  ú Oî+ ´>  ¦ ˜  É º ”£ Ði à e6` ˜%. §¦  r ë : HHf Af ¸9 A ‰ ½ñ”  ‘ 7\" 0\" •{ô þ s^ ~&d` s H Ç ‚+ Ó ¦ x Œ 6 # 87 BEC–_ „s ¸ {#> l 0K" "[ Ð  ú 9Q  Af éþ  ˜ H¶ t –  •x r ~ ß œñŒ  hÉ ¯ %¼ ç_ © 6s &“ s aÜ9, s\ 0K @Âì_  A /Ò ¦ r ´> «f h© 9¸ hÉ    O z+\" &{y x• &“ >\ 6ô. sX> H œ  r ¦ xÇ  BÃÇ ‰f éþ ß  O¦ ©ñŒ ~ l^\" "[ ç_ o Y“  6 Ìô H¶ t–  œ •x r ¶ t – æ[ É éþ ß t fß ÒQ ) º “ "[ ç_ Ø\ _K"ë sÀ#t> . BÄ – a r “ úÉ r¸f BÃÇ ‰ ⺠£Œh ú  ±“ :•\" ~ l^_ Ä 7y&“ X8\ _ Ìô ¤• ]¤ Ç ætr §  É £ ú ð&) ô Ø[“ 6õ °s ³‰. [13] ³a Vint (ri − rj ) = 4π 2 a δ (ri − rj ). m (3) Rb "– sÀ#” BEC >\ e#"_ { ¶ éÐ ÒQ   ”Qf 9   œ © - ߦ { \t\ >í “, GP ~&d\" Ãu&ܖ >í ¦– Ó  ½ñ”f ºh¼Ð ß  – ǯ  ú § Ÿ¼Ð‹  ½ñ” » í úÐ ô °õ q“K 4ܖ+, s ~&d_ Ä6$` ·˜ § Ó  x ¦ ˜ 9   ô. GP ~&d“ Õ ^_ 4¸$ܖ “K { Ç Ó r ½ñ”É ª ‰ Ÿúí¼Ð  9 ¤š  º Ò h Ò úÉ âºß 3h¼Ð  à Š& Å ´“ Ä\ë K$&ܖ Ûo  §r –  ¦ 9 @Âì_ >í\ e# Ãu&“ Kë` ]/  ì /Ò ߁ ”Q ºh ß jB ø r –  –¦ N H Í  A  þ ‰ ½ñ”É ©/h¼Ð ßß AÐ  €, ‚+ s^ ~&d“ @&ܖ çéô þI– “K Ó r œ  ––Ç +  /Ò 9º / 3h¼Ð  º ” : @Âì_ {Ã\ @K K$&Ü Ò Ã el M r  ¦  H  ë\ BEC\ @ô ñ$&“ sK\ 0 > ½ ÷ m  / íh  p +   Ç &  ¦x Ér   Ÿ Œ ü¾þ H 1 3 º ”¼ , s\ :K #Q Óo|[` < ~> %` à eÜo ¦x t Ót¦ ’  ¦   /)  [email protected]. a Œf   ßÍ  #l" a S - íø Uss. H –ê ´  ” d (1)\ d(2)ü d (3)` @{ “, [email protected]°` 2ô Ê ” <”   ¦  /9¦ /ú [ ê ¯¦ Ç H œ H ¦ x  î©    ü úÉ ” 3 º ” ¨ç{ \ 6 € A< °“ d` %` à e r ¦ ¦   . i 2 ∂ Ψ=− ∂t 2m i=1 2 N 2 iΨ + Vext Ψ + g0 |Ψ|2 Ψ (4) Œf #l" g0 = 4π a/mܖ" Ø– “ô ½ Ã\ ¼Ðf tÐ  + ©º æ[ Ç Ë œ ¦ p  é ßø   . a "_ íê Uss9, m“ "_ || · H¶ –Í ´ r¶ É é 9¾ Ó   ”É s. s d“ 1961¸ Grossü Pitaevskii yy ĕô r   < •• ŒŒ »¸ Ç  ”¼Ð dܖ Gross-Pitaevskii(GP) ~&ds ԏ. t Ó  ½ñ” ;  ¦2 • †r  t æ[ Ç + Ó Œ ÓÉ 9þ tÐ  A ½¼Ðf  ” } ½“ {[_ Ø– “ô q‚þ †Ü–" s d` ¦ ¦  1 ú£  Ûl ~t ·6` _pô.  §§¦ Ç  ”  A D s d` Ûl 0K p² Purdue ÙÓo ÕÒ\" ¦ ¦ G ˜t þü ªf ¨ H N >_ {– ½$ Ù` ɍ X 6ô ~Z` &6 h 9Ð ¨í) þ Ò <   ½O h  a ˜¦ H xÇ Ó¦ x Œ þ ‰ ½ñ” ¨·  ” # ‚+ s^ ~&d` ½K  e [12]. s /d A Ó ¦ p   B” N ‰f A ” ^>\" 0_ d (1)_ K–" A< °“ +I_ K H   Ðf ü úÉ þ  r A ¦   ñi \ & %. Ψ(r1 , r2 , · · ·) ≈ ˜ Ψ(x, y, z ) . (xyz )(N −1)/2 (5) II. ŽŽ” Ò] T= UXÅ Þ«¢ ù ÒÖX Åk k êNk sËÄ N>_ ˜”Ü– sÀ#” >_ Kxžmî“ 6õ h Ð>¼Ð ÒQ  9ÐßÉ £ r   –r § ú /Q °s  ?#”.  2 N 2 i i=1 H=− 2m + Vext + i<j Vint (ri − rj ). (1) Œf #l" Vext “ üÂ\" Kt ¨ç (J[s9, r É @ҁf  î $> H H ™ Vint  {[ ç_  6\ _ (J[s. H  t – œ •x  9þ ß ©ñŒ  Ç $> ô ™ Vext  "[` ¿# ¿l 0K €¯ô ü ( H ¶ t¦  éþ ºQ º A 9¹ @Ò  Ç ™ $>Ðf  éþ x¸ 9 © » J[–", s "[_ •\ {ñ s Ätrv H ¶ t 9 ¦ & œ H ɦ Ç i+  «\f  A p½í  í %½` ô. z+" s\ 0K q1~$ l Ÿ ´> H¦ xÓ ‰ \ S(anisotropic magnetic trap) u\ 6ô. q1~ œ ¦ xÇ ©   p½ xÓ í  í\ © $ l ŸS u x, y ~¾Ü– @g&stë, z ‰œ H ӆ ½Ó¼Ð /Ahß H  – ÓÓ ½¾¼Ð /Ah ß ¸o $> ¸ª ~†Ü– [email protected]&“ éí › (J[_ —€`  H  –H ™ œ¦  t9, A< °s ³‰. ü ú ð&) ³a Vext = 1 2 N 2 22 22 m ωx x2 + ωy yi + ωz zi . i i=1 Œf #l" x, y , z  >\ lÕ½ à e [ >_ Ã–" H ¦ tÉ   ü+ º ” j h ºÐf H  •• § r a ¦ –¤Ç ŒŒ £ úÉ '” ßá yy 6õ °“ ›>d` ë7ô. N N N 2 yi , i=1 x2 = i=1 x2 , i y2 = z2 = i=1 2 zi (6) ¤ £ ŒQ 9 œ ü < 9¹ º h 7, # {_ ©I\ lÕ  X €¯ô Í >  ¦ tH  ǁ H h A  A j î”   >_ 0u m 0u ]Y_ ¨çe` _pô. s L H¦ Ç j  þ  jè  é ] s +I_ K þ™ ì_ "o, A r ¶ (2) ˜ ˜ δ < Ψ|H |Ψ >= 0. (7) Œf #l" ωx = ωy = ωz s. M\  ωx ü ωy  €ß  :  < •ç –   ¸ ¼Ð  7f h>h¼Ð [ _ s l• ٖ s ë\" >Z&ܖ 2 HH H  LÇ å /ô. ¦ –¤  ßá ÐÒ ü úÉ ½ñ” 3 \ ë7Kô z–Â' A< °“ ~&d` % ÇH´ r Ó ¦  ¦  º ” ` à e.  ˜ ˜ H Ψ = E Ψ, (8)  HH ¨7 ƒ½ë ELTB ~&d` s6K >íô BEC I_· · · – ^¿% 1 Ó ¦ x ½ñ”   ß –Ç œ ©  x ”ºò p -325- Œf #l" 2 IV. w–©  M Š¥ Œ † 2m m 22 2 2 + ωx x + ωy y 2 + ωz z 2 2 2 (N − 1)(N − 3) 1 1 1 + + 2+ 2 2m 4 x2 y z g + xyz ∂2 ∂2 ∂2 + 2+ 2 ∂x2 ∂y ∂z r Of - ¦»úÉ £ úÉ ' ß ìZ\"_ \t “Ä°“ 6õ °“ ›>\ ë ¯r § r a ¦ – ¤Ç á 7ô. E= < ψ |H |ψ > ≥ E0 < ψ |ψ > (14) H=− (9)  s9 g = g0 (2π )− 2 3 Γ(N/2) Γ(N/2 − 1/2) 3 N (N − 1) 2 (10)   ” lp þ ‰ ½ñ” s. s d` 11ô + s^ ~&d(equivalent linear ¦ xxÇ ‚A Ó  two-body equation, ELTB)s ô.   Ç |ψ > β _ <Ãsٖ, 0_ \t [email protected]°• β _ <  ʺ¼Ð A - /ú¸  Ê † ¯ † º f A Òp” jèú ßá ú Ãs. " 0 Â1d_ þ™°` ë7rv β ° x ¯¦ –¤ H¯ ¦Ô   1¼ ª úf - ú Œ© - ` ¹Ü€, Õ °\"_ \t °s {I_ \t ¯ ¯ •œ ¯ H Ç ¯¦  c ú  ú ú ¨ ¯¼Ð /) 9Ðß °õ ô °` °> | ܖ [email protected]. Kxžmî a  –  /ú ß Af ºþ ” _ [email protected]°` >í l 0K" ÁrÙ d (9)_ t ¯¦ – H ¡~    •Ó Œ ½ /ú 9¹ ß  úÉ  } †_ [email protected]°s €¯ . të s °“ β _ Gamma ¯ – ¯r Ê <º¼Ð ª  3h¼Ð Ò Q>  †Ãsٖ Õ pì` K$&ܖ Àl #§.  r¦   f º £ úÉ   i " Äo 6õ °“ \ 6 %. H § r H ¦ x  ψ 1 ψ xyz = ψ 1 ψ (xyz )2 3 2 III. Ä0®z ” w–]• ÅØÉ ìZnŽº ŒX Š¥K¤ ¢ ¤  ” d (9) 0u "&ܖ y\  µß  † H  A éh¼Ð  Œ  1– ½ ¶ ™ Ïí H Ó ¦Ê  í< : 1  ú f  7H ` Ÿ† l Më\ ~> Ûot ·. " s  H¦ §H Hë f ªi<f ú ú9 O  Œ  \" €%†\" ¸ ·” ìZ` 6 #  H œ Æ ˜ ˜  r¦ x H   ¨Ц   A º ”  K\ ½K˜“ ô. s\ 0K, Ă d (9)_  ¦ Ç ¦   Œ Ó º  ¨¦ ª  ú 9Ðß t} ½` Árô K\ ½ “, Õ K\ 8 Kxžmî • †¦ Ǧ ¦x  –   Ð úÐ  Œ Ó º ” _ 1 K– ¸l– ô. t} ½` Ár €, d H š Ç • †¦  (9) Ã ìo&9, y ~†\ @K 6õ °“ pì H r  º ÷ • ÓÓ Œ ½¾ / £ úÉ  § r r Ó    ½ñ” 3Q ~&ds %#”. − 2 d2 m2 (N − 1)(N − 3) 1 + ωx x2 + 2 2m dx 2 2m 4 x2 φ(x) = Ex φ(x). (11) 2 m3 ωx ωy ωz 1 β− 3 2 1 2 (15)   s  β  ô>\", Õ >íu“ HH HÇ  f ª ß – 1 ψ ψ xyz 3 2 = ≈ m3 ωx ωy ωz 3 2 Γ(β ) Γ β+1 2 β− 2 3 3 m3 ωx ωy ωz (16) < ú 9¼Ð %É ” ú º ” jÐ  ü ¸ {u ٖ a“ e` · à e. z]– ⠍ ˜ ~r H ¦ ˜  ´ H (N − 1)/2˜ °sٖ, N ≈ 100 s\" ¿ ° Ð  ú¼Ð H¯ ©f º ú œ H ¯   _ s 1 % s – ”. H Ð Œ • j ú -  h<ºÐ ¨Ð ” s] 8 \t\ β _ B>†Ã– ½K˜. d(14)– x ¦ Ê  Ð Ò Â' E (β ) = 1 β −1/2 (N −1)(N −3) + β2 + 4 · 1 (ωx + ωy + ωz ) 2 m3 ωx ωy ωz 3 1 2  ”É 3h¼Ð  s d“ K$&ܖ Ûo9, K–" 6_ Ûs\ % r  ¦ Ðf £  3 § ¦ ¦ H  . 1 φ(u) = Cx uβ exp − u2 . 2 (12) β− 1 2 +g β− 1 −3/2 2 (17) Œf #l" u = mωx / x "s \ Ãs9, β = H é H   ¶ O º (N − 1)/2s“ Cx  ½ ©Ãs. " d (9)\ ¦ H  ©o œº f ”   f Œ Ó º ª  £ ú ÒQ " t} ½` Ár € Õ Ûs 6õ °s Å#” • †¦  ¦H §   . 1 ψ (u, v, t) = C (uvt) exp − (u2 + v 2 + t2 ) . 2 β  hÐ hº s. D–î B>Ã α = β − 1/2\ •{ # r  ¦  ¸9Œ E (α) = 1 α N 2 ·1 2 −1 2 + α(α + 2) 1 2 (ωx + ωy + ωz ) m3 ωx ωy ωz 3 +g (13) α−3/2 (18) Œf <  < ø Ð <  /£& é #l" v ü t uü ðt– y ü z \ @6÷ "s H Í xH¶ H  O º £ \ Ãs. 7, v = mωy / y s9, t = mωz / z s ¤    j  <º  Of hºÐ  . s] s †Ã\ β \ ìZ\"_ B>Ã–  Ê ¦ ¦ r  H « <ºÐ  Œ ” r+ †Ã– 6 # d (9)_ {I_ \t “Ä >Ê x   Œ© - ¦» •œ ¯¦ ú ¨Ð °` ½K˜. ¦  ¨¦  ” jèú 9 ú  & ` ½ “, s ds þ™°` t€ pì°s 0s ÷  ¯¦  r¯ Q  ¸   Œ # ô ›|` s6 # α ë7  ›|` ½ Ç H ¦ x –¤ H ¦ ßá ¸  ¨ É + º ” ½ à e.  α 1 2 α− 2 N −1 2 2 3g = ωx + ωy + ωz m3 ω x ω y ω z 5 1 2 (19) -326- ÇGt Æ Dü<r hü ô²Óo†t “DÓo”, Volume 50, Number 5, 2005¸ 5 t 4 Z Table 1. Energies per particle in the ground state for the total number of particles. ELTB values are calculated from the ELTB equation with the variational methods and GP values are obtained numerically from the GP equation. GP values are from the reference [10]. N 100 200 500 1000 2000 5000 10000 15000 20000 ELTB value 2.67 2.89 3.40 4.04 4.96 6.77 8.73 10.17 11.36 GP value 2.66 2.86 3.30 3.84 4.61 6.12 7.76 8.98 9.98 Percent value(%) 99 99 97 95 92 90 88 88 88 Table 2. Kinetic energies per particle((E/N )kin ), external potential energies per particle((E/N )trap ), and interaction energies per particle((E/N )pot ) from ELTB equation are shown for the varying N , the total number of particles in a system. The values in a parenthesis are the corresponding values from the numerical calculations of GP equation in Ref. [10]. N 100 200 500 1000 2000 5000 10000 15000 20000 (E/N )kin 1.05 0.94 0.77 0.64 0.51 0.37 0.28 0.24 0.22 (1.06) (0.98) (0.86) (0.76) (0.66) (0.54) (0.45) (0.41) (0.38) (E/N )trap 1.39 1.54 1.89 2.30 2.88 3.99 5.18 6.06 6.77 (1.39) (1.52) (1.81) (2.15) (2.64) (3.57) (4.57) (5.31) (5.91) (E/N )pot 0.23 0.40 0.74 1.11 1.58 2.42 3.27 3.88 4.37 (0.21) (0.36) (0.63) (0.93) (1.32) (2.01) (2.74) (3.26) (3.68)  ßá  ¨Œ  ” s\ ë7  α\ ½ # r d (18)\ @{ €, ¦ –¤ H ¦   /9  Eβ 5 =g 2 + 2 1 3 2 (N/2 − 1)2 (ωx + ωy + ωz ) +1 β − 1/2 m3 ωx ωy ωz 1 2 −3 β−  ªÔÐ / s\ ÕAᖠ ?€ Fig. 1s . Õa\" ˜ ¦   ) ªËf Ð a > ¶  éº ”   ú º3 ú  € "à &f\  ì °õ ÃuK$ °_   r ¯ ¯ (20)  hÐ   & – &”. s g ∝ N (N − 1) N \ ß> _”  H  ¼ > r ¼Ð ú 9ºŸ    Œ Ó Ù– N °s &|Ã2 \ 6ô t} ½ g/xyz _ ¯  ¤ H ¦ xÇ •†  ò  : q  ´õ &tl Më\ Òl õs9, \ t · H t H H¦   ú § ¦ ºh¼Ð ¨ ª  Qþ ¯¼Ð /) “ Ãu&ܖ ½ € Õ  ×#[ ܖ [email protected].   H¦ t a ¦ ¦  3 º ” ` %` à e.  V. + ÇØ s]  éº  >_ "à N ` 100Â' 20000t r&9  ¶ ¦ Ò  o   •œ Œ©f 9 © - ú ¨:  {I\"_ { { \t °` ½K‘ õ\ Table œ ¯¦ r ¦ 1\  Í. #l" GP°sê GP ~&d` Ãu&  Ç Œf x ¯Í úø Ó ¦ ½ñ” ºh  ¼Ð   ܖ ó õs9 [10], ELTB°sê ELTB ~&d`  r ¯Í úø Ó ¦  ½ñ”  r¦ x O  Œ 3h¼Ð   þ úÉ ìZ` s6 # K$&ܖ ó õs. ÑìÖ °“  r ˜r¦ ¯r GP°_ ELTB °\ @ô q\ %–   ¯s. >í ¯ ú ¯ ú /  Ð ·  ß Ç¦ p –   ) \ 6 Rb_ ßê Us a = 100a0 –", a0  ˜# xa  –ø  íÍ ´ H Ðf H  ÐQ ͧ ø£ ¸ éþ º A  ) í\  ìt2s. ¢ô "[` ¿l 0K 6 ŸS ( Ç ¶ t¦ xa ‰ √ √ ™ x H $> lº J[_ ”1Í ωz = 8ωx = 8ωy = (2π )220 rad/sec  s9, \t ωx – ½  °s. "à q“ - H Ð ©o ) ú éº §  a¯ ¶  ¦ h h : & &` M(100>\" 1000> s) ÃuK$ °õ  hf h   º3 ú  H ¯  r O  ß ú ¸ ìZ\ _ô >í °s š 5 % ?\" ¸ {u “ e Ç –¯ /f ú 9¦ ” ˜  ¼ Ü9, ՘ " Ã\ @K"• š  12 %\ t ªÐ H é º /f¸ ¸ ¶ ¦  Å §H ú  ·. s Purdue ÕÒ\" 1 " >\ &6K˜€` H ªf é  h Ќ ¨ ¶ x ¤¦ : ¸ M š  4.5 % &•%~ õü {uô. [14] sô   ñ¸i < 9 Ç Q Ç  H  9º úÉ âº õ {à ´“ Ä\ 88 % &•_ scaling` §r H ñ¸ ¦  x ŸŒ §h X ú  º ” ½O jB+ : # q“& ñSô °\ s\ à e ~Z` ]/½  &‰Ç ¯ ¦ H Ó¦ NÉ º ” 9 © - úÉ 9 º ”   à e. { { \t °“ { à &f\  ‚  œ ¯r    A þh¼Ð £  Ð ª £  Ÿ Q¼ ¯ +&ܖ 7 l ˜ Õ 7_ ;s ×#׍  x H x ¤¦ H ¦˜  ú º ”<  ` · à eX, s GP ~&d` Ãu&ܖ ó õ H H Ó ¦ ½ñ” ºh¼Ð  ¯  r r Ó¦  úÉ â¾ Ð °“ †` ˜“. Table 2ü Fig. 1\" é0 { {_ \t\ î1 < f ßA 9 © - l H–  œ ¦ rx - \t(Ekin /N )ü, ü ŸS \t(Etrap /N ), Ø– < @Ò í\ - ‰ æ[ tÐ Ç œ •x  ©ñŒ - “ô  6 \t(Epot /N )– ¾# ˜€. F Ð ºQ Ќ ‹ñ ¤ c – H ¯r ߁ ” úÉ î\ e °“ GP ~&d` Ãu&ܖ ó °Ü–", Ó ¦ ½ñ” ºh¼Ð  ú¼Ðf  r¯ § AŒ ,QÒ3 q“\ 0 # V#Å%. Fig. 1\" ˜rx yy_ ¦   f Ð ŒŒ •• -¸ º  ú þQú ¯ ú º ” ªQ \t• ¿ õ ¸ [#´ ` · à e. Õ  ˜ t H ¦ ˜   , ELTB ~&d\ ìZ` &6 # %“ õ GP Ó  r¦ x ½ñ” O h Œ 3É  r  H Fig. 1. Kinetic energies(Ekin ), external trap energies (Etrap ), interaction energies from collision(Epot ), and their total energies(Etot ) per particles are shown for the total number of particles in a system. All energies are scaled by the total particle number(N )  HH ¨7 ƒ½ë ELTB ~&d` s6K >íô BEC I_· · · – ^¿% 1 Ó ¦ x ½ñ”   ß –Ç œ ©  x ”ºò p -327- ¯ úÐ í\ - ©ñŒ - ”Qf ½ °˜ ŸS \t  6 \t\ e#" † ‰ œ •x  HÓ œ ©8¼   ß9, î1 \t\ e#"  6` · à e rx l - ”Qf 8 Œ£ ú º ”  H •§¦ ˜   X/h¼Ð í\ - ©ñŒ - ¸ . ]@&ܖ ŸS \t  6 \t_ š  H‰ œ •x  8 ¼ß ©/h ¸Ð rl - /Ä   ßtë, @& š – 1 \t @| 50 –œ  H îx Ì %–  ß. s õ 1 \t 1 †Ã_ Ð © ¼   rl - l <º œ   H îx xÊ 2 pì :K %#t \ l“.  ¦ Ÿ 3Q ¯ ) r` x  H a j  úRЌp O ŸŒ 3É ß s] t ¶(˜€1s ìZ` : # %“ >í ˜ ¤w r¦ x r – ¯r úÉ °“ GP ~&d` Ãu&ܖ ó °õ °“ '1€d` Ó ¦ ½ñ” ºh¼Ð  ú úÉ Ÿlª”  r ¯ r xœ¦ Ð ˜s9, š 12 % ?\" ¸ [#´. Äo %# ¸ /f ú þQú º 3Q ˜ t H  p N · B”  /d (19)ü (20)“ ¢„y D–î õ–", çß < r a É - hÐ Ðf ߖ r –é  9ºü 9¾ @ҁf  í\ $> ¸ª > {Ã< ||, üÂ\" K” ŸS (J[_ —€  Ó  ‰ ™ œ – ˜  ß ú 1 Œ© - ú ¨+ º ”  ë ·€ ~> {I_ \t °` ½½ à e. s •œ ¯¦ É  Ћ ” «f ¸÷ úÉ 8 úÉ 9Ð Ò –+ f z+\" r•&t ·“  ´“ {– sÀ  ´> §r §r  Q f #” >\"_ BEC I_ { \t\ Æ&    œ © Œ - Òñ  • ¦ HH  jB<¼Ð‹ ú¼Ð «  jB+ º \ ]/†Ü–+ ·Ü–_ z+\ lï` ]/½ à ¦ NÊ ¡ ´> r¦ NÉ ¦ ” ¯¼Ð /) e` ܖ [email protected]. a p cý k P8 ò > - ߁ O Q Ò¦ 7 ºñ \t >í\ ìZ_ sn#\ År“ ë Ã& – r ¦ HH ¦  ¸üÒ /<§ ü< lº q” Œ ` •<Œ “ @†“ Óo†õ 1Ä ‚Ò_a y  Æ tÆ x t ™ ¼n  ¨ /<§ é¼Ð ºŸ& ×wm. s ƒ½ “ @†“_ t"ܖ Ã'÷   H Æ ¶  v 3þ %_m. p cŠU YwÔ Øô [1] S. N. Bose, Z. Phys. 26, 178 (1924). [2] A. Einstein, Sitzungsber. Kgl. Preuss. Akad. Wiss. 261 (1924). [3] A. Einstein, Sitzungsber. Kgl. Preuss. Akad. Wiss. 3 (1925). [4] F. London, Nature, ...
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