article1 -  HH ¨7 ƒ½ë Sae Mulli (The Korean...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 $> ¸ª ~†Ü– [email protected]&“ éí › (J[_ —€`  H  –H ™ œ¦  t9, A< °s ³‰. ü ú ð&) ³a Vext = 1 2 N 2 22 22 m ωx x2 + ωy yi + ωz zi . i i=1 Œf #l" x, y , z  >\ lÕ½ à e [ >_ Ã–" H ¦ tÉ   ü+ º ” j h ºÐf H  •• § r a ¦ –¤Ç ŒŒ £ úÉ '” ßá yy 6õ °“ ›>d` ë7ô. N N N 2 yi , i=1 x2 = i=1 x2 , i y2 = z2 = i=1 2 zi (6) ¤ £ ŒQ 9 œ ü < 9¹ º h 7, # {_ ©I\ lÕ  X €¯ô Í >  ¦ tH  ǁ H h A  A j î”   >_ 0u m 0u ]Y_ ¨çe` _pô. s L H¦ Ç j  þ  jè  é ] s +I_ K þ™ ì_ "o, A r ¶ (2) ˜ ˜ δ < Ψ|H |Ψ >= 0. (7) Œf #l" ωx = ωy = ωz s. M\  ωx ü ωy  €ß  :  < •ç –   ¸ ¼Ð  7f h>h¼Ð [ _ s l• ٖ s ë\" >Z&ܖ 2 HH H  LÇ å /ô. ¦ –¤  ßá ÐÒ ü úÉ ½ñ” 3 \ ë7Kô z–Â' A< °“ ~&d` % ÇH´ r Ó ¦  ¦  º ” ` à e.  ˜ ˜ H Ψ = E Ψ, (8)  HH ¨7 ƒ½ë ELTB ~&d` s6K >íô BEC I_· · · – ^¿% 1 Ó ¦ x ½ñ”   ß –Ç œ ©  x ”ºò p -325- Œf #l" 2 IV. w–©  M Š¥ Œ † 2m m 22 2 2 + ωx x + ωy y 2 + ωz z 2 2 2 (N − 1)(N − 3) 1 1 1 + + 2+ 2 2m 4 x2 y z g + xyz ∂2 ∂2 ∂2 + 2+ 2 ∂x2 ∂y ∂z r Of - ¦»úÉ £ úÉ ' ß ìZ\"_ \t “Ä°“ 6õ °“ ›>\ ë ¯r § r a ¦ – ¤Ç á 7ô. E= < ψ |H |ψ > ≥ E0 < ψ |ψ > (14) H=− (9)  s9 g = g0 (2π )− 2 3 Γ(N/2) Γ(N/2 − 1/2) 3 N (N − 1) 2 (10)   ” lp þ ‰ ½ñ” s. s d` 11ô + s^ ~&d(equivalent linear ¦ xxÇ ‚A Ó  two-body equation, ELTB)s ô.   Ç |ψ > β _ <Ãsٖ, 0_ \t [email protected] β _ <  ʺ¼Ð A - /ú¸  Ê † ¯ † º f A Òp” jèú ßá ú Ãs. " 0 Â1d_ þ™°` ë7rv β ° x ¯¦ –¤ H¯ ¦Ô   1¼ ª úf - ú Œ© - ` ¹Ü€, Õ °\"_ \t °s {I_ \t ¯ ¯ •œ ¯ H Ç ¯¦  c ú  ú ú ¨ ¯¼Ð /) 9Ðß °õ ô °` °> | ܖ [email protected]. Kxžmî a  –  /ú ß Af ºþ ” _ [email protected]` >í l 0K" ÁrÙ d (9)_ t ¯¦ – H ¡~    •Ó Œ ½ /ú 9¹ ß  úÉ  } †_ [email protected] €¯ . të s °“ β _ Gamma ¯ – ¯r Ê <º¼Ð ª  3h¼Ð Ò Q>  †Ãsٖ Õ pì` K$&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, 141, 643 (1938). [5] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science 269, 198 (1995). [6] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [7] C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); C. C. Bradley, C. A. Sackett and R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997). [8] F. Pereira Dos Santos et al., Phys. Rev. Lett. 86, 3459 (2001). [9] M. R. Matthews et al., Phys. Rev. Lett. 83, 2498 (1999). [10] B. P. Anderson, P. C. Haljan, C. E. Wieman and E. A. Cornell, Phys. Rev. Lett. 85, 2857 (2000). [11] L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. 40, 646 (1961)[Sov. Phys. JETP 13, 451 (1961)]; E. P. Gross, Nuovo Cimento 20, 454 (1961); J. Math. Phys. 4, 195 (1963). [12] Alexander L. Zubarev, Yeong E. Kim, Phys. Lett. A 263, 33 (1999); Yeong E. Kim and Alexander L. Zubarev, J. Phys. B: At. Mol. Opt. Phys. 33, 3905 (2000); Yeong E. Kim and Alexander L. Zubarev, Phys. Rev. A 66, 053602 (2002). [13] Alexander L. Fetter and John Dirk Walecka, Quantum Theory of Many-Particle Systems (McGrawHill Book Company, New Yokr, 1971), p. 314. [14] Yeong E. Kim and Alexander L. Zubarev, J. Phys. B: At. Mol. Opt. Phys. 33, 55 (2000). -328- ÇGt Æ Dü<r hü ô²Óo†t “DÓo”, Volume 50, Number 5, 2005¸ 5 t 4 Z Energies per Particle in BEC from the ELTB Equation Dooyoung Kim, Guanghao Jin, Jeongyun Kim and Jin-Hee Yoon∗ Department of Physics, Inha University, Inchon 402-751 (Received 9 December 2004, in final form 26 April 2005) 87 In 1995, Bose-Einstein condensates (BEC) in Rb were observed by the Joint Institute for Lab- oratory Astrophysics (JILA) group at Massachusetts Institute of Technology (MIT) and a series of experiments observing BEC in alkali metal gases ensued. For the transition to BEC, the temperature of the sample be lowered, and at the same time, the number density of atoms should reach a certain amount. In this research, we tested the equivalent linear two-body (ELTB) equation, which was derived from Gross-Pitaevskii (GP) equation, by calculating the energy per particle in the lowest energy state. To solve the ELTB equation analytically, we used a variational method. Our results are consistent with the numerical results from the GP equation, especially when the number of atoms is more than 10,000. Therefore, we can conclude that the ELTB equation describes BEC very well. PACS numbers: 30 Keywords: Bose-Einstein condensates ∗ E-mail: [email protected] ...
View Full Document

This note was uploaded on 03/22/2011 for the course PHYSICS 116 taught by Professor Scopel during the Spring '11 term at Seoul National.

Ask a homework question - tutors are online