N E N T O N G D A M O D E TlT CAC HE CO N G O VAO LA CAC D O N M O D E KET H d P VA N E N P H U T H U O C
T H A M s6 BIEN D A N G
vo TINH Trudng Dgi hgc Su phgm - Dai hgc Hui
DOAN THI MY LIEN Tnidng THPT Vinh Xudn, ThUa Thien Hui
Tom tat: Trong mot moi trudng phi tuyen, mSi lien he giiJa nen tong da raode tijf cac photon ddn raode 6 ngo vao vdi nen thSng thudng cua photon c6 tan so ting d ngo ra dupc thiet lap thong qua cac phUdng trinh chuyen dpng Heisenberg. Xen tdng da mode tdng quat vdi cac trang thai kit hdp phy thuoc tham so bien dang va trang thai neu phu thupc tham s6 biln dang se dupc trinh bay trong bai bao nay.
1 GlCl THIEU
Nara 1963 R. I. Glauber va Sudarshan da dUa ra khai niSm trgng thdi kit hOp [5]. DHy la trang thdi cd dd b i t dinh tUPng ilng vdi gidi han lupng til chuin suy ra tir he thiic bit dinh Heisenberg. Nftra 1970, D. Stoler da mo ta b^ng ly thuyet mpt trang th&i dac biet ma sau nay dUdc gpi la trgng thdi nen [7]. TVong trang thai n^y, mdt trong hai bien dp true giao cd dd bat dinh nho hdn gidi han lUdng tii chuin va la trang thai rad dau cho mot ldp cSc tr^ng thai phi cd diln, trang thai kit hop phu thuOc tham s6 biln dang [8] cung nSra trong ldp trang t h ^ nay. Vl nguyen t ^ neu thanh phan dUdc nen hoan toan cua trudng CO mang tin hieu thi tin hieu dd cd t h i dUOc thu lai ma khdng hi nhilu. Vi vSy cac trang thai nen khdng nhiing cd j ' nghia quan trpng trong linh vUc quang lupng tii ma cdn dupc ap dung rpng rai trong cac linh vuc khac cila vat ly Do dd, nd dUdc nhieu nha khoa hpc quan tam nghidn cihi. Yuen [9] da nghien cihi cac trang thai n^n mpt mode tdng quat, Caves va Schumaker ^4] da khao sat cac trang thai nen hai mode mot each chi tilt. Xam 1999 nen tdng da mode da dUdc Nguyen Ba An va Vd Tinh idiao sat vdi c ^ photon ddn mode kit hdp va dOn mode nen [3j. X'guygn Viet Cudng [2] khao sat nen tdng da raode vdl cac trang thai dac biet. Bai bao nay la md rpng cdng trinh tr^n vl nen tong da mode tit cac h§ CO ngd vao IS. cac ddn mode kit hop va nen phy thupc tham s& bien dang.
T^p chi Khoa hgc va Giao due, TrUflng Dai hgc Su pham Hue ISSN 1859-1612, So 01(17)/2011: tr. 21-28
2 DIEU KIEX XEX TOXG TONG QUAT
_ Z 9 j £ ^ - DOAN THI MY LIEX
Xet cac mode khong tuong quan thi dieu kign d l h§ da mode c6 nen tdng dUdc cho bffi bilu thiic sau [1], [3].
V =R{e-v. [ fj <c«) - n <<?;>']}+[ n <j*.> ~ i l (<;;> f l (c,)]. (1)
j = i j - i j = i i = i j = i TVong dd, C'^, Cj lan lupt la toan tii sinh, huy boson, hj la toan tii sd boson. Diia vao (1) ta se khao sat nen tdng da mode vdi cac trang thai kit hpp phu thuoc tham so biln dang va trang thai nen phy thudc tham sd biln dang. Xiu 1' < 0 thi he cd nen tdng, cdn nlu V > 0 thi he khong dudc nen tdng.
3 TRAXG T H A I KET HQP PHU THUOC THAM S 6 BlfiX DAXG
TVong khdng gian Fock thi trang thai kit hdp phu thudc tham s6 bien dang dUdc djnh nghia nhu sau [8]:
n=0 V I'^Jg-
trong do, a = re'' la s6 phiic d}c trung cho trang thai |a) , r va 9 lin luot la bien dp ket h(3p va pha: ki hieu
|Q|2m _ „
X«(W')=EST='B,|l-9)9|an: K
9 - 1
Gpi C va C lin lupt la toan tii huy ra sinh boson, sii dung (2) ta tinh dupc mpt sd gia trj trung binh sau:
.ic)^r-'^.-AH^^^wr^^^^, ,3,
XEX T O N G D A MODE TlT CAC HE CO XGO VAO LA CAC DOX MODE... 23
4 TRAXG THAI XEX PHU THUOC THAM s6 BIEX DAXG
Trang thai nen phu thugc tham s5 biln dang dUdc tao thaiih b3Jig each lay todn tut Saiz) [6] tac dyng len trang thdi \P,z) , nghia la:
TVong dd /3 = pe''^ la sd phiic dac trUng cho trang thai |/3, z) , p,x 1 ^ ^^d^ 1^ ^i^n dp va pha kit hpp. Sa{z) dUdc dinh nghia nhu sau:
S„{z)=exp{^a'-'-S.+^). (8)
Sii dung (9), ta tinh dUdc mdt sd gia tri trung binh sau:
..(C>. =V.-\\P\^)Y,l/S\^".L^^p^Jpco,hs-l3'exp{,,l>)sinhs) (9)
•AC*),, = X,-'(l/5ft E ^'^^"']Jln]J[n + lW.i^'""'"' " ' 8 « ^ J ' ( - ' * ) « " ' ' 0 ' ^ ° '
fe±lM(3.2e„,„^. + ^^e:
I |nl,!|n + 2],! V
' exp(-i4>)sinhsa,shs -2\3 \^(\P\^) f " 131'" i " , ^ , , .exp{-ii,)sinhacoshs
•AC-'), = ^:\\p?)T.\m'"\\J,w,2l i^"""''''+fieM-2i<t>)sinhh)
».(ft>, = \f3\\'m'')'Z\l^\''"r^^,(coshh + sinhh) + 3inhh
r ('2) - KWm'') E l g l ' " < / ' r T i l " ! t ^}coshssinhs X (l3--'exp{i,i,)A-l3'cxp{-i9)).
5 XEX T 6 X G D A M O D E Ttr TR.4XG THAI KfiT HOP PHU THUOC THAM s 6 BIEX DAXG
TVong tntcJug hpp nay cac mode cl ngo vho dgu thupc trang thai kgt hpp phn thupc tham s6 biln d^ng \a) . Thay cac tri trmig biuh da tinh td bieu thiic (3) den (6) vao bilu thiJc (1), roi x§t trudng hpp ctj = a = re'^ ta thu dupc bilu thiic sau:
24 VO TINH - DOAN THJ MY LIEX
.,. .-({'^-(^i-^-"( |^-'(^\?;^V w f i f
(n + l)(n+2)^-''^^'l/^Vnii^l ^^
+ «7'(--')E- i + i ] . N-Hr'jE'- W,![n+iy
(13)
a) Khao sat su phy thupc ciia dieu kien nen Vi vao cic tham s6
I ;
'; 1 1 1
Hinh 1: Vdi ^ = 0,6 = 0. £)5 thi hdm Vi x 10"^^ duoc khdo sdt theo cdc tham s6 vdi N = 4, 5, 6. Hinh a) khdo sdt theo r khi g = 0.6. Hinh b) khdo sdt theo q khi r = 2 (cdc iham sd duoc chgn di khdo .saf theo thii tu tang d&n tuong iing vdi net gach, net ch&m ch&m vd net liin).
Tit do thi hinh 1 ta thay nen tdng luon co thi xay ra iing vdi hai khoang giS. tr; cua r va q, mpt khoang ting vdi gia tri nho cua r, q cbn mdt khoang ihig vdi gid trj Ida hdn. Neu miic dd nen tdng ciia he dUOc ti'nh theo gia tri am ciia hara dupc khao sat thi ham cd gia tri am cang ldn tiic la miic dp nen cang Idn. Tudng ilng vdi mot s6 ddn mode nhit dinh thi miic dp nen tdng hiu nhu khdng thay dli theo gia tri r, q trong khoang thii nhit nhUng lai tang theo gia tri r. q troug khoang gia tri thii hai. Khi N tang thi hai khoang gid tri ciia r. q dl cd nen tdng tang len.
b) Khao sat su phu thudc ciia dieu kien nen tong Vi tbeo thiia sd pha 0
Tix dd thi hinh 2a) ta thay he ludn dat nen tdng cite dai tai 5 = 0. Miic dp nen tdng cUc dai tang khi N tang. Xgoai ra giiia L^ va 6 cd mdi lien he chat che vdi nhau, khi gia tri ;^
thay ddi thi miic dd nen tdng cue dai theo 6 cung thay ddi theo (hinh 2b)). Vay vdi rapt gia tri ^ cho trudc ta c6 thi chon gia tri 9 dl he dat nen tdng cue dai.
XEN TOXG DA MODE TlJ C.AC HE CO XGO \\\0 LA CAC DON MODE .
.--/.
Hinh 2: Vdi r = 1, q = 0.6. Hinh a) la di thi ham Vi ( x l O " " ) , 14 ( x l O - ' " ) , Vj ( X 1 0 - " ) khdo sdt theo 6v6iN=6, 7. 8va^ = 0. Hinh b) la do thi ham 14 ( x IO"") khdo sdt theo 6 vdi .p = 0,7r/6,7r/3 vd X ~ 6 (cdc tham so duoc chon de khdo sdt theo tha til tdng dan tuong iing vdi net gach, net ch&m chdm vd net hen).
6 NEN TOXG DA MODE TL' C.\C TRAXG THAI KET HOP PHU THUOC THAM S6 BIEX DAXG VA XEX PHU THUOC TH.A.-M SO BIEN DAXG
W
/... , ;
; i
Hinh 3: Trin hinh a) Id do thi hdm Vo (xlO""™) khdo sdt theo r khi K = 1, 2, 3 vd q = 0.9, s = 2. p = 2. Tren hinh b) Id do tht hdm V2 (xlO"^^) khdo sdt theo p khi g = 0.7, 0.71. 0.72, r = 1, s = 0.8 (cdc tham sd duoc chon di khdo sdt theo thii tu tdng d&n tU&ng iing VO2 net ga,ch, net ch&m cham vd net liin).
Trong trudng hdp nay ta xet K mode d trang thai n6n phu thudc thara sd biln dang \3. 2) va N - K mode con l^i 6 trang thai kit hop phu thudc tham sd biln dang |a) . Thay cac gia tri trung binh tii bilu thiic (9) din (12) vao (1) sau do ta xet trudng hpp, cac mode kit hpp phu thupc tham so biln dang la giong nhau Oj = a. cac raode nen phu thupc thara sd
VO TIXH - DOAX THI MY LIEX
biln dang la gilng nhau ,Jj = 3 va « = ./> = X = 0 ta thu duoc:
>)^K"'(p^) V p^"r^^ pjsinhscosfcs - sinftscoshsj
I ' h, V ["l.!l"+21,!J
r °* / n-I-1 / M^'*'
+ { [sinhh + / X - ' (fi') f ; p ^ ' i ^ i i ^ (cosA^s + sinft^s)
- K'(P^)EP^" Ji;;p^(-''»-'^"'")T
i-'^^''^'>]:rw^ri
(14)
a) Kbao sat anh hudng ciia cac tham sl din dilu kien nen tong V2 TCr dd thi khao sat dieu kien nen hieu UT theo A" d do thi hinh 3, ta thay iing vdi mpt gia tri ciia K cd hai kho&ng gia tri ciia r thoa man dilu kien nen tdng Khi K t5ng thi ca hai khoSng gia tri r dl cd nen tdng tang len. Khi q tang thi khoang gia trj p tang. Ludn cd nen cue dai theo r, miic dp nen cilc dai giam khi q. s tang.
b) Khao sat dilu kien nen \ 2 theo hudng nen ip
Ta thiy he Iudn d^t uen ting cUc dai tai tp = 0. Cac gia tri cua tham so trong trudng hdp iiay khdng nhiing anh hudiig din kha uang co nen ting hay khdng ma con anh hudng den miic dp ueu tdug. Cu thi. khi gia tri ciia p, s tang thi raiic dp uen tong cUc dai theo hUdug ip giam va khoang gia tri ciia .,:; thoa man dilu kien nen tdng cting giam.
XEX TOXG DA MODE Til CAC HE CO XGO VAO LA CAC DOX MODE...
*)
• 1
'%.
Hinh 4: Vdi N = 8, K = 6, ip = 0. Hinh a) la de thi hdm V2 (xlO"'") khdo sdt theo rkhis = 0.5, p = 1 vd q = 0.62, 0.6,1, O.64. Hinh b) Id do thi ham V2 (xlO-'=) khdo sdt theo r khi q = 0.8, p = 1.5 vd s = 0.5, 0.6. 0.7 (cac tham sd duac chan de'khdo sdt theo thii tu tdng ddn tuang iing vdi net gach, net ch&m ch&m vd net liin).
a) b)
/•'/
, , • • • ' /
/
•<:. \
;\
\ • ' • • . . .
Hinh 5: Vdi N = 8, K = 6 vd r = 0.5, g = 0.87. Tren hinh a) la dd thj hdm V^
(xlO~^^) khdo sdt theo tp khi s = 1.77 vd p = 0.51, 0.52, 0.53. Tren hinh b) Id do thi hdm V2 (xlQ-^'*) khdo sdt '.p khi p = 0.52, vd s = 1.75, 1,76, 1.77 (cdc iham sd duoc chon de khdo sdt theo th-d iu tang ddn tuong Ung vdi net gach, net chdm cham vd net liin).
7 KfiTLUAX
Cac bilu thiic ciia dilu kien nen tdng da mode tdng quat vdi cac trang thai kit hpp phu thupc tham sd biln d^ng va trang thai nen phu thupc tham so biln dang dUdc thilt lap trong trudng hpp cu thi: c&c mode khdng tUdng quan, d ciing mpt trang thai thi gidng nhau. Cac kit qua khao sat nen tdng da mode tdng quat vdi cac trang thdi ddn mode kit hdp phu thupc tham so bien dang va trang thai nin phu thupc tham so bien dang cho thiy, nen tdng da raode ludn xay ra nlu chpn cac tham sd diu vao phii hpp, miic dp nen tang hay giara phu thudc manh vao gia tri cac tham sd do. Do \ay se cd uen ddn mode co tin s6 tdng d ngd ra.
VO TINH - DOAX THI M i ' LIEN
TAI LIEU THAM KHAO
[1] Vo Tinh (2001). Mgt s6 hieu itng trong he photon-exciton-biexciton d bdn ddn kich thich guanj Luan an tiln si Vat ly, TVubng DHSP Ha Xoi.
[2] Xguyin Viet CutJng (2008). Nen tdng da mode vdt cdc trung thdi kk hap dac biit Luan van Thac si Vat Iy, Trutmg DHSP Hui - DH Hue.
[3] Nguyen Ba An and Vo Tinh (2000). General multimode sum-squeezing. Physics Letters A, 261, pp. 34-39.
[4] Caves M C. and Schumaker L. B. (1985). New formalism for two photon quantum optics. I. Quadrature phases and squeezed states. Phys. Rev. A, 3 1 , pp. 12-20.
[5] Glauber R. J.(1963). Coherent and incoherent states of the Radiation field. Phys. Rev.
B, 131(6), pp. 2766-2788.
[6] M. O and Zubairy M.S. (2000). Quantum optics. Cambridge University Pres, Cam- bride.
[7] Stoler (1970-1971). Equivalence classes of minimum-uncertainty. Phys. Rev. lett D, 1, pp. 37-45.
|8] Quesne C 2001. Ann. Phys.. XY 293 147
[9] Yuen (1976). Two photon coherent states of the radiation field. Phys. Rev,13, pp.
240-15S.
Title: MULTIMODE SUM-SQUEEZING FROM COHERENT STATES AND Q - DE- FORMED SQUEEZING
Abstract: In a nonhneax medium the relation between multi-mode sum-squeezing of input single-mode photons and normal squeezing of output sum frequency photon is established by Heisenberg motion equations. A general multi-mode sum-squeezing frpm q - deformed cpherent states and q - defprmed squeezing states is presented in this paper.
TS. VO TIXH
Khoa \'at l,v. TVucfng Dai hoc Su pham - Dai hpc Hui ThS. D O - \ N THI .MY LIEX
IVuang THPT \'inh Xuan, ThUa Thien Hue
