NEN TONG DA MODE BAC CAO HILLERY TlT HE CO NGO VAO LA CAC DON MODE KET H d P , DON MODE
K £ T H O P T H E M P H O T O N V A D d N MODE NEN
vo TINH Tnldng Dgi hoc Su phgm - Dai hgc Hui
NGUYSN ANH BANG Tnldng THPT Thanh Tuyin, Binh DUdng
T o m t a t : Tfl Hamiltonian ciia m p t he tUOng t a c gom cac p h o t o n va cac nguydn tfl ciia mdi t r u d n g p h i t u \ e n , cac phuong t r i n h chuygn dpng Heisenberg c u a c ^ to&n tfl sinb buy h a t p h o t o n dUdc t h i l t l a p . T h d n g q u a vifc giai h§ phudng t r i n h nay vdi p h e p g i n d u n g bac hai theo thdi gian b e va t i n h phUdng sai bien dp t r u c giao bac cao, mdi liSn he gifla nen tOng bien dp tri^c giao d a mode bac cao Hillery tfl cac p h o t o n d ngd vao vdi n e n bien dp tri/c giao b a c cao Hillery cua photon cd t i n so tong d ngd r a dupc binh t h a n h . Cung t r o n g bai bao nay, dieu kien nen tdng d a m o d e b a c cao Hillery tdng q u a t dupc r u t r a va tfl dd nen tong d a m o d e b ^ cao Hillery dupc khao sat vdi cac p h o t o n d traiig thai k i t hpp, nen k i t h p p va k i t hpp th§m p h o t o n .
1 G I 6 I T H I E U
Nen tdng vk uen hieu d a m o d e tdng q u a t bac n h a t d a dupc khao sat bdi cac t a c gia NguyOu B a A n vh. Vd H n h [5], [6 . [7]. T i l p t h e o d o cac t a c gia Vd T i n h va P b a m Thi H a n h T h a o d a k h a o s a t nen t 6 n g d a mode b§f cao tfl he cac don mode k i t hop va ddu m o d e nen [3].
Cac k i t q u a nghien cflu cho t h i y b a n c h i t cd hoc lupng tfl cfla a n h sang dupc bpc 1$ trijc t i l p t h d n g q u a cac t r g n g thai nen d a m o d e bac cao. Viec nghign cflu cac photon nen co t a n sd la t d n g cua cac t a n sd photon b a n d a u 6 t r a n g thai k i t hpp, t r a n g t h a i pbi cd d i l n t r o n g mdi t r u d n g phi tuyen khdng nhflng cd y nghia q u a n t r p n g t r o n g linh vflc cdng ngh§
m a cdn cd ddng gdp rat ldn trong Hnh vi^c khoa hpc co ban, md rdng t i m higu b i i t c u a con ngudi sau hdn nfla vg ban chat c u a t r u d n g dien tfl va ttrong tac cua no vdi I'at c h a t . Bai b a o na\' t r i n h b a y khSo sat md rdng cdng t r i n h [3] vdi cac t r a n g t h a i d ngd vao la c^
d d n m o d e k i t hpp, ddn mode k i t hpp t h g m p h o t o n va d d n m o d e nen.
Tap chi Khoa hoc va Giao dye, TrvrcJng D^i hgc Su pham Hug ISSN 1859-1612. S6 03(23)/2012. tr. 21-29
VO TINH - NGUYEN ANH B A N G
2 NEN TONG DA MODE BAC CAO HILLERY
Nen tdng da mode bac cao Hillerj- tdng quat da dUdc nghien cflu d [3]. Ndi dung cua nen tdng da mode b ^ cao Hillery cd thg dUdc tdm tat nhu sau: Xet qu^ trinb v^t ly xay ra trong mdi trudng phi tuyen, trong do N photon vdi t i n sd u}i,u}2,u}3,...,uj^' kit hpp vdi nhau dg tao thanh mot photon cd tSn sd tdng ils = uii-{-IJJ2 + ^3 + ••• + -^'.v. Hamiltonian flng vdi su sinh ra mot tin sd tdng nhu thi cd dang [2]
iis = Yl^^^i + ilsns + 9s{csCi...CN + h.c), (1)
trong dd Hj = c^Cj,fts = c^cs vdi c~Cj va cJ, cs theo thfl tfl la cac toan tfl sinh, buy flng vdi cdc mode Uj va Qs- H&ng s& tuong tac gs dupc gia thilt la tht(c. ^'l cac photon dao dong trong miln quang hpc vdi tin so cao cd 10^^ Hz ndn thanb pbin bien thien nhajib dirpc tdch rigng ra va vilt
Cjit) = C,(t)e-^'\ cs(t) - Cs{t)e-'^'', (2) trong dd cac toan tfl Cj{t), Cs{t) biln thien ch§m theo thdi gian vi thdng thudng gs <
u^.Ms-
Todn tfl bien dp tri^c giao ciia tin sd tdng Us luy thfla k dupc dinh nghia
{3}
Tfl dd ta tinh dupc giao hoan tfl
[Xcs.i:{l>.t),Xc,.i,(,p+^,t)\ = J F o , ( t , t ) , (4)
trong d6 Fc,{k,t) = [ e | ( ( ) , C j ' ( t ) ] . DiSn ki?n ds co nen bifn do luy tliiia k kiSu Hillery theo phuong 9 la
1,
(5) Toan tu "tap the" luy thiXa k Qs.k{^:t) duoc dinh nghia nhu sau
Qs.dv, i) ^ 2 C^-Ht) n Ci(t)e-<^ + C+<'-"(() n c;(()e'>
L j - i j=i Tir do ta tinh duoc giao hoan
[Os,i(l»,t),Qs.i- ( v + ^ , t ) ] = ^Fs{k,N,t),
(6)
(7)
NEN TONG DA MODE BAG CAO HILLERY TU HE CO NGO VAO..
^^^?
trong dd
N
Fs(KN,t)^Cl-Ht)Cp''-'\t)Fs(NA)+Fcs{k-l,t)]ln^(t). (8)
j = i
Trang thai "t&p the" cua cac mode Uj difpc gpi la nen tdng da mode bac cao Hillery theo hudng tp neu VQs,k{^,t] thda man digu kien
vQ„fe„-K|kMyM<„, p)
trong dd phudng sai
VQs,t{v,t) = (Q|,t(p,t)) - {Qs.t(f,t)y (10) Mai lien hg gifla nen Hillery don mode cd tin sd tdng vdi nen tdng da mode bac cao Hillery
dUdc rut ra bing each dung Hamiltonian (1) dl thilt lap phudng trinh chuyen ddng cho cac todn tfl cin quan tam, bg phuang trinh thu dupc cd dang
4 ( t ) s ^ = _i55 J^ CHt)Cs[t). (11)
t-i.i-jti
^ dC,(t) dt
k=l
(12) Lay dao ham (12) theo thdi gian mdt lin nfla rdi van dung (12) vao ket qua tmh dao ham cho ta kit qua
C;s{t) = -9lCs(t)Fs{N,t). (13) Trong phep gin dung thdi gian ngan, sU phu thudc thdi gian cua nghigm Cs(t) dudi dang
khai trign Taylor din bac hai cd dang (quy udc Cs(Q) = Cs. -.)
Cs(t) =Cs~ igst n Cj - -g%CCsFs{N). (14) Vdi dieu kien bo qua s6 hang bac hai trd Ign cua thdi gian va thdi dilm ban diu cac mode khdng tUdng quan vdi nhau ta viet dUdc
C%{t) = ( l - ^»l«'Fs(W)) Cf - ikgstCl-^ n Cj. (15)
eft) = ( l - \slt^Fs(N)^ C J ' + j i s s t C j " - " n c ; . (16)
v o TJNH - NGUYEN ANH B.4NG
Khi do m6i hen he giiia nen Hillery don mode t i n so t6ng vdi nen tong da mode b§c cao Hillery duoc thilt lap
V sVXc,.U^.t]~\{FcAk,t))\
= kVsi' [vQs.k ( v + I ) - i |(FS(*,iV.«))j] . (17)
Phuong trinh (17) cho thiy moi ben he chat che gifla nen ddn mode Hillery cd t i n sS tdng d thdi digm t > 0 vdi nen tdng da mode bac cao Hillery d thdi di§m t — 0. Tbeo dd khdng CO nen tdng da mode bac cao Hillery tbi cung khdng tdn tai nen ddn mode Hillery cua mode Cs.
Qua tinh toan ta dUdc bilu thflc cua dilu kign nen tdng da mode bac cao Hillery:
V =2Re<
2 iasr-"[n(%>-n (<?;)(<?.)
)
(18)Di/a vho (18) ta se khao sat nen tong da mode bac cao vdi cac he dac biet. Nlu V < 0 thi h? cd nen tong. Cdn khdng, he khdng dUdc nen tdng.
3 NEN T O N G DA MODE BAC CAO HILLERY TIT HE CO NGO VAO LA CAC DON MODE KET HOP. DON MODE KET HOP THEM PHOTON VA DON MODE NEN
a) Trudng hpp t i t ca cac mode deu d trang thai kit hpp them photon
Mot mode kit hdp thgm photon duoc md ta bdi sd phflc ttp = rpC^^'' vk sd nguygn m. Theo dd 1-ec to trang thai kit bpp them photon duoc xac dinh bdi
\ap, m) - "•""•lap)
[m!L„(-|Q,|2)]i/2' (19)
Mx) = E d
(-3;)"m!^ ( 7 i ! ) 2 ( m - n ) ! Ik da thflc Laguerre bac m (m la sd nguyen dudug) theo x.
Sfl dimg vecta trang thai kit hpp them photon ta tuih dupc mot sd gia tri trung binh d trang thai nay nhu sau:
NEN TONG DA MODE BAG CAO HILLERY TU HE CO NGO VAO..
(Cp> = o; ' ' i „ ( - | a p R ' (Cp) = a,- Y,(m + l-i)L,(-\a^f) (CD = 4"° r ! _ , „ 121 • («») = -
i m ( - l a p P ) • + l ) L „ - i ( - a,\^)
(20)
W - i O p P ) ' ' " " i „ ( - | Q p | - ) Xet trucing hop cac mode ket hop them photon 1& gi6ng nhau dp = rpe"'p va tham so kgt hdp QS = Tse''i!. Thay (20) vao (18) ta duoc
Vl = 2rfMTl'''cos[2[-^ + (k-l)i>s + «i»p
Y,{m + l-t)L,{-rl)y. f;L,{^rDyN.
( ^
l ) i „ + l ( - r ^ ) i™(-'-|)^ 5 I i . ( - ' - p ) s 2AIs
im{-'-?) (21)
Ket qua khao sat 6 hinh 1 cho thiy rang nlu k cang tang thi chu ky nen theo i?s cang giam, hay n6i each khac la xac suat co nen tang lgn.
i" '^''^
Hinh 1: DS thi cua ham i i x 10 ^ khdo sdt theo Us va rp vdi i?p = 0; y = 0; -V = 5: m = 1: r s = 2. Hinh (a) k = 2. Hinh (b) k = 3.
b) TVudng hop CO L mode 6 trang thai kit hop them photon va cac mode con lai 6 trang thai kit hop
Mot mode kit hop dugc mo ta bdi s6 phiic Q, = T,exp{i^q},q = 1.2 A'. Theo do vec td trang thai kit h0p duoc xac dinh bdi
|a,> = exp I - j l ^ s H =>T (o-qa*)\0). (22)
VO TINH - NGUYEN .\NH BANG Sfl dyng vecta trang thai kit hpp ta tinh dUdc mdt sd gid tri trung binh d trang thai nky nhu sau:
(C+) = o;, ( C , ) = a „ (Cj>=o?, (n,) = K p . (23) Xet trudng hdp cac mode kit hdp thdm photon la giong nbau ap — rpe'^p, cac mode kit hpp la gidng nhau a^ = rge"'" va tbam sd kit hdp as = rge*''^. Thay (20) va (23) vao (18) ta dUdc
V2 = 2 r f - " r J ( ' « - « | r f c o 5 [ 2 ( - v > + ( f c - l ) t f s + itfp + ( . V - i ) . J , ) ] f;(m + 1 - i)L,{~rl) i ,'ZLi(-rl). 21,
L.,(-rl) L,„(-rl)
[ i.„(-r|) ') '' [ L„(-r?) ) ] (24) Kit qua khao sat ham V2 theo r, va Vp a hinh 2a cho thay he co nen teng da mode vdi cac gia trj ciia r„ Tp thi'ch hdp. Khao sat nen tOng d hinh 2b, ta thay vdi v? = 0 miic do ngn t6ng la ldn nhat, khi ^ tang tii 0 dgn 7r/4 thi do nen teng cang giam. Khi ip = 7r/2 thi hg khdng c6 nen teng.
Hinh 2: Do thi ciio ham 1', x lO^' khao sdt trong tnang hop i>p = 0;A' = 5;m = 1; r s = 5; L = 3; 1), = 0; r , = 2. Htnh (a) Khio sdt theo r„ r,. ^ = 0. Omh (b) khao sd( tteo rp,r, = 1.5. ^- = a ; | ; | . (Cdc tham s6 duac chpn ba gid tri de'khdo sdt theo tha tu tdng din tuang Ung vdi dudng gach dit, gach ngdn va liin net).
) va cac mode con Iai d trang thai kgt c) TVudng hdp cd L mode d trang thai nen kit t
hdp thgm photon
Mot mode nen duoc mo ta bdi hai s6 phfc Q, = r,exp{iij,) va z = sea:p{ix) theo do vecto trang thai nen ket hop nia cac don mode bi nen la
|aj-2> = Dc,(aj)Sc,(z)|0). (25)
NEN TONG DA MODE BAC CAO HILLERY TlT HE CO NGO VAO .
Sfl dung vectd trang thdi nen ket hdp tinh dUdc mpt sd gia tri trung btnh d trang thdi nay nhu sau:
(C;)=o-, (C,) = o„ {Cj) =
Xet trudng hop cac mode kit hdp them photon la giong nhau ctp — rpC^^", cac mode nen kit hop la giong nbau Qj — rjC^', tham so kgt hop as — TsC^^^ va tham so nen 2 — se''^.
Thay (20) ™ (26) vao (IS) ta duoc Vs
i ™ ( - r | ) N-L
2{N-L).
f;(m+l-i)L.(-rJ) „_^ ^Ei.(-r=) i" ° i - ( - i ) j ""••
( r ^ . „ . . ) ^ ( ^ I . ^ ^ W ^ - l )
A^L,-(-r^) 2,jv_i)
_2(N-L).21 / » 0 \ I (27)
Ket qua khao sat d hinh 3a cho thiy rang trong khoang gia tri cua TJ dang khao sat thi khi sd mode d trang thai nen kit bpp L cang Idn tbi mflc do nen tdng cang ldn vdi cung mdt gia tri xdc dinh r-j. Ta biit rSag khi sd mode nen ket hpp tang thi s6 mode d trang thai ket bdp them photon giam. Vay trong he ma ta dang xet, khi sd mode d trang thai kit hpp thgm photon giam thi dp nen lai tang.
Khao sat d hinh 3b cbo thiy rang khi m nhan cdc gid tri tang din thi mflc dp nen tdng dat cue dai dia phUdng ciing tang din. Nhu vky mflc do nen tdng dat cUc dai dia phudng cfla he se ldn nlu m nhan cac gia tri ldn.
4 KET LUAN
Nhu vay dUa v^o dilu kign nen tdug da mode bac cao HiUery tdng qudt, ta thilt lap dUdc dilu kien nen t5ng da mode cho cdc trang thai kit hdp them photon, kit hap va kit hdp them photon, ket hdp thgm photon va nen kit hdp trong trUdng hpp cu thg: cdc mode khdng tudng quan, d ctrng mot trang thai thi giong nhau. Kit qua khao sat nen tdng da mode bdc cao Hillery cfla cac photon kit hdp. kit hdp them photon va ndn ket hdp d ngd vho bing phUdng pbap tinh s5 va do thi cho thiy nen tdng da mode bSc cao Hillery phu thupc chat che vao bien do kit hpp cfla trang thdi kit hpp rg, bien dp nen s. bien dp nen kit hpp cua trang thai kit hop rj, bien dp kit hpp rs cua mode cs va cdc pha tUdng flng
\ O TINH - NGUYEN ANH BANG
005 000 0 05 0.10
lb)
x:i>
\
\ \
* •
.. "
/ / / /
0 0 0,1 0.2 0.3 0.4 0-5 0.6 0 7
Hinh 3: Do th cua hdm V3 x 10"^ khdo sdt trong tnldng hap •dp = Q;'dj = 0;ip = 0; X = 0; i3s = 0.2: A' = 6; r s = 4: k = 3; rp = 1.5; s = 0.4. Hinh (a) khdo sdt theo r^, m =1, L = 2, 3, 4- Hinh (h) khdo sdt theo s, rj = 2, L = 3. m = 1, 3, 5.(Cdc tham sd duac chgn ba gia tri de khdo sdt theo thU tu tdng ddn tuang Ung vdi dudng hin net. gach ngdn vd gach ddi).
cua chflng. Tuy nhien trong qud trinh khao sdt ta vin rut ra dUpc mpt so kit qua cu thi nbil: (i) Tham s6 thgm photon m cang ldn thi mflc dd nen tdng dat cue dai dia phUdng cua hg cang ldn; (ii) Nen tdng dat cue dai khi gdc pha cua toan tfl tap thi tp = kt:; (iii) Bign dp ket hpp ciia trang thai kit hdp them photon Vp cang ldn thi mflc dd nen tdng cang tang.
Tdm lai, qua kit quS, khao sat cho thiy chi cin 6 ngd vao cd mpt mode d trang thai phi cd dien, cdn cac mode cdn lai d trang thdi kit hdp thi ludn ludn cd nen tdng da mode bac cao hay d ngd ra cd nen tdng bign do true giao cfla ddn mode tin sd tdng kilu Hillery
TAI LIEU THAM KHAO
[1] Vd Tinh. Pham Dieu Quj-nh Chau(2010). Nen tdng da mode tfl be cd ngd vao la cac ddn mode ket hpp them photon, ddn mode kit hdp va ddn mode nen. Tap chi Khoa hoc va Gido due Tnldng DH Su pham-DH Hui, s6 02(14), tr. 22-28.
[2] Vd Tinh (2001) Moi s6 hieu -Ccng trong he photon-exciton-biexciton d bdn ddn kich thich quang, Lugn an tiln si Vat ly Ti-udng DHSP Ha Npi.
[3] Vd Tinh, Phgm Thi Hanh Thao(2010). Nen tdng da mode bac cao Hillery. Tg,p chi Khoa hgc vd Gido dye Trudng DH Su pham-DH Hui, so 02(14), tr 22-28.
[4] Agar^-al G. S. and Tara K. (1991). Nonclassical properties of States generated by excitations on a coherent state, Phys. Rev. A. 43(1), pp. 492-497.
NEN TONG DA MODE BAC CAO HILLERY TlJ HE CO NGO \ A 0 . . . 29
[5] Nguyen Ba An and Vo Tinb (1999)."General multimode sum-squeezing". Physics Let- ters A, 261, pp. 34-39
[6] Nguyen Ba An and Vo Tinh (2000). General multimode difference-squeezing. Physics Letters A. 270, pp. 27-40.
[7] Nguyen Ba An and Vo Tinh (2000). Multimode difference-squeezing. J. Phys. A:
Mathematics & General, 33, pp. 2951-2962.
[8] Hillery M. (1989). Sum and Difference squeezing of the electromagnetic field, Phys.
Rev. A, 40(8), pp. 3147-3155.
[9] Kumar A. and Gupta S. P. (1997). Sum squeezing in four-wave sum frequency gener- ation, Optics communication, 136, pp. 441-446.
[10] Stoler (1970-1971). Equivalence classes of minimum-uncertainty. Phys. rev. lett D, I, pp. 37-45.
Title: THE GENERAL HILLERY HIGHER-ORDER MULTIMODE SUM-SQUEEZING FROM PHOTON-ADDED COHERENT STATES, COHERENT STATES AND SQUEEZED STATES
Abstract: From the Hamiltonian of an interacting system including photons and atoms in nonlinear medium, tbe Heisenberg equations of motion for creation (annihilation) op- erators are set up. The relation between tbe Hillery higher-order multimode quadrature amplitude sum-squeezing of input photons to the HiUery high-order squeezing of output sum-frequency photon is established by solving these equations with short time approxima- tion method and computing tbe photon quadrature ampUtude variance. Building Hillery higher-order multimode sum-squeezing condition, using HiUery higher-order multimode sum-squeezing condition to study from coherent states, squeezed states and photon-added coherent states is also presented in this paper.
TS, VO TINH
Khoa Vat ly. Trudng Dgi hpc Su pham - Dai hoc Hui ThS. NGLTEN ANH B A N G
Trudng THPT Thanh Tuyen - Binh DUdng
