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ST 1 bTU A I IICUPImiddotICTIES Of 1I0TnN~

IN T IlIC EEmiddotI EVEI

II S TW U-M OI H MO I -

Submitted to Jour nal de PhysI que

1985

1 INTRODUCTION

During ten last years the photon antibunching effect has attracted considerable interest since it ha s no counterpart in classical electrodynamics and hence provides new evishydence of the wave-particle dualism Photon antibunching i s characterized by a nonclassical state of the field in which the variance of the number of photons is less than the mean number of photons ie the photons exhibit sub-Poissonian statistics Jbile experiments thus far are restricted to the study of resonance fluorescence from single atoms 1 1-3 various physical mechanism that generate such a state have been shown to be theoretically possibl e (see ref s 4-20

l and reviews 2122 l Among them are second harmon ic generation 4-7 and three-wave parametric interaction 18 in a nonlinear medium resonance fluorescence from two-level atoms 9-15 degenerate four-wave mixing 16 two-photon absorption l I7 18

free-electron-laser geshy

neration 11920 Recently the generation of squeezed sta tes another class of nonclassical states 28~ has al so become the object of active researches (see i 28-30 and refs therein)

In this paper the production of sub-Poissonian nonc l assi shycal states by using two-mode processes in a three- level system will be discussed On the basis of the exactly soluble model studied in23 we shall examine the time behaviour of t he norshymally ordered variances of the photon numbers and of the crO SE correlation between the modes It will be shmrn t hat und er s u j shytable conditions photon antibunching and ant i cor r e l atiCln net shyween the modes are possible in the sy s t em

The remainder of the paper is or ga ni zed as f(ll I IlWR Til tJl shy

tion 2 the model Hamiltonian t he neceSRnrv we study tlli

elf til

UUII

lIuod II ClIlI

d

and suIts are given In section 3 the normally order ed var i ances tion 4 the cross correlation Finally the conclusions ar

2 MODEL HAMILTONI AN ANn liAS 1 r lrlIlA) 11111

The l ambda - conI i gu r ltlt i llll 1111 he re is shown i n F jg 1 Till IIpptr t he level s I and 2 hy lIipule tion 1-2 i s f orbiddln fhe

2

3 Fig l EneYgy level structure of the lambda configuYation emitter inteYshyacting with the two-mode resonant radiation field

2 tion of the emitter with the two-mode resonant radi a tion field in the dipole

middot 23 and rotat1ng wave approx lmations 1S

3 A +A A

H = I nO R + I nu) a a + J = 1 J JJ a = 1 a a a

( I ) 2 A A A + A

+t I 8a (a a R 3a + a a R a 3 ) bull a=1

Here the operator Rjj =Ii gt lti I describes the population of l e shyel i wi t h the correspond i ng energy tO j and state vector I j gt R

imiddot = li ltil is the op~rator of transition from level j to leshy

vel i The operators R jJ

i j = 123 obey the folloing re shylations

3

RjJ Rkf = RiPO jk I Rjj

= 1 (2) i= 1

A A A A

The photon operators a a+ describe two modes of the r adiashytion field with the reonan~e frequencies w =0 3-0 and ga ar e the corresponding parameters of emitter~ode coupling

We assume that the emitter is initially on the lowest leshyvel I In t he papers 23

1 the time-dependent two-mode photon- d istr ibu ti on f unction P (n 1n2) has been fou nd explicitly and r i go r ou s l y It r eads

t

P (n 11 ) - P (nln )R (t) +P(n +ln - l)R 1 1 (t) +I 2 Z 1 1 2 2n 1+ n2 shyn 1 n2

(3) I P(n l + ln2 )[1 - R In +1 n (t) -R 20 + 1 n (t)]

1 2 1 2

wi ll ~

4g2g22nl(n2+1) i2(t g 2n +g2(n + 1 )]It (I) I shy 1 sn -v 11 2 21 2 2rg 2n + g2 (n + 1)]

1 1 2 2 (4)

g~ n~ Bin2[ty g~ n l + gi(n 2+ 1)] f ~ 2n + g 2(n + 1)] 2

2 2

It bull ( I) 4g2g~nl(n2+1) 4( y gln + g2(n +1)1 1 SID 2 1 1 2 21 1II ~

~ ~(~2-+ 1U2 jjl i6 ilctttYi (J - ~ at

t k~~131bull J ~ - 0-- 1

3

and t he funct i ou P(n t n z) =P t =o (0 1 0 2) i s t he ini t i al pho ton d i s t r i bution In general case f or an arbi t ra r y i n i t ial field state described by a dens i ty matrix we have

r

P (n l nZ ) Flt n2n l l ~ID l D z gt pmiddot (5)

Starting from eq s (3- 5 ) we shall examine phot on stat i s tics of th e mod es i n the r ema i nd er of t his pape r

3 NORHALLY ORDERED VARI ANCES OF THE NUHBERS OF THE PHOTONS I N THE HODES

From eq (3 ) one easily find s the nor ma l l y or dered var i ance V[ sJ of the phot on number i n t he signal mode mod e 2)

V[ sl et) lta~2 (t) a~(traquo -lti(t)2(traquo 2

= l P (0 1deg2)(0 2 -deg ) - I P (0 0 ) n 12= ( 6 2nn I nn l 122

I 2 I 2

=V[sJ (O) + 2 1 P (n 1 o Z) R 2n ( t) (D2 - n ) -1 I P(0 I D )Rlnn ( t)12 n 2 2 nl n2 1 2 n l n2 1 2

The expres s i on of t he cor r e spond ing quan t i ty V[ p) for the pump mode (mode 1) i s fo und f rom eq (3) to be

V[ p] (t) ~ lt -r2 ( t) ~~ (traquo - lt~i(t) I (traquo 2

I P (0 l 02 )(D~ - deg1 ) - I 1 P (n l O2 ) II 112 I n2 n 1middotn2 (7)

= V[pJ (0) 2 I P (n 1oz)R 1 (t)(n -ii1) -I Pen 02 ) R (t) 12-t 1lf1 n In n n 1 t 11lel 2 - t 2

Here n1 ~ I P(olo2)ol and O co I P(n 1n2)o2 denote the mean2l n2 ~ ne

number s o f the i ni t ia l photons i n t he pump and signal modes respec t ively

As i s well known t he quantities V[ s] and V[ ] c haracterize the photon s t a t i s t ical proper ti es o f t he mOdel They are proshypor t ional t o t he exc e ss coincidence counting rates to be meashysured i n a Hanbur y Brown and TW1 ls- t ype 2425 1 experimen t The s i gn pl us or minu s of V[ 5J (or V[ p] ) shows the photon s t a t is shyt ic s o f t he s ignal (or pump) mode i s super- or sub- Poissonian and ind i Cate s t herefor e whether bunching (+) or antibuncbing (- ) occur s 12 122 Now l1e cons i der some consequences opound the expr es s ions (6) and (7)

4

Fig 2 Temporal behavi our of the normally order ed variance V[sJ (t) of the photon nwnber

01

o in t he s ignal-mode f ie ld produ shy~ - - - - - - - - - - - - shyced from the vacuwn by Glaubershymiddot01 coherent pwnping Calculations

02 ar e perfoYmed f or g I = g 2 g 0

P(n l 02) = an oP(n 1 ) J pe n I) = - -03 - - ~1 shy= exp(-n 1)n 1 1 01 and = 12n l-04 Time i s measur ed -in units of

I g o l)o 10 20

got-

Fir st of all note that if the signal mode 2 is initially in the vacuum state ie P(ol fi 2) = ( 2op(n1) from (6) we have

2](t) = -I L P(n )R deg (t ) 1 lt O (8)Vrs 1 zn n 1 I

This implies that the signal light f ield generated from the inishytial vacuum by arbitrary light pumping has sub-Poissonian phoshyt on statistics and hence exhibits photon antibunching for a ll times t gt O In fig2 we plot V[sl versus time got for the case of coherent punping with gl = g2 E go P (n

1 )= exp(-nl)ilnl ol

and n = 12 The figure shows distincly the Rabi nonlinear osshyl cillations collapse and revival of their envelopes ~see IZ6

1

and below) Le t us now assume that bot h the signal and pump modes are

initiall y in Gl auber states Iz gt and Izegt respectively This1means the initial distribution P(ol n2) of the photon numbers in the modes is given by

- _n _n - 2 - 2 F(n n ) = expl-(o - II )1 n 1 o 2 I (n n I deg = Iz I n = z I (9)

12 12 12 1 2 I 1 2 2

Substituting (9) into (6) and (7) we perform a numerical exashymination for Vlsi and V[pl I n fi g3 we plot these quantities a s functions of time for g2 = 2g 1 2g o il 2 = 9 ii l = 4 For very short times (0 lt got lt 025) the value of V[sl is positive whilst the value of V[p l is negative As time goes on (got lt 3) we obser ve osc il l at ions of bot h t he va lue s and their corresshyponding s i gns (see fig3a) Fr om the figure it is clear t hat sub-Poi s sonian photon statatistics and antibunching in the mode s occur f or some fin i te i n t er vals Such a time behaviour of photon stati s t ics is conne c ted with the so-called Rabi osshycillations 121 2Z 261 For longer times (got gt 3) f s] and f p] reach

5

-- ----- -----

Fig J Time dependence of t he norrna U y orviered vari ances Vs(t) 0 8

and -P l(t) of the photon numshybers m the signa l and pump ~ 0 I

modes respectively Both the i i elds of the modes are initi - poundr

aU y i n the Glauber coherent 0

s tates wi t h nl = 4 D) =9 ~ g 1 = 0 5 g 2 = go middot Ti me- i s measuroed ~ i n units of l go (a J Short ~ -0 1

times go t lt 5 In this case t he z basic oseillations occur at the ~ 0 Rabi- froequeneies and are sub- if B

j ect to Gauss ian collapse leshy

o

Vip i 0

0 )

2 I 5BUlting in the s teady behaviovy 0 (bJ Long t i mes t he range of t 1me 9 t shy

is eight t imes l arogergt than in ra j The l ong t i me behaviourgt is c luzrac t eroi zed by rgtevival

4 b)lD 08

~ 06 laquo Vep) 100 0 1 ~g 02

5 0 CD Ves]L -02 gt Z -04

2i -05 f shy~ middot08 CL

0 10 20 30 10

got shy

the s te ady val ues Thi s corr e spond s to the Cummi ngs coll apse r egi on I Z6 ~ The negat ive steady va lue Vr~J pr edi c t s ant i hunc hi n8 i n t he s i gna l mode The st ationary beh~v iour of t he variances cont i nue s un t i l t he appea r ance of revivals Z6 lt see fig 3b where they s tar t o sc i llating afa i n Thus we see t hat i n t he cons i shyder ed ca se when g~ n2 g l ii l =9raquo 1 ant i bunching in t he s i gnal mode t a kes p lace bo t h in the Rabi osci lla t ion region and du shyring the collapse are r eviv a l An anal yt i c approxima t e de scrip shyt ion of t he above- ob t ained numer ica l results can be given i n the following

Du e to the olell-known properti e s of the t wo-d i mension Poisshysonian distr i bu t i on (9 )

6

0 P ( n I O2) = Ii t ( 0 I - 1 11 2) o 2P (n I II 2) = ii 2 P ( 11 I 11 2 - 1 ) (10) 1

from eqs (6) and (7) we have

V ( t) =21~ [R (t) -R (t ) J - [R (t ) 1 ls1 2 2n 1 n2 + 1 2 n In 2 2 n 1 n 2

(I I ) ______ 2

[R (t)] + 1V[ j ( t) = 20 [ R l t ) - R (t)]P 1111 + 111 I n l 11 2 I n In 2

1 2

Here f or convenience the notation ( ) -= ~ P (n l n2 )(middot middot )has been introduc ed n l n 2

In t he case n 1 li 2 raquo 1 the summations over and n 2 i n (J I)n 1 c an be y erformed appr oximate ly by u s ing saddl e-point techn i shy

26 ques i f one not i ces that the we i ghting fac tor P~l O2) wil l peak a t (iiI ( 2) wi th r e latively narrow dispers i on This means t ha t the expr ess ions in (II ) can be c hanged to integral s of f a s t o~cil l a t ing func tions whic h aft er tr i v ial t r ans f ormations can be writ t en a s

2 2- - u2 + 2 2 2 V [ s 1 (t) = g ig 2 n 1 n 2 (4F ( t 2 g 2) - F (2 t 2 g 2 ) ] shy

(W 2 + g~ ) 2 _ 4 4 - 2 - 2 ( 12 ) 0 + 1 2 gg o ( n+ 1) _ 2 _ _ [ 4F (t g 2) _ F (2 t g~)l _ 1 2 1 2 [ 4F(t g2)_F(2t g 2)1 ~

W4 R 2 2

Z - 2 g o 2

v[p] (1)= 7 t 4 g (02 + l ) F( t g )+ g ~DI F(2t g )]2

g~ii l (i11 + 1) 2 - shy- [ 4g2 ( n2 + 1)F (t g2+ g2 )+ g~ 0 + 1 )F (2t g2+ g2)1

2 2 2 1 2 ti l 2 (W + g 1)

g2-deg 2 - 2 2- 2 -

1 - 2~ [ 4 g e(0 2 + 1) F (t g2 )+ g t n I F (2 t g2) J 2 + 1

2- 2 shywhere Wyglo + g~(o +l)and1 ~ 2

(I ~F(t g ~ 1 - J d x I rd x 2 51 (x l x 2 ) cos [ L vi g~ 1 + g~x 2 + g 2 1 2 deg 0 s

with the kernel p(X 1 x~) obtained from P( n1o2)by using t_ne

Euler-Maclaurin-fortnulae (see 26 )

m - Ih - - shyJ(x x2) =0 l4~xlx2)exp t -(nl 10 2 ) X I + X 2 - x 1ln ( x 1n ( 2 1 )

- x 2ln(x 2 iiz) l

9

Note that the functio n F(tg2)approaehe s un i ty i n the l i mit t oo

Then we c a n easily find from eqs ( 12) the ana lytical approx i shybull 0 0 h

mat~ express10ns for t h e steady val ues Vl ~ and VIp ] of t e var1ance s They re ad

VO

2 2 - II 2 + 1 [1s = 30 1 g 2 -n I -n ltgt _ ____ nE + 2 ~ 2- - - ---- shy

9 4 4- 2 g n 2 - - -- + g(n 2 + 1)] 2 [ g l Ut ~~c 1) 2 g en 2 )1

2shy

e l

2 - 2 - 441 gi n I + g 2 ( n 2 + 1)]

2- 2 - 2 -- 2 ( 15) o 2- - g J ill 1 4g 2 (n 2 + 1)

VIp = g l n l n 1 2=-----2-=---2 (n + 1) ~ (~ll +~~~3 (n3~_ IIg t D 1 + g 2 (n 2 + 1 ) 1 I~ (n l d) + g~ (n2+ 1 )] 2

4 - 2 4 - 2 2g in I + 2g 2 (n 2 + 1) I ------ --- ------- t 1

ltgt 2 - 4shy4 g ~ D l g (n + 1)1

I I 2 2

o ~e plot the quan t i t y Vi sl a s a f unc t i on o f no i n fi g 4a (for I = 10 g I = g 2 = g o) and a s a f uncti o n of n l~ i n fi g 4 b (fo r - ) fl n2 = 0 10 gl = g2 ~ go bull From th~ f1 g_ures 1ve see that VI s1 1S negat i ve for v ery small v a lues n 2 ( n

2 == 0 ) a nd for large va l u es

n as compared wi t h n l (g ~n2 raquo g~nl ) By using the first equashytion in (15) we c a n e as i ly get the est i mate

2shyo 3g n

_ I_ IVIs 1 - 2-

lt 0 ( 16 ) g 2 n 2

2- 2shyfor t he case g 2n 2 g n O~ the other hand f or the c a se when

lthe in i tial st a t e o f t11e signa l mode f i e l d is v acu um or n e a r shyvacuum ie n2 Oand whe n iJ I raquol we find s t rai~ht-forwa rdly from eq (11 ) the eva l uat i o n

9 4( 4n 2 o g lry I

V - ----~--- lt 0 ( 17)l sI 2-- 244r g Jl g I

1 1 2

(wh i c h i s also i n compl iance wi th t h e f i rs t equat i on i n (15) in t he limi t no ~ 0 ) The i nequal i t i e s i n eqs ( 16) and (l l)

co ri1Jl that photon s t a t istics of the s i 8nal mode field i n t h e they iy reg~me will Je sub-~oi s sonial1 i f e i the r g2 raquo d ered L(ii I n 2 raquo1)or = 0 0 1 raquo 1 Th i s c onf i rms the numerica ln 2 mod e tak obta ined above se e figs 2 3 a nd 4 ring the Iort times and l ar ge i nit i al pho to n number s n I n2

t i on of t~~e sadd ~~yoint method for eva lu a ting the i nt e gral t he follorhg get

4- 4- 2 Due to t he [ (gl n 1 + g2 n2) t 1 2- 2-- 2

sonian di s t r i bUlP - 2- -2----Z-JCOS [ tj g 1n l + g 2 n 2 + g 1 (18) 6 8(g l n 1 +g2 D2 + g )

0 1 VIO I (n 1 )

01 -0 1

OH I VI~ I ( il l ) b) ---ilz =0

o -02

-01 -03

-02 -OL

-0 3 50 100

50 100 150 200 2 shy92

n 2 g

2 n1

Fig 4 S t eady value V[o J OJ t he noma Uy oYdeYed voshyr i anCe of t he photon ntinbeY i n t he s i gna l mode (a ) The depend enc e of V[~ l uponon 2 in the case g l = g2 1= 10 (b) The dependence of vr s 1 upon ill in the case s when g 1 = g 2 n2 = 0 and 1O

The r i ght-hand s ide of e q ( 18) be hav es lik e damped c osine osshyc i ll ations which are ter minated by the Gauss i an env elope

2exp [_t 2r2 1 Here t he collapse time T i s defi n ed as c

2- 2- 2) c

2 g l n 1t- g zD2 + g 11 2 r (g ) = 2 ------ - - -- (19) c ( ~ 4shy

g n f g n I 1 2 2

The approximate expr ession (18 ) toget her with eqs ( 12 ) give a d e sc r ip t i on o f the behav iou r of t h e v a r i a nc es V[ s1 V l rd i n the s ho rt time r egime gs TC ( see f i g 3a ) In pa rticul a r for ea r l y t i me s t~ 0 we can obtain

1 2 2 - - 4 1 4- 2 4 VI (tl - g g 11 n t + V[pl (t) ~ - g n t + ( 20)

81 2 1 2 I 2 3 1 1

Empha s ize t hat the a pp r o x i mate express i on (1 8) is no t v a lid i n t h e l on~ time re gio n Therefore it cannot desc r ibe the rev i val of t h e e s c i 11 a t i o n envelopes of VI 1 a nd Vr l wh ich o c c urs whe n l rbull see f ig 3b s p

Ap p l yi ng t he tong- t i me approx imate method developed on the bas i s of t he sad d le-poin t method by Eberl y and h is coworker s f 6 1 1 ( ) 111 re _ ~ t o Clt1 eu a te t h e 1nt egral 13 we can get 1n the spec i a l c a se ~ 1 a ~ 2 g o the foll owi ng ev aluat i on

2 2 2 2F (t g ) = 1 - r (I j ) exp l -- if (t g ) cos cent (t g ) ( 21 )

wher e t h e no tat i on

9

and t he funct i ou P(n t n z) =P t =o (0 1 0 2) i s t he ini t i al pho ton d i s t r i bution In general case f or an arbi t ra r y i n i t ial field state described by a dens i ty matrix we have

r

P (n l nZ ) Flt n2n l l ~ID l D z gt pmiddot (5)

Starting from eq s (3- 5 ) we shall examine phot on stat i s tics of th e mod es i n the r ema i nd er of t his pape r

3 NORHALLY ORDERED VARI ANCES OF THE NUHBERS OF THE PHOTONS I N THE HODES

From eq (3 ) one easily find s the nor ma l l y or dered var i ance V[ sJ of the phot on number i n t he signal mode mod e 2)

V[ sl et) lta~2 (t) a~(traquo -lti(t)2(traquo 2

= l P (0 1deg2)(0 2 -deg ) - I P (0 0 ) n 12= ( 6 2nn I nn l 122

I 2 I 2

=V[sJ (O) + 2 1 P (n 1 o Z) R 2n ( t) (D2 - n ) -1 I P(0 I D )Rlnn ( t)12 n 2 2 nl n2 1 2 n l n2 1 2

The expres s i on of t he cor r e spond ing quan t i ty V[ p) for the pump mode (mode 1) i s fo und f rom eq (3) to be

V[ p] (t) ~ lt -r2 ( t) ~~ (traquo - lt~i(t) I (traquo 2

I P (0 l 02 )(D~ - deg1 ) - I 1 P (n l O2 ) II 112 I n2 n 1middotn2 (7)

= V[pJ (0) 2 I P (n 1oz)R 1 (t)(n -ii1) -I Pen 02 ) R (t) 12-t 1lf1 n In n n 1 t 11lel 2 - t 2

Here n1 ~ I P(olo2)ol and O co I P(n 1n2)o2 denote the mean2l n2 ~ ne

number s o f the i ni t ia l photons i n t he pump and signal modes respec t ively

As i s well known t he quantities V[ s] and V[ ] c haracterize the photon s t a t i s t ical proper ti es o f t he mOdel They are proshypor t ional t o t he exc e ss coincidence counting rates to be meashysured i n a Hanbur y Brown and TW1 ls- t ype 2425 1 experimen t The s i gn pl us or minu s of V[ 5J (or V[ p] ) shows the photon s t a t is shyt ic s o f t he s ignal (or pump) mode i s super- or sub- Poissonian and ind i Cate s t herefor e whether bunching (+) or antibuncbing (- ) occur s 12 122 Now l1e cons i der some consequences opound the expr es s ions (6) and (7)

4

Fig 2 Temporal behavi our of the normally order ed variance V[sJ (t) of the photon nwnber

01

o in t he s ignal-mode f ie ld produ shy~ - - - - - - - - - - - - shyced from the vacuwn by Glaubershymiddot01 coherent pwnping Calculations

02 ar e perfoYmed f or g I = g 2 g 0

P(n l 02) = an oP(n 1 ) J pe n I) = - -03 - - ~1 shy= exp(-n 1)n 1 1 01 and = 12n l-04 Time i s measur ed -in units of

I g o l)o 10 20

got-

Fir st of all note that if the signal mode 2 is initially in the vacuum state ie P(ol fi 2) = ( 2op(n1) from (6) we have

2](t) = -I L P(n )R deg (t ) 1 lt O (8)Vrs 1 zn n 1 I

This implies that the signal light f ield generated from the inishytial vacuum by arbitrary light pumping has sub-Poissonian phoshyt on statistics and hence exhibits photon antibunching for a ll times t gt O In fig2 we plot V[sl versus time got for the case of coherent punping with gl = g2 E go P (n

1 )= exp(-nl)ilnl ol

and n = 12 The figure shows distincly the Rabi nonlinear osshyl cillations collapse and revival of their envelopes ~see IZ6

1

and below) Le t us now assume that bot h the signal and pump modes are

initiall y in Gl auber states Iz gt and Izegt respectively This1means the initial distribution P(ol n2) of the photon numbers in the modes is given by

- _n _n - 2 - 2 F(n n ) = expl-(o - II )1 n 1 o 2 I (n n I deg = Iz I n = z I (9)

12 12 12 1 2 I 1 2 2

Substituting (9) into (6) and (7) we perform a numerical exashymination for Vlsi and V[pl I n fi g3 we plot these quantities a s functions of time for g2 = 2g 1 2g o il 2 = 9 ii l = 4 For very short times (0 lt got lt 025) the value of V[sl is positive whilst the value of V[p l is negative As time goes on (got lt 3) we obser ve osc il l at ions of bot h t he va lue s and their corresshyponding s i gns (see fig3a) Fr om the figure it is clear t hat sub-Poi s sonian photon statatistics and antibunching in the mode s occur f or some fin i te i n t er vals Such a time behaviour of photon stati s t ics is conne c ted with the so-called Rabi osshycillations 121 2Z 261 For longer times (got gt 3) f s] and f p] reach

5

-- ----- -----

Fig J Time dependence of t he norrna U y orviered vari ances Vs(t) 0 8

and -P l(t) of the photon numshybers m the signa l and pump ~ 0 I

modes respectively Both the i i elds of the modes are initi - poundr

aU y i n the Glauber coherent 0

s tates wi t h nl = 4 D) =9 ~ g 1 = 0 5 g 2 = go middot Ti me- i s measuroed ~ i n units of l go (a J Short ~ -0 1

times go t lt 5 In this case t he z basic oseillations occur at the ~ 0 Rabi- froequeneies and are sub- if B

j ect to Gauss ian collapse leshy

o

Vip i 0

0 )

2 I 5BUlting in the s teady behaviovy 0 (bJ Long t i mes t he range of t 1me 9 t shy

is eight t imes l arogergt than in ra j The l ong t i me behaviourgt is c luzrac t eroi zed by rgtevival

4 b)lD 08

~ 06 laquo Vep) 100 0 1 ~g 02

5 0 CD Ves]L -02 gt Z -04

2i -05 f shy~ middot08 CL

0 10 20 30 10

got shy

the s te ady val ues Thi s corr e spond s to the Cummi ngs coll apse r egi on I Z6 ~ The negat ive steady va lue Vr~J pr edi c t s ant i hunc hi n8 i n t he s i gna l mode The st ationary beh~v iour of t he variances cont i nue s un t i l t he appea r ance of revivals Z6 lt see fig 3b where they s tar t o sc i llating afa i n Thus we see t hat i n t he cons i shyder ed ca se when g~ n2 g l ii l =9raquo 1 ant i bunching in t he s i gnal mode t a kes p lace bo t h in the Rabi osci lla t ion region and du shyring the collapse are r eviv a l An anal yt i c approxima t e de scrip shyt ion of t he above- ob t ained numer ica l results can be given i n the following

Du e to the olell-known properti e s of the t wo-d i mension Poisshysonian distr i bu t i on (9 )

6

0 P ( n I O2) = Ii t ( 0 I - 1 11 2) o 2P (n I II 2) = ii 2 P ( 11 I 11 2 - 1 ) (10) 1

from eqs (6) and (7) we have

V ( t) =21~ [R (t) -R (t ) J - [R (t ) 1 ls1 2 2n 1 n2 + 1 2 n In 2 2 n 1 n 2

(I I ) ______ 2

[R (t)] + 1V[ j ( t) = 20 [ R l t ) - R (t)]P 1111 + 111 I n l 11 2 I n In 2

1 2

Here f or convenience the notation ( ) -= ~ P (n l n2 )(middot middot )has been introduc ed n l n 2

In t he case n 1 li 2 raquo 1 the summations over and n 2 i n (J I)n 1 c an be y erformed appr oximate ly by u s ing saddl e-point techn i shy

26 ques i f one not i ces that the we i ghting fac tor P~l O2) wil l peak a t (iiI ( 2) wi th r e latively narrow dispers i on This means t ha t the expr ess ions in (II ) can be c hanged to integral s of f a s t o~cil l a t ing func tions whic h aft er tr i v ial t r ans f ormations can be writ t en a s

2 2- - u2 + 2 2 2 V [ s 1 (t) = g ig 2 n 1 n 2 (4F ( t 2 g 2) - F (2 t 2 g 2 ) ] shy

(W 2 + g~ ) 2 _ 4 4 - 2 - 2 ( 12 ) 0 + 1 2 gg o ( n+ 1) _ 2 _ _ [ 4F (t g 2) _ F (2 t g~)l _ 1 2 1 2 [ 4F(t g2)_F(2t g 2)1 ~

W4 R 2 2

Z - 2 g o 2

v[p] (1)= 7 t 4 g (02 + l ) F( t g )+ g ~DI F(2t g )]2

g~ii l (i11 + 1) 2 - shy- [ 4g2 ( n2 + 1)F (t g2+ g2 )+ g~ 0 + 1 )F (2t g2+ g2)1

2 2 2 1 2 ti l 2 (W + g 1)

g2-deg 2 - 2 2- 2 -

1 - 2~ [ 4 g e(0 2 + 1) F (t g2 )+ g t n I F (2 t g2) J 2 + 1

2- 2 shywhere Wyglo + g~(o +l)and1 ~ 2

(I ~F(t g ~ 1 - J d x I rd x 2 51 (x l x 2 ) cos [ L vi g~ 1 + g~x 2 + g 2 1 2 deg 0 s

with the kernel p(X 1 x~) obtained from P( n1o2)by using t_ne

Euler-Maclaurin-fortnulae (see 26 )

m - Ih - - shyJ(x x2) =0 l4~xlx2)exp t -(nl 10 2 ) X I + X 2 - x 1ln ( x 1n ( 2 1 )

- x 2ln(x 2 iiz) l

9

Note that the functio n F(tg2)approaehe s un i ty i n the l i mit t oo

Then we c a n easily find from eqs ( 12) the ana lytical approx i shybull 0 0 h

mat~ express10ns for t h e steady val ues Vl ~ and VIp ] of t e var1ance s They re ad

VO

2 2 - II 2 + 1 [1s = 30 1 g 2 -n I -n ltgt _ ____ nE + 2 ~ 2- - - ---- shy

9 4 4- 2 g n 2 - - -- + g(n 2 + 1)] 2 [ g l Ut ~~c 1) 2 g en 2 )1

2shy

e l

2 - 2 - 441 gi n I + g 2 ( n 2 + 1)]

2- 2 - 2 -- 2 ( 15) o 2- - g J ill 1 4g 2 (n 2 + 1)

VIp = g l n l n 1 2=-----2-=---2 (n + 1) ~ (~ll +~~~3 (n3~_ IIg t D 1 + g 2 (n 2 + 1 ) 1 I~ (n l d) + g~ (n2+ 1 )] 2

4 - 2 4 - 2 2g in I + 2g 2 (n 2 + 1) I ------ --- ------- t 1

ltgt 2 - 4shy4 g ~ D l g (n + 1)1

I I 2 2

o ~e plot the quan t i t y Vi sl a s a f unc t i on o f no i n fi g 4a (for I = 10 g I = g 2 = g o) and a s a f uncti o n of n l~ i n fi g 4 b (fo r - ) fl n2 = 0 10 gl = g2 ~ go bull From th~ f1 g_ures 1ve see that VI s1 1S negat i ve for v ery small v a lues n 2 ( n

2 == 0 ) a nd for large va l u es

n as compared wi t h n l (g ~n2 raquo g~nl ) By using the first equashytion in (15) we c a n e as i ly get the est i mate

2shyo 3g n

_ I_ IVIs 1 - 2-

lt 0 ( 16 ) g 2 n 2

2- 2shyfor t he case g 2n 2 g n O~ the other hand f or the c a se when

lthe in i tial st a t e o f t11e signa l mode f i e l d is v acu um or n e a r shyvacuum ie n2 Oand whe n iJ I raquol we find s t rai~ht-forwa rdly from eq (11 ) the eva l uat i o n

9 4( 4n 2 o g lry I

V - ----~--- lt 0 ( 17)l sI 2-- 244r g Jl g I

1 1 2

(wh i c h i s also i n compl iance wi th t h e f i rs t equat i on i n (15) in t he limi t no ~ 0 ) The i nequal i t i e s i n eqs ( 16) and (l l)

co ri1Jl that photon s t a t istics of the s i 8nal mode field i n t h e they iy reg~me will Je sub-~oi s sonial1 i f e i the r g2 raquo d ered L(ii I n 2 raquo1)or = 0 0 1 raquo 1 Th i s c onf i rms the numerica ln 2 mod e tak obta ined above se e figs 2 3 a nd 4 ring the Iort times and l ar ge i nit i al pho to n number s n I n2

t i on of t~~e sadd ~~yoint method for eva lu a ting the i nt e gral t he follorhg get

4- 4- 2 Due to t he [ (gl n 1 + g2 n2) t 1 2- 2-- 2

sonian di s t r i bUlP - 2- -2----Z-JCOS [ tj g 1n l + g 2 n 2 + g 1 (18) 6 8(g l n 1 +g2 D2 + g )

0 1 VIO I (n 1 )

01 -0 1

OH I VI~ I ( il l ) b) ---ilz =0

o -02

-01 -03

-02 -OL

-0 3 50 100

50 100 150 200 2 shy92

n 2 g

2 n1

Fig 4 S t eady value V[o J OJ t he noma Uy oYdeYed voshyr i anCe of t he photon ntinbeY i n t he s i gna l mode (a ) The depend enc e of V[~ l uponon 2 in the case g l = g2 1= 10 (b) The dependence of vr s 1 upon ill in the case s when g 1 = g 2 n2 = 0 and 1O

The r i ght-hand s ide of e q ( 18) be hav es lik e damped c osine osshyc i ll ations which are ter minated by the Gauss i an env elope

2exp [_t 2r2 1 Here t he collapse time T i s defi n ed as c

2- 2- 2) c

2 g l n 1t- g zD2 + g 11 2 r (g ) = 2 ------ - - -- (19) c ( ~ 4shy

g n f g n I 1 2 2

The approximate expr ession (18 ) toget her with eqs ( 12 ) give a d e sc r ip t i on o f the behav iou r of t h e v a r i a nc es V[ s1 V l rd i n the s ho rt time r egime gs TC ( see f i g 3a ) In pa rticul a r for ea r l y t i me s t~ 0 we can obtain

1 2 2 - - 4 1 4- 2 4 VI (tl - g g 11 n t + V[pl (t) ~ - g n t + ( 20)

81 2 1 2 I 2 3 1 1

Empha s ize t hat the a pp r o x i mate express i on (1 8) is no t v a lid i n t h e l on~ time re gio n Therefore it cannot desc r ibe the rev i val of t h e e s c i 11 a t i o n envelopes of VI 1 a nd Vr l wh ich o c c urs whe n l rbull see f ig 3b s p

Ap p l yi ng t he tong- t i me approx imate method developed on the bas i s of t he sad d le-poin t method by Eberl y and h is coworker s f 6 1 1 ( ) 111 re _ ~ t o Clt1 eu a te t h e 1nt egral 13 we can get 1n the spec i a l c a se ~ 1 a ~ 2 g o the foll owi ng ev aluat i on

2 2 2 2F (t g ) = 1 - r (I j ) exp l -- if (t g ) cos cent (t g ) ( 21 )

wher e t h e no tat i on

9

-- ----- -----

Fig J Time dependence of t he norrna U y orviered vari ances Vs(t) 0 8

and -P l(t) of the photon numshybers m the signa l and pump ~ 0 I

modes respectively Both the i i elds of the modes are initi - poundr

aU y i n the Glauber coherent 0

s tates wi t h nl = 4 D) =9 ~ g 1 = 0 5 g 2 = go middot Ti me- i s measuroed ~ i n units of l go (a J Short ~ -0 1

times go t lt 5 In this case t he z basic oseillations occur at the ~ 0 Rabi- froequeneies and are sub- if B

j ect to Gauss ian collapse leshy

o

Vip i 0

0 )

2 I 5BUlting in the s teady behaviovy 0 (bJ Long t i mes t he range of t 1me 9 t shy

is eight t imes l arogergt than in ra j The l ong t i me behaviourgt is c luzrac t eroi zed by rgtevival

4 b)lD 08

~ 06 laquo Vep) 100 0 1 ~g 02

5 0 CD Ves]L -02 gt Z -04

2i -05 f shy~ middot08 CL

0 10 20 30 10

got shy

the s te ady val ues Thi s corr e spond s to the Cummi ngs coll apse r egi on I Z6 ~ The negat ive steady va lue Vr~J pr edi c t s ant i hunc hi n8 i n t he s i gna l mode The st ationary beh~v iour of t he variances cont i nue s un t i l t he appea r ance of revivals Z6 lt see fig 3b where they s tar t o sc i llating afa i n Thus we see t hat i n t he cons i shyder ed ca se when g~ n2 g l ii l =9raquo 1 ant i bunching in t he s i gnal mode t a kes p lace bo t h in the Rabi osci lla t ion region and du shyring the collapse are r eviv a l An anal yt i c approxima t e de scrip shyt ion of t he above- ob t ained numer ica l results can be given i n the following

Du e to the olell-known properti e s of the t wo-d i mension Poisshysonian distr i bu t i on (9 )

6

0 P ( n I O2) = Ii t ( 0 I - 1 11 2) o 2P (n I II 2) = ii 2 P ( 11 I 11 2 - 1 ) (10) 1

from eqs (6) and (7) we have

V ( t) =21~ [R (t) -R (t ) J - [R (t ) 1 ls1 2 2n 1 n2 + 1 2 n In 2 2 n 1 n 2

(I I ) ______ 2

[R (t)] + 1V[ j ( t) = 20 [ R l t ) - R (t)]P 1111 + 111 I n l 11 2 I n In 2

1 2

Here f or convenience the notation ( ) -= ~ P (n l n2 )(middot middot )has been introduc ed n l n 2

In t he case n 1 li 2 raquo 1 the summations over and n 2 i n (J I)n 1 c an be y erformed appr oximate ly by u s ing saddl e-point techn i shy

26 ques i f one not i ces that the we i ghting fac tor P~l O2) wil l peak a t (iiI ( 2) wi th r e latively narrow dispers i on This means t ha t the expr ess ions in (II ) can be c hanged to integral s of f a s t o~cil l a t ing func tions whic h aft er tr i v ial t r ans f ormations can be writ t en a s

2 2- - u2 + 2 2 2 V [ s 1 (t) = g ig 2 n 1 n 2 (4F ( t 2 g 2) - F (2 t 2 g 2 ) ] shy

(W 2 + g~ ) 2 _ 4 4 - 2 - 2 ( 12 ) 0 + 1 2 gg o ( n+ 1) _ 2 _ _ [ 4F (t g 2) _ F (2 t g~)l _ 1 2 1 2 [ 4F(t g2)_F(2t g 2)1 ~

W4 R 2 2

Z - 2 g o 2

v[p] (1)= 7 t 4 g (02 + l ) F( t g )+ g ~DI F(2t g )]2

g~ii l (i11 + 1) 2 - shy- [ 4g2 ( n2 + 1)F (t g2+ g2 )+ g~ 0 + 1 )F (2t g2+ g2)1

2 2 2 1 2 ti l 2 (W + g 1)

g2-deg 2 - 2 2- 2 -

1 - 2~ [ 4 g e(0 2 + 1) F (t g2 )+ g t n I F (2 t g2) J 2 + 1

2- 2 shywhere Wyglo + g~(o +l)and1 ~ 2

(I ~F(t g ~ 1 - J d x I rd x 2 51 (x l x 2 ) cos [ L vi g~ 1 + g~x 2 + g 2 1 2 deg 0 s

with the kernel p(X 1 x~) obtained from P( n1o2)by using t_ne

Euler-Maclaurin-fortnulae (see 26 )

m - Ih - - shyJ(x x2) =0 l4~xlx2)exp t -(nl 10 2 ) X I + X 2 - x 1ln ( x 1n ( 2 1 )

- x 2ln(x 2 iiz) l

9

Note that the functio n F(tg2)approaehe s un i ty i n the l i mit t oo

Then we c a n easily find from eqs ( 12) the ana lytical approx i shybull 0 0 h

mat~ express10ns for t h e steady val ues Vl ~ and VIp ] of t e var1ance s They re ad

VO

2 2 - II 2 + 1 [1s = 30 1 g 2 -n I -n ltgt _ ____ nE + 2 ~ 2- - - ---- shy

9 4 4- 2 g n 2 - - -- + g(n 2 + 1)] 2 [ g l Ut ~~c 1) 2 g en 2 )1

2shy

e l

2 - 2 - 441 gi n I + g 2 ( n 2 + 1)]

2- 2 - 2 -- 2 ( 15) o 2- - g J ill 1 4g 2 (n 2 + 1)

VIp = g l n l n 1 2=-----2-=---2 (n + 1) ~ (~ll +~~~3 (n3~_ IIg t D 1 + g 2 (n 2 + 1 ) 1 I~ (n l d) + g~ (n2+ 1 )] 2

4 - 2 4 - 2 2g in I + 2g 2 (n 2 + 1) I ------ --- ------- t 1

ltgt 2 - 4shy4 g ~ D l g (n + 1)1

I I 2 2

o ~e plot the quan t i t y Vi sl a s a f unc t i on o f no i n fi g 4a (for I = 10 g I = g 2 = g o) and a s a f uncti o n of n l~ i n fi g 4 b (fo r - ) fl n2 = 0 10 gl = g2 ~ go bull From th~ f1 g_ures 1ve see that VI s1 1S negat i ve for v ery small v a lues n 2 ( n

2 == 0 ) a nd for large va l u es

n as compared wi t h n l (g ~n2 raquo g~nl ) By using the first equashytion in (15) we c a n e as i ly get the est i mate

2shyo 3g n

_ I_ IVIs 1 - 2-

lt 0 ( 16 ) g 2 n 2

2- 2shyfor t he case g 2n 2 g n O~ the other hand f or the c a se when

lthe in i tial st a t e o f t11e signa l mode f i e l d is v acu um or n e a r shyvacuum ie n2 Oand whe n iJ I raquol we find s t rai~ht-forwa rdly from eq (11 ) the eva l uat i o n

9 4( 4n 2 o g lry I

V - ----~--- lt 0 ( 17)l sI 2-- 244r g Jl g I

1 1 2

(wh i c h i s also i n compl iance wi th t h e f i rs t equat i on i n (15) in t he limi t no ~ 0 ) The i nequal i t i e s i n eqs ( 16) and (l l)

co ri1Jl that photon s t a t istics of the s i 8nal mode field i n t h e they iy reg~me will Je sub-~oi s sonial1 i f e i the r g2 raquo d ered L(ii I n 2 raquo1)or = 0 0 1 raquo 1 Th i s c onf i rms the numerica ln 2 mod e tak obta ined above se e figs 2 3 a nd 4 ring the Iort times and l ar ge i nit i al pho to n number s n I n2

t i on of t~~e sadd ~~yoint method for eva lu a ting the i nt e gral t he follorhg get

4- 4- 2 Due to t he [ (gl n 1 + g2 n2) t 1 2- 2-- 2

sonian di s t r i bUlP - 2- -2----Z-JCOS [ tj g 1n l + g 2 n 2 + g 1 (18) 6 8(g l n 1 +g2 D2 + g )

0 1 VIO I (n 1 )

01 -0 1

OH I VI~ I ( il l ) b) ---ilz =0

o -02

-01 -03

-02 -OL

-0 3 50 100

50 100 150 200 2 shy92

n 2 g

2 n1

Fig 4 S t eady value V[o J OJ t he noma Uy oYdeYed voshyr i anCe of t he photon ntinbeY i n t he s i gna l mode (a ) The depend enc e of V[~ l uponon 2 in the case g l = g2 1= 10 (b) The dependence of vr s 1 upon ill in the case s when g 1 = g 2 n2 = 0 and 1O

The r i ght-hand s ide of e q ( 18) be hav es lik e damped c osine osshyc i ll ations which are ter minated by the Gauss i an env elope

2exp [_t 2r2 1 Here t he collapse time T i s defi n ed as c

2- 2- 2) c

2 g l n 1t- g zD2 + g 11 2 r (g ) = 2 ------ - - -- (19) c ( ~ 4shy

g n f g n I 1 2 2

The approximate expr ession (18 ) toget her with eqs ( 12 ) give a d e sc r ip t i on o f the behav iou r of t h e v a r i a nc es V[ s1 V l rd i n the s ho rt time r egime gs TC ( see f i g 3a ) In pa rticul a r for ea r l y t i me s t~ 0 we can obtain

1 2 2 - - 4 1 4- 2 4 VI (tl - g g 11 n t + V[pl (t) ~ - g n t + ( 20)

81 2 1 2 I 2 3 1 1

Empha s ize t hat the a pp r o x i mate express i on (1 8) is no t v a lid i n t h e l on~ time re gio n Therefore it cannot desc r ibe the rev i val of t h e e s c i 11 a t i o n envelopes of VI 1 a nd Vr l wh ich o c c urs whe n l rbull see f ig 3b s p

Ap p l yi ng t he tong- t i me approx imate method developed on the bas i s of t he sad d le-poin t method by Eberl y and h is coworker s f 6 1 1 ( ) 111 re _ ~ t o Clt1 eu a te t h e 1nt egral 13 we can get 1n the spec i a l c a se ~ 1 a ~ 2 g o the foll owi ng ev aluat i on

2 2 2 2F (t g ) = 1 - r (I j ) exp l -- if (t g ) cos cent (t g ) ( 21 )

wher e t h e no tat i on

9

Note that the functio n F(tg2)approaehe s un i ty i n the l i mit t oo

Then we c a n easily find from eqs ( 12) the ana lytical approx i shybull 0 0 h

mat~ express10ns for t h e steady val ues Vl ~ and VIp ] of t e var1ance s They re ad

VO

2 2 - II 2 + 1 [1s = 30 1 g 2 -n I -n ltgt _ ____ nE + 2 ~ 2- - - ---- shy

9 4 4- 2 g n 2 - - -- + g(n 2 + 1)] 2 [ g l Ut ~~c 1) 2 g en 2 )1

2shy

e l

2 - 2 - 441 gi n I + g 2 ( n 2 + 1)]

2- 2 - 2 -- 2 ( 15) o 2- - g J ill 1 4g 2 (n 2 + 1)

VIp = g l n l n 1 2=-----2-=---2 (n + 1) ~ (~ll +~~~3 (n3~_ IIg t D 1 + g 2 (n 2 + 1 ) 1 I~ (n l d) + g~ (n2+ 1 )] 2

4 - 2 4 - 2 2g in I + 2g 2 (n 2 + 1) I ------ --- ------- t 1

ltgt 2 - 4shy4 g ~ D l g (n + 1)1

I I 2 2

o ~e plot the quan t i t y Vi sl a s a f unc t i on o f no i n fi g 4a (for I = 10 g I = g 2 = g o) and a s a f uncti o n of n l~ i n fi g 4 b (fo r - ) fl n2 = 0 10 gl = g2 ~ go bull From th~ f1 g_ures 1ve see that VI s1 1S negat i ve for v ery small v a lues n 2 ( n

2 == 0 ) a nd for large va l u es

n as compared wi t h n l (g ~n2 raquo g~nl ) By using the first equashytion in (15) we c a n e as i ly get the est i mate

2shyo 3g n

_ I_ IVIs 1 - 2-

lt 0 ( 16 ) g 2 n 2

2- 2shyfor t he case g 2n 2 g n O~ the other hand f or the c a se when

lthe in i tial st a t e o f t11e signa l mode f i e l d is v acu um or n e a r shyvacuum ie n2 Oand whe n iJ I raquol we find s t rai~ht-forwa rdly from eq (11 ) the eva l uat i o n

9 4( 4n 2 o g lry I

V - ----~--- lt 0 ( 17)l sI 2-- 244r g Jl g I

1 1 2

(wh i c h i s also i n compl iance wi th t h e f i rs t equat i on i n (15) in t he limi t no ~ 0 ) The i nequal i t i e s i n eqs ( 16) and (l l)

co ri1Jl that photon s t a t istics of the s i 8nal mode field i n t h e they iy reg~me will Je sub-~oi s sonial1 i f e i the r g2 raquo d ered L(ii I n 2 raquo1)or = 0 0 1 raquo 1 Th i s c onf i rms the numerica ln 2 mod e tak obta ined above se e figs 2 3 a nd 4 ring the Iort times and l ar ge i nit i al pho to n number s n I n2

t i on of t~~e sadd ~~yoint method for eva lu a ting the i nt e gral t he follorhg get

4- 4- 2 Due to t he [ (gl n 1 + g2 n2) t 1 2- 2-- 2

sonian di s t r i bUlP - 2- -2----Z-JCOS [ tj g 1n l + g 2 n 2 + g 1 (18) 6 8(g l n 1 +g2 D2 + g )

0 1 VIO I (n 1 )

01 -0 1

OH I VI~ I ( il l ) b) ---ilz =0

o -02

-01 -03

-02 -OL

-0 3 50 100

50 100 150 200 2 shy92

n 2 g

2 n1

Fig 4 S t eady value V[o J OJ t he noma Uy oYdeYed voshyr i anCe of t he photon ntinbeY i n t he s i gna l mode (a ) The depend enc e of V[~ l uponon 2 in the case g l = g2 1= 10 (b) The dependence of vr s 1 upon ill in the case s when g 1 = g 2 n2 = 0 and 1O

The r i ght-hand s ide of e q ( 18) be hav es lik e damped c osine osshyc i ll ations which are ter minated by the Gauss i an env elope

2exp [_t 2r2 1 Here t he collapse time T i s defi n ed as c

2- 2- 2) c

2 g l n 1t- g zD2 + g 11 2 r (g ) = 2 ------ - - -- (19) c ( ~ 4shy

g n f g n I 1 2 2

The approximate expr ession (18 ) toget her with eqs ( 12 ) give a d e sc r ip t i on o f the behav iou r of t h e v a r i a nc es V[ s1 V l rd i n the s ho rt time r egime gs TC ( see f i g 3a ) In pa rticul a r for ea r l y t i me s t~ 0 we can obtain

1 2 2 - - 4 1 4- 2 4 VI (tl - g g 11 n t + V[pl (t) ~ - g n t + ( 20)

81 2 1 2 I 2 3 1 1

Empha s ize t hat the a pp r o x i mate express i on (1 8) is no t v a lid i n t h e l on~ time re gio n Therefore it cannot desc r ibe the rev i val of t h e e s c i 11 a t i o n envelopes of VI 1 a nd Vr l wh ich o c c urs whe n l rbull see f ig 3b s p

Ap p l yi ng t he tong- t i me approx imate method developed on the bas i s of t he sad d le-poin t method by Eberl y and h is coworker s f 6 1 1 ( ) 111 re _ ~ t o Clt1 eu a te t h e 1nt egral 13 we can get 1n the spec i a l c a se ~ 1 a ~ 2 g o the foll owi ng ev aluat i on

2 2 2 2F (t g ) = 1 - r (I j ) exp l -- if (t g ) cos cent (t g ) ( 21 )

wher e t h e no tat i on

9

2 deg1 + deg2f(t g ) [1 + ( g~t )2]-14--gshy

4W 0 2

2 - - g ot 0 1+ deg 242-1v(t g )= 2(n + n2)sin~ -- [1 +(1 - -3- g ot ) J (22)

4Wo 4Wo 2 2

2 - - got gotep(t g ) = Wot + (O t + n2 )sm-- - --- (n + D ) shy

1 22Wo 2Wo 1 deg 1 + deg 2

- - arc tan [ 3 g4 ot 1

2 4W o

has been introduced and

2 - - 2 Wo g (n + deg ) + g (2 3)o 1 2

The approximate expression (21 ) is ra t he r good f or all time s 26~ It ind i cate s t he periodicity of the revi val s The interval TR be t ween t he rev ivals of the oscil l a tion e nve l ope of F(t g~ can be found fr om t he second eq in (22 ) to be

2 2 2 TR (g ) = (417 g 0 ) Wo(g ) (24 )

4 CROSS CORRELATION

More i n t eresting is the cr os s correlation be t ween the t wo modes Its ma gni tude can be character ized by

~ + ~+ - - ~+ - - + ~ Vcro6 s (t) = lt a 1 (t)a2 (l ) 8 2 (t)8 1(traquo - lt 8 1(t)8 1 (traquo lta 2 (t) a (traquo = 2

(25) = I P (n n ) n n - I l P (n n ) 0 I I I P (n D ) n

t 1 2 1 2 t 1 2 1 1 2 2 02n 111 2 deg 1deg 2 1

The qua nti t y i s propor tional to the excess coinc i denceVero86 counting ra t e t o be measured in ao exper i ment of t he Hanbury Brown-Twiss-type with two d if f erent light beams 22271 vIe speak o ~ an ticorre l at ion whe~ VC~06S becomes negat i ve Aoticomiddot~Telashyt l on has been observed 10 llght scatterlng f rom nonspherlcal parti cl es in d i l u t e solution27~We consider now the time beshyhaviour of Vcross in the mode l (I)

Ut i lizing eq(3) and the def ini t ion (25) we can get

Veross (t) = Vcross (0) +l P ( n l D2 ) (OCi)R211 U (t)+ (o2-nJR1 (l)shy 12 12 12

(26)1 l P(U 1D 2)R 1 n (1)11 I P(D 1n )R20 (t)2I]I n 2 1 2 I]1 n 1 2

2

10

Equa t ion (26) t oge ther wi t h egs (4) allow us to describe th e t ime evolut i on of the cross correlation i n the general case of the i nit ia l field state (5 ) Le t us assume t ha t the i ni shytia l phot on dis t r i bution P~I n2 ) i s given by (9) This means t hat the s i gna l and pump f ield s are coher ent and sta t i s t ica l shyly independ en t at the begi nning of the i nteraction Then due to t he pr oper ties ( 10) we f ind f r om (26) t he r esult

V (t) = n I R (t ) - R (t) J + n rR t) shycross J 2 1 + 12 2 1 2 2 1 1 _ +1

(27) - R (l)l - R (t (t ) R 2 n 1 n 21deg 1 2 In 1 0 2

Her e ( ) i nd i cat es a s i n sec tion 3 averagi n8 over the di stribution (9)

02

-II 0 lII o 0-02

gt

-04

~_ L~ 2 4 6 8 10 12 14 16 18

go t - shyFig 5 Time evo lu~ion of the cross-correlati on nction Vcro s s (l) Ro-th t hr f ields of the modes ar i ni t ially in t he Glmtber coherent states wi th 0 1 =1 ~ u2 = 3 ~g t=~2 =2g TimR is measured i n units of tgoo

A numerica l computation of t he formul a (2 7 ) has been per shy ormed for the CllSC (o r ~xample when gl =2g 2 = 2g o nl =1 2

= 3 The resulLA nT~ p lot t ed in f i g 5 We see from then2 f i gure t hat ~nticnrrclaLion occurs f or some fin ite i ntervals o f t imes in the R~b i oRc il lat ion reg i on and du r ing t he st eashydy r e gi me The RIICCf SS ivl presence s of corre l ation and anti shycorrelat i on in 1I1l Rn lgt i oscill a tion region are noted For large pho t on numb~rs ii 1 bull ii~1 hy using t he approx imate f orshymulas

11

2 0- _

glg~nl(n2+1) 2 R D (t) = ---------- I 4F (tmiddot g ) - F(2tmiddot l) I

n l n2 2- 2- 2 2 2 2 [ g 1n 1 + g 2( n 2 + 1)]

2 2 _ glg2 (ill +1)(02+1) 2 2 2 2 R (t) - --------------14F(tg +g )-F(2tg +g )1

2n l + 1 n 2 2[ g~(nl +1) + g~(02 + 1)1 2 1 2 1 2

2 2- (Ii 2 4-2 2 (28) _ 4g 1g2n n2 +1)F(t g ) + glll l F (2t g2 ) (t) - 1 - l ____________R ______________z I n n 2 - 212 2[g2n +g (n + 1)]

1 1 2 2 2 2 - - 2 4- 2

(t) 1 __~~_L~~n 2 + l~_~_~~~1~~_=~)__ l~n +1 2shy2 2rg n + g2 (Ii + 2) ] 2

1 1 2 2

obtained from (4) we can appl y the same analytical description he have used in the previous section to (27) The collapse and revival of Vc ross are easily described by utilizing the a pshyprox imate expressions (18) and (21) Due to that the function t(t g2) approache s unity in the collapse region see (18) and in the region of infin itel y large time s see (21) the steady value of Vcoss is quickly found to be

2- 2 - 2 - 2 - o - 1 2- - gl n l +4g 2(n 2 +1) gl 1l 1 +4g 2(n 2 +2)

Vc ro s s - 2- g 1 11 2r~~~1--~-~fn-2-)]2- - [g ~lt-~~(~~--)j-2 +

r i1 1__+ __1 ____ ___ __ n1 1 + l- galgi11(n2 + 1) ~ -g2-(~---1-) g2 (li + 1)] 2 - -2----2--=------2 ~ shy

2 L 1 1 2 2 (gl 11 1 +g2(n t 1)1 J 2

4- 2+ 2g 4( 0 + 1)~- gin 1 __~_~_____

3 2g 2n (n + 1) ---- 2 _ 4 (29)--gl 2 1 2 (21i +g(n + 1)

4 111 1 2 2

2- 2shyHence for the case gl 11 1 raquog2n2one ge t s

0 13g2-11 22V - -------- lt 0 (30)Gross 2shy

4g 1 n I

The inequality in eq (30 ) indicates the anticorrelation between the signal and pump mode s i n the steady regime This analytical prediction is in compl i ance wi th t he numerical result obtained above see fig S where g 2n g 2n =16) In the opposite limit

2- 2 - 1 1 2 2 case g n laquoI Il one has 1 1 2 2

2shy

VO gl 111 2--- gt 0cross 2- (31)

g2 11 2

that indicates the correlation bezween the modes

12

Finally I nr very short times t 0 we find f rom eqs (27) and (4 ) t hl rn r11lu I I

V (I) _1_ il it t 4 cross 12 J g 2 1 2 32)

which descr i bes the a ppea rance of the correlation bezwe en the mod es due t o t he i nterac t ion with the emi t t er see fig5

5 CONCLU SIONS

In this paper we hav e studied the charact eris tic s of the photon s t at i stica l proper ties in a so luble t hree- leve l pl us two-mode l The normally ordered v ariances o f t he photon numshybers and the cros s- correla tion fun ction hav e been ca lcul a t ed It has heen s110wn t ha t the s t ates of the signa l mode f i eld produced from it s ini ti al vacuum are sub-Po i sson ian for all t ime s Thi s p r oduc t i on of sub-Poissonian states does not deshypend on t he ini tia l sta t e of t he pump mode field

The quantum co l l a pse and r eviv a l-behaviour of the pho t on sta t i s t ica l charac t eri stic s i n t he l oss less model cons idered here have been no t ed

For t he case vhen both t he sina l -mode and pump-mode f ie lds are i nitial ly in t he Glauber coherent stat es t he t i me behashyviour of t he pho t on- number v ar i ances and of the CrOSS correshyla tion between the mod es has been numerically examined and analy t icall y approx i mat ed The t heor e t ica l investigat i ons show tha t under suitabl e condit ions the occurrence of photon an t i shybunch ing an d t he presence of anticorrelat i on between t he mod es are potent ially possible in t he system It has been estabshylished t hat

(i ) In t h e region of short t imes (t he Rabi oscillation region ) pho t on statis tics of t he s ignal-mode fi e ld i s subshyPoissonian for some f i ni te i n terval s

(ii) Photon statistics of t he signal-mode f ield in the steady regime will be sub-Poissonisn if either g~02 raquo g~D 1

(ill 2raquo 1) or 02 ~ O(ii 1 raquo 1) (ii i) Due to the interaction wi th the emitter the correshy

lations hetwCcn the lTIodes come into existence The anticorshyrelation between thl ffimlls wi II occlIr ill llH steady regime if

2- - - shyg1Dl raquoJiII (11 II I )

All the ns1Il Igt I I- v 1t~ II ]gt1 01 i Itd wi r IInut 1JIj np perturshybation theory Ind tlll (llr I Ii(l ll l ppIOXilllllli(lr1

13

REFERENCES

I Kimble HJ Dagenais M Mandel L PhysRevLett 1977 39 p691

2Dagenais M Mandel L PhysRev 1978 A18 p2217 3 Short R Mandel L PhysRevLett 1983 51 p384 4 Stoler D PhysRevLett 1974 33 p1397 5 Drummond P D UcNeil K J Halls D F Opt Corom

1979 28 p 255 6 Mostowski J Rzazewski K PhysLett 1978 A66 p275 7 Halls DF Tindle CT JPhys 197 2 A5 p543 8 Paul H Brunner ~ AnnPhysLeipz 198138 p89 9 Carmichael HJ Talls DF J Phys 1976 B9 pL43

ibid 1976 B9 p1199 10 Kimble HJ Handel L PhysRev 1976 A13 p2123 II Jakeman E Pike ER Vaughan JH JPhys 1977 AIO

p L257 12 Carmi chael HJ et al J Phys 1978 All pLI21 13 Agarwal G S et al PhysRev 1977 A15 p 1613 14 Hiegand M JPhys 1983 B16 p1133 15 Ficek Z Tanas R Kielich S PhysRev 1984 A29

p2004 16 Yuen HP Slapiro JH OptLett 19794 p334 17 Simaan HD Loudon R JPhys 1975 A8 p539 18 Tornau N Bach A OptComm 1974 I I p46 19 Becker H Zubairy MS PhysRev 1982 A25 p2200 20 Lee CT PhysRev 1985 A3 1 p 1213 21 Halls DF Nature 1979 280 p451 22 Paul H Rev ModPhys 1982 54 p1061 23 Bogolubov NN (Jr) Fam Le Kien Shumovsky AS

PhysLett 1984 lOlA p 201 PhysLett 1985 107A p456

24 Hanbury Brown R Twiss RQ Nature Lond 1956 177 p27

25 Glauber R J Optical Coherence and Photon Statistics In CDeWitt ABlandin and CCohen-Tannoudji (Eds) Quantum Optics and Electronics Gordon and Breach New York 1965

26 Eberly JH Naro zhny NB Sanche z-Mondragon J J PhysRev 1981 A23 p 236

27 Griffin HG Pusey P N PhysRevLett 1979 43 p 1100 28 Hillerv M PhysRev 1985 A31 p 338 29 Hillery M Zubairy MS Hodkiewic z K PhysLett

1984 103A p 259 30 Meystre P Zubairy MS Phys Lett 1982 89A p390

Received by Publishing Department on Hay 29 1985

14

hpl 1oIN I I ( raquol lI E17-85-402 011111 1(

lHII~ III I I hill I I II ltT II I Ihll T O IIDB

a I)l~XY lh1JtIJ I H I 111 y lUIIJJIJ 11 11 I I C Jl C1 e Ml

1 V Ill IJ1 X J PI I I pile T ill-gt II ( 10 T IIC1lilJ e CKHX CB OHC T II IPoTOHOB

11 Tll I 111 I JI III II Moi1 Tl gtY IH) III1 P II O H ~lI YXMOAOBOH MOAeJIH BbNHcneshy

II hl illll~I I JI I II Y IIN P JJI)I I I IJIJIC OapHltllHH lJHCen IPOrOHOB H CPYHKnHR

h 1111( I -- 1114 JlltIllJl P IYJlh JIJThI nonyq eHbl 6e3 HCnOn1gt30 aaHHfI

Iilllllill 11 I~t Y IIW IIII II I p a c Le Ilne ll lUI KoppenfiTopoB Y C TaHoBneHO

I TO I ll r l IlMI II I II II P JUI II CI U Ullllbllot1 MOAe reHepHpye~lOro H3 Bashy

KYYMI I f1 1 II1IJ IHlO T Cn cy()-nyac c o HoBbIMH AaHo onHcaHHe Bpeshy

Nl lI l1l l l 11 (1 III IUIII CI d TI IC rHlleCKHX x apaKTepHcTHK IPOTOH OB I)JUI

f1yI II 111 111 KO ll P L~ II T JlblX MOA nOKa3aHo ITO npH onpeshy

[llHtIIlIlIJ( VtJI Il II IIII X l lNCIOT MecTo a HTlu pynnHpoBKa ltJOTOHOB

H II I III(II1) J IIIIIIIII ~l e lliAY HOAaMH B CHCTeMe

PlIlillTfl 1II I Il IlI It lIa Il nafiopa T opHH TeOpeTHIeCKOH ltJH3HKH OHJIH

TIpel1Jlll1t 1 1Ill~lhH ItO IO HHCTHTYT8 RAepHbIX HccnenOB8HHA bull llyl5Ha 1985

Bogolu b(w N N bull bull Jr Fam Le Kien EI7-85-402 Shumovsky A5

S tat is t i ~ a l Properties of Photons i n a Three-Leve l Plus Two-Hode Model

The characteristics of the photon statistical propershyt i es a re studied in a soluble three-level plus two-mode model The model consists of a single lambda-configuration t hree- level emitter interacting with two modes of the quanshyt i zed radiation field the emitter being initially in t he l owest level The normally ordered variances of the photon numbers and the cross-correlation function are cal culated Al l the results are obtained withou t using per t urbation theory and decorrelation approximations It is s hown t hat the states of the signal-mode f ield produced f r om i t s ini shytia l vacuum ar e sub-Poissoni an for all times For the ca se when both the s igna l -mode and pump-mod ( fi elds ar e i ni tially i n t he Gl auber cohe r ent s t a t CR the t empor nl behnv i our of t b l phnlnn Sl n t i at i la l ( b lTn(lImiddotriR Ii I~ 11r J(~(l i bedUnder uitlIhl rlllldil illnfl pllOlplI 1IIIIi lllllll 1I1 111 (lml a nt i correl ation

tn l rOI~tiuUy Jlnlhi hle i n til YS lell

iflll 1111 hen II THI lnrllltcI lit t il t LJhor a tory If IIIt tH I 1 jmiddot11 1 lhvtl iiI IIIlK

1985

2 0- _

glg~nl(n2+1) 2 R D (t) = ---------- I 4F (tmiddot g ) - F(2tmiddot l) I

n l n2 2- 2- 2 2 2 2 [ g 1n 1 + g 2( n 2 + 1)]

2 2 _ glg2 (ill +1)(02+1) 2 2 2 2 R (t) - --------------14F(tg +g )-F(2tg +g )1

2n l + 1 n 2 2[ g~(nl +1) + g~(02 + 1)1 2 1 2 1 2

2 2- (Ii 2 4-2 2 (28) _ 4g 1g2n n2 +1)F(t g ) + glll l F (2t g2 ) (t) - 1 - l ____________R ______________z I n n 2 - 212 2[g2n +g (n + 1)]

1 1 2 2 2 2 - - 2 4- 2

(t) 1 __~~_L~~n 2 + l~_~_~~~1~~_=~)__ l~n +1 2shy2 2rg n + g2 (Ii + 2) ] 2

1 1 2 2

obtained from (4) we can appl y the same analytical description he have used in the previous section to (27) The collapse and revival of Vc ross are easily described by utilizing the a pshyprox imate expressions (18) and (21) Due to that the function t(t g2) approache s unity in the collapse region see (18) and in the region of infin itel y large time s see (21) the steady value of Vcoss is quickly found to be

2- 2 - 2 - 2 - o - 1 2- - gl n l +4g 2(n 2 +1) gl 1l 1 +4g 2(n 2 +2)

Vc ro s s - 2- g 1 11 2r~~~1--~-~fn-2-)]2- - [g ~lt-~~(~~--)j-2 +

r i1 1__+ __1 ____ ___ __ n1 1 + l- galgi11(n2 + 1) ~ -g2-(~---1-) g2 (li + 1)] 2 - -2----2--=------2 ~ shy

2 L 1 1 2 2 (gl 11 1 +g2(n t 1)1 J 2

4- 2+ 2g 4( 0 + 1)~- gin 1 __~_~_____

3 2g 2n (n + 1) ---- 2 _ 4 (29)--gl 2 1 2 (21i +g(n + 1)

4 111 1 2 2

2- 2shyHence for the case gl 11 1 raquog2n2one ge t s

0 13g2-11 22V - -------- lt 0 (30)Gross 2shy

4g 1 n I

The inequality in eq (30 ) indicates the anticorrelation between the signal and pump mode s i n the steady regime This analytical prediction is in compl i ance wi th t he numerical result obtained above see fig S where g 2n g 2n =16) In the opposite limit

2- 2 - 1 1 2 2 case g n laquoI Il one has 1 1 2 2

2shy

VO gl 111 2--- gt 0cross 2- (31)

g2 11 2

that indicates the correlation bezween the modes

12

Finally I nr very short times t 0 we find f rom eqs (27) and (4 ) t hl rn r11lu I I

V (I) _1_ il it t 4 cross 12 J g 2 1 2 32)

which descr i bes the a ppea rance of the correlation bezwe en the mod es due t o t he i nterac t ion with the emi t t er see fig5

5 CONCLU SIONS

In this paper we hav e studied the charact eris tic s of the photon s t at i stica l proper ties in a so luble t hree- leve l pl us two-mode l The normally ordered v ariances o f t he photon numshybers and the cros s- correla tion fun ction hav e been ca lcul a t ed It has heen s110wn t ha t the s t ates of the signa l mode f i eld produced from it s ini ti al vacuum are sub-Po i sson ian for all t ime s Thi s p r oduc t i on of sub-Poissonian states does not deshypend on t he ini tia l sta t e of t he pump mode field

The quantum co l l a pse and r eviv a l-behaviour of the pho t on sta t i s t ica l charac t eri stic s i n t he l oss less model cons idered here have been no t ed

For t he case vhen both t he sina l -mode and pump-mode f ie lds are i nitial ly in t he Glauber coherent stat es t he t i me behashyviour of t he pho t on- number v ar i ances and of the CrOSS correshyla tion between the mod es has been numerically examined and analy t icall y approx i mat ed The t heor e t ica l investigat i ons show tha t under suitabl e condit ions the occurrence of photon an t i shybunch ing an d t he presence of anticorrelat i on between t he mod es are potent ially possible in t he system It has been estabshylished t hat

(i ) In t h e region of short t imes (t he Rabi oscillation region ) pho t on statis tics of t he s ignal-mode fi e ld i s subshyPoissonian for some f i ni te i n terval s

(ii) Photon statistics of t he signal-mode f ield in the steady regime will be sub-Poissonisn if either g~02 raquo g~D 1

(ill 2raquo 1) or 02 ~ O(ii 1 raquo 1) (ii i) Due to the interaction wi th the emitter the correshy

lations hetwCcn the lTIodes come into existence The anticorshyrelation between thl ffimlls wi II occlIr ill llH steady regime if

2- - - shyg1Dl raquoJiII (11 II I )

All the ns1Il Igt I I- v 1t~ II ]gt1 01 i Itd wi r IInut 1JIj np perturshybation theory Ind tlll (llr I Ii(l ll l ppIOXilllllli(lr1

13

REFERENCES

I Kimble HJ Dagenais M Mandel L PhysRevLett 1977 39 p691

2Dagenais M Mandel L PhysRev 1978 A18 p2217 3 Short R Mandel L PhysRevLett 1983 51 p384 4 Stoler D PhysRevLett 1974 33 p1397 5 Drummond P D UcNeil K J Halls D F Opt Corom

1979 28 p 255 6 Mostowski J Rzazewski K PhysLett 1978 A66 p275 7 Halls DF Tindle CT JPhys 197 2 A5 p543 8 Paul H Brunner ~ AnnPhysLeipz 198138 p89 9 Carmichael HJ Talls DF J Phys 1976 B9 pL43

ibid 1976 B9 p1199 10 Kimble HJ Handel L PhysRev 1976 A13 p2123 II Jakeman E Pike ER Vaughan JH JPhys 1977 AIO

p L257 12 Carmi chael HJ et al J Phys 1978 All pLI21 13 Agarwal G S et al PhysRev 1977 A15 p 1613 14 Hiegand M JPhys 1983 B16 p1133 15 Ficek Z Tanas R Kielich S PhysRev 1984 A29

p2004 16 Yuen HP Slapiro JH OptLett 19794 p334 17 Simaan HD Loudon R JPhys 1975 A8 p539 18 Tornau N Bach A OptComm 1974 I I p46 19 Becker H Zubairy MS PhysRev 1982 A25 p2200 20 Lee CT PhysRev 1985 A3 1 p 1213 21 Halls DF Nature 1979 280 p451 22 Paul H Rev ModPhys 1982 54 p1061 23 Bogolubov NN (Jr) Fam Le Kien Shumovsky AS

PhysLett 1984 lOlA p 201 PhysLett 1985 107A p456

24 Hanbury Brown R Twiss RQ Nature Lond 1956 177 p27

25 Glauber R J Optical Coherence and Photon Statistics In CDeWitt ABlandin and CCohen-Tannoudji (Eds) Quantum Optics and Electronics Gordon and Breach New York 1965

26 Eberly JH Naro zhny NB Sanche z-Mondragon J J PhysRev 1981 A23 p 236

27 Griffin HG Pusey P N PhysRevLett 1979 43 p 1100 28 Hillerv M PhysRev 1985 A31 p 338 29 Hillery M Zubairy MS Hodkiewic z K PhysLett

1984 103A p 259 30 Meystre P Zubairy MS Phys Lett 1982 89A p390

Received by Publishing Department on Hay 29 1985

14

hpl 1oIN I I ( raquol lI E17-85-402 011111 1(

lHII~ III I I hill I I II ltT II I Ihll T O IIDB

a I)l~XY lh1JtIJ I H I 111 y lUIIJJIJ 11 11 I I C Jl C1 e Ml

1 V Ill IJ1 X J PI I I pile T ill-gt II ( 10 T IIC1lilJ e CKHX CB OHC T II IPoTOHOB

11 Tll I 111 I JI III II Moi1 Tl gtY IH) III1 P II O H ~lI YXMOAOBOH MOAeJIH BbNHcneshy

II hl illll~I I JI I II Y IIN P JJI)I I I IJIJIC OapHltllHH lJHCen IPOrOHOB H CPYHKnHR

h 1111( I -- 1114 JlltIllJl P IYJlh JIJThI nonyq eHbl 6e3 HCnOn1gt30 aaHHfI

Iilllllill 11 I~t Y IIW IIII II I p a c Le Ilne ll lUI KoppenfiTopoB Y C TaHoBneHO

I TO I ll r l IlMI II I II II P JUI II CI U Ullllbllot1 MOAe reHepHpye~lOro H3 Bashy

KYYMI I f1 1 II1IJ IHlO T Cn cy()-nyac c o HoBbIMH AaHo onHcaHHe Bpeshy

Nl lI l1l l l 11 (1 III IUIII CI d TI IC rHlleCKHX x apaKTepHcTHK IPOTOH OB I)JUI

f1yI II 111 111 KO ll P L~ II T JlblX MOA nOKa3aHo ITO npH onpeshy

[llHtIIlIlIJ( VtJI Il II IIII X l lNCIOT MecTo a HTlu pynnHpoBKa ltJOTOHOB

H II I III(II1) J IIIIIIIII ~l e lliAY HOAaMH B CHCTeMe

PlIlillTfl 1II I Il IlI It lIa Il nafiopa T opHH TeOpeTHIeCKOH ltJH3HKH OHJIH

TIpel1Jlll1t 1 1Ill~lhH ItO IO HHCTHTYT8 RAepHbIX HccnenOB8HHA bull llyl5Ha 1985

Bogolu b(w N N bull bull Jr Fam Le Kien EI7-85-402 Shumovsky A5

S tat is t i ~ a l Properties of Photons i n a Three-Leve l Plus Two-Hode Model

The characteristics of the photon statistical propershyt i es a re studied in a soluble three-level plus two-mode model The model consists of a single lambda-configuration t hree- level emitter interacting with two modes of the quanshyt i zed radiation field the emitter being initially in t he l owest level The normally ordered variances of the photon numbers and the cross-correlation function are cal culated Al l the results are obtained withou t using per t urbation theory and decorrelation approximations It is s hown t hat the states of the signal-mode f ield produced f r om i t s ini shytia l vacuum ar e sub-Poissoni an for all times For the ca se when both the s igna l -mode and pump-mod ( fi elds ar e i ni tially i n t he Gl auber cohe r ent s t a t CR the t empor nl behnv i our of t b l phnlnn Sl n t i at i la l ( b lTn(lImiddotriR Ii I~ 11r J(~(l i bedUnder uitlIhl rlllldil illnfl pllOlplI 1IIIIi lllllll 1I1 111 (lml a nt i correl ation

tn l rOI~tiuUy Jlnlhi hle i n til YS lell

iflll 1111 hen II THI lnrllltcI lit t il t LJhor a tory If IIIt tH I 1 jmiddot11 1 lhvtl iiI IIIlK

1985

REFERENCES

I Kimble HJ Dagenais M Mandel L PhysRevLett 1977 39 p691

2Dagenais M Mandel L PhysRev 1978 A18 p2217 3 Short R Mandel L PhysRevLett 1983 51 p384 4 Stoler D PhysRevLett 1974 33 p1397 5 Drummond P D UcNeil K J Halls D F Opt Corom

1979 28 p 255 6 Mostowski J Rzazewski K PhysLett 1978 A66 p275 7 Halls DF Tindle CT JPhys 197 2 A5 p543 8 Paul H Brunner ~ AnnPhysLeipz 198138 p89 9 Carmichael HJ Talls DF J Phys 1976 B9 pL43

ibid 1976 B9 p1199 10 Kimble HJ Handel L PhysRev 1976 A13 p2123 II Jakeman E Pike ER Vaughan JH JPhys 1977 AIO

p L257 12 Carmi chael HJ et al J Phys 1978 All pLI21 13 Agarwal G S et al PhysRev 1977 A15 p 1613 14 Hiegand M JPhys 1983 B16 p1133 15 Ficek Z Tanas R Kielich S PhysRev 1984 A29

p2004 16 Yuen HP Slapiro JH OptLett 19794 p334 17 Simaan HD Loudon R JPhys 1975 A8 p539 18 Tornau N Bach A OptComm 1974 I I p46 19 Becker H Zubairy MS PhysRev 1982 A25 p2200 20 Lee CT PhysRev 1985 A3 1 p 1213 21 Halls DF Nature 1979 280 p451 22 Paul H Rev ModPhys 1982 54 p1061 23 Bogolubov NN (Jr) Fam Le Kien Shumovsky AS

PhysLett 1984 lOlA p 201 PhysLett 1985 107A p456

24 Hanbury Brown R Twiss RQ Nature Lond 1956 177 p27

25 Glauber R J Optical Coherence and Photon Statistics In CDeWitt ABlandin and CCohen-Tannoudji (Eds) Quantum Optics and Electronics Gordon and Breach New York 1965

26 Eberly JH Naro zhny NB Sanche z-Mondragon J J PhysRev 1981 A23 p 236

27 Griffin HG Pusey P N PhysRevLett 1979 43 p 1100 28 Hillerv M PhysRev 1985 A31 p 338 29 Hillery M Zubairy MS Hodkiewic z K PhysLett

1984 103A p 259 30 Meystre P Zubairy MS Phys Lett 1982 89A p390

Received by Publishing Department on Hay 29 1985

14

hpl 1oIN I I ( raquol lI E17-85-402 011111 1(

lHII~ III I I hill I I II ltT II I Ihll T O IIDB

a I)l~XY lh1JtIJ I H I 111 y lUIIJJIJ 11 11 I I C Jl C1 e Ml

1 V Ill IJ1 X J PI I I pile T ill-gt II ( 10 T IIC1lilJ e CKHX CB OHC T II IPoTOHOB

11 Tll I 111 I JI III II Moi1 Tl gtY IH) III1 P II O H ~lI YXMOAOBOH MOAeJIH BbNHcneshy

II hl illll~I I JI I II Y IIN P JJI)I I I IJIJIC OapHltllHH lJHCen IPOrOHOB H CPYHKnHR

h 1111( I -- 1114 JlltIllJl P IYJlh JIJThI nonyq eHbl 6e3 HCnOn1gt30 aaHHfI

Iilllllill 11 I~t Y IIW IIII II I p a c Le Ilne ll lUI KoppenfiTopoB Y C TaHoBneHO

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S tat is t i ~ a l Properties of Photons i n a Three-Leve l Plus Two-Hode Model

The characteristics of the photon statistical propershyt i es a re studied in a soluble three-level plus two-mode model The model consists of a single lambda-configuration t hree- level emitter interacting with two modes of the quanshyt i zed radiation field the emitter being initially in t he l owest level The normally ordered variances of the photon numbers and the cross-correlation function are cal culated Al l the results are obtained withou t using per t urbation theory and decorrelation approximations It is s hown t hat the states of the signal-mode f ield produced f r om i t s ini shytia l vacuum ar e sub-Poissoni an for all times For the ca se when both the s igna l -mode and pump-mod ( fi elds ar e i ni tially i n t he Gl auber cohe r ent s t a t CR the t empor nl behnv i our of t b l phnlnn Sl n t i at i la l ( b lTn(lImiddotriR Ii I~ 11r J(~(l i bedUnder uitlIhl rlllldil illnfl pllOlplI 1IIIIi lllllll 1I1 111 (lml a nt i correl ation

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1985