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Squeezing in MultiphotonAbsorptionPaulina Marian aa Department of Physics , Polytechnical Institute ofBucharest , Splaiul Independenţei 313, Bucharest, 77206,RomaniaPublished online: 01 Mar 2007.

To cite this article: Paulina Marian (1990) Squeezing in Multiphoton Absorption, Journal ofModern Optics, 37:3, 285-293, DOI: 10.1080/09500349014550351

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JOURNAL OF MODERN OPTICS, 1990, VOL . 37, NO. 3, 285-293

Squeezing in multiphoton absorption

PAULINA MARIAN

Department of Physics, Polytechnical Institute of Bucharest,Splaiul Independenlei 313, Bucharest 77206, Romania

(Received 11 April 1989; revision received 27 July 1989and accepted 26 August 1989)

Abstract . The conditions for obtaining squeezing by single-beam multiphotonabsorption are examined . For an initial coherent light the amount of squeezing iscalculated in the second order of interaction time using the master-equationformalism. A limit formula for the amount of squeezing is found in the case of aninitial strong coherent beam . The results are compared with the exact numericalcalculations and some remarks are made on the short-time approximation inmultiphoton absorption .

1. IntroductionThe multiphoton absorption as a nonlinear phenomenon was intensively studied

in the last two decades . For the two-photon case the general solutions of the masterequation were derived and the antibunching effect was investigated [1] . This purequantum mechanical effect was predicted in two-photon absorption by McNeil andWalls [2] . A similar prediction was made for k-photon absorption [3] and asymptoticformulae for strong initial fields were given [4] .

A significant piece of progress in the study of multiphoton absorption was thederiving of the exact solution of the master equation [5] . The statistical properties,which are determined by the diagonal matrix elements of the density operator werelargely investigated and for initial chaotic and coherent fields the antibunchingeffect was calculated .

The generation of squeezed light by two-photon absorption was analysed byLoudon who considered the model of an initial coherent beam passing once throughthe absorber [6] . The amount of squeezing found in this process in the short-timeapproximation is comparable to that calculated for the steady-state two-photonabsorption in a cavity driven by coherent light [7] .

Squeezing was considered in the same case of an initial coherent beam formultiphoton absorption (k = 2, 3, 4) in [8] and [9], using the exact solutions of themaster equation [5] . These solutions are complicated mathematical quantities andthe evaluation of the squeezing was performed numerically. Examination of theseresults [8, figure 1] reveals the important place that two-photon absorption hasamong the multiphoton processes. The squeezing in two-photon absorption of acoherent beam is very persistent and exhibits a slow time variation . These remarksare also confirmed by the numerical calculation of the squeezing in two-photonabsorption performed in [10] and [11] . These works clearly show that the short-timeapproximation [6] gives much less squeezing than the numerical calculations, thetimes at which maximum squeezing occurs being also much shorter than thenumerical results .

The present work deals with the investigation of squeezing by multiphotonabsorption in the model proposed by Loudon [6] for the two-photon case . We use the

0950-0340/90 $3 .00 Q 1990 Taylor & Francis Ltd .

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master equation for the reduced density operator of the unabsorbed field and work inthe short-time approximation. The squeezing parameters calculated in this short-time approximation are compared with the exact numerical results obtained in [9] forthree- and four-photon cases .

In section 2 the short-time approximate solutions of the master equation arederived and the expectation values which are necessary for the evaluation ofsqueezing are established .

The variances in the two quadratures of the unabsorbed field are calculated anddiscussed in section 3 . For an initial strong coherent field the maximum of squeezingis found . A comparison is then made between the numerical results concerning thesqueezing parameters [9] and the results in the short-time approximation. Someconclusions on the utility of this short-time approximation in multiphotonabsorption are given .

In the Appendix are presented some details concerning the calculation of theexpectation values in section 2 .

2 . The master equation and its short-time approximate solutionsWe consider the system of a single-mode radiation field coupled with Ntwo-level

non-interacting atoms . The frequency of the field allows k-photon absorption, allother transitions being negligible . We assume the N atoms in the lower state so thatthe k-photon emission is ignored . The master equation for the reduced densityoperator of the field under the Markoff approximation is [12, 13] :

a t=akpa+k _ z a+kakp - 2 Pa

+ka k

(1)at

where

r=NJt.

(2)

J is the coupling constant of the two levels depending on the matrix element forthe k-photon transition . In the basis of the Fock states the matrix elements of thedensity operator satisfy the equation

a <mIPIm+µ>[(m+(m+k)!(m+k)!] 2

<m+kIPIm+k+ z>

2C(m-k),+ (m+Pµ)k)~J<mIPIm+µ~

(3)

The expectation values involved in the calculation of squeezing are

00<a+ a> _ Y m <mI PI m),

(4)m=0

00<a +">= Y <mIpa+ "lm>, µ=1,2 .

(5)

Using the notation [1]

we obtain in equations (3)-(5)

m=0

C(m

+µ)1

J1/2<mIPIm+µ> = Y'm(U),

(6)m!

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Squeezing in multiphoton absorption

287

a

(m+k)! ' ~~

1

m!

(m+µ)!atm(µ)= m~ 'Ym+k(µ)-2 (m-k)!+(m+µ-k)!](7)

00(a+a>= E m).(0),

m=0M

<a + -u>= m Vm(µ) •

(9)M=0

The properties of the field after passage through the absorber can be found using thesolutions of equation (7) .

Assuming short interaction time, the condition t << 1 holds and the solution of (7)may be written as

Y' m(µ) 40)(,U) + ta0m

+ t2 (a20m\

(10)at ),=O 2 at //Jt = °

4(°"(µ) is the quantity (6) evaluated before interaction . From the master equationm(7) we obtain

(at0 = (mm k)! 2f(m_ k)!+(m+,uµ)k)! ~~fimo~(µ)

(11)

(a2

(m+2k)!~)~~0~ate `Ym(µ)

= m'

'Ym+2k(µ)s=0

- 1 C(m+k)!+(m+µ+ k)! + m! + (m+µ)!2

m!

(m+µ)!

(m-k)! (m+µ-k)!

X (m+k)!0m°+k(µ)-1-1

m! + (m+µ)! 2 a(0)(µ) •

(12)m!

41(m-k)! (m+µ-k)!)

At this point we can remark that the off-diagonal matrix elements of the densityoperator at the moment t are vanishing if the density operator of the initial field isdiagonal in the Fock space . So

0(M0)(µ)=01//m(µ)=0, µ00 .

(13)

Squeezing is determined by the off-diagonal matrix elements of the densityoperator in the Fock space . If the initial field is either chaotic,

(0)=> <n>nPch

r

n+1 In><nIR=01(1 +<n>)

J,

or in a pure number state, p;,° ) =ln><nl, it does not exhibit squeezing after passagethrough a multiphoton absorber . This applies to all orders of interaction time, owingto the structure of the exact solutions of (7) [5] . On the other hand the discussion ofantibunching involves only the diagonal matrix elements of the density operator .Therefore multiphoton absorption yields antibunching in the case of the above-mentioned initial fields . For a review see [14] .

In the following we assume an initial coherent state of the field with meanphoton-number Joel'

(8)

Pc°b = 1'001, (14a)

omol(µ)=m1 lal2m+ "`exp[ - Ial 2 -iµarg(a)] . (14b)

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Owing to the relation

(m+k)! ,~,L -k(µ)={a{2k010%U) .m!

Y'

equation ( 10) becomes

0,„(µ)={1+rC{al2k-iCm! + (m+ µ)! ~~

(

2 (m-k)i (m+µ-k)!

i 2

4k 1 2k (m+k)! (m+µ+k)!

M!

+ 2 IaI

2101

m! + (m+µ)! + (m-k)!

(m+µ)!

41

!

(m+µ)! )2

+ (m+µ-k)!1 +-((mm-!k)

+ (m + µ - k)! J~m°~(µ)

(16 )

After straightforward but lengthy calculations (see the Appendix for details) theexpectation values (8) and (9) are obtained

<a +2) exp [2i arg (a)]={a{2-2 lal2k-2k(k-1 +2{ale)

+41-k{alak-2(k-1+2IaI2)+k!I2{2k

CCk(«I21

IaI2

x 1F1 (-k, 1 ; - Ial 2)+2k(k+1)1F1( - k,2; -IaI 2 )

+(k+1)(k+2)i«I 2 1F1( - k, 3 ; -Ial2)]} •

(19)

1F1 is the confluent hypergeometric function [15]

1F1(fl>Y;z)=1+

$(~+1} . . .(~+n-1)z"

(20)n=1 y(y+1) . . . (y+n-1) n!

The expectation values (17)-(19) are polynomials with respect to {al due to the wellknown relation [15] :

(p + k)!LP(z) = k! p ~ 1F1( - k,p+ 1 ; z),

where LP(z) is a Laguerre polynomial .

<a+a)=Ial2-ik{al2k+2

k{al 2k [ - Ial 2k +kl 1F1(-k,1 ;-IaI 2 )], (17)

2<a+>exp[iarg(a)]=IaI-2 lal zk-1_4 klalak-1

+ 8 k!lal 2k + 1[(k+1) 1F1(-k, 2;-Ia{ 2)-C1- Ia22)

iFi(-k,1 ; (18)

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3. Evaluation of squeezingFor a single-mode radiation field characterised by the annihilation operator a the

squeezing condition reads

Squeezing in multiphoton absorption 289

squeezing in the first quadrature means Q + <0 or squeezing in the secondquadrature means Q _ < 0 .

The calculation of the parameters Q ± is straightforward using the results (17)-(19)

Q±=

T4-kzlalak-z + Ialzk+z k! 1+ II2J1F1( -k, 1 ;_Ial 2 )

_IaI 2k+ 2 (k+1)! 1F1(-k,2;-Ialz) ± cos (2 arg (a))

xI-2k(k-1)Ialzk-z_

4k(2k-1)Ialak-2+s k1Ialzk

z- 2kk(k-1)

z x

IaI

+IaI 2

1F1(- k, 1 ; - IaI)+2(k+1)(k - 1 0( 12)

x 1F1 (-k, 2; -IaI2)+(k+

21)(k+ 2)

IaI 2 1 F1(-k, 3 ; -IaI 2 ) .

(26)

We denote by Q the most negative value of both Q + and Q_

Q

arg(a) -=nn) -- Q arg (a) -= 2n2 1 n

(27)= Q + (-~

Using a simple algebra of the functions 1F1 , [ 15], the parameter Q may be put in amore suitable form :

2

2k-2Q=-2 k(k-1)Ialzk-z+ 4k~Ialzk I(3k-1)C-(kI

-1) i +1F1(-k+1,1 ; -IaI 2 )

+ IaI2-[(2k-1)1F1(-k+1,1 ; -Iaj2)-(ik-1)1F1(-k+2,1 ; -Ialz)]

(28)

V(X1)<4 or V(X2)<4 i (21)

V(Xj)=(X;)-(Xj>2, j=1,2, (22)

X_ a+a+

X2a- a+

(23)1

2

, 2i

Using (a+ )=(a)*, (a+z)=(az)* the variances (22) are

(24 a)V(X1)=4+2[Re V(a+)+(a+a)-(a)(a+)],

V(X2)=a+i[ - Re V(a+)+(a+a)-(a)(a+)] . (24b)

If we introduce the parameters

(25)Qt = ±Re V(a +)+(a+ a)-(a+)(a),

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The minimum value of Q with respect to T is

Qmin= -k2(k- 1)2lal2k-4

({4k![(3k-1)(1F1(-k+1, 1 ; -I«I2) l

al2k-2

l

(k

+ lal2-, iF,(-k+l, 1 ; -IaI2)(2k-1)-( 32-1)iF1(-k+2, 1 ; -I«I 2))]}

For strong initial fields (IaI2 >> 1), the result (29) yields

k-1(30)

4(3k-2) -

The value of Qmin decreases with k but the amount of squeezing is small even atlarge k .

- 12 ti Qmin'1< - 16 •

The time at which maximum squeezing occurs is from (28)

Tm=[(k-2)!]-'I IF 1(-k+1, 1 ; -I«I2)[I2I2(3k-1)+(k-1)(2k-1)]

-1-(k-1)(

3k_1 11 F1( - k+2,1 ; -IaI2)- _

(k 1)!I«I2k

(32)

In the table we present the values of the parameters Qmin and Tm in the short-time calculation (ST) (29) and (32), in exact numerical calculation from [9] and theratio Qmin/Qmin(ST) for three- and four-photon absorption processes with Joel' = 10,20 and 30 .

The discrepancies between the exact numerical results and the short-timeapproximate ones seem to be higher than in the two-photon absorption case [10, 11] .The ratio Qmin/Qmin(ST) is a slowly increasing function of k while the parameters Tm

and Tm(ST) decrease when the initial mean photon-number increases .We may conclude that the short-time calculations (up to the second order in T)

give the correct sense of variation of the squeezing parameters Tm and Qmin but the

The minimum values of the squeezing parameter Q min from numerical calculations [9] and theshort-time approximation Qmin(ST) in three- and four-photon absorption processeswith Jul' = 10, 20 and 30. Same for the times at which maximum squeezing occurs .

(29)

k IaI 2Qmin

[9] Qmin(ST)

Tm

[9]Tm(ST) Qmin

Qmin(ST)

3 10 -0 .18215 -0 .06485 2.5 x 10 -3 4 . 32 x 10-4 2.8120 -0 .18840 -0 .06800 0.8 x 10 -3 1 . 13 x 10-4 2.7730 -019050 -0.06911 0.5 x 10 -5 0 . 51 x 10-4 2.75

4 10 -0 .18825 -0 .05737 1 .5 x 10 -4 1 . 912 x 10-5 3 .2820 -0 .19295 -0 .06533 0. 5 x 10 -4 0. 272 x 10-5 2 . 9530 -019360 -0.06834 2. 5 x 10 -6 0. 082 x 10-5 2. 83

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Squeezing in multiphoton absorption 291

amount of squeezing is less by approximately a factor of 2.7-3 than the numericalvalues .

However, due to their closed analytical form, the short-time parameters (29) and(32) may be used for a first rapid evaluation of the squeezing when a large number ofphotons are absorbed or in the case of very intense initial fields when the numericalefforts are much increased .

The expectation values (A 18) give also a good idea about the complexity of thefunctions involved in the exact formulae [5, equations (31) and (34)] .

Appendix The calculation of the expectation values of the field operatorsWe have to calculate the summations (8) and (9) using the matrix elements of the

density operator (16) .

(a) The mean photon-numberWe use the matrix elements (14b) and (16) for µ=0

m ~;,,° ~(0 ) = ~a~ 2 ,

(A 1)M=O

a0

m!EOM Cm°)(0)(m-k)! = lal 2k(k+ lal 2 ) .

(A 2)

In the following we repeatedly make use of the definition and some properties of theconfluent hypergeometric function 1F1 [15, chap . 13] .

1F'1(#'Y;z)=1+

A(P+1) . . .(fl+n-1)z"-

(A3)"=1 Y(Y+l) . . .(y+n-1) n!'

d"

fl(fl+1) . . .(/J+n-1)

(~

dz"1F1(fl,Y;z)°Y(Y+1) . . .(y+n-1) 1F1(P+n,y+n ;z),

(A 4)

1F1($, y ; z)=exp z 1F1(Y - fl, Y ; -z) .

(A 5)

From (A 3)-(A 5) we obtain in a straightforward way

00

Em i/4 (0) (mMIk)!=(k+l)l l 2 1F1( - k,2; - lal2) .

(A6)M=O

In the summation

_ m! 2S1m=°

mo(°)m (0)(m-k)!

(A7)

we use the substitution n = m -k, (A 3) and (A 4) to get

S1=Ia121`k![k1F1( - k,1 ; - 10(1 2)+lal2(k+1)1F1( -k,2 ; - lal 2 )] •

(A 8)

With the results (A 1) and (A 2), (A 6) and (A 8) we easily obtain the mean photon-number (17) .

(b) The expectation values <a +, >Making use again of (A 3)-(A 5) we get the results

000m°)(µ) = lal" exp [- iu arg (a)],

(A 9)M=O

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°° ~~~

m!0'Ym°~(U) (m-k)! - Ial

zk+" exp [- iµ arg (a)],

(A 10)

WmO~(µ) (m+µ)! = Ia{2k-µ k! IF1(-µ,k+1-

a 2 ex

ar aM= O

(m+µ-k)!

(k-/.l)! µ;-~ ) p[ - µ g()].

(A 11)

(E 4mo)(u) m+' k)!

=IaI"k!1F1( - k,1; -IaI 2)ex

1 ar

A 12)M=O

m.

P [- µ g (a)]

(

)

Y_Omo~(µ)(m+µ+k)! -Ialµ(k+µ)1

1F1(-k,µ+1,-IaI2)exp[-iµarg(a)] . (A13)M=O

(m+µ)!

µ!

00

m=0'Y4O)(~)C(mmlk)iJ2=laI2k+µk!1F1(-k, 1 ; -IaI 2)exp[ -1 arg(a)] • (A14)

~Gm°~(µ)

m!(m+µ)!

=IaI2k+"(k+µ)!m=o

(m-k)!(m+µ-k)!

U!

x 1F1(-k,µ+1 ;-Joel 2)exp[-iµarg(a)] . (A 15)

Y_ gl .no'(µ)[ (m+µ)!

J2=II" I p[ - µarg(a)-IaI2 ]a k ex

1M=O

(m+µ- k)!

d°X

d(I 2)" [laI2k 1F 1(k+1,1 ; IaI 2 )] .oel

This last summation may be also put in the form

~Gm°'(u)[(m+µ)!

J2=IaI2k +"k! !

m=o

(m+µ-k)!

µ

(A 16)

C

(k+µ- n)!

1

x n=o n!(k-n)![(µ-n)!]21«l2"1F1(-k,µ+1-n ; -{a{ 2 )

x exp [-ip arg (a)] .

(A 17)

In deriving (A 17) we have used (A 4)-(A 5) .Introducing these results in (9) and using (16), we get the expectation values

<a+ "> in the short-time approximation (up to the second order in the interactiontime)

z<a+"> exp [ 1µ arg (a)] = Joel"+ 2 +IaI 2k4 Ia I2k+µ

X [1-Joel-

k!2µ(k-µ)1F1(-µ,k+1-µ; - IaI 2 )

J2

µ

+ s Ialzk+"kf-1F1 (-k,1 ; -IaI2)+µ!"~

(n'(+k~~

1

[(µ n)!] 2

x IaI -2n 1F1 (-k, 1 +µ-n; _I(XI2)] .

(A 18)

With µ=1, 2 we obtain from (A 18) the results (18) and (19) .

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Squeezing in multiphoton absorption

293

Finally we mention two recurrence relations of the hypergeometric functions IF,that we have used to derive (28)

k1F1( -k, l ; -IaI2)=(2k-1)iFi(-k+1,1 ; -Ia I2)

+IaI2 1 F1(-k+1,1 ; - IaI 2) - (k - 1)1F1(- k+2,1 ; - IaI 2 ),(A 19)

kIaI2IF1(-k+1,2; -IaI2)=(k-1+ IaI 2)IF,( - k+1,1 ; -IaI 2 )

-(k-1)1F1(-k+2,1 ; -IaI 2 ) .

(A20)

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[8] ZUBAIRY, M . S ., RAZMI, M . S. K ., IQBAL, S . and IDRESS, M ., 1983, Phys . Lett . A, 98,168 .[9] GARCIA-FERNANDEZ, P., SAINZ DE Los TERREROS, L., BERMEJO, F. J . and SANTORO, J .,

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Optica Acta, 33, 945 .[12] SHEN, Y. R ., 1967, Phys . Rev., 155, 921 .[13] AGARWAL, G . S ., 1970, Phys . Rev. A, 1, 1445 .[14] PAUL, H., 1982, Rev. mod. Phys ., 54, 1061 .[15] ABRAMOWITZ, M ., and STEGUN, I. A., 1965, Handbook of Mathematical Functions

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