Simplified description of unsaturated two-photon absorption. II. Squeezing and enhanced photon antibunching

Volume

5 8, number

OPTICS COMMUNICATIONS

1

SIMPLIFIED DESCRIPTION OF UNSATURATED TWO-PHOTON II. SQUEEZING AND ENHANCED PHOTON ANTIBUNCHING

1 May 1986

ABSORPTION.

A BANDILLA Zentralrnstrtut fw Optrk und Spektroskople, Akademe Received

1 October

1985, revised manuscript

received

der Wmenschaften der DDR, 1199 BerlmAdlershof, 6 January

GDR

1986

The summation of the short-time expansion for two-photon absorption 1s extended to phase-dependent expectation values m order to calculate squeezmg and the mfluence of squeezmg on enhancement of antlbunchmg The treatment 1s well sated for great photon numbers and arbitrary absorption strengths and dehvers simple analytlcal expressions It 1s shown that squeezmg 1s responsible for the posslblhty of enhanced photon antlbunchmg

1. Introduction Although many authors emphasize the practrcal importance of squeezed field states, it seems that squeezmg IS for the trme bemg more an mterestmg nonclassrcal phenomenon whrch partially characterrzes quantum mechamcally a radiation field without a classical analog Another pomt 1s the proposed measurement of squeezmg whrch works only mdrrectly vra detection of photon antrbunchmg 111a homodyne mterference expenment [l-3] or by analyzmg the noise-power spectrum [4] An addrtronal disadvantage 111the measurement 1s that the coherent reference srgnal must be much stronger than the squeezed field, so the correspondmg antrbtmchmg 1s only a very small effect In addition, m the case of resonance fluorescence from one atom the photon antrbunchmg m the mterference field whrch 1s due to squeezmg 1s a noticeable smaller effect than the drrect observable non-porssoman photon statrstrcs, as was shown by &Iandel [3] Here we find for two-photon absorption practically no reductron What 1s more rmportant, squeezmg m twophoton absorption can be shown to be responsible for the possrbrhty of enhanced photon antrbunchmg It means that the photon number Gi), whrch characterizes antrbunchmg rn its l/Wdependence can be drastically drmuushed by destructrve mterference with an equally intense reference beam, resultmg 111an en0 0304018/86/$03.50 0 Elsevrer Scrence Pubhshers B V (North-Holland Physics Pubhshmg Drvrsron)

hancement of the effect [5] One-photon absorptron 1s unable to do the same In thrs paper, we enlarge first the srmphfied description of two-photon absorptron, treated 111ref [6] for the mam diagonal of the density matnx, to the nondragonal elements m order to give simple expressions at hrgh photon numbers for possrble squeezmg Tlus 1s done by summing the short-trme expansion. Squeezmg as unsymmetnc behavror of phase and amphtude fluctuations 1s expected for hght after twophoton absorption due to reduced mtensrty fluctuations which create augmented phase fluctuatrons and has been shown to exrst for such light 111refs [7] and [8 ] Of course, thus phenomenon 1s already mherent m the mvestrgatron of enhanced antrbunchmg for twophoton absorbed hght [9] In the present paper we show exphcrtly that squeezmg 1s a necessary assumption for an enhancement of photon antrbunchmg Thrs opens a new posnbrlrty to measure squeezmg 111mtense radratron fields after two. photon absorption, or generally, after multrphoton absorption

2. Exact solution of the master equation for unsaturated two-photon absorption In order for the paper to be self-contamed we brrefly sketch the solutron of the master equatron for un63

saturated two-photon absorption [9,10] 2dpF/dT = @2p,,t2

1 May 1986

OPTICS COMMUNICATIONS

Volume 5 8, number 1

- at2a2pF - pFat2a2,

(2 1)

where pF 1sthe reduced density operator of the light field mode, at and a the correspondmg creation and annAulatlon operator, respectwely, and T 1s a dlmenslonless tune parameter Usmg the generatmg function method, a separation ansatz yields for the generatmg function [9]

where the first part of (2 8) represents the stationary value of Gb, 1,T) With (2.7) and (2 4) mserted m (2.2) and the special case (2 8) we possess a complete solution of (2 1) for mltlally coherent hght

3 Summation of the short-time expansion of
Havmg m mmd to determme the squeezmg of the two-photon absorbed field mode and also the photon statistics after mterference with a coherent reference signal, we need m addltlon to (at (T)a(T)) and (at2(T)a2(T)>, already calculated m ref [6], the followmg expectation values (a+CT)), $&,

T) = [(n + &!/n’] 1’2(&F(T)ln

+ p),

(2 3)

and fi(.Y) = C,“6Q,

A, = n(n + P - 1) + d&P - I),(2 4)

where Cgb) IS a Gegenbauer polynomud and P charaCterlZeS the off-dlagonahty Of pFn,n+,,(T) = (nl pF(T)I n t cc) We assume mltlally coherent hght , i.e ,

GOI,cc,T) = ew[b120- l)la*‘, a = Ial exp(iq,)

(2 5)

(a+2(T)),

and their complex conjugates However, we observe the relations [9 ] (at2(T)) = exp(-21qa) emT(a+(03a(T)),

(3 1)

and = -(d/dT)
(3.2)

therefore it ISsufficient to denve a snnple expression for tit(T)) (m the mteractlon picture) by summmg the short-tnne expansion This wti be done m this section. With the relation C,“(1) = (2/n)

and use Some’s formula [ 111

,

(n > 0)

(3.3)

it follows from (2 8)

00 eZYzu= 2’Qo) nEO(n + u)Z,+&)C,O(_Y) =

(2 6)

m order to determme the coefficients b,O In (2 6) we denote by Z,+,(z) the modfied Bessel function From (2 2), (2 5) and (2 6) we obtam for a#0

= o* exp(-lol12) &(M2)

C

b,O= exp(-lar12)~*~(2/I~12)“I’(u)(n + u)Zn+,(la12),

(27) andforu=Oorp=lwefoundmref Gti, 1,

exp(--n2T) = 1 - n2T/1f + (?I~T)~/~I - + .

[9]

(3.5)

and try to sum m (3 4) by apphcatlon of Somne’s formula (2 6) First we obtam from the generatmg function of Bessel functions [ 111

79= exp(-la12)a* Zo(la12) [

(2 8)

64

Now we develop

2a* ewt-Id)

1 May 1986

OPTICSCOMMUNICATIONS

Volume58, number 1

n~lInt142)

= a* - a* exp(-la12)lo(la12)

(3 6)

With (3 6) the stationary term of (3 4) 1slalled By mductlon we can derive the followmg relation

creased to the half for t = 1 Absorption to about 10% of the mltlal photons IS aclueved for 4 near 10 Therefore we must only assume lct12S 1 to reach a very good approximation m (3 10) The next higher contnbutlon for (3 10) ISof the order of l/larl 3 Now it is easy to get from (3 10) by dlfferentlatlon, as mdlcated m (3 2),

(J2(T)&y

?.+ = [(21- l)‘nCI,_~(l)]I,v

= Q*lQ12 + (y* t2 - 4E (l&/2

+ i(v - l)v(2v - 1)[(21- l)‘nc;_~(l)]~=v_l

+ (3.7 j

( 1 8 (ltD7i2+0$31L)

For (3 1) we have to use results of ref [6], namely (3 IO),

which suffices m most cases for us and uses = “-‘,+1;-

c;_,(l)

l)=(n;y-l)

(3 12)

(3.8)

Applymg now (2 6) on (3.4) with (3 5) and (3 7) we receive

which gives for (3 1) (at2(T)> = exp(_21qa)emT

5

n=l

x

(2v - 1)’ + L ( 2yro

x

+

\ 1

2v-‘r(Y - 1)

(3 9)

Note the decreasmg powers of la12, the mltlal photon number, m (3 9) With (3 9) we find for
+a*-

E2

1-+2~-_E3;;~;~+_

g1a12

a* =(l +&S

t2 --a*

1

6(1 t[)3

= exp(-21qa)

(Y- l)V(2Y - 1)

1(.X12 6

(2v-- 3)’

10112 +‘2(3 +‘) ’ +t

n2”1,(l~12) = exp(la12)(la12)v

c;= 2lal2T < 1

lal2 (1 t Q’2 ’ (3 10)

As m ref [6], the convergence regon of (3 10) can be enlarged by analytlcal contmuatlon to t > 0 In order to see that (3.10) ISan expansion m decreasing powers of la12, the parameter t = 21cu12Tfieedsa separate consideration Accordmg to the results of ref. [6], the photon number (
where we assumed T Q 1 m order to have a reasonable photon number (cf [6] ) Summarlzmg, we conclude that by summmg the short-tune expansion (3.5) for all expectation values we obtam new expansions, but now m powers of l/ loJ2, which converge very fast for great mltlal photon numbers Therefore we need only to calculate a few terms of these new expansions For lllustratlon we show m fg 1 the photon number, the dlsperslon and the modulus of the expectation value (3 13), the last one is mamly responsible for the generation of squeezmg. The exact curves and approximate results comclde excellently because G(O)) = 1000 s 1 Note, however, that there are quantum-mechamcally lmportant differences between Gzt(Z’Ja(7’))and IGzt2(T))Iwhich cannot be shown m the figure, but are contamed m the formulas To give an Idea of these very small, but for the calculation of antlbunchmg and squeezmg declslve dlstmctlons, we write (3 12) and (3 13) for the asymptotic regon (,$+ -)

= 1/2T + +,

(3 12a)

and 65

Volume 58, number 1

1 May 1986

OPTICS COMMUNICATIONS

4 =+++(at(T)a(T))(l

two-photon absorptw

=f__

t eeT) - I(at(T))12

1 E3+3tZ+3E, l2

T
(1t#

(4 5)

Eq (4 5) tells us that the amphtude al IS111a squeezed state Analogously one gets for ur, = lr/2

08

&$=at#(ltQ 07

(4.6)

For strong two-photon absorption we have g + = and (4 7)

Ir

10

10

lo-

lo-'

Fig 1 Photon number, dlsperslon of the photon number and the modulus of (at 2(T), (eq (3 13)) m dependence of T The full hnes are calculated accordmg to the exact solution Circles (photon number and dlspernon) and crosses (I(at2(7’)>) illustrate the almost perfect comcldence between the approxlmate results of tlus paper and the exact results Relatively great differences occur only m the stationary state, whch IS more or less of academic mterest Note the absolute scabng m T Physically reahstlc values can be expected only for T < 1

(3 13a) (at’(T)) = exp(-2ip (Y)[ 1/2T + f - “1 2 ’ where 1/2T1s the hmit of laj2/(1 + t) for t + =

4. Squeezing According

a=fJ1

tq,

and enhanced photon antibunching

to the decomposltlon fzt=a1

-3,

(4 1)

The factor 214 shows that two-photon absorbed hght ISnot m a squeezed coherent state, where the product of uncertamtles IS preserved at the mmlmum value The loss of mnumum uncertamty ISdue to fluctuatlons, mtroduced by the lrreverslble absorption process Now, we will consider the role of squeezmg 111the proposed enhancement of antlbunchmg by destructive mterference [5,9] As 111ref [9] we observe with the help of a polarizer (fg 2) the mterference between a coherent beam and a two-photon absorbed beam (m lmear polarlzatlon) However, contrary to the usual proposals for the measurement of squeezmg we suppose both beams to be of nearly equal strength Note that these two beams are rather mtense, because for achevmg a noticeable two-photon absorption we need a very strong radiation due to the small cross section Therefore the genume quantum properties of the two-photon absorbed hght are very small Then we calculate the normal-order variance of the

of the anmh&tlon and creation operator, respectlvely, we consider the uncertamtles A&

= (Ial, - bl$12)

= a(
.,Tydent

coherent beam

(4 2)

Squeezmg ISpresent when one of the amphtudes al and a2 has an uncertamty Aa,2<$,

r=1,2

(43)

Relation (4 3) 1sequivalent to a(Gzta) - i(at )I*) k i Re((a*) - *)< 0

-----b’=

(4 4)

Insertmg the results for two-photon absorbed coherent hght m (4 2), we find for qa = 0 (compare (2 5)) 66

a:@kos8-a$n8

Fig 2 Spatial onentatlon of the different potiatlon components The duectlon (1) IS given, e g , by a magnetic field The component (2) remams m the coherent state la cos ej2

OPTICSCOMMUNICATIONS

Volume 5 8, number 1

Eq (4 11) demonstrates, that squeezmg of al (&T < $) shows up m (4 8) under the condition (4 9) as antlbunchmg. Now we will drop the assumption (4 9) and consider all terms m (4 8) In order to slmphfy our dlscusslon, we confine ourselves to the asymptotic regon (t = 2M2(sm2e)T% l), already mdlcated m fg 1 and by way of examples dlscussed m (3 12a), (3 13a) and (4 7) Let us start with the last term m (4 8), which gives by means of (4 11) and (4 5)

resultmg field (fig 2) * b+ = at,(zJ cos e - “J sm 8 (cf equation

(3.4) of ref [9] )

(bt2b2> - (b+M2

+ 2 cos E sm 0 cos e [Q(Gzj(T)%; - (fz~2(7)a,(T),) +

c0s2f sm2e

(T)fzI (T))

41ar12cos2e sm28(&zT -a)

+cc]

[~~QI~(Gz~(T)u~(~)

(4 8)

The notations for (4 8) are made clear m fig 2. Accordmgly the initial states for both modes are generated by decomposltlon of a coherent beam mto two coherent modes (ICYsm ejl for mode (1) and Ia! cos ej2 for mode (2)) These two states are mdependent, but nevertheless have a defitlte phase relation which we need for consldermg the mterference The expectatlon values of the coherent reference beam are expressed m (4 8) by different powers of (a cos e) Usually one assumes that the reference mode 1s very mtense m comparison with the signal

Ial cos2e9 k?p)ap)).

(4 9)

Under this condltlon only the last term m (4 8) surmves a dlvlslon by the resultmg photon number, because both are proportional to lcu12cos2e This last term transforms an exlstmg squeezing m an antlbunchmg of the photon statistics Let us choose the phase difference between the two mterfermg modes as -vq

(4 10)

=o,

where the indices 1 and 2 mdlcate mode (1) and (2) (cf fig 2) Then we find for the last term m (4 8) (except the factor cos20) c0s2e

sm2e [ ]

=41ai2

c~~2~~~2e(~~

=41a12

- i(af(T))~~)

+ (cx*~Gz;(T), + c c ) - (G~t,(n)~&~ + c c )])

cpa1

1 May 1986

(4 1 la)

COS~E~III~~(-&)

From (4.11 a) we see that under (4 10) squeezmg is present in (4 The nuddle term m (4 8) has (3 10) and (3 11) for the hnut t 2

COSE

=2

sm

e COS~[

_]

cos ESII~B ~~~e[d+q)$

The first term was determmed c0s2e[

the phase relation 1 la) to be calculated with --, 00, glvmg

+C

c.].

(4 12)

already m [6] to be (4 13)

1 =c0s2e(-',)(af(~)(a1(T))

Combmmg (4 11 a), (4 12) and (4.13), we obtam for (4.8) (b+W) =

- (b+z#

c0s2e[j~(af(zy~0~e

--*COS

~~11lei~](4

14)

The mmus m (4 14) slgnahes photon antlbunchmg and note, that the modulus contams the difference of the amplitudes. This modulus occurs also m the resultmg photon number [9] (b+b)=

i(a~(zy~0~e

-~*cos~sIII~~~+~~~~~~,

(4 15) and shows the posslbtity of reducmg the resultmg photon number By that the quantity ((bt2bZ) - (b+b)2)/(b+z$,

-$)

(4 11)

* Note that a:(T) and 17:are creation operators of mode (1) and (2), respectwely, and must not be confused wth the hernutean amphtudes a1 and (12,mtroduced m eq. (4.1).

which 1s measured m an Hanbury Brown-Twlss expenment , can be enhanced and we receive an effectively greater photon antibunchmg. Some remarks must be made. First, III (4 4) some small corrections are neglected 67

Volume 58, number 1

OPTICS COMMUNICATIONS

whch do not alter the prmaple, and their unnnportance was already shown m ref [9] Second, and more essentil for tlxs paper, the presence of squeezmg m (4 1 la) 1svital for this form of (4.14), allowmg the enhancement of antlbunchmg Another phase difference (4.10) destroys the squeezmg m (4 1la) and therefore also (4.14) Espeaally, when the middle term (4.12) 1sforced to vamsh (I&~ = n/2), squeezmg m (4 11a) disappears and the - %par? photon number (btb) cannot be made small m this manner Thud, we must emphasue that tlus reduction of the photon number (4.15) 1stotally different from a usual one-photon absorption process, because here the photon number ISdnmmshed due to the presence of a reference hght beam Note also, that (4 15) cannot go to zero because the two-photon absorbed hght ISno longer m a coherent state! Eventually, we must say that our proposal concerns only very mtense nonclassxal beams, as emergmg from two-photon absorptlon Therefore It ISnot helpful m resonance fluorescence, where the mtenslty ISusually low

68

1 May 1986

References [I] H P Yuen and J H Shapiro, IEEE Tram Inf Theory 24 (1978) 657,26 (1980) 78 [2] J H Shapuo, H P Yuen and J A Machado Mata, IEEE Trans Inf Theory 25 (1979) 179 [3] L Mandel, Phys Rev Lett 49 (1982) 136 [4] See, e g , B Yurke, Phys Rev A 32 (1985) 300 [5] A BandlIla and H -H Rltze, Optws Comm 28 (1979) 126 [6] A Band&t, submltted for pubhcatlon m Optics Comm [7] M S Zubauy, M S K Razml, Saleem Iqbal and M Idress, Phys Lett 98 A (1983) 168 [ 81 R Loudon, Optics Comm 49 (1984) 67 [ 9 ] A Band& and H -H fitze, Optics Comm 32 (1980) 195 [IO] H D Slmaan and R Loudon, J Phys A Math Gen 8 (1978) 435 [ 1 I] W Magnus, F Oberhettmger and R P Som, Formulas and theorems for the special functions of mathematxxl physxs (Sprmger-Verlag, 1966)