1C/96/269

United Nations Educational Scientific and Cultural Organizationand

international Atomic Energy Agency

INTERNATIONAL CENTRE EOR THEORETICAL PHYSICS

HIGHER-ORDER AMPLITUDE SQUEEZINGOF PHOTONS PROPAGATING THROUGH A SEMICONDUCTOR

Nguyen Ba An*international Centre for Theoretical Physics, Trieste. Italy.

ABSTRACT

Photon amplitude Kth power squeezing is studied when the coherent photon propa-

gates through a semiconductor containing the exciton. If the exciton is prepared initially

in a coherent stale, the phol.on may become amplitude Kth power squeezed, it is shown

that, in the short-time limit, the photon squeezing in the P direction does not appear

at. all while that in l.he A" direction is possible for all the amplitude powers K. ]i\ l.he

latter case, the amount of squeezing is larger for higher power A'. Dependences on all the

system parameters a.s well a.s on l.he output light, det.ed.ion moment, are investigated in

detail.

M1RAMAUE- TRIESTE

December 1996

"Pernmnenl. address: Institute of Physics, P.O. "Box 429 Ro Ho, Hanoi "10000, Vietnam.

I. INTRODUCTION

Advanced optical measurement technologies, e.g., gravitational wave detection inter-

ferometers or active laser gyroscopes, have now been approaching the shot noise limit set

by t.Tic: e.|iia.ntnm mechanics. The new sl.ral.e;gy is circumventing the sl.a.nelarel e.|iia.ntnm

limit, t.o a.chie;ve an arbitrarily desired precision in e;xpe;rimenl.a.l measurements. The cir-

e:nmve;nt appears possible in squeezed stales. Those are slates in which fluclualions in one

observable become smaller than fluctuations in a coherent state, at the cost of sacrificing

fluctuations in the other conjugate partner observable to meet the lleisenberg uncertainty

principle.

The conventional quadrature a.niplilude squeezing, or shortly a.niplilude squeezing,

deals with the variance of linear conibinalions of field operators'. Hillery2 introduced for

the first time the concept of squeezing associated with the variables describing the real

and imaginary parts of the square of the complex amplitude of the field. Recently, much

research has been devoted to the so-called amplitude Klh power squeezing which contains

t.he conventional squeezing when K = 1 and t.he Hillery's when K = 2 as particular cases.

Tn general, one spea.ks of the higher-order a.niplilude squeezing when K = 2, 3, . . . > I. Tt

is wort.h noting that this t.ype of higher-order squeezing should not. be identified with t.haf,

defined earlier by Hong and Mandel3. Of particular interest is the situation under which

the conventional squeezing is absent but the higher-order squeezing takes place4. On the

ot.her hand, when squeezings of different, powers K coexist, it is essential to elucidate

t.he K- dependence;. Since; techniques for measuring higher-order correlations in quantum

optics are; improving, the; highe;r-order squeezing wenilel be e;xpe;riment.a.lly observeel an el

find its various applications in the near future.

Many physical nonlinear systems have been shown5"9 to display squeezing with A' > 1.

A generic connection between the conventional and higher-order squeezings was found in

t.he Mfk harme>nie: generation proce;sK2'(>: Tf t.he Mfk harme)nie: is squeezeel in the; e:on-

ve;ntiona.l sense. the;n the; [inidament.a.1 she)ulel be; Mth powe;r sqne;e;ze;d. Tt. alse) seemK

that squeezing of all powers K might occur at the same time, for instance, in Hainan

scatterings8. This however is true only for a short initial period of the time evolution.

Tins paper st.udies pliolons pa.ssing t.lirough a (init.e size semiconductor. We KIIOW that.,

due siiTndta.neously to bot.li the exdt.on-pliot.on and exciton- excilon interactions inside

the semiconductor, the output light may in general become squeezed to any power A'.

However, there exist time intervals during which any A'"" power squeezing is possible.

And, there also exist time intervals during which either only squeezing of certain powers

K appeai'K or no squeezing occurs at. all for any K. The excilon-indiiced TviechanisiTi of

generation of squeezed photons in t.lie conventional sense wa.s proposed in Ref. 10. Tlie

present paper is a. natural generalization of Ref. 10 to the liigher-order squeezing ca.se.

We organize our paper as follows. Section 11 formulates the problem starting with an

effective bosonic iiamiltonian describing the interacting photon-exciton system. Section

111 solves t.lie Hani il Ionian for the lime evolut.ion in the secula.r a.pproxima.lion using t.lie

pola.ril.on piclure. Seclion IV deriveK expressions for the pliot.on amplitude Kfh power

squeezing and analyzers the squeezing clia.rad.erislic in t.lie para.meler space;. Conclusion

is the final section.

II. FORMULATION

Consider for simplicity a. two-band semiconductor with direct band gap and allowed

inlerband dipole t.ransit.ion. Under excita.lion the seniiconduclor is looked upon a.s a

system of interacting electron-hole pairs. This fermionic system can be bosonized into an

excitonic one as was done, e.g., in Ref. 10. The coupled light-semiconductor system can

be described by an effective Iiamiltonian in terms of exciton operators a, a+ and photon

operators c, c+

H = Ea+a + LOC+C — q(a+c -f c+a) + fa+a+aa (1)

with E and u> the exciton and photon energies (ti = 1), g and / the exciton-photon and

exciton-exciton interactions whose analytic expressions are given in Hef. 10. Suppose that

at. / = 0 the light liit.s t.lie semicondnctor and lea.ves il at. I, = T > 0. The interaction

3

process takes place of course during the interval [0. T]. It is expected that input classical

pTiot.oTiN at. / < 0 may evolve iiij.o a.mplit.iiele; Kfk power Kqueez.eel one;s whie:h are delected

a.K the; output, light. be;hind l.lie sernie:ondiid.or a! /• > T. The onpnj. photon is said j.o be

amplitude Klh power squeezed in P or in A' directions if

or

(3)

. X(K,T) = l '

4

re;spee;t.ively. where; Ax = x— < x > with x any e>pe;rator an el < x > l.he quan tum a.ve;ragc

of x,

and

We Khali solve; (1) [or l.he t ime c:v<>liit.i<>n in the; next. Ke;ction. Tn l.hiK see:t.ie>ii we (i) prove;

l.hat. the; amp l i t ude Kqueezeel s la te of any powe;r K IK a ne)Tie;laKsie;al sl.a.te and (ii) ele;rive;

exp l id t expreKsions [or F(K,T).

(i) Using t h e C.ilauber-Sudarshan representa t ion of the densi ty ma t r ix 1 1 ' 1 2 we can wr i te

the variances as

<(AP(K,i))2>= [ < F(h ••'•}> I + f \^zK(t)-Q<cK(i)>yV(z(i))(l2z(i). (1)

t)Y >= < i ^ t ) > + I {&,"(*) _ JR < cK(i) >Yv(z(i))<tiz(i) (5)

where; z(l.) is a ce)Tnplex number c.lm.rae:t.e;riz.ing l.he e:olie;renl. slate al. lime; / and V l.he

weight function in the expansion of the density matrix in terms of coherent states. Clearly

from (1) and (5) that, for any squeezed state, for which either (2) or (3) holds, the function

V CriTine)!. generally be inl.e;rprel.e;d us a probability <lisl.inbnl.ioTi Kine:e it IK not. nori-Tiegative.

As a reKiill, the; ri.mplil.nele; Kih powe;r Kqueez.eel pliotons defineel above are; TioriclrisKicril [or

any K.

4

(ii) Aiming at the calculation of the amplitude Kth power squeezing, it is desirable to

derive for t.Tic: function F(KJ.) forms which are convenient, in application t.o an arbitrary

K. This l.ask was in fact l.ried in Ref. 7 bnl. the expression [blind there (sec (2.7) in Ref.

7) looks too cumbersome and by no means convenient, it turns out that much simpler

formulae can be obtained. Indeed, by virtue of the commutation relation c(t)e+R(t) =

c+K~1(t) (nc(t) + A') with nc(t) = e+(t)c(t) the photon number operator., we are able to

derive by mathematical induction t.vvo simple formulae Tor F(K. /) . The first, formula is

J= l J=0

and the second formula is

where K^ = K(K — 1 ) . . . ('A' — q + 1). Formula (6) is useful to get F(K, t) in terms of

nr\t). For the first five amplitude powers, we get immediately from (6) (try the same by

using (2.7) in Ref. 7)

F{5, t) = 5 [o<(t)

Formula (7), on the other hand, is helpful in evaluating expectation values because it is

normally ordered.

I I I . P H O T O N T I M E EVOLUTION

To KOIVC (1) for the photon time evolution we transform it. int.o t.lie polariton repre-

sentation in which it reads111

u=\

+f E E E E w w - t f (<K(O««(<K(O • (S)

TTic: l,wo-braTic:h pola.rij.(ni o p e r a t o r s f.v,,,(/). a + ( / ) a r e r e l a t e d t.o t.Tic: pTiolon o p e r a t o r s as

TTic: polaritori energy •(}„ and t.Tic: t.raTis[brTnat.ion coefficient.* u,., and -(;„ arc: given b y m

+ E + (-iy,_ 1"~2

uv = 1 +-1/2

The Iieisenberg equation of motion of the polariton is solvable exactly in the secular

approximation10'13 which retains only resonant terms in (8). Let us consider the near

resonant situation when S = u> — E is very small, i.e.. \S/E\ <C 1 . Then the secular

approximation works well and t.Tic: polarit.on lime evohilion can explicitly be: derived in

t.Tic: form

= exp <J —i <},, + 5 1 - / ^ a M °> l ) "'•''

/. 1 . (10)

We: Tiavc: adopted the abbrevia t ions h = h(l, = 0) with b any opera tor and j \ n i = 'l.fvf/o'^.

Tnsc:rting (10) inlo (9) givc:s j.}i<; tiTnc:-d<;p<;nd<;nce of t}i<; photon opera tor .

IV. PHOTON AMPLITUDE KTH POWER SQUEEZING

To examine the squeezing in A and P directions we introduce the following functions

S{P.KA) =< F[KJ.)> (11)

and

where : . . . : denotes a. normal ordering of t.Tic: operators KanclwicTiecl bet.ween the colons.

Tf - 1 < S(X, K, I.) < 0 ( -1 < S{P, K, I) < 0) the pliol.on is Haid l.o be sqn<x:>«:d in llic .Y

direction.

Making use of (1), (5), (9) and (10) we have obtained

A' K -iK-i-i i+j

x

of2A'

f -;

I'I—L

— 2 5R < aK

j i+

2

> ^ < > (H)

K — q K — q

X < ath~

(K-q-i)\(K-q-j)H\j\

A s K<;<;TI f r on i ( 1 3 ) l.o ( 1 5 ) . w<; Ticcd t o c a i c u l a U ; O K ; ri.v<;ragc o f t.Tic: p r o d u c t , o f p o l a r i t o n

operators < a~lk(t)ap(t)apl(t)a'2(t) > with /;. q, p and r any integers including zeros.

Assume that the photon and the exciton are initially (at t = 0) in their coherent states

<:Tia.™d.criz<;d respectively by complex mnnberH zc = \/~N\:cx]>(i0c) ri.n<] za = ^ /A^c jxpf i^ )

with Ar... ft.... Ar,j rin<] ^!t r<;a.l. Using proper t ies of j.}i<; diKplaccrncTil. op<;ral.or D[,(z) =

v,xp(zb+ — z'"b) a n d t h e o r t h o g o n a l i t . y c o n d i l i o n for t.Tic: t r a i l N f o r m a . l . i o n <:o<;ffi<:icnl.sl u v

7

u-iV-2 = 0. we are able to express the initial photon-exciton system in terms of the polaritons

a.s

za,zc> = Da(za)D,(z,)\Q >

= /}u1(zi)/?cva(22)|0 > = \Z!,Z2 > (16)

where; £r1-2 = »i,2~c + ^1.2^i- T'IK; averaging ov<;r l.he photon and excilon sta.U; can then

equivalently be carried out over the polariton state. Namely,

Snbslil.iiJ.iTig (10) ink) (17) yields al'ler lengOiy and I.edioiiK opera.loric nianipnla.UonH

X C?Jt,p(^i: 0, flUqf 12; -111; - r / l 2 ; 0

x C?9,r(22; fc/12, /22, - p / 1 2 , -I22,0; /)

vvlierc

m, n, .s, w: I.) = z4^ exp {i [k(l + m(k - 1 )/

[1 - exp[/(/ + mk + n

~ 1J/2)

(19)

with ;•' complex, q, k integers and all the remaining arguments real.

Figures 1 and 2 draw the squeeze functions f 11) and (12) versus Et for the conventional

amplitude squeezing A' = 1 (short-dashed), amplitude-squared squeezing A' = 2 (long-

daslied) and a.Tnplil.nde-cnbed squeezing K = 3 (solid). The parameters nscxl for t}i<;se

figures arc ujE = 1, gjE = 0.04, f/E = 0.004, Na = Ar, = 100 and 0a = 0,, =

0. As followed from (19) and (11) to (18). the fnnclion S varies in lime by m<;anK

of trigonometrical functions. As a consequence of this, there exist three types of time

domains. The first type is time domains within which squeezing does not occur at all.

The second type is lime domains wil.hin which only squeezing of cerlain powers K can

arise. And, l.he l.hird type is lime domains wilhin which sqn<;<;zing of any ampliliidc; power

K is possible. The duration and posilion of such lime domains are sensitively delermined

8

by the system parameters. For a fixed set of parameters one can adjust the semiconductor

optical size L = vgT (vg t.Tic: group ve;le>e:it.y of light ITIHICIC: the; Ke;mice>nehie:t.e>r), by which

t.he light, Txjrnn propri.gri.tcK, U) ele;tee:t. l.Tit- squeezing of desireel amplitude powers K. A

generic property can be obtained analytically in the short-time limit £ T < 1 when both

:c\ and z-i are real. In this case, by virtue of (19) and (11) to (18). it is easy to derive

(20)

and

S(X. A. t) = -x2{K)t2 + 0{t3) (21)

where p(K) and x(K) are some cumbersome real functions of A' which increase for growing

K. Equations (20) an el (21) imply that, in the; short-time limit., squeezing in t.he P

(lirect.ioii is absolutely imposKii)le while squeezing in the; A" clirect.ion always happe;ns for

all amplit.uele; powers as ve;rifie;d from Figs. 1 ri.n<l 2. Figure 2 also con firmK t.hat., in t.he

short- time limit, the amount of squeezing in the X direction increases with the amplitude

power A. We stress, however, that the above generic property only holds for real ;c\ and

z-2. If either of z\ and ;:•>_ or both are complex, the situation changes qualitatively. To

Kee this we plot., in Fig. 3, S(X, Kj.) verKiiK El, with same parri.m<;ters i-i.s in Fig. 2 but

ftc = TT/4. i.e.. both z\ ri.n<] z2 are compl<;x. Tt is manifeKJ. t.hat., in the short-time limit, t.he

A = 1,2 amplitude squeezing is impossible but the A' = 3.1 amplitude squeezing appears

and. the A = 5 amplitude squeezing is again impossible, ... in contrast to (21). More

information on the A'-dependence of the amplitude squeezing is obtained from Fig. A

where; t.he out.put light is cletect.ed at. T = Q.Qbf E (Kee the figure; caption). An alt.ernri.tive

way t.o st.udy the dependence; on be>th lime; and ri.mplit.nele; powe;r is to ple>t t.he fnne:t.ie)n

S versus K for varieniK ele;tee:t.ie)n moment.s. This is ele)ne in Fig. 5 where; the; detection is

made at the moments ET = 0.02. 0.05, 0.1, 0.15 and 0.2. Up to A' = 10 the following

conclusions can be drawn. For short times (ET = 0.02, 0.05). amplitude squeezing arises

for all penvers K in ae:e:ordance; wit.h the; ge;neric pre>pe;rt.y eKJ.ablis}ie;d ri.T>e)ve; in t.he she>r(-

t.ime; limit. Fe.|. (21). Fe>r a le)nger time (ET = 0.1), squeezing with K < 10 st.ill e>e:e:nrK

9

but there appears a tendency of disappearance of squeezing for A' > 10. Tor even longer

limes. there exist, T-dependent cril.ica.l powers Kj such t.Tiat. squeezing l.a.kes place only

for K < KT (IKTC KT = 7 lor ET = 0.15 am] KT = 5 lor ET = 0.2).

it is known that the photon-exciton interaction is strongest in the exact resonance

u; = E. The squeezing effect is thus also governed by the resonance condition. Figure 6

plots S(X. K. t) as a function of detuning LO/E for different A' and t. The long- dashed,

sTiort-d ashed and solid curves correspond l.o {K, El,} = {1, 0.65}, {2, 1.35} and {3, 0.9},

respectively. The values of El are chosen at. minium in Fig. 2. Figure 6 transparently

shows that. Tor all powers K the squeezing effect, is HJ.roTigeHJ. at l.lie resonance rind de-

creases away from the resonance.

in general, disregarding the detection moment, the minima of the squeeze function S

are deeper for larger exciton (photon) average number Na (N.S). This was indeed justified10

for t.Tie conventional squeezing. Yet., for a given semiconductor opt.ica.1 length L, t.lie

<]<;tect.ion inoTvi<;nt T is fix<;d. T'}i<;n. it. is more rea.sorm.ble t.o investigal.e the dependence

of the squeezing on the initial exciton average number at a given moment of detection T.

The dependence of squeezing on the initial exciton average number Na is plotted in Fig.

7 using the parameters like in Fig. 1 but LO/E = 0.9 and ET = 0.6. The Avdependence

is quil.e nontrivia.1 a.s recognized from Fig. 7. It. is worth not.icing t.Tiat., for Na = 0,

amplitnde-ciiTxjd Kqueez.ing is a.bsent but. amplitiide-Kquar<;d and conventional Kqueez.ings

are present. The presence of squeezing for Na = 0 and LO/E < 1 could cause a surprise

because no real excitons are available in this case. Here the role of the virtual exciton is

played.

Finally, we discuss on the role of t.Tie exciton-exciton int.eract.ion. Tf t}i<;re is no excit.on-

excit.on irit.<;ract.ion. i.e., /' = 0, i.}i<; phot.on-excit.on Kyst.<;in remains conpl<;d but no aqucvy-

ing can be generat.ed from j.}i<; initially nori-s<.|ii<;<;z<;d st.a.teK because of lack of rionliri<;a.ri(.y.

In this case, however, if the exciton is initially prepared in a squeezed state, then the

squeezing of exciton can be transferred to the photon during the course of time evolution

and vice versa.1'1. For initially non-squeezed excil.ons the condition / ^ 0 is necessary for

t.Tie pTiot.on t.o be squeezed. Rut. for a. given detection moment T. f ^ 0 is not. a sufficient,

10

condition. Also, it is not true that for a given T the greater / the better the squeezing.

Tlie TIOTItrivial dependence OTI JjE is KTIOWTI in Fig. S for K = 2 and E'T = 1.35.

V. CONCLUSION

in conclusion, we have shown that photon amplitude Klh power squeezing can be

generated from an initially non-squeezed exciton-photon system. The photon squeezing

cliara.cterislic depends in a. complicated manner on all t.Tic: involved parameters gj E, f j E,

^/E, Na, Nc. 0a, 8C and E'T. Tf all t}i<; pa.ramct.<;rK but, GTK; ar<; fix<;d. t}i<; squeezing

seems to depend ''periodically'* on the varying parameter. In general, the choice of '* best"

parameters is difficult. However, in the short-time limit 1£T -C 1. the photon squeezing

in the X direction is possible for all the amplitude powers A' and the higher A' the larger

t.Tic: amount of squeezing.

ACKNOWLEDGMENTS

This work was supported by SAREC, IAEA and UNESCO. The author would like to

thank 1CTP and its Condensed Matter Croup for hospitality at Trieste.

REFERENCES

1 R. London and P. L. KnighL J. Mod. Opl. 34: 709 (1987).

2M. llillery. Opt. Comm. 62. 135 (1987).

S C. K. Hong arid L. Mamld, Phys. Rev. Lell. 54, 323 (1985); Phys. Rev. A32, 794

(1985).

1 A. V. Clii/bov, J. W. Hans and K. Yeong, Pliys. Rev. A52, 1698 (1995).

5 Zhi-ming Zhang. Lei-Xu, Jin-lin Chai and Fu-li Li, Phys. Lett. A150. 27 (1990).

fi You-bang Zlian, Pliys. L<:1.1.. A160, 498 (1991): 503 (1991).

7Si-de Du and Chan-de Cong, Phys. Lett. A168. 296 (1992).

8 A. Kmvmr and P. S. GnpUi, Qiii-niJiiTii ScJiniclasK. Opl. 7. 835 (1995).

!) Jisno Wi-nig. Qiiimnei Sui and Clnia.nkui Wa.ng, Quanlinn ScrniclaKs. Opl.. 7. 917 (1995).

10 Nguyen Ba An. Phys. Rev. B48, 11732 (1993); Quantum Opt. 4. 397 (1992).

11 R. J. Glauber, Pliys. Rev. Loll. 10, 84 (1963); Phys. Rev. 131, 2766 (1963).

12 E. C. C. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).

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14 Nguyen Ba An. Mod, Phys. Lett. B5, 567 (1991).

FIGURES

FIG. 1. (Conventional amplitude squeezing (short-dashed curve), amplitude-squared squeez-

ing (long-da.shed curve) and amplitude-cubed squeezing (solid curve) in the P direction a.s a,

function of l<:t. The parameters used are u/K = 1. g/l-J = 0.0-1; ///:• = 0.00-1, A:a = ;\c = 100

and 6a = 6t = 0. The horizontal straight line passing through 0 in this as well as in all other

figures is traced just to guide the eye.

FIG. 2. Same as in Fig. 1 but for the squeezing in the A" direction.

FIG. -i. S(X,K,l) as a function of El with the parameters -JJ/E = 1, g/E = 0.04,

///<; = 0.00-1, A',, = /V, = 100, Ba = 0 and Bc = r,jA. 'The curves with decreasing length of

flashing correspond to A' = 1, 2, 3 and -1, respectively. 'The solid curve is for A' = 5.

PIG. -1. 5(.V, /\, t) detected at 7 = 0.05//:' as a function of K with the same parameters a.s

in Fig. ;{. The squeezing "'selects'" the amplitude power A': up to A' = 10, only the K = 3, 4, 7

and 8 amplitude squeezing occurs while K = 1, 2, 5, fi, 9 and 10 squeezing does not appear.

PIG. 5. 5(.V, K.t) as a. function of A' for different detection moments. 'The curves with

increasing length of dashing correspond to i-t = 0.02, 0.05, 0.1, 0.15 and 0.2, respectively. The

used parameters are a.s in Pig. 1.

FIG. 6. S(X,K,T) versus detuning u;/E for {K, ET} = {1. 0.65} (long- dashed curve),

{2, 1.35} (short-dashed curve) and {3, 0.9} (solid curve). The other parameters are as in Fig.

1.

FIG. 7. S(X. K,T) as a function of the initial exciton average number Na for K = 1

(long-dashed). A' = 2 (short-dashed) and A' = 3 (solid). The other parameters are as in

Pig. 1 but u/fC = 0.9 and ICT = 0.G.

PIG. 8. 5(.V, /\,7') as a. function of the scaled exciton-exciton interaction ///'. ' for K = 2

and i'Vr = 1.35. The other parameters are a.s in Pig. 1 but, g = 10/.

0.5 1.5

Et

Fig.l

0.5 1.5

Et

Fig-2

14

0.02

0.015

0.01

£ 0.005

M 0

S-0.005CO

- 0 . 0 1

- 0 . 0 1 5

- 0 . 0 2

0.2 0.4 0.6

Et

Fig. 3

0.8

8

K

Fig.A

10

15

8 10

K

Fig.5

0 . 9 6 0 . 9 8 1.02 1 .04

DETUNING

Fig-6

16

0 200 400 600 800 1000

EXCITON AVERAGE NUMBER

Fig.7

0.5

- 0 . 2 5 -

0 0.002 0.004 0.006

EXCITON-EXCITON INTERACTION

ig.8

17