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  • 1C/96/269

    United Nations Educational Scientific and Cultural Organization and

    international Atomic Energy Agency

    INTERNATIONAL CENTRE EOR THEORETICAL PHYSICS

    HIGHER-ORDER AMPLITUDE SQUEEZING OF 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:vliit.in 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

    = [ < F(h ••'•}> I + f \^z K(t)-QyV(z(i))(l2z(i). (1)

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

  • (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

  • u=\

    +f E E E E w w - t f (

  • 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«:d in llic .Y

    direction.

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

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

    x

    of 2A'

    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

  • 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

  • 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