A N N A L E N D E R PHYSIK 7.Folge. Band 38.1981. Heft 2, S. 89-168 Anticorrelations in Nondegenerate Parametric Three-Wave Interaction By H. PAULand W. BRUNNER Zentralinstitut fur Optik und Spekt,roskopieder Akademie der IVissenschaften drr DDR, BerlinAdlershof Dedicated to Prof.Dr. Gustav Richter on the Occasion of the 70th Anniversary of his Birthday Abstract. It is shown that the anticorrelations in the photon numbers of the signal and the idler wave, respectively, generated in parametric three-wave interaction under suitable conditions, violate an inequality well-known in classical communication theory. Hence, it must be concluded that these anticorrelations reflect a typical quantum mechanical feature of light. In fact, this nonclassical behaviour is responsible for the occurrence of photon antibunching in the superposition field made up of both the signal and the idler wave, as will be pointed out in some detail. Antikorrelationon bei der nichtentarteten parametrischen Drei-Wellen-Wt?chselwirkung Inhaltsubersicht. Es wird gezeigt, d a l die Antikorrelationen zwischen den Wotonenzahlen in der Signal- und der Idlerwelle, die bci parametrischer Drei-Wellen-Wechselwirkungunter geeigneten Bedingungen erzeugt werden, eine aus der klassischen Kommunikationstheorie bekannte Ungleichung verletzen. Daraus mul3 man schlielen, daB d i m Antikorrelationcn einen typisch quantenmechanischen Wesenszug des Lichtes wiedergeben. I n der Tat ist, wie im rinzelnen ausgefiihrt wird, dieses nichtklassische Verhalten verantwortlich fur das Auftreten von Photonen-,4ntibunching in dem durch Superposition von Signal- und Idlerwelle entstehcnden Gesamtfeld. 1. Introduction During the last years the photon antibunching effect has attracted considerable interest, since it has no counterpart in classical optics and, hence, provides new evidence of the wave-particle dualisin. While experiments thus far are restricted t o the study of resonance fluorescence froin single atoms [ l , 21, various physical nicchanisms that generate antibunching have been analyzed theoretically (see e.g. [3- lo]). Among them is the process of degenerate parametric three-wave interaction. It has been shown 141 t h a t a weak fundamental wave will acquire antibunching properties in the course of interaction with a strong coherent harmonic which is simultaneously irradiated into the nonlinear crystal, provided the phases of the two waves are adjusted such that the fundamental wave experiences maximum attenuation, diic t o amplification of the harmonic. Similarly, photon antibunching may be detected idso in nondegenerate parametric three-wave interaction provided the superposition field made up of both the signal and the idler wave, is made the object of investigation[ll, 12].The occurrence of antibunching properties in this case is intimately connected with the presence of anticorrelations between the signal and the idler wave, which have been predicted in [13, 141. 7 Ann. Physik. 7. Folge, Bd. 38 H.PAULand W. BRUNNER 90 The main purpose of the present paper is t o point out that those anticorrelations are, in fact, quantum mechanical in nature. We will show in detail that they violate a n in- equality well-known in classical (statistical) coimiunication theory, i.e. that they are stronger than can be understood in classical terms. Moreover, the connection of the anticorrelations with the antihnnching properties of the superposition field will be elucidated. 2. Basic Equations I n contrast to references [ I Y , 141. in which all three modes were quantized, we use the so-called parametric approximation, i.e. we assume the pump wave t o be intense enough t o allow for a classical description in forin of a plane wave with a constant amplitude not altered by the interaction process. In this way the short time approxiination t o which the treatment in [13, 141 had to be restricted, can be avoided. We start from the well-known equations of motion for the photon annihilation operators q8 and qi, corresponding t o the signal and the idler wave, respectively, which read in the interaction picture [15] Here y, taken positive, denotes the effective coupling constant, which is proportional to both the nonlinear susceptibility of the (nonlinear) medium and the amplitude of the pump wave. The phase factor E is defined by the phase of the pump wave vpaccording to the relation The solutions t o eqs. (1)) (2) are given by [15] + &*WPi+ q"t) = c(t) QI q d t ) = c ( t ) qi + E*s(t) q,+ 7 (4) - (5) Here and in what follows we adopt the convention that operators written without any argument refer t o the initial state, and we use the following abbreviations c(t) = cosh (yt), s ( t ) = sinh (yt). (6) With the solutions (4), (5) at hand, it is a straight-forward matter to evaluate products of operators whose expectation values are needed in the calculation of photon statistical properties. Using the well-known commutation relations between photon creation and annihilation operators we arrive a t the following normally ordered expressions for the relevant terms q,C ( t ) q,(t) = c2q:q, + s2[qi+qi + 11 + &scq'qi + E*scqsfqi+ 7 (7) + s"qi'2qf + 4q:qi + 21 + &2s2c2q:q; + &*"%2q,"2qi+2 + 2&s3c[qi'q: + 2qJ + ?&*S3CQsf[Qi+2Qj+ 2qifI (8) + 2&c3sq,+q;qi + .'&*c3sqsf~q8qi' + 4c2s2q,+q8[qi+qi + 13 q,f2(t) q?(t) = c 4 q y q ; Q8 Anticorrelations in Parametric Three-Wave Interaction 91 (t,he corresponding expressions for the idler wave are obtained by interchanging the subscripts i and 8 on the right-hand sides of eqs. (7)) (S)), q,fq?qfl8 + c4p;fq8q?qi S4[q$q8 f ' 1 [q?qi f ' 1 &*282C2q:2qi+2 €s"[q,+q, qi+qi + + + f E*'3cq$q$[q2q8 f q?qi f 3i &*8c3q,4qi+[q,+q8 + qcqi 11 + + ' 2 2 2 2 2 + 31 q8qi q8qi (9) + E'c3[dq8 f qzqi f ' 1 48qi + + &"."Czq;q: + ,*2s2c2q&?qifz + .2c2[q,+2q: + Pi'": + 2q:q,qicqi + 3q:q's + 3qifqi + 11 - We assume the signal and the idler wave t o be initially in GLAUBERstates la8>and lai>, respectively. This implies that the two fields are statistically independent a t the beginning of the interaction. Hence, the expectation values of the operators on the righthand sides of eqs. (7)-(9) factorize. Moreover, due t o the well-known properties of the GLAUBERstates [16] qjx> = q+ = a*@ 9 (10) the calculation of the expectation values for the operator conibinations ( 7 ) - (9) simply reduces t o the replacement q: -+ a , q8 -+ ae,qi+ -+ cw : ,t qi --f u i . 3. Photon Statistics in the Signal and the Idler Wave I n the wag described before, one easily finds the mean photon number in the signal wave to be + + + (11) . F i , ( t ) z (a,+(t) qs(t)> = c2 la,l2 82(jai12 1 ) 2~ l a 8 1 /ail C O ~ @ , while the quantity characteristic of the photon statistical properties of the signal wave reads q f ( t ) >- (q:(t) q8(t))2 2 8 C I a,I ai I cos 0 + 82 Here, the angle 0 stands for I + - qli 72 -2 ' = 2s2{c2 a, 12 0 = i p p - ip, I I m p } + s4. (12) (13) cxr, where ips and qi are the phases of the complex numbers a: and respectively. The corresponding relations for the idler wave are obtained from eqs. (11) and (12) by interchanging and mi. We should like to remind the reader that the sign of the quantity A, (or Aj) which is proportional t o the excess coincidence counting rate t o be measured in a HANBURY BROWN and TWISStype [17] experiment, indicates whether bunching (+) or antibunching (-) occurs. It is obvious from eq. ( 1 2 ) that d,(t) is always positive, i.e. the signal (and, similarly, the idler wave), exhibits bunching. This result is, in fact, not surprising, since the elementary process - the "fusion"of a pump photon from both a signal and a n idler photon - has the character of one-photon absorption for the signal and the idler wave, respectively. 4. Cross Correlations More interesting are the cross correlations between the two waves. Their magnitude can be characterized by the quantity Acrow(t) 7' = <!I,+ ( t ) !d(4 qdt) q,(tD - <a,+( t ) q,(t)> <qif (0 q&)) (14) 92 R. PAUL and W. BRUNNEB Utilizing eqs. (9) and (11)we obtain the simple result We speak of anticorrelations, when A,,,,, becomes negative. Obviously, the specification 0 = z will provide the best opportunity to achieve this. It is evident from eq. (11)that this choice corresponds to maximum attenuation of both the signal and the idler wave. Let us assume, for simplicity, that the initial mean photon numbers are the same in the signal and the idler wave, lasl2 = 1 0 1 ~ 1 ~ . According t o eq. (11)and the corresponding equation for iii(t), this remains so for all times. Moreover, the fluctuations are also equal in both beams, A&) = A&). Then the ratio Ac,oss(t)/i,s(t)is practically independent of Zs(0) for, say G,(O) = I o I , ~ ~2 100, in the early stage of the interaction process. Its temporal behaviour is represented in Fig. 1for different initial values E,(O). One recogni- Fig. 1. Evolution of cross correlations between the signal and the idler wave for different values of the initial mean photon number z,(O) = Zi(0) zes that AermB(t)/G8(t)is negative for not too large values of t ; this means that indeed anticorrelations are generated between the signal and the idler wave. After a certain time has elapsed, the anticorrelations are converted, however, into correlations. This can be explained by the fact that at this time the contributions due t o spontaneous processes (amplified parametric fluorescence signals in the signal and the idler wave) become dominant. Physically, it is clear that this will happen the earlier the smaller i,,(O). This feature is displayed in Fig. 1. We should like to emphasize that the occwrrence of antic*orreliLtionsbetween two waves is in no way a matter of surprise in cblassical wave theory. For exaniple. iint,icorrelations have recently been observed in light scattering froin rionsphericttl pnrt ic'les Anticorrelations in Parametric Three-Wave Interaction 93 in dilute solution [HI. The experimental technique consisted in measuring the cross correlation of signals from two spatially separated detectors each of which received light ’ from the same small number of scatterers. Denoting the (fluctuating)deviation of the intensity in the beam 1or 2, respectively, from its average value by - - i , = I, - I,, i, = I , - I,, we can write the classical analog of the cross correlation (14) as A Ccross(t) l w ((ii(t) iz(t)>>, where the double bracket indicates ensemble averaging. As can easily be proved by means of Schwarz’s inequality, the quantity (17) satisfies the following inequality well-known in classical statistical communication theory (see e-g. t191) 2 What is really exciting from the classical point of view is the fact that the anticorrelations produced in nondegenerate parametric three-wave interaction are stronger, in the early stage of the interaction process, than those allowed inthe framework of a classical description, namely, they violate the inequality IAcrowI 5A, (19) I Fig. 2. Comparison of cros8 and autocorrelations. Curve a represents -Ac,&# and curve bA,/Z,, versus t,ime. The initial values of the mean photon numbers are QO) = QO) = 10 that corresponds to (18) in case A, = A P This is seen from Fig. 2, where both the quantities -Across/%8 and A,F8 have been plotted for a typical case. Hence, the anticorrelations in question are, in fact, incompatible with classical wave theory and, therefore, reflect a specific quantum mechanical feature of the radiation field. This nonclassical behaviour manifests itself more directly in antibunching properties displayed by the superposition field made up by the signal and the idler wave, as will be shown in the following section. H. PAULand W. BRL-KSER 94 5. Antibunching It is well known since the pioneering work of GLAUBER [ 2 0 ] that the coincidelice counting rate in a HANBURY BROWN and TWISStype [ 1 5 ] experiment is proportional t o the quantity (20) Here E ( + ) ,E ( - ) denote the positive and negative frequency part, respectively, of t'he operator for the electric field strength, at a given point r in space, and z is the response time of the (broad-band) detectors. For simplicity, the field has been assumed t o he linearly polarized. Considering now the fieldcomposed of thesignal and the idler wave, which are both idealized a s single-mode states of the radiation field, we need to take into account in eq. ( 2 0 ) only the contributions corresponding t o those two modes. (Note that, due to the normal ordering of the operators in eq. ( 2 0 ) , non-excited niodes do not contribute t o the quantity R ( t ) ) This . means, we may put E(*)(r, t ) = @*)(r, t ) + @*)(r, t ) , 111) where E!+),E i - ) ,in the considered case of running plane waves, in explicit terml;. rend (22) (LO, circular frequency and k, wave vector of the signal wave, V mode volume). Siinilnr relations hold for the idler wave. We now make the essential assumption that the frequency difference Ato = I w g - rot is large enough to satisfy the condition I d M Z 1 . (25) Physically, this means that the detectors average out the short-time fluctuation* clue t o the beating of the two modes. I n fact, it is only in such circumstances that the superposition field can be expected to exhibit antibunching. since otherwise the aforementioned fluctuations will give rise t o strong bunching. For the sake of simplicity let us further assume that the value of w,/oi is still close t o unity which is compatible with (23). (Even when z i s chosen, hy order of magnitude. as small as 10V2 s, as it has been demanded from an analysis of the correlation length of the antihunching phenomenon in nondegenerate parametric three-wave interaction [21], both requirements are fulfilled, e.g., for o,= (1015 1013) s-1 and m i = lof5 s-l). We thus may neglect the difference between o,and mi,which comes into play through the square root in eqs. ( 2 2 ) , and utilizing the assumption ( 2 3 ) we find from eq. (20, tlliit the coincidence counting rate, apart from a constant factor, is given by + 2(t)= <41+2(t)4 : w + ( 4 Z 2 ( t )4 w + 2(4i!-(t) 4dt) 4,(t)>* (24) From eq. (24) follows the excess coincidence counting rate (with respect to randoni coincidences) to be proportional to d(t) +A i + ZAcro,, where the above notation has been used. (5) Anticorrelationa in Parametric Three-Wave Iiiteract,ion 95 Specializing, as before, t o the case E,(O) = Ei(0),we have A (t) = 2 ( 4 + (26) ~cr0.9,)* Due t o the fact stated in the previous section, that the cross correlation term is negative and violates the inequality (19) in the early stage of the interaction process: A ( t ) becoines indeed negative which is a n indication of antibunching. Hence the latter effect, in case of nondegenerate parametric three-wave interaction, clearly has its root in the magnitude of the anticorrelations which is incompatible with classical wave theory. 1 A convenient measure of the antibunching effect is the quantity A/(%, Zi) = -A/Z8, 2 whose explicit value follows froin eqs. (26),.(12), (14) and (11)as (cf. also [12]) + A _ Piz, sinh (yt) (2Z,(O) [sinh (3yt) - cosh (3yt)]+ E,(O) [cosh (2yt) - sinh (2yt)l 1 [sinh (3yt)- sinh(yt)]) 2 + sinh2 ( y t ) (27) Making a quantitative comparison with degenerate parametric three-wave interaction [4],one observes that eq. (27),at given t o t a l number of photons in the initial state, differs from the corresponding expression applying t o the degenerate case only in that the noise contributions are larger by a factor of 2. Since the amplified noise is relevant, in the early stage of the interaction process, for small initial photon nuinbers only, one can say that the magnitude of the antibunching effect is practically the same in the 10 and not too large times. degenerate and the nondegenerate case, for 5, Finally, we should like t o remark that the restriction (23) can be avoided, when the signal and the idler wave are linearly polarized in mutually orthogonal directions. I n this case it has recently been shown [22] that the antibunching effect will be displayed by the interference field produced by passing the two beams (whose frequencies are assumed t o coincide) through a n appropriately orientated analyzer. Acknowledgement. The authors are greateful to Dr. A. BANDILLA for stiinulating discussions. References [l] H. J. KIMBLE,M. DAGENAIS and L. MAXDEL, Phys. Rev. Lett. 39, 691 (19ii). [Z] M. DAGENAIS and L. M&DF.L, Phys. Rev. A 18, 2217 (1978). [3] N. CHANDRAand H. F'RAKASH,Phys. Rev. A 1,1696 (1970). [4] D. STOLER,Phys. Rev. Lett. 33, 1397 (1974). and R. LOUDON, J. Phys. A 8, 539 (1975). [5] H. D. SINAAN [6] H. PAUL,U. Mom, and W. BRUNNER, Opt. Commun. 17, 145 (1976). [7] A. BANDILLA and H.-H. RITZE,Ann. Physik Leipz. 33, 207 (1976). [8] U. M o m and H. PAUL,Ann. Physik Leipz. 36, 461 (1978). and H.-H. RITZE,Opt. Commun. 88,126 (1979). [9] A. BANDILLA [lo] H.-H. RITZEand A. BANDILLA, Opt. Commun. 29,126 (19i9). [ll] L. WTA and J. PE~~INA, Czech. J. Phys. B 27, 831 (1977). Czech. J. Phys. B 28, 392 (19782, [12] L. MI~TAand J. PE~~INA, [13] T. V. TRONG and F.-J. SCHUTTE,Ann. Physik Leipz. 36, 216 (1978). and J. PEKINA,Czech. J. Phys. B 98, 1183 (1958). [14] V. PE~~INOVA and H. PAUL,Progress in Optics, ed. by E. WOLF.Vol. 15. p. 21. North-Holland [15] W. BRUNXER Publishing Company, Amsterdam 1977. H. PAULand W. BRUNKER 96 [16] R. J. GLAUBER,Phys. Rev. 131, 2766 (1963). [17] R.HAXBURPBROWSand R. Q. Twrss, Nature Lond. 177, 27 (1956). Phys. Rev. Lett. 43, 1100 (1979). [18] W. G . GRIFFIK and P. N. PUSEY, [19] D. MIDDLETOS,An Introduction t o Statistical Communication Theory, p. 188. McGraw-Hill, New Tork 1960. [10] R. J. GLIUBER,Optical Coherence and Photon Statistics, in: C. DEWITT, A. BLANDIX and C. COHES-TASNOUDJI(Eds.), Quantum Optics and Electronics, Gordon and Breach, Xew York 1966. [21] H.PAULand W.BRUPITER, Opt. Acta 27, 263 (1980). [ E l A. B A K D ~ Land A H.-H. RITZE.Opt. Commun. 34,190 (1980). Bei der Redaktion eingegangen am 10.September 1980. Anschr. d. Verf.: Prof. Dr.H. PAUL und Prof. Dr. W. BRUNNER Zentralinstitut fur Optik und Spektroskopie der Akademie der Wissenschaften der DDR DDR-1199 Berlin. Rudower Chaussee G

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