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Squeezed and Correlated States of Quantum Fields And View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server SQUEEZED AND CORRELATED STATES OF QUANTUM FIELDS AND MULTIPLICITY PARTICLE DISTRIBUTIONS V.V. Do donov, I.M. Dremin, O.V. Man'ko, V.I. Man'ko, and P.G. Polynkin Leb edev Physical Institute, Leninsky Prosp ekt, 53, 117924, Moscow, Russia Abstract The primary aim of the present pap er is to attract the attention of particle physicists to new developments in studying squeezed and cor- related states of the electromagnetic eld as well as of those working on the latest topic to new ndings ab out multiplicity distributions in quantum chromo dynamics. The new typ es of nonclassical states used in quantum optics as squeezed states, correlated states, even and o dd coherent states (Schrodinger cat states) for one{mo de and multimode interaction are reviewed. Their distribution functions are analyzed ac- cording to the metho d used rst for multiplicity distributions in high processed by the SLAC/DESY Libraries on 28 Feb 1995. 〉 energy particle interactions. The phenomenon of oscillations of particle distribution functions of the squeezed elds is describ ed and confronted to the phenomenon of oscillations of cumulant moments of some dis- tributions for squeezed and correlated eld states. Possible extension PostScript of the metho d to elds di erent from the electromagnetic eld (gluons, pions, etc.) is sp eculated. HEP-PH-9502394 1 1. INTRODUCTION The nature of any source of radiation (of photons, gluons or other entities) can b e studied by analyzing multiplicity distributions, energy sp ectra, var- ious correlation prop erties, etc. A particular example is provided by the coherent states of elds which give rise to Poisson distribution. However, in most cases one has to deal with various distributions revealing di erent dynamics. We try to describ e some of them app earing in electro dynamics and in chromo dynamics but start rst with coherent states. As incomplete as it is, this review, we hop e, will stimulate physicists working on di erent topics to think over similarity of their problems and to lo ok for common solutions. The coherent states for photons have b een intro duced in [1]. They are widely used in electro dynamics and quantum optics [2, 3]. The coherent states of some nonstationary quantum systems were constructed in [4]-[7], and this construction was based on nding some new time{dep endent in- tegrals of motion. The integral of motion which is quadratic in p osition and momentum was found for classical oscillator with time{dep endent fre- quency a long time ago by Ermakov[8]. His result was rediscovered by Lewis [9]. The two time{dep endentintegrals of motion which are linear forms in p osition and momentum for the classical and quantum oscillator with time{ dep endent frequency,aswell as corresp onding coherent states were found in [5]; for a charge moving in varying in time uniform magnetic eld this was done in [4] (in the absense of the electric eld), [10,11] (nonstation- ary magnetic eld with the \circular" gauge of the vector p otential plus uniform nonstationary electric eld), and [12] (for the Landau gauge of the time-dep endentvector p otential plus nonstationary elecrtic eld). For the multimo de nonstationary oscillatory systems such new integrals of motion, b oth of Ermakov's typ e (quadratic in p ositions and momenta) and linear in p osition and momenta, generalizing the results of [5]were constructed in [7]. The approach of constructing the coherent states of the parametric systems based on nding the time{dep endentintegrals of motion was reviewed in [13] and [14]. Recently the discussed time{dep endentinvariants were obtained using No ether's theorem in [15]-[17]. The coherent states [1] are considered (and called) as classical states of the eld. This is related to their prop erty of equal noise in b oth quadrature comp onents (dimensionless p osition and momentum) corresp onding to the noise in the vacuum state of the electromagnetic eld, and to the minimiza- tion of the Heisenb erg uncertainty relation [18]. These two prop erties yield the Poisson particle distribution function in the eld coherent state, which is the characteristic feature of the classical state. On the other hand, the so{called squeezed states of the photons were considered in quantum optics for the one{mo de eld [19]-[23]. The main prop erty of these states is that the quantum noise in one of the eld quadra- ture comp onents is less than for the vacuum state. The imp ortant feature of the nonclassical states is the p ossibility to get statistical dep endence of the eld quadrature comp onents if their correlation co ecient is not equal to zero. For the classical coherent state and for the particle numb er state there is no such correlation. The correlated states p ossessing the quadra- ture statistical dep endence were intro duced in [24]by usage of the pro cedure of minimization of the Schrodinger{Rob ertson uncertainty relation [25, 26]. The nonclassical states of another typ e, namely,even and o dd coherent states, were intro duced in [27] and called as the Schrodinger cat states [28]. The even and o dd coherent states are very simple even and o dd sup erp osi- tions of the usual coherent states. The particle distribution in these states di ers essentially from the Poissonian statistics of the classical states. Its most striking feature is the oscillations of the particle distribution functions whichischaracteristic prop erty of nonclassical light [29]-[31]. They are es- p ecially strong for even and o dd coherent states [27,32], moreover, they are very sensitive to the correlation of the quadrature comp onents [31]. These phenomena are typical not only for the one{mo de photons pro duction, but also for the multimo de case [33]-[35]. The multimo de generalizations of the Schrodinger cat states were studied in [32, 35]. Let us turn now to high energy particle interactions. Several years ago the exp erimentalists of UA5 Collab oration in CERN noticed [36] a shoul- der in the multiplicity distribution of particles pro duced in pp collisions at energies ranging from 200 to 900 GeV in the center of mass system. It lo oked like a small wiggle over a smo oth curve and was immediately as- crib ed by theorists to pro cesses with larger number of Pomerons exchanged in the traditional schemes. More recently, several collab orations studying + e e collisions at 91 GeV in CERN rep orted (see, e.g., [37, 38]) that they failed to t the multiplicity distributions of pro duced particles by smo oth curves (the Poisson and Negative Binomial distributions were among them). Moreover, subtracting such smo oth curves from the exp erimental ones they found steady oscillatory b ehaviour of the di erence. It was ascrib ed to the pro cesses with di erentnumb er of jets. In such circumstances one is tempted to sp eculate ab out the alterna- tive explanation when considering p ossible similarity of these ndings to typical features of squeezed and correlated states. Indeed, we know that the usual coherent states app eared imp ortant for the theory of particle pro- duction [39]. Moreover, the squeezed states, b eing intro duced initially for solving the problems of quantum optics, now b egin to p enetrate to di erent other branches of physics, from solid state physics [40]-[42] to cosmology and gravitation [43]-[46]. Thus why they could not arise in particle physics? In slightly di erent context the ideas ab out squeezed states in particle physics were promoted in [47]-[49]. No attempts to use analogy with other typ es of nonclassical states likeSchrodinger cat states or correlated states are known. In the ab ove approach to exp erimental data, however, the form of oscil- lations dep ends on the background subtracted. The new sensitive metho d was prop osed in [50,51] (for the review see [52]). It app eared as a bypro duct of the solution of the equations for generating functions of multiplicity dis- tributions in quantum chromo dynamics (QCD). It app eals to the moments of the multiplicity distribution. According to QCD, the so{called cumulant moments (or just cumulants), describ ed in more details b elow, should reveal the oscillations as functions of their ranks, while they are identically equal to zero for the Poisson distribution and are steadily decreasing functions for Negative Binomial distribution so widely used in phenomenological analysis. Exp erimental data show the oscillatory b ehaviour of cumulants (see [53]) of multiplicity distributions in high energy inelastic pro cesses initiated by various particles and nuclei, even though some care should b e taken due to the high multiplicity cut-o of the data. When applied to the squeezed states, the metho d demonstrates [54] the oscillations of cumulants in slightly squeezed states and, therefore, can b e useful for their detection. As wehave already p ointed out ab ove, the aim of this article is to review the prop erties of the nonclassical states (squeezed, correlated, even and o dd coherent ones) b oth for one- and multimo de cases in the context of p ossible applications of them to high energy physics, as well as to acquaint with new metho ds of analysis of multiplici ty distributions. However, we will not sp eculate here ab out any application in more details, leaving the topic for future publications. The article is organized as follows. In sections 2 and 3 we describ e the quantities used in analysis of multiplicity distributions for one{dimensional and multidimensional cases, resp ectively.
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