CERN-TH/95-37 DFTT-14/95 VON NEUMANN AND SHANNON-WEHRL ENTROPY FOR SQUEEZED STATES AND COSMOLOGICAL PARTICLE PRODUCTION M. Gasp erini Dipartimento di FisicaTeorica, Via P.Giuria 1, 10125 Turin, Italy and M. Giovannini Theory Division, CERN, CH-1211, Geneva 23, Switzerland Abstract We show that the e ective coarse graining of a two-mo de squeezed density matrix, implicit in the Wehrl approaches to a semiclassical phase-space distribution, leads to results in agreement with previous di erent de nitions of entropy for the pro cess of pair pro duction from the vacuum. We also present, in this context, a p ossible processed by the SLAC/DESY Libraries on 19 Feb 1995. 〉 interpretation of the entropy growth as an ampli cation (due to the squeezing) of our lack of knowledge ab out the initial conditions, which gives rise to an e ective decoherence of the squeezed density matrix. PostScript ||||||||| To app ear in the \String gravity and physics at the Planck energy scale " (World Scienti c, Singap ore, 1995) CERN-TH/95-37 February 1995 HEP-TH-9502112 VON NEUMANN AND SHANNON-WEHRL ENTROPY FOR SQUEEZED STATES AND COSMOLOGICAL PARTICLE PRODUCTION M. Gasp erini and M. Giovannini Dipartimento di FisicaTeorica, UniversitadiTorino, Via P.Giuria 1, 10125 Turin, Italy ABSTRACT We show that the e ective coarse graining of a two-mo de squeezed density matrix, implicit in the Wehrl approaches to a semiclassical phase-space distribution, leads to results in agreement with previous di erent de - nitions of entropy for the pro cess of pair pro duction from the vacuum. We also present, in this context, a p ossible interpretation of the entropy growth as an ampli cation (due to the squeezing) of our lackofknowledge ab out the initial conditions, which gives rise to an e ective decoherence of the squeezed density matrix. Some years ago it was suggested that the in quantum pro cess of pair pro duction from the vacuum, induced by the action of a time-varying classical background, the growth of the average numb er of quanta of a given eld should b e naturally asso ciated to the growth of entropy of that eld [1,2]. More recently, suchan entropy growth was quanti ed on the grounds of a squeezed state approach to the pro cess of particle pro duction [3-7]. With this approach one can easily compute, in particular, the total entropy S stored in the cosmological p erturbations ampli ed by in ation, and one obtains [4,6] 3=2 H f S ' S (1) cmb M p o where S is the thermal black-body entropy of the 3K photon microwave back- cmb ground, H the nal curvature scale of in ation, and M the Planck mass. f p The use of the squeezed state formalism to compute entropy pro duction might seem to give rise to a puzzle, since the squeezed evolution is unitary. The solu- tion of the puzzle is that, of course, there is no loss of information in principle asso ciated to the evolution of the initial state into a nal squeezed quantum state 1 (and so no p ossible information paradox, like in the case of black-hole dynamics). The loss of information o ccurs however in practice, as a consequence of the pro cess of measure of the observables characterizing the nal squeezed state. There are indeed variables (related to the so-called \sup er uctuant" op erators [8]) which are ampli ed by the squeezed evolution and b ecome thus available for macroscopic ob- servations, while other variables (the \sub uctuant" ones [8]) are corresp ondingly \squeezed" and b ecome unobservable. A simple and intuitiveway to illustrate this situation is to consider a semi- classical description of the squeezing pro cess as a parametric ampli cation of the wave function, \hitting" the e ective p otential barrier of a Schro edinger-like equa- tion (with time-likevariable) [9,10]. The nal amplitude of the wave scattered by the barrier may b e larger or smaller than the initial amplitude, dep ending on the phase with which the waveenters the barrier. What is macroscopically detected, however, is the average ampli cation of the wave, out of an initial random distri- bution of phases. This p ointwas clearly stressed already in the rst pap er on the cosmological ampli cation of tensor p erturbations [11]. A macroscopic observa- tion of the quantum uctuations of the metric, ampli ed by the cosmic evolution, traces out over phases. In order to compute the entropy asso ciated to a squeezing pro cess, this macro- scopic loss of information has to b e taken into account in the form of a suitable \coarse graining" of the density matrix of the nal (pure) squeezed state. Sucha coarse graining was originally p erformed in a Fock basis, i.e. with resp ect to the eigenstates of the numb er op erator, byaveraging over phases [3-5]. The number op erator, however, is only a particular example of sup er uctuantvariable which grows in the case of a pair pro duction pro cess, while the conjugate variable (the phase) is corresp ondingly squeezed. This suggests the use, in general, of the eigen- states of the sup er uctuant op erator as the natural basis for the coarse graining of a squeezed density matrix [6,7]. A p ossible di erent basis, that of coherent states with real amplitude, has b een suggested however in [12]. It is imp ortant to stress that all these di erent approaches [3-7,12] to the entropy of a squeezed state exactly lead to the same result when the entropy growth S , for eachmodek,isevaluated in the large squeezing regime, namely k S ' 2r ; r >> 1 (2) k k k where r is the mo dulus of the (complex) squeezing parameter [8]. In the previ- k ously quoted pap ers, however, the decoherence pro cess which leads to a reduced 2 density matrix (from which to compute the quantum Von Neumann entropy) red is imp osed \by hand", by setting the o -diagonal terms of the matrix to zero in some chosen basis; at b est, like in the case of the numb er basis, it is justi ed in a semiclassical way using a sto chastic approach and a sort of random-phase approx- imation [3,4]. The problem is that it is formally imp ossible in the numb er basis, as well as in the more general sup er uctuant basis, to compute the quantum von 2 ) obtained Neumann entropy(Tr ln ) from a reduced matrix ( 6= red red red red by tracing out phases, or another sub uctuantvariable. Indeed, a sub uctuant op erator do es not commute with the conjugate sup er uctuant one, so that b oth op erators cannot b e simultaneously included in the complete set of observables characterizing a pure squeezed state. In view of these preliminary remarks, we can say that the aim of this pap er is twofold. First of all wewant to compute the semiclassical Shannon-Wehrl entropy [13,14] for a two-mo de squeezed vacuum state, in order to show that also in that approach one recovers the result (2) for the large squeezing limit. This con rms that the entropy of a highly squeezed state is very robust to the particular scheme of coarse graining implemented (as discussed also in [15,16]). In the second place wewant to present a p ossible justi cation to the reduction of the density matrix in a sup er uctuant basis [6,7], byinterpreting such a reduc- tion as the consequence of our lackofknowledge ab out the initial con guration which is the starting p oint of the squeezed evolution. To this purp ose we shall represent the initial state, in general, as a statistical mixture of coherent states. This representation automatically de nes a quantum von Neumann entropy asso- ciated to the initial mixture. This entropyhowever is constant, as the squeezed evolution is unitary, and then it do es not contribute to the di erence S between nal and initial entropy.However, if the statistical distribution satis es (at least approximately) a principle of equipartition of the probabilities among the states of the mixture, then the o -diagonal terms of the density matrix turn out to b e suppressed. An e ective decoherence is thus pro duced, whose asso ciated entropy in the sup er uctuant basis is eventually ampli ed by the squeezed evolution. The arguments presented in this pap er are intended to apply to any pro cess of pair pro duction, describ ed by a Bogoliub ov transformation connecting the particle y y ~ ~ fb; b g and anti-particle fb; b g annihilation and creation jini op erators to the jouti y y ones, fa; a ; a;~ a~ g . Such a transformation reads, for eachmodek, y ~ a =c (k)b +c (k)b k + k k y y ~ ~a =c (k)b +c (k)b (3) k + k k 3 2 2 where the Bogoliub ov co ecients c (k ) satisfy jc j jc j = 1. By putting + 2i k c (k ) = cosh r ; c (k )=e sinh r (4) + k k the transformation (3) can b e rewritten as a unitary transformation, y y y y ~ a = b ; a~ = b (5) k k k k k k k k generated by the two-mo de squeezing op erator [8] y y 2i k ~ ~ = exp z b b z b b ; z = r e (6) k k k k k k k k k Our discussion can thus b e applied in general to any dynamic situation in which the initial vacuum state j0i evolves towards a nal two-mo de squeezed vacuum state, jz i = j0i,characterized by a non-vanishing value of the squeezing parameter k r = jz j (related to the exp ectation value of the numb er of pro duced particles by k k y y 2 2 hn i = hz jb b jz i = h0ja a j0i = jc (k )j = sinh r ).