# Wave-Particle Duality || Duality of Fluctuations, Fields, and More

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<ul><li><p>CHAPTER 4 </p><p>DUALITY OF FLUCTUATIONS, FIELDS, AND MORE </p><p>PETER E. GORDON </p><p>1. INTRODUCTION </p><p>Much of the contemporary literature on duality is an attempt to eliminate the concept. That is, it consists of efforts to account for phenomena without introduc-ing the wave-particle duality. </p><p>Surdin(l) questions whether the "photon concept is really necessary?" by substituting wave packets. Marshall(2) tries for a duality without particles while Bach's(3) particles simulate wave behavior. Garuccio's(4) enhanced photon detec-tors produce the effect of duality without the basic property. The list of these ambitious and well-developed strategies could be extended. </p><p>Another approach is to dissolve the mystery or dilemma of duality by physically unifying the subject. Chief among these ideas is the coexisting wave and particle of de Broglie and more recently Selleri. (5) </p><p>Our approach is to accept duality and to attempt to broaden the concept. Our idea of duality is as a dynamic property whose consequences have not yet been fully explored. In fact, after extending its applicability, we advance duality as a worldview. </p><p>In the next three sections, we try to add to the properties of light to which we apply the wave-particle duality. In Section 2 we generalize Einstein's original result for the fluctuations of blackbody radiation to apply to any field. We do this by separating the operators for the fluctuations into wave and particle parts according to criteria drawn from the quantum theory of coherence. We define </p><p>PETER E. GORDON Physics Department, University of Massachusetts at Boston, Boston, Massa-chusetts 02125, USA. </p><p>Wave-Particle Duality, edited by Franco Selleri. Plenum Press, New York, 1992. </p><p>69 </p><p>F. Selleri (ed.), Wave-Particle Duality Plenum Press, New York 1992</p></li><li><p>70 PETER E. GORDON </p><p>orders of wave fluctuation which we relate to mth-order coherence. We apply this to the wave-particle fluctuations of nonclassical states and relate the wave fluctuation to squeezed states. </p><p>In Section 3 we inquire as to what extent the fields themselves can be separated into wave and particle parts. We reach certain conclusions which we apply to nonclassical states and to the vacuum. </p><p>In Section 4 we look at the two radiation processes, spontaneous and stimulated, asking what this duality has to do with the wave-particle duality. We propose a model for stimulated emission in terms of the nonclassical states discussed earlier and compare the results of this model with those of recent experiments. We offer yet another argument relating spontaneous emission and the vacuum fluctuations. For each radiation process we examine the stimulation process separately from the emission process. We conclude this section with a hypothesis regarding radiation. </p><p>In Section 5 we analyze a number of experiments proposed and performed with the object of detecting the wave. We try to show in each experiment that the wave is unobservable and propose this as a principle; that half of the duality, the wave, is not directly observable. </p><p>In Section 6 we look at the consequences of this unobservability and its rela-tion to measurement, and the relation of wave-particle duality to measurement. </p><p>We close by presenting duality as a worldview. Einstein's reality may be difficult to resurrect. Duality offers us an alternative. </p><p>2. DUALITY OF FLUCTUATIONS </p><p>The first clear indication of the wave-particle nature of light was due to Einstein. He showed that the fluctuations in blackbody radiation, with which he and Planck were concerned at the time, could be divided into two parts: one part is derivable from the wave properties of light, the other is due to its particle properties. </p><p>We will begin by deriving Einstein's original separation using the methods of the quantum theory of coherence. We will generalize Einstein's result from blackbody radiation. We will do this by showing that the operators for the fluctuations can be separated into wave and particle parts. Our results will then hold for any field. </p><p>2.1. Generalized Wave-Particle Fluctuations </p><p>One reason for generalizing the separation of the fluctuation is that dualism is not confined to blackbody radiation and so the separation of the fluctuation should not be either. We demonstrate this separation of the fluctuations to second, third, and fourth orders explicitly. By defining orders of "wave" fluctuation we give new definitions of mth-order coherence and mth-order coherent states. It is </p></li><li><p>DuALITY OF FLUCTUATIONS, FIElDS, AND MORE 71 </p><p>interesting to see that the concepts of dualism can be used to reproduce quantita-tive results of contemporary theory. </p><p>The fluctuations in number may be expressed as </p><p>(~n)2 = n + nz (1) </p><p>Einstein(6) showed that the first term would arise from a system consisting of independent particles and the second from a system of waves. </p><p>We would like to generalize Einstein's result using the methods of coherent states. (7) </p><p>We will separate the operators for the fluctuation into two parts according to the following definitions: </p><p>1. The first, or "particle" part, will give the Poisson fluctuation. 2. The second, or "wave" part, will give a zero result when applied to a </p><p>coherent state. </p><p>This is a reasonable definition in view of the association between the coherent state and the classical wave. The definite-valued wave gives zero "wave" fluctua-tion, i.e., no fluctuation beyond the Poisson. An operator for the fluctuation which gives zero on the coherent state will therefore be called a wave fluctuation. </p><p>In general, the wave fluctuation will be nonzero. Its value will depend on the nature of the state. We will see that for blackbody or Gaussian radiation it gives the usual result. Notice that the wave fluctuation is defined with reference to the field, i.e., to the coherent state. The fluctuations are in terms of the number operator. We will discuss fluctuations of the field in the next section. </p><p>The simplest case is the second-order fluctuation in n, the case considered by Einstein: </p><p>(~ii)2 = (a+a - (n2 = a+a + a+a+aa - 2a+ (n) + (n)2 = a+a + W(2) (2) </p><p>The first term gives the Poisson fluctuation while the second, which we have labeled WZ, gives zero for a coherent state, </p><p>(3) </p><p>while giving of course (n}2 for the well-known wave fluctuation of the Bose-Einstein distribution described by the Gaussian density operator </p><p>(4) </p></li><li><p>72 PETER E. GORDON </p><p>We note for future reference the third- and fourth-order Poisson fluctuations: </p><p>( A )3 = -~n Poisson n (an)4poisson = 3n2 + n </p><p>For the third-order case we have </p><p>Using </p><p>we calculate </p><p>(an)3 = a+a + [2(n)3 - 3(n)(a+)2a2 + (a+)3a3] + [3(a+)2a2 - 3(n)2] = a+a + W(3) </p><p>(5) (6) </p><p>(7) </p><p>where we have bracketed terms of the same order to show that W(3) goes to zero in the limit of a coherent state. </p><p>The wave fluctuation is of course nonzero for general states and may be calculated. </p><p>As an example, the third-order wave fluctuation for II') or blackbody radiation [the distribution Eq. (4)] may be calculated from </p><p>(I'IW(3)(a+ ,a)h) = Tr{W(3)(a+ ,a)p} = ('TT(n)-IW(3)(a* ,a) . exp -laI2j(n)d2a </p><p>which holds because W(3) (a+ ,a) is normal ordered. We then get </p><p>(9) </p><p>In fact, these methods can be used to obtain a quantitative measure of the wave fluctuations of any field. </p><p>The fourth-order Poisson fluctuations Eq. (6) differ from the second and third orders so that it is worthwhile to see whether this separation holds in the fourth order. </p><p>We make use of an expansion for normal-ordered moments. (8) </p></li><li><p>DuALITY OF FLUCfUATIONS, FIELDS, AND MORE 73 </p><p>(10) </p><p>[The coefficients (for the normal-ordered moment) are called homogeneous product sums and are equal to the product of the first f!.. integers taken v at time, repetitions allowed.] First we expand </p><p>(a+a - {n4 = (a+a)4 - 4(a+a)3{n) + 6(a+a)2 {n)2 - 3{n)4 </p><p>Using Eq. (10), we have </p><p>(LlTi)4 = [(a+)4a4 - 4{n)(a+)3a3 + 6{n)2(a+)2a2 - 3{n)4] + [6(a+)3a3 - 12{n)(a+)2a2 + 6{n)2a+] + [7(a+)a2 - 7{n)2] + 3{n)2 + (n) = 3{n)2 + (n) + W(4) (11) </p><p>We have again grouped terms of common order to show that W(4) gives zero for a coherent state leaving the fourth-order Poisson fluctuation. </p><p>Note that these results are for operators and therefore apply to any field, not just blackbody radiation. </p><p>2.2. Nonclassical States </p><p>Now we would like to apply these methods to nonclassical states, states with a limited number of photons. This will later allow us to consider the dualistic properties of the vacuum. </p><p>The concept of wave fluctuation may be used to define mth-order coherence(9) and mth-order coherent states. </p><p>We recently explored properties of second-order coherent states. (10) The states satisfy the second-order coherence condition </p><p>{IIi: (a+a)2: III) = (IIla+alII)2 (12) where the dots denote normal ordering. </p><p>For a state containing up to two photons in a given mode we have states of the form </p><p>2 </p><p>III) = I Cnln) n=O </p><p>(13) </p><p>Assuming the normalization condition </p><p>Ilc 12 = 1 n n </p><p>(14) </p></li><li><p>74 PETER E. GoRDON </p><p>as well as Eq. (12) we obtain (up to a phase factor) a one-parameter family of states in which </p><p>[ (n)2 J1I2 ICol = 1 + T - (n) Icll = [(n) - (n)2]112 </p><p>V2 Ic21 = T(n) </p><p>(15) </p><p>(16) </p><p>(17) </p><p>with (n) denoting the average number of photons in a given mode. We note that Eq. (16) implies that the value of (n) is limited to </p><p>0:0:;; (n) :0:;; 1 (18) </p><p>The second-order coherent state with maximal (n) is </p><p>III) = ViIO) + VI12) (19) Calculation shows that although </p><p>(20) </p><p>we have </p><p>(21) </p><p>Wave fluctuations may therefore be used to define orders of coherent states: mth-order coherent state has zero wave fluctuation up to mth-order W</p></li><li><p>DuALITY OF FLucruATlONS, FIELDs, AND MORE </p><p>'JYpe of behavior Photon bunching Coherent states Squeezed states </p><p>3. DUALITY OF FIELDS </p><p>(W(2 > 0 (W(2 = 0 (W(2 < 0 </p><p>Fluctuations </p><p>Above Poisson Poisson Below Poisson </p><p>75 </p><p>Having separated the fluctuations it is natural to attempt to extend this separation to the fields themselves. To what extent can the fields be separated into wave and particle parts? </p><p>The duality of fields would, like the duality of fluctuations, show wave and particle parts being displayed simultaneously. We do not, however, have in mind any spatial separation oflight into waves and particles. Photons in any case cannot be localized in the same sense that a particle in nonrelativistic quantum mechanics can be located. About the difficulty of placing the wave in space and time we will have more to say later. So, although particles and waves are often described visually, we do not hold out hopes for a visualizable model of duality. </p><p>We will treat in turn classical and nonclassical fields because, as we will see, our method of separating classical fields does not apply to nonclassical ones. </p><p>3.1. Classical Fields </p><p>Classical fields can be described by a positive definite probability function P(o.). For these fields the convolution theorem allows us to separate P(o.) into two parts: </p><p>1. A coherent state with </p><p>(23) </p><p>where </p><p>a = f aP(o.)d2o. (24) is the mean value of the total field. </p><p>2. A field P(o. + a) for which the mean value vanishes and is commonly known as an unphased field: </p><p>f aP(o. + a)d2o. = 0 (25) In the terms of the convolution theorem, we have </p><p>P(o.) = f 8(2)(0. - a - o.')P(o.' + a)d2o.' (26) </p></li><li><p>76 PETER E. GoRDON </p><p>The coherent state, P(a) = 8(2)(a - (i), is as close as we can get to a classical wave in quantum theory. It is in fact a classical wave plus the (unavoidable) vacuum fluctuations. For these fields the amplitudes add as in classical wave theory; that is, it will interfere as a classical wave. </p><p>For unphased fields on the other hand, the intensities add. When combining a number of such fields we can add the number of "particles" in each unphased field to get the number in the total field. </p><p>Of course the unphased field is not exactly a set of classical particles any more than the coherent state is exactly a classical wave. Second-order correlations for this state will differ from those for the Gaussian, as an example. </p><p>This separation of the field allows us to discuss the fluctuations of the field (aa)2. All of our results on the duality of fluctuations in the last section were in terms of the number operator. The field was brought in only to define the wave fluctuations wen). </p><p>The fluctuation of the field operator for the coherent part of the field is zero, because the coherent part of the field is a delta function in a. </p><p>For the unphased field we have for the mean number of equivalent particles </p><p>(n)unPhased = f lal2P(a + (i)d2a = f la + al2p(a + (i)d2a - a2 = la21 - la21 = (aa)2 (27) </p><p>The fluctuations of the field do not divide up neatly into wave and particle parts like the fluctuations for the number operator. But the above result does show dualistic properties: </p><p>The means square fluctuation (the variance) of the amplitude of the total field is equal to the average number of particles in the unphased part of the field. </p><p>Let us do a number of "checks" on this conclusion. Consider the number of particles in the other part of the field: </p><p>(28) </p><p>Combining the number of particles in each part of the field </p><p>(n)unPhased + (n)coherent = la21 - lal2 + lal2 = la21 (29) Since </p><p>(30) </p><p>we have </p><p>(n)unPhased + (n)coherent = (n)total field </p></li><li><p>DuALITY OF FLUCTUATIONS, FmLos, AND MORE 77 </p><p>Now consider the special case </p><p>(n)unphased = 0 = (~a)2 (31) </p><p>We have: </p><p>1. No fluctuation of the field 2. Total field = la) (because (n)unPhased = 0) </p><p>Now let us introduce a result in order to draw another conclusion. Hillery, (13) and earlier Aharonov et al. ,(14) showed that for a classicalP(a) any pure state must be a coherent state la). </p><p>In terms of our results we see that in order for a classical state to have particle properties it must be a mixture. In fact, we can say that particles are the mixture properties of states. That is, insofar as a classical state can be said to consist of particles it must be a mixture. </p><p>We will demonstrate the same property in nonclassical states. A particular example of unphased fields is P(lal) which does not depend on </p><p>the phase of a. Such stationary fields have density matrices in the number representation which reduce to diagonal form. They are thus equivalent to a mixture of number states. Laser light is often described as a convolution of the Gaussian P(a) [Eq. (4), another mixture of number states] and the coherent state. </p><p>This then is the extent to which fields with positive-definite P(a) can be separated into wave and particle parts. </p><p>3.2. Nonclassical Fields </p><p>Not all superpositions of the number states can be described by a positive-definite P(a). We met up with such a state in Eq. (19). These nonclassical states have regions where P(a) is negative but which cannot be observed because of the vacuum fluctuations. That is, the negative values are restricted to a range so small that they are always averaged out by the classical observation. We will elaborate on this point after discussing the vacuum in the next section. </p><p>Because of this property the diagonal of the density matrix of the field (alpla) is better suited f...</p></li></ul>