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

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    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.

    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.

    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)

    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.

    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

    PETER E. GORDON Physics Department, University of Massachusetts at Boston, Boston, Massa-chusetts 02125, USA.

    Wave-Particle Duality, edited by Franco Selleri. Plenum Press, New York, 1992.


    F. Selleri (ed.), Wave-Particle Duality Plenum Press, New York 1992


    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.

    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.

    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.

    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.

    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.

    We close by presenting duality as a worldview. Einstein's reality may be difficult to resurrect. Duality offers us an alternative.


    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.

    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.

    2.1. Generalized Wave-Particle Fluctuations

    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


    interesting to see that the concepts of dualism can be used to reproduce quantita-tive results of contemporary theory.

    The fluctuations in number may be expressed as

    (~n)2 = n + nz (1)

    Einstein(6) showed that the first term would arise from a system consisting of independent particles and the second from a system of waves.

    We would like to generalize Einstein's result using the methods of coherent states. (7)

    We will separate the operators for the fluctuation into two parts according to the following definitions:

    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

    coherent state.

    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.

    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.

    The simplest case is the second-order fluctuation in n, the case considered by Einstein:

    (~ii)2 = (a+a - (n2 = a+a + a+a+aa - 2a+ (n) + (n)2 = a+a + W(2) (2)

    The first term gives the Poisson fluctuation while the second, which we have labeled WZ, gives zero for a coherent state,


    while giving of course (n}2 for the well-known wave fluctuation of the Bose-Einstein distribution described by the Gaussian density operator



    We note for future reference the third- and fourth-order Poisson fluctuations:

    ( A )3 = -~n Poisson n (an)4poisson = 3n2 + n

    For the third-order case we have


    we calculate

    (an)3 = a+a + [2(n)3 - 3(n)(a+)2a2 + (a+)3a3] + [3(a+)2a2 - 3(n)2] = a+a + W(3)

    (5) (6)


    where we have bracketed terms of the same order to show that W(3) goes to zero in the limit of a coherent state.

    The wave fluctuation is of course nonzero for general states and may be calculated.

    As an example, the third-order wave fluctuation for II') or blackbody radiation [the distribution Eq. (4)] may be calculated from

    (I'IW(3)(a+ ,a)h) = Tr{W(3)(a+ ,a)p} = ('TT(n)-IW(3)(a* ,a) . exp -laI2j(n)d2a

    which holds because W(3) (a+ ,a) is normal ordered. We then get


    In fact, these methods can be used to obtain a quantitative measure of the wave fluctuations of any field.

    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.

    We make use of an expansion for normal-ordered moments. (8)



    [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

    (a+a - {n4 = (a+a)4 - 4(a+a)3{n) + 6(a+a)2 {n)2 - 3{n)4

    Using Eq. (10), we have

    (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)

    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.

    Note that these results are for operators and therefore apply to any field, not just blackbody radiation.

    2.2. Nonclassical States

    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.

    The concept of wave fluctuation may be used to define mth-order coherence(9) and mth-order coherent states.

    We recently explored properties of second-order coherent states. (10) The states satisfy the second-order coherence condition

    {IIi: (a+a)2: III) = (IIla+alII)2 (12) where the dots denote normal ordering.

    For a state containing up to two photons in a given mode we have states of the form


    III) = I Cnln) n=O


    Assuming the normalization condition

    Ilc 12 = 1 n n


  • 74 PETER E. GoRDON

    as well as Eq. (12) we obtain (up to a phase factor) a one-parameter family of states in which

    [ (n)2 J1I2 ICol = 1 + T - (n) Icll = [(n) - (n)2]112

    V2 Ic21 = T(n)




    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

    0:0:;; (n) :0:;; 1 (18)

    The second-order coherent state with maximal (n) is

    III) = ViIO) + VI12) (19) Calculation shows that although


    we have