Appl. Phys. A 87, 585–592 (2007)

Applied Physics A

DOI: 10.1007/s00339-007-3974-0

Materials Science & Processing

Coherence

b. lengeler

2. Physikalisches Institut, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany

Received: 10 October 2006/Accepted: 22 January 2007 Published online: 13 March 2007 • © Springer-Verlag 2007

The concept of coherence is widely used in different areas of physics like in optics, in quantum optics or in neutron and X-ray scattering, however with subtle differences in meaning for the different communities. In quantum optics it is mainly the source of photons and its characterization in terms of coherence functions which is of concern. In a scattering experiment, on the other hand, the source is supposed to be characterized and it is the internal degrees of freedom of the sample which are studied via their influence on the detected interference pattern. It is one of the purposes of this paper to clarify the different concepts and to show how they are interrelated. The paper is organized as follows. First, we will discuss what interferes in a physical event. This will be treated according to the Feynman formulation of quantum mechanics in terms of probability amplitudes and we will describe nine rules on how to calculate with these amplitudes. Then we will discuss what destroys interference. The main part of the paper treats a number of applications from quantum optics and X-ray and neutron scattering. These include quantum beats, Hanbury Brown and Twiss interferometry, entangled states, Einstein–Podolsky–Rosen paradox and speckle from coherently illuminated samples.

ABSTRACT

PACS 42.25.Hz;

1

42.25.Kb

What interferes in a physical event?

In the Feynman lectures (volume III) [1] are given a number of rules on how to deal with probability amplitudes. We will illustrate this method by a Young double slit experiment and by X-ray scattering by a single crystal (Fig. 1). However, the particles which are scattered can be any microscopic particle (photon, neutron, electron, alpha-particle, etc.). Rule 1: First, we define the physical event by fixing the initial and final states. In the present case, one photon of well defined energy propagates from the source S to the detector D and arrives at time t in D. Rule 2: Secondly, we must identify all alternatives how the event can happen. In the double slit experiment the photon can take path 1 by slit 1 and it can take path 2 by slit 2. In the X-ray u Fax: +49 241 80 22306, E-mail: [email protected]

scattering experiment the photon can pass by each electron in the sample (assuming only single scattering in the first Born approximation). Rule 3: Thirdly, we assign to each alternative j a probability amplitude (amplitude) which in bracket notation is written as ϕ j =
final|initialj

(1)

We say, for the double slit experiment, the photon has an amplitude to pass by slit 1 and it has an amplitude to pass by slit 2. This is completely different from saying that the photon passes with a certain probability by slit 1 and with another probability by slit 2. The probability amplitude is the basic new concept from quantum mechanics. Rule 4: When an alternative occurs in sequential steps the corresponding partial amplitudes have to be multiplied. In the double slit experiment, we write ϕ j =
D|Sj =
D| j
j|S

j = 1, 2 ,

(2)

where
j|S is the amplitude for the propagation of the photon from source S to slit j , and
D| j is the amplitude for propagation from the slit to the detector. Rule 5: The total amplitude for the event is the sum of the amplitudes for all, undistinguished alternatives (superposition principle).

ϕ= ϕ j =
D|S =
D| j
j|S . (3) j

j

The emphasis is on “all” and on “undistinguished”. In other words, all alternatives must be considered, and only those amplitudes, which belong to undistinguished alternatives have to be superposed coherently (addition of amplitudes). It is always possible, at least in principle, to find out which alternative was realized. If this information is only available by performing an additional experiment, then the alternative is undistinguished. Physically, the insertion of the closure relation
| j
j| = 1 (4) j

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This is for example the case for a classical electromagnetic wave with many photons in the same state. However, for clarity reasons it is useful to first consider the fate of an individual particle in the event, and only then repeat the event many times under identical conditions. 2

FIGURE 1 In a Young double slit experiment and in a scattering experiment a particle propagates from the source S to the detector D

means the sum over all, undistinguished alternatives, i.e. the alternatives define the base to be inserted. The superposition principle is the center-piece of quantum mechanics. As a consequence, the Schroedinger and the Dirac equations, which are the equations for calculating with the probability amplitudes, are linear equations in these amplitudes. Rule 6: According to Born, the probability P for the event is given by the modulus squared of the total amplitude 
2 
2     2    P = |ϕ| =  ϕj  = 
D| j
j|S . (5) j

j

The mixed terms in this expression describe interference, as indicated in Fig. 1. Rule 7: For a photon with energy E = hω = hc/λ = hck to show up via path 1 in a detector pixel z at time t , it must be emitted from the source at the retarded time t − u 1 /c (Fig. 1). The amplitude for this to happen is ϕ1 (t) = C exp [−iω(t − u 1 /c)] = C exp [−i(ωt − ku 1)] . (6)

This amplitude has the form of a complex wave. It describes the wave character of a propagating quantum mechanical particle. Note that we need the amplitudes of the same photon to be emitted at different retarded times from the source in order for it to be detected at time t in the pixel z . These two amplitudes interfere resulting in the typical oscillating probability   2 2 2 πdz P(z) = |ϕ1 + ϕ2 | = 4|C| cos . (7) λB Here, d is the distance of the slits and B is the distance between the slits and the detector. When the photon is detected, it is detected as a particle, i.e. only one pixel will react. As a consequence, when the event occurs only once, the probability of the event and the intensity are completely different. Only when the event is repeated many times under identical conditions, will the probability and the intensity merge. This was illustrated beautifully in a double slit experiment with electrons by Tonomura [2]. In many experimental situations the gradual merging of probability and intensity is not pursued.

What destroys interference?

All real interference experiments show more or less pronounced smearing of the interference pattern. There are basically two reasons for this smearing: first, it may be known which alternative is taken in the event, and secondly, the event may be repeated under non-identical conditions. The first case is treated by the following rule. Rule 8: When it is known, without further experiment, which alternative is taken by the particle, then this alternative does not contribute to interference (incoherent superposition of amplitudes). The intensity distribution of these events does not follow the interferential probability distribution. Let us consider a double slit experiment, but this time with monoenergetic electrons. When the electrons are not watched then we observe interference, as in Fig. 1. On the other hand, it is possible to observe by which slit an electron passes. For that purpose we can use a light source behind the screen, which emits so many photons that no electron passes unobserved. In that case the interference is lost. If part of the electrons is observed and others are not observed, then we observe an interference pattern on a background. An important consequence of Rule 8 is the following: amplitudes for alternatives with different final states must never be added coherently. This is due to the fact that the different final states distinguish the alternatives unambiguously without need for further test. As an example, we consider a double slit experiment with electrons where the propagating electrons are watched by Compton scattering (Fig. 2). In this version the photon detector D1 observes only photons coming from slit 1 and correspondingly for detector D2 . Here, there are two particles involved in our event, namely an electron and a photon. There is another rule needed for dealing with this situation. Rule 9: When in an event two particles do something simultaneously, but independently, then the corresponding amplitudes have to be multiplied. In the case of our double slit experiment with watched electrons (Fig. 2) the amplitudes for the two alternatives are
D|1
1|S
D1 |1
1|L
D|2
2|S
D2 |2
2|L

(8)

with the interpretation that a photon propagates from L to D1 ( D2 ) via slit 1 (2) and at the same time an electron propagates from S to D via slit 1 (2). The final states are obviously different ( D1 and D versus D2 and D), hence, the amplitudes have to be superposed incoherently and the probability for the event shows no interference P = |
D|1
1|S
D1 |1
1|L|2 + |
D|2
2|S
D2 |2
2|L|2 . (9)

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3

Applications

The rules of how to calculate with probability amplitudes, which have been outlined in the previous sections, will be illustrated in a number of applications from different fields of physics. 3.1

FIGURE 2

A double slit experiment with observed electrons (version 1)

Rule 8 is a formulation of the uncertainly principle: it is impossible to design an apparatus able to determine which alternative the particle takes, without destroying interference at the same time. In the case of the observed electron in our double slit experiment, the electron leaves a trace, namely a photon in detector D1 or D2 . The question arises if any trace left in the event will destroy interference. The answer is that the trace must be localized enough in order to identify an alternative unambiguously. For instance, Compton scattered photons in X-ray diffraction do not contribute to interference (Bragg diffraction) since it is known (in principle) which electron did the scattering. On the other hand, in inelastic coherent neutron scattering (which is used, for instance, for determining phononic dispersion curves) the neutron leaves a trace in the crystal, the phonon, but the trace is as large as the whole crystal. Therefore, the amplitudes must be superposed coherently. We return to the second possibility of how interference might be lost: collecting intensity for non-identical events as described by a quantum mechanical mixture. Interference can only be observed if the event is repeated many times. However, in real experiments there are always uncontrolled degrees of freedom. As a consequence the event is repeated under non-identical conditions leading to (partial) washing out of the interference. This situation is described in quantum mechanics by a density operator. Examples of uncontrolled degrees of freedom in a scattering experiment might be the following: The energy and momentum of the incoming and outgoing particles are not well defined. The sample may show disorder. The detector may show inadequate temporal and spatial resolution. As a consequence, interference can be lost in the source, in the sample, in the detector and in their mutual arrangement. Table 1 gives a summary for the different situations how interference can be lost.

Quantum beats

Interference has primarily nothing to do with waves. Sometimes the probability amplitudes have the form of complex waves, in other cases they do not, as illustrated by quantum beats. Consider an atom or a nucleus with two close energy levels E 1 and E 2 and a short laser pulse of duration δt which excites fluorescence (Fig. 3). We assume the laser pulse so short that its energy spread δE = h/δt is larger than the difference ∆E = E 2 − E 1 . The event is the absorption of a photon by the two-level system and the emission of a fluorescence photon. There are two alternatives for the event to happen: the excitations via level 1 and via level 2. The corresponding amplitudes are written according to Rule 4 as a product of two factors:     i λt ϕ j = a exp − (E j − E f )t exp − . (10) h 2 Here, a is the amplitude for photoabsorption, assumed (for simplicity) to be equal for the levels 1 and 2. The second factor (two exponentials) is the amplitude for the emission of a (fluorescence) photon of energy ( E j − E f ). This amplitude is assumed to be damped in time with a rate γ = 1/τ . When the laser pulse is so short that δE > ∆E then the two alternatives are undistinguished. As a consequence, the total amplitude ϕ = ϕ1 + ϕ2 will give a probability for the emission of a fluorescence photon at time t :    ∆E 2 2 P(t) = |ϕ(t)| = 2 |a| 1 + cos t exp (−γt) . (11) h The probability shows oscillations in time superposed on a decaying background. A single coherent excitation contributes to the intensity in one point t . By repeating the event many times the full curve of probability is observed. Quantum beats are used for determining the separation of close energy levels in atoms and nuclei. 3.2

Spatial Hanbury Brown and Twiss (HBT) interferometer, entangled states

In a typical scattering experiment there is only one particle involved in the event “a particle propagates via a sample from the source to the detector”. It is also possible that more than one particle is involved in the event, as illustrated

System in pure state

System in qm mixture

Probability for event

Always in the same initial and final states Always the same

Very frequent repetition of event

Intensity and probability merge

Different states for different repetitions Different for different repetitions Intensity reveals washed-out probability distribution

Repetition of event

TABLE 1

How interference is lost

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The additional alternatives 3 and 4 are distinguished from another and from the alternatives 1 and 2, since the final states are different. In this case it would be wrong to say that waves interfere with one another or that two particles interfere with one another. The only correct formulation is: what interfere are the amplitudes for all undistinguished alternatives of the same physical event. 3.3

Entanglement of non-identical particles

Entanglement implies at least two particles, which are involved in the event and at least two undistinguished alternatives for the event to happen. However, the particles do not have to be identical, as illustrated in the following version of a double slit experiment with observed electrons (Fig. 4). The photon detectors D1 and D2 are now arranged in such a way that each sees both slits, so that it is not known at which slit the photon is scattered. We consider first the event when one electron propagates from S to D and one photon from L to D1 . There are two undistinguished alternatives: electron and photon pass by slit 1 and electron and photon pass by slit 2. As a consequence the photon and the electron are in an entangled state with amplitude FIGURE 3

Coherent excitation in a double level system

by the Hanbury Brown and Twiss interferometer [3]. We consider two photon sources 1 and 2 and two detectors a and b. First, we assume that one photon from each source propagates to the detectors, one for each detector. There are two alternatives for this to happen: (1) Alternative 1: photon 1 to detector a and photon 2 to detector b with the amplitude
a|1
b|2; (2) Alternative 2: photon 1 to detector b and photon 2 to detector a with the amplitude
b|1
a|2. If the photons are identical (same energy and same polarization) then the total amplitude is ϕ =
a|1
b|2 +
b|1
a|2 .

(12)

The probability P = |ϕ|2 for the event contains a coherent superposition of amplitudes and hence there is interference. This interferometry, invented by HBT for measuring the diameter of nearby stars, has played a vital role in the development of quantum optics [3]. Two points are noteworthy in this regard:

ϕ1 =
D|1
1|S
D1 |1
1|L +
D|2
2|S
D1 |2
2|L .

(14)

Since the total amplitude contains a coherent superposition there is interference. A second and different event, in this case, is when an electron propagates from S to D and a photon propagates from L to D2 . It has a similar amplitude ϕ2 with D1 replaced by D2 . 3.4

Entanglement and non-locality: Einstein–Podolsky–Rosen (EPR) paradox [1, 4]

Two particles in an entangled state merge into one new unit. It is no longer allowed to speak of two separate particles and, as a consequence, they do not have to communicate with one another, even if they are separated far from one another. This should be illustrated by the so-called EPR paradox in the decay of positronium in the singlet ground state with total spin S = 0 and total angular momentum L = 0.

1. The total amplitude cannot be factorized in an amplitude for particle 1 and an amplitude for particle 2. The two particles are said to be entangled. They form a unit, i.e. it is no longer allowed to speak of two separate particles. Entanglement is ubiquitous in quantum mechanics and plays an important role in quantum computing and teleportation. 2. If the experiment would have be done with identical fermions then the total amplitude (12) would have had a minus sign instead of a plus sign. We now extend the HBT experiment by allowing both photons to go into the same detector. In that case the probability is P = |
a|1
b|2 +
b|1
a|2|2 + |
a|1
a|2|2 + |
b|1
b|2|2 .

(13)

FIGURE 4

A double slit experiment with observed electrons (version 2)

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The positronium is similar to a hydrogen atom but with the proton replaced by a positron. In the singlet ground state it decays with a lifetime of 10−10 s by the emission of 2γ quanta of 511 keV. The conservation of momentum and angular momentum implies that the two gamma quanta are emitted under 180◦ and that both photons are right-hand circular polarized (RHC) or both left-hand circularly polarized (LHC). It never happens that one is RHC and the other LHC (Fig. 5). In addition, the parity of the positronium is −1, so that the state transforms under inversion I like I |00 −1 = − |00 −1 |00 −1 ≡ |S = 0, L = 0, I = − 1 .

(15)

We consider now a coincidence experiment with two detectors sensitive to linear polarization, x and y, as indicated in Fig. 6. There are four different events in this case with the initial state |00 −1 and four final states |α1 α2  with α standing for x or y polarized. There are two alternatives in each event: both photons are RHC or both photons are LHC. The closure relation reads |R1 R2 
R1 R2 | + |L 1 L 2 
L 1 L 2 | = 1 .

(16)

The total amplitude for an event is ϕ =
α1 α2 |00 − 1 =
α1 α2 |R1 R2 
R1 R2 |00 − 1 +
α1 α2 |L 1 L 2 
L 1 L 2 |00 − 1.

(17)

Noticing that I2 = 1 , I† = I I|L 1 L 2  =|R1 R2  ,

I|R1 R2  =L 1 L 2 

(18)

(see Fig. 5 for the third and fourth relation) we find


L 1 L 2 |00 − 1 = L 1 L 2 |I 2 |00 − 1 −1 = −
R1 R2 |00 − 1 = √ . 2

(19)

Hence 1 ϕ = √ {
α1 |R1 
α2 |R2  −
α1 |L 1 
α2 |L 2 } . 2

(20)

Since the amplitude cannot be factorized the two photons are in an entangled state. Using the relations between linear and circular polarized light we get 1 i
x|R =
x|L = √ ,
y|R = −
y|L = √ . 2 2

(21)

The corresponding probabilities for the four different events can now easily be calculated and are summarized in Table 2 (second column). We observe when detector D1 detects an x - (or y-) polarized photon, detector D2 detects an y- (x -)polarized photon. It never happens that both detectors detect photons with the same linear polarization. This observation has triggered many discussions about non-local behaviour, about communication with a speed higher than the speed of light. However, if one accepts the rules for calculating with probability amplitudes then there is no EPR paradox. The two entangled photons form a unit and they do not have to communicate with one another. To say that the two photons have to communicate implies that one denies the entanglement. In that case, the amplitudes for the two alternatives (the two gamma rays are both RHC or both LHC) are added incoherently giving a probability 1 1 PEPR = |
α1 α2 |R1 R2 |2 + |
α1 α2 |L 1 L 2 |2 . 2 2

(22)

This results in a probability of 1/4 for all four events, as listed in the third column of Table 2. If this argumentation were correct, whereas the experimental result gives P1 = P2 = 1/2 and P3 = P4 = 0, then one would have indeed a paradox with the need for superluminal communication. In reality, the EPR argumentation contains a mistake: the two amplitudes have been added incoherently, whereas, according to the rules of quantum mechanics, they should be added coherently because they are undistinguished. The interference of the amplitudes increases the probabilities P1 and P2 (for
xy| and
yx| ) to 0.5 and decreases those for P3 and P4 (for
xx| and
yy| ) to 0. As a consequence, quantum mechanics does not violate causality.

FIGURE 5

Decay of positronium into two RHC or two LHC photons

Final state


x1 y2 |
y1 x2 |
x1 x2 |
y1 y2 | Coincidence measurement of the positronium decay with x-, ypolarization sensitive detectors

FIGURE 6

TABLE 2

ium

Entangled state

Non-entangled state qm mixture (wrong)

 P1 = 12 P2 = 1 2 P3 = 0 P4 = 0

 P1 = 14 P2 = 14 P3 = 14 P4 = 1 4

Probabilities for linear polarized light in the decay of positron-

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One more point is worth mentioning in connection with entangled states. It concerns the stability of entangled states relative to decoherence by coupling to the environment. First, there are very stable entangled states, like, e.g. in the hydrogen molecule H2. The energy E of the ground state is given by E =
a(1)b(2) + a(2)b(1)|H|a(1)b(2) + a(2)b(1)

(23)

where H is the Hamiltonian for the protons and the electrons and their Coulomb interaction. The two electrons are in an entangled state with two undistinguished alternatives: “electron 1 at proton a and electron 2 at proton b”, and “electron 1 at proton b and electron 2 at proton a”. The resulting interference terms give the main contribution to the chemical bond (here 4.5 eV). This strong bond makes the hydrogen molecule insensitive to most external influences. However, since the electron pair is confined to a small region in space (about 1 Å3 ) the system is not appropriate for teleportation. On the other hand, for widely separated partners in a pair of entangled photons (as in the case of the two photons from positronium or for a down converted photon pair), the weak (or no) bond between the partners makes them prone to coupling to the environment with the loss of entanglement (decoherence). The whole problem in teleportation with entangled photon pairs is to keep the coupling to the environment small and to prevent decoherence [5, 6]. 3.5

Speckle from a coherently illuminated sample

We consider a crystalline sample with defects (Fig. 7a) and we would like to do a scattering experiment on this sample with X-ray photons, neutrons, etc., as probing particles. In light of what has been said up to now we can consider the sample as a multislit system with internal degrees of freedom. It is the purpose of a scattering experiment to probe these (static and dynamic) degrees of freedom. Let us consider monochromatic X-ray photons as probing particles. As event we have the propagation of one photon from the source to a detector pixel via the sample. The scattering of the photon by each atom j with atomic form factor f j ( K ) defines in the first Born approximation the different alternatives. Interference can be lost by identification of an alternative (e.g. by Compton scattering at one electron) or by repetition of the event under non-identical conditions. We now consider a second sample (Fig. 7b) with the same average periodic lattice and with the same average deviation from the periodic lattice (same concentration of defects). Nevertheless the samples A and B are not identical. The different

configuration of the defects gives the samples individuality. The question arises: can this individuality be detected in a scattering experiment. Standard X-ray scattering (with noncoherent illumination of the sample) gives the same Bragg peaks and the same diffuse scattering for both samples A and B but it is not able to detect the individuality of the samples A and B. This can only be detected by coherent illumination. For simplicity reasons we consider only static disorder in our samples A and B. The total amplitude for a photon with energy hω and incoming and outgoing momenta hk1 , and hk2 , to be detected at time t is ϕ = exp {−iωt + ik(u 1 + u 2 )}


N 1



f j (K ) exp −i K rj .

(24) Here, u 1 and u 2 are the distances from the source and from the detector to the origin rj = 0 in the sample. The term hK = h(k2 − k1 ) is the momentum transfer. Since the scattering pro

cess is elastic, k1 = k2 = k . The exponential exp −i K rj is the contribution of the retardation to the amplitudes specific for atom j . It is usual to express the probability for the event in terms of the scattering function S( K ) with   
N

2 NS(K ) =  f j (K ) exp −i K rj  . 1

(25)

The exponential prefactor in (24) is the same for all alternatives and has dropped out in the modulus squared. The scattering function S(K ) can be written in the following way: NS(K ) =


N  
 f j (K )2 + 1

j =n



f j fn∗ exp −i K (rj − rn ) .

(26) There are three important quantities linked to the momentum transfer hK : 1. The term 1/K is the probing length along the direction of K . Scattering centers in that range contribute to interference, whereas for centers far separated ( K (rj − rn )  1) the interference terms (n = j) disappear. 2. The range ∆K of K -values swept in an event gives the spatial resolution in the scattering experiment. It is important to note that the range ∆K includes only those momentum transfers which correspond to all, undistinguished alternatives for the propagation of our photon. Let us illustrate this for the lateral resolution dt of a microscope (Fig. 8). Here, the range of momentum transfer is     ∆K = (k2 − k1 ) − k 2  − k 1 ≈ k 1 − k1 (27) ∆K = 2k1 sin α = 4π λ sin α and dt =

FIGURE 7 Two sample areas with the same average lattice, with the same defect concentration but with different configurations of the defects

2π λ λ = = . ∆K 2 sin α 2 N.A.

(28)

The last expression in terms of the numerical aperture N.A. is the usual expression known in optics for the lateral resolution dt . The larger the numerical aperture, N.A., the

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FIGURE 8

FIGURE 9

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591

Lateral resolution and numerical aperture N.A. in a microscope

Uncertainly ellipsoids in a scattering experiment

more alternatives contribute to the image and the better the resolution. 3. The uncertainty in K , specified by the volume VδK of the uncertainty ellipsoid gives the coherence volume Vc =  (2π)3 VδK (Fig. 9). There is an ellipsoid of uncertainty for the incoming and for the outgoing wave vector, due to poorly defined direction and energy of the photons. Both contribute to an uncertainty ellipsoid of the momentum transfer with volume VδK . When the coherence volume Vc is larger than the illuminated sample volume, Vill , then speckle is observed. If, on the other hand, Vill  Vc , the speckle will disappear by configurational averaging. A simple simulation will illustrate this behaviour. We consider a square lattice of 100 × 100 atoms ( N = 10 000) with 1% of statistically distributed substitutional defects. The atomic form factor of the defects (α = 2) may be 1.5 f , with f being the form factor of the host atoms (α = 1). The average form factor is
2 1
f= fj = cα f α =
f j  j α=1 conf N fj = f + δ fj

(29)

Inserting these expressions into the scattering function gives  2   1
S (K ) = N  f  δ K,G + δ f j δ f n∗ exp −i K rj − rn j,n N (30)

FIGURE 10 Scattered intensity for two defect configurations showing the same Bragg peak (1,1), the same diffuse scattering but different speckle (full line and broken line)

The first term represents Bragg scattering by lattice planes defined by the reciprocal lattice vector G . The second term describes diffuse scattering plus speckle. Figure 10 gives the scattered intensity for two different defect configurations. The central peak is the Bragg peak (1,1) (with many side lobes as a consequence of the small number N ). The background is speckle plus diffuse scattering. If averaged over many configurations, the speckle is washed-out and only the diffuse scattering (called Laue scattering in this case) is left over besides the Bragg peak. This configurational average is what occurs if the illuminated volume Vill is much larger than the coherence volume. Until recently, this was the case in almost all X-ray and neutron scattering experiments. Only with the advent of highly brilliant synchrotron radiation sources of the third generation has coherent X-ray illumination become feasible. It is used to study diffusional and other dynamic processes in disordered systems and extends speckle spectroscopy with laser light into the range of atomic resolution [7]. For a statistical distribution of the defects the configurational average can easily be evaluated:  2
S (K ) = N  f  δ K,G

  1
+ δ f j δ f n∗ exp −i K rj − rn j,n N   2 

 2  with δ f j δ f n∗ = δ f j  δ jn = | f |2 −  f  δ jn . (31)

The Kronecker delta δ jn expresses the fact that in this case all correlations vanish except for the self-correlations. In the present case, the Laue diffuse scattering is basically a constant showing the weak K -dependence of the form factor. In summary, the Bragg and Laue scattering are identical for the two defect configurations, whereas the speckle is not. In the present example, we have considered a static defect configuration. If the defect configuration varies in time the speckle pattern also varies in time. The temporal variations of the intensity and of intensity correlations for a given K -value are the basis of X-ray speckle spectroscopy [7, 8].

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1 R.P. Feynman, R.B. Leighton, M.L. Sands, in: The Feynman Lectures on Physics, vol. III (Addison-Wesley, Boston, 1965) 2 A. Tonomura, in: The Quantum World Unveiled by Electron Waves (World Scientific, London, 1998) 3 R. Loudon, in: The Quantum Theory of Light, 2nd edn. (Oxford University Press, New York, 1991)

4 A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935) 5 N. Gershenfeld, in: The Physics of Information Technology (Cambridge University Press, Cambridge, 2000) 6 M.A. Nielsen, I.L. Chuang, in: Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000) 7 T. Thurn-Albrecht, G. Meier, P. Müller-Buschbaum, A. Patkowski, W. Steffen, G. Grübel. D.L. Abernathy, O. Diat, M.G. Koch, M.T. Retz, Phys. Rev. E 59, 642 (1999) 8 B. Lengeler, Naturwissenschaften 88, 249 (2001)