Neutron interferometry. Hofer Joachim

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2 Contents 1 Introduction Foundations of neutron optics Fundamental techniques Superposition and quantum phase shift of matter waves in an interferometer The refractive index The Fermi pseudopotential The double-slit experiment Neutron double slit experiment Neutron interferometers Mach-Zehnder-type interferometers Aharonov-Bohm effects Scalar Aharonov-Bohm effect Original idea for charged particles Scalar Aharonov-Bohm effect for neutrons Neutron interferometry in condensed matter physics Measurement of the coherent scattering length Partial beam path detection 9

3 1 Introduction 1.1 Foundations of neutron optics The foundations of neutron optics are the same as for quantum mechanics in general, namely the equivalence hypothesis of particle motion and wave behaviour, displayed in the de Broglie relation: mv = h λ = k. This relation shows the connection between a particle (mass m, velocity v) and a matter wave with wavelength λ. In the matter wave view the particles are now described by the Schrödinger equation: ψ(r, t) i = Hψ(x, t). t For a single particle in a potential V this equation takes the form ψ(x, t) i = [ 2 + V (x, t)]ψ(x, t). t 2m By assuming that the wave function is separable and that the potential is time-independent we find the solution ψ(x, t) = ψ(x)e i E t, ψ(x) = c 1 e ikx + c 2 e ikx, with k 2 = 2m (E V ). 2 After the neutron was discovered in 1932 (by James Chadwick), it was soon verified that it s motion is determined by quantum mechanics (Halban and Preiswerk, Mitchell and Powers, both 1936). Since the stationary Schrödinger equation is of the same form as the stationary classical wave equation (Helmholtz equation), we can expect that the light optical phenomena which are consequences of the stationary equation (e.g. reflection, refraction, diffraction and interference) also occur for neutron waves. For the examples mentioned above this has already been experimentally verified. The time evolution is expected to be different since the time derivatives in the wave equations are different (of first order in the Schrödinger equation, of second order in the classical wave equation). Neutron optics also differs from standard quantum optics in a number of ways, e.g. neutrons have a rest mass and are fermions, while photons have no rest mass and are bosons. One of the most important (and most interesting) facts in quantum mechanics (and thus in neutron optics) is that by gaining path information one destroys the wavelike behaviour. So a neutron should not be considered as a particle traversing one of the possible paths, but rather as a wave traversing all of the paths (of course this is just my interpretation 1 ). 1.2 Fundamental techniques In this section I want to introduce a few methods which will be used throughout this article Superposition and quantum phase shift of matter waves in an interferometer In an interferometer the incident beam is split into two (or more) separate beams. The beams travel along different paths where they are exposed to different potentials (which results in different phases). At some point the beams are brought together again and allowed to interfere. The resulting beam is the superposition of the separated beams: ψ = ψ I + ψ II. We are now looking for a way to express the phase of a matter wave. For simplicity we first regard a plane wave ψ(x, t) = e i(kx ωt) = e i 1 (px Et) with the phase φ(x, t) = px Et. 1 If you are interested in this topic, compare to Feynmans path integral approach to quantum mechanics and to the de Broglie-Bohm interpretation.

4 Now we compute the time derivative of the phase: dφ dt = pv E L is the Lagrangian, which is related to the Hamiltonian by a Legendre transform L = pv H. The total phase accumulated between a source point (x 0,t 0 ) and the point (x,t) is given by φ(x, t) = 1 t t 0 L dt = 1 x = L. x 0 p dx 1 t t 0 H dt. Note that in general a particle cannot be described by a plane wave. For a more precise derivation of the phase one has to consider the general expression for a particle wave packet, but the end result would be the same. Actually all the arguments used above can be applied to the wave packet as well, except that the wave packet is not in an energy eigenstate (but the expectation value is the classical energy of the particle). We now consider a simple neutron interferometer as shown in Figure 1. A neutron source (or, more gen- Figure 1: Schematic composition of a (Mach-Zehnder-type) neutron interferometer erally, a particle source) is located at the point (x 0,t 0 ). From now on we consider the neutrons as matter waves moving along the interferometer. At some point the incident beam is split into two parts moving on different paths (I/II). The beam travelling on path I experiences a phase shift (by interaction with a potential V), while the beam travelling on path II does not. After that the two beams are brought together again, they interfere with each other so that the exit beam consists of a superposition of their wave functions. We assume that the incident beam is a plane wave: ψ in = e ikx. Because of symmetry reasons we can restrict the calculation to one dimension, so x simply denotes the pathlength. ψ I = re ikx, ψ II = te ikx. t and r denote the transmission/reflection coefficient of the first beam splitter. Note that the "particle mirrors" (at the upper left and lower right corner of figure 1) are assumed to be ideal, i.e. their reflection coefficient is one. We also assume perfect beam splitters ( ψ I 2 = ψ II 2 = 1 2 ), which implies r r = t t = 1 2. The reflection and transmission coefficients are thus given by 1 2 times a phase factor. We now furthermore assume that both beam splitters have the same reflection and transmission coefficients and see that the exit wave is given by: ψ out = rte i(kx+φ V ) + tre ikx, where φ V denotes the phase shift caused by the potential V. If the pathlengths are equal (x = x ), we can write ψ out = rte ikx (1 + e iφ V ).

5 We now calculate the phases of the two waves traversing the two different paths: The phase difference is thus given by: φ I (x, t) = 1 φ II (x, t) = 1 x x which is a path integral around the interferometer The refractive index x 0 p I dx 1 x 0 p II dx 1 φ = φ II φ I, t t 0 H I dt, t t 0 H II dt. We want to calculate the index of refraction for neutron matter waves. A neutron moving in a time-independent potential can be described by ψ(x, t) = ψ(x)e i E t. The time independent Schrödinger equation is a Helmholtz equation and can be written in the form ψ(x) + K 2 (x)ψ(x) = 0 with the wavevector 2m K(x) = (E V (x)). (1) 2 We now define the refractive index as the ratio of the wavevector inside the potential to the free space wavevector k = 2m E: 2 n(x) = K(x) = 1 V (x) k E. (2) For V << E this is approximately n(x) 1 V (x) 2E. (3) The Fermi pseudopotential Neutrons interact with their environment via all four forces of nature (electromagnetism, gravitation, strong nuclear force, weak nuclear force). But unlike in standard quantum optics the neutrons interaction with matter is dominated by the strong nuclear force. Since the range of the strong nuclear force is approximately the nuclear radius and thereby much smaller then typical de Broglie wavelengths for neutrons, we can use the Fermi pseudopotential. The fermi pseudopotential for a single nucleus at the origin is given by V (r) = 2π 2 m bδ(r), where b j denotes the scattering length. For multiple nuclei located at the points r j one gets the expression V (r) = j 2π 2 m b jδ(r r j ). Note that in a magnetic material the neutron also interacts with the magnetic field (neutrons may be neutral, but they still have a spin and thereby an associated magnetic moment). The neutron furthermore interacts with the electric field surrounding nuclei and electrons in matter and with the gravitational field (if present). For a non-magnetic macroscopic medium averaging the fermi pseudopotential over the volume leads to the effective optical potential: V optical = 2π 2 m b cn, (4) where b c denotes the coherent scattering length, which is simply the average of the b j, and N denotes the particle density. Absorption effects or nuclear reactions are described by an imaginery term of the potential, i.e. the scattering length becomes complex. This leads to a complex refractive index (Equation 2). Most isotopes have a positive scattering length, thereby the effective optical potential of most materials is repulsive, even though the individual nuclei always provide an attractive potential for neutrons. If the energy of the neutron is less than the potential, the wavevector of the wave inside the potential is imaginery. This leads to total reflection at all angles with the wave function decreasing exponentially inside the medium.

6 2 The double-slit experiment The double slit experiment is one of the most fundamental experiments in quantum optics. It perfectly shows the wave properties of matter, even of macroscopic particles. The wave function is obtained by solving the Schrödinger function using appropriate boundary conditions. Then, using Borns probability interpretation of the wave function, one gets: ψ = ψ A + ψ B, ψ 2 = ψ A 2 + ψ B 2 + 2Re(ψ Aψ B ). ψ 2 is the probability density, i.e. the probability for a particle to be found in the interval [x, x + dx] is given by ψ(x) 2 dx. The interference term coincides with the experimental results and can t be explained by classical assumptions. ψ A 2 / ψ B 2 is the probability density if slit B / slit A is closed. 2.1 Neutron double slit experiment In this section I want to introduce you to the double slit experiment with neutrons performed 1988 by Anton Zeilinger et al. The experiment was conducted on a neutron optical bench (see Figure 2) with very cold neutrons. The slits S1, S2, S3 and the prism were used to select a specific wavelength band out of the source Figure 2: Left side: experimental setup, right side: object slit S5. beam. Slit S3 also defined the width of the coherent wave front in the object plane. The exit slit S4 was used to measure the intensity distribution in the image plane (i.e. the slit was moved step by step and in each step the neutrons were counted for a specific time). The neutrons were counted in a BF3 detector (a tube filled with boron trifluoride gas, which undergoes an alpha interaction with neutrons). The measured intensity distribution (Figure 3) was in perfect accordance to the theoretical predictions. Figure 3: Double slit diffraction pattern, the solid curve represents the theoretical predictions.

There are several realisations of Mach-Zehnder-type Figure 4: General schema of a Mach-Zehnder interferometer. interferometers in neutron optics.

7 3 Neutron interferometers 3.1 Mach-Zehnder-type interferometers Mach-Zehnder-type interferometers are neutron interferometers, which are topologically identical to the Mach- Zehnder interferometer of classical optics (Figure 4). There are several realisations of Mach-Zehnder-type Figure 4: General schema of a Mach-Zehnder interferometer. interferometers in neutron optics. I am going to introduce the standard version of a perfect crystal interferometer called the standard triple Laue case interferometer (which is the best setup for most applications). Perfect crystal interferometers use Bragg diffraction for beam separation (amplitude division). Figure 5 shows the principle of a LLL interferometer. I won t go into details about the necessary crystal perfection and it s production, such informations can be found in [1]. The three crystal planes are assumed to be identical, i.e. their reflection Figure 5: Symmetric LLL interferometer. and transmission coefficients are the same. Then, by denoting the incident wave as ψ 0, one gets the intensities (for details about the calculations refer to the section "Fundamental techniques" at the beginning of this article): where the constants A and B are given by I 0 = ψ I + ψ II 2 = trrψ 0 e iφ1 + rrtψ 0 e iφ2 2 = A(1 + cos( φ)), (5) I H = trtψ 0 e iφ I + rrrψ 0 e iφ II 2 = B A cos φ, (6) A = ψ 0 2 r 4 t 2, B = ψ 0 2 ( t 4 r 2 + r 6 ). Note that in this case the reflection and transmission coefficients strongly depend on the wavevectors of the wave packet (respectively the momentum of the neutron). As a matter of fact there are components of the wave which experience zero transmission (or zero reflection). According to equations 5 and 6 these components do not contribute to the interference pattern, but to the constant part of the H-beam (and to the beams leaving the interferometer at the middle crystal). This is in correlation to a typical interference pattern as shown in figure 6. One can also see that I 0 + I H is constant, which is expected because of particle conservation (the material used in the interferometer has zero neutron absorption). A detailed calculation of the reflection/transmission coefficients is done in [1, Chapter 10]. Due to imperfections of the experimental setups and the neutron beam

8 Figure 6: Interference pattern obtained with a perfect crystal interferometer. D denotes the optical path difference between the beams. the measured results do not completely coincide with the predicted behaviour. The measured intensity in the forward beam can be described by I meas = A + B cos( φ + φ 0 ). A, B and φ 0 are parameters characteristic of each setup. 4 Aharonov-Bohm effects The Aharonov-Bohm effects are phenomena in which a particle is affected by an electric/magnetic field (it really works for all kinds of fields which are induced by a potential and interact with the particle, in theory even for gravitational fields), even though the particle is confined to an area in which the respective field is zero (the wave function has to be negligible in the field, otherwise the effect could be interpreted as interaction of the wave function with the field). The effects show that the electromagnetic potentials are not just mathematical constructs, but have a physical meaning. 4.1 Scalar Aharonov-Bohm effect The scalar Aharonov-Bohm effect (also electric Aharonov-Bohm effect) was originally predicted for charged particles (e.g. electrons). I am going to explain the effect for charged particles first and after that I ll explain how the principles can be applied to neutrons Original idea for charged particles Imagine an experimental setup as shown in figure 7. An incoming (electron) beam is split into two parts, both Figure 7: Principle of the electric AB effect.

9 beams travel to Faraday cages. In one Faraday cage a time dependent electric potential V is present, in the other cage no potential is present. The time-dependent potential is given by 0 t < t 1 V (x) = V 0 t 1 t t 2 0 t > t 2 where at t 1 the electron is already in the cage and at t 2 the electron is still in the cage. Classically we don t expect any effect, because there is no force acting on the electrons: F = E + v B = 0 t. For a quantum mechanical description we have to solve the time dependent Schrödinger equation ( 2 2m + V )ψ = i t ψ. The potential is time dependent and therefore the energy is not conserved. For simplicity we use the plane wave ansatz and distinguish between the three time intervals: t < t 1 (V = 0) : ψ 1 = e i(k1x ω1t) = 2 k 2 1 2m = ω 1. t 1 t t 2 (V = V 0 ) : ψ 2 = e iφ2 e i(k2x ω2t) t > t 2 (V = 0) : ψ 1 (x, t 1 ) = ψ 2 (x, t 1 ) x = k 2 = k 1. 2 k 2 2 2m + V 0 = ω 2 2 k 2 2 2m + V 0 = 2 k 2 1 2m + V 0 = ω 1 + V 0 = ω 2 = ω 1 + V0 φ 2 = V0 t 1. ψ 3 = e iφ3 e i(k3x ω3t) ψ 2 (x, t 2 ) = ψ 3 (x, t 2 ) x = k 3 = k 2 = k 1 = ω 3 = ω 2 V0 = ω 1. The phase difference is therefore given by: φ = φ 3 = V0 (t 2 t 1 ). We can see that a phase shift occurs even though there is no electric field inside the Faraday cage and classically no interaction is expected. This effect has not yet been experimentally verified for charged particles. Note that the scalar AB-effect is non-dispersive, i.e. the phase shift is independent of the wavevector k and therefore no wave packet spreading occurs. So it is possible to observe phase shifts much greater than the coherence length of the beam Scalar Aharonov-Bohm effect for neutrons Neutrons are, as the name suggests, neutral, but they do possess a spin and therefore they interact with a magnetic field. One now can simply replace the electric potential of the last section with a magnetic potential of the same form. Note that the magnetic potential is given by and therefore one gets the resulting phase shift V = ±µb(t) φ = ± µb 0 (t 2 t 1 ).

10 5 Neutron interferometry in condensed matter physics Neutron interferometry is (in condensed matter physics) mainly used to make accurate measurements of nuclear scattering properties of elements and isotopes. All measurements with a neutron interferometer are based on the intensity modulation caused by the phase shift of a matter sample. 5.1 Measurement of the coherent scattering length In the preceding sections we have already seen that an interference pattern is generally of the form I = A + B cos(φ + λ), where A is the mean counting rate including the non-interfering background and B ist the amplitude of the interfering part. φ denotes the phase shift caused by the effective potential of the sample and λ denotes the phase shift of the empty interferometer (e.g. because of pathlength difference) and the phase shift caused by any other potentials, if present. These quantities are depending on the respective setup and have to be determined experimentally before any measurements can be done. We consider again the symmetric LLL interferometer shown in Figure 8, where the phase shifter is a slab shaped matter sample. By using equation 3 and equation 4 as well as the relation E = 2 k 2, one gets the refractive index By using k 1 = k 2 = nk 0 n = 1 V 2E = 1 b cnλ 2 2π. 2m = (2π)2 2 2mλ 2 = k = (n 1)k 0 we see that the phase shift in one of the sub-beams is given by Dj 0 (n 1)k 0 dx = (n 1)k 0 D j = λb c ND j, where D j denotes the respective pathlength in the matter sample. The overall phaseshift is therefore given by 1 φ = λb c N D = λb c ND 0 ( cos(θ B δ) 1 cos(θ B + δ) ). Out of the interference pattern obtained by rotating the matter sample one now gets the coherent scattering Figure 8: Symmetric LLL interferometer. length of the material. The accuracy for bc b c is typically of the order of several parts in 1000 [1, Chapter 3]. 6 Partial beam path detection It is well known that, in any interference experiment, the interference pattern is destroyed if path information is available. However, one can ask oneself, what happens if partial path information is available. One way to

11 achieve this is to attenuate one of the sub-beams in the interferometer by absorbing some of the neutrons in one path. The absorbed neutrons can t contribute to the interference pattern and therefore one would simply expect a smaller amplitude in the interference pattern. This is in fact the case, but as we will see there s more to it than just that. In particular one has to distinguish between stochastic and deterministic absorption. Stochastic absorption means that there is no way to predict whether a neutron will be absorbed or not. Deterministic absorption means that (in principle) it is known with certainty what will happen at any point in the absorber region at any time (if a neutron happens to be there). Examples are a partially absorbing foil (stochastic) and a rotating periodic chopper (deterministic). Consider now again a symmetric triple Laue interferometer. In the last section we saw that the phase shift caused by the phase plate is given by The wave function at the 0-detector is therefore φ = λb c N D. ψ O = ψ 0L + ψ 0R e iφ and, by assuming that ψ 0L = ψ 0R, the intensity is given by I O = 2 ψ 0R 2 (1 + cos(φ)). Imagine now that in addition to the phase plate an absorber is inserted in the left beam (Figure 9). Consider Figure 9: Experimental setup for stochastic and deterministic absorption. first the stochastic case, i.e. the absorber is a partially absorbing foil. The absorption leads to a complex index of refraction, therefore the phase difference is complex as well. The wave function at the 0-detector is now given by ψ 0L = ψ 0L e i(γ+iλ) = ψ 0L a e iγ, where a = e 2λ is the transmission probability of a neutron along the left beam. The real part of the phase shift can (in this case) be neglected, i.e. ψ 0L = a ψ 0L. A detailed description of the complex refractive index and the phase shift can be found in [3] or [1, Chapter 4]. The intensity at the 0-detector is now given by I O = ψ 0L + ψ 0R e iφ 2 = ψ 0R 2 [1 + a + 2 a cos(φ)]. We now replace the foil with the periodic chopper. The transmission probability is then given by a = t open t open + t closed = t open T. We now have to distinguish between the case when the chopper is open, and the case when it is closed (an ideal beam chopper is assumed i.e. the beam is either completely undisturbed or completely absorbed). The intensity is then given by the weighted sum of the intensities in the respective cases: I O = a ψ 0L + ψ 0R e iφ 2 + (1 a) ψ 0R 2 = ψ 0R 2 [1 + a + 2a cos(φ)].

12 Figure 10: Interference patterns for stochastic and deterministic absorption. Graphs are drawn such that the interference patterns without absorber appear with the same size in both cases. Note that for equal transmission probabilities the mean intensity is the same for stochastic and deterministic absorption (as one would expect because the same number of neutrons is absorbed), but the amplitude of the interference pattern is different. Figure 10 shows the interference patterns obtained by Summhammer, Rauch and Tuppinger in One way to interpret this results is to link the amplitude of the interference pattern to the amount information one has about the possible path of the neutron. In the case of a stochastic absorber one knows, if a neutron is detected, that it has either gone along the right path, or, with a reduced probability, along the left path. It is interesting that even though the probability for the left path can be decreased to a arbitrarily small number, there is always an interference pattern left (but for a extremely small probabilty the amplitude of the interference pattern might not be noticable). If the neutron has a 99% probability of going the right path the amplitude of the interference pattern is still about 0.01 = 0.1 = 10% of the value it has in the situation without the absorber. In the case of an ideal chopper the neutron has either equal chance for both paths (if the chopper is open) or a 100% chance for the right path (if the chopper is closed). Therefore one has more detailed information about the possible path of the neutron (if the chopper is closed, it is known with certainty what happens in the absorber region). This results in the lower amplitude of the interference pattern in the deterministic case. So one can see, that "partial lack of information about each neutron s path is not equivalent with no lack of information for a fraction of the neutrons and total lack of information for the others" (A. Zeilinger).

13 References [1] Helmut Rauch, Samuel A. Werner Neutron Interferometry Oxford University Press [2] Anton Zeilinger et. al. Single- and double-slit diffraction of neutrons Rev. Mod. Phys. 60, (1988) [3] Summhammer, Rauch, Tuppinger Stochastic and deterministic absorption in neutron-interference experiments Phys. Rev. A 36, 4447 (1987)

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