Multilevel interference of a neutron wave

Multilevel interference of a neutron wave

S. V. Grigoriev,1,*Yu. O. Chetverikov,1S. V. Metelev,1and W. H. Kraan2 1_{Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188350, Russia} 2

Department Radiation, Radionuclides & Reactors, Faculty of Applied Sciences, TU-Delft, 2629 JB Delft, The Netherlands 共Received 20 June 2006; published 6 October 2006兲

We present an analytical and numerical analysis of neutron multilevel interference phenomena generated when a neutron passes through a series of N resonant coils operated at the successive conditions ប共␻₀ + n⌬␻兲=2␮_n共B₀+ n⌬B兲 with n=0,1, ... ,N−1. Each coil produces spin flip with probability␳ between 0 and 1; thus the number of waves for the neutron is doubled after each coil, finally giving 2Ninterfering neutron waves. The phase difference between any pair is a multiple of a time dependent “phase quantum”⌬⌽共t兲. The analysis predicts for each number N a highly regular pattern for the quantum mechanical probability to find the neutron spin in one specific state as a function of␳ and ⌬⌽. These patterns evolve in time and show revivals after a time T determined by the step⌬␻ according to T=2␲/⌬␻. For some adjustments of the system an analytical solution is obtained. Application of multilevel interference in high-resolution neutron modulated-intensity-by-zero-effort–type spectrometers is discussed.

DOI:10.1103/PhysRevA.74.043605 PACS number共s兲: 03.75.Dg, 42.87.Bg

I. INTRODUCTION

Multipath interference in optics 关1,2兴 and multimode

in-terference in dynamical systems关3兴 has recently emerged as

an extremely active field of research. Dynamical systems with a broad spectrum of excitations, when all the levels are populated, reveal rich interference patterns in both time and space 关4,5兴. Particularly, large scale interference leads to

well-ordered long-range regularities共such as quantum reviv-als关5兴兲 in the time-space probability distribution of the wave

function. Therefore it is of great interest to prepare a wave packet in a controlled way and measure its multimode or multipath interference.

In our previous paper 关6兴 we studied multipath

interfer-ence of a neutron during passage through N resonant coils in a dc field B0, each flipping the neutron spin with a probabil-ity ␳ between 0 and 1, interspaced by regions of length L with a homogeneous magnetic field B1. For this study we used Ramsey’s resonance method of the “separated oscillat-ing fields.” The same configuration was described in Refs. 关7,8兴 for two coils only. It was found that after the first

reso-nant coil the neutron wave is split into two waves for the two different spin states. In the subsequent region with field B1, these waves collect opposite phase shifts. In the next reso-nance coil each neutron wave is split again, thus making four waves. Hence, after N resonance coils we have 2Ninterfering waves. Each pair contributes to a highly regular pattern for the quantum mechanical共QM兲 probability to find the neutron spin in a specific state 共e.g., “up”兲, in a two-dimensional space subtended by the “axes” spin flip probability␳and line integral共B1− B0兲L. We derived an analytical expression for this probability as a function of these parameters and the number N. This expression was testified both by computer calculations and by neutron experiments. The experimental data were consistent with theory. We point out that the pat-tern in this 2D space is stationary, since all interfering waves correspond to states with the same energy and wave vector.

In the present paper we discuss again a set of N resonant coils in series, but now operated at successively increasing frequencies␻0+ i⌬␻共i=0¯N兲. Again, the neutron wave is split into 2N_{waves, however, each with different energy and}

wave vector. We will see that the phase difference between any pair of waves is a multiple of a phase quantum⌬␻t, i.e.,

depending on time. Moreover, an energy spectrum of equi-distant levels occupied according to a binomial distribution is created. The resulting multilevel interference pattern is not stationary, but evolves in time, giving revivals on a time scale T = 2␲/⌬␻. The shape of the pattern is determined by the number of resonant coils N and the spin flip probability of one coil␳.

This phenomenon has much in common with the neutron resonant spin echo 共NRSE兲 method recently developed 关9–11兴, based on earlier works on the resonant interaction of

neutrons with time-dependent magnetic fields关12–14兴. Thus,

a high resolution spectrometer for quasielastic neutron scat-tering was proposed on the basis of the NRSE method with the resonant coils of different frequencies 关15兴. It has

re-ceived the name modulation of intensity by zero effort 共MIEZE兲 and produces a sinusoidal intensity modulation of the incoming beam. The first scattering experiment on MIEZE had been performed and proved the possibilities of this technique 关16兴. Furthermore, a combination of several

MIEZE setups with one common detector position was pro-posed to get a periodic signal of arbitrary time shape关17兴. In

this scheme sharp signals well separated in time are possible. However, each MIEZE setup needs its own device for polar-ization analysis, limiting in practice their number to values around 5. In the present paper we lift out the analyzers and demonstrate that multilevel MIEZE may be realized in an easier way.

We give a theoretical treatment of multimode interference of neutron waves. The concept is developed in Sec. II and we describe how 2N neutron waves appear in an experiment with N resonant coils. We derive analytical expressions for the interference in the case of identical and nonidentical coils. Numerical calculations for the interference are given in Sec. III. Section IV presents both a short discussion and final conclusion.

II. NEUTRON MULTIWAVE RESONANCE INTERFERENCE

A. Identical resonant devices

The succession of magnetic fields B0and B1, which is N times repeated, is shown in Fig. 1共a兲. rf coils in the path sections with field B0 are operated at the resonance fre-quency ␻0. To understand interference between neutron waves in this configuration of fields we need to solve the Schrödinger equation, in which each rf coil is described as a 共2⫻2兲 matrix Cˆ 关given in Eq. 共18兲 below兴 operating on the

2D complex spinor

␺共rជ,t兲 =

冉

␣共t1兲exp共ik0x +␻t兲

␤共t1兲exp共ik0x +␻t兲

冊

, 共

冑

␣

2_+␤2_{= 1 at any time兲} representing the spin state of the neutron at entrance time t into the coil after the neutron entered the first coil at t1. Before presenting the mathematics of its solution in Sec. III, we give an intuitive description关7–12兴.

A valid solution is a plane neutron wave with wave num-ber k0, traveling along the x axis through the field configu-ration defined above. When the neutron enters the field B0, the wave number k0 changes to k1. By energy conservation, the total energyប␻does not change and the resulting wave number k1 differs from k0 in first approximation as

k1= k0+␮nB0/共បv兲, 共1兲 where mn,␮n, andv are the mass, the magnetic moment, and

the velocity of the neutron, respectively.

When spin flip occurs, a photon of energy ប␻0 is ex-changed between the neutron state and the rf field, i.e., the neutron spin state with momentum k1 gains or loses an amount of potential energy ⌬E=2␮nB0. When the neutron passes the field boundary from B0to B1, this potential energy is released as a kinetic energy change. We assume that the spin flip in the rf field was not complete but only partial, with a probability ␳= 1 / 2, for all resonance coils. Then at this boundary the neutron wave is split into 2 plane waves with wave numbers k₊and k₋corresponding to the spin states

共

1₀

兲

共up兲 and

共

0

1

兲

共down兲, respectively. Again, by energy conser-vation, the total energy corresponding to each state does not change at the transition from B0to B1, so the wave numbers

k+_{and k}−_{in the first approximation are}

k_±= k₁±␮n共B0− B1兲

បv . 共2兲

So, the initial neutron wave is split into a “nonflipped” and “flipped” part with wave vectors k−and k+ and energiesប␻ andប共␻+␻0兲. After the next coil each of these waves is split again into two waves with equal amplitudes—and so on. Thus, after N coils the initial wave is split into two groups of 2N−1 _{neutron waves with amplitudes 共1/2兲}N _{of the initial}

wave. In the first group the neutron was flipped an odd num-ber of times and therefore these waves have spin state “down.” The energy of the states corresponding to these waves isប共␻+␻0兲. They are located at the upper k level of diagram Fig.1共b兲. In the other group the neutron was flipped an even number of times 共or not flipped at all兲, so these waves have spin state “up.” In the states corresponding to these waves the neutron has the same energyប␻as initially. These waves are at the lower k level. In 共k,x兲 space 关Fig.

1共c兲兴 we can follow the phase shifts ⌬␸ of the individual waves relative to the phase value␸0= k₁x. Each wave in each

group has its specific path in this diagram.

At any position after the system of resonance coils the phase difference for an arbitrary pair is m⌬␾, where m = 0 , 1 , . . . , N and

⌬␾=

冕

0

l

关k+共x

⬘

兲 − k−共x

⬘

兲兴dx

⬘

共3兲 is the line integral over one path section with field B1. Thus, ⌬␾is a quantum of phase. The amplitude of the wave with a given phase shift m⌬␾共m=0,1... ,N兲 is determined by three factors.

共i兲 The spin flip probability of one rf coil ␳= sin2

冉

2␮n

ប Brf␶/2

冊

⬅ sin2␰ 共4兲 共so ␳ depends on the amplitude of the rf field Brf and the residence time␶, which is proportional to the neutron wave-length␭ and the length of the rf coil l兲.

共ii兲 The number of flipping events m. FIG. 1. 共a兲 System with many 共N=6兲 identical resonant coils

共iii兲 The number Am of pathways in 共k,x兲 space having

this particular phase shift m⌬␾ after N coils 共binomial dis-tribution兲:

Am= N! m!共N − m兲!.

Thus, the waves with the spin state “up” can be summarized as ␹1=

兺

m=1 N−1 Am 2N共sin␰兲 N−m_共cos␰兲m_{exp共im⌬␾兲 共5兲}

and the waves with the spin state “down” as

␹2=

兺

m=1 N−1

Am

2N共sin␰兲

m_共cos␰兲N−m_{exp共− im⌬␾兲. 共6兲}

According to quantum mechanics the probability R for the neutron spin to collapse into the spin state “up” or “down” is equal to 兩␹1兩2 _{or 兩␹2兩}2_{, respectively, with the polarization} component Pzzalong the z axis given as

Pzz=兩␹1兩2−兩␹2兩2. 共7兲

This problem was considered in detail in Sec. IV of Ref. 关6兴 and the quantitative solution was obtained through the

matrix method. The analytical expression for the probability

R is

R =␳sin

2_共N␥/2兲

sin2共␥/2兲 , 共8兲

where the angle␥is given by

cos共␥/2兲 =

冑

1 −␳cos共⌬␾/2兲. 共9兲 In this expression ⌬␾=共␮n/ 2ប兲共B1− B0兲L/v is identical to

the phase quantum Eq.共3兲.

B. Nonidentical resonance devices

Equations共8兲 and 共9兲 imply that for identical resonance

devices one obtains a stationary interference pattern in a 2D space subtended by the axes: field B1and spin flip probabil-ity␳. However, in some cases, mentioned below, it is impor-tant that a time evolution occurs.

For this purpose, let us take N resonant coils, adjusted such that they have the successive resonance conditions ful-filled

ប共␻0+ i⌬␻兲 = 2␮n共B0+ i⌬B兲, 共10兲

where i共i=1, ... ,N兲 is the number of the coil. So, the reso-nance frequency and magnetic field increase from one coil to the next by ⌬␻ and⌬B, respectively 关Fig.2共a兲兴. Again we suppose that each coil has flip probability␳⬍1, so a neutron wave entering the system is doubled after each coil and 2N waves exist at the end of the system. As in the case of iden-tical coils, each wave has its own path in the共k,x兲 diagram 关Fig.2共b兲兴.

To find an analytical expression for the behavior of the neutron wave, similar to the case of identical coils in Eqs.

共7兲–共9兲, we must split the problem in two steps. First, the

path through the system of coils 0⬍x⬍N共l+L兲. Here the neutron wave, with energy␻, is N times flipped and split into waves with wave vectors k0− i⌬k 共i=1, ... ,N兲, and now also with different energies 共␻+ i⌬␻兲, illustrated in Fig. 2共c兲. Each wave acquires its specific phase as a result of its history through the system, which is a multiple of the phase shift inside one resonance coil ⌬␾=⌬␻␶= 2l⌬k. Here ␶ is the residence time in one coil. So we create a phenomenon of “multilevel” interference.

The second step is the space after the system of coils:

x⬎N共l+L兲. Here these waves interfere and produce a pattern

evolving in time and space. We want to express the interfer-ence analytically in the same way as in the case of identical coils. However, the phase of the individual waves contains the phase history of the first step, which is different for all 2N

waves.

B

a

B0 SF1: ρ=1/2 SF2: ρ=1/2 SF3: ρ=1/2 SF4: ρ=1/2 SF5: ρ=1/2 SF6: ρ=1/2 l

b

k

₁

k

₀

k

₊

k

-x

x

20 ₂1 ₂2 ₂3 ₂4 ₂5 N waves

c

ω+ω0

ω

x

20 ₂1 ₂2 ₂3 ₂4 ₂5 N waves ω-2∆ω ω+ω0+ 2∆ω ω ω+ω0+ 4∆ω ω-4∆ω ω-5∆ω ω+ω0+ 5∆ω 1 4 6 6 4 1 1 4 6 6 4 1 1 5 10 10 5 1

To eliminate the effect of these phase shifts we adjust the frequency step ⌬␻ and the time ␶ such that we fulfill the condition

⌬␾=⌬␻␶= 2⌬kl = n2␲, 共11兲 where n is an integer number. Then, at the end of the system of coils the phase differences between all N waves are mul-tiples of 2␲. As in the previous case, the amplitude of the wave with a given energy and wave vector is determined by 共i兲 the spin-flip probability of one coil␳= sin2_{␰,共ii兲 the} num-ber of flipping events m, and共iii兲 the number of waves Amat

a particular energy level共binomial distribution兲.

Under the condition 共11兲 the waves with the spin state

“up” after the system can be summarized as

␹1=

兺

m=1 N−1 Am 2N共sin␰兲 N−m_共cos␰兲m_{exp关im⌬⌽共t,x兲兴 共12兲}

and the waves with the spin state “down” as

␹2=

兺

m=1 N−1

Am

2N共sin␰兲

m_共cos␰兲N−m_{exp关− im⌬⌽共t,x兲兴. 共13兲}

In full analogy to the case of identical coils 关Eq. 共8兲兴, the

quantum mechanical probability R can be analytically ex-pressed as

R =␳sin

2_{关共N␥共x,t兲/2兴}

sin2关␥共x,t兲/2兴 , 共14兲 where␳is the spin flip probability of the neutron in one coil 关Eq. 共4兲兴 and the angle ␥共x,t兲 is given by

cos关␥共x,t兲/2兴 =

冑

1 −␳cos关⌬⌽共x,t兲/2兴. 共15兲 The “phase quantum”⌬⌽ now depends on time t and coor-dinate x:

⌬⌽共t,x兲 = ⌬k共x − x0兲 − ⌬␻共t − t0兲, 共16兲 where x₀is the coordinate of the exit of the system and t₀the time when the neutron leaves it. For nonidentical resonant devices⌬⌽ is no longer a quantum of phase, but rather the evolution of time t. We see that the phase differences after the system increase both in time and as observed at increas-ing distance behind the system. We notice that the phase will reproduce after a “revival time”兩t − t0兩revgiven by

兩t − t0兩rev= 2␲ ⌬␻.

This holds at any place behind the system at increasing time

t.

III. NUMERICAL EXPERIMENT

In order to testify the theoretical consideration done in the previous section and to formally solve the Schrödinger equa-tion for the full system of N共non兲identical resonance coils, a computer calculation was performed. The solution for a neu-tron entering one coil at time t1and leaving it at t1+␶共where ␶= l /v兲 can be written 关18兴

⌿共t1+␶兲 = Cˆ共t1,␶,␻0兲⌿共t1兲, 共17兲 where the initial spin state of the neutron is ⌿共t1兲 =␣共t兲

共

1₀

兲

+␤共t兲

共

₁0

兲

with ␣= 1 and ␤= 0. The matrix

Cˆ 共t₁,␶,␻0兲 is a 2⫻2 matrix of the form

Cˆ 共t1,␶,␻0兲 =

冉

cos共␰兲exp共i␻0␶/2兲 − i sin共␰兲exp关i␻0共t1+␶/2兲兴

− i sin共␰兲exp关− i␻0共t1+␶/2兲兴 cos共␰兲exp共− i␻0␶/2兲

冊

. 共18兲

Here we remind the reader that␰=共2␮n/ប兲Brf␶/ 2.

Then after N resonance coils the neutron wave function can be written

⌿共t1+ N␶兲 = Cˆ共tN,␶,␻0+ N⌬␻兲Cˆ关tN−1,␶,␻0+共N − 1兲⌬␻兴 ¯

⫻Cˆ共t2,␶,␻0+⌬␻兲Cˆ共t1,␶,␻0兲⌿共t1兲, 共19兲 where ti= t1+共i−1兲␶共i=1,2, ... ,N兲. Thus, the calculation in-volves successive multiplication of the matrices Cˆ 关Eq. 共18兲兴

describing the action of one resonant coil.

The polarization component Piis found by evaluating the

well known expression

Pi⬅ 具␴i典 = 具⌿*共t1+ N␶兲兩␴i兩⌿共t1+ N␶兲典, 共20兲

where i = x , y , z and␴iare the corresponding Pauli matrices.

From this we obtain the quantum-mechanical共QM兲 probabil-ity R according to R =共1− Pzz兲/2.

As seen from Eqs. 共18兲 and 共19兲, the pattern of the QM

probability R depends on parameters of the system that one can vary.

共i兲 Obviously it is ruled by the number of resonant coils

N. This is the first parameter.

共ii兲 The second one is ␰, which determines the spin flip probability of the resonant coil according to Eq. 共4兲: ␳

= sin2␰. We vary␰from 0 to 2␲.

共iii兲 The third parameter is the frequency step ⌬␻. It is important that its value, combined with the residence time␶, is adjusted such that we fulfill the condition⌬␻␶= n2␲关Eq. 共11兲兴. Then, the resultant patterns can be also described by

Eqs.共14兲–共16兲.

In Fig.3 we show the QM probability R for systems of

⌬⌽共x,t兲 关see Eq. 共16兲兴 at x=x0共just after the last coil兲, so the

phase depends only on time t. For all N the parameter␰was taken as␲/ 4, so ␳ becomes 1 / 2. To define a specific time scale, we set the parameter⌬␻= 200 kHz. For this value we

can readily satisfy Eq.共11兲 for some wavelength in the

ther-mal spectrum with a length of the rf coils equal to a few cm. As seen, the revival time comes at 2␲/⌬␻=␲0.01 ms for this choice. For N = 2, with Eqs. 共14兲–共16兲 we get R

= cos2_{共⌬␻t / 2兲. As N increases, we get sophisticated patterns} with narrower main maxima and with secondary maxima. The points 共squares兲 obtained by the computational tech-nique fall on the lines on the basis of Eqs.共14兲–共16兲.

c

b

0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.5 1.0 ρ

=1/2, N=10

R

time (ms) 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.5 1.0

R

ρ

=1/2, N=6

time (ms) 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.5 1.0

R

ρ

=1/2, N=2

time (ms)

a

FIG. 3. The QM probability R关according to Eq. 共14兲, lines兴 to

measure the neutron spin state “up” as a function of time t at the exit of systems consisting of N = 2 , 6 , 10 nonidentical resonance coils with spin flip probability␳=1/2. The condition ⌬␻␶=2␲ 关Eq. 共11兲兴 is fulfilled. To define a time scale, the frequency step is set

⌬␻=200 kHz. We notice a periodicity 共revival time兲 equal to 2␲/⌬␻=␲⫻0.01 ms. Squares: the same result obtained with the numerical approach关Eqs. 共18兲–共20兲兴.

c

b

0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.5 1.0

R

time (ms)

N = 6,

ρ

= 1/15

0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.5 1.0

N = 6,

ρ

= 1/2

R

time (ms)

0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.5 1.0

R

N = 6,

ρ

= 1/4

time (ms)

a

Figure4shows the same as Fig.3, now for fixed N = 6 but various values for the parameter␰in the spin flip probability ␳= sin2_{␰: ␰= i␲/共2N兲 with i=1,2,3. Experimentally, these} curves can be observed by taking appropriate values for the amplitude Brf in the rf coils关see Eq. 共4兲兴. Again, the points 共squares兲 obtained by the computational technique fall on the lines on basis of Eqs.共14兲–共16兲.

Our computational technique enables us also to investi-gate the QM probability R when the condition 共11兲 is not

fulfilled. Figure5shows R as a function of time t for N = 6 and for the value of the frequency ⌬␻␶= 2␲/ j with j = 1 , 2 , 3 , 4 , 5. It is seen that the function R is periodic in time with the period of T = 2␲/ j⌬␻. The shape of the function within one period changes significantly with increase of j. Thus, it shows an arbitrary waving behavior with the shifted phase, which is difficult to describe in an analytic way in simple expressions.

IV. CONCLUDING REMARKS

In this paper we give, first, a theoretical description of polarized neutron multilevel experiments in a system of N nonidentical resonant coils with spin flipping probability be-tween 0 and 1. A large number of neutron waves with differ-ent wave vectors and energies are obtained. These waves interfere and each pair contributes to highly regular patterns in quantum mechanical probability to find the neutron in a specific spin state. Behind the system this pattern evolves in time and in space. For the specific adjustment of the system of resonators共⌬␻␶= 2␲兲 we derived an analytical expression for this probability, for arbitrary values of the flipping prob-ability␳ of one resonator and a phase quantum equal to the line integral of the field between the resonators, which in practice becomes equal to time t. This expression was testi-fied by computer calculations.

Secondly, such a system of resonators may be used for multilevel MIEZE but with no restrictions on the number of the resonators because one can use only one single analyzer at the exit of this device. For practical purposes it is not convenient to work with N resonators, whose frequencies increase from one to another by⌬␻. Then, the last Nth reso-nator will have共a rather high兲 frequency␻0+ N⌬␻with the corresponding dc field B0+ N⌬B. It is also difficult to keep the difference in frequency between two neighboring coils equal to ⌬␻. Fortunately for experimentalists, multilevel splitting will occur also when the resonant devices in the system are operated in the following way: all odd resonant devices at frequency␻0in the dc field B0=␻0共ប/2␮n兲 and all

even resonant devices at ␻0+⌬␻ in the dc field B₀+⌬B =共␻0+⌬␻兲共ប/2␮n兲.

In this case the same analysis of Sec II B may be applied with exactly the same results as given by Eqs.共14兲–共16兲. It

will simplify significantly the experimental efforts to realize multilevel interference. Therefore we conclude that the prob-lem of multilevel interference may be of interest both from theoretical and experimental points of view.

0 5 10 15 20 25 0.00 0.25 0.50 0.75 1.00 1.25 ρ=1/2 N=6 ∆ωτ=2π/5 time (µs) R 0 5 10 15 20 25 0.00 0.25 0.50 0.75 1.00 1.25 ρ=1/2 N=6 ∆ωτ=π/2 time (µs) R 0 5 10 15 20 25 0.00 0.25 0.50 0.75 1.00 1.25 time (µs) ρ=1/2 N=6 ∆ωτ=2π/3 R 0 5 10 15 20 25 0.00 0.25 0.50 0.75 1.00 1.25 time (µs) ρ=1/2 N=6 ∆ωτ=π R 0 5 10 15 20 25 0.00 0.25 0.50 0.75 1.00 1.25 ρ=1/2 N=6 ∆ωτ=2π R time (µs)

ACKNOWLEDGMENTS

The work was supported by INTAS Foundation Grant No. INTAS-03-51-6426兲, RFFR 共Project No. 05-02-16558兲. The research project was partially supported by the European

Commission under the 6th Framework Programme through the Key Action: Strengthening the European Research Area, Research Infrastructures, Contract No. RII3-CT-2003-505925.

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