arXiv:hep-ph/0306198 21 Jun 2003

Measurement of quantum states of neutrons in the Earth’s

gravitational field

V. V. Nesvizhevsky,1,* H. G. Bo¨rner,1 A. M. Gagarski,2

A. K. Petoukhov,1,2 G. A. Petrov,2 H. Abele,3 S. Bae.ler,3,7

G. Divkovic,3 F. J. Rue.,3 Th. Sto¨ferle,3 A. Westphal,3

A. V. Strelkov,4 K. V. Protasov,5 and A. Yu. Voronin6,5

1ILL, Grenoble, France 2PNPI, Gatchina, Russia 3University

of Heidelberg, Germany 4JINR, Dubna, Russia 5ISN, Grenoble,

France 6FIAN, Moscow, Russia 7University of Mainz, Germany

      ~Received 1 November 2002; published 13 May 2003!

The lowest stationary quantum state of neutrons in the Earth’s gravitational

 field is identified in the measurement of neutron transmission between

 a horizontal mirror on the bottom and an absorber/scatterer on top.

Such an assembly is not transparent for neutrons if the absorber height is

 smaller than the ‘‘height’’ of the lowest quantum state.

DOI: 10.1103/PhysRevD.67.102002 PACS number~s!: 03.65.Ta

I. INTRODUCTION

Quantum stationary states of matter in electromagnetic

and strong fields are well known. Examples are eigenstates

of electrons in atoms, and the ones of protons and neutrons

in nuclei. It is straightforward to expect the existence of

quantum states of matter in a gravitational field as well. In

the laboratory the gravitational field alone does not create a

potential well, as it can only confine particles by forcing

them to fall along gravity field lines. We need a second

‘‘wall’’ to create the well. The analytical solution of the

Schro¨dinger equation, which describes the quantum states of

a particle with mass m above a horizontal reflecting mirror, is

known and can be found in textbooks @1–3#. However, up to

the present experiment no such quantum effect had been observed.

It is very difficult to measure because gravitational

quantum effects are negligible for macroscopic objects and

the electromagnetic interaction dominates for charged particles.

Therefore one should use neutral elementary particles

with a long lifetime. Such a scenario with slow neutrons was

proposed in Ref. @4# and later discussed in Ref. @5#. Recently

a high precision neutron gravitational spectrometer has been

built and installed at the Institute Laue-Langevin. It is optimized

for this scenario and related studies @6#. We will discuss

here an experiment, which allowed to observe for the

first time the lowest quantum state of neutrons in the Earth’s

gravitational field. The results of this experiment were summarized

in Ref. @7#. Here we will give a detailed account of

all experimental aspects concerning this work. Additional

discussions, analytical solutions, and related publications can

be found elsewhere, for instance, in Refs. @8–22#.

II. BASIC PRINCIPLES

In the quasiclassical approximation the value of the nth

energy level En (n51,2,3,...) is calculated using the Bohr-

Sommerfeld formula:

En

5A3 S9.m

8 D.F p.\.g.Sn2

1

4 DG2

. ~1!

This approximation works extremely well for linear potentials

and provides almost an exact solution even for the

ground state (n51). Only acceleration in the Earth’s gravitational

field g, the neutron mass m5940 MeV/c2, and the

Planck constant \56.61310216 eV/s define the energy values

En . The energies of the four lowest levels are 1.4 peV,

2.5 peV, 3.3 peV, and 4.1 peV, respectively. The spatial neutron

density distribution, which corresponds to the lowest

levels, is shown in Fig. 1. The probability cn

2(z) of finding

neutrons at the height z has n maxima and (n21) minima

with the probability zero in each minimum as for any ordinary

standing wave.

*Corresponding author. Email address: nesvizhevsky@ill.fr

FIG. 1. Quantum states of neutrons are formed in the potential

well between the Earth’s gravitational field on top and a horizontal

mirror on bottom. The probability of finding neutrons at a height z,

corresponding to the nth quantum state, is proportional to the

square of the neutron wave function cn

2(z). The vertical axis z gives

an idea about the spatial scale for this phenomenon.

PHYSICAL REVIEW D 67, 102002 ~2003!

0556-2821/2003/67~10!/102002~9!/$20.00 .2003 The American Physical

Society 67 102002-1

The basic idea of this experiment is to let neutrons

‘‘flow’’—with a certain horizontal velocity—between a mirror

below and an absorber/scatterer above. The absorber acts

as a selector for the vertical velocity component. Then one

measures the number N of transmitted neutrons as a function

of the absorber height Dh. This measurement should allow

the identification of the neutron quantum states because a

clear ‘‘signature’’ should appear: the classical dependence

NCL(Dh) is modified into a stepwise quantum-mechanical

dependence NQM(Dh) at small absorber heights Dh. This is

described below.

III. EXPERIMENTAL SETUP

The experimental setup is shown in Fig. 2. Details are

described in the text and in Ref. @6#.

Neutrons of broad velocity and angular distribution arrive

at the entrance of the setup. The spectrum of the horizontal

velocity components of neutrons was measured. The results

are described below and shown in Fig. 3. The angular distribution

of neutrons at the entrance to the mirrors/absorber is

illustrated in Fig. 4. The vertical angular divergence (;4

31022) is defined by relative positions of titanium collimating

plates and the front edge of the bottom mirror. The arrangement

of the absorber and the mirror reduces the range

of incoming vertical velocities by an order of magnitude. The

neutron transmission between the bottom mirror and the absorber

was measured as a function of the inclination angle of

the complete setup. This showed that there is no correlation

and demonstrated high uniformity of the phase space density

of incoming neutrons. The upper cutoff in the vertical velocity

component is defined by the absorber height. Only neutrons

that can penetrate ~between the bottom mirrors and the

absorber! are those which have a very small vertical velocity

component at the front edge of the bottom mirror ~see Fig.

4!. This allows a simple determination of the horizontal velocity

component. The lower cutoff in the horizontal velocity

component defines a parabolic trajectory to start at the edge

of the lower titanium collimator plate. The higher cutoff is

done by the edge of the higher collimator plate. In actual

measurements the heights of these two titanium plates were

adjusted in such a way that there was no other cutoff in the

spectrum of horizontal velocity components. This spectrum

is shown in Fig. 3. The horizontal velocity component of

5–15 m/s is much bigger than the vertical velocity of , 5

cm/s. Therefore, in a vertical slice, one can find several neu-

FIG. 2. The experimental setup. The gravitational levels are expected

to be formed above the bottom mirrors ~6!. The mirrors and the

detector ~7! are placed inside a vacuum chamber ~5!, which has a

30- mm-thick Al entrance window. Between the exit of the neutron guide

~1! and the vacuum chamber entrance ~separated by a distance of ;1.5 cm!

 an adjustable input collimator ~9! is installed. The separation between the

neutron guide and the experimental setup allows to mechanically isolate the

setup against parasitic vibrations, originating from the neutron guide. The

 complete setup is mounted on a standard pressurized antivibration table ~2!.

A system of three interconnected pneumatic valves provides leveling of the

‘‘floating’’ tabletop. A polished flat granite slab ~3! is mounted to the

‘‘floating’’ table top on three active legs which consist of piezotranslators ~4!.

 They are connected in a closed loop with precise inclinometers ~11!,

installed on the granite surface. Length variation of the piezotranslators

 enables automatic leveling of the granite with an absolute precision better

than 10 mrad. A magnetic shield ~8! insures screening of external magnetic

 field gradients. The entrance neutron guide ~1! ends in a piece which has

at its exit a horizontal opening of 13 cm length and of 1 cm height. A thin

 Al foil of thickness 30 mm is glued to its exit. The entrance collimator ~9!

has two thick horizontal titanium plates with length and height larger than

 the size of the exit window of the neutron guide. The height position of

every plate can be adjusted independently with the accuracy of ;10 mm.

This titanium collimator, in combination with the absorber and the bottom

 mirror ~6!, allows for selection of neutron velocity components ~see

Fig. 4 below!.