Quantum states of neutrons in the earth's gravitational field

Institute Laue-Langevin, Grenoble 

09.06.05 INSTITUT MAX VON LAUE - PAUL LANGEVIN 

V.V.Nesvizhevsky

Institute Laue-Langevin, Grenoble 

The biggest worldwide research reactor 

France, Germany, England, Italy, Swiss, 

Autrish, Russia, Spain, Chekh. Rep. 

Dominant part of all World research in the 

field of fundamental physics of particles and 

fields with neutrons 

Grenoble 

“2.5” out of 7 European 

Research Centers 

(EUROFORUM): 

ILL, ESRF, ½ EMBL 


V.V.Nesvizhevsky

Plan of this presentation 

1. Short Introduction: Ultra Cold Neutrons - UCN. 

First experiment of storage of UCN in 1968 in Joint Institute for Nuclear Research in Dubna: 

V.I.Luschikov, Yu.N.Pokotilovski, A.V.Strelkov and F.L.Shapiro (1969). JETP Letters 9: 40-45. 

09.06.05 

INSTITUT MAX VON LAUE - PAUL LANGEVIN 

V.V.Nesvizhevsky


2. Quantum states of neutrons in the Earth’s gravitational field above a 

mirror. 

- General analytical solution of the Schrödinger equation for an object in a gravitational 

field above a mirror is given in textbooks on quantum mechanics. 

- An experiment with UCN was proposed in 1976 : V.I.Luschikov (1977), Physics Today: 42- 

51; V.I.Luschikov and A.I.Frank (1978), JETP Letters 28(9): 559-561. 

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3. Experimental results. 

- Observation and study: H.Abele, S.Bäßler, H.G.Börner, A.M.Gagarski, V.V.Nesvizhevsky, 

A.K.Petukhov, K.V.Protasov, A.V.Strelkov, A.Yu.Voronin, A.Westphal et al : Nature 415: 297-299 

(2002); Physical Review D 87: 102002 (2003); Europ.Phys.Journ. C 40(4):479-491 (2005). 

Institute Laue-Langevin, Grenoble, France; 

Petersburg Nuclear Physics Institute, Gatchina, Russia; 

Laboratory of Sub-Atomic Physics and Cosmology, Grenoble, France; 

Lebedev Institute, Moscow, Russia; 

Mainz University, Germany; 

Heidelberg University, Germany; 

DESI, Hamburg, Germany; 

University Joseph-Furrier, Grenoble, France; 

Joint Institute for Nuclear research, Dubna, Russia 

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V.V.Nesvizhevsky


4. Prospects and applications. 

- Search for additional (spin-independent or spin-dependent) short-range forces at the 

distances of 1nm - 10µm 

- Improvement of the upper limit for the neutron electric charge 

- Verification of some extensions of the quantum mechanics, such the fundamental loss of 

quantum phase coherence due to interaction of quantum systems with gravitational field (with the 

observation time of up to 10 3 s), or logarithmic terms in the Schrödinger equation. 

surface 

- This method provides a rare opportunity to measure distribution of hydrogen above/below 

- This phenomenon allows to solve the problem of neutron-tight valve for UCN traps 

- It provides a convenient tool to study such phenomena as neutron localization (Andersentype), 

or interaction of waves with rough surfaces. 

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1. Effective Fermi-potential and storage of UCN in traps 

Nuclei in matter 

Usually: 

~99.99 % - elastic reflection 

~10 -4 - inelastic reflection at 

phonons to the thermal energy 

range 

~10 -5 - inelastic reflection at surface 

nanoparticles to the UCN energy 

range 

~10 -5 - absorption 

Neutron 

UCN 

V 

E 

H 

UCN 

UCN 

Earth ' 

field s 

UCN 

T ~ 1mK 

~ (1÷ 

6) m / s 

~10 

−7 

~ 1m 

eV 

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1. UCN production 

UCN (Ultra Cold Neutrons) – soft fraction of spectrum of cold or thermal neutrons 

Uranium + water + neutrons + light 

Very low background ! 

Curved neutron guide 

to extract UCN 

Liquid-deuterium 

source of cold 

neutrons 


V.V.Nesvizhevsky

1. UCN production 

New reactor in 1995 ! 


V.V.Nesvizhevsky

2. Quantum states of neutrons in the gravitational field 

Nature 415, 297-299 

(17 January 2002) 

Valery V. Nesvizhevsky * , Hans G. Börner * , 

Alexander K.Petoukhov * ‡ , Hartmut Abele † , Stefan 

Baeßler † , Frank J.Rue ß † , Thilo Stöferle † , Alexander 

Westphal † , Alexei M. Gagarski ‡ , Guennady A. 

Petrov ‡ & Alexander V. Strelkov § 

* Institute Laue-Langevin, Grenoble, France; 

† University of Heidelberg, Germany; 

‡ Petersburg Nuclear Physics Institute, Gatchina, 

Russia; 

§ Joint Institute for Nuclear Research, Dubna, Russia. 

09.06.05 


V.V.Nesvizhevsky

2. How to observe any quantum states of matter in 

a gravitational field ? 

Neutron above a mirror in the 

Earth’s gravitational field 

1) Electric neutrality (usually any gravitational 

interaction in laboratory conditions is much weaker 

that other interactions) 

2) Long lifetime 

3) Small mass 

⎛ h 

⎜∆E 

≈ 

⎝ ∆τ 

⎛ h ⎞ 

⎜∆v 

⋅∆x 

≈ ⎟ 

⎝ m ⎠ 

4) Energy (temperature) of UCN is extremely small 

and not equal to the installation temperature 

⎞ 

⎟ 

⎠ 

Quantum state energy in the Bohr- 

Sommerfeld approximation : 

E 

n 

≈ 

3 

⎛ 9 ⋅m 

⎜ 

⎝ 8 

n 

⎞ 

⎟⋅ 

⎠ 

⎛ 

⎜π 

⎝ 

⋅h 

⋅ g 

⎛ 

⋅⎜n 

− 

⎝ 

1 

4 

⎞⎞ 

⎟⎟ 

⎠⎠ 

2 


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2. Probability to observe a neutron above a mirror 

Height above a mirror 

in microns 

The precise solution of the 

corresponding Schrödinger 

equation 


V.V.Nesvizhevsky


How the 

experiment with 

neutrons is related 

to the falling down 

of an apple in the 

gravitational field ? 

Higher probability to observe 

neutrons (an apple) at some 

heights and zero probability – 

for a pure quantum state – to 

observe them somewhere in 

between 


V.V.Nesvizhevsky


Neutrons spend longer 

time at the top of its 

“trajectory” and the spacing 

between the maxima is 

bigger at the top as well 


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2. General scheme of the experiment 

V horizont ~4-15 m/s 

V vertic ~2 cm/s 

∆E 

≈ 

h 

∆τ 

Selection of vertical and horizontal velocity components 


V.V.Nesvizhevsky

2. «Table-top experiment» 

Some parameters of the 

experimental installation and 

characteristic parameters of 

the phenomenon: 

10 9 

8 

Shutter 

control 

-Effective temperature of 

neutrons is ~20 nK 

1 

3 

5 

6 

7 

4 

11 

Amplifier 

Amplifier 

ADC 

DAC 

COMPUTER 

-Background suppression is 

a factor of ~10 8 -10 9 

-Absolute horizontal leveling 

precision is ~10 -6 rad 

-Parallelism of the bottom 

mirror and the 

absorber/scatterer is ~10 -6 

2 

1. Neutron guide 

2. Anti-vibration table 

3. Polished granite stone 

4. Piezo translators 

5. Vacuum chamber 

6. Mirrors and absorber 

7. Detector 

8. Anti-magnetic shielding 

9. Input collimator 

10. Neutron shutter 

11. Inclinometers 


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2. Measurement 

∆v 

⋅∆x 

≈ 

h 

m 


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3. Calibration of distances 

The distance between a mirror and a scatterer/absorber 

is measured using the capacitors method 

Calibration : 

Long-focus microscope 

Calibrated wires-spacers 

Mechanical devices (“comparators”) 


V.V.Nesvizhevsky

3. Test of the mirrors quality 

Why ? 

Non-specular reflections would cause false non-transparency of the slit mirror/scatterer 

Neutron trajectories 

mirror 

mirror 

Collimator 

Neutron 

detector 

General scheme to verify the quality of our mirrors 


V.V.Nesvizhevsky

3. Test of the mirrors quality 

F ( ∆z) 

= α ⋅ ∆z 

⋅ (1 − 

L⋅ϕ 

/ ∆z 

K n . s. 

) 

mirror −3 

K n . s. 

< 8⋅10 


V.V.Nesvizhevsky 

µm

3. Surface of scatterers 

3.0µm 

3.0µm 

3.0µm 

3.0µm 

Image 

1 

2 

3 

4 

Hauteur maxi. 

1.9 

1.95 

2.1 

1.85 

Distance moyenne 

5.0 

4.6 

5.64 

4.4 

3.0µm 

3.0µm 

Study of scatterer’s surfaces using an 

atomic force microscope or optical 

microscope 

3.0µm 

3.0µm 


V.V.Nesvizhevsky

3. Measurement of neutron horizontal velocity components 

V hor 

= 

L ⋅ 

g 

2 ⋅ ∆l 

Scatterer/absorber 

Collimators 

τ 

L 

= 

V 

g ⋅τ 

∆l 

= 

L 2 

hor 

Neutron trajectories 

2 

L 

Bottom mirror 

General scheme to select horizontal velocity components and 

to measure their spectrum 


V.V.Nesvizhevsky

3. Measurement of neutron horizontal velocity components 

Average horizontal velocity 

component along the neutron 

beam axis is ~6.5 m/s for the 

broad initial spectrum 


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3. Theoretical description 

The model of tunneling through gravitational barrier 

ξ >> 1 

3 

4 

2 

D ( ξ) 

≈ Exp[ 

− ⋅ξ ], Γn 

( ξ ) = ωn 

⋅ D( 

ξ) 

3 

ω 

n 

≈ 

n+ 

( E 

1 

− En) / h 

⎧1, 

ξ < 0 

⎪ 

D( 

ξ ) = ⎨ 

⎪An 

⋅ Exp[ 

− 

⎩ 

4 

3 

⋅ξ 

3 

2 

], ξ 

> 0 

P n 

( ξ ) Exp( 

−Γ ( ξ ) ⋅τ 

) 

= 

n 

P ( ξ ) 

n 

= Exp( 

−Γ ( ξ ) ⋅ 

n 

L 

V 

hor 

) 

⎛ ⎛ 

⎜ ⎜ L 

F( ∆z, 

V ∑⎜ 

hor ) = βn 

⋅ Exp⎜−α 

⋅ 

⎜ 

n 

⎜ V 

⎝ ⎝ 

hor 

⋅C 

2 

n 

⎛ 

⎜ 

⋅ Exp⎜− 

⎜ 

⎝ 

4 ⎛ ∆z 

− z 

⋅⎜ 

3 

⎝ z0 



n 

⎞ 

⎟ 

⎠ 

3 

2 

⎞⎞⎞ 

⎟⎟⎟ 

⎟⎟⎟ 

⎟ ⎟ 

⎟ 

⎠ 

⎠⎠

Probability to observe neutron 

versus height 

3. Theoretical description 

Penetrability of the gravitational barrier 

Height above the classical turning 

point for any quantum state 

µm 


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3. Results 

Narrow spectrum; soft fraction; comparison to the theoretical model 

z 

exp 

2 

= 21.3 ± 2.2syst ± 0.7stat, 

µ 

m 

z 

exp 

1 

= 12.2 ± 1.8syst ± 0.7stat 

, µ 

m 

z 

z 

teor 

1 

teor 

2 

= 13.7µ 

m 

= 24.0µ 

m 

µmµm 


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2. Results 

Narrow spectrum; soft fraction; comparison to the theoretical model 


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3. Results 

The penetrating neutron flux versus the horizontal velocity component 

Expected : 

0.2 µm/ms 

Measured : 

0.16±0.04 µm/ms 

µm 


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3. Measurements with a position-sensitive detector 

Position-sensitive detector 

with the spatial resolution 

of 1-2 µm 


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3. An example of such a position-sensitive detector 

DX 

15 mm 

120 mm 

Picture of developed detector with tracks 

Fission 

fragment 

UCN 

neutrons 

Plastic 

Supermirror coating 

Uranium-235 

~0.2 µm ~0.5 µm 


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3. Measurements with such a position-sensitive detector 

Preliminary results of measurement of 

density of neutron standing wave above a 

mirror using a position-sensitive detector 

with the spatial resolution of ~ 1.5 µm 

A few lowest quantum states 


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Pure quantum states 

1st quantum state 

4th quantum state 

0 10 20 30 40 50 

Z, micron 

0 10 20 30 40 50 

Z, micron 

2nd quantum state 

0 10 20 30 40 50 

Z, micron 

3rd quantum state 

0 10 20 30 40 50 

Z, micron 


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Pure quantum states 

st initial One quantum quantum state state , negative before the step15 

microns 

β 

nk 

( ∆z 

) = ψ ( z + ∆z 

) ⋅ψ 

( z) 

⋅ dz 

step 

∞ 

∫ 

−∞ 

n 

step 

Z, micron 

0 10 20 30 40 50 

step 

+2 initialTwo quantum states , before negativethe step 15 microns 

Z, micron 

0 10 20 30 40 50 


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3 

4 

1 

2 


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4. Resonance transitions between quantum states 

How to excite such a 

resonance transition : 

- Oscillations of a bottom mirror – due to nuclear forces; 

- Oscillations of a mass – due to gravitational forces; 

- Oscillations of electro-magnetic forces … 

Quantum trap 

Resonance 

transition 

Probability of 

transition 

E 

i 

− E 

j 

= h ⋅w 

ν ≈ 260Hz 

21 

ij 

δE 

E 

≈ 

min 

10 −18 

δE 

2 

min 

− E 

1 

eV 

≈ 10 − 

6 

Frequency of perturbation, Hz 


V.V.Nesvizhevsky

4. Resonance transitions between quantum states 

Estimation of the lifetime of a neutron quantum state 

Scatterer 

Scatterer 

Neutron beam 

Mirror 

Neutron 

detector 

Scheme of an experiment to estimate the lifetime of neutrons in 

the quantum states 


V.V.Nesvizhevsky

4. Applications in fundamental physics 

Additional short-range forces 

Why additional forces? 

-Light particles 

-Additional spatial dimensions 

( − / λ) 

V ( z) 

= −V0 ⋅ Exp z 

V0 = 2 ⋅π 

⋅ G ⋅αG 

⋅ m⋅ 

ρm 

⋅ λ 

2 

G ⋅ m1 

⋅ m2 

V ( r ) = − ⋅ 

G 

− 

12 

r 

( 1+ 

α ⋅ exp( r / )) 

12 

λ 

12 


V.V.Nesvizhevsky

Non-existence of additional bound state: 

(attractive interaction) 

Shift of the characteristic size of the 

neutron wave function : 

a G 



V 

2 

0 

⋅ m ⋅ λ < . 72 

Boundary for additional 

short-range forces 

0 ⋅ h 

2 

V.V. N. and K.V.P. Class. 

Quantum Grav. 21 4557– 

4566 (2004) 

Neutron 

experiments 

M=3 

Casimir 

forces, 

M=2 

Macroscopic 

measurements of 

gravitational forces 

M=2 

1 nm 10 µm 



?, m

Why neutrons ? 

-Due to their electric 

neutrality: small false 

effects 

a G 

-Wavelength in the range 

from 1 ? (thermal 

neutrons) to 10 µm 

(quantum states in the 

gravitational field) 

-Significant methodical 

progress in this field 

allows us to carry out 

precision experiments 



1 nm 10 µm 

“Conservative” perspectives: 

-high-density coatings (tungsten, gold) – 

factor 4-5 

-Precision measurement of the wavefunctions 

shape – factor 10 2 at 10 µm 

-Resonance transitions between quantum 

states – factor 10 3 -10 5 at 10 µm 

-Dedicated experiments with very cold 

neutrons – factor 10 2 -10 4 at 1 nm 



?, m

4. More examples 

- Neutrons in neutron guides 

- Particles/atoms in the top turning point of their classical trajectory 

in the gravitational field: atomic/neutron fountains 

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4. Scales of temperature and energy in neutron physics 

Ultra cold 

nanoparticles 

???????? ??? 

Cold 

moderators 

Reactor 

moderators 

20 nK 

10 -3 ? 

10 1 10 3 

10 -12 10 -7 10 -3 10 -1 10 7 Neutron energy, eV 

VUCN 

Fission 

Quantum states of neutrons in the 

Earth’s gravitational field 

09.06.05 


V.V.Nesvizhevsky

Quantum states of neutrons in the earth's gravitational field

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