Neutrons: Wave-Particle Properties

Coherence Properties

Basics of Neutron Interferometry

Dephasing - Decoherence

Weak Measurements

Active Neutron Apparatus Interaction (Fizeau)

Résumé

Helmut Rauch

Atominstitut, TU-Wien; Austria

Non-locality and destructive interference

of matter waves

CONNECTION

de Broglie

Schrödinger

&

boundary conditions

H(r ,t) i(r ,t)

t

Wave Properties Particle Properties

m = 1.674928(1) x 10-27 kg

s = 1

2

= - 9.6491783(18) x 10-27 J/T

= 887(2) s

R = 0.7 fm

= 12.0 (2.5) x10-4 fm3

u - d - d - quark structure

m ... mass, s ... spin, ... magnetic moment, c ... Compton wavelength, B ...

... -decay lifetime, R ... (magnetic) confine- deBroglie wavelength, c ...

ment radius, ... electric polarizability; all other -B ________________ coherence length, p ... packet

measured quantities like electric charge, magnetic two level system length, d ... decay length, k.…

monopole and electric dipole moment are com- momentum width, t ... chopper

patible with zero B ________________ opening time, v ... group velocity,

……phase.

ch

m.c = 1.319695(20) x 10-15 m

For thermal neutrons

= 1.8 Å, 2200 m/s

Bh

m.v

Bh

m.v = 1.8 x 10-10 m

c1

2 k

10-8 m

p v t . 10-2 m

d v . 1.942(5) x 106 m

0 2 (4)

The Neutron

effeffceff

2II

0I

00

DkkDNbkD)n1(sdk

cosBAI

Neutron Interferometry

reactor

Interferometer

set-up S18Interferometer

set-up S18Instutut Laue-Langevin, Grenoble

reactor

Interferometer family

I0 = c| trr + rrt |2

16000

14000

12000

10000

8000

6000

4000

2000

0

inte

nsity (

co

un

ts/1

0s)

-800 -700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600

ΔD / µm

D:DATA:Cycle 155:ifm30:adjust:nosample_3_wide.dat H beam O beam

High order interferences

0 10 20 -10 -20 -30

Δ[Å]

λ = 1.92(2) Å

State presentations

Schrödinger Equation:

t

)t,r(i)t,r()t,r(V)t,r(

m2

2

Wave Function (Eigenvalue solution in free space):

kde)t,k()2()t,r(3)trk(i2/3

Spatial distribution: Momentum distribution:

2)t,r()t,r(

2)t,k()t,k(g

Coherence Function:

;'rr

'tt Stationary situation: :)0(

kde)k(g)2()()0()(3rki2/3

and others (Wigner

function etc.)

Partial

waves fill

the whole

space

dxekax ikx

)()(

k1

k2

k3

Wave-Packet

SPATIAL VERSUS MOMENTUM MODULATION

044

)(

42

)(

2

2

0

cos2

)()()(

2

22

2

2

2

2

x

x

xx

x

xx

x

x

x

eeee

xxxI

044

)(

4

2

)(

2

2

0

cos2

)()()(

2

22

2

2

2

2

x

x

xx

x

xx

x

x

x

ee

ee

xxxI

)cos(12/)(exp[)( 0

0

22

00k

kkkkkI

Spatial distribution

Momentum distribution

)cos(1exp)( 00

2

)(

0

2

20

k

kkI k

kk

WIGNER FUNCTION

d2

x2

xe4

1xk,W

*ik

Definition:

Properties: 2

xdkxk,W

2

kdxx,kW

Interferometric Gaussian packets:

0221/42II,I

ixkx/2xexpx4x

0III xxx

H. Rauch, M. Suda, Appl.Phys.B60 (1994) 181

Spatial

distribution

Momentum

distribution

Spatial

distribution Momentum

distribution

More information – less mystics -

due to post-selection !

Post-selection

methods

4 cm

6 c

m

contrast internal phase

Position Post-Selection

Verification of

Schrödinger cat-

like states

k

kcos1)k(g)k(I 00

D.L. Jacobson, S.A. Werner, H. Rauch, Phys.Rev A49 (1994) 3196

Momentum

Post-selection

3500

3000

2500

2000

1500

1000

500

inte

nsity [

counts

/360s]

2.742.722.70

wavelength [Å]

1 = 0 Å

1 = 490 Å1.0

0.8

0.6

0.4

0.2

0.0norm

aliz

ed c

ohere

nce f

unction

8006004002000

2 [Å]

1 = 0 Å

1 = 490 Å

M.Baron, H.Rauch, M.Suda, J.Opt.B5 (2003) S341

Wave Packet Structure

H. Rauch, H. Wölwitsch, R. Clothier, S.A. Werner (1992) Phys. Rev. A46, 49

D.L. Jacobson, B.E. Allman, M. Zawisky, S.A. Werner, H. Rauch (1996) J.Jap.Phys.Soc. A65, 94

Time-Postselection

EPR-Photon Experiment

|k1-k2|=nδk

0

0.1

0.5

1

2

gletsin,

constkkkk 00

r)kk(2cos1)r,k,k(I

χ = π/2

χ = 0 χ = π

Wave – Lattice Interaction

χ = 3π/2

Quantum potential

( , ) ( , ) ( , ) / r t R r t eiS r t

R

tR

S

m

22 0

S

t mS V

m

R

R

1

2 202

2 2

( )

Qm

R

R

2 2

2

( , ) ( , ) ( , ) r t R r t r t 2 2 mv r t S r t

( , ) ( , )

Bohm D. (1952a) Phys. Rev. 85, 166 Bohm D. (1952b) Phys. Rev. 89, 180

C. Dewdney, P.R. Holland, A. Kyprianidis (1986) Phys. Rev. A119, 259

Quantum potential and trajectories

Weak

Measurements

Absorbing phase shifter

e e

i i i"a

2/)( NDinca NDincaeI

Ia

)(

0

closedopen

open

tt

ta

stochastic deterministic

.cos2)1(2

0

2

00 aaIIIII

sto

cos2)1(

)1(

2

0

2

00

2

0det

aa

aaI

I

IIIII

Absorption

results

2/)( NDinca

2/)(

0

)()( 2 eaeae

NDinca

0

2)(2

0

eaa

0aa

Small a-case:

J. Summhammer, H. Rauch, D. Tuppinger, Phys.Rev. A 36

(1987) 4447

Event by event simulation

•

H. De Raedt, F. Jin, K. Michielsen, Quantum Matter 1 (2012) 20

Double Loop Visibility

minmax

minmax

II

IIV

fM. Suda, H. Rauch, M. Peev,

J.Opt.B:Quantum Semiclass.Opt. 6(2004)345

Δf Td

Vstoch

df

fd

stoT

TV

f

2/cos4

2/cos422

ectionodyne

kTifV fdsto f

dethom

)2/(cos41 0

2

2

Stimulated Coherence

df

fd

stoT

TV

f

2/cos4

2/cos422

12/cos4

2/cos4

0

2

0

2det

df

fd

Tk

kTV

f

deterministic stochastic

Δf Td

Active neutron-apparatus

interaction

Dynamic Spin-Superposition

(One photon exchange)

Zero-field Larmor precession

Photon exchange but no path information:

ΔNΔϕ>1/2 Δvr = 2mB/mv < Δvbeam

Δt < Δl/Δvr < 2π/ωr; ΔlΔk < 1/2

G. Badurek, H. Rauch,

J.Summhammer,Phys.

Rev.Lett. 51 (1983) 1015

one photon exchange:

0

)sin(

)cos(

),()(

t

t

P

zee

r

r

tii

rr

On-resonance→ single photon

exchange

l

vB n

21

02 Br

H.Weinfurter, G.Badurek, H.Rauch, D.Schwahn, Z.Phys. B72(1988)195

tB

B

tB

cos

0)(

1

0ωj = ω0 + j ω

a J Bj j ( ) 1

sin( / )2

I z e a z b z eif j j

ij t

j

22

1 (

B t

B B t

( )

cos

0

0

0 1

B1=62.8G

ω/2π=7534 Hz

)cos(2 tja j

j

j

j

Noise field and dephasing (Multi photon exchange)

tij

j

jjfjezbza

Multi-photon exchange: results

J. Summhammer, K.A. Hamacher, H. Kaiser, H. Weinfurter, D.L.

Jacobson, S.A. Werner, Phys.Rev.Lett. 75 (1995) 3206

ν = 7534 Hz → ΔE= 3.24.10-11eV

<< ΔEbeam = 10-4eV

Mono- and multi-mode noise

N

i

iii tBtB1

)cos()(

),(),(ˆ2

),(),( 22

trtrBm

ktrH

t

tri

zLxxtBBtrB ˆ))()().((),( 0

Formalism zLxxtBBtrB ˆ))()().((),( 0

N

i

iii tBtB1

)cos()(

ti

n

xikn

NnnIIInn

NeeeJJtx

)()........(),( 11

),.........(),........(),........(),.........(

2sin2

2

22

1111

0

02

2

0

2

0

NNNN

i

i

i

i

ii

i

ii

nn

nnn

v

LT

BTT

nm

Bm

kkn

0

)(2

0 ........Re1).(),(2

1).(

1

v

xwith

eJJetxetxtxI

i

ii

n

tni

nn

i

III

i

Ii

N

m

m

tim

mfexctxI

)(),(0

evenn

mnn

ni

Nnnmm

i

i

f

NeJJc

;

10 cos)().......(1

mT

i

m

M

i

i dttIT

tIM

I0

0

1

00 )(1

)(1

),(),(ˆ2

),(),( 22

trtrBm

ktrH

t

tri

dttT

CmT N

i

iii

m

0 1

cos1

Multi-frequency photon exchange

ω1= 1 kHz, B1= 40 G ω1= 2 kHz; ω2= 3 kHz;

B1= 15 G; B2= 14.2 G

G. Sulyok, H. Lemmel, H. Rauch, Phys.Rev. A85 (2012) 033624

one frequency two frequencies

f1= 3kHz, f2=5kHz, f3=7kHz,

f4=11kHz, f5=13kHz

Bi= 4 G

Bi= 11 G

G. Sulyok, H. Lemmel, H. Rauch, Phys.Rev. A85 (2012) 033624

Five-mode case

Dephasing at low order

20000

15000

10000

5000

0

Inte

nsity (

co

un

ts/1

0s)

-100 0 100

D (m)

H-Beam

+ B = 9G

O-Beam

+ B = 9G

20000

15000

10000

5000

0

Inte

nsity (

co

un

ts/1

0s)

-100 0 100

D (m)

H-Beam

+ B = 9G

O-Beam

+ B = 9G

20000

15000

10000

5000

0

Inte

nsi

ty (

cou

nts

/10

s)

-100 0 100

D (m)

H-Beam

+ B = 9G

O-Beam

+ B = 9G

Magnetic noise fields

M.Baron, H.Rauch, M.Suda, J.Opt.B5 (2003) S244

Neutron Fizeau effect

mkV x 222

0

xx kmw

LkKk xx

Lkkkknk xy 222

)()( kk

w

wcmk

c

w

w

wkkk

2

22

01

qkk

Galiilean:

/: mqwith

Einsteinian:

22

021)(

k

kmV

k

Kkn

x

x

Lkx 112

xxxx

xx

vw

kLV

v

wLk

1212

0

Phase shift depends on velocity

of boundary surface !!

Neutron Fizeau effect measurements

wx=3.9 cm/s

A.G. Klein et al. Phys.Rev.Lett. 46

(1981) 1551

wx=2.4 m/s

U. Bonse, A. Rumpf

Phys.Rev.Lett. 56 (1986) 2441

Fizeau-Wheel

cww NbDttgvDtvDtLx 2;/)(

Al: Dλ=167.8 µm; α=30º; vw=10m/s

→ ΔE = 2.47x10-11eV = 24,7 peV

/)( Etiti ee

tiDttgivDtDtDiNbeeee wc

/2/)(2)(

DtgvE w /2

• Quantum Contextuality and Kochen-Specker phenomenon

Yuji Hasegawa

I 0

II 0

Neutron ( )

( )

:

where :

:

i i i

r

c

L

r

E

e e e E

N b D phase shifter

t spin rotator

t zero field precession

Triply entangled states

Phase shifter ( )

Spin rotator ( )

Zero field precession ( )

I 0

II 0

Neutron ( )

( )i i i

r

E

e e e E

:

where :

:

c

L

r

N b D phase shifter

t spin rotator

t zero field precession

0/23/2

0

/2

3/2

Mermin`s Inequality: Measurements

Y. Hasegawa, R. Loidl, G. Badurek S. Sponar, H. Rauch, Phys.Rev.Lett. 103 (2009) 040403

Compton frequency

as an internal clock ?

initiated by: H. Müller, A. Peters, S. Chu, Nature 463 (2010) 926

h

mcc

c

c

2

static: time dependent:

F. Mezei, Z. Physik 255 (1972) 146 R. Gähler, Golub, J.Phys. France 49 (1988) 1195

MHzB

L 12 0

Larmor interferometry

COW-Experiment

(Colella, Overhauser, Werner)

Eo

ko

m

k

mmgH

2 2

2

2 2

2( )

k k kom gH

ko

( ) sin

2

2

cow II I k S

cowm

hgAo 2

2

2 sin

R. Colella, A.W. Overhauser, S.A. Werner, Phys.Rev.Lett. 34 (1974) 1472

E0= 20 meV; mgH ~ 1.003 neV

mchC

Peters A., Chung K.Y., Chu S. Nature

400 (1999) 849

Müller H., Peters A., Chu S. Nature 463

(2010) 926

Collela R., Overhauser A.W., Werner

S.A. Phys.Rev.Lett. 34 (1975) 1053

Hzmc

C

252

10

Use of Compton Frequency

Gravity phase shift

sin2 02

2

02

2

gAh

m

v

LmgH

mc

Umc

sin2

22

.

02

2

2

0

222

gAh

m

m

kmgz

m

k

dsk

)/1( 2

0 mcU

)( mgHU

zrrandrGMg

czmmgz

r

GMmmcLgr

2

2

22

/

1

2

1

Schwarzschild metric for motion

zrrandrGMg

zmmgzr

GMmLcl

2

2

/

2

1

classical motion

Müller H., Peters A., Chu S. Nature 463 (2010) 926

debate with: Wolf P., Blanchhet L., Borde C.J., Raynaud S., Salomon C., Cohen-Tannoudji,

Class.Quantum Grav. 28 (2011) 145017

ddmc c

21

• Advanced post-selection makes quantum experiments better understandable and less mystic.

• Quantum physics gives more correlations, entanglements an contextual effects than classical physics.

• For the interpretation one should keep in mind that the initial or boundary conditions are also based on statistical measurements of an equally prepared ensemble and, therefore, it does not seem so surprising that only statistical predictions about the outcome of an experiment can be made.

• Thus, we do not know everything at the beginning so we also do not know everything at the end.

Résumé

Thank You

• Geometrical

Phases

Berry Phase (adiabatic & cyclic evolution)

Non-adiabatic evolution Non-adiabatic & non-cyclic evolution

[ Berry; Proc.R.S.Lond. A 392, 45 (1984)]

[Samuel & Bhandari, PRL 60, 2339 (1988)] [Aharonov & Anandan, PRL 58, 1593 (1987)

(for 2-level systems) Ω

Geometric

Phase

Results:

S. Filipp, Y. Hasegawa, R. Loidl and H. Rauch, Phys.Rev. A72 (2005) 021602

Cancelling dynamical phase, if

Non-adiabatic & Non-cyclic

Phase

Variance of geometric phase (g2) tends to 0 for

increasing time of evolution in a magnetic field.

G. De Chiara and G.M. Palma, PRL 91, 090404 (2003)

R.S. Whitney, Y. Gefer, Phys.Rev.Lett. 90(2003)190402

Dephasing -

Decoherencing

Fizeau-Wheel

t

tgvvv

vVt

v

tgv

tgvmv

tgmv

V

v

tgvt

wnn

w

n

w

wn

nn

w ..

12

2)(

2

0

22

0

12

Lk x

Fizeau

cww NbDttgvDtvDtLx 2;/)(

Al: V0=5.415x10-8 eV; vn=1000m/s; vw=10m/s

→ ΔE = 5.4x10-12eV = 5.4 peV

/)( Etiti ee 2

2

0

2

0

)(n

w

wnn

w

v

vV

tgvvv

vVE

Ultra-cold

neutrons

Ultra-cold neutrons

at ILL

S. Filipp, J. Klepp, Y. Hasegawa, Ch. Plonka, P. Geltenbort, U. Schmidt, H. Rauch, Phys.Rev.Lett.102 (2009) 030404

Experimental

set-up

Compensation of the

dynamical phase

Compensation in the case of

noise fields

F. Filipp, J. Klepp, Y. Hasegawa, Ch. Plonka, P. Geltenbort, U. Schmidt, H. Rauch, Phys.Rev.Lett.102 (2009) 030404

Rubustness of the

geometric phase

Visualisation of the

robustness of geometric

phases

Dynamical Phase ( ): dt

dGeometric Phase ( ):

Stability calculations

Wave-Latice Interaction