Quantum revivals — connections to bouncing neutrons?

Continuum approx n. E more than 10 n−4 n+2 n+4 n−2 n. E− function. (continuous). Levels. Lots of pretty mathematics. 1st dervative µclassical oscillations ج ذ.
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Institut Laue Langevin, Grenoble, France. Les Houches, Fevrier 2010

Quantum revivals — connections to bouncing neutrons? Robert S. Whitney REVIEWING A REVIEW: R.W. Robinett,

\Quantum wavepa ket revivals", Physi s Reports

392, 1-119 (2004)

What is a “quantum revival”? Quantum revival : wavepacket spreads out and then ... unexpe tedly recombines

T l O N iva rev

after many periods

quantum version of Poincare’s ´ return??

after ~160 periods

R

L A IV V E

Outline

| Two types of revival problem.

[a] initial-state localized in position [b] initial-state localized in energy

| Simple picture of neutron expts

) localized in energy

} Theory of revivals } Mathematical dangers! | Revivals in Neutron experiment  Observable?? YES

...but must take theory to lower levels + finite lifetimes

 Useful tool??

NOT SURE

Neutron expt: initial-state localized in position or energy

Initial state localized in position: 1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111

microns

1111111111111111111111111 00000000 00000000000000000 00000000 11111111 00000000000000000 0000000011111111111111111 11111111 00000000000000000 11111111111111111 mirror 00000000 11111111 00000000000000000 11111111111111111 00000000 11111111 00000000000000000 11111111111111111 00000000 11111111 00000000000000000 0000000011111111111111111 11111111 00000000000000000 11111111111111111 00000000 11111111 00000000000000000 11111111111111111 00000000 11111111 00000000000000000 11111111111111111 mask 00000000 11111111 00000000000000000 0000000011111111111111111 11111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 fraction nm 00000000000000000 11111111111111111 00000000000000000 11111111111111111

Initial state localized in energy: HIGH energy

approx 50 levels

LOWER energy + energy filtering

LOWER energy

} occupied

Neutron expt: final state

Simplest case: only 2 levels phase shift= π x even integer

phase shift= (E 1−E 0)t

incoming state 111111111111 000000000000 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111

flat wavefunct

1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 + 0000000000 1111111111 0000000000 1111111111 + 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 _

1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111

many "bounces"

1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 + 0000000000 1111111111 0000000000 1111111111 + 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 _

1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111

phase shift= π x odd integer 111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 + 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 + 000000000 111111111

1111111111 0000000000 0000000000 1111111111 _ 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111

Revivals for few levels not in Robinett’s review — he only considers many-levels

Overlap initial and final wavefunction:

ε1 ε2

i

(t)



t os + t

Trev = 1/ 2ε − 1

4/9

4/9 1/9

t

t −4/9

+ = 

=

1 (1 + 2 ) 2 1 (1 2 ) 2

Tcl=1/ε+

l

2 1 2 i  t i  t jA(t)j = 9 1 + e 1 + e 2 1 = 9 1 + 4 os2  t + 4 os 

En

iva

h 0j

rev

A(t) =

1/9

t

) classical oscillation period

cf. oscillator where freq = level-spacing

) REVIVAL period

for oscillator 

=0

Revivals for many levels

E− function

Levels

(continuous)

Continuum approx more than 10

En n−4 n−2

Lots of pretty mathematics 1st dervative classical oscillations T l 2nd derivative REVIVALS with period Trev

) )

3−level Trev = 1/ 2ε −

n+2 n+4

 T l

many−level 1

Tcl

Trev rev

iva

rev iva

l

l

1

Tcl=1/ε+

n

1/9

1/9

t

t

analogy: 2-slit interference

! grating interference

Revivals and fractional revivals A 0

D

C

B

Trev /4

Trev /2

Trev

time

Tcl

initial oscillation

t=0

not in 3-level model

Tcl

A

t= Trev

REVIVAL

fractional revivals in continuum theory

t=T cl

Tcl

B fractional revival fractional revival

t= Trev /2 C t= Trev /4 D

Mathematical dangers [1℄

For neutrons — only 50 levels above mirror

}

use N levels in initial-state for N = 10 Is continuum model good enough? Errors go like 1=N or not? Can we do better?

} }

[2℄

We know: Hamiltonian

...but what about Revivals

) spectrum ) E-function ) Revivals

) E-function ) spectrum ) Hamiltonian

Revivals reveal neutron-mirror interaction?? Danger:

i.e. trapping potential

Difficult & uniqueness unknown dependence on expt uncertainties?? cf. hearing sound of drum (2D — non-unique)

Mathematical dangers [3℄

No unique definition of E-functions?? physically reasonable E−function

mathematically allowed E−function

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

finite lifetime

} Is this a formal problem or a pra ti al one? } Will uncertainity introduced by life-times make it worse?

Revivals as useful tool?? harmonic osc anharmonic osc 1 α=1/800 β=0

Very small changes in spectrum

anharmonic osc 2 α=1/800 β=2 x 10 −6 anharmonic osc 1 initial oscillation

t=0

anharmonic osc 2

t=T cl

t=2T cl

t= Trev

t= Trev+T cl

A

REVIVAL B fractional revival fractional revival

t= Trev /2 C t= Trev /4 D

Trev

 800T l

Trev

 2933T l

Ex iting or Frightening??

Summary

| Two types of revival problem:

[a] initial-state localized in position [b] initial-state localized in energy

( easiest for neutron expt

} Theory: Robinett, Physi s Reports 392, 1-119 (2004) ...

but

helpful picture is beating in 3-level model

| Revivals in Neutron experiment  Observable?? YES  Useful tool?? NOT SURE

extremely sensitive to expt parameters

| NEED theory for intermediate NÆ of levels & finite lifetimes