1

Tunneling of atoms, nuclei and molecules Carlos Bertulani (Texas A&M University-Commerce, USA)

Collaborators A.B. Balantekin (Wisconsin) V. Flambaum (Sydney) M. Hussein (Sao Paulo) D. de Paula (Rio) V. Zelevinsky (Michigan)

Int. Workshop on Critical Stability, Santos, October 13, 2014

Resonant tunneling (point particle + double-hump barrier)

quasi-bound state

Well-known. Used in device technology. E.g. Resonant Diode Tunneling device.

V

GaAs

x

E

EF emitter collector EF

V

2

3 3

Resonant tunneling (with single barrier) (composite particle + single-step barrier)

quasi-bound state

Poorly known. Occurs in atomic, molecular and nuclear systems. E.g. fusion of loosely-bound nuclei.

(a)

(b)

4

Feshbach Resonances

5

€

H = −!2

2m∂2

∂x12 +

∂2

∂x22

$

% &

'

( ) +V(x1) +V(x2) +U x1 − x2( )

Step-barrier V + square-well U

€

V(x) =V0, − a /2 ≤ x ≤ a /20, otherwise

$ % &

€

U(x) =0, − d /2 ≤ x ≤ d /2∞, otherwise

% & '

barrier interaction between particles

U

V

Schroedinger equation

Step barrier

Square-well

CB, Flambaum, Zelevinsky, JPG 34, 1 (2007)

6

€

x =x1 + x22

"

# $

%

& ' , y = x1 − x2 +

d2

€

−!2

2m∂2

∂x2−!2

2m∂2

∂y2+U(y −

d2) +W(x,y)

$

% & &

'

( ) ) Ψ x,y( ) = EΨ x,y( )

(b) d/2 < a

change of variables

c.m. relative

barrier for c.m. motion

x

y

0

(a) d/2 > a

W=V0 W=V0

W= 0

x

y

0 W=2V0

7

Problem equivalent to a single particle (c.m. coordinate x) tunneling through 2 barriers.

Zakhariev, Sokolov, Ann. d. Phys. 14, 229 (1964) Saito, Kayanuma, J. Phys. Condens. Matter 6 (1994) 3759

x

y

0

8

Tunneling of Molecules ( )rVzZUzZU

mm+⎟⎠

⎞⎜⎝

⎛ ++⎟⎠

⎞⎜⎝

⎛ −++=224

22 pPH

€

p2

m+V r( )

"

# $

%

& ' φn r( ) = εnφn r( )

€

Ψ Z,r( ) = ψ Z( )φn r( )n=0

∞

∑

€

Unm Z( ) =4m!2

U Z +z2

"

# $

%

& ' +U Z − z

2"

# $

%

& '

)

* +

,

- . ∫ φn

* r( )φm r( )dr

€

d2

dZ2+ kn

2"

# $

%

& ' ψn Z( ) − Unm Z( )

m∑ ψm Z( ) = 0

€

kn2 =

4m!2

E − εn( )

€

ψnl Z( ) = eik nZδnl +12ikn

eik n Z−Z'( )Unm Z'( )ψml Z'( )dZ'−∞

∞

∫m=0

∞

∑

€

Rnl Z( ) =12ikn

eik nZ'Unm Z'( )ψml Z'( )dZ'−∞

∞

∫m=0

∞

∑

€

Tnl Z( ) = δnl +12ikn

e− ik nZ'Unm Z'( )ψml Z'( )dZ'−∞

∞

∫m=0

∞

∑

basis expansion

effective potential

effective equation for cm motion

general solution

= Z/a

effective potential

Um

i

U00

U11

U01

9

€

R l =knkln=0

∞

∑ Rnl2, Tl =

knkln=0

∞

∑ Tnl2

€

Pn→ l =knkl

Rnl2

+ Tnl2( )

€

U0a!c

=10, mU0

!2a = 6

Step barrier parameters

€

V0b!c

= 2, mV0!2

b = 4

Square-well particle-particle interaction

Reflection and transmission probabilities

Transition probability

Goodvin, Shegelski, PRA 72, 042713 (2005)

a

U0

b

V0

Transmission

Transition

k = ~

T1 P0 ! 1

10

Goodvin, Shegelski, PRA 72, 042713 (2005)

By moving to its ground state the excited molecule increases its chance of transmission.

Trans- mission

Transition

The opening of the higher channels decreases the probability of transmission.

11

€

k0 +2mU0a!2

tan k0 r2( ) = 0

For d + d ! 4He

With a and U0 simulating Coulomb barrier For large Z’s, A’s

!  Many resonances possible, if <r2> large (loosely-bound)

Problem: strong interactions are too strong!

Tunneling of a (two-particle) nucleus

€

U0a→δ Z( )valid for

delta barrier T 0

12 Electron screening in fusion reactions

€

ΔE = E'−EAdiabatic model:

Rolfs, 1995

€

σlabfusion ~ σbare(E + ΔE)

~ exp πη(E) ΔEE

&

' ( )

* + σbare(E)

Electron screening enhancement

Small effects

Vacuum polarization

Balantekin, CB, Hussein, NPA 627 (1997)324

Z1Z2 = 1

•  Thermal motion, lattice vibrations, beam energy spread

•  Nuclear breakup channels (in weakly-bound nuclei)

•  Dynamics of tunneling

all ≤ 1%

Not a solution! (we need ~ 100%) 13

Bang, PRC 53 (1996) R18 Langanke, PLB 369 (1996) 211

H + He

Golser, Semrad, PRL 14 (1991) 1831

Mainly charge-exchange

E’ = E – Sp.∆x

Wrong extrapolation of stopping power

€

Sp = −dEdx

Data has to be corrected for stopping power:

Very few data on stopping at ultra-low energies:

14

Simplest test

CB, de Paula, PRC 62, 045802 (2000) PLB 585, 35 (2004)

Charge exchange (pickup) Projectile slows down to carry electron with

p + H

P + D

e-

Elliptic coordinates

€

ξ =r1 + r2R

; η =r1 − r2R

; φ

Ψ = F(ξ)G(η)eimφ

€

ddξ

ξ2 −1( ) dFdξ$

% &

'

( ) +

R2ξ2

2E + 2Rξ − m2

ξ2 −1$

% &

'

( ) F ξ( ) = 0

ddη

1−η2( ) dGdη$

% &

'

( ) −

R2ξ2

2E + 2Rξ +

m2

η2 −1$

% &

'

( ) G η( ) = 0

change of variables

Two-center wfs (molecular orbitals):

15

Expansion basis: molecular orbitals for p+H

He+ H+H+

!,3210φδπσ

λ

letterCodeofValue

€

lzΦs = ±λΦs

16

(Hellman, Feynmann relation)

For Ep < 10 keV, only 1sσ and 2pσ 2-level problem - resonant exchange

Coupled-channels calculation

b (a.u.)

€

i! ddtam t( ) = Em t( )am t( ) − i! an t( ) m d

dtn

n∑

€

m ddtn =

mdVp /dt nEn t( ) −Em t( )

,

€

Pexch ≈12

+12cos

1!

E2p t( ) −E1σ t( )[ ]dt−∞

∞

∫' ( )

* + ,

17

H+ + He collisions (two-active electrons)

Slater-type orbitals

F . C = S . C . E

Two-center basis for two-electrons Hartree-Fock equations

t. d. coupled-channels equations

€

φ =Nrn−1 e−ξr Ylm θ,φ( )

€

Φi = c jiAφ i

A + c jiBφ i

B[ ]i=1

n

∑

€

Fµν =Hµν + Pλρλρ

∑ µν | λρ( ) − 12 µρ | λν( )'

( ) *

+ ,

€

Hµν = φµ

* 1( ) −12∇12 −

1r1AA

∑'

( )

*

+ , φν

* 1( ) dτ1∫∫ , Pλ ρ = 2 cλ icρ ii=1

occ

∑

€

µν | λρ( ) = φµ1( ) φ

ν1( ) 1r12

∫∫ φλ2( ) φ

ρ2( )dτ1dτ2 , Sµν = φ

µ1( ) φ

ν1( )∫ dτ1

18

Damping of resonant exchange H(1s) ⇔ He(1s2s)

Landau-Zener effect + dissipation

€

P = e−Γ Δt coll cos2H12 a2v

%

& ' (

) * 19

Stopping power at very low energies

p + H

P + D

e-

Threshold effect

He: 1s2 ! 1s2s: 19.8 eV

€

EP ≥µ2

4MPme

ΔE ≥ 8 keVCB, PLB 585, 35 (2004)

CB, de Paula, PRC 62, 045802 (2000)

20

21

Baron Muenchhausen escaping from a swamp by pulling himself up by his own hair. G.A. Buerger (1786).

Virtual particles enhance tunneling

QFT: a "physical" particle consists of a "naked" particle "dressed" in a cloud of short-lived "virtual" particles.

22

€

Π r,r ';E( ) = gkk ,λ∑

2 r ˆ p ⋅ ekλ( )eik ⋅r n n ˆ p ⋅ ekλ*( )e− ik ⋅r r '

E −En −ωk − i0n∑

= M r,r ';E( ) +Γ r,r';E( )

€

HΨ r( ) + Π r,r';E( )Ψ r '( )d3r∫ = EΨ r( )

Quantum Muenchhausen

Non-relativistic reduction:

Flambaum , Zelevinsky, PRL 83, 3108 (1999)

€

M +Γ

€

M

Hagino, Balantekin PRC 66, 055801 (2002) Small effect for tunneling of stable nuclei. Effect for loosely-bound nuclei (and molecules) unexplored.

M = self-energy, mass renormalization, virtual photons Γ = decay width, real photons

Self-energy:

23

Neutron stars and neutron-rich nuclei

Neutron-rich nucleus (light with N/Z ~ 1.5)

Stable nucleus N/Z ~ 1.5

24

Loosely-bound nuclei

Tetraneutron?

10/26/2002

Marques et al, PRC 65, 044006 (2002)

halo

25

Tetraneutron as a dineutron–dineutron molecule CB, Zelevinsky, J. Phys. G 29 (2003) 2431

€

Ψ r1,r2,r3,r4( ) = A ψ R( )φa r1,r2( )φb r3,r4( ){ }

€

H = TR + Ta + Tb + viji< j∑ (rij)

€

P2

2mN

+U1 R( ) +U2 R( )"

# $

%

& ' ψ R( ) = Eψ R( )

R 1

2

3

4

- Effective potential repulsive -  No margin for pocket or state -  no tetraneutron in singlet, or triplet state! -  Confirmed by Pieper, PRL 90 252501 (2003)

With realistic NN potentials vij

Antissimetrization (Pauli-principle)

Effective wave equation

26

n

n 9Li

11Li

Fusion of halo nuclei

€

H = −!2

2m∇12 −!2

2m∇22 +VA r1A( ) +VB r1B( ) +VA r2A( ) +VB r2B( )

€

Ψ± ≅N ΨA r1A( ) ± c±ΨB r1B( )[ ]

€

I = ΨA VB r1B( ) ΨA

€

L = ΨA VB r1B( ) ΨB

€

O = ΨA |ΨB

€

E(R) =ΨHΨΨ |Ψ

=S2n 1+ O2( ) + 2OJ+ 2I

1+ O2

Model Hamiltonian

Landau approximation

Covalent bond

Loosely bound nuclei are like Rydberg states in atoms

CB, Balantekin, PLB 314, 275 (1993)

27

€

9Li+11Li

€

V(R) = E(R) −S2n

€

σ Ecm( ) =πk2

2l+1( )Tll∑ Ecm( )

CB, Balantekin, PLB 314, 275 (1993)

Effective covalent potential Fusion cross section

More on fusion with weakly-bound nuclei :

Canto, Gomes, Donangelo, Hussein, Phys. Rep. 424, 1 (2005)

28

Data: Kasagi, PRL 79, 371 (1997)

Bremsstrahlung in α-decay

Interference between photons emitted within and outside the barrier tunneling time

γ

29

Time-dependent analysis

€

dE ω( ) =8πω2

3m2c3Z2e2 Ψ r,t( ) −i!∇Ψ r,t( )[ ]d3r∫

2

€

dPdE γ

=1

E γ

dE ω( )dE γ

C(out)

QM (out) QM (all space)

Eα » 0

Eα ~ 0

C(out)

QM (out)

QM (all space)

CB, de Paula, Zelevinsky, PRC 60, 031602(R) (1999)

Power emitted:

Solving for t.d. w.f. ! power emitted:

Not well understood. Not verified experimentally.

30

€

H =px2

2mx

+py2

2my

+12k x − y( )2 +

12qx2

Composite particles seeing different potentials

Normal modes: 0,:0 22 →+=→ −+ ωµ

ωx

x mqkm

xy m

qkm →=→ −+22 ,:0 ω

µω

q < 0

Equal masses and k >> q:

€

ω− =ω0 1+q8k

$

% &

'

( ) , ω0

2 =q2m

Tunneling probability:

€

P = exp2πE!ω−

%

& '

(

) * = exp −

2πE!ω0

%

& '

(

) * × exp

πEmω0

2!k%

& '

(

) *

enhancement

€

mx /my <<1: ω− ≈q

mx +my

1+q8kmy

mx

%

& '

(

) *

Zelevinsky, Flambaum, JPG 34, 355 (2005)

Only one particle sees barrier:

Finite size always enhance tunneling probability

31

Tunneling of Cooper pairs

€

H =px2 + py

2

2m+12k x − y( )2 +

12q x2 + y2( )

Normal modes:

€

ω+2 =

2k + qm

, ω−2 =

qm

Does not depend on k ! no finite size effect

€

E = E ∞( ) +! 2k /m

2−! 2k + q( ) /m

2

Energy transfer from internal to c.m. motion

Adiabatic approx. not valid for tunneling through a Josephson junction (more complicated)

Zelevinsky, Flambaum, JPG 34, 355 (2005)

Both particles see barrier:

32

€

H = −px2

2mx

+py2

2my

+V x − y( ) +U x( ) +U y( )

€

x,y→r,R

€

Ψ r,R( ) =ψ(R)φ(r,R)

€

α R( ) = φ∂φ∂R

€

ψ R( ) = u R( )exp − α R'( )dR'R∫&

' ( ) * +

ϕ normalized

€

u" R( ) +2M!2 E − ˜ U R( )[ ] u R( ) = 0

€

˜ U R( ) = ε R( ) −E0 +!2

2Mα2 R( ) + α' R( ) +β R( )[ ]

€

β R( ) = φ∂2φ∂R2

Composite particle fusion enhancement CB, Flambaum, Zelevinsky, JPG 34, 1 (2007)

Particles see different barriers:

change of variables Adiabatic approximation

33 33

€

E0 = −2.225 MeVr0 = 2 fm

CB, Flambaum, Zelevinsky, JPG 34, 1 (2007)

Features probably seen in experiments. But not disentangled from uncertainties in potentials, polarization effects, etc.

Deuteron tunneling through barrier step.

34

1D

2D

Laser

Interference

Optical Lattices

35

€

a(B) = aB=0 1−Δ

B −B0( )$

% & &

'

( ) )

B0

Δ = B-field width at the resonance

Feshbach Resonances

36

Probability distribution of strongly bound (Eb=10 kHz) rubidium molecules in optical lattice with size D = 2 µm. From top to bottom the time is t = 0, t = 10 ms, t = 25 ms and t = 100 ms.

Diffusion of Strongly Bound Molecules

T. Bailey, CB, E. Timmermans, PR A 85, 033627 (2012)

37

Escape probability of strongly bound rubidium molecules in optical lattice with D=2 µm, for increasing potential barrier heights. Barrier heights increase by 1.5 from the solid to dashed-dotted, from dashed-dotted to long-dashed, and from long-dashed to dotted curve.

Diffusion of Strongly Bound Molecules

38

Integrity probability of loosely bound rubidium molecules in an optical lattice D = 2 µm, and potential barrier height V0=5 kHz. The binding energies were parametrized in terms of the barrier height, with E4=V0/20, E3=V0/5, E2=V0/2, and E1=V0/1.2.

Diffusion of Loosely Bound Molecules

39

Blue histograms are the relative probability of finding a molecule in its ground state at a given position along the lattice. Red histograms give the relative probability of finding individual atoms after the dissociation.

Diffusion of Loosely Bound Molecules

t1 = 200 ms

t2 = 400 ms

40

Spreading position width of bound molecules, σM(t), shown by solid line and of dissociated atoms, σA(t), shown by dotted line. The dashed curve is a fit to the asymptotic time dependence σM(t) ~ t1/2. No clear asymptotic dependence is found for the dissociated atoms.

Diffusion of Loosely Bound Molecules

€

D = 0.156 !2

mbσ02

Compared to Einstein diffusion coefficient

€

D =kTb

€

T ~ !2

mσ02

41

Conclusions:

“Although a [large] number of theoretical works have studied tunneling phenomena in various situations, quantum tunneling of a composite particle, in which the particle itself has an internal structure, has yet to be clarified.” Saito and Kayanuma, J. Phys.: Condens. Matter 6 (1994) 3759 •  20 years after: still remains open to imagination and creativity.