Download Unexpected features of quantum degeneracies in the many

Unexpected features of quantum degeneracies in the many-body continuum
M. Ploszajczak (GANIL)
with
J. Okolowicz (INP-Krakow), J. Dukelsky (CSIC-Madrid)
closed quantum
system
open quantum
system
continuum coupling effects
in binding: 0.1-3 MeV
Instability of SM eigenstates at
channel threshold
- segregation of time scales in the continuum
- alignment of near-threshold states with decay channels
- instability of SM eigenstates at channel threshold
- level degeneracies
Level degeneracy is the subtlest quantal effect that appears at
the highest energy resolution
How accidental is a degeneracy?
J.von Neumann, E.Wigner (1929)
Hermitian problems
Generic situation: avoided level crossings
"2
Level crossing if "1 # " 2
!
!
"1
!
but...
consider
H ( R* ) " {H ( R)} , ( R " X,Y,...)
"E ( R) # E 2 ( R) $ E1 ( R) =
{
2
{
[H11'(R) $ H 22' (R)] + 4 H12' (R)
2
2 1/ 2
}
" #E ( X,Y ) = A( X $ X * ) + 2B( X $ X * )(Y $ Y * ) + C (Y $ Y * )
!
!
!
H (R
*
H ( R)
)
M.V. Berry (1984)
"n (C ) = "n e ±i#(C ) / 2
!
!
2 1/ 2
}
=0
Molecular physics:
M.Vaz Pires et al., J. Chem Phys. 69 (1978) 3242
Nuclear physics:
D.L.Hill, J.A.Wheeler, Phys. Rev. 89 (1952) 1102
R.S. Nikam, P. Ring, Phys. Rev. Lett. 58 (1987) 980
Non-Hermitian problems
VH
Level repulsion
Width clustering
!
VNH
vs
Level clustering
Width repulsion
! VH
VNH
"2
"
!
"!1
!
!
E
" = v in + iw ex
!
!
!
*
- Exceptional point "1 = " 2 ("1 = "1 ) is a generic situation in NH problems
- Defect of the vector space: DH = n "1
- Topological phase: "1 # i"2 , "2 # $i"1
!
!
!
C. Dembowski et al., PRL 86 (2001) 787(microwave cavity)
!
!
Exceptional points in the scattering continuum
- Do they exist in the low-energy scattering continuum?
- Do they appear for realistic values of continuum coupling
parameters?
Shell Model Embedded in the Continuum
K. Bennaceur et al. Nucl. Phys. A671 (2000) 203
J. Rotureau et al., PRL 95 (2005) 042503; NP A767 (2006) 13
Q = [ A]
SM
SMEC
P = [ A "1] # [1]
T = [ A " 2] # [2]
HTT
H PT
... = ...
HQP
HQQ
!
!
H PP
HQQ " HQQ ( E )
!
!
!
HQQ ( E ) = HQQ + HQP GP ( E ) H PQ
!
!
( +)
!
HQQ
H PP
closed quantum
!
system
!
!
HQT
HTT
! open quantum
system
Open QS solution in Q :
!
HQQ "# = E# "#
biorthogonal basis
*
#
"#˜ HQQ = E "#˜
"# = $ a#i %Ai
!
*
; "#˜ = "#
i
"
!
The scattering function (in I=Q+P+...)
˜$ ;
" = # + % c$ "
c
E
c
E
$
!
c" =
!
#" HQP $ Ec
E % E"
˜ # $ 1+ GP( +) H PQ "#
"
(
)
!
!
Inside of the interaction region :
!
"Ec ~ $ c# "#
#
eigenfunctions of HQQ ( E )
!
!
O
#$
= "#˜ O "$
!
For bound states: E" ( E ) is real and E" ( E ) = E
For unbound states: physical resonances
poles of S-matrix
!
!
!
"#˜ "$ = %#$
E " = Re(E" ( E )) E = E"
#" = $2Im(E" ( E )) E = E"
Exceptional points in the scattering continuum appear due to an
interplay between Hermitian and non-Hermitian parts of HQQ ( E ) :
'
+ VQQ ( E;V0 )
HQQ ( E ) = HQQ + HQP GP( +) ( E ) H PQ = HQQ
!
scattering energy
!
E=0: 1st N-emission
threshold
!
continuum coupling
constant
Single-root solutions of a discriminant:
" (# )
det [HQQ ( E;V0 ) $ EI] = 0
"E
(
!
!
!
)
(" = 0,1)
give positions ( E;V0 ) µ of singularities, where µ = 1,2,..., M " n ( n #1)
! vector space
and n is a dimension of the
M = n ( n "1) in non-integrable quantum models
!
!
J. Dukelsky et al., arXiv:0811.4105
Conclusions
- EPs (double-poles of the S-matrix) can appear in the low-energy
continuum for realistic continuum Shell Model Hamiltonians
(SMEC, Gamow Shell Model)
- Number of EPs (M=n(n-1)) is typical for non-integrable systems
- Several EPs ‘escape’ from a physical region of (E;V0 )-parameters
- The non-resonant continuum is important for the spectroscopy
of weakly bound nuclei (energy shifts and degeneracy,
!
modification of effective NN correlations
(spectroscopic
factors), binding systematics, manifestation of clustering,
exceptional points, segregation of times scales,...)
Experimental observation of EPs would provide a stringent test
of configuration mixing involving nucleons in discrete and
continuum states