Suppression of chaos and occurrence of rotational bands at high energies

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1 0/ Suppression of chaos an occurrence of rotational bans at high energies Base on Occurrence of high-lying rotational bans in the interacting boson moel M.Macek, J.Dobeš, P.Cejnar Physical Review C 8 ( Regularity-inuce separation of intrinsic an collective ynamics M.Macek, J.Dobeš, P.Stránský, P.Cejnar Physical Review Letters 05 ( Pavel Cejnar Institute of Particle an Nuclear Physics Faculty of Mathematics an Physics Charles University, Prague, Czech Republic mff.cuni.cz Primosten 0

2 0/ Program: Aiabatic separation of intrinsic an collective motions Pavel Cejnar A few wors on chaos in nuclei Aiabatic separation an chaos in IBM Institute of Particle an Nuclear Physics Faculty of Mathematics an Physics Charles University, Prague, Czech Republic Conclusions an outlook mff.cuni.cz Primosten 0

3 Program: Aiabatic separation of intrinsic an collective motions Pavel Cejnar A few wors on chaos in nuclei Aiabatic separation an chaos in IBM Institute of Particle an Nuclear Physics Faculty of Mathematics an Physics Charles University, Prague, Czech Republic Conclusions an outlook mff.cuni.cz Primosten 0

4 03/ We all know this book! The first eition in 980 An essential reference for all practitioners of almost any many-boy technique

5 04/ Page 508 an below: The aiabatic approximation

6 05/ Aiabatic theorem of quantum mechanics Hamiltonian H(λ riven by λ= t/τ from λ=0 (at t=0 to λ= (at t=τ. System evolves from Ψ(t=0 Φ i (0, the i th eigenstate of H(0, to an unknown final state Ψ(t=τ. If the change is very slow, the final state is almost precisely the i th eigenstate of H(. So Ψ(t=τ Φ i ( with Φ i ( Ψ ( t = τ for λ [ min Ei± ( λ E i ( λ ] 0 λ 0 λ τ >> max Ψ i ± ( λ H Ψi ( λ /

06/ Aiabatic theorem of quantum mechanics Hamiltonian H(λ riven by λ= t/τ from λ=0 (at t=0 to λ= (at t=τ. System evolves from Ψ(t=0 Φ i (0, the i th eigenstate of H(0, to an unknown final state Ψ(t=τ.

So Ψ(t=τ Φ i ( with Φ i ( Ψ ( t = τ for τ Ψ i λ H Ψ λ >> max ± ( 0 λ Some applications: Lanau-Zener mechanism Lanau (93, Zener (93, Stueckelberg (93, Majorana (93 Non-aiabatic passage through an

7 06/ Aiabatic theorem of quantum mechanics Hamiltonian H(λ riven by λ= t/τ from λ=0 (at t=0 to λ= (at t=τ. System evolves from Ψ(t=0 Φ i (0, the i th eigenstate of H(0, to an unknown final state Ψ(t=τ. If the change is very slow, the final state is almost precisely the i th eigenstate of H(. So Ψ(t=τ Φ i ( with Φ i ( Ψ ( t = τ for τ Ψ i λ H Ψ λ >> max ± ( 0 λ Some applications: Lanau-Zener mechanism Lanau (93, Zener (93, Stueckelberg (93, Majorana (93 Non-aiabatic passage through an avoie level crossing Geometric phase Pancharatnam (956, Berry (984 Phase acquire along a close path in parameter space aroun a egeneracy point Aiabatic quantum computation Fahri et al. (000 System riven to an unknown groun state carrying the result of some computation Born-Oppenheimer approximation Born, Oppenheimer (97, Eckart (935 Separation of intrinsic an collective wave functions of a molecule or nucleus bans of rotational & vibrational states built on iniviual intrinsic ( electronic or nucleonic states of the molecule (nucleus λ i ( / [ min E ( λ ( λ ] 0 λ i± E i λ

8 07/ Separation of intrinsic, vibrational, an rotational excitations in molecules & nuclei Jungclaus et al. (00 Herzberg (950

08/ Separation of intrinsic, vibrational, an rotational excitations in molecules & nuclei Questions: How far in excitation energy can such bans be followe?

9 08/ Separation of intrinsic, vibrational, an rotational excitations in molecules & nuclei Questions: How far in excitation energy can such bans be followe? How goo is the escription of collective bans base on the aiabatic separation? Are there some requirements on the given intrinsic state to support the existence of well istinguishe collective bans? Hints: Various intrinsic states may show ifferent susceptibility to external perturbations (like the Coriolios term. In this respect, a very important role may be playe by the egree of chaos in the intrinsic ynamics (chaos => instability.

10 Program: Aiabatic separation of intrinsic an collective motions Pavel Cejnar A few wors on chaos in nuclei Aiabatic separation an chaos in IBM Institute of Particle an Nuclear Physics Faculty of Mathematics an Physics Charles University, Prague, Czech Republic Conclusions an outlook mff.cuni.cz Primosten 0

09/ Classical & quantum chaos Classical sensitivity to initial conitions

Quantum no genuinely quantum efinition of chaos link to quantum-classical

(Wigner Nuclei show signatures of quantum chaos: n & p resonances Bohigas et

11 09/ Classical & quantum chaos Classical sensitivity to initial conitions practical loss of preictability quasi ergoic trajectories in the phase space Quantum no genuinely quantum efinition of chaos link to quantum-classical corresponence Bohigas (984: Chaos on quantum level affects statistical properties of energy spectra. Chaotic systems yiel correlations consistent with the GOE or GUE. (Wigner Nuclei show signatures of quantum chaos: n & p resonances Bohigas et al. (983 low-energy levels Von Egiy et al. (987 ω=0.6 P( s s α ω ω e = Γ( Nearest-neighbor spacing ω+ α ω s ω+ ω+ ω+ s i = E E i i E E i i Broy ω=0 ω=

0/ Origin of chaos in nuclei Many-boy ynamics Complex

Collective ynamics Nuclear vibrations an rotations

Vorkapic, Dieperink 98899; Alhassi, Whelan, Novoselsky

99; Canetta, Maino 000; Cejnar, Jolie, Macek, Heinze,

(998; PRL 93,350(004; PRC 75,06438(007; PRC 80,0439(009;

12 0/ Origin of chaos in nuclei Many-boy ynamics Complex interactions of all particles in the nucleus Shell-moel approach: Zelevinsky et al. 990s-000s Single-particle ynamics Nucleonic motions in eforme nuclear potentials Arvieu et al. 987; Rozmej, Arvieu 99; Heiss, Nazmitinov, Rau Collective ynamics Nuclear vibrations an rotations Interacting Boson Moel Iachello, Arima 975 Paar, Vorkapic, Dieperink 98899; Alhassi, Whelan, Novoselsky 99093; Mizusaki et al. 99; Canetta, Maino 000; Cejnar, Jolie, Macek, Heinze, Casten, Dobeš, Stránský PLB 40,4(998; PRE 58,387 (998; PRL 93,350(004; PRC 75,06438(007; PRC 80,0439(009; PRC 8,04308(00; PRL 05,07503(00 Geometric Moel Bohr 95 Cejnar, Stránský, Kurian, Hruška 0040 PRL 93,050(004; PRC 74,04306(006; PRE 79, 0460(009; PRE 79,0660(009

13 / Geometric moel classical chaos map for J = 0 Stránský et al. 00 JPconf.39,000 H ' J π γ 3 4 = + π β + Aβ + Bβ cos3 γ + C β x, y, z ( β, γ K + β = I V T T rot vib eforme spherical γ-soft Regular phase space fraction f reg = Ω reg / Ω tot

$009 PRE 79,0460 PRE 79,0660 f reg classical regular fraction ω ajunct of Broy parameter mass parameter$

14 / Geometric moel comparison of classical & quantal measures for J = 0 Stránský et al. 009 PRE 79,0460 PRE 79,0660 f reg classical regular fraction ω ajunct of Broy parameter mass parameter quantization type quantization type

15 3/ Interacting boson moel comparison with the GM H n = η n = + Macek et al. 007 PRC 75,06438 Macek et al. 009 PRC 80,0439 Macek 00 issertation ( η ( Q N χ + + Qχ= s + s + χ[ SU(3 > control parameter => multi-imensional chaotic map. Nevertheless, there exist regions of almost full compatibility with the geometric moel. Q + χ ( ] η 0 U(5 spherical eforme χ 0 O(6 J = 0, E = 0 B χ

16 4/ Interacting boson moel classical chaos map for J = 0 f reg H n = η n N χ χ = + Q s s [ χ= + + χ ] ( η = 0.5 η = 0. 7 Macek et al. 007 PRC 75,06438 Macek et al. 009 PRC 80,0439 Macek 00 issertation f reg ( η ( Q Q SU(3 η 0 U(5 spherical eforme χ 0 O(6 E E χ χ f reg = Ω reg / Ω tot

17 4/ Interacting boson moel classical chaos map for J = 0 f reg n N χ χ = + Q s s [ χ= + + χ ] ( η = 0.5 η = 0. 7? H = η n Macek et al. 007 PRC 75,06438 Macek et al. 009 PRC 80,0439 Macek 00 issertation f reg ( η Change of stability of some high-energy β an γ vibrations ( Q Q SU(3 η 0 U(5 spherical eforme χ 0 Arc of Regularity? O(6 E E ω β = ω γ Change of stability of low-energy β an γ vibrations χ f reg = Ω reg / Ω tot ω β = ω γ χ

18 Program: Aiabatic separation of intrinsic an collective motions Pavel Cejnar A few wors on chaos in nuclei Aiabatic separation an chaos in IBM Institute of Particle an Nuclear Physics Faculty of Mathematics an Physics Charles University, Prague, Czech Republic Conclusions an outlook mff.cuni.cz Primosten 0

19 5/ SU(3 limit of IBM home of rotational bans H = κ ( Q χ = 7 Q 7 κ'( L L (λ,μ = the SU(3 quantum numbers efine the intrinsic K = missing label of SU(3>O(3 reuction state l =K,K+,K+, K+max{λ,μ} enumerates states in rotational bans χ = SU(3 U(5 spherical eforme O(6 (λ,μ Iachello, Arima 987 E Hamiltonian eigenstates ( κ κ' ( κ ( λ µ λµ 3λ 3µ 3 l = l 8 ( λ, µ, K,l Below we only focus on K = 0 bans. N = 6

20 6/ Away from SU(3 limit SU(3 quasiynamical symmetry Macek et al. 00 PRL 05,07503 H n = η n ( η ( Q Q N χ χ = + Q s s [ χ= + + χ ] ( N=30 SU(3 η 0 U(5 spherical eforme χ 0 O(6 low energy meium energy li = ( λ, µ, K A li ( λ, µ, K ( λ, µ, K, actual amplitue eigenstate probability l P = i λ, µ A li ( ( λ, µ, K K l SU(3 eigenstate l i i th state with angular momentum l Quasiynamical symmetry Rowe et al

21 6/ Away from SU(3 limit SU(3 quasiynamical symmetry Macek et al. 00 PRL 05,07503 H n = η n ( η ( Q Q N χ χ = + Q s s [ χ= + + χ ] ( N=30 SU(3 η 0 U(5 spherical eforme χ 0 O(6 low energy meium energy l i = ( λ, µ, K A li ( λ, µ, K ( λ, µ, K, l actual amplitue eigenstate probability l P = i λ, µ A li ( ( λ, µ, K K SU(3 eigenstate meium energy l i i th state with angular momentum l Quasiynamical symmetry Rowe et al

22 6/ Away from SU(3 limit SU(3 quasiynamical symmetry Macek et al. 00 PRL 05,07503 H n = η n ( η ( Q Q N χ χ = + Q s s [ χ= + + χ ] ( N=30 SU(3 η 0 U(5 spherical eforme χ 0 O(6 low energy meium energy l i = ( λ, µ, K A li ( λ, µ, K ( λ, µ, K, l actual amplitue eigenstate probability l P = i λ, µ A li ( ( λ, µ, K K SU(3 eigenstate high energy!!! meium energy l i i th state with angular momentum l Quasiynamical symmetry Rowe et al

23 7/ Away from SU(3 limit search for rotational bans Macek et al. 00 PRL 05,07503 H n = η n ( η ( Q Q N χ χ = + Q s s [ χ= + + χ ] ( η = 0.5, χ = 0.9 η = 0.5, χ = 0.5 SU(3 η 0 U(5 spherical eforme χ 0 O(6 ( xk x ( yk y Pearson correlation coefficient π ( x, y = n k s (for statistical vectors an x sy x y As the vectors x an y take probabilities from the SU(3 ecomposition: 0 l i j x { P λ, µ }, y { P( λ, µ } π (0 (perfect anticorrelation 0 (null correlation + (perfect correlation Search for the l j states that maximize correlation with a given 0 i state with E(l > > E(l <. These set of states form natural caniates for rotational bans. Calculate: C(,4 i = max j[ π (0i, j ]maxk[ π (0i,4k ], l ( i j N=30

8/ Away from SU(3 limit search for rotational bans Macek et al. 00 PRL 05,07503 H n = η n ( η ( Q Q N χ χ = + Q s s [ + + + χ= + + χ ] ( η = 0.5, χ = 0.9 η = 0.5, χ = 0.5 SU(3 η 0 U(5 spherical 7 0.

24 8/ Away from SU(3 limit search for rotational bans Macek et al. 00 PRL 05,07503 H n = η n ( η ( Q Q N χ χ = + Q s s [ χ= + + χ ] ( η = 0.5, χ = 0.9 η = 0.5, χ = 0.5 SU(3 η 0 U(5 spherical eforme χ 0 O(6 N=30 For members of the same rotational ban we require: with E( l j E(0i = l( l + In particular, Il I E(4k E(0i R(4 / i 3.33 E( E(0 j i l I l' Macek et al. 00 PRC 8,04308

+ + χ ] ( η = 0.5, χ = 0.9 η = 0.5, χ = 0.5 H Q Q SU(3 η 0 U(5 spherical 7 0.

25 9/ Away from SU(3 limit search for rotational bans Macek et al. 00 PRL 05,07503 n = η n ( η ( N χ χ = + Q s s [ χ= + + χ ] ( η = 0.5, χ = 0.9 η = 0.5, χ = 0.5 H Q Q SU(3 η 0 U(5 spherical eforme χ 0 O(6 N=30 f reg = Ω reg / Ω tot

26 Macek et al. 00 PRC 8, / R(4/ C(,4 N=30 χ =.3 χ = 0.9 χ = 0.5 η = 0.5 l=4 η = 0.5 η = 0.5 l=3 +n 0 l l= l=0 f reg

27 Macek et al. 00 PRC 8,04308 / Away from SU(3 limit search for rotational bans Alaga rule B E ( l, K l', K' B B E E H n = η n ( η ( Q Q N χ χ = + Q s s [ χ= + + χ ] ( = ( lk K l' K' ( l ( l, K, K l', K' = l', K' K' T ( lk K l' ( l K K l' E K K' K' SU(3 η 0 U(5 spherical eforme χ 0 meium energies N=30 high energies intra-ban inter-ban intra-ban inter-ban O(6

28 Program: Aiabatic separation of intrinsic an collective motions Pavel Cejnar A few wors on chaos in nuclei Aiabatic separation an chaos in IBM Institute of Particle an Nuclear Physics Faculty of Mathematics an Physics Charles University, Prague, Czech Republic Conclusions an outlook mff.cuni.cz Primosten 0

/ A tentative criterion for a statistically enhance occurrence of (aiabatically separate rotational bans within the IBM: increase regularity of intrinsic (vibrational motions. Why so?

29 / A tentative criterion for a statistically enhance occurrence of (aiabatically separate rotational bans within the IBM: increase regularity of intrinsic (vibrational motions. Why so? The form of wave function makes a typical chaotic state very vulnerable to external perturbations (ergoicity => large overlaps, mixing. In contrast, regular states are more resistant. This simple criterion may be applicable in general many-boy systems. It shoul be further teste in various moels. β γ Thanks to collaborators: Michal Macek (now Hebrew Uni Jerusalem Pavel Stránský (now UNAM Mexico Jan Dobeš (NPI Řež near Prague GM: a regular intrinsic state Thank you for attention! GM: a chaotic intrinsic state

Regular & Chaotic. collective modes in nuclei. Pavel Cejnar. ipnp.troja.mff.cuni.cz

Regular & Chaotic. collective modes in nuclei. Pavel Cejnar. ipnp.troja.mff.cuni.cz Pavel Cejnar Regular & Chaotic collective modes in nuclei Institute of Particle and Nuclear Physics Faculty of Mathematics and Physics Charles University, Prague, Czech Republic cejnar @ ipnp.troja.mff.cuni.cz