ELASTIC ENHANCEMENT FACTOR AS AN INDICATOR OF ...

ELASTIC ENHANCEMENT FACTOR AS AN INDICATOR OF PECULIARITIES OF INTERNAL DYNAMICS IN OPEN QUANTUM SYSTEMS. Valentin V. Sokolov Budker Institute of Nuclear Physics and Novosibirsk State Technical University. – Typeset by FoilTEX –

Yad Hashmona, Israel; 22-27 October 2014

October 15, 2014

In collaboration with: Yaroslav A. Kharkov, and Oleg V. Zirov.

– Typeset by FoilTEX –

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Enhancement factor is the typical ratio F(β) of the cross sections of elastic and inelastic chaotic resonance processes.

Nuclear physics (β = 1), many-electron atoms in strong magnetic field (β = 2) as well as macroscopic analog devices.

What the enhancement factor can tell us about dynamics of the open system excited on the intermediate stage?


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1. J.J. Verbaarschot, H.A. Weidenm¨ uller and M.R. Zirnbauer, Phys. Rep. 129, 367 (1985). (GOE) 2. Y.V. Fyodorov, D.V. Savin and H-J. Sommers, Journ. Phys. A, 38, 10731 (2005). (GUE) 3. J.J.M. Verbaarschot, Ann. Phys. (NY) 168, 368 (1986). 4. N. Lehmann, D. Saher, V.V. Sokolov, H.-J. Sommers, Nucl. Phys. A 582, 223, (1995). 5. N. Lehmann, D. Savin, V.V. Sokolov, H.-J. Sommers, Pysica D 86, 572, (1995). 6. G.L. Celardo, F.M. Izrailev, V.G. Zelevinsky and G.P. Berman, Phys. Rev. E 76, 031119 (2007). 7. H. Kunz and B. Shapiro, Phys. Rev. E, 58, 400, (1998). – Typeset by FoilTEX –

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I. VERBAARSCHOT’S REGIME: Very large number M ≫ 1 of very weak, T ≪ 1, channels. ∫ ∞ F(β)(η) = 1 + δ1β + η K(β)(η) = 2 + δ1β − η dse−ηs B2(β)(s) , 0

where

⟨Q ⟩

∫

2

K(β)(η) =

2

⟨Q⟩

−1=

∞

ds e−ηs

[

] 1 − B2(β)(s) .

0

is the time delay variance, B2(β)(s) - the Dyson’s spectral binary form factor. At last, the parameter η = M T is a ratio η = tH/tW of two, the Heisenberg and the dwell, characteristic times.


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i.VIEW FROM INSIDE: Spectral

statistics of the internal

Hamiltonian H. The global scale: mean level density and mean level spacing ⟩ 1 ∑ 1 ⟨ 1 ρ=⟨ δ(E − En)⟩ = − ℑ T r = . π E + i0 − H d n

The Heisenberg time: tH = 2πℏρ =


2πℏ d

.

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Local scale - universal spectral fluctuations. Density-density two-point correlation function : (β) C2 (ε)

=

1 ρ2

∑ ∑ ε ⟨ n δ(E + 2 − En) n′ δ(E − 2ε − En′ )⟩ − 1 (β)

= δ (ρ ε) − Y 2 (ρ ε) . (β)

The Dyson’s binary form factor B 2 (s) is defined by the Furrier transformation ∫ Y

(β) 2 (ρ ε)

=2

∞

(β)

B 2 (s) cos (2πρεs) ds . 0

The internal dynamics depend on presence or absence of the timereversal symmetry. – Typeset by FoilTEX –

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ii. VIEW FROM OUTSIDE: Non-Hermitian Effective Hamilto-

nian H=H−

i AA† 2

The slow-varying (global) reference quantities: the mean scattering matrix and transmission coefficients ⟨S ⟩ = ⟨S ⟩ δ ; ab

aa

ab

aa 2 1 − ⟨S ⟩ = T a .

Equivalent channels: T a = T b = ... = T .


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The characteristic time scale is set now by the mean Wigner delay time 1 ⟩ Γ 2 ℏ⟨ ∑ 2ℏ 1 n 2 ⟨Q⟩ = = − ⟨ℑ T r ⟩ M n (E − En)2 + 41 Γ2n M E−H

and is determined by the resonance spectrum of the intermediate open system. The mean delay of a spatially small wave packet satisfies the following relation: M ⟨Q⟩ = 2πℏρ = tH . – Typeset by FoilTEX –

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Finally, the dwell (or Weisskopf) time equals tW = ℏ/ΓW

1 = ⟨Q⟩/T = tH . MT

The ratio η = M T = tH /tW measures the degree of openness of the intermediate system. If η ≫ 1( tH ≫ tW ) enhancement factor carries no information on spectral properties of the internal system. Only when η ≪ 1, ( tH ≪ tW ), this factor does convey information on the character of the internal dynamics. – Typeset by FoilTEX –

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EXPLICIT ANALYTICAL EXPRESSIONS for the both symmetry classes are available: ( ( ) ) (( )) η 1 F(1)(η) = 2+ 2 1 − e−η − (2 + η) e−η − (2 − η) e 2 Ei(−η/2) , 2η −η 1 − e F(2)(η) = 1 + . η

In both the cases dF(β)(η) 1 =− . dη 2 η=0 – Typeset by FoilTEX –

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3.0

2.5

2.0

1.5

0

10

20

30

40

50

60

Figure 1: F(β) versus η. Top curve β = 1, bottom β = 2.


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1.04

1.03

1.02

1.01

10

20

30

Figure 2: The ratio


40

50

60

F(1) . F(2) +1

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Enhancement Factor from regular to chaotic regime. Nuclear physics β = 1 (alternatively, many-electron atoms in strong magnetic field β = 2): shell-model reference basis + residual two-body interaction controlled by an interaction strength parameter κ.

Transient matrix Ensembles: from Poisson to Wigner-Dyson level statistics: – Typeset by FoilTEX –

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1. The joint probability distribution P (E1, E2, ..., EN |κ) of the eigenenergies of the internal Hamiltonian H changes, depending on the parameter of chaoticity κ, from Poissonion distribution of fully independent levels (κ = 0) to that of highly correlated levels what is typical of the GO or GU Ensembles (κ → ∞). 2. The mean level density ρ is supposed to be independent of the parameter κ.


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The relation ∫ F(β)(η|κ) = 1+δ1β +K(β)(η|κ) = 2+δ1β −η

∞

dse−ηs B2(β)(s|κ) .

0

holds at any value of the interaction strength κ. Limiting cases: B2(β)(s|0) ≡ 0 ⇒ No level correlations; B2(β)(s|∞) = B2(β)(s) ⇒ maximal correlations. Correspondingly: F(β)(η|0) ≡ 2 + δ1β ; F(β)(η|∞) ⇒ GE (β). – Typeset by FoilTEX –

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In the transition region 0 < κ < ∞, the binary form factor B2(β)(s|κ) is known currently only for systems with broken time-reversal symmetry, β = 2. (However, the pattern in the case β = 1 can be expected to be qualitatively similar.) Then: B2(s|0) ≡ 0;

B2(s|∞) = (1 − s)θ(1 − s).

Besides, ∫∞ 0


dsB2(s|κ ̸= 0) ≡ 12 .

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This results in F(η|0) ≡ 2; F(η|∞) = 1 +

1−e−η η

(GUE);

The slope at the point η = 0 remains universal and equals dF(η|κ ̸= 0) 1 ≡− . dη 2 η=0


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FHΗÈΚL 2.0

1.8

1.6

1.4

0

5

10

15

20

Η

Figure 3: F versus η for (from top to bottom) κ = 0.5, 5, 50. Black points - exact numerics; blue lines - small κ approximation; red lines - large κ approximation; green line indicates the slope at the point η = 0. – Typeset by FoilTEX –

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As the openness η grows, the enhancement factor F(η|κ) goes down only until some minimal value Fmin > 1 and then goes up. This happens at a critical openness ηc(κ) fixed by the (η|κ) = 0 . Corresponding value of the chaoticity condition dFdη parameter κ(Fmin) can be, finally, extracted from the equation F(ηc(κ)|κ) = Fmin.

So, we conclude that the

enhancement factor directly measures the degree of internal chaos. – Typeset by FoilTEX –

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II. MACROSCOPIC ANALOGUE DEVICES: channels.

Few open

Under simplifying suggestion of equivalence, Ta = Tb = ... = T , of all M open channels the enhancement factor reads: (β) FM (T )

= 1 + δ1β + (1 − T )

(β)

JM (T ) (β) PM (T )

.

The functions J(β) and P(β) are represented by the well known 3- (β = 1) or 2- (β = 2) fold integrals.


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For the systems with broken TRS (β = 2) the factor FM (T ) can be calculated analytically at any given number of channels M and is represented in the form of a ratio of two (M − 1)order polynomials. For example, F2(T ) =

4−T 2+T

,

F4(T ) = 3

8−6T +4T 2−T 3 12+6T −4T 2+T 3

and so on... The mentioned above connection between enhancement factor and delay time variance that looks as (2) KM (T )

=

M +1

2 1−(1−T ) T2 M 2−1

does not exist anymore! – Typeset by FoilTEX –

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2.0

1.8

1.6

1.4

1.2

0.2

0.4

0.6

0.8

1.0

Figure 4: FM versus T . From top to bottom M = 2, 3, 4, 9, 10.

The larger is the number of channels the faster enhancement factor decays when T grows. – Typeset by FoilTEX –

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TWO NON-EQUIVALENT CHANNELS: 5

4

3

2

0.0

0.2

0.4

0.6

0.8

1.0

Figure 5: F(T + ∆, T − ∆) at different values of ∆: from bottom to top ∆ = 0, 0.994T, 0.99985T , 0.9999985T . (Analytical solution.) – Typeset by FoilTEX –

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THE INFLUENCE OF ABSORPTION is taken into account by introducing the overall exponentially decaying factor e−γτ . Analytical solution is still possible in the case of an analogue 2D system with no TRS and only two equivalent open channels (that is of practical interest). (T,γ) F2(T, γ) = 1 + (1 − T ) N D(T,γ)

where


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N (T, γ) = T (2T 2 − T γ + γ 2) + γ 3 eγ/T Ei(−γ/T ) , and ( ( )) D(T, γ) = T T 3 + γ 2 − T γ(1 + γ) + 2T 2 2(γ − 1) + eγ3γ + −1 (

)

γ (1 − T )γ − 3 T (2 − γ coth(γ/T )) eγ/T Ei(−γ/T ) . 2

2

No analytical results are known in the case of GOE.


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3.0

F2(T, γ)

2.5 2.0 1.5 1.0 0.0

0.2

0.4

0.6

0.8

1.0

T Figure 6: F2(T, γ) for β = 1 (up) and β = 2 (down): γ = 0, 0.03, 1, 3 (from top to bottom. The influence of absorption is stronger in the case of T-invariant systems.) – Typeset by FoilTEX –

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CONCLUSION:

There is strong qualitative distinction between properties of the elastic enhancement factors in the cases of complicated mesoscopic systems and analogue macroscopic devices!


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References: 1. P.A. Moldauer, Phys. Rev. 123, 968 (1961). 2. A. M¨ uller and H.L. Harney, Phys. Rev. C 35, 1228 (1986). 3. J.J.M. Verbaarschot, Ann. Phys. (NY) 168, 368 (1986). 4. C.H. Lewenkopf and H.A. Weidenm¨ uller, Ann. Phys. (NY) 212, 53 (1991). 5. N. Lehmann, D. Saher, V.V. Sokolov, H.-J. Sommers, Nucl. Phys. A 582, 223, (1995). 6.

N. Lehmann, D. Savin, V.V. Sokolov, H.-J. Sommers,


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Pysica D 86, 572, (1995). 7. Y.V. Fyodorov and H.-J. Sommers, Phys. Rev. Lett. 76, 4709, (1996). 8. Y.V. Fyodorov, D.V. Savin and H.-J. Sommers, J. Phys. A: Math. Gen. 38, 10731, (2005). 9. G.L. Celardo, F.M. Izrailev, V.G. Zelevinsky and G.P. Berman, Phys. Rev. E 76, 031119 (2007). 10. S. Sorathia, PHD thesis, Puebla, Mexico (2010). 11. Z. Pluhaˇr, H.A. Weidenm¨ uller et al., Ann. Phys. (NY) 243, 1 (1995); – Typeset by FoilTEX –

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12. B. Dietz, T. Friedrich et al., Phys. Rev. E, 81, 036205 (2010). 13. M. Lawniczak, S. Bauch, O. Hul, L. Sirko, Phys. Rev. E, 81, 046204 (2011). 14. E. Wigner, Phys. Rev. 98, 145 (1955). 15. E.T. Smith, Phys. Rev. 118, 349; 119, 2098 (1960). 16. J.J. Verbaarschot, H.A. Weidenm¨ uller and M.R. Zirnbauer, Phys. Rep. 129, 367 (1985). (GOE) 17. Y.V. Fyodorov, D.V. Savin and H-J. Sommers, Journ. Phys. A, 38, 10731 (2005). (GUE) – Typeset by FoilTEX –

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18. Y.V. Fyodorov and H.-J. Sommers, Phys. Rev. Lett. 76, 4709, (1996). (GUE) 19. Vladimir Zelevinsky, Alexander Volya, Pysics Report 391, 311 (2004). 20. T. Guhr, Phys. Rev. Lett. 76, 4709, (1996). 21. H. Kunz and B. Shapiro, Phys. Rev. E, 58, 400, (1998). 22. Y.A. Kharkov, Magister Thesis, Novosibirsk State University, (2011) 23. Yaroslav A. Kharkov, Valentin V. Sokolov, Physics Letters B, 718, 1562, (2013.) – Typeset by FoilTEX –

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Thank you very much for your attention!


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