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Restoration of symmetry in time-dependent calculations. Josephson - - PowerPoint PPT Presentation

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Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. Guillaume SCAMPS Center for computational sciences, Tsukuba University November 6th 2017 Collaboration : Y. Hashimoto, T.

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Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 1 / 25

Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation.

Guillaume SCAMPS

Center for computational sciences, Tsukuba University November 6th 2017 Collaboration : Y. Hashimoto, T. Nakatsukasa

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Recent evolution of the mean-field dynamics

TDHF

No pairing correlations

TDHF+BCS

Simplified pairing correlations Computational time ×1.5

TDHFB

Full pairing Computational time ×1000 TDHF+BCS :

  • S. Ebata, T. Nakatsukusa, et al., Phys. Rev. C 82, 034306 (2010).
  • G. Scamps, D. Lacroix, Phys. Rev C 87, 014605 (2013).

Time-dependent Hartree-Fock-Bogoliubov (TDHFB) :

  • I. Stetcu, A. Bulgac, P. Magierski, and K. J. Roche, Phys. Rev. C 84, 051309(R) (2011).
  • Y. Hashimoto, Phys. Rev. C 88, 034307 (2013).

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 2 / 25

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SLIDE 3

Collision between two superfluid nuclei described with TDHFB with a Gogny force

HFB breaks the particle-number symmetry → qp-vacuum states have define gauge angles

20O+20O

20O 20O

at t=0, rotation of the gauge angle : U → eiϕU θ(z) V → e−iϕV θ(z)

Results :

  • 2
  • 1

1 4 6 8 10 12 14 16 18 20 22 Relative distance R (fm) Relative momentum

Pz/ h

/

1/ fm ( ) 45 90 135

Ecm= 11.41 MeV

  • Y. Hashimoto, G. Scamps, Phys.
  • Rev. C 94, 014610 (2016)

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 3 / 25

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SLIDE 4

Evolution of two TDHFB calculation at the vicinity of the barrier

κ(r, ↑, r, ↓) = |κ(r, ↑, r, ↓)|e2iϕ(r)

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 4 / 25

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SLIDE 5

Evolution of two TDHFB calculation at the vicinity of the barrier

κ(r, ↑, r, ↓) = |κ(r, ↑, r, ↓)|e2iϕ(r)

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 4 / 25

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Josephson transfer

20O+20O

20O 20O

Transfer

Js ∝ sin (2ϕ) .

Josephson effect

The transfer of nucleons depends

  • n the relative gauge angle.
  • Y. Hashimoto, G. Scamps, Phys.
  • Rev. C 94, 014610 (2016)

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 5 / 25

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Problem

Question

The relative gauge angle is not a parameter of the reaction. Does those results of TDHFB are spurious ?

Projection method

We should restore the symmetry of the relative gauge angle |Ψ(t = 0) = ˆ PNL−NR (NL − NR)|φ, ˆ PNL−NR(NL − NR) = 1 2π

2π

eiϕ[(ˆ

NL−ˆ NR)−(NL−NR)]dϕ

We have to consider an evolution of a mixture of HFB states 1 2

N N

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 6 / 25

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SLIDE 8

Approximation

Starting point

|Ψ(t = 0) =

M

  • n=1

cn|φn(t = 0), |φn(t = 0) = ei 2nπ

M

ˆ NL|φ(t = 0).

Assumption during the evolution

|Ψ(t) =

M

  • n=1

cn|φn(t), |φn(t) evolves with the TDHFB equation

  • f motion

Evolution Observables

O = Ψ(t)|ˆ O|Ψ(t) Ψ(t)|Ψ(t) We assume an evolution of a set of TDHFB trajectories with fixed coefficients.

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 7 / 25

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Multi-nucleon transfer in the sub-barrier regime

Motivations :

Montanari et al., PRL 113 (2014)

Transfer under the barrier

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 8 / 25

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Calculation of the transfer probabilities

Projection method for TDHF

PL(N) = Ψ(t)| ˆ PL(N)|Ψ(t) ˆ PL(N) = 1 2π

2π

eiϕ(ˆ

NL−N)dϕ

  • C. Simenel, PRL 105 (2010).

Projection method with pairing

PL(N) = Ψ(t)|ˆ PL(N)ˆ P(Ntot)|Ψ(t) Ψ(t)|ˆ P(Ntot)|Ψ(t)

  • G. Scamps, D. Lacroix, PRC 87, (2013).

*only when one of the fragment is superfluid

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 9 / 25

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SLIDE 11

Triple projection method

Triple projection method

PL(N, t) = Ψ(t)| ˆ PL(N)|Ψ(t) Ψ(t)|Ψ(t) |Ψ(t) = 1 (2π2)

2π

dϕ1

2π

dϕ2eiϕ1(ˆ

N−Ntot) ˆ

UTDHFB(t0, t)eiϕ2(ˆ

NL−NL)|φ(t = 0)

TDPHFB

Û t=0 t=tf

TDHFB

Û t=0 t=tf Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 10 / 25

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Triple projection : Pfaffian method

Evolution Overlap

  • G. F. Bertsch and L. M. Robledo, PRL 108, 042505 (2012)

Optimized Pfaffian calculation : M. Wimmer, ACM Trans. Math Softw. 38, 30 (2012).

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 11 / 25

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Test on toy model

Simple model

V (t) = V0exp(−αt2)

Exact solution

Time-dependent Multi-configuration method

  • K. Dietrich, Phys. Let. B 32, 6 (1970).

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 12 / 25

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Toy model

Result

0.0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

P2

  • 15
  • 10
  • 5

5 10 15

t [10−22 s]

Exact

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 13 / 25

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SLIDE 15

Toy model

Result

  • 0.02

0.0 0.02 0.04 0.06 0.08 0.1 0.12

P2

  • 15
  • 10
  • 5

5 10 15

t [10−22 s]

TDHFB Exact x10

Conclusion

Spurious result with the TDHFB evolution

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 13 / 25

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Stationary condition

Assumption during the evolution

|Ψ(t) =

M

  • n=1

cn|φn(t), |φn(t) evolves with the TDHFB equation of motion

Stationary condition

Without interaction : φn(t + dt)|φn(t) ≃ 0

Important point

We need to impose stationary condition in the TDHFB equation of motion

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 14 / 25

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Modification of the equation of motion

i ∂ ∂t

U(t)

V (t)

  • = H

U(t)

V (t)

  • ,

H =

  • h

∆ −∆∗ −h∗

  • (1)

H =

h − δλL,R(t) − ǫk(t)

∆ −∆∗ −h∗ + δλL,R(t) − ǫk(t)

  • (2)

δλL,R(t) = λL,R(t) − λL,R(t = 0)

0.0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

P2

  • 15
  • 10
  • 5

5 10 15

t [10−22 s]

TDHFB eq. (1) TDHFB eq. (2) Exact

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 15 / 25

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SLIDE 18

Comparison

Results

10−5 10−4 10−3 10−2 10−1

P2

10−3

2 5

10−2

2 5

10−1

2 5

1

−V0 [MeV]

TDHFB exact

Conclusion

The method works well in the toy model. We can expect a predictive power in realistic calculations.

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 16 / 25

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TDHFB with Gogny interaction

z x y

x and y direction : Harmonic

  • scillator basis nx + ny ≤ 4

z direction : Lagrange mesh nz= 46 Nbase = 2760 In comparison, a full cartesian mesh is about 100 000 degrees of freedom Cost of the calculation : one collision done in one day with 20 CPUs

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 17 / 25

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Dispersion of the trajectories

Evolution of the set of TDHFB trajectories

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 18 / 25

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SLIDE 21

Dispersion of the trajectories

Evolution of the set of TDHFB trajectories

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 18 / 25

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SLIDE 22

Effect of the dispersion of the trajectories

Non stable results

0.0 0.05 0.1 0.15 0.2 0.25

P2

200 400 600 800 1000 1200 1400

t [fm/c]

Overlap

0.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

|φ1(t)|φ2(t)|2

200 400 600 800 1000 1200 1400

t [fm/c]

TDHFB Intrinsic frame Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 19 / 25

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SLIDE 23

20O + 20O

Results

0.0 0.005 0.01 0.015

P2

200 400 600 800 1000 1200 1400

t [fm/c]

Ec.m.=9.31 MeV Ec.m.=9.21 MeV Ec.m.=8.91 MeV

Comparison with TDHF

10−4 10−3 10−2 10−1 1

PL(NL)

8 9 10 11 12 13 14 15 16

NL

TDHFB TDHF

Pair transfer as a function of the distance of closest approach

10−4 10−3 10−2

P2

9.5 10.0 10.5 11.0 11.5 12.0

D [fm]

Important point

This calculation predicts no individual transfer.

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 20 / 25

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SLIDE 24

Asymmetric reaction 14O + 20O

P1 P2 P1

2/4

10−6 10−5 10−4 10−3 10−2

P1, P2

9.5 10.0 10.5 11.0 11.5 12.0

D [fm] Expected value for uncorrelated pair P2 = P2

1/4 (K. Hagino, G. Scamps, PRC 92 (2015)).

Important point

Large enhancement factor.

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 21 / 25

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Summary and outlook

Summary

Restoration of the symmetry for TDHFB Evolution of a set of HFB states Prescription to obtain a correct behavior of the probabilities Method tested on a toy model and applied on realistic calculation

Prospects

Study of larger systems for comparison with experimental data Development of a theory to make the self-consistent evolution of a projected state

  • G. Scamps, and Y. Hashimoto, PRC 96, 031602(R) (2017).

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 22 / 25

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SLIDE 26

Thank you

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 23 / 25

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SLIDE 27

Nucleus-Nucleus potential

Frozen potential

8.8 9 9.2 9.4 9.6 9.8 10 8 9 10 11 Relative distance R (fm) 45 90 135 V (R) ( MeV ) FD

0° 90°

U → eiϕU θ(z) V → e−iϕV θ(z) ρ = V ∗V t → ρ κ = V ∗Ut → κe2iϕ

Trajectory

κ = κ(l) + κ(r)e2iϕ Epair = 1 4

  • αβγδ

v αβγδκ∗

αβκγδ

∆E ∝ sin(ϕ)2

Important point

The Nucleus-nucleus potential depends on the relative gauge angle

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 24 / 25

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SLIDE 28

Nucleus-Nucleus potential

Frozen potential

8.8 9 9.2 9.4 9.6 9.8 10 8 9 10 11 Relative distance R (fm) 45 90 135 V (R) ( MeV ) FD

0° 90°

U → eiϕU θ(z) V → e−iϕV θ(z) ρ = V ∗V t → ρ κ = V ∗Ut → κe2iϕ

κ density

90

  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.0 0.1 0.2 0.3 0.4 0.5

Real(κ) [fm−1]

  • 8
  • 6
  • 4
  • 2

2 4 6 8

z [fm]

κ = κ(l) + κ(r)e2iϕ Epair = 1 4

  • αβγδ

v αβγδκ∗

αβκγδ

∆E ∝ sin(ϕ)2

Important point

The Nucleus-nucleus potential depends on the relative gauge angle

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 24 / 25

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SLIDE 29

Triple projection : Pfaffian method

  • G. F. Bertsch and L. M. Robledo, PRL 108, 042505 (2012)

PB(N, t) = 1 N 1 (2π)4 2π ei(ϕ1−ϕ4)Ni −iϕ2N−iϕ3Ntot Ψ(ϕ1, t)|eiϕ2 ˆ NB eiϕ3 ˆ Ntot |Ψ(ϕ4, t)dϕ1dϕ2dϕ3dϕ4 N = 1 (2π)3 2π ei(ϕ1−ϕ4)Ni −iϕ3Ntot Ψ(ϕ1, t)|eiϕ3 ˆ Ntot |Ψ(ϕ4, t)dϕ1dϕ3dϕ4. Ψ(ϕ1, t)|eiϕ2 ˆ NB eiϕ3 ˆ Ntot |Ψ(ϕ4, t) = (−1)n det C∗ det C′

n

α vαv′ α pf

  • V T U

V T eiϕ3 (1 + Θ(z)eiϕ2 )V ′∗ −V ′†eiϕ3 (1 + Θ(z)eiϕ2 )V U′†V ′∗

  • .

Optimized Pfaffian calculation : M. Wimmer, ACM Trans. Math Softw. 38, 30 (2012).

Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 25 / 25