Quantum Hydrodynamics Analysis of Quantum Synchronization ...acesar/slidesanconet/marcati.pdf ·...

Quantum Hydrodynamics Analysisof Quantum Synchronization

over Quantum Networks

Pierangelo Marcati(joint work with P. Antonelli, S.-Y. Ha, D. Kim)

GranSasso Science Institute andUniversita degli Studi dell’Aquila - L’Aquila, Italy

[email protected]

Classical References

Winfree, A. T.: Biological rhythms and the behavior of populations ofcoupled oscillators. J. Theor. Biol. 16, 15-42 (1967).

T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen and O. Shochet, Novel typeof phase transition in a system of self-driven particles, Phys. Rev. Lett. 75(1995) pp. 1226–1229.

Kuramoto, Y.: Chemical Oscillations, waves and turbulence.Springer-Verlag, Berlin. 1984.

Kuramoto, Y.: Lecture notes in theoretical physics. 30, 420 (1975).

Strogatz S.,From Kuramoto to Crawford: exploring the onset ofsynchronization in populations of coupled oscillators, Physica D 143(2000), 1-20.

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, [email protected])Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks2 / 37

Some applications

Computer science Parallel computing or GPUs

Cryptography (including Quantum)

Digital Music

ITC

Multimedia

Neuroscience

Physics


Synchro

Sinusoidally coupled nonlinear oscillators active rotors on the circle S1. Letxi = e

√−1θi be the position of the i-th rotor.

xi determined by phase θi. In the absence of coupling,

dθidt

= Ωi, i.e., θi(t) = θi(0) + Ωit,

Ωi random variable (the natural phase-velocity (frequency))Kuramoto derived a coupled phase model heuristically from the complexGinzburg-Landau system.

dθidt

= Ωi −K

N

N∑j=1

sin(θi − θj), t > 0, i = 1, · · · ,N, (1)

subject to initial data θi(0) = θi0.We say that the system SYNCHRONIZE if

limt→∞

‖θi − θj‖ = 0


Kuramoto studied - the mean-field limit N →∞ - of the system (1)Unimodal frequency distribution function g(Ω),g(−Ω) = g(Ω), spt(g) bounded ,

∫g(Ω)dΩ = 1.

(one-humped , symmetric w.r.t. mean Ωpc := 1

N

∑Ni=1 Ωi)

e.g. g(Ω) = γπ[γ2+(Ω−Ω0)2)

, having width γ > 0 and mean Ω0,

Continuous dynamical phase transition at a critical value of the couplingstrength Kcr = 2

πg(0) , in the mean-field limit.

Asymptotic order parameter r∞ ∈ [0, 1] ( phase synchronization inmean-field limit)

r∞(K) := limt→∞

limN→∞

∣∣∣ 1

N

N∑i=1

e√−1θi(t)

∣∣∣,If K > Kcr then r∞ > 0

If θi(t = 0) uniformly distributed on [0, 2π), r∞ = 0

If θi = θc for i = 1, · · · , N , then r∞ = 1.(completely synchronizedconfiguration)


Ha

Let θ = (θ1, · · · , θN ) and the natural frequency set as follows.

Dθ(t) := max1≤i,j≤N

|θi(t)− θj(t)|, t ≥ 0, DΩ := maxi,j|Ωi − Ωj |.

Theorem (Ha S.-Y., Ha T. Y. and Kim J.-H. 2010)

SupposeΩi = Ωj , i 6= j, K > 0, D0 := Dθ(0) < π,

and let θ = θ(t) be the smooth solution to the system (1)-(??) with initialphase θ0. Then

e−KtD0 ≤ Dθ(t) ≤ e−KαtD0, t ≥ 0, (2)

where α depends on the diameter of the initial phase configuration

α :=sinD0

D0.


We also recall the estimate of existence of a trapping region fornon-identical oscillators from as follows.

Lemma (Choi Y.-P., Ha S.-Y., Jung, S. and Kim, Y- 2012)

Let θ = θ(t) be the global smooth solution to (1)-(??) satisfying

0 < D0 < π, DΩ > 0, K > Ke :=DΩ

sinD0.

Then we have

(i) supt≥0

Dθ(t) ≤ D0 < π.

(ii) ∃ t0 > 0 such that supt≥t0

Dθ(t) ≤ D∞,

where D∞ is defined by

D∞ := arcsin[DΩ

K

]∈(

0,π

2

).


Quantum Lohe Model

Phase synchronization cannot occur in quantum systems with constantlinear interactions

We consider N quantum oscillators (”nodes” ) connected by a quantumnetwork and Nonlinear interactions

Wavefunctions at each node distributed over quantum channels to allother connected nodes, by means of quantum teleportationLocal evolution given by free evolution plus a nonlinear interactionThe Kuramoto system can be generalized to non-Abelian model wherevariables are n× n complex matrices Ui in U(n) where U∗i denotes theHermitian conjugate of Ui and Hi is a prescribed constant n× nHermitian matrix, i = 1....N .

iUiU∗i = Hi +

ik

2N

N∑0

aij(UjU∗i − UiU∗j )


Choi,Ha version of Lohe Model (Mathematicians notations)

χjk = χjk‖ψj‖‖ψk‖

i∂tψj = −1

2∆ψj + Vjψj +

iK

2N

N∑j=1

χjk

(ψk −

〈ψj , ψk〉‖ψj‖2

ψj

)Consider for the moment N = 2, we have

i∂tψ1 =− 1

2∆ψ1 + V1ψ1 + i

K

4χ12

(ψ2 −

〈ψ1, ψ2〉‖ψ1‖2

ψ1

)i∂tψ2 =− 1

2∆ψ2 + V2ψ2 + i

K

4χ12

(ψ1 −

〈ψ2, ψ1〉‖ψ2‖2

ψ2

).


References on Quantum

S.-H. Choi, S.-Y. Ha, Quantum synchronization of the Schrodinger-Lohemodel, J. Phys. A: Math. Theor. 47 (2014), 355104.

P. Antonelli and P. Marcati, On the finite energy weak solutions to asystem in Quantum Fluid Dynamics, Comm. Math. Phys. 287 (2009), no2, 657–686.

P. Antonelli and P. Marcati, The Quantum Hydrodynamics system in twospace dimensions, Arch. Rat. Mech. Anal. 203 (2012), 499–527.


From NLS to QHD via WKB

i~∂tψ = − ~2

2m∆ψ + V ψ + f ′(|ψ|2)ψ

−∆V = |ψ|2

Energy

E[ψ] =

∫~2

2|∇ψ|2 + f(|ψ|2) +

1

2|∇V |2 dx

WKB ansatz: ψ =√ρeiS/~, then (ρ, S) satisfy

∂tρ+ div(ρ∇S) = 0

∂tS +1

2|∇S|2 + f ′(ρ) + V =

~2

2

∆√ρ

√ρ


WKB ansatz

∂tρ+ div(ρ∇S) = 0

∂tS +1

2|∇S|2 + f ′(ρ) + V =

~2

2

∆√ρ

√ρ

u = ∇S ⇒ ∂tu+ (u · ∇)u+∇f ′(ρ) +∇V =~2

2∇(

∆√ρ

√ρ

)J = ρu = ρ∇S ⇒ (ρ, J) solves (QHD) & E[ψ] = E[ρ, J ]

~2|∇ψ|2 = ~2|∇√ρ|2 + ρ|∇S|2 = ~2|∇√ρ|2 +|J |2

ρ


Mathematical problems of WKB

vacuum: WKB ansatz ψ = |ψ|eiS/~ valid only if ψ(t, x) 6= 0; S not defined in ψ = 0regularity issue: the nodal set ψ = 0 may have dimH = 1(Federer,Ziemer)

irrotationality: ∇∧ u = 0, no vortices are taken into account in theWKB description

CONCLUSION: The phases analysis is ill-posed one has to deal with thewave functions and the the first two moments (observables)


Madelung transformation

Moments associated to the wave function. ψ

ρ := |ψ|2 mass densityJ := ~(ψ∇ψ) current density.

∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

~2

2ρ∇(

∆√ρ

√ρ

)





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ− ~2 div(∇√ρ⊗∇√ρ)





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,~2ρ(∇ψ ⊗∇ψ)





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,

~2ρ(∇ψ ⊗∇ψ) = ~2ρ

((ψ∇ψ)⊗ (ψ∇ψ)

|ψ|2

)





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,

~2ρ(∇ψ ⊗∇ψ) =1

ρ

[~2ρ(ψ∇ψ)⊗ ρ(ψ∇ψ)

− ~2(ψ∇ψ)⊗ (ψ∇ψ)]





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,

~2ρ(∇ψ ⊗∇ψ) =1

ρ

[~2ρ(ψ∇ψ)⊗ ρ(ψ∇ψ)

+~2(ψ∇ψ)⊗ (ψ∇ψ)]





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,

~2ρ(∇ψ ⊗∇ψ) =1

ρ

[~2(

1

2∇ρ)⊗ (

1

2∇ρ) + J ⊗ J

]





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

Formally,

~2ρ(∇ψ ⊗∇ψ) = ~2∇√ρ⊗∇√ρ+J ⊗ Jρ

.





∂tρ+ div J = 0,

∂tJ + ~2 div(ρ(∇ψ ⊗∇ψ)

)+∇P (ρ) + ρ∇V =

~2

4∇∆ρ

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V =

Formally,

~2ρ(∇ψ ⊗∇ψ) = ~2∇√ρ⊗∇√ρ+J ⊗ Jρ

.

Problem: vacuum


Polar Factorization

Let ψ ∈ H1(R3), define

P (ψ) =φ ∈ L∞ s.t. ‖φ‖L∞ ≤ 1, ψ = φ|ψ| a.e. R3

.

φ ∈ P (ψ), implies |φ| = 1,√ρ dx-a.e. in R3,

φ is uniquely defined√ρ dx−a.e. in R3.

(Lieb, Loss, Thm. 6.19), ψ ∈W 1,1loc , ∇ψ = 0 a.e. in ψ = 0.

We call (any) φ ∈ P (ψ) polar factor associated to ψ.


Stability lemma

Lemma

Let φ ∈ L∞(R3), ψ = |ψ|φ a.e. and ‖φ‖L∞(R3) ≤ 1 then

∇√ρ = Re(φ∇ψ) a.e., Λ := ~Im(φ∇ψ) a.e.

~2ρ(∇ψ ⊗∇ψ) = ~2∇√ρ⊗∇√ρ+ Λ⊗ Λ a.e.

H1−stability: ψn ⊂ H1(R3), ψn → ψ in H1,

∇√ρn → ∇√ρ, Λn → Λ in L2(R3).

Remark

However φn φ weak ∗ in L∞.

Remark

∇√ρ and Λ =√ρ ∇S make sense only in the previous sense


Finite energy weak solutions

The pair (ρ, J) is a finite energy weak solution to the QHD system in[0, T )× R3, if there exist locally integrable functions

√ρ,Λ, s.t.

√ρ ∈ L2

loc([0, T );H1loc(R3)), Λ ∈ L2

loc([0, T );L2loc(R3));

∀ η, ζ ∈ C∞0 ([0, T )× R3), ρ := (√ρ)2, J :=

√ρΛ satisfy∫

ρ∂tη + J · ∇η dxdt+

∫ρ0η(0) dx = 0,∫ (

J · ∂tζ + Λ⊗ Λ : ∇ζ + P (ρ) div ζ − ρ∇V · ζ − ~2

4ρ∆ div ζ

+ ~2∇√ρ⊗∇√ρ : ∇ζ)dxdt+

∫J0 · ζ(0) dx = 0;

generalized irrotationality condition, ∇∧ J = 2∇√ρ ∧ Λ. In thesmooth case ρ∇∧ u = ∇∧ J − 2∇√ρ ∧ Λ


Finite Energy Weak Solutions - II

Proposition

Let f(ρ) ∼ ργ , with 1 ≤ γ < 3. Given ψ0 ∈ H1(R3), letρ0 := |ψ0|2, J0 := Im(ψ0∇ψ0) .Then there exists a global finite energyweak solution to the QHD system with initial data (ρ0, J0). Furthermore,the energy is conserved.

Proof.

Consider ψ ∈ C([0,∞);H1(R3)) solution to the Cauchy problem for NLSequation with ψ(0) = ψ0, define

√ρ := |ψ|,Λ := Im(φ∇ψ), then use the

polar factorization Lemma.


Remarks

The only constraint on the initial data is that they are associated to aprescribed wave function in H1(R3);

no regularity assumptions;

no smallness;

no boundedness away from zero.


QHD with Interactions

∂tρ+ div J = 0

∂tJ + div

(J ⊗ Jρ

)+∇P (ρ) + ρ∇V + K =

~2

2ρ∇(

∆√ρ

√ρ

)−∆V = ρ

K in the case semiconductor device (Bløtekjær, Baccarani, Wordeman)takes the form K = 1

τ JEnergy:

E[ρ, J ] =

∫R3

~2

2|∇√ρ|2 +

1

2

K · Jρ

+ f(ρ) +1

2|∇V |2 dx,

dissipates along the flow of solutions

E(t) +

∫ t

0

∫R3

K · Jρ

dxdt′ = E(0).


Mathematical Tools, Dispersion and Local Smoothing

Strichartz estimates (Strichartz, Ginibre-Velo, Keel-Tao)(q, r) are admissible if 2 ≤ q ≤ ∞, 2 ≤ r ≤ 6 and 1

q = 32

(12 −

1r

)‖e

i2t∆f‖Lq

tLrx. ‖f‖L2

‖∫ t

0e

i2

(t−s)∆F (s) ds‖LqtL

rx. ‖F‖

Lq′t L

r′x

Local smoothing estimates (Constantin-Saut, Sjolin, Vega)

‖ei2t∆f‖

L2([0,T ];H1/2loc (R3))

. ‖f‖L2

‖∫ t

0e

i2

(t−s)∆F (s) ds‖L2([0,T ];H

1/2loc (R3))

. ‖F‖L1tL

2x


Theorem (Existence)

Let ψ0 ∈ H1(R3) and let ρ0 := |ψ0|2, J0 := ~Im(ψ0∇ψ0). Then for any0 < T <∞ there exists a finite energy weak solution (ρ, J) to the QHDsystem with collisions in [0, T ]×R3 with initial data (ρ0, J0). The solutionsatisfies

√ρ ∈ L∞(R+ : H1(R3)),Λ ∈ L∞(R+;L2(R3)) ∩ L2(R+;L2(R3))

and √ρ ∈ Lq([0, T ];W 1,r(R3)),Λ ∈ Lq([0, T ];Lr(R3)),

for any 0 < T <∞, where (q, r) is any arbitrary (Strichartz) admissiblepair for Schrodinger in R3.


Fractional step (operator splitting)

solve the QHD without collisions (NLS)

update with collisions / interactions

∂tJ +K = 0⇒ Jnew = Function(τ, Jold)⇒ difficult part

φnew = Function(τ, φold), ψnew = Function(τ, φold)√ρold


Potentials Decomposition

Theorem (Ortner, Suli )

(decomposition of function in W 1,6)

Let V ∈ LqtW1,6x ,2 ≤ q ≤ +∞ then

V = V∞(C∞x unbounded) + Vp(∈ LqtW 1,6x )

in particular

(i) V∞(t, ·) ∈ C∞x , ;

(ii) ‖Vp‖LqtW

1,6x≤ C‖∇V ‖Lq

tL6x;

(iii) ‖∇Vq‖LqtL

6x

+ ‖∇Vq‖L∞t,x≤ C‖∇V ‖Lq

tL6x;

(iv) |V∞(t, x)| ≤ C|x|5/6‖∇V ‖LqtL

6x

;

(v) ‖∂αV∞‖LqL6x≤ C‖∇V ‖LqL6

x, for any α ∈ N3, |α| ≥ 1.


Time dependent potentials (Antonelli, D’Amico and M.)

i∂tψ = −1

2∆ψ + V∞ψ + Vpψ + |ψ|2(γ−1)ψ

ψ(0) = ψ0.

One has Global Well Posedness in Σ(R3) by extending to LqtD. Fujiwara, A construction of the fundamental equation for theSchrodinger equation, Journ. Anal. Math. 35 (1979), 41-96where

Σ(R3) = ψ ∈ H1(R3) : | · |ψ ∈ L2(R3).

The parametrix given by the oscillatory integral

(E(t, s)φ)(x) :=( −i

2π(t− s)

) d2

∫Rd

eiS(t,s,x,y)

φ(y)dy

the phase function S(t, s, x, y) is the classical action along the classicalpaths (Feynman).


back to Schrodinger - Lohe model

i∂tψ1 =− 1

2∆ψ1 + V ψ1 + iKχ12

(ψ2 −

〈ψ1, ψ2〉‖ψ1‖2L2

ψ1

)i∂tψ2 =− 1

2∆ψ2 + V ψ2 + iKχ12

(ψ1 −

〈ψ2, ψ1〉‖ψ2‖2L2

ψ2

),

(3)

Let ρk := |ψk|2, Jk = Im(ψk∇ψk), rk :=∫ρk dx = ‖ψk‖2L2

ρkj = Re(ψkψj)(k 6= j), rkj = Re(〈ψk, ψj〉) =∫ρkj dx

Equation for the mass densities

∂tρk + div Jk =2KχkjRe

ψk

(ψj −

〈ψk, ψj〉‖ψk‖L2

ψk

)=2Kχkj

(ρkj −

rkjrkρk

),


Densities correlations

Since ddtr1 = d

dtr2 = 0, we assume r1 = r2 = 1.

∂tρkj =Re

(− i

2∆ψ1 + iV ψk +Kχkj(ψj − 〈ψj , ψk〉ψk

)ψ2

+ ψk

(i

2∆ψj − iV ψj +Kχ− kj(ψk − 〈ψ2, ψk〉ψ2

)=− 1

2Imψk∆ψj −∆ψkψj

+Kχkj (ρk + ρj − 2(rkjρkj − skjσkj)) ,

σkj := Im(ψ1ψ2), skj := Im(〈ψj , ψk〉) =∫σkj dx.

∂tρkj + div Jkj = Kχ12 (ρk + ρj − 2(rkjρkj − skjσkj)) ,

where the momentum is given by Jkj = 12 Im(ψk∇ψj + ψj∇ψk).


Sigma correlations

∂tσ12 =Im

(− i

2∆ψ1 + iV ψ1 +Kχ12(ψ2 − 〈ψ2, ψ1〉ψ1

)ψ2

+ ψ1

(i

2∆ψ2 − iV ψ2 +Kχ− 12(ψ1 − 〈ψ2, ψ1〉ψ2

)=− 1

2Re(∆ψ1ψ2 − ψ1∆ψ2

)+Kχ12Im(−2〈ψ2, ψ1〉ψ1ψ2).

∂tσ12 + divG12 = −2Kχ12 (r12σ12 + s12ρ12) .


Density correlations closure

∂tρ1 + div J1 =2Kχ12(ρ12 − r12ρ1)

∂tρ2 + div J2 =2Kχ12(ρ12 − r12ρ2)

∂tρ12 + div J12 =Kχ12(ρ1 + ρ2 − 2(r12ρ12 − s12σ12))

∂tσ12 + divG12 =− 2Kχ12(r12σ12 + s12ρ12).

(4)

Then d

dtr12 =2Kχ12(1 + s2

12 − r212)

d

dts12 =− 4Kχ12s12r12.

We have, as r12(0) 6= −1 r12(t)→ 1, s12(t)→ 0, and furthermore

1− r12(t) . e−2Kχ12t. s12(t) . e−4Kχ12t (5)


Phase portrait

y

x´ = 2*(1+y²-x²)y´ = -4*x*y

-1,4

-1,2

-1

-0,8

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

x

-1,4 -1,2 -1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1 1,2 1,4


Total Energy Bounds

E(t) =

∫1

2|∇ψ1|2 +

1

2|∇ψ2|2 + V (|ψ1|2 + |ψ2|2) dx.

d

dtE(t) =− 2Kχ12r12E(t) + 2Kχ12

∫Re∇ψ1 · ∇ψ2 + 2V ψ1ψ2

dx

=− 2Kχ12r12E(t)− 2Kχ12

∫1

2|∇(ψ1 − ψ2)|2 + V |ψ1 − ψ2|2 dx+

2Kχ12E(t) ≤ 2Kχ12(1− r12(t))E(t).

By Gronwall’s inequality we get that

E(t) ≤ e2Kχ12

∫ t0 (1−r12(s)) dsE(0)


First Variation Energy

Consider

E(t) =

∫1

2|∇(ψ1 − ψ2)|2 + V |ψ1 − ψ2|2 dx,

then

d

dtE(t) = −2Kχ12(1 + r12)E(t)− 4Kχ12s12

∫1

2Im(∇ψ1 · ∇ψ2) + V σ12 dx.

E(t) ≤e−2Kχ12tE(0) + 2Kχ12C

∫ t

0e−2Kχ12(t−s)e−cs ds

≤e−2Kχ12tE(0) + 2Kχ12C′ ∣∣e−2Kχ12t − e−ct

∣∣hence E(t) converges to zero exponentially as t→∞.


Synchronization

Since we know limt→∞ r12 = 1, limt→∞ s12 = 0 then

limt→∞‖ψ1 − ψ2‖L2 = 0

By the previous inequality on the energy of the first variation

limt→∞‖ψ1 − ψ2‖H1 = 0

To show the full hierarchy, let

ρd := |ψ1 − ψ2|2, Jd := Im((ψ1 − ψ2)∇(ψ1 − ψ2))

ρa :=1

2|ψ1 − iψ2|2, Ja :=

1

2Im(ψa∇ψa)

ρa =1

2(ρ1 + ρ2) + σ12, Ja =

1

2(J1 + J2) +G12


Full Hierarchy

The whole set of hydrodynamic equations is now closed, and reads


Existence of the interacting dynamics

Theorem (Existence)

For any initial datum ρ..., J...., there exists a globally in time finiteenergy weak solution of the full Hierarchy system

Remark (NonUniqueness)

The same non uniqueness phenomena proved by Camillo De Lellis andLaszlo Szekelyhidi for the Euler system was proved for QHD byDonatelli,Feireisl,M. (CPDE 2015)


Thank You !


Quantum Hydrodynamics Analysis of Quantum Synchronization ...acesar/slidesanconet/marcati.pdf ·...

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