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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

pierangelo.marcati@gssi.infn.it

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, Italypierangelo.marcati@gssi.infn.it)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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks3 / 37

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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks4 / 37

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)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks5 / 37

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.

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks6 / 37

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

).

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks7 / 37

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 )

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks8 / 37

Lohe-Kuramoto

When k = 0 the decoupled system is given by the finite-dimensionalanalog of the Schrodinger equation iUi = HiUi with free dynamicsUi(t) = eiHitUi(0)We generalize to complex n-vectors |ψi〉 (with n = 2 for qubit models)

i~∂t |ψi〉 = Hi |ψi〉+ i~k2N

N∑0

aij(|ψj〉 − |ψi〉 〈ψj |ψi〉)

Let |ψi〉 = |ψi〉〈ψi|ψi〉 , H int

i = i ~k2N

∑N0 aij(|ψj〉〈ψi| − |ψi〉〈ψj |), then

i~∂t |ψi〉 = (Hi +H inti )|ψi〉

In the quantum analog the wavefunction |ψi〉 of the quantum system atthe i-th node is an element of a Hilbert space Hi, and such wavefunctionscan be taken orthonormal with respect to the node, i.e. 〈ψj |ψi〉 = δij ,

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks9 / 37

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

).

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks10 / 37

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.

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks11 / 37

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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks12 / 37

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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks13 / 37

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)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks14 / 37

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

)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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 =

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ− ~2 div(∇√ρ⊗∇√ρ)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,~2ρ(∇ψ ⊗∇ψ)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,

~2ρ(∇ψ ⊗∇ψ) = ~2ρ

((ψ∇ψ)⊗ (ψ∇ψ)

|ψ|2

)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,

~2ρ(∇ψ ⊗∇ψ) =1

ρ

[~2ρ(ψ∇ψ)⊗ ρ(ψ∇ψ)

− ~2(ψ∇ψ)⊗ (ψ∇ψ)]

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,

~2ρ(∇ψ ⊗∇ψ) =1

ρ

[~2ρ(ψ∇ψ)⊗ ρ(ψ∇ψ)

+~2(ψ∇ψ)⊗ (ψ∇ψ)]

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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

4∇∆ρ−~2 div(∇√ρ⊗∇√ρ)

Formally,

~2ρ(∇ψ ⊗∇ψ) =1

ρ

[~2(

1

2∇ρ)⊗ (

1

2∇ρ) + J ⊗ J

]

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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 =

Formally,

~2ρ(∇ψ ⊗∇ψ) = ~2∇√ρ⊗∇√ρ+J ⊗ Jρ

.

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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 =

Formally,

~2ρ(∇ψ ⊗∇ψ) = ~2∇√ρ⊗∇√ρ+J ⊗ Jρ

.

Problem: vacuum

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks15 / 37

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.

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks16 / 37

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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks17 / 37

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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks18 / 37

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.

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks19 / 37

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.

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks20 / 37

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).

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks21 / 37

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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks22 / 37

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.

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks23 / 37

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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks24 / 37

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.

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks25 / 37

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).

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks26 / 37

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

),

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks27 / 37

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).

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks28 / 37

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) .

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks29 / 37

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)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks30 / 37

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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks31 / 37

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)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks32 / 37

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→∞.

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks33 / 37

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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks34 / 37

Full Hierarchy

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

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks35 / 37

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)

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks36 / 37

Thank You !

Pierangelo Marcati (joint work with P. Antonelli, S.-Y. Ha, D. Kim) ( GranSasso Science Institute and Universita degli Studi dell’Aquila - L’Aquila, Italypierangelo.marcati@gssi.infn.it)Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks37 / 37