Quantum Hydrodynamics Analysis of Quantum Synchronization over Quantum Networks

Quantum Hydrodynamics Analysis of Quantum Synchronization

over Quantum Networks

Pierangelo Marcati

(joint work withP. Antonelli, S.-Y. Ha, D. Kim) GranSasso Science Institute and

Universit`a degli Studi dell’Aquila - L’Aquila, Italy

Classical References

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

T. Vicsek, A. Czir´ok, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of 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 of synchronization in populations of coupled oscillators, Physica D 143

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. Let xi= e

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

x_i determined by phase θ_i. In the absence of coupling, dθi

dt = 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 complex Ginzburg-Landau system.

dθ_i

dt = Ω_i−K N

N

X

j=1

sin(θ_i− θ_j), t > 0, i = 1, · · · , N, (1) subject to initial data θ_i(0) = θ_i0.

We say that the system SYNCHRONIZE if

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

g(−Ω) = g(Ω), spt(g) bounded , R g(Ω)dΩ = 1.

(one-humped , symmetric w.r.t. mean Ω^pc:= _N¹ PN i=1Ωi) e.g. g(Ω) = _π[γ2+(Ω−Ω^γ 0)²), having width γ > 0 and mean Ω0,

Continuous dynamical phase transition at a critical value of the coupling strength Kcr = _πg(0)² , in the mean-field limit.

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

r^∞(K) := lim

t→∞ lim

N →∞

1 N

N

X

i=1

e

√−1θ_i(t)  ,

∞

Ha

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

D_θ(t) := max

1≤i,j≤N|θ_i(t) − θ_j(t)|, t ≥ 0, D_Ω:= max

i,j |Ω_i− Ω_j|.

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

Ω_i = Ω_j, i 6= j, K > 0, D₀ := D_θ(0) < π,

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

e^−KtD₀ ≤ D_θ(t) ≤ e^−KαtD₀, t ≥ 0, (2) where α depends on the diameter of the initial phase configuration

We also recall the estimate of existence of a trapping region for non-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, D_Ω > 0, K > K_e:= D_Ω sin D0

. Then we have

(i) sup

t≥0

D_θ(t) ≤ D₀ < π.

(ii) ∃ t0> 0 such that sup

t≥t0

Dθ(t) ≤ D^∞, where D^∞is defined by

Quantum Lohe Model

Phase synchronization cannot occur in quantum systems with constant linearinteractions

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

Wavefunctions at each node distributed over quantum channels to all other connected nodes, by means of quantum teleportation

Local evolution given by free evolution plus a nonlinear interaction The Kuramoto system can be generalized tonon-Abelian model where variables are n × n complex matrices Ui in U (n) where U_i^∗ denotes the Hermitian conjugate of Ui and Hi is a prescribed constant n × n Hermitian matrix, i = 1....N .

ik ^N

Lohe-Kuramoto

Whenk = 0 the decoupled system is given by the finite-dimensional analog of the Schr¨odinger equation i ˙Ui= HiUi with free dynamics Ui(t) = e^iHitUi(0)

We generalize to complex n-vectors |ψ_ii (with n = 2 for qubit models)

i~∂t|ψ_ii = H_i|ψ_ii + i~k 2N

N

X

0

aij(|ψji − |ψ_ii hψ_j|ψ_ii)

Let d|ψ_ii = _hψ^|ψii

i|ψ_ii, H_i^int = i_2N^~k P_N

0 a_ij( d|ψ_jidhψ_i| − d|ψ_ii dhψ_j|), then i~∂t|ψd_ii = (H_i+ H_i^int) d|ψ_ii

Choi,Ha version of Lohe Model (Mathematicians notations)

˜

χ_jk = χ_jkkψ^kψjk

kk

i∂_tψ_j = −1

2∆ψ_j + V_jψ_j+ iK 2N

N

X

j=1

˜ χ_jk



ψ_k−hψ_j, ψ_ki kψ_jk² ψ_j



Consider for the moment N = 2, we have











i∂tψ1= − 1

2∆ψ1+ V1ψ1+ iK 4χ˜12



ψ2−hψ₁, ψ2i kψ₁k² ψ1



i∂_tψ₂= − 1

2∆ψ₂+ V₂ψ₂+ iK 4χ˜₁₂



ψ₁−hψ₂, ψ₁i kψ₂k² ψ₂

 .

References on Quantum

S.-H. Choi, S.-Y. Ha, Quantum synchronization of the Schr¨odinger-Lohe model, J. Phys. A: Math. Theor. 47 (2014), 355104.

P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in Quantum Fluid Dynamics, Comm. Math. Phys. 287 (2009), no 2, 657–686.

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

From NLS to QHD via WKB







i~∂tψ = −~²

2m∆ψ + V ψ + f⁰(|ψ|²)ψ

− ∆V = |ψ|² Energy

E[ψ] = Z ~²

2 |∇ψ|²+ f (|ψ|²) + 1

2|∇V |²dx WKB ansatz: ψ =√

ρe^iS/~, then (ρ, S) satisfy







∂_tρ + div(ρ∇S) = 0

∂tS +1

2|∇S|²+ f⁰(ρ) + V = ~² 2

∆√

√ ρ ρ

WKB ansatz







∂tρ + div(ρ∇S) = 0

∂_tS +1

2|∇S|²+ f⁰(ρ) + V = ~² 2

∆√

√ ρ ρ u = ∇S⇒ ∂_tu + (u · ∇)u + ∇f⁰(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 ansatzψ = |ψ|e^iS/~ valid only if ψ(t, x) 6= 0

; S not defined in {ψ = 0}

regularity issue: the nodal set {ψ = 0} may have dimH= 1(Federer, Ziemer)

irrotationality: ∇ ∧ u = 0, no vorticesare taken into account in the WKB description

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

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂tρ + div J = 0,

∂tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V = ~²

2 ρ∇ ∆√

√ ρ ρ



Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂tρ + div J = 0,

∂tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂_tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂_tρ + div J = 0,

∂_tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂_tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =~²

4 ∇∆ρ−~²div(∇√

ρ ⊗ ∇√ ρ)

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂_tρ + div J = 0,

∂_tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂_tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =~²

4 ∇∆ρ−~²div(∇√

ρ ⊗ ∇√ ρ)

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂_tρ + div J = 0,

∂_tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂_tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =~²

4 ∇∆ρ−~²div(∇√

ρ ⊗ ∇√ ρ)

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂_tρ + div J = 0,

∂_tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V =~² 4 ∇∆ρ

∂_tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V = ~²

4∇∆ρ − ~²div(∇√

ρ ⊗ ∇√ ρ)

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂_tρ + div J = 0,

∂_tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂_tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =~²

4 ∇∆ρ−~²div(∇√

ρ ⊗ ∇√ ρ)

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂tρ + div J = 0,

∂tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =~²

4 ∇∆ρ−~²div(∇√

ρ ⊗ ∇√ ρ) Formally,

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂tρ + div J = 0,

∂tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =~²

4 ∇∆ρ−~²div(∇√

ρ ⊗ ∇√ ρ) Formally,

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂_tρ + div J = 0,

∂_tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂_tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =~²

4 ∇∆ρ−~²div(∇√

ρ ⊗ ∇√ ρ) Formally,

~²ρ(∇ ¯ψ ⊗ ∇ψ) = ~²ρ (ψ∇ ¯ψ) ⊗ ( ¯ψ∇ψ)

|ψ|²



Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂tρ + div J = 0,

∂_tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂_tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =~²

4 ∇∆ρ−~²div(∇√

ρ ⊗ ∇√ ρ) Formally,

2 1h

2

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂_tρ + div J = 0,

∂tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =~²

4 ∇∆ρ−~²div(∇√

ρ ⊗ ∇√ ρ) Formally,

~²ρ(∇ ¯ψ ⊗ ∇ψ) = 1 ρ h

~²ρ( ¯ψ∇ψ) ⊗ ρ( ¯ψ∇ψ)

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂_tρ + div J = 0,

∂_tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂_tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V =~²

4 ∇∆ρ−~²div(∇√

ρ ⊗ ∇√ ρ) Formally,

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂_tρ + div J = 0,

∂_tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂_tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V = Formally,

~²ρ(∇ ¯ψ ⊗ ∇ψ)=~²∇√

ρ ⊗ ∇√

ρ + J ⊗ J ρ .

Madelung transformation

Moments associated to the wave function. ψ ρ := |ψ|² mass density

J := ~( ¯ψ∇ψ) current density.

∂tρ + div J = 0,

∂tJ +~²div ρ(∇ ¯ψ ⊗ ∇ψ)

+ ∇P (ρ) + ρ∇V = ~² 4 ∇∆ρ

∂tJ +div J ⊗ J ρ



+ ∇P (ρ) + ρ∇V = Formally,

√ √ J ⊗ J

Polar Factorization

Let ψ ∈ H¹(R³), define

P (ψ) =φ ∈ L^∞s.t. kφkL^∞ ≤ 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_loc^1,1, ∇ψ = 0 a.e. in {ψ = 0}.

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

Stability lemma

Lemma

Let φ ∈ L^∞(R³), ψ = |ψ|φ a.e. and kφk_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: {ψ_n} ⊂ H¹(R³), ψ_n→ ψ in H¹,

∇√

ρ_n→ ∇√

ρ, Λ_n→ Λ in L²(R³).

Remark

However φ_n* φ weak ^∗ in L^∞.

Finite energy weak solutions

The pair (ρ, J ) is afinite 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 ); H_loc¹ (R³)),Λ ∈ L²_loc([0, T ); L²_loc(R³));

∀ η, ζ ∈ C₀^∞([0, T ) × R³), ρ := (√

ρ)², J :=√

ρΛ satisfy Z

ρ∂_tη + J · ∇η dxdt + Z

ρ₀η(0) dx = 0, Z

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

4 ρ∆ div ζ + ~²∇√

ρ ⊗ ∇√

ρ : ∇ζ
dxdt + Z

J₀· ζ(0) dx = 0;

generalized irrotationality condition,∇ ∧ J = 2∇√

ρ ∧ Λ. In the

√

Finite Energy Weak Solutions - II

Proposition

Let f (ρ) ∼ ρ^γ, with 1 ≤ γ < 3. Given ψ0 ∈ H¹(R³), let

ρ₀:= |ψ₀|², J₀:= Im( ¯ψ₀∇ψ₀) .Then there exists a global finite energy weak 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 NLS equation 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 a prescribed 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 ⊗ J ρ



+ ∇P (ρ) + ρ∇V +K= ~²

2 ρ∇ ∆√

√ ρ ρ



− ∆V = ρ

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

Energy:

E[ρ, J ] = Z

R³

{~² 2 |∇√

ρ|²+1 2

K · J

ρ + f (ρ) +1

2|∇V |² }dx, dissipates along the flow of solutions

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 −¹_r
ke^2it∆f k_L^q

tL^r_x . kf kL²

k Z t

0

e^2i(t−s)∆F (s) dsk_L^q

tL^r_x . kF k_L^q0˜

t L^˜r0_x

Local smoothing estimates (Constantin-Saut, Sj¨olin, Vega) ke^2it∆f k

L²([0,T ];H_loc^1/2(R³)). kf kL²

k Z t

0

e^2i(t−s)∆F (s) dsk_{L2([0,T ];H}1/2

loc(R³)). kF kL¹_tL²_x

Theorem (Existence)

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

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

and √

ρ ∈ L^q([0, T ]; W^1,r(R³)), Λ ∈ L^q([0, T ]; L^r(R³)),

for any 0 < T < ∞, where (q, r) is any arbitrary (Strichartz) admissible pair for Schr¨odinger in R³.

Fractional step (operator splitting)

solve the QHD without collisions (NLS) update with collisions / interactions

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

√

Potentials Decomposition

Theorem (Ortner, S¨uli )

(decomposition of function in ˙W^1,6) Let ˜V ∈ L^q_tW˙x^1,6,2 ≤ q ≤ +∞ then

V = V˜ ∞(C_x^∞unbounded) + V_p(∈ L^q_tW_x^1,6) in particular

(i) V∞(t, ·) ∈ C_x^∞, ; (ii) kV_pk_Lq

tWx^1,6 ≤ Ck∇ ˜V k_L^q

tL⁶_x; (iii) k∇V_qk_L^q

tL⁶_x+ k∇V_qk_L^∞

t,x ≤ Ck∇ ˜V k_L^q

tL⁶_x;

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 L^q_t D. Fujiwara, A construction of the fundamental equation for the Schr¨odinger equation, Journ. Anal. Math. 35 (1979), 41-96 where

Σ(R³) = {ψ ∈ H¹(R³) : | · |ψ ∈ L²(R³)}.

The parametrix given by the oscillatory integral (E(t, s)φ)(x) :=

 −i

2}π(t − s)

^d₂ Z

R^d

eiS(t,s,x,y) } φ(y)dy

back to Schr¨ odinger - Lohe model











i∂tψ1= − 1

2∆ψ1+ V ψ1+ iKχ12



ψ2−hψ₁, ψ2i kψ₁k²_L2 ψ1



i∂tψ2= − 1

2∆ψ2+ V ψ2+ iKχ12



ψ1−hψ₂, ψ1i kψ₂k²_L2

ψ2

 ,

(3)

Let ρk := |ψk|², Jk= Im( ¯ψk∇ψ_k), rk:=R ρ_kdx = kψkk²_L2

ρ_kj = Re( ¯ψ_kψj)(k 6= j), r_kj = Re(hψ_k, ψji) =R ρ_kjdx Equation for the mass densities

∂_tρ_k+ div J_k=2Kχ_kjRe

 ψ¯_k



ψ_j−hψ_k, ψ_ji kψ_kk_L2

ψ_k



Densities correlations

Since _dt^dr1 = _dt^dr2 = 0, we assume r1 = r2= 1.

∂tρkj =Re

 

−i

2∆ ¯ψ1+ iV ¯ψk+ Kχkj( ¯ψj− hψ_j, ψki ¯ψk

 ψ2

+ ¯ψ_k i

2∆ψj− iV ψ_j+ Kχ − kj(ψ_k− hψ₂, ψ_kiψ₂

 

= −1

2Imψ¯_k∆ψ_j− ∆ ¯ψ_kψ_j + Kχ_kj(ρ_k+ ρ_j − 2(r_kjρ_kj− s_kjσ_kj)) , σ_kj := Im( ¯ψ1ψ2), s_kj := Im(hψj, ψ_ki) =R σ_kjdx.

∂tρkj+ div Jkj = Kχ12(ρk+ ρj− 2(r_kjρkj− s_kjσkj)) ,

Sigma correlations

∂_tσ₁₂=Im

 

−i

2∆ ¯ψ₁+ iV ¯ψ₁+ Kχ₁₂( ¯ψ₂− hψ₂, ψ₁i ¯ψ₁

 ψ₂ + ¯ψ₁ i

2∆ψ₂− iV ψ₂+ Kχ − 12(ψ₁− hψ₂, ψ₁iψ₂

 

= − 1

2Re ∆ ¯ψ1ψ2− ¯ψ1∆ψ2
+ Kχ₁₂Im(−2hψ2, ψ1i ¯ψ1ψ2).

∂_tσ₁₂+ div G₁₂= −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+ div G₁₂= − 2Kχ₁₂(r₁₂σ₁₂+ s₁₂ρ₁₂).

(4)

Then 



 d

dtr12=2Kχ12(1 + s²₁₂− r₁₂² ) 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

-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) = Z 1

2|∇ψ₁|²+1

2|∇ψ₂|²+ V (|ψ₁|²+ |ψ₂|²) dx.

d

dtE(t)= − 2Kχ₁₂r₁₂E(t) + 2Kχ₁₂ Z

Re∇ ¯ψ₁· ∇ψ₂+ 2V ¯ψ₁ψ₂ dx

= − 2Kχ₁₂r₁₂E(t) − 2Kχ₁₂ Z 1

2|∇(ψ₁− ψ₂)|²+ V |ψ₁− ψ₂|²dx+

2Kχ12E(t) ≤2Kχ12(1 − r12(t))E(t).

By Gronwall’s inequality we get that E(t) ≤ e^2Kχ12

Rt

0(1−r12(s)) dsE(0)

First Variation Energy

Consider

E(t) =˜ Z 1

2|∇(ψ₁− ψ₂)|²+ V |ψ1− ψ₂|²dx, then

d

dtE(t) = −2Kχ˜ 12(1 + r12) ˜E(t) − 4Kχ12s12

Z 1

2Im(∇ ¯ψ1· ∇ψ₂) + V σ12dx.

E(t) ≤e˜ ^−2Kχ12tE(0) + 2Kχ˜ ₁₂C Z _t

0

e^{−2Kχ12(t−s)}e^−csds

≤e^−2Kχ12tE(0) + 2Kχ˜ 12C⁰

e^−2Kχ12t− e^−ct 

Synchronization

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

t→∞lim kψ₁− ψ₂k_L2 = 0

By the previous inequality on the energy of the first variation

t→∞lim kψ₁− ψ₂k_H1 = 0 To show the full hierarchy, let

ρ_d:= |ψ1− ψ₂|², J_d:= Im(( ¯ψ1− ¯ψ2)∇(ψ1− ψ₂))

ρa:= 1

2|ψ₁− iψ₂|², Ja:= 1

2Im( ¯ψa∇ψ_a)

1 1

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 finite energy weak solution of the full Hierarchy system

Remark (NonUniqueness)

The same non uniqueness phenomena proved by Camillo De Lellis and L´aszl`o Sz´ekelyhidi for the Euler system was proved for QHD by Donatelli, Feireisl, M. (CPDE 2015)

Thank You !