Quantum data-bus

Optimal dynamics for quantum information transfer and effective entangling gate through

homogeneous quantum wires

Leonardo Banchi

Università degli studi di Firenze, Florence, Italy

with

T. J. G. Apollaro, A. Cuccoli, R. Vaia, P. Verrucchi, and

A. Bayat, S. Bose

Lviv - April 7, 2011

Overview

Goal−→information transmission between distant parts Spin chain data-bus

Tool−→Optimal dynamics

induce a coherent (ballistic) dynamics in a dispersive chain

Results:

High quality state transmission

High quality generation of long-distance entanglement

Quantum data-bus

Spin chain data-bus

S. Bose, PRL, 2003

S. Bose, Contemporary Physics, 2007

Register A Register B

Quantum data-bus

|ψ i

Spin chain data-bus

S. Bose, PRL, 2003

S. Bose, Contemporary Physics, 2007

Register A Register B

Quantum data-bus

Spin chain data-bus

S. Bose, PRL, 2003

S. Bose, Contemporary Physics, 2007

Register A Register B

State transmission

A Γ B

Out of equilibrium initial state:|Ψ_ini=|ψ_Ai |ψ_Γi |ψ_Bi

State of B at timet: ρB(t) =Tr

AΓ

e^−itHAΓB|Ψ_ini hΨ_in|e^itHAΓB

Measure of the transmission quality: minimum fidelity F(t) =min

|ψ_AihψA|ρB(t)|ψAi

State transmission

A Γ B

Out of equilibrium initial state:|Ψ_ini=|ψ_Ai |ψ_Γi |ψ_Bi State of B at timet:

ρB(t) =Tr

AΓ

e^−itHAΓB|Ψ_ini hΨ_in|e^itHAΓB

Measure of the transmission quality: minimum fidelity F(t) =min

|ψ_AihψA|ρB(t)|ψAi

State transmission

A Γ B

Out of equilibrium initial state:|Ψ_ini=|ψ_Ai |ψ_Γi |ψ_Bi State of B at timet:

ρB(t) =Tr

AΓ

e^−itHAΓB|Ψ_ini hΨ_in|e^itHAΓB

Measure of the transmission quality: minimum fidelity F(t) =min

|ψ_Aihψ_A|ρB(t)|ψ_Ai

Optimal dynamics

Requirement: quadratic Hamiltonian in terms oflocalcreation and annihilation operators

H_AΓB =X

ij

M_ija^†_ia_j+N_ija^†_ia^†_j +h.c.

Normal modes and dispersion relationωk

H_AΓB=X

k

ωkd_k^†dk

Density of excitations in the initial state

ρ(k) =hΨ_in|d^†_kdk|Ψ_ini

Optimal dynamics

Induce a coherent (ballistic) dynamics in a dispersive chain

=⇒ Makeρ(k)mostly peaked around the linear zone ofωk through a minimalengineering of the interactions

Optimal dynamics

Requirement: quadratic Hamiltonian in terms oflocalcreation and annihilation operators

H_AΓB =X

ij

M_ija^†_ia_j+N_ija^†_ia^†_j +h.c.

Normal modes and dispersion relationωk

H_AΓB=X

k

ωkd_k^†dk

Density of excitations in the initial state

ρ(k) =hΨ_in|d^†_kdk|Ψ_ini

Optimal dynamics

Induce a coherent (ballistic) dynamics in a dispersive chain

=⇒ Makeρ(k)mostly peaked around the linear zone of ωk through a minimalengineering of the interactions

Consequences of the optimal dynamics

Optimal dynamics

Induce a coherent (ballistic) dynamics in a dispersive chain

=⇒ Makeρ(k)mostly peaked around the linear zone of ωk through a minimalengineering of the interactions

All relevant excitations (with highρ(k)) have constant group velocityvk =∂kωk

High quality “wave-packet” transmission F(t^∗)'1

Ballistic transport with arrival timet^∗ 'vN

Optimal dynamics with the XY model

A Γ B

XX model

H_Γ=J

N−1

X

n=1

σ^x_nσ^x_n+1+σ^y_nσ^y_n+1

H_AΓ=J₀ σ₀^xσ₁^x+σ₀^yσ^y₁ H_ΓB =J₀ σ_N^xσ_N+1^x +σ^y_Nσ^y_N+1

Optimal dynamics by settingJ₀to a properly tuned value

Banchi, Apollaro, Cuccoli, Vaia, Verrucchi, PRA, 2010

Optimal dynamics with the XX model

Dispersion relation (J ≡1)

ωk=cosk Density of excitations

ρ(k)≈ 1

∆²+cot²k ∆ = J₀² 2−J²₀ peaked around the inflection pointk=π/2with width∆ Normal modes (n=1, . . . ,N+2)

kn = πn+ϕk_n

N+3 ϕk =2k−2 cot⁻¹

2−J²₀ J²₀ cotk

Banchi, Apollaro, Cuccoli, Vaia, Verrucchi, in preparation

Optimal dynamics with the XX model

Group velocity of the “wave-packet” close to the inflection point

vk =∂kωk 'const.×

1+ 2

NJ₀⁶ −1 2

k²+O(k⁴)

Optimal values ofJ₀^opt.: J^opt.₀ '1.05N⁻¹⁶

Non-weak coupling regime and ballistic transport

N 25 51 101 251 501 1001 2501 5001

J₀^opt. 0.63 0.56 0.49 0.42 0.37 0.33 0.28 0.25 t^∗ 32.5 60.2 112 266 520 1025 2533 5041

Movie

Coherent transport: N = 50

|Ψ_ini=|↑i|Ω_Γi|↑i

Wave packet propagation displayed by the magnetizationS^z(x,t)

J₀ =J₀^opt.'0.56J J₀=J

State transmission quality

F(t∗)

N

It works with very long channels!

F(t) =min

|ψ_Aihψ_A|ρB(t)|ψ_Ai

State transmission quality

F(t∗)

N

It works with very long channels!

F(t) =min

|ψ_Aihψ_A|ρB(t)|ψ_Ai

Entangling quantum gates between distant qubits

|↑i_A|↑i_B −→e^iφ↑↑|↑i_A|↑i_B

|↑i_A|↓i_B −→e^iφ↑↓|↓i_A|↑i_B

|↓i_A|↑i_B −→e^iφ↓↑|↑i_A|↓i_B

|↓i_A|↓i_B −→e^iφ↓↓|↓i_A|↓i_B

Optimal dynamics effectively generates an entangling quantum gate Fixing

|Ψ_ini= (|↑i+|↓i)_A⊗ |ψi_Γ⊗(|↑i+|↓i)_B AandBget maximally entangled att^∗

Banchi, Bayat, Verrucchi, Bose, PRL in print

Interacting models and works in progress...

H=

N

X

n=0

Jn

σ_n^xσ_n+1^x +σ_n^yσ_n+1^y + ∆σ_n^zσ^z_n+1

whereJN =J₀andJn=J forn6=0,N

Mean field approach for∆'0: mapping to a quadratic free-fermionic Hamiltonian

Numerical optimization: findingJ₀^opt

Complex scattering for strong∆

Scattering can favour transmission

Optimize the initial state of the chain as well

Bayat, Banchi, Bose, Verrucchi, in preparation

Interacting models and works in progress...

H=

N

X

n=0

Jn

σ_n^xσ_n+1^x +σ_n^yσ_n+1^y + ∆σ_n^zσ^z_n+1

whereJN =J₀andJn=J forn6=0,N

Mean field approach for∆'0: mapping to a quadratic free-fermionic Hamiltonian

Numerical optimization: findingJ₀^opt Complex scattering for strong∆

Scattering can favour transmission

Optimize the initial state of the chain as well

Bayat, Banchi, Bose, Verrucchi, in preparation

Conclusions

General idea for quadratic Hamiltonians

High quality information transmission between distant parts Experimental feasibility (unmodulated interactions)

Cold atoms

Banchi, Bayat, Verrucchi, Bose, PRL in print Other possible applications in solid state