Quantum control using diabatic and adibatic

transitions

Diego A. Wisniacki

University of Buenos Aires

Colaboradores-ReferenciasColaborators

Gustavo Murgida (UBA)

Pablo Tamborenea (UBA)

Short version ---> PRL 07, cond-mat/0703192

APS ICCMSE

Outline

Introduction

The system: quasi-one-dimensional quantum dot with 2 e inside

Landau- Zener transitions in our system

The method: traveling in the spectra

Results

Final Remarks

Introduction

∣ initial⟩ ∣ final ⟩

∣ final ⟩≈∣ target ⟩ Desired state

t0

tf

Introduction

H

iE

i

Main idea of our work

To travel in the spectra of eigenenergies

Ei

Control parameter

Introduction

To navigate the spectra

Introduction

To navigate the spectra

Introduction

To navigate the spectra

Introduction

To navigate the spectra

The system

Quasi-one-dimensional quantum dot:

Confining potential: doble quantum well filled with 2 e

Lz

Lz≫L

x yLy

Lx

Colaboradores-ReferenciasThe system

H=−ℏ2

2 m ∂

2

∂ z1

2 ∂2

∂ z2

2V z

1V z

2V

C∣z1− z

2∣−e z1 z

2E t

Time dependent electric field

Coulombian interaction

The Hamiltonian of the system:

Note: no spin term-we assume total spin wavefunction: singlet

The system

PRE 01 Fendrik, Sanchez,Tamborenea

Interaction induce chaos

Nearest neighbor spacing distribution

System: 1 well, 2 e

Colaboradores-ReferenciasThe system

We solve numerically the time independent Schroeringer eq.

Electric field is considered as a parameter

Characteristics of the spectrum (eigenfunctions and eigenvalues)

The system

Spectra lines

Avoided crossings

Colaboradores-ReferenciasThe systemCero slopedelocalized

Positive slope e¯ in the right dot

Negative slope e¯ in the left dot

Landau-Zener transitions in our model

LZ model

∣1 ⟩ ,∣2 ⟩

H =1

2

1=E

0

1

2=E

0

2

Linear functions

hyperbolas

Landau-Zener transitions in our model

LZ model

∣ t−∞ ⟩=∣1 ⟩

P1t∞=exp−2 2 / ℏ v 1− 2Probability to remain in the state 1

P2 t∞=1−exp−2 2 / ℏ v 1− 2

Probability to jump to the state 2

t =v tif

Landau-Zener transitions in our model

LZ model

2 / ℏ v 1−2≫1Adibatic transitions

Diabatic transitions 2 / ℏ v 1−2≪1

v≪2 / ℏ1−2

v≫2 / ℏ1−2

Colaboradores-ReferenciasLandau-Zener transitions in our model

E(t)

We study the prob. transition in several ac. For example:

Full system 2 level systemLZ prediction

P1

E=0.07, 0.27, 0.53,1.07, 4.27kV

cm ps∣1⟩

∣2 ⟩ ∣1⟩

∣2 ⟩

Colaboradores-ReferenciasLandau-Zener transitions in our model

We study the prob. transition in several ac. For example:

Full system 2 level system

The method: navigating the spectrum

We use adiabatic and rapid transitions to travel in the spectra

Choose the initial state and the desired final state in the spectra

Find a path in the spectra

Avoid adiabatic transitions in very small avoided crossings

If it is posible try to make slow variations of the parameter

Results

First example: localization of the e¯ in the left dot EPL 01 Tamborenea, Metiu

(sudden switch method)

PRRt =∫R dz1∫R dz 2∣ z1, z2, t ∣

2

LL

Results

First example: localization of the e¯ in the left dot

EPL 01 Tamborenea, Metiu (sudden switch method)

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults Second example: complex path

Colaboradores-ReferenciasResults

Third example: more complex path

Results

∣⟨target∣ T

f⟩∣=0.91

Colaboradores-ReferenciasResults

Forth example: target state a coherent superposition

∣ target ⟩= a1∣R R ⟩ a

2∣L L ⟩a

3∣R L ⟩ ∣a 1∣

2=∣a2∣

2=∣a 3∣

2=1 /3

Colaboradores-ReferenciasResults

Forth example: target state a coherent superposition

∣target⟩= 1

3[∣R R ⟩∣L L ⟩∣R L ⟩ ] ∣a 1∣

2=∣a2∣

2=∣a 3∣

2=1 /3

Colaboradores-ReferenciasResults

Forth example: target state a coherent superposition

∣target⟩= 1

3[∣R R ⟩∣L L ⟩∣R L ⟩ ] ∣a 1∣

2=∣a2∣

2=∣a 3∣

2=1 /3

Colaboradores-ReferenciasResults

Forth example: target state a coherent superposition

∣target⟩= 1

3[∣R R ⟩∣L L ⟩∣R L ⟩ ] ∣a 1∣

2=∣a2∣

2=∣a 3∣

2=1 /3

Colaboradores-ReferenciasResults

Forth example: target state a coherent superposition

∣target⟩= 1

3[∣R R ⟩∣L L ⟩∣R L ⟩ ] ∣a 1∣

2=∣a2∣

2=∣a 3∣

2=1 /3

Colaboradores-ReferenciasResults

Forth example: target state a coherent superposition

∣target⟩= 1

3[∣R R ⟩∣L L ⟩∣R L ⟩ ] ∣a 1∣

2=∣a2∣

2=∣a 3∣

2=1 /3

Colaboradores-ReferenciasResults

Forth example: target state a coherent superposition

∣target⟩= 1

3[∣R R ⟩∣L L ⟩∣R L ⟩ ] ∣a 1∣

2=∣a2∣

2=∣a 3∣

2=1 /3

Colaboradores-ReferenciasResults

Forth example: target state a coherent superposition

∣target⟩= 1

3[∣R R ⟩∣L L ⟩∣R L ⟩ ]

∣a 1∣2=∣a2∣

2=∣a 3∣

2=1 /3

Colaboradores-ReferenciasThe method: questions

We need well defined avoided crossings

a/R

Stadium billiard

Is our method generic?

Is our method experimentally possible?

LZ transitions

Sanchez, Vergini DW PRE 96

Colaboradores-ReferenciasFinal Remarks

We found a method to control quantum systems

Our method works well: ∣⟨target∣ T

f⟩∣≈0.9

With our method it is posible to travel in the spectra of

the system

We can control several aspects of the wave function

(localization of the e¯, etc).

Colaboradores-ReferenciasFinal Remarks

We can also obtain a combination of adiabatic states

Control of chaotic systems

Decoherence??? Next step???.