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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
H
iE
i
Main idea of our work
To travel in the spectra of eigenenergies
Ei
Control parameter
Introduction
H
iE
i
Main idea of our work
To travel in the spectra of eigenenergies
Ei
Control parameter
Introduction
H
iE
i
Main idea of our work
To travel in the spectra of eigenenergies
Ei
Control parameter
Introduction
H
iE
i
Main idea of our work
To travel in the spectra of eigenenergies
Ei
Control parameter
The system
Quasi-one-dimensional quantum dot:
Confining potential: doble quantum well filled with 2 e
Lz
Lz≫L
x yLy
Lx
The system
Quasi-one-dimensional quantum dot:
Confining potential: doble quantum well filled with 2 e
Lz
Lz≫L
x yLy
Lx
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)
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
We study the prob. transition in several ac. For example:
Colaboradores-ReferenciasLandau-Zener transitions in our model
We study the prob. transition in several ac. For example:
∣1⟩
∣2 ⟩ ∣1⟩
∣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
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 Is our method generic?
Colaboradores-ReferenciasThe method: questions
We need well defined avoided crossings
a/R
Stadium billiard
Is our method generic?
LZ transitions
Sanchez, Vergini DW PRE 96
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).