Dissipative dynamics of spins in quantum dots A.O. Caldeira Universidade Estadual de Campinas...

Dissipative dynamics of spins in quantum dots

A. O. Caldeira

Universidade Estadual de CampinasCampinas, BRAZIL

Collaborators

Harry Westfahl Jr. – LNLS – BRAZILFrederico Borges de Brito – UNICAMP – BRAZIL Gilberto Medeiros-Ribeiro – LNLS – BRAZIL Maya Cerro – UNICAMP/LNLS – BRAZIL

O que é dissipação quântica?

Movimento dissipativo+

Mecânica quântica

Uma breve preparação

Movimento dissipativo

Movimento dissipativo

Movimento em um meio viscoso

Dissipação + Flutuações

O movimento Browniano

A Mecânica Quântica através de alguns exemplos

O tunelamento de uma partícula quântica

A Mecânica Quântica através de alguns exemplos

O tunelamento coerente de uma partícula quântica

Mecânica QuânticaX

Dissipação

• A mecânica quântica se aplica a sistemas nas escalas atômicas e sub-atômicas: sistemas isolados ou sujeitos a interações externas controladas.

• A dissipação ocorre em sistemas macroscópicos sujeitos à influência (incontrolável) do ambiente onde estão inseridos.

Onde os dois efeitos podem ser simultaneamente observados?

?

Quântico (microscópico)

Clássico(macroscópico)

md 910 md 610

mdm 69 1010

mdm 69 1010 Sistemas meso e nanoscópicos

Superconducting Quantum Interference Devices (SQUIDs):

O paradigma

H

Sistemas magnéticos

Partículas magnéticas

Tunelamento coerente de partículas magnéticas(103 104 spins por partícula)

Sistemas de dois níveis

Vários sistemas aqui apresentados envolvem sobreposições de duas configurações

ba

Dispositivos e qubits

Dissipação destrói a coerência necessária para ofuncionamento do processador quântico: descoerência

Spin eletrônico em pontos quânticos

Um possível candidato a qubit NANO ???

Introduction

• Main goal – Study of the possibility of implementation of solid state

qubits: spins in self assembled quantum dots

• Candidates and drawbacks – Photons » non-interacting entities – Optical Cavity » weak atom-field coupling – ion traps » short phonon lifetime – NMR » low signal – Superconducting devices » decoherence – Spins in quantum dots » ?

Quantum bits (DiVincenzo '01)

• Well defined two level system– Single electron spin

• Quantum dots: Coulomb blockade + Pauli exclusion

• Addressing– Well defined energy splittings

• g-factor (Landé) engineering

• Reset– Energy splitting » kT

• Electronic Zeeman frequency

• Gates– Resonant EM Field

• Microcavity

• Long decoherence times– Isolated from dissipative channels

• Strong electronic confinement

Quantum bits (DiVincenzo '01)

• Well defined two level system– Single electron spin

• Quantum dots: Coulomb blockade + Pauli exclusion

Self Assembled Quantum Dots

STM scans of self-assembled island formation through epitaxial growth of Ge on a Sisubstrate. Left scans: 50nm x 50nm. Right scan: 35nm x 35nm (Courtesy of G.Medeiros-Ribeiro)

Dots

Model

Electronic confinement

Electronic confinement

• Coulomb Blockade

1e 2e

5e3e 4e 6e

s-shell p-shell

by G. Medeiros-Ribeiro

Quantum bits (DiVincenzo '01)

• Addressing– Well defined energy splittings

• g-factor (Landé) engineering

• Reset– Energy splitting » kT

• Electronic Zeeman frequency

Addressing and resetting

...+dVψψg+dVψψg=g BBAAeff

SRL- strain reducing layer

gA gA

gAgA

gB gC

samples A, D

sample C

• g-factor engineering

G. Medeiros-Ribeiro, E. Ribeiro, H. W. Jr., Appl. Phys. A, 2003; cond-mat/0311644

Quantum bits (DiVincenzo '01)

• Gates– Resonant EM Field

• Microcavity

• Long decoherence times– Isolate from dissipative channels

• Strong electronic confinement

• Magnetic moment (red vector) in a magnetic field (brown vector) : the conservative dynamics

– Precession of the moment around the external field direction

SBgμ=dtSd

B

0

S

Dissipative spin dynamics

Dissipative spin dynamics

• Relaxation dynamics– Landau-Lifshits damping (yellow arrow) drives the system towards a

collinear state

S

SSBλ

SBgμ=dtSd

B

0

0λ

λ

• Noise– Fluctuating terms (green arrow) to our equations of motion

S

Sb+SSBλ

SBgμ=dtSd

i

B

λ

Dissipative spin dynamics

Microscopic dissipative spin dynamics

• Quantum noise and dissipation– Damping and Noise from microscopic interaction with lattice

phonons

Static Field:

Noise+Fluctuations: Phonons

Oscillating Field (microcavity):

,BgμΔ B 0

,ΩtcosBgμtε B 1

Microscopic dissipative spin dynamics

• Quantum dissipation formulation:

xz

y

• Noise and dissipation

• Bloch-Redfield equations– Linear differential equations of motion (quantum average of

components)

Determined by noise time correlation function

xz

yJ

Microscopic dissipative spin dynamics

yz

yzyzyyyzxy

xzxzxxxyx

σΔ=dtdσ

tAσtΓσtΓσΔσtε=dtdσ

tAσtΓσtΓσtε=dtdσ

/

/

/

tA,tΓ,tΓ iijii

Fluctuating magnetic field (noise)

Electrons:

Phonons:

Spin-Orbit Interaction:

Orbital degrees of freedom:2D Harmonic Oscillator states

OpticalAcoustic

e-Ph interaction: PiezoelectricDeformation

PotentialMagneto

-elastic

RashbaDresselhaus

Fluctuating magnetic fields RB

Dissipation Mechanism

• No bath• No spin-orbit interaction

• No bath• Spin-Orbit interaction

Dissipation Mechanism

Dissipation Mechanism

• Orbital contact with the phonon bath

• Non-interacting spin and orbit

Dissipation Mechanism

• Orbital contact with the phonon bath• Spin-orbit interaction

– Indirect spin entanglement with the bath

• Lateral– - LQD (Hanson et al ’03)– - VQD (Fujisawa et al ’02)– - SAQD (Medeiros-Ribeiro et al ’99)

• Vertical (frozen):

Electronic Confinement

meVω 10

meVω 500

0ωω

meVω 50

Spin-Orbit Hamiltonian:

,BgμΔ xB ,ωmγβ c parameters: 0ω

yyy†y

zzz†zxSO

Pβσ++aaω+

Pβσ+aaω+σΔ

H

2

1

2

1

2

0

0

Acoustic Phonon Bath

65

3

102

355

=δ

=δ● “orbital” bath spectral function

piezoelectric– deformation potential

ωωθωω

δωm=ωJ D

s

D

sDs

2

3=s5=s

● GaAs

● InAs

65

3

105

149

=δ

=δ

Approximate form of the Hamiltonian

2

22

2

220

**

2

2

1

2

2

1

2

qm

Cqm

m

p

qmm

pHH

aa

aaaa

a a

a

SOphe

Effective Bath of Oscillators

Equivalent Hamiltonian:

Laplace transform of the equations of motion for the spin:

)()(ˆ)(ˆ zFzzK

)(ˆlim0

iKIm=ωJ eff

allows us to define an effective spectral function

Spin Orbit Phonon bath

– B is the generalized incomplete beta function

As seen by the spins...

where

D

ss

DD ωω

φδ+ωω

ω

ω=ωZ 1

22

0

0102

s,x,B+s,x,Bπ

x=xφ s

s

s

s

D

s

+s

D

s

eff

ωω

δ+ωZ

ωω

δβm

=ωJ 2

22

2

2

Bath resonance

Behaviour of the effective bath spectral function

Piezoelectric coupling:

H. W. Jr. et al.Phys. Rev B. 70 (2004)

6

23

2

5

32

D

Deff

ωω

δ+ωZ

ωω

δβm

=ωJ

Dissipative Mechanism)1( sδ• weak coupling

)1( sδ

Bath resonance

• strong coupling

210 sπ

δω~Ω s

s

s

s δ

)π(sω~Ω

2

20

Effective spectral function

• Low frequency limit ( and )

Always super-ohmic– See also (Khaetskii & Nazarov '01)

sΩω s

sDD δωω

ωω

∕1

0 1

– Can be ohmic!

24

0

2

+s

D

Dseff ω

ωωω

δβmωJ

• High frequency limit ( )

2222

42

s

Ds

eff ωω

δsπ

βmωJ

Ds ωωΩ

yz

yzyzyyyzxy

xzxzxxxyx

σΔ=dtdσ

tAσtΓσtΓσΔσtε=dtdσ

tAσtΓσtΓσtεdtdσ

/

/

/

Microscopic dissipative spin dynamics

xz

y

• General expression for the microscopic spin dynamics:

The Bloch-Redfield equation

Microscopic dissipative spin dynamics

• General expression for the coefficients

ijU is the free spin time evolution operator

Microscopic dissipative spin dynamics

• Long time asymptotic behavior– (No driving) Damped precession around the static field

direction

2coth

211

)/2(1

ΔΔJ=

Tt sxx

0)()(0)( ttAt xzy

)(2

1)/2( JtAx

yz

zyzyyyzy

xxxxx

σΔ=dtdσ

σΓσΓσΔ=dtdσ

AσΓdtdσ

/

/

/

2coth

2

1

1)/2(

1

ΔΔJ

=T

txx

s

Microscopic dissipative spin dynamics

Microscopic dissipative spin dynamics

Driven spin dynamics

ztt ˆcos)( 0 • Transverse external field

• Useful parameters for the model

detuning

effective field amplitude

dephasing

ΔΩ Δ5.0Ω

Driven spin dynamics

Peaks: ||0 S

Driven spin dynamics

Peak: S

• Two distinct time regimes

Driven spin dynamics

R

• Short time dynamics

• Long time dynamics

Long time dynamics

s

Resonant dynamics s

Resonant dynamics

• Very long decoherence (relaxation) times– Good for keeping quantum information – Bad for reseting

T 1

• Very long decoherence (relaxation) times– Good for keeping quantum information – Bad for reseting

Resonant dynamics

Short time dynamics

Resonant dynamics s

• Very long decoherence (relaxation) times– Good for keeping quantum information – Bad for reseting

Resonant dynamics

• Very long decoherence (relaxation) times– Good for keeping quantum information – Bad for reseting

Resonant dynamics

Resonance dominated

ΔΩ Resonant dynamics

Resonant dynamics

Bulk valuesΔΩ

Off-resonance dynamics

Bulk values0.8ΔΩ

Bath assisted cooling

– A high degree of polarization can be achieved in short times with a sequence of (ns) short pulses

A. E. Allahverdyan et al., Phys. Rev. Lett. 93 (2004)

• Reset pulses– Use the large dissipation mechanism (cooling)– Reset times O(ns)

Summary

• Indirect dissipation mechanism: Spin Orbit Phonon• Non-perturbative approach reveals a new resonance and new

regimes of dissipation• Perturbative regime only valid for large confinement energies

(SAQD)• Solution of the Bloch-Redfield equations reveals two

dynamical regimes• Short time dynamics dominated by the bath resonance

Thanks:HP-Brazil, FAPESP, CNPq