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