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
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