Microscopic dissipative spin dynamics

Dissipative dynamics of spins in quantum dots
A. O. Caldeira
Universidade Estadual de Campinas
Campinas, BRAZIL
Collaborators
Harry Westfahl Jr. – LNLS – BRAZIL
Frederico Borges de Brito – UNICAMP – BRAZIL
Gilberto Medeiros-Ribeiro – LNLS – BRAZIL
Maya Cerro – UNICAMP/LNLS – BRAZIL
Uma breve preparação
O que é dissipação quântica?
Movimento dissipativo
+
Mecânica quântica
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 Quântica
X
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)
d 10 9 m
10 9 m  d 10 6 m
Clássico
(macroscópico)
d 10 m
6
Sistemas meso e nanoscópicos
10 9 m  d 10 6 m
H
Superconducting Quantum Interference Devices (SQUIDs):
O paradigma
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
 a   b 
Dispositivos e qubits
Dissipação destrói a coerência necessária para o
funcionamento do processador quântico: descoerência
Um possível candidato a qubit
Spin eletrônico em pontos quânticos
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 Si
substrate. Left scans: 50nm x 50nm. Right scan: 35nm x 35nm (Courtesy of G.
Medeiros-Ribeiro)
Electronic confinement
Dots
Model
Electronic confinement
• Coulomb Blockade
1e
2e
3e 4e 5e 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
• g-factor engineering
samples
A, D
gB
sample C




geff = g A ψ ψ dVA + g B ψ ψ dVB +...
gC
gA
gA
gA
gA
SRL- strain reducing layer
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
Dissipative spin dynamics
• Magnetic moment (red vector) in a magnetic field (brown
vector) : the conservative dynamics
– Precession of the moment around the external field direction

S

 
dS
= gμB B0  S
dt
Dissipative spin dynamics
• Relaxation dynamics
– Landau-Lifshits damping (yellow arrow) drives the system towards a
collinear state

λ

S

 
dS
= gμ B B0  S
dt
  
 λB0  S  S






λ
Dissipative spin dynamics
• Noise
– Fluctuating terms (green arrow) to our equations of motion

λ

S

 
dS
= gμ B B  S
dt
    
 λB  S  S + bi  S
Microscopic dissipative spin dynamics
• Quantum noise and dissipation
– Damping and Noise from microscopic interaction with lattice
phonons
Static Field:
Δ  gμB B0 ,
Oscillating Field
(microcavity): εt   gμB B1cosΩt ,
Noise+Fluctuations: Phonons
Microscopic dissipative spin dynamics
• Quantum dissipation formulation:
• Noise and dissipation
x
y
z
Microscopic dissipative spin dynamics
• Bloch-Redfield equations
– Linear differential equations of motion (quantum average of
components)
Γ ii t , Γ ij t , Ai t 
Determined by noise time
correlation function J
x
z
y
dσ x / dt = ε t  σ y  Γ xx t  σ x  Γ xz t  σ z  Ax t 
dσ y / dt =  ε t  σ x  Δ σ z  Γ yy t  σ y  Γ yz t  σ z  Ay t 
dσ z / dt =  Δ σ y
Fluctuating magnetic field (noise)
Orbital degrees of freedom:
2D Harmonic Oscillator states
Electrons:
e-Ph interaction:
Piezoelectric
Acoustic
Phonons:
Spin-Orbit
Interaction:
Deformation
Potential
Optical
Dresselhaus
Fluctuating magnetic fields
Rashba
BR
Magnetoelastic
Dissipation Mechanism
• No bath
• No spin-orbit interaction
Dissipation Mechanism
• No bath
• Spin-Orbit interaction
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
Electronic Confinement
• Lateral
– ω0  1meV - LQD (Hanson et al ’03)
– ω0  5meV - VQD (Fujisawa et al ’02)
– ω0  50meV - SAQD (Medeiros-Ribeiro et al ’99)
ω  ω0
• Vertical (frozen):
Spin-Orbit
Hamiltonian:
parameters:
H SO
Δ
1
 †
  σ x + ω0  a z a z +   βσ z Pz
2
2

1

+ ω0  a†y a y +  + βσ y Py
2


Δ  gμB Bx , β  γc m ω , ω0
Acoustic Phonon Bath
Approximate form of the Hamiltonian
p2 1 * 2 2
H e ph  H SO  *  m 0 q
2m 2
p
Ca
1
2

 maa  qa 
2
2
m
2
m

a
a
a a

2
a
●
“orbital” bath spectral function
●
ω
J s ω = m ω δ   θ ωD  ω
 ωD 
–
δ5 = 2 106
2
D s
piezoelectric s = 3
deformation potential
s=5
GaAs
δ3 = 355
s


q 

2
●
InAs
δ3 = 149
δ5 = 5 106
Effective Bath of Oscillators
Laplace transform of the equations of motion for the spin:
Kˆ ( z ) ˆ ( z )  F ( z )
allows us to define an effective spectral function


J eff ω = lim Im Kˆ (  i )
 0
Equivalent Hamiltonian:
Spin  Orbit  Phonon bath
s+ 2
 ω
m β δs  
ωD 

J eff ω =
2s
2
2 ω 
Z ω + δs  
 ωD 
As seen by the spins...
where

2
 ω0   ω 
Z ω =     
 ωD   ωD 
2
2

 ω 
1+ δs φs   
 ωD  



x s 2
s
φs  x  = 
B x, s,0+  1 B x, s,0
π
– B is the generalized incomplete beta function
Bath resonance
H. W. Jr. et al.
Phys. Rev B. 70 (2004)
Behaviour of the effective bath spectral
function
Piezoelectric coupling:
5
 ω
m β δ3  
ωD 

J eff ω =
6
2
2 ω 
Z ω + δ3  
 ωD 

2
Bath resonance
Dissipative Mechanism
• weak coupling (δs 1)
δs 

Ωs ~ ω0 1 

 π s  2  
• strong coupling (δs  1)
Ωs ~ ω0
(s  2 )π
2δs
Effective spectral function
1∕ s
• Low frequency limit
( ω  Ωs
and
 ωD 
J eff ω  m β δs  
 ω0 

–
2
4
 ω0 1 
ω

 
ωD
 ωD δs 
ω
 
 ωD 
s+ 2
Always super-ohmic
See also (Khaetskii & Nazarov '01)
• High frequency limit
(
Ωs  ω  ωD
)
π s  2   ω 
J eff ω  m β
 
4δs
 ωD 

–
Can be ohmic!
2
2
2
s 2
)
Microscopic dissipative spin dynamics
• General expression for the
microscopic spin dynamics:
x
The Bloch-Redfield equation
dσ x / dt
z
y
 ε t  σ y  Γ xx t  σ x  Γ xz t  σ z  Ax t 
dσ y / dt =  ε t  σ x  Δ σ z  Γ yy t  σ y  Γ yz t  σ z  Ay t 
dσ z / dt =  Δ σ y
Microscopic dissipative spin dynamics
• General expression for the coefficients
U ij
is the free spin time evolution operator
Microscopic dissipative spin dynamics
• Long time asymptotic behavior
– (No driving) Damped precession around the static field
direction
 (t )  0  Ay (t )  xz (t )  0
1
1
Δ 

xx (t  2 /  s )  =
J  Δ coth  
T1 2
 2 
Ax (t  2 / )  
dσ x / dt
1
J ()
2
  Γ xx σ x  Ax
dσ y / dt =  Δ σ z  Γ yy σ y  Γ yz σ z
dσ z / dt =  Δ σ y
1
xx (t  2 / )  =
T1
1
 Δ 

J  Δ coth 

2
 2 

s

Microscopic dissipative spin dynamics
Microscopic dissipative spin dynamics
Driven spin dynamics
• Transverse external field

 (t )   0 cos t zˆ
• Useful parameters for the model
detuning
effective field amplitude
dephasing
Driven spin dynamics
ΩΔ
Ω  0.5Δ
Peaks:
 0   |   S |
Driven spin dynamics
Peak:    S
Driven spin dynamics
• Two distinct time regimes
• Long time dynamics
R

• Short time dynamics


Resonant dynamics
     s
Long time dynamics
s
Resonant dynamics
• Very long decoherence (relaxation) times
– Good for keeping quantum information
– Bad for reseting
T1
Resonant dynamics
• Very long decoherence (relaxation) times
– Good for keeping quantum information
– Bad for reseting
Resonant dynamics
     s
Short time dynamics
Resonant dynamics
• 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
Resonance dominated
Resonant dynamics
ΩΔ
Resonant dynamics
ΩΔ
Bulk values
Off-resonance dynamics
Ω  0.8Δ
Bulk values
Bath assisted cooling
A. E. Allahverdyan et al.,
Phys. Rev. Lett. 93 (2004)
• Reset pulses
– Use the large dissipation mechanism (cooling)
– Reset times O(ns)
–
A high degree of polarization can be achieved in short times with a
sequence of (ns) short pulses
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