APS March Meeting 2020 Arian Vezvaee Virginia Tech

1. APS March Meeting 2020
Driven dynamics of an electron coupled to spin-3/2 nuclei in
quantum dots
Arian Vezvaee, G. Sharma, S. E. Economou, E. Barnes
March 4th, 2020
Department of Physics, Virginia Tech, Blacksburg, VA 24061

2. Charged quantum dots and nuclear spin problem
Optical pumping of the electron
interacting with nuclear spin bath:
Dynamic Nuclear Polarization
1

3. Charged quantum dots and nuclear spin problem
• Most theoretical frameworks formulate the
problem with spin-1/2 → only a
qualitative description of the nuclear bath.
• InAs/GaAs: I = 1/2 (As with I = 3/2).
2

4. Charged quantum dots and nuclear spin problem
• Most theoretical frameworks formulate the
problem with spin-1/2 → only a
qualitative description of the nuclear bath.
• InAs/GaAs: I = 1/2 (As with I = 3/2).
• Strained induced quadrupolar interactions.
2

5. Charged quantum dots and nuclear spin problem
• Most theoretical frameworks formulate the
problem with spin-1/2 → only a
qualitative description of the nuclear bath.
• InAs/GaAs: I = 1/2 (As with I = 3/2).
• Strained induced quadrupolar interactions.
• Models with spin-3/2
– Huang and Hu, PRB 81 205304 (2010):
Fermi’s golden rule
– Yang and Sham, PRB 88 235304 (2013):
Fokker-Planck equation
2

6. Charged quantum dots and nuclear spin problem
• Most theoretical frameworks formulate the
problem with spin-1/2 → only a
qualitative description of the nuclear bath.
• InAs/GaAs: I = 1/2 (As with I = 3/2).
• Strained induced quadrupolar interactions.
• Models with spin-3/2
– Huang and Hu, PRB 81 205304 (2010):
Fermi’s golden rule
– Yang and Sham, PRB 88 235304 (2013):
Fokker-Planck equation
• Our framework: comprehensive study of a
periodically driven electron spin coupled to
a nuclear spin I > 1/2.
2

7. Experimental setup and general formalism
• Full Hamiltonian of the system:
H(t) = H0,e + Hc (t) + Hres + H0,n + Hhf + HQ
• Pulsed experiments
Greilich et al.
Science 317 1896 (2007)
Hc (t + TR ) = Hc (t)
|T¯x, 3/2i
|¯xi
e
|Tx, 3/2i
|xi
+e
B
Optical axis: x
Magnetic field: z
3

8. Experimental setup and general formalism
• Full Hamiltonian of the system:
H(t) = H0,e + Hc (t) + Hres + H0,n + Hhf + HQ
• Pulsed experiments
Greilich et al.
Science 317 1896 (2007)
Hc (t + TR ) = Hc (t)
• Dynamical maps approach
PRL 107 047601 (2011)
PRB 89 165301 (2014)
|T¯x, 3/2i
|¯xi
e
|Tx, 3/2i
|xi
+e
B
Optical axis: x
Magnetic field: z
3

9. Experimental setup and general formalism
• Full Hamiltonian of the system:
H(t) = H0,e + Hc (t) + Hres + H0,n + Hhf + HQ
• Pulsed experiments
Greilich et al.
Science 317 1896 (2007)
Hc (t + TR ) = Hc (t)
• Dynamical maps approach
PRL 107 047601 (2011)
PRB 89 165301 (2014)
|T¯x, 3/2i
|¯xi
e
|Tx, 3/2i
|xi
+e
B
Optical axis: x
Magnetic field: z
• Markovian approximation: the electron spin dynamics reaches its
steady state much faster than the nuclear spin.
• Double perturbative expansion in the hyperfine flip-flop, and
quadrupolar interaction term.
3

10. Overview of Our Theoretical Framework
4

11. Overview of Our Theoretical Framework
4

12. Overview of Our Theoretical Framework
4

13. Overview of Our Theoretical Framework
4

14. Overview of Our Theoretical Framework
4

15. Overview of Our Theoretical Framework
4

16. Electron dynamics
• Kraus operator
ρ =
j
Ej ρE†
j
• Spin Vector representation: Se, = Tr(σ ρ) for σx , σy , σz , σ0 ≡ 12×2.
• Precession of the electron
THEORY OF DYNAMIC NUCLEAR POLARIZATION AND . . .
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
number of periods
Se
from the
Zeeman
period. W
polarize
is thus
through
leading
frequenc
It sho
Fig. 3 w
A/ωe
close to
Se,x
e,yS
Se,z
5

17. Adding a single nuclei (spin-3/2)
• Evolution of the electron and a single nuclear spin.
6

18. Adding a single nuclei (spin-3/2)
• Evolution of the electron and a single nuclear spin.
• Nuclear Spin Vector representation: Sn, = Tr(ˆλ ρ4×4)
• Markovian approximation S ≈ S
(∞)
e ⊗ Sn
• Double perturbative expansion in hyperfine and quadrupolar interactions:
Sn = S(0,0)
n + S(1,0)
n + S(0,1)
n + ...
6

19. Theoretical Framework
7

20. Flip rates
• Notation:{+3/2, +1/2, −1/2, −3/2} ←→ {++, +, −, −−}
• wj
i : i → j
•
dp(m)
dt
=
n=m
wm
n p(n) −
n=m
wn
mp(m).
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0
0.5
1.0
1.5
2.0
2.5
Electron Zeeman Frequency ωe (GHz)
Flip-rates(MHz)
w+
++
w++
+
w-
+
w+
-
w--
-
w-
--
8

21. Theoretical Framework
9

22. Nuclear spin polarization distribution for I > 1/2
• Magnetization of a spin-1 bath (toy model):
N = N+1 + N0 + N−1
m = N
z-component
m = 0
+1
0
1
+1
0
1
10

23. Nuclear spin polarization distribution for I > 1/2
• Magnetization of a spin-1 bath (toy model):
N = N+1 + N0 + N−1
m = N
• Configuration (m)(N+1, N0, N−1)
z-component
m = 0
+1
0
1
+1
0
1
10

24. Nuclear spin polarization distribution for I > 1/2
• Magnetization of a spin-1 bath (toy model):
N = N+1 + N0 + N−1
m = N
• Configuration (m)(N+1, N0, N−1)
• Configurations (0)(1, 0, 1), (0)(0, 2, 0):
same magnetization m = 0.
• Degeneracy of configurations D(m, N)
z-component
m = 0
+1
0
1
+1
0
1
10

25. Nuclear spin polarization distribution for I > 1/2
• Magnetization of a spin-1 bath (toy model):
N = N+1 + N0 + N−1
m = N
• Configuration (m)(N+1, N0, N−1)
• Configurations (0)(1, 0, 1), (0)(0, 2, 0):
same magnetization m = 0.
• Degeneracy of configurations D(m, N)
• Another parameter beyond magnetization to
distinguish the configurations:
P(m) → P(m, N0)
• P(m) → N0
P(m, N0)
z-component
m = 0
+1
0
1
+1
0
1
10

26. Kinetic equation for spin-1
d
dt
P(m, N0) = F(m, N0) + G(m + 1, N0 − 1) − G(m, N0) − F(m + 1, N0 + 1)
F(m, N0) = w0
−1(m − 1)P(m − 1, N0 − 1)N−(m − 1, N0 − 1) − w−1
0 P(m, N0)N0
G(m, N0) = w1
0 (m − 1)P(m − 1, N0 + 1)(N0 + 1) − w0
1 P(m, N0)N+(m, N0).
11

27. Kinetic equation for spin-1
d
dt
P(m, N0) = F(m, N0) + G(m + 1, N0 − 1) − G(m, N0) − F(m + 1, N0 + 1)
F(m, N0) = w0
−1(m − 1)P(m − 1, N0 − 1)N−(m − 1, N0 − 1) − w−1
0 P(m, N0)N0
G(m, N0) = w1
0 (m − 1)P(m − 1, N0 + 1)(N0 + 1) − w0
1 P(m, N0)N+(m, N0).
Lattice interpretation
F(m, N0), G(m, N0): net rate of
population transfer among nodes
(configurations)
(m, N0)
(m + 1, N0 1)
F
(m 1, N0 + 1)
F
(m 1, N0 1)
G
(m + 1, N0 1)
G
11

28. Lattice interpretation and Polarization
Lattice interpration for N = 3 nuclear spin with I = 1:
-0.000557175
-0.00161241 -0.00161241
-0.00647193
P(1, 2)
P(0, 3)
P(-1, 2)
0
-c(1)
c(1)
-c(1)
c(1)-c(2)
c(2)
-c(3)
c(3)-c(2)
c(2)
-c(3)
c(3)
0
P(2, 1)P(3, 0)
P(1, 0)
P(1, 2)
P(0, 1)
P(0, 3)
P(-1, 0)
P(-1, 2)
P(-2, 1) P(-3, 0)
3
2
1
2
12

29. Lattice interpretation and Polarization
Lattice interpration for N = 3 nuclear spin with I = 1:
-0.000557175
-0.00161241 -0.00161241
-0.00647193
P(1, 2)
P(0, 3)
P(-1, 2)
0
-c(1)
c(1)
-c(1)
c(1)-c(2)
c(2)
-c(3)
c(3)-c(2)
c(2)
-c(3)
c(3)
0
P(2, 1)P(3, 0)
P(1, 0)
P(1, 2)
P(0, 1)
P(0, 3)
P(-1, 0)
P(-1, 2)
P(-2, 1) P(-3, 0)
3
2
1
2
Generalization to higher spins:
multi-dimensional lattices
-300 -200 -100 0 100 200 300
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Magnetization m
P(m) N=100
12

30. Conclusion
• Vezvaee, Sharma, Economou,
and Barnes, in preparation
arxiv:2003.xxx
• A fully quantum theoretical
framework for generation of
DNP in setups with periodic
pulse sequences such as the
mode-locking
• Polarization formulation in
terms of large sparse linear
systems
• Straightforward generalization
to any nuclear spin
-300 -200 -100 0 100 200 300
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Magnetization m
P(m)
N=100
z-component
m = 0
+1
0
1
+1
0
1
13

31. 14