Spin-orbit effects on the full dynamics of double quantum dot qubit states

E. CotaCentro de Nanociencias y Nanotecnología,

Universidad Nacional Autónoma de México , Ensenada, MEXICO

S. E. Ulloa, J.E. RolónDepartment of Physics and Astronomy, Ohio University, USA

G. PlateroInstituto de Ciencia de Materiales de Madrid – CSIC, Madrid, SPAIN

APS March Meeting , Boston, MA, 29 February 2012

Combining rotations around two axes

0X S T

0Y S i T

Z S

Petta et al (2005)

, , , ,( ) 2, ( ) 2ext nuc l nuc r nuc l nuc rB B B B dB B B

J.M. Taylor et al. PRB 76, 035315 (2007)

Double dot:

Model

Electron spin Hamiltonian for a single QD: ˆˆ

eff e ext nucH B B S

, , , ( )l rhf tot hf eff hf effH H H J S S

,ˆ ˆ ˆ ˆ( ) ( ) ( )l r l r

hf tot eH B S S dB S S J S S

T

T

0T

S

02S

02S

0 ( ) 2 ( 0)

( 1)

( 1)

s

s

s

T m

T m

T m

( ) 2S

Singlet : ( 0, 0)sS m ( 1)S Triplet:

Energy levels for finite external magnetic field(in )eV 5 .25 5 0z z c soB dB t t

( )eV

E

ZEZE

T

S

T

Tunneling Hamiltonian:

† †' '

,

ˆ ˆ ˆ ˆ ˆL RT L R R LH t a a t a a

spin indices

Defining

1 2 1 2

, 0,2

x y z

iT T T T T

one can write, in basis

02 02ˆ . .

, , ; , ,

T c

x y z x y z

H it T S t S S H c

T T T T t t t t

Spin-orbit interaction introduces non-spin conserving tunnelingmatrix elements (Danon & Nazarov, PRB 80, 0413019 (R) (2009))

0 02, , , ,S T T T S

0 2 2

2 0 0 2

0 0 0

2 0 0 2

2 2

z c

z x y

z z

z x y

c x y z x y

dB dB dB t

dB B it t

dB it

dB B it t

t it t it it t

Total Hamiltonian in basis

HFI couples and triplet statesSSOI couples and triplet states02S

0 02, , , ,S T T T S

x y z sot t t t Let

ˆ ˆ ˆ,iH

Let be the relevant subspace onto which we want to constrain the dynamics of the system

: projection operator onto ; 1Q P P

P

P

0

0

( )( )

( )

PPG z P

z PH P PR z P

QR z V V V

z QH Q QVQ

projected Green’s functiononto P

level-shift operator

Bloch-Feshbach Projection MethodConstruct effective operators acting within relevant subspace

0( ) ( )effH z PH P PR z P

The poles of give the eigenvalues of ( )PG z P effH

Summary

We calculate numerically the probability of measurement of in double quantum dot system, including hyperfine and spin-orbit interactions, using the complete basis.

When probabilities show oscillations, depending on the value of , corresponding to new frequencies due to coupling mainly to

Frequency of oscillations of strongly dependent on values of and detuning

Feshbach projection method allows determination of (through identification of the frequency of )

( )P S

0,sot sot

T

B ( )P S

sotT

P(Z)

0.40.60.81.0

tso

=0 t

so=.5

S02

S02

0.0

0.1

0.2 T

+

T-

T+, T

-

0.0

0.2

0.4

0.6T

0

T0

0 10 20 300.0

0.1

0.2S

S

rot

(ns)0 20 40 60 80

-400

-200

0

time(ns)

0.0

0.2

0.4

0.6

0.8

1.0

S02T

0

S

Bz=5 dBz=.25 t

c=5 t

so=0

ss

rot

20rot