weak measurement weak values : applications to mesoscopic electronic systems · 2008-01-16 · weak...
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WEAK MEASUREMENT & WEAK VALUES :APPLICATIONS TO
MESOSCOPIC ELECTRONIC SYSTEMS
special thanks toY. Aharonov (Tel Aviv)V. Spitalnik (WIS)support:
Schwerpunkt
Project SPINTRONICS (DFG) ;MINERVA
with
ALESSANDRO ROMITO (Weizmann Inst.)YAROSLAV BLANTER (Delft)
Super Conductivity & MagnetismHSINCHU TAIWAN 2007
Org: Vinokour;Galperin;Lin;Rosenstein
outline…
ultra fast introduction to WEAK VALUES
spin based q-bit (2e): protocol
non-standard values
WEAK VALUES: (WEAK VALUES: (AharonovAharonov, Bergmann, , Bergmann, LebowitzLebowitz;;AharonovAharonov, Albert, , Albert, VaidmanVaidman))
not easily accessible correlationsnot easily accessible correlations
simultaneous access to nonsimultaneous access to non--commuting variablescommuting variables
nonnon--standard expectation values (even complex !)standard expectation values (even complex !)
potential applications in metrologypotential applications in metrology
…………
weak measurement, weak value:
time
post selection(eigenvalue of )
weak measuremetof
preparation
A
B
ˆ ˆ[ , ] 0A B ≠
2
ˆ ˆ ˆ| | | | |
ˆ| || | |
|
nfin
nfin
n nin in in fin fin in
nfin inn
fin in nfin in
B B B
B
Ψ
Ψ
≡ Ψ Ψ = Ψ Ψ ⋅ Ψ Ψ =
Ψ Ψ= Ψ Ψ ⋅
Ψ Ψ
∑
∑
0
preselected state
post-selected state
weakly measur
ˆ ed observable
in
fin
B
Ψ
Ψ0
0
ˆ| |ˆ|
fin in
weakfin in
BB
Ψ Ψ=
Ψ Ψ
0
0
ˆ| |ˆ|
fin in
weakfin in
BB
Ψ Ψ=
Ψ Ψ
example 1: Spin 1/2example 1: Spin 1/2
time
post selection(eigenvalue of )
weak measuremetof
preparation
A
B
12zS = +
12xS = +
x zˆ ˆ(S +S )measuring
2
example 1: Spin 1/2example 1: Spin 1/2
x zˆ ˆ1 (S +S ) 1| [ ] |
2 221 1|2 2
x z
x z
S S
S S
= + = +=
= + = +
2 1 !2 2
>
ˆ| |ˆ|
fin in
weakfin in
BB
Ψ Ψ=
Ψ Ψ
1 1 [- , ]2 2+
BEYOND THE RANGE OF
WEAK VALUES WITH QUANTUM DOTS
J. R. Petta, C. Marcus et al (Harvard) . Science 309
(2005)
gates
dot 1
dot 2
quantum point contact
CHARGE BASED WEAK MEASUREMENT
GVGV 1 2
1 2G GV Vε = −
total #electrons =1
quantumpoint
contact
Iweak
measurementstrong
measurement
decoherencet~ 1 p sec
CHARGE BASED WEAK MEASUREMENT
outline…
ultra fast introduction to WEAK VALUES
spin based q-bit (2e): protocol
non-standard values
DOUBLE DOT--SPIN
Marcus, Yacoby …..e.g. Nature vol. 435 p. 925 (2005)
(1,1)S
(1,1)S (0,2)S
(0,2)S
ε
(1,1)T
INCLUDE SPIN DEGREES OF FREEDOM
(1,1)S
(1,1)S (0,2)S
(0,2)S
ε
2t
(1,1)T0
DOUBLE DOT--SPIN
include tunneling
(1,1) and S(0,2) mix
T(1,1) and (0, 2) do
(0, 2
no
) !
t
!
S
T
T high energy
(1,1)S
(1,1)S (0,2)S
(0,2)S
ε
2t
(1,1)T0DOUBLE DOT--SPIN
effect of nuclear field
mixing triplet & singlet
g S T
sg nuclear
nuclear t
E BB E
⎛ ⎞⎜ ⎟⎝ ⎠
x
( )gS ε
0T
↓↑ ↑↓ y
(0, 2)S=
DOUBLE DOT--SPIN
(1,1)S
(1,1)S (0,2)S
(0,2)S
ε
2t
(1,1)T0
COMPLETE CONTROLof MOTION on BLOCH SPHERE
The Real Stuff ---
A Protocol
we include the spin degrees of freedom we measure charge states (0,2) vs. (1,1) we manipulate pseudo-spin states (rotation on Bloch sphere)
ε
t
pre-selectedW.M.post
selectedstrong
measurmentAε
BεCε
ε
t
200n sec
TIME SCALES
1n sec
10n – 5 secμ 10n – 5 secμ
1-10n sec
(0, 2) (0,1) (0, 2)→ →
5-10 secμAε
BεCε
outline…
ultra fast introduction to WEAK VALUES
spin based q-bit (2e): protocol
non-standard values
ε
tθ δ
θ
δ
S > 1
S < 0
W.M. postselected
strongmeasurment
pre-selectedAε
BεCε
SUMMARYSUMMARY
time
post selection(eigenvalue of )
weak measuremetof
preparation
A
B
3 step protocol
double QD: INCLUDE SPIN DEGREES OF FREEDOM
(1,1)S
(1,1)S (0,2)S
(0,2)S
ε
2t
(1,1)T0
θ
δ
S > 1
S < 0
NON STANDARD VALUES OF S
