This document discusses metrology using single electrons in metallic and superconducting islands. It covers theories of electron transport in these systems and their environments. Applications include using these systems for metrology and achieving error rates low enough for metrological standards. Solid state entanglers are also discussed as a way to generate entanglement for metrology applications.

1. METROLOGY WITH SINGLE ELECTRONS
VLADIMIR BUBANJA
1

2. Single Electronics
2

3. Single Electronics
M.C. Esher
3

4. Single Electronics
M.C. Esher
4

5. Outline
• Metallic islands
• Superconducting islands
• Solid state entanglers
Summary
5

6. Outline
• Metallic islands
• Superconducting islands
• Solid state entanglers
Summary
6

7. 7

8. 8

9. 9

10. Hamiltonian of the system:
H H 0 + HT ,
=
=H0 ∑
i=S ,I ,D
H i + H env
∑ (l 
H i = + eVi )cl†,i cl ,i ,(i =
l
S , D)
=HI ∑α
α cα cα + Q 2 / 2CΣ
†
H env = ∑ ωα bα bα
†
α
H T = H T 2 , H Ti = H i− ,
HT 1 + H i+ +
=H1+ ∑
= ( H i+ )†
Tα p cα (t )c p (t )e − iϕ1 ( t ) , H i−
α,p
†
10

11. Inelastic cotunneling
γ ∝ ∫ d ε1 ∫ d ε 2 Γ1 (ε1 + eV1 ) Re[ D(ε1 , ε 2 )] Γ 2 (ε 2 + eV2 )
Γi (eV ) ∝ ∫ d ε1 ∫ d ε 2 f (ε1 )[1 − f (ε 2 )] P(ε1 − ε 2 + eV )
P( E ) ∝ ∫ dt exp( J (t ) + iEt / )
d ω Re[ Z (ω )] β ω
J (t ) ∫ ω RK [coth 2 (cos(ωt ) − 1) − isin(ωt )]
11

12. Odintsov, Bubanja and Schön, Phys. Rev. B, 46, 6875
12

13. Odintsov, Bubanja and Schön, Phys. Rev. B, 46, 6875
Zorin et al., J. Appl. Phys. 88, 2665.
13

14. EU Project COUNT: R-pump
4j-2p
Tr-3p T+ T- G2 Drain G3
G1 Sou G2 G4 T+ G T- G3
T+ G1
G3 G1
T- 5j+Tr
(Pad No.1)
4j+Tr Sou
G
G Pad No.2) G2 Source
T+
5j 4j
4j
G1 3j 4j
T+
SL_KPN3
T+
Source
G1
3j+Tr 4j+Tr G
G
(Pad No.4) (Pad No.3))
Sou T-
G2 T-
G2 T- G T+ G2 Sou G1 G3
3j-2p Drain G2 Source G1 G3 4j+Tr
(Pad No.5)
Lotkhov et al, Appl. Phys. Lett. 78, 946 (2001)
14

15. 005
.7
00
.5
005
.2
-.7 -.5 -.2
005 00 005 005
.2 00
.5 005
.7
-.2
005
-.5
00
-.7
005
15

16. Quantum Metrological Triangle
h
V =n f f
2e I = ef
R-pump
Josephson Effect SET LNE, France
V Quantum Hall
I
Effect
1 h
= = 1,2,...)
V 2
I (n
ne
16

17. 005
.7
00
.5
005
.2
-.7 -.5 -.2
005 00 005 005
.2 00
.5 005
.7
-.2
005
-.5
00
-.7
005
17

18. Elastic cotunneling
18

19. 19

20. 20

21. 21

22. 22

23. 23

24. Current through the system:
I = 〈 S −1 (t , −∞) I (t ) S (t , −∞)〉
ˆ
1+ 2 z
1  eV 
I=  
(2π ) 2 Γ(2 + 2 z )e 4ν Ω 0  Ω 0  ∫ d d  F ( ) F ( )
α β α β
×∫ d x1dx 2 g1 (x1 ) g 2 (x 2 ) ∫ dte
i ( α −β ) t / 
P(x1 ,0; x 2 ,| t |)
 2C1C2 z E1 +  
F ( ) = 1, 2 +
[1 − f ( )]U , 
 (C1 + C2 ) Ω 0 
2
 2C1C2 z E2 −  
− f ( )U 1, 2 + , 
 (C1 + C2 ) Ω 0 
2
Bubanja, Phys. Rev. B 78, 155423 (2008)
24

25. Outline
• Metallic islands
• Superconducting islands
• Solid state entanglers
Summary
25

26. Hybrid SET transistor
N S N
A
CG
VL VR
VG
26

27. Pekola et al, Nature Physics 4, 120 (2008)
27

28. 28

29. Motivation:
Averin and Pekola, Phys. Rev. Lett. 101, 066801 (2008).
Achievable error rates: 10-6 – 10-7.
Therefore NISIN transistor is not suitable for metrology.
29

30. 30

31. Z(ω)
31

32. Nucleon pairing
32

33. There is a quasiparticle on the island when gate voltage is adjusted so that:
33

34. Ec
Δ
34

35. 35

36. 36

37. 37

38. In the resolvent formalism current can be expressed as:
38

39. Conclusion: promising for metrology, 10-8 can be achieved!
Bubanja, Phys. Rev. B 83, 195312 (2011)
39

40. Outline
• Metallic islands
• Superconducting islands
• Solid state entanglers
Summary
40

41. NS interface
• Andreev reflection
• Crossed Andreev reflection
41

42. Nonlinear optics
Regular mirror Phase conjugating mirror
42

43. 43

44. J. Feinberg, Opt. Lett. 7, 486 (1982)
44

45. e
N I N e S
e
h
45

46. Bogoliubov-de Gennes approach
1  e
= ∑ ∫ d r Ψσ (r )[
H BCS 3
( ∇ − A(r )) 2 + U (r ) − µ ]Ψσ (r )
†
σ 2m i c
+ ∫ d 3 r [∆(r ) Ψ † (r )Ψ † (r ) + ∆* (r )Ψ ↓ (r )Ψ ↑ (r )]
↑ ↓
∆(r ) = (r ) Ψ ↓ (r )Ψ ↑ (r )
−g
{Ψσ (r ), Ψσ ' (r ')}+ δσ ,σ 'δ (r − r ')
†
=
46

47. ∆( x) = ∆ Θ( x); U ( x) = U 0δ ( x); Z = mU 0 /  2 k F
A Z=0 A Z=1
B
B
E/∆ E/∆
A: Probability of Andreev reflection
B: Probability of ordinary reflection
Blonder, Tinkham, and Klapwijk: Phys. Rev. B 25, 4515 (1982)
47

48. Cross-correlations measurement
Solid-state entangler
Wei & Chandrasekhar, Nature Physics 6, 494 (2010)
48

49. Experimental and theoretical results of voltage noise power at 0.4K, 0.3K, and 0.25K
Wei & Chandrasekhar, Nature Physics 6, 494 (2010)
49

50. Circuit influence on entanglement current
50

51. 51

52. 52

53. Bubanja and Iwabuchi, Phys. Rev. B 84, 094501 (2011)
53

54. Outline
• Metallic islands
• Superconducting islands
• Solid state entanglers
Summary
54

55. Summary
• We have developed theories of electron transport in:
semiconducting QD’s, metallic, superconducting islands,
and carbon nanotubes taking into account charging as well
as the effects of the electromagnetic environment.
• Applications include most accurate SET devices and their
use in metrology, computing and sensing.
55