1. HRI Winter School 2015, Allahabad Interferometry in the Quantum Hall Effect Regime Lecture 2: Quantum Electronics Tools
exercises
Go over equations
Naïve calc for HBT
Weizmann Institute of Science, Israel
Department of Cond. Mat. Physics
Itamar Sivan

2. Plan for the day
What is quantum electronics? Formulation of quantum Kirchhoff rules Applying to real systems (MZI, HBT)

3. Quantum electronics
What is quantum electronics?

4. Resistivity 𝑇 𝑇 5 Temperature dependence, depends on temperature
𝜌(𝑇)
Temperature
Constant

5. Quantum electronics
Ideal conductor – no scattering

6. Quantum electronics Ψ 𝑖 (𝑥)= 𝑑𝜀 1 2𝜋ℏ𝑣 𝜀 𝑎 𝑖 𝜀 𝑒 𝑖𝑘 𝜀 𝑥
Ψ 𝑖 (𝑥)= 𝑘 𝑎 𝑖,𝑘 1 𝐿 𝑒 𝑖𝑘𝑥
Ψ 𝑖 (𝑥)= 𝑑𝜀 1 2𝜋ℏ𝑣 𝜀 𝑎 𝑖 𝜀 𝑒 𝑖𝑘 𝜀 𝑥
𝐼 𝑖 (𝑥)= 𝑒ℏ 2𝑚 Ψ 𝑖 † 𝑥 1 𝑖 𝜕 Ψ 𝑖 𝑥 𝜕𝑥 + 1 −𝑖 𝜕 Ψ 𝑖 † 𝑥 𝜕𝑥 Ψ 𝑖 𝑥
𝑎 𝑖 † 𝜀′ 𝑎 𝑗 𝜀 = 𝑓 𝑖 𝜀 ∙ 𝛿 𝑖𝑗 𝛿 𝜀− 𝜀 ′
𝑓 𝑖 𝜀 = 1 1+ 𝑒 (𝜀− 𝜇 𝑖 )/𝑘𝑇
−∞ +∞ 𝑑𝜀∙ 𝑓 𝑖 𝜀 − 𝑓 𝑗 𝜀 = 𝜇 𝑖 - 𝜇 𝑗 ≡𝑒𝑉

7. Quantum electronics Ideal conductor with a single scatterer
𝑆= 𝑟 𝑡 𝑡′ 𝑟′
S is Unitary:
𝑆 † 𝑆=𝑆 𝑆 † =𝕀
𝑟 2 = 𝑟′ 2 = 1− 𝑡 2 = 1− 𝑡′ 2
𝑟 2 ≡𝑅 𝑟 ′ ′ 2 ≡𝑅′
𝑡 2 ≡𝑇 𝑡′ 2 ≡𝑇′

8. Quantum electronics 𝐺 0 = 𝑒 2 ℎ 𝐼 𝑖 = 𝑒 2 ℎ 𝑉 𝐼 𝑖 =𝑇 𝑒 2 ℎ 𝑉 𝐺=𝑇∙ 𝐺 0
Ideal 1D conductor:
Ideal 1D conductor with scatterer:
𝐺 0 = 𝑒 2 ℎ
𝐼 𝑖 = 𝑒 2 ℎ 𝑉
𝐼 𝑖 =𝑇 𝑒 2 ℎ 𝑉
𝐺=𝑇∙ 𝐺 0
Can we understand it ‘classically’?
𝑅= ℎ 𝑒 2 ∙ 1 𝑇 = ℎ 𝑒 2 + ℎ 𝑒 2 1−𝑇 𝑇

9. Quantum electronics
Formulation of quantum Kirchhoff rules

10. Quantum electronics Multi-channel scattering
𝑆= 𝑟 11 𝑡 12 𝑡 21 𝑟 ⋯ 𝑡 1𝑁 ⋮ 𝑡 𝑁1 ⋱ 𝑟 𝑁𝑁
S is Unitary:
𝑆 † 𝑆=𝑆 𝑆 † =𝕀
𝑟 𝑖𝑖 2 ≡ 𝑅 𝑖𝑖
𝑡 𝑖𝑗 2 ≡ 𝑇 𝑖𝑗
𝑟 𝑖𝑖 2 =1− 𝑗 𝑡 𝑖𝑗 2

11. Quantum electronics The quantum of conductance 𝐼 = 𝑒 2 ℎ 𝑉
Landauer’s formula: 𝐼 = 𝑒 2 ℎ 𝑇∙𝑉
Landauer-Buttiker (Quantum Kirchhoff): 𝐼 𝛼 = 𝛽 𝐺 𝛼𝛽 𝑉 𝛽 𝐺 𝛼𝛽 = 𝑒 2 ℎ 1− 𝑅 𝛼𝛼 𝑒 2 ℎ 𝑇 𝛼𝛽 , 𝛼≠𝛽

12. Quantum electronics
Applying QKR to real systems (MZI, HBT)

13. Quantum electronics
Quantum Point contact

14. A QPC device

15. Quantum electronics
Mach-Zehnder interferometer

16. The Mach-Zehnder interferometer
Interference is observed as function of the path difference 1 2 3
𝐴 𝐷1,𝐿 =𝑡𝑡 𝑒 𝑖𝑘 𝑥 𝑢
+ 𝑟𝑟 𝑒 𝑖𝑘 𝑥 𝑑
𝐴 𝐷1,𝐿 2 = 𝑇 2 + 𝑅 2 +2𝑅𝑇 cos 𝑥 𝑑 − 𝑥 𝑢

17. The Mach-Zehnder interferometer
X 𝐵 = 𝜙

18. The Mach-Zehnder interferometerS
BS1 M1 M2
BS2 D2
J. Yang et al., nature 2003

19. Quantum electronics
Hanbury-Brown & Twiss interferometer

20. The Mach-Zehnder interferometer

21. The Mach-Zehnder interferometer
X 𝐵 = 𝜙

22. The Mach-Zehnder interferometer
𝐴 𝐷1,𝑆 =𝑡𝑡
+ 𝑟𝑟 𝑒 𝑖2𝜋𝐴𝐵
𝐴 𝐷1,𝑆 2 = 𝑇 2 + 𝑅 2 +2𝑅𝑇 cos 2𝜋𝐴𝐵

23. The Hanbury-Brown & Twiss interferometer

24. actual sample

25. actual sample (no bridges)
path length ~8µm

26. The Hanbury-Brown & Twiss interferometer
upper Mach-Zehnder
lower Mach-Zehnder
( 0.75 /hour, 1.00 /mV )
( 0.69 /hour, 0.73 /mV )
Neder et al., Nature 2007

27. The Hanbury-Brown & Twiss interferometer
( 1.41 /mV, 1.72 /hour )
Neder et al., Nature 2007

28. The Hanbury-Brown & Twiss interferometer
2.0
1.5
fringes / mV
1.0
0.5
0.5
1.0
1.5
fringes per hour
Neder et al., Nature 2007

29. Zoom out Defined quantum electronics
Formulated the quantum Kirchhoff rules
Applyed to real systems (MZI, HBT)

30. Course plan Overview of the field & Motivation Formalism for
Mesoscopic problems
I -> G -> see oscillating -> diphase it -> formulate in classical terms -> how can we retrieve a purely classical result?
Controlled dephasing
& Quantum Erasers
Interaction effects