1. Physics Department, Beijing Normal University
Improved master equation to quantum transport: From Born to self-consistent Born approximation
Xin-Qi Li （李新奇）
Physics Department, Beijing Normal University
In collaboration with
Dr J.S. Jin (Hangzhou Normal University)
Prof. Y.J. Yan (HKUST)

2. Outline: ◆ Master equation approach to quantum transport
- n-dependent ME
- quantum noise, FCS & LD analysis
◆ ME under self-consistent Born approximation
- basic idea/observation and formulation
- recover the exact result of noninteracting transport
- restore the nonequilibrium Kondo effect
- n-dependent SCBA-ME and applications
◆ Summary

3. Part (I):
Number-Resolved M.E. Approach
to Quantum Transports

4. Landauer-Buttiker (Scattering) Theory
Approaches to Mesoscopic Transports
Landauer-Buttiker (Scattering) Theory
Nonequilibrium Green’s Function Approach

5. Remarks: simplicity; classical feature; quantum coherence;
Rate Equation Approach
Remarks: simplicity; classical feature; quantum coherence;
zero temperature and large bias voltage
RE/ME approach is becoming popular … e.g., Jauho’s group
and others …

6. S Master Equation Approach to Quantum Transports Quantum Dissipation
Environment
Quantum Dissipation
Quantum Transport
X.Q. Li et al:
PRL 94, (2005)
PRB 71, (2005)
PRB 76, (2007)

7. Current
Noise spectrum: MacDonald’s formula

8. Full counting statistics
Levitov, Lee, & Lesovik:
J. Math. Phys. 37, 4845(1996)
Generating function:
Current
Zero-frequency noise
Fano factor

10. Large Deviation (LD) Analysis
( FCS )
( LD )
LD References:
Science 323, 1309 (2009)
PRL 104, (2010)

11. Noise and Counting Statistics in Mesoscopic Transports
Quantum noise (contains additional information …)
FCS: a fascinating theoretical tool for quantum transport
arXiv:

12. K. Ensslin & A.C. Gossard etal, PRL 96, 076605 (2006)
Counting Statistics of Single Electron Transport in a Quantum Dot

13. Science 310,
1634 (2006)

14. QPC/SET测量固态（电荷）量子比特 Gurvitz: PRB 56, 15215 (1997) Non-trivial points:
- signal-to-noise ratio
quantum efficiency of meas.
quantum trajectory under meas.
Schoen: PRB (98); RMP(00); PRL(02)

15. Application to qubit measurements:
Xin-Qi Li et al : Quantum measurement of a solid-state qubit: A unified quantum master equation approach, Phys. Rev. B 69, (2004).
Xin-Qi Li et al : Spontaneous Relaxation of a Charge Qubit under Electrical Measurement, Phys. Rev. Lett. 94, (2005).
X.N. Hu et al: Quantum measurement of an electron in disordered potential,
Phys. Rev. B 73, (2006).
J.S. Jin et al: Quantum coherence control of solid-state charge qubit by means of a sub-optimal feedback algorithm, PRB 73, (2006).
S.K. Wang et al: Continuous weak measurement and feedback control of a solid-state charge qubit: physical unravelling of non-Lindblad master equation, Phys. Rev. B 75, (2007).
H.J. Jiao et al: Quantum measurement characteristics of double-dot single electron transistor, Phys. Rev. B 75, (2007).
H. J. Jiao et al: Weak Measurement of Qubit Oscillations with Strong Response Detectors: Violation of the Fundamental Bound Imposed on Linear Detectors,
Phys. Rev. B 79, (2009).

16. Application to Mesoscopic Transports:
1） Single and double QDs: Coulomb staircase, noise spectrum
X.Q. Li et al, PRB 71, (2005)
J. Y. Luo et al, PRB 76, (2007)
2） QD coupled to FM electrodes: spin-dependent current & fluctuations
J. Y. Luo et al, J. Phys.:Condens.Matter 20, (2008)
3）Transport through parallel quantum dots: counting statistics,
magnetic field switching of current, giant fluctuations of current,
harmonic decomposition of the interference pattern
S.K. Wang et al, PRB 76, (2007)
F. Li, X.Q.Li, W.M. Zhang, and S.A. Gurvitz:
Europhys. Lett. 88, 37001(2009)
F. Li et al, Physica E 41, 521 (2009)
F. Li et al, Physica E 41, 1707(2009)
J. Li et al, Large-Deviation Analysis for Counting Statistics
in Mesoscopic Transports, Phys. Rev. B 84, (2011)

17. Part (II) Quantum Transport Master Equation under
Self-Consistent Born Approximation

19. No level-broadening effect !
More general
multilevel systems,

20. In this simple example, the “broadening factor” can be generated
by modifying the system free propagator with an effective one,
including the tunneling self-energy

21. Self-Consistent Born Approximation

22. G G Dyson Equation = + = + Self-Consistent Born Approximation

23. Born Approx
Self-Consistent
Born Approx

24. Violation of the Quantum Regression Theorem !
Born Approx
Self-Consistent
Born Approx
Violation of the Quantum Regression Theorem !

25. Steady State

26. Non-interacting system

27. Example F. Li, X.Q. Li, W.M. Zhang and
S. Gurvitz, Europhys. Lett. 88, 37001(2009).
S.A. Gurvitz, Phys. Rev. B 44, (1991).
Example

28. Achievements of the SCBA M.E.
Noninteracting case:
1) exact
2) valid for arbitrary voltages
Interacting case:
1) recover the nonequilibrium Kondo effect
2) contain the cotunneling process

29. Example: transport through Coulomb island
State basis

30. For noninteracting system
The above result is
exactly the same
as in the book:

31. Nonequilibrium Kondo Effect
D. C. Ralph and R. A. Buhrman,
Phys. Rev. Lett. 72, 3401 (1994).

32. Coulomb Staircase Simplified solution under Hartree-Fock
approximation: eV

33. Comparison with some other approaches
( 1 ) Bloch rate equation & single-electron wavefunction approach
( 2 ) Exact master equation (available only for noninteracting case)

34. ( 3 ) 2nd-order VN equation approach
It cannot predict the challenging Kondo effect
( 4 ) The well-established nGF approach

35. Particle-number-resolved SCBA-ME

36. Example (I): noninteracting single-level quantum dot
Parameters:

37. Example (II): Anderson-Kondo quantum dot
( temperature effect)
( bandwidth effect)

38. Summary ◆ Master equation approach to quantum transport
- n-dependent ME
- quantum noise, FCS & LD analysis
◆ ME under self-consistent Born approximation
- basic idea/observation and formulation
- recover the exact result of noninteracting transport
- restore the nonequilibrium Kondo effect
- noise spectrum, FCS (under working)