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)

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

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

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

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 …

S Master Equation Approach to Quantum Transports Quantum Dissipation Environment Quantum Dissipation Quantum Transport X.Q. Li et al: PRL 94, 066803 (2005) PRB 71, 205304 (2005) PRB 76, 085325 (2007)

Current Noise spectrum: MacDonald’s formula

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

Large Deviation (LD) Analysis ( FCS ) ( LD ) LD References: Science 323, 1309 (2009) PRL 104, 160601 (2010)

Noise and Counting Statistics in Mesoscopic Transports Quantum noise (contains additional information …) FCS: a fascinating theoretical tool for quantum transport arXiv:0705.4083

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

Science 310, 1634 (2006)

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)

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, 085315 (2004). Xin-Qi Li et al : Spontaneous Relaxation of a Charge Qubit under Electrical Measurement, Phys. Rev. Lett. 94, 066803 (2005). X.N. Hu et al: Quantum measurement of an electron in disordered potential, Phys. Rev. B 73, 035320 (2006). J.S. Jin et al: Quantum coherence control of solid-state charge qubit by means of a sub-optimal feedback algorithm, PRB 73, 233302 (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, 155304 (2007). H.J. Jiao et al: Quantum measurement characteristics of double-dot single electron transistor, Phys. Rev. B 75, 155333 (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, 075320 (2009).

Application to Mesoscopic Transports: 1） Single and double QDs: Coulomb staircase, noise spectrum X.Q. Li et al, PRB 71, 205304 (2005) J. Y. Luo et al, PRB 76, 085325 (2007) 2） QD coupled to FM electrodes: spin-dependent current & fluctuations J. Y. Luo et al, J. Phys.:Condens.Matter 20, 345215 (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, 125416 (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, 115319 (2011)

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

No level-broadening effect ! More general multilevel systems,

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

Self-Consistent Born Approximation

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

Born Approx Self-Consistent Born Approx

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

Steady State

Non-interacting system

Example F. Li, X.Q. Li, W.M. Zhang and S. Gurvitz, Europhys. Lett. 88, 37001(2009). S.A. Gurvitz, Phys. Rev. B 44, 11924 (1991). Example

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

Example: transport through Coulomb island State basis

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

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

Coulomb Staircase Simplified solution under Hartree-Fock approximation: eV

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

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

Particle-number-resolved SCBA-ME

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

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

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)