1. Hierarchical Quantum Master Equation Theory and Efficient Propagators YiJing Yan Hong Kong University of Science and Technology University of Science and Technology of China Physics and Chemistry in Quantum Dissipative Systems 9-11 August 2010, Kyoto Supported by (HK$) RGC of HKSAR & (RMB ￥ ) NNSF of China

2. Key quantity: Reduced density operator   (t) = Tr env  total (t) bath system (t)(t) TT I(t)I(t) Decoherence quantum transport

3. Outline Introduction: Nonperturbative QDT HEOM-QDT: Exact formalism Numerical methodology development in conjunction with the Kubo’s stochastic Liouville equation (also classified as the Zusman equation) and the development of the modified Zusman equation and so on Summary

4. Introduction: Exact QDT Influence functional path integral theory Feynman & Vernon 1963 (second-quantization with Grassman variables) Exact formalism for Gaussian system-bath coupling Hierarchical equations of motion (HEOM) Y. Tanimura & R. Kubo (89); Y. Tanimura (90); … J.S. Shao, (04) R.X. Xu, P.Cui, X.Q. Li, Y. Mo, & YJY (05); R.X. Xu & YJY (07) (JS. Jin, X. Zheng & YJY, 2008) Mathematically equivalent but HEOM formalism is more suitable for numerical performance / applications

5. stochastic bath operators Formally exact and numerically tractable for with arbitrary system H(t) arbitrary system dissipative mode Q j bath can be either bosonic or fermionic, canonical or grand canonical, at arbitrary temperature arbitrary time-dependent voltage or chemical potential Required only Gaussian JCP 122:041103 (2005); PRE 75:031107 (2007) JCP 128:234703 (2008); 129:184112 (2008); 130:164708 (2009) HEOM-QDT

6. Quantum Transport Setup (t)(t) TT  = L or R bath: fermion & grand canonical system: fermion

7. HEOM: Generic Form n = ( n 0,n 1,…,n K ), ( n j = 0 or 1 for fermion ) New J. Phys. 10 (08) 093016; JCP (08) 128:234703; 129 :184112; (09) 130:124508; 164708  {  } of either commutator or anticommutator due to the Grassman parity as the fermion system coupling mode  (t)  j (t), …,  k (t)  jj’ (t), …,  jk (t), …,  kk’ (t) tier 0 1 2 3 … relates to current exact if no e-e interaction

8. Dynamics Coulomb Blockade electrode Two-lead case: adiabatic charging eq

9. Response to ramp-up voltage T = 0.2 meV Temperature effects   =0.5,   = 2.5, U = 4

10. Kondo Problem Anderson model T  T Kondo L R U Dynamic Kondo transport response to a  (t)-pulse voltage

11. For harmonic systems, n max = 2, in the RWA Hierarchical construction for the combined effects of anharmonicity, s-b coupling strengthe, and memory Exact and nonperturbative, always converged Extremely Expensive HEOM for Dissipative Dynamics # of auxiliary {  n }: “Full CI” e.g. with (n k  0 for boson bath)

12. Numerical Methodology Development in conjunction with the stochasic Liouville equation (SLE) the modified SLE and so on

13.  - Controlled Filtering Algorithm C(t) of (K+1) = 20 terms;L: anchor level of HEOM versus > 10 12 Q. Shi, L.P. Chen, G.J. Nan, R.X.Xu, YJY (09) JCP 130:084105; 164518

14. Q. Shi, L. Chen, G. Nan, R.X. Xu, YJY, (09) JCP 130:164518 HEOM-based theory versus Stochastic Liouville equation (Zusman equation) Construction of  -Controlled Filtering Scheme

15. SLE via HEOM semiclassical Diffusive  R. Kubo (62), (69), Tanimura (06) Zusman (80), (83) Shi, Chen, Nan, Xu & Yan (09) Scaling and filtering algorithm

16. SLE vs Modified SLE

17. SLE vs Modified SLE (2) ZE is subject to severe positivity violation especially for  V > 1 Modified ZE has the positivity range at least of  < 25,  < 100, |  E o |< 25,  V < 100 (as tested) (09) JCP 131:214111

18. Numerical Methodology K= N + N’ N: # of poles in Bose/Fermi function decomposition N’: # of poles in spectral density “Basis-set” Issue

19. Sum-over-poles Schemes of Fermi / Bose function MSD (Matsubara spectrum decomposition): e.g. Bose function PFD (Partial fraction decomposition): Croy & Saalmann, PRB 80, 073102 (2009) J. Xu, RX Xu, YJY, Chem. Phys. 370, 109 (2010)

20. Location of PFD Poles

21. Superiority of PFD over MSD But the most recent one (PSD) based on Padé approximant is the best of all

22. Superiority of Padé Spectrum Decomposition over PFD

23. Normal form of PSD deviation Same for Fermi and Bose function deviation N-independent as N > 8 or x > x 1/e

24. Characteristic Validity Length

25. PSD-Poles All Pure Imaginary J. Hu, R. X. Xu & YJY, JCP, submitted T. Ozaki (07), PRB 75:035123 eigenvalues of matrix:

26. HEOM with White-Noise Residue Correction A. Ishizaki & Y. Tanimura (05) J. Phys. Soc. Jpn Y. Tanimura (06) J. Phys. Soc. Jpn R.X. Xu & YJY (07) PRE

27. HEOM with [N/N]-PSD J.J. Ding, J. Hu, B.L. Tian, R.X. Xu & YJY, (in preparation)

28. Summary Time

29. Summary HEOM-QDT as a fundamental formalism in quantum mechanics of dissipative systems Numerical methodology advancement making HEOM formalism a practical tool  Efficient filtering  [ N-1 / N ]- and [ N / N ]-PSD [ 0 / 0 ]-PSD  Modified SLE

30. Acknowledgment Prof. Rui-Xue Xu(UST of China) Prof. Qiang Shi(Inst. Chem., CAS, Beijing) Dr. Jian Xu(USTC) Dr. Jie Hu(HKUST) Dr. Jinshuang Jin(HKUST, Hang Zhou Normal Univ.) Dr. Xiao Zheng(HKUST, Duke Univ) Ms. Bao-Ling Tian(USTC) Ms. Jin-Jin Ding(USTC) Mr. Meng Luo(HKUST) Supported by (HK$) RGC of HKSAR & (RMB ￥ ) NNSF of China