The twin-formulation of quantum statistical mechanics is employed to describe a new methodology for the solution of the equations of motion of the reduced density matrix in their hierarchical formulation. It is shown that the introduction of tilde operators and of their algebra in the dual space greatly simplifies the application of numerical techniques for the propagation of the density matrix. The application of tensor-train representation of a vector to solve complex quantum dynamical problems within the framework of the twin-formulation is discussed. Next, applications of the hierarchical equations of motion to a dissipative polaron model are presented showing the validity and accuracy of the new approach.
REFERENCES
1.
S.
Mukamel
, Principles of Nonlinear Optical Spectroscopy
(Oxford University Press
, USA
, 1995
).2.
A. G.
Redfield
, in Advances in Magnetic and Optical Resonance
, Advances in Magnetic Resonance Vol. 1, edited by J. S.
Waugh
(Academic Press
, 1965
), pp. 1
–32
.3.
A.
Kühl
and W.
Domcke
, J. Chem. Phys.
116
, 263
(2002
).4.
A. F.
Izmaylov
, D.
Mendive–Tapia
, M. J.
Bearpark
, M. A.
Robb
, J. C.
Tully
, and M. J.
Frisch
, J. Chem. Phys.
135
, 234106
(2011
).5.
J. S.
Endicott
, L.
Joubert-Doriol
, and A. F.
Izmaylov
, J. Chem. Phys.
141
, 034104
(2014
).6.
R.
Kubo
and Y.
Toyozawa
, Prog. Theor. Phys.
13
, 160
(1955
).7.
M. F.
Gelin
, D.
Egorova
, and W.
Domcke
, Phys. Rev. E
84
, 041139
(2011
).8.
M. F.
Gelin
, L. Z.
Sharp
, D.
Egorova
, and W.
Domcke
, J. Chem. Phys.
136
, 034507
(2012
).9.
Y.
Tanimura
and R.
Kubo
, J. Phys. Soc. Jpn.
58
, 101
(1989
).10.
Y.
Tanimura
, J. Phys. Soc. Jpn.
75
, 082001
(2006
).11.
Y.
Tanimura
, J. Chem. Phys.
142
, 144110
(2015
).12.
S. R.
White
, Phys. Rev. Lett.
102
, 190601
(2009
).13.
E.
Jeckelmann
and S. R.
White
, Phys. Rev. B
57
, 6376
(1998
).14.
Y.
Yao
, K.-W.
Sun
, Z.
Luo
, and H.
Ma
, J. Phys. Chem. Lett.
9
, 413
(2018
).15.
M. H.
Beck
, A.
Jackle
, G. A.
Worth
, and H. D.
Meyer
, Phys. Rep.
324
, 1
(2000
).16.
H.
Wang
and M.
Thoss
, J. Chem. Phys.
119
, 1289
(2003
).17.
U.
Manthe
, J. Chem. Phys.
128
, 164116
(2008
).18.
I.
Burghardt
, K.
Giri
, and G. A.
Worth
, J. Chem. Phys.
129
, 174104
(2008
).19.
R.
Borrelli
and A.
Peluso
, J. Chem. Phys.
144
, 114102
(2016
).20.
R.
Borrelli
and M. F.
Gelin
, Chem. Phys.
481
, 91
(2016
).21.
N.
Zhou
, L.
Chen
, Z.
Huang
, K.
Sun
, Y.
Tanimura
, and Y.
Zhao
, J. Phys. Chem. A
120
, 1562
(2016
).22.
N.
Zhou
, Z.
Huang
, J.
Zhu
, V.
Chernyak
, and Y.
Zhao
, J. Chem. Phys.
143
, 014113
(2015
).23.
H.
Wang
, X.
Song
, D.
Chandler
, and W. H.
Miller
, J. Chem. Phys.
110
, 4828
(1999
).24.
H.
Wang
and M.
Thoss
, J. Chem. Phys.
124
, 034114
(2006
).25.
Y.
Fujihashi
, L.
Chen
, A.
Ishizaki
, J.
Wang
, and Y.
Zhao
, J. Chem. Phys.
146
, 044101
(2017
).26.
H.
Wang
, J. Phys. Chem. A
119
, 7951
(2015
).27.
O.
Vendrell
and H.-D.
Meyer
, J. Chem. Phys.
134
, 044135
(2011
).28.
A.
Raab
, I.
Burghardt
, and H.
Meyer
, J. Chem. Phys.
111
, 8759
(1999
).29.
A.
Raab
, Chem. Phys. Lett.
319
, 674
(2000
).30.
R.
Borrelli
and M. F.
Gelin
, J. Chem. Phys.
145
, 224101
(2016
).31.
R.
Borrelli
and M. F.
Gelin
, Sci. Rep.
7
, 9127
(2017
).32.
F.
Gelin Maxim
and B.
Raffaele
, Ann. Phys.
529
, 1700200
(2017
).33.
Y.
Takahashi
and H.
Umezawa
, Int. J. Mod. Phys. B
10
, 1755
(1996
).34.
D. S.
Kosov
, J. Chem. Phys.
131
, 171102
(2009
).35.
R.
Borrelli
, Chem. Phys.
515
, 236
(2018
).36.
L.
Chen
and Y.
Zhao
, J. Chem. Phys.
147
, 214102
(2017
).37.
M.
Schmutz
, Z. Phys. B: Condens. Matter Quanta
30
, 97
(1978
).38.
U.
Fano
, Rev. Mod. Phys.
29
, 74
(1957
).39.
T.
Arimitsu
and H.
Umezawa
, Prog. Theor. Phys.
77
, 32
(1987
).40.
M.
Suzuki
, Int. J. Mod. Phys. B
05
, 1821
(1991
).41.
I.
Oseledets
, SIAM J. Sci. Comput.
33
, 2295
(2011
).42.
C.
Lubich
, I.
Oseledets
, and B.
Vandereycken
, SIAM J. Numer. Anal.
53
, 917
(2015
).43.
H.-P.
Breuer
and F.
Petruccione
, The Theory of Open Quantum Systems
(Oxford University Press
, USA
, 2002
).44.
M.
Suzuki
, J. Phys. Soc. Jpn.
54
, 4483
(1985
).45.
K.
Blum
, Density Matrix Theory and Applications
(Plenum
, New York
, 1981
).46.
A.
Pereverzev
and E. R.
Bittner
, J. Chem. Phys.
125
, 104906
(2006
).47.
F.
Shibata
and T.
Arimitsu
, J. Phys. Soc. Jpn.
49
, 891
(1980
).48.
Y.
Tanimura
and R.
Kubo
, J. Phys. Soc. Jpn.
58
, 1199
(1989
).49.
R.
Kubo
, J. Phys. Soc. Jpn.
17
, 1100
(1962
).50.
A.
Ishizaki
, T. R.
Calhoun
, G. S.
Schlau-Cohen
, and G. R.
Fleming
, Phys. Chem. Chem. Phys.
12
, 7319
(2010
).51.
F. C.
Jones
and T. J.
Birmingham
, Plasma Phys.
17
, 15
(1975
).52.
Y.
Tanimura
, Phys. Rev. A
41
, 6676
(1990
).53.
A.
Ishizaki
and G. R.
Fleming
, J. Chem. Phys.
130
, 234111
(2009
).54.
Q.
Shi
, L.
Chen
, G.
Nan
, R.-X.
Xu
, and Y.
Yan
, J. Chem. Phys.
130
, 084105
(2009
).55.
E. J.
O’Reilly
and A.
Olaya-Castro
, Nat. Commun.
5
, 3012
(2014
).56.
A.
Ishizaki
and Y.
Tanimura
, J. Phys. Soc. Jpn.
74
, 3131
(2005
).57.
C.
Lubich
, T.
Rohwedder
, R.
Schneider
, and B.
Vandereycken
, SIAM J. Matrix Anal. Appl.
34
, 470
(2013
).58.
I.
Oseledets
and E.
Tyrtyshnikov
, SIAM J. Sci. Comput.
31
, 3744
(2009
).59.
S.
Holtz
, T.
Rohwedder
, and R.
Schneider
, Numer. Math.
120
, 701
(2011
).60.
P. S.
Thomas
and T.
Carrington
, J. Phys. Chem. A
119
, 13074
(2015
).61.
Q.
Shi
, Y.
Xu
, Y.
Yan
, and M.
Xu
, J. Chem. Phys.
148
, 174102
(2018
).62.
J.
Haegeman
, C.
Lubich
, I.
Oseledets
, B.
Vandereycken
, and F.
Verstraete
, Phys. Rev. B
94
, 165116
(2016
).63.
J.
Haegeman
, T. J.
Osborne
, and F.
Verstraete
, Phys. Rev. B
88
, 075133
(2013
).64.
M. L.
Wall
and L. D.
Carr
, New J. Phys.
14
, 125015
(2012
).65.
J. J.
García-Ripoll
, New J. Phys.
8
, 305
(2006
).66.
R.
Borrelli
, A.
Capobianco
, A.
Landi
, and A.
Peluso
, Phys. Chem. Chem. Phys.
17
, 30937
(2015
).67.
A.
Landi
, R.
Borrelli
, A.
Capobianco
, and A.
Peluso
, J. Phys. Chem. Lett.
10
, 1845
(2019
).68.
A.
Troisi
and G.
Orlandi
, Phys. Rev. Lett.
96
, 086601
(2006
).69.
L.
Chen
, R.
Borrelli
, and Y.
Zhao
, J. Phys. Chem. A
121
, 8757
(2017
).70.
Y.
Tanimura
, J. Chem. Phys.
137
, 22A550
(2012
).71.
A.
Landi
, R.
Borrelli
, A.
Capobianco
, A.
Velardo
, and A.
Peluso
, J. Chem. Theory Comput.
14
, 1594
(2018
).72.
A.
Landi
, R.
Borrelli
, A.
Capobianco
, A.
Velardo
, and A.
Peluso
, J. Phys. Chem. C
122
, 25849
(2018
).73.
E. J.
Heller
, J. Chem. Phys.
64
, 63
(1976
).74.
A.
McLachlan
, Mol. Phys.
8
, 39
(1964
).75.
76.
M.
Tsuchimoto
and Y.
Tanimura
, J. Chem. Theory Comput.
11
, 3859
(2015
).© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.
