Hierarchy equations
approach to quantum dissipative dynamics
 Derivation of hierarchal equations of motion (master equation and
FokkerPlanck equation) 
 Inclusion of low temperature correction terms 
 Derivation of multistate quantum FokkerPlanck equation and its
applications 
 Initial systembath coherence and imaginary HEOM 
Representative references
 Y. Tanimura and R. Kubo, Time evolution of a quantum
system in contact with a nearly GaussianMarkoffian noise bath, J.
Phys. Soc. Jpn. 58, 101114 (1989).
[OPEN SELECT]

 Y. Tanimura and P. G. Wolynes, Quantum and classical
FokkerPlanck equations for a GaussianMarkovian noise bath,
Phys. Rev. A43,
41314142 (1991). 

Y. Tanimura, Nonperturbative expansion method for a
quantum system coupled to a harmonicoscillator bath,
Phys.
Rev. A41, 66766687 (1990).


Y. Tanimura and Y. Maruyama, GaussianMarkovian
quantum FokkerPlanck approach to nonlinear spectroscopy of a displaced
Morse potentials system: dissociation, predissociation and optical Stark
effects,
J. Chem. Phys. 107, 17791793 (1997).

 Y. Tanimura, Stochastic Liouville, Langevin, FokkerPlanck, and
master equation approaches to quantum dissipative systems,
J. Phys. Soc. Jpn. 75, 082001 (2006). [full
text PDF: OPEN SELECT]

 Y. Tanimura, Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities,
J. Chem. Phys. 141,
044114 (2014). [Open Access] 

Y. Tanimura, RealTime and ImaginaryTime Quantum Hierarchal FokkerPlanck Equations,
J. Chem. Phys.
142, 144110 [20 pages] (2015)
(PDF)
[Open Access] 
 T. Ikeda and Y. Tanimura, LowTemperature Quantum
FokkerPlanck and Smoluchowski Equations and Their Extension to Multistate
Systems, J.
Chem. Theory Comput. 15 (2019). (Error correction)

 Y. Tanimura,
Perspective: Numerically "Exact" Approach to Open Quantum Dynamics: The Hierarchical Equations of
Motion (HEOM), J. Chem. Phys. 153,
020901 (2020).
[Open Access] 
Complete list of the related subject (applications of
HEOM)
Lecture Note
Program Codes
 Markovian2005(GaussianMarkovian
master equation for a spin system coupled to 3D anisotropic baths) 
 nonMarkovian2009
(with low temperature correction terms, Jan 12, 2011 version) 

nonMarkovian2009+2D (with low temperature correction terms+ subroutines
for linear and 2D correlation spectra, Oct. 3rd, 2012 version) 
 onMarkovian2012+2D
(Drude+Brownian mode) 
 TanimuranFP15 (realtime and imaginarytime quantum hierarchal
FokkerPlanck equations, March 21 , 2015 version) 
 GPUHEOM (HEOM code
for CUDA)
M. Tsuchimoto and Y.
Tanimura, J. Chem. Theory Comput.,11, 3859 (2015). 

Lowtemperature corrected quantum FokkerPlanck and multistate quantum
FokkerPlanck Eqs. T. Ikeda and Y. Tanimura, J. Chem. Theory Comput. 15
(2019). 
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