HEOM

Hierarchical equations of motion approach to quantum dissipative dynamics

	Derivation of hierarchal equations of motion (master equation and Fokker-Planck equation)
	Inclusion of low temperature correction terms
	Derivation of multistate quantum Fokker-Planck equation and its applications
	Initial system-bath coherence and imaginary HEOM

Representative references

	Y. Tanimura and R. Kubo, Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath, J. Phys. Soc. Jpn. 58, 101-114 (1989). [OPEN SELECT]
	Y. Tanimura and P. G. Wolynes, Quantum and classical Fokker-Planck equations for a Gaussian-Markovian noise bath, Phys. Rev. A43, 4131-4142 (1991).
	Y. Tanimura, Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath, Phys. Rev. A41, 6676-6687 (1990).
	Y. Tanimura and Y. Maruyama, Gaussian-Markovian quantum Fokker-Planck approach to nonlinear spectroscopy of a displaced Morse potentials system: dissociation, predissociation and optical Stark effects, J. Chem. Phys. 107, 1779-1793 (1997).
	Y. Tanimura, Stochastic Liouville, Langevin, Fokker-Planck, 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, Real-Time and Imaginary-Time Quantum Hierarchal Fokker-Planck Equations, J. Chem. Phys. 142, 144110 [20 pages] (2015) (PDF) [Open Access]
	T. Ikeda and Y. Tanimura, Low-Temperature Quantum Fokker-Planck 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

Lecture note on reduced hierarchy equations of motion: Theory (Zip file, March 25 2025)

Lecture note on reduced hierarchy equations of motion: Applications (Zip file, March 25 2025)

Lecture Mpegs

	Quantum noise cannot be Markovian: Hierarchical equations of motion approach: PART I (Intensive Lectures)
	Quantum noise cannot be Markovian: Hierarchical equations of motion approach: PART II (Seminars)
	Nonequilibrium thermodynamics described as state variables (MPEG 185M) (PPT lecture note)
	Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (HEOM) (Mpeg 661M)- at ISSP 2023.10.7 &12

　

Program Codes

	Markovian2005(Gaussian-Markovian 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 (real-time and imaginary-time quantum hierarchal Fokker-Planck equations, March 21 , 2015 version)
	GPU-HEOM (HEOM code for CUDA) M. Tsuchimoto and Y. Tanimura, J. Chem. Theory Comput.,11, 3859 (2015).
	Low-temperature corrected quantum Fokker-Planck and multi-state quantum Fokker-Planck Eqs. T. Ikeda and Y. Tanimura, J. Chem. Theory Comput. 15 (2019).
	S. Koyanagi and Y. Tanimura, Classical and quantum thermodynamics in a non-equilibrium regime: Application to thermostatic Stirling engine, J. Chem. Phys. 161, 114113 (2024).(PDF)
	Thermodynamic Quantum Fokker-Planck Equation (T-QFPE). Ver, 1,1 (Oct. 21, 2024);S. Koyanagi and Y. Tanimura, J. Chem. Phys. 161, 112501 (2024).(PDF)ierarchical Equations of Motion for the Multiple Baths (HEOM-MB). Ver. 1.0 (Oct. 17, 2024); J. Chem. Phys. 161, 162501 (2024).(PDF)

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