Hierarchical equations of motion

Hierarchical Equations of Motion

HEOMs are developed to describe the time evolution of the density matrix ρ ( t ) for an open quantum system. It is a non-perturbative, non-Markovian approach to propagating in time a quantum state. Motivated by the path integral formalism presented by Feynman and Vernon, Kubo and Tanimura derive the HEOM from a combination of statistical and quantum dynamical techniques.

Using a two level spin-boson system Hamiltonian

H ^ = H ^ A ( a ^ + , a ^ − ) + V ( a ^ + , a ^ − ) ∑ j c j x ^ j + ∑ j [   p ^ 2 2 + 1 2 x ^ j 2 ]

Characterising the bath phonons by the spectral density J ( ω ) = ∑ j c j 2 δ ( ω − ω j )

By writing the density matrix in path integral notation and making use of Feynman-Vernon influence functional, all the bath coordinates in the interaction terms can be grouped into this influence functional which in some specific cases can be calculated in closed form. Assuming a high temperature heat bath with the Drude spectral distribution J ( ω ) = ℏ η γ 2 ω / π ( γ 2 + ω 2 ) and taking the time derivative of the path integral form density matrix the equation and writing it in hierarchal form yields

∂ ∂ t ρ ^ N = − i ( H ^ A + N γ ) ρ ^ N − 1 γ ℏ 2 V ^ × Θ ^ ρ ^ N + i N ℏ Θ ^ ρ ^ N − 1

where Θ destroys system excitation and hence can be referred to as the relaxation operator.

Θ ^ = − n γ β ( V ^ × − i β ℏ γ 2 V ^ ∘ )

The second term in Θ ^ is the temperature correction term with the inverse temperature β = 1 / k B T and the "Hyper-operator" notation is introduced.

A ^ × ρ ^ = A ^ ρ ^ − ρ ^ A ^

A ^ ∘ ρ ^ = A ^ ρ ^ + ρ ^ A ^

As with the SLE in hierarchal form, the counter n can go up to infinity which is a problem numerically, however Kubo and Tanimura provide a method by which the infinite hierarchy can be truncated to a finite set of N differential equations where N is determined by some constraint sensitive to the characteristics of the system i.e. frequency, amplitude of fluctuations, bath coupling etc. The "Terminator" defines the depth of the hierarchy. A simple relation to eliminate the ρ ^ n + 1 term is found.   ρ ^ N + 1 = − Θ ^ ρ ^ N / ℏ γ

The statistical nature of the HEOM approach allows information about the bath noise and system response to be encoded into the equation of motion doctoring the infinite energy problem of Kubo's SLE by introducing the relaxation operator ensuring a return to equilibrium.

Low Temperature

HEOM can be derived for a variety of spectral distributions i.e. Brownian, Lorentzian as well as combinations of such distributions at any temperature. In the case of the Drude distribution at low temperature the hierarchy is extended. By modifying the correlation function that describes the noise correlation function strongly non-Markovian and non-perturbative system-bath interactions can be dealt with.

∂ ∂ t ρ ^ n , j 1 , . . , j K = − [ i ℏ H ^ A × + n γ + ∑ k = 1 K ( j k ν k − 1 ν k ℏ 2 V ^ × Θ ^ k ) + Γ ^ 0 ] ρ ^ n , j 1 , . . , j K − i ℏ V ^ × [ ρ ^ ( n + 1 ) , j 1 , . . , j K + ∑ k = 1 K ρ ^ n , j 1 , . . , ( j k + 1 ) , . . , j K ] − i n ℏ Θ ^ 0 ρ ^ ( n − 1 ) , j 1 . . j K − ∑ k = 1 K i j k ℏ Θ ^ k ρ ^ n , j 1 , . . , ( j k − 1 ) , . . , j K In this equation, only ρ ^ 0 , 0 , . . . , 0 contains all order of system bath interactions with other elements ρ ^ n , j 1 . . . j K being auxiliary terms, moving deeper into the hierarchy, the order of interactions decreases, which is contrary to usual perturbative treatments of such systems. Θ ^ k = c k V ^ × where c k is a constant determined in the correlation function.

Γ ^ 0 ≡ η β ℏ 2 ( 1 − β γ n c 0 ) V ^ × V ^ ×

This Γ ^ 0 term arises from the Matsubara cut-off term introduced to the correlation function and thus holds information about the memory of the noise.

Below is the terminator for the HEOM

∂ ∂ t ρ ^ n , j 1 , . . . , j K ≃ − ( i ℏ H ^ A × − ∑ k = 1 K 1 ν k ℏ 2 V ^ × Θ ^ k + Γ ^ 0 ) ρ ^ n , j 1 , . . . , j K

Performing a Wigner transformation on this HEOM, the quantum Fokker-Planck equation with low temperature correction terms emerges.