Bosonic Bath

Overview

The BosonBath object describes the interaction between the system ($s$) and a exterior bosonic environment ($b$), which can be modeled by

\[H_{sb}=V_{s}\sum_k g_k (b_k + b_k^\dagger),\]

in terms of the coupling strength $g_k$ and the annihilation (creation) operator $b_k (b_k^\dagger)$ associated to the $k$-th mode of the bosonic environment. Here, $V_s$ refers to the system-interaction operator. In particular, $V_s$ must be a Hermitian operator which can act on both bosonic and fermionic systems degree of freedom. In the fermionic system case, $V_s$ must have even parity to be compatible with charge conservation.

The effects of a bosonic environment (initially in thermal equilibrium and linearly coupled to the system) are completely encoded in the two-time correlation functions, namely

\[C(t_1, t_2) =\frac{1}{2\pi}\int_{0}^{\infty} d\omega J(\omega)\left[n(\omega)e^{i\omega (t_1-t_2)}+(n(\omega)+1)e^{-i\omega (t_1-t_2)}\right],\]

where $J(\omega)=2\pi\Sigma_k |g_k|^2 \delta(\omega-\omega_k)$ is the spectral density of the bath and $n(\omega)=\{\exp(\omega/k_B T)-1\}^{-1}$ represents the Bose-Einstein distribution.

A more practical representation can be found by expressing the correlation function as a sum of exponential terms (Exponent), namely

\[C(t_1, t_2)=\sum_i \eta_i e^{-\gamma_i (t_1-t_2)}.\]

This allows us to define an iterative procedure which leads to the hierarchical equations of motion (HEOM).

Construct BosonBath (with real and imaginary parts are combined)

One can construct the BosonBath object with the coupling operator Vs::QuantumObject and the two lists η::AbstractVector and γ::AbstractVector which corresponds to the exponential terms $\{\eta_i\}_i$ and $\{\gamma_i\}_i$, respectively.

bath = BosonBath(Vs, η, γ)

Construct BosonBath (with real and imaginary parts are separated)

When $\gamma_i \neq \gamma_i^*$, a closed form for the HEOM can be obtained by further decomposing $C(t_1, t_2)$ into its real (R) and imaginary (I) parts as

\[C(t_1, t_2)=\sum_{u=\textrm{R},\textrm{I}}(\delta_{u, \textrm{R}} + i\delta_{u, \textrm{I}})C^{u}(t_1, t_2)\]

where $\delta$ is the Kronecker delta function and $C^{u}(t_1, t_2)=\sum_i \eta_i^u \exp(-\gamma_i^u (t_1-t_2))$

In this case, the BosonBath object can be constructed by the following method:

bath = BosonBath(Vs, η_real, γ_real, η_imag, γ_imag)

Here, η_real::AbstractVector, γ_real::AbstractVector, η_imag::AbstractVector and γ_imag::AbstractVector correspond to the exponential terms $\{\eta_i^{\textrm{R}}\}_i$, $\{\gamma_i^{\textrm{R}}\}_i$, $\{\eta_i^{\textrm{I}}\}_i$ and $\{\gamma_i^{\textrm{I}}\}_i$, respectively.

Print Bosonic Bath

One can check the information of the BosonBath by the print function, for example:

print(bath)

BosonBath object with 4 exponential-expansion terms

Calculate the correlation function

To check whether the exponential terms in the BosonBath is correct or not, one can call correlation_function to calculate the correlation function $C(t)$, where $t=t_1-t_2$:

c_list = correlation_function(bath, tlist)

Here, c_list is a list which contains the value of $C(t)$ corresponds to the given time series tlist.

Exponent

HierarchicalEOM.jl also supports users to access the specific exponential term with brackets []. This returns an Exponent object, which contains the corresponding value of $\eta_i$ and $\gamma_i$:

e = bath[2] # the 2nd-term
print(e)

Bath Exponent with types = "bRI", η = 1.5922874021206546e-6 + 0.0im, γ = 0.3141645167860635 + 0.0im.

The different types of the (bosonic-bath) Exponent:

• "bR" : from real part of bosonic correlation function $C^{u=\textrm{R}}(t_1, t_2)$
• "bI" : from imaginary part of bosonic correlation function $C^{u=\textrm{I}}(t_1, t_2)$
• "bRI" : from combined (real and imaginary part) bosonic bath correlation function $C(t_1, t_2)$

One can even obtain the Exponent with iterative method:

for e in bath
    println(e)
end

Bath Exponent with types = "bRI", η = 4.995832638723504e-5 - 2.5e-6im, γ = 0.005 + 0.0im.

Bath Exponent with types = "bRI", η = 1.5922874021206546e-6 + 0.0im, γ = 0.3141645167860635 + 0.0im.

Bath Exponent with types = "bRI", η = 1.0039844180003819e-6 + 0.0im, γ = 0.6479143347831898 + 0.0im.

Bath Exponent with types = "bRI", η = 3.1005439801387293e-6 + 0.0im, γ = 1.8059644711829272 + 0.0im.