HierarchicalEOM.jl: An efficient Julia framework for hierarchical equations of motion in open quantum systems

TL;DR

An open-source software package (HierarchicalEOM.jl), characterized by a notable speed and accessibility to new users, that achieves a significant speedup with respect to the corresponding method in the Quantum Toolbox in Python (QuTiP), upon which this package is founded.

Abstract

The hierarchical equations of motion (HEOM) approach can describe the reduced dynamics of a system simultaneously coupled to multiple bosonic and fermionic environments. The complexity of exactly describing the system-environment interaction with the HEOM method usually results in time-consuming calculations and a large memory cost. Here, we introduce an open-source software package called HierarchicalEOM$.$jl: a Julia framework integrating the HEOM approach. HierarchicalEOM$.$jl features a collection of methods to compute bosonic and fermionic spectra, stationary states, and the full dynamics in the extended space of all auxiliary density operators (ADOs). The required handling of the ADOs multi-indexes is achieved through a user-friendly interface. We exemplify the functionalities of the package by analyzing a single impurity Anderson model, and an ultra-strongly coupled charge-cavity system interacting with bosonic and fermionic reservoirs. HierarchicalEOM$.$jl achieves a significant speedup with respect to the corresponding method in the Quantum Toolbox in Python (QuTiP), upon which this package is founded.

Figures (7)

• Figure 1: The ecosystem of the HierarchicalEOM.jl package.(a) Users should specify the system Hamiltonian $H_\textrm{s}(t)$, coupling operators ($V_\textrm{s}$ or $d_\textrm{s}$), and the bath correlation function $C(t)$. For the exponent $\{\eta_k,\gamma_k\}$, users can either specify the physical parameters characterizing the spectral density of the bath by built-in functions, or directly providing a list of exponents. (b) Construction of the bath-object which includes the system coupling operator and a list of exponents characterizing the bath correlation function. (c) Construction of the HEOM Liouvillian superoperator (HEOMLS) matrix $\hat{\mathcal{M}}$ which defines the hierarchical equations of motion from the system Hamiltonian and the bath-objects. (d) Computation of the dynamics and stationary states for all auxiliary density operators using $\hat{\mathcal{M}}$. (e) The hierarchy dictionary translates the index of each ADO into the corresponding multi-index ensembles together with the exponents of the bath, and vice-versa. (f) The hierarchy dictionary allows a high-level interpretation of the ADOs to compute some physical properties. (g) Logo of HierarchicalEOM.jl package.
• Figure 2: Schematic illustration of a light-matter system (an electron coupled to a bosonic mode in a cavity) interacting with bosonic and fermionic reservoirs. Here, $g$ is the coupling strength between the electron and cavity, $\Delta$ is the coupling strength between the cavity and the bosonic reservoir, and $\Gamma_\textrm{L}=\Gamma_\textrm{R}=\Gamma$ are the coupling strengths between the electron and its fermionic reservoirs.
• Figure 3: The Kondo resonance from equilibrium to non-equilibrium. In ($\textbf{a}$), the black dashed, red dash-dotted, orange dotted, green solid, and blue dashed curves represent the density of states $A(\omega)$ for bias voltage $\Phi$ at $0$, $1$, $2$, $3$, and $4~\textrm{mV}$, respectively. One can observe that the density of states of the electronic system exhibits two Hubbard peaks and an additional central peak at equilibrium (i.e., bias voltage $\Phi=0$). By increasing the bias voltage $\Phi$ to non-equilibrium, the central peak is suppressed and splits into two. In ($\textbf{b}$), the black curve shows the effects of the Kondo resonance on the differential conductance as a function of bias voltage $\Phi$. The conductance $G$ has a peak at $\Phi=0$ as the Kondo resonance acts as a transport channel for the electron.
• Figure 4: Effects of the electron-cavity coupling on the spectra of the charge and cavity system. In (a), the black dashed, red dotted, and blue solid curves represent the density of states $A(\omega)$ for the charge-cavity coupling strength $g$ at $0$, $0.25\Gamma$, and $0.5\Gamma$, respectively. The resonance in the density of states of the charge system is reduced and shifts towards lower energies as $g$ increases. In (b), we plot the power spectral density $S(\omega)$ of the cavity and uses two different methods [HEOM (red dotted and blue solid curves for $g$ at $0.25\Gamma$ and $0.5\Gamma$, respectively) and perturbative master equation (abbreviated as ME, which is represented by orange dash-dotted and green dashed curves for $g$ at $0.25\Gamma$ and $0.5\Gamma$, respectively)] to describe the interaction between the cavity and the bosonic reservoir. Both approaches show a single peak at $\omega=\omega_{\textrm{c}}$ for $g\in(0, 0.5]$. The Born-Markov approximation used to derive the ME may result in an inaccurate estimation of the magnitude of the $S(\omega)$ compared to the HEOM, especially in the high-frequency range.
• Figure 5: Effect of the electron-cavity coupling on the conductance between the charge system and fermionic reservoirs. The black dashed, red dotted, and blue solid curves represent the conductance $G$ for the charge-cavity coupling strength $g$ at $0$, $0.25\Gamma$, and $0.5\Gamma$, respectively. The conductance depends on the absolute value of the bias voltage $\Phi$ and reaches its maximum when the average energy of the charge system is equal to the Fermi-level of one of the fermionic reservoirs, i.e., when $e\left|\Phi\right|/2=\epsilon$. As the electron-cavity coupling $g$ increases, both the conductance $G$ and the average energy of the charge system decrease, resulting in a wider splitting of the conductance peaks.