Matrix-product-operator approach to the nonequilibrium steady state of driven-dissipative quantum arrays

2015/04/30 by Eduardo Mascarenhas, H. Flayac, Hugo Flayac +1 · 7 citations
Computer Science · Mathematics · Physics and Astronomy · #Ansatz #Degenerate energy levels #Density matrix #Density matrix renormalization group #Dissipative system #Hamiltonian (control theory) #Mathematical optimization #Mathematics #Matrix multiplication #Matrix product state #Non-equilibrium thermodynamics #Operator (biology) #Physics #Quantum #Quantum Information and Cryptography #Quantum and electron transport phenomena #Quantum many-body systems #Quantum mechanics #Quantum system #Statistical physics #cond-mat.mes-hall #quant-ph

paper · pdf · doi:10.1103/physreva.92.022116

published as Phys. Rev. A 92, 022116 (2015)

arxiv created 2015/07/14 · openalex publication_date 2015/08/19 · arxiv updated 2015/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We develop a numerical procedure to efficiently model the nonequilibrium steady state of one-dimensional arrays of open quantum systems based on a matrix-product operator ansatz for the density matrix. The procedure searches for the null eigenvalue of the Liouvillian superoperator by sweeping along the system while carrying out a partial diagonalization of the single-site stationary problem. It bears full analogy to the density-matrix renormalization-group approach to the ground state of isolated systems, and its numerical complexity scales as a power law with the bond dimension. The method brings considerable advantage when compared to the integration of the time-dependent problem via Trotter decomposition, as it can address arbitrarily long-ranged couplings. Additionally, it ensures numerical stability in the case of weakly dissipative systems thanks to a slow tuning of the dissipation rates along the sweeps. We have tested the method on a driven-dissipative spin chain, under various assumptions for the Hamiltonian, drive, and dissipation parameters, and compared the results to those obtained both by Trotter dynamics and Monte Carlo wave function methods. Accurate and numerically stable convergence was always achieved when applying the method to systems with a gapped Liouvillian and a nondegenerate steady state.

Citations

• Matrix Product Density Operators: Simulation of Finite-Temperature and Dissipative Systems
• Signatures of the superfluid-insulator phase transition in laser-driven dissipative nonlinear cavity arrays
• Quantum trajectories and open many-body quantum systems
• Many-body physics with ultracold gases
• Nonequilibrium phase diagram of a driven and dissipative many-body system
• Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems
• The density-matrix renormalization group in the age of matrix product states
• Coupling superconducting qubits via a cavity bus
• Density Matrix Renormalization Group for Dummies

Cited by

• Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems
• Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems
• Pitfalls in the theory of carrier dynamics in semiconductor quantum dots: Single-particle basis versus the many-particle configuration basis
• Optimal stochastic unraveling of disordered open quantum systems: Application to driven-dissipative photonic lattices
• Spontaneous Beliaev-Landau scattering out of equilibrium
• Quantum stabilization of photonic spatial correlations
• Suppressing decoherence in quantum plasmonic systems by the spectral-hole-burning effect
• Phase diagram of incoherently driven strongly correlated photonic lattices