TY - JOUR
T1 - Efficient choice of coloured noises in stochastic dynamics of open quantum systems
AU - Matos, Daniel
AU - Lane, Matt
AU - Ford, Ian J.
AU - Kantorovich, Lev
PY - 2020/11/10
Y1 - 2020/11/10
N2 - The Stochastic Liouville-von Neumann (SLN) equation describes the dynamics of an open quan- tum system reduced density matrix coupled to a non-Markovian harmonic environment. The in- teraction with the environment is represented by complex coloured noises which drive the system, and whose correlation functions are set by the properties of the environment. We present a number of schemes capable of generating coloured noises of this kind that are built on a noise amplitude reduction procedure [H. Imai et al, Chemical Physics cbf446, 134 (2015)], including two analytically optimised schemes. In doing so, we pay close attention to the properties of the correlation func- tions in Fourier space, which we derive in full. For some schemes the method of Wiener filtering for deconvolutions leads to the realisation that weakening causality in one of the noise correlation functions improves numerical convergence considerably, allowing us to introduce a well controlled method for doing so. We compare the ability of these schemes, along with an alternative optimised scheme [K. Schmitz and J. T. Stockburger, EPJ ST 227, 1929 (2019)], to reduce the growth in the mean and variance of the trace of the reduced density matrix, and their ability to extend the region in which the dynamics is stable and well converged for a range of temperatures. By numerically optimising an additional noise scaling freedom, we identify the scheme which performs best for the parameters used, improving convergence by orders of magnitude and increasing the time accessible by simulation.
AB - The Stochastic Liouville-von Neumann (SLN) equation describes the dynamics of an open quan- tum system reduced density matrix coupled to a non-Markovian harmonic environment. The in- teraction with the environment is represented by complex coloured noises which drive the system, and whose correlation functions are set by the properties of the environment. We present a number of schemes capable of generating coloured noises of this kind that are built on a noise amplitude reduction procedure [H. Imai et al, Chemical Physics cbf446, 134 (2015)], including two analytically optimised schemes. In doing so, we pay close attention to the properties of the correlation func- tions in Fourier space, which we derive in full. For some schemes the method of Wiener filtering for deconvolutions leads to the realisation that weakening causality in one of the noise correlation functions improves numerical convergence considerably, allowing us to introduce a well controlled method for doing so. We compare the ability of these schemes, along with an alternative optimised scheme [K. Schmitz and J. T. Stockburger, EPJ ST 227, 1929 (2019)], to reduce the growth in the mean and variance of the trace of the reduced density matrix, and their ability to extend the region in which the dynamics is stable and well converged for a range of temperatures. By numerically optimising an additional noise scaling freedom, we identify the scheme which performs best for the parameters used, improving convergence by orders of magnitude and increasing the time accessible by simulation.
M3 - Article
JO - PHYSICAL REVIEW E
JF - PHYSICAL REVIEW E
SN - 1539-3755
ER -
TY - JOUR
T1 - Exactly thermalized quantum dynamics of the spin-boson model coupled to a dissipative environment
AU - Lane, M. A.
AU - Matos, D.
AU - Ford, I. J.
AU - Kantorovich, L.
PY - 2020/6/1
Y1 - 2020/6/1
N2 - We present an application of the extended stochastic Liouville-von Neumann equation (ESLN) method introduced earlier [G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 95, 125124 (2017)2469-995010.1103/PhysRevB.95.125124; G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 97, 224310 (2018)]2469-995010.1103/PhysRevB.97.224310, which describes the dynamics of an exactly thermalized open quantum system reduced density matrix coupled to a non-Markovian harmonic environment. Critically, the combined system of the open system fully coupled to its environment is thermalized at finite temperature using an imaginary-time evolution procedure before the application of real-time evolution. This initializes the combined system in the correct canonical equilibrium state rather than being initially decoupled. Here we apply our theory to the spin-boson Hamiltonian and develop a number of competing ESLN variants designed to reduce the numerical divergence of the trace of the open-system density matrix. We find that a careful choice of the driving noises is essential for improving numerical stability. We also investigate the effect of applying higher-order numerical schemes for solving stochastic differential equations, such as the Stratonovich-Heun scheme, and conclude that stochastic sampling dominates convergence with the improvement associated with the numerical scheme being less important for short times but required for late times. To verify the method and its numerical implementation, we first consider evolution under a fixed Hamiltonian and show that the system either remains in, or approaches, the correct canonical equilibrium state at long times. Additionally, evolution of the open system under nonequilibrium Landau-Zener (LZ) driving is considered and the asymptotic convergence to the LZ limit was observed for vanishing system-environment coupling and temperature. When coupling and temperature are nonzero, initially thermalizing the combined system at a finite time in the past was found to be a better approximation of the true LZ initial state than starting in a pure state.
AB - We present an application of the extended stochastic Liouville-von Neumann equation (ESLN) method introduced earlier [G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 95, 125124 (2017)2469-995010.1103/PhysRevB.95.125124; G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 97, 224310 (2018)]2469-995010.1103/PhysRevB.97.224310, which describes the dynamics of an exactly thermalized open quantum system reduced density matrix coupled to a non-Markovian harmonic environment. Critically, the combined system of the open system fully coupled to its environment is thermalized at finite temperature using an imaginary-time evolution procedure before the application of real-time evolution. This initializes the combined system in the correct canonical equilibrium state rather than being initially decoupled. Here we apply our theory to the spin-boson Hamiltonian and develop a number of competing ESLN variants designed to reduce the numerical divergence of the trace of the open-system density matrix. We find that a careful choice of the driving noises is essential for improving numerical stability. We also investigate the effect of applying higher-order numerical schemes for solving stochastic differential equations, such as the Stratonovich-Heun scheme, and conclude that stochastic sampling dominates convergence with the improvement associated with the numerical scheme being less important for short times but required for late times. To verify the method and its numerical implementation, we first consider evolution under a fixed Hamiltonian and show that the system either remains in, or approaches, the correct canonical equilibrium state at long times. Additionally, evolution of the open system under nonequilibrium Landau-Zener (LZ) driving is considered and the asymptotic convergence to the LZ limit was observed for vanishing system-environment coupling and temperature. When coupling and temperature are nonzero, initially thermalizing the combined system at a finite time in the past was found to be a better approximation of the true LZ initial state than starting in a pure state.
UR - http://www.scopus.com/inward/record.url?scp=85092227979&partnerID=8YFLogxK
U2 - 10.1103/PhysRevB.101.224306
DO - 10.1103/PhysRevB.101.224306
M3 - Article
AN - SCOPUS:85092227979
VL - 101
JO - Physical Review B (Condensed Matter and Materials Physics)
JF - Physical Review B (Condensed Matter and Materials Physics)
SN - 1098-0121
IS - 22
M1 - 224306
ER -
TY - JOUR
T1 - Exactly Thermalised Quantum Dynamics of the Spin-Boson Model coupled to a Dissipative Environment
AU - Lane, Matt
AU - Matos, Daniel
AU - Ford, Ian J.
AU - Kantorovich, Lev
PY - 2020/4/8
Y1 - 2020/4/8
N2 - We present an application of the Extended Stochastic Liouville-von Neumann equations (ESLN) method introduced earlier [PRB 95, 125124 (2017); PRB 97, 224310 (2018)] which describes the dynamics of an exactly thermalised open quantum system reduced density matrix coupled to a non- Markovian harmonic environment. Critically, the combined system of the open system fully coupled to its environment is thermalised at finite temperature using an imaginary time evolution procedure before the application of real time evolution. This initialises the combined system in the correct canonical equilibrium state rather than being initially decoupled. Here we apply our theory to the spin-boson Hamiltonian and develop a number of competing ESLN variants designed to reduce the numerical divergence of the trace of the open system density matrix. We find that a careful choice of the driving noises is essential for improving numerical stability. We also investigate the effect of applying higher order numerical schemes for solving stochastic differential equations, such as the Stratonovich-Heun scheme, and conclude that stochastic sampling dominates convergence with the improvement associated with the numerical scheme being less important for short times but required for late times. To verify the method and its numerical implementation, we first consider evolution under a fixed Hamiltonian and show that the system either remains in, or approaches, the correct canonical equilibrium state at long times. Additionally, evolution of the open system under non-equilibrium Landau-Zener (LZ) driving is considered and the asymptotic convergence to the LZ limit was observed for vanishing system-environment coupling and temperature. When coupling and temperature are non-zero, initially thermalising the combined system at a finite time in the past was found to be a better approximation of the true LZ initial state than starting in a pure state.
AB - We present an application of the Extended Stochastic Liouville-von Neumann equations (ESLN) method introduced earlier [PRB 95, 125124 (2017); PRB 97, 224310 (2018)] which describes the dynamics of an exactly thermalised open quantum system reduced density matrix coupled to a non- Markovian harmonic environment. Critically, the combined system of the open system fully coupled to its environment is thermalised at finite temperature using an imaginary time evolution procedure before the application of real time evolution. This initialises the combined system in the correct canonical equilibrium state rather than being initially decoupled. Here we apply our theory to the spin-boson Hamiltonian and develop a number of competing ESLN variants designed to reduce the numerical divergence of the trace of the open system density matrix. We find that a careful choice of the driving noises is essential for improving numerical stability. We also investigate the effect of applying higher order numerical schemes for solving stochastic differential equations, such as the Stratonovich-Heun scheme, and conclude that stochastic sampling dominates convergence with the improvement associated with the numerical scheme being less important for short times but required for late times. To verify the method and its numerical implementation, we first consider evolution under a fixed Hamiltonian and show that the system either remains in, or approaches, the correct canonical equilibrium state at long times. Additionally, evolution of the open system under non-equilibrium Landau-Zener (LZ) driving is considered and the asymptotic convergence to the LZ limit was observed for vanishing system-environment coupling and temperature. When coupling and temperature are non-zero, initially thermalising the combined system at a finite time in the past was found to be a better approximation of the true LZ initial state than starting in a pure state.
M3 - Article
JO - Physical Review B (Condensed Matter and Materials Physics)
JF - Physical Review B (Condensed Matter and Materials Physics)
SN - 1098-0121
ER -