In this tutorial, I will review some recent progresses in non-Markovian dynamics of quantum open systems. I will be focused with non-Markovian quantum trajectories and non-Markovian master equation approaches. In particular, I will show how to derive an exact non-Markovian master equation from the corresponding quantum trajectories. I will also show how to use quantum trajectories to derive the well-known Kraus operators for pure dephasing noise. The applications of quantum trajectories to quantum information and decoherence will be briefly reviewed.In this tutorial, I will review some recent progresses in non-Markovian dynamics of quantum open systems. I will be focused with non-Markovian quantum trajectories and non-Markovian master equation approaches. In particular, I will show how to derive an exact non-Markovian master equation from the corresponding quantum trajectories. I will also show how to use quantum trajectories to derive the well-known Kraus operators for pure dephasing noise. The applications of quantum trajectories to quantum information and decoherence will be briefly reviewed.

OSA Workshop on Entanglement and Quantum DecoherenceJan 1, 20080 citations

Quantum jumps revisited: An overview of quantum trajectory theory

H. J. Carmichael

Springer eBooksApr 7, 20089 citations

Quantum trajectories for time-dependent adiabatic master equations

Ka-Wa Yip, Tameem Albash, Daniel A. Lidar

We describe a quantum trajectories technique for the unraveling of the quantum adiabatic master equation in Lindblad form. By evolving a complex state vector of dimension $N$ instead of a complex density matrix of dimension ${N}^{2}$ , simulations of larger system sizes become feasible. The cost of running many trajectories, which is required to recover the master equation evolution, can be minimized by running the trajectories in parallel, making this method suitable for high performance computing clusters. In general, the trajectories method can provide up to a factor $N$ advantage over directly solving the master equation. In special cases where only the expectation values of certain observables are desired, an advantage of up to a factor ${N}^{2}$ is possible. We test the method by demonstrating agreement with direct solution of the quantum adiabatic master equation for 8-qubit quantum annealing examples. We also apply the quantum trajectories method to a 16-qubit example originally introduced to demonstrate the role of tunneling in quantum annealing, which is significantly more time consuming to solve directly using the master equation. The quantum trajectories method provides insight into individual quantum jump trajectories and their statistics, thus shedding light on open system quantum adiabatic evolution beyond the master equation.We describe a quantum trajectories technique for the unraveling of the quantum adiabatic master equation in Lindblad form. By evolving a complex state vector of dimension $N$ instead of a complex density matrix of dimension ${N}^{2}$, simulations of larger system sizes become feasible. The cost of running many trajectories, which is required to recover the master equation evolution, can be minimized by running the trajectories in parallel, making this method suitable for high performance computing clusters. In general, the trajectories method can provide up to a factor $N$ advantage over directly solving the master equation. In special cases where only the expectation values of certain observables are desired, an advantage of up to a factor ${N}^{2}$ is possible. We test the method by demonstrating agreement with direct solution of the quantum adiabatic master equation for 8-qubit quantum annealing examples. We also apply the quantum trajectories method to a 16-qubit example originally introduced to demonstrate the role of tunneling in quantum annealing, which is significantly more time consuming to solve directly using the master equation. The quantum trajectories method provides insight into individual quantum jump trajectories and their statistics, thus shedding light on open system quantum adiabatic evolution beyond the master equation.

arXiv:1710.03431Physical Review AFeb 28, 201823 citations

The Quantum Jump Approach and Quantum Trajectories

Gerhard C. Hegerfeldt

Lecture notes in physicsJan 1, 200311 citations

Trajectory integration of the quantum hydrodynamic equations of motion

Corey Jason Trahan

Recently, in an effort to solve more realistic problems in quantum dynamics, much attention has been directed into numerically integrating the quantum hydrodynamic equations of motions (QHEM), as opposed to directly solving the time-dependent Schrödinger equation (TDSE). Such efforts have been provoked by the many numerical drawbacks encountered when solving the TDSE on a fixed-grid. In this dissertation, one trajectory method for integrating the QHEM is reviewed, and two novel trajectories methods are described. The first of these, the quantum trajectory method (QTM), was introduced in 1999 and has been used to solve many problems in quantum dynamics since then. However, severe numerical problems are encountered when this method is applied to problems that form wave function nodes. To get around this problem, new methods for numerically integrating the QHEM are needed. In the first novel method described, the arbitrary Lagrangian-Eulerian (ALE) method, particle trajectories are governed by a predetermined equation of motion that is user-supplied. The ALE method remedies inflation and compression problems encountered in the pure Lagrangian QTM. In the second new method discussed, the derivative propagating method (DPM), single quantum trajectories can be calculated one at a time, as opposed to the ensemble propagation of the QTM and ALE method. Using these two methods, new solutions to the QHEM are obtained where the QTM fails. In addition to solving the QHEM, the DPM is also used to solve the classical Klein-Kramers equation in this dissertation. This equation governs the Markovian phase space evolution of a system coupled to an environment such as a heat bath. This marks the first time single trajectories have been used to solve both the QHEM and the Klein-Kramers equations.Recently, in an effort to solve more realistic problems in quantum dynamics, much attention has been directed into numerically integrating the quantum hydrodynamic equations of motions (QHEM), as opposed to directly solving the time-dependent Schrödinger equation (TDSE). Such efforts have been provoked by the many numerical drawbacks encountered when solving the TDSE on a fixed-grid. In this dissertation, one trajectory method for integrating the QHEM is reviewed, and two novel trajectories methods are described. The first of these, the quantum trajectory method (QTM), was introduced in 1999 and has been used to solve many problems in quantum dynamics since then. However, severe numerical problems are encountered when this method is applied to problems that form wave function nodes. To get around this problem, new methods for numerically integrating the QHEM are needed. In the first novel method described, the arbitrary Lagrangian-Eulerian (ALE) method, particle trajectories are governed by a predetermined equation of motion that is user-supplied. The ALE method remedies inflation and compression problems encountered in the pure Lagrangian QTM. In the second new method discussed, the derivative propagating method (DPM), single quantum trajectories can be calculated one at a time, as opposed to the ensemble propagation of the QTM and ALE method. Using these two methods, new solutions to the QHEM are obtained where the QTM fails. In addition to solving the QHEM, the DPM is also used to solve the classical Klein-Kramers equation in this dissertation. This equation governs the Markovian phase space evolution of a system coupled to an environment such as a heat bath. This marks the first time single trajectories have been used to solve both the QHEM and the Klein-Kramers equations.

Jan 1, 20030 citations

We describe a quantum trajectories technique for the unraveling of the quantum adiabatic master equation in Lindblad form. By evolving a complex state vector of dimension

N

instead of a complex density matrix of dimension

{N}^{2}

, simulations of larger system sizes become feasible. The cost of running many trajectories, which is required to recover the master equation evolution, can be minimized by running the trajectories in parallel, making this method suitable for high performance computing clusters. In general, the trajectories method can provide up to a factor

N

advantage over directly solving the master equation. In special cases where only the expectation values of certain observables are desired, an advantage of up to a factor

{N}^{2}

is possible. We test the method by demonstrating agreement with direct solution of the quantum adiabatic master equation for 8-qubit quantum annealing examples. We also apply the quantum trajectories method to a 16-qubit example originally introduced to demonstrate the role of tunneling in quantum annealing, which is significantly more time consuming to solve directly using the master equation. The quantum trajectories method provides insight into individual quantum jump trajectories and their statistics, thus shedding light on open system quantum adiabatic evolution beyond the master equation.