Time Evolution and Quantum System Dynamics

Table of contents

• Introduction

• Time Evolution Solutions

  • Solution

  • Accessing data in solutions

  • Multiple trajectories solution

• Schrödinger Equation Solver

  • Unitary evolution

  • Example: Spin dynamics

• Lindblad Master Equation Solver

  • Von Neumann equation

  • The Lindblad master equation

  • Example: Dissipative Spin dynamics

  • Example: Harmonic oscillator in thermal bath

  • Example: Two-level atom coupled to dissipative single-mode cavity

• Monte Carlo Solver

  • Monte Carlo wave-function

  • Example: Two-level atom coupled to dissipative single-mode cavity (MC)

  • Running trajectories in parallel

• Stochastic Solver

  • Stochastic Schrödinger equation

  • Stochastic master equation

  • Example: Homodyne detection

• Solving Problems with Time-dependent Hamiltonians

  • Generate QobjEvo

  • QobjEvo fields (attributes)

  • Using parameters

Introduction

Although in some cases, we want to find the stationary states of a quantum system, often we are interested in the dynamics: how the state of a system or an ensemble of systems evolves with time. QuantumToolbox provides many ways to model dynamics.

There are two kinds of quantum systems: open systems that interact with a larger environment and closed systems that do not. In a closed system, the state can be described by a state vector. When we are modeling an open system, or an ensemble of systems, the use of the density matrix is mandatory.

The following table lists the solvers provided by QuantumToolbox for dynamic quantum systems and the corresponding type of solution returned by the solver:

Equation	Function Call	Problem	Returned Solution
Unitary evolution, Schrödinger equation	sesolve	sesolveProblem	TimeEvolutionSol
Lindblad master eqn. or Von Neuman eqn.	mesolve	mesolveProblem	TimeEvolutionSol
Monte Carlo evolution	mcsolve	mcsolveProblem mcsolveEnsembleProblem	TimeEvolutionMCSol
Stochastic Schrödinger equation	ssesolve	ssesolveProblem ssesolveEnsembleProblem	TimeEvolutionStochasticSol
Stochastic master equation	smesolve	smesolveProblem smesolveEnsembleProblem	TimeEvolutionStochasticSol

Solving dynamics with pre-defined problems

QuantumToolbox provides two different methods to solve the dynamics. One can use the function calls listed above by either taking all the operators (like Hamiltonian and collapse operators, etc.) as inputs directly, or generating the problems by yourself and take it as an input of the function call, e.g., sesolve(prob).