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Time-evolution equations

In quantum mechanics, time evolution describes how a quantum state changes under a Hamiltonian. Pure states are typically described by the Schrödinger equation, while mixed states are described by master equations. For closed systems, this is the von Neumann equation, whereas open systems require the Lindblad master equation to capture decoherence and dissipation.

In QruiseML, the time-evolution equation is specified through the first argument of Problem.

Imaginary unit

In the following time-evolution equations, \(i\) is the imaginary unit, with \(i^2=-1\).

Schrödinger equation

Pure states in a closed quantum system can be simulated by solving the Schrödinger equation, which describes the unitary evolution of the wavefunction, \(|\psi(t)\rangle\), over time:

\[ i\frac{\partial}{\partial t} |\psi(t)\rangle = H(t) |\psi(t)\rangle. \]

Here, \(\frac{\partial}{\partial t}\) is the partial derivative with respect to time \(t\), and \(H(t)\) is the Hamiltonian operator.

To simulate a closed quantum system, set

Problem("Schroedinger", ...)

Master equations

In QruiseML, master-equation dynamics are specified using "MasterEquation" as the first argument in Problem. The resulting equation depends on whether collapse operators (c_ops) are provided:

Collapse operators

Collapse operators describe dissipative processes acting on a quantum system. Common examples include energy relaxation (\(L=\sigma_-\)), dephasing (\(L=\sigma_z\)), and cavity loss (\(L=a\)).

Von Neumann equation

Mixed states in a closed quantum system can be simulated by solving the von Neumann equation, which describes the time evolution of the system's density matrix, \(\rho(t)\):

\[ i\frac{\partial \rho(t)}{\partial t} = \left[ H(t), \rho(t) \right] . \]

Here, \(\frac{\partial}{\partial t}\) is the partial derivative with respect to time \(t\), and \(H(t)\) is the Hamiltonian operator.

In QruiseML, the von Neumann equation is chosen using "MasterEquation" in Problem without including collapse operators:

Problem("MasterEquation", ...)

Lindblad equation

Open quantum systems can be simulated using the Lindblad master equation, which includes both unitary evolution and dissipative processes. It describes the time evolution of the system's density matrix, \(\rho(t)\):

\[ i\frac{\partial}{\partial t}\rho_t = [H(t), \rho_t] + \sum_i \gamma_i \left(L_i \rho_t L_i^\dagger - \frac{1}{2}\left\{L_i^\dagger L_i, \rho_t \right\}\right), \]

where \(\frac{\partial}{\partial t}\) is the partial derivative with respect to time \(t\), \(H(t)\) is the Hamiltonian operator, \(\gamma_i\) are decay rates (associated with decoherence or dissipation), and \(L_i\) are the corresponding collapse operators.

In QruiseML, the Lindblad equation is chosen using "MasterEquation" in Problem and providing one or more collapse operators through c_ops:

Problem("MasterEquation", ..., c_ops=(c1, c2, ...))