Benchmarks

A cavity mode, modeled as a Fock space with a certain cutoff, is driven on resonance with a classical laser. Since photon loss is included, the system is an open system and evolves according to a master equation $$\dot{\rho} = i \big[ \rho, H \big] + \kappa\big(a\rho a^\dagger - \frac{1}{2} a^\dagger a \rho - \frac{1}{2} \rho a^\dagger a \big)$$ with the Hamiltonian $$H = \Delta_c a^\dagger a + \eta (a + a^\dagger).$$

This benchmark measures the time it takes to create all necessary operators and states, perform a time-evolution according to a master equation and calculate the expectation values of the number operator at certain times. The Schrödinger time-evolution neglects the loss, obviously.

An atom, modeled as a dipole, is resonantly coupled to a cavity mode, modeled as a Fock space with a certain cutoff. Photon loss and incoherent driving by a thermal bath are included and taken into account by the master equation $$\dot{\rho} = i \big[ \rho, H \big] + \kappa \left( 1 + \bar n \right) \, \mathcal{D} \left[ a \right] \rho + \kappa \bar n \, \mathcal{D} \left[ a^\dagger \right] \rho + \gamma \, \mathcal{D} \left[ \sigma^- \right] \rho$$ with the Hamiltonian $$H = \omega_c \, a^\dagger a + \frac{\omega_a}{2} \, \sigma^z + g \big( a^\dagger \sigma^- + a \sigma^+ \big).$$

Again, we create all necessary objects, perform a time evolution and calculate the expectation value of the number operator in the photon mode at particular times. For the Schrödinger equation the incoherent dynamics are neglected.

A Gaussian wave packet propagates in a harmonic potential while simultaneously being subject to friction, which is accounted for by the master equation $$\dot{\rho} = i \big[ \rho, H \big] + \nu \mathcal{D} \left[ \sqrt{\frac{\omega m}{2}} \, x + \frac{i}{\sqrt{2 \omega m}} \, p \right] \rho$$ and the Hamiltonian $$H = \frac{p^2}{2m} + \frac{m \omega^2}{2} \, x^2.$$

We measure the time it takes to create all necessary objects, perform a time evolution and calculate the expectation value of the position operator at certain times.