Master Equation Solver: Vacuum Rabi oscillations¶

Authors: J.R. Johansson and P.D. Nation

Slight modifications: C. Staufenbiel (2022)

This notebook demonstrates how to simulate the quantum vacuum rabi oscillations in the Jaynes-Cumming model, using the Master Equation Solver qutip.mesolve. We also consider the dissipative version of the Jaynes-Cumming model, i.e., the cavity and the atom are coupled to an environment.

For more information on the theory behind the Master Equation Solver see the documentation.

Package import¶

Introduction¶

The Jaynes-Cumming model is the simplest possible model of quantum mechanical light-matter interaction, describing a single two-level atom interacting with a single electromagnetic cavity mode. The Hamiltonian for this system is (in dipole interaction form)

$H = \hbar \omega_c a^\dagger a + \frac{1}{2}\hbar\omega_a\sigma_z + \hbar g(a^\dagger + a)(\sigma_- + \sigma_+)$

or with the rotating-wave approximation

$H_{\rm RWA} = \hbar \omega_c a^\dagger a + \frac{1}{2}\hbar\omega_a\sigma_z + \hbar g(a^\dagger\sigma_- + a\sigma_+)$

where $\omega_c$ and $\omega_a$ are the frequencies of the cavity and atom, respectively, and $g$ is the interaction strength.

In this example we also consider the coupling of the Jaynes-Cummings model to an external environment, i.e., we need to solve the system using the Master Equation Solver qutip.mesolve. The coupling to the environment is described by the collapse operators (as described in the docs). Here, we consider two collapse operators for the cavity $C_1, C_2$, describing creation and annihilation of photons, and one collapse operator for the atom $C_3$.

$C_1 = \sqrt{\kappa (1+\langle n \rangle)} \; a$

$C_2 = \sqrt{\kappa \langle n \rangle}\; a^\dagger$

$C_3 = \sqrt{\gamma} \; \sigma_-$

where $\langle n \rangle$ is the average number of photons in the environment. By setting $\langle n \rangle=0$ we remove the creation of photons and only consider the annihilation of photons.

Problem parameters¶

Here we use units where $\hbar = 1$:

Setup the operators, the Hamiltonian and initial state¶

Here we define the initial state and operators for the combined system, which consists of the cavity and the atom. We make use of the tensor product, where the first part refers to the cavity and the second part to the atom. We define the atom to be in the excited state and the cavity in its ground state.

The initial state consists of the cavity ground state and the atom in the excited state. We define the collapse operator for the cavity/atom in the combined system and the Hamiltonian with and without the rotating-wave-approach.

Create a list of collapse operators that describe the dissipation¶

We create a list of collapse operators c_ops, which is later passed on to qutip.mesolve. For each of the three processes one collapse operator is defined.

Evolve the system¶

Here we evolve the system with the Lindblad master equation solver qutip.mesolve, and we request that the expectation values of the operators $a^\dagger a$ and $\sigma_+\sigma_-$ are returned by the solver by passing the list [a.dag()*a, sm.dag()*sm] as the fifth argument to the solver.

Visualize the results¶

Here we plot the excitation probabilities of the cavity and the atom (these expectation values were calculated by the mesolve above). We can clearly see how energy is being coherently transferred back and forth between the cavity and the atom.

Non-zero temperature¶

Above we set $T = 0$ and thereby discarded the photon creation by the environment. We can activate this term by setting the corresponding variable to a positive value and perform the same calculation as above. In comparison to the previous plot, we see that the cavity has more energy than the atom.

Software version:¶

Testing¶