03: Jaynes-Cummings Model and the Rotating Wave Approximation#

In this tutorial we will construct the Jaynes-Cummings Hamiltonian (with and without the RWA) and see how the system evolves under the Schrodinger equation (that is, without dissipation) . We will use this to investigate the limits of the RWA in the JCM.

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 full Hamiltonian for the system (in dipole interaction form) is given by

\[H = H_{\rm atom} + H_{\rm cavity} + H_{\rm interact}\]

The atom Hamiltonian we use in this case is

\[\frac{1}{2} \hbar \omega_{a} \sigma_z\]

where \(\omega_a\) is the system frequency.

Note that the Hamiltonian for the atom may take numerous forms. Any Hermitean operator on a two-level state is possible, but it is useful to nomalize the operator so that the difference between its eigenvalues is \(1\) so that \(\omega_a\) has consistent units.

The cavity Hamiltonian is given by

\[H_{\rm cavity} = \hbar \omega_c a^\dagger a\]

where \(\omega_c\) is \(\omega_a\) is the frequencies of the cavity and \(a\) and \(a^\dagger\) are the annihilation and creation operators of the cavity respectively.

The interaction Hamiltonian is given by

\[H_{\rm interact} = \hbar g(a^\dagger + a)(\sigma_- + \sigma_+)\]

or with the rotating-wave approximation

\[H_{\rm interact-RWA} = \hbar g(a^\dagger\sigma_- + a\sigma_+)\]

where \(\sigma_-\) and \(\sigma_+\) are the annihilation and creation operators for the atom respectively.

Note that in this notebook we will work in units where \(\hbar=1\).

Tasks#

Imports#

%matplotlib inline
import matplotlib.pyplot as plt

import qutip
import numpy as np

Construct the Hamiltonian#

  • add variables for the atom and cavity parameters

  • create the operators for the JCM Hamiltonian

  • combine into the JCM Hamiltonian (no RWA)

  • look at the energy eigenvalues and eigenstates of the Hamiltonian

Here are some example parameter values to start with:

\( \omega_c = 2 \pi \\ \omega_a = 2 \pi \\ g = 0.05 \cdot 2 \pi \\ \)

Solve the Schrodinger equation#

  • create the initial state of the system (use the state with no photons and the spin system in its excited state)

  • evolve the system for some time, saving the result.

If you need to remind yourself of how sesolve, remember that you can type qutip.sesolve? into a notebook cell to bring up the documentation.

Visualise the evolution#

  • create expectation operators for observing the state of the system.

  • add these to sesolve

  • plot the expectation values together on a set of axes

  • change the value of g and see how it affects the period

Two good expectation operators to use are the projectors on the light and matter sub-systems.

Compare the RWA and non-RWA#

  • construct another Hamiltonian that uses the RWA

  • evolve the system under the RWA Hamiltonian.

  • add the results to the plot

  • experiment with parameters to determine where the RWA non-RWA diverges