Meaning of 'O_i' and 'Δ_i' terms in PulseBackend Hamiltonians

Question

For OpenPulse enabled backends, the Hamiltonian can be retrieved via its configuration. The configuration holds a dictionary containing for example the Hamiltonian as a LaTeX string. Example code:

from qiskit.providers.fake_provider import FakeBelem
backend = FakeBelem()
print(backend.configuration().hamiltonian['h_latex'])

When rendered, the Hamiltonian (for fake backend FakeBelem) is defined as seen in this picture:

The Hamiltonian for other backends, for example the real ibmq_guadalupe look similar, just with more coupling(J)- and control channel(U)-Terms.

What is the meaning of the $\Delta_i$ and $O_i$ symbols (marked red, first line) in these kind of Hamiltonians?

Answer 1

After some research and feedback from the IBM slack channel, I figured out the meaning:

• $\Delta_i$ represents the anharmonicity of the transmon qubit and has units of frequency
• $O_i$ is the number operator, which represents the total number of particles in the system and is unitless

With these definitions, the term in the red box can be described as a frequency shift, if the number operator $O_i > 1$, taking into account the possible higher state transitions possible for transmons, for example: $|1\rangle \rightarrow |2\rangle$.

If $O_i \le 1$ the term evaluates to zero indicating qubit (only two states) behaviour.