[1]:

import quanguru as qg
import numpy as np

16 - Jaynes-Cummings model of light-matter interaction#

Before demonstrating the Jaynes-Cummings (JC) Hamiltonian in QuanGuru, we provide some background for the JC Hamiltonian in this tutorial.

The JC Hamiltonian is commonly written as

$H_{JC} = \hbar\omega_{c} a^{\dagger}a + \frac{1}{2}\hbar\omega_{q}\sigma_{z} + \hbar g(a^{\dagger}\sigma_{-} + a\sigma_{+})$

where $\sigma_{\pm} = (\sigma_{x} \pm i\sigma_{y})/2$ are raising/lowering operators for a two-level system, $\sigma_{\mu}$ are the Pauli spin operators with $\mu\in\{x,y,z\}$ , $a^{\dagger}$ and $a$ are the creation and annihilation operators for the field mode, and $\omega_{c}$ , $\omega_{q}$ , and $g$ are the cavity-field, qubit, and coupling (angular-) frequencies, respectively. Note that the above Hamiltonian is written in a common notation where the order of the sub-system Hilbert spaces is implicitly defined by the ordering in the coupling. This means, for example, that the composite form of the number operator is written explicitly as $(a^{\dagger}a)\otimes 1_{2,2}$ (similarly, $a^{\dagger}\otimes\sigma_{-}$ and $1_{d,d}\otimes\sigma_{z}$ , where $d$ is the truncation dimension for the cavity operators and $\otimes$ is the tensor product).

Here, the eigenstates of the qubit are

Excited state : $|e\rangle = \left[\begin{array}{ll} 1 \\ 0 \end{array}\right] \text{ (with eigenvalue }\frac{1}{2}\omega_{q})$
Ground state: $|g\rangle = \left[\begin{array}{ll} 0 \\ 1 \end{array}\right] \text{ (with eigenvalue }-\frac{1}{2}\omega_{q})$

Together with the $\sigma_{\pm} = (\sigma_{x} \pm i\sigma_{y})/2$ definition, we have

$\sigma_{+} = \left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right]$ so that $\sigma_{+}|e\rangle = 0$ and $\sigma_{+}|g\rangle = |e\rangle$
$\sigma_{-} = \left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right]$ so that $\sigma_{-}|e\rangle = |g\rangle$ and $\sigma_{-}|g\rangle = 0$

Finally, the matrix representations of JC Hamiltonian (with $d=10$ ) for two different cases of sub-system Hilbert space orders are

$H_{JC} = \hbar\omega_{c} a^{\dagger}a\otimes 1_{2,2} + \frac{1}{2}\hbar\omega_{q}1_{d,d}\otimes\sigma_{z} + \hbar g(a^{\dagger}\otimes\sigma_{-} + a\otimes\sigma_{+})$
$H_{JC} = \hbar\omega_{c}1_{2,2} \otimes a^{\dagger}a + \frac{1}{2}\hbar\omega_{q}\sigma_{z}\otimes 1_{d,d} + \hbar g(\sigma_{-}\otimes a^{\dagger} + \sigma_{+}\otimes a)$

In above convention, we see that the zero-excitation state $|0, g\rangle$ (or, $|g, 0\rangle$ in the qubit first order of sub-spaces) appears at $(2, 2)$ (or, at $(d+1, d+1)$ in the qubit first order of sub-spaces) index of the matrix representation. There are some other counter-intuitive details in above convention, and an alternative convention for JC Hamiltonian is to write the qubit term with a minus ( $- \frac{1}{2}\hbar\omega_{q}\sigma_{z}$ ) so that the excited/ground state definition is switched

Excited state : $|g\rangle = \left[\begin{array}{ll} 1 \\ 0 \end{array}\right] \text{ (with eigenvalue }-\frac{1}{2}\omega_{q})$
Ground state: $|e\rangle = \left[\begin{array}{ll} 0 \\ 1 \end{array}\right] \text{ (with eigenvalue }\frac{1}{2}\omega_{q})$

However, in this alternative case, we also need to make some changes for the $\sigma_{\pm}$ operators, otherwise, together with the $\sigma_{\pm} = (\sigma_{x} \pm i\sigma_{y})/2$ definition, we have

$\sigma_{+} = \left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right]$ so that $\sigma_{+}|g\rangle = 0$ and $\sigma_{+}|e\rangle = |g\rangle$
$\sigma_{-} = \left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right]$ so that $\sigma_{-}|g\rangle = |e\rangle$ and $\sigma_{-}|e\rangle = 0$

Here, we could introduce some unconventional definitions for $\sigma_{\pm}$ by switching the $\pm$ in their definition to $\mp$ , but we have a better alternative that is to replace $\sigma_{-}$ and $\sigma_{+}$ with 2-dimensional truncations of $a$ and $a^{\dagger}$ , respectively. Then, the matrix representations of JC Hamiltonian (with $d=10$ ) for two different cases of sub-system Hilbert space orders are

$H_{JC} = \hbar\omega_{c} a^{\dagger}a\otimes 1_{2,2} - \frac{1}{2}\hbar\omega_{q}1_{d,d}\otimes\sigma_{z} + \hbar g(a^{\dagger}\otimes a_{2,2} + a\otimes a_{2,2}^{\dagger})$
$H_{JC} = \hbar\omega_{c}1_{2,2} \otimes a^{\dagger}a + \frac{1}{2}\hbar\omega_{q}\sigma_{z}\otimes 1_{d,d} + \hbar g(a_{2,2}\otimes a^{\dagger} + a_{2,2}^{\dagger}\otimes a)$

and, in the first case, we have $|0, g\rangle$ at $(0, 0)$ .

15 - Time dependent Hamiltonian 2

16 - Jaynes-Cummings Hamiltonian (in QuanGuru)