Schrieffer–Wolff transform
These notes derive the effective dispersive Hamiltonian for a two-level qubit coupled to two resonator modes via a Tavis–Cummings interaction, using the Schrieffer–Wolff (SW) transformation. We start from the full qubit–resonator Hamiltonian, define an anti-Hermitian generator in the dispersive regime, and obtain the cross-Kerr and resonator–resonator coupling terms to second order in $g_i / \Delta_i$.
1. Hamiltonian
We take an effective two-level system (qubit) in the basis $\{\lvert e\rangle, \lvert g\rangle\}$. The Pauli operators are
\[ \hat{\sigma}_z = \lvert e\rangle\langle e\rvert - \lvert g\rangle\langle g\rvert \equiv \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \tag{1} \]
\[ \hat{\sigma}_+ = \lvert e\rangle\langle g\rvert \equiv \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad \hat{\sigma}_- = \lvert g\rangle\langle e\rvert \equiv \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}. \tag{2} \]
These obey the usual commutation relations
\[ [\hat{\sigma}_+,\hat{\sigma}_-] = \hat{\sigma}_z, \qquad [\hat{\sigma}_z,\hat{\sigma}_\pm] = \pm 2 \hat{\sigma}_\pm. \tag{3} \]
The two resonators are modeled as bosonic modes with annihilation and creation operators $\hat{b}_i$, $\hat{b}_i^\dagger$ ($i = 1,2$), satisfying
\[ [\hat{b}_i,\hat{b}_j^\dagger] = \delta_{ij}, \qquad [\hat{b}_i,\hat{b}_j^\dagger \hat{b}_j] = \delta_{ij}\hat{b}_j. \tag{4} \]
The number operator for each resonator mode is
\[ \hat{n}_i = \hat{b}_i^\dagger \hat{b}_i. \tag{5} \]
The free Hamiltonian for the qubit and the two resonators is
\[ \hat{H}_0 = \frac{\hbar \omega_q}{2}\,\hat{\sigma}_z + \sum_{i=1}^2 \hbar\omega_i \hat{b}_i^\dagger \hat{b}_i, \tag{6} \]
where $\omega_q$ is the qubit frequency and $\omega_1, \omega_2$ are the resonator frequencies. The capacitive (transverse) coupling between the qubit and the resonators is
\[ \hat{H}_\mathrm{int} = \sum_{i=1}^2 \hbar g_i \big(\hat{b}_i + \hat{b}_i^\dagger\big) \big(\hat{\sigma}_+ + \hat{\sigma}_-\big). \tag{7} \]
In the regime where all bare frequencies are much larger than the coupling strengths, $\omega_q,\omega_i \gg g_i$, we can apply the rotating wave approximation (RWA). The interaction Hamiltonian becomes a Tavis–Cummings interaction,
\[ \hat{H}_\mathrm{int} = \sum_{i=1}^2 \hbar g_i \big( \hat{b}_i \hat{\sigma}_+ + \hat{b}_i^\dagger \hat{\sigma}_- \big). \tag{8} \]
The total Hamiltonian is therefore
\[ \hat{H} = \hat{H}_0 + \hat{H}_\mathrm{int} = \frac{\hbar \omega_q}{2}\,\hat{\sigma}_z + \sum_{i=1}^2 \hbar\omega_i \hat{b}_i^\dagger \hat{b}_i + \sum_{i=1}^2 \hbar g_i \big( \hat{b}_i \hat{\sigma}_+ + \hat{b}_i^\dagger \hat{\sigma}_- \big). \tag{9} \]
2. Schrieffer–Wolff transformation
We define the detuning of each resonator mode from the qubit as
\[ \Delta_i = \omega_q - \omega_i. \tag{10} \]
We work in the dispersive regime
\[ \lvert \Delta_i \rvert \gg g_i \quad\Longleftrightarrow\quad \left\lvert \frac{g_i}{\Delta_i} \right\rvert \ll 1, \tag{11} \]
which allows us to treat the coupling perturbatively via a Schrieffer–Wolff transformation. We choose an anti-Hermitian generator
\[ \hat{S} = \sum_{i=1}^2 \lambda_i \big( \hat{b}_i \hat{\sigma}_+ - \hat{b}_i^\dagger \hat{\sigma}_- \big), \tag{12} \]
and perform the unitary transformation
\[ \hat{H}_\mathrm{SW} = e^{\hat{S}} \hat{H} e^{-\hat{S}} = \hat{H} + [\hat{S},\hat{H}] + \frac{1}{2}[\hat{S},[\hat{S},\hat{H}]] + \cdots. \tag{13} \]
We choose $\hat{S}$ such that it cancels the first-order interaction:
\[ [\hat{S},\hat{H}_0] = - \hat{H}_\mathrm{int}. \tag{14} \]
With this choice, the SW-transformed Hamiltonian to second order becomes
\[ \hat{H}_\mathrm{SW} = \hat{H}_0 + \frac{1}{2} [\hat{S},\hat{H}_\mathrm{int}]. \tag{15} \]
3. Commutator $[\hat{S},\hat{H}_0]$ and choice of $\lambda_i$
First we evaluate the commutator
\[ \begin{aligned}{} [\hat{S},\hat{H}_0] &= \sum_{i=1}^2 \lambda_i \Big[ \hat{b}_i\hat{\sigma}_+ - \hat{b}_i^\dagger \hat{\sigma}_-, \hat{H}_0 \Big] \\ &= \sum_{i=1}^2 \lambda_i \Big( [\hat{b}_i\hat{\sigma}_+,\hat{H}_0] - [\hat{b}_i^\dagger \hat{\sigma}_-,\hat{H}_0] \Big). \end{aligned} \tag{16} \]
Using the commutation relations (3) and (4), we find
\[ \begin{aligned}{} [\hat{b}_i\hat{\sigma}_+,\hat{H}_0] &= \left[ \hat{b}_i\hat{\sigma}_+, \frac{\hbar\omega_q}{2}\hat{\sigma}_z + \sum_{j=1}^2 \hbar\omega_j \hat{b}_j^\dagger \hat{b}_j \right] \\ &= \frac{\hbar\omega_q}{2} \hat{b}_i [\hat{\sigma}_+,\hat{\sigma}_z] + \sum_{j=1}^2 \hbar\omega_j [\hat{b}_i,\hat{b}_j^\dagger \hat{b}_j]\hat{\sigma}_+ \\ &= \frac{\hbar\omega_q}{2} \hat{b}_i(-2\hat{\sigma}_+) + \hbar\omega_i \hat{b}_i \hat{\sigma}_+ \\ &= -\hbar(\omega_q - \omega_i) \hat{b}_i\hat{\sigma}_+ = -\hbar\Delta_i \hat{b}_i\hat{\sigma}_+, \end{aligned} \tag{17} \]
Similarly,\[ \begin{aligned}{} [\hat{b}_i^\dagger \hat{\sigma}_-,\hat{H}_0] &= \left[ \hat{b}_i^\dagger \hat{\sigma}_-, \frac{\hbar\omega_q}{2}\hat{\sigma}_z + \sum_{j=1}^2 \hbar\omega_j \hat{b}_j^\dagger \hat{b}_j \right] \\ &= \frac{\hbar\omega_q}{2} \hat{b}_i^\dagger [\hat{\sigma}_-,\hat{\sigma}_z] + \sum_{j=1}^2 \hbar\omega_j [\hat{b}_i^\dagger,\hat{b}_j^\dagger \hat{b}_j]\hat{\sigma}_- \\ &= \frac{\hbar\omega_q}{2} \hat{b}_i^\dagger(2\hat{\sigma}_-) - \hbar\omega_i \hat{b}_i^\dagger \hat{\sigma}_- \\ &= \hbar(\omega_q - \omega_i) \hat{b}_i^\dagger \hat{\sigma}_- = \hbar\Delta_i \hat{b}_i^\dagger \hat{\sigma}_-, \end{aligned} \tag{18} \]
Putting the above together, we obtain,
\[ \begin{aligned}{} [\hat{S},\hat{H}_0] &= \sum_{i=1}^2 \lambda_i \big( -\hbar\Delta_i \hat{b}_i\hat{\sigma}_+ - \hbar\Delta_i \hat{b}_i^\dagger \hat{\sigma}_- \big) \\ &= -\hbar \sum_{i=1}^2 \lambda_i \Delta_i \big( \hat{b}_i\hat{\sigma}_+ + \hat{b}_i^\dagger \hat{\sigma}_- \big). \end{aligned} \tag{19} \]
Comparing with Eq. (8), the condition (14) is satisfied if
\[ \lambda_i = \frac{g_i}{\Delta_i}. \tag{20} \]
With this choice, the SW Hamiltonian is given by Eq. (15). The truncation to second order is valid in the dispersive limit (11).
4. Evaluating $\frac{1}{2}[\hat{S},\hat{H}_\mathrm{int}]$
We now compute
\[ \begin{aligned}{} [\hat{S},\hat{H}_\mathrm{int}] &= \sum_{i,j=1}^2 \lambda_i g_j \Big[ \hat{b}_i\hat{\sigma}_+ - \hat{b}_i^\dagger \hat{\sigma}_-, \hat{b}_j\hat{\sigma}_+ + \hat{b}_j^\dagger \hat{\sigma}_- \Big] \\ &= \sum_{i,j=1}^2 \lambda_i g_j \left( [\hat{b}_i\hat{\sigma}_+,\hat{b}_j^\dagger \hat{\sigma}_-] - [\hat{b}_i^\dagger \hat{\sigma}_-,\hat{b}_j\hat{\sigma}_+] \right). \end{aligned} \tag{21} \]
The individual commutators are
\[ \begin{aligned}{} [\hat{b}_i\hat{\sigma}_+,\hat{b}_j^\dagger \hat{\sigma}_-] &= [\hat{b}_i,\hat{b}_j^\dagger]\hat{\sigma}_+\hat{\sigma}_- + \hat{b}_j^\dagger \hat{b}_i [\hat{\sigma}_+,\hat{\sigma}_-] \\ &= \delta_{ij}\hat{\sigma}_+\hat{\sigma}_- + \hat{b}_j^\dagger \hat{b}_i \hat{\sigma}_z, \end{aligned} \tag{22} \]
\[ \begin{aligned}{} [\hat{b}_i^\dagger \hat{\sigma}_-,\hat{b}_j\hat{\sigma}_+] &= [\hat{b}_i^\dagger,\hat{b}_j]\hat{\sigma}_-\hat{\sigma}_+ + \hat{b}_j \hat{b}_i^\dagger [\hat{\sigma}_-,\hat{\sigma}_+] \\ &= -\delta_{ij}\hat{\sigma}_-\hat{\sigma}_+ - \hat{b}_j \hat{b}_i^\dagger \hat{\sigma}_z. \end{aligned} \tag{23} \]
Adding them,
\[ \begin{aligned}{} \frac{1}{2}[\hat{S},\hat{H}_\mathrm{int}] &= \frac{1}{2} \sum_{i,j=1}^2 \hbar \lambda_i g_j \Big\{ (\hat{\sigma}_+\hat{\sigma}_- + \hat{\sigma}_-\hat{\sigma}_+)\delta_{ij} + (\hat{b}_j^\dagger \hat{b}_i + \hat{b}_j \hat{b}_i^\dagger)\hat{\sigma}_z \Big\}. \end{aligned} \tag{24} \]
For $i=j$,
\[ \begin{aligned}{} \frac{1}{2}[\hat{S},\hat{H}_\mathrm{int}]_{i=j} &= \sum_{i=1}^2 \frac{\hbar \lambda_i g_i}{2} (\hat{\sigma}_+\hat{\sigma}_- + \hat{\sigma}_-\hat{\sigma}_+) + \sum_{i=1}^2 \hbar \lambda_i g_i \hat{b}_i^\dagger \hat{b}_i \hat{\sigma}_z. \end{aligned} \tag{25} \]
The first sum is a constant shift ($\hat{\sigma}_+\hat{\sigma}_- + \hat{\sigma}_-\hat{\sigma}_+ = \hat{\mathbb{1}}$), while the second sum describes a dispersive (cross-Kerr) shift of the resonator frequencies conditioned on the qubit state. For $i\neq j$,
\[ \begin{aligned}{} \frac{1}{2}[\hat{S},\hat{H}_\mathrm{int}]_{i\neq j} &= \frac{\hbar \lambda_1 g_2}{2} \big( \hat{b}_1^\dagger \hat{b}_2 + \hat{b}_2 \hat{b}_1^\dagger \big)\hat{\sigma}_z + \frac{\hbar \lambda_2 g_1}{2} \big( \hat{b}_1^\dagger \hat{b}_2 + \hat{b}_2 \hat{b}_1^\dagger \big)\hat{\sigma}_z \\ &= \hbar \frac{\lambda_1 g_2 + \lambda_2 g_1}{2} \big( \hat{b}_1^\dagger \hat{b}_2 + \hat{b}_2 \hat{b}_1^\dagger \big)\hat{\sigma}_z. \end{aligned} \tag{26} \]
Define the effective resonator–resonator coupling
\[ J = \hbar \frac{\lambda_1 g_2 + \lambda_2 g_1}{2}, \tag{27} \]
so that
\[ \frac{1}{2}[\hat{S},\hat{H}_\mathrm{int}]_{i\neq j} = J \big( \hat{b}_1^\dagger \hat{b}_2 + \hat{b}_2 \hat{b}_1^\dagger \big)\hat{\sigma}_z. \tag{28} \]
Finally, introducing the dispersive shifts
\[ \chi_i = \lambda_i g_i = \frac{g_i^2}{\Delta_i}, \tag{29} \]
the Schrieffer–Wolff transformed Hamiltonian can be written as
\[ \begin{aligned}{} \hat{H}_\mathrm{SW} &= \sum_{i=1}^2 \hbar\omega_i \hat{b}_i^\dagger \hat{b}_i + \frac{\hbar \tilde{\omega}_q}{2} \hat{\sigma}_z + \sum_{i=1}^2 \hbar\chi_i \hat{b}_i^\dagger \hat{b}_i \hat{\sigma}_z \\ &\quad + J \big( \hat{b}_1^\dagger \hat{b}_2 + \hat{b}_2 \hat{b}_1^\dagger \big)\hat{\sigma}_z, \end{aligned} \tag{30} \]
where the Lamb-shifted qubit frequency is
\[ \tilde{\omega}_q = \omega_q + \chi_1 + \chi_2. \]
Introducing the vector notation
\[ \hat{\mathbf{b}} = \begin{pmatrix} \hat{b}_1 \\[0.2em] \hat{b}_2 \end{pmatrix}, \qquad \hat{\mathbf{b}}^\dagger = \begin{pmatrix} \hat{b}_1^\dagger & \hat{b}_2^\dagger \end{pmatrix}, \tag{31} \]
we can rewrite the interaction terms compactly as
\[ \hat{H}_\mathrm{SW} = \sum_{i=1}^2 \hbar\omega_i \hat{b}_i^\dagger \hat{b}_i + \frac{\hbar \tilde{\omega}_q}{2} \hat{\sigma}_z + \hat{\mathbf{b}}^\dagger \begin{pmatrix} \chi_1 & J \\ J & \chi_2 \end{pmatrix} \hat{\mathbf{b}}\,\hat{\sigma}_z. \tag{32} \]