Two-photon coupling via Josephson element II: Interaction renormalizations and cross-Kerr coupling

TL;DR

This work develops a comprehensive theory of interactions between a resonator and a phase-qubit atom coupled through a symmetric dc SQUID, focusing on the two-photon resonance regime. By deriving an approximate quantum Hamiltonian and employing Schrieffer-Wolff transformations, the authors obtain a renormalized two-photon coupling $\tilde{g}_2$, a cross-Kerr coupling $\tilde{K}$, and an anharmonicity $\tilde{\varXi}_\mathrm{a}$, all dressed by nonresonant interactions and vacuum fluctuations. The analysis reveals that cross-Kerr and optomechanical-type renormalizations can be comparable to bare couplings, and that the two-photon interaction can reach tens of MHz with realistic rf-SQUID parameters and seven metastable atom levels. Numerical estimates for a phase-qubit rf SQUID demonstrate how coupler bias, resonator/atom frequencies, and bias fluxes control resonance conditions and interaction strengths, with practical implications for photodetection, nondemolition readout, and quantum information processing. The work also discusses the validity of the bosonic-atom approximation, potential quasispin alternatives, and environmental effects, outlining future directions for experimental verification and application-driven optimizations.

Abstract

We study the interactions mediated by symmetric superconducting quantum interference device (SQUID), their renormalizations, and applicability of the anharmonic oscillator model for a coupled phase qubit. The coupling SQUID can switch between single- or two-photon interaction in situ. We consider a coupled resonator and an rf SQUID. The latter dwells in the vicinity of its metastable well holding a number of anharmonic energy states and acts as an artificial atom known as the phase qubit. Apart from the linear and two-photon couplings, interactions of optomechanical type and a cross-Kerr coupling arise. Near the two-photon resonance, we calculate the renormalizations due to nonresonant interactions, which are more prominent with the higher Josephson energy of the coupler. We interpret the renormalizations by depicting some of the virtual processes involved. That also allows us to determine the minimal amount of metastable states in the phase qubit for the renormalization formulas to hold.

Figures (6)

• Figure 1: A resonator and an rf SQUID in the regime of phase qubit interact through the SQUID coupler. The system evolves around the equilibrium phase differences $\phi_\mathrm{r}^\mathrm{min}$ and $\phi_\mathrm{a}^\mathrm{min}$. The respective varying departures are $\varphi_\mathrm{r}$ and $\varphi_\mathrm{a}$. Conjugated to them, dynamic variables $n_\mathrm{r}$ and $n_\mathrm{a}$ are the Cooper pair number at the full capacitances. $\delta = \phi_\mathrm{r}^\mathrm{min} - \phi_\mathrm{a}^\mathrm{min}$ is the equilibrium phase difference over the coupler. $\Phi_\mathrm{c}$ and $\Phi_\mathrm{e}$ are the coupler and rf SQUID bias fluxes. Cartoons show potentials and energy levels of the resonator (left) and the phase qubit (right).
• Figure 2: Some corrections to the bare linear and cross-Kerr couplings. (I) Inductive single-photon corrections arise due to combination of the optomechanical interaction $G_2$ with the energy quadratic in the atom variables (1, red) and the atom cubic nonlinearity $X_\mathrm{a}$ (2, blue). (II) Cross-Kerr corrections arise due to optomechanical coupling $g_2$ with the energy quadratic in the resonator variables (3, green) combined with the atom cubic nonlinearity. Each part of the process contributes to its full amplitude $\sim n_\mathrm{r} n_\mathrm{a}$. First index in a state label corresponds to the number of the resonator energy level, whereas the second one labels the atom level. Dashed lines correspond to an alternative process path that climbs the atom energy ladder as high as possible. The wavy lines stand for creation or annihilation of a resonator photon, while the straight lines denote the atom processes. Each diagram can be read from left to right and vice versa, as indicated by the arrows at the start and end of the respective processes. Loops contract creation and annihilation, such that only vacuum parts contribute---i.e. the unity in $bb^\dag = 1 + b^\dag b$. For brevity, vacuum fluctuations are equal in the atom and resonator, $\varphi_{\mathrm{a},\mathrm{zpf}} = \varphi_{\mathrm{r},\mathrm{zpf}} = \varphi_\mathrm{zpf}$ and $n_{\mathrm{a},\mathrm{zpf}} = n_{\mathrm{r},\mathrm{zpf}} = n_\mathrm{zpf}$. Each virtual process brings a relevant factor (in corresponding colors and positions) to the full amplitude.
• Figure 3: Some corrections to the two-photon coupling. (a) Virtual transitions $0e \leftrightarrow 1e$ and $1e \leftrightarrow 2g$ mediated by the number-conserving linear coupling $g_-$ (4$-$ black) and the optomechanical coupling $G_2$ (1, red). (b) Virtual transitions $2g \leftrightarrow 1e$ with the number-conserving linear coupling and $1e \leftrightarrow 0e$ with the atom cubic nonlinearity $X_\mathrm{a}$ (2, blue) and the number-nonconserving part of the linear inductive coupling $g_1^i$ (5, orange). (c) Virtual transitions $0e \leftrightarrow 1e$ and $1e \leftrightarrow 2g$ mediated by the two--atom-excitation coupling $G_2$ and the number-nonconserving linear inductive interaction $g_+$ (4$+$ black). Other notations as in Fig. \ref{['figDiagramsLinearAndCrossKerr']}.
• Figure 4: (Top panel) Double-well potential (thick black) of the uncoupled rf SQUID as given by Eq. \ref{['eqAtomPotential']}. Other curves show the potential in the harmonic (dotted), cubic (thin dashed), and quartic (thin solid) approximations. The energy origin is at the bottom of the left well. Shaded rectangle highlights the region shown in the other panel. (Bottom panel) Metastable well hosting levels $|{g}\rangle$, $|{e}\rangle$, $|{f}\rangle$, $|{h}\rangle$, and three others. Solid lines indicate the energies of metastable states and several over-the-barrier states, selected among numerical solutions of the stationary Schrödinger equation for the full potential. Dashed lines indicate the energy levels according to the perturbation theory as in Eq. \ref{['eqFreqs']}. Parameters: $E^\mathrm{a}_C/E^\mathrm{a}_L = 3.950 \times 10^{-4}$, $E^\mathrm{a}_\mathrm{J}/E^\mathrm{a}_L = 3.3424$, and $\Phi_\mathrm{e} = 0.7150\Phi_0$.
• Figure 5: Dependence of the resonator and atom frequencies on the flux biases: the coupler bias $\Phi_\mathrm{c}$ in the upper panel and the atom bias $\Phi_\mathrm{e}$ in the lower panel. Resonator frequency $\tilde{\omega}_\mathrm{r}$ according to Eq. \ref{['eqFreqs']}; atom frequency $\tilde{\omega}_\mathrm{a}$ determined numerically as described in the text. Blue area is the region from $2\tilde{\omega}_\mathrm{r} - \tilde{g}_2$ to $2\tilde{\omega}_\mathrm{r} + \tilde{g}_2$, where $\tilde{g}_2$ is the two-photon coupling strength. Gray shade marks the minimal region of the $0e \leftrightarrow 2g$ resonance, i.e. where the detuning is smaller than $\tilde{g}_2$. Circuit parameters as in the text. Thin lines show the frequencies when the atom Josephson energy departs from the required value by the indicated fraction.