Engineering Purely Nonlinear Coupling between Superconducting Qubits Using a Quarton

Strong nonlinear coupling of superconducting qubits and/or photons is a critical building block for quantum information processing. Because of the perturbative nature of the Josephson nonlinearity, linear coupling is often used in the dispersive regime to approximate nonlinear coupling. However, this dispersive coupling is weak and the underlying linear coupling mixes the local modes, which, for example, distributes unwanted self-Kerr nonlinearity to photon modes. Here, we use the quarton to yield purely nonlinear coupling between two linearly decoupled transmon qubits. The quarton’s zero ${\varphi }^2$ potential enables an ultrastrong gigahertz-level cross-Kerr coupling, which is an order of magnitude stronger compared to existing schemes, and the quarton’s positive ${\varphi }^4$ potential can cancel the negative self-Kerr nonlinearity of qubits to linearize them into resonators. This ultrastrong cross-Kerr coupling between bare modes of qubit-qubit, qubit-photon, and even photon-photon is ideal for applications such as single microwave photon detection, ultrafast two-qubit gates, and readout.

Figure 1

The quarton as a purely nonlinear element. (a) Schematic plot of the nonlinear ($K$) vs linear ($1/2L$) landscape of inductive superconducting elements with centrosymmetric potentials ($U$). The flux qubit has a negative and a positive inductance branch with strength dependent on ratio $\alpha$. The quarton (red) is a special flux qubit with $\alpha =0.5$ that has no linear potential. The gray region is energetically unstable (see Supplemental Material [32]). (b) Schematic line scale of the relative nonlinearity of the elements in (a). The quarton (red spider symbol) is at infinity. The respective potentials $U(\varphi )$’s are plotted below. The tilted quarton (light red spider symbol with apostrophe) is a quarton with small linear inductive potential.

Figure 2

Quarton-mediated purely nonlinear light and/or matter interactions. (a) Canonical circuit of quarton (red) nonlinearly coupling two transmons (blue) indexed $a$, $b$, and its spring-mass analog with Josephson energies as nonlinear spring coefficients $E_{J,a}$, $E_Q$, $E_{J,b}$. Strong nonlinear coupling between (b) matter-matter modes when ($E_{J,a}\ne E_Q\ne E_{J,b}$) quarton induces positive qubit nonlinearities, (c) light-matter modes when ($E_{J,a}=E_Q\ne E_{J,b}$) quarton cancels qubit $a$’s self-Kerr nonlinearity, (d) light-light modes when ($E_{J,a}=E_Q=E_{J,b}$) quarton cancels both qubits’ self-Kerr nonlinearity. Photon annihilation operators, $\widehat{a}$, $\widehat{b}$; qubit operators, ${\widehat{\sigma }}_{z,ab}$.

Figure 3

Purely nonlinear coupling (both ${\varphi }_a^2{\varphi }_b^2$ and ${\varphi }_{a,b}^3{\varphi }_{b,a}$) between qubits with tunable frequencies less than 9 GHz, mediated by (a) tilted quarton (red) with $C_J=5\mathrm{fF}$ vs (b) $C$-shunt SQUID [22] (green). (c) Nonlinear coupling $\chi$ (for ${\widehat{a}}^{†}\widehat{a}{\widehat{b}}^{†}\widehat{b}$ in ${\varphi }_a^2{\varphi }_b^2$ and similar for ${\varphi }_{a,b}^3{\varphi }_{b,a}$) scales linearly with $E_Q$ for tilted quarton, allowing for order of magnitude improvement over $C$-shunt SQUID at large $E_Q$. $C$-shunt SQUID’s linear coupling cancellation relies on RWA, which is invalid for large $E_Q$ (light green). For tilted quarton, (d) simultaneous self-Kerr $({\widehat{a}}^{†2}{\widehat{a}}^2,{\widehat{b}}^{†2}{\widehat{b}}^2)$ cancellation is possible with $E_{J,a}=E_{J,b}$, which is also used in (c). (e) Same qubits flux tuned to $E_{J,a}\ne E_{J,b}$ leads to self-Kerr cancellation of only one mode at a time.

Figure 4

Purely cross-Kerr (${\varphi }_a^2{\varphi }_b^2$)-type nonlinear coupling between qubits with tunable frequencies less than 7 GHz, mediated by (a) QRM (red) vs (b) JRM [26] (green). (c) Cross-Kerr strength $\chi$ (for ${\widehat{a}}^{†}\widehat{a}{\widehat{b}}^{†}\widehat{b}$ in ${\varphi }_a^2{\varphi }_b^2$) scales linearly with $E_Q$ for QRM, allowing for order of magnitude improvement over JRM at large $E_Q$. For QRM, (d) simultaneous self-Kerr $({\widehat{a}}^{†2}{\widehat{a}}^2,{\widehat{b}}^{†2}{\widehat{b}}^2)$ cancellation is possible with $E_{J,a}=E_{J,b}$, which is also used in (c). (e) Same qubits flux tuned to $E_{J,a}\ne E_{J,b}$ leads to self-Kerr cancellation of only one mode at a time.