Figure 1

(a) Schematic representation of the superconducting ring resonator, comprising two SC transmission line (TL) segments coupled by two SQUIDs that can be externally controlled by two different applied magnetic fluxes. ac modulation of ${\mathrm{SQUID}}_1$ will accomplish the action of beam splitting between the two fundamental modes of the ring resonator. The device can optionally be tuned by a stub on the midline as shown. (b) Equivalent circuit, consisting of two parametrically coupled $LC$ resonators. The modulation of the ${\mathrm{SQUID}}_1$ impedance is represented by the time-dependent inductor $K(t)$. The $LC$ circuit parameters are chosen so that the impedance of the two parallel resonators matches the two-terminal impedance, looking down into the $AB$ port, of the ring resonator, in a frequency band including the two fundamental modes. (c) The $AB$-port impedance of the resonator and of the equivalent circuit. The unmodulated fundamental mode frequencies ${\omega }_{1,2}$ are given by the two zeros of this impedance.Reuse & Permissions

Figure 2

To good approximation, the quantum evolution is confined to the Bloch sphere defined by the $|01⟩$ and $|10⟩$ states. Counterrotating terms in the Hamiltonian at frequency $2\Delta \omega$ perturb the simple Bloch-sphere (red) evolution by terms of order $f/\Delta \omega$. Bloch-Siegert oscillations (blue; with cusp) cause nutation of the Bloch vector superimposed on regular precession in the rotating frame. The device parameters for the ring resonator are as given in the text.Reuse & Permissions

Figure 3

Fidelity of the beam-splitting gate as a function of pulse duration, using a naive rotating wave approximation (RWA), a rotating wave approximation including some additional slow-rotating terms perturbatively, capturing the occurrence of Bloch-Siegert oscillations (BSO), in comparison with the numerically exact solution. Note that since the BSO result reproduces the exact result with large precision, the two lines lie essentially on top of each other in the main panel and can only be distinguished in the magnified plot (inset). Because of its simple perturbative nature, the BSO fidelity can (and does) exceed 1. The fidelity can be further optimized by carefully adjusting the resonator frequencies and thus the phase of the BSO. The device parameters for the ring resonator are as given in the text, except that stub tuning is used to raise the mode splitting to $\Delta \omega \approx 260\mathrm{MHz}$.Reuse & Permissions