Selective coupling of superconducting charge qubits mediated by a tunable stripline cavity

We theoretically investigate selective coupling of superconducting charge qubits mediated by a superconducting stripline cavity with a tunable resonance frequency. The frequency control is provided by a flux-biased dc superconducting quantum interference device attached to the cavity. Selective entanglement of the qubit states is achieved by sweeping the cavity frequency through the qubit-cavity resonances. The circuit is able to accommodate several qubits and allows one to keep the qubits at their optimal points with respect to decoherence during the whole operation. We derive an effective quantum Hamiltonian for the basic, two-qubit-cavity system, and analyze appropriate circuit parameters. We present a protocol for performing Bell inequality measurements, and discuss a composite pulse sequence generating a universal control-phase gate.

Figure 1

Sketch of the device: charge qubits [single Cooper pair boxes (SCB)] coupled capacitively $\left(C_c\right)$ to a stripline cavity integrated with a dc SQUID formed by two large Josephson junctions (JJ); cavity eigenfrequency is controlled by magnetic flux $\Phi$ through the SQUID.

Figure 2

Equivalent circuit for the device in Fig. 1: chain of $LC$ oscillators represents the stripline cavity, ${\varphi }_1$ and ${\varphi }_N$ are superconducting phase values at the ends of the cavity, ${\varphi }_j$ and ${\varphi }_l$ are local phase values where the qubits are attached; attached dc SQUID has effective flux-dependent Josephson energy $E_{Js}\left(f\right)$ and capacitance $C_s$; control line for tuning the SQUID is shown at the right; SCB qubits are coupled to the cavity via small capacitances $C_{c1}$ and $C_{c2}$.

Figure 3

Solution of dispersion equation (11) for first mode, $kd\le \pi ∕2$ $(d\le \lambda ∕4)$, for large (a) and small (b) Josephson energies of the SQUID [$(2e∕\hslash {)}^2{L}_{cav}{E}_s=16$ and 4, respectively]; inset shows corresponding spatial distributions of the phase $\varphi ∕\varphi \left(0\right)$ in the cavity.

Figure 4

Protocol for creating a Bell pair: The cavity frequency is sequentially swept through resonances with both qubits; at the first resonance the oscillator is entangled with qubit 1, at the next resonance the oscillator swaps its state onto qubit 2 and ends up in the ground state. A Bell measurement is performed by applying Rabi pulses to noninteracting qubits, and projecting on the qubit eigenbasis $\{\mid g⟩,\mid e⟩\}$ by measuring quantum capacitance.

Figure 5

Pulse sequence producing (trivial) diagonal gate: during time $T_1$, qubit 1 swaps its state onto the oscillator, then the oscillator interacts with qubit 2 before swapping its state back onto qubit 1; free evolution during time $T_3$ is added to annihilate the two-photon state in the cavity.

Figure 6

Correct pulse sequence for performing a control-phase gate: time intervals $T_1$, $T_4$, $T_8$, and $T_9$ are single-photon $\pi$ pulses, whereas $T_2$ and $T_6$ are two-photon $\pi$ pulses; free evolution during times $T_3,T_5$, and $T_7$ is added to annihilate excited photon states, and create a nontrivial phase shift.

Figure 7

Gate circuit for constructing a CNOT gate using the control-phase gate: a $z$-axis rotation is applied to qubit 1, and Hadamard gates $\mathrm{H}$ are applied to the second qubit.