Scalable superconducting qubit circuits using dressed states

We study a coupling (decoupling) method between a superconducting qubit and a data bus that uses a controllable time-dependent electromagnetic field (TDEF). As in recent experiments, the data bus can be either an $LC$ circuit or a cavity field. When the qubit and the data bus are initially fabricated, their detuning should be made far larger than their coupling constant, so these can be treated as two independent subsystems. However, if a TDEF is applied to the qubit, then a “dressed qubit” (i.e., qubit plus the electromagnetic field) can be formed. By choosing appropriate parameters for the TDEF, the dressed qubit can be coupled to the data bus and, thus, the qubit and the data bus can exchange information with the assistance of the TDEF. This mechanism allows the scalability of the circuit to many qubits. With the help of the TDEF, any two qubits can be selectively coupled to (and decoupled from) a common data bus. Therefore, quantum information can be transferred from one qubit to another.

Figure 1

(Color online) A three-junction flux qubit is coupled to an $LC$ circuit by their mutual inductance $M$. The dc bias magnetic flux through the qubit is ${\Phi }_e$. A time-dependent magnetic field ${\Phi }_e\left(t\right)$ can be applied to the qubit, so the qubit can be coupled to the $LC$ circuit. Further details are explained in the text.

Figure 2

(Color online) The dependence of the eigenvalues $E$ and $G$ of Eq. (9) on the detuning frequency $\Delta ∕2\pi =({\omega }_q-{\omega }_c)∕2\pi$ and the amplitude of the Rabi frequency $\mid \lambda \mid ∕2\pi$. Here, the eigenvalues have been rescaled as frequencies, i.e., $E∕h$ and $G∕h$. The gap between these two surfaces corresponds to the frequency $\Omega ∕2\pi$ of the dressed qubit.

Figure 3

(Color online) The frequency $\Omega ∕2\pi$ of a dressed qubit (blue curve) and the frequency difference $(\omega -{\omega }_c)∕2\pi$ (dark red line); both vs the frequency ${\nu }_c={\omega }_c∕2\pi$ of the TDEF for the qubit frequency ${\omega }_q∕2\pi =2\mathrm{GHz}$. The frequency of the $LC$ circuit is $\omega ∕2\pi =4\mathrm{GHz}$; and the Rabi frequency of the qubit associated with the TDEF is $\mid \lambda \mid ∕2\pi =0.2\mathrm{GHz}$. The crossing point denotes the value of ${\nu }_c\approx 2.96\mathrm{GHz}$ when the condition $\Omega =\omega -{\omega }_c$ is satisfied.

Figure 4

(Color online) The $N$-flux qubits are coupled to an $LC$ circuit by their mutual inductances $M_m$ $(m=1,\dots ,N)$. The bias magnetic flux through the $m\text{th}$ qubit is ${\Phi }_e^{\left(m\right)}$. A TDEF can be applied to any one of the qubits (e.g., ${\Phi }_e^{\left(m\right)}\left(t\right)$ through the $m\text{th}$ qubit) such that the qubit can be coupled to the $LC$ circuit with the help of the TDEF.

Figure 5

(Color online) To couple qubit to the $LC$ circuit, the frequency ${\nu }_{\mathrm{ex}}$, applied to the qubit, is plotted as a function of the coupling constant ${\lambda }^{\prime }$ between the qubit and the external microwave. As an example, four different points are marked in the curve to show the required frequencies of external microwave when different coupling constants ${\lambda }^{\prime }$ are given.