Figure 1

(a) and (c) System-pulse resonant interaction for qubit system 1. In (a), the pulse is resonant with the $|1\rangle \leftrightarrow |2\rangle$ transition; while in (c) the pulse is resonant with the $|0\rangle \leftrightarrow |1\rangle$ transition. (b) System-cavity resonant interaction for qubit system 1. The pulse is resonant with the $|1\rangle \leftrightarrow |2\rangle$ transition. (d) System-cavity-pulse off-resonant Raman coupling for $n-1$ qubit systems ($2,3,\dots ,n$) in a cavity. For simplicity, we here only draw a figure for qubit system $j$ interacting with the cavity mode and a classical pulse ($j=2,3,\dots ,n$). ${\delta }_j={\Delta }_{c,j}-{\Delta }_j$ is the detuning of the cavity mode with the pulse, and ${\Delta }_j={\omega }_{31}^j-{\omega }_j$ is the detuning between the pulse frequency ${\omega }_j$ and the $|1\rangle \leftrightarrow |3\rangle$ transition frequency ${\omega }_{31}^j$ of qubit system $j.$ Note that $g_j$, ${\delta }_j,$ ${\Delta }_{c,j},$ and ${\Delta }_j$ may be different for qubit systems ($2,3,\dots ,n$) due to nonidentical level spacings of the qubit systems which are caused by the nonuniformity of the qubit system parameters. The coupling constant $g_j$ may also vary with qubits due to nonexact placement of qubits in a cavity. The Rabi frequency of the pulse applied to qubit system $j$ is denoted by ${\Omega }_j$.Reuse & Permissions

Figure 2

(a) System-cavity-pulse off-resonant Raman coupling for $n$ qubit systems ($1,2,\dots ,n$) in a cavity. For simplicity, we here only draw a figure for qubit system $j$ interacting with the cavity mode and a classical pulse ($j=1,2,\dots ,n$). ${\delta }_j={\Delta }_{c,j}-{\Delta }_j$ is the detuning of the cavity mode with the pulse, ${\Delta }_{c,j}={\omega }_{20}^j-{\omega }_c$ is the detuning between the cavity mode frequency ${\omega }_c$ and the $|0\rangle \leftrightarrow |2\rangle$ transition frequency ${\omega }_{20}^j$ of qubit system $j,$ and ${\Delta }_j={\omega }_{21}^j-{\omega }_j$ is the detuning between the pulse frequency ${\omega }_j$ and the $|1\rangle \leftrightarrow |2\rangle$ transition frequency ${\omega }_{21}^j$ of qubit system $j.$ The blue arrow dashed line represents a second pulse applied to the qubit system $j$ to cancel the Stark shift of the level $|1\rangle$ induced by the first pulse. (b) System-cavity off-resonant interaction for each of $n$ three-level qubit systems ($1,2,\dots ,n$) in Ref. [17]. $g^{\prime }$ is the nonresonant coupling strength between the cavity mode and the $|0\rangle \leftrightarrow |2\rangle$ transition. $\delta ={\omega }_{20}-{\omega }_c$ is the detuning of the cavity mode frequency with the $|0\rangle \leftrightarrow |2\rangle$ transition frequency.Reuse & Permissions

Figure 3

(a) Sketch of the setup for six superconducting qubit systems and a (gray) standing-wave quasi-one-dimensional coplanar waveguide resonator. The two blue curved lines represent the standing wave magnetic field, which is in the $z$ direction. The green dot represents the qubit system 1, which is placed at an antinode of the standing wave magnetic field to achieve the maximal qubit-cavity coupling constant $g$. The red dots represent the qubit systems $2,3,\dots ,$ and $n$, which are placed at locations where the magnetic fields are the same. Each qubit system could be a superconducting charge-qubit system shown in (b), flux-biased phase-qubit system in (c), and flux-qubit system in (d). The superconducting loop of each qubit system, which is a large square for (b) and (d) while a large circle for (c), is located in the plane of the resonator between the two lateral ground planes (i.e., the $x$-$y$ plane). Each qubit system is coupled to the cavity mode through a magnetic flux ${\Phi }_c$ threading the superconducting loop, which is created by the magnetic field of the cavity mode. A classical magnetic pulse is applied to each qubit system through an ac flux ${\Phi }_e$ threading the qubit superconducting loop, which is created by an ac current loop (i.e., the red dashed-line loop) placed on the qubit loop. The frequency and intensity of the pulse can be adjusted by changing the frequency and intensity of the ac loop current. $E_J$ is the Josephson junction energy ($0.6<\alpha <0.8$) and $V_g$ is the gate voltage. $\lambda$ is the wavelength of the resonator mode, and $L$ is the length of the resonator.Reuse & Permissions