Multifrequency modes in superconducting resonators: Bridging frequency gaps in off-resonant couplings

A superconducting quantum interference device (SQUID) inserted in a superconducting waveguide resonator imposes current and voltage boundary conditions that makes it suitable as a tuning element for the resonator modes. If such a SQUID element is subject to a periodically varying magnetic flux, the resonator modes acquire frequency sidebands. In this work we calculate the multifrequency eigenmodes of resonators coupled to periodically driven SQUIDs and we use the Lagrange formalism to propose a theory for their quantization. The elementary excitations of a multifrequency mode can couple resonantly to physical systems with different transition frequencies and this makes the resonator an efficient quantum bus for state transfer and coherent quantum operations in hybrid quantum systems. As an example of the application of our multifrequency modes, we determine their coupling to transmon qubits with different frequencies and we present a bichromatic scheme for entanglement and gate operations.

Figure 1

(a) Superconducting resonator, interrupted by a SQUID at $x=0$. The SQUID is modulated by an external magnetic field $\Phi \left(t\right)$ yielding a time-dependent Josephson energy $E_J\left(t\right)$. (b) Circuit diagram of a resonator interrupted by an inline SQUID. Here $L_T$ and $C_T$ denote, respectively, the inductance and capacitance per length of the waveguide such that each inductor (capacitor) has the inductance (capacitance) $L_T\Delta x$ $\left(C_T\Delta x\right)$. The resonator is described in the limit of $\Delta x\to 0$. The inline SQUID is characterized by the time-dependent Josephson energy $E_J\left(t\right)$ and its capacitance $C$.

Figure 2

Mode functions when we modulate the SQUID with a frequency ${\omega }_d=2\pi \times 2.0$ GHz and $\delta E_J/E_{J,0}=0.4$ for a mode with $kd=4.614$ $(\omega =2\pi \times 7.343$ GHz). The green solid line shows the central frequency component $u_{\omega }\left(x\right)$, while the red dashed and the blue dash-dotted lines show the sideband components $u_{-}\left(x\right)$ and $u_{+}\left(x\right)$, respectively. The parameters of the resonator are $d=1.2$ cm, $E_{J,0}/\hslash =2\pi \times 715$ GHz, $v=1.2\times {10}^8$ m/s, and $L_T=50\Omega /v$. At $x=0$ we have inserted a vertical line to elucidate the discontinuities in the mode functions across the SQUID.

Figure 3

Classical resonances for the odd modes of a resonator calculated by Eq. (15) with the same parameters as in Fig. 2. We modulate the SQUID with varying strengths at a frequency ${\omega }_d=2\pi \times 2.0$ GHz. We show the variation of the central frequency of (a) the first mode, (b) the third mode, and (c) the fifth mode. The even modes are not considered since they do not experience the modulation of the SQUID.

Figure 4

Coupling strength of the two lowest states of a transmon to the resonator mode. The frequencies are such that ${\omega }_{10}={\omega }_{-}=\omega -{\omega }_d$, where $\omega$ is the frequency of the first mode in a resonator with $d=0.25$ cm and $\omega =2\pi \times 10.82$ GHz in the absence of modulation. The transmon is located at $x_t=0.1d$ and has $E_J/E_C=80$ and $\beta =2/3$. The thick (green) line is calculated by Eq. (29) for the modulation frequency set to ${\omega }_d=2\pi \times 6$ GHz, while the thin (blue) line shows Eq. (29) for ${\omega }_d=2\pi \times 0.5$ GHz. The dashed (red) line shows the result of the quasistatic approximation (7), which is independent of the modulation frequency.

Figure 5

(a) Setup with two transmon qubits placed at $x_I=0.1d$ and $-0.1d$. (b) Time evolution of two-qubit density-matrix elements obtained by a simulation of the master equation with the Hamiltonian in Eq. (34), a resonator decay rate of $\kappa =2\pi \times 0.2$ MHz, and transmon decay and dephasing lifetimes $T_1=10$ $\mu \mathrm{s}$ and $T_2=5$ $\mu \mathrm{s}$. The transmon qubits have different excitation frequencies ${\Omega }_1=2\pi \times 6.0$ GHz and ${\Omega }_2=2\pi \times 6.5$ GHz and the resonator frequency is $2\pi \times 10.82\mathrm{GHz}$. For the parameters described in the text we obtain the Hamiltonian (34) with $G=2\pi \times 2.5$ MHz and $\delta =2\pi \times 10$ MHz. In the simulation we have also included a Kerr term with magnitude given in Eq. (33).