Inductively shunted transmon qubit with tunable transverse and longitudinal coupling

We present the design of an inductively shunted transmon qubit with flux-tunable coupling to an embedded harmonic mode. This circuit construction offers the possibility to flux-choose between pure transverse and pure longitudinal coupling, that is, coupling to the ${\sigma }_x$ or ${\sigma }_z$ degree of freedom of the qubit. While transverse coupling is the coupling type that is most commonly used for superconducting qubits, the inherently different longitudinal coupling has some remarkable advantages both for readout and for the scalability of a circuit. Being able to choose between both kinds of coupling in the same circuit provides the flexibility to use one for coupling to the next qubit and one for readout, or vice versa. We provide a detailed analysis of the system's behavior using realistic parameters, along with a proposal for the physical implementation of a prototype device.

Figure 1

Inductively shunted transmon qubit with the possibility to flux-choose between longitudinal and transverse coupling to an embedded resonator. The qubit mainly consists of a single Josephson junction (depicted in blue).

Figure 2

The frequencies of qubit and resonator as a function of the (reduced) flux through the coupling loops ${\phi }_x$. Solid lines show accurate numerical results, dashed lines show the predictions using the formulas from Sec. 2, and the lighter color curve shows results at a flux ${\phi }_{Xb}=\pi$ in the big loop. While the resonator frequency is not affected by the large-loop flux-biasing, the qubit frequency experiences a drop.

Figure 3

The four most important coupling terms as a function of the (reduced) flux through the coupling loops ${\phi }_x$. Solid lines show accurate numerical results, dashed lines show the predictions using the formulas from Sec. 2, and the lighter color curves show results at a flux ${\phi }_{Xb}=\pi$ in the big loop. The longitudinal coupling is almost doubled due to the large-loop flux-biasing, while the point where the transverse coupling disappears is considerably shifted. The longitudinal coupling always disappears exactly at multiples of ${\phi }_x=k\pi$. (a) Longitudinal ($g_{zx}$) and transverse coupling ($g_{xx}$). (b) Unwanted coupling terms; note the change of scale.

Figure 4

The relative anharmonicities of qubit and resonator as a function of the (reduced) flux through the coupling loops ${\phi }_x$. Solid lines show accurate numerical results, and dashed lines show the predictions using the formulas from Sec. 2. Note that in this plot, the predictions are indistinguishable from the numerical results. The lighter color curve shows results at a flux ${\phi }_{Xb}=\pi$ in the big loop.

Figure 5

Relative anharmonicity of the qubit as a function of the (reduced) fluxes through the coupling loops ${\phi }_x$ and through the big loop ${\phi }_{Xb}$. While the anharmonicity is negative in a large range around ${\phi }_{Xb}=0$ (dark blue region), it changes sign when approaching ${\phi }_{Xb}=\pi$ and has a steep maximum at ${\phi }_{Xb}={\phi }_x=\pi$.

Figure 6

The four most important coupling terms as a function of the (reduced) flux through the coupling loops ${\phi }_x$ for a coupling junction array of $k=9$ junctions per array. Solid lines show accurate numerical results, dashed lines show the predictions using the formulas from Sec. 2, and the lighter color curves show what happens at a flux ${\phi }_{Xb}=\pi$ in the big loop. The longitudinal coupling is almost doubled due to the large-loop flux-biasing, while the point where the transverse coupling disappears is considerably shifted. The longitudinal coupling always disappears exactly at multiples of ${\phi }_x=k\pi$. (a) Longitudinal ($g_{zx}$) and transverse coupling ($g_{xx}$). (b) Unwanted coupling terms; note the change of scale.

Figure 7

The relative anharmonicities of qubit and resonator as a function of the (reduced) flux through the coupling loops ${\phi }_x$ for a coupling junction array with $k=9$ junctions per array. Solid lines show accurate numerical results, and dashed lines show the predictions using the formulas from Sec. 2. Note that in this plot, the predictions are indistinguishable from the numerical results. The lighter color curve shows results at a flux ${\phi }_{Xb}=\pi$ in the big loop. The smaller plot on the right shows again the relative anharmonicity of the resonator; note the change of scale.

Figure 8

Adapted qubit-resonator system with additional inductances $L_{ai}$ in the coupling branches.

Figure 9

Detail of a single coupling branch from Fig. 8. The phase difference across the junction array ${\phi }_d$ has no dynamics on its own but depends on the phase difference $\phi$ across the whole device.

Figure 10

$\phi$ as a function of the dependent variable ${\phi }_d$ (referring to Fig. 9) for different parameters. While the blue curve with $k\gamma /\beta >1$ is invertible, the magenta one with $k\gamma /\beta <1$ is not.

Figure 11

The frequencies of qubit and resonator as a function of the (reduced) flux through the coupling junctions ${\phi }_x$ for the adapted circuit with $k=5$ junctions per array. The lighter color curve shows results at a flux ${\phi }_{Xb}=\pi$ in the big loop.

Figure 12

The four most important coupling terms as a function of the (reduced) flux through the coupling junctions ${\phi }_x$ for the adapted circuit with $k=5$ junctions per array. The lighter color curves show what happens at a flux ${\phi }_{Xb}=\pi$ in the big loop. Due to the large-loop flux-biasing, all coupling terms get slightly bigger (the longitudinal coupling is almost doubled), while the point where the transverse coupling disappears is considerably shifted. (a) Longitudinal ($g_{zx}$) and transverse ($g_{xx}$) coupling. (b) Unwanted coupling terms; note the change of scale.

Figure 13

The relative anharmonicities of qubit and resonator as a function of the (reduced) flux through the coupling loops ${\phi }_x$ for the adapted circuit with $k=5$ junctions per array. The lighter color curve shows results at a flux ${\phi }_{Xb}=\pi$ in the big loop. While the resonator anharmonicity is unchanged, the qubit anharmonicity is now positive and boosted to up to 18%. The smaller plot on the right shows again the relative anharmonicity of the resonator; note the change of scale.

Figure 14

Proposal for the physical implementation of the inductively shunted transmon qubit with tunable transverse and longitudinal coupling. (a) Electrical schematic of the qubit-resonator circuit coupled to the control and readout microwave environment. The qubit dynamics is dominated by the Josephson junction with energy $E_{Jq}$ and capacitance $C_q$ (colored in blue). The frequency of the resonator mode is given by the equivalent inductance formed by $L$, $L_a$ (colored in red), the inductances of the Josephson junction arrays (colored in green), and the shunting capacitances $C$. (b) Finite-element model used to simulate the resonator coupling to the microwave drives. The color scale indicates the magnitude of the computed electric field at the surface of the thin-film superconducting electrodes, for a total energy stored in resonator mode of $1\mathrm{J}$. The $+$ and $-$ symbols represent the polarity of the electric field. The capacitors $C$ and $C_q$ are implemented using so-called finger capacitors, while $C_c$ and $C_g$ are given by the stray field coupling to the rectangular waveguide sample holder shown in panel (c). The inductive elements of the circuit are introduced in the model as lumped elements connecting the pads (shown in the insets below). The table shows the resulting linewidth values $\kappa$ for three different frequencies of the resonator mode, chosen in the pass-band of the waveguide. (c) Finite-element model used to simulate the 3D waveguide sample holder. Recently, a similar sample holder geometry has been used to perform multiplexed quantum readout [28]. The qubit-resonator circuit is deposited on a sapphire substrate, which is indicated by the green rectangle. The electric-field magnitude along the waveguide is frequency-dependent, and its profile is schematically shown for 6, 7, and $8\mathrm{GHz}$. The impedance and the mode profile between the waveguide and the coaxial cable connected to the input port are matched using the tuning screws. (d) Direct extension of the proposed physical implementation for two capacitively coupled qubit-resonator systems (compare Ref. [18]). The resonators are designed to have different eigenmode frequencies, and they can be individually addressed using the collective waveguide mode represented by the direction of the $\vec{E}$ field.