Planar Multilayer Circuit Quantum Electrodynamics

Experimental quantum information processing with superconducting circuits is rapidly advancing, driven by innovation in two classes of devices, one involving planar microfabricated (2D) resonators, and the other involving machined three-dimensional (3D) cavities. We demonstrate that circuit quantum electrodynamics can be implemented in a multilayer superconducting structure that combines 2D and 3D advantages. We employ standard microfabrication techniques to pattern each layer, and rely on a vacuum gap between the layers to store the electromagnetic energy. Planar qubits are lithographically defined as an aperture in a conducting boundary of the resonators. We demonstrate the aperture concept by implementing an integrated, two-cavity-mode, one-transmon-qubit system.

Figure 1

Qubit-resonator coupling in different cQED approaches. (a) In-plane coupling in 2D. The electric field lines of the resonator (blue) are aligned with the dipole moment of the qubit (red), both of which are in the plane of qubit fabrication. (b) Out-of-plane coupling in a multilayer planar device. The resonator is now represented as a section of a multilayer whispering-gallery-mode resonator [23], consisting of two superconducting thin-film rings deposited on different sapphire substrates that are separated by an electrically thin vacuum gap. The qubit is defined by an aperture carved directly from the conducting boundary of the resonator. The orange and blue arrows represent the resonator surface-current density and electric field lines, respectively.

Figure 2

Photograph of chip-stack elements. Thin-film Al rings are patterned in a single $e$-beam lithography step along with the Josephson junction on a sapphire substrate. The boxed region shows a magnified optical micrograph of the embedded aperture qubit (false colored in red) and the location of the Josephson junction (red cross). The axis of symmetry, represented over substrate layer 2, defines the angular position $\theta$ around the ring.

Figure 3

Electric and magnetic contributions to qubit-resonator coupling, as calculated using a HFSS. (a) Electric field amplitude $|\mathbf{E}|$ in the resonator WG modes as a function of $\theta$, for one photon of energy. The angular dependence of the surface-current-density $j_s$ follows that of $E$, but is shifted by 90°. (b) Surface-charge-density amplitude $\sigma$ shown in color scale with overlayed white signs to indicate the relative charge polarity. The charges in the island and corresponding image charges in the opposite layer below determine the electric contribution to the qubit-resonator coupling. (c) Qubit-mode surface-current amplitude $j_s$ shown in color scale with overlayed white arrows to represent the direction of the flow. The narrow rails on each side of the aperture are equivalent to a shared inductance between the qubit and resonator and determine the magnetic contribution to the qubit-resonator coupling. (d) Qubit-resonator coupling rate $g$ of the aperture qubit to the WG resonator modes as a function of the qubit position around the ring $\theta$. The right vertical axis also shows the equivalent cross-Kerr $\chi$ for a $\mathrm{\Delta }=1\mathrm{GHz}$ detuning. The coupling $g$ is the algebraic sum of the electric and magnetic contributions, which interfere constructively or destructively as a function of $\theta$. For the readout (storage), maximal constructive (destructive) interference occurs at about 45°, while near $\theta =0$, 180° the coupling is capacitive (inductive).

Figure 4

(a) Aperture transmon energy relaxation. A single exponential fits the data with a $T_1=70\mu \mathrm{s}$. The population inversion is defined as $(P_e-P_e^{\mathrm{th}})/(P_g^{\mathrm{th}}-P_{\mathrm{e}}^{\mathrm{th}})$, where $P_e^{\mathrm{th}}$ and $P_g^{\mathrm{th}}$ are the thermal populations. (b) Storage parity relaxation from which we infer an energy relaxation lifetime of $T_1^S=45\mu \mathrm{s}$. Inset: pulse sequence for the measurement, see Appendix pp4.

Figure 5

(a) HFSS calculation of the surface currents and electric fields, which are contained within the $100\text{−}\mu \mathrm{m}$ vacuum gap separating the two layers, for the TEM $D^{\parallel }$ WG mode. The orthogonal $D^{\perp }$ mode closely resembles the $D^{\parallel }$, up to a 90° rotation. Depending on the configuration of the input-output ($I\text{−}O$) coupling pins [see (c)], these modes can be used either as a qubit readout mode (low coupling-quality-factor $Q_c$) or storage for quantum states (high $Q_c$). (b) Simulated $Q_c$ as a function of the coupling coax pin depth inside the sample holder at $\theta =0°$. The inset shows a not-to-scale cross-sectional representation of the chip stack in a sample holder (black). We can selectively couple to $D^{\parallel }$ ($Q_c^R={10}^4$) while remaining uncoupled from $D^{\perp }$ ($Q_c^S>{10}^8$), as indicated by the vertical gray line, which corresponds to the nominal parameters of the measured device.

Figure 6

We measure $T_2^R=8\mu \mathrm{s}$ (inset) and $T_2^E=20\mu \mathrm{s}$. By dynamically decoupling the qubit from low-frequency noise we observe an order-of-magnitude improvement in the coherence time of the aperture transmon exceeding $T_1$. The dashed line is a guide for the eye.

Figure 7

Using the dispersive cross-Kerr interaction with the readout mode ${\chi }_{sr}/2\pi =0.25\mathrm{kHz}$ (pulse sequence in inset), we measure a storage lifetime $T_1^S=45\mu \mathrm{s}$ for $\overline{n}>10$ photons.

Figure 8

Calibration of the photon-number parity measurement in the storage is achieved with a qubit-state revival experiment. For small storage-mode displacements ${\overline{n}}_s\sim 0$ (blue), the decay is dominated by intrinsic qubit decoherence. For increasing displacements, up to ${\overline{n}}_s=3$ (red), the apparent increase in decoherence is due to the large qubit-cavity interaction rate, and we observe qubit-state revivals at integer multiples of $2t_p=2\pi /{\chi }_{qs}$.