Experimental realization of a $\cos(2\varphi)$ transmon qubit

Abstract

Superconducting circuits with embedded symmetries are good candidates to robustly protect quantum information from dominant error channels. The $\cos(2\varphi)$ qubit, consisting of an island shunted to ground through a tunneling element that selectively transmits pairs of Cooper pairs, leverages charge-parity symmetry to protect from charge-induced errors. In this experiment, we observe a doublet of states of opposite Cooper-pair parity split by $13.6~\mathrm{MHz}$. Operating in a soft-transmon regime, this splitting is two orders of magnitude smaller than in previous implementations, pushing charge-induced losses well beyond the measured coherence times. Despite the low transition frequency, we demonstrate coherent qubit control, single-shot readout, and resolve quantum jumps. Charge protection of the qubit is evidenced by a $100-$fold suppression of the island charge matrix element compared to the unprotected plasmon transition, placing dielectric loss limits above $10~\mathrm{ms}$. The measured $T_1 = 70~μ\mathrm{s}$ and $T_2^\mathrm{echo}= 2.5~μ\mathrm{s}$ are instead limited by $1/f$ flux noise in the tunnelling element's loop. This experiment shows that pushing Cooper-pair pairing in the transmon regime sets high limits on charge-induced losses while preserving coherent control and single-shot readout of the low-frequency qubit. We identify flux noise as the dominant remaining limitation, calling for gradiometric designs or novel $4e$-tunneling elements.

Figures (14)

• Figure 1: Device implementation. (a) Lumped-element circuit of a capacitor shunted by a KITE loop that transfers pairs of Cooper pairs (blue). (b) Optical false-color micrograph of the device. Pairs of Cooper pairs tunnel in and out of the central island (brown) through the KITE (dark-field optical micrograph in blue inset) composed of chains of JJs (SEM image in yellow inset) and small JJs (SEM image in orange inset). A flux line (green) threads magnetic flux through the KITE loop, while a charge line (red) carries radio-frequency drives and controls the offset charge. A quarter wave-length resonator (purple) is capacitively coupled to the island for dispersive readout. While the KITE is in aluminium, all other metallic components are in niobium on silicon. (c) Four lowest energy levels and wavefunctions confined in the double-well potential (black line) of Hamiltonian \ref{['eq:H1M']} at $\mathcal{N}_g=0, \varphi_\mathrm{ext}=\pi$. The polar representation on a circle communicates the compactness of the degree of freedom $\varphi$. (d) Flux dispersion of the two lowest energy levels of Hamiltonian \ref{['eq:H1M']} at $\mathcal{N}_g=0$. (e) Same as panel (c) in the discrete charge representation $\mathcal{N}$. (f) Charge dispersion of the ground state doublet at $\varphi_\mathrm{ext}=\pi$. The avoided crossing at $\mathcal{N}_g=0.5$ stems from the KITE asymmetry. The case of perfect symmetry is overlaid in dashed lines for clarity.
• Figure 2: Readout mode spectroscopy and single shot measurements. (a) Reflected signal phase (color) versus probe frequency (y-axis) and external flux (x-axis). Anti-crossings (arrows) appear when a circuit transition (insets) collides with the readout frequency. The regions of frequency collision are broadened by charge dispersion, and are encoded in the arrow widths. (b,c) Two-dimensional histograms of the measured $(I,Q)$ values at resonance, $N_g=0$ and $\varphi_\mathrm{ext}=\pi$, integrated over $10~\mu\mathrm{s}$ after post-selection on even (b) or odd (c) quasiparticle number. Labels indicate the state associated to each distribution. (d) Measured readout-signal phase (y-axis) versus time (x-axis), after post-selection on even quasiparticle number. The telegraphic signal transits between three zones corresponding to $\ket{0-}$ (green), $\ket{0+}$ (cyan), and higher plasmon states denoted $\ket{\ge 1}$ (orange).
• Figure 3: Device spectroscopy. (left) Transition frequencies (y-axis) from $\ket{0+}$ (cyan) and $\ket{0-}$ (green) versus external flux (x-axis), calculated from the three-mode Hamiltonian (\ref{['sec:fit']}) at $N_g=0$. (center, right) Two-tone spectroscopy data around the first 4 plasmon doublets in the vicinity of $\pi$. We measure the transition probability out of the ground state doublet (color) from $\ket{0+}$ (center) and $\ket{0-}$ (right) versus external flux (x-axis) and the frequency of the probe tone (y-axis). For the ground state doublet (lowest frame), we instead represent the transition probability from $\ket{0+}$ to $\ket{0-}$. The three-mode model (solid lines) reproduces the spectra over the full measured bandwidth. The one-mode model (dashed lines), valid at low energies, captures the two lowest doublets with a $3\%$ error. Theory curves are shown on half of the data for clarity.
• Figure 4: Rabi chevrons and Ramsey spectroscopy. (a) Measured transition probability (color) from $\ket{0+}$ to $\ket{0-}$ in the presence of a Rabi drive of varying duration (y-axis) and frequency (x-axis). (b) Transition probability from $\ket{0+}$ to $\ket{0-}$ (y-axis) versus delay time (x-axis) between the two $\pi/2$ pulses of the Ramsey sequence. The data (open circles) fit an exponentially decaying envelope with a Ramsey decay time $T_2^\mathrm{Ramsey}=1.44~\mu\mathrm{s}$. The signal oscillations arise from the combined digital and physical detunings of the $\pi/2$ pulses with respect to the bare qubit frequency.
• Figure 5: Analysis of device losses. (a) Island charge matrix element (x-axis) for the doublet and plasmon transitions (y-axis). Theoretical predictions from Eq. \ref{['eq:H1M']} (dashed bars) agree with the measurements (colored bars), and remaining discrepancies are attributed to uncertainty in the attenuation of the input lines. (b,c) Decay time (y-axis) of the doublet (b) and the plasmon (c) versus external flux (x-axis). The data (circles) are compared to calculated contributions from Eqs. \ref{['eq:diel']},\ref{['eq:1of']} (solid lines) with the relevant end states and associated transition frequencies. The dielectric-induced loss limit (red hatched zone) is two orders of magnitude larger for the protected doublet than the unprotected plasmon. (d,e) Echo decay time (y-axis) versus external flux (x-axis) for the doublet (d) and plasmon (e) transitions. We fit the data (circles) to a $1/f$ flux noise model (solid line). (f,g) Echo decay time (y-axis) versus offset charge (x-axis) for the doublet (f) and plasmon (g) transitions. We fit the data (circles) to a $1/f$ charge noise model (dashed lines) for three values of the noise amplitude (color). Error bars are estimated from a bootstrap method (\ref{['sec:t1']}).