Controlled Parity of Cooper Pair Tunneling in a Hybrid Superconducting Qubit

Abstract

Superconducting quantum circuits derive their nonlinearity from the Josephson energy-phase relation. Besides the fundamental $\cosφ$ term, this relation can also contain higher Fourier harmonics $\cos(kφ)$ corresponding to correlated tunneling of $k$ Cooper pairs. The parity of the dominant tunneling process, i.e.~whether an odd or even number of Cooper pairs tunnel, results in qualitatively different properties, and controlling this opens up a wide range of applications in superconducting technology. However, access to even-dominated regimes has remained challenging and has so far relied on complex multi-junction or all-hybrid architectures. Here, we demonstrate a simple "harmonic parity qubit" (HPQ); an element that combines two aluminum-oxide tunnel junctions in parallel to a gate-tunable InAs/Al nanowire junction forming a SQUID, and use spectroscopy versus flux to reconstruct its energy-phase relation at 85 gate voltage points. At half flux quantum, the odd harmonics of the Josephson potential can be suppressed by up to two orders of magnitude relative to the even harmonics, producing a double-well potential dominated by even harmonics with minima near $\pmπ/2$. The ability to control harmonic parity enables supercurrent carried by pairs of Cooper pairs and provides a new building block for Fourier engineering in superconducting circuits.

Figures (11)

• Figure 1: (a) Superconducting qubit circuits based on Josephson elements that transfer one, two, or three Cooper pairs (left to right). The panels below show the corresponding Josephson potentials (black) and the wavefunctions of the two lowest states (blue and orange), and higher states (gray). Higher-order Cooper-pair tunneling gives rise to higher Josephson harmonics $\cos(k\phi)$, leading to different qubit properties. (b) Circuit realized in this work. The Josephson element is a SQUID with two SIS junctions in series on the left branch and a gate-tunable S-Sm-S nanowire junction on the right. The harmonic contents of the two SQUID branches interfere as a function of the magnetic flux $\Phi_\mathrm{e}$ threading the loop. At $\Phi_\mathrm{e}=\Phi_0/2$, the gate voltage controls whether an even or odd number of Cooper pairs dominates the tunneling process. (c) False-colored scanning electron microscope image of the Harmonic Parity Qubit (HPQ) device. A $\lambda/4$ resonator (brown) is capacitively coupled to the qubit island (blue) for readout. (d) Zoom of the SQUID loop showing the two SIS junctions (left) and the S-Sm-S junction (right). (e) SEM of the InAs/Al nanowire junction. The junction is defined by removing $\sim200nm$ of the epitaxial Al shell (red) covering the nanowire. The electrostatic gate (green) is patterned beneath the junction and counter-etched to leave a vacuum gap between the gate and the nanowire.
• Figure 2: Two-tone spectroscopy of (a) a transmon, (b) a double-junction transmon, (c) the Harmonic parity qubit at $V_g = 9.8V$ versus external flux $\Phi_\text{e}$ (see main text for details). In (a),(b) the signal trace corresponds to an average of qubit spectroscopy measurements over different drive powers. Dashed lines are fits to the identified transition frequencies used to extract the Hamiltonian parameters. (d)-(f) Harmonics of the potential in absolute value using parameters extracted from the fit. Each potential is decomposed into harmonics via a Fourier series $U(\phi)=\sum_{k=1}^{\infty} {c}_k \cos(k\phi)$. The transmon transitions are well described by a $\cos(\phi)$ term, while the double-junction transmon shows appreciable higher harmonics. For the HPQ at half flux quantum, the first harmonic is strongly suppressed, and $\cos(2\phi)$ dominates. (g)-(i) show the potential calculated using fit parameters. The transmon potential is sinusoidal, while the double-junction transmon shows a more parabolic well. The HPQ potential at half flux quantum shows a double well dominated by a $\cos(2\phi)$-term.
• Figure 3: Spectroscopy of the HPQ as a function of external flux $\Phi_\text{e}$ for three different gate voltages: (a) $V_g = -7V$, (b) $V_g = -0.2V$, and (c) $V_g = 7.2V$. Solid points denote simulation of the resonator frequency when the qubit is in the ground (black) and when the qubit is in the excited state (pink), using parameters from the qubit spectroscopy fit. (d)-(f) Two-tone qubit spectroscopy as a function of flux at the corresponding $V_g$. Black dashed lines indicate transitions that have been used for the fit, while gray dashed lines indicate other transitions we have identified but not used for the fit, some of which arise from the resonator being thermally excited. (g)-(i) Potential (top-left), harmonics (top-right), and charge-number wavefunction amplitudes for ground $\ket{0}$ (bottom-left) and first excited $\ket{1}$ (bottom-right) states calculated at half flux quantum using parameters extracted from the fit, showing the transition to qubit states identified by opposite Cooper pair parity.
• Figure 4: (a) Harmonic content versus gate voltage. Top: The sum of even (blue) and odd (red) coefficients $c_k$ as a function of gate voltage. The odd components $c_\mathrm{odd}$ are progressively suppressed with increasing $V_g$ and change sign, implying $c_\mathrm{odd}=0$ when it crosses zero, while $c_\mathrm{even}$ remains large and varies slowly. Bottom: Individual even (blue) and odd (red) coefficients, each normalized to its value at $V_g=-7V$. All odd components decrease, whereas all the even components increase. In particular, $c_1$ crosses zero at multiple gate points. All coefficients are extracted by fitting the qubit spectrum at each gate. (b) Harmonic parity ratio $|c_\mathrm{even}/c_\mathrm{odd}|$ versus gate voltage. Tunability of $|{c}_\text{even}/{c}_\text{odd}|$ spans two orders of magnitude across the measured gate range and formally diverges where $|{c}_\text{odd}|$ crosses zero. (c) Phase point $\phi_\mathrm{min}$ where the qubit potential is minimum for each $V_g$. Dashed vertical lines demarcate three regimes based on $\phi_\mathrm{min}$: a $\cos(\phi)$-dominated regime, a comparable $\cos(\phi)$ and $\cos(2\phi)$ regime, and an even-harmonic dominated regime consistent with a $\pi$-periodic potential. Insets show the potential at representative gate voltages (left to right) $V_g=-6.0V$, $V_g=2.4V$, $V_g=9.2V$.
• Figure S1: Fitting routine. (a)/(d) Wide-range/narrow-range two-tone spectroscopy as a function of applied flux. (b),(e) Same data with extracted transition frequencies overlaid: $f_{01}$ (black markers), $f_{12}$ (red markers), and $f_{02}$ (orange markers). These are the transitions used in the fit. (c),(f) Global fit of the qubit Hamiltonian to the extracted points. Black/orange/red dashed curves show the fitted transitions, while gray dashed lines show transitions we identified that were not used for the fit.