Microwave quantum diode

Quantum engineering, bridging quantum mechanics and engineering, has rapidly developed over recent decades. Superconducting two-level systems (qubits) are key building blocks whose eigenenergies, non-linearity, and coupling strengths are readily engineered. Their large non-linearity enables selective control, making them attractive for many applications, though other microscopic two-level systems also offer advantages. Quantum devices require low-temperature operation and isolation from noise; ferrite-based circulators/isolators provide non-reciprocal routing but are bulky and require strong magnetic fields, hindering scalability and proximity to quantum circuits. Ferrite-free approaches have been explored, including devices based on artificial atoms' nonlinearity, dc-SQUIDs, Josephson-junction arrays, semiconductor mixer circuits (with potential on-chip extensions via Josephson mixers), quantum Hall effect-based mesoscopic circulators (requiring large magnetic fields), and passive on-chip Josephson-element circulators operating in a charge-sensitive regime. Charge sensitivity can be mitigated by careful parameter tuning. The device reported here operates in a parameter regime insensitive to charge fluctuations or charge-parity switching, requires only small magnetic fields, and demonstrates a proof-of-concept compact on-chip microwave diode suitable for microwave readout in superconducting quantum circuits. The strong non-reciprocity is relevant for circuit quantum thermodynamics to manage heat flow, the compact size favors multi-channel readout, and rectification is tunable with small magnetic fields.

The work situates itself among efforts to realize non-reciprocal microwave components without bulky ferrites. Prior art includes: ferrite circulators/isolators with high isolation and low insertion loss but large footprint and magnetic-field requirements; ferrite-free devices exploiting nonlinearity of artificial atoms and dc-SQUIDs; Josephson-junction arrays and traveling-wave parametric amplifier schemes enabling wideband isolation by frequency conversion; semiconductor-mixer-based isolators potentially adaptable on-chip via Josephson mixers; mesoscopic circulators leveraging the quantum Hall effect (requiring strong magnetic fields not compatible with superconducting circuits); and passive on-chip Josephson-element circulators operating in charge-sensitive regimes. Limitations often include parameter regimes leading to charge sensitivity or the need for strong magnetic fields. The present work leverages a superconducting flux qubit’s intrinsic nonlinearity and asymmetric coupling to two resonators to achieve non-reciprocity without ferrites, strong magnetic fields, or charge-sensitive operation, aiming for compactness and scalability.

Device design: A three-junction superconducting flux qubit is inductively coupled to two superconducting coplanar resonators of different lengths and coupling strengths. Junctions 2 and 3 have equal critical currents Ic, while junction 1 is smaller by asymmetry parameter α=0.6 (Ic1=αIc). The qubit has two islands with total capacitances CG1=C1+C2+Cg1 and CG2=C2+C3+Cg2. The inductive couplings to resonators are realized by aluminum wires with local inductances L1 (left, weaker coupling) and L2 (right, stronger coupling), yielding coupling constants gj∝Lj. Nominal resonator design frequencies are f1=6.5 GHz (left, weakly coupled) and f2=7.5 GHz (right, strongly coupled). From the theoretical model, couplings are estimated as g1/(2π)=−89 MHz and g2/(2π)=155 MHz. The two resonators are also mutually coupled via the qubit inductances, producing a hybrid mode. Both resonators couple to feedlines through nominally equal small coupling capacitors (lumped-element approximation validated by SONNET EM simulations), creating nominally symmetric outbound coupling to ports 2 and 4. Measurement setup: The chip is wire-bonded on a gold-coated PCB and measured in a cryo-free dilution refrigerator at 15 mK. Two identical input lines provide RF drive with attenuation stages: 6 dB (300 K), 15 dB (60 K), 10 dB (3 K), 10 dB (Still), and 15 dB (50 mK). Outputs from ports 3 and 4 connect via equal-length coaxial cables to a two-channel coaxial microwave switch (Radiall R577 433002) at the mixing chamber. Under DC control, the switch routes either port to the output chain while terminating the other in 50 Ω, minimizing asymmetries. The output chain includes two isolators at the mixing chamber, a 42 dB low-noise HEMT at 4 K, and 52 dB room-temperature amplification. A global DC magnet coil at the mixing chamber provides flux bias; on-chip bias lines could alternatively be used. One-tone and two-tone spectroscopy are performed; the main rectification results use one-tone spectroscopy at the qubit degeneracy point (Φ=0.5Φ0). Measurement definitions: The transmission Sxy is the ratio of the signal amplitude at output port X to input port Y. Rectification ratio is defined as R=(S42^2−S31^2)/(S42^2+S31^2). The hybrid mode frequency is fh=6.761 GHz; at low powers it is dispersively shifted by x=22 MHz to fh+x≈6.784 GHz due to qubit coupling. With increasing drive power, the mode decouples and the resonance moves toward fh. Strong nonlinearity (quantum Duffing behavior) emerges once the drive exceeds port-dependent thresholds Pl (l=1,2): P1=−112 dBm for drive from port 1/resonator 1, and P2=−117 dBm for drive from port 2/resonator 2 (5 dB difference). Above threshold, single Lorentzian lines in |S31|^2 and |S42|^2 split into two, with peak positions following a theoretical expression derived in Supplementary Note 2, matching observed dashed lines in the data. The measured ratio P1/P2=3.2 (≈5 dB). Calibration and error mitigation: To remove line-specific and output-chain asymmetries, an in-situ background calibration is performed by measuring off-resonance transmissions at various flux values to obtain baseline transmissions S01,bg=S11+S04 and S02,bg=S21+S03 (with S03≈S04 due to symmetric connections). Calibrated device transmissions are S31=S01−S01,bg and S42=S02−S02,bg. At Φ=0.5Φ0 and high power (−74 dBm), calibrated S31 and S42 match within 1.5 dB over 4–5.2 GHz, while in 5.2–7.8 GHz a pronounced diode effect up to 15 dB is observed near the hybrid mode. At the lowest input power (−134 dBm), to suppress noise-induced artefacts in R, only data with |S42|^2+|S31|^2 above a set threshold are used. Estimated maximum error in R due to background noise is ≈35% when R≈0, decreasing rapidly for nonzero R; at higher powers errors are below ±10%. Fabrication: Devices are fabricated on high-resistivity Si (675 µm) with a 30 nm Al2O3 ALD dielectric. A 200 nm Nb layer is sputtered and patterned (e-beam lithography on 300 nm positive resist; post-bake at 150 °C; CF4+O2 RIE; Al2O3 as etch stop). Josephson junctions are defined by Dolan-bridge bilayer (PMMA/MMA) e-beam lithography, developed in MIBK:IPA and methyl-glycol:methanol. Prior to metal deposition, in-situ Ar plasma cleans native oxide. Two 30 nm Al evaporations at +18° and −18° with in-situ oxidation form tunnel barriers. Lift-off is done in hot acetone. Test 3-junction SQUIDs exhibit room-temperature resistance ≈2.6 kΩ. Analysis and enhancements: Theory predicts P1/P2∝(f1κh1)/(f2κh2), where κh1, κh2 are hybrid-mode leakage rates via coupling capacitors CK1 and CK2, suggesting stronger rectification by making CK1≠CK2 and increasing coupling asymmetry (e.g., via length-dependent inductance engineering). Bandwidth is limited by fixed resonator frequencies; replacing with tunable (SQUID-based) resonators can increase and tune the rectifying bandwidth.

• Demonstration of a compact, on-chip microwave quantum diode using a superconducting flux qubit asymmetrically coupled to two resonators, operated at the qubit degeneracy point (Φ=0.5Φ0).
• Clear non-reciprocity evidenced by different drive thresholds for nonlinear response: P1=−112 dBm (port 1) vs P2=−117 dBm (port 2), a 5 dB (≈4.3 fW) difference. Measured P1/P2=3.2.
• High rectification near the qubit–resonator avoided crossing: at −99 dBm input power, rectification ratio R>90% over a 50 MHz band (6.81–6.86 GHz); R>60% over a 250 MHz band (approximately 6.67–6.91 GHz). Similar strong rectification observed at −114 dBm.
• Hybrid mode frequency fh=6.761 GHz with low-power dispersive shift x=22 MHz to ≈6.784 GHz; linewidth κh/(2π)=1.1 MHz. With increasing power, hybrid-mode splitting observed consistent with a quantum Duffing oscillator model; theoretical peak positions match experiment.
• Calibrated wideband data show S31 and S42 agree within ≈1.5 dB in 4–5.2 GHz at high power (−74 dBm) and differ by up to ≈15 dB in 5.2–7.8 GHz near the hybrid mode, confirming non-reciprocity.
• Insertion loss and isolation at Φ=0.5Φ0 (from Table 1; insertion loss values include an added 3 dB due to equal power splitting): • −134 dBm: insertion loss 9.6 dB (resonance and max), isolation 4.9 dB (resonance and max). • −114 dBm: insertion loss 16.6 dB (resonance), 29.4 dB (max); isolation 5.3 dB (resonance), 19 dB (max). • −99 dBm: insertion loss 25.7 dB (resonance), 30 dB (max); isolation 3.1 dB (resonance), 15.7 dB (max).
• Rectification is tunable with applied magnetic flux; stronger effects appear near hybrid-mode resonances.

The observed diode behavior arises from the intrinsic nonlinearity of the flux qubit that controls the transmission of the coupled resonator–qubit–resonator system. Because the qubit couples asymmetrically to the two resonators/ports (via different inductive couplings and resonator parameters), the power required to drive the system into the strongly nonlinear regime differs depending on the propagation direction. Above port-dependent thresholds, single Lorentzian transmission peaks split, and for suitable powers and flux, the peaks in |S31|^2 and |S42|^2 overlap only weakly, yielding strong rectification (R approaching unity). The theoretical model captures the power-dependent splitting and explains the measured ratio P1/P2, linking it to asymmetries in leakage rates through coupling capacitors. This suggests strategies to enhance rectification further by deliberately engineering unequal couplers and stronger coupling asymmetry (e.g., via length-dependent inductances). The device provides strong non-reciprocity over tens to hundreds of MHz around the avoided crossing and is flux tunable, enabling control via small magnetic fields. While insertion losses are currently high, this proof-of-concept demonstrates a compact, ferrite-free path to on-chip isolation and directionality compatible with superconducting quantum circuits. Replacing fixed-frequency resonators with tunable (SQUID-based) ones could broaden and tune the rectifying bandwidth. The architecture is scalable to multiple readout channels and is relevant for quantum information processing, microwave readout chains, optomechanics, and circuit quantum thermodynamics where directional heat and signal flow are desired.

This work introduces a flux-tunable, compact on-chip microwave diode based on a superconducting flux qubit asymmetrically coupled to two resonators. Experiments at millikelvin temperatures show a 5 dB asymmetry in the drive power required to reach strong nonlinearity, and strong transmission rectification: R>90% over a 50 MHz band and >60% over a 250 MHz band near the qubit–resonator avoided crossing. Measurements agree with a theoretical model of power-dependent mode splitting. The approach offers a ferrite-free, small-footprint route to non-reciprocal components suitable for scalable cryogenic quantum systems, with flux tunability and compatibility with multi-channel integration. Future work will target reducing insertion loss, enhancing coupling asymmetry (e.g., via unequal couplers), and implementing tunable resonators to expand bandwidth. The architecture could enable tunable photonic quantum heat valves and rectifiers and be applied in quantum information readout and optomechanical platforms.

• High insertion loss in the current implementation (up to ~30 dB max), partly due to equal power splitting at capacitive couplers (+3 dB accounted for), limits immediate practical deployment without further optimization.
• Rectification bandwidth is constrained by fixed resonator frequencies; broader tunable bandwidth requires integrating tunable (SQUID-based) resonators.
• At the lowest input power (−134 dBm), signal-to-noise ratio is limited; rectification estimates near vanishing R can have up to ~35% error due to background noise, though errors drop below ~10% at higher powers.
• Despite careful in-situ calibration, residual background transmission differences up to ~10% remain.
• The device requires small magnetic fields for flux bias (though much smaller than ferrite devices), which may impose integration considerations in some systems.