Flux-tunable regimes and supersymmetry in twisted cuprate heterostructures

Van der Waals (vdW) heterostructures, realized through the exfoliation and assembly of single atomic layers, are artificial quantum materials having widely tunable electronic and optical properties. Within these structures, the interplay of topological effects, strong correlation, and confinement can be precisely controlled by adjusting the interlayer twist angle, yielding a wealth of interesting phenomena, including unconventional superconductivity,¹ topological ferromagnetic order,² and correlated insulating states,³ just to mention a few.

Recently, the development of innovative fabrication techniques allowed the isolation of atomically thin Bi₂Sr₂CaCu₂O_8+δ (Bi-2212) crystals^4–6 preserving superconductivity and near-perfect lattice structure and paved the way to the realization^7–9 of vdW heterostructures between twisted cuprate layers showing a strong dependence of the Josephson energy on the twist angle. The applied stacking technologies freeze the chemistry of the cuprate crystals below 200 K (−73 °C) in ultra-pure argon atmosphere and preserve the intrinsic and spatially competing striped orders made of oxygen interstitials,¹⁰ incommensurate local lattice distortions, and charge modulations,¹¹ as found in pristine cuprate single crystals.^12,13 In these junctions, where detrimental disorder is reduced to a minimum, the d-wave nature of the superconducting state has significant effects on the junction characteristics. Particularly, within a narrow range of twist angles close to 45°, it results in a strong suppression of single Cooper pair tunneling,⁹ consequently making the contribution of two-Cooper-pair tunneling dominant. In this regime, the Josephson energy has a leading $cos ( 2 φ ̂ )$ dependence on the superconducting phase difference $φ ̂$⁠, and the junction¹⁴ hosts a peculiar superconducting state where time-reversal symmetry can be spontaneously broken, in agreement with the experimental work in Ref. 7. Further interesting topological phases were predicted at lower twist angles^15,16 away from optimal doping or in more complex trilayer structures.¹⁷ 

Very recently, it was proposed to utilize such twisted vdW cuprate junctions to realize superconducting quantum devices.^18,19 Specifically, the circuit design proposed in Ref. 18, nicknamed “flowermon,” consists of a single vdW cuprate junction with a twist angle $θ$ in the range $42 ° – 48 °$⁠, shunted by a capacitor and coupled to a control and readout resonator in a circuit QED architecture.²⁰ The flowermon exploits the peculiar $cos ( 2 φ ̂ )$ dependence of the Josephson energy, stemming from the twisted d-wave nature of the order parameter, to encode a qubit inherently protected against capacitive noise.

Capacitive fluctuations, arising from charge noise or stray electric fields, stand out as one of the most critical sources of noise limiting the coherence of many currently used superconducting qubits such as the transmon.²¹ Over the years, significant research efforts focused on understanding and characterizing dielectric properties of materials^22–24 to reduce capacitive losses as well as on the development of alternative qubit designs exploiting external magnetic fluxes or gates to drive the qubit to regimes with vanishing sensitivity to this kind of noise. Notable examples are the rhombus chain,^25–28 the $0 – π$⁠,^29–31 the bifluxon,³² the blochnium,³³ the KITE,^34,35 and semiconductor-superconductor^36–39 qubits, which can be employed also for hybrid topological protection schemes.^40,41 A crucial difference between these qubits and the flowermon is that in the latter, protection originates from the d-wave nature of the order parameter, while in the former, it is achieved through circuit engineering. In this regard, the concept of flowermon is closely linked to the pioneering theoretical^25,42,43 and experimental^44,45 research, which first explored the suppression of tunneling in d-wave based Josephson junctions to realize superconducting qubits.

In this Letter, we present a quantum device illustrated in Fig. 1(a), comprising two twisted cuprate junctions integrated in a superconducting quantum interference device (SQuID) loop and threaded by an external magnetic flux. By adjusting the external flux and the twist angle, this device can be tuned into various regimes hosting: a symmetric, “twist-based,” double-well potential, a “plasmonic” potential, and a “flux-biased” double-well potential, as illustrated in Fig. 1(b). The structure of the low-energy spectrum changes across the different regimes leading to distinct sensitivities to charge and flux noise fluctuations. The high-tunability of the device also enables the realization of a supersymmetric Hamiltonian where the spectrum has one non-degenerate ground state, and all other states are degenerate in pairs. Supersymmetric spectra arise in superconducting circuits due to the non-trivial interplay of different tunneling mechanisms, involving rhombus elements,⁴⁶ Majorana quasi-particles,⁴⁷ or the charging energy spectrum of a single junction.⁴⁸ Here, we show that supersymmetry marks the transition between the plasmonic and the flux-biased regimes, triggering significant changes in the coherence properties of the circuit.

As a result, the parameter Y controls the symmetry of the low-energy eigenstates and the structure of the spectrum. This dependence gives rise to the regimes introduced in Fig. 1(b) and underpins significant changes in the system dynamics and susceptibility to external fluctuations.

As the external flux increases above a threshold value corresponding to $Y = 1$⁠, indicated by the green curve in Fig. 1(b), the system enters the “plasmonic” regime (see the supplementary material for more details). In this regime, the low-energy eigenstates are confined within a single well centered around $φ = 0$⁠, and the charge relaxation matrix element becomes finite analogous to what happens in the transmon.²¹ It is interesting to note that at the transition between the twist-based and the plasmonic regimes, the Josephson potential becomes quartic, thus realizing the so-called “quarton” regime, which could also show peculiar coherence properties.^50,53,54 At even higher flux values, when $Y < − 1$ [dashed black curve in Fig. 1(b)], a second local minimum centered around $φ = π$ appears in the potential. As long as this minimum is at high energy, the structure of the low-energy spectrum does not change. Correspondingly, the system remains in the plasmonic regime that, thus, includes the whole orange-shaded region in Fig. 1(b).

For flux values above the supersymmetry point, $Φ x > Φ x SUSY$⁠, the ground state remains a Gaussian around the $φ = 0$ well, while the first excited state becomes a Gaussian centered at the $φ = π$ well. This structure is the hallmark of the flux-biased regime. In this regime, the ground and first excited states are both symmetric under the parity operator $K ̂$⁠, and the matrix element $n 01$ vanishes exactly. For similar reasons, however, the flux-derivative of the qubit frequency becomes finite, yielding a finite $Γ φ$ as shown in Fig. 3(c). This problem is general to all flux-biased $cos ( 2 φ ̂ )$ qubits shunted by a large capacitor, as thoroughly illustrated in Ref. 55.

In all regimes, the dephasing induced by charge noise and the relaxation induced by flux noise are suppressed by the large shunt capacitance and symmetries. (See the supplementary material for more details.)

We now consider the effect of the asymmetries between the junctions described by the parameters d, $d κ$⁠, and $d θ$⁠. Since the asymmetries in the tunneling energies, d and $d κ$⁠, are mostly due to geometric factors, we assume for simplicity $d κ = d$⁠. Moreover, in evaluating the coherence properties, we focus on charge-induced relaxation and flux-induced dephasing, as these are the dominant loss mechanisms, similar to the case of symmetric junctions. (See the supplementary material for more details.) There are two interesting features to highlight in the results shown in Figs. 4(c) and 4(d). First, the asymmetries d and $d κ$ do not undermine the inherent protection against charge and flux noise characteristics of the twist-based regime, especially when compared to an asymmetric transmon. Second, $d θ$ is much more detrimental as it leads to a strong enhancement of both flux-induced dephasing (around $ϕ x ∼ 0$⁠) and charge-induced relaxation (at higher $ϕ x$⁠). The different effects of $d = d κ$ and $d θ$ on the decoherence rates can be understood considering that they yield contributions with different strengths and symmetries to the Hamiltonian. Specifically, the corrections induced by d and $d κ$ are proportional to $cos 2 θ$ and $E κ / E J$ and are strongly suppressed at high twisting angles and for $E κ / E J ≪ 1$⁠. On the other hand, according to Eq. (6), the twist angle asymmetry introduces a contribution proportional to $E J sin 2 θ$⁠, which is large in the twist-based regime. This term does not have a well-defined symmetry and may yield contributions to both dephasing and relaxation. In Figs. 4(a) and 4(b), we also note that introducing small $d = d κ$ breaks the supersymmetry but preserves the quasi-degeneracy of the low-energy states. Conversely, for $d θ ≠ 0$⁠, the breaking of the supersymmetry appears at lower energy, and it is significantly larger. To further elucidate the overall effect of the asymmetries across the different regimes in Fig. 5, we plot the total decoherence time, $T 2$⁠, for different values of the parameters d, $d κ$⁠, and $d θ$⁠. The decoherence time (see the supplementary material for more details), $T 2$⁠, accounts for the effects of both transverse coupling to capacitive fluctuations and longitudinal coupling to flux noise. From this plot, we observe that a twist-angle asymmetry of $d θ ∼ 5 %$ completely hinders the protection offered by the twist-based regime. With a more precise fabrication process^7,9 that achieves $d θ ∼ 0.2 %$⁠, corresponding to an absolute error of approximately $0.2 °$ at $θ ∼ 43 °$⁠, the coherence properties are slightly improved.

In conclusion, we developed a device based on a SQuID loop of two junctions formed by twisted cuprate heterostructures. By manipulating the external flux in the loop, the device can be tuned to substantially different regimes, and it features the interplay of different mechanisms of protection against decoherence. At low values of the external flux, the circuit maintains the protection against charge noise offered by a single-junction flowermon with the added benefit of tunable energy levels as well as the possibility to manipulate the qubit by flux.^56–58 This protection can be traced back to the inherent d-wave nature of the junctions, which preserves the double-well structure and symmetry of the potential even in the presence of external flux. At flux values close to $Φ x = Φ 0 / 2$⁠, the circuit develops a double-well potential by a more conventional flux-biased mechanism. This regime also shows significant protection from charge noise, but dephasing due to flux fluctuations in the loop is not suppressed. The critical flux at which the circuit enters the flux-biased regime is a special point, where the spectrum exhibits a supersymmetric structure. Here, we demonstrated that the onset of supersymmetry marks a change in the symmetry properties of the excited state and yields sharp modifications in the system's coupling to external noise fluctuations and, consequently, in the decoherence rates. Note that supersymmetry is not specific to the device proposed. For example, a rhombus circuit element with a suitably tuned flux may yield a $cos 2 φ$ term, thus leading to a supersymmetric spectrum at specific values of the energy scales. Nevertheless, in this case, the supersymmetry condition is only approximately fulfilled because of quantum phase fluctuations.⁴⁶ 

The role of imperfections in junction fabrication was also investigated leading to the discovery that the system is robust to the asymmetry between the energy of the junctions but highly sensitive to the asymmetry in the twist angles. The twist-angle asymmetry strongly hinders the protection offered by the twist-based regime against flux noise induced dephasing similar to what occurs, e.g., in semiconductor-based $cos 2 φ$ qubits.³⁷ In other words, in twist-based qubits, the first harmonic of the Josephson effect is suppressed not by destructive interference effects, but due to the inherent d-wave structure of the order parameter. This naturally provides additional protection against flux noise, but it also explains why $d θ$ is highly detrimental and should be minimized by suitably engineering the fabrication process. We note that the flowermon circuit proposed in Ref. 18 completely avoids this problem though being less tunable.

The SQuID circuit considered here, therefore, offers the opportunity to explore fundamental problems in quantum physics and to develop devices for quantum technology. Furthermore, it contributes to illustrate how integrating unconventional materials and heterostructures into quantum superconducting circuits can unveil intriguing phenomena opening research pathways and triggering further progresses in fabrication technology. Further recent work in this direction concerns the integration of ferromagnetic junctions in superconducting circuits.^53,59 The possibility to experimentally realize the device proposed in the present work critically depends on the ability to fabricate high-quality twisted interfaces and to integrate them in quantum superconducting nanocircuits. A first advancement in this direction is discussed in Refs. 60 and 61, which demonstrate experimentally how to decouple the circuit fabrication from the fabrication of an atomically thin vdW cuprate, through circuit integration within a silicon nitride membrane.

See the supplementary material for a more detailed derivation of the results, a comparison between 0 – π and ± π / 2 qubits, and additional plots and figures illustrating the effects of the different noise sources and the symmetries of the eigenstates.

V.B. and A.C. acknowledge financial support from PNRR MUR project PE0000023-NQSTI financed by the European Union—Next Generation EU. This work was partially supported by the Deutsche Forschungsgemeinschaft (Nos. DFG 512734967, DFG 492704387, and DFG 460444718), co-funded by the European Union (Nos. ERC-CoG, 3DCuT, 101124606 and ERC-StG, cQEDscope, 101075962). The authors are deeply grateful to Bernard van Heck, Kornelius Nielsch, Francesco Tafuri, Domenico Montemurro, and Davide Massarotti, for support and illuminating discussions.

The authors have no conflicts to disclose.

Alessandro Coppo: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Luca Chirolli: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Nicola Poccia: Conceptualization (equal); Writing – review & editing (equal). Uri Vool: Conceptualization (equal); Formal analysis (equal); Writing – review & editing (equal). Valentina Brosco: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.