Hybrid ferromagnetic transmon qubit: Circuit design, feasibility, and detection protocols for magnetic fluctuations

Halima Giovanna Ahmad, Valentina Brosco, Alessandro Miano, Luigi Di Palma, Marco Arzeo, Domenico Montemurro, Procolo Lucignano, Giovanni Piero Pepe, Francesco Tafuri, Rosario Fazio, and Davide Massarotti

Phys. Rev. B 105, 214522 – Published 24 June 2022

Abstract

We propose to exploit currently available tunnel ferromagnetic Josephson junctions to realize a hybrid superconducting qubit. We show that the characteristic hysteretic behavior of the ferromagnetic barrier provides an alternative and intrinsically digital tuning of the qubit frequency by means of magnetic field pulses. To illustrate functionalities and limitation of the device, we discuss the coupling to a readout resonator and the effect of magnetic fluctuations. The possibility to use the qubit as a noise detector and its relevance to investigate the subtle interplay of magnetism and superconductivity is envisaged.

Received 16 December 2021
Revised 25 May 2022
Accepted 25 May 2022

DOI:https://doi.org/10.1103/PhysRevB.105.214522

Physics Subject Headings (PhySH)

Josephson effect

Josephson junctions Superconducting qubits

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Halima Giovanna Ahmad ^1,2,3,*, Valentina Brosco ^4,5, Alessandro Miano ^1,†, Luigi Di Palma ¹, Marco Arzeo ², Domenico Montemurro ^1,3, Procolo Lucignano ¹, Giovanni Piero Pepe¹, Francesco Tafuri ^1,6, Rosario Fazio ^7,1, and Davide Massarotti ^8,3

¹Dipartimento di Fisica “Ettore Pancini,” Università di Napoli “Federico II,” Monte S. Angelo, I-80126 Napoli, Italy
²Seeqc, Strada Vicinale Cupa Cinthia, 21, I-80126 Napoli, Italy
³CNR-SPIN, UOS Napoli, Monte S. Angelo, via Cinthia, I-80126 Napoli, Italy
⁴Institute for Complex Systems, National Research Council and Dipartimento di Fisica, Università “La Sapienza,” P.le A. Moro 2, 00185 Rome, Italy
⁵Research Center Enrico Fermi, Via Panisperna 89a, 00184 Rome, Italy
⁶CNR–Istituto Nazionale di Ottica (CNR-INO), Largo Enrico Fermi 6, 50125 Florence, Italy
⁷Abdus Salam ICTP, Strada Costiera 11, I-34151 Trieste, Italy
⁸Dipartimento di Ingegneria Elettrica e delle Tecnologie dell'Informazione, Università degli Studi di Napoli Federico II, via Claudio, I-80125 Napoli, Italy

^*halimagiovanna.ahmad@unina.it
^†Present address: Department of Applied Physics, Yale University, New Haven, CT 06520, USA.

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Vol. 105, Iss. 21 — 1 June 2022

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Images

Figure 1
In (a), ferro-transmon circuit design: the readout (RO) resonator is capacitively coupled to the qubit through $C_{g}$ . The qubit is schematized as a hybrid SQUID in parallel with a bias capacitor $C_{b}$ . In the SQUID, there are a standard tunnel SIS JJ and a tunnel ferromagnetic SFS JJ. Blue and red arrows indicate magnetic fields applied along the $z$ -axis and the $x$ -axis, respectively. In (b), $I_{c} (H)$ modulation in a tunnel $S I_{f} S$ JJ with a $3.5 nm$ GdN barrier and area of $49 μ m^{2}$ , normalized to the maximum $I_{c}^{HI} = 350 nA$ . Blue and red curves in (b) refer to the down and up magnetic field pattern, respectively. We highlight in each plot the low- and high- $I_{c}$ level states (LO and HI) and the working point (dashed black line). In (c), an example is shown of a magnetic field pulse sequence $H_{p}$ (black line), and maximum digital tuning of the critical current $I_{c} / I_{c}^{HI}$ (red line) for the JJ is shown in (b). The blue and magenta dashed boxes refer to LO and HI levels, respectively. The time in (c) is normalized to a magnetic pulse timescale $τ$ (see the text). (d) Calculated total Josephson energy $E_{J}$ of a hybrid SQUID composed of a SIS JJ with $E_{J}^{SIS} = 10 GHz$ , i.e., $I_{c}^{SIS} \sim 30 nA$ , and a $S I_{f} S$ JJ with GdN barrier. $I_{c}^{SFS}$ is the critical current of the SFS JJ in the hybrid SQUID in (a), which is fixed here to $350 nA$ . The 3D-plot shows the dependence of $E_{J}$ on an external flux $Φ_{Z}$ (in units of the quantum magnetic flux $Φ_{0}$ ) and the magnetic field pulsed sequence $H_{p} (t)$ in (c), with time $t$ normalized to $τ$ .
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Figure 2
Transmission parameter $S_{21} (ω)$ in arbitrary units for the ferro-transmon readout resonator as a function of $I_{c}^{SFS}$ , the critical current in the tunnel-SFS JJ. In (a) and (b), we explore the hybrid DC-SQUID configuration, with SIS JJ critical current $I_{c}^{SIS} = 30 nA$ , and the ferro-transmon configuration with a single tunnel-SFS-JJ, respectively. We fix $C_{b} = 70 fF$ in (a) and $C_{b} = 90 fF$ in (b), and $C_{g} = 4 fF$ in both configurations. The color scale in (a) and (b) refers to the $S_{21}$ of a notch-type resonator in a log-scale [58], with load and coupling quality factor $Q_{l} = 1.7 \times 10^{4}$ and $Q_{c} = 2 \times 10^{4}$ , respectively, and no impedance mismatch at input and output ports of the feedline. In (c) and (d), we show the readout resonator frequency dispersive shift in the high (low) level states (LO in blue and HI in red, respectively) of the SFS JJ for the configurations in panels (a) and (b), respectively. HI and LO levels in (c) and (d) are highlighted in panels (a) and (b) with the red and blue vertical dashed lines, respectively, and reported in Table 1 for completeness.
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Figure 3
Relaxation time $T_{1}$ due to Purcell (red dashed curve) and dielectric (orange curve) effects, magnetization (blue dashed-dotted curve), and flux noise (green dotted curve) fluctuations for the ferro-transmon in the hybrid DC-SQUID configuration as a function of the qubit frequency $ω_{01}$ for $g = 80 MHz$ , $E_{J}^{SFS} (HI) = 32 GHz$ , $E_{J}^{SIS} (HI) = 14 GHz$ , $E_{c} = 260 MHz$ , $κ = 400 kHz$ , and $τ_{c} = 1 ps$ . In (a), we fix $ϕ_{Z} = π / 4$ and we change $ϕ_{L} \in [0, π]$ , while in (b) we fix $Φ_{L} \sim π / 12$ and we change $ϕ_{Z} \in [0, π / 2]$ . The gray curve represents the total relaxation time $T_{1}$ , given by the sum of all the relaxation contributions.
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Figure 4
Gaussian decay rate $σ_{M_{i}}$ [Eq. (29)] along the up and down magnetization curves as a function of the local magnetic field. The red and blue dots correspond to $E_{J}^{SFS} = 32 GHz$ and $E_{J}^{SFS} = 20 GHz$ , and we set $E_{c} = 0.261 GHz$ , ${E_{J}}^{SIS} = 15 GHz$ , and $A_{M_{i}} \sim 1 μ T$ . The dependence of $E_{J}^{SFS}$ on the magnetic field is estimated from the Fraunhofer-like curves given in Fig. 1.
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Figure 5
Measurement protocol for the investigation of low- and high-frequency spectral noise functions of the intrinsic magnetic fluctuations (green boxes) and external magnetic field fluctuations (orange boxes). The protocol is divided into two phases: phase (1) concerns standard qubit characterization; phase (2) summarizes the study of relaxation phenomena through relaxation time measurements and dephasing time measurements for noise spectroscopy. In phase (2) we also report standard microwave pulsed sequences for the qubit drive (QD) and the readout (RO) for the relaxation, Ramsey, and echo sequences, respectively [4].
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Figure 6
Behavior of the decaying exponent of $ρ_{01} (t)$ for different $1 / f$ noise's exponents $ν = 1.5, 1.25, 1, 0.75, 0.5$ and infrared $ω_{ir} = 0.01 ω_{0}$ . The gray dashed lines represent the corresponding analytic approximations defined by Eqs. (29, 30).
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