Fast flux control of 3D transmon qubits using a magnetic hose

In many quantum systems used for quantum information processing, quantum optics and quantum enhanced measurements, magnetic fields are essential to mediate interactions or tune parameters.^1–3 For individual quantum systems like qubits, the typical wish list is a combination of large field strength and the ability to change the field rapidly and apply it only locally, while, at the same time, to not introduce additional decoherence channels. Very often, it is hard for experiments to simultaneously meet these criteria as, e.g., fast switching times are limited by eddy or super-currents introduced by the metallic boundaries protecting the quantum system from decoherence.

One of the most promising platforms for realizing a quantum computer is superconducting qubits. Magnetic fields play an important role in this architecture to tune the frequency of individual qubits, change the coupling between pairs of qubits,⁴ and apply parametric drives for quantum limited amplifiers^5,6 and novel coupling schemes.^4,7 So-called flux bias lines⁸ are widely used in planar superconducting qubit architectures, but so far, it has proven to be difficult to implement a similar technology in a 3D architecture.⁹ All experimental effort introducing control lines into waveguide cavities allows only for the application of DC magnetic fields¹⁰ and even reduces the quality factor of the cavity to 10⁵ or less.^11,12

In this paper, we demonstrate magnetic field control using a magnetic hose, on time scales of less than 100 ns inside a superconducting 3D waveguide cavity, with a quality factor exceeding 10⁶. The magnetic hose and its working principle were presented for the first time in Ref. 13. There, the authors could already demonstrate experimentally that a DC magnetic field is transferred four times more efficiently in open space as compared to, e.g., a ferromagnetic rod. We adapted this technology to work with AC fields and modified it further to work with a superconducting cavity. The hose is designed as an effective microwave filter to not compromise the energy relaxation time of the qubit and still allow for fast flux control of 3D transmon qubits.

Our system, illustrated in Fig. 1, consists of two transmon qubits, fabricated on the same sapphire substrate. The sapphire is placed in the rectangular aluminum cavity through a slit in the cavity wall and is held in place and thermalized by a copper clamp on the outside of the cavity.

The magnetic hose consists of a set of concentric cylindrical layers. Half of the layers are made of ferromagnetic material (with a large magnetic permeability of $μr≈104$⁠) and the other half of superconducting material (with a permeability effectively $μr→0$⁠). These two types of layers are alternated forming a magnetic metamaterial structure. This structure exhibits an extremely anisotropic effective magnetic permeability, with very large permeability along the axial direction of the hose, $μr||→∞$⁠, and a very small permeability in the (radial) perpendicular direction, $μr⊥→0$⁠. Intuitively, the ferromagnetic layers provide large permeability components in all directions, but the interlayered superconducting layers shield any radial magnetic field component. The design of this metamaterial structure,¹³ which results from concepts in transformation optics,^15,16 allows us to transfer static magnetic fields over long distances. To that effect, a dense structure of alternating layers is essential for approaching the ideal material and enhancing the transport.

Following this idea, we built a magnetic hose consisting of six ferromagnetic layers made of mu-metal and six superconducting layers made of aluminum. The ferromagnetic layers are 10 mm long, while the aluminum layers are 15 mm. The ferromagnetic shells are not inserted into the cavity volume to avoid losses due to dissipation, while the aluminum layers are extended into the cavity to improve the magnetic field transfer to the qubit. The aluminum shells have a thickness of 150 μm, and the mu-metal shells are 100 μm thick. All shells are wrapped around a 1-mm-thick central mu-metal wire. The outermost layer of the hose is directly machined out of the same block of aluminum as the cavity and has a length of 5 mm. This construction prevents losses from currents crossing an otherwise unavoidable gap between the hose and the cavity wall. The elongated part of the outermost shell acts as a $λ/4$ resonator with a resonance frequency around 10 GHz. Any electro-magnetic wave below this resonance is reflected and does not reach the ferromagnetic core of the hose. Thus, the magnetic hose does not provide any additional loss channel for microwaves at the qubit and cavity frequencies, which was verified by finite element simulations.

The discoidal coil at the back end of the hose has 18 turns, resulting in a diameter of 3.2 mm and an expected self-inductance of 523 nH in free space, which is reduced to 504 nH due to the outermost superconducting shell surrounding it. The coil design is chosen such that the cutoff frequency is about 100 MHz and does not limit the system response time, while still being able to generate a full $Φ0$ at the qubit location. More details on the assembly and hose efficiency can be found in the supplementary material.

A magneto-static simulation shows that the magnetic hose can route the magnetic field through a hole into a superconducting box as depicted in Fig. 2. Besides guiding the magnetic field into the cavity, the magnetic hose also routes it back to the outside. As a result, the total magnetic flux through the hole is always zero and flux quantization is preserved.

Implications of flux quantization also extend to the superconducting layers forming the hose itself. For this reason, a magnetic hose can transport the magnetic field only if all superconducting layers are cut along their length, preventing super-currents to flow in closed circles [see the inset of Fig. 1(b)]. Since ferromagnetic layers also have some electrical conductivity, the cut is also extended to them in order to minimize circular eddy currents. The cut introduced in the hose layers acts as a rectangular waveguide with a cutoff frequency of more than 60 Ghz.

In summary, the original hose design has been adapted to work with 3D architectures by carefully designing a hose extension that can provide enough field intensity at the qubit location while avoiding the introduction of losses. Moreover, a cut along the shells circumvents flux quantization and combined with a low inductance coil allows quick variations of the magnetic flux.

The simulation suggests a magnetic field transfer $TB(x,y,z)=Bx(x,y,z)/Bcoil$ of around $TB(5 mm,0, 0)$ = 0.37% at the hose end and $TB(6 mm,0, 0)$ = 0.08% at the central qubit. The fields at the qubit centers are 22.4 nT at the central and 5.8 nT at the side qubit. The ratio is compatible with the measured magnetic crosstalk between the qubits. By comparing this simulation with the one where only the hose is removed, we evaluate an improvement of 10⁵ for the field transferred at the central qubit. More details can be found in the supplementary material. We want to emphasize that similar fields at the qubit location could only be created by putting the same coil as used for the hose inside the cavity. This though would lead to enhanced losses and reduce the coherence of the qubit and the cavity. The total field estimated from the experiment at the qubit location is 240 nT for a current of 1 mA, 10 times more than that expected from simulation. We attribute this difference to a deviation of the assembled structure from the simulated one. We have seen in simulations that the absolute magnetic field strength is sensitive to small changes in the design parameters. However, relative field strengths are scaling correctly and improvements in the design can be predicted well.

After assessing the functionality of the hose through numerical simulations, we prepare the experiment (see the supplementary material) with the aim of verifying the qubit resonance frequency tunability by applying a DC voltage to the coil. In this experiment, we use an Arbitrary Waveform Generator (AWG) channel to provide the above-mentioned coil excitation. The flux map for both transmon qubits is shown in Fig. 3. The maximum coil current that we can apply in the experiment before heating up the cryostat is about 3 mA. The measurements have never shown hysteresis, which is expected as the generated fields $≤$ 100 μT are much smaller than the saturation fields of the mu-metal $≈$ 800 mT.

Spectroscopy is performed using a saturation pulse with variable frequency, followed by a readout pulse at the cavity resonance frequency, for each value of the AWG output voltage. The qubit resonance frequencies are extracted from the measurements and fitted. The flux map also demonstrates the local control capabilities of the hose as one qubit is more affected by the injected magnetic flux than the other. From the fit, we can extract that the hose is around five times more effective on the center qubit, which agrees well with numerical simulations. This factor can be further improved by using an optimized smaller hose design with a funnel-shaped tip and optimizing the placement of the qubits. The data point density has been increased in the avoided crossing region to measure the qubit–qubit interaction,¹⁸ which is J/2 $π≈$ 120 MHz.

In order to assess the fast flux capabilities of our system, we perform another experiment using the sequence shown in Fig. 4(a). To excite the qubit, we use a Gaussian amplitude-modulated excitation pulse, with a standard deviation of around 16.6 ns and a total length of 100 ns, generated by the AWG and up-mixed with an in-phase quadrature mixer. The pulse length is a trade-off between time and frequency resolution for this particular experiment. Another AWG channel is used to excite the coil with pulses added to an optional DC offset. During the measurements, the flux pulse and the readout pulse have a fixed delay. The frequency and the delay of the excitation pulse are changed linearly, to track the qubit resonance frequency during the flux pulse.

The first measurement [Fig. 4(b), blue] is obtained using a square flux pulse for the coil excitation [dashed line in Fig. 4(a)]. The measurement result shows a smooth change in the qubit resonance frequency, with a double exponential of the form $A(e−t/τ1+ e−t/τ2)$⁠. The extracted time constants are τ₁ = (25 ± 2) ns and τ₂ = (231 ± 10) ns.

In a second set of measurements, we speed up the system response through a simple pulse shape, adding an overshoot and a double exponential decay [continuous line in Fig. 4(a)] to the flux pulse (see the supplementary material). This sequence is inspired by the second method suggested in the supplementary material of Ref. 19. Using this sequence, we obtain the result shown in Fig. 4(b) (red line). The change in the qubit resonance frequency now occurs with an initial rise time of τ₁ = (14.1 ± 0.3) ns and a slower tail τ₂ = (100 ± 2) ns. The frequency of the qubit remains stable for 500 ns after the jump, with frequency fluctuation within 600 kHz, limited by the measurement precision. A further speed-up could be achieved by a better pulse premodulation, following the methods suggested in Ref. 19.

After checking the hose capabilities to apply a magnetic field, we estimate the effect of the hose on the system by measuring T₁, $T2*$⁠, qubit population, and the internal quality factor of the cavity. The decay time of the center qubit, as a function of frequency, ranges from $T1=$ (3.6 ± 0.4) μs at the high sweet spot up to $T1=$ (57 ± 8) μs toward low frequencies. This behavior is a combination of the Purcell effect of the cavity and the hose modes, which would limit T₁ to around 40 μs at 8 GHz plus an additional frequency independent loss as, e.g., observed in Ref. 20. This frequency independent loss leads to a scaling of $T1∼1/ω$ close to what we observe in the experiment.

The coherence time varies from about $T2*=$(4.0 ± 0.2) μs at the high sweet spot to $T2*=$ (230 ± 30) ns in the most flux-sensitive region of the flux map. An echo sequence shows an improvement by about a factor of 5–20 with the maximum value at the lower sweet spot of T_2E = (35 ± 2) μs. More details about the qubit coherence can be found in the supplementary material.

The qubit population has been estimated using the method shown in Ref. 21. The results obtained on several samples are in the range of 60 mK–80 mK (i.e., 0.5% to 3% excited-state population), comparable to the results obtained in other experiments with similar parameters.^21,22 This result and the amount of dephasing measured at the sweet spot as well as the obtained T₁ times are very typical for the qubits build in our facilities and are compatible with other experiments done in our lab without magnetic hoses.

In order to measure the internal quality factor Q_int of the cavity with the hose, a reflection measurement using a microwave circulator is done. The measured internal quality factor $Qint$ = 1.2 × 10⁶ is very typical for 3D waveguide cavities. Simulations show that $Qint$ can be compromised by the dissipation of surface current crossing seams between the cavity wall and the outermost shell of the hose.²³ Experimentally, with such a seam present, we measured Q_int = 25 × 10³ and Q_int = 3 × 10⁵ with indium sealing. To maximize Q_int, we avoid such a seam by machining the outermost hose shell and the cavity from the same aluminum block.

The measured flux crosstalk of 20% between the qubits is, for this particular magnetic hose design, larger than that for typical 2D flux-bias lines,²⁴ which can rely on effective shielding from superconducting ground planes. Crosstalk can be reduced using appropriate flux decoupling sequences²⁵ or by reducing the dimensions of the hose using, e.g., microfabrication techniques (see the supplementary material). Furthermore, we want to emphasize that 2D architectures could benefit from applying the magnetic hose in order to circumvent complicated wiring schemes. This would simplify chip design and reduce the unimpeded flow of supercurrents along the ground plane of the superconducting chip.²⁴ 

The maximal amount of magnetic field being transferred through the hose is limited by the used ferromagnetic and superconducting materials. For weak fields, the material can be assumed to deviate very little from their ideal behavior, but higher fields would induce nonlinear responses of the materials, which one would need to analyze carefully. On one hand, the superconducting shells are not expected to significantly limit the hose performance since radial components of the magnetic field (the ones that the superconductor shields) are much smaller than axial components. Furthermore, high-Tc superconductors like Yttrium-Barium-Copper-Oxid are able to shield magnetic fields of >1 T completely.²⁶ Ferromagnetic shells, on the other hand, are expected to be the main limiting factor for high-field transfer. Numerical calculations indicate that, though, fields on the order of hundreds of mT can be transported through hoses made of high-field-saturation ferromagnetic materials like soft irons or electrical steels.²⁷ 

The experimental work presented in this paper shows the potential of using a magnetic hose to control various quantum systems. The hose is most applicable in scenarios where a local applied magnetic field is required, combined with the ability to generate the magnetic field at a distant location and route it to the quantum systems of interest. Most obviously, it would be ideally suited to individually control qubits in superconducting circuit-like architectures^28–30 without compromising coherence times. Moreover, due to the ability to apply time-varying signals, one could also use the hose to drive parametric amplifiers^6,31 or tunable couplers.⁴ The magnetic hose could also allow new designs of nuclear magnetic resonance (NMR) sensors based on nitrogen vacancy (NV) centers^32,33 or even enable magnetic coupling between NV centers.³⁴ 

In summary, we have shown that a magnetic hose can be used to change the magnetic flux inside an aluminum waveguide cavity in less than 100 ns without compromising coherence times and quality factors. As such, the hose is ideally suited to implement fast flux bias lines in 3D³⁰ and waveguide²⁸ circuit QED architectures.

See the supplementary material for more information about hose design, field transfer, setup, qubit parameters, flux noise analysis, and description of the measurements.