We implement a broadly tunable phase shifter for microwaves based on superconducting quantum interference devices (SQUIDs) and study it both experimentally and theoretically. At different frequencies, a unit transmission coefficient, |*S*_{21}| = 1, can be theoretically achieved along a curve where the phase shift is controllable by the magnetic flux. The fabricated device consists of three equidistant SQUIDs interrupting a transmission line. We model each SQUID embedded at different positions along the transmission line with two parameters, capacitance and inductance, the values of which we extract from the experiments. In our experiments, the tunability of the phase shift varies from 0.07 × *π* to 0.14 × *π* radians along the full-transmission curve with the input frequency ranging from 6.00 GHz to 6.28 GHz. The reported measurements are in good agreement with simulations, which is promising for future design work of phase shifters for different applications.

Recent progress in superconducting microwave electronics has inspired research on a more complete toolbox for quantum engineering.^{1} Here, superconducting circuits with Josephson junctions exhibit a solid and scalable technology platform stemming from their mature lithographic fabrication processes.^{2} During the recent decades, fascinating superconducting rf components for the toolbox have been demonstrated, such as Josephson parametric amplifiers,^{3,4} kinetic inductance travelling-wave amplifiers,^{5,6} switches,^{7,8} circulators,^{9,10} isolators,^{11} beam splitters,^{12,13} phase shifters,^{14,15} and photon detectors.^{16–18} In the future, these may be integrated into monolithic circuits for sophisticated quantum signal processing.

To further improve the ability to process quantum microwave information, a quickly tunable, compact, and lossless phase shifter for microwave photons operating over a broad frequency band is a highly desirable tool, not only to tailor propagating single-photon states^{19} but also to tune the phase of on-chip coherent microwave sources.^{20} If such sources are further augmented with quantum-circuit refrigerators,^{21,22} bulky room temperature signal generators could be replaced by devices on a single chip. Such a tool would be highly desirable for scaling up a quantum computer.^{23}

Interestingly, the transfer of quantum states between distant stationary qubits has been achieved utilizing propagating microwave photons.^{24} Such photons are also required for far-field microwave quantum communication. The utilization of a tunable phase shifter in such schemes provides opportunities for the detailed control of the quantum states of the propagating photons. For example, the phase shifter would allow for the preparation of an arbitrary squeezing angle of squeezed states for secure communication.^{25}

Previously, long arrays of superconducting quantum interference devices (SQUIDs) have been demonstrated to implement a transmission line with a tunable speed of light, and consequently, a tunable phase shift for microwaves.^{26,27} However, the characteristic impedance of the transmission line naturally changes with the phase shift here, which provides an additional challenge in the above-described single-photon applications that essentially call for the absence of reflections from the phase shifter. In this letter in contrast, we experimentally realize a tunable phase shifter based on three equidistant SQUIDs separated by coplanar-waveguide transmission lines (CPWTLs), a scenario theoretically shown in Ref. 14 to provide vanishing reflections for a finite range of phase shifts. Adopting differential flux bias lines instead of single-ended flux bias lines used in Ref. 14 decreases the cross coupling of the SQUID fluxes. Consequently, we can tune the operation frequency of the device by 280 MHz, rather than operating at a single frequency point.^{14} We utilize our theoretical model complemented by numerical computations to control the phase shifter such that it provides essentially unit transmission throughout this frequency range of interest. Thus, our work is a significant step toward an extended toolbox of superconducting microwave components.

The considered phase shifter is composed of three SQUIDs connected by two CPWTLs of equal length *d* = 6.41 mm, as shown in Fig. 1(a). The use of CPWTLs provides a possibility for achieving a broad bandwidth and unit transmission in a finite frequency range in contrast to a phase shifter where microwaves reflect from a resonator.^{15} In our theoretical model, each SQUID is treated as a parallel connection of a tunable inductor and a capacitor, thus forming an *LC* oscillator. The inductance of an ideal SQUID, *L*_{k}, can be modulated by applying an external magnetic flux as *L*_{k} = Φ_{0}/(4*πI*_{c}| cos(*π*Φ_{k}/Φ_{0})|), where Φ_{0} is the magnetic flux quantum, *I*_{c} is the critical current of the SQUID, and Φ_{k} (*k* = 1, 2) is the external magnetic-flux threading the loop of SQUID *k*. In contrast to Ref. 14, we have not assumed all SQUIDs to be identical, but allow for the center SQUID in our model to have a different capacitance, *C*_{2}, from that of the side SQUIDs, *C*_{1}. Note that the two side SQUIDs are tuned identically.

Let us consider the quantum scattering of microwaves from the leftmost *LC* oscillator in Fig. 1(e). In the Heisenberg picture, the quantum network theory^{28} yields

where $\xe2L1L$ and $\xe2L1R$ denote annihilation operators of the left-moving wave on the left and on the right sides of the left resonator, respectively, and $\xe2R1L$ and $\xe2R1R$ denote the corresponding right-moving operators. Considering Kirchhoff’s current law and the expansion of the charge operators in terms of the annihilation operators, one can write

Similar equations and boundary conditions can be obtained by analyzing the middle and right oscillators in Fig. 1(e). The CPWTLs between the three oscillators generate a delay that converts into a phase change of the propagating signal, *ϕ* = *ωd*/$v$, where *ω* is the angular frequency of the microwave radiation and $v$ is its speed. This can be written as

We solve Eq. (1) utilizing the boundary conditions Eqs. (2) and (3) to obtain the transmission coefficient^{14}

where

Assuming that the SQUID inductances are arbitrarily tunable, we may choose

where *θ* is a free real-valued parameter fixing our choice of the flux points. There is only a single free parameter since we have chosen the parametrization of the inductance such that microwave reflections from the circuit vanish. These equations can be used to determine the range of inductances and critical currents needed to achieve a desired phase shift at a given signal frequency. In fact, the insertion of Eq. (7) into Eq. (4) yields *S*_{21} = *e*^{i(θ+2ϕ)}. Thus, the device exhibits full transmission and a phase shift of *θ* in addition to 2*ϕ* arising from the transmission line of length 2*d*. The tunability range of *θ* depends on the ability to implement the inductances according to Eq. (7). In particular, vanishing or negative inductances are not feasible in the implementation described in Fig. 1.

To implement the above theoretical scheme, we fabricate a sample adopting shadow evaporation and load it into a dilution refrigerator operating at 15 mK. On-chip differential bias lines are utilized to produce a bias magnetic field at each SQUID with low crosstalk. The device is reciprocal and symmetric with respect to the left and right SQUIDs, which renders it convenient to integrate the phase shifter with other on-chip components for future applications.

The power level of the probe signal at the device is kept below −90 dBm in order to keep the SQUIDs in the linear regime. Note that this is well above the single-photon level. Details of the measurement setup are given in the supplementary material.

We begin the characterization of our device by first focusing on a single SQUID at a time. Namely, we measure the transmission coefficient of the device as a function of the magnetic-flux bias of each SQUID at a time, ideally leaving the other two SQUIDs at a constant magnetic field. Figure 2(a) shows the transmission amplitude |*S*_{21}| as a function of the flux bias of the left SQUID for three periods at five different frequencies ranging from 6.00 GHz to 6.28 GHz. The amplitude changes very gently around integer flux quanta but exhibits sharp drops in the vicinity of half-integer flux values where the SQUID inductance ideally diverges. In fact, we observe two drops in the transmission amplitude around each half-integer flux point since the parallel *LC* oscillator achieves its maximum impedance at finite inductance where the *LC* resonance matches with the probe frequency.

Figure 2(b) shows the transmission amplitude from 0.4 × Φ_{0} to 0.6 × Φ_{0} to demonstrate that the experimental results are in good agreement with simulations where we use the SQUID capacitance and the critical current *I*_{c} as fitting parameters. Consequently, we obtain 1.5 *μ*A for the critical current of the left SQUID, which yields ∼0.44 nH of inductance. Thus, at zero flux, the magnitude of the transmission coefficient is close to unity since the impedance of a zero-flux SQUID is given by 1/[(1/*iωL*_{1}) + *iωC*_{1}] = *i*18.7 Ω for the obtained capacitance *C*_{1} = 180 fF at the frequencies of interest. In Figs. 2(c)–2(f), we apply this method to the middle and right SQUIDs and obtain similar results as for the left SQUID. In our model, we set the critical current and capacitance of the right SQUID equal to those of the left SQUID, 1.5 *μ*A and 180 fF, respectively. The critical current of the middle SQUID is 1.24 *μ*A and the capacitance is 170 fF.

In Ref. 14, this type of a phase shifter was challenging to operate at different frequencies because of the inductive crosstalk. The applied bias current induced unwanted currents to the ground plane near all SQUIDs. To eliminate this effect, we redesigned the flux bias lines to be differential and removed the ground plane from its vicinity.

Consequently, we show in Fig. 3, the phase of the scattering parameter as a function of the magnetic flux threading the middle SQUID and both side SQUIDs at five different frequencies that match those of Fig. 2. The fluxes through the side SQUIDs are tuned to keep their inductances equal to each other. Using bilinear interpolation in the flux plane, we extract the transmission coefficient along the full-transmission curve that follows Eq. (7) for both the magnitude |*S*_{21}| (see the supplementary material) and the phase Arg(*S*_{21}). Figures 3(c), 3(f), 3(i), 3(l), and 3(o) show the measured and simulated phases along the curve at five frequencies. The clear modulation of the phase is observable along the curve, which indicates that we may tune the phase at will while keeping the transmission through the phase shifter close to unity.

Figure 4 summarizes the tunability of the phase shift in a dense frequency grid from 6.00 GHz to 6.28 GHz. Here, we define the tunability as max_{s}Arg(*S*_{21}) − min_{s}Arg(*S*_{21}) along the full-transmission curve parametrized by the parameter *s* defined in Fig. 3. We observe that in the whole frequency range considered, the tunability is over 0.07 × *π* radians. The largest tunability is 0.14 × *π* radians at 6.013 GHz. The tunability can be further optimized by fine-tuning the parameters of the transmission lines and of the SQUIDs in the design and fabrication steps. Such parameters include the sizes of the junctions that determine the critical currents and the capacitances of the junctions. Theoretically, an ideal three-SQUID device may achieve a maximum phase shift tunability approaching 2*π* at the optimal frequency.^{14}

At a given flux point on the curve of unit transmission, the transmission amplitude is relatively close to unity in the full studied frequency band from 4 GHz to 8 GHz, whereas the phase shift resulting from the phase shifter changes linearly by the amount of *π* (see the supplementary material for data). This results in a phase error of roughly 10^{−3} rad/MHz. Thus, also at a given flux point, the phase shifter works accurately in a relatively broad frequency band.

In conclusion, we implemented a phase shifter composed of three equidistant SQUIDs in CPWTLs. We presented an extension to the quantum theory of the phase shifter by allowing the parameters of the middle SQUID to be different from those of the identical side SQUIDs. The undesired coupling from each flux bias line to the two distant SQUIDs was reduced by adopting differential flux bias lines. Consequently, by tuning the magnetic fluxes through the SQUIDs, we managed to observe significant phase shifts throughout a 280-MHz bandwidth from 6 GHz to 6.28 GHz. The experiments were found to be in good agreement with classical-circuit simulations, which provided us with estimates of the parameters of the SQUIDs.

This tunable phase shifter exhibits potential for applications in quantum microwave signal generation and processing. In the future, we aim to optimize the phase shifter for operation in a broader frequency range and for a larger tunability of the phase shift for specific applications with the help of our theoretical model given realistic critical currents and capacitances. In addition, the phase shifter can be integrated with other microwave components, such as a quantum-circuit refrigerator^{21,22} and a microwave source^{20} to achieve a tunable single-chip source. Thus, this work paves the way for advanced cryogenic microwave devices and expands the quantum-engineering toolbox.^{1}

See the supplementary material for the measurement setup and amplitude of transmission.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This research was financially supported by the European Research Council under Grant No. 681311 (QUESS); by the European Commission through H2020 program project QMiCS (Grant Agreement No. 820505, Quantum Flagship); by the Academy of Finland under its Centers of Excellence Program Grant Nos. 312300, 312059, 308161, 314302, 316551, 318937, and 319579; and by the Alfred Kordelin Foundation, the Emil Aaltonen Foundation, the Vilho, Yrjö, and Kalle Väisälä Foundation, the Jane and Aatos Erkko Foundation, the Technology Industries of Finland Centennial Foundation and Finnish Cultural Foundation. We thank Jan Goetz and Juha Hassel for useful discussions and the provision of facilities and technical support by Aalto University at OtaNano–Micronova Nanofabrication Center.

## REFERENCES

*et*et al,

*et*et al,

*et*et al,