Broadband Tunable Phase Shifter For Microwaves

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 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 from $0.07\times\pi$ to $0.14\times\pi$ radians along the full-transmission curve with the input frequency ranging from 6.00 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.

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 tool would be highly desirable for scaling up a quantum computer [23].
Interestingly, 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. Utilization of a tunable phase shifter in such schemes provides opportunities for 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].
In this letter, we experimentally realize a tunable phase shifter based on three equidistant SQUIDs in a transmission line. We improve upon our previous design [14] by adopting differential flux bias lines instead of single-ended flux bias lines to greatly decrease the cross coupling of the SQUID fluxes, from which the previous design suffered. Consequently, we demonstrate that we can tune the operating frequency of the device by 280 MHz. 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 towards an extended toolbox of superconducting microwave components.

II. THEORETICAL MODEL
The considered phase shifter is composed of three SQUIDs connected by two coplanarwaveguide transmission lines (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 external magnetic flux as 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 , than that of the side SQUIDS, C 1 .
Let us consider quantum scattering of microwaves from the leftmost LC oscillator in Fig. 1(e). In the Heisenberg picture, the quantum network theory [26] yields whereâ L L1 andâ R L1 denote annihilation operators of the left-moving wave on the left and on the right side of the left resonator, respectively, andâ L R1 andâ R R1 denote the corresponding right-moving operators. Considering Kirchhoffs current law and Fourier expansion of the charge operators with annihilation operators of signal quanta, one can writeŝ 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 which 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 aŝ We solve Eq. (1) utilizing the boundary conditions Eq. (2) and Eq. (3) to obtain the transmission coefficient where Assuming that the SQUID inductances are arbitrarily tunable, we may choose where θ is a free real-valued parameter fixing . In particular, vanishing or negative inductances are not feasible in the implementation described in Fig. 1.

III. EXPERIMENTAL RESULTS
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 Supplementary Materials. The capacitance and I c of the middle SQUID are 170 fF and 1.24 µA, respectively.

A. Characterization
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

B. Phase shift and its tunability
In Ref. [14], this type of a phase shifter was challenging to operate at different frequencies because of inductive crosstalk. The applied bias current induced unwanted currents to the ground 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  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 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  The shading defines the region of achievable phase shifts, i.e., the tunablity region of the phase shifter.
Supplementary Materials for data). This results in a phase error of roughly 10 −3 rad/MHz.
Thus also in a given flux point, the phase shifter works accurately in a relatively broad frequency band.

IV. CONCLUSION
In conclusion, we implemented a phase shifter composed of three equidistant SQUIDs in a transmission line. We presented an extension to the 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 an improved design. 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 lager tunability of the phase shift. 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].