Sharp-selectivity in-line topology low temperature superconducting bandpass filter for superconducting quantum applications

This paper presents a new class of sharp-selectivity low-temperature superconducting filter that incorporates lumped element resonant couplings. Dependent on a novel synthesis approach, the proposed filter exhibits great advantages such as: (1) a very simple in-line topology (without any cross coupling), (2) extremely compact size based on lumped inductor-capacitor (LC) elements, and (3) multiple transmission zeros (TZs) independently generated and controlled (via each resonant coupling). To facilitate the physical implementation, a group of lumped element circuit models are detailed, where series LC units are adopted for both the resonators and the resonant couplings. Considering an in-line topology here, the entire filter layout is then designed by cascading the lumped models one after another. For verification, a 5th-order bandpass filter centered at 5 GHz, with 500 MHz bandwidth and 3 TZs, is designed, simulated, and tested at cryogenic temperature (4.2 K). Moreover, preliminary simulations of the presented filter in series with an on-chip rapid single-flux-quantum microwave pulse generator are discussed for superconducting quantum applications.


Introduction
Microwave filters play an important role in a variety of superconducting analog and digital systems and therefore have attracted much attention in recent years. In [1][2][3][4], one very popular topic concerns the design of high-temperature superconducting (HTS) filters based on distributed resonant structures. By replacing normal metal layers with HTS thinfilm materials, these filters provide an extremely low insertion-loss performance that is crucially demanded in modern wireless applications.
Recently, low temperature superconducting (LTS) microwave circuits have been extensively investigated for superconducting quantum computing systems [5][6][7][8][9]. Specifically, we have been developing an on-chip rapid singleflux-quantum microwave pulse generator (RSFQ MPG) [10,11] to control superconducting qubits in a scalable quantum computing system [12]. Figure 1 illustrates the schematic of an RSFQ MPG. The clock (CLK) generator outputs a single-flux-quantum (SFQ) pulse train, the duration time of which is controlled by the RSFQ switch. Then, the multi-flux-quantum (MFQ) driver converts an SFQ pulse into an MFQ pulse to amplify the final output level. Here, the microwave filter functions as a key component to convert the MFQ pulse train into a microwave pulse, and should satisfy the following requirements: (1) on-chip implementation with extremely compact size, which indicates that lumped LC structures are preferred; and (2) sharp frequency selectivity as well as uneven group delay performance, which can control the rise and fall time of the microwave pulse. Especially, for quantum annealing machines using nonlinear oscillators [13][14][15], it is required to very slowly ramp up microwave pulses because the ramp-up time corresponds to the temperature scheduling.
Relying on the same idea, a group of lumped or quasilumped LTS filters are designed and associated with RSFQ circuitry for qubit controls in [16][17][18][19]. More efforts, dealing with the implementation techniques of LTS filters, are reported in [20] and [21]. These proposed filters, based on conventional Chebyshev characteristics, however, require a very high order and massive elementary units. Thus, a favorable mechanism to meet the frequency selectivity and group delay performance for our purpose remains a problem.
In this work, we propose a new class of LTS lumped element bandpass filter possessing generalized Chebyshev characteristics [22], where the desired frequency selectivity and group delay requirements are achieved with a small number of transmission zeros (TZs). Note that, without cross coupling (compared with literatures [1][2][3][4]), the TZs here are individually generated via resonant couplings (also called frequency-variant couplings), figure 2. As a result, the filter can be structured by a simple in-line topology as figure 3 shows, where the configuration simplicity as well as TZ independence are quite beneficial for RSFQ MPG realization.
In the following, a synthesis theory to determine the presented in-line prototype is described in section 2. Series type LC unit models for constructing the resonators and couplings are detailed in section 3. In section 4, a 5th-order bandpass filter, centered at 5 GHz with 10% bandwidth and three finite TZs, is demonstrated for validation. The simulated waveform by utilizing such a filter with an RSFQ MPG is also provided to show its availability for superconducting quantum applications. Finally, section 5 concludes the paper.

Synthesis theory
Direct synthesis technique refers to a deterministic procedure to identify the topology as well as all the coupling coefficients of an Nth-order filter network. This usually involves the admittance matrix transformations of the filter lowpass prototype by [22] [ As a result, undesired couplings from an original prototype (with admittance matrix [Y 0 ]) are annihilated and we finally derive a practical filter topology (with admittance matrix [Y K+1 ]) that is very suitable for physical implementation. By means of diverse synthesis techniques in literature (e.g. [22][23][24]), a variety of practical topologies comprising cross-coupled constant couplings have been proposed. These topologies provide basic design principles for many filter constructions, such as [1][2][3][4].
Particularly, the in-line topology shown in figure 3 is considered as the most attractive solution due to its simplicity; and we have accomplished the relevant synthesis procedure very recently in [25,26]. Without any cross-coupling involved (e.g., in [1][2][3][4]), here the TZs are separately generated via a set of resonant couplings between adjacent resonators. Accordingly, the admittance matrix (named as [Y final ]) is determined in terms of (1), as figure 4 shows. Note that, [Y final ] identifies all the coupling coefficients required for the filter realization. Due to an in-line topology, it is apparent in figure 4 that non-zero coupling coefficients only exist in the diagonal arrows.

Lumped element models
To physically realize the derived in-line filter in the bandpass domain, a lumped element transformation technique by applying series LC units is introduced in this paper. As a   [22][23][24] in the lowpass Ω-domain, bandpass domain, and their equivalent circuit models. Note that f 0 refers to the center frequency, while B n represents the fractional bandwidth of a bandpass filter. result, all the capacitances and inductances can be obtained at one time. Regarding the prototype in figure 3, here we consider three elementary blocks for the filter construction, i.e., resonators connected by a constant coupling (block A), resonators connected by a resonant coupling (block B), and terminal resonators connected to the source/load (block C). In the following, the fractional bandwidth of the filter in the bandpass domain is assumed as B n , while the center frequency is f 0 . In addition, χ i (i=1, 2 K, N) gives the reactance-slope parameter of each LC resonator [27], which are manually selected during design.

Block A: resonators connected by a constant coupling
The lumped model for block A is illustrated in figure 5. Note that two resonators, p and k, are modeled by series LC resonant units separately. Moreover, the constant coupling is realized via a traditional T-shape K-inverter comprising both positive and negative inductances (i.e., L pk ) [27], where the negative parameters are thereafter absorbed into the resonators at both sides (see right side of figure 5). The eventual inductances and capacitances of block A are thus obtained as: where the coupling coefficients M 0,pp and M 0,pk are derived from the former section. Accordingly, L kk and C kk can be obtained in the same manner.

Block B: resonators connected by a resonant coupling
The lumped model for a resonant coupling is however never discussed in the literature. Specifically, here we propose a new kind of K-inverter comprising series LC resonant units, as shown in figure 6. By absorbing all the negative parameters into the resonators (see right side of figure 6), the ultimate circuit parameters are determined by:

Block C: terminal resonators connected to source/load
On the other hand, the terminal couplings at source and load in figure 7 are processed with a particular mechanism. Considering the source and load impedances as Z S and Z L (suppose Z S =Z L =50 Ω in this work), respectively, the de-normalized coupling coefficients in the bandpass domain are calculated directly as [28]: It is then interesting to notice that, by selecting χ 1 and χ N as the couplings k 0,S1 and k 0,NL will become equal to the 50 Ω terminal impedances. It implies that these terminal couplings . Note that four types of couplings (described in [22]) are involved and denoted in different blankets.   can be removed without influence on the filter response, which further miniaturizes the whole filter area [29].

Physical implementation
The obtained LC network is implemented by using the AIST Nb 2.5 kA cm −2 Standard Process 2 (STP2) [30]. Particularly, the inductors are constructed by using spiral coils on M4 layer, and the capacitors are realized by parallel plates on M3 and M4 layers. The relevant layouts are initially determined from a 3D extractor InductEx (v5.06) [31], and then adjusted under ANSYS Electromagnetic Suites (v19.2). Specifically, figure 9(a) gives the layout view of the L 11 & C 11 resonant unit. As the desired lumped parameters are already known in figure 8, the layout can thus be adjusted by fitting its reactance the same with that of the lumped element model, which is illustrated in figure 9(b). The same process is utilized to other sub-networks (e.g., L 22 & C 22 , L 12 & C 12 , etc) as well.
Since an in-line topology is concerned, the whole layout of the 5th-order filter can then be constructed with sub-networks (namely the resonant units) in series. Figure 10 gives the micrograph of the fabricated filter. Note that, an M1 layer is placed outside of the lumped elements to shunt C 12 , C 23 , C 34 , and L 45 to ground. Note that the total size of the filter is 1650 μm×350 μm, which is 53% reduced from our previous lowpass filter designed in [11].

Experimental results
The filter is then measured by a vector network analyzer (Keysight, P9374A) at a 4.2 K cryogenic temperature. The tested result is compared with EM simulation in figure 11. Imperfection of in-band return loss and insertion loss are mainly attributed to the frequency response of the cryogenic probe during measurement. Nevertheless, a sharp out-of-band selectivity (with 3 TZs at almost desired locations) can be obtained in both simulation and measurement, therefore revealing the validity of the presented approach.  Furthermore, we verify the functionality of the filter combined with an on-chip RSFQ MPG using the Josephson circuit simulator JSIM [32], whose output waveforms are illustrated in figure 12. Particularly, figure 12(a) provides the original 5 GHz MFQ pulse train generated by the MFQ driver shown in figure 1, where two SFQ pulses are generated every 200 ps. In the simulations, '0 ps' refers to the time we rise the bias voltage so that the SFQ circuit is ready for propagating SFQ pulses. By turning on the RSFQ switch at 200 ps, the SFQ pulses appear after a few picoseconds delay at around 300 ps. Note that, the amplitude of SFQ pulses, i.e. 0.45 mV, is adjustable by changing the parameter of Josephson Junctions in the RSFQ MPG for different applications. The MFQ pulse train is then converted to diverse 5-GHz microwave pulses in figures 12(b) and (c) through alternative filters (one is a lowpass filter utilized in our previous work [11], and the other is the 5th-order bandpass filter in this work). It is noticed that an output microwave with much slower rise time can be acquired when the proposed sharp-selectivity bandpass filter is applied, thus demonstrating the effectiveness of this work for quantum computing applications.    [11], and (c) with the proposed 5th-order bandpass filter (BPF) example.

Conclusion
A new kind of lumped element LTS filter is proposed to achieve sharp selectivity and uneven group delay performance in this paper. Without any cross-coupling nor extra structure, multiple TZs are generated based on a group of resonant couplings individually, which results in a simple inline topology for compact filter implementation on a chip. Complete design principles, including a direct synthesis method and a lumped circuit transformation technique, are introduced successively. Validity of the presented approach is revealed by a 5th-order example by using AIST STP2 process, where both simulated and tested results are given. In addition, the simulated output waveform by applying the filter with an RSFQ MPG is also provided, which infers the effectiveness of such filter in superconducting quantum systems in the future.