# Multiplexing Superconducting Qubit Circuit for Single Microwave Photon Generation

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## Abstract

We report on a device that integrates eight superconducting transmon qubits in \(\lambda /4\) superconducting coplanar waveguide resonators fed from a common feedline. Using this multiplexing architecture, each resonator and qubit can be addressed individually, thus reducing the required hardware resources and allowing their individual characterisation by spectroscopic methods. The measured device parameters agree with the designed values, and the resonators and qubits exhibit excellent coherence properties and strong coupling, with the qubit relaxation rate dominated by the Purcell effect when brought in resonance with the resonator. Our analysis shows that the circuit is suitable for generation of single microwave photons on demand with an efficiency exceeding 80%.

## Keywords

Superconducting qubit Transmon Superconducting resonator Single photon generation## 1 Introduction

A new paradigm of information processing based on the laws of quantum physics has triggered intensive research into studying physical systems that can be used as the building blocks of a future quantum processor. Remarkable progress towards realising quantum information processing elements has been achieved by using both natural and artificial atoms as qubits, and in arranging them into more complex circuits [1]. Artificial atoms are engineered quantum systems that have a number of advantages in comparison with their natural counterparts. First, they are fabricated using existing well-developed nanofabrication methods of conventional electronics and therefore can be placed at will whilst having controllable custom-designed features. Second, their size is macroscopic (\(\gtrsim \)1 \(\mu \)m), which simplifies the task of coupling multiple qubits together into integrated quantum circuits. The large size of artificial atoms results in their large dipole moment which enables strong coupling of individual qubits to electromagnetic field. Third, their energy levels are tunable by external fields, which simplifies control of the quantum states and inter-qubit couplings. As a downside, the undesired coupling to the environment is strong, leading to shorter coherence times of artificial as compared to natural atoms. This requires careful design of the experimental apparatus to protect fragile quantum states.

Superconducting quantum devices containing Josephson junctions can behave like atoms and are primary candidates to being the building blocks of the quantum processor [2]. Whilst there has been enormous progress in the field in the past 15 years or so [3], the superconducting qubit circuits require further optimisation in order to meet the stringent requirements on coherence for large-scale quantum information processing [4]. Nonetheless, the presently available circuits are already good enough for doing quantum optics and atomic physics experiments on a chip [5, 6]. A key approach to this is coupling a superconducting qubit to a microwave resonator, thus forming a circuit quantum electrodynamics (cQED) architecture [7, 8], which is a solid-state analogue of the cavity QED approach [9] used to study the interaction of natural atoms with photons. In circuit QED, the field confinement produced by the very small mode volume in combination with the macroscopic size of the qubit results in strong qubit–photon coupling, where quantum excitations can transfer between the artificial atom and the resonator and back many times before decay processes become appreciable.

Circuit QED architecture is uniquely suitable for manipulating microwave radiation at the single photon level [8] and for generation of single microwave photons on demand [10, 11]. An alternative approach to the single microwave photon generation is based on an artificial superconducting atom directly coupled to an open-end transmission line, a 1D analogue of the 3D half-space [12]. Here we report on a design comprising eight resonators, each housing a superconducting transmon qubit [13] that exhibits good coherence properties. The design allows for efficient frequency multiplexed testing and assessment of the superconducting qubits and is suitable for single microwave photon generation. This involves time-domain control of the quantum state in the superconducting circuit, which is a well-developed technique commonly used in qubit experiments. The key feature of our design is that it allows for multiplexed generation of single photons with an efficiency of >80% using quantum state manipulation and dynamic tuning of the circuit parameters. This provides the means of coupling solid-state qubits with each other for long-distance communications as well as linking stationary and flying qubits.

## 2 \(\lambda /4\) Resonator Design

The resonators are designed as coplanar waveguides in superconducting niobium metal film, having a centreline width *W* \(=\)20 \(\mu \)m and centreline to groundplane spacing of *S* = 10 \(\mu \)m, achieving a characteristic impedance \(Z_0 \simeq 50\,\varOmega \) on our sapphire substrate. We calculate from the geometry of our transmission lines a capacitance per unit length of \(c_r \simeq 153\;\mathrm {pF/m}\), an inductance per unit length \(l_r \simeq 402\,\mathrm {nH/m}\), and a phase velocity of \(\beta = 53.3\;\mathrm {radians/m/GHz}\) so that a \(\lambda /4\) resonator at 7 GHz has a length of 4220 \(\mu \)m. The eight \(\lambda /4\) resonators are designed with unloaded operating frequencies of 7.0, 7.1, 7.2 \(\ldots \) 7.7 GHz by varying the length of a reference design (see Fig. 1a). The resonators are inductively coupled to a common feedline running diagonally across the chip (see Fig. 1b). The coupling strength between the feedline and the resonators is adjusted by a short, \(\delta x \sim 400\,\mu \)m, section of the resonator parallel to the feedline with centre to centre distance of \(44\,\mu \)m.

## 3 Qubit Design

*r*promoting long relaxation times. At the same time, the value of

*r*determines the maximum qubit operation speed through the level anharmonicity \(|\nu _{01} - \nu _{12}|\), with faster operation times favouring smaller values of \(r < 100\) [13]. Since in our case the qubit forms part of a single photon source where both parameters are important, we selected a compromise value as \(r = 35\) which allows for operation times of the order of tens of ns whilst maintaining long relaxation times (>1 \(\mu \)s).

The final important property of the qubit for our purposes is the state-dependent dispersive shift \( \chi / 2 \pi \) of the resonator [18]. When the qubit and resonator are well detuned from one another, the state of the qubit \(|{0}\rangle \) or \(|{1}\rangle \) ‘pulls’ the resonator frequency by \(\pm \chi / 2 \pi \), providing a means to read out the qubit state [19].

The self-capacitance \(C_{{\varSigma }}\) of the device is the sum of the tunnel junction capacitance and the geometric capacitance. We estimate the junction capacitance using a typical value of the specific capacitance for Al ultrasmall tunnel junctions of 45 fF/\(\mu \)m\(^2\) [22], which for our junction area of \(100 \times 100\,\hbox {nm}^2\) gives \(\simeq 0.45\,\hbox {fF}\). Clearly, the junction capacitance contributes only about 1% to \(C_{{\varSigma }}\) and we can safely neglect it. This in turn yields \(C_g = 3.2 \, \mathrm {fF}\).

## 4 Device Fabrication

The samples were fabricated on a single-crystal c-axis oriented sapphire wafer. The fabrication process involved two major steps: patterning of the niobium groundplane, followed by deposition of the transmon structure and tunnel junctions. Both stages of the device fabrication utilised electron beam lithography at 100 kV.

*1. Patterning of the groundplane* The feedline, \(\lambda /4\) resonators and qubit shunt capacitors were formed by etching a 100 nm niobium metal film deposited by DC magnetron sputtering onto the sapphire wafer. The metal-coated wafer was spin-coated with AR6200(EL11) resist at 6000 rpm for 60 s, and the resist was baked on a hotplate at 150\(^\circ \)C for 9 min before the resonator and feedline structures were exposed in an e-beam writer to a dose of 350 \(\mu \)C/cm\(^2\) using a 58 nA beam current and proximity correction routine. The exposed wafer was developed in AR600-546 developer for 3 min before immersing in isopropyl alcohol to halt the development process. The resulting wafer was dried using N\(_2\) gas, and the resist was reflow baked for 5 min at 150\(^\circ \)C. An SF\(_6\)+O\(_2\) reactive ion etching process was then used to transfer the pattern from the resist into the niobium film. The reactive ion etching chamber was pre-conditioned for 10 min before the samples were loaded and etched for 3 min 15 s, resulting in pattern transfer into the niobium film. The e-beam resist was removed by immersing in AR600-71 solvent for 5 min, followed by sonicating in acetone for 1 min and rinsing in isopropyl alcohol for 1 min, before drying in N\(_2\) gas.

The patterned wafer was then coated with a protective layer of AZ5214E photoresist before dicing into 8\(\times \)8 mm chips using a hubless resin blade. Following dicing, the protective resist was removed in acetone followed by isopropyl alcohol and N\(_2\) drying.

*2. Deposition of transmon tunnel junctions* The chips were prepared for qubit junction deposition by spin-coating with two layers of MMA(8.5)MAA EL11 resist, at 4000 rpm for 60 s, and then baking each layer for 2 min at 160\(^\circ \)C, to create a 1 \(\mu \)m film of MMA(8.5)MAA resist, followed by a single layer of 950 PMMA A4 spun at 4000 rpm for 60 s and baked for 10 min at 160\(^\circ \)C to produce a 200 nm film. To avoid excessive charging during e-beam exposure, the resist structure was coated with an “E-spacer 300z” conductive layer by spinning at 4000 rpm for 60 s. The qubit patterns were exposed using a beam current of 1.1 nA on fine structures and 10 nA on coarse structures. The exposed resist was developed by rinsing in flowing deionised water for 30 s to remove the E-spacer layer, followed by immersion in 1:4 vol/vol methyl isobutyl ketone/isopropyl alcohol developer for 20 s, and then immersion in a 1:2 vol/vol mixture of methyl glycol/methanol for 20 s, followed by immersion in isopropyl alcohol for 10 s to halt development, before blowing dry with N\(_2\) gas.

The resulting resist structure was used to deposit the qubit junctions in an electron beam evaporation chamber equipped with a tilting sample stage, argon ion milling capability and O\(_2\) inlet needle valve. The sample surface was first cleaned by ion milling for 5 min using 1 kV argon ions with a nominal ion beam current density of 100 \(\mu \)A/cm\(^2\). The sample position under the ion beam was adjusted so as to clean the sample surface whilst preserving the resist mask. A 20 nm aluminium film was evaporated whilst the stage was tilted at \(-17^\circ \). The aluminium was then oxidised in 4 mbar O\(_2\) for 4 min before a second 20 nm aluminium layer was deposited with the stage tilted to +17\(^\circ \). The resulting chips were subjected to a lift-off process in warm acetone at 52\(^\circ \) for 30 minutes, followed by a cleaning rinse in fresh acetone, isopropanol and drying with N\(_2\) gas. The normal state resistance of the parallel combination of two nominally 100\(\times \)100 nm\(^2\) tunnel junctions was measured and found to be 8.2 k\(\varOmega \).

## 5 Device Characterisation

### 5.1 Measurement Setup and Resonator Characterisation

Initial characterisation of the sample was performed by measuring the feedline transmission \(S_{21}\) around 6.8–7.8 GHz, which identified the \(\lambda /4\) resonators. We chose to focus the study on the resonator at 7.5 GHz. The transmission past this resonator at an incident power in the single photon regime is shown in Fig. 4a and corresponds to an external quality factor \(Q_\mathrm{c} \simeq 5500\), or a resonator decay rate \(\kappa / 2 \pi \simeq 1.4\,\mathrm {MHz}\). The resonance lineshape is well fit using the method of [23] by assuming the resonator is overcoupled to the feedline, with an internal quality factor of at least \(Q_\mathrm{i} \gtrsim 38600\). The true internal quality factor is difficult to measure in our device due to the deliberate overcoupling of the resonators, however values in the literature can be optimised to yield \(Q_\mathrm{i} \gtrsim 1 \times 10^6\) [24].

### 5.2 Dispersive Shift Measurement

The presence of the qubit was confirmed by the dispersive shift of the resonator frequency due to the interaction of the resonator and the nonlinear element, which was measured by single-tone spectroscopy, sweeping the frequency and power of a probe tone in a continuous wave experiment, whilst measuring transmission through the feedline with a vector network analyser. As shown in Fig. 4b, at input powers below \(-110\,\)dBm the qubit is in the dressed ground state \(|{0}\rangle \) and experiences a shift towards lower frequencies. At input powers above \(\gtrsim -90\,\)dBm, the resonator becomes populated with many photons and the resonance frequency returns to that of the bare resonator [21, 25, 26]. With the qubit–resonator detuning \(\varDelta _0 = 990\,\)MHz, we observed a shift between the dressed ground state \(|{0}\rangle \) and the bare resonator of 3.9 MHz.

### 5.3 Qubit–Resonator Coupling: ‘*g*’ Measurement

### 5.4 Qubit Characterisation

#### 5.4.1 Rabi Oscillations

To characterise the qubit, we measured coherent Rabi oscillations around the qubit maximal gap where the energy bands are first order insensitive to the magnetic field (the “sweet spot”), by sweeping a variable length pulse from 8.502 to 8.520 GHz, with durations from 0 to 1000 ns. We inferred the qubit state by measuring the dispersive shift of the resonator through \(\pm \chi / 2 \pi \) [18]. The Rabi oscillations are presented in Fig. 6a and Fourier transformed in Fig. 6b. The oscillations exhibit a typical pattern [27], with the lowest oscillation frequency, \(\varOmega = 6.17\,\)MHz, at the drive frequency of 8.512 GHz when the drive frequency matches the qubit gap. At zero detuning, the probability for the qubit to be in the excited state is close to unity. When the drive frequency is detuned by \(\varDelta _0\) to lower or higher frequency from the qubit sweet spot, the Rabi oscillation frequency increases as \(\sqrt{\varOmega ^2 + \varDelta _0^2}\). Also, as expected, when the drive frequency is detuned from the sweet spot, the probability of the excited state decreases as \(\varOmega ^2 \big / ( \varOmega ^2 + \varDelta _0^2 ) \) and also the qubit decoheres faster. From these data, we found that for our microwave amplitude at zero detuning, it takes about 80 ns to transfer the qubit from the ground to excited state. For comparison, an optimised state of the art system can accomplish the excitation in \(\sim \) 1 ns [28]. The pulse that performs this excitation is called the \(\pi \)-pulse.

#### 5.4.2 \(T_1\) and \(T_2\) Measurements

To characterise the coherence properties of the transmon qubit, we measured the qubit energy relaxation time \(T_1\) by exciting the qubit from the ground to first excited state using a \(\pi \) pulse at the qubit frequency \(\omega _a\), before measuring the resonator transmission at the frequency \(\omega _r = 2 \pi f_0\), then varying the delay time \(\tau \) between the control and readout pulses as indicated in the inset of Fig. 7a. The observed decay of the excited qubit state is described by an exponential relaxation, giving \(T_1 = 4.72 \pm 0.06\,\mu \)s, which compares favourably with the transmon relaxation times measured in earlier experiments [27, 29, 30, 31], but which is a factor of 30 below the state of the art [32, 33, 34].

The qubit dephasing time was first measured using the Ramsey fringe visibility technique of two \(\pi /2\) pulses separated by a variable delay \(\tau \), to include the influence of low-frequency noise that may adversely affect the photon source during operation. We found \(T_{2,\mathrm {Ramsey}} = 6.38 \,\mu \mathrm {s}\) presented in Fig. 7bi, by fitting a pair of sinusoids under an exponentially decaying envelope to the data recorded whilst the microwave excitation was detuned by 100 kHz from the qubit centre frequency. Fourier transforming the Ramsey fringe data (Fig. 7bii) reveals two oscillating components of equal magnitude, separated in frequency by 554 kHz. We interpret these two components as arising from quasiparticle tunnelling events that take place on a characteristic timescale slow compared to each shot of the experiment, but fast compared to the \(10^4-10^5\) realisations that are averaged to provide the detected signal [32]. The visible beating in our device is due to our choice of a relatively large \(E_\mathrm{C}\) in comparison with other qubits in the literature.

## 6 Discussion

The dynamic protocol calls for the qubit to first be excited by a \(\pi \)-pulse whilst detuned from the \(\lambda /4\) resonator, followed by use of the flux tuning line to bring the qubit into resonance with the \(\lambda /4\) structure for a duration \(1/(2g) = 58\,\mathrm {ns}\), effectively swapping the excitation from the qubit to the resonator. In our experiment, the flux tuning line used to modulate \(E_\mathrm{J}\) has a bandwidth of 1 GHz, meaning that we can bring the qubit and resonator into resonance with one another in \(\sim \)1 ns. The time taken to tune the qubit is therefore negligible compared to the \(\pi \)-pulse duration (80 ns) and vacuum Rabi swap (58 ns).

*g*and \(\varOmega \) are the vacuum Rabi and driven Rabi angular frequencies, respectively. For our device, we find that \(\eta _\mathrm {dynamic} \approx 85 \%\).

The efficiency of the photon source when dynamic tuning is employed is limited principally by the loss mechanisms of the \(\lambda /4\) resonator into which the photon is emitted. Our photon source efficiency could be improved by increasing \(Q_\mathrm{i}\). The best internal quality factor reported for 2D superconducting resonators at low excitation power is \(Q_\mathrm{i} \sim 2 \times 10^6\) [35].

## 7 Conclusions

We have fabricated and characterised a superconducting qubit multiplexing circuit that appears to be promising for single microwave photon generation. We note that the resonators’ frequencies and quality factors are close to the designed ones. The qubit parameters, such as the maximal energy gap, which exceeds the resonator frequency by about 1 GHz, are also close to the desired ones. Finally, the qubit–resonator and resonator-feedline couplings appeared to be consistent with expectations. The measured qubit coherence times and resonator intrinsic quality factors are consistent with those reported by other groups, but can be further improved to produce a single microwave photon source with an efficiency approaching 100%.

## Notes

### Acknowledgements

This work was carried out within the project EXL03 MICROPHOTON of the European Metrology Research Programme (EMRP). EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. REG and YuAP acknowledge support from the UK’s EPSRC under Grant EP/K01675X/1. YuAP acknowledges partial support by the Royal Society (Grant WM110105). JS, OPS and JP acknowledge support from Academy of Finland Grants 284594 and 272218.

### Author Contributions

The device design was discussed by all authors. Iterations of device fabrication and measurement were performed by REG and SEdG using facilities at Lancaster University and Chalmers University of Technology, and by JS, REG, and OPS using facilities at Aalto University. REG, SEdG and YuAP wrote the paper with comments from all authors.

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