Approaching ultra-strong coupling in Transmon circuit-QED using a high-impedance resonator

TL;DR

This paper demonstrates a superconducting Transmon qubit coupled to a high-impedance resonator, achieving near ultra-strong coupling, which opens pathways for exploring new quantum regimes and enhancing quantum information processing.

Contribution

The study introduces a high-impedance resonator coupled to a Transmon qubit, reaching a coupling ratio close to the ultra-strong regime, and suggests potential for even stronger coupling with advanced architectures.

Findings

Achieved a large vacuum Rabi splitting of 910 MHz.

Reached a coupling ratio of 0.071, near ultra-strong coupling.

Potential to reach a coupling ratio of 0.45 with improved design.

Abstract

In this experiment, we couple a superconducting Transmon qubit to a high-impedance $645\ \Omega$ microwave resonator. Doing so leads to a large qubit-resonator coupling rate $g$, measured through a large vacuum Rabi splitting of $2g\simeq 910$ MHz. The coupling is a significant fraction of the qubit and resonator oscillation frequencies $\omega$, placing our system close to the ultra-strong coupling regime ($\bar{g}=g/\omega=0.071$ on resonance). Combining this setup with a vacuum-gap Transmon architecture shows the potential of reaching deep into the ultra-strong coupling $\bar{g} \sim 0.45$ with Transmon qubits.

Equations

$$2g\lesssim \sqrt{{\omega }_r{\omega }_a}.$$

$$\hat{H}=\hslash {\omega }_r{\hat{a}}^{†}\hat{a}+\hslash {\omega }_a{\hat{b}}^{†}\hat{b}-\frac{E_c}{2}{\hat{b}}^{†}{b}^{†}\hat{b}\hat{b}+\hslash g(\hat{a}+{\hat{a}}^{†})(\hat{b}+{\hat{b}}^{†}),$$

$$g\simeq \sqrt{\frac{{\omega }_a{\omega }_r}{4(1+\frac{C_J}{C_c})(1+\frac{C_r}{C_c})}}.$$

$$\begin{array}{rl}{C}_{J,\text{eff}}& =\frac{C_J{C}_r+C_J{C}_c+C_c{C}_r}{C_r+C_c},\\ C_{r,\text{eff}}& =\frac{C_J{C}_r+C_J{C}_c+C_c{C}_r}{C_J+C_c}.\end{array}$$

$$\frac{g}{\sqrt{{\omega }_a{\omega }_r}}=\frac{1}{2}\sqrt{\frac{1}{1+\frac{C_J}{C_c}}\frac{1}{1+\frac{C_r}{C_c}}}\le \frac{1}{2}.$$

$$C_{J,\text{eff}}=C_{r,\text{eff}}\simeq C_J+C_r,$$

Full text

I Introduction

Cavity QED is a study of the light-matter interaction between atoms and the confined electro-magnetic field of a cavity Raimond et al. (2001). For an atom in resonance with the cavity, a single excitation coherently oscillates with vacuum Rabi frequency $g$ between the photonic and atomic degree of freedom if $g$ exceeds the rate at which excitations decay into the environment (the strong coupling condition). Spectroscopically, this is observed as a mode-splitting (vacuum Rabi splitting) with distance $2g$. If the coupling is small with respect to the resonator (atomic) frequency $\omega_{r}$ ($\omega_{a}$), $g\ll\omega_{r},\omega_{a}$, and the frequencies respect the condition $|\omega_{a}-\omega_{r}|\ll|\omega_{r}+\omega_{a}|$, the interaction is faithfully described with the Jaynes-Cummings (JC) model Jaynes and Cummings (1963). As the coupling becomes a considerable fraction of $\omega_{r}$ or $\omega_{a}$, typically $\bar{g}=g/\omega_{r,a}\sim 0.1$, the JC model no longer applies and the interaction is better described by the Rabi model Rabi (1936); Braak et al. (2016); Rossatto et al. (2016); Xie et al. (2017). This ultra-strong coupling (USC) regime shows the breakdown of excitation number conservation, however excitation parity remains conserved for arbitrarily large $\bar{g}$ Casanova et al. (2010). The key prediction for the deep-strong coupling (DSC) regime, where $\bar{g}\sim 1$, is a symmetry breaking of the vacuum (i. e. qualitative change of the ground state) similar to the Higgs mechanism Garziano et al. (2014). The prospect of probing these new facets of light-matter interaction, in addition to potential applications in quantum information technologies Romero et al. (2012); Stassi and Nori (2017), has spurred many experimental efforts to reach increasingly large coupling rates.

Experimentally, strong coupling has been achieved in systems with atoms Thompson et al. (1992); Raimond et al. (2001); McKeever et al. (2003); Birnbaum et al. (2005), and various solid-state implementations, including superconducting circuits with different types of qubits Wallraff et al. (2004); Clarke and Wilhelm (2008); Petersson et al. (2012), and semiconductor systems Reithmaier et al. (2004). A different category of experiments using an ensemble of $N$ emitters benefit from a $\sqrt{N}$ enhancement of the coupling and therefore strong coupling has been observed in a wide variety of systems Herskind et al. (2009); Putz et al. (2014); Zhu et al. (2011). With such ensembles USC has been shown in the optical and THz frequency domain Anappara et al. (2009); Scalari et al. (2012); Gambino et al. (2014); Schwartz et al. (2011). The only platform that observed higher coupling rates with a single emitter uses a superconducting circuit with a flux qubit. Pioneered by the experiments of Refs. Niemczyk et al. (2010); Forn-Díaz et al. (2010), experiments in the DSC regime have now been achieved with flux qubits coupled to resonators Yoshihara et al. (2017) as well as an electro-magnetic continuum Forn-Díaz et al. (2016). Additionally, the U/DSC coupling regime of the Rabi model was the subject of recent analog quantum simulations Langford et al. (2016); Braumüller et al. (2016).

Here we explore coupling strengths at the edge of the USC regime in circuit QED using a superconducting Transmon qubit Koch et al. (2007) coupled to a microwave cavity that has a high characteristic impedance. When the Transmon and fundamental mode of the cavity are resonant, we spectroscopically measure a coupling $g/2\pi=455$ MHz, corresponding to $\bar{g}=0.071$. With the prospect of maximizing the Transmon analogue of the dipole moment bos , we show how this system could approach its theoretical upper limit SI ; jaa

[TABLE]

As in previous implementations, the cavity is inherently a multi-mode system. In addition to this first deviation from the Rabi model, this architecture differentiates itself from previous implementations of an ultra-strong Rabi interaction by the weak anharmonicity of the Transmon. Its higher excitation levels become increasingly relevant with higher couplings and in the USC regime it cannot be considered a two-level system. The system studied here is therefore not a strict implementation of the Rabi model, but is still expected to bear many of the typical USC features and a proposal has been made to measure them Andersen and Blais (2017).

II Setup

Our device, depicted in Fig. 1, consists of a high-impedance superconducting $\lambda/2$ microwave resonator Pozar (2009) capacitively over-coupled to a $50\ \Omega$ feedline on one end and coupled to a Transmon qubit on the other. The resonator is a $1\ \mu$m wide, $180$ nm thick and $\sim 6.5$ mm long meandering conductor. It is capacitively connected to a back ground plane through the $275\ \mu$m Silicon substrate as well as through vacuum/Silicon to the side ground planes.

The Transmon is in part coupled to ground through a vacuum gap capacitor, see Fig. 1(b). Its bottom electrode constitutes one island of the Transmon, the other plate is a suspended multi-layer graphene flake. The diameter of this capacitor is $15\ \mu$m with a gap of $150$ nm. This device was designed to couple the mechanical motion of the suspended multi-layer graphene to the Transmon qubit, where the coupling is mediated by a DC voltage offset LaHaye et al. (2009); Pirkkalainen et al. (2015). Here we characterize the system at zero DC voltage, where the coupling to the motion is negligible. To enable tunability of the qubit frequency, a SQUID loop is incorporated such that the Josephson energy $E_{J}$ can be modified using an external magnetic field. $E_{J}(\phi)$ is a function of the flux through the SQUID loop $\phi$ following $E_{J}(\phi)=E_{J,\text{max}}\cos(\pi\phi/\phi_{0})$ where $\phi_{0}=h/2e$ is the superconducting flux quantum.

We fabricate our devices in a three-step process. First we define our microwave resonators on a $275\ \mu$m Silicon substrate using reactive ion etching of molybdenum-rhenium alloy Singh et al. (2014). Subsequently, $\mbox{Al/AlO}_{x}\mbox{/Al}$ Josephson junctions are fabricated using aluminum shadow evaporation Dolan (1977). Finally, we stamp a multi-layer graphene flake on the $15\mu$m diameter opening in the ground plane using deterministic dry viscoelastic stamping technique Castellanos-Gomez et al. (2014). From room temperature resistance measurements, optical and SEM images we observe that the flake is suspended, though folded, and that it does not short the qubit to ground.

This device implements the circuit shown in Fig. 1 SI . Following circuit quantization Devoret (1997), we find that the dynamics of the system are governed by the Hamiltonian

[TABLE]

where $\hat{a}$ ($\hat{b}$) is the annihilation operator for resonator (Transmon) excitations. The bare resonator and Transmon frequencies are given by $\omega_{r}=1/\sqrt{L_{r}C_{r,\text{eff}}}$, $\omega_{a}=\sqrt{8E_{J}E_{c}}-E_{c}$, the charging energy is given by $E_{c}=e^{2}/2C_{J,\text{eff}}$ and the coupling strength

[TABLE]

The dependence on the flux $\phi$ is omitted in the expression of the coupling strength and Transmon frequency for clarity. It is important to distinguish the capacitances $C_{c},C_{J},C_{r}$ from the effective capacitances

[TABLE]

The former correspond to the physical circuit elements, whereas the latter lead to the correct eigen-frequencies of the resonator and Transmon, defined as the oscillation rate of charges through the inductance $L_{r}$ and Josephson junction respectively. Using finite-element simulation software, the qubit is designed such that its capacitance to ground $C_{J}=51$ fF and its coupling capacitor is $C_{c}=9$ fF. The parameters of other circuit elements will be extracted from the data. We will denote the lowest three eigen-states of the Transmon by $\ket{g}$, $\ket{e}$ and $\ket{f}$ with increasing energies.

We characterize our device at a temperature of 15 mK, mounted in a radiation-tight box. From a vector network analyzer we send a microwave tone that is heavily attenuated before being launched on the feedline of the chip. The transmitted signal is send back to the vector network analyzer through a circulator and a low-noise HEMT amplifier. This setup is detailed in the supplementary information SI . It allows us to probe the absorption of our device and thus the energy spectrum of the Hamiltonian (2). At high driving power we measure the bare cavity resonance Bishop et al. (2010) to have a total line-width of $\kappa=2\pi\times 29.3$ MHz and a coupling coefficient of $\eta=\kappa_{\text{c}}/\kappa=0.96$, giving the ratio between the coupling rate $\kappa_{\text{c}}$ and total dissipation rate $\kappa=\kappa_{\text{c}}+\kappa_{\text{i}}$, where $\kappa_{\text{i}}$ is the internal dissipation rate.

III Results

With a current biased coil, we can control the magnetic field and tune the effective $E_{J}$ to bring the qubit in resonance with the cavity. Where the Transmon and resonator frequencies cross, we measure a vacuum Rabi splitting which gives an estimate of the coupling rate $2g/2\pi\simeq 910$ MHz as shown in Fig. 2.

In Fig. 3, we show the result of performing two-tone spectroscopy to probe the qubit frequency Blais et al. (2004); Wallraff et al. (2005). When the qubit is detuned from the cavity, the resonator acquires a frequency shift which is dependent on the state of the qubit. Hence, probing the transmission of the feedline at the cavity resonance (shifted by the qubit in the ground state), while exciting the qubit with another microwave tone, will cause the transmission to change by a value $\delta|S_{21}|$ due to the qubit-state dependent shift. In Fig. 3(a) we measure the spectral response of the qubit for different magnetic fields. As the magnetic flux through the SQUID loop tunes the qubit frequency we track the ground to first exited state transition as a function of magnetic field. Since the probe power is kept constant during this experiment a clear power broadening of the qubit is visible, because more of the power is delivered to the qubit as it is closer to the cavity in frequency. The $38$ MHz line-width of this resonance translates to very short coherence times ($T_{1}\sim 40$ ns) compared to typical implementations Houck et al. (2008). Purcell losses contribute less than $2$ MHz to this line-width and the full origin of this high dissipation remains unknown. The secondary faint resonance corresponds to the spectral response of the first to second exited state transition of the Transmon due to some residual occupation of the first excited state. The difference in frequency between both transitions provides an estimate of the charging energy (or equivalently the anharmonicity of the Transmon), $E_{c}/h\sim 370$ MHz. In reality what we measure is a quantity that is dressed by the interaction with the cavity and diverges from the bare value of $E_{C}$.

We fit a numerical diagonalization of the Hamiltonian detailed in the supplementary information SI to the acquired data, obtaining the fits shown as dashed lines in Figs. 2,3. We thereby obtain the Hamiltonian parameters $E_{c}/h=300$ MHz, $g/2\pi=455$ MHz (on resonance), $\omega_{r}/2\pi=6.367$ GHz and $E_{J,\text{max}}/h=46$ GHz. Combined with our knowledge of the capacitances $C_{J}$ and $C_{c}$, we extract the following values for the circuit elements of the resonator: $C_{r}=57.1$ fF and $L_{r}=9.65$ nH. If we assume that the parallel LC oscillator corresponds to the fundamental mode of a $\lambda/2$ resonator, then the resonators effective impedance $Z_{r}=\sqrt{L_{r}/C_{r}}=411\ \Omega$ is related to the characteristic impedance of the transmission line through $Z_{0}=\pi Z_{r}/2$ SI , yielding a value $Z_{0}=645\ \Omega$.

IV Towards higher coupling in Transmon systems: a proposal

In the circuit of Fig. 1(d), the coupling rate is limited following

[TABLE]

The highest couplings are therefore achieved by maximizing two capacitance ratios: $C_{c}/C_{J}$ and $C_{c}/C_{r}$. In the language of cavity QED with natural atoms, maximizing the first ratio is equivalent to increasing the dipole moment of the atom which is done by using Rydberg atoms Raimond et al. (2001). Maximizing the second ratio increases the vacuum fluctuations of the cavities electric field as performed in alkali-atom experiments in a very small optical cavity Thompson et al. (1992).

In the regime $C_{c}/C_{J},C_{c}/C_{r}\gg 1$ the effective capacitances of Eq. (4) are approximated by

[TABLE]

these capacitances being the quantities to minimize to increase the coupling. Maximizing the coupling whilst keeping the resonator and Transmon frequencies constant therefore requires a large increase in the inductances. In other words, the higher the impedance of the resonator and the higher the ratio $E_{c}/E_{J}$ in the Transmon, the higher the coupling.

The highest ratio of $E_{c}/E_{J}$ for which we remain in the Transmon regime is $\sim 1/20$ Koch et al. (2007). Combined with a typical choice of the Transmon frequency $\omega_{a}/2\pi=6$ GHz, compatible with most microwave experimental setups, we obtain a value of the Transmons total capacitance $C_{J,\text{eff}}=C_{r,\text{eff}}\simeq C_{J}+C_{r}\simeq 20$ fF. Choosing $C_{J}=C_{r}=10$ fF maximizes Eq. 5. Fixing the resonator frequency to $\omega_{r}=\omega_{a}$ leads to a value for resonators characteristic impedance: $722\ \Omega$ if a $\lambda/4$ resonator is used, $1.44$ k$\Omega$ for a $\lambda/2$ resonator and $918\ \Omega$ for a lumped element resonator. The coupling achieved now depends on the value of the coupling capacitor. For $C_{c}=200$ fF for example, $\bar{g}=0.45$ and the system is deep in the USC regime.

In Ref. bos , the USC regime was reached by increasing the first capacitance ratio of Eq. (5), $C_{c}/(C_{J}+C_{c})\simeq 0.9$ through the use of a vacuum-gap coupling capacitor. In this work, the large coupling is reached by increasing the second ratio $C_{c}/(C_{r}+C_{c})\simeq 0.136$ above usual values through the use of a high impedance resonator whilst the first capacitance ratio remains modest $C_{J}/(C_{J}+C_{c})\simeq 0.15$. Combining both approaches into a single device represented schematically in Fig. 4 would allow experimentally, reaching deep into the USC regime $\bar{g}=0.45$ by matching the circuit parameters presented previously. The values $C_{J}=10$ fF and $C_{c}=200$ fF can be easily achieved experimentally reproducing the vacuum-gap Transmon architecture of Ref. bos with a smaller gap and larger capacitive plate, maybe even by replacing the vacuum gap by a dielectric. The use of a $\lambda/4$ resonator rather than a $\lambda/2$ is preferable as it decreases the impedance needed as well as increases the frequency spacing between the fundamental and higher modes. Moving to a $\lambda/4$ resonator makes the current architecture sufficient in terms of resonator impedance. The impedance could be further increased using a high kinetic inductance based resonator Samkharadze et al. (2016) or by using an array of Josephson junctions Masluk et al. (2012).

This proposal is however limited by the underlying assumption that only a single mode of the resonator participates in the dynamics of the system. However for larger coupling rates, the higher modes no longer play a weak perturbative role Gely et al. (2017). Exploring the exact consequences of this fact on the observable USC phenomena that can be observed is outside the scope of this work, as is determining alternatives to probing the system spectroscopically to show for example the non-trivial ground state that one would expect in this regime. For a detailed study of these topics, we refer the reader to Ref. Andersen and Blais (2017).

V Conclusion

We have shown that it is possible to enhance the coupling between a microwave resonator and a Transmon qubit by increasing the impedance of the resonator to $645\ \Omega$ compared to typical $50\ \Omega$ implementations. In doing this we reach a coupling rate of $g/2\pi=455$ MHz at resonance, which is close to the ultra-strong coupling regime ($\bar{g}=0.071$). We have shown that by optimizing this strategy through sources of high inductance, combined with a vacuum-gap Transmon architecture, we have the potential of reaching far into the ultra-strong coupling regime.

Acknowledgments The authors thank Alessandro Bruno, Leo DiCarlo, Nathan Langford, Adrian Parra-Rodriguez and Marios Kounalakis for useful discussions.