Tunable coupling of transmission-line microwave resonators mediated by an rf SQUID
- Friedrich Wulschner^{1, 2},
- Jan Goetz^{1, 2},
- Fabian R Koessel^{1, 2},
- Elisabeth Hoffmann^{1, 2},
- Alexander Baust^{1, 2, 3},
- Peter Eder^{1, 2, 3},
- Michael Fischer^{1, 2, 3},
- Max Haeberlein^{1, 2},
- Manuel J Schwarz^{1, 2, 3},
- Matthias Pernpeintner^{1, 2, 3},
- Edwar Xie^{1, 2, 3},
- Ling Zhong^{1, 2, 3},
- Christoph W Zollitsch^{1, 2},
- Borja Peropadre^{4},
- Juan-Jose Garcia Ripoll^{5},
- Enrique Solano^{2, 6, 7},
- Kirill G Fedorov^{1, 2},
- Edwin P Menzel^{1, 2},
- Frank Deppe^{1, 2, 3}View ORCID ID profile,
- Achim Marx^{1} and
- Rudolf Gross^{1, 2, 3}Email authorView ORCID ID profile
https://doi.org/10.1140/epjqt/s40507-016-0048-2
© Wulschner et al. 2016
Received: 14 March 2016
Accepted: 8 July 2016
Published: 28 July 2016
Abstract
We realize tunable coupling between two superconducting transmission line resonators. The coupling is mediated by a non-hysteretic rf SQUID acting as a flux-tunable mutual inductance between the resonators. We present a spectroscopic characterization of the device. In particular, we observe couplings \(g/2\pi\) ranging between −320 MHz and 37 MHz. In the case of \(g \simeq 0\), the microwave power cross transmission between the two resonators is reduced by almost four orders of magnitude as compared to the case where the coupling is switched on.
Keywords
1 Introduction
In circuit quantum electrodynamics, the controllable interaction of circuit elements is a highly desirable resource for quantum computation and quantum simulation experiments. The most common method is a static capacitive or inductive coupling between cavities and/or qubits. In such a system, exchange of excitations can be controlled by either tuning the circuit elements in and out of resonance or using sideband transitions [1–4]. While this approach has proven to be useful for few coupled circuit elements, it seems impracticable for larger systems, where it is hard to provide sufficient detunings between all circuit elements [5]. Therefore, one may alternatively use tunable coupling elements such as qubits [6–9] or SQUIDs [10–15]. One particular example for an interesting application of such actively coupled circuit elements are quantum simulations of bosonic many-body Hamiltonians [16–20]. In such a scenario, the bosonic degrees of freedom can be represented by networks of (possibly nonlinear) superconducting resonators. For this quantum simulator, a tunable coupler would constitute an important control knob. A more general scope of this device is the controllable routing of photonic states on a chip, which is interesting for quantum information as well as quantum simulation experiments.
In this work, we experimentally investigate the case of two nearly frequency-degenerate superconducting transmission line resonators coupled by an rf SQUID acting as a tunable mutual inductance in the spirit of Refs. [21–23]. Although such a setup looks similar to the case of a flux qubit mediated coupling [6], there are important conceptual differences resulting in performance advantages. In a flux qubit coupler [24, 25], the resonator-resonator coupling is limited to twice the dispersive qubit-resonator shift (typically a few MHz). Efforts to increase the maximum coupling by relaxing the dispersive coupling assumption have contributed to the limited isolation of 2.6 d Bbetween the resonators in the off-state of the coupler in Ref. [6]. This conceptual disadvantage obviously outweighs the potential quantum switch properties [24, 25] of the flux qubit coupler for many practical applications. In contrast, couplers between superconducting qubits based on the classical phase dynamics of an rf SQUID have shown large couplings [11, 13, 14] and good isolation properties [14]. Compared to our previous work [6], we achieve two significant improvements: First, the range of achievable coupling strengths between the resonators is increased from \(g/(2\pi) \in [-28.7\, \mathrm{M} \mathrm{Hz} , 8.4\, \mathrm{M} \mathrm{Hz} ]\) to \(g/(2\pi) \in [-302\, \mathrm{M} \mathrm{Hz} , 37\, \mathrm{M} \mathrm{Hz} ]\). Second, comparing the signal transmission between both resonators for the coupled (\(g \gg0\)) and decoupled (\(g \simeq 0\)) case, the signal isolation is increased from \(2.6\, \mathrm{d} \mathrm{B} \) to \(38.5\, \mathrm{d} \mathrm{B} \). Especially the increased isolation of the device discussed in the present work is a key prerequisite for several applications both in quantum simulation and quantum computation setups. The manuscript is structured as follows. After briefly discussing the relevant theory in Section 2 we introduce the sample and measurement setup in Section 3. In Section 4, we present a spectroscopic characterization of the rf SQUID coupler followed by a summary and conclusions in Section 5. The appendix contains a short discussion of the additional feature of parametric amplification observed in our device.
2 System Hamiltonian
3 Sample and measurement setup
Figure 1(a) shows the layout of the sample chip. In the resonator design, we omit the second groundplane to reduce the direct geometric coupling between the two resonators. The rf SQUID is galvanically connected to both center strips of the resonators over a length of \(200\,\mu \mathrm{m} \). The sample is fabricated as follows. First, a \(100\, \mathrm{n} \mathrm{m} \) thick niobium layer is sputter deposited onto a \(250\,\mu \mathrm{m} \) thick, thermally oxidized silicon wafer. The resonators and the SQUID loop are patterned using optical lithography and reactive ion etching. The Josephson junction of the SQUID is fabricated in a \(\mathrm{Nb}/\mathrm {AlO}_{\mathrm{x}}/\mathrm{Nb}\) trilayer process with \(\mathrm{SiO}_{2}\) as insulating layer between top and bottom electrode [32].^{1} The resonators have a characteristic impedance of \(Z_{0} = 64\,\Omega \) and the resonance frequencies \(\omega_{\mathrm{A}}/2\pi= 6.482\, \mathrm{G} \mathrm{Hz} \) and \(\omega_{\mathrm{B}}/2\pi= 6.461\, \mathrm{G} \mathrm{Hz} \).^{2} The SQUID loop has dimensions of \(200\,\mu \mathrm{m} \times 100\,\mu \mathrm{m} \) and a screening parameter \(\beta = 0.934\) to maximize the coupling according to Equation (4) while keeping the SQUID monostable. The sample is mounted inside a gold plated copper box, which is attached to the base temperature stage of a dilution refrigerator operating at \(26\, \mathrm{m} \mathrm{K} \). A superconducting solenoid attached to the top of the sample box is used to generate the external flux applied to the rf SQUID. As depicted in Figure 1(b), one port of each resonator is connected to an attenuated input microwave line, whereas the remaining ports are connected to output lines containing microwave amplifiers.
4 Resonator spectroscopy
Next, we analyze the properties of our device in the coupled and decoupled state in more detail. Because of the small detuning of the resonators, the coupled modes are not necessarily symmetric and antisymmetric superpositions of the uncoupled modes. This is also seen in the spectroscopy of the single resonators (see Figure 2), where the modes have different intensities. The mode mixing can be estimated from the eigenvectors of the Hamiltonian in Equation (9). For \(g/2\pi= +37\, \mathrm{M} \mathrm{Hz} \) and \(g/2\pi= -320\, \mathrm{M} \mathrm{Hz} \) we obtain the mixing ratios \(63:37\) and \(52:48\), respectively. Hence, in the latter case, our sample satisfies the condition \(\vert g\vert \gg \vert \Delta \vert \), where the detuning becomes insignificant. In the decoupled case near \(\Phi_{\mathrm{ext}} = 0.468 \Phi_{0}\) and \(\Phi_{\mathrm{ext}} = 0.532 \Phi _{0}\), the off-diagonal elements in the Hamiltonian of Equation (9) vanish and the modes are pure excitations of resonator A or B.
5 Conclusions
In conclusion, we present a flux-tunable coupling between two superconducting resonators based on a SQUID containing a single Josephson junction. Spectroscopically, we measure negative and positive couplings ranging from \(-320\, \mathrm{M} \mathrm{Hz} \) to \(37\, \mathrm{M} \mathrm{Hz} \). Furthermore, the observed suppression of the cross-transmission of up to \(38.5\, \mathrm{d} \mathrm{B} \) proves the ability to effectively turn off the coupling and is an important improvement over previous work [6], where still 27.5% (\(2.6\, \mathrm{d} \mathrm{B} \) change in cross-transmission) of the signalpower was transmitted to the uncoupled resonator for \(g \simeq0\). With the achieved performance, our coupler can be considered as a useful tool for quantum computation with a controlled nearest neighbor interaction or to route information on a chip in a controllable way. Regarding quantum simulation experiments [16–19, 38, 39], our device could be especially useful because it allows one to change both amplitude and sign of the coupling constant. For such future experiments, a fast flux antenna enabling a non-adiabatic change of the coupling strength can be integrated onto the chip straightforwardly.
Declarations
Acknowledgements
The authors acknowledge support from the German Research Foundation through SFB 631 and FE 1564/1-1; the EU projects CCQED, PROMISCE and SCALEQIT; the doctorate program ExQM of the Elite Network of Bavaria; the Spanish MINECO projects FIS2012-33022, FIS2012-36673-C03-02, and FIS2015-69983-P; the CAM Research Network QUITEMAD+; the Basque Government IT472-10 and UPV/EHU UFI 11/55. ES acknowledges support from a TUM August-Wilhelm Scheer Visiting Professorship and hospitality of Walther-Meißner-Institut and TUM Institute for Advanced Study.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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