Resonant and off-resonant microwave signal manipulation in coupled superconducting resonators 1, 2, 1 1 1 1, Mathieu Pierre, ∗ Sankar Raman Sathyamoorthy, Ida-Maria Svensson, G¨oranJohansson, and Per Delsing y 1Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden 2Laboratoire National des Champs Magn´etiquesIntenses (LNCMI), Universit´ede Toulouse, INSA, CNRS UPR 3228, EMFL, FR-31400 Toulouse, France We present an experimental demonstration as well as a theoretical model of an integrated circuit designed for the manipulation of a microwave field down to the single-photon level. The device is made of a superconducting resonator coupled to a transmission line via a second frequency-tunable resonator. The tunable resonator can be used as a tunable coupler between the fixed resonator and the transmission line. Moreover, the manipulation of the microwave field between the two resonators is possible. In particular, we demonstrate the swapping of the field from one resonator to the other by pulsing the frequency detuning between the two resonators. The behavior of the system, which determines how the device can be operated, is analyzed as a function of one key parameter of the system, the damping ratio of the coupled resonators. We show a good agreement between experiments and simulations, realized by solving a set of coupled differential equations. In quantum technology the interaction between quan- so far28. Furthermore, it is becoming possible to simulate tum states of light and various degrees of freedom of mat- complex quantum systems, such as many-body states of ter can be controlled in a variety of systems. Among condensed matter, using arrays of superconducting res- them, macroscopic superconducting circuits cooled to onators and qubits. For this purpose, tunable couplings millikelvin temperatures are developing as a platform are essential to implement arbitrary Hamiltonians. Even to manipulate microwave photons and artificial atoms. more interestingly, dynamic processes can be studied if They are easy to engineer because they are integrated the couplings can be tuned fast enough, on the timescale electrical circuits. This forms the field of circuit quan- of the processes under study. tum electrodynamics (circuit-QED)1,2. Since resonators are either capacitively or inductively Using electrical circuits for building quantum systems coupled to transmission lines, a first approach to make allows for a precise design of Hamiltonian parameters the coupling tunable is to use a tunable circuit element, within a wide range3. Furthermore, some parameters can such as a tunable inductance, for instance a supercon- also be made tunable in situ, for instance, the resonance ducting quantum interference device (SQUID)29. To al- frequencies of resonators and the transition frequency of low for more complex manipulations of the microwave artificial atoms, also known as quantum bits4{7. signals, a second approach is based on a dual resonator It is also essential for many experiments and appli- architecture. A high quality factor resonator, dedicated cations to have tunable couplings, or equivalently, life- to the storage of microwave radiation, which can be times or linewidths. Tunable couplings have already been viewed as a quantum node, is connected to a transmis- demonstrated between qubits8{13, between qubits and sion line via a low quality factor resonator. This low-Q resonators14{18, and between resonators19{21. resonator permits the fast transfer, storage or retrieval, In this work we focus on the tunable coupling between of the quantum information encoded in the microwave ra- a resonator and a transmission line. This function is re- diation. It has already been shown how parametric pro- quired in several types of applications. First, in quantum cesses can be used for the coherent manipulation of the communication22, it is envisioned that “flying" qubits are microwave signals, either by coupling the two resonators sent over long distances in the form of photons23 propa- with a Josephson ring modulator30, a flux-driven Joseph- gating between nodes acting as quantum memories and son junction circuit31, or with a superconducting qubit32. processors. These nodes could be implemented as mi- In our work, we use a similar dual resonator architecture, crowave resonators coupled to qubits or other types of but our approach for the coherent control is different. We quantum systems. It has been shown that the trans- made the low-Q resonator frequency-tunable, and the res- arXiv:1802.09034v2 [cond-mat.mes-hall] 19 Mar 2019 fer efficiency can be increased if one can adjust the cou- onators are simply capacitively coupled. plings at both ends of the transmission chain24,25. Ad- In a previous article, we demonstrated the storage of justing the coupling between the transmission line and microwaves in a superconducting resonator by switch- the terminating resonator to the temporal and spectral ing on and off this tunable coupler33. We showed that properties of the incoming wave packet can result in microwaves can be released from the storage resonator full absorption26, which can be viewed as an impedance through the frequency-tunable low-Q coupling resonator matching condition for the resonator27. Inversely, a res- at a varying rate. We presented a sample that was en- onator with tunable coupling can also be used to emit mi- gineered to show a high on/off coupling ratio. The goal crowave photons contained in an arbitrary wave packet. of the current article is to extend this work by present- This has only been achieved with more complex schemes ing a generic model for this coupled-resonator circuit, 2 valid in a large range of parameters and supported by (a) g experimental data in good agreement with the theory. κ κib κia Vin We show that the behavior of each sample is governed ωb(Φ) ωa by a single parameter, a ratio between coupling rates, b†, b a†, a which corresponds to the damping ratio of the coupled Vout Cout Cc resonator system. We present an experimental compar- ison of two samples operating in two distinct regimes. Coupling resonator Storage resonator One of the sample corresponds to the results already pre- sented in our previous work33. It is optimized for direct addressing of the storage resonator, which is done in the off-resonant coupling of the two resonators. For the sec- ond sample, we show that the storage resonator can be addressed through a swapping procedure exploiting the resonant coupling of the two resonators. Coupling resonator I. SYSTEM AND MODEL SQUID A. The measured system The system under study is composed of two microwave resonators (see Fig.1). The resonators are coupled through a coupling capacitance Cc, permitting the trans- fer of energy between them. One of the resonators fea- tures a tunable resonance frequency. This allows to con- trol the energy exchange between the two resonators, by changing their detuning. The frequency tunability Storage resonator is based on a superconducting quantum interference de- (b) 1 mm vice (SQUID). It behaves as a tunable, nonlinear, and nondissipative inductance embedded in the resonator4,5. The tunable resonator has been engineered so that its FIG. 1. (a) Model of the system under study. A storage res- range of reachable resonance frequency crosses the res- onator with frequency !a=(2π) is coupled to a transmission onance frequency of the second resonator, which is con- line via a coupling resonator with tunable frequency !b=(2π). (b) Optical microscope image of the corresponding supercon- stant. In addition, the tunable resonator is also coupled ducting integrated circuit (sample I). to a transmission line, which allows us to excite the sys- tem and probe it through microwave reflectometry. It will therefore be referred to as the coupling resonator, or resonator B. The other resonator contains no SQUID the system. Note that the Hamiltonian may be time and thus has a fixed resonance frequency and a long life- dependent, as, in addition to the time-dependent drive, time. It is therefore suitable for microwave storage for the resonance frequency of resonator B !b can be rapidly 33 instance , and will be referred to as the storage res- tuned in the experiment. onator, or resonator A. The coupling resonator is capacitively coupled to a transmission line, which makes the system open and dis- sipative. In addition, both resonators have finite intrinsic B. Theoretical model lifetimes, 1/κia and 1/κib. To describe the evolution of the quantum state of the system, we use the Lindblad The theoretical model of the system is depicted in master equation34{36, which gives the time evolution of Fig.1(a). In the rotating wave approximation, valid be- the density matrix ρ = ρa ρb: cause the coupling rate g between the resonators is much ⊗ i h i smaller than the resonator resonance frequencies !a and ρ_ = H;^ ρ + κ [^b]ρ + κia [^a]ρ + κib [^b]ρ, (2) !b, the Hamiltonian of the coupled resonators is −¯h D D D where denotes the Lindblad superoperator, defined as H^ =¯h!aa^ya^ + ¯h!b^by^b + ¯hg(^a^by +a ^y^b) D 1 [^x]ρ =xρ ^ x^y 2 x^yx;^ ρ . Solving this equation gives theD time evolution− of the average photon number in each + i¯hpκ V ∗(t)^b V (t)^by ; (1) in − in resonator, which cannot be measured directly in the ex- periment. For instance, for the storage resonator, wherea ^ and ^b are the field ladder operators for resonators A and B, respectively, and V (t) the input field driving n = a^ya^ = T r(^ayaρ^ ): (3) in h ai h i 3 The classical response of the system to the input field (κ ; κ κ, see TableI), this yields, for the storage ia ib Vin is given by the equations of motion for the expectation resonator field, values of the resonator fields A = a^ and B = ^b .