##### Quantum behavior of the Duffing oscillator at the dissipative phase transition

Q. M. Chen, M. Fischer, Y. Nojiri, M. Renger, E. D. Xie, M. Partanen, S. Pogorzalek, K. G. Fedorov, A. Marx, F. Deppe, R. Gross

Nature Communications 14 (1), 2896 (2023).

The non-deterministic behavior of the Duffing oscillator is classically attributed to the coexistence of two steady states in a double-well potential. However, this interpretation fails in the quantum-mechanical perspective which predicts a single unique steady state. Here, we measure the non-equilibrium dynamics of a superconducting Duffing oscillator and experimentally reconcile the classical and quantum descriptions as indicated by the Liouvillian spectral theory. We demonstrate that the two classically regarded steady states are in fact quantum metastable states. They have a remarkably long lifetime but must eventually relax into the single unique steady state allowed by quantum mechanics. By engineering their lifetime, we observe a first-order dissipative phase transition and reveal the two distinct phases by quantum state tomography. Our results reveal a smooth quantum state evolution behind a sudden dissipative phase transition and form an essential step towards understanding the intriguing phenomena in driven-dissipative systems. Classical mechanics predicts a bistability in the dynamics of the Duffing oscillator, a key model of nonlinear dynamics. By performing quantum simulations of the model, Chen et al. explain the bistability by quantum metastable states with long lifetimes and reveal a first-order dissipative phase transition.

##### Propagating quantum microwaves: towards applications in communication and sensing

M. Casariego, E. Z. Cruzeiro, S. Gherardini, T. Gonzalez-Raya, R. Andre, G. Frazao, G. Catto, M. Moettoenen, D. Datta, K. Viisanen, J. Govenius, M. Prunnila, K. Tuominen, M. Reichert, M. Renger, K. G. Fedorov, F. Deppe, H. van der Vliet, A. J. Matthews, Y. Fernandez, R. Assouly, R. Dassonneville, B. Huard, M. Sanz, Y. Omar

Quantum Science and Technology 8 (2), 23001 (2023).

The field of propagating quantum microwaves is a relatively new area of research that is receiving increased attention due to its promising technological applications, both in communication and sensing. While formally similar to quantum optics, some key elements required by the aim of having a controllable quantum microwave interface are still on an early stage of development. Here, we argue where and why a fully operative toolbox for propagating quantum microwaves will be needed, pointing to novel directions of research along the way: from microwave quantum key distribution to quantum radar, bath-system learning, or direct dark matter detection. The article therefore functions both as a review of the state-of-the-art, and as an illustration of the wide reach of applications the future of quantum microwaves will open.

##### Scattering coefficients of superconducting microwave resonators. II. System-bath approach

Q. M. Chen, M. Partanen, F. Fesquet, K. E. Honasoge, F. Kronowetter, Y. Nojiri, M. Renger, K. G. Fedorov, A. Marx, F. Deppe, R. Gross

Physical Review B 106 (21), 214506 (2022).

We describe a unified quantum approach for analyzing the scattering coefficients of superconducting mi-crowave resonators with a variety of geometries, and demonstrate its consistency with the classical approach [Q.-M. Chen et al., Phys. Rev. B 106, 214505 (2022)]. We also generalize the result to a chain of resonators with time delays, and reveal several transport properties similar to a photonic crystal and can be used to design high-quality resonators. These results form a firm theoretical ground for analyzing the scattering coefficients of an arbitrary resonator network. They set a step forward to designing and characterizing superconducting microwave resonators in a complex superconducting quantum circuit.

##### Scattering coefficients of superconducting microwave resonators. I. Transfer matrix approach

Q. M. Chen, M. Pfeiffer, M. Partanen, F. Fesquet, K. E. Honasoge, F. Kronowetter, Y. Nojiri, M. Renger, K. G. Fedorov, A. Marx, F. Deppe, R. Gross

Physical Review B 106 (21), 214505 (2022).

We describe a unified classical approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. To fill the gap between experiment and theory, we also consider the influences of small circuit asymmetry and the finite length of the feedlines, and describe a procedure to correct their influences in typical experiments. We show that, similar to the transmission coefficient of a hanger-type resonator, the reflection coefficient of a necklace- or cross-type resonator also contains a so-called reference point that can be used to characterize the internal quality factor of the resonator. Our results provide a comprehensive understanding of superconducting microwave resonators from the design concepts to the characterization details.

##### Flow of quantum correlations in noisy two-mode squeezed microwave states

M. Renger, S. Pogorzalek, F. Fesquet, K. Honasoge, F. Kronowetter, Q. Chen, Y. Nojiri, K. Inomata, Y. Nakamura, A. Marx, F. Deppe, R. Gross, K. G. Fedorov

Physical Review A 106 (5), 52415 (2022).

We study nonclassical correlations in propagating two-mode squeezed microwave states in the presence of noise. We focus on two different types of correlations, namely, quantum entanglement and quantum discord. Quantum discord has various intriguing fundamental properties which require experimental verification, such as the asymptotic robustness to environmental noise. Here, we experimentally investigate quantum discord in propagating two-mode squeezed microwave states generated via superconducting Josephson parametric ampli-fiers. By exploiting an asymmetric noise injection into these entangled states, we demonstrate the robustness of quantum discord against thermal noise while verifying the sudden death of entanglement. Furthermore, we investigate the difference between quantum discord and entanglement of formation, which can be directly related to the flow of locally inaccessible information between the environment and the bipartite subsystem. We observe a crossover behavior between quantum discord and entanglement for low noise photon numbers, which is a result of the tripartite nature of noise injection. We demonstrate that the difference between entanglement and quantum discord can be related to the security of certain quantum key distribution protocols.

##### Open-Air Microwave Entanglement Distribution for Quantum Teleportation

T. Gonzalez-Raya, M. Casariego, F. Fesquet, M. Renger, V. Salari, M. Mottonen, Y. Omar, F. Deppe, K. G. Fedorov, M. Sanz

Physical Review Applied 18 (4), 44002 (2022).

Microwave technology plays a central role in current wireless communications, including mobile com-munication and local area networks. The microwave range shows relevant advantages with respect to other frequencies in open-air transmission, such as low absorption losses and low-energy consumption, and in addition, it is the natural working frequency in superconducting quantum technologies. Entangle-ment distribution between separate parties is at the core of secure quantum communications. Therefore, understanding its limitations in realistic open-air settings, especially in the rather unexplored microwave regime, is crucial for transforming microwave quantum communications into a mainstream technology. Here, we investigate the feasibility of an open-air entanglement distribution scheme with microwave two -mode squeezed states. First, we study the reach of direct entanglement transmission in open air, obtaining a maximum distance of approximately 500 m with parameters feasible for state-of-the-art experiments. Subsequently, we adapt entanglement distillation and entanglement swapping protocols to microwave technology in order to reduce the environment-induced entanglement degradation. The employed entan-glement distillation helps to increase quantum correlations in the short-distance low-squeezing regime by up to 46%, and the reach of entanglement increases by 14% with entanglement swapping. Importantly, we compute the fidelity of a continuous-variable quantum teleportation protocol using open-air-distributed entanglement as a resource. Finally, we adapt this machinery to explore the limitations of quantum com-munication between satellites, where the impact of thermal noise is substantially reduced and diffraction losses are dominant.

##### Tuning and amplifying the interactions in superconducting quantum circuits with subradiant qubits

Q. M. Chen, F. Kronowetter, F. Fesquet, K. E. Honasoge, Y. Nojiri, M. Renger, K. G. Fedorov, A. Marx, F. Deppe, R. Gross

Physical Review A 105 (1), 12405 (2022).

We propose a tunable coupler consisting of N fixed-frequency qubits, which can tune and even amplify the effective interaction between two superconducting quantum circuits. The tuning range of the interaction is proportional to N, with a minimum value of zero and a maximum that can exceed the physical coupling rates between the coupler and the circuits. The effective coupling rate is determined by the collective magnetic quantum number of the qubit ensemble, which takes only discrete values and is free from collective decay and decoherence. Using single-photon pi-pulses, the coupling rate can be switched between arbitrary choices of the initial and final values within the dynamic range in a single step without going through intermediate values. A cascade of the couplers for amplifying small interactions or weak signals is also discussed. These results should not only stimulate interest in exploring the collective effects in quantum information processing, but also enable development of applications in tuning and amplifying the interactions in a general cavity-QED system.

##### Mechanical frequency control in inductively coupled electromechanical systems

T. Luschmann, P. Schmidt, F. Deppe, A. Marx, A. Sanchez, R. Gross, H. Hübl

Scientific Reports 12 (1), 1608 (2022).

Nano-electromechanical systems implement the opto-mechanical interaction combining electromagnetic circuits and mechanical elements. We investigate an inductively coupled nano-electromechanical system, where a superconducting quantum interference device (SQUID) realizes the coupling. We show that the resonance frequency of the mechanically compliant string embedded into the SQUID loop can be controlled in two different ways: (1) the bias magnetic flux applied perpendicular to the SQUID loop, (2) the magnitude of the in-plane bias magnetic field contributing to the nano-electromechanical coupling. These findings are quantitatively explained by the inductive interaction contributing to the effective spring constant of the mechanical resonator. In addition, we observe a residual field dependent shift of the mechanical resonance frequency, which we attribute to the finite flux pinning of vortices trapped in the magnetic field biased nanostring.

##### Experimental quantum teleportation of propagating microwaves

K. G. Fedorov, M. Renger, S. Pogorzalek, R. Di Candia, Q. M. Chen, Y. Nojiri, K. Inomata, Y. Nakamura, M. Partanen, A. Marx, R. Gross, F. Deppe

Science Advances 7 (52), eabk0891 (2021).

The field of quantum communication promises to provide efficient and unconditionally secure ways to exchange information, particularly, in the form of quantum states. Meanwhile, recent breakthroughs in quantum computation with superconducting circuits trigger a demand for quantum communication channels between spatially separated superconducting processors operating at microwave frequencies. In pursuit of this goal, we demonstrate the unconditional quantum teleportation of propagating coherent microwave states by exploiting two-mode squeezing and analog feedforward over a macroscopic distance of d = 0.42 m. We achieve a teleportation fidelity of F = 0.689 +/- 0.004, exceeding the asymptotic no-cloning threshold. Thus, the quantum nature of the teleported states is preserved, opening the avenue toward unconditional security in microwave quantum communication.

##### Beyond the standard quantum limit for parametric amplification of broadband signals

M. Renger, S. Pogorzalek, Q. Chen, Y. Nojiri, K. Inomata, Y. Nakamura, M. Partanen, A. Marx, R. Gross, F. Deppe, K. G. Fedorov

Npj Quantum Information 7 (1), 160 (2021).

The low-noise amplification of weak microwave signals is crucial for countless protocols in quantum information processing. Quantum mechanics sets an ultimate lower limit of half a photon to the added input noise for phase-preserving amplification of narrowband signals, also known as the standard quantum limit (SQL). This limit, which is equivalent to a maximum quantum efficiency of 0.5, can be overcome by employing nondegenerate parametric amplification of broadband signals. We show that, in principle, a maximum quantum efficiency of unity can be reached. Experimentally, we find a quantum efficiency of 0.69 +/- 0.02, well beyond the SQL, by employing a flux-driven Josephson parametric amplifier and broadband thermal signals. We expect that our results allow for fundamental improvements in the detection of ultraweak microwave signals.

##### RF Antenna Design for 3D Quantum Memories

F. Deppe, E. Xie, K. G. Fedorov, G. Andersson, J. Muller, A. Marx, R. Gross, Ieee

International Symposium of the Applied-Computational-Electromagnetics-Society (ACES) (2021).

A quantum memory has to meet the conflicting requirements of strong coupling for fast readout and weak coupling for long storage. Multimode rectangular superconducting 3D cavities are known to satisfy both properties. Here, we systematically study the external coupling to the two lowest-frequency modes of an aluminum cavity. First, we introduce a general analytical scheme to describe the capacitive coupling of the antenna pin and validate this model experimentally. On this basis, we engineer an antenna which is overcoupled to the first mode, but undercoupled to the second mode.

##### Lokales Quantennetzwerk für Alice und Bob

F. Deppe, K.G. Fedorov, A. Marx

Akadmie Aktuell Heft 2 (74), 36-38 (2021).

Vom wissenschaftlichen Nischenthema zum international anerkannten Forschungsfeld: Quantenmikrowellen eröffnen viele Anwendungsperspektiven, für die sich auch die Industrie interessiert.

##### In situ tunable nonlinearity and competing signal paths in coupled superconducting resonators

M. Fischer, Q. M. Chen, C. Besson, P. Eder, J. Goetz, S. Pogorzalek, M. Renger, E. Xie, M. J. Hartmann, K. G. Fedorov, A. Marx, F. Deppe, R. Gross

Physical Review B 103 (9), 94515 (2021).

We have fabricated and studied a system of two tunable and coupled nonlinear superconducting resonators. The nonlinearity is introduced by galvanically coupled dc superconducting quantum interference devices. We simulate the system response by means of a circuit model, which includes an additional signal path introduced by the electromagnetic environment. Furthermore, we present two methods allowing us to experimentally determine the nonlinearity. First, we fit the measured frequency and flux dependence of the transmission data to simulations based on the equivalent circuit model. Second, we fit the power dependence of the transmission data to a model that is predicted by the nonlinear equation of motion describing the system. Our results show that we are able to tune the nonlinearity of the resonators by almost two orders of magnitude via an external coil and two on-chip antennas. The studied system represents a basic building block for larger systems, allowing for quantum simulations of bosonic many-body systems with a larger number of lattice sites.

##### Erste Demonstration von Quantenüberlegenheit

M.J. Hartmann, F. Deppe

Physik in unserer Zeit 52, 12 (2021).

Mit dem Sycamore-Quantenprozessor von Google gelang zum ersten Mal überzeugend ein Experiment, in dem ein Quantensystem ein Problem besser löst als derzeit verfügbare herkömmliche Supercomputer. Die Hardware basiert auf der Technologie der supraleitenden Quantenschaltkreise. Ihr wird schon länger ein besonders großes Skalierungspotenzial hin zu mehr Quantenbits bescheinigt. Der verwendete Chip besitzt 53 Qubits. Sie sind in einem zweidimensionalen quadratischen Gitter angeordnet und durch Nächste-Nachbar-Wechselwirkung gekoppelt. Somit stellt das Experiment einen großen technologischen Fortschritt für das gesamte Feld der Quantenwissenschaften und -technologien dar. Obwohl der praktische Nutzen derzeit noch gering erscheint, sind die Arbeiten des Google-Teams ein wichtiger Schritt hin zu skalierbarem Quantenrechnen. Damit erscheint erstmals eine fehlerkorrigierte, supraleitende Quantencomputer-Architektur in nicht allzu ferner Zukunft möglich.

##### Sideband-resolved resonator electromechanics based on a nonlinear Josephson inductance probed on the single-photon level

P. Schmidt, M. T. Amawi, S. Pogorzalek, F. Deppe, A. Marx, R. Gross, H. Hübl

Communications Physics 3 (1), 233 (2020).

Light-matter interaction in optomechanical systems is the foundation for ultra-sensitive detection schemes as well as the generation of phononic and photonic quantum states. Electromechanical systems realize this optomechanical interaction in the microwave regime. In this context, capacitive coupling arrangements demonstrated interaction rates of up to 280Hz. Complementary, early proposals and experiments suggest that inductive coupling schemes are tunable and have the potential to reach the single-photon strong-coupling regime. Here, we follow the latter approach by integrating a partly suspended superconducting quantum interference device (SQUID) into a microwave resonator. The mechanical displacement translates into a time varying flux in the SQUID loop, thereby providing an inductive electromechanical coupling. We demonstrate a sideband-resolved electromechanical system with a tunable vacuum coupling rate of up to 1.62kHz, realizing sub-aNHz(-1/2) force sensitivities. The presented inductive coupling scheme shows the high potential of SQUID-based electromechanics for targeting the full wealth of the intrinsically nonlinear optomechanics Hamiltonian. Recently, inductively-coupled optomechanical systems have been realized. They represent an important step forward towards achieving strong light-matter interaction, offer extreme sensitivity to mechanical displacement, and allow to study quantum phenomena on a single quantum level. In this work, a superconducting device is inductively coupled to a microwave resonator forming an electromechanical system operating at the single-photon level.

##### Secure quantum remote state preparation of squeezed microwave states

S. Pogorzalek, K. G. Fedorov, M. Xu, A. Parra-Rodriguez, M. Sanz, M. Fischer, E. Xie, K. Inomata, Y. Nakamura, E. Solano, A. Marx, F. Deppe, R. Gross

Nature Communications 10, 2604 (2019).

Quantum communication protocols based on nonclassical correlations can be more efficient than known classical methods and offer intrinsic security over direct state transfer. In particular, remote state preparation aims at the creation of a desired and known quantum state at a remote location using classical communication and quantum entanglement. We present an experimental realization of deterministic continuous-variable remote state preparation in the microwave regime over a distance of 35 cm. By employing propagating two-mode squeezed microwave states and feedforward, we achieve the remote preparation of squeezed states with up to 1.6 dB of squeezing below the vacuum level. Finally, security of remote state preparation is investigated by using the concept of the one-time pad and measuring the von Neumann entropies. We find nearly identical values for the entropy of the remotely prepared state and the respective conditional entropy given the classically communicated information and, thus, demonstrate close-to-perfect security.