The scattering coefficients of superconducting microwave resonators: II. System-bath approach

  1. Qi-Ming Chen,
  2. Matti Partanen,
  3. Florian Fesquet,
  4. Kedar E. Honasoge,
  5. Fabian Kronowetter,
  6. Yuki Nojiri,
  7. Michael Renger,
  8. Kirill G. Fedorov,
  9. Achim Marx,
  10. Frank Deppe,
  11. and Rudolf Gross
We describe a unified quantum approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. We also generalize the method
to a chain of resonators in either hanger- or necklace-type, and reveal interesting transport properties similar to a photonic crystal. It is shown that both the quantum and classical analyses provide consistent results, and they together form a solid basis for analyzing the decoherence effect in a general microwave resonator. These results pave the way for designing and applying superconducting microwave resonators in complex circuits, and should stimulate the interest of distinguishing different decoherence mechanisms of a resonator mode beyond free energy relaxation.

The scattering coefficients of superconducting microwave resonators: I. Transfer-matrix approach

  1. Qi-Ming Chen,
  2. Meike Pfeiffer,
  3. Matti Partanen,
  4. Florian Fesquet,
  5. Kedar E. Honasoge,
  6. Fabian Kronowetter,
  7. Yuki Nojiri,
  8. Michael Renger,
  9. Kirill G. Fedorov,
  10. Achim Marx,
  11. Frank Deppe,
  12. and Rudolf Gross
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 them in typical measurement results. We show that, similar to the transmission coefficient of a hanger-type resonator, the reflection coefficient of a necklace- or bridge-type resonator does also contain a reference point which can be used to characterize the electrical properties of a microwave resonator in a single measurement. Our results provide a comprehensive understanding of superconducting microwave resonators from the design concepts to the characterization details.

Tuning and Amplifying the Interactions in Superconducting Quantum Circuits with Subradiant Qubits

  1. Qi-Ming Chen,
  2. Florian Fesquet,
  3. Kedar E. Honasoge,
  4. Fabian Kronowetter,
  5. Yuki Nojiri,
  6. Michael Renger,
  7. Kirill G. Fedorov,
  8. Achim Marx,
  9. Frank Deppe,
  10. and Rudolf Gross
We propose a tunable coupler consisting of N off-resonant and fixed-frequency qubits that can tune and even amplify the effective interaction between two general circuit components.
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 in the system. 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 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

  1. Thomas Luschmann,
  2. Philip Schmidt,
  3. Frank Deppe,
  4. Achim Marx,
  5. Alvaro Sanchez,
  6. Rudolf Gross,
  7. and Hans Huebl
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: (i) the bias magnetic flux applied perpendicular to the SQUID loop, (ii) 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.

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

  1. Michael Fischer,
  2. Qi-Ming Chen,
  3. Christian Besson,
  4. Peter Eder,
  5. Jan Goetz,
  6. Stefan Pogorzalek,
  7. Michael Renger,
  8. Edwar Xie,
  9. Michael J. Hartmann,
  10. Kirill G. Fedorov,
  11. Achim Marx,
  12. Frank Deppe,
  13. and Rudolf Gross
We have fabricated and studied a system of two tunable and coupled nonlinear superconducting resonators. The nonlinearity is introduced by galvanically coupled dc-SQUIDs. 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 the basic building block for larger systems, allowing for quantum simulations of bosonic many-body systems with a larger number of lattice sites.

Quantum Fourier Transform in Oscillating Modes

  1. Qi-Ming Chen,
  2. Frank Deppe,
  3. Re-Bing Wu,
  4. Luyan Sun,
  5. Yu-xi Liu,
  6. Yuki Nojiri,
  7. Stefan Pogorzalek,
  8. Michael Renger,
  9. Matti Partanen,
  10. Kirill G. Fedorov,
  11. Achim Marx,
  12. and Rudolf Gross
Quantum Fourier transform (QFT) is a key ingredient of many quantum algorithms. In typical applications such as phase estimation, a considerable number of ancilla qubits and gates are
used to form a Hilbert space large enough for high-precision results. Qubit recycling reduces the number of ancilla qubits to one, but it is only applicable to semi-classical QFT and requires repeated measurements and feedforward within the coherence time of the qubits. In this work, we explore a novel approach based on resonators that forms a high-dimensional Hilbert space for the realization of QFT. By employing the perfect state-transfer method, we map an unknown multi-qubit state to a single resonator, and obtain the QFT state in the second oscillator through cross-Kerr interaction and projective measurement. A quantitive analysis shows that our method allows for high-dimensional and fully-quantum QFT employing the state-of-the-art superconducting quantum circuits. This paves the way for implementing various QFT related quantum algorithms.

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

  1. Philip Schmidt,
  2. Mohammad T. Amawi,
  3. Stefan Pogorzalek,
  4. Frank Deppe,
  5. Achim Marx,
  6. Rudolf Gross,
  7. and Hans Huebl
as well as the generation of phononic and photonic quantum states [3-10]."]Electromechanical systems realize this optomechanical interaction in the microwave regime. In this context, capacitive coupling arrangements demonstrated interaction rates of up to 280 Hz [11]. Complementary, early proposals [12-15] and experiments [16,17] suggest that inductive coupling schemes are tunable and have the potential to reach the vacuum 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 1.62 kHz, realizing sub-aN Hz-1/2 force sensitivities. Moreover, we study the frequency splitting of the microwave resonator for large mechanical amplitudes confirming the large coupling. The presented inductive coupling scheme shows the high potential of SQUID-based electromechanics for targeting the full wealth of the intrinsically nonlinear optomechanics Hamiltonian.

Challenges in Open-air Microwave Quantum Communication and Sensing

  1. Mikel Sanz,
  2. Kirill G. Fedorov,
  3. Frank Deppe,
  4. and Enrique Solano
Quantum communication is a holy grail to achieve secure communication among a set of partners, since it is provably unbreakable by physical laws. Quantum sensing employs quantum entanglement
as an extra resource to determine parameters by either using less resources or attaining a precision unachievable in classical protocols. A paradigmatic example is the quantum radar, which allows one to detect an object without being detected oneself, by making use of the additional asset provided by quantum entanglement to reduce the intensity of the signal. In the optical regime, impressive technological advances have been reached in the last years, such as the first quantum communication between ground and satellites, as well as the first proof-of-principle experiments in quantum sensing. The development of microwave quantum technologies turned out, nonetheless, to be more challenging. Here, we will discuss the challenges regarding the use of microwaves for quantum communication and sensing. Based on this analysis, we propose a roadmap to achieve real-life applications in these fields.

Ultrawide-range photon number calibration using a hybrid system combining nano-electromechanics and superconducting circuit quantum electrodynamics

  1. Philip Schmidt,
  2. Daniel Schwienbacher,
  3. Matthias Pernpeintner,
  4. Friedrich Wulschner,
  5. Frank Deppe,
  6. Achim Marx,
  7. Rudolf Gross,
  8. and Hans Huebl
We present a hybrid system consisting of a superconducting coplanar waveguide resonator coupled to a nanomechanical string and a transmon qubit acting as nonlinear circuit element.
We perform spectroscopy for both the transmon qubit and the nanomechanical string. Measuring the ac-Stark shift on the transmon qubit as well as the electromechanically induced absorption on the string allows us to determine the average photon number in the microwave resonator in both the low and high power regimes. In this way, we measure photon numbers that are up to nine orders of magnitude apart. We find a quantitative agreement between the calibration of photon numbers in the microwave resonator using the two methods. Our experiments demonstrate the successful combination of superconducting circuit quantum electrodynamics and nano-electromechanics on a single chip.

Scalable 3D quantum memory

  1. Edwar Xie,
  2. Frank Deppe,
  3. Daniel Repp,
  4. Peter Eder,
  5. Michael Fischer,
  6. Jan Goetz,
  7. Stefan Pogorzalek,
  8. Kirill G. Fedorov,
  9. Achim Marx,
  10. and Rudolf Gross
Superconducting 3D microwave cavities offer state-of-the-art coherence times and a well controlled environment for superconducting qubits. In order to realize at the same time fast
readout and long-lived quantum information storage, one can couple the qubit both to a low-quality readout and a high-quality storage cavity. However, such systems are bulky compared to their less coherent 2D counterparts. A more compact and scalable approach is achieved by making use of the multimode structure of a 3D cavity. In our work, we investigate such a device where a transmon qubit is capacitively coupled to two modes of a single 3D cavity. The external coupling is engineered so that the memory mode has an about 100 times larger quality factor than the readout mode. Using an all-microwave second-order protocol, we realize a lifetime enhancement of the stored state over the qubit lifetime by a factor of 6 with a Z-fidelity of 82%. We also find that this enhancement is not limited by fundamental constraints.