Solomon equations for qubit and two-level systems

  1. Martin Spiecker,
  2. Andrei I. Pavlov,
  3. Alexander Shnirman,
  4. and Ioan M. Pop
We model and measure the combined relaxation of a qubit, a.k.a. central spin, coupled to a discrete two-level system (TLS) environment. We present a derivation of the Solomon equations
starting from a general Lindblad equation for the qubit and an arbitrary number of TLSs. If the TLSs are much longer lived than the qubit, the relaxation becomes non-exponential. In the limit of large numbers of TLSs the populations are likely to follow a power law, which we illustrate by measuring the relaxation of a superconducting fluxonium qubit. Moreover, we show that the Solomon equations predict non-Poissonian quantum jump statistics, which we confirm experimentally.

Observation of Josephson Harmonics in Tunnel Junctions

  1. Dennis Willsch,
  2. Dennis Rieger,
  3. Patrick Winkel,
  4. Madita Willsch,
  5. Christian Dickel,
  6. Jonas Krause,
  7. Yoichi Ando,
  8. Raphaël Lescanne,
  9. Zaki Leghtas,
  10. Nicholas T. Bronn,
  11. Pratiti Deb,
  12. Olivia Lanes,
  13. Zlatko K. Minev,
  14. Benedikt Dennig,
  15. Simon Geisert,
  16. Simon Günzler,
  17. Sören Ihssen,
  18. Patrick Paluch,
  19. Thomas Reisinger,
  20. Roudy Hanna,
  21. Jin Hee Bae,
  22. Peter Schüffelgen,
  23. Detlev Grützmacher,
  24. Luiza Buimaga-Iarinca,
  25. Cristian Morari,
  26. Wolfgang Wernsdorfer,
  27. David P. DiVincenzo,
  28. Kristel Michielsen,
  29. Gianluigi Catelani,
  30. and Ioan M. Pop
An accurate understanding of the Josephson effect is the keystone of quantum information processing with superconducting hardware. Here we show that the celebrated sinφ current-phase
relation (CφR) of Josephson junctions (JJs) fails to fully describe the energy spectra of transmon artificial atoms across various samples and laboratories. While the microscopic theory of JJs contains higher harmonics in the CφR, these have generally been assumed to give insignificant corrections for tunnel JJs, due to the low transparency of the conduction channels. However, this assumption might not be justified given the disordered nature of the commonly used AlOx tunnel barriers. Indeed, a mesoscopic model of tunneling through an inhomogeneous AlOx barrier predicts contributions from higher Josephson harmonics of several %. By including these in the transmon Hamiltonian, we obtain orders of magnitude better agreement between the computed and measured energy spectra. The measurement of Josephson harmonics in the CφR of standard tunnel junctions prompts a reevaluation of current models for superconducting hardware and it offers a highly sensitive probe towards optimizing tunnel barrier uniformity.

A quantum Szilard engine for two-level systems coupled to a qubit

  1. Martin Spiecker,
  2. Patrick Paluch,
  3. Niv Drucker,
  4. Shlomi Matityahu,
  5. Daria Gusenkova,
  6. Nicolas Gosling,
  7. Simon Günzler,
  8. Dennis Rieger,
  9. Ivan Takmakov,
  10. Francesco Valenti,
  11. Patrick Winkel,
  12. Richard Gebauer,
  13. Oliver Sander,
  14. Gianluigi Catelani,
  15. Alexander Shnirman,
  16. Alexey V. Ustinov,
  17. Wolfgang Wernsdorfer,
  18. Yonatan Cohen,
  19. and Ioan M. Pop
The innate complexity of solid state physics exposes superconducting quantum circuits to interactions with uncontrolled degrees of freedom degrading their coherence. By using a simple
stabilization sequence we show that a superconducting fluxonium qubit is coupled to a two-level system (TLS) environment of unknown origin, with a relatively long energy relaxation time exceeding 50ms. Implementing a quantum Szilard engine with an active feedback control loop allows us to decide whether the qubit heats or cools its TLS environment. The TLSs can be cooled down resulting in a four times lower qubit population, or they can be heated to manifest themselves as a negative temperature environment corresponding to a qubit population of ∼80%. We show that the TLSs and the qubit are each other’s dominant loss mechanism and that the qubit relaxation is independent of the TLS populations. Understanding and mitigating TLS environments is therefore not only crucial to improve qubit lifetimes but also to avoid non-Markovian qubit dynamics.

Operating in a deep underground facility improves the locking of gradiometric fluxonium qubits at the sweet spots

  1. Daria Gusenkova,
  2. Francesco Valenti,
  3. Martin Spiecker,
  4. Simon Günzler,
  5. Patrick Paluch,
  6. Dennis Rieger,
  7. Larisa-Milena Pioraş-Ţimbolmaş,
  8. Liviu P. Zârbo,
  9. Nicola Casali,
  10. Ivan Colantoni,
  11. Angelo Cruciani,
  12. Stefano Pirro,
  13. Laura Cardani,
  14. Alexandru Petrescu,
  15. Wolfgang Wernsdorfer,
  16. Patrick Winkel,
  17. and Ioan M. Pop
We demonstrate flux-bias locking and operation of a gradiometric fluxonium artificial atom using two symmetric granular aluminum (grAl) loops to implement the superinductor. The gradiometric
fluxonium shows two orders of magnitude suppression of sensitivity to homogeneous magnetic fields, which can be an asset for hybrid quantum systems requiring strong magnetic field biasing. By cooling down the device in an external magnetic field while crossing the metal-to-superconductor transition, the gradiometric fluxonium can be locked either at 0 or Φ0/2 effective flux bias, corresponding to an even or odd number of trapped fluxons, respectively. At mK temperatures, the fluxon parity prepared during initialization survives to magnetic field bias exceeding 100Φ0. However, even for states biased in the vicinity of 1Φ0, we observe unexpectedly short fluxon lifetimes of a few hours, which cannot be explained by thermal or quantum phase slips. When operating in a deep-underground cryostat of the Gran Sasso laboratory, the fluxon lifetimes increase to days, indicating that ionizing events activate phase slips in the grAl superinductor.

Energy-participation quantization of Josephson circuits

  1. Zlatko K. Minev,
  2. Zaki Leghtas,
  3. Shantanu O. Mundhada,
  4. Lysander Christakis,
  5. Ioan M. Pop,
  6. and Michel H. Devoret
Superconducting microwave circuits incorporating nonlinear devices, such as Josephson junctions, are one of the leading platforms for emerging quantum technologies. Increasing circuit
complexity further requires efficient methods for the calculation and optimization of the spectrum, nonlinear interactions, and dissipation in multi-mode distributed quantum circuits. Here, we present a method based on the energy-participation ratio (EPR) of a dissipative or nonlinear element in an electromagnetic mode. The EPR, a number between zero and one, quantifies how much of the energy of a mode is stored in each element. It obeys universal constraints—valid regardless of the circuit topology and nature of the nonlinear elements. The EPR of the elements are calculated from a unique, efficient electromagnetic eigenmode simulation of the linearized circuit, including lossy elements. Their set is the key input to the determination of the quantum Hamiltonian of the system. The method provides an intuitive and simple-to-use tool to quantize multi-junction circuits. It is especially well-suited for finding the Hamiltonian and dissipative parameters of weakly anharmonic systems, such as transmon qubits coupled to resonators, or Josephson transmission lines. We experimentally tested this method on a variety of Josephson circuits, and demonstrated agreement within several percents for nonlinear couplings and modal Hamiltonian parameters, spanning five-orders of magnitude in energy, across a dozen samples.

Quantum non-demolition dispersive readout of a superconducting artificial atom using large photon numbers

  1. Daria Gusenkova,
  2. Martin Spiecker,
  3. Richard Gebauer,
  4. Madita Willsch,
  5. Francesco Valenti,
  6. Nick Karcher,
  7. Lukas Grünhaupt,
  8. Ivan Takmakov,
  9. Patrick Winkel,
  10. Dennis Rieger,
  11. Alexey V. Ustinov,
  12. Nicolas Roch,
  13. Wolfgang Wernsdorfer,
  14. Kristel Michielsen,
  15. Oliver Sander,
  16. and Ioan M. Pop
Reading out the state of superconducting artificial atoms typically relies on dispersive coupling to a readout resonator. For a given system noise temperature, increasing the circulating
photon number n¯ in the resonator enables a shorter measurement time and is therefore expected to reduce readout errors caused by spontaneous atom transitions. However, increasing n¯ is generally observed to also increase these transition rates. Here we present a fluxonium artificial atom in which we measure an overall flat dependence of the transition rates between its first two states as a function of n¯, up to n¯≈200. Despite the fact that we observe the expected decrease of the dispersive shift with increasing readout power, the signal-to-noise ratio continuously improves with increasing n¯. Even without the use of a parametric amplifier, at n¯=74, we measure fidelities of 99% and 93% for feedback-assisted ground and excited state preparation, respectively.

Reducing the impact of radioactivity on quantum circuits in a deep-underground facility

  1. Laura Cardani,
  2. Francesco Valenti,
  3. Nicola Casali,
  4. Gianluigi Catelani,
  5. Thibault Charpentier,
  6. Massimiliano Clemenza,
  7. Ivan Colantoni,
  8. Angelo Cruciani,
  9. Luca Gironi,
  10. Lukas Grünhaupt,
  11. Daria Gusenkova,
  12. Fabio Henriques,
  13. Marc Lagoin,
  14. Maria Martinez,
  15. Giorgio Pettinari,
  16. Claudia Rusconi,
  17. Oliver Sander,
  18. Alexey V. Ustinov,
  19. Marc Weber,
  20. Wolfgang Wernsdorfer,
  21. Marco Vignati,
  22. Stefano Pirro,
  23. and Ioan M. Pop
As quantum coherence times of superconducting circuits have increased from nanoseconds to hundreds of microseconds, they are currently one of the leading platforms for quantum information
processing. However, coherence needs to further improve by orders of magnitude to reduce the prohibitive hardware overhead of current error correction schemes. Reaching this goal hinges on reducing the density of broken Cooper pairs, so-called quasiparticles. Here, we show that environmental radioactivity is a significant source of nonequilibrium quasiparticles. Moreover, ionizing radiation introduces time-correlated quasiparticle bursts in resonators on the same chip, further complicating quantum error correction. Operating in a deep-underground lead-shielded cryostat decreases the quasiparticle burst rate by a factor fifty and reduces dissipation up to a factor four, showcasing the importance of radiation abatement in future solid-state quantum hardware.

State preparation of a fluxonium qubit with feedback from a custom FPGA-based platform

  1. Richard Gebauer,
  2. Nick Karcher,
  3. Daria Gusenkova,
  4. Martin Spiecker,
  5. Lukas Grünhaupt,
  6. Ivan Takmakov,
  7. Patrick Winkel,
  8. Luca Planat,
  9. Nicolas Roch,
  10. Wolfgang Wernsdorfer,
  11. Alexey V. Ustinov,
  12. Marc Weber,
  13. Martin Weides,
  14. Ioan M. Pop,
  15. and Oliver Sander
We developed a versatile integrated control and readout instrument for experiments with superconducting quantum bits (qubits), based on a field-programmable gate array (FPGA) platform.
Using this platform, we perform measurement-based, closed-loop feedback operations with 428ns platform latency. The feedback capability is instrumental in realizing active reset initialization of the qubit into the ground state in a time much shorter than its energy relaxation time T1. We show experimental results demonstrating reset of a fluxonium qubit with 99.4% fidelity, using a readout-and-drive pulse sequence approximately 1.5μs long. Compared to passive ground state initialization through thermalization, with the time constant given by T1= 80μs, the use of the FPGA-based platform allows us to improve both the fidelity and the time of the qubit initialization by an order of magnitude.

Implementation of a transmon qubit using superconducting granular aluminum

  1. Patrick Winkel,
  2. Kiril Borisov,
  3. Lukas Grünhaupt,
  4. Dennis Rieger,
  5. Martin Spiecker,
  6. Francesco Valenti,
  7. Alexey V. Ustinov,
  8. Wolfgang Wernsdorfer,
  9. and Ioan M. Pop
The high kinetic inductance offered by granular aluminum (grAl) has recently been employed for linear inductors in superconducting high-impedance qubits and kinetic inductance detectors.
Due to its large critical current density compared to typical Josephson junctions, its resilience to external magnetic fields, and its low dissipation, grAl may also provide a robust source of non-linearity for strongly driven quantum circuits, topological superconductivity, and hybrid systems. Having said that, can the grAl non-linearity be sufficient to build a qubit? Here we show that a small grAl volume (10×200×500nm3) shunted by a thin film aluminum capacitor results in a microwave oscillator with anharmonicity α two orders of magnitude larger than its spectral linewidth Γ01, effectively forming a transmon qubit. With increasing drive power, we observe several multi-photon transitions starting from the ground state, from which we extract α=2π×4.48MHz. Resonance fluorescence measurements of the |0>→|1> transition yield an intrinsic qubit linewidth γ=2π×10kHz, corresponding to a lifetime of 16μs. This linewidth remains below 2π×150kHz for in-plane magnetic fields up to ∼70mT.

Microscopic charging and in-gap states in superconducting granular aluminum

  1. Fang Yang,
  2. Tim Storbeck,
  3. Thomas Gozlinski,
  4. Lukas Grünhaupt,
  5. Ioan M. Pop,
  6. and Wulf Wulfhekel
Following the emergence of superconducting granular aluminum (grAl) as a material for high-impedance quantum circuits, future development hinges on a microscopic understanding of its
phase diagram, and whether the superconductor-to-insulator transition (SIT) is driven by disorder or charging effects. Beyond fundamental relevance, these mechanisms govern noise and dissipation in microwave circuits. Although the enhancement of the critical temperature, and the SIT in granular superconductors have been studied for more than fifty years, experimental studies have so far provided incomplete information on the microscopic phenomena. Here we present scanning tunneling microscope measurements of the local electronic structure of superconducting grAl. We confirm an increased superconducting gap in individual grains both near and above the Mott resistivity ρM≈400 μΩcm. Above ρM we find Coulomb charging effects, a first indication for decoupling, and in-gap states on individual grains, which could contribute to flux noise and dielectric loss in quantum devices. We also observe multiple low-energy states outside the gap, which may indicate bosonic excitations of the superconducting order parameter.