Gradient-Descent Quantum Process Tomography by Learning Kraus Operators

We perform quantum process tomography (QPT) for both discrete- and continuous-variable quantum systems by learning a process representation using Kraus operators. The Kraus form ensures that the reconstructed process is completely positive. To make the process trace preserving, we use a constrained gradient-descent (GD) approach on the so-called Stiefel manifold during optimization to obtain the Kraus operators. Our ansatz uses a few Kraus operators to avoid direct estimation of large process matrices, e.g., the Choi matrix, for low-rank quantum processes. The GD-QPT matches the performance of both compressed-sensing (CS) and projected least-squares (PLS) QPT in benchmarks with two-qubit random processes, but shines by combining the best features of these two methods. Similar to CS (but unlike PLS), GD-QPT can reconstruct a process from just a small number of random measurements, and similar to PLS (but unlike CS) it also works for larger system sizes, up to at least five qubits. We envisage that the data-driven approach of GD-QPT can become a practical tool that greatly reduces the cost and computational effort for QPT in intermediate-scale quantum systems

Wigner negativity in the steady-state output of a Kerr parametric oscillator

The output field from a continuously driven linear parametric oscillator may exhibit considerably more squeezing than the intracavity field. Inspired by this fact, we explore the nonclassical features of the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporal wave packet mode description. Utilizing a new numerical method, we have access to the density matrix of arbitrary wave packet modes. Remarkably, we find that even though the steady-state cavity field is always characterized by a positive Wigner function, the output may exhibit Wigner negativity, depending on the properties of the selected mode

Numerical study of Wigner negativity in one-dimensional steady-state resonance fluorescence

In a numerical study, we investigate the steady-state generation of nonclassical states of light from a coherently driven two-level atom in a one-dimensional waveguide. Specifically, we look for states with a negative Wigner function, since such nonclassical states are a resource for quantum information processing applications, including quantum computing. We find that a waveguide terminated by a mirror at the position of the atom can provide Wigner-negative states, while an infinite waveguide yields strictly positive Wigner functions. Moreover, our paper reveals a connection between the purity of a quantum state and its Wigner negativity. We also analyze the effects of decoherence on the negativity of a state

Steady-State Generation of Wigner-Negative States in One-Dimensional Resonance Fluorescence

In this work we demonstrate numerically that the nonlinearity provided by a continuously driven two-level system allows for the generation of Wigner-negative states of the electromagnetic field confined in one spatial dimension. Wigner-negative states, also known as Wigner nonclassical states, are desirable for quantum information protocols beyond the scope of classical computers. Focusing on the steady-state emission from the two-level system, we find the largest negativity at the drive strength where the coherent reflection vanishes

PT-symmetric circuit QED

A parity-time (PT)-symmetric system emerging from a quantum dynamics is highly desirable in order to understand the possible implications of PT symmetry in the next generation of quantum technologies. In this work, we address this need by proposing and studying a circuit-QED architecture that consists of two coupled resonators and two qubits (each coupled to one resonator). By means of external driving fields on the qubits, we are able to tune gains and losses in the resonators. Starting with the quantum dynamics of this system, we show the emergence of the PT symmetry via the selection of both driving amplitudes and frequencies. We engineer the system such that a non-number-conserving dipole-dipole interaction emerges, introducing an instability at large coupling strengths. The PT symmetry and its breaking, as well as the predicted instability in this circuit-QED system, can be observed in a transmission experiment

Observation of Three-Photon Spontaneous Parametric Down-Conversion in a Superconducting Parametric Cavity

Spontaneous parametric down-conversion (SPDC) has been a key enabling technology in exploring quantum phenomena and their applications for decades. For instance, traditional SPDC, which splits a high-energy pump photon into two lower-energy photons, is a common way to produce entangled photon pairs. Since the early realizations of SPDC, researchers have thought to generalize it to higher order, e.g., to produce entangled photon triplets. However, directly generating photon triplets through a single SPDC process has remained elusive. Here, using a flux-pumped superconducting parametric cavity, we demonstrate direct three-photon SPDC, with photon triplets generated in a single cavity mode or split between multiple modes. With strong pumping, the states can be quite bright, with flux densities exceeding 60 photons per second per hertz. The observed states are strongly non-Gaussian, which has important implications for potential applications. In the single-mode case, we observe a triangular star-shaped distribution of quadrature voltages, indicative of the long-predicted "star state." The observed state shows strong third-order correlations, as expected for a state generated by a cubic Hamiltonian. By pumping at the sum frequency of multiple modes, we observe strong three-body correlations between multiple modes, strikingly, in the absence of second-order correlations. We further analyze the third-order correlations under mode transformations by the symplectic symmetry group, showing that the observed transformation properties serve to "fingerprint" the specific cubic Hamiltonian that generates them. The observed non-Gaussian, third-order correlations represent an important step forward in quantum optics and may have a strong impact on quantum communication with microwave fields as well as continuous-variable quantum computation

Robust Preparation of Wigner-Negative States with Optimized SNAP-Displacement Sequences

Hosting nonclassical states of light in three-dimensional microwave cavities has emerged as a promising paradigm for continuous-variable quantum information processing. Here we experimentally demonstrate high-fidelity generation of a range of Wigner-negative states useful for quantum computation, such as Schrodinger-cat states, binomial states, Gottesman-Kitaev-Preskill states, as well as cubic phase states. The latter states have been long sought after in quantum optics and have never been achieved experimentally before. We use a sequence of interleaved selective number-dependent arbitrary phase (SNAP) gates and displacements. We optimize the state preparation in two steps. First we use a gradient-descent algorithm to optimize the parameters of the SNAP and displacement gates. Then we optimize the envelope of the pulses implementing the SNAP gates. Our results show that this way of creating highly nonclassical states in a harmonic oscillator is robust to fluctuations of the system parameters such as the qubit frequency and the dispersive shift

Propagating Wigner-Negative States Generated from the Steady-State Emission of a Superconducting Qubit

We experimentally demonstrate the steady-state generation of propagating Wigner-negative states from a continuously driven superconducting qubit. We reconstruct the Wigner function of the radiation emitted into propagating modes defined by their temporal envelopes, using digital filtering. For an optimized temporal filter, we observe a large Wigner logarithmic negativity, in excess of 0.08, in agreement with theory. The fidelity between the theoretical predictions and the states generated experimentally is up to 99%, reaching state-of-the-art realizations in the microwave frequency domain. Our results provide a new way to generate and control nonclassical states, and may enable promising applications such as quantum networks and quantum computation based on waveguide quantum electrodynamics

Tripartite Genuine Non-Gaussian Entanglement in Three-Mode Spontaneous Parametric Down-Conversion

We show that the states generated by a three-mode spontaneous parametric down-conversion (SPDC) interaction Hamiltonian possess tripartite entanglement of a different nature to other paradigmatic three-mode entangled states generated by the combination of two-mode SPDC interactions. While two-mode SPDC generates Gaussian states whose entanglement can be characterized by standard criteria based on two-mode quantum correlations, these criteria fail to capture the entanglement generated by three-mode SPDC. We use criteria built from three-mode correlation functions to show that the class of states recently generated in a superconducting-circuit implementation of three-mode SPDC ideally have tripartite entanglement, contrary to recent claims in the literature. These criteria are suitable for triple SPDC but we show that they fail to detect tripartite entanglement in other states which are known to possess it, which illustrates the existence of two fundamentally different notions of tripartite entanglement in three-mode continuous-variable systems

Ultrastrong coupling in two-resonator circuit QED

We report on ultrastrong coupling between a superconducting flux qubit and a resonant mode of a system comprised of two superconducting coplanar stripline resonators coupled galvanically to the qubit. With a coupling strength as high as 17.5% of the mode frequency, exceeding that of previous circuit quantum electrodynamics experiments, we observe a pronounced Bloch-Siegert shift. The spectroscopic response of our multimode system reveals a clear breakdown of the Jaynes-Cummings approximation. In contrast to earlier experiments, the high coupling strength is achieved without making use of an additional inductance provided by a Josephson junction