This chapter investigates the coupling of a two-level system to a nonlinear resonator, one of the simplest extensions to the well-known Jaynes–Cummings model. In the experiment, the two-level system is a superconducting transmon qubit, and the resonator is a coplanar resonator with a Kerr nonlinearity induced by a Josephson junction. Effects arising from the resonator nonlinearity (bistability, parametric amplification, squeezing) qualitatively modify the dispersive qubit-resonator coupling. This makes it possible to design a qubit readout which has both a high fidelity and the potential of being QND. Motivated by fundamental and practical issues concerning the quantum limit of such a measurement, the chapter also studies spectroscopically the quantum back action exerted by the intraresonator field onto the qubit. Qualitative agreement is obtained with analytical results and quantitative agreement with numerical simulations.
This chapter investigates the coupling of a two-level system to a nonlinear resonator, one of the simplest extensions to the well-known Jaynes–Cummings model. In the experiment, the two-level system is a superconducting transmon qubit, and the resonator is a coplanar resonator with a Kerr nonlinearity induced by a Josephson junction. Effects arising from the resonator nonlinearity (bistability, parametric amplification, squeezing) qualitatively modify the dispersive qubit-resonator coupling. This makes it possible to design a qubit readout which has both a high fidelity and the potential of being QND. Motivated by fundamental and practical issues concerning the quantum limit of such a measurement, the chapter also studies spectroscopically the quantum back action exerted by the intraresonator field onto the qubit. Qualitative agreement is obtained with analytical results and quantitative agreement with numerical simulations.
One of the most studied model systems in quantum optics is a two-level atom strongly coupled to a single mode of the electromagnetic field stored in a cavity, a research field named cavity quantum electrodynamics or CQED. CQED has recently received renewed attention due to its implementation with superconducting artificial atoms and coplanar resonators in the so-called circuit quantum electrodynamics (cQED) architecture. In cQED, the couplings can be much stronger than in CQED due to the design flexibility of superconducting circuits and to the enhanced field confinement in one-dimensional cavities. This enabled the realization of fundamental quantum physics and quantum information processing experiments with a degree of control comparable to that obtained in CQED. The purpose of this chapter is to investigate the situation where the resonator to which the atom is coupled is made nonlinear with a Kerr-type nonlinearity, causing its energy levels to be nonequidistant. The system is then described by a nonlinear Jaynes-Cummings Hamiltonian. This considerably enriches the physics since a pumped nonlinear resonator displays bistability, parametric amplification, and squeezing. The interplay of strong coupling and these nonlinear effects constitutes a novel model system for quantum optics that can be implemented experimentally with superconducting circuits. This chapter is organized as follows. In a first section we present the system consisting of a superconducting Kerr nonlinear resonator strongly coupled to a transmon qubit. In the second section, we describe the response of the sole nonlinear resonator to an external drive. In the third section, we show how the resonator bistability can be used to perform a high-fidelity readout of the transmon qubit. In the last section, we investigate the quantum backaction exerted by the intracavity field on the qubit.
We study the backaction of a driven nonlinear resonator on a multilevel superconducting qubit. Using unitary transformations on the multilevel Jaynes-Cummings Hamiltonian and quantum optics master equation, we derive an analytical model that goes beyond linear response theory. Within the limits of validity of the model, we obtain quantitative agreement with experimental and numerical data, both in the bifurcation and in the parametric amplification regimes of the nonlinear resonator. We show in particular that the measurement-induced dephasing rate of the qubit can be rather small at high drive power. This is in contrast to measurement with a linear resonator where this rate increases with the drive power. Finally, we show that, for typical parameters of circuit quantum electrodynamics, correctly describing measurement-induced dephasing requires a model going beyond linear response theory, such as the one presented here.
This chapter analyzes the high excitation nonlinear response of the Jaynes–Cummings model in quantum optics when the qubit and cavity are strongly coupled. It studies the response in the parameter ranges appropriate for transmon qubits in the circuit quantum electrodynamics architecture. It focuses on the situation where the system behaves essentially as a nonlinear quantum oscillator and explores quantum and semi-classical dynamics. One of the central motivations is that under strong excitation tones, the nonlinear response can lead to qubit quantum state discrimination and the chapter then presents results for the cases when the qubit and cavity are on resonance or far off-resonance (dispersive).
Josephson junctions are a fundamental building block of superconducting quantum circuits. They provide the nonlinearity required to realize quantum bits and enable a wide range of nonlinear phenomena central to quantum technologies. This thesis explores two superconducting systems in which this nonlinearity plays a central role. In the first part, we investigate a superconducting resonator terminated by a superconducting quantum interference device (SQUID), an arrangement of two Josephson junctions connected in parallel. The SQUID provides a tunable Kerr nonlinearity and allows the resonator to be driven parametrically. Under the combined action of the parametric drive, Kerr nonlinearity, and dissipation to the environment, the state of the resonator can undergo abrupt changes as the pump frequency or amplitude is varied. We interpret these changes within the framework of dissipative phase transitions (DPTs). By implementing a finite-component thermodynamic limit, we experimentally and theoretically investigate both first- and second-order DPTs in this system. We characterize their steady-state and dynamical signatures, including squeezing below vacuum, phase coexistence, hysteresis, spontaneous symmetry breaking, and critical slowing down. Finally, we exploit the enhanced susceptibility near the second-order critical point to implement a frequency-estimation protocol. We show that the estimation precision scales quadratically with the effective system size, demonstrating that finite-component DPTs can provide a quantum advantage in a sensing protocol. In the second part, we investigate a novel architecture for the readout of superconducting qubits. Conventional dispersive readout relies on a transverse interaction between a transmon and a resonator, which can lead to Purcell decay and measurement-induced state transitions, ultimately limiting the readout fidelity. To overcome these limitations, we implement a nonlinear coupling by galvanically connecting the transmon and resonator through a Josephson junction. This interaction provides a strong cross-Kerr coupling for clear state discrimination, but also introduces a detrimental transverse interaction. We experimentally show that the exchange part of the transverse interaction can be suppressed by adding a capacitor in parallel with the coupling junction while preserving the strong cross-Kerr coupling. In addition, we show that the same circuit acts as an intrinsic Purcell filter, protecting the qubit against Purcell decay. The coupling junction also induces a Kerr nonlinearity in the resonator, which we exploit to implement a bifurcation readout with a 99.4 % fidelity in a 68 ns integration time. This performance is achieved without a dedicated Purcell filter or a near-quantum-limited amplifier. Overall, this thesis demonstrates the engineering of nonlinear superconducting circuits to investigate dissipative phase transitions, exploit criticality for enhanced quantum sensing, and develop a novel qubit readout architecture enabling fast, high-fidelity measurement.
We demonstrate a qubit readout scheme that exploits the Jaynes-Cummings nonlinearity of a superconducting cavity coupled to transmon qubits. We find that, in the strongly driven dispersive regime of this system, there is the unexpected onset of a high-transmission "bright" state at a critical power which depends sensitively on the initial qubit state. A simple and robust measurement protocol exploiting this effect achieves a single-shot fidelity of 87% using a conventional sample design and experimental setup, and at least 61% fidelity to joint correlations of three qubits.
Recent experiments have realized a familiar model of quantum optics, dispersive optical bistability, in the context of microwave coplanar resonators with internal Josephson junctions that are used to achieve large effective Kerr nonlinearities with very low dissipation. A Josephson junction qubit constitutes an effective atom that may be coupled to such a nonlinear resonator to provide a very useful probe that is sensitive to the quantum noise in the nonlinear cavity. In particular, I will explain how sidebands on the qubit excitation spectrum are sensitive to the effective temperature, oscillation frequency and damping rate of the intracavity field fluctuations.