We measure the lifetime (T₁) and coherence (T₂) of two-level defect states (TLSs) in the insulating barrier of a Josephson phase qubit and compare to the interaction strength between the two systems. We find for the average decay times a power-law dependence on the corresponding interaction strengths, whereas for the average coherence times we find an optimum at intermediate coupling strengths. We explain both the lifetime and the coherence results using the standard TLS model, including dipole radiation by phonons and anticorrelated dependence of the energy parameters on environmental fluctuations.

The analysis of wave-packet dynamics may be greatly simplified when viewed in phase space. While harmonic oscillators are often used as a convenient platform to study wave packets, arbitrary state preparation in these systems is more challenging. Here, we demonstrate a direct measurement of the Wigner distribution of complex photon states in an anharmonic oscillator--a superconducting phase circuit, biased in the small anharmonicity regime. We apply our method on nondispersive wave packets to explicitly show phase locking in states prepared by a frequency chirp. This method requires a simple calibration, and is easily applicable in our system out to the fifth level.

We measure the state dynamics of a tunable anharmonic quantum system, the Josephson phase circuit, under the excitation of a frequency-chirped drive. At small anharmonicity, the state evolves like a wavepacket -a characteristic response in classical oscillators; in this regime we report exponentially enhanced lifetimes of highly excited states, held by the drive. At large anharmonicity, we observe sharp steps, corresponding to the excitation of discrete energy levels. The continuous transition between the two regimes is mapped by measuring the threshold of these two effects.Ever since the laws of quantum mechanics were formulated, there has been an ongoing effort to explain the emergence of classical laws in experimental systems. The first explanation by Bohr states that these systems operate in the limit of large quantum numbers [1], in which case they may be described by a wavepacket that on the average follows the classical equations of motion [2]. In addition, coupling to uncontrolled, external degrees of freedom (decoherence), is often related to the emergence of classicality [3]. Recent experiments and calculations have demonstrated the quantum to classical transition in oscillators, via noise saturation at low temperature due to zero point fluctuations [4, 5], and harmonic behavior at high temperatures in a cavity-QED system [6].In a classical anharmonic oscillator, such as a pendulum, the energy expectation can be deterministically increased to large values if the driving force is frequencychirped and its amplitude is sufficiently large. This phenomenon is commonly known as autoresonance [7]. The physical mechanism behind this effect is adiabatic, nonlinear phase-locking between the system and the drive, yielding a controllable excitation as the system's resonance frequency follows the drive frequency as a function of time. This effect is utilized in a wide variety of systems [8, 9], and recently in Josephson-based oscillators [5, 10]. In a quantum anharmonic oscillator, the expected time evolution under a similar drive is sequential excitation of single energy levels of the system, or "quantum ladder climbing" [11]. In practice, for a given anharmonicity the drive itself introduces some mixing between the energy levels due to power broadening and finite bandwidth, which may wash out ladder climbing and lead to a classical behavior in a quantum system [12,13]. In this letter, we measure the dynamics in these two distinct regimes in the same system by varying the drive parameters and the system's anharmonicity.Our system, the Josephson phase circuit (JPC, see Fig. 1a), is a superconducting oscillator with a nonlinear inductor formed by a Josephson junction. It can be described energetically by a double-well potential that depends on the phase difference δ across the junction. We tune the potential by means of an external magnetic flux bias [14] to vary the anharmonicity and measure the state. Traditionally, the circuit is operated as a twolevel system (qubit) [14,15], or a d-level system (qudit) [16], by ...

Multi-level control of quantum coherence exponentially reduces communication and computation resources required for a variety of applications of quantum information science. However, it also introduces complex dynamics to be understood and controlled. These dynamics can be simplified and made intuitive by employing group theory to visualize certain four-level dynamics in a 'Bell frame' comprising an effective pair of uncoupled two-level qubits. We demonstrate control of a Josephson phase qudit with a single multi-tone excitation, achieving successive population inversions between the first and third levels and highlighting constraints imposed by the two-qubit representation. Furthermore, the finite anharmonicity of our system results in a rich dynamical evolution, where the two Bell-frame qubits undergo entangling-disentangling oscillations in time, explained by a Cartan gate decomposition representation. The Bell frame constitutes a promising tool for control of multi-level quantum systems, providing an intuitive clarity to complex dynamics.

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