No abstract

We observe a crossover from strong to weak chaos in the spatiotemporal evolution of multiple site excitations within disordered chains with cubic nonlinearity. Recent studies have shown that Anderson localization is destroyed, and the wave packet spreading is characterized by an asymptotic divergence of the second moment m2 in time (as t 1/3 ), due to weak chaos. In the present paper, we observe the existence of a qualitatively new dynamical regime of strong chaos, in which the second moment spreads even faster (as t 1/2 ), with a crossover to the asymptotic law of weak chaos at larger times. We analyze the pecularities of these spreading regimes and perform extensive numerical simulations over large times with ensemble averaging. A technique of local derivatives on logarithmic scales is developed in order to quantitatively visualize the slow crossover processes.

In the absence of nonlinearity all eigenmodes of a chain with disorder are spatially localized (Anderson localization). The width of the eigenvalue spectrum and the average eigenvalue spacing inside the localization volume set two frequency scales. An initially localized wave packet spreads in the presence of nonlinearity. Nonlinearity introduces frequency shifts, which define three different evolution outcomes: (i) localization as a transient, with subsequent subdiffusion; (ii) the absence of the transient and immediate subdiffusion; (iii) self-trapping of a part of the packet and subdiffusion of the remainder. The subdiffusive spreading is due to a finite number of packet modes being resonant. This number does not change on average and depends only on the disorder strength. Spreading is due to corresponding weak chaos inside the packet, which slowly heats the cold exterior. The second moment of the packet grows as t;{alpha}. We find alpha=1/3.

Hybrid quantum systems based on spin ensembles coupled to superconducting microwave cavities are promising candidates for robust experiments in cavity quantum electrodynamics (QED) and for future technologies employing quantum mechanical e ects [1][2][3][4] . At present, the main source of decoherence in these systems is inhomogeneous spin broadening, which limits their performance for the coherent transfer and storage of quantum information [5][6][7] . Here we study the dynamics of a superconducting cavity strongly coupled to an ensemble of nitrogen-vacancy centres in diamond. We experimentally observe how decoherence induced by inhomogeneous broadening can be suppressed in the strong-coupling regime-a phenomenon known as 'cavity protection' 5,7 . To demonstrate the potential of this e ect for coherent-control schemes, we show how appropriately chosen microwave pulses can increase the amplitude of coherent oscillations between the cavity and spin ensemble by two orders of magnitude.The processing of quantum information requires special devices that can store and manipulate quantum bits. Hybrid quantum systems 2 combine the advantages of different systems to overcome their individual physical limitations. In this context superconducting microwave cavities have emerged as ideal tools for realizing strong coupling to qubits 3,4,[8][9][10][11][12] for the transfer of excitations on the singlephoton level 13,14 . For the storage of quantum information the negatively charged nitrogen-vacancy (NV) centres in diamond show great potential, especially owing to their long coherence times (up to one second 15 ) and to the combination of microwave and optical transitions which makes them an easily accessible and controllable qubit 16 . Coherently passing quantum information between such a spin and a cavity requires that they are strongly coupled to each other. As has recently been shown 4,10-12 , this limit can be reached by collective coupling to a large spin ensemble, in which case the coupling strength is increased by the square root of the ensemble size. However, this collective coupling comes with a considerable downside: in a solid-state environment a spin is always prone to inhomogeneous broadening. In particular, for an ensemble of NV centres, magnetic dipolar interaction with excess nuclear and electron spins in the diamond crystal leads to an inhomogeneous broadening of the spin transition 17 , which acts as the dominant source of decoherence. Several approaches, including echo-type refocusing techniques 18,19 , have been suggested to overcome this limitation. Here we will concentrate on recent theoretical proposals which rely on the specific shape of the inhomogeneous spectral In our homodyne detection measurements, the input microwave signal is split into two paths, both serving as a reference signal as well as for testing and controlling our experiment. Outside the cryostat both signal paths are combined by a frequency mixer and the quadratures I and Q are recorded with a fast analog-to-digital converter (ADC) ...