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  • Typical convergence of an adaptive frequency oscillator (Eqs. (1)–(3)) driven by a harmonic signal (I(t)=sin(2πt)) and different coupling constants K. The coupling constant determines the convergence speed and the amplitude of oscillations around the frequency of the driving signal in steady state — the higher K the faster the convergence and the larger the oscillations. ... (a) Typical convergence of an adaptive frequency oscillator (Eqs. (1)–(3)) driven by a harmonic signal (I(t)=sin(2πt)). The frequencies converge in an oscillatory fashion towards the frequency of the input (indicated by the dashed line). After convergence it oscillates with a small amplitude around the frequency of the input. The coupling constant determines the convergence speed and the amplitude of oscillations around the frequency of the driving signal in steady state. In all figures, the top right panel shows the driving signals (note the different scales). (b)–(f) Non-harmonic driving signals. We depict representative results on the evolution of ωdωF=ω−ωFωF vs. time. The dashed line indicates the zero error between the intrinsic frequency ω and the base frequency ωF of the driving signals. (b) Square pulse I(t)=rect(ωFt), (c) Sawtooth I(t)=st(ωFt) (d) Chirp I(t)=cos(ωct) ωc=ωF(1+12(t1000)2). (Note that the graph of the input signal is illustrative only since the change in frequency takes much longer than illustrated.) (e) Signal with two non-commensurate frequencies I(t)=12[cos(ωFt)+cos(22ωFt)], i.e. a representative example how the system can evolve to different frequency components of the driving signal depending on the initial condition ωd(0)=ω(0)−ωF. (f) I(t) is the non-periodic output of the Rössler system. The Rössler signal has a 1/f broad-band spectrum, yet it has a clear maximum in the frequency spectrum. In order to assess the convergence we use ωF=2πfmax, where fmax is found numerically by FFT. The oscillator convergences to this frequency. ... (N=10000, K=0.1) — (a) The FFT (black line) of the Rössler signal (for t=[99800,100000]) in comparison with the distribution of the frequencies of the oscillators (grey bars, normalized to the number of oscillators) at time 105 s. The spectrum of the FFT has been discretized into the same bins as the statistics of the oscillators in order to allow for a good comparison with the results from the full-scale simulation. (b) Time-series of the output signal O(t) (bold line) vs the teaching signal T(t) (dashed line). ... Adaptive frequency oscillator... The structure of the dynamical system that is capable to reproduce a given teaching signal T(t). The system is made up of a pool of adaptive frequency oscillators. The mean field produced by the oscillators is fed back negatively on the oscillators. Due to the feedback structure and the adaptive frequency property of the oscillators it reconstructs the frequency spectrum of T(t) by the distribution of the intrinsic frequencies. ... Coupled oscillators... Frequency analysis... (a) (N=1000, K=200) — T(t) is a non-stationary input signal (cf. text), in contrast to Figs. 4 and 5 the histogram of the distribution of the frequency ωi is shown for every 5 s, the grey level corresponds to the number of oscillators in the bins (note the logarithmic scale). The thin white line indicates the theoretical instantaneous frequency. Thus, it can be seen that the distribution tracks very well the non-stationary spectrum, however about 4% of the oscillators diverge after the cross-over of the frequencies. (b) This plots outlines the maximum tracking performance of the system for non-stationary signal. The input signal has a sinusoidal varying frequency. The frequency response of the adaptation is plotted (see text for details). As comparison we plot the first-order transfer function HK∞ and the vertical line indicates ωs=1. (c) The grey area shows the region where the frequency response of the adaptation is H>22. While for slower non-stationary signals the upper bound is a function of K, the bound becomes independent of K for ωs>1 (red dashed line).
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  • Nonlinear oscillator... Visibility dynamics in different timescales for initial state |Ψ2〉 and Γ=0. The timescale τp is associated to the decay of the envelope of the oscillations with characteristic time τr1. A very subtle increase in the amplitudes of the oscillations can be observed around t=τr2. The timescale of the fastest oscillations of the dynamics is τo. ... Visibility dynamics for initial state |Ψ2〉 and Γ varying from 0 to 0.1. For each value of Γ, the unit of time is chosen as the corresponding τp in (a), τr1 in (b), τr2 in (c), and τo in (d). For the majority of values of Γ investigated, the initial dynamics is flattened around t=2τp. Except for very small values of Γ, τr1 and τr2 are associated to partial revivals. When Γ increases, the number of fast initial oscillations decreases, but their characteristic durations are given by τo, which does not vary with Γ. ... Predictability dynamics in different timescales for initial state |Ψ2〉. The timescale τp is associated to the decay of the envelope of the oscillations with characteristic time τo. Revivals can be observed around the first multiples of τr. ... Predictability dynamics in different timescales for initial state |Ψ1〉. The timescale τp is associated to the rise and decay of the oscillations with characteristic time τo. Revivals occur in the region around τr and its first multiples. ... Visibility dynamics in different timescales for initial state |Ψ2〉 and Γ=0.1. The timescale τp is associated to the rise and decay of the initial dynamics. Both τr1 and τr2 are related to partial revivals. There are no oscillations besides the revivals and the initial rise and decay; the timescale of their duration is given by τo.
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  • Behavior of periodically oscillating flame with large-scale oscillation for Vac=5kV, fac=20Hz, and U0=11m/s. ... Oscillation frequency in terms of AC frequency for U0=11.0m/s and Vac=5kV. ... Low frequency... Average amplitude of large-scale oscillation with AC frequency for U0=11.0m/s and Vac=5kV. ... Oscillation... Phase diagrams of HL and dHL/dt for various oscillation modes. ... Edge height of lifted flame together with oscillation amplitude with AC frequency for Vac=5kV and U0=11.0m/s.
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  • Equivalent circuit of the linearized qubit–SQUID system. ϕm and ϕp are the two independent variables of a DC SQUID. ϕm correpsonds to the circulating current of the SQUID, and ϕp couples with the ramping current of the SQUID. The capacitances of the inner oscillator loop and the external oscillator loop are Cm=2CJ and Cp, the shunt capacitance outside the SQUID. Flux of the three loops, q=q0σz, ϕm, and ϕp, are chosen as independent variables in the calculation. Each of the inductances in the three loops interacts by mutual inductances as are indicated by the paired dots near the inductances. ... The measuring circuit of the DC SQUID which surrounds the qubit. CJ and I0 are the capacitance and critical current of each of the junctions, and ϕi are the gauge-invariant phases of the junctions. The qubit is represented symbolically by a loop with an arrow indicating the magnetic moment of the |0〉 state. The SQUID is shunted by a capacitor Csh and the environmental impedance Z0(ω).
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  • Qubit... (Color online.) Contour plots of the normalized decay rate γ(τ)/γ0 of the qubit only in the cavity bath, versus the time interval τ between successive measurements, and the central frequency ωcav of the cavity mode. (a) The width of the cavity frequency is λ=10−4Δ, and accordingly the cavity quality factor Q=104. (b) The width of the cavity frequency λ=5×10−3Δ, corresponding to the cavity quality factor Q=2×103. The region 1⩽γ(τ)/γ0⩽1.05 is shown as light magenta. The QZE region corresponds to γ(τ)/γ01. Evidently, a transition from the QZE to the AZE is observed by varying the central frequency of the cavity mode at finite τ (τ>0.6Δ−1 when Q=104, and τ>2.6Δ−1 when Q=2×103). ... (Color online.) Time dependence of the probability for the qubit at its excited state. In the resonant case, the parameters are ωcav=Δ=100g and τ=0.1g−1. In the detuning case, the cavity mode frequency is varied to ωcav=80g. Note that the successive measurements slow down the decay rate of excited state in the resonant case, which is the QZE. While in the detuning case, the measurements speed up the qubit decay rate, which is the AZE. ... The normalized effective decay rate γ(τ)/γ0 of the qubit for two quality factors Q when τ=5Δ−1, in the presence of both the cavity bath and the low-frequency qubitʼs intrinsic bath. ... (Color online.) (a) Sketch of a qubit with the spontaneous dissipation rate γ coupled to a cavity with the loss rate κ via a coupling strength g. (b) and (c) schematically show the bath density spectrum of the qubit environment: (b) the Ohmic qubitʼs intrinsic bath (green dashed) and the Lorentzian cavity bath (red solid), (c) the low-frequency qubitʼs intrinsic bath (green dashed) and the Lorentzian cavity bath (red solid). ... (Color online.) (a) Superconducting circuit model of a frequency-tunable transmission line resonator, which is archived by changing the boundary condition, coupled with a qubit. (b) Superconducting circuit model (1) of the effective tunable inductors, which are consisted of a series array of SQUIDs (2).
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  • (a) Spin maser oscillation signal observed in a time span of 24 hours. (b) Transient pattern in the initial spin maser oscillation. (c) Steady state oscillation after the transient settled. The signals shown in the ordinates represent the beat between the spin detection signal and a 36.12 Hz fixed frequency reference signal for a lock-in amplifier. ... (a) Fourier spectrum obtained when the newly introduced current source was used for the oscillator operation. The measurement time was 3×104 seconds. (b) Fourier spectrum obtained when the previous current source was used for the oscillator operation. The measurement time was 1×104 seconds. ... Nuclear spin oscillator... Steady state oscillations observed at three low frequencies: (a) ν0=17.7 Hz, (b) ν0=8.9 Hz, (c) ν0=2.5 Hz. The strength of the static field adopted in the individual operations are indicated in the respective panels. ... Frequency precision... Frequency precision of the spin oscillation. The abscissa represents the standard deviation of the frequency ν determined by fitting a function ϕ(t)=2πνt+ϕ0 to the observed precession phases ϕ from t=0 to t=Tm. Solid, dotted and dashed lines are the presentation of three cases with power laws σν∝Tm−3/2, σν∝Tm−1, and σν∝Tm−1/2 respectively.
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  • The period of oscillation T0 in a QR as a function of the transverse and longitudinal effective confinement lengths of the QR lp and lv. ... The period of oscillation T0 in a QR as a function of the electron–phonon coupling strength α and the Coulomb bound potential β. ... Qubit... The period of oscillation T0 in a QR as a function of the ellipsoid aspect ratio e′ and the electron–phonon coupling strength α.
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  • Schematic diagram of a cryogenic crystal oscillator based on a SiGe HBT. ... Oscillator model for the phase noise analysis. ... Oscillator mounting inside the cryocooler. ... Time and Frequency Department, FEMTO-ST Institute, Besançon, France... Schematic diagram of a cryogenic crystal oscillator based on a MOSFET transistor. ... Low noise oscillators... PSD of two HBT-based liquid helium oscillators, for different bias voltages.
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  • Microwave spectroscopy of a one-electron double dot. (a) Charge occupancy of the left dot, M, as a function of ε for several microwave frequencies. (b) One-half of the resonance peak splitting as a function of f for several values of VT. Solid lines are best fits to the experimental data using the theory outlined in the text. Inset: Two-level system energy level diagram. (c) Amplitude of the resonance, expressed as Mmax(τ)/Mmax(τ=5ns), as a function of chopped cw period, τ, with f=19GHz. Theory gives a best fit T1=16ns (solid line, see text). Inset: Single photon peak shown in a plot of M as a function of ε for τ=5ns and 1μs. (d) Power dependence of the resonance for f=24GHz. Widths are used to extract the ensemble-averaged charge dephasing time T2*. At higher microwave powers multiple photon processes occur. Curves are offset by 0.3 for clarity. ... Rabi oscillation... Spin qubit... Charge qubit
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  • Cross-frequency coupling... Steps applied to compute both CFC and CFD. (A) High frequency power at frequency v is estimated from the original signal by applying a sliding Hanning tapered time window followed by a Fourier transformation (red line). After that, both the original signal and the power envelope of the high frequency signal are divided into segments. Within each segment, the original signal and the power envelope of the high frequency signal are Fourier-transformed and cross-spectra between them are computed. (B) CFC and CFD quantification. CFC is quantified by coherence and CFD is calculated from the PSI between the phase of slow oscillation fi and power of fast oscillation vj. The red segment indicates the frequency range over which the PSI is calculated. The PSI is calculated for the bandwidth β. ... Statistical assessment of the CFC and CFD when controlling for multiple comparisons over frequencies. (A) Observed CFC/D and clustering threshold. All observed CFC/D values were pooled together (i.e. all frequency by frequency bins) and the threshold is set at the 99.5th percentile of the resulting distribution (right panel). Contiguous CFC/D values exceeding the threshold formed a cluster (left panel). The summed CFC/D values from a given cluster were considered the cluster score. (B) Circular shifted CFC/D and the cluster reference distribution. Random number of the Fourier-transferred phase segment sequences was circular shifted with respect to the amplitude envelope segments and the CFC/D values were recomputed 1000 times. For each randomization, the CFC/D contiguous values exceeding the threshold were used to form reference clusters (e.g., cluster1, cluster2, and so on in the left panel) and the respective cluster scores were calculated. The resulting 1000 maximum cluster scores formed the cluster-level reference distribution. For the observed cluster score, the p value was determined by considering the fraction of cluster scores in the reference distribution exceeding the observed cluster score (right panel). ... Neuronal oscillations... Phase spectra between low frequency signal and high frequency envelope. The red curves represent the envelope of high frequency signals. Fig. 1 is adapted from Schoffelen et al. (2005). Left panel: The low frequency signal is leading the high frequency envelope by 10ms. This constant lead translates into a phase-lead that linearly increases with frequency (e.g., 0.25rad for 4Hz, 0.50rad for 8Hz and 0.75rad for 12Hz). Right panel: The low frequency signal is lagging the high frequency envelope by 10ms. This constant lag translates into a phase-lag that linearly decreases with frequency (e.g., −0.25rad for 4Hz, −0.50rad for 8Hz and −0.75rad for 12Hz). ... Cross-frequency directionality
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