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Qubit dynamics in Bloch ball picture. North pole corresponds to the excited (antisymmetric) energy eigenstate |1〉 and south pole corresponds to the ground (symmetric) state |0〉. Initially the electron is localized in one of the dots. Quality of Rabi oscillations Q=40. The effect of image charge potential: (a) K=0 and (b) K=0.4. ... Quality of qubit Rabi oscillations vs. distance to a metal surface. Centers of quantum dots are located 100nm apart. Lines and points correspond to analytical and numerical solutions, respectively. ... Quality of qubit Rabi oscillations vs. the distance between quantum dots. Qubit is located 50nm far from the metal surface. Lines and points correspond to analytical and numerical solutions, respectively. ... The moving charge in the qubit drags charges in metal that indispensably entails Joule loss: d is a double dot separation and D is a distance to the metal surface.
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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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The relational curve of the oscillating period T and the electron–LOP coupling constant α. ... Qubit... The relational curve of the oscillating period T and the confinement length R.
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Visualization of the ground state |0〉 and the coherent pointer-states |L〉 and |R〉 of the oscillator in the potential V(x).
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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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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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The phase of the oscillator after action of a linearly-chirped pulsed force as a function of the chirp strength. In this case Ω0=4, σ=5. ... The amplitude of a harmonic oscillator after the action of a pulsed force with a Gaussian envelope and a linear chirp in dependence on the chirp strength, ΔΩ. In this case Ω0=5 and σ=5, 10, and 20; A=1 in all figures. ... The amplitude of the oscillator after the action of a force with an asymmetrical Gaussian envelope, σ1=5, Ω0=5, σ2=10 and 20. ... The amplitude of the oscillator vs. ΔΩ in the case of a periodical chirp in the force. The parameters of the force are: Ω0=5, σ=20, b=4. ... Classical oscillator... Frequency chirp
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Frequency modulation... Frequency modulated Ca2+ oscillations. (A) A computer generated (in silico) oscillating wave with the parameters: period (T), frequency (f), full duration half maximum (FDHM), and duty cycle is depicted. (B) Oscillating wave frequency modulated by agonist concentration. (C) Oscillating wave frequency modulated by the different agonists X, Y, and Z. Three single cell Ca2+ recordings of a Fluo-4/AM-loaded neuroblastoma cell (D), HeLa cell (E), and cardiac cell (F) with the parameters T, f, FDHM, and duty cycle stated. Scale bars are 100s. ... Frequency decoders and host cells. Illustration showing the frequencies and periods that modulate the different frequency decoders and host cells. ... Frequency decoding
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Summary of resonance frequencies found in all 10 babies (1 and 2 represents first and second run, respectively) ... A screen from our phase analysis program, showing phase analysis performed at four points of the respiratory cycle: top of breath, mid-inspiration, mid-expiration and bottom of breath. Corresponding points from the driving trace and the mouth pressure trace are matched and the phase difference calculated. In this case, the phase difference at the top of breath is 0° at an oscillating frequency of 20 Hz. ... high-frequency oscillation
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Adaptive frequency oscillator... The left plot of this figure represents the evolution of ω(t) when the adaptive Hopf oscillator is coupled to the z variable of the Lorenz attractor. The right plot represents the z variable of the Lorenz attractor. We clearly see that the adaptive Hopf oscillators can correctly learn the pseudo-frequency of the Lorenz attractor. See the text for more details. ... Plots of the frequency of the oscillations of the Van der Pol oscillator according to ω. Here α=50. There are two plots, for the dotted line the oscillator is not coupled and for the plain line the oscillator is coupled to F=sin30t. The strength of coupling is ϵ=2. We clearly see basins of phase-locking, the main one for frequency of oscillations 30. The other major basins appear each 30n (dotted horizontal lines). We also notice small entrainment basins for some frequencies of the form 30pq. For a more detailed discussion of these results refer to the text. ... We show the adaptation of the Van der Pol oscillator to the frequencies of various input signals: (a) a simple sinusoidal input (F=sin(40t)), (b) a sinusoidal input with uniformly distributed noise (F=sin(40t)+uniform noise in [−0.5,0.5]), (c) a square input (F=square(40t)) and (d) a sawtooth input (F=sawtooth(40t)). For each experiment, we set ϵ=0.7 and α=100 and we show three plots. The right one shows the evolution of ω(t). The upper left graph is a plot of the oscillations, x, of the system, at the beginning of the learning. The lower graph shows the oscillations at the end of learning. In both graphs, we also plotted the input signal (dashed). In each experiment, ω converges to ω≃49.4, which corresponds to oscillations with a frequency of 40 rad s−1 like the input and thus the oscillator correctly adapts its frequency to the frequency of the input. ... Frequency spectra of the Van der Pol oscillator, both plotted with ω=10. The left figure is an oscillator with α=10 and on the right the nonlinearity is higher, α=50. On the y-axis we plotted the square root of the power intensity, in order to be able to see smaller frequency components. ... This figure shows the convergence of ω for several initial frequencies. The Van der Pol oscillator is perturbed by F=sin(30t), with coupling ϵ=0.7, α=50. We clearly see that the convergence directly depends on the initial conditions and as expected the different kinds of convergence correspond to the several entrainment basins of Fig. 7.
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