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Top: Rabi oscillations of the switching probability p (5×104 events) measured just after a resonant microwave pulse of duration τ. Solid line is a fit used to determine the Rabi frequency. Bottom: test of the linear dependence of the Rabi frequency with Uμw. ... (A) Calculated transition frequency ν01 as a function of φ and Ng. (B) Measured center transition frequency (symbols) as a function of reduced gate charge Ng for reduced flux φ=0 (right panel) and as a function of φ for Ng=0.5 (left panel), at 15mK. Spectroscopy is performed by measuring the switching probability p (105 events) when a continuous microwave irradiation of variable frequency is applied to the gate before readout. Continuous line: best fits used to determine circuit parameters. Inset: Narrowest line shape, obtained at the saddle point (Lorentzian fit with a FWHM Δν01=0.8MHz).
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The same as in Fig. 4 for U=2 and for the time-dependent energy levels ε1 and ε2 presented in the inset in the left panel—they oscillate harmonically with frequency ω=1 and the pulse envelope has a Gaussian shape of duration τ=30 centered at t0=100. ... The same as in Fig. 3 but for U=0 (upper panels) and for U=2 (lower panels) for the time-dependent energy levels ε1 and ε2 presented in the inset, in the upper left panel—they oscillate harmonically around the values ε=±1 with frequency ω=0.1, and the pulse envelope has a Gaussian shape of duration τ=30 centered at t0=92. The energy levels of the right qubit have constant values ε3=ε4=1. ... Coupled qubits... Occupancy probability n1(t=∞) of the first QD of the left qubit (qubits are in the perpendicular configuration) as a function of the frequency ω of the time-dependent V1(t) displayed in the inset—it oscillates harmonically with ω=0.5 and the pulse envelope has a Gaussian shape of duration τ=30, V2=1, U1=U2=2, εi=0, n1(0)=n3(0)=1. ... Occupation probability n1(t) of the first QD in the left qubit (the left panel) and n4(t) of the second QD in the right qubit (the right, panel) as the functions of time for U=10. The energy levels ε1 and ε2 of the left qubit oscillate harmonically around the values ε=±2 with amplitude Δ=2, frequency ω=0.05 (in V/ℏ units, see the inset in the left panel) and energy levels of the right qubit having constant values, ε3=−ε4=2. The qubits are in the linear configuration. ... Schematic representation of two interacting qubits formed by two DQDs with one excess electron in each qubit. The broken lines correspond to the Coulomb interaction U between the electrons localized on the neighboring QDs of both qubits and V denotes the interdot tunneling matrix element. ... Charge oscillations
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The schematic figure for the projected quantum levels in IJJ composed of two junctions, the switching dynamics, and the transition between two quantum states caused by the irradiation of the microwave whose frequency is Ω2. ... Rabi-oscillation... The schematic figure for the quantum levels for IJJ, which are projected onto the potential barrier of the single Josephson junction without the coupling. The energy levels of the out-of-phase and the in-phase oscillations have the highest and the lowest eigen-energies, respectively.
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Two-electron state amplitude in a dimer, with both molecules subject to a periodic force. After a full revolution the two electronic states each change their sign, leaving the total state invariant. Frequencies: ω1=1, ω2=1, G1=−100, G2=−200, G=40 (near adiabatic limit). Thick line: first, initially excited component. Medium thick line: second and third components. Thin line: fourth component. ... Two-electron state amplitudes in a dimer. The thick line shows the time dependent amplitude of the first (initially excited component), the thin line that of the second component in Eq. (5). Frequencies: ω1=1, ω2=4, G1=−40, G2=−80, G=16 (near adiabatic limit) ... Non-adiabaticity effects in the real part of the initially excited component, as a function of time. The frequencies on the two dimers are ω1=1 and ω2=2. The values of the coupling parameters are as follows. Thick line: G1=−80, G2=−160, G=40 (near adiabatic limit). Thin line: G1=−8, G2=−16, G=4 (non-adiabatic case)
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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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Average PTO power as a function of oscillating frequency for straight (♦: solid line) and bent leg (□: broken line) tines (oscillation angle β=+27°). ... Subsoiler draft signals with time for the control and the range of oscillating frequencies. ... Dominant frequency of draft signal over the oscillating frequency range. ... Proportion of cycle time for cutting and compaction phases versus oscillating frequency (oscillation angle β=+27°). ... Dominant frequency of torque signal over the oscillating frequency range. ... Frequency... Oscillating tine
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Current-composed quantity Q(t) (solid lines) and the far-removed qubit QD occupancy, n3 (dashed lines), as a function of time for the horizontal qubit-detector connection, U13=U24=0, 2 or 4, respectively. μL=−μR=20, ΓL=5, ΓR=10, U12=U34=5 and the other parameters are the same as in Fig. 2. The lines for U13=U24=2 (4) are shifted by −1 (−2) for better visualisation. ... The sketch of the qubit-detector systems discussed in the text. Double quantum dot (1 and 4) between the left and right electron reservoirs stands for the qubit charge detector. Qubit is represented by two coupled quantum dots (2 and 3) occupied by a single electron. Straight black (zig-zag red) lines correspond to the tunnel matrix elements V14, V23 (Coulomb interactions, e.g. U14, U24) between the appropriate states. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.) ... The asymptotic pulse-induced current I(τ) against the time interval (pulse length) τ – for details see the text – and the charge occupation of the far-removed qubit QD, n3, (dashed lines) for the qubit-detector system schematically shown in Fig. 1b. The upper (bottom) panel corresponds to ΓL=5, ΓR=10 (ΓL=5, ΓR=1). μL=−μR=20 and the other parameters are the same as in Fig. 2. The current lines are multiplied by −2 for better visualisation. ... Current-composed quantity Q(t) (solid lines) and the charge occupation of the far-removed qubit QD, n3, (dashed lines) as a function of time for the qubit-detector system schematically shown in Fig. 1b. The upper (bottom) panel corresponds to (ΓL,ΓR)=(5,1) ((ΓL,ΓR)=(5,10)). The other parameters are: μL=−μR=2 or μL=−μR=20, ε1,2,3,4=0, U24=5, U14=50, n2(t<10)=0, n3(t<10)=1. The lines for μL=−μR=20 are shifted by −1 for better visualisation. ... Upper panel: Current-composed quantity Q(t) (solid lines) and the far-removed qubit QD occupancy, n3, (dashed lines) as a function of time for different qubit-detector connections shown in Fig. 1d (U12=5), Fig. 1c (U12=U24=5) and Fig. 1b (U24=5)—the upper, middle and lower curves, respectively. The bottom panel depicts the corresponding left (solid lines) and right (dashed lines) currents, IL(t), IR(t), flowing in the system for the above three qubit-wire connections. μL=−μR=2, ΓL=5, ΓR=10 and the other parameters are the same as in Fig. 2. The lines in the upper panel for U12=U24=5 and for U24=5 are shifted by −1 and −2, respectively, and by −0.15 and −0.3 in the bottom panel. Note different scales in the vertical axis of both panels.
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Qubits in solids... Schematic diagram of qubits addressed in a frequency domain. The ions whose 3H4(1)± 3 2–1D2(1) transitions are resonant with a common cavity mode are employed as qubits. ... Basic scheme of the concept of the frequency-domain quantum computer. The atoms are coupled to a single cavity mode. Lasers with frequencies of νk and νl are directed onto the set of atoms and interact with the kth and lth atoms selectively.
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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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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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