### 11731 results for qubit oscillator frequency

Contributors: M. Hebbache

Date: 2014-01-01

Quantum computing requires a set of universal quantum gates. The standard set includes single quantum bit (**qubit**) gates and the controlled-NOT (CNOT) which is the analog of the classical XOR gate. It flips the state of the target **qubit** conditioned on the state of the control **qubit**. We investigated the possibility of implementing a CNOT logic gate using magnetically coupled impurity spins of diamond, namely the electron spin-1 carried by the nitrogen-vacancy color center and the electron spin-12 carried by a nearby nitrogen atom in substitutional position (P1 center). It is shown that a 96ns gate time with a high-fidelity can be realized by means of pulsed electron spin resonance spectroscopy....(Color online) |A|2 is the probability of finding the spin system in the state |⇓↓〉. It **oscillates** at the high **frequency** D (=2.88GHz). The **frequency** of the beats is χ/2 (=16.7MHz). The amplitude of **oscillations** is also modulated by an additional cosine wave signal of **frequency** χ (see text). |C|2 is the probability of finding the spin system in the state |0↓〉. It **oscillates** at the low **frequency** χ. It is almost zero in the time interval 90–100ns. The probability of finding spin system in the state |⇑↓〉, |B|2, has the same **oscillations** than |A|2 but it is anti-phase (see Fig. 3).
...Ideal truth table and schematic representation of a two-**qubit** CNOT gate irradiated by a sequence of two microwave π/2-pulses of equal width t and a variable waiting time between pulses τ. In the text, x and y are the states of two impurity spins of diamond, namely the spin-12 carried by the P1 center and the spin-1 carried by the NV−1 color center. The symbol ⊕ is the addition modulo 2, or equivalently the XOR operation.
...(Color online) NV−1 Rabi **oscillations**. Control **qubit** down: blue, red and green lines correspond, respectively, to the time evolution of |A|2, |B|2 and |C|2, i.e., the probabilities of finding the spin system in the state |⇓↓〉, |⇑↓〉 and |0↓〉. Control **qubit** up: red, blue and green lines represent, respectively, |A′|2, |B′|2 and |C′|2, i.e., the probabilities of finding the spin system in the state |⇓↑〉, |⇑↑〉 and |0↑〉, i.e., |A′|2=|B|2, |B′|2=|A|2 and |C′|2=|C|2 (see text). Fig. 4 gives details in the interval 60–120ns. They can also be revealed by a zoom in.
... Quantum computing requires a set of universal quantum gates. The standard set includes single quantum bit (**qubit**) gates and the controlled-NOT (CNOT) which is the analog of the classical XOR gate. It flips the state of the target **qubit** conditioned on the state of the control **qubit**. We investigated the possibility of implementing a CNOT logic gate using magnetically coupled impurity spins of diamond, namely the electron spin-1 carried by the nitrogen-vacancy color center and the electron spin-12 carried by a nearby nitrogen atom in substitutional position (P1 center). It is shown that a 96ns gate time with a high-fidelity can be realized by means of pulsed electron spin resonance spectroscopy.

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Contributors: Kouichi Ichimura

Date: 2001-09-01

A quantum computer where quantum bits (**qubits**) are defined in **frequency** domain and interaction between **qubits** is mediated by a single cavity mode is proposed. In this quantum computer, **qubits** can be individually addressed regardless of their positions. Therefore, randomly distributed systems in space can be directly employed as **qubits**. An application of nuclear spins in rare-earth ions in a crystal for the quantum computer is quantitatively analyzed....**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.
... A quantum computer where quantum bits (**qubits**) are defined in **frequency** domain and interaction between **qubits** is mediated by a single cavity mode is proposed. In this quantum computer, **qubits** can be individually addressed regardless of their positions. Therefore, randomly distributed systems in space can be directly employed as **qubits**. An application of nuclear spins in rare-earth ions in a crystal for the quantum computer is quantitatively analyzed.

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Contributors: Gholamhossein Shahgoli, John Fielke, Jacky Desbiolles, Chris Saunders

Date: 2010-01-01

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**...Based on the published benefits of oscillatory tillage, a subsoiler was developed at the University of South Australia, which had two deep working oscillatory tines and could be fitted with four shallow leading tines for increased loosening efficiency. A series of field trials were conducted in a sandy-loam soil to determine the most efficient setting of the tine's oscillatory motion and to compare the effect of using straight or bentleg tines. The tines were **oscillated** with an amplitude at the tip of ±69mm and an **oscillation** angle of 27° using a forward speed of 3km/h. The **frequency** of **oscillation** was varied from 1.9 to 8.8Hz. Analysis showed that the underside of the **oscillating** tine pushed rearward on the soil during part of the **oscillation** cycle, this decreased the draft in comparison to rigid tillage from 25.8 to 9.3kN. Increasing **oscillation** **frequency**, increased the PTO power requirement from 2.5kW at 1.9Hz to 26.3kW at 8.8Hz. The peaks and troughs in draft and torque were able to be aligned with the various phases of the **oscillating** tillage. An optimum **oscillation** **frequency** of 3.3Hz (velocity ratio of 1.5) was observed for minimum power to operate the **oscillating** subsoiler. Whilst at this setting, the combined draft and PTO power was similar to the draft power of rigid tillage, but when considering the higher losses due to tractive efficiency and lower PTO power losses, the **oscillating** tillage would be expected to require around 27% less engine power than rigid tillage....**Oscillating** tine ... Based on the published benefits of oscillatory tillage, a subsoiler was developed at the University of South Australia, which had two deep working oscillatory tines and could be fitted with four shallow leading tines for increased loosening efficiency. A series of field trials were conducted in a sandy-loam soil to determine the most efficient setting of the tine's oscillatory motion and to compare the effect of using straight or bentleg tines. The tines were **oscillated** with an amplitude at the tip of ±69mm and an **oscillation** angle of 27° using a forward speed of 3km/h. The **frequency** of **oscillation** was varied from 1.9 to 8.8Hz. Analysis showed that the underside of the **oscillating** tine pushed rearward on the soil during part of the **oscillation** cycle, this decreased the draft in comparison to rigid tillage from 25.8 to 9.3kN. Increasing **oscillation** **frequency**, increased the PTO power requirement from 2.5kW at 1.9Hz to 26.3kW at 8.8Hz. The peaks and troughs in draft and torque were able to be aligned with the various phases of the **oscillating** tillage. An optimum **oscillation** **frequency** of 3.3Hz (velocity ratio of 1.5) was observed for minimum power to operate the **oscillating** subsoiler. Whilst at this setting, the combined draft and PTO power was similar to the draft power of rigid tillage, but when considering the higher losses due to tractive efficiency and lower PTO power losses, the **oscillating** tillage would be expected to require around 27% less engine power than rigid tillage.

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Contributors: Dong-Qi Liu, Gang-Qin Liu, Yan-Chun Chang, Xin-Yu Pan

Date: 2014-01-01

Detection and manipulation of the **qubit**. (a) Fluorescence image of nanodiamond prepared on the CPW transmission line. NV S1 is circled. The inset is a photo of CPW with 20μm gaps fabricated on a silica glass. (b) CW ODMR spectrum for NV S1. The inset is energy levels of NV center. A 532nm laser is used to excite and initialize the NV center. Fluorescence is collected by a confocal microscope. (c) Rabi **oscillation** of NV S1. Rabi **oscillation** period is about 62ns. (d) Hahn echo and CPMG control pulse sequences. πx (πy) implies the direction of microwave magnetic fields parallel to x (y).
...Spectral density of the spin bath. (a) NV S1, (b) NV S2. All values of spectral density S(ω) of the spin bath are extracted from the CPMG data (blue points). Each blue data point represents a specific probed **frequency** ω=πn/t, in which n is the number of control pulses and t is the specific duration. The red points are the average values at a certain **frequency**. The mean spectral density is fit to the Lorentzian function (Eq. (3)) (green line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
...Overcoming the spin **qubit** decoherence is a challenge for quantum science and technology. We investigate the decoherence process in nanodiamonds by Carr–Purcell–Meiboom–Gill (CPMG) technique at room temperature. We find that the coherence time T2 scales as nγ. The elongation effect of coherence time can be represented by a constant power of the number of pulses n. Considering the filter function of CPMG decoupling sequence as a δfunction, the spectrum density of noise has been reconstructed directly from the coherence time measurements and a Lorentzian noise power spectrum model agrees well with the experiment. These results are helpful for the application of nanodiamonds to nanoscale magnetic imaging....Characterization of lifetime of NV center spins. (a) Ramsey interference of NV S1 (circle) and NV S2 (diamond). The **oscillation** in Ramsey signal originates from the beating among different transitions corresponding to the host three 14N nuclear spin states. The **oscillation** **frequency** of Ramsey signal is equal to microwave detuning from spin resonance. Solid lines ~exp[−(t/T2⁎)m] fit the experimental data points, where m is a free parameter. (b) Comparison of Hahn echo coherence time T2 of NV S1 (circle) and NV S2 (diamond). The solid lines are fits to ~exp[−(t/T2)p], in which p is a fit parameter.
... Overcoming the spin **qubit** decoherence is a challenge for quantum science and technology. We investigate the decoherence process in nanodiamonds by Carr–Purcell–Meiboom–Gill (CPMG) technique at room temperature. We find that the coherence time T2 scales as nγ. The elongation effect of coherence time can be represented by a constant power of the number of pulses n. Considering the filter function of CPMG decoupling sequence as a δfunction, the spectrum density of noise has been reconstructed directly from the coherence time measurements and a Lorentzian noise power spectrum model agrees well with the experiment. These results are helpful for the application of nanodiamonds to nanoscale magnetic imaging.

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Contributors: S Lee, R Blowes, A.D Milner

Date: 1998-09-01

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**...In an **oscillating** system driven by a sine wave pump, the resonance **frequency** of the respiratory system can be determined using phase analysis. At resonance **frequency**, when elastance and inertance cancel out, flow becomes in-phase with resistance. In premature infants with respiratory distress syndrome, owing to surfactant deficiency, localized areas of hyperinflation and collapse develop, resulting in complex changes in overall pulmonary mechanics. We investigated the effect of measuring resonance **frequency** of the respiratory system by phase analysis at different points of the respiratory cycle: end of inspiration, end of expiration, mid-inspiration and mid-expiration. Ten ventilated premature infants with respiratory distress syndrome were studied, gestational age ranged from 24 to 30 weeks (mean 27.6 weeks) and birth weight ranged from 0.7 to 1.505 kg (mean 0.984 kg). Results: The resonance **frequency** was consistently higher when measured at the end of inspiration compared with the end of expiration. The expected trend of phase variation, that is, negative below the resonance **frequency** and positive above, was most consistently found when analysis was done at the end of inspiration. Conclusions: These findings were most likely a result of the complexity of pulmonary mechanics in the surfactant-deficient lungs, rendering the single compartment model we based our theory on inadequate. However, phase analysis performed at the end of inspiration seemed to produce the most reliable and consistent results. ... In an **oscillating** system driven by a sine wave pump, the resonance **frequency** of the respiratory system can be determined using phase analysis. At resonance **frequency**, when elastance and inertance cancel out, flow becomes in-phase with resistance. In premature infants with respiratory distress syndrome, owing to surfactant deficiency, localized areas of hyperinflation and collapse develop, resulting in complex changes in overall pulmonary mechanics. We investigated the effect of measuring resonance **frequency** of the respiratory system by phase analysis at different points of the respiratory cycle: end of inspiration, end of expiration, mid-inspiration and mid-expiration. Ten ventilated premature infants with respiratory distress syndrome were studied, gestational age ranged from 24 to 30 weeks (mean 27.6 weeks) and birth weight ranged from 0.7 to 1.505 kg (mean 0.984 kg). Results: The resonance **frequency** was consistently higher when measured at the end of inspiration compared with the end of expiration. The expected trend of phase variation, that is, negative below the resonance **frequency** and positive above, was most consistently found when analysis was done at the end of inspiration. Conclusions: These findings were most likely a result of the complexity of pulmonary mechanics in the surfactant-deficient lungs, rendering the single compartment model we based our theory on inadequate. However, phase analysis performed at the end of inspiration seemed to produce the most reliable and consistent results.

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Contributors: Ludovic Righetti, Jonas Buchli, Auke Jan Ijspeert

Date: 2006-04-15

Nonlinear **oscillators** are widely used in biology, physics and engineering for modeling and control. They are interesting because of their synchronization properties when coupled to other dynamical systems. In this paper, we propose a learning rule for **oscillators** which adapts their **frequency** to the **frequency** of any periodic or pseudo-periodic input signal. Learning is done in a dynamic way: it is part of the dynamical system and not an offline process. An interesting property of our model is that it is easily generalizable to a large class of **oscillators**, from phase **oscillators** to relaxation **oscillators** and strange attractors with a generic learning rule. One major feature of our learning rule is that the **oscillators** constructed can adapt their **frequency** without any signal processing or the need to specify a time window or similar free parameters. All the processing is embedded in the dynamics of the adaptive **oscillator**. The convergence of the learning is proved for the Hopf **oscillator**, then numerical experiments are carried out to explore the learning capabilities of the system. Finally, we generalize the learning rule to non-harmonic **oscillators** like relaxation **oscillators** and strange attractors....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.
... Nonlinear **oscillators** are widely used in biology, physics and engineering for modeling and control. They are interesting because of their synchronization properties when coupled to other dynamical systems. In this paper, we propose a learning rule for **oscillators** which adapts their **frequency** to the **frequency** of any periodic or pseudo-periodic input signal. Learning is done in a dynamic way: it is part of the dynamical system and not an offline process. An interesting property of our model is that it is easily generalizable to a large class of **oscillators**, from phase **oscillators** to relaxation **oscillators** and strange attractors with a generic learning rule. One major feature of our learning rule is that the **oscillators** constructed can adapt their **frequency** without any signal processing or the need to specify a time window or similar free parameters. All the processing is embedded in the dynamics of the adaptive **oscillator**. The convergence of the learning is proved for the Hopf **oscillator**, then numerical experiments are carried out to explore the learning capabilities of the system. Finally, we generalize the learning rule to non-harmonic **oscillators** like relaxation **oscillators** and strange attractors.

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Contributors: Mina Amiri, Jean-Marc Lina, Francesca Pizzo, Jean Gotman

Date: 2016-01-01

Examples of a spike without HFOs (left) and a spike with HFOs (right), as defined with the Analytic Morse wavelet in the time–**frequency** domain.
...High **Frequency** **Oscillations**...Parameter selection for the Analytic Morse Wavelet; top: time–**frequency** presentation for different values of n (m=40), bottom: raw signal and filtered signal (80–250Hz). Blue lines represent HFO interval marked visually.
...Examples of detection errors. Left: HFO without isolated blob but having **oscillation** in the raw signal. Right: HFO without visible **oscillation** in the raw signal but representing an isolated peak. Blue lines show the HFO interval marked by reviewers.
...Time–**frequency**...To demonstrate and quantify the occurrence of false High **Frequency** **Oscillations** (HFOs) generated by the filtering of sharp events. To distinguish real HFOs from spurious ones using analysis of the raw signal. ... To demonstrate and quantify the occurrence of false High **Frequency** **Oscillations** (HFOs) generated by the filtering of sharp events. To distinguish real HFOs from spurious ones using analysis of the raw signal.

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Contributors: Hongjing Ma, Weiqing Liu, Ye Wu, Meng Zhan, Jinghua Xiao

Date: 2014-08-01

Spatial **frequencies** distributions...Ragged **oscillation** death...The phase synchronization domains (areas enclosed by the red lines) and the OD regions (black areas) in the parameter space of ε-δω for a ring of coupled Rossler systems with different **frequency** distributions: (a) G={1,2,3,4,5,6,7,8}, (b) G={1,4,3,6,2,8,5,7}, and (c) G={1,2,3,6,8,4,7,5}. N=8. The ragged OD sates are clear in (b) and (c) within a certain interval of δω indicated by two vertical dashed lines. In all three insets, the values of ωj are plotted for given ω0=0 and δω=N. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
...The bifurcation diagram and the largest Lyapunov exponent λ of the coupled Rossler **oscillators** versus the coupling strength ε with the same spatial arrangement of natural **frequencies** as in Fig. 1(a)–(c), respectively for δω=0.58. The bifurcation diagram is realized by the soft of XPPAUT [33] where the black dots are fixed points and the red dots are the maximum and minimum values of x1 for the stable periodic solution while the blue dots means the max/min values of x1 for the unstable periodical states. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
...The critical curves of OD domain from analysis in N coupled Landau–Stuart **oscillators** for different N’s: (a) N=2, (b) N=3, and (c)–(e) N=4 for G={1,2,3,4},G={1,2,4,3}, and G={1,3,2,4}, respectively. The ragged OD domain is clear in (d). The numerical results with points within the domains perfectly verify the analytical results.
...The OD regions in the parameter space of ε-δω for a ring of coupled Rossler systems with different **frequency** distributions: (a) G={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}, (b) G={26,16,25,18,5,14,10,4,6,7,21,12,23,8,1,15,9,29,28,11,2,20,27,30,3,13,17,22,24,19}, and (c) G={19,22,18,13,10,28,7,15,17,8,30,12,26,11,20,9,27,21,25,6,29,1,23,5,3,24,16,14,4,2}. N=30. In all three insets, the values of ωj are plotted for given ω0=0 and δω=N.
...Coupled nonidentical **oscillators**...In this paper, the effect of spatial **frequencies** distributions on the **oscillation** death in a ring of coupled nonidentical **oscillators** is studied. We find that the rearrangement of the spatial **frequencies** may deform the domain of **oscillation** death and give rise to a ragged **oscillation** death in some parameter spaces. The usual critical curves with shape V in the parameter space of **frequency**-mismatch vs coupling-strength may become the shape W (or even shape WV). This phenomenon has been not only numerically observed in coupled nonidentical nonlinear systems, but also well supported by our theoretical analysis. ... In this paper, the effect of spatial **frequencies** distributions on the **oscillation** death in a ring of coupled nonidentical **oscillators** is studied. We find that the rearrangement of the spatial **frequencies** may deform the domain of **oscillation** death and give rise to a ragged **oscillation** death in some parameter spaces. The usual critical curves with shape V in the parameter space of **frequency**-mismatch vs coupling-strength may become the shape W (or even shape WV). This phenomenon has been not only numerically observed in coupled nonidentical nonlinear systems, but also well supported by our theoretical analysis.

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Contributors: Fang Yuan, Daotong Chong, Quanbin Zhao, Weixiong Chen, Junjie Yan

Date: 2016-07-01

The condensation **oscillation** of submerged steam was investigated theoretically and experimentally at the condensation **oscillation** regime. It was found that pressure **oscillation** **frequency** was consistent with the bubble **oscillating** **frequency** and there was a quasi-steady stage when bubble diameters remained constant. A thermal-hydraulic model for the condensation **oscillation** regime was proposed based on potential flow theory, taking into account the effects of interface condensation and translatory flow. Theoretical derivations indicated that **oscillation** **frequencies** were mainly determined by bubble diameters and translatory velocity. A force balance model was applied to the calculation of bubble diameters at quasi-steady stage, and the **oscillation** **frequencies** were predicted with the calculated diameters. Theoretical analysis and experimental results turned out that **oscillation** **frequencies** at the condensation **oscillation** regime decreased with the increasing steam mass flux and pool temperature. The predicted **frequencies** corresponded to the experimental data well with the discrepancies of ±21.7%....Dominant **frequencies** of 10mm nozzle.
...Condensation regime map by Cho et al. [1] (C–chugging, TC—transitional region from chugging to CO, CO—condensation **oscillation**, SC—stable condensation, BCO—bubble condensation **oscillation**, IOC—interfacial **oscillation** condensation).
...Condensation **oscillation**...**Frequencies** at different test conditions—250kgm−2s−1.
...**Frequency**...Prediction accuracy of simultaneous equations for **oscillation** **frequency**.
...**Frequencies** at different test conditions—300kgm−2s−1.
... The condensation **oscillation** of submerged steam was investigated theoretically and experimentally at the condensation **oscillation** regime. It was found that pressure **oscillation** **frequency** was consistent with the bubble **oscillating** **frequency** and there was a quasi-steady stage when bubble diameters remained constant. A thermal-hydraulic model for the condensation **oscillation** regime was proposed based on potential flow theory, taking into account the effects of interface condensation and translatory flow. Theoretical derivations indicated that **oscillation** **frequencies** were mainly determined by bubble diameters and translatory velocity. A force balance model was applied to the calculation of bubble diameters at quasi-steady stage, and the **oscillation** **frequencies** were predicted with the calculated diameters. Theoretical analysis and experimental results turned out that **oscillation** **frequencies** at the condensation **oscillation** regime decreased with the increasing steam mass flux and pool temperature. The predicted **frequencies** corresponded to the experimental data well with the discrepancies of ±21.7%.

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Contributors: Ch. Wunderlich, Ch. Balzer

Date: 2003-01-01

Illustration of a linear ion trap including an axial magnetic field gradient. The static field makes individual ions distinguishable in **frequency** space by Zeeman-shifting their internal energy levels (solid horizontal lines represent **qubit** states). In addition, it mediates the coupling between internal and external degrees of freedom when a driving field is applied (dashed horizontal lines stand for vibrational energy levels of the ion string, see text).
...Rabi **oscillations** on the optical E2 transition S1/2-D5/2 in Ba + . A fit of the data (solid line) yields a Rabi **frequency** of 71.4 × 2πkHz and a transversal relaxation time of 100 μs (determined by the coherence time of the ir light used to drive the E2 resonance).
...Illustration of the coupled system ‘**qubit** ⊗ harmonic **oscillator**’ in a trap with magnetic field gradient. Internal **qubit** transitions lead to a displacement dz of the ion from its initial equilibrium position and consequently to the excitation of vibrational motion. In the formal description the usual Lamb–Dicke parameter is replaced by a new effective one (see text).
...This chapter discusses quantum measurements and new concepts for experiments with trapped ions. Quantum mechanics is a tremendously successful theory playing a central role in natural sciences even beyond physics, and has been verified in countless experiments, some of which were carried out with very high precision. Quantum theory predicts correlations between two or more quantum systems once an entangled state of these systems has been generated. The chapter introduces experiments with 171Yb+ ions demonstrating the precise manipulation of hyperfine states of single ions essentially free of longitudinal and transverse relaxation. A new concept for ion traps is described that allows for experiments requiring individual addressing of ions and conditional dynamics with several ions even with radiation in the radio **frequency** (rf) or microwave (mw) regime. It is shown how an additional magnetic field gradient applied to an electrodynamic trap individually shifts ionic **qubit** resonances making them distinguishable in **frequency** space. Thus, individual addressing for the purpose of single **qubit** operations becomes possible using long-wavelength radiation. At the same time, a coupling term between internal and motional states arises even when rf or mw radiation is applied to drive **qubit** transitions. Thus, conditional quantum dynamics can be carried out in this modified electrodynamic trap and in such a new type of trap all schemes originally devised for optical QIP in ion traps can be applied in the rf or mw regime, too....(a) Relevant energy levels and transitions in 138Ba + . (b) Schematic drawing of major experimental elements. OPO: Optical parametric **oscillator**; YAG: Nd:YAG laser; LD: laser diode; DSP: Digital signal processing system allows for real time control of experimental parameters; AOM: Acousto-optic modulators used as optical switches and for tuning of laser light; PM: Photo multiplier tube, serves for detection of resonance fluorescence. All lasers are **frequency** and intensity stabilized (not shown).
...Schematic drawing of the resonances of **qubits** j and j + 1 with some accompanying sideband resonances. The angular **frequency** vN corresponds to the Nth axial vibrational mode, and the **frequency** separation between carrier resonances is denoted by δω.
... This chapter discusses quantum measurements and new concepts for experiments with trapped ions. Quantum mechanics is a tremendously successful theory playing a central role in natural sciences even beyond physics, and has been verified in countless experiments, some of which were carried out with very high precision. Quantum theory predicts correlations between two or more quantum systems once an entangled state of these systems has been generated. The chapter introduces experiments with 171Yb+ ions demonstrating the precise manipulation of hyperfine states of single ions essentially free of longitudinal and transverse relaxation. A new concept for ion traps is described that allows for experiments requiring individual addressing of ions and conditional dynamics with several ions even with radiation in the radio **frequency** (rf) or microwave (mw) regime. It is shown how an additional magnetic field gradient applied to an electrodynamic trap individually shifts ionic **qubit** resonances making them distinguishable in **frequency** space. Thus, individual addressing for the purpose of single **qubit** operations becomes possible using long-wavelength radiation. At the same time, a coupling term between internal and motional states arises even when rf or mw radiation is applied to drive **qubit** transitions. Thus, conditional quantum dynamics can be carried out in this modified electrodynamic trap and in such a new type of trap all schemes originally devised for optical QIP in ion traps can be applied in the rf or mw regime, too.

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