### 21982 results for qubit oscillator frequency

Contributors: T.P. Orlando, Lin Tian, D.S. Crankshaw, S. Lloyd, C.H. van der Wal, J.E. Mooij, F. Wilhelm

Date: 2002-03-01

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 SQUID used to measure the flux state of a superconducting flux-based **qubit** interacts with the **qubit** and transmits its environmental noise to the **qubit**, thus causing the relaxation and dephasing of the **qubit** state. The SQUID–**qubit** system is analyzed and the effect of the transmittal of environmental noise is calculated. The method presented can also be applied to other quantum systems....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).
... The SQUID used to measure the flux state of a superconducting flux-based **qubit** interacts with the **qubit** and transmits its environmental noise to the **qubit**, thus causing the relaxation and dephasing of the **qubit** state. The SQUID–**qubit** system is analyzed and the effect of the transmittal of environmental noise is calculated. The method presented can also be applied to other quantum systems.

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Contributors: Yun-Fei Liu, Jing-Lin Xiao

Date: 2008-09-01

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.
...In this paper, we study the influence of LO phonon (LOP) on the charge **qubit** in a quantum dot (QD), and find that the eigenenergies of the ground and first excited states are reduced due to the electron–LOP interaction. At the same time, the time evolution of the electron probability density is obtained, the dependence of the **oscillating** period on electron–LOP coupling constant is found, the relation of between the **oscillating** period and the confinement length of the QD is calculated. Finally, we consider the effects of the electron–LOP coupling constant on pure dephasing factor under considering the correction of electron–LOP interaction for the wave functions. Our results suggest that electron–LOP interaction has very important effects on charge **qubit**. ... In this paper, we study the influence of LO phonon (LOP) on the charge **qubit** in a quantum dot (QD), and find that the eigenenergies of the ground and first excited states are reduced due to the electron–LOP interaction. At the same time, the time evolution of the electron probability density is obtained, the dependence of the **oscillating** period on electron–LOP coupling constant is found, the relation of between the **oscillating** period and the confinement length of the QD is calculated. Finally, we consider the effects of the electron–LOP coupling constant on pure dephasing factor under considering the correction of electron–LOP interaction for the wave functions. Our results suggest that electron–LOP interaction has very important effects on charge **qubit**.

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Contributors: S. Filippov, V. Vyurkov, L. Fedichkin

Date: 2011-11-01

**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.
...A charge-based **qubit** is subject to image forces originating in nearby metal gates. Displacement of charge in an **oscillating** **qubit** indispensably results in moving charges in metal. Therefore, Joule loss is one more source of **qubit** decoherence. We have estimated the quality of Rabi **oscillations** for a realistic double-quantum-dot as Q∼100. This kind of decoherence cannot be suppressed by lowering temperature as it is evoked by surface roughness scattering of electrons which is almost insensitive to temperature. Possibilities to avoid such a decoherence are briefly discussed. The effect of energy dissipation and image charge potential on **qubit** dynamics is studied by means of a specific local-in-time non-Markovian master equation....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.
... A charge-based **qubit** is subject to image forces originating in nearby metal gates. Displacement of charge in an **oscillating** **qubit** indispensably results in moving charges in metal. Therefore, Joule loss is one more source of **qubit** decoherence. We have estimated the quality of Rabi **oscillations** for a realistic double-quantum-dot as Q∼100. This kind of decoherence cannot be suppressed by lowering temperature as it is evoked by surface roughness scattering of electrons which is almost insensitive to temperature. Possibilities to avoid such a decoherence are briefly discussed. The effect of energy dissipation and image charge potential on **qubit** dynamics is studied by means of a specific local-in-time non-Markovian master equation.

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Contributors: F.K. Wilhelm, S. Kleff, J. von Delft

Date: 2004-01-26

Visualization of the ground state |0〉 and the coherent pointer-states |L〉 and |R〉 of the **oscillator** in the potential V(x).
...In the spin-boson model, the properties of the **oscillator** bath are fully characterized by the spectral density of **oscillators** J(ω). We study the case when this function is of Breit–Wigner shape and has a sharp peak at a **frequency** Ω with width Γ≪Ω. We use a number of approaches such as the weak-coupling Bloch–Redfield equation, the non-interacting blip approximation (NIBA) and the flow-equation renormalization scheme. We show, that if Ω is much larger than the **qubit** energy scales, the dynamics corresponds to an ohmic spin-boson model with a strongly reduced tunnel splitting. We also show that the direction of the scaling of the tunnel splitting changes sign when the bare splitting crosses Ω. We find good agreement between our analytical approximations and numerical results. We illuminate how and why different approaches to the model account for these features and discuss the interpretation of this model in the context of an application to quantum computation and read-out. ... In the spin-boson model, the properties of the **oscillator** bath are fully characterized by the spectral density of **oscillators** J(ω). We study the case when this function is of Breit–Wigner shape and has a sharp peak at a **frequency** Ω with width Γ≪Ω. We use a number of approaches such as the weak-coupling Bloch–Redfield equation, the non-interacting blip approximation (NIBA) and the flow-equation renormalization scheme. We show, that if Ω is much larger than the **qubit** energy scales, the dynamics corresponds to an ohmic spin-boson model with a strongly reduced tunnel splitting. We also show that the direction of the scaling of the tunnel splitting changes sign when the bare splitting crosses Ω. We find good agreement between our analytical approximations and numerical results. We illuminate how and why different approaches to the model account for these features and discuss the interpretation of this model in the context of an application to quantum computation and read-out.

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Contributors: Haiteng Jiang, Ali Bahramisharif, Marcel A.J. van Gerven, Ole Jensen

Date: 2015-09-01

It is well established that neuronal **oscillations** at different **frequencies** interact with each other in terms of cross-**frequency** coupling (CFC). In particular, the phase of slower **oscillations** modulates the power of faster **oscillations**. This is referred to as phase–amplitude coupling (PAC). Examples are alpha phase to gamma power coupling as observed in humans and theta phase to gamma power coupling as observed in the rat hippocampus. We here ask if the interaction between alpha and gamma **oscillations** is in the direction of the phase of slower **oscillations** driving the power of faster **oscillations** or conversely from the power of faster **oscillations** driving the phase of slower **oscillations**. To answer this question, we introduce a new measure to estimate the cross-**frequency** directionality (CFD). This measure is based on the phase-slope index (PSI) between the phase of slower **oscillations** and the power envelope of faster **oscillations**. Further, we propose a randomization framework for statistically evaluating the coupling measures when controlling for multiple comparisons over the investigated **frequency** ranges. The method was firstly validated on simulated data and next applied to resting state electrocorticography (ECoG) data. These results demonstrate that the method works reliably. In particular, we found that the power envelope of gamma **oscillations** drives the phase of slower **oscillations** in the alpha band. This surprising finding is not easily reconcilable with theories suggesting that feedback controlled alpha **oscillations** modulate feedforward processing reflected in the gamma band....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 ... It is well established that neuronal **oscillations** at different **frequencies** interact with each other in terms of cross-**frequency** coupling (CFC). In particular, the phase of slower **oscillations** modulates the power of faster **oscillations**. This is referred to as phase–amplitude coupling (PAC). Examples are alpha phase to gamma power coupling as observed in humans and theta phase to gamma power coupling as observed in the rat hippocampus. We here ask if the interaction between alpha and gamma **oscillations** is in the direction of the phase of slower **oscillations** driving the power of faster **oscillations** or conversely from the power of faster **oscillations** driving the phase of slower **oscillations**. To answer this question, we introduce a new measure to estimate the cross-**frequency** directionality (CFD). This measure is based on the phase-slope index (PSI) between the phase of slower **oscillations** and the power envelope of faster **oscillations**. Further, we propose a randomization framework for statistically evaluating the coupling measures when controlling for multiple comparisons over the investigated **frequency** ranges. The method was firstly validated on simulated data and next applied to resting state electrocorticography (ECoG) data. These results demonstrate that the method works reliably. In particular, we found that the power envelope of gamma **oscillations** drives the phase of slower **oscillations** in the alpha band. This surprising finding is not easily reconcilable with theories suggesting that feedback controlled alpha **oscillations** modulate feedforward processing reflected in the gamma band.

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Contributors: S.K. Ryu, Y.K. Kim, M.K. Kim, S.H. Won, S.H. Chung

Date: 2010-01-01

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.
...The **oscillation** behavior of laminar lifted flames under the influence of low-**frequency** AC has been investigated experimentally in coflow jets. Various **oscillation** modes were existed depending on jet velocity and the voltage and **frequency** of AC, especially when the AC **frequency** was typically smaller than 30Hz. Three different **oscillation** modes were observed: (1) large-scale **oscillation** with the **oscillation** **frequency** of about 0.1Hz, which was independent of the applied AC **frequency**, (2) small-scale **oscillation** synchronized to the applied AC **frequency**, and (3) doubly-periodic **oscillation** with small-scale **oscillation** embedded in large-scale **oscillation**. As the AC **frequency** decreased from 30Hz, the **oscillation** modes were in the order of the large-scale **oscillation**, doubly-periodic **oscillation**, and small-scale **oscillation**. The onset of the **oscillation** for the AC **frequency** smaller than 30Hz was in close agreement with the delay time scale for the ionic wind effect to occur, that is, the collision response time. **Frequency**-doubling behavior for the small-scale **oscillation** has also been observed. Possible mechanisms for the large-scale **oscillation** and the **frequency**-doubling behavior have been discussed, although the detailed understanding of the underlying mechanisms will be a future study....**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.
... The **oscillation** behavior of laminar lifted flames under the influence of low-**frequency** AC has been investigated experimentally in coflow jets. Various **oscillation** modes were existed depending on jet velocity and the voltage and **frequency** of AC, especially when the AC **frequency** was typically smaller than 30Hz. Three different **oscillation** modes were observed: (1) large-scale **oscillation** with the **oscillation** **frequency** of about 0.1Hz, which was independent of the applied AC **frequency**, (2) small-scale **oscillation** synchronized to the applied AC **frequency**, and (3) doubly-periodic **oscillation** with small-scale **oscillation** embedded in large-scale **oscillation**. As the AC **frequency** decreased from 30Hz, the **oscillation** modes were in the order of the large-scale **oscillation**, doubly-periodic **oscillation**, and small-scale **oscillation**. The onset of the **oscillation** for the AC **frequency** smaller than 30Hz was in close agreement with the delay time scale for the ionic wind effect to occur, that is, the collision response time. **Frequency**-doubling behavior for the small-scale **oscillation** has also been observed. Possible mechanisms for the large-scale **oscillation** and the **frequency**-doubling behavior have been discussed, although the detailed understanding of the underlying mechanisms will be a future study.

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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: A.G. Khachatryan, F.A. van Goor, K.-J. Boller

Date: 2006-12-18

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 motion of a classical (harmonic) **oscillator** is studied in the case where the **oscillator** is driven by a pulsed **oscillating** force with a **frequency** varying in time (**frequency** chirp). The amplitude and phase of the **oscillations** left after the pulsed force in dependence on the profile and strength of chirp, **frequency** and duration of the force is investigated....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 ... The motion of a classical (harmonic) **oscillator** is studied in the case where the **oscillator** is driven by a pulsed **oscillating** force with a **frequency** varying in time (**frequency** chirp). The amplitude and phase of the **oscillations** left after the pulsed force in dependence on the profile and strength of chirp, **frequency** and duration of the force is investigated.

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Contributors: Erik Smedler, Per Uhlén

Date: 2014-03-01

**Frequency** modulation...Calcium (Ca2+) **oscillations** are ubiquitous signals present in all cells that provide efficient means to transmit intracellular biological information. Either spontaneously or upon receptor ligand binding, the otherwise stable cytosolic Ca2+ concentration starts to **oscillate**. The resulting specific oscillatory pattern is interpreted by intracellular downstream effectors that subsequently activate different cellular processes. This signal transduction can occur through **frequency** modulation (FM) or amplitude modulation (AM), much similar to a radio signal. The decoding of the oscillatory signal is typically performed by enzymes with multiple Ca2+ binding residues that diversely can regulate its total phosphorylation, thereby activating cellular program. To date, NFAT, NF-κB, CaMKII, MAPK and calpain have been reported to have **frequency** decoding properties....**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 ... Calcium (Ca2+) **oscillations** are ubiquitous signals present in all cells that provide efficient means to transmit intracellular biological information. Either spontaneously or upon receptor ligand binding, the otherwise stable cytosolic Ca2+ concentration starts to **oscillate**. The resulting specific oscillatory pattern is interpreted by intracellular downstream effectors that subsequently activate different cellular processes. This signal transduction can occur through **frequency** modulation (FM) or amplitude modulation (AM), much similar to a radio signal. The decoding of the oscillatory signal is typically performed by enzymes with multiple Ca2+ binding residues that diversely can regulate its total phosphorylation, thereby activating cellular program. To date, NFAT, NF-κB, CaMKII, MAPK and calpain have been reported to have **frequency** decoding properties.

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Contributors: J.R. Petta, A.C. Johnson, J.M. Taylor, A. Yacoby, M.D. Lukin, C.M. Marcus, M.P. Hanson, A.C. Gossard

Date: 2006-08-01

We demonstrate high-speed manipulation of a few-electron double quantum dot. In the one-electron regime, the double dot forms a charge **qubit**. Microwaves are used to drive transitions between the (1,0) and (0,1) charge states of the double dot. A local quantum point contact charge detector measures the photon-induced change in occupancy of the charge states. Charge detection is used to measure T1∼16ns and also provides a lower bound estimate for T2* of 400ps for the charge **qubit**. In the two-electron regime we use pulsed-gate techniques to measure the singlet–triplet relaxation time for nearly-degenerate spin states. These experiments demonstrate that the hyperfine interaction leads to fast spin relaxation at low magnetic fields. Finally, we discuss how two-electron spin states can be used to form a logical spin **qubit**....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** ... We demonstrate high-speed manipulation of a few-electron double quantum dot. In the one-electron regime, the double dot forms a charge **qubit**. Microwaves are used to drive transitions between the (1,0) and (0,1) charge states of the double dot. A local quantum point contact charge detector measures the photon-induced change in occupancy of the charge states. Charge detection is used to measure T1∼16ns and also provides a lower bound estimate for T2* of 400ps for the charge **qubit**. In the two-electron regime we use pulsed-gate techniques to measure the singlet–triplet relaxation time for nearly-degenerate spin states. These experiments demonstrate that the hyperfine interaction leads to fast spin relaxation at low magnetic fields. Finally, we discuss how two-electron spin states can be used to form a logical spin **qubit**.

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