### 21982 results for qubit oscillator frequency

Contributors: Peihao Huang, Hang Zheng

Date: 2010-11-19

Diagrammatic sketch of a **qubit** coupled with structured environments. The environment in the 1st case consists of a two level system coupled to a bath. The environment in 2nd case is a damped harmonic **oscillator**.
...(1st case) P(t) as a function of time for the on-resonance case (ΔA=ΔB), where the decoherence is enhanced with T. Inset (a): Fourier analysis of P(t). One can see that two **frequencies** are dominating the dynamics and the peaks locate at ΔA±g0. Inset (b): The effective spectral density Jeff(ω). Here, it is not πJeff(ΔA) but πJeff(ΔA±g0) indicates the damping rate γA.
...The dynamics of a **qubit** in two different environments are investigated theoretically. The first environment is a two level system coupled to a bosonic bath. And the second one is a damped harmonic **oscillator**. Based on a unitary transformation, we find that the decoherence of the **qubit** can be reduced with increasing temperature T in the first case, which agree with the results in Ref. [8], whereas, it cannot be reduced with T in the second case. In both cases, the **qubit** dynamics are changed substantially as the coupling increases or finite detuning appears....(2nd case) P(t) as a function of time, where the decoherence is enhanced with T. Inset (a): Fourier analysis of the main plot. One sees that two **frequencies** are dominating the dynamics and the splitting of the peaks increases with temperature. Inset (b): The effective spectral density Jeff(ω). The square, triangle and circle points correspond to the dominant **frequencies** of P(t) in different temperatures, respectively. One can see that smaller Jeff’s, which characterize long time dynamics, are almost the same for three different temperatures. This is the reason why the damping rate of P(t) is almost not changing with different temperatures.
... The dynamics of a **qubit** in two different environments are investigated theoretically. The first environment is a two level system coupled to a bosonic bath. And the second one is a damped harmonic **oscillator**. Based on a unitary transformation, we find that the decoherence of the **qubit** can be reduced with increasing temperature T in the first case, which agree with the results in Ref. [8], whereas, it cannot be reduced with T in the second case. In both cases, the **qubit** dynamics are changed substantially as the coupling increases or finite detuning appears.

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Contributors: Ping Yang, Ping Zou, Zhi-Ming Zhang

Date: 2012-10-01

We present a system composed of two flux **qubits** and a transmission-line resonator. Instead of using the rotating wave approximation (RWA), we analyze the system by the adiabatic approximation methods under two opposite extreme conditions. Basic properties of the system are calculated and compared under these two different conditions. Relative energy-level spectrum of the system in the adiabatic displaced **oscillator** basis is shown, and the theoretical result is compared with the numerical solution....(Color online.) Schematic diagram of the displaced **oscillator** basis. The horizontal axis x′=x2mω0ℏ. All three wells maintain the same harmonic character, and usual eigenstates as well. The equilibrium position of the left (or the right) well is shifted by a specific constant. The shift direction is to the left (or right) when the **qubits** are in |+〉=|e1,e2〉 (or |−〉=|g1,g2〉). The middle potential well which is double degenerate corresponds to non-displaced case in which the states of the two **qubits** are opposite, i.e., |0〉 (|g1,e2〉 or |e1,g2〉), and the equilibrium position is higher than the others. The eigenstates which have the same value of n in the left well are degenerate with that in the right well.
...(Color online.) (a) Schematic diagram of the structure. The two light blue squares are improved three-junction flux **qubits** fabricated to the center conductor. (b) Schematic graph of the system. Two identical **qubits** (i.e. parameters Δ, ϵ, energy-level splitting Eq and coupling strength g for both **qubits** are of the same value) viewed as a two-level system with ground state |g〉 and excited state |e〉, are coupled to a harmonic **oscillator** whose characteristic **frequency** is ω0.
...(Color online.) Comparison between the displaced **oscillator** adiabatic approximation method and the numerical solution for the lowest two levels. ℏω0/Eq=10. The black solid lines stand for the lowest two energy levels calculated by adiabatic approximation. The green dashed line and the red dashed line correspond to the lowest two energy levels obtained by the numerical solution. (a) θ=0. (b) θ=π/6. (c) θ=π/4. (d) θ=π/3.
... We present a system composed of two flux **qubits** and a transmission-line resonator. Instead of using the rotating wave approximation (RWA), we analyze the system by the adiabatic approximation methods under two opposite extreme conditions. Basic properties of the system are calculated and compared under these two different conditions. Relative energy-level spectrum of the system in the adiabatic displaced **oscillator** basis is shown, and the theoretical result is compared with the numerical solution.

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Contributors: A.-B.A. Mohamed, A. Joshi, S.S. Hassan

Date: 2016-03-01

Several quantum-mechanical correlations, notably, quantum entanglement, measurement-induced nonlocality and Bell nonlocality are studied for a two **qubit**-system having no mutual interaction. Analytical expressions for the measures of these quantum-mechanical correlations of different bipartite partitions of the system are obtained, for initially two entangled **qubits** and the two photons are in their vacuum states. It is found that the **qubits**-fields interaction leads to the loss and gain of the initial quantum correlations. The lost initial quantum correlations transfer from the **qubits** to the cavity fields. It is found that the maximal violation of Bell’s inequality is occurring when the quantum correlations of both the logarithmic negativity and measurement-induced nonlocality reach particular values. The maximal violation of Bell’s inequality occurs only for certain bipartite partitions of the system. The **frequency** detuning leads to quick **oscillations** of the quantum correlations and inhibits their transfer from the **qubits** to the cavity modes. It is also found that the dynamical behavior of the quantum correlation clearly depends on the **qubit** distribution angle. ... Several quantum-mechanical correlations, notably, quantum entanglement, measurement-induced nonlocality and Bell nonlocality are studied for a two **qubit**-system having no mutual interaction. Analytical expressions for the measures of these quantum-mechanical correlations of different bipartite partitions of the system are obtained, for initially two entangled **qubits** and the two photons are in their vacuum states. It is found that the **qubits**-fields interaction leads to the loss and gain of the initial quantum correlations. The lost initial quantum correlations transfer from the **qubits** to the cavity fields. It is found that the maximal violation of Bell’s inequality is occurring when the quantum correlations of both the logarithmic negativity and measurement-induced nonlocality reach particular values. The maximal violation of Bell’s inequality occurs only for certain bipartite partitions of the system. The **frequency** detuning leads to quick **oscillations** of the quantum correlations and inhibits their transfer from the **qubits** to the cavity modes. It is also found that the dynamical behavior of the quantum correlation clearly depends on the **qubit** distribution angle.

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Contributors: Eugene M. Chudnovsky

Date: 2004-05-01

**Qubit**...Fundamental conservation laws mandate parameter-free generic mechanisms of decoherence of quantum **oscillations** in double-well systems. We consider two examples: tunneling of the magnetic moment in nanomagnets and tunneling between macroscopic current states in SQUIDs. In both cases the decoherence occurs via emission of phonons and photons at the **oscillation** **frequency**. We also show that in a system of identical **qubits** the decoherence greatly increases due to the superradiance of electromagnetic and sound waves. Our findings have important implications for building elements of quantum computers based upon nanomagnets and SQUIDs. ... Fundamental conservation laws mandate parameter-free generic mechanisms of decoherence of quantum **oscillations** in double-well systems. We consider two examples: tunneling of the magnetic moment in nanomagnets and tunneling between macroscopic current states in SQUIDs. In both cases the decoherence occurs via emission of phonons and photons at the **oscillation** **frequency**. We also show that in a system of identical **qubits** the decoherence greatly increases due to the superradiance of electromagnetic and sound waves. Our findings have important implications for building elements of quantum computers based upon nanomagnets and SQUIDs.

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Contributors: M Thorwart, E Paladino, M Grifoni

Date: 2004-01-26

Flux **qubit**...Example of the dynamics for the symmetric case ε=0, where the **oscillator** **frequency** is in resonance with the TSS **frequency**, i.e., Ω=Δ0. Parameters are: g=0.18Δ0, κ=0.014 (→α=0.004), kBT=0.1ℏΔ0. QUAPI parameters are M=12, K=1, Δt=0.06/Δ0.
...Sz(ω) for two values of the **oscillator** **frequency** Ω. Parameters are: ε=0, g=0.07Δ0, κ=0.014, kBT=0.1ℏΔ0.
...Main: Dephasing rates corresponding to peak 1 and peak 2 in the Figs. 1 and 3 as a function of the HO **frequency** Ω. The parameters are: ε=0, g=0.07Δ0, κ=0.014, kBT=0.1ℏΔ0. Inset: Same for stronger damping κ=0.02 with α=0.01=const. (like in [15]). This implies that with varying Ω also g is changed.
...We investigate the dynamics of the spin-boson model when the spectral density of the boson bath shows a resonance at a characteristic **frequency** Ω but behaves Ohmically at small **frequencies**. The time evolution of an initial state is determined by making use of the mapping onto a system composed of a quantum mechanical two-state system (TSS) which is coupled to a harmonic **oscillator** (HO) with **frequency** Ω. The HO itself is coupled to an Ohmic environment. The dynamics is calculated by employing the numerically exact quasiadiabatic path-integral propagator technique. We find significant new properties compared to the Ohmic spin-boson model. By reducing the TSS-HO system in the dressed states picture to a three-level system for the special case at resonance, we calculate the dephasing rates for the TSS analytically. Finally, we apply our model to experimentally realized superconducting flux **qubits** coupled to an underdamped dc-SQUID detector. ... We investigate the dynamics of the spin-boson model when the spectral density of the boson bath shows a resonance at a characteristic **frequency** Ω but behaves Ohmically at small **frequencies**. The time evolution of an initial state is determined by making use of the mapping onto a system composed of a quantum mechanical two-state system (TSS) which is coupled to a harmonic **oscillator** (HO) with **frequency** Ω. The HO itself is coupled to an Ohmic environment. The dynamics is calculated by employing the numerically exact quasiadiabatic path-integral propagator technique. We find significant new properties compared to the Ohmic spin-boson model. By reducing the TSS-HO system in the dressed states picture to a three-level system for the special case at resonance, we calculate the dephasing rates for the TSS analytically. Finally, we apply our model to experimentally realized superconducting flux **qubits** coupled to an underdamped dc-SQUID detector.

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Contributors: P. Parafiniuk, R. Taranko

Date: 2008-09-01

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** ... We have studied the electron dynamics in different geometrical arrangements of the two coupled double quantum dot structures. Applying the equation of motion method for appropriate correlation functions the occupation probabilities of different quantum dots of the considered system has been theoretically investigated. The numerical calculations were performed for different forms of the time-dependent tunneling amplitudes and quantum dot energy levels. We found, among others, that under some conditions for the tunneling amplitudes changed in the form of Gaussian pulses it is possible to localize the electron in a controlled manner on the given dot of the considered system.

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Contributors: A.J. Fotue, N. Issofa, M. Tiotsop, S.C. Kenfack, M.P. Tabue Djemmo, A.V. Wirngo, H. Fotsin, L.C. Fai

Date: 2016-02-01

a) Ground state energy E0 and (b) first excited state energy E1 as a function of the cyclotron **frequency** ωC for α = 7.0; F = 105.5; l0 = 0.45.
...In this paper, we examine the time evolution of the quantum mechanical state of a magnetopolaron using the Pekar type variational method on the electric-LO-phonon strong coupling in a triangular quantum dot with Coulomb impurity. We obtain the Eigen energies and the Eigen functions of the ground state and the first excited state, respectively. This system in a quantum dot is treated as a two-level quantum system **qubit** and numerical calculations are done. The Shannon entropy and the expressions relating the period of **oscillation** and the electron-LO-phonon coupling strength, the Coulomb binding parameter and the polar angle are derived....(a) ground state energy E0 and (b) first excited state energy E1 as a function of the cyclotron **frequency** ωC for F = 105.0; l0 = 0.45; β = 0.8;.
...Transition **frequency** ω as a function of the cyclotron **frequency** ωc for (a) F = 105.0; l0 = 0.45; β = 0.8; ϑ = π/2; φ = 2π, (b) α = 7.0; l0 = 0.45; β = 0.8; ϑ = π/2; φ = 2π, (c) α = 7.0; F = 105.5; β = 0.8; ϑ = π/2; φ = 2π, (d) α = 7.0; F = 105.5; l0 = 0.45; ϑ = π/2; φ = 2π.
...a) ground state energy E0 and (b) first excited state energy E1 as a function of the cyclotron **frequency** ωC for α = 7.0; l0 = 0.45; β = 0.8.
...Period of **oscillation** τ as a function of the cyclotron **frequency** ωC for (a) F = 105.0; l0 = 0.45; β = 0.8; ϑ = π/2; φ = 2π, (b) α = 7.0; l0 = 0.45; β = 0.8; ϑ = π/2; φ = 2π, (c) α = 7.0; F = 105.5; β = 0.8; ϑ = π/2; φ = 2π, (d). α = 7.0; F = 105.5; l0 = 0.45; ϑ = π/2; φ = 2π.
... In this paper, we examine the time evolution of the quantum mechanical state of a magnetopolaron using the Pekar type variational method on the electric-LO-phonon strong coupling in a triangular quantum dot with Coulomb impurity. We obtain the Eigen energies and the Eigen functions of the ground state and the first excited state, respectively. This system in a quantum dot is treated as a two-level quantum system **qubit** and numerical calculations are done. The Shannon entropy and the expressions relating the period of **oscillation** and the electron-LO-phonon coupling strength, the Coulomb binding parameter and the polar angle are derived.

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Contributors: R. Taranko, T. Kwapiński

Date: 2015-01-01

The sketch of the **qubit**–detector systems considered in the paper. The **qubit** (two coupled quantum dots: x and y) is coupled electrostatically via U parameter with one of the detector QDs. Panels A, B and C correspond to the single-QD, double-QD and triple-QD detectors, respectively.
...**Qubit** QD occupations, nx(t), versus time for the DQD (TQD) detector – curves a–c (d, e) and for different initial conditions. Curves a and d: **qubit** is ‘frozen’ in the state nx=0,ny=1 until t=40 when the occupancies of all detector QDs achieve their steady state values. Curves b and e: **qubit** is ‘frozen’ in the state nx=0,ny=1 and also n2=n3=0 until t=40 when the occupancy of the first detector QD, n1, achieves its steady state value. Curve c: all couplings in the **qubit**–detector system are switched on at t=40 (i.e. nx=0,ny=1, n1=n2=0 for t<40). The other parameters: Vxy=4, U=4, Vij=0.5, Γ=1, εi=0 and μL=−μR=20.
...The time evolution of a charge **qubit** coupled electrostatically with different detectors in the forms of single, double and triple quantum dot linear systems in the T-shaped configuration between two reservoirs is theoretically considered. The correspondence between the **qubit** quantum dot **oscillations** and the detector current is studied for different values of the inter-dot tunneling amplitudes and the **qubit**–detector interaction strength. We have found that even for a **qubit** coupled with a single QD detector, the coherent beat patterns appear in the **oscillations** of the **qubit** charge. This effect is more evident for a **qubit** coupled with double or triple-QD detectors. The beats can be also observed in both the detector current and the detector quantum dot occupations. Moreover, in the presence of beats the **qubit** **oscillations** hold longer in time in comparison with the beats-free systems with monotonously decaying **oscillations**. The dependence of the **qubit** dynamics on different initial occupations of the detector sites (memory effect) is also analyzed....Charge **qubit**...The nearby **qubit** QD occupation, nx(t), as a function of time for the triple-QDs detector shown in Fig. 1C for different values of the **qubit** tunneling amplitude Vxy=1,2 and 4, respectively. The upper (bottom) panel corresponds to μL=−μR=1 (μL=−μR=10). The other parameters are εi=0, V12=V23=1, Vxy=4, U=4 and the initial conditions as in Fig. 2.
...Nearby **qubit** QD occupation, nx(t), as a function of time for the triple-QD detector (see Fig. 1C) for different values of U parameter: U=0,2,3,4 and 6, respectively. The bias voltage μL=−μR=10, other parameters and initial conditions as in Fig. 6.
...Nearby **qubit** QD occupation, nx(t), as a function of time for different forms of the detector depicted in Fig. 1. The upper (bottom) panel corresponds to the ΓL=ΓR=Γ=1 (Γ=0.2). The tunneling coupling between QDs is V=1 for the detector and Vxy=4 for the **qubit**, energy levels of all QDs are equal to εi=0, μL=−μR=10 and U=4. The **qubit** was ‘frozen’ in the configuration nx=0, ny=1 for t<15, i.e. until the detector QD occupancies and currents jL and j12 achieved their stationary values. The curves B and C are shifted down by 1 and 2 for clarity.
...**Qubit** dynamics ... The time evolution of a charge **qubit** coupled electrostatically with different detectors in the forms of single, double and triple quantum dot linear systems in the T-shaped configuration between two reservoirs is theoretically considered. The correspondence between the **qubit** quantum dot **oscillations** and the detector current is studied for different values of the inter-dot tunneling amplitudes and the **qubit**–detector interaction strength. We have found that even for a **qubit** coupled with a single QD detector, the coherent beat patterns appear in the **oscillations** of the **qubit** charge. This effect is more evident for a **qubit** coupled with double or triple-QD detectors. The beats can be also observed in both the detector current and the detector quantum dot occupations. Moreover, in the presence of beats the **qubit** **oscillations** hold longer in time in comparison with the beats-free systems with monotonously decaying **oscillations**. The dependence of the **qubit** dynamics on different initial occupations of the detector sites (memory effect) is also analyzed.

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### Deviations from reversible dynamics in a **qubit**–**oscillator** system coupled to a very small environment

Contributors: A. Vidiella-Barranco

Date: 2014-02-15

Plot of the linear entropy ζ (as a function of t) of qubit1 initially in state |e〉, the **oscillator** initially in the number state |1〉 and qubit2 initially in a maximally mixed state, p=0.5 for a longer time-scale, with λ1=1.0 and λ2=0.1.
...Plot of the linear entropy ζ (as a function of t) of qubit1 initially in state |e〉 and the **oscillator** initially in a binomial state with M=7 and q=0.85. In this case qubit2 is decoupled, λ2=0.0 and λ1=1.0.
...In this contribution it is considered a simple and solvable model consisting of a **qubit** in interaction with an **oscillator** exposed to a very small “environment” (a second **qubit**). An isolated **qubit**–**oscillator** system having the **oscillator** initially in one of its energy eigenstates exhibits Rabi **oscillations**, an evidence of coherent quantum behaviour. It is shown here in which way the coupling to a small “environment” disrupts such regular behaviour, leading to a quasi-periodic dynamics for the **qubit** linear entropy. In particular, it is found that the linear entropy is very sensitive to the amount of mixedness of the “environment”. For completeness, fluctuations in the **oscillator** energy are also taken into account....Plot of the linear entropy ζ (as a function of t) of a **qubit** initially in state |e〉 and the **oscillator** initially in the mixed state ρosc(0)=f|0〉〈0|+(1−f)|1〉〈1| with λ=1.0 and f=0.5.
...Plot of the linear entropy ζ (as a function of t) of qubit1 initially in state |e〉 and the **oscillator** initially in a binomial state with M=11 and q=0.95. In this case qubit2 is coupled to the **oscillator**, with λ2=0.1, λ1=1.0, and p=0.5.
...Plot of the linear entropy ζ (as a function of t) of qubit1 initially in state |e〉 and the **oscillator** initially in a binomial state with M=100 and q=0.1. In this case qubit2 is decoupled, λ2=0.0 and λ1=1.0.
... In this contribution it is considered a simple and solvable model consisting of a **qubit** in interaction with an **oscillator** exposed to a very small “environment” (a second **qubit**). An isolated **qubit**–**oscillator** system having the **oscillator** initially in one of its energy eigenstates exhibits Rabi **oscillations**, an evidence of coherent quantum behaviour. It is shown here in which way the coupling to a small “environment” disrupts such regular behaviour, leading to a quasi-periodic dynamics for the **qubit** linear entropy. In particular, it is found that the linear entropy is very sensitive to the amount of mixedness of the “environment”. For completeness, fluctuations in the **oscillator** energy are also taken into account.

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Contributors: R. Taranko, P. Parafiniuk

Date: 2008-06-01

Schematic representation of the systems of a single **qubit** and two **qubits** interacting with the single electron transistor. The wavy lines correspond to the Coulomb interaction of electrons localized on the nearest quantum dots.
...The asymptotic current JL(t=∞) tunneling between the left lead and the SET QD against the pulse duration τ which determines the coherent **qubit** evolution (for details, see the text). The curve A (B, C, D, E) corresponds to the system of one (two) **qubit** coupled with the SET. The curve B (C) describes the case when the electron tunneling amplitude V inside one (both) **qubit** has a shape of the rectangular pulse of duration τ. The curve D (E) corresponds to the case when the pulse durations in the first and second **qubits** are equal τ and 2τ, respectively. εi=0,i=1,2,3,4,5; U1=U2=2, V=1 and μL=-μR=5.
...The system of one and two **qubits** coupled electrostatically with the quantum dot placed between the two electron reservoirs (single electron transistor, SET) has been studied. The **qubit** charge **oscillations** and the current flowing through the SET were calculated using the equation of motion method for appropriate correlation functions. The calculations were done for constant and harmonically **oscillating** **qubit** energy levels. We also show that the SET current flowing in response to the pulse of the tunneling amplitude between the **qubit** quantum dots versus the pulse duration exhibits the damped oscillatory time-dependence similar to the **qubit** charge **oscillations**....The probability n4(t) of finding the electron in the nearby **qubit** QD against the time in the case of two **qubits** coupled with the SET. The thin, thick and broken curves correspond to the bias voltage Vb=0,2 and 4, respectively. μL=-μR, U1=0.5, U2=1 and the other parameters are as in Fig. 5.
...The probability n3(t) of finding the electron in the nearby **qubit** QD and the current tunneling between the left lead and the SET QD against the time, the upper and lower panels, respectively. The left (right) panels correspond to one **qubit** (two **qubits**) coupled with the SET. The thin solid line describes the **qubits** with the constant electron energy levels, ε2=-ε3=ε4=-ε5=-1, and the thick, solid and broken lines correspond to harmonically driven energy levels with the amplitude Δ=1 and 2, respectively. ε1=0, μL=2, μR=-2, U1=U2=5, n3(0)=n5(0)=1, n1(0)=n2(0)=n4(0)=0.
...The probabilities n3(t) and n5(t) (the upper panels) of finding electrons in the far-removed **qubit** QDs against the time for the case of two **qubits** coupled with the SET. In the lower panel the current JL(t) tunneling between the left lead and the SET QD against the time is displayed. ε1=2, ε2=ε3=ε4=ε5=0, V1=V2=1, U1=U2 and μL=-μR=0.5.
...Charge **qubit** ... The system of one and two **qubits** coupled electrostatically with the quantum dot placed between the two electron reservoirs (single electron transistor, SET) has been studied. The **qubit** charge **oscillations** and the current flowing through the SET were calculated using the equation of motion method for appropriate correlation functions. The calculations were done for constant and harmonically **oscillating** **qubit** energy levels. We also show that the SET current flowing in response to the pulse of the tunneling amplitude between the **qubit** quantum dots versus the pulse duration exhibits the damped oscillatory time-dependence similar to the **qubit** charge **oscillations**.

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