(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.
Contributors:A.-B.A. Mohamed, A. Joshi, S.S. Hassan
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.
Contributors:Ping Yang, Ping Zou, Zhi-Ming Zhang
(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.
Contributors:Eugene M. Chudnovsky
Contributors:R. Taranko, T. Kwapiński
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.
... 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
Contributors:M Thorwart, E Paladino, M Grifoni
Flux qubit... Example of the dynamics for the symmetric case ε=0, where the oscillatorfrequency 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 oscillatorfrequency 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 ). This implies that with varying Ω also g is changed.
Contributors:A.J. Fotue, N. Issofa, M. Tiotsop, S.C. Kenfack, M.P. Tabue Djemmo, A.V. Wirngo, H. Fotsin, L.C. Fai
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.
... (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.
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.
... 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.
Contributors:R. Taranko, P. Parafiniuk
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 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
Contributors:P. Parafiniuk, R. Taranko
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 qubitoscillate 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