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The evolutions of reduced density matrix elements ρ12 (below) and ρ11 (up) in SB and SIB models in low-**frequency** bath. The parameters are the same as in Fig. 1.
... The spectral density functions Johm(ω) (b) and Jeff(ω) (a) versus the **frequency** ω of the bath modes, where Δ=5×109Hz,λκ=1,ξ=0.01,Ω0=10Δ,T=0.01K,Γ=2.6×1011Hz.
... The evolutions of reduced density matrix elements of ρ12 (below) and ρ11 (up) in SIB model in medium-**frequency** bath in different values of Ω0, the other parameters are the same as in Fig. 1.
... The response functions of the Ohmic bath in (a) low and (c) medium **frequencies** and effective bath in (b) low and (d) medium **frequencies**. The parameters are the same as in Fig. 1. The cut-off **frequencies** for the two cases are taken according to Fig. 2.
... The sketch map on the low-, medium-, and high-**frequency** baths.

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The response functions of the Ohmic bath and effective bath, where Δ=5×109Hz, λκ=1050, ξ=0.01, Ω0=10Δ, T=0.01K, Γ=2.6×1011, the lower-**frequency** and high-**frequency** cut-off of the baths modes ω0=11Δ, and ωc=100Δ.

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A contour plot indicating location of two-dimensional potential energy minima forming a symmetric double well potential when the cantilever equilibrium angle θ0=cos−1[Φo/2BxA], ωi=2π×12000 rad/s, Bx=5×10−2 T. The contour interval in units of **frequency** (E/h) is ∼4×1011 Hz.
... A superconducting-loop-**oscillator** with its axis of rotation along the z-axis consists of a closed superconducting loop without a Josephson Junction. The superconducting loop can be of any arbitrary shape.
... A contour plot indicating location of a two-dimensional global potential energy minimum at (nΦ0=0, θn+=π/2) and the local minima when the cantilever equilibrium angle θ0=π/2, ωi=2π×12000 rad/s, Bx=5.0×10−2 T. The contour interval in units of **frequency** (E/h) is ∼3.9×1011 Hz.
... The potential energy profile of the superconducting-loop-**oscillator** when the intrinsic **frequency** is 10 kHz. (a) For external magnetic field Bx=0, a single well harmonic potential near the minimum is formed. (b) Bx=0.035 T. (c) For Bx=0.045 T, a double well potential is formed.
... A schematic of the flux-**qubit**-cantilever. A part of the flux-**qubit** (larger loop) is projected from the substrate to form a cantilever. The external magnetic field Bx controls the coupling between the flux-**qubit** and the cantilever. An additional magnetic flux threading through a dc-SQUID (smaller loop) which consists of two Josephson junctions adjusts the tunneling amplitude. The dc-SQUID can be shielded from the effect of Bx.

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(Color online) Upper panel: adiabatic energies during a LZ sweep of a **qubit** coupled to two **oscillators**. Parameters: γ=0.25ℏv and Ω2=100ℏv, both as in Fig. 4; ℏΩ1=80ℏv. Lower panel: probability P↑→↑(t) that the system stays in the initial state |↑00〉 (solid), and corresponding exact survival final survival probability P↑→↑(∞) of Eq. (20) (dotted).
... (Color online) LZ dynamics of a **qubit** coupled to one **oscillator**, far outside the RWA regime: γ=ℏΩ=0.25ℏv. The red solid curve is the survival probability P↑→↑(t) when starting in the initial state |↑0〉. The dotted black line is the exact survival probability P↑→↑(∞) based on Eq. (16). The dashed purple curve depicts the average photon number in the **oscillator** if the **qubit** would be measured in state |↓〉; the dash-dotted blue curve at the bottom shows the analogous average photon number in case the **qubit** would be measured |↑〉.
... (Color online) Upper panel: adiabatic energies during a LZ sweep of a **qubit** coupled to two **oscillators** with large energies, and with detunings of the order of the **qubit**–**oscillator** coupling γ. Parameters: γ=0.25ℏv and ℏΩ2=100ℏv, as before; ℏΩ1=96ℏv. Lower panel: probability P↑→↑(t) that the system stays in the initial state |↑00〉 (solid), and corresponding exact survival final survival probability P↑→↑(∞) of Eq. (20) (dotted).
... (Color online) Upper panel: adiabatic energies during a LZ sweep of a **qubit** coupled to two **oscillators**. Parameters: γ=0.25ℏv, ℏΩ1=90ℏv, and Ω2=100ℏv. Viewed on this scale of **oscillator** energies, the differences between exact and avoided level crossings are invisible. Lower panel: for the same parameters, probability P↑→↑(t) that the system stays in the initial state |↑00〉 (solid), and corresponding exact survival final survival probability P↑→↑(∞) of Eq. (20) (dotted).
... (Color online) Sketch of adiabatic eigenstates during LZ sweep of a **qubit** that is coupled to one **oscillator**. Starting in the ground state |↑0〉 and by choosing a slow LZ sweep, a single photon can be created in the **oscillator**. Due to cavity decay, the one-photon state will decay to a zero-photon state. Then the reverse LZ sweep creates another single photon that eventually decays to the initial state |↑0〉. This is a cycle to create single photons that can be repeated.

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Power spectral density for the low **frequency** **oscillator** at the resonance point (Φdc=0.00015Φ0) for the three spontaneous decay rates shown in Fig. 2: γ=0.005,0.05,0.5 per cycle. The other parameters are given in the text.
... (a) Close-up of the time-averaged (Floquet) energies of the single photon resonance (500 MHz), solid lines, with the time-independent energies given dotted lines. (b) The output power of the low **frequency** **oscillator** at 300 MHz, as a function of the static magnetic flux bias: γ=0.005 per cycle (solid line), γ=0.05 per cycle (crosses), γ=0.5 per cycle (circles). The other parameters are given in the text.
... Schematic diagram of persistent current **qubit** [6] inductively coupled to a (low **frequency**) classical **oscillator**. The insert graph shows the time-averaged (Floquet) energies as a function of the external bias field Φx1 for the parameters given in the text.
... Persistent current **qubit**

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Solid lines: synchronization degree D (and in-phase current quadrature 〈X〉) as functions of F for several values of the detection efficiency ηeff. Dashed and dotted lines illustrate the effects of the energy mismatch (ε≠0) and the **frequency** mismatch (Ω≠Ω0).

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Probe current **oscillations** in the first (a) and the second (b) **qubit** when the system is driven non-adiabatically to the double-degeneracy point X for the case EJ1=9.1GHz and EJ2=13.4GHz. Right panels show the corresponding spectra obtained by Fourier transformation. Arrows and dotted lines indicate theoretically expected position of the peaks.
... EJ1 dependence of the spectrum components of Fig. 6. Solid lines: dependence of Ω+ε and Ω−ε obtained from Eq. (6) using EJ2=9.1GHz and Em=14.5GHz and varying EJ1 from zero to its maximum value of 13.4GHz. Dashed lines: dependence of the **oscillation** **frequencies** of both **qubits** in the case of zero coupling (Em=0).
... Schematic diagram of the two-coupled-**qubit** circuit. Black bars denote Cooper pair boxes.
... Probe current **oscillations** in the first (a) and the second (b) **qubit** when the system is driven non-adiabatically to the points R and L, respectively. Right panels show the corresponding spectra obtained by the Fourier transform. Peak position in the spectrum gives the value of the Josephson energy of each **qubit**, indicated by arrow. In both cases, the experimental data (open triangles and open dots) can be fitted to a cosine dependence (solid lines) with an exponential decay with 2.5ns time constant.
... Solid-state **qubits**

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Time evolution of the reduced probability inversion P˜′1(2)−P˜1(2) in the coupling region for ΩR=Tc (a), ΩR>Tc (b), and ΩRoscillation with the same ΩR in each case is shown in gray line.
... Dependence of instantaneous tunneling currents on Rabi **frequency** ΩR and tunneling rate Tc at t=tp (a), t=1.25tp (b), t=1.5tp (c), and t=2tp (d).
... Rabi **oscillation**... (a) Schematic representation of a four-dot structure with an electron in Rabi **oscillation** and another one in quantum tunneling. (b) Time-average current spectrum as functions of ℏω and ε3 for Tc=ΩR=0.4GHz. (c) Schematic diagrams of FLIP operation.

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Comparison between simulated evolution of a **qubit's** Rabi **oscillations** and processed measurement signal for p¯=0.5, Δp=0.1 and τ=TR/16. Dashed curve: |c1|2 over time (in units of the Rabi period TR) in the presence of weak measurements. Dotted curve: |c1|2 over time in the absence of measurements. The solid curve corresponds to the evolution of the estimate g based on the measurement results.
... Power spectrum of |c1|2 in the presence of measurements. It assumes its maximum at the **frequency** ΩR of the undisturbed Rabi **oscillations**.

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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.
... (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.

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