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  • Schematic of the Josephson-junction qubit structure that enables measurements of the two non-commuting observables of the qubit, σ z and σ y , as required in the QND Hamiltonian ( 2). For discussion see text.... Spin representation of the QND measurement of the quantum coherent oscillations of a qubit. The oscillations are represented as a spin rotation in the z - y plane with frequency Δ . QND measurement is realized if the measurement frame (dashed lines) rotates with frequency Ω ≃ Δ .
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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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  • Population difference for zero static bias. Further parameters are Δ / Ω = 0.5 , ℏ β Ω = 10 and g / Ω = 1.0 . The adiabatic approximation and VVP are compared to numerical results. The first one only covers the longscale dynamics, while VVP also returns the fast oscillations. With increasing time small differences between numerical results and VVP become more pronounced. Fig::P_e=0_D=0.5_g=1.0... As a first case, we consider in Fig. Fig::PF_e=Sqrt0.5_D=Sqrt0.5_g=1.0 a weakly biased qubit ( ε / Ω = 0.5 ) being at resonance with the oscillator ( Δ b = Ω ). For a coupling strength of g / Ω = 1.0 , we find a good agreement between the numerics and VVP. The adiabatic approximation, however, conveys a slightly different picture: Looking at the time evolution it reveals collapse and rebirth of oscillations after a certain interval. This feature does not survive for the exact dynamics. Like in the unbiased case, the adiabatic approximation gives only the first group of frequencies between the quasidegenerate subspaces, and thus yields a wrong picture of the dynamics. In order to cover the higher frequency groups, we need again to go to higher-order corrections by using VVP. For the derivation of our results we assumed that ε is a multiple of the oscillator frequency Ω , ε = l Ω . In this case we found that the levels E ↓ , j 0 and E ↑ , j + l 0 form a degenerate doublet, which dominates the long-scale dynamics through the dressed oscillations frequency Ω j l . For l being not an integer those doublets cannot be identified unambiguously anymore. For instance, we examine the case ε / Ω = 1.5 in Fig. Fig::PF_e=1.5_D=0.5_g=1.0. Here, it is not clear which levels should be gathered into one subspace: j and j + 1 or j and j + 2 . Both the dressed oscillation frequencies Ω j 1 and Ω j 2 influence the longtime dynamics.... Fourier transform of the population difference in Fig. Fig::P_e=0_D=0.5_g=1.0. The left-hand graph shows the whole frequency range. The lowest frequency peaks originate from transitions between levels of a degenerate subspace and are determined through the dressed oscillation frequency Ω j 0 . Numerical calculations and VVP predict group of peaks located around ν / Ω = 0 , 1.0 , 2.0 , 3.0 . The first group at ν / Ω = 0 is shown in the middle graph. One can identify frequencies Ω 0 0 and Ω 2 0 , which fall together, and Ω 1 0 . The small peak comes from the frequency Ω 3 0 . This first gr... Population difference and Fourier spectrum for a biased qubit ( ε / Ω = 0.5 ) at resonance with the oscillator Δ b = Ω in the ultrastrong coupling regime ( g / Ω = 1.0 ). Concerning the time evolution VVP agrees well with numerical results. Only for long time weak dephasing occurs. The inset in the left-hand figure shows the adiabatic approximation only. It exhibits death and revival of oscillations which are not confirmed by the numerics. For the Fourier spectrum, VVP covers the various frequency peaks, which are gathered into groups like for the unbiased case. The adiabatic approximation only returns the first group. Fig::PF_e=Sqrt0.5_D=Sqrt0.5_g=1.0... respectively. Concerning the population difference, we see a relatively good agreement between the numerical calculation and VVP for short timescales. In particular, VVP also correctly returns the small overlaid oscillations. For longer timescales, the two curves get out of phase. The adiabatic approximation only can reproduce the coarse-grained dynamics. The fast oscillations are completely missed. To understand this better, we turn our attention to the Fourier transform in Fig. Fig::F_e=0_D=0.5_g=1.0. There, we find several groups of frequencies located around ν / Ω = 0 , ν / Ω = 1.0 , ν / Ω = 2.0 and ν / Ω = 3.0 . This can be explained by considering the transition frequencies in more detail. We have from Eq. ( VVEnergies)... with ζ k , j l = 1 8 ε ↓ , k 2 - ε ↓ , j 2 + ε ↑ , j + l 2 - ε ↑ , k + l 2 being the second-order corrections. For zero bias, ε = 0 , the index l vanishes. The term k - j Ω determines to which group of peaks a frequency belongs and Ω j 0 its relative position within this group. The latter has Δ as an upper bound, so that the range over which the peaks are spread within a group increases with Δ . The dynamics is dominated by the peaks belonging to transitions between the same subspace k - j = 0 , while the next group with k - j = 1 yields already faster oscillations. To each group belong theoretically infinite many peaks. However, under the low temperature assumption only those with a small oscillator number play a role. For the used parameter regime, the adiabatic approximation does not take into account the connections between different manifolds. It therefore covers only the first group of peaks with k - j = 0 , providing the long-scale dynamics. For ε = 0 , the dominating frequencies in this first group are given by Ω 0 0 = | Δ e - α / 2 | , Ω 1 0 = | Δ 1 - α e - α / 2 | and Ω 2 0 = | Δ L 2 0 α e - α / 2 | , where Ω 0 0 and Ω 2 0 coincide. A small peak at Ω 3 0 = | Δ L 3 0 α e - α / 2 | can also be seen. Notice that for certain coupling strengths some peaks vanish; like, for example, choosing a coupling strength of g / Ω = 0.5 makes the peak at Ω 1 0 vanish completely, independently of Δ , and the Ω 0 0 and Ω 2 0 peaks split. The JCM yields two oscillation peaks determined by the Rabi splitting and fails completely to give the correct dynamics, see the left-hand graph in Fig. Fig::F_e=0_D=0.5_g=1.0. Now, we proceed to an even stronger coupling, g / Ω = 2.0 , where we also expect the adiabatic approximation to work better. From Fig. Fig::EnergyVSg_e=0_W=1_D=0.5 we noticed that at such a coupling strength the lowest energy levels are degenerate within a subspace. Only for oscillator numbers like j = 3 , we see that a small splitting arises. This splitting becomes larger for higher levels. Thus, only this and higher manifolds can give significant contributions to the long time dynamics; that is, they can yield low frequency peaks. Also the adiabatic approximation is expected to work better for such strong couplings . And indeed by looking at Figs. Fig::P_e=0_D=0.5_g=2.0 and Fig::F_e=0_D=0.5_g=2.0, we notice that both the adiabatic approximation and VVP agree quite well with the numerics. Especially the first group of Fourier peaks in Fig. Fig::F_e=0_D=0.5_g=2.0 is also covered almost correctly by the adiabatic approximation. The first manifolds we can identify with those peaks are the ones with j = 3 and j = 4 . This is a clear indication that even at low temperatures higher oscillator quanta are involved due to the large coupling strength. Also frequencies coming from transitions between the energy levels from neighboring manifolds are shown enlarged in Fig. Fig::F_e=0_D=0.5_g=2.0. The adiabatic approximation and VVP can cover the main structure of the peaks involved there, while the former shows stronger deviations. If we go to higher values Δ / Ω 1 , the peaks in the individual groups become more spread out in frequency space, and for the population difference dephasing already occurs at a shorter timescale. For Δ / Ω = 1 , at least VVP yields still acceptable results in Fourier space but gets fast out of phase for the population difference.... Fourier spectrum of the population difference in Fig. Fig::P_e=0_D=0.5_g=2.0. In the left-hand graph a large frequency range is covered. Peaks are located around ν / Ω = 0 , 1.0 , 2.0 , 3.0 etc. Even the adiabatic approximation exhibits the higher frequencies. The upper right-hand graph shows the first group close to ν / Ω = 0 . The two main peaks come from Ω 3 0 and Ω 4 0 and higher degenerate manifolds. Frequencies from lower manifolds contribute to the peak at zero. The adiabatic approximation and VVP agree well with the numerics. The lower right-hand graph shows the second group of peaks around ν / Ω = 1.0 . This group is also predicted by the adiabatic approximation and VVP, but they do not fully return the detailed structure of the numerics. Interestingly, there is no peak exactly at ν / Ω = 1.0 indicating no nearest-neighbor transition between the low degenerate levels. Fig::F_e=0_D=0.5_g=2.0... Population difference and Fourier spectrum for ε / Ω = 1.5 , Δ / Ω = 0.5 and g / Ω = 1.0 . Van Vleck perturbation theory is confirmed by numerical calculations, while results obtained from the adiabatic approximation deviate strongly. In Fourier space, we find pairs of frequency peaks coming from the two dressed oscillation frequencies Ω j 1 and Ω j 2 . The spacings in between those pairs is about 0.5 Ω . The adiabatic approximation only returns one of those dressed frequencies in the first pair. Fig::PF_e=1.5_D=0.5_g=1.0... Fourier transform of the population difference in Fig. Fig::P_e=0_D=0.5_g=1.0. The left-hand graph shows the whole frequency range. The lowest frequency peaks originate from transitions between levels of a degenerate subspace and are determined through the dressed oscillation frequency Ω j 0 . Numerical calculations and VVP predict group of peaks located around ν / Ω = 0 , 1.0 , 2.0 , 3.0 . The first group at ν / Ω = 0 is shown in the middle graph. One can identify frequencies Ω 0 0 and Ω 2 0 , which fall together, and Ω 1 0 . The small peak comes from the frequency Ω 3 0 . This first group of peaks is also covered by the adiabatic approximation. The other groups come from transitions between different manifolds. The adiabatic approximation does not take them into account, while VVP does. A blow-up of the peaks coming from transitions between neighboring manifolds is given in the right-hand graph. In the left-hand graph additionally the Jaynes-Cummings peaks are shown, which, however, fail completely. Fig::F_e=0_D=0.5_g=1.0
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  • Our methodology can be used to predict dynamics of nanomechanical resonators connected to either quantum dots or superconducting qubits. The criterion for the single term approximation to be valid is readily met by current experiments such as those presented in Refs.  and their parameters yield near perfect agreement between numerical and analytic results. Most experiments operate in a regime where the qubit dynamics are not greatly perturbed by the presence of the oscillator, which has a much lower frequency ( ϵ ≈ Δ ≈ 10 GHz, ω = 1 GHz). In Figure fig:1, we chose ϵ ≈ Δ ≈ 100 MHz, because this better demonstrates the effect of the oscillator on the qubit. These parameters can be achieved experimentally using the same qubit design but with an oscillating voltage applied to the CPB bias gate . However, we stress the accuracy of our method is not restricted to this regime.... fig:2 Main panel: comparison of dynamics calculated from truncating ( Sin5) at N M A X = ± 10 (red) and a numerically exact approach (blue). Lower left: Fourier transform of the dynamics. Lower right: the numerical weight of the n t h term in the series expansion of ( Sin5), showing there are still only two dominant frequencies at n = 0 and n = - 1 . Parameters: ω = 0.5 , g = 0.1 , ϵ = 0 , Δ = 0.5 , T = 1 ~ m K , ℏ = 1 and k b = 1 .... Figure fig:3 demonstrates this idea, showing that by measuring Ω and fitting it to our expression ( eqn:rho3), we can obtain submilli-Kelvin precision in the experimentally relevant regime of 20-55 mK. At low temperatures the single term frequency plateaus, causing the accuracy to break down. In the higher temperature limit, we also see a deviation from the diagonal, this is to be expected as we leave the regime of validity described by ( eqn:crit). Naturally accuracy in this region could be improved by retaining higher order terms in ( Sin5), but this would become a more numeric than analytic approach. The upper inset shows the dependence of the accuracy of the prediction on the number of points (at a separation of 1ns) sampled from the dynamics. The accuracy increases initially as more points improve the fitted value of Ω , however after a certain length the accuracy is diminished by long term envelope effects in the dynamics not captured by the single term approximation. We note that the corresponding analysis in the frequency domain would not be equally affected by the long time envelope, however a large number of points in the FFT is then required in order to obtain the desired accuracy. The lower inset of Figure fig:3 shows the direct dependence of Ω on the temperature. The temperature range with steepest gradient and hence greatest frequency dependence on temperature varies with the coupling strength; thus the device could be specifically designed to have a maximal sensitivity in the temperature range of the most interest.... Figure fig:1 shows a comparison of the dynamics predicted using these expressions and a numerically exact approach. The latter are obtained by imposing a truncation of the oscillator Hilbert space at a point where the dynamics have converged and any higher modes have an extremely low occupation probability. Our zeroth order approximation proves to be unexpectedly powerful, giving accurate dynamics well into the strong coupling regime ( g / ω = 0.25 ) and even beyond this it still captures the dominant oscillatory behaviour, see Figure fig:1. Stronger coupling increases the numerical weight of higher frequency terms in the series, causing a modulation of the dynamics. The approximation starts to break down at ( g / ω = 0.5 ). The equations ( eqn:rho0) and ( eqn:rho1) are obviously unable to capture the higher frequency modulations to the dynamics or any potential long time phenomena like collapse and revival, but these are unlikely to be resolvable in experiments in any case. Nonetheless, it is worth pointing out that even in this strong coupling case the base frequency of the qubit dynamics is still adequately captured by our single term approximation.... fig:1 Comparison of the single term approximation (red) and a numerically exact approach (blue) for different coupling strengths. Uncoupled Rabi oscillations are also shown as a reference (green). Left: the population ρ 00 t in the time-domain. Right: the same data in the frequency domain. The full numerical solution was Fourier transformed using Matlab’s FFT algorithm. Other parameters are ω = 1 GHz, g = 0.1 GHz, ϵ = Δ = 100 MHz and T = 10 mK.... fig:3 Demonstration of qubit thermometry: T i n is the temperature supplied to the numerical simulation of the system and T o u t is the temperature that would be predicted by fitting oscillations with frequency ( eqn:rho3) to it. The blue line is the data and red line shows the effect of a 10kHz error in the frequency measurement; the grey dashed line serves as a guide to the eye. The lower inset shows the variation of the qubit frequency Ω with temperature. The upper inset shows the dependence of the absolute error in the prediction against the signal length (see text). Other parameters are: ω = 1 GHz, g = 0.01 GHz, ϵ = 0 , Δ = 100 MHz... Including extra terms in the series expansion ( Sin5) makes the time dependence of the qubit dynamics analytically unwieldy, because the rational function form of the series leads to a complex interdepence of the positions of the poles in ( eqn:rsol1). However, if the values of the parameters are known the series can truncated at ( ± N M A X ) to give an efficient numerical method to obtain more accurate dynamics, extending the applicability of our approach beyond the regime described by ( eqn:crit). This is demonstrated in Fig. fig:2, where the dynamics are clearly dominated by two frequencies – an effect that could obviously never be captured by a single term approximation. There is a qualitative agreement between the many terms expansion and full numerical solution, particularly at short times. We would not expect a perfect agreement in this case because the simulations are of the dynamics in the large tunnelling regime ( Δ = 0.5 ), and the polaron transform makes the master equation perturbative in this parameter. Nonetheless, the rapid convergence of the series is shown in Fig. fig:2; N M A X = 5 - 10 is sufficient to calculate ρ 00 t and ρ 10 t with an accuracy only limited by the underlying Born Approximation. The asymmetry of the amplitudes of the terms in the series expansion of ( Sin5) is due to the exponential functions in the series.
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  • (a) Principle of the detection scheme. After the Rabi pulse, a microwave pulse at the plasma frequency resonantly enhances the escape rate. The bias current is maintained for 500 n s above the retrapping value. (b) Resonant activation peak for different Rabi angle. Each curve was offset by 5 % for lisibility. The Larmor frequency was f q = 8.5 ~ G H z . Pulse 2 duration was 10 ~ n s . (c) Resonant activation peak without (full circles) and after (open circles) a π pulse. The continuous line is the difference between the two switching probabilities. (d) Rabi oscillation measured by DC current pulse (grey line, amplitude A = 40 % ) and by resonant activation method with a 5 ~ n s RAP (black line, A = 62 % ), at the same Larmor frequency. fig4... The parameters of our qubit were determined by fitting spectroscopic measurements with the above formulae. For Δ = 5.855 ~ G H z , I p = 272 ~ n A , the agreement is excellent (see figure fig1c). We also determined the coupling constant between the SQUID and the qubit by fitting the qubit “step" appearing in the SQUID’s modulation curve (see insert of figure fig1c) and found M = 20 ~ p H . We first performed Rabi oscillation experiments with the DCP detection method (figure fig1b). We chose a bias point Φ x , tuned the microwave frequency to the qubit resonance and measured the switching probability as a function of the microwave pulse duration τ m w . The observed oscillatory behavior (figure fig2a) is a proof of the coherent dynamics of the qubit. A more detailed analysis of its damping time and period will be presented elsewhere ; here we focus on the amplitude of these oscillations.... (insert) Typical resonant activation peak (width 40 ~ M H z ), measured after a 50 ~ n s microwave pulse. Due to the SQUID non-linearity, it is much sharper at low than at high frequencies. (figure) Center frequency of the resonant activation peak as a function of the external magnetic flux (squares). It follows the switching current modulation (dashed line). The solid line is a fit yielding the values of the shunt capacitor and stray inductance given in the text. fig3... (a) Rabi oscillations at a Larmor frequency f q = 7.15 ~ G H z (b) Switching probability as a function of current pulse amplitude I without (closed circles, curve P s w 0 I ) and with (open circles, curve P s w π I b ) a π pulse applied. The solid black line P t h 0 I b is a numerical adjustment to P s w 0 I b assuming escape in the thermal regime. The dotted line (curve P t h 1 I b ) is calculated with the same parameters for a critical current 100 n A smaller, which would be the case if state 1 was occupied with probability unity. The grey solid line is the sum 0.32 P t h 1 I b + 0.68 P t h 0 I b . fig2... We then measure the effect of the qubit on the resonant activation peak. The principle of the experiment is sketched in figure fig4a. A first microwave pulse at the Larmor frequency induces a Rabi rotation by an angle θ 1 . A second microwave pulse of duration τ 2 = 10 n s is applied immediately after, at a frequency f 2 close to the plasma frequency, with a power high enough to observe resonant activation. In this experiment, we apply a constant bias current I b through the SQUID ( I b = 2.85 μ A , I b / I C = 0.85 ) and maintain it at this value 500 ~ n s after the microwave pulse to keep the SQUID in the running state for a while after switching occurs. This allows sufficient voltage to build up across the SQUID and makes detection easier, similarly to the plateau used at the end of the DCP in the previously shown method. At the end of the experimental sequence, the bias current is reduced to zero in order to retrap the SQUID in the zero-voltage state. We measured the switching probability as a function of f 2 for different Rabi angles θ 1 . The results are shown in figure fig4b. After the microwave pulse, the qubit is in a superposition of the states 0 and 1 with weights p 0 = c o s 2 θ 1 / 2 and p 1 = s i n 2 θ 1 / 2 . Correspondingly, the resonant activation signal is a sum of two peaks centered at f p 0 and f p 1 with weights p 0 and p 1 , which reveal the Rabi oscillations.... We show the two peaks corresponding to θ 1 = 0 (curve P s w 0 , full circles) and θ 1 = π (curve P s w π , open circles) in figure fig4c. They are separated by f p 0 - f p 1 = 50 ~ M H z and have a similar width of 90 ~ M H z . This is an indication that the π pulse efficiently populates the excited state (any significant probability for the qubit to be in 0 would result into broadening of the curve P s w π ), and is in strong contrast with the results obtained with the DCP method (figure fig2b). The difference between the two curves S f = P s w 0 - P s w π (solid line in figure fig4c) gives a lower bound of the excited state population after a π pulse. Because of the above mentioned asymmetric shape of the resonant activation peaks, it yields larger absolute values on the low- than on the high-frequency side of the peak. Thus the plasma oscillator non-linearity increases the sensitivity of our measurement, which is reminiscent of the ideas exposed in . On the data shown here, S f attains a maximum S m a x = 60 % for a frequency f 2 * indicated by an arrow in figure fig4c. The value of S m a x strongly depends on the microwave pulse duration and power. The optimal settings are the result of a compromise between two constraints : a long microwave pulse provides a better resonant activation peak separation, but on the other hand the pulse should be much shorter than the qubit damping time T 1 , to prevent loss of excited state population. Under optimized conditions, we were able to reach S m a x = 68 % .... A typical resonant activation peak is shown in the insert of figure fig3. Its width depends on the frequency, ranging between 20 and 50 ~ M H z . This corresponds to a quality factor between 50 and 150 . The peak has an asymmetric shape, with a very sharp slope on its low-frequency side and a smooth high-frequency tail, due to the SQUID non-linearity. We could qualitatively recover these features by simple numerical simulations using the RCSJ model . The resonant activation peak can be unambiguously distinguished from environmental resonances by its dependence on the magnetic flux threading the SQUID loop Φ s q . Figure fig3 shows the measured peak frequency for different fluxes around Φ s q = 1.5 Φ 0 , together with the measured switching current (dashed line). The solid line is a numerical fit to the data using the above formulae. From this fit we deduce the following values C s h = 12 ± 2 p F and L = 170 ± 20 p H , close to the design. We are thus confident that the observed resonance is due to the plasma frequency.... Finally, we fixed the frequency f 2 at the value f 2 * and measured Rabi oscillations (black curve in figure fig4d). We compared this curve to the one obtained with the DCP method in exactly the same conditions (grey curve). The contrast is significantly improved, while the dephasing time is evidently the same. This enhancement is partly explained by the rapid 5 ~ n s RAP (for the data shown in figure fig4d) compared to the 30 ~ n s DCP. But we can not exclude that the DCP intrinsically increases the relaxation rate during its risetime. Such a process would be in agreement with the fact that for these bias conditions, T 1 ≃ 100 ~ n s , three times longer than the DCP duration.... (a) AFM picture of the SQUID and qubit loop (the scale bar indicates 1 ~ μ m ). Two layers of Aluminium were evaporated under ± ~ 20 ~ ∘ with an oxidation step in between. The Josephson junctions are formed at the overlap areas between the two images. The SQUID is shunted by a capacitor C s h = 12 ~ p F connected by Aluminium leads of inductance L = 170 ~ p H (solid black line). The current is injected through a resistor (grey line) of 400 ~ Ω . (b) DCP measurement method : the microwave pulse induces the designed Bloch sphere rotations. It is followed by a current pulse of duration 20 ~ n s , whose amplitude I b is optimized for the best detection efficiency. A 400 ~ n s lower-current plateau follows the DCP and keeps the SQUID in the running-state to facilitate the voltage pulse detection. (c) Larmor frequency of the qubit and (insert) persistent-current versus external flux. The squares and (insert) the circles are experimental data. The solid lines are numerical adjustments giving the tunnelling matrix element Δ , the persistent-current I p and the mutual inductance M . fig1
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  • Figure  fig:Ttrans shows the numerically obtained coherence times and whether the decay is predominantly Gaussian or Markovian. For the large oscillator damping γ = ϵ , the conditions for the validity of the (Markovian) Bloch-Redfield equation stated at the end of Sec.  sec:analytics-g2 hold. Then we observe a good agreement of the numerically obtained T ⊥ * and Eq. ... (Color online) Dephasing time for purely quadratic qubit-oscillator coupling ( g 1 = 0 ), resonant driving at large frequency, Ω = ω 0 = 5 ϵ , and various values of the oscillator damping γ . The driving amplitude is A = 3.5 γ , such that always n ̄ = 6.125 . Filled symbols mark Markovian decay, while stroked symbols refer to Gaussian shape. The solid line depicts the value obtained for γ = ϵ in the Markov limit. The corresponding numerical values are connected by a dashed line which serves as guide to the eye.... (Color online) Typical time evolution of the qubit operator σ x (solid line) and the corresponding purity (dashed) for Ω = ω 0 = 0.8 ϵ , g 1 = 0.02 ϵ , γ = 0.02 ϵ , and driving amplitude A = 0.06 ϵ such that the stationary photon number is n ̄ = 4.5 . Inset: Purity decay shown in the main panel (dashed) compared to the decay given by Eq.  P(t) together with Eq.  Lambda(t) (solid line).... Figure  fig:timeevolution depicts the time evolution of the qubit expectation value σ x which exhibits decaying oscillations with frequency ϵ . The parameters correspond to an intermediate regime between the Gaussian and the Markovian dynamics, as is visible in the inset.... (Color online) Dephasing time for purely linear qubit-oscillator coupling ( g 2 = 0 ), resonant driving, Ω = ω 0 , and oscillator damping γ = 0.02 ϵ . The amplitude A = 0.07 ϵ corresponds to the mean photon number n ̄ = 6.125 . Filled symbols and dashed lines refer to predominantly Markovian decay, while for Gaussian decay, stroked symbols and solid lines are used.
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  • fig:dqdqpc Schematic of an isolated DQD qubit and capacitively coupled low-transparency QPC between source (S) and drain (D) leads.... In simple dyne detection (see schematic in Fig.  fig:rfcircuit), the output signal V o u t t is amplified, and mixed with a local oscillator (LO). The LO for homodyne detection of the amplitude quadrature is V L O t ∝ cos ω 0 t , where the LO frequency is the same as the signal of interest (or very slightly detuned). The resulting low-frequency beats due to mixing the signal with the LO are easily detected.... Equivalent circuit for continuous monitoring of a charge qubit coupled to a classical L C oscillator with inductance L and capacitance C . We consider the charge-sensitive detector that loads the oscillator circuit to be a QPC (see Fig. fig:dqdqpc for details). Measurement is achieved using reflection with the input voltage, V i n t , and the output voltage, V o u t t , being separated by a directional coupler. The output voltage is then amplified and mixed with a local oscillator, L O , and then measured. fig:rfcircuit... The choice e > 0 corresponds to defining current in terms of the direction of electron flow. That is, in the opposite direction to conventional current. The DQDs are occupied by a single excess electron, the location of which determines the charge state of the qubit. The charge basis states are denoted | 0 and | 1 (see Fig.  fig:dqdqpc). We assume that each quantum dot has only one single-electron energy level available for occupation by the qubit electron, denoted by E 1 and E 0 for the near and far dot, respectively.... The two conjugate parameters we use to describe the oscillator state are the flux through the inductor, Φ t , and the charge on the capacitor, Q t . The dynamics of the oscillator are found by analyzing the equivalent circuit of Fig.  fig:rfcircuit using the well-known Kirchhoff circuit laws. Doing this we find that the classical system obeys the following set of coupled differential equations... Consider the equivalent circuit of Fig.  fig:rfcircuit. The oscillator circuit consisting of an inductance L and capacitance C terminates the transmission line of impedance Z T L = 50 Ω . The voltages (potential drops) across the oscillator components can be written as
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  • Numerical and analytical evaluation of the entanglement dynamics between the two qubits for ω 0 = 0.15 ω , β = 0.16 and α = 3 . Entanglement between the qubits exhibits collapse and revival. The analytic expression agrees well with the envelope of the numerically evaluated entanglement evolution.... Here, within the adiabatic approximation, we extend the examination to the two-qubit case. Qualitative differences between the single-qubit and the multi-qubit cases are highlighted. In particular, we study the collapse and revival of joint properties of both the qubits. Entanglement properties of the system are investigated and it is shown that the entanglement between the qubits also exhibits collapse and revival. We derive what we believe are the first analytic expressions for the individual revival signals beyond the RWA, as well as analytic expression for the collapse and revival dynamics of entanglement. In the quasi-degenerate regime, the invalidity of the RWA in predicting the dynamical evolution will clearly be demonstrated in Sec. s.collapse_rev (see Figs. f.collapse_revival_double and f.collapse_revival_single).... The three potential wells corresponding to the states | 1 , 1 | N 1 (left), | 1 , 0 | N 0 (middle) and | 1 , - 1 | N -1 (right). The factor Δ X z p is the zero point fluctuation of a harmonic oscillator. For an oscillator of mass M and frequency ω the zero point fluctuation is given by Δ X z p = ℏ / 2 M ω .... Collapse and revival dynamics for ω 0 = 0.1 ω , β = 0.16 and α = 3 . The first two panels show analytic evaluations of (a.) one-qubit and (b.) two-qubit probability dynamics, and (c.) shows that the two-qubit analytic formula matches well to the corresponding numerical evolution. In each case the initial state is a product of a coherent oscillator state with the lowest of the S x states. Note the breakup in the main revival peak of the two-qubit numerical evaluation, which comes from the ω - 2 ω beat note, not included in the analytic calculation, and not present for a single qubit.... When squared, the probability shows two frequencies of oscillation, 2 Ω N ω and 2 2 Ω N ω . Since three new basis states are involved, we could expect three frequencies, but two are equal: | E N + - E N 0 | = | E N - - E N 0 | . This is in contrast to the single-qubit case where only one Rabi frequency determines the evolution . We show below in Fig. f.col_rev the way differences between one and a pair of qubits can be seen.... If the average excitation of the oscillator, n ̄ = α 2 , is large one can evaluate the above sum approximately (see Appendix) and obtain analytic expressions and graphs of the evolution, as shown in Fig. f.col_rev. As expected, because of the double frequency in ( eqnP(t)), the revival time for S t 2 ω 0 is half the revival time for S t ω 0 . Thus there are two different revival sequences in the time series. Appropriate analytic formulas, e.g., ( a.Phi), agree well with the numerically evaluated evolution even for the relatively weak coherent excitation, α = 3 .... Entanglement dynamics between the qubits. (a) ω 0 = 0.1 ω , β = 0.16 , (b) ω 0 = 0.15 ω , β = 0.16 , (c) ω 0 = 0.1 ω , β = 0.2 and (d) ω 0 = 0.15 ω , β = 0.2 . For all the figures, α = 3 .... Collapse and revival dynamics for P 1 2 , - 1 2 α t , given ω 0 = 0.15 ω , β = 0.16 and α = 3 . Note the single revival sequence. Also, note that there are no breakups in the revival peaks in contrast to the two-qubit case (Fig. f.collapse_revival_double). The RWA fails to describe the dynamical evolution even for the single qubit case.... With recent advances in the area of circuit QED, it is now possible to engineer systems for which the qubits are coupled to the oscillator so strongly, or are so far detuned from the oscillator, that the RWA cannot be used to describe the system’s evolution correctly . The parameter regime for which the coupling strength is strong enough to invalidate the RWA is called the ultra-strong coupling regime . Niemczyk, et al. and Forn-Díaz, et al. have been able to experimentally achieve ultra-strong coupling strengths and have demonstrated the breakdown of the RWA. Motivated by these experimental developments and the importance of understanding collective quantum behavior, we investigate a two-qubit TC model beyond the validity regime of RWA. The regime of parameters we will be concerned with is the regime where the qubits are quasi-degenerate, i.e., with frequencies much smaller than the oscillator frequency, ω 0 ≪ ω , while the coupling between the qubits and the oscillator is allowed to be an appreciable fraction of the oscillator frequency. In this parameter regime, the dynamics of the system can neither be correctly described under the RWA, nor can the effects of the counter rotating terms be taken as a perturbative correction to the dynamics predicted within the RWA by including higher powers of β . For illustration, systems are shown in Fig. f.model for which the RWA is valid, or breaks down, because the condition ω 0 ≈ ω is valid, or is violated. The regime that we will be interested in, for which ω 0 ≪ ω , is shown on the right.... There is only one revival sequence for the single qubit system as a consequence of having only one Rabi frequency in the single qubit case. The analytic and numerically exact evolution of P 1 2 , - 1 2 α t is plotted in Fig. f.collapse_revival_single. The single revival sequence is evident from the figure. A discussion on the multiple revival sequences for the K -qubit TC model, within the parameter regime where the RWA is valid, can be found in .
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  • (color online). Rabi oscillations for a squbit-fluctuator system. The probability p 1 t to find the squbit in state | 1 is obtained from numerical integration of Eq. ( eq:mastereq) (solid line) and the analytical solution Eq. ( eq:pdyn-3) (dashed line) which is valid for b x / J ≫ 1 . For weak decoherence of the fluctuator, γ / J 1 , damped single-frequency oscillations are restored. The fluctuator leads to a reduction of the first maximum in p 1 t to ∼ 0.8 [(a) and (b)] and ∼ 0.9 [(c) and (d)], respectively. The parameters are (a) b x / J = 1.5 , γ / J = 0.5 ; (b) b x / J = 1.5 , γ / J = 1.5 ; (c) b x / J = 3 , γ / J = 0.5 ; (d) b x / J = 3 , γ / J = 1.5 .... exhibits single-frequency oscillations with reduced visibility [Fig.  Fig2(b)]. Part of the visibility reduction can be traced back to leakage into state | 2 . More subtly, off-resonant transitions from | 1 to | 2 induced by the driving field lead to an energy shift of | 1 , such that the transition from | 0 to | 1 is no longer resonant with the driving field, which also reduces the visibility. For b x / 2 ℏ Δ ω = 1 / 3 , corresponding to b x / h = 150 ~ M H z in Ref. ... Energy shifts induced by AC driving field. – Exploring the visibility reduction at a time-scale of 10 n s requires Rabi frequencies | b x | / h 100 M H z . We show next that, in this regime, transitions to the second excited squbit state lead to an oscillatory behavior in p 1 t with a visibility smaller than 0.7 . For characteristic parameters of a phase-squbit, the second excited state | 2 is energetically separated from | 1 by ω 21 = 0.97 ω 10  . Similarly to | 0 and | 1 , the state | 2 is localized around the local energy minimum in Fig.  Fig2(a). For adiabatic switching of the AC current, transitions to | 2 can be neglected as long as | b x | ≪ ℏ Δ ω = ℏ ω 10 - ω 21 ≃ 0.03 ℏ ω 10 . However, for b x comparable to ℏ Δ ω , the applied AC current strongly couples | 1 and | 2 because 2 | φ ̂ | 1 ≠ 0 , where φ ̂ is the phase operator. For typical parameters, b x / ℏ Δ ω ranges from 0.05 to 1 , depending on the irradiated power . Taking into account the second excited state of the phase-squbit, the squbit Hamiltonian in the rotating frame is 
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  • Parameters in calculations and measurements in units of GHz. In the first column, cal: δ ω Ω R 0 stands for the calculation to study the shift of the resonant frequency, and cal: Γ R s t δ ω m w stands for the calculation to study the decay of Rabi oscillations due to quasistatic flux noise. “Optimal" in the last column means that at each ε m w , ω m w is chosen to minimize dephasing due to quasistatic flux noise.... (Color online) Rabi oscillation curves with different Rabi frequencies Ω R measured at different static flux bias ε . At each Ω R , δ ω m w is chosen to minimize dephasing due to quasistatic flux noise. The red lines are the fitting curves.  In the measurements shown in the middle and bottom panels, only parts of the oscillations are monitored so that we can save measurement time while the envelopes of Rabi oscillations are captured. The inset is a magnification of the data in the bottom panel together with the fitting curve.... In the Rabi oscillation measurements, a microwave pulse is applied to the qubit followed by a readout pulse, and P s w as a function of the microwave pulse length is measured. First, we measure the Rabi oscillation decay at ε = 0 , where the quasistatic noise contribution is negligible. Figure  GRfR1p5(d) shows the measured 1 / e decay rate of the Rabi oscillations Γ R 1 / e as a function of Ω R 0 . For Ω R 0 / 2 π up to 400 MHz, Γ R 1 / e is approximately 3 Γ 1 / 4 , limited by the energy relaxation, and S Δ Ω R 0 is negligible. For Ω R 0 / 2 π from 600 MHz to 2.2 GHz, Γ R 1 / e > 3 Γ 1 / 4 . A possible origin of this additional decoherence is fluctuations of ε m w , δ ε m w : Ω R 0 is first order sensitive to δ ε m w , which is reported to be proportional to ε m w itself.  Next, the decay for the case ε ≈ Δ is studied. To observe the contribution from quasistatic flux noise, the Rabi oscillation decay as a function of ω m w is measured, where the contribution from the other sources is expected to be almost constant. Figure  GRfR1p5(b) shows Γ R 1 / e at ε / 2 π = 4.16 GHz as a function of δ ω m w while keeping Ω R / 2 π between 1.5 and 1.6 GHz. Besides the offset and scatter, the trend of Γ R 1 / e agrees with that of the simulated Γ R s t . This result indicates that numerical calculation properly evaluates δ ω m w minimizing Γ R s t . Finally, the decay for the case ε ≈ Δ as a function of ε m w , covering a wide range of Ω R , is measured (Fig.  Rabis).... (Color online) Power spectrum density of flux fluctuations S n φ ω extracted from the Rabi oscillation measurements in the first ( ε / 2 π = 4.16 GHz) and second cooldowns. The PSDs obtained from the spin-echo and energy relaxation measurements in the second cooldown are also plotted. The black solid line is the 1/ f spectrum extrapolated from the FID measurements in the second cooldown. The purple dashed line is the estimated Johnson noise from a 50  Ω microwave line coupled to the qubit by a mutual inductance of 1.2 pH and nominally cooled to 35 mK. The pink dotted line is a Lorentzian, S n φ m o d e l ω = S h ω w 2 / ω 2 + ω w 2 , and the orange solid line is the sum of the Lorentzian and the Johnson noise. Here the parameters are S h = 3.6 × 10 -19 r a d -1 s and ω w / 2 π = 2.7 × 10 7 H z .... Josephson devices, decoherence, Rabi oscillation, $1/f$ noise... (Color online) (a) Numerically calculated shift of the resonant frequency δ ω (black open circles) and the Bloch–Siegert shift δ ω B S (blue line). (b) Numerically calculated decay rate Γ R s t (black open circles) and Rabi frequency Ω R (red solid triangles) as functions of the detuning δ ω m w from ω 01 . The purple solid line is a fit based on Eq. ( fRfull). The measured 1/ e decay rates Γ R 1 / e at ε / 2 π = 4.16 GHz for the range of Rabi frequencies Ω R / 2 π between 1.5 and 1.6 GHz (blue solid circles) are also plotted. (c) Calculated Rabi frequency Ω R , based on Eq. ( fRfull), as a function of ε for the cases (i)  ω m w = ω 01 + δ ω (black solid line) and (ii)  ω m w / 2 π = 6.1  GHz (red dashed line). The upper axis indicates ω 01 , corresponding to ε in the bottom axis. (d) The measured 1 / e decay rate of the Rabi oscillations, Γ R 1 / e , at ε = 0 and as a function of Ω R 0 . The red solid line indicates 3 4 Γ 1 obtained independently.... The condition, ∂ Ω R / ∂ ε = 0 , is satisfied when ε = 0 or δ ω m w = δ ω - Ω R 0 2 / ω 01 . For Ω R 0 / 2 π = 1.52 GHz and ω 01 / 2 π = 6.400 GHz, the latter condition is calculated to be δ ω m w / 2 π = - 295  MHz, slightly different from the minimum of Γ R s t seen in Fig.  G R f R 1 p 5 (b). The difference is due to the deviation from the linear approximation in Eq. ( fRfull), Ω R 0 ∝ ε m w / ω 01 . Figure  GRfR1p5(c) shows the calculation of Ω R as a function of ε , based on Eq. ( fRfull). The Rabi frequency Ω R 0 at the shifted resonance decreases as ε increases, while Ω R , for a fixed microwave frequency of ω m w / 2 π = 6.1  GHz, has a minimum of approximately ω 01 / 2 π = 6.4 GHz. Here in the first order, Ω R is insensitive to the fluctuation of ε .... In Fig.  GRfR1p5(a), δ ω as a function of Ω R 0 is plotted together with the well-known Bloch–Siegert shift,  δ ω B S = 1 4 Ω R 0 2 ω 01 , obtained from the second-order perturbation theory. Fixed parameters for the calculation are Δ / 2 π = 4.869 and ε / 2 π = 4.154  GHz ( ω 01 / 2 π = 6.400  GHz). We find that δ ω B S overestimates δ ω when Ω R 0 / 2 π 800  MHz. The deviation from the Bloch–Siegert shift is due to the component of the ac flux drive that is parallel to the qubit’s energy eigenbasis; this component is not averaged out when Ω R is comparable to ω m w .
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