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tau0 The tangle as a function of α in the symmetric case W = 0 for different values of the **qubit** tunnelling amplitude D . One can appreciate that the result of Eq. ( tangl) is indeed reached asymptotically.... Concerning the asymmetric case, our results for the ground state entanglement appear similar to those found by Costi and McKenzie in Ref. , where the interaction of a **qubit** with an ohmic environment was numerically analyzed. It turns out that, for a bath with finite band-width, the entanglement displays a behavior analogous to that reported in Figs. ( tau10)-( tau01), when plotted with respect to the value of the impedance of the bath. Here, instead, we concentrated on the dependence of the tangle on the coupling strength between the **qubit** and the environmental **oscillator**. Unfortunately, the coupling strength is not easily related to the coefficient of the spectral density used in Ref. , and therefore one cannot make a precise comparison between th... pot The lower adiabatic potential for D = 10 and α = 2 . The dashed line refers to the symmetric, W = 0 , case (dashed line), while the solid line refers to W = 1 . The case of frozen **qubit** ( W = D = 0 ) would have given a pair of independent parabolas instead of the adiabatic potentials U l , u of Eq. ( udq).... As we have shown, the procedure is easily extended to the asymmetric case and this is important since the entanglement changes dramatically for any finite (however small) value of the asymmetry in the **qubit** Hamiltonian. As mentioned in section sect2 above, this is due to the fact the this term modifies the symmetry properties of the Hamiltonian, so that the form of the ground state changes radically and the same occurs to the reduced **qubit** state. For example, for a large enough interaction strength, the **qubit** state is a complete mixture if W = 0 , while it becomes the lower eigenstate of σ z if W 0 . As a result, for large α , there is much entanglement if W = 0 , while the state of the system is factorized and thus τ = 0 if W 0 . This is seen explicitly in Fig. ( tau10). Furthermore, from the comparison of Figs. ( tau10), ( tau01), and ( tau0), one can see that, with increasing α , the tangle increases monotonically in the symmetric case, while it reaches a maximum before going down to zero if W 0 . This is due to the fact that, in the first case, the ground state of the system becomes a Schrödinger cat-like entangled superposition, approximately given by — 12 { — + —- - — - —+ } , for 1 , schroca where | φ ± are the two coherent states for the **oscillator** defined in Eq. ( due1), centered in Q = ± Q 0 , respectively, and almost orthogonal if α ≫ 1 . In the presence of asymmetry, on the other hand, the **oscillator** localizes in one of the wells of its effective potential and this implies that, for large α , the ground state is given by just one of the two components superposed in Eq. ( schroca). This is, clearly, a factorized state and therefore one gets τ = 0 . Since τ is zero for uncoupled sub-systems (i.e., for very small values of α ), weather W = 0 or not, and since, for W 0 , it has to decay to zero for large α , it follows that a maximum is present in between. In fact, for intermediate values of the coupling, there is a competition between the α -dependences of the two non zero components of the Bloch vector. In particular, the length | b → | is approximately equal to one for both small and large α ’s, see Figs. ( asx)-( asz), but the vector points in the x direction for α ≪ 1 and in the z direction for α ≫ 1 . The maximum of the tangle in the asymmetric case occurs near the point in which b x ≈ b z . For the symmetric case, we were also able to derive analytically the sharp increase of the entanglement at α = 1 . This behavior appears to be reminiscent of the super-radiant transition in the many **qubit** Dicke model, which, in the adiabatic limit, shows exactly the same features described here, and which can be described along similar lines. Finally, we would like to comment on the relationship of this work with those of Refs. and . The approach proposed by Levine and Muthukumar, Ref. , employs an instanton description for the effective action. This has been applied to obtain the entropy of entanglement in the symmetric case, in the same critical limit described above. It turns out that this description is equivalent to a fourth order expansion of the lower adiabatic potential U l . This approximation, although retaining all the distinctive qualitative features discussed above, gives slight quantitative changes in the results. Concerning the asymmetric case, our results for the ground state entanglement appear similar to those found by Costi and McKenzie in Ref. , where the interaction of a **qubit** with an ohmic environment was numerically analyzed. It turns out that, for a bath with finite band-width, the entanglement displays a behavior analogous to that reported in Figs. ( tau10)-( tau01), when plotted with respect to the value of the impedance of the bath. Here, instead, we concentrated on the dependence of the tangle on the coupling strength between the **qubit** and the environmental **oscillator**. Unfortunately, the coupling strength is not easily related to the coefficient of the spectral density used in Ref. , and therefore one cannot make a precise comparison between the two results. At least qualitatively, however, we can say that the ground state quantum correlations induced by the coupling with an ohmic environment are already present when the **qubit** is coupled to a single **oscillator** mode. 99 weiss U. Weiss, Quantum Dissipative Systems, 2 nd ed., World Scientific 1999. yuma see, e.g., Yu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). levine G. Levine and V. N. Muthukumar, Phys. Rev. B 69, 113203 (2004). martinis R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and J. M. Martinis, Phys. Rev. Lett. 93 077003 (2005); P. R. Johnson, W. T. Parsons, F. W. Strauch, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, Phys. Rev. Lett. 94, 187004 (2005). pino E. Paladino, L. Faoro, G. Falci, and R. Fazio, Phys. Rev. Lett. 88, 228304 (2002); G. Falci, A. D’Arrigo, A. Mastellone, and E. Paladino, Phys. Rev. Lett. 94, 167002 (2005) hines A.P. Hines, C.M. Dawson, R.H. McKenzie and G.J. Milburn, Phys. Rev. A 70, 022303 (2004). blais A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004); A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M.H. Devoret, S. M. Girvin and R. J. Schoelkopf, Phys. Rev. Lett. 95, 060501 (2005). prb03 F. Plastina and G. Falci, Phys. Rev. B 67, 224514 (2003). costi T.A. Costi and R.H. McKenzie, Phys. Rev. A 68, 034301 (2003). ent1 A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002); T. J. Osborne, and M. A. Nielsen Phys. Rev. A 66, 032110 (2002). ent2 G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003); L. A. Wu, M. S. Sarandy, and D. A. Lidar, Phys. Rev. Lett. 93, 250404 (2004). ent3 T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and V. Tognetti, Phys. Rev. Lett. 94, 147208 (2005). ent4 N. Lambert, C. Emary, and T. Brandes, Phys. Rev. Lett. 92, 073602 (2004). crisp M.D. Crisp, Phys. Rev. A 46, 4138 (1992). Irish E.K. Irish, J. Gea-Banacloche, I. Martin, and K. C. Schwab, Phys. Rev. B 72, 195410 (2005). Rungta V. Coffman, J. Kundu, and W.K. Wootters, Phys. Rev. A 61, 052306 (2000); T. J. Osborne, Phys. Rev. A 72, 022309 (2005), see also quant-ph/0203087. Wallraff A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M.H. Devoret, S. M. Girvin and R. J. Schoelkopf, Phys. Rev. Lett. 95, 060501 (2005). Nakamura Y. Nakamura, Yu.A. Pashkin and J.S. Tsai, Phys. Rev. Lett. 87, 246601 (2001). armour A.D. Armour, M.P. Blencowe and K.C. Schwab, Phys. Rev. Lett. 88, 148301 (2002). Grajcar M. Grajcar, A. Izmalkov and E. Ilxichev, Phys. Rev. B 71, 144501 (2005). Chiorescu I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C.J.P.M. Harmans and J.E. Mooij, Nature 431, 159 (2004).... wf Normalized ground state wave function for the **oscillator** in the lower adiabatic potential, for D = 10 and α = 2 and with W = 0 (dashed line) and W = 0.1 (solid line).

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To study the fidelity of the population transfer between | a and | s , in Fig. subfig:a2s15P, we choose the anti-symmetric state as the initial condition. Applying a continuous-wave TDMF with Rabi **frequency** Ω 0 = 15 γ 0 and detuning δ = 0 , the symmetric state | s reaches its maximum population of 0.90 at time 0.1 γ 0 -1 . After this maximum, the population continues to **oscillate** between | a and | s due to the applied field. This **oscillation** is damped by an overall decay as we include damping with a rate γ 0 . The corresponding concurrence is shown in in Fig. subfig:a2s15C. The concurrence **oscillates** at twice the **frequency** of the population **oscillation**, since both | s and | a are maximally entangled. The local maximum values of the entanglement occur at times n π / 2 δ 2 + Ω 0 2 where either | s or | a is occupied.... In Fig. subfig:figg2sP, we show results for the population of | s for continuous driving. It can be seen that the population reaches a maximum value, but afterwards exhibits rapid **oscillations** at **frequency** 2 2 Ω 0 , while the amplitude of the subsequent maxima in the population decays as an exponential function exp - γ 0 + γ 12 t until the system approaches its stationary state. This result can be understood by reducing the system to a two-state system only involving in the states | s and | g . The numerical results can be well fitted by the solution of this two-state approximation,... (Color online) Time evolution of the population (solid red line) in the antisymmetric state | a . The idea is to prepare the anti-symmetric state while the two **qubits** are non-degenerate, and only afterwards render the two **qubits** degenerate. For this, the parameters are chosen such that γ 12 = 0.9986 γ 0 , λ = 50 γ 0 , δ = 50 γ 0 , Ω 0 = 50 γ 0 . The two **qubit** transition **frequencies** are adjusted via time-dependent bias fluxes, such that the **frequency** difference Δ t (dashed black line) changes from 18 γ 0 to zero as a cosine function during the time period 120 γ 0 -1 to 160 γ 0 -1 . The driving field (dash-dotted blue line) is turned off from its initial value Ω 0 in the period 165 γ 0 -1 to 175 γ 0 -1 .... Our model system consists of two flux **qubits** coupled to each other through their mutual inductance M and to a reservoir of harmonic **oscillators** modeled as an LC circuit, see Fig. fig:system.... An example for this is shown in Fig. fig:figPadtn. Initially, the two **qubits** have a **frequency** difference Δ t = 0 = Δ 0 = 18 γ 0 . Applying a continuous TDMF Ω during 0 ≤ γ 0 t ≤ 165 allows to populate the antisymmetric state, as can be seen in Fig. fig:figPadtn. After a certain time ( γ 0 t = 120 in our example), the bias fluxes are continuously adjusted such that the two **qubits** become degenerate, Δ γ 0 t ≥ 160 = 0 . It can be seen from Fig. fig:figPadtn that a preparation fidelity for the antisymmetric state in the degenerate two-**qubit** system of about F = 0.94 is achieved. Finally, the TDMF is switched off as well in the time period 165 ≤ γ 0 t ≤ 175 , demonstrating that it is not required to preserve the population in the antisymmetric state. It should be noted that this scheme does not rely on a delicate choice and control of parameters, as it is the case, e.g., for state preparation via special-area pulses.... fig:BellSCRAPdtn(Color online) Creation of superpositions of Bell states controlled by the static detuning δ 0 . The populations ρ ± in states | Φ + (dashed red line) and | Φ - (dash-dotted green line) exhibit periodic **oscillations** as a function of δ 0 . The maximum concurrence C is larger than 0.95 (solid blue line).... fig:BellSCRAP(Color online) Creation of Bell states | Φ ± using the SCRAP technique from the ground state. The solid red line denotes population ρ + in | Φ + , while the dashed green line shows population ρ - in | Φ - . The concurrence (dash-dotted blue line) has a maximum value of 0.94 . The black dash double dotted line indicates the applied SCRAP pulse Rabi **frequency**, while the thick blue line shows the time-dependent Stark detuning.... fig:SCRAPg2s(Color online) Robust populating the symmetric state from the ground state via the SCRAP technique for γ 12 = 0.9986 γ 0 , and λ = 50 γ 0 . The solid black line shows the population of the desired symmetric state, while the thick solid blue line and the dash-dotted red line are the time-dependent Rabi **frequency** and detuning required for SCRAP.... fig:systemTwo superconducting flux **qubits** interacting with each other through their mutual inductance M and damped to a common reservoir modeled as an LC circuit. The individual bias fluxes are varied dynamically in order to control the **qubit** transition **frequencies** around the optimum point. In the figure, crosses indicate Josephson junctions, whereas the bottom circuit loop visualizes the bath.... where Φ 0 = h / 2 e is the flux quantum, the system in Fig. fig:system can be described by the total Hamiltonian H = H Q + H B . The two-**qubit** Hamiltonian H Q in two-level approximation and rotating wave approximation is given by

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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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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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(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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(color) Numerically optimized free-evolution period T for some of clock protocols considered here, when the **oscillator** noise has an Allan deviation of 1 Hz.... (color) Probability (P) of measuring each basis state as a function of the atom-**oscillator** phase difference ( φ ). Shown are the various protocols for two and five atoms. Each differently colored curve corresponds to a basis state that ψ 1 is projected onto after free evolution. Vertical text near the curves’ peaks indicates the optimized phase estimate ( φ E s t ). In the simulations, the **frequency** corrections are φ E s t / 2 π T . Shaded in the background is the Gaussian distribution whose variance φ 2 represents the atom-**oscillator** phase differences that occur in the simulation. Also listed is the optimized probe period T , squeezing parameter κ where applicable, and long-term **frequency** variance of the clock extrapolated to 1 second. For long-term averages of n seconds, the variance is f 2 / n .... Numerical simulations of the clock protocols considered here are summarized in Figure figPerf. Ramsey’s protocol defines the standard quantum limit (SQL), and it is evident that entangled states of two or more **qubits** can reduce clock instability, although the GHZ states yield no gain for the noise model considered here, as has been noted previously . The spin-squeezed states suggested by André et al. yield the best performance for 3 to 15 **qubits**, and improve upon the SQL variance by a factor of N -1 / 3 . For more **qubits**, the protocol of Bu... (color online) Long-term statistical variance of entangled clocks that contain different numbers of **qubits**, compared to the standard quantum limit (SQL). The most stable clocks found by the large-scale search are shown as black points. Each point is based on several hours of runtime on NISTxs computing cluster, where typically 2000 processor cores were utilized in parallel. Also shown is the simulated performance of analytically optimized clock protocols. Approximately 15 **qubits** are required to improve upon the SQL by a factor of two.

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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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fig3 Experimental device: a transmon **qubit** coupled to a nonlinear superconducting cavity. (a) Circuit diagram of the device; the anharmonic resonator is formed from a meander inductor embedded with a Josephson junction and an interdigitated capacitor. The resonator is isolated from the 50 Ω environment by coupling capacitors C c and coupled to a transmon **qubit** characterized by Josephson and charging energy scales E J and E c respectively. The coupling rate is g . (b) Scanning electron micrograph of the the device with the resonator and **qubit** junctions (lower and upper insets).... fig2 Energy levels and the effective nonlinearity λ of the strongly coupled system. (a) The measured coefficient of nonlinear response for a strongly coupled system versus **qubit**-cavity detuning for the **qubit** prepared in the ground state in the low excitation regime. Theory curves for the model Eq. ( eq:1) with N l = 2 (blue), N l = 3 (gray), and for a model of coupled nonlinear classical **oscillators** (red) are shown. The arrows indicate the locations of avoided crossings of the level pairs | 1 , 3 ↔ | 0 , 4 and | 1 , 4 ↔ | 0 , 5 . (Inset) The transmission of the resonator when driven with tone at ω d that occupied the resonator with = 0.4 (black) and = 10 (gray) off-resonant photons. (b) Energy levels of the **qubit**-**oscillator** model with N l = 7 show the avoided crossings in the 4 and 5 excitation manifold. (c) Quantum trajectory simulation of the system exhibits a general trends of increasing effective nonlinearity λ with diminishing **qubit**-cavity detuning ( Δ ) with abrupt reductions associated with avoided crossings in the 4 and 5 excitation manifold. For these simulations, the **qubit** energy levels were modeled as a Duffing nonlinearity.... fig2 Measured autoresonance and threshold sensitivity on the level structure and initial **qubit** state. (a) Color plot shows S | 1 versus **qubit** detuning. The dashed line indicates the AR threshold, V | 1 . AR measurements were not taken for small values of the detuning as indicated by the hatched region. (b) Color plot of S | 0 with V | 0 indicated as a solid black line. The AR threshold, V | 1 , is also plotted for comparison as a dashed black line. The two arrows indicate the location of avoided crossings in the 4 and 5 excitation manifold.... fig3duffing (a) The transmitted magnitude for chirp sequences with drive voltages that were above (black), near (dashed), and below (gray) the AR threshold. (Inset) Pulse sequence: the **qubit** manipulation pulse was applied immediately before the start of the chirp sequence. (b) The average transmitted magnitude near 400 ns versus drive voltage shows S | 0 (black) and S | 1 (red) for Δ / 2 π = 0.59 GHz.

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For maximum generality, we first define a minimal model needed to describe the splitting of Fig. fig:Splitting. To this end, we restrict ourselves to the lowest two states of the phase **qubit** circuit (the **qubit** subspace) and disregard the longitudinal coupling ∝ τ z . Within the rotating wave approximation (RWA) the Hamiltonian reads... (color online) (a) Analytically obtained transition spectrum of the Hamiltonian ( eq:H4Levels) in the minimal model for Ω q / h = 40 MHz and v ⊥ / h = 25 MHz. Dashed-dotted lines show the transition **frequencies** while the gray-scale intensity of the thicker lines indicates the weight of the respective Fourier-components in the probability P . The system shows a symmetric response as a function of the detuning δ ω . Two of the four lines are double degenerate. (b) The same as (a) but including the second order Raman process with Ω f v = v ⊥ Ω q / Δ . The two degenerate transitions in (a) split and the symmetry of the response is broken. Inset: Schematic representation of the structure of the Hamiltonian ( eq:H4Levels). We denote the ground and excited states of the **qubit** as and and those of the TLF as and . Arrows indicate the couplings between **qubit** and fluctuator v ⊥ and to the microwave field Ω q and Ω f v .... The sample investigated in this study is a phase **qubit** , consisting of a capacitively shunted Josephson junction embedded in a superconducting loop. Its potential energy has the form of a double well for suitable combinations of the junction’s critical current (here, I c = 1.1 μ A) and loop inductance (here, L = 720 pH). For the **qubit** states, one uses the two Josephson phase eigenstates of lowest energy which are localized in the shallower of the two potential wells, whose depth is controlled by the external magnetic flux through the **qubit** loop. The **qubit** state is controlled by an externally applied microwave pulse, which in our sample is coupled capacitively to the Josephson junction. A schematic of the complete **qubit** circuit is depicted in Fig. fig:Splitting(a). Details of the experimental setup can be found in Ref. ... (color online) (a) Schematic of the phase **qubit** circuit. (b) Probability to measure the excited **qubit** state (color-coded) vs. bias flux and microwave **frequency**, revealing the coupling to a two-level defect state having a resonance **frequency** of 7.805 GHz (indicated by a dashed line).... superconducting **qubits**, Josephson junctions, two-level
fluctuators, microwave spectroscopy, Rabi **oscillations**
... (color online) (a) Experimentally observed time evolution of the probability to measure the **qubit** in the excited state, P t , vs. driving **frequency**; (b) Fourier-transform of the experimentally observed P t . The resonance **frequency** of the TLF is indicated by vertical lines. (c) Time evolution of P and (d) its Fourier-transform obtained by the numerical solution of Eq. ( eq:master_eq) as described in the text, taking into account also the next higher level in the **qubit**. (As the anharmonicity Δ / h ∼ 100 MHz in our circuit is relatively small, this required going beyond the second order perturbation theory and analyze the 6-level dynamics explicitly). The **qubit**’s Rabi **frequency** Ω q / h is set to 48 MHz.... Spectroscopic data taken in the whole accessible **frequency** range between 5.8 GHz and 8.1 GHz showed only 4 avoided level crossings due to TLFs having a coupling strength larger than 10 MHz, which is about the spectroscopic resolution given by the linewidth of the **qubit** transition. In this work, we studied the **qubit** interacting with a fluctuator whose energy splitting was ϵ f / h = 7.805 GHz. From its spectroscopic signature shown in Fig. fig:Splitting(b), we extract a coupling strength v ⊥ / h ≈ 25 MHz. The coherence times of this TLF were measured by directly driving it at its resonance **frequency** while the **qubit** was kept detuned. A π pulse was applied to measure the energy relaxation time T 1 , f ≈ 850 ns, while two delayed π / 2 pulses were used to measure the dephasing time T 2 , f * ≈ 110 ns in a Ramsey experiment. To read out the resulting TLF state, the **qubit** was tuned into resonance with the TLF to realize an iSWAP gate, followed by a measurement of the **qubit**’s excited state.... where δ ω = ϵ q - ϵ f . The level structure and the spectrum of possible transitions in the Hamiltonian ( eq:H4Levels) is illustrated in Fig. fig:Transitionsa. The transition **frequencies** in the rotating frame correspond to the **frequencies** of the Rabi **oscillations** observed experimentally.... Figure fig:DataRabi(a) shows a set of time traces of P taken for different microwave drive **frequencies**. Each trace was recorded after adjusting the **qubit** bias to result in an energy splitting resonant to the chosen microwave **frequency**. The Fourier transform of this data, shown in Fig. fig:DataRabi(b), allows us to distinguish several **frequency** components. **Frequency** and visibility of each component depend on the detuning between **qubit** and TLF. We note a striking asymmetry between the Fourier components appearing for positive and negative detuning of the **qubit** relative to the TLF’s resonance **frequency**, which is indicated in Figs. fig:DataRabi(a,b) by the vertical lines at 7.805 GHz. We argue below that this asymmetry is due to virtual Raman-transitions involving higher levels in the **qubit**.

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Phase **qubit** coupled to a tank circuit.... In our method a resonant tank circuit with known inductance L T , capacitance C T and quality factor Q T is coupled with a target Josephson circuit through the mutual inductance M (Fig. fig1). The method was successfully applied to a three-junction **qubit** in classical regime, when the hysteretic dependence of ground-state energy on the external magnetic flux was reconstructed in accordance to the predictions of Ref.

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