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  • Figure   fig:FreqVsAmpl(a) shows linecuts of the experimental (black dots) and numerical (full red lines) chevrons. The linecuts are taken at the frequency ω F C corresponding to the maximum-visibility sideband oscillations, indicated by the full and dashed vertical lines in Fig. 3. The agreement between the experiments and simulations is excellent. In particular, the decay rate of the oscillations can be explained by the separately measured loss of the qubit and cavity and roughly corresponds to κ + γ 1 / 2 , where γ 1 is the bare transmon relaxation rate. This is expected for oscillations between states | e 0 and | g 1 . It also indicates that for these powers, the visibility loss can be completely attributed to damping. The lack of experimental points at pulse widths < 30 n s is a technical limit of the present configuration of our electronics that can be improved in future experiments.x x... (color online) (a) Schematic of energy levels in a combined qubit-resonator system, showing first-order red sideband transition. (b) Optical microscope image with inset showing expanded view of one of the qubits. The terminations of the flux-bias lines for both qubits are visible, and they are used for both dc bias and FC signals. (c) Schematic of qubit-cavity layout and signal paths.... We used a sample consisting of two asymmetric transmon qubits capacitively coupled to the voltage antinodes of a coplanar waveguide resonator [Fig. fig:schem(b, c)]. The cavity had a bare fundamental resonance frequency ω r / 2 π = 8.102 G H z and decay rate κ / 2 π = 0.37 M H z . Qubit-state measurements were performed in the high-power limit . The qubits, labeled Q1 and Q2, were designed to be identical, with mutual inductances to their bias lines of 1 p H for Q2 and 2 p H for Q1. The qubits were excited by microwave pulses sent through the resonator, and the flux lines were used for dc flux biasing of the qubits as well as the high-speed flux modulation pulses for exciting sideband transitions. The dc flux lines included cryogenic filters before connecting to a bias-T for joining to the ac flux line, which had 20 / 6 / 10 d B of attenuation at the 4 K / 0.7 K / 0.03 K plates. The distribution of cold attenuators and the flux-bias mutual inductances were chosen as a compromise to allow for a sufficient flux amplitude for high-speed modulation of the qubit energy levels with negligible Joule heating of the refrigerator while avoiding excessive dissipation coupled to the qubits from the flux-bias lines.... (color online) Spectroscopy vs. flux for Q2 showing g-e (solid blue points) and e-f (hollow red points) transition frequencies. Blue and red lines correspond to numerical fits. Heavy black line shows bare cavity resonance frequency. Vertical dashed line indicates flux bias point for sideband measurements described in subsequent figures along with ac flux drive amplitude, 2 Δ Φ = 70.9  m Φ 0 , corresponding to 2 Δ ω g e / 2 π = 572  MHz, used in Figs. 3(c), 4(c).... Figure   fig:FreqVsAmpl(d) shows the sideband oscillation frequency Ω / 2 π extracted from the experimental linecuts (blackxdots) as a function of the flux-modulation amplitude Δ Φ . As expected from Eq. ( eq:H:t), whose prediction is given by the solid black line, the dependence of Ω with Δ Φ is linear at low amplitude and deviates at larger amplitudes. Beyond this simple model with only two transmon levels, quantitative agreement is found between the measured data and numerical simulations (full red line). For the numerical simulations, the link between the theoretical flux modulation amplitude Δ Φ and applied power is made by taking advantage of the linear dependence of Ω with Δ Φ at low power. Because of this, it is possible to convert the experimental flux amplitude from arbitrary units to m Φ 0 using only the lowest drive amplitude for calibration.... (color online) (a-c) Experimental data showing sideband oscillations as a function of pulse duration vs. flux-drive frequency. The amplitude of the flux pulse is reduced by (a) 10 d B , (b) 4 d B relative to (c). (d-f) Corresponding numerical simulations of sideband oscillations vs. drive frequency. Vertical white lines running through each plot indicate the frequency slices used in Fig. fig:FreqVsAmpl.... (color online) (a),(b),(c) Sideband oscillations corresponding to the white slices in Fig. fig:chevron(a-c). Experimental points correspond to black dots; numerical simulations (not fits) indicated by red lines. (d) Sideband oscillation frequency vs. flux drive amplitude (lower horizontal axis) or corresponding frequency modulation amplitude (upper horizontal axis). The dashed line shows a linear fit to the low frequency data points, while the red solid line indicates the theoretical dependence from the numerical simulations. The full black line shows the analytical sideband oscillation frequency from Eq. ( eq:H:t).
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  • The first-qubit oscillation frequency f d as a function of time t (normalized by the energy relaxation time T 1 ) for C x = 0 (solid line) and C x = 6 fF (dashed line), assuming N l 1 = 1.355 and parameters of Eq. ( 2.16). Dash-dotted horizontal line, ω r 1 / 2 π = 15.3 GHz, shows the long-time limit of f d t . Two dotted horizontal lines show the plasma frequency for the second qubit: ω l 2 / 2 π = 10.2 GHz for N l 2 = 10 and ω l 2 / 2 π = 8.91 GHz for N l 2 = 5 . The arrow shows the moment t c of exact resonance in the case N l 2 = 5 .... The circuit schematic of a flux-biased phase qubit and the corresponding potential profile (as a function of the phase difference δ across the Josephson junction). During the measurement the state | 1 escapes from the “left” well through the barrier, which is followed by oscillations in the “right” well. This dissipative evolution leads to the two-qubit crosstalk.... The oscillating term in Eq. ( 3.11a) describes the beating between the oscillator and driving force frequencies, with the difference frequency increasing in time, d t ~ 2 / d t = α t - t c , and amplitude of beating decreasing as 1 / t ~ (see dashed line in Fig.  f4a). Notice that F 0 = 1 / 4 , F ∞ = 1 , and the maximum value is F 1.53 = 1.370 , so that E 0 is the long-time limit of the oscillator energy E 2 , while the maximum energy is 1.37 times larger:... The second qubit energy E 2 (in units of ℏ ω l 2 ) in the oscillator model as a function of time t (in ns) for (a) C x = 5 fF and T 1 = 25 ns and (b) C x = 2.5 fF and 5 fF and T 1 = 500 ns, while N l 2 = 5 . Dashed line in (a) shows approximation using Eq. ( 3.10). The arrows show the moment t c when the driving frequency f d (see Fig.  f3) is in resonance with ω l 2 / 2 π = 8.91 GHz.... mcd05, a short flux pulse applied to the measured qubit decreases the barrier between the two wells (see Fig.  f0), so that the upper qubit level becomes close to the barrier top. In the case when level | 1 is populated, there is a fast population transfer (tunneling) from the left well to the right well. Due to dissipation, the energy in the right well gradually decreases, until it reaches the bottom of the right well. In contrast, if the qubit is in state | 0 the tunneling essentially does not occur. The qubit state in one of the two potential minima (separated by almost Φ 0 ) is subsequently distinguished by a nearby SQUID, which completes the measurement process.... Now let us consider the effect of dissipation in the second qubit. ... Dots: Rabi frequencies R k , k - 1 / 2 π for the left-well transitions at t = t c , for N l = 10 , C x = 6 fF, and T 1 = 25 ns. Dashed line shows analytical dependence 1.1 k GHz.... 2.16 Figure f2 shows the qubit potential U δ for N l = 10 (corresponding to φ = 4.842 ), N l = 5 ( φ = 5.089 ), and N l = 1.355 ( φ = 5.308 ); the last value corresponds to the bias during the measurement pulse (see below). The qubit levels | 0 and | 1 are, respectively, the ground and the first excited levels in the left well.... Solid lines: log-log contour plots for the values of the error (switching) probability P s = 0.01 , 0.1, and 0.3 on the plane of relaxation time T 1 (in ns) and coupling capacitance C x (in fF) in the quantum model for (a) N l 2 = 5 and (b) N l 2 = 10 . The corresponding results for C x , T T 1 in the classical models are shown by the dashed lines (actual potential model) and the dotted lines [oscillator model, Eq. ( bound1)]. The numerical data are represented by the points, connected by lines as guides for the eye. The scale at the right corresponds to the operation frequency of the two-qubit imaginary-swap quantum gate.... 3.17 in the absence of dissipation in the second qubit ( T 1 ' = ∞ ) for N l 2 = 5 and 10, while T 1 = 25 ns. (In this subsection we take into account the mass renormalization m → m ' ' explicitly, even though this does not lead to a noticeable change of results.) A comparison of Figs.  f4(a) and f7 shows that in both models the qubit energy remains small before a sharp increase in energy. However, there are significant differences due to account of anharmonicity: (a) The sharp energy increase occurs earlier than in the oscillator model (the position of short-time energy maximum is shifted approximately from 3 ns to 2 ns); (b) The excitation of the qubit may be to a much lower energy than for the oscillator; (c) After the sharp increase, the energy occasionally undergoes noticeable upward (as well as downward) jumps, which may overshoot the initial energy maximum; (d) The model now explicitly describes the qubit escape (switching) to the right well [Figs.  f7(b) and f7(c)]; in contrast to the oscillator model, the escape may happen much later than initial energy increase; for example, in Fig.  f7(b) the escape happens at t ≃ 44 ns ≫ t c ≃ 2.1 ns.
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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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  • The profile of coherent quantum oscillations in an unbiased qubit dephased by the non-Gaussian noise with characteristic amplitude v 0 = 0.15 Δ and correlation time τ = 300 Δ -1 obtained by direct simulation of qubit dynamics with noise. Solid line is the exponential fit of the oscillation amplitude at large times. Dashed line is the initial 1 / t decay caused by effectively static distribution of v .... The rate γ of exponential qubit decoherence at long times t ≫ τ for ε = 0 and (a) Gaussian and (b) a model of the non-Gaussian noise with characteristic amplitude v 0 and correlation time τ . Solid lines give analytical results: Eq. ( e7) in (a) and Eq. ( e16) in (b). Symbols show γ extracted from Monte Carlo simulations of qubit dynamics. Note different scales for γ in parts (a) and (b). Inset in (b) shows schematic diagram of qubit basis states fluctuating under the influence of noise v t .
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  • An important feature of the qubit relaxation in the presence of driving is that the stationary distribution over the qubit states differs from the thermal Boltzmann distribution. If the oscillator-mediated decay is the dominating qubit decay mechanism, the qubit distribution is determined by the ratio of the transition rates Γ e and Γ g . One can characterize it by effective temperature T e f f = ℏ ω q / k B ln Γ e / Γ g . If the term in curly brackets in the numerator of Eq. ( eq:resonant_power_spectrum) is dominating, T e f f ≈ 2 T , but if the field parameters are varied so that this term becomes comparatively smaller T e f f increases, diverges, and then becomes negative, approaching -2 T . Negative effective temperature corresponds to population inversion. The evolution of the effective temperature with the intensity of the modulating field is illustrated in Fig.  fig:effect_temp.... Apart from the proportionality to r a 2 , the attractor dependence of Γ 1 is also due to the different curvature of the effective potentials around the attractors. For weak oscillator damping κ ≪ ν a , the parameter ν a in Eq. ( eq:resonant_power_spectrum) is the frequency of small-amplitude vibrations about attractor a . It sets the spacing between the quasienergy levels, the eigenvalues of the rotating frame Hamiltonian H S r close to the attractor. The function Re  N + - ω has sharp Lorentzian peaks at ω = ± ν a with halfwidth κ determined by the oscillator decay rate. The dependence of ν a on the control parameter β is illustrated in Fig.  fig:nu_a.... Left panel: Squared scaled attractor radii r a 2 as functions of the dimensionless field intensity β for the dimensionless friction κ / | δ ω | = 0.3 . Right panel: The effective frequencies ν a / | δ ω | for the same κ / | δ ω . Curves 1 and 2 refer to small- and large amplitude attractors.... The scaled decay rate factors for the excited and ground states, curves 1 and 2, respectively, as functions of scaled difference between the qubit frequency and twice the modulation frequency; Γ 0 = ℏ C Γ r a 2 / 6 γ S . Left and right panels refer to the small- and large-amplitude attractors, with the values of β being 0.14 and 0.12, respectively. Other parameters are κ / | δ ω = 0.3 , n ̄ = 0.5 .... The scaled decay rates Γ e , g as functions of detuning ω q - 2 ω F are illustrated in Fig.  fig:decay_spectra. Even for comparatively strong damping, the spectra display well-resolved quasi-energy resonances, particularly in the case of the large-amplitude attractor. As the oscillator approaches bifurcation points where the corresponding attractor disappears, the frequencies ν a become small (cf. Fig.  fig:nu_a) and the peaks in the frequency dependence of Γ e , g move to ω q = 2 ω F and become very narrow, with width that scales as the square root of the distance to the bifurcation point. We note that the theory does not apply for very small ω q - 2 ω F | where the qubit is resonantly pumped; the corresponding condition is m ω 0 Δ q δ C r e s 2 r a 2 / ℏ ω q 2 ≪ T 1 T 2 -1 + ω q - 2 ω F 2 T 1 T 2 . For weak coupling to the qubit, Γ e ≪ κ , it can be satisfied even at resonance.... Picot08, and was increasing with the driving strength on the low-amplitude branch (branch 1 in the left panel of Fig.  fig:nu_a), in qualitative agreement with the theory. It is not possible to make a direct quantitative comparison because of an uncertainty in the qubit relaxation rates noted in Ref. ... The effective scaled qubit temperature T e f f * = k B T e f f / ℏ ω q as function of the scaled field strength β in the region of bistability for the small- and large-amplitude attractors, left and right panels, respectively; ω q - 2 ω F / | δ ω | = - 0.2 and 0.1 in the left and right panels; other parameters are the same as in Fig.  fig:decay_spectra.... The decay rate of the excited state of the qubit Γ e ∝ R e ~ N + - ω q - 2 ω F sharply increases if the qubit frequency ω q coincides with 2 ω F ± ν a , i.e., ω q - 2 ω F resonates with the inter-quasienergy level transition frequency. This new frequency scale results from the interplay of the system nonlinearity and the driving and is attractor-specific, as seen in Fig.  fig:nu_a. In the experiment, for ω q close to 2 ω 0 , the resonance can be achieved by tuning the driving frequency ω F and/or driving amplitude F 0 . This quasienergy resonance destroys the QND character of the measurement by inducing fast relaxation.
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  • (Color online) (a) Nonlinear response A of the detector coupled to the qubit prepared in its ground state | ↓ (orange solid line) and in its excited state | ↑ (black dashed line) for the same parameters as in Fig.  fig2. The quadratic qubit-detector coupling induces a global frequency shift of the response by δ ω e x = 2 g . (b) Discrimination power D ω e x of the detector coupled to the qubit for the same parameters as in a). fig3... (Color online) (a) Asymptotic population difference P ∞ of the qubit states, and (b) the corresponding detector response A as a function of the external frequency ω e x for the same parameters as in Fig.  fig2. fig4... (Color online) (a) Relaxation rate Γ of the nonlinear quantum detector, (b) the measurement time T m e a s , and (c) the measurement efficiency Γ m e a s / Γ as a function of the external frequency ω e x . The parameters are the same as in Fig.  fig2. fig5... For a fixed value of g , the shift between the two cases of the opposite qubit states is given by the frequency gap δ ω e x ≃ 2 g . Figure fig3 (a) shows the nonlinear response of the detector for the two cases when the qubit is prepared in one of its eigenstates: | ↑ (orange solid line) and | ↓ (black dashed line).... (Color online) (a) Amplitude A of the nonlinear response of the decoupled quantum Duffing detector ( g = 0 ) as a function of the external driving frequency ω e x . (b) The corresponding quasienergy spectrum ε α . The labels N denote the corresponding N -photon (anti-)resonance. The parameters are α = 0.01 Ω , f = 0.006 Ω , T = 0.006 Ω , and γ = 1.6 × 10 -4 Ω . fig1... Before turning to the quantum detection scheme, we discuss the dynamical properties of the isolated detector, which is the quantum Duffing oscillator. A key property is its nonlinearity which generates multiphoton transitions at frequencies ω e x close to the fundamental frequency Ω . In order to see this, one can consider first the undriven nonlinear oscillator with f = 0 and identify degenerate states, such as | n and | N - n (for N > n ), when δ Ω = α N + 1 / 2 . For finite driving f > 0 , the degeneracy is lifted and avoided quasienergy level crossings form, which is a signature of discrete multiphoton transitions in the detector. As a consequence, the amplitude A of the nonlinear response signal exhibits peaks and dips, which depend on whether a large or a small oscillation state is predominantly populated. The formation of peaks and dips goes along with jumps in the phase of the oscillation, leading to oscillations in or out of phase with the driving. A typical example of the nonlinear response of the quantum Duffing oscillator in the deep quantum regime containing few-photon (anti-)resonances is shown in Fig.  fig1(a) (decoupled from the qubit), together with the corresponding quasienergy spectrum [Fig.  fig1(b)]. We show the multiphoton resonances up to a photon number N = 5 . The resonances get sharper for increasing photon number, since their widths are determined by the Rabi frequency, which is given by the minimal splitting at the corresponding avoided quasienergy level crossing. Performing a perturbative treatment with respect to the driving strength f , one can get the minimal energy splitting at the avoided quasienergy level crossing 0 N as... (Color online) Nonlinear response A of the detector as a function of the external driving frequency ω e x in the presence of a finite coupling g = 0.0012 Ω to the qubit (black solid line). The blue dashed line indicates the response of the isolated detector. The parameters are the same as in Fig.  fig1 and ϵ = 2.2 Ω and Δ = 0.05 Ω , in correspondence to realistic experimental parameters . fig2... Notice that g and α depend on the external flux ϕ e x , i.e., they are tunable in a limited regime with respect to the desired oscillator frequency Ω , where the coupling term is considered as a perturbation to the SQUID ( g oscillator to dominate. The dependence of the dimensionless ratios α / Ω and g / Ω is shown in Fig.  fig0.
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  • (Color online) (a) The qubit frequency as a function of parameter β L for fixed L = 50  pH and several values of capacitance C = 0.1 , 0.3, 1.0 and 3.0 pF (from top to bottom), corresponding to the values of the ratio E L / E c ≈ 1.7 × 10 4 , 5.1 × 10 4 , 1.7 × 10 5 and 5.1 × 10 5 . (b) Anharmonicity parameter δ as a function of parameter β L for the same as in (a) inductance L and capacitance values (from top to bottom).... (Color online) Position of the lowest six levels (solid lines) in the potential Eq. ( U-phi) for φ e = π as a function of parameter β L for typical values of L and C , yielding E J / E c ∼ E L / E c ≈ 5.1 × 10 4 . With an increase of β L , the spectrum crosses over from that of the harmonic oscillator type (left inset) to the set of the doublets (right inset), corresponding to the weak coupling of the oscillator-type states in two separate wells. The spectrum in the central region β L ≈ 1 is strongly anharmonic. The dashed line shows the bottom energy of the potential U φ φ e = π , which in the case of β L > 1 is equal to - Δ U ≈ - 1.5 E L β L - 1 2 / β L (in other words, Δ U is the height of the energy barrier in the right inset) . The dotted (zero-level) line indicates the energy in the symmetry point φ = 0 , i.e. at the bottom of the single well ( β L ≤ 1 ) or at the top of the energy barrier ( β L > 1 ). The black dot shows the critical value β L c at which the ground state energy level touches the top of the barrier separating the two wells.... where Î is the operator of supercurrent circulating in the qubit loop. The dependence of the reverse inductance L J Φ e = Φ 0 / 2 n calculated numerically in the two lowest quantum states ( n = 0 and 1) for L = 50  pH and the same set of capacitances C as in Fig.  frequency-anharmonicity is shown in Fig.  inductance-L01. One can see that the ratio of the geometrical to Josephson inductances L / L J takes large and very different values that can be favorably used for the dispersive readout, ensuring a sufficiently large output signal. Note that for β L 1 the inductance L J n = 1 changes the sign to positive.... Figure f-shift shows this relative frequency shift versus parameter β L . One can see that for the rather conservative value of dimensionless coupling κ = 0.05 , the relative frequency shift can achieve the easily measured values of about 10 % . The efficiency of the dispersive readout can be improved in the non-linear regime with bifurcation . With our device this regime can be achieved in the resonance circuit including, for example, a Josephson junction (marked in the diagram in Fig. 1 by a dashed cross). Due to the high sensitivity of the amplitude (phase) bifurcation to the threshold determined by the effective resonance frequency of the circuit, one can expect a readout with high fidelity even at a rather weak coupling of the qubit and the resonator (compare with the readout of quantronium in Ref. ). Further improvement of the readout can be achieved in the QED-based circuit including this qubit .... Such a large, positive anharmonicity is a great advantage of the quartic potential qubit allowing manipulation within the two basis qubit states | 0 and | 1 not only when applying resonant microwave field, ν μ w ≈ ν 10 , but also when applying control microwave signals with large frequency detuning or using rather wide-spectrum rectangular-pulse control signals. The characteristic qubit frequency ν 10 = Δ E 0 / h and the anharmonicity factor δ computed from the Schrödinger equation for the original potential Eq. ( U-phi) in the range 0.9 ≤ β L ≤ 1.02 are shown in Fig.  frequency-anharmonicity. One can see that the significant range in the tuning of the qubit frequency within the range of sufficiently large anharmonicity ( ∼ 20 - 50 % ) is attained at a rather fine (typically ± 1 - 2 % ) tuning of β L around the value β L = 1 . Such tuning of β L is possible in the circuit having the compound configuration shown in Fig. 1b. For values of β L > 1 , the symmetric energy potential has two minima and a barrier between them. The position of the ground state level depends on β L and the ratio of the characteristic energies E J / E c = β L E L / E c . The value of β L at which the ground state level touches the top of the barrier sets the upper limit β L c for the quartic qubit (marked in Fig.  levels-beta by solid dot). At β L > β L c , the qubit energy dramatically decreases and the qubit states are nearly the symmetric and antisymmetric combinations of the states inside the two wells (see the right inset in Fig. 2). Although the qubit with such parameters has very large anharmonicity and can be nicely controlled by dc flux pulses , its readout can hardly be accomplished in a dispersive fashion.... (a) Electric diagram of the qubit coupled to a resonant circuit and (b) possible equivalent compound (two-junction SQUID) circuit of the Josephson element included into the qubit loop. Capacitance C includes both the self-capacitance of the junction and the external capacitance. Due to inclusion in the resonant circuit of a Josephson junction JJ’, the resonator may operate in the nonlinear regime, enabling a bifurcation-based readout.... (Color online) The resonance frequency shift in the circuit due to excitation of the qubit with the inductance value L = 50  pH and the set of capacitances C , decreasing from top to bottom. The dimensionless coupling coefficient κ = 0.05 .... (Color online) The values of the Josephson inductance of the quartic potential qubit in the ground (solid lines) and excited (dashed lines) states calculated for the geometric inductance value L = 50  pH and the set of capacitances C , increasing from top to bottom for both groups of curves.
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  • (Color online) (a) The occupation of the excited state on resonance versus microwave amplitude in units of the corresponding Rabi frequency. The lines are calculations for the following parameters; red solid Γ = 0.055 n s -1 and σ ν = 0.235 n s -1 , blue dotted line σ ν = 0 with Γ = 0.055 n s -1 , purple dashed line σ δ = 0 and Γ = 0.20 n s -1 . (b) The width of the spectroscopic peak from the Gaussian fits as a function of microwave amplitude in units of Rabi frequency. The lines are calculations for the following parameters; red solid Γ = 0.055 n s -1 and σ ν = 0.235 n s -1 , green dashed line σ ν = 0 with Γ = 0.055 n s -1 , blue dotted line σ ν = 0 and Γ = 0.585 n s -1 .... The effects of flux noise on the decay of the Rabi oscillation, of course, becomes much more pronounced for δ ≠ 0 . Figure rabidetunefig shows such Rabi oscillations at various detunings over a range of 0 ≤ δ 10 9 s -1 . At long microwave pulse times (not shown) the occupation of the excited state reaches the equilibrium values discussed in Sec secintraspec. Near resonance, the value of ρ 11 for long pulse lengths is determined mainly by the noise amplitude, while for large δ it is set by δ . The data cannot be fit to solutions of the Bloch equations using only the phenomenological decay constants. As in the spectroscopy, static Gaussian detuning noise must be included to fit the data over the whole range of detuning. The lines in Figure rabidetunefig correspond to calculations with an initial Rabi frequency of 0.59 n s -1 , Γ = 0.075 n s -1 and σ ν = 0.22 n s -1 . The detuning frequency is taken from φ x using the conversion from Fig.  spec2dfig. For the range of δ in Fig.  rabidetunefig Δ E 01  is in a region with relatively few spectral anomalies. However for δ < 0 , on the other side of resonance towards splittings in the spectroscopy, the data generally do not agree with theory suggesting that the spectroscopic splittings have a strong effect on the coherence, as expected.... The data for resonant Rabi oscillations, shown in Fig. rabifig. The solid line is a fit using Eq. fin including a 0.5 ns time delay for the rise time for the microwave pulse and averaged over quasi-static Gaussian noise in φ x equal to that obtained from the fits to the spectroscopy data in Fig. specfig. This gives a Rabi frequency f r a b i = 119 M H z and decay rate, η = 0.042 n s -1 , for the oscillations. Equation equ:rabidecay, with δ = 0 , together with our previously measured value for the decay rate of the first excited state Γ 1 (which gives the rate Γ in the equations above), and the observed Rabi decay rate η discussed below, imply Γ ν = 0.01 n s -1 .  Even though Ω R a b i is not affected by flux noise to first order for δ = 0 , the amplitude of the flux noise in our qubit is large enough to cause a measurable effect. If the flux noise were neglected, it would be necessary to increase η to 0.058 n s -1 in order to account for the observed decay. In addition to increasing the decay rate, the low frequency noise also reduces the steady state occupation of the excited state , i.e. its value for long pulses, as calculated from Eq. av. The ordinate of Fig. specfig has been calibrated using this value and is consistent with the estimate obtained  from the calculated tunneling rates from the two levels involved during the readout pulse.... (Color online) (a) Schematic and (b) micrograph of an rf SQUID qubit and the readout magnetometer, (c) a cross section of the wafer around the junctions and (d) micrograph giving a detailed view of junctions.... The components that create and combine the microwave pulses and readout pulse are shown at the top of Fig.  fig:circuitdiag. The pulse generator is capable of producing measurement pulses with rise times as short as 200 ps. Two microwave mixers are used to modulate the envelope of the continuous microwave signal produced by the microwave source giving an on/off ratio of 10 3 . The output of the mixers is then amplified 20 dB and coupled through variable attenuators that set the amplitude of the microwave pulses applied to the qubit. Finally, these microwave pulses are combined with the video pulses using a hybrid coupler. The video pulses enter on the directly coupled port while the microwave pulse are coupled using the indirectly coupled port. The video pulses used for the data shown in the following sections are around typically 5 ns with a rise time of 0.5 ns. The directionality and frequency response of the coupler allows the two signals to be combined with a minimum amount of reflections and loss of power. This signal is then coupled to the qubit through a coaxial line that is filtered by a series of attenuators at 1.4 K (20 dB), 600 mK (10 dB) and at the qubit temperature (10 dB) followed by a lossy microstrip filter that cuts off around 1 GHz.... The measurement sequence for coherent oscillations is very similar to that used to measure the lifetime of the excited state except that the duration of the (generally shorter) microwave pulse is varied and the microwave pulse is immediately followed by the readout flux pulse. This signal, on the high frequency line, is illustrated in the lower inset in Figure rabifig. Figure rabifig shows an example of the Rabi oscillations when δ = 0 and the microwave frequency, f x r f = 17.9 G H z . This bias point lies in a "clean" range of the spectrum ( 17.6 - 18 G H z ) as seen in Fig. spec2dfig, which should be the best region for observing coherent oscillations between the ground and excited states. Most of the time domain data, including the lifetime measurements of Fig.  decayfig, have been taken in this frequency range. Each data point in Fig. spec2dfig corresponds to the average of several thousand measurements for a given pulse length.... This solution, shows that when driven with microwaves, the population of the excited states oscillates in time, demonstrating the phenomenon of Rabi oscillations, which have been seen in a number of different superconducting qubits . Ω ,  the frequency of the oscillations for δ = 0 , is ideally proportional to the amplitude of the microwaves excitation φ x r f . This linear dependence is accurately seen in our qubit at low power levels (see Fig. rabifig upper inset) providing a convenient means to calibrate the amplitude of the microwaves incident on the qubit in terms of Ω .... (Color online) Rabi oscillations for detunings going from top to bottom of 0.094, 0.211, 0.328, 0.562, 0.796 and 1.269 n s -1 with the corresponding fits using Γ = 0.075 n s -1 and σ ν = 0.22 n s -1... (Color online) The occupation of the excited state as a function of detuning for microwave powers corresponding to attenuator settings of 39 (squares), 36 (circles), 33 dB (triangles). Lines are fits using Eq.  av at microwave amplitudes corresponding to the measured Rabi frequency for each attenuator setting (0.017, 0.024, 0.034 n s -1 ) with Γ = 0.055 n s -1 convoluted with static Gaussian noise with σ ν = 0.235 n s -1 at the angular frequencies of the Rabi oscillations that correspond to these microwave powers... (Color online) The occupation of the excited state as a function of the length of the microwave pulse demonstrating Rabi oscillations. The line is a fit to Eq. fin for δ = 0 averaged over quasi-static noise with σ ν = 0.22 n s -1 . This gives f r a b i = 119 MHz and decay time T 2 ˜ = 24 n s -1 . The upper inset shows Rabi frequency as a function of amplitude of applied microwaves in arbitrary units. The line is a linear fit to the lower microwave amplitude data. The lower inset show the measurement pulse sequence.... Decoherence \and Superconducting Qubits \and Flux Qubit \and SQUIDs
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  • (a) Detail of the qubit spectroscopy near Δ U / ℏ ω p = 3.55 , showing splittings of strengths S ≈ 44 MHz and 24 MHz. (b) Tunneling probability versus measurement delay time τ D after application of π -pulse. Solid (dashed) line is taken at a well depth of solid (dashed) arrow in (a), corresponding to a resonant (off-resonant) bias. Inset illustrates how the qubit probability amplitude first moves to state | 1 g and then oscillates between | 1 g and | 0 e . (c) and (d) Tunneling probability (gray scale) versus well depth and τ D for experimental data (c) and numerical simulation (d). The peak oscillation periods are observed to correspond to the spectroscopic splittings.... Spectroscopy of ω 10 obtained using the current-pulse measurement method, as a function of well depth Δ U / ℏ ω p . For each value of Δ U / ℏ ω p , the grayscale intensity is the normalized tunneling probability, with an original peak height of 0.1 - 0.3 . Insets: A given splitting in the spectroscopy of magnitude S comes from a critical-current fluctuator coupled to the qubit with strength h S / 2 . On resonance, the qubit-fluctuator eigenstates are linear combinations of the states | 1 g and | 0 e , where | g and | e are the two states of the fluctuator.... (a) Room temperature measurement of the fast current pulse. (b) Tunneling probability versus δ I m a x with the qubit in state | 0 (solid circles) and in an equal mixture of states | 1 and | 0 (open circles). Fit to data is shown by the solid line. The plateau, being less than 0.5, corresponds to a maximum measurement fidelity of 0.63.... (a) Schematic of the qubit circuitry. For the qubit used in Fig. 2, the Josephson critical-current and junction capacitance are I 0 ≈ 10 μ A and C ≈ 2 pF; in Figs. 3 and 4, each of these values is about 5 times smaller. (b) Potential energy landscape and quantized energy levels for I φ = I d c prior to the state measurement. (c) At the peak of δ I t , the qubit well is much shallower and state | 1 rapidly tunnels to the right hand well.
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  • Energy-level diagram of a qubit and transitions created by a bichromatic field at double resonance ( ω 0 = ω , ω 1 = ω r f ).
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