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  • Reconstructed process matrix for a) Û and b) two repetitions of Û . The map E i j produces a matrix E k , l for each element i j . Hence elements of the matrix E are labeled by m = 4 i - 1 + k , n = 4 j - 1 + l , where the factor 4 results from the size of the two-qubit state space. For example, the 11 00 ( i = 1 , j = 4 ) element of an input density matrix is mapped to E 11 00 , a 4 × 4 block of E given by m ∈ 1 4 and n ∈ 13 16 . The position of each peak is in agreement with the theoretical prediction.... \centering Time and Frequency Division, National Institute of Standards and Technology, Boulder, CO 80305, U.S.A. ... Hybrid qubit storage in the 9 B e + 2s 2 S 1 / 2 hyperfine levels. The states are labeled using the total angular momentum quantum numbers F and M F . 1 , 0 are the qubit states used for single qubit gates and transport, and 1 G , 0 G are used for two-qubit gates. For detection, the 1 , - 1 and 2 , 2 states are used. At the applied magnetic field ( B ≃ 0.011964 T), the frequencies for transitions between pairs of states with the same F are well resolved.... a) Schematic of the qubit ion trajectories (solid red and dotted blue lines) and gate operations used to implement Û . The single qubit rotations are “ π / 2 ” ≡ R π / 2 0 (eq. eq:rotation). The two-qubit gate implements G ̂ = D 1 i i 1 . b) Full sequence used to perform process tomography on Û and Û 2 . This sequence is repeated 350 times for each setting of preparation/analysis.
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  • (Color online) Upper panel: Rabi frequency | Ω R | in units of c = n p h g c 2 / L for μ = - 100 ϵ L and κ = 5 ϵ L , where ϵ L = 2 m L 2 -1 . A large photon linewidth κ has been chosen to highlight the essential features. The crosses denote the Rabi frequency and damping determined numerically from Eq. ( eq:Dgamma2). Solid green and red lines correspond to the solutions for the limits Ω | μ | , see Eq. ( eq:GammaOmega_metallic), respectively. Lower panel: The ratio between Rabi frequency and damping, Ω R / Γ R , determines the fidelity of qubit rotations.... These functions are plotted in Fig.  fig:plots. The Rabi frequency is, as expected, exponentially suppressed in the length of the CR. However, as the photon frequency Ω approaches the critical value | μ | , the prefactor 1 - Ω / | μ | qubit state for a time t * = π / 4 Ω R . In the presence of damping, the fidelity of such an operation can be estimated as... (Color online) Upper panel: A semiconductor nanowire (along the x axis) hosting Majorana fermions is embedded in a microwave stripline cavity (along the y axis). The red lines show the amplitude of the electric field E → r → . Dark blue (light yellow) sections of the wire indicate topologically nontrivial (trivial) regions. MBSs (stars) exist at the edges of nontrivial (topological superconductor, TS) regions. The MBSs γ 1 and γ 2 can be braided using a T -junction . Lower panel: Band structure of the individual sections of the wire. The four MBSs γ 1 , 2 , 3 , 4 encode one logical qubit. The central MBSs γ 2 and γ 3 are tunnel-coupled ( t c ) to a topologically trivial, gapped central region (CR, light yellow) with length L . All energies are small compared to the induced gap Δ .
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  • Single-qubit-gate parameters used for programming the quantum processor to produce the states in Figure 2 of the main text.... Components for universal computation. Circuit diagrams for arbitrary unitary transformations on a, one and b, two qubits. The operations for each circuit are implemented from left to right with each line representing one qubit. Part a also indicates the decomposition employed for R θ φ . The dashed box in part b contains the three degrees of freedom α , β , δ that determine the two-qubit operation’s local equivalence class. The brackets highlight the decomposition of the two-qubit operation U as described in the text: U = C ⊗ D ⋅ V ⋅ A ⊗ B .... Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA
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  • Structured bath/weak coupling: We now turn to the structured spectral density given by Eq.( eq:density). The main features of the corresponding system [Eq.( eq:system)] can already be understood by analyzing only the coupled two-level-harmonic oscillator system (without damping, i.e. Γ = 0 ). For ε = 0 this system exhibits two characteristic frequencies, close to Ω and Δ , associated with the transitions 1 and 2 in Fig.  fig:correlation_weak(c). These should also show up in the correlation function C ω ; and indeed Fig.  fig:correlation_weak(a) displays a double-peak structure with the peak separation somewhat larger than Δ - Ω , due to level repulsion. The coupling to the bath will in general lead to a broadening of the resonances and an enhancement of the repulsion of the two energies. Due to the very small coupling ( α = 0.0006 ) peak positions of C ω in Fig.  fig:correlation_weak can with very good accuracy be derived from a second order perturbation calculation for the coupled two-level-harmonic oscillator system, yielding the following transition frequencies [depicted in inset (c) of Fig.  fig:correlation_weak]: ω 1 , + - ω 0 , + = Ω - g 22 Δ 0 / Δ 2 0 - Ω 2 ≈ 0.987 Ω and ω 0 , - - ω 0 , + = Δ 0 + g 22 Δ 0 / Δ 2 0 - Ω 2 ≈ 1.346 Ω . With the two peaks we associate two different dephasing times, τ Ω and τ Δ , as shown in inset (a) and (b) of Fig.  fig:correlation_weak.... Spin-spin correlation function as a function of frequency for experimentally relevant parameters discussed in Ref. : α = 0.0006 , Δ 0 = 4 GHz, ε = 0 (this is the so-called “idle state”), Ω = 3 GHz, Γ = 0.02 , and ω c = 8 GHz. The sum rule is fulfilled with an error of less than 1 %. (a) Blow up of the peak region reveals a double peak; (b) blow up of the larger peak, (c) term scheme of a two level system coupled to an harmonic oscillator, drawn for Δ 0 ≫ Ω ; ( α = 0.0006 corresponds to g / Ω ≈ 0.06 .) -0.9cm... Stronger coupling to bath: Figure  fig:times_nobias(b) shows τ Δ , τ Ω , and τ w for a larger coupling strength of α = 0.01 . Figure  fig:correlation_strong(a) shows one of the calculated correlation functions. Note that the stronger coupling α leads to a larger separation, or “level repulsion”, between the Δ - and Ω -peaks than in Fig.  fig:correlation_weak. The inset of Fig.  fig:times_nobias(b) shows the renormalized tunneling matrix element Δ ∞ as a function the initial matrix element Δ 0 . Very importantly, for Δ 0 Ω , Δ increases during the flow, whereas for Δ 0 Ω , it decreases . This behavior can be understood from the fact that f ω l in Eq.( eq:flow1) changes sign at ω = Δ : If the weight of J ω under the integral in ( eq:flow1) is larger for ω > Δ 0 , which is the case if Δ Δ 0 ]. Note also, that the upward renormalization towards larger Δ ∞ in the inset of Fig.  fig:times_nobias(b) is stronger than the downward one towards smaller values, i.e., the renormalization is not symmetric with respect to Δ 0 = Ω . The reason for this asymmetry lies in the fact that f ω l has a larger weight for ω Δ . Also τ Δ and even τ w = 1 / J Δ ∞ in Fig.  fig:times_nobias(b) show an asymmetric behavior with a steep increase at Δ 0 ≈ Ω : dephasing times for Δ 0 > Ω are larger than for Δ 0 Ω than for Δ 0 Ω , dephasing times can be significantly enhanced (as compared to Δ 0 oscillator in (b).... Spin-spin correlation function for the structured bath [Eq.( eq:density)] as a function of frequency . The maximum height of the middle peak in (b) is ≈ 7.2 . -0.9cm
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  • Left: Schematic illustration of an experimental arrangement for measuring the phase dependence of the population of the excited state | 1 : (a) The microwave field couples the ground state ( | 0 ) to the excited state ( | 1 ). A third level, | 2 , which can be coupled to | 1 optically, is used to measure the population of | 1 via fluorescence detection. (b) The microwave field is turned on adiabatically with a switching time-constant τ s w , and the fluorescence is monitored after a total interaction time of τ . Right: Illustration of the Bloch-Siegert Oscillation (BSO): (a) The population of state | 1 , as a function of the interaction time τ , showing the BSO superimposed on the conventional Rabi oscillation. (b) The BSO oscillation (amplified scale) by itself, produced by subtracting the Rabi oscillation from the plot in (a). (c) The time-dependence of the Rabi frequency. Inset: BSO as a function of the absolute phase of the field with fixed τ .
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  • Power out of the resonator as a function of frequency for different input powers at zero flux bias.... V in, rms / V out, rms versus V out, rms 2 at Φ a = 0 and the low power resonance frequency for zero flux bias (f=8.2423Ghz). The circles show the experimental data, and the line shows the expected theoretical result for parameters determined by fits to figure 3 and figure 4.... Measured resonance frequency as a function of flux bias.
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  • To initial entangled state | I 0 of the two qubits with control field mode in its coherent state, its evolution involves on the various parameters in Eq.( t0-1). It depends on the number N in an extremely nonlinear and intricate way. The kind perplexity makes it hard to study the Rabi model, never the less, it also provide the opportunity to preserve the entanglement of the two qubits by careful choice of the appropriate parameters. Because the coherent state has the probability of Poisson distribution, which will be approximated by a Gauss distribution if the average number | α | 2 is large enough. The intricacy of the Rabi model could be utilized to make the quantity Y ~ N , + 2 L ~ N , + 4 = B N in Eq.( t0-1) extremely small when N is in the neighborhood of | α | 2 by some selection of the appropriate parameters, which will guaranty the initial state of the qubits unchanging. This is shown in Fig.( fig8). It can be easy to see that whenever we select the parameters appropriate, for example, the parameters as | α | 2 = 55 ,   a = - 0.6 ,   β = 0.5599 ,   ω 0 ω = 0.24 , the Bell state | I 0 = | 1 , 0 = 1 2 | ↑ ↑ - | ↓ ↓ have the probability about 1 - 0.005 = 99.5 100 to remain unchanged.... Figs.( fig11-3) show the general trait for the qubit remaining in its initial states | 1 , ± for different parameter a = 0.2 ,   0 ,   - 0.2 . Obviously, P N 1 t is influenced by four parameters β , a , N , ω 0 ω . In Ref., it is shown that the coupling strength β ranges from 0.01 to 1 for the application of adiabatic approximation (weak coupling will not be discussed here). From Eqs.( e0)-( t0),( p1- omega00), we see that the parameter a will come to action apparently whenever β ≈ 0.01 - 0.6 . As stated before, the qubits is equivalent to a two qubits system and the non-equal-energy-level parameter a represents the coupling strength between the two qubits. This shows the coupling of the two qubits changes their dynamics considerately in the range of β ≈ 0.1 - 0.6 for the adiabatic approximation method to be applied, and this is our limit on the coupling parameter β... Schematic diagram of P N 1 t with the four parameters as N = 2 ,   ω 0 ω = 0.25 ,   β = 0.2 , a = 0.2 ,   0 ,   - 0.2 from the top to bottom respectively. The apparent difference in these three figures strongly implies that the parameter a influences the qubits dynamically.... The parameter a is connected with the inter-qubit coupling strength κ as a = ± κ in symmetric and asymmetric transition cases respectively. Study also shows that the parameter a negative is favorable for | T α t approaching zero, as Fig.( fig8) exhibits. So the inter-qubit coupling is in favor of preservation of the initial entanglement, especial with asymmetrical transition case ( a < 0 ).
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  • These environmentally-induced Rabi oscillations are a clear signature of the non-Markovian behavior produced by the RLC environment, and are completely absent in the RL environment because the energy from the qubits is quickly dissipated without being temporarily stored. In the RL environment the decay in time of ρ 11 t has the characteristic non-oscillatory Markovian behavior. These environmentally-induced Rabi oscillations are generic features of circuits with resonances in the real part of the admittance. The frequency of the Rabi oscillations Ω R a = π κ Ω 3 / 2 Γ is independent of the resistance since Ω R a ≈ Ω π L 2 C / L 1 2 C 0 , and has the value of Ω R a = 2 π f R a ≈ 360 × 10 6  rad/sec in Fig.  fig:seven. This effect is similar to the so-called circuit quantum electrodynamics which has been of great experimental interest recently ... In Fig.  fig:four, T 1 is plotted for the phase qubit as a function of the qubit frequency ω 01 in the case of spectral densities describing an RLC [Eq. ( eqn:spectral-density-isolation)] or Drude [Eq. ( eqn:sd-drude)] isolation network at fixed temperatures T = 0 (main figure) and T = 50 mK (inset), with J i n t ω = 0 corresponding to R 0 ∞ . In the limit of low temperatures k B T / ℏ ω 01 ≪ 1 , the relaxation time becomes... fig:four (Color-online) T 1 (in seconds) as a function of qubit frequency ω 01 . The solid (red) curves describe the phase qubit with RLC isolation network (Fig.  fig:three) with parameters R = 50  ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and qubit parameters C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 . The dashed curves correspond to an RL isolation network with the same parameters, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 × 10 9 rad/sec.... As an illustration of the qualitative results discussed in this section, we show in Fig.  fig:six the frequency shift (renormalization) of the phase qubit with RLC isolation network described in Fig.  fig:three. We make the identification E = ω 01 and δ E = δ ω 01 . Near resonance ω 01 ≈ Ω , we find a frequency renormalization of about 2 % which is due to the term δ E r e s .... (Color-online) Schematic drawing of a phase qubit with an RLC isolation circuit. The phase qubit is shown inside the solid (red) box, the RLC isolation circuit is shown inside the dashed box to the left, and the internal admittance circuit is shown inside the dashed box to the right.... fig:six Renormalization of energy splitting for the phase qubit with RLC isolation network (Fig.  fig:three) for the parameters R = 50  ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and qubit parameters C 0 = 4.44 pF, R 0 = 5000  ohms and L 0 = 0 , T = 0 , and Ω = 141 × 10 9 rad/sec.... (Color-online) Flux qubit measured by a dc-SQUID gray (blue) line. The qubit corresponds to the inner SQUID loop with critical current I c and capacitance C J for both Josephson junctions denoted by the large × symbol. The inner SQUID is shunted by a capacitance C s , and environmental resistance R and is biased by a ramping current I b . The dc-SQUID loop has junction capatitance C 0 and critical current I c 0 .... fig:five (Color-online) T 1 (in nanoseconds) as a function of qubit frequency ω 01 . The solid (red) curves describe a phase qubit with RLC isolation network (Fig.  fig:three) with same parameters of Fig.  fig:two except that R 0 = 5000  ohms. The dashed curves correspond to an RL isolation network with the same parameters of the RLC network, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 × 10 9 rad/sec.
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  • (Color online) (a) Schematic representation of qubit projective measurement, where a current pulse allows a qubit in the excited state | e to tunnel to the right well ( R ), while a qubit in the ground state | g stays in the left well ( L ). (b) Readout circuit, showing lumped-element L R - C R readout resonator inductively coupled to the qubit, with Josephson junction effective inductance L J and capacitance C , with loop inductance L . Qubit control is through the differential flux bias line ( F B ). The readout resonator is capacitively coupled through C c to the readout line, in parallel with the other readout resonators. The readout line is connected through a cryogenic circulator to a low-noise cryogenic amplifier and to a room temperature microwave source. (c) Photomicrograph of four-qubit sample. F B 1 - 4 are control lines for each qubit and R R is the resonator readout line. Inset shows details for one qubit and its readout resonator. Scale bar is 50 μ m in length; fig.setup... (Color online) (a) Phase of signal reflected from readout resonator, as a function of the probe microwave frequency (averaged 900 times), for the qubit in the left ( L , blue) and right ( R , red) wells. Dashed line shows probe frequency for maximum visibility. (b) Reflected phase as a function of qubit flux bias, with no averaging. See text for details. fig.phase... (Color online) Setup for frequency-multiplexed readout. Multiplexed readout signals I p and Q p from top FGPA-DAC board are up-converted by mixing with a fixed microwave tone, then pass through the circulator into the qubit chip. Reflected signals pass back through the circulator, through the two amplifiers G 1 and G 2 , and are down-converted into I r and Q r using the same microwave tone, and are then processed by the bottom ADC-FPGA board. Data in the shadowed region are the down-converted I r and Q r spectra output from the ADC-FPGA board; probe signals from the FPGA-DAC board have the same frequency spectrum. D C indicates the digital demodulation channels, each processed independently and sent to the computer. fig.measure... With the bias points chosen for each qubit, we demonstrated the frequency-multiplexed readout by performing a multi-qubit experiment. To minimize crosstalk, we removed the coupling capacitors between qubits used in Ref. 6. In this experiment, we drove Rabi oscillations on each qubit’s | g ↔ | e transition and read out the qubit states simultaneously. We first calibrated the pulse amplitude needed for each qubit to perform a | g → | e Rabi transition in 10 ns. The drive amplitude was then set to 1, 2/3, 1/2 and 2/5 the calibrated Rabi transition amplitude for qubits Q 1 to Q 4 respectively, so that the Rabi period was 20 ns, 30 ns, 40 ns and 50 ns for qubits Q 1 to Q 4 . We then drove each qubit separately using an on-resonance Rabi drive for a duration τ , followed immediately by a projective measurement and qubit state readout. This experiment yielded the measurements shown in Fig.  fig.rabi(a)-(d) for qubits Q 1 - Q 4 respectively.... (Color online) (a)-(d) Rabi oscillations for qubits Q 1 - Q 4 respectively, with the qubits driven with 1, 2/3, 1/2 and 2/5 the on-resonance drive amplitude needed to perform a 10 ns Rabi | g → | e transition. (e) Rabi oscillations measured simultaneously for all the qubits, using the same color coding and drive amplitudes as for panels (a)-(d). fig.rabi... We demonstrated the multiplexed readout using a quantum circuit comprising four phase qubits and five integrated resonators, shown in Fig.  fig.setup(c). The design of this chip is similar to that used for a recent implementation of Shor’s algorithm,, but here the qubits were read out off a single line using microwave reflectometry, replacing the SQUID readout used Ref 6. . This dramatically simplifies the chip design and significantly reduces the footprint of the quantum circuit. We designed the readout resonators so that they resonated at frequencies of 3-4 GHz (far de-tuned from the qubit | g ↔ | e transition frequency of 6-7 GHz), with loaded resonance linewidths of a few hundred kilohertz. This allows us to use frequency multiplexing, which has been successfully used in the readout of microwave kinetic inductance detectors as well as other types of qubits. Combined with custom GHz-frequency signal generation and acquisition boards, this approach provides a compact and efficient readout scheme that should be applicable to systems with 10-100 qubits using a single readout line, with sufficient measurement bandwidth for microsecond-scale readout times.... The calibration of the readout process was done in two steps. We first optimized the microwave probe frequency to maximize the signal difference between the left and right well states. This was performed by measuring the reflected phase φ as a function of the probe frequency, with the qubit prepared first in the left and then in the right well. In Fig.  fig.phase(a), we show the result with the qubit flux bias set to 0.15 Φ 0 , where the difference in L J in two well states was relatively large. The probe frequency that maximized the signal difference was typically mid-way between the loaded resonator frequencies for the qubit in the left and right wells, marked by the dashed line in Fig.  fig.phase(a). We typically obtained resonator frequency shifts as large as ∼ 150 kHz for the qubit between the two wells, as shown in Fig.  fig.phase(a), significantly larger than the resonator linewidth.... With the probe frequency set in the first step, the flux bias was then set to optimize the readout. As illustrated in Fig.  fig.phase(b), the optimization was performed by measuring the resonator’s reflected phase as a function of qubit bias flux, at the optimal probe frequency, 3.70415 GHz in this case. The qubit was initialized by setting the flux to its negative “reset” value (position I), where the qubit potential has only one minimum. The flux was then increased to an intermediate value Φ , placing the qubit state in the left well, and the reflection phase measured with a 5 μ s microwave probe signal (blue data). The flux was then set to its positive reset value (position V), then brought back to the same flux value Φ , placing the qubit state in the right well, and the reflection phase again measured with a probe signal (red data). Between the symmetry point III ( Φ = 0.5 ) and the regions with just one potential minimum ( Φ ≤ 0.1 or Φ ≥ 0.9 ), the qubit inductance differs between the left and right well states, which gives rise to the difference in phase for the red and blue data measured at the same flux. This difference increases for the flux bias closer to the single-well region, which can give a signal-to-noise ratio as high as 30 at ambient readout microwave power. The optimal flux bias was then set to a value where the readout had a high signal-to-noise ratio (typically > 5), but with a potential barrier sufficient to prevent spurious readout-induced switching between the potential wells. Several iterations were needed to optimize both the probe frequency and flux bias.... With each qubit individually characterized, we then excited and measured all four qubits simultaneously, as shown in Fig.  fig.rabi(e). There is no measurable difference between the individually-measured Rabi oscillations in panels (a)-(d) compared to the multiplexed readout in panel (e).
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  • (color online) Decaying qubit oscillations with initial state | ↑ in a weakly probed CPB with 6 states for α = Z 0 e 2 / ℏ = 0.08 , A = 0.1 E J / e , E C = 5.25 E J and N g = 0.45 , so that E e l = 2.1 E J and ω q b = 2.3 E J / ℏ . (a) Time evolution of the measured difference signal Q ̇ ∝ ξ o u t - ξ i n (in units of 2 e E J / ℏ ) of the full CPB and its lock-in amplified phase φ o u t (frequency window Δ Ω = 5 E J / ℏ ), compared to the estimated phase φ h f 0 ∝ σ x 0 in the qubit approximation. The inset resolves the underlying small rapid oscillations with frequency Ω = 15 E J / ℏ in the long-time limit. (b) Power spectrum of Q ̇ for the full CPB Hamiltonian (solid) and for the two-level approximation (dashed).... Although later on we focus on the dynamics of a superconducting charge qubit as sketched in Fig.  fig:setup, our measurement scheme is rather generic and can be applied to any open quantum system. We employ the system-bath Hamiltonian... eq:7 allows one to retrieve information about the coherent qubit dynamics in an experiment. Figure  fig:oscillation(a) shows the time evolution of the expectation value Q ̇ t for the initial state | ↑ ≡ | 1 , obtained via numerical integration of the master equation ... CPB in the presence of the ac driving which in principle may excite higher states. The driving, due to its rather small amplitude, is barely noticeable on the scale chosen for the main figure, but only on a refined scale for long times; see inset of Fig.  fig:oscillation(a). This already insinuates that the backaction on the dynamics is weak. In the corresponding power spectrum of Q ̇ depicted in Fig.  fig:oscillation(b), the driving is nevertheless reflected in sideband peaks at the frequencies Ω and Ω ± ω q b . In the time domain these peaks correspond to a signal cos Ω t - φ o u t t . Moreover, non-qubit CPB states leads to additional peaks at higher frequencies, while their influence at frequencies ω Ω is minor. Experimentally, the phase φ o u t t can be retrieved by lock-in amplification of the output signal, which we mimic numerically in the following way : We only consider the spectrum of ξ o u t in a window Ω ± Δ Ω around the driving frequency and shift it by - Ω . The inverse Fourier transformation to the time domain provides φ o u t t which is expected to agree with φ h f 0 t and, according to Eq. ... (color online) (a) Fidelity defect δ F = 1 - F and (b) time-averaged trace distance between the driven and the undriven density operator of the CPB for various driving amplitudes as a function of the driving frequency. All other parameters are the same as in Fig.  fig:oscillation.... In order to quantify this agreement, we introduce the measurement fidelity F = φ o u t σ x 0 , where f g = ∫ d t f g / ∫ d t f 2 ∫ d t g 2 1 / 2 with time integration over the decay duration. Thus, the ideal value F = 1 is assumed if φ o u t t and σ x t 0 are proportional to each other, i.e. if the agreement between the measured phase and the unperturbed expectation value σ x 0 is perfect. Figure  fig:fidelity(a) depicts the fidelity as a function of the driving frequency. As expected, whenever non-qubit CPB states are excited resonantly, we find F ≪ 1 , indicating a significant population of these states. Far-off such resonances, the fidelity increases with the driving frequency Ω . A proper frequency lies in the middle between the qubit doublet and the next higher state. In the present case, Ω ≈ 15 E J / ℏ appears as a good choice. Concerning the driving amplitude, one has to find a compromise, because as A increases, so does the phase contrast of the outgoing signal... eq:7, to reflect the unperturbed time evolution of σ x 0 with respect to the qubit. Although the condition of high-frequency probing, Ω ≫ ω q b , is not strictly fulfilled and despite the presence of higher charge states, the lock-in amplified phase φ o u t t and the predicted phase φ 0 h f t are barely distinguishable for an appropriate choice of parameters as is shown in Fig.  fig:oscillation(a).
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