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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.Data Types:- Image

- (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 .Data Types:- Image

- 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, USAData Types:- Image
- Tabular Data

- 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 . -0.9cmData Types:- Image

- 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 .Data Types:- Image

- 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.Data Types:- Image

- 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 ).Data Types:- Image

- 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.Data Types:- Image

- (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| g ↔ | e transition and read out the**qubit**’s**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).Data Types:- Image

- (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).Data Types:- Image

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