Filter Results
20446 results
In the example figure (Fig.  fig:qubosc1d2d), the control bias is varied from left to right for a low frequency oscillator circuit (1.36GHz). For each bias point the simulation is reinitialised, the stochastic time evolution of the system density matrix is simulated over 1500 oscillator cycles. Then the oscillator and qubit charge expectation values are extracted to obtain the power spectrum for each component, with a frequency resolution of 4.01MHz. The power spectra for each time series are collated as an image such that the power axis is now represented as a colour, and the individual power spectra are vertical ‘slices’ through the image. The dominant frequency peaks become line traces, therefore illustrating the various avoided crossings, mergeings and intersections. The example figure shows the PSD ‘slice’ at Bias = 0.5187 , the broadband noise is readily apparent and is due to the discontinuous quantum jumps in the qubit. The bias oscillator peak (1.36GHz) is most prominent in the oscillator PSD, as would be expected, but it is also present in the qubit PSD. It should also be noted that most features are present in both the qubit and oscillator, including the noise which is generated by the quantum jumps and the quantum state diffusion processes. Interestingly, the qubit PSD is significantly stronger than the oscillator PSD, however, a larger voltage is generated by the smaller charge due to the extremely small island capacitance, V q = q / C q .... fig:mwRamp (Color online) Oscillator PSD as a function of the applied microwave drive frequency f m w , for microwave amplitudes A m w = 0.0050 (A) and A m w = 0.0100 (B). It is important to notice that there are now two frequency axes per plot, a drive (H) and a response (V). Of particular interest is the magnified section which shows clearly the distinct secondary splitting in the sub-GHz regime. This occurs due to a high frequency interaction seen in the upper plots, where the lower Rabi sideband of the microwave drive passes through the high frequency oscillator signal. The maximum splitting occurs when the Rabi amplitude is a maximum, hence this is observed for a very particular combination of bias and drive, which is beneficial for charactering the qubit. Most importantly, this would not be observed with a conventional low frequency oscillator configuration as the f m w - f o s c separation would be too large for the Rabi frequency. ( κ = 5 × 10 -5 ).... Fig.  fig:mwRamp is presented in a similar manner as Fig.  fig:BiasRamp. However there are now two frequency axes: the horizontal axis represents the frequency of the applied microwave drive field, and the vertical axis is the frequency response. It should be remembered that the microwave frequency axis is focused near the qubit transition frequency ( f q u b i t ≈ 3.49GHz) and the diagonally increasing line is now the microwave frequency.... Autler Townes effect, charge qubit, characterisation, frequency spectrum... fig:QubitOscEnergy A two level qubit is coupled to a many level harmonic oscillator, investigated for two different oscillator energies. Firstly, the oscillator resonant frequency is set to 1.36GHz, this more resembles the conventional configuration such that the fundamental component of the oscillator does not drive the qubit. However, we also investigate the use of a high frequency oscillator of 3.06GHz which can excite this qubit. In addition, qubit is constantly driven by a microwave field at 3.49GHz to generate Rabi oscillations and in this paper we examine the relation between these three fields.... fig:qubosc1d2d (Color online) Oscillator and Qubit power spectra slices for Bias = 0.5187, using the low frequency oscillator circuit f o s c = 1.36 GHz. The solid lines overlay the energy level separations found in Fig.  fig:EnergyLevel. ( κ = 5 × 10 -5 ). As one would expect, the bias oscillator peak at 1.36GHz is clearly observed in the oscillator PSD, but only weakly in the qubit PSD. Likewise the qubit Rabi frequency is found to be stronger in the qubit PSD. However it is important to note that the qubit dynamics such as the Rabi oscillations are indeed coupled to the bias oscillator circuit and so can be extracted. In addition, it is recommended to compare the layout of the most prominent features with Fig.  fig:BiasRamp.... fig:BiasRamp (Color online) Oscillator PSD as a function of bias, for microwave amplitudes A m w = 0.0025 (A) and A m w = 0.0050 (B). The red lines track the positions (in frequency) of significant power spectrum peaks (+10dB to +15dB above background), the overlaid black and blue lines are the qubit energy and microwave transition (Fig.  fig:EnergyLevel). Unlike Fig.  fig:qubosc1d2d, in these figures the 3.06GHz oscillator circuit can now drive the qubit (Fig.  fig:EnergyLevel) and so creates excitations which mix with the microwave driven excitations creating a secondary splitting centred on f m w - f o s c (430MHz). This feature contains the Rabi frequency information in the sidebands of the splitting, but now in a different and controllable frequency regime. In addition, the intersection of the two differently driven excitations (illustrated in the magnified sections), opens the possibility of calibrating the biased qubit against a fixed engineered oscillator circuit, using a single point feature. ( κ = 5 × 10 -5 ).... In a previous paper , a method was proposed by which the energy level structure of a charge qubit can be obtained from measurements of the peak noise in the bias/control oscillator, without the need of extra readout devices. This was based on a technique originally proposed for superconducting flux qubits but there are many similarities between the two technologies. The oscillator noise peak is the result of broadband noise caused by quantum jumps in the qubit being coupled back to the oscillator circuit. This increase in the jump rate becomes a maximum when the Rabi oscillations are at peak amplitude, this should only occur when the qubit is correctly biased and the microwave drive is driving at the transition frequency. Therefore by monitoring this peak as a function of bias, we can associate a bias position with a microwave frequency equal to that of the energy gap, hence constructing the energy diagram (Fig.  fig:EnergyLevel).... fig:Jumps (Color online) (A) Oscillator power spectra when the coupled qubit is driven at f m w = 5.00 GHz. An increase in bias noise power ( f o s c = 1.36 GHz) can be observed when Rabi oscillations occur, the more frequent quantum jump noise couples back to the oscillator. (B) Bias noise power peak position changes as a function of f m w , the microwave drive frequency. Therefore, it is possible to probe the qubit energy level structure by using the power increase in the oscillator which is already in place, eliminating the need for additional measurement devices. However, it should be noted that the surrounding oscillator harmonics may mask the microwave driven peak. ( κ = 1 × 10 -3 ).
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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 QND character of the qubit measurement is studied by repeating the measurement. A perfect QND setup guarantees identical outcomes for the two repeated measurement with certainty. In order to fully characterize the properties of the measurement, we can initialize the qubit in the state | 0 , then rotate the qubit by applying a pulse of duration τ 1 before the first measurement and a second pulse of duration τ 2 between the first and the second measurement. The conditional probability to detect the qubit in the states s and s ' is expected to be independent of the first pulse, and to show sinusoidal oscillation with amplitude 1 in τ 2 . Deviations from this expectation witness a deviation from a perfect QND measurement. The sequence of qubit pulses and oscillator driving is depicted in Fig.  Fig1b). The conditional probability P 0 | 0 to detect the qubit in the state "0" twice in sequence is plotted versus τ 1 and τ 2 in Fig.  Fig1c) for Δ = 0 , and in Fig.  Fig1d) for Δ / ϵ = 0.1 . We anticipate here that a dependence on τ 1 is visible when the qubit undergoes a flip in the first rotation. Such a dependence is due to the imperfections of the mapping between the qubit state and the oscillator state, and is present also in the case Δ = 0 . The effect of the non-QND term Δ σ X results in an overall reduction of P 0 | 0 .... (Color online) Conditional probability to obtain a) s ' = s = 1 , b) s ' = - s = 1 , c) s ' = - s = - 1 , and d) s ' = s = - 1 for the case Δ t = Δ / ϵ = 0.1 and T 1 = 10 ~ n s , when rotating the qubit around the y axis before the first measurement for a time τ 1 and between the first and the second measurement for a time τ 2 , starting with the qubit in the state | 0 0 | . Correction in Δ t are up to second order. The harmonic oscillator is driven at resonance with the bare harmonic frequency and a strong driving together with a strong damping of the oscillator are assumed, with f / 2 π = 20 ~ G H z and κ / 2 π = 1.5 ~ G H z . Fig6... In Fig.  Fig5 we plot the second order correction to the probability to obtain "1" having prepared the qubit in the initial state ρ 0 = | 0 0 | , corresponding to F 2 t , for Δ t = Δ / ϵ = 0.1 . We choose to plot only the deviation from the unperturbed probability because we want to highlight the contribution to spin-flip purely due to tunneling in the qubit Hamiltonian. In fact most of the contribution to spin-flip arises from the unperturbed probability, as it is clear from Fig.  Fig3. Around the two qubit-shifted frequencies, the probability has a two-peak structure whose characteristics come entirely from the behavior of the phase ψ around the resonances Δ ω ≈ ± g . We note that the tunneling term can be responsible for a probability correction up to ∼ 4 % around the qubit-shifted frequency.... We now investigate whether it is possible to identify the contribution of different mechanisms that generate deviations from a perfect QND measurement. In Fig.  Fig7 we study separately the effect of qubit relaxation and qubit tunneling on the conditional probability P 0 | 0 . In Fig.  Fig7 a) we set Δ = 0 and T 1 = ∞ . The main feature appearing is a sudden change of the conditional probability P → 1 - P when the qubit is flipped in the first rotation. This is due to imperfection in the mapping between the qubit state and the state of the harmonic oscillator, already at the level of a single measurement. The inclusion of a phenomenological qubit relaxation time T 1 = 2 ~ n s , intentionally chosen very short, yields a strong damping of the oscillation along τ 2 and washes out the response change when the qubit is flipped during the first rotation. This is shown in Fig.  Fig7 b). The manifestation of the non-QND term comes as a global reduction of the visibility of the oscillations, as clearly shown in Fig.  Fig7 c).... (Color online) Comparison of the deviations from QND behavior originating from different mechanisms. Conditional probability P 0 | 0 versus qubit driving time τ 1 and τ 2 starting with the qubit in the state | 0 0 | , for a) Δ = 0 and T 1 = ∞ , b) Δ = 0 and T 1 = 2 ~ n s , and c) Δ = 0.1 ~ ϵ and T 1 = ∞ . The oscillator driving amplitude is f / 2 π = 20 ~ G H z and a damping rate κ / 2 π = 1.5 ~ G H z is assumed. Fig7... For driving at resonance with the bare harmonic oscillator frequency ω h o , the state of the qubit is encoded in the phase of the signal, with φ 1 = - φ 0 , and the amplitude of the signal is actually reduced, as also shown in Fig.  Fig3 for Δ ω = 0 . When matching one of the two frequencies ω i the qubit state is encoded in the amplitude of the signal, as also clearly shown in Fig.  Fig3 for Δ ω = ± g . Driving away from resonance can give rise to significant deviation from 0 and 1 to the outcome probability, therefore resulting in an imprecise mapping between qubit state and measurement outcomes and a weak qubit measurement.... (Color online) Schematic description of the single measurement procedure. In the bottom panel the coherent states | α 0 and | α 1 , associated with the qubit states | 0 and | 1 , are represented for illustrative purposes by a contour line in the phase space at HWHM of their Wigner distributions, defined as W α α * = 2 / π 2 exp 2 | α | 2 ∫ d β - β | ρ | β exp β α * - β * α . The corresponding Gaussian probability distributions of width σ centered about the qubit-dependent "position" x s are shown in the top panel. Fig2... The combined effect of the quantum fluctuations of the oscillator together with the tunneling between the qubit states is therefore responsible for deviation from a perfect QND behavior, although a major role is played, as expected, by the non-QND tunneling term. Such a conclusion pertains to a model in which the qubit QND measurement is studied in the regime of strong projective qubit measurement and qubit relaxation is taken into account only phenomenologically. We compared the conditional probabilities plotted in Fig.  Fig6 and Fig.  Fig7 directly to Fig. 4 in Ref. [... (Color online) a) Schematics of the 4-Josephson junction superconducting flux qubit surrounded by a SQUID. b) Measurement scheme: b1) two short pulses at frequency ϵ 2 + Δ 2 , before and between two measurements prepare the qubit in a generic state. Here, ϵ and Δ represent the energy difference and the tunneling amplitude between the two qubit states. b2) Two pulses of amplitude f and duration τ 1 = τ 2 = 0.1 ~ n s drive the harmonic oscillator to a qubit-dependent state. c) Perfect QND: conditional probability P 0 | 0 for Δ = 0 to detect the qubit in the state "0" vs driving time τ 1 and τ 2 , at Rabi frequency of 1 ~ G H z . The oscillator driving amplitude is chosen to be f / 2 π = 50 ~ G H z and the damping rate κ / 2 π = 1 ~ G H z . d) Conditional probability P 0 | 0 for Δ / ϵ = 0.1 , f / 2 π = 20 ~ G H z , κ / 2 π = 1.5 ~ G H z . A phenomenological qubit relaxation time T 1 = 10 ~ n s is assumed. Fig1
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Parameters in calculations and measurements in units of GHz. In the first column, cal: δ ω Ω R 0 stands for the calculation to study the shift of the resonant frequency, and cal: Γ R s t δ ω m w stands for the calculation to study the decay of Rabi oscillations due to quasistatic flux noise. “Optimal" in the last column means that at each ε m w , ω m w is chosen to minimize dephasing due to quasistatic flux noise.... (Color online) Rabi oscillation curves with different Rabi frequencies Ω R measured at different static flux bias ε . At each Ω R , δ ω m w is chosen to minimize dephasing due to quasistatic flux noise. The red lines are the fitting curves.  In the measurements shown in the middle and bottom panels, only parts of the oscillations are monitored so that we can save measurement time while the envelopes of Rabi oscillations are captured. The inset is a magnification of the data in the bottom panel together with the fitting curve.... In the Rabi oscillation measurements, a microwave pulse is applied to the qubit followed by a readout pulse, and P s w as a function of the microwave pulse length is measured. First, we measure the Rabi oscillation decay at ε = 0 , where the quasistatic noise contribution is negligible. Figure  GRfR1p5(d) shows the measured 1 / e decay rate of the Rabi oscillations Γ R 1 / e as a function of Ω R 0 . For Ω R 0 / 2 π up to 400 MHz, Γ R 1 / e is approximately 3 Γ 1 / 4 , limited by the energy relaxation, and S Δ Ω R 0 is negligible. For Ω R 0 / 2 π from 600 MHz to 2.2 GHz, Γ R 1 / e > 3 Γ 1 / 4 . A possible origin of this additional decoherence is fluctuations of ε m w , δ ε m w : Ω R 0 is first order sensitive to δ ε m w , which is reported to be proportional to ε m w itself.  Next, the decay for the case ε ≈ Δ is studied. To observe the contribution from quasistatic flux noise, the Rabi oscillation decay as a function of ω m w is measured, where the contribution from the other sources is expected to be almost constant. Figure  GRfR1p5(b) shows Γ R 1 / e at ε / 2 π = 4.16 GHz as a function of δ ω m w while keeping Ω R / 2 π between 1.5 and 1.6 GHz. Besides the offset and scatter, the trend of Γ R 1 / e agrees with that of the simulated Γ R s t . This result indicates that numerical calculation properly evaluates δ ω m w minimizing Γ R s t . Finally, the decay for the case ε ≈ Δ as a function of ε m w , covering a wide range of Ω R , is measured (Fig.  Rabis).... (Color online) Power spectrum density of flux fluctuations S n φ ω extracted from the Rabi oscillation measurements in the first ( ε / 2 π = 4.16 GHz) and second cooldowns. The PSDs obtained from the spin-echo and energy relaxation measurements in the second cooldown are also plotted. The black solid line is the 1/ f spectrum extrapolated from the FID measurements in the second cooldown. The purple dashed line is the estimated Johnson noise from a 50  Ω microwave line coupled to the qubit by a mutual inductance of 1.2 pH and nominally cooled to 35 mK. The pink dotted line is a Lorentzian, S n φ m o d e l ω = S h ω w 2 / ω 2 + ω w 2 , and the orange solid line is the sum of the Lorentzian and the Johnson noise. Here the parameters are S h = 3.6 × 10 -19 r a d -1 s and ω w / 2 π = 2.7 × 10 7 H z .... Josephson devices, decoherence, Rabi oscillation, $1/f$ noise... (Color online) (a) Numerically calculated shift of the resonant frequency δ ω (black open circles) and the Bloch–Siegert shift δ ω B S (blue line). (b) Numerically calculated decay rate Γ R s t (black open circles) and Rabi frequency Ω R (red solid triangles) as functions of the detuning δ ω m w from ω 01 . The purple solid line is a fit based on Eq. ( fRfull). The measured 1/ e decay rates Γ R 1 / e at ε / 2 π = 4.16 GHz for the range of Rabi frequencies Ω R / 2 π between 1.5 and 1.6 GHz (blue solid circles) are also plotted. (c) Calculated Rabi frequency Ω R , based on Eq. ( fRfull), as a function of ε for the cases (i)  ω m w = ω 01 + δ ω (black solid line) and (ii)  ω m w / 2 π = 6.1  GHz (red dashed line). The upper axis indicates ω 01 , corresponding to ε in the bottom axis. (d) The measured 1 / e decay rate of the Rabi oscillations, Γ R 1 / e , at ε = 0 and as a function of Ω R 0 . The red solid line indicates 3 4 Γ 1 obtained independently.... The condition, ∂ Ω R / ∂ ε = 0 , is satisfied when ε = 0 or δ ω m w = δ ω - Ω R 0 2 / ω 01 . For Ω R 0 / 2 π = 1.52 GHz and ω 01 / 2 π = 6.400 GHz, the latter condition is calculated to be δ ω m w / 2 π = - 295  MHz, slightly different from the minimum of Γ R s t seen in Fig.  G R f R 1 p 5 (b). The difference is due to the deviation from the linear approximation in Eq. ( fRfull), Ω R 0 ∝ ε m w / ω 01 . Figure  GRfR1p5(c) shows the calculation of Ω R as a function of ε , based on Eq. ( fRfull). The Rabi frequency Ω R 0 at the shifted resonance decreases as ε increases, while Ω R , for a fixed microwave frequency of ω m w / 2 π = 6.1  GHz, has a minimum of approximately ω 01 / 2 π = 6.4 GHz. Here in the first order, Ω R is insensitive to the fluctuation of ε .... In Fig.  GRfR1p5(a), δ ω as a function of Ω R 0 is plotted together with the well-known Bloch–Siegert shift,  δ ω B S = 1 4 Ω R 0 2 ω 01 , obtained from the second-order perturbation theory. Fixed parameters for the calculation are Δ / 2 π = 4.869 and ε / 2 π = 4.154  GHz ( ω 01 / 2 π = 6.400  GHz). We find that δ ω B S overestimates δ ω when Ω R 0 / 2 π 800  MHz. The deviation from the Bloch–Siegert shift is due to the component of the ac flux drive that is parallel to the qubit’s energy eigenbasis; this component is not averaged out when Ω R is comparable to ω m w .
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(Color online) Rabi-type oscillations of occupation probabilities of | ρ ρ ' states for strongly coupled qubits with the initial state ψ 0 = | 00 + | 10 / 2 . Here the parameters are J / h = 5 GHz, ω 0 / 2 π =4GHz, and Ω 0 / 2 π = 600 MHz at the degeneracy point where E = E in Fig. weak(a).... In Fig. PC we show the Rabi-type oscillation for strongly coupled qubits. While the P 00 ( P 01 ) is reversed from 0.5 (0) to 0 (0.5) at Ω t = (odd) π , we can observe that the probabilities P 10 and P 11 remain their initial values 0.5 and 0, respectively. In this case the parameters need not satisfy the commensurate condition of Eq. ( condition) for the CNOT gate operation.... The scheme for CNOT gate operation in this study uses the non-Rabi oscillations for | 10 and | 11 states which are commensurate with the Rabi oscillation for | 00 and | 01 states. In Fig. TwoRabi we display the numerical results obtained from the Hamiltonian in Eqs. ( tilH0) and ( tilH1), which show such commensurate mode oscillations. The initial state, | ψ 0 = | 00 + | 10 / 2 , is driven by an oscillating field with the resonant frequency ω = ω 0 < ω 1 .... The commensurate oscillations of resonant and non-resonant modes enable the high fidelity CNOT gate operation by finely tuning the oscillating field amplitude for any given values of qubit energy gap and coupling strength between qubits. While for a sufficiently strong coupling the CNOT gate can be achieved for any given parameter values, for a weak coupling a relation between the parameters should be satisfied for the fidelity maxima. For a sufficiently weak coupling compared to the qubit energy gap, J / ℏ ω 0 ≪ 1 , we have α 1 ≈ 1 and β 1 ≈ 0 , resulting in the expression for g in Eq. ( ga). For J / ℏ ω 0 ≪ 1 , Eq. ( ga) immediately gives rise to the relation g / J ≪ 1 and thus g / ℏ ω 0 ≪ 1 after some manipulation. This means that for a weak coupling J / ℏ ω 0 ≪ 1 the numerical results are well fit with the RWA as shown in Table table, because the RWA is good for g / ℏ ω 0 ≪ 1 . As a result, the high performance CNOT gate operation can be achieved as shown in Fig. dF.... Let us consider a concrete example for comprehensive understanding. For superconducting flux qubits, g = m B is the coupling between the amplitude B of the magnetic microwave field and the magnetic moment m , induced by the circulating current, of the qubit loop. In order to adjust the value of g , actually we need to vary the microwave amplitude B , because the qubit magnetic moment is fixed at a specified degeneracy point. The Rabi-type oscillation occurs between the transformed states | 0 = | + | / 2 and | 1 = | - | / 2 . The states of qubits can be detected by shifting the magnetic pulse adiabatically . Since these qubit states are the superposition of the clockwise and counterclockwise current states, | and | , the averaged current of qubit states vanishes at the degeneracy point in Fig. weak(a). Thus, one can apply a finite dc magnetic pulse to shift the qubits slightly away from the degeneracy point to detect the qubit current states.... (Color online) (a) Energy levels E ρ ρ ' of coupled qubits, where ρ , ρ ' ∈ 0 1 . E s s ' with s , s ' ∈ are shown as thin dotted lines. The distance between two degeneracy points corresponds to the coupling strength between two qubits. (b) Occupation probabilities of | ρ ρ ' states during Rabi-type oscillations at the lower degeneracy point where E = E . Here we use the parameter values such that coupling strength J / h = 0.6GHz, qubit energy gap ω 0 / 2 π =4GHz, and Rabi frequency Ω 0 / 2 π = 600 MHz. The initial state is chosen as ψ 0 = | 00 + | 10 / 2 and the CNOT gate is expected to be achieved at Ω t = (odd) π .... The values of g / h for the main fidelity maxima ( n = 1 ) obtained from numerical calculation and from the RWA of Eq. ( g) for various coupling J and qubit energy gap ω 0 . For small ω 0 and large J the oscillations are far from the Rabi oscillation. Here, the unit of all numbers is GHz.... (Color online) (a) Commensurate oscillations of occupation probability of coupled-qubit states with the initial state, | ψ 0 = | 00 + | 10 / 2 for g / h = 0.265 GHz. The non-resonant oscillation modes ( P 10 and P 11 ) are commensurate with the resonant modes ( P 00 and P 01 ). At Ω t = (odd) π , P 10 and P 11 recover their initial values, thus the CNOT gate operation is achieved. Here Ω 0 = g / ℏ , J / h = 0.5 GHz, and ω 0 / 2 π = 4.0GHz. (b) Higher order commensurate modes for smaller g / h = 0.122 GHz with the same J and ω 0 .... Figure weak(a) shows the energy levels E s s ' as a function of κ b , where we choose κ a such that | E s s ' - E - s s ' | ≫ t q a and thus t q a can be negligible. In the figure there are two degeneracy points; lower degeneracy point where E = E and upper degeneracy point where E = E . By adjusting the variable κ b , the coupled-qubit states can be brought to one of these degeneracy points. Here the distance between these degeneracy points is related to the coupling strength between two qubits.
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tab1 Typical settings of the controllable experimental parameters ( V k and Φ k ) and the corresponding time evolutions Û j t of the qubit-bus system. Here, C g k and 2 ε J k are the gate capacitance and the maximal Josephson energy of the k th SQUID-based charge qubit. ζ k is the maximum strength of the coupling between the k th qubit with energy ε k and the bus of frequency ω b . The detuning between the qubit and the bus energies is ℏ Δ k = ε k - ℏ ω b . n = 0 , 1 is occupation number for the number state | n of the bus. The various time-evolution operators are: Û 0 t = exp - i t H ̂ b / ℏ ,  Û 1 k t = exp - i t δ E C k σ ̂ x k / 2 ℏ ⊗ Û 0 t ,  Û 2 k = Â t cos λ ̂ n | 0 k 0 k | - sin λ ̂ n â / n ̂ + 1 | 0 k 1 k | + â † sin ξ ̂ n / n ̂ | 1 k 0 k | + cos ξ ̂ n | 0 k 0 k | , and Û 3 k t = Â t exp - i t ζ k 2 | 1 k 1 k | n ̂ + 1 - | 0 k 0 k | n ̂ / ℏ Δ k , with Â t = exp - i t 2 H ̂ b + E J k σ ̂ z k / 2 ℏ , λ ̂ n = 2 ζ k t n ̂ + 1 / ℏ , and ξ ̂ n = 2 ζ k t n ̂ / ℏ .... A pair of SQUID-based charge qubits, located on the left of the dashed line, coupled to a large CBJJ on the right, which acts as an information bus. The circuit is divided into two parts, the qubits and the bus. The dashed line only indicates a separation between these. The controllable gate voltage V k k = 1 2 and external flux Φ k are used to manipulate the qubits and their interactions with the bus. The bus current remains fixed during the operations.
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Discrimination time as function of the coupling strength between qubit and oscillator. Here ℏ Ω / k B T = 2 , κ / Ω = 0.025 and ℏ ν / k B T = 1.95 .... In this section we present a different measurement protocol. It is based on the short time dynamics illustrated as follows: for the qubit initially in the state 1 / 2 | ↑ + | ↓ the probability distribution of momentum is plotted in Fig.  probability (a) and (b).... As one can see for the parameters of Fig.  comp, in the WQOC regime the measurement time is longer than the dephasig time. Their difference decreases as we increase Δ due to the onset of the strong coupling plateau in the dephasing rate, approaching the quantum limit where the measurement time becomes comparable to the dephasing time. Note that, for superstrong coupling either between qubit and oscillator or between oscillator and bath, corrections of the order κ / Ω ↓ 2 of the dephasing rate gain importance. These corrections are not treated in our Born approximation. Therefore the regions where the dephasing rate becomes lower than the measurement rate, in violation with the quantum limitation of Ref. , should be regarded as a limitation of our approximation.... Dephasing rate dependence on driving: dependence on Δ for different driving strengths F 0 ( κ / Ω = 10 -4 and ν = 2 Ω ). Top inset: dependence of the decoherence rate on F 0 for different values of κ ( Δ / Ω = 5 ⋅ 10 -2 and ν = 2 Ω ). Bottom inset: dependence of the decoherence rate on driving frequency ν for different vales of κ ( Δ / Ω = 0.5 ). Here ℏ Ω / k B T = 2 and F ¯ 0 is the dimensionless force F 0 ℏ / k B T m k B T .... Probability density of momentum P p 0 t (a), snapshots of it at different times (b) and expectation value of momentum for the two different qubit states (c). Here ℏ Ω / k B T = 2 , Δ / Ω = 0.45 , κ / Ω = 0.025 and ℏ ν / k B T = 1.9 and p ¯ 0 is the dimensionless momentum p 0 / k B T m .... We consider a simplified version of the experiment described in Ref. . The circuit consists of a flux qubit drawn in the single junction version, the surrounding SQUID loop, an ac source, and a shunt resistor, as depicted in Fig.  circuit. We note here that we later approximate the qubit as a two-level system. The qubit used in the actual experiment contains three junctions. An analogous but less transparent derivation would, after performing the two-state approximation, lead to the same model, parameterized by the two-state Hamiltonian, the circulating current, and the mutual inductance, in an identical way .... The dependence on the driving frequency has also been analyzed in Fig.  antrieb. Here we observe two peaks at Ω ↑ and Ω ↓ . At ν = Ω the classical driven and undamped trajectory ξ t diverges. In terms of the calculation this means that the Floquet modes are not well-defined when the driving frequency is at resonance with the harmonic oscillator — we have a continuum instead. Physically this means that at t = 0 our oscillator has the frequency Ω because it has not yet "seen" the qubit, and we are driving it at resonance, and by amplifying the oscillations of x ̂ which is subject to noise we amplify the noise seen by the qubit. The dephasing rate is also expected to diverge. The peaks at Ω ↑ and Ω ↓ show the same effect after the qubit and the oscillator become entangled. The dephasing rate drops again for large driving frequencies to the value obtained in the case without driving.... Simplified circuit consisting of a qubit with one Josephson junction (phase γ , capacitance C q and inductance L q ) inductively coupled to a SQUID with two identical junctions (phases γ 1 , 2 , capacitance C S ) and inductance L S . The SQUID is driven by an ac bias I B t and the voltage drop is measured by a voltmeter with internal resistance R . The total flux through the qubit loop is Φ q and through the SQUID is Φ S .... In Fig.  probability one can see that the two peaks corresponding to the two states of the qubit split already during the transient motion of p ̂ t , much faster than the transient decay time. If the peaks are well enough separated, a single measurement of momentum gives the needed information about the qubit state, and has the advantage of avoiding decoeherence effects resulting from a long time coupling to the environment. Nevertheless the parameters we need to reduce the discrimination time also enhance the decoherence rate.
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