Filter Results
20444 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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(Color online) FC driving of a transmon with an external flux. The transmon is modelled using the first four levels of the Hamiltonian given by Eq. ( eqn:duffing), using parameters E J / 2 π = 25 GHz and E C / 2 π = 250 MHz. We also have g g e / 2 π = 100 MHz and ω r / 2 π = 7.8 GHz, which translates to Δ g e / 2 π ≃ 2.1 GHz. a) Frequency of the transition to the first excited state obtained by numerical diagonalization of Eq. ( eqn:duffing). As obtained from Eqs. ( eqn:hamonic:1) to ( eqn:hamonic:4), the major component in the spectrum of ω g e t when shaking the flux away from the flux sweet spot at frequency ω F C also has frequency ω F C . However, when shaking around the sweet spot, the dominant harmonic has frequency 2 ω F C . Furthermore, the mean value of ω g e is shifted by G . b) Rabi frequency of the red sideband transition | 1 ; 0 ↔ | 0 ; 1 . The system is initially in | 1 ; 0 and evolves under the Hamiltonian given by Eq. ( eqn:H:MLS) and a flux drive described by Eq. ( eqn:flux:drive). Full red line: analytical results from Eq. ( eqn:rabi:freq) with m = 1 and φ i = 0.25 . Dotted blue line: m = 2 and φ i = 0 . Black dots and triangles: exact numerical results. c) Geometric shift for φ i = 0.25 (full red line) and 0 (dotted blue line). d) Increase in the Rabi frequency for higher coupling strengths with φ i = 0.25 and Δ φ = 0.075 . e) Behavior of the resonance frequency for the flux drive. As long as the dispersive approximation holds ( g g c r i t / 2 π = 1061 MHz), it remains well approximated by Eq. ( eqn:resonance), as shown by the full red line. The same conclusion holds for the Rabi frequency. fig:transmon... (Color online) Average error with respect to the perfect red sideband process | 1 ; 0 ↔ | 0 ; 1 . A gaussian FC pulse is sent on the first qubit at the red sideband frequency assuming the second qubit is in its ground state. Full red line: average error of the red sideband as given by Eq. ( eqn:FUV:simple) when the second qubit is excited. Blue dashed line: population transfer error 1 - P t , with P t given by Eq. ( eqn:pop:transfer). Black dots: numerical results for the average error. We find the evolution operator after time t p for each eigenstate of the second qubit. The fidelity is extracted by injecting these unitaries in Eq. ( eqn:trace). The qubits are taken to be transmons, which are modelled as 4-level Duffing oscillators (see Section  sec:Duffing) with E J 1 = 25 GHz, E J 2 = 35 GHz, E C 1 = 250 MHz, E C 2 = 300 MHz, yielding ω 01 1 = 5.670 GHz and ω 01 2 = 7.379 GHz, and g 01 1 = 100 MHz. The resonator is modeled as a 5-level truncated harmonic oscillator with frequency ω r = 7.8 GHz. As explained in Section  sec:transmon, the splitting between the first two levels of a transmon is modulated using a time-varying external flux φ . Here, we use gaussian pulses in that flux, as described by Eq. ( eqn:gaussian) with τ = 2 σ , σ = 6.6873 ns, and flux drive amplitude Δ φ = 0.075 φ 0 . The length of the pulse is chosen to maximize the population transfer. fig:FUV... This method is first applied to simulate a R 01 1 pulse by evolving the two-transmon-one-resonator system under the Hamiltonian of Eq. ( eqn:H:MLS), along with the FC drive Hamiltonian for the pulse. The simulation parameters are indicated in Table  tab:sequence. To generate the sideband pulse R 01 1 , the target qubit splitting is modulated at a frequency that lies exactly between the red sideband resonance for the spectator qubit in states | 0 or | 1 , such that the fidelity will be the same for both these spectator qubit states. We calculate the population transfer probability for | 1 ; 0 ↔ | 0 ; 1 after the pulse and find a success rate of 99.2% for both initial states | 1 ; 0 and | 0 ; 1 . This is similar to the prediction from Eq. ( eqn:pop:transfer), which yields 98.7%. The agreement between the full numerics and the simple analytical results is remarkable, especially given that with | δ ± / ϵ n | = 0.23 the small δ ± ≪ ϵ n assumption is not satisfied. Thus, population transfers between the transmon and the resonator are achievable with a good fidelity even in the presence of Stark shift errors coming from the spectator qubit (see Section  sec:SB).... In Fig.  fig:transmonb), the Rabi frequencies predicted by the above formula are compared to numerical simulations using the full Hamiltonian Eq. ( eqn:H:MLS), along with a cosine flux drive. The geometric shifts described by Eq. ( eq:G) are also plotted in Fig.  fig:transmonc), along with numerical results. In both cases, the scaling with respect to Δ φ follows very well the numerical predictions, allowing us to conclude that our simple analytical model accurately synthesizes the physics occurring in the full Hamiltonian. It should be noted that, contrary to intuition, the geometric shift is roughly the same at and away from the sweet spot. This is simply due to the fact that the band curvature does not change much between the two operation points. However, as expected from Eqs. ( eqn:hamonic:1) to ( eqn:hamonic:4), the Rabi frequencies are much larger for the same drive amplitude when the transmon is on average away from its flux sweet spot. In that regime, large Rabi frequencies ∼ 30 -40 MHz can be attained, which is well above dephasing rates in actual circuit QED systems, especially in the 3D cavity . However, the available power that can be sent to the flux line might be limited in the lab, putting an upper bound on achievable rates. Furthermore, at those rates, fast rotating terms such as the ones dropped between Eq. ( eq:eps:n) and ( eq:V) start to play a role, adding spurious oscillations in the Rabi oscillations that reduce the fidelity. These additional oscillations have been seen to be especially large for big relevant ε m ω / Δ ~ j , j + 1 n ratios, i.e. when the qubit spends a significant amount of time close to resonance with the resonator and the dispersive approximation breaks down.... We have also defined ω ' p = 8 E C E J Σ cos φ i , the plasma frequency associated to the operating point φ i . This frequency is illustrated by the black dots for two operating points on Fig.  fig:transmona). In addition, there is a frequency shift G , standing for geometric, that depends on the shape of the transmon energy bands. As is also illustrated on Fig.  fig:transmona), this frequency shift comes from the fact that the relation between ω j , j + 1 and φ is nonlinear, such that the mean value of the transmon frequency during flux modulation is not its value for the mean flux φ i . To fourth order in Δ φ , it is... In words, the infidelity 1 - F U V is minimized when the Rabi frequency that corresponds to the FC drive is large compared to the Stark shift associated to the spectator qubit. The average fidelity corresponding to the gate fidelity Eq. ( eqn:FUV:simple) is illustrated in Fig.  fig:FUV as as a function of S 2 (red line) assuming the second qubit to be in its excited state. We also represent as black dots a numerical estimate of the error coming from the spectator qubit’s Stark shift. The latter is calculated with Eqs. ( eqn:trace) and ( eqn:avg:fid). Numerically solving the system’s Schrödinger equation allows us to extract the unitary evolution operator that corresponds to the applied gate. Taking U to be that evolution operator for the spectator qubit in state | 0 and V the operator in state | 1 , we obtain the error caused by the Stark shift shown in Fig.  fig:FUV. The numerical results closely follow the analytical predictions, even for relatively large dispersive shifts S 2 .... Schemes for two-qubit operations in circuit QED. ϵ is the strength of the drive used in the scheme, if any. ∗ There are no crossings in that gate provided that the qubits have frequencies separated enough that they do not overlap during FC modulations. tab:gates... Amplitude of the gaussian pulse over time. Δ φ ' is such that the areas A + and 2 A - are equal. Then, driving the sideband at its resonance frequency for the geometric shift that corresponds to the flux drive amplitude Δ φ ' allows population inversion. fig:gaussian... (Color online) Sideband transitions for a three-level system coupled to a resonator. Applying an FC drive at frequency Δ i , i + 1 generates a red sideband transitions between states | i + 1 ; n and | i ; n + 1 , where the numbers represent respectively the MLS and resonator states. Similarly, driving at frequency Σ i , i + 1 leads to a blue sideband transition, i.e. | i ; n ↔ | i + 1 ; n + 1 . Transitions between states higher in the Fock space are not shown for reasons of readability. This picture is easily generalized to an arbitrary number of levels. fig:MLS:sidebands... Table  tab:gates summarizes theoretical predictions and experimental results for recent proposals for two-qubit gates in circuit QED. These can be divided in two broad classes. The first includes approaches that rely on anticrossings in the qubit-resonator or qubit-qubit spectrum. They are typically very fast, since their rate is equal to the coupling strength involved in the anticrossing. Couplings can be achieved either through direct capacitive coupling of the qubits with strength J C  , or through the 11-02 anticrossing in the two-transmon spectrum which is mediated by the cavity . The latter technique has been successfully used with large coupling rates J 11 - 02 and Bell-state fidelities of ∼ 94 % . However, since these gates are activated by tuning the qubits in and out of resonance, they have a finite on/off ratio determined by the distance between the relevant spectral lines. Thus, the fact that the gate is never completely turned off will make it very complicated to scale up to large numbers of qubits. Furthermore, adding qubits in the resonator leads to more spectral lines that also reduce scalability. In that situation, turning the gates on and off by tuning qubit transition frequencies in and out of resonance without crossing these additional lines becomes increasingly difficult as qubits are added in the resonator, an effect known as spectral crowding.
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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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quantum degenerate regime We now move to exploring what happens when the detuning between the qubit and the cavity is reduced such that the anharmonicity of quantum ladder of states becomes much larger than the corresponding linewidth κ (see Fig.  gino:fig:context). In order to describe the response of the system to external drive in this regime it is important to take into account the quantum dynamics on the lower anharmonic part of the ladder. When the system is initialized in the ground state, there is a range of drive strengths for which the system will remain blockaded from excitations out of the ground state. However, since the anharmonicity of the JC ladder decreases with excitation number, the transition frequency for excitations between adjacent levels ultimately approaches the bare cavity frequency. Qualitatively, when the excitation level n is such that the anharmonicity becomes smaller than the linewidth κ , we expect the state dynamics to be semiclassical, similar to a driven-damped harmonic oscillator... gino:chirp_figure (Color) Readout control pulse (a) Time trace of the drive amplitude: a fast initial chirp frequency chirp( 10 n s ) can selectively steer the initial state, while the qubit is detuned from the cavity ( ω q - ω c / 2 π ≈ 2 g ). It is followed by a slow displacement to increase contrast and lifetime of the latching state, while the qubit is resonant with the cavity ( κ / 2 π = 2.5 MHz ). The drive amplitude ramp is limited so that the photon blockade photon blockadeis not broken, but the contrast is enhanced by additional driving at the highest drive amplitude. (b) A diagram of transition frequencies shows how the drive frequency chirps through the JC ladder frequencies of the (+) manifold, and how the manifold changes due to the time dependent qubit frequency. (c) Wave packet snapshots at selected times (indicated by bullet points on panel (b)) of the chirping drive frequency of panel (b) conditioned on the initial state of the qubit. (d) The temporal evolution of the reduced density matrix | ρ m n | (the x , y axes denote the quantum numbers m , n of the cavity levels) of the cavity with the control pulse (a) when the qubit initial state is superposition 1 2 | 0 + | 1 . The resonator enters a mesoscopic state of superposition around t = t c due to the entanglement with the qubit and the quantum state sensitivity of the protocol. At later times the off-diagonal parts of this superposition dephase quickly due to the interaction with the environment and the state of the system is being completely projected around t = 3 t c .... strong!driving( t qubit being detuned. Due to the interaction with the qubit, the cavity behaves as nonlinear oscillator with its set of transition frequencies depending on the state of the qubit (see the two distinct sets of lines in Fig.  gino:chirp_figure(b)). The cavity responds with a ringing behavior which is different for the two cases (see Fig.  gino:chirp_figure(c)). The ringing due to the pulse effectively maps the | ↓ and | ↑ to the dim and bright state basins, respectively (see Fig. 3(c)). Since κ t c ≪ 1 , an initial superposition α | ↑ + β | ↓ maps into a coherent superposition of the dim and bright states. Next, (2) a much weaker long pulse transfers the initially created bright state (for initial | ↑ ) to even brighter and longer lived states ( t c t c effects a projection of the pointer state. In designing such a pulse sequence we have the following physical considerations: (a) the initial fast selective chirp... (Color) Symmetry breaking. State-dependent transition frequency ω n , q = 2 π E n + 1 , q - E n , q versus photon number n , where E n , q denotes energy of the system eigenstate with n photons and qubit state q : (a) for the JC model, parameters as in Figs.  gino:fig:latch000 and  gino:fig:densclass; (b) for the model extended to 2 qubits, δ 1 / 2 π = - 1.0 G H z , δ 2 / 2 π = - 2.0 G H z , g 1 / 2 π = g 2 / 2 π = 0.25 G H z . Here, χ 2 denotes the 0-photon dispersive shift dispersive regimeof the second qubit; (c) for the model extended to one transmon qubit  koch charge-insensitive 2007, tuned below the cavity, / 2 π = 7 G H z E C / 2 π = 0.2 G H z , E J / 2 π = 30 G H z , g / 2 π = 0.29 G H z . (For the given parameters, δ 01 / 2 π = - 0.5 G H z , δ 12 / 2 π = - 0.7 G H z , defining δ i j = E j - E i - , with E i the energy of the i th transmon level.) In all panels, the transition frequency asymptotically returns to the bare cavity frequency. In (a) the frequencies within the σ z = ± 1 manifolds are (nearly) symmetric with respect to the bare cavity frequency. For (b), if the state of one (‘spectator’) qubit is held constant, then the frequencies are asymmetric with respect to flipping the other (‘active’) qubit. In (c), the symmetry is also broken due the existence of higher levels in the weakly anharmonic transmon.... The solution of eqn  gino:eq:classic is plotted in Fig.  gino:fig:densclass for the same parameters as in Fig.  gino:fig:latch000b. For weak driving the system response approaches the linear response of the dispersively shifted cavity. Above the lower critical amplitude ξ C 1 the frequency response bifurcates, and the JC oscillator enters a region of bistability... Qubit state measurement in circuit QED Circuit QEDcan operate in different parameter regimes and relies on different dynamical phenomena of the strongly coupled transmon-resonator system strong!coupling. The dispersive readout is the least disruptive to the qubit state and it is realized where the cavity and qubit are strongly detuned. The high power readout operates in a regime where the system response can be described using a semi-classical model and yields an relatively high fidelity fidelitywith simple measurement protocol. When the cavity and qubit are on resonance (the quantum degenerate regime quantum degenerate regime) it is theoretically predicted that the photon blockade photon blockade can also be used to realize a high fidelity readout. gino:fig:context... matt-pc in Fig.  gino:fig:latch000, where we show the average heterodyne amplitude a as a function of drive frequency and amplitude. Despite the presence of 4 qubits in the device, the fact that extensions beyond a two-level model would seem necessary since higher levels of the transmons... Solution to the semiclassical equation  gino:eq:classic, using the same parameters as Fig.  gino:fig:latch000b. (a) Amplitude response as a function of drive frequency and amplitude. The region of bifurcation bifurcationis indicated by the shaded area, and has corners at the critical points C 1 , C 2 . The dashed lines indicate the boundaries of the bistable region for a Kerr oscillator (Duffing oscillator) Duffing oscillator, constructed by making the power-series expansion of the Hamiltonian to second order in N / . The Kerr bistability bistability Kerr region matches the JC region in the vicinity of C 1 but does not exhibit a second critical point. (b) Cut through (a) for a drive of 6.3 ξ 1 , showing the frequency dependence of the classical solutions (solid line). For comparison, the response from the full quantum simulation of Fig.  gino:fig:latch000b is also plotted (dashed line) for the same parameters. (c) Cut through (a) for driving at the bare cavity frequency, showing the large gain available close to C 2 (the ‘step’). Faint lines indicate linear response. (d) Same as (c), showing intracavity amplitude on a linear scale. gino:fig:densclass... Transmitted heterodyne amplitude a as a function of drive detuning (normalized by the dispersive shift dispersive regime χ = g 2 / δ ) and drive amplitude (normalized by the amplitude to put n = 1 photon in the cavity in linear response, ξ 1 = κ / 2 ). Dark colors indicate larger amplitudes. (a) Experimental data  matt-pc, for a device with cavity at 9.07 G H z and 4 transmon qubits transmonat 7.0 , 7.5 , 8.0 , 12.3 G H z . All qubits are initialized in their ground state, and the signal is integrated for the first 400 n s ≃ 4 / κ after switching on the drive. (b) Numerical results for the JC model of eqn  gino:eq:master, with qubit fixed to the ground state and effective parameters δ / 2 π = - 1.0 G H z , g / 2 π = 0.2 G H z , κ / 2 π = 0.001 G H z . These are only intended as representative numbers for circuit QED Circuit QEDand were not optimized against the data of panel (a). Hilbert space is truncated at 10,000 excitations (truncation artifacts are visible for the strongest drive), and results are shown for time t = 2.5 / κ .... (Color) Symmetry breaking. State-dependent transition frequency ω n , q = 2 π E n + 1 , q - E n , q versus photon number n , where E n , q denotes energy of the system eigenstate with n photons and qubit state q : (a) for the JC model, parameters as in Figs.  gino:fig:latch000 and  gino:fig:densclass; (b) for the model extended to 2 qubits, δ 1 / 2 π = - 1.0 G H z , δ 2 / 2 π = - 2.0 G H z , g 1 / 2 π = g 2 / 2 π = 0.25 G H z . Here, χ 2 denotes the 0-photon dispersive shift
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The two possible flux values at the readout spot leads to two possible frequencies for the tunable cavity coupled to the qubit loop. Similar microwave readout schemes have been used with other rf-SQUID phase qubits . For our circuit design, the size of this frequency difference is proportional to the slope d f c / d φ c of the cavity frequency versus flux curve at a particular cavity flux φ c = Φ c / Φ o . The transmission of the cavity can be measured with a network analyzer to resolve the qubit flux (or circulating current) states. The periodicity of the rf SQUID phase qubit can be observed by monitoring the cavity’s resonance frequency while sweeping the qubit flux. This allows us to observe the single-valued and double-valued regions of the hysteretic rf SQUID. In Fig.  Fig4(a), we show the cavity response to such a flux sweep for design A . Two data sets have been overlaid, for two different qubit resets ( φ q = ± 2 ) and sweep directions (to the left or to the right), allowing the double-valued or hysteretic regions to overlap. There is an overall drift in the cavity frequency due to flux crosstalk between the qubit bias line and the cavity’s rf SQUID loop that was not compensated for here. This helps to show how the frequency difference in the overlap regions increases as the slope d f c / d φ c increases.... (Color online) (a) Qubit spectroscopy (design A ) overlaid with cavity spectroscopy at two frequencies, f c =  6.58 GHz and 6.78 GHz. (b) Zoom-in of the split cavity spectrum in (a) when f c =  6.78 GHz with corresponding fit lines. (c) Zoom-in of the split cavity spectrum in (a) when f c =  6.58 GHz with corresponding fit lines. (d) Cavity spectroscopy (design B ) while sweeping the qubit flux with f c =  7.07 GHz showing a large normal-mode splitting when the qubit is resonant with the cavity. All solid lines represent the uncoupled qubit and cavity frequencies and the dashed lines show the new coupled normal-mode frequencies. Notice in (d) the additional weak splitting from a slot-mode just below the cavity, and in (c) and (d), qubit tunneling events are visible as abrupt changes in the cavity spectrum.... (Color online) (a) Cavity spectroscopy (design A ) while sweeping the cavity flux bias with the qubit far detuned, biased at its maximum frequency. The solid line is a fit to the model including the junction capacitance. (b) Zoom-in near the maximum cavity frequency showing a slot-mode. (c) Line-cut on resonance along the dashed line in (b) with a fit to a skewed Lorentzian (solid line).... In general, rf SQUID phase qubits have lower T 2 * (and T 2 ) values than transmons, specifically at lower frequencies, where d f 01 / d φ q is large and therefore the qubit is quite sensitive to bias fluctuations and 1/f flux noise . For example, 600 MHz higher in qubit frequency, at f 01 = 7.98 GHz, Ramsey oscillations gave T 2 * = 223  ns. At this location, the decay of on-resonance Rabi oscillations gave T ' = 727  ns, a separate measurement of qubit energy decay after a π -pulse gave T 1 = 658  ns, and so, T 2 ≈ 812  ns, or T 2 ≈ 3.6 × T 2 * , a small, but noticeable improvement over the lower frequency results displayed Fig.  Fig6. The current device designs suffer from their planar geometry, due to a very large area enclosed by the non-gradiometric rf SQUID loop (see Fig.  Fig1). Future devices will require some form of protection against flux noise , possibly gradiometric loops or replacing the large geometric inductors with a much smaller series array of Josephson junctions .... (Color online) Coupling rate 2 g / 2 π (design A ) as a function of cavity frequency ω c / 2 π . The solid red (blue) line is the prediction from Eq. ( eq:g) (including L x and C J ’s). The (dotted) dashed line is the prediction for capacitive coupling with C = 15  fF ( C = 5  fF). The solid circles were measured spectroscopically (see text). At lowest cavity frequency, the solid ⋆ results from a fit to the Purcell data, discussed later in section  TCQEDC. The gray region highlights where the phase qubit (design A ) remains stable enough for operation (see text).... Next, we carefully explore the size of the dispersive shifts for various cavity and qubit frequencies. In order to capture the maximum dispersive frequency shift experienced by the cavity, we applied a π -pulse to the qubit. A fit to the phase response curve allows us to extract the cavity’s amplitude response time 2 / κ , the qubit T 1 , and the full dispersive shift 2 χ . Changing the cavity frequency modifies the coupling g and the detuning Δ 01 , while changes to the qubit frequency change both Δ 01 and the qubit’s anharmonicity α . In Fig.  Fig9(a), we show the phase qubit’s anharmonicity as a function of its transition frequency ω 01 / 2 π extracted from the spectroscopic data shown in Fig.  Fig5 from section  QBB for design A . The solid red line is a polynomial fit to the experimental data, used to calculate the three-level model curves in Fig.  Fig9(b–d), while the blue line is a theoretical prediction of the relative anharmonicity (including L x , but neglecting C J ) using perturbation theory and the characteristic qubit parameters extracted section  QBB. In Fig.  Fig9(b–d), we find that the observed dispersive shifts strongly depend on all of these factors and agree well with the three-level model predictions . For comparison, in Fig.  Fig9(b), we show the results for the two-level system model (bold dashed line) when f c = 6.58  GHz, which has a significantly larger amplitude for all detunings (outside the “straddling regime”). Notice that it is possible to increase the size of the dispersive shifts for a given | Δ 01 | / ω 01 by decreasing the cavity frequency f c , which increases the coupling rate 2 g / 2 π (as seen in Fig.  Fig2 in section  TCQED). Also, notice that decreasing the ratio of | Δ 01 | / ω 01 also significantly increases the size of the dispersive shifts, even when the phase qubit’s relative anharmonicity α r decreases as ω 01 increases. Essentially, the ability to reduce | Δ 01 | helps to counteract any reductions in α r . These results clearly demonstrate the ability to tune the size of the dispersive shift through selecting the relative frequency of the qubit and the cavity. This tunability offers a new flexibility for optimizing dispersive readout of qubits in cavity QED architectures and provides a way for rf SQUID phase qubits to avoid the destructive effects of tunneling-based measurements.... (Color online) (a) Time domain measurements (design A ). Rabi oscillations for frequencies near f 01 = 7.38 GHz. (b) Line-cut on-resonance along the dashed line in (a). The fit (solid line) yields a Rabi oscillation decay time of T ' = 409  ns. (c) Ramsey oscillations versus qubit flux detuning near f 01 = 7.38  GHz. (d) Line-cut along the dashed line in (c). The fit (solid line) yields a Ramsey decay time of T 2 * = 106  ns. With T 1 = 600  ns, this implies a phase coherence time T 2 = 310  ns.... (Color online) (a) Pulse sequence. (b) Rabi oscillations (design A ) for various pulse durations obtained using dispersive measurement at f 01 = 7.18 GHz, with Δ 01 = + 10 g . (c) A single, averaged time trace along the vertical dashed line in (b). (d) Rabi oscillations extracted from the final population at the end of the drive pulse, along the dashed diagonal line in (b). (e) Zoom-in of dashed box in (b) showing Rabi oscillations observed during continuous driving.... We can explore the coupled qubit-cavity behavior described by Eq. ( eq:H) by performing spectroscopic measurements on either the qubit or the cavity near the resonance condition, ω 01 = ω c . Fig.  Fig7(a) shows qubit spectroscopy for design A overlaid with cavity spectroscopy for two cavity frequencies, f c =  6.58 GHz and 6.78 GHz. Fig.  Fig7(d) shows cavity spectroscopy for design B with the cavity at its maximum frequency of f c m a x = 7.07  GHz while sweeping the qubit flux bias φ q . In both cases, when the qubit frequency f 01 is swept past the cavity resonance, the inductive coupling generates the expected spectroscopic normal-mode splitting.... The weak additional splitting just below the cavity in Fig.  Fig7(d) is from a resonant slot-mode. We can determine the coupling rate 2 g / 2 π between the qubit and the cavity by extracting the splitting size as a function of cavity frequency f c from the measured spectra. Three examples of fits are shown in Fig.  Fig7(b–d) with solid lines representing the bare qubit and cavity frequencies, whereas the dashed lines show the new coupled normal-mode frequencies. For design A ( B ), at the maximum cavity frequency of 6.78 GHz (7.07 GHz), we found a minimum coupling rate of 2 g m i n / 2 π = 78  MHz (104 MHz). Notice that the splitting size is clearer bigger in Fig.  Fig7(c) than for Fig.  Fig7(b) by about 25 MHz. The results for the coupling rate 2 g / 2 π as a function of ω c / 2 π for design A were shown in Fig.  Fig2 in section  TCQED. Also visible in Fig.  Fig7(c–d) are periodic, discontinuous jumps in the cavity spectrum. These are indicative of qubit tunneling events between adjacent metastable energy potential minima, typical behavior for hysteretic rf SQUID phase qubits . Moving away from the maximum cavity frequency increases the flux sensitivity, with the qubit tunneling events becoming more visible as steps. This behavior is clearly visible in Fig.  Fig7(c) and was already shown in Fig.  Fig4 in Sec.  QBA and, as discussed there, provides a convenient way to perform rapid microwave readout of traditional tunneling measurements . Next, we describe dispersive measurements of the phase qubit for design A . These results agree with the tunneling measurements across the entire qubit spectrum.
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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 =  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 =  t exp - i t ζ k 2 | 1 k 1 k | n ̂ + 1 - | 0 k 0 k | n ̂ / ℏ Δ k , with  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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(color online) Decoherence suppression by the auxiliary chaotic setup. (a) the evolution of the coherence C x y = S ̂ x 2 + S ̂ y 2 of the state of the qubit, where the red asterisk curve and the black triangle curve represent the ideal trajectory without any decoherence and the trajectory under natural decoherence and without corrections; and the green curve with plus signs and the blue solid curve denote the trajectories with I 0 / ω q = 5 and 30 . With these parameters, the dynamics of the Duffing oscillator exhibits periodic and chaotic behaviors. τ = 2 π / ω q is a normalized time scale. (b) and (c) are the energy spectra of δ q t with I 0 / ω q = 5 (the periodic case) and 30 (the chaotic case). The energy spectrum S δ q ω is in unit of decibel (dB). (d) the normalized decoherence rates Γ / ω q versus the normalized driving strength I 0 / ω q .
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