Contributors:Griffith, E. J., Ralph, J. F., Greentree, Andrew D., Clark, T. D.
In the example figure (Fig. fig:qubosc1d2d), the control bias is varied from left to right for a low frequencyoscillator 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 frequencyoscillator 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 frequencyoscillator 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 frequencyoscillator 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 frequencyoscillator 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 ).
Contributors:Wineland, D. J., Barrett, M., Britton, J., Chiaverini, J., DeMarco, B., Itano, W. M., Jelenkovi'c, B., Langer, C., Leibfried, D., Meyer, V., Rosenband, T., Schätz, T.
Hence, the dipole force on the | ↓ state is twice that on the | ↑ state and in the opposite direction. (This same displacement operator has been used previously to create Schrödinger cat states of a single ion (Myatt et al. 2000).) To implement this gate on two ions, the Raman transition beams were separated in frequency by 3 ω z + δ , where 3 ω z is the stretch mode frequency for two ions aligned along the z axis and δ is a small detuning ( ≪ ω z ). The separation of the ions was adjusted to be an integer multiple of 2 π / Δ k so that the optical-dipole force on each ion was in the same direction if the ions were in the same spin state but in opposite directions if the spin states was different (Eq. ( dipole_force)). This had the effect that the application of the laser beams to the | ↓ | ↑ and | ↑ | ↓ states caused excitation on the stretch mode but the motion was not excited when the ions were in the | ↓ | ↓ or | ↑ | ↑ states. The detuning δ and duration of the displacement pulses ( 2 π / δ ) were chosen to make one complete (circular) path in phase space with an area that gave a phase shift of π / 2 on the | ↓ | ↑ and | ↑ | ↓ states (Fig. Didi_gate). Under these conditions, the overall transformation was: | ↓ | ↓ → | ↓ | ↓ , | ↓ | ↑ → e i π / 2 | ↓ | ↑ , | ↑ | ↓ → e i π / 2 | ↑ | ↓ , and | ↑ | ↑ → | ↑ | ↑ = e i π e - i π | ↑ | ↑ . Therefore, this operator acts like the product of an operator that applies a π / 2 phase shift to the | ↑ state on each ion separately (a non-entangling gate) and π phase gate between the two ions. The π / 2 phase shifts can be removed by applying additional single qubit rotations or accounted for in software. (In an algorithm carried out by a series of single-qubit rotations and two-qubit phase gates, the extra phase shifts can be removed by appropriately shifting the phase of subsequent or prior single-qubit rotations.)... Schematic representation of relevant energy levels for stimulated-Raman transitions (not to scale). Shown are two ground-state hyperfine levels ( | ↓ and | ↑ ) for one ion, two (of possibly many) excited levels ( | i and | i ' ), and the harmonic oscillator levels for one mode of motion. Typically, ω z ≪ ω 0 ≪ Δ i , Δ i ' , ω F ≪ ω o p t where ω o p t is an optical frequency.... Time and Frequency Division\\National Institute of Standards and Technology\\Boulder,
CO, 80305-3328, USA\\
$\dag$ permanent address: Institute of Physics, Belgrade,
... Schematic representation of the displacements of the axial stretch-mode amplitude in phase space for the four basis states of the two spin qubits. The detuning and amplitude of the displacements are chosen to give a π / 2 phase shift on the | ↓ | ↑ and | ↑ | ↓ states while the | ↓ | ↓ and | ↑ | ↑ states are unaffected because the optical dipole forces for these states do not couple to the stretch mode.
Contributors:Kofman, A. G., Zhang, Q., Martinis, J. M., Korotkov, A. N.
The first-qubitoscillationfrequency 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.
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 oscillatorfrequency 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
Contributors:Beaudoin, Félix, da Silva, Marcus P., Dutton, Zachary, Blais, Alexandre
(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.
Contributors:Ozeri, R., Itano, W. M., Blakestad, R. B., Britton, J., Chiaverini, J., Jost, J. D., Langer, C., Leibfried, D., Reichle, R., Seidelin, S., Wesenberg, J. H., Wineland, D. J.
A list of atomic constants of several of the ions considered for quantum information processing. Here I is the nuclear spin, γ is the natural linewidth of the P 1 / 2 level , ω 0 is the frequency separation between the two qubit states set by the hyperfine splitting of the S 1 / 2 level , ω f is the fine-structure splitting , λ 1 / 2 and λ 3 / 2 are the wavelengths of the transitions between the S 1 / 2 and the P 1 / 2 and P 3 / 2 levels , respectively. The branching ratio of decay from the P levels to the D and the S levels is f .... Table Table3 lists ϵ R ∞ for different ion species for ω t r a p / 2 π =5 MHz and a single-circle ( K =1) gate. With the exception of 9 Be + , the error due to photon recoil in a two ion-qubit gate is below 10 -4 . For this error heavier ions benefit due to their smaller recoil.... Relevant energy levels (not to scale) in an ion-qubit, with nuclear spin I . The P 1 / 2 and P 3 / 2 excited levels are separated by an angular frequency ω f . The S 1 / 2 electronic ground state consists of two hyperfine levels F = I - 1 / 2 and F = I + 1 / 2 . The relative energies of these two levels depends on the sign of the hyperfine constant A h f and can vary between ion species (in this figure, A h f is negative). The qubit is encoded in the pair of m F = 0 states of the two F manifolds separated by an angular frequency ω 0 . Coherent manipulations of the qubit levels are performed with a pair of laser beams that are detuned by Δ from the transition to the P 1 / 2 level, represented by the two straight arrows. The angular frequency difference between the two beams equals the angular frequency separation between the qubit levels ω b - ω r = ω 0 . Some ion species have D levels with energies below the P manifold. Wavy arrows illustrate examples of Raman scattering events.... Gates are assumed to be driven by pairs of Raman beams detuned by Δ from the transition between the S 1 / 2 and the P 1 / 2 levels (See Fig. Levels). We further assume that the Raman beams are linearly polarized and Raman transitions are driven by both σ + photon pairs and σ - photon pairs. The two beams in a Raman pair are designated as red Raman ( r ) and blue Raman ( b ) by their respective frequencies. In the following we also assume that Δ is much larger than the hyperfine and Zeeman splitting between levels in the ground and excited states.... Here P t o t a l - g a t e is the probability that one of the ions scattered a photon during the two-qubit gate, and P t o t a l is the one-qubit gate scattering probability given in Eq. ( photon per pulse). Since both the Raman and the total scattering probabilities increase by the same factor as compared to the one-qubit gate, the ratio of the two errors ϵ D / ϵ S will remain the same as given by Eq. ( StoD error_ratio). Table Table2 lists ϵ D / ϵ S for the different ions when ϵ S = 10 -4 . Notice that for ϵ S = 10 -4 some ions require | Δ 0 | 2 ω f / 3 f . For those ions ϵ D is no longer negligible compared to ϵ S .... Most ion species considered for QIP studies have a single valence electron, with a 2 S 1 / 2 electronic ground state, and 2 P 1 / 2 and 2 P 3 / 2 electronic excited states. Some of the ions also have D levels with lower energy than those of the excited state P levels. Ions with a non zero nuclear spin also have hyperfine structure in all of these levels. A small magnetic field is typically applied to remove the degeneracy between different Zeeman levels. Here we consider qubits that are encoded into a pair of hyperfine levels of the 2 S 1 / 2 manifold. Figure Levels illustrates a typical energy level structure.... Schematic of Raman laser beam geometry assumed for the two-qubit phase gate. The gate is driven by two Raman fields, each generated by a Raman beam pair. Each pair consists of two perpendicular beams of different frequencies that intersect at the position of the ions such that the difference in their wave vector lies parallel to the trap axis. One beam of each pair is parallel to the magnetic field which sets the quantization direction. The beams’ polarizations in each pair are assumed to be linear, perpendicular to each other and to the magnetic field. Wavy arrows illustrate examples of photon scattering directions.... NIST Boulder, Time and Frequency division, Boulder,
... A list of different errors in a two-qubit phase gate due to spontaneous photon scattering. The error due to Raman scattering back into the S 1 / 2 manifold, ϵ S , is calculated assuming Gaussian beams with w 0 =20 μ m, a gate time τ g a t e =10 μ s, ω t r a p / 2 π = 5 MHz, a single circle in phase space ( K =1), and 10 mW in each of the four Raman beams. P 0 is the power in milliwatts needed in each of the beams, and Δ 0 / 2 π is the detuning in gigahertz for ϵ S = 10 -4 . The ratio between errors due to Raman scattering to the D and S manifolds, ϵ D / ϵ S , is given when ϵ S = 10 -4 . The asymptotic value of ϵ D in the | Δ | ≫ ω f limit is ϵ D ∞ . The Lamb-Dicke parameter η for the above trap frequency is also listed for different ions.... A list of errors in a single-qubit gate ( π rotation) due to spontaneous photon scattering. The error due to Raman scattering back into the S 1 / 2 manifold ϵ S is calculated with the same parameters as Fig. Power_vs_error: Gaussian beams with w 0 = 20 μ m, a single ion Rabi frequency Ω R / 2 π = 0.25 MHz ( τ π = 1 μ sec), and 10 mW in each of the Raman beams. P 0 is the power (in milliwatts) needed in each of the beams, and Δ 0 / 2 π is the detuning (in gigahertz) for ϵ S = 10 -4 . The ratio between errors due to Raman scattering to the D and S manifolds, ϵ D / ϵ S , is given when ϵ S = 10 -4 . The asymptotic value of ϵ D in the | Δ | ≫ ω f limit is ϵ D ∞ .... Assume that the ratio of the beam waist to the transition wavelength is constant for different ion species. In this case, the power needed to obtain a given Rabi frequency and to keep the error below a given value would scale linearly with the optical transition frequency. A more realistic assumption might be that the Raman beam waist is not diffraction limited and is determined by other experimental considerations, such as the inter-ion distance in the trap or beam pointing fluctuations. In this case, assuming that w 0 is constant, the required power would scale as the optical transition frequency cubed. Either way, ion species with optical transitions of longer wavelength are better suited in the sense that less power is required for the same gate speed and error requirements. In addition, high laser power is typically more readily available at longer wavelengths. Finally, we note that the error is independent of the fine-structure splitting as long as we have sufficient power to drive the transition. The transition wavelengths of different ions are listed in Table Table0.
Contributors:Yoshihara, Fumiki, Nakamura, Yasunobu, Yan, Fei, Gustavsson, Simon, Bylander, Jonas, Oliver, William D., Tsai, Jaw-Shen
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 .
Contributors:Hanneke, D., Home, J. P., Jost, J. D., Amini, J. M., Leibfried, D., Wineland, D. J.
Single-qubit-gate parameters used for programming the quantum processor to produce the states in Figure 2 of the main text.... Components for universal computation. Circuit diagrams for arbitrary unitary transformations on a, one and b, two qubits. The operations for each circuit are implemented from left to right with each line representing one qubit. Part a also indicates the decomposition employed for R θ φ . The dashed box in part b contains the three degrees of freedom α , β , δ that determine the two-qubit operation’s local equivalence class. The brackets highlight the decomposition of the two-qubit operation U as described in the text: U = C ⊗ D ⋅ V ⋅ A ⊗ B .... Time and Frequency Division, National Institute of Standards and Technology, Boulder,
Colorado 80305, USA
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.
Contributors:Whittaker, J. D., da Silva, F. C. S., Allman, M. S., Lecocq, F., Cicak, K., Sirois, A. J., Teufel, J. D., Aumentado, J., Simmonds, R. W.
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 qubitfrequency, 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 qubitfrequencies. 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 qubitfrequency 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 qubitfrequency 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.