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**frequencies**...**frequencies**....**frequency**... network of stochastic**oscillators**...**QUBIT**...**frequencies**);Data Types:- Document

**qubits**Data Types:- Document

- IEEE Transactions on Ultrasonics, FerroElectrics, and
**Frequency**Control...**oscillator**Data Types:- Other

- 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, Yugoslavia ... 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.Data Types:- Image
- Tabular Data

- 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 ).Data Types:- Image
- Tabular Data

- 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.Data Types:- Image
- Tabular Data

- 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. Fig1Data Types:- Image
- Tabular Data

- (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 spectatorStark 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**’s**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.Data Types:- Image
- Tabular Data

- 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, Colorado 80305 ... 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.Data Types:- Image
- Tabular Data

- 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 theenergy eigenbasis; this component is not averaged out when Ω R is comparable to ω m w .**qubit**’sData Types:- Image
- Tabular Data