### 56121 results for qubit oscillator frequency

Contributors: Fedorov, A., Feofanov, A. K., Macha, P., Forn-Díaz, P., Harmans, C. J. P. M., Mooij, J. E.

Date: 2010-04-09

fig:rabi_sym_2(color online). (a) MW **frequency** vs f ϵ . The dotted white line is obtained from Eq. ( total H1) with I p = 400 nA and Δ = ν o s c . The observed vacuum Rabi splitting is maximal due to fully transverse coupling of the **qubit** to the **oscillator** η = π / 2 . (b) Vacuum Rabi **oscillations** for different values of f ϵ . In the experiment f ϵ was controlled by the amplitude of the current pulse I ϵ while Δ was tuned to ν o s c by changing the external magnetic field B . The inset shows ν R extracted from data (red circles) and estimated from Eq. ( Rabi freq general) (blue line). The color indicates the switching probability of the SQUID minus 0.5....fig:spectrum1(color online). (a) Schematic representation of the control and measurement pulses to perform spectroscopy. (b) Diagram of Landau-Zener transitions transferring the excitation of the **oscillator** to the **qubit**. (c) MW **frequency** vs f ϵ (controlled by the amplitude of the current pulse I ϵ ). The color indicates the switching probability of the SQUID minus 0.5. The white dotted line is obtained from Eq. ( total H1) with Δ = 2.04 GHz, I p = 420 nA. The vacuum Rabi splitting of 180 MHz corresponds to the effective **qubit**-**oscillator** coupling strength reduced by sin η ....A flux **qubit** biased at its symmetry point shows a minimum in the energy splitting (the gap), providing protection against flux noise. We have fabricated a **qubit** whose gap can be tuned fast and have coupled this **qubit** strongly to an LC **oscillator**. We show full spectroscopy of the **qubit**-resonator system and generate vacuum Rabi **oscillations**. When the gap is made equal to the **oscillator** **frequency** $\nu_{osc}$ we find the strongest **qubit**-resonator coupling ($g/h\sim0.1\nu_{\rm osc}$). Here being at resonance coincides with the optimal coherence of the symmetry point. Significant further increase of the coupling is possible....fig:rabi_sym(color online). Vacuum Rabi **oscillations** (a) and MW **frequency** (b) vs magnetic f α . In the experiment the **qubit** was kept in its symmetry point ( ϵ = 0 ) by appropriately adjusting the amplitude of the current pulse I ϵ while Δ was changed by f α with use of external magnetic field B (a) or by applying the current pulse I α for fixed B (b). The color scale shows the switching probability of the SQUID minus 0.5. (c) **Frequency** of the vacuum Rabi **oscillations** extracted from data (a) and theoretical estimation (blue line) from Eq. ( Rabi **frequency**) as a function of f α . The minimum in ν R determines the bare **qubit**-**oscillator** coupling 2 g and corresponds to the resonance conditions Δ = ν o s c . (d) Single trace of the vacuum Rabi **oscillations** for Δ ≃ ν o s c ....fig:scheme(color online). (a) Circuit schematics: the tunable gap flux **qubit** (green) coupled to a lumped element superconducting LC **oscillator** (red) and controlled by the bias lines I ϵ , I ϵ , d c , I α (black). The SQUID (blue) measures the state of the **qubit**. The gradiometer loop (emphasized by a dashed line) is used to trap fluxoids. (b) Scanning Electron Micrograph (SEM) of the sample. (c) Energy diagram of the **qubit**-**oscillator** system. The minimum of energy splitting of the **qubit** Δ is reached at the symmetry point when one fluxoid is trapped in the gradiometer loop and the difference in magnetic fluxes f ϵ Φ 0 is 0 controlled by I ϵ and I ϵ , d c . By controlling the flux f α Φ 0 with I α and uniform field B one can tune Δ in resonance with **oscillator** **frequency** ν o s c . ... A flux **qubit** biased at its symmetry point shows a minimum in the energy splitting (the gap), providing protection against flux noise. We have fabricated a **qubit** whose gap can be tuned fast and have coupled this **qubit** strongly to an LC **oscillator**. We show full spectroscopy of the **qubit**-resonator system and generate vacuum Rabi **oscillations**. When the gap is made equal to the **oscillator** **frequency** $\nu_{osc}$ we find the strongest **qubit**-resonator coupling ($g/h\sim0.1\nu_{\rm osc}$). Here being at resonance coincides with the optimal coherence of the symmetry point. Significant further increase of the coupling is possible.

Data types:

Contributors: Hausinger, Johannes, Grifoni, Milena

Date: 2010-09-08

Figure Fig::SpectrumVSg shows the quasienergy spectrum against the coupling strength g . For simplicity, we study the unbiased case ε = 0 , which implies m = L = 0 and hence gaps with Ω 0 , 0 n , K = | Δ 0 L K 0 α e - α 2 | ≡ Ω K . Thus, for g = 0 and Δ ≠ 0 , the twofold degeneracy of the unperturbed case is lifted by a gap of width Δ 0 . For g ≠ 0 , the gap size is further determined by the Laguerre polynomial, so that additional degeneracies can occur at the zeros of L K 0 α . When choosing the driving amplitude A such that Δ 0 = 0 the twofold degeneracy is kept for arbitrary g and K . Because the dressing by the Bessel function does not depend on g or the **oscillator** level, we reach the remarkable conclusion that the coherent destruction of tunneling (CDT), predicted for a driven **qubit** , might occur also for a **qubit**-**oscillator** system in the ultrastrong coupling limit. In Fig. Fig::DressedOsc, the dressed **oscillation** **frequencies** are plotted against the dimensionless coupling g / Ω . Next to an exponential decay, they exhibit zeros that depend through the Laguerre polynomial characteristically on the **oscillator** quantum number K . Hence, because the ** qubit’s** dynamics involves several

**oscillator**levels, we predict that suppression of tunneling cannot be reached by just tuning the coupling g . The dynamics. To prove the statements above, we calculate the survival probability of the

**qubit**P ↓ ↓ t : = ↓ | ρ ̂ r e d t | ↓ , where ρ ̂ r e d is obtained by tracing out the

**oscillator**degrees of freedom from the density operator of the

**qubit**-

**oscillator**system:...We introduce an approach to studying a driven

**qubit**-

**oscillator**system in the ultrastrong coupling regime, where the ratio $g/\Omega$ between coupling strength and

**oscillator**

**frequency**approaches unity or goes beyond, and simultaneously for driving strengths much bigger than the

**qubit**energy splitting (extreme driving). Both

**qubit**-

**oscillator**coupling and external driving lead to a dressing of the

**qubit**tunneling matrix element of different nature: the former can be used to suppress selectively certain

**oscillator**modes in the spectrum, while the latter can bring the

**qubit**'s dynamics to a standstill at short times (coherent destruction of tunneling) even in the case of ultrastrong coupling....In Fig. Fig::Dynam1(c) we are with g / Ω = 1.0 already deep in the ultrastrong coupling regime. The

**frequency**Ω 1 is now different from zero, and additionally Ω 3 appears. The lowest peak belongs to the

**frequencies**Ω 0 , Ω 2 , and Ω 4 , which are equal for g / Ω = 1.0 , see Fig. Fig::DressedOsc. A complete population inversion again takes place. Our results are confirmed by numerical calculations. For g = 0.5 , 1.0 , the latter yield additionally fast

**oscillations**with Ω and ω ex . Furthermore, Ω 1 is shifted in Fig. Fig::Dynam1(c) slightly to the left, so that concerning the survival probability the analytical and numerical curves get out of phase for longer times. To include also the

**oscillations**induced by the driving and the coupling to the quantized modes, connections between the degenerate subspaces need to be included in the calculation of the eigenstates of the full Hamiltonian ....(Color online) Quasienergy spectrum of the

**qubit**-

**oscillator**system against the static bias ε for weak coupling g / ω ex = 0.05 . Further parameters are Δ / ω ex = 0.2 , Ω / ω ex = 2 , A / ω ex = 2.0 . The first six

**oscillator**states are included. Numerical calculations are shown by red (light gray) triangles, analytical results in the region of avoided crossings by black dots. A good agreement between analytics and numerics is found. Blue (dark gray) squares represent the case Δ = 0 . Fig::QuasiEnEpsAnaDfinite...(Color online) Size of the avoided crossing Ω K against the dimensionless coupling strength g / Ω for an unbiased

**qubit**( ε = 0 ). Further, Δ / Ω = 0.4 , ω ex / Ω = 5.3 and A / Ω = 8.0 . Ω K vanishes at the zeros of the Laguerre polynomial L K 0 α . The dashed lines (a), (b), (c) represent g / Ω = 0.1 , 0.5 , 1.0 , respectively, as considered in Fig. Fig::Dynam1. Fig::DressedOsc...(Color online) Coherent destruction of tunneling in a driven

**qubit**-

**oscillator**system. The same parameters as in Fig. Fig::Dynam1 are used except that A / Ω = 12.7 , which leads to Δ 0 = 0 . Three coupling strengths are examined: g / Ω = 0.1 (a), 0.5 (b) and 1.0 (c). The analytical calculations (black, dashed lines) predict complete localization for all three cases. Also the numerics (red curves) shows strong localization for short timescales with fast

**oscillations**overlaid. For long times this localization vanishes (see inset in (a)). Fig::DynamCDT...While tuning the coupling g to a zero of a Laguerre polynomial corresponding to a dominant

**oscillator**mode yields a reduction of tunneling, tuning the driving amplitude A to a zero of a Bessel function can yield almost complete localization at short times. As already noticed in Fig. Fig::SpectrumVSg, this phenomenon is independent of the coupling strength g . We choose in Fig. Fig::DynamCDT the driving amplitude A , so that Δ 0 = 0 . This is the same condition as found for CDT in a driven

**qubit**. Analogously, our analytical solution now predicts localization for arbitrary coupling strength g . All dressed

**oscillation**

**frequencies**Ω K vanish. However, third-order corrections in Δ will give small contributions to Δ 0 . Hence, a numerical exact solution yields

**oscillations**of P ↓ ↓ t with a long period. On a short timescale and for ω ex ≫ Δ also the numerical solution appears to be strongly localized, while for long times, the inset in Fig. Fig::DynamCDT (a) shows complete population inversion for the numerics. In conclusions, we developed a powerful formalism to investigate analytically a

**qubit**-

**oscillator**system in the ultrastrong coupling and extreme driving regime, a situation which is in close experimental reach and offers excellent control possibilities. Our approach relies on perturbation theory with respect to a single parameter only, the

**qubit**tunneling matrix element Δ , and thus goes beyond the driven Jaynes-Cummings model, with no rotating-wave approximation being applied. We acknowledge financial support under DFG Program SFB631. We thank Sigmund Kohler for helpful remarks....(Color online) Dynamics of the

**qubit**for ε = 0 , Δ / Ω = 0.4 , ω ex / Ω = 5.3 , A / Ω = 8.0 , and temperature ℏ Ω k B T -1 = 10 . The graphs show the Fourier transform F ν of the survival probability P ↓ ↓ t (see the insets). We study the different coupling strengths indicated in Fig. Fig::DressedOsc, g / Ω = 0.1 (a), 0.5 (b) and 1.0 (c). Analytical results are shown by black curves, numerics by dashed orange curves. Fig::Dynam1...Additional crossings occur independent of ε if driving and

**oscillator**

**frequency**are commensurable, Ω / ω ex = j / N with integers j , N > 0 , resulting in infinite many degenerate states. We avoid such a situation by choosing incommensurable

**frequencies**or high values for j and N , so that only high-photon processes are affected.. Note that for L ≠ 0 there are always L nondegenerate levels. For L > 0 those are the first L spin-down states (positive slope), while for L < 0 the first L spin-up states (negative slope). At finite Δ avoided crossings occur in the energy spectrum at the sites of the resonances (red triangles and black dots in Fig. Fig::QuasiEnEpsAnaDfinite). To explain the origin of these avoided crossings we express H ̂ in the basis ( CoupledEigenstates) yielding the off-diagonal elements ... We introduce an approach to studying a driven

**qubit**-

**oscillator**system in the ultrastrong coupling regime, where the ratio $g/\Omega$ between coupling strength and

**oscillator**

**frequency**approaches unity or goes beyond, and simultaneously for driving strengths much bigger than the

**qubit**energy splitting (extreme driving). Both

**qubit**-

**oscillator**coupling and external driving lead to a dressing of the

**qubit**tunneling matrix element of different nature: the former can be used to suppress selectively certain

**oscillator**modes in the spectrum, while the latter can bring the

**qubit**'s dynamics to a standstill at short times (coherent destruction of tunneling) even in the case of ultrastrong coupling.

Data types:

Contributors: Kofman, A. G., Zhang, Q., Martinis, J. M., Korotkov, A. N.

Date: 2006-06-02

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**. ...We analyze the crosstalk error mechanism in measurement of two capacitively coupled superconducting flux-biased phase **qubits**. The damped **oscillations** of the superconducting phase after the measurement of the first **qubit** may significantly excite the second **qubit**, leading to its measurement error. The first **qubit**, which is highly excited after the measurement, is described classically. The second **qubit** is treated both classically and quantum-mechanically. The results of the analysis are used to find the upper limit for the coupling capacitance (thus limiting the **frequency** of two-**qubit** operations) for a given tolerable value of the measurement error probability....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. ... We analyze the crosstalk error mechanism in measurement of two capacitively coupled superconducting flux-biased phase **qubits**. The damped **oscillations** of the superconducting phase after the measurement of the first **qubit** may significantly excite the second **qubit**, leading to its measurement error. The first **qubit**, which is highly excited after the measurement, is described classically. The second **qubit** is treated both classically and quantum-mechanically. The results of the analysis are used to find the upper limit for the coupling capacitance (thus limiting the **frequency** of two-**qubit** operations) for a given tolerable value of the measurement error probability.

Data types:

Contributors: Bennett, Douglas A., Longobardi, Luigi, Patel, Vijay, Chen, Wei, Averin, Dmitri V., Lukens, James E.

Date: 2008-11-14

(Color online) (a) The occupation of the excited state on resonance versus microwave amplitude in units of the corresponding Rabi **frequency**. The lines are calculations for the following parameters; red solid Γ = 0.055 n s -1 and σ ν = 0.235 n s -1 , blue dotted line σ ν = 0 with Γ = 0.055 n s -1 , purple dashed line σ δ = 0 and Γ = 0.20 n s -1 . (b) The width of the spectroscopic peak from the Gaussian fits as a function of microwave amplitude in units of Rabi **frequency**. The lines are calculations for the following parameters; red solid Γ = 0.055 n s -1 and σ ν = 0.235 n s -1 , green dashed line σ ν = 0 with Γ = 0.055 n s -1 , blue dotted line σ ν = 0 and Γ = 0.585 n s -1 ....The effects of flux noise on the decay of the Rabi **oscillation**, of course, becomes much more pronounced for δ ≠ 0 . Figure rabidetunefig shows such Rabi **oscillations** at various detunings over a range of 0 ≤ δ 10 9 s -1 . At long microwave pulse times (not shown) the occupation of the excited state reaches the equilibrium values discussed in Sec secintraspec. Near resonance, the value of ρ 11 for long pulse lengths is determined mainly by the noise amplitude, while for large δ it is set by δ . The data cannot be fit to solutions of the Bloch equations using only the phenomenological decay constants. As in the spectroscopy, static Gaussian detuning noise must be included to fit the data over the whole range of detuning. The lines in Figure rabidetunefig correspond to calculations with an initial Rabi **frequency** of 0.59 n s -1 , Γ = 0.075 n s -1 and σ ν = 0.22 n s -1 . The detuning **frequency** is taken from φ x using the conversion from Fig. spec2dfig. For the range of δ in Fig. rabidetunefig Δ E 01 is in a region with relatively few spectral anomalies. However for δ < 0 , on the other side of resonance towards splittings in the spectroscopy, the data generally do not agree with theory suggesting that the spectroscopic splittings have a strong effect on the coherence, as expected....The data for resonant Rabi **oscillations**, shown in Fig. rabifig. The solid line is a fit using Eq. fin including a 0.5 ns time delay for the rise time for the microwave pulse and averaged over quasi-static Gaussian noise in φ x equal to that obtained from the fits to the spectroscopy data in Fig. specfig. This gives a Rabi **frequency** f r a b i = 119 M H z and decay rate, η = 0.042 n s -1 , for the **oscillations**. Equation equ:rabidecay, with δ = 0 , together with our previously measured value for the decay rate of the first excited state Γ 1 (which gives the rate Γ in the equations above), and the observed Rabi decay rate η discussed below, imply Γ ν = 0.01 n s -1 . Even though Ω R a b i is not affected by flux noise to first order for δ = 0 , the amplitude of the flux noise in our **qubit** is large enough to cause a measurable effect. If the flux noise were neglected, it would be necessary to increase η to 0.058 n s -1 in order to account for the observed decay. In addition to increasing the decay rate, the low **frequency** noise also reduces the steady state occupation of the excited state , i.e. its value for long pulses, as calculated from Eq. av. The ordinate of Fig. specfig has been calibrated using this value and is consistent with the estimate obtained from the calculated tunneling rates from the two levels involved during the readout pulse....(Color online) (a) Schematic and (b) micrograph of an rf SQUID **qubit** and the readout magnetometer, (c) a cross section of the wafer around the junctions and (d) micrograph giving a detailed view of junctions....The components that create and combine the microwave pulses and readout pulse are shown at the top of Fig. fig:circuitdiag. The pulse generator is capable of producing measurement pulses with rise times as short as 200 ps. Two microwave mixers are used to modulate the envelope of the continuous microwave signal produced by the microwave source giving an on/off ratio of 10 3 . The output of the mixers is then amplified 20 dB and coupled through variable attenuators that set the amplitude of the microwave pulses applied to the **qubit**. Finally, these microwave pulses are combined with the video pulses using a hybrid coupler. The video pulses enter on the directly coupled port while the microwave pulse are coupled using the indirectly coupled port. The video pulses used for the data shown in the following sections are around typically 5 ns with a rise time of 0.5 ns. The directionality and **frequency** response of the coupler allows the two signals to be combined with a minimum amount of reflections and loss of power. This signal is then coupled to the **qubit** through a coaxial line that is filtered by a series of attenuators at 1.4 K (20 dB), 600 mK (10 dB) and at the **qubit** temperature (10 dB) followed by a lossy microstrip filter that cuts off around 1 GHz....The measurement sequence for coherent **oscillations** is very similar to that used to measure the lifetime of the excited state except that the duration of the (generally shorter) microwave pulse is varied and the microwave pulse is immediately followed by the readout flux pulse. This signal, on the high **frequency** line, is illustrated in the lower inset in Figure rabifig. Figure rabifig shows an example of the Rabi **oscillations** when δ = 0 and the microwave **frequency**, f x r f = 17.9 G H z . This bias point lies in a "clean" range of the spectrum ( 17.6 - 18 G H z ) as seen in Fig. spec2dfig, which should be the best region for observing coherent **oscillations** between the ground and excited states. Most of the time domain data, including the lifetime measurements of Fig. decayfig, have been taken in this **frequency** range. Each data point in Fig. spec2dfig corresponds to the average of several thousand measurements for a given pulse length....We report measurements of coherence times of an rf SQUID **qubit** using pulsed microwaves and rapid flux pulses. The modified rf SQUID, described by an double-well potential, has independent, in situ, controls for the tilt and barrier height of the potential. The decay of coherent **oscillations** is dominated by the lifetime of the excited state and low **frequency** flux noise and is consistent with independent measurement of these quantities obtained by microwave spectroscopy, resonant tunneling between fluxoid wells and decay of the excited state. The **oscillation's** waveform is compared to analytical results obtained for finite decay rates and detuning and averaged over low **frequency** flux noise....This solution, shows that when driven with microwaves, the population of the excited states **oscillates** in time, demonstrating the phenomenon of Rabi **oscillations**, which have been seen in a number of different superconducting **qubits** . Ω , the **frequency** of the **oscillations** for δ = 0 , is ideally proportional to the amplitude of the microwaves excitation φ x r f . This linear dependence is accurately seen in our **qubit** at low power levels (see Fig. rabifig upper inset) providing a convenient means to calibrate the amplitude of the microwaves incident on the **qubit** in terms of Ω ....(Color online) Rabi **oscillations** for detunings going from top to bottom of 0.094, 0.211, 0.328, 0.562, 0.796 and 1.269 n s -1 with the corresponding fits using Γ = 0.075 n s -1 and σ ν = 0.22 n s -1...(Color online) The occupation of the excited state as a function of detuning for microwave powers corresponding to attenuator settings of 39 (squares), 36 (circles), 33 dB (triangles). Lines are fits using Eq. av at microwave amplitudes corresponding to the measured Rabi **frequency** for each attenuator setting (0.017, 0.024, 0.034 n s -1 ) with Γ = 0.055 n s -1 convoluted with static Gaussian noise with σ ν = 0.235 n s -1 at the angular **frequencies** of the Rabi **oscillations** that correspond to these microwave powers...(Color online) The occupation of the excited state as a function of the length of the microwave pulse demonstrating Rabi **oscillations**. The line is a fit to Eq. fin for δ = 0 averaged over quasi-static noise with σ ν = 0.22 n s -1 . This gives f r a b i = 119 MHz and decay time T 2 ˜ = 24 n s -1 . The upper inset shows Rabi **frequency** as a function of amplitude of applied microwaves in arbitrary units. The line is a linear fit to the lower microwave amplitude data. The lower inset show the measurement pulse sequence....Decoherence \and Superconducting **Qubits** \and Flux **Qubit** \and SQUIDs ... We report measurements of coherence times of an rf SQUID **qubit** using pulsed microwaves and rapid flux pulses. The modified rf SQUID, described by an double-well potential, has independent, in situ, controls for the tilt and barrier height of the potential. The decay of coherent **oscillations** is dominated by the lifetime of the excited state and low **frequency** flux noise and is consistent with independent measurement of these quantities obtained by microwave spectroscopy, resonant tunneling between fluxoid wells and decay of the excited state. The **oscillation's** waveform is compared to analytical results obtained for finite decay rates and detuning and averaged over low **frequency** flux noise.

Data types:

Contributors: Serban, I., Dykman, M. I., Wilhelm, F. K.

Date: 2009-07-29

An important feature of the **qubit** relaxation in the presence of driving is that the stationary distribution over the **qubit** states differs from the thermal Boltzmann distribution. If the **oscillator**-mediated decay is the dominating **qubit** decay mechanism, the **qubit** distribution is determined by the ratio of the transition rates Γ e and Γ g . One can characterize it by effective temperature T e f f = ℏ ω q / k B ln Γ e / Γ g . If the term in curly brackets in the numerator of Eq. ( eq:resonant_power_spectrum) is dominating, T e f f ≈ 2 T , but if the field parameters are varied so that this term becomes comparatively smaller T e f f increases, diverges, and then becomes negative, approaching -2 T . Negative effective temperature corresponds to population inversion. The evolution of the effective temperature with the intensity of the modulating field is illustrated in Fig. fig:effect_temp....Apart from the proportionality to r a 2 , the attractor dependence of Γ 1 is also due to the different curvature of the effective potentials around the attractors. For weak **oscillator** damping κ ≪ ν a , the parameter ν a in Eq. ( eq:resonant_power_spectrum) is the **frequency** of small-amplitude vibrations about attractor a . It sets the spacing between the quasienergy levels, the eigenvalues of the rotating frame Hamiltonian H S r close to the attractor. The function Re N + - ω has sharp Lorentzian peaks at ω = ± ν a with halfwidth κ determined by the **oscillator** decay rate. The dependence of ν a on the control parameter β is illustrated in Fig. fig:nu_a....Left panel: Squared scaled attractor radii r a 2 as functions of the dimensionless field intensity β for the dimensionless friction κ / | δ ω | = 0.3 . Right panel: The effective **frequencies** ν a / | δ ω | for the same κ / | δ ω . Curves 1 and 2 refer to small- and large amplitude attractors....The scaled decay rate factors for the excited and ground states, curves 1 and 2, respectively, as functions of scaled difference between the **qubit** **frequency** and twice the modulation **frequency**; Γ 0 = ℏ C Γ r a 2 / 6 γ S . Left and right panels refer to the small- and large-amplitude attractors, with the values of β being 0.14 and 0.12, respectively. Other parameters are κ / | δ ω = 0.3 , n ̄ = 0.5 ....The scaled decay rates Γ e , g as functions of detuning ω q - 2 ω F are illustrated in Fig. fig:decay_spectra. Even for comparatively strong damping, the spectra display well-resolved quasi-energy resonances, particularly in the case of the large-amplitude attractor. As the **oscillator** approaches bifurcation points where the corresponding attractor disappears, the **frequencies** ν a become small (cf. Fig. fig:nu_a) and the peaks in the **frequency** dependence of Γ e , g move to ω q = 2 ω F and become very narrow, with width that scales as the square root of the distance to the bifurcation point. We note that the theory does not apply for very small ω q - 2 ω F | where the **qubit** is resonantly pumped; the corresponding condition is m ω 0 Δ q δ C r e s 2 r a 2 / ℏ ω q 2 ≪ T 1 T 2 -1 + ω q - 2 ω F 2 T 1 T 2 . For weak coupling to the **qubit**, Γ e ≪ κ , it can be satisfied even at resonance....We investigate the relaxation of a superconducting **qubit** for the case when its detector, the Josephson bifurcation amplifier, remains latched in one of its two (meta)stable states of forced vibrations. The **qubit** relaxation rates are different in different states. They can display strong dependence on the **qubit** **frequency** and resonant enhancement, which is due to quasienergy resonances. Coupling to the driven **oscillator** changes the effective temperature of the **qubit**....Picot08, and was increasing with the driving strength on the low-amplitude branch (branch 1 in the left panel of Fig. fig:nu_a), in qualitative agreement with the theory. It is not possible to make a direct quantitative comparison because of an uncertainty in the **qubit** relaxation rates noted in Ref. ...The effective scaled **qubit** temperature T e f f * = k B T e f f / ℏ ω q as function of the scaled field strength β in the region of bistability for the small- and large-amplitude attractors, left and right panels, respectively; ω q - 2 ω F / | δ ω | = - 0.2 and 0.1 in the left and right panels; other parameters are the same as in Fig. fig:decay_spectra....The decay rate of the excited state of the **qubit** Γ e ∝ R e ~ N + - ω q - 2 ω F sharply increases if the **qubit** **frequency** ω q coincides with 2 ω F ± ν a , i.e., ω q - 2 ω F resonates with the inter-quasienergy level transition **frequency**. This new **frequency** scale results from the interplay of the system nonlinearity and the driving and is attractor-specific, as seen in Fig. fig:nu_a. In the experiment, for ω q close to 2 ω 0 , the resonance can be achieved by tuning the driving **frequency** ω F and/or driving amplitude F 0 . This quasienergy resonance destroys the QND character of the measurement by inducing fast relaxation. ... We investigate the relaxation of a superconducting **qubit** for the case when its detector, the Josephson bifurcation amplifier, remains latched in one of its two (meta)stable states of forced vibrations. The **qubit** relaxation rates are different in different states. They can display strong dependence on the **qubit** **frequency** and resonant enhancement, which is due to quasienergy resonances. Coupling to the driven **oscillator** changes the effective temperature of the **qubit**.

Data types:

Contributors: Xian-Ting Liang

Date: 2007-12-05

The response functions of the Ohmic bath and effective bath, where Δ=5×109Hz, λκ=1050, ξ=0.01, Ω0=10Δ, T=0.01K, Γ=2.6×1011, the lower-**frequency** and high-**frequency** cut-off of the baths modes ω0=11Δ, and ωc=100Δ.
...Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** coupled to an Ohmic bath directly and via an intermediate harmonic **oscillator** (IHO). Here, we suppose the **oscillation** **frequencies** of the bath modes are higher than the IHO’s. When we choose suitable parameters the **qubits** in the two models may have almost same decoherence and relaxation times. However, the decoherence and relaxation times of the **qubit** in the **qubit**-IHO-bath model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the **oscillation** **frequency** of the IHO. ... Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** coupled to an Ohmic bath directly and via an intermediate harmonic **oscillator** (IHO). Here, we suppose the **oscillation** **frequencies** of the bath modes are higher than the IHO’s. When we choose suitable parameters the **qubits** in the two models may have almost same decoherence and relaxation times. However, the decoherence and relaxation times of the **qubit** in the **qubit**-IHO-bath model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the **oscillation** **frequency** of the IHO.

Data types:

Contributors: Zorin, A. B., Chiarello, F.

Date: 2009-08-27

(Color online) (a) The **qubit** **frequency** as a function of parameter β L for fixed L = 50 pH and several values of capacitance C = 0.1 , 0.3, 1.0 and 3.0 pF (from top to bottom), corresponding to the values of the ratio E L / E c ≈ 1.7 × 10 4 , 5.1 × 10 4 , 1.7 × 10 5 and 5.1 × 10 5 . (b) Anharmonicity parameter δ as a function of parameter β L for the same as in (a) inductance L and capacitance values (from top to bottom)....(Color online) Position of the lowest six levels (solid lines) in the potential Eq. ( U-phi) for φ e = π as a function of parameter β L for typical values of L and C , yielding E J / E c ∼ E L / E c ≈ 5.1 × 10 4 . With an increase of β L , the spectrum crosses over from that of the harmonic **oscillator** type (left inset) to the set of the doublets (right inset), corresponding to the weak coupling of the **oscillator**-type states in two separate wells. The spectrum in the central region β L ≈ 1 is strongly anharmonic. The dashed line shows the bottom energy of the potential U φ φ e = π , which in the case of β L > 1 is equal to - Δ U ≈ - 1.5 E L β L - 1 2 / β L (in other words, Δ U is the height of the energy barrier in the right inset) . The dotted (zero-level) line indicates the energy in the symmetry point φ = 0 , i.e. at the bottom of the single well ( β L ≤ 1 ) or at the top of the energy barrier ( β L > 1 ). The black dot shows the critical value β L c at which the ground state energy level touches the top of the barrier separating the two wells....We propose a superconducting phase **qubit** on the basis of the radio-**frequency** SQUID with the screening parameter value $\beta_L = (2\pi/\Phi_0)LI_c \approx 1$, biased by a half flux quantum $\Phi_e=\Phi_0/2$. Significant anharmonicity ($> 30%$) can be achieved in this system due to the interplay of the cosine Josephson potential and the parabolic magnetic-energy potential that ultimately leads to the quartic polynomial shape of the well. The two lowest eigenstates in this global minimum perfectly suit for the **qubit** which is insensitive to the charge variable, biased in the optimal point and allows an efficient dispersive readout. Moreover, the transition **frequency** in this **qubit** can be tuned within an appreciable range allowing variable **qubit**-**qubit** coupling....where Î is the operator of supercurrent circulating in the **qubit** loop. The dependence of the reverse inductance L J Φ e = Φ 0 / 2 n calculated numerically in the two lowest quantum states ( n = 0 and 1) for L = 50 pH and the same set of capacitances C as in Fig. **frequency**-anharmonicity is shown in Fig. inductance-L01. One can see that the ratio of the geometrical to Josephson inductances L / L J takes large and very different values that can be favorably used for the dispersive readout, ensuring a sufficiently large output signal. Note that for β L 1 the inductance L J n = 1 changes the sign to positive....Figure f-shift shows this relative **frequency** shift versus parameter β L . One can see that for the rather conservative value of dimensionless coupling κ = 0.05 , the relative **frequency** shift can achieve the easily measured values of about 10 % . The efficiency of the dispersive readout can be improved in the non-linear regime with bifurcation . With our device this regime can be achieved in the resonance circuit including, for example, a Josephson junction (marked in the diagram in Fig. 1 by a dashed cross). Due to the high sensitivity of the amplitude (phase) bifurcation to the threshold determined by the effective resonance **frequency** of the circuit, one can expect a readout with high fidelity even at a rather weak coupling of the **qubit** and the resonator (compare with the readout of quantronium in Ref. ). Further improvement of the readout can be achieved in the QED-based circuit including this **qubit** ....Such a large, positive anharmonicity is a great advantage of the quartic potential **qubit** allowing manipulation within the two basis **qubit** states | 0 and | 1 not only when applying resonant microwave field, ν μ w ≈ ν 10 , but also when applying control microwave signals with large **frequency** detuning or using rather wide-spectrum rectangular-pulse control signals. The characteristic **qubit** **frequency** ν 10 = Δ E 0 / h and the anharmonicity factor δ computed from the Schrödinger equation for the original potential Eq. ( U-phi) in the range 0.9 ≤ β L ≤ 1.02 are shown in Fig. **frequency**-anharmonicity. One can see that the significant range in the tuning of the **qubit** **frequency** within the range of sufficiently large anharmonicity ( ∼ 20 - 50 % ) is attained at a rather fine (typically ± 1 - 2 % ) tuning of β L around the value β L = 1 . Such tuning of β L is possible in the circuit having the compound configuration shown in Fig. 1b. For values of β L > 1 , the symmetric energy potential has two minima and a barrier between them. The position of the ground state level depends on β L and the ratio of the characteristic energies E J / E c = β L E L / E c . The value of β L at which the ground state level touches the top of the barrier sets the upper limit β L c for the quartic **qubit** (marked in Fig. levels-beta by solid dot). At β L > β L c , the **qubit** energy dramatically decreases and the **qubit** states are nearly the symmetric and antisymmetric combinations of the states inside the two wells (see the right inset in Fig. 2). Although the **qubit** with such parameters has very large anharmonicity and can be nicely controlled by dc flux pulses , its readout can hardly be accomplished in a dispersive fashion....(a) Electric diagram of the **qubit** coupled to a resonant circuit and (b) possible equivalent compound (two-junction SQUID) circuit of the Josephson element included into the **qubit** loop. Capacitance C includes both the self-capacitance of the junction and the external capacitance. Due to inclusion in the resonant circuit of a Josephson junction JJ’, the resonator may operate in the nonlinear regime, enabling a bifurcation-based readout....(Color online) The resonance **frequency** shift in the circuit due to excitation of the **qubit** with the inductance value L = 50 pH and the set of capacitances C , decreasing from top to bottom. The dimensionless coupling coefficient κ = 0.05 ....(Color online) The values of the Josephson inductance of the quartic potential **qubit** in the ground (solid lines) and excited (dashed lines) states calculated for the geometric inductance value L = 50 pH and the set of capacitances C , increasing from top to bottom for both groups of curves. ... We propose a superconducting phase **qubit** on the basis of the radio-**frequency** SQUID with the screening parameter value $\beta_L = (2\pi/\Phi_0)LI_c \approx 1$, biased by a half flux quantum $\Phi_e=\Phi_0/2$. Significant anharmonicity ($> 30%$) can be achieved in this system due to the interplay of the cosine Josephson potential and the parabolic magnetic-energy potential that ultimately leads to the quartic polynomial shape of the well. The two lowest eigenstates in this global minimum perfectly suit for the **qubit** which is insensitive to the charge variable, biased in the optimal point and allows an efficient dispersive readout. Moreover, the transition **frequency** in this **qubit** can be tuned within an appreciable range allowing variable **qubit**-**qubit** coupling.

Data types:

Contributors: Volkov, P. A., Fistul, M. V.

Date: 2013-05-31

The typical solution consisting of the single instanton "step" on i -th **qubit** (solid green (thick) line) and tails on other **qubits** (solid red (thin) lines)....We report a theoretical study of coherent collective quantum dynamic effects in an array of N **qubits** (two-level systems) incorporated into a low-dissipation resonant cavity. Individual **qubits** are characterized by energy level differences $\Delta_i$ and we take into account a spread of parameters $\Delta_i$. Non-interacting **qubits** display coherent quantum beatings with N different **frequencies**, i.e. $\omega_i=\Delta_i/\hbar$ . Virtual emission and absorption of cavity photons provides a long-range interaction between **qubits**. In the presence of such interaction we analyze quantum correlation functions of individual **qubits** $C_i(t)$ to obtain two collective quantum-mechanical coherent **oscillations**, characterized by **frequencies** $\omega_1=\bar{\Delta}/\hbar$ and $\omega_2=\tilde{\omega}_R$, where $\tilde{\omega}_R$ is the resonant **frequency** of the cavity renormalized by interaction. The amplitude of these **oscillations** can be strongly enhanced in the resonant case when $\omega_1 \simeq \omega_2$. ... We report a theoretical study of coherent collective quantum dynamic effects in an array of N **qubits** (two-level systems) incorporated into a low-dissipation resonant cavity. Individual **qubits** are characterized by energy level differences $\Delta_i$ and we take into account a spread of parameters $\Delta_i$. Non-interacting **qubits** display coherent quantum beatings with N different **frequencies**, i.e. $\omega_i=\Delta_i/\hbar$ . Virtual emission and absorption of cavity photons provides a long-range interaction between **qubits**. In the presence of such interaction we analyze quantum correlation functions of individual **qubits** $C_i(t)$ to obtain two collective quantum-mechanical coherent **oscillations**, characterized by **frequencies** $\omega_1=\bar{\Delta}/\hbar$ and $\omega_2=\tilde{\omega}_R$, where $\tilde{\omega}_R$ is the resonant **frequency** of the cavity renormalized by interaction. The amplitude of these **oscillations** can be strongly enhanced in the resonant case when $\omega_1 \simeq \omega_2$.

Data types:

Contributors: Cooper, K. B., Steffen, Matthias, McDermott, R., Simmonds, R. W., Oh, Seongshik, Hite, D. A., Pappas, D. P., Martinis, John M.

Date: 2004-05-31

(a) Detail of the **qubit** spectroscopy near Δ U / ℏ ω p = 3.55 , showing splittings of strengths S ≈ 44 MHz and 24 MHz. (b) Tunneling probability versus measurement delay time τ D after application of π -pulse. Solid (dashed) line is taken at a well depth of solid (dashed) arrow in (a), corresponding to a resonant (off-resonant) bias. Inset illustrates how the **qubit** probability amplitude first moves to state | 1 g and then **oscillates** between | 1 g and | 0 e . (c) and (d) Tunneling probability (gray scale) versus well depth and τ D for experimental data (c) and numerical simulation (d). The peak **oscillation** periods are observed to correspond to the spectroscopic splittings....Spectroscopy of ω 10 obtained using the current-pulse measurement method, as a function of well depth Δ U / ℏ ω p . For each value of Δ U / ℏ ω p , the grayscale intensity is the normalized tunneling probability, with an original peak height of 0.1 - 0.3 . Insets: A given splitting in the spectroscopy of magnitude S comes from a critical-current fluctuator coupled to the **qubit** with strength h S / 2 . On resonance, the **qubit**-fluctuator eigenstates are linear combinations of the states | 1 g and | 0 e , where | g and | e are the two states of the fluctuator....(a) Room temperature measurement of the fast current pulse. (b) Tunneling probability versus δ I m a x with the **qubit** in state | 0 (solid circles) and in an equal mixture of states | 1 and | 0 (open circles). Fit to data is shown by the solid line. The plateau, being less than 0.5, corresponds to a maximum measurement fidelity of 0.63....(a) Schematic of the **qubit** circuitry. For the **qubit** used in Fig. 2, the Josephson critical-current and junction capacitance are I 0 ≈ 10 μ A and C ≈ 2 pF; in Figs. 3 and 4, each of these values is about 5 times smaller. (b) Potential energy landscape and quantized energy levels for I φ = I d c prior to the state measurement. (c) At the peak of δ I t , the **qubit** well is much shallower and state | 1 rapidly tunnels to the right hand well....We have detected coherent quantum **oscillations** between Josephson phase **qubits** and microscopic critical-current fluctuators by implementing a new state readout technique that is an order of magnitude faster than previous methods. The period of the **oscillations** is consistent with the spectroscopic splittings observed in the **qubit**'s resonant **frequency**. The results point to a possible mechanism for decoherence and reduced measurement fidelity in superconducting **qubits** and demonstrate the means to measure two-**qubit** interactions in the time domain. ... We have detected coherent quantum **oscillations** between Josephson phase **qubits** and microscopic critical-current fluctuators by implementing a new state readout technique that is an order of magnitude faster than previous methods. The period of the **oscillations** is consistent with the spectroscopic splittings observed in the **qubit**'s resonant **frequency**. The results point to a possible mechanism for decoherence and reduced measurement fidelity in superconducting **qubits** and demonstrate the means to measure two-**qubit** interactions in the time domain.

Data types:

Contributors: Thrailkill, Z. E., Lambert, J. G., Ramos, R. C.

Date: 2009-09-20

The second case where ω R 1 > ω R 2 > ω R 3 is shown in Fig. fig3. This is very similar to the first case, however, there is a slight overall increase in the state transfer rate. This is because the detuning is still relative to R2, but the detuning of R3 is now slightly less. As the **qubits** approach resonance with the resonators from below, the dispersive coupling strength becomes slightly larger because the **frequency** of R3 is a little closer to the **frequency** of the **qubits**. The small differences in resonator **frequencies** also cause slight non-uniformities in the high **frequency** **oscillations** or ripples that become more pronounced at lower detuning. This interference is not a factor if the **qubits** are sufficiently detuned. Regardless of these artifacts, it is clear that an array of resonators that is used for memory storage can also be used to dispersively couple **qubits**....In both cases, at large detuning of -2000 MHz, the excitation appears to smoothly **oscillate** into **qubit** 2, as shown in Fig. fig2 and fig3. As the detuning decreases, two things happen: First, the time it takes for the excitation to move into **qubit** 2 gets shorter, indicating an increase in the effective coupling strength between the two **qubits**. Second, small ripples, or **oscillations**, begin to appear. These **oscillations** are due to the detuning becoming small enough that the dispersive limit approximation starts to break down. This result shows us that using three identical resonators in parallel to dispersively couple **qubits** is effectively similar to using only one resonator with three times the coupling strength. In general, an array of n resonators can be replaced by a single resonator with n times the coupling strength. The dispersive coupling can be increased by reducing the detuning, but at the cost of having significant **oscillations** between the **qubits** and resonators....figure2(a) Two **qubits** (Q1, Q2) dispersively coupled to an array of three resonators (R1, R2, R3) via identical coupling capacitors C c . (b),(c) For these two simulations, the system is initialized with a single excitation in Q1, and the two **qubits** maintain equal **frequencies** as they are simultaneously detuned from the resonators. Each vertical cut represents the population in R2 over time at a particular detuning Δ Q , R 2 , of the **qubits** from R2. In (b), the **frequencies** of all three resonators are equal ( ω R 1 = ω R 2 = ω R 3 ). At large detuning ( Δ Q , R 2 = -2000 MHz), the excitation smoothly **oscillates** between the two **qubits** without significant interference from the resonators. As the magnitude of the detuning decreases, the effective coupling between the two **qubits** strengthens, thus the **oscillation** of the excitation becomes more frequent. Also, the direct coupling of the **qubits** to the resonators strengthens, causing the small ripples. In (c), the **frequencies** of R1 and R3 are set slightly above and below R2, respectively ( ω R 1 > ω R 2 > ω R 3 ). The excitation **oscillates** slightly faster than in (b) because the small offsets in **frequency** of R1 and R3 increases the coupling bandwidth, resulting in a small increase in coupling between the **qubits** over the same range of detuning. The offset of R1 and R3 from R2 also causes the ripples to be non-uniform at smaller detuning....We consider an array of resonators used as a memory register. To accomplish this, one must be able to transfer an excitation from a **qubit** to a specific resonator, without coupling to the other resonators in the array. This is implemented by designing resonators that are sufficiently detuned from each other. To determine the amount of detuning required to avoid crosstalk between resonators, we examine a single **qubit** coupled to an array of two resonators as shown in Fig. block1. The **qubit** (Q) and resonator 1 (R1) are fixed at the same **frequency** while resonator 2 (R2) is detuned. All **qubit**-resonator couplings are presumed to be identical with a strength of g i j = 110 M H z for all i , j , i.e., all coupling capacitances C c are equal. This value is typical in experiments such as in Ref. . We begin with Q in the excited state and let the system evolve over time while recording the population in R1, as we increase the detuning Δ R 2 , R 1 , of R2. The result of this simulation is shown in Fig. fig1....The result of the simulation is shown in Fig. fig4 where the population in R2 is plotted versus the detuning Δ Q 2 , Q 1 , of Q2 from Q1, in the range of -2000 MHz to 2000 MHz. As per design, at zero detuning, the excitation is transferred after a time of 16 ns. This time can be chosen based on the desired time scale by adjusting the **qubit**-**qubit** and **qubit**-resonator coupling strengths. At large detuning, i.e., beyond ± 1.5 GHz, the coupling between Q1 and R2 becomes dispersive up to about 30 ns, showing that dispersive coupling can be weak enough to isolate the active **qubit**. Q2 can be detuned further to reduce the dispersive coupling if the desired time scale is longer, e.g., to perform operations on Q1. Thus, this simulation shows that a control **qubit** can be effectively used to turn coupling on and off between a **qubit** and an array of resonators....figure3 (a) Placing a control **qubit** (Q2) between the active **qubit** (Q1) and the resonator array (R1, R2, R3) allows coupling to be turned on and off. (b) The system is initialized with an excitation in Q1, Q1 is in-resonance with R2, and the **frequencies** of R1 and R3 are set slightly above and below R2, respectively ( ω R 1 > ω R 2 > ω R 3 ). Each vertical cut represents the population in R2 over time at a particular detuning Δ Q 2 , Q 1 , of Q2 from Q1. At zero detuning, the excitation readily **oscillates** in and out of the resonator R2. As Q2 is detuned further away, the coupling between Q1 and R2 becomes weaker, resulting in slower **oscillations** of the excitation. The detuning of the control **qubit** Q2 can be chosen based on the desired time scale, e.g. the time required to manipulate Q1....figure1(a) Schematic of a **qubit** (Q) coupled to an array of two resonators (R1, R2) via identical coupling capacitors C c . The **qubit** is characterized as having capacitance C J and critical current I c , with bias current I b . The resonators are characterized by inductance L i and capacitance C i , for i = 1 , 2 . (b) For this simulation, the system is initialized with a single excitation in Q, and Q is in-resonance with R1. Each vertical cut represents the population in R1 over time at a particular detuning Δ R 2 , R 1 , of R2 from R1. The **qubit**-resonator coupling strength is 110 MHz. At zero detuning, the excitation **oscillates** between the **qubit** and the two resonators; since the two resonators are identical here, each resonator is only half populated. As R2 is detuned, the excitation **oscillates** between Q and R1 with minimal population in R2. After more **oscillations**, R2 will accumulate some population, even at large detuning, which causes the appearance of ripples....Josephson junction-based **qubits** have been shown to be promising components for a future quantum computer. A network of these superconducting **qubits** will require quantum information to be stored in and transferred among them. Resonators made of superconducting metal strips are useful elements for this purpose because they have long coherence times and can dispersively couple **qubits**. We explore the use of multiple resonators with different resonant **frequencies** to couple **qubits**. We find that an array of resonators with different **frequencies** can be individually addressed to store and retrieve information, while coupling **qubits** dispersively. We show that a control **qubit** can be used to effectively isolate an active **qubit** from an array of resonators so that it can function within the same **frequency** range used by the resonators....Next, we investigate the behavior of a system consisting of two **qubits** dispersively coupled to an array of three resonators, as shown in Fig. block2. We consider two cases: (1) All three resonators are designed with the same resonant **frequency**, and (2) the resonant **frequencies** of the three resonators are slightly detuned so that ω R 1 > ω R 2 > ω R 3 . We demonstrate how information in the form of an excitation is transferred dispersively from one **qubit** to another through an array of resonators. In both cases, the system is initialized with a single excitation in Q1, and both **qubits** are held in resonance with each other as the magnitude of their detuning from R2 is decreased from, say, -2000 MHz to -400 MHz. ... Josephson junction-based **qubits** have been shown to be promising components for a future quantum computer. A network of these superconducting **qubits** will require quantum information to be stored in and transferred among them. Resonators made of superconducting metal strips are useful elements for this purpose because they have long coherence times and can dispersively couple **qubits**. We explore the use of multiple resonators with different resonant **frequencies** to couple **qubits**. We find that an array of resonators with different **frequencies** can be individually addressed to store and retrieve information, while coupling **qubits** dispersively. We show that a control **qubit** can be used to effectively isolate an active **qubit** from an array of resonators so that it can function within the same **frequency** range used by the resonators.

Data types: