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  • 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).
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  • Decay envelopes of the observed oscillations (red points), and relative fitting curves (blue continuous lines).... The frequency Ω / 2 π given by eq.  omega for ϕ x = 0 as a function of ϕ c = 2 π Φ c / Φ 0 (blue curve). Dashed orizontal lines mark the range of oscillation frequencies observed (about 10-20 GHz), corresponding to the range of used top values of the applied pulse (defined by the vertical dashed lines). In the inset it is sketched the flux pulse used for the qubit manipulation, changing the potential from the two-well “W” case to the single-well “V” case (in red).... Some experimental oscillations observed for different pulse height. The measured frequency is indicated in the top right part of each plot.... Red points are the decay rate γ I (left panel) and γ I I (right panel) obtained by fitting of the experimental decay curves in oscillations with eq. envelope. The blue line in the left panel is the fit of the data points with eq. gammaI (Section 3). In the right panel the blue line is the average value of the scattered values of γ I I
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  • A probe signal composed of one microwave tone per qubit to be read out is sent through the common transmission line. The interaction between each qubit and resonator leads to a state-dependent dispersive shift, Δ ω r = g ~ 2 / ω q - ω r σ z , of the resonator frequency, where g ~ is the effective coupling between the resonator and qubit, ω q and ω r are the angular resonance frequencies of the qubit and resonator, and σ z is ± 1 depending on the state of the qubit. The composite signal probes all resonators at the same time, storing the information on the state of all qubits in the transmitted tones. Detection of the transmitted amplitude and phase of each of the tones provides a simultaneous non-destructive measurement of the states of all qubits. The probe signal is generated by mixing a reference microwave tone in the band of the resonators and a multi-tone DAC output using an IQ mixer, see Fig.  fig:setup. By using the I and Q quadratures, we address the upper and lower sidebands of the mixing product individually to effectively double the bandwidth of the system. The mixer output is combined with the qubit manipulation signal through a directional coupler. A strongly attenuated line transmits the combined signal to the sample, which is attached to the mixing chamber stage of a dilution refrigerator. Two cryogenic circulators and a high-pass filter at 30 mK are used to prevent reflections and noise from traveling from the cryogenic amplifier back to the sample. A chain of amplifiers provides 80 dB gain to boost the transmitted probe signal to levels sufficient for the detection stage, which employs an identical IQ mixer to convert the signal back to baseband frequencies. The local oscillator inputs of both mixers are fed from the same reference microwave source, resulting in a homodyne detection with a fixed phase offset. An additional high-pass filter between the local oscillator ports of the two mixers prevents leakage of the baseband signal. After digitizing both quadratures, the amplitude and phase of all components of the probe signal are extracted via FFT. The maximum number of devices that can be probed with the described technique is defined by the frequency separation between resonators and the bandwidth of the acquisition board. The frequencies of the resonators on our chip are spaced at intervals of 150 MHz and the acquisition board has a bandwidth somewhat below 500 MHz, allowing for the simultaneous detection of up to six devices.... In the next set of experiments, we tuned the uniform flux coil and two compact local coils placed above the sample to bias three qubits at their symmetry points. Limiting the number of qubits to three was necessary because of the lack of additional (on-chip) coils and not due to the readout technique itself. After setup of the readout pulse to probe circuits number #2, 3 and 5, we performed a spectroscopy of all three qubits simultaneously. A continuous microwave excitation signal of varying frequencies was applied to the sample and a pulsed three-tone probe signal was applied every 10 μ s . When the excitation frequency matches the gap between the ground and first excited states of a qubit, the instantaneous dispersive shift of the center frequency of the corresponding resonator switches between positive and negative, thus changing the mean amplitude and phase of the transmitted probe tone. Figure fig:multi_spec shows the spectra of three qubits measured in parallel.... Finally, we performed simultaneous manipulation with time resolved measurements on three qubits. Here, we used individual microwave excitations for every qubit, which were added together via a power combiner. We note that the complete excitation chain could be replaced by a reference source and a mixer controlled by a single arbitrary waveform generator with sufficient bandwidth to drive all qubits, similar to FDM readout tone generation. Measurement data are reported in Fig.  fig:rabi. All three qubits were simultaneously driven by individual excitation tones and the readout was performed in parallel using the FDM protocol. Every qubit can be Rabi-driven at a different power. Left panels of Fig.  fig:rabi present Rabi oscillations at three different powers for all qubits. The measured linear power dependences of Rabi oscillations reported on the right panels in Fig.  fig:rabi are in excellent agreement with theory.... The transmission spectrum of the sample, measured with a vector network analyzer is reported in Fig.  fig:tm6q(a). The seven absorption peaks correspond to the seven readout resonators. Close to each peak its bare resonance frequency as well the identification number of the device are printed. The inset of Fig.  fig:tm6q(b) shows the transmission at frequencies around the resonance of device #3 vs. the magnetic flux bias. The two points at which the dispersive frequency shift changes its sign correspond to avoided level crossings of the qubit and resonator. We demonstrate FDM by measuring the maximum possible number of devices simultaneously. Figure  fig:tm6q(b) shows the transmitted amplitude of the six probe tones versus the external uniformly applied magnetic flux. Each curve is shown aligned with the corresponding transmission peak to the left in Fig.  fig:tm6q(a). The amplitude of the transmitted signal is constant as long as the qubit remains far detuned from the resonator. The amplitude changes drastically around two distinct fluxes, again indicating anti-crossings between the qubit and the corresponding resonator. There is a minimum between these two peaks, because the readout frequencies were set on resonance, with the dispersive shifts at the symmetry points of the qubits taken into account. The readout frequency of device #3 is shown as a dashed line in the inset.... (color online). Simultaneous manipulation and detection of three qubits. Left plots: Rabi oscillations at several powers; traces are vertically offset for better visibility; curves with the same color/offset (blue-bottom, green-center, red-top) are measured simultaneously using the FDM technique described in the main text. Right plots: Rabi oscillation frequency versus power of the excitation tone; the error bars are smaller than the size of the dots.... (color online). Multiplexed spectroscopy of qubits #2, 3 and 5. The qubit manipulation microwave excites qubits when its frequency matches the transition between their ground and excited states. The state of all three qubits is continuously and simultaneously monitored by the multi-tone probe signal. The horizontal axis reports the uniform bias flux applied to the chip.... (color online). (a) Transmission spectrum of the sample with all qubits far detuned from the resonances. (b) FDM readout of six flux qubits. The main plot shows the transmission amplitude at the resonance frequencies of devices #1 to 6 vs. the magnetic flux generated by the uniform field coil, measured using FDM. The inset shows the transmission amplitude at several frequencies close to resonance #3, measured with a network analyzer. The dashed line indicates the probe frequency used for this device in the main plot. The curves in the main plot are normalized and shifted vertically for better visibility. The offset along the horizontal axis is due to magnetic field non-uniformity, which is likely due to the screening currents generated in the superconducting ground plane.... superconducting flux qubit, qubit register, dispersive readout, frequency division multiplexing, microwave resonators, Rabi oscillations ... Experimental setup used for FDM readout. The qubit manipulation signal is generated by a single microwave source for spectroscopy and three additional microwave sources, DAC channels and mixers for pulsed excitation of the qubits.
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  • The oscillations in these FID experiments decay due to voltage noise from DC up to a frequency of approximately 1 / t . As the relaxation time, T 1 is in excess of 100 μ s in this regime, T 1 decay is not an important source of decoherence (Fig. S4). The shape of the decay envelope and the scaling of coherence time with d J / d ϵ (which effectively changes the magnitude of the noise) reveal information about the underlying noise spectrum. White (Markovian) noise, for example, results in an exponential decay of e - t / T 2 * where T 2 * ∝ d J / d ϵ -2 is the inhomogeneously broadened coherence time . However, we find that the decay is Gaussian (Fig. t2stard) and that T 2 * (black line in Fig. t2stare) is proportional to d J / d ϵ -1 (red solid line in Fig. t2stare) across two orders of magnitude of T 2 * . Both of these findings can be explained by quasistatic noise, which is low frequency compared to 1 / T 2 * . In such a case, one expects an amplitude decay of the form exp - t / T 2 * 2 , where T 2 * = 1 2 π d J / d ϵ ϵ R M S and ϵ R M S is the root-mean-squared fluctuation in ϵ (Eq. S3). From the ratio of T 2 * to d J / d ϵ -1 , we calculate ϵ R M S = 8 μ V in our device. At very negative ϵ , J becomes smaller than Δ B z , and nuclear noise limits T 2 * to approximately 90ns, which is consistent with previous work . We confirm that this effect explains deviations of T 2 * from d J / d ϵ -1 by using a model that includes the independently measured T 2 , n u c l e a r * and Δ B z (Eq. S1) and observe that it agrees well with measured T 2 * at large negative ϵ (dashed red line in Fig. t2stare).... Two level quantum systems (qubits) are emerging as promising candidates both for quantum information processing and for sensitive metrology . When prepared in a superposition of two states and allowed to evolve, the state of the system precesses with a frequency proportional to the splitting between the states. However, on a timescale of the coherence time, T 2 , the qubit loses its quantum information due to interactions with its noisy environment. This causes qubit oscillations to decay and limits the fidelity of quantum control and the precision of qubit-based measurements. In this work we study singlet-triplet ( S - T 0 ) qubits, a particular realization of spin qubits , which store quantum information in the joint spin state of two electrons. We form the qubit in two gate-defined lateral quantum dots (QD) in a GaAs/AlGaAs heterostructure (Fig. pulsesa). The QDs are depleted until there is exactly one electron left in each, so that the system occupies the so-called 1 1 charge configuration. Here n L n R describes a double QD with n L electrons in the left dot and n R electrons in the right dot. This two-electron system has four possible spin states: S , T + , T 0 , and T - . The S , T 0 subspace is used as the logical subspace for this qubit because it is insensitive to homogeneous magnetic field fluctuations and is manipulable using only pulsed DC electric fields . The relevant low-lying energy levels of this qubit are shown in Fig. pulsesc. Two distinct rotations are possible in these devices: rotations around the x -axis of the Bloch sphere driven by difference in magnetic field between the QDs, Δ B z (provided in this experiment by feedback-stabilized hyperfine interactions), and rotations around the z -axis driven by the exchange interaction, J (Fig. pulsesb) . A S can be prepared quickly with high fidelity by exchanging an electron with the QD leads, and the projection of the state of the qubit along the z -axis can be measured using RF reflectometery with an adjacent sensing QD (green arrow in Fig. pulsesa).... The device used in these measurements is a gate-defined S - T 0 qubit with an integrated RF sensing dot. a The detuning ϵ is the voltage applied to the dedicated high-frequency control leads pictured. b, The Bloch sphere that describes the logical subspace of this device features two rotation axes ( J and Δ B Z ) both controlled with DC voltage pulses. c, An energy diagram of the relevant low-lying states as a function of ϵ . States outside of the logical subspace of the qubit are grayed out. d, J ϵ and d J / d ϵ in three regions; the 1 1 region where J and d J / d ϵ are both small and S - T 0 qubits are typically operated, the transitional region where J and d J / d ϵ are both large where the qubit is loaded and measured, and the 0 2 region where J is large but d J / d ϵ is small and large quality oscillations are possible. pulses... Ramsey oscilllations reveal low frequency enivronmental dynamics. a, The pulse sequence used to measure exchange oscillations uses a stabilized nuclear gradient to prepare and readout the qubit and gives good contrast over a wide range of J . b, Exchange oscillations measured over a variety of detunings ϵ and timescales consistently show larger T 2 * as d J / d ϵ shrinks until dephasing due to nuclear fluctuations sets in at very negative ϵ . c, Extracted values of J and d J / d ϵ as a function of ϵ . d, The decay curve of FID exchange oscillations shows Gaussian decay. e, Extracted values of T 2 * and d J / d ϵ as a function of ϵ . T 2 * is proportional to d J / d ϵ -1 , indicating that voltage noise is the cause of dephasing of charge oscillations. f, Charge oscillations measured in 0 2 . This figure portrays the three basic regions we can operate our device in: a region of low frequency oscillations and small d J / d ϵ , a region of large frequency oscillations and large d J / d ϵ , and a region where oscillations are fast but d J / d ϵ is comparatively small. t2star... Since we observe J to be approximately an exponential function of ϵ , ( d J / d ϵ ∼ J ), we expect and observe the quality (number of coherent oscillations) of these FID oscillations, Q ≡ J T 2 * / 2 π ∼ J d J / d ϵ -1 , to be approximately constant regardless of ϵ . However, when ϵ is made very positive and J is large, an avoided crossing occurs between the 1 1 T 0 and the 0 2 T 0 state, making the 0 2 S and 0 2 T 0 states electrostatically virtually identical. Here, as ϵ is increased, J increases but d J / d ϵ decreases(Fig. pulsesd), allowing us to probe high quality exchange rotations and test our charge noise model in a regime that has never before been explored.... Spin-echo measurements reveal high frequency bath dynamics. a, The pulse sequence used to measure exchange echo rotations. b, A typical echo signal. The overall shape of the envelope reflects T 2 * , while the amplitude of the envelope as a function of τ (not pictured) reflects T 2 e c h o . c, T 2 e c h o and Q ≡ J T 2 e c h o / 2 π as a function of J . A comparison of the two noise models: power law and a mixture of white and 1 / f noise. Noise with a power law spectrum fits over a wide range of frequencies (constant β ), but the relative contributions of white and 1 / f noise change as a function of ϵ . d, A typical echo decay is non-exponential but is well fit by exp - τ / T 2 e c h o β + 1 . e, T 2 e c h o varies with d J / d ϵ in a fashion consistent with dephasing due to power law voltage fluctuations. echo... Using a modified pulse sequence that changes the clock frequency of our waveform generators to achieve picosecond timing resolution (Fig. S1)), we measure exchange oscillations in 0 2 as a function of ϵ and time (Fig. t2stare) and we extract both J (Fig. t2starc) and T 2 * (Fig. t2stard) as a function of ϵ . Indeed, the predicted behavior is observed: for moderate ϵ we see fast oscillations that decay after a few ns, and for the largest ϵ we see even faster oscillations that decay slowly. Here, too, we observe that T 2 * ∝ d J d ϵ -1 (Fig. t2stard), which indicates that FID oscillations in 0 2 are also primarily dephased by low frequency voltage noise. We note, however, that we extract a different constant of proportionality between T 2 * and d J / d ϵ -1 for 1 1 and 0 2 . This is expected given that the charge distributions associated with the qubit states are very different in these two regimes and thus have different sensitivities to applied electric fields. We note that in the regions of largest d J / d ϵ (near ϵ = 0 ), T 2 * is shorter than the rise time of our signal generator and we systematically underestimate J and overestimate T 2 * (Fig. S1).... The use of Hahn echo dramatically improves coherence times, with T 2 e c h o (the τ at which the observed echo amplitude has decayed by 1 / e ) as large as 9 μ s , corresponding to qualities ( Q ≡ T 2 e c h o J / 2 π ) larger than 600 (Fig. echoc). If at high frequencies (50kHz-1MHz) the voltage noise were white (Markovian), we would observe exponential decay of the echo amplitude with τ . However, we find that the decay of the echo signal is non-exponential (Fig. echod), indicating that even in this relatively high-frequency band being probed by this measurement, the noise bath is not white.
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  • Rabi oscillation of the double SQUID manipulated as a phase qubit by applying microwave pulses at 19 GHz. The oscillation frequency changes from 540 MHz to 1.2 GHz by increasing the power of the microwave signal by 10 dB.... Probability of measuring the state | L as a function of the pulse duration. The coherent oscillation shown here has a frequency of 14 GHz and a coherence time of approximately 1.2 ns.... Measurement of the relaxation time T 1 for the double SQUID operated as a phase qubit.... Measured oscillation frequencies versus amplitude of the short flux pulse (full dots). The solid curve is a numerical simulation using the measured parameters of the circuit.... Measured Rabi oscillation frequency versus the normalized amplitude of the microwave signal (solid dots). The dashed line is a linear fit taking into account slightly off-resonance microwave field, while the fit represented by the solid line considers a population of higher excited states.
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  • We use a small-inductance superconducting loop interrupted by three Josephson junctions (a 3JJ qubit) , inductively coupled to a high-quality superconducting tank circuit  (Fig.  fig1). This approach is similar to the one in entanglement experiments with Rydberg atoms and microwave photons in a cavity . The tank serves as a sensitive detector of Rabi transitions in the qubit, and simultaneously as a filter protecting it from noise in the external circuit. Since ω T ≪ Ω / ℏ , the qubit is effectively decoupled from the tank unless it oscillates with frequency  ω T . That is, while wide-band (i.e., fast on the qubit time scale) detectors up to now have received most theoretical attention (e.g., ), we use narrow-band detection to have sufficient sensitivity at a single frequency even with a small coupling coefficient; cf. above Eq. ( S). The tank voltage is amplified and sent to a spectrum analyzer. This is a development of the Silver–Zimmerman setup in the first RF-SQUID magnetometers , and is effective for probing flux qubits . As such, it was used to determine the potential profile of a 3JJ qubit in the classical regime .... We plotted S V , t ω for different HF powers P in Fig.  fig3. As P is increased, ω R grows and passes  ω T , leading to a non-monotonic dependence of the maximum signal on  P in agreement with the above picture. This, and the sharp dependence on the tuning of ω H F to the qubit frequency, confirm that the effect is due to Rabi oscillations. The inset shows that the shape is given by the second line of Eq. ( S) for all curves.... Measurement setup. The flux qubit is inductively coupled to a tank circuit. The DC source applies a constant flux Φ e ≈ 1 2 Φ 0 . The HF generator drives the qubit through a separate coil at a frequency close to the level separation Δ / h = 868 MHz. The output voltage at the resonant frequency of the tank is measured as a function of HF power.... The Al qubit inside the Nb pancake coil.... (a) Comparing the data to the theoretical Lorentzian. The fitting parameter is g ≈ 0.02 . Letters in the picture correspond to those in Fig.  fig3. (b) The Rabi frequency extracted from (a) vs the applied HF amplitude. The straight line is the predicted dependence ω R / ω T = P / P 0 . The good agreement provides strong evidence for Rabi oscillations.... The spectral amplitude of the tank voltage for HF powers P a qubit modifying the tank’s inductance and hence its central frequency, and in principle similarly for dissipation in the qubit increasing the tank’s linewidth ; these are inconsequential for our analysis.... Without an HF signal, the qubit’s influence at ω T is negligible. Thus, the “dark” trace in Fig.  fig3 is a quantitative measure of  S b .
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  • Contrary to the shift produced by the linear coupling term, the sign of this frequency shift now depends on ϵ . Since g 2 is negative (see figure fig:couplings), δ ν 0 2 actually has the same sign as ϵ . We also note that the quadratic term has no effect on the qubit when ϵ = 0 , since at that point the average flux generated by both qubit states | 0 and | 1 averages out to zero so that the SQUID Josephson inductance is unchanged.... We will now discuss quantitatively the behaviour of g 1 and g 2 for actual sample parameters : I C = 3.4 μ A , M = 6.5 p H , I p = 240 n A , Δ = 5.5 G H z , ν p = 3.1 G H z , L J = 300 p H , f ' / 2 = 1.45 π . We will restrict ourselves to a range of bias conditions relevant for our conditions, supposing that I b varies between ± 300 n A and that f ' / 2 varies by d f ' = ± 4 ⋅ 10 -3 π around 1.45 π . We chose such an interval for f ' because it corresponds to changing the qubit bias point ϵ by ± 2 G H z around 0 . The constants g 1 and g 2 are plotted in figure fig:couplings as a function of I b for two different values of f ' ( g 1 is shown as a full line, g 2 as a dashed line, and the two different values of f ' are symbolized by gray for d f ' = - 2 π 4 ⋅ 10 -3 and black for d f ' = 0 ). It can be seen that the coupling constants only weakly depend on the value of the flux in this range, so that we will neglect this dependence in the following and consider that g 1 and g 2 only depend on the bias current I b . Moreover we see from figure fig:couplings that the approximations made in equation eq:g1g2approx are justified in this range of parameters since g 1 is closely linear in I b and g 2 nearly constant. We also note that g 1 = 0 for I b = 0 . This fact can be generalized to the case where the SQUID-qubit coupling is not symmetric and the junctions critical current are dissimilar : in certain conditions these asymmetries can be compensated for by applying a bias current I b * for which g 1 I b * = 0 . At the current I b * , the qubit is effectively decoupled from the measuring circuit fluctuations to first order.... (a) qubit biased by Φ x and SQUID biased by current I b . (b) Simplified electrical scheme : the SQUID-qubit system is seen as an inductance L J connected to the shunct capacitor C s h through inductance L s h . Φ a is the flux across the two inductances L J and L s h in series.... Qubit frequency ν q as a function of the bias ϵ for Δ = 5.5 G H z (minimum frequency in the figure). The dashed line indicates the phase-noise insensitive bias point ϵ = 0 where d ν q / d ϵ = 0... Frequency shift per photon δ ν 0 as a function of I b and ϵ . The white regions correspond to -15 M H z and the black to + 35 M H z . The solid line ϵ m I b indicates the bias conditions for which δ ν 0 = 0 . The dashed line indicates the phase noise insensitive point ϵ = 0 ; the dotted line indicates the decoupling current I b = I b * .... The hamiltonian eq:qubit_hamiltonian yields a qubit transition frequency ν q = Δ 2 + ϵ 2 . The corresponding dependence is plotted in figure fig:nuq for realistic parameters. An interesting property is that when the qubit is biased at ϵ = 0 (dashed line in figure fig:nuq), it is insensitive to first order to noise in the bias variable ϵ .... We stress that these biasing conditions are non-trivial in the sense that they do not satisfy an obvious symmetry in the circuit. This point is emphasized in figure fig:deltanu0 where we plotted as a dashed line the bias conditions ϵ = 0 for which the qubit is insensitive to phase noise (due to flux or bias current noise) ; and as a dotted line the decoupling current conditions I b = I b * for which the qubit is effectively decoupled from its measuring circuit. The ϵ m I b line shares only one point with these two curves : the point I b * ϵ which is optimal with respect to flux, bias current, and photon noise. For the rest, the three lines are obviously distinct. This makes it possible to experimentally discriminate between the various noise sources limiting the qubit coherence by studying the dependence of τ φ on bias parameters.... The flux-qubit is a superconducting loop containing three Josephson junctions threaded by an external flux Φ x ≡ f Φ 0 / 2 π . It is coupled to a DC-SQUID detector shunted by an external capacitor C s h whose role is to limit phase fluctuations across the SQUID as well as to filter high-frequency noise from the dissipative impedance. The SQUID is threaded by a flux Φ S q ≡ f ' Φ 0 / 2 π . The circuit diagram is shown in figure fig1a. There, the flux-qubit is the loop in red containing the three junctions of phases φ i and capacitances C i ( i = 1 , 2 , 3 ). It also includes an inductance L 1 which models the branch inductance and eventually the inductance of a fourth larger junction . The two inductances K 1 and K 2 model the kinetic inductance of the line shared by the SQUID and the qubit. The SQUID is the larger loop in blue. The junction phases are called φ 4 and φ 5 and their capacitances C 4 and C 5 . The critical current of the circuit junctions is written I C i ( i = 1 to 5 ). The SQUID loop also contains two inductances K 3 and L 2 which model its self-inductance. The SQUID is connected to the capacitor C s h through superconducting lines of parasitic inductance L s . The phase across the stray inductance and the SQUID is denoted φ A . The whole circuit is biased by a current source I b in parallel with a dissipative admittance Y ω . Since our goal is primarily to determine the qubit-plasma mode coupling hamiltonian, we will neglect the admittance Y ω .
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  • frequencies... frequencies.... frequency... network of stochastic oscillators... QUBIT... frequencies);
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  • 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.... 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.
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  • The average energy H of the system as a function of the driving frequency ω . The main peak ( ω 0 ≈ 0.6 ) corresponds to the resonance. The left peak at ω 0 / 2 is the nonlinear effect of the excitation by a subharmonic, similar to a multiphoton process in the quantum case. The right peak at 2 ω 0 is the first overtone and it has no quantum counterpart. Here ϕ e d = π ; ϕ e a = 0.05 ; γ = 10 -3 .... The dependence of the pseudo-Rabi frequency on the driving amplitude ϕ e a for ω = 0.6 , γ = 10 -3 . The solid line, Ω = 0.35 ϕ e a 2 + ω - 0.63 2 1 / 2 , is the best fit to the calculated data.... (Color online) The potential profile of Eq. ( eq_potential) with α = 0.8 , ϕ e d = π . The arrows indicate quantum (solid) and classical (dotted) oscillations.... A quantitative difference between this effect and true Rabi oscillations is in the different scale of the resonance frequency. To induce Rabi oscillations between the lowest quantum levels in the potential ( eq_potential), one must apply a signal in resonance with their tunneling splitting, which is exponentially smaller than ω 0 . Still, this is not a very reliable signature of the effect, since the classical effect can also be excited by subharmonics, ∼ ω 0 / n , as we can see in Fig.  fig4.... In the presence of the external field ( eq_external) the system will undergo forced oscillations around one of the equlibria. For α = 0.8  , which is close to the parameters of the actual devices , the values of the dimensionless frequencies become ω θ ≈ 0.612 , and ω χ ≈ 0.791 . Solving the equations of motion ( eq_motion) numerically, we see the appearance of slow oscillations of the amplitude and energy superimposed on the fast forced oscillations (Fig.  fig2), similar to the classical oscillations in a phase qubit (Fig. 2 in  ). The dependence of the frequency of these oscillations on the driving amplitude shows an almost linear behaviour (Fig.  fig3), which justifies the “Pseudo-Rabi” moniker.... The key observable difference between the classical and quantum cases, which would allow to reliably distinguish between them, is that the classical effect can also be produced by driving the system at the overtones of the resonance signal, ∼ n ω 0 (Fig.  fig4). This effect can be detected using a standard technique for RF SQUIDs . The current circulating in the qubit circuit produces a magnetic moment, which is measured by the inductively coupled high-quality tank circuit. For the tank voltage V T we have... where τ T = R T C T is the RC-constant of the tank, ω T = L T C T -1 / 2 its resonant frequency, M the mutual inductance between the tank and the qubit, and I q t the current circulating in the qubit. The persistent current in the 3JJ loop can be determined directly from ( eq_I). Its behaviour in the presence of an external RF field is shown in Fig.  fig2c. Note that the sign of the current does not change, which is due to the fact that the oscillations take place inside one potential well (solid arrow in Fig.  fig1), and not between two separate nearby potential minima like in the quantum case. (Alternatively, this would also allow to distinguish between the classical and quantum effects by measuring the magnetization with a DC SQUID.)... (a) Driven oscillations around a minimum of the potential profile of Fig. fig1 as a function of time. The driving amplitude is ϕ e a = 0.01 , driving frequency ω = 0.612 , and the decay rate γ = 10 -3 . Low-frequency classical beat oscillations are clearly seen. (b) Low-frequency oscillations of the persistent current in the 3JJ loop. (c) Same for the energy of the system.
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