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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 .... 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 .
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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 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.
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(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.... 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.
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The first-qubit oscillation frequency f d as a function of time t (normalized by the energy relaxation time T 1 ) for C x = 0 (solid line) and C x = 6 fF (dashed line), assuming N l 1 = 1.355 and parameters of Eq. ( 2.16). Dash-dotted horizontal line, ω r 1 / 2 π = 15.3 GHz, shows the long-time limit of f d t . Two dotted horizontal lines show the plasma frequency for the second qubit: ω l 2 / 2 π = 10.2 GHz for N l 2 = 10 and ω l 2 / 2 π = 8.91 GHz for N l 2 = 5 . The arrow shows the moment t c of exact resonance in the case N l 2 = 5 .... The circuit schematic of a flux-biased phase qubit and the corresponding potential profile (as a function of the phase difference δ across the Josephson junction). During the measurement the state | 1 escapes from the “left” well through the barrier, which is followed by oscillations in the “right” well. This dissipative evolution leads to the two-qubit crosstalk.... The oscillating term in Eq. ( 3.11a) describes the beating between the oscillator and driving force frequencies, with the difference frequency increasing in time, d t ~ 2 / d t = α t - t c , and amplitude of beating decreasing as 1 / t ~ (see dashed line in Fig.  f4a). Notice that F 0 = 1 / 4 , F ∞ = 1 , and the maximum value is F 1.53 = 1.370 , so that E 0 is the long-time limit of the oscillator energy E 2 , while the maximum energy is 1.37 times larger:... The second qubit energy E 2 (in units of ℏ ω l 2 ) in the oscillator model as a function of time t (in ns) for (a) C x = 5 fF and T 1 = 25 ns and (b) C x = 2.5 fF and 5 fF and T 1 = 500 ns, while N l 2 = 5 . Dashed line in (a) shows approximation using Eq. ( 3.10). The arrows show the moment t c when the driving frequency f d (see Fig.  f3) is in resonance with ω l 2 / 2 π = 8.91 GHz.... mcd05, a short flux pulse applied to the measured qubit decreases the barrier between the two wells (see Fig.  f0), so that the upper qubit level becomes close to the barrier top. In the case when level | 1 is populated, there is a fast population transfer (tunneling) from the left well to the right well. Due to dissipation, the energy in the right well gradually decreases, until it reaches the bottom of the right well. In contrast, if the qubit is in state | 0 the tunneling essentially does not occur. The qubit state in one of the two potential minima (separated by almost Φ 0 ) is subsequently distinguished by a nearby SQUID, which completes the measurement process.... Now let us consider the effect of dissipation in the second qubit. ... Dots: Rabi frequencies R k , k - 1 / 2 π for the left-well transitions at t = t c , for N l = 10 , C x = 6 fF, and T 1 = 25 ns. Dashed line shows analytical dependence 1.1 k GHz.... 2.16 Figure f2 shows the qubit potential U δ for N l = 10 (corresponding to φ = 4.842 ), N l = 5 ( φ = 5.089 ), and N l = 1.355 ( φ = 5.308 ); the last value corresponds to the bias during the measurement pulse (see below). The qubit levels | 0 and | 1 are, respectively, the ground and the first excited levels in the left well.... Solid lines: log-log contour plots for the values of the error (switching) probability P s = 0.01 , 0.1, and 0.3 on the plane of relaxation time T 1 (in ns) and coupling capacitance C x (in fF) in the quantum model for (a) N l 2 = 5 and (b) N l 2 = 10 . The corresponding results for C x , T T 1 in the classical models are shown by the dashed lines (actual potential model) and the dotted lines [oscillator model, Eq. ( bound1)]. The numerical data are represented by the points, connected by lines as guides for the eye. The scale at the right corresponds to the operation frequency of the two-qubit imaginary-swap quantum gate.... 3.17 in the absence of dissipation in the second qubit ( T 1 ' = ∞ ) for N l 2 = 5 and 10, while T 1 = 25 ns. (In this subsection we take into account the mass renormalization m → m ' ' explicitly, even though this does not lead to a noticeable change of results.) A comparison of Figs.  f4(a) and f7 shows that in both models the qubit energy remains small before a sharp increase in energy. However, there are significant differences due to account of anharmonicity: (a) The sharp energy increase occurs earlier than in the oscillator model (the position of short-time energy maximum is shifted approximately from 3 ns to 2 ns); (b) The excitation of the qubit may be to a much lower energy than for the oscillator; (c) After the sharp increase, the energy occasionally undergoes noticeable upward (as well as downward) jumps, which may overshoot the initial energy maximum; (d) The model now explicitly describes the qubit escape (switching) to the right well [Figs.  f7(b) and f7(c)]; in contrast to the oscillator model, the escape may happen much later than initial energy increase; for example, in Fig.  f7(b) the escape happens at t ≃ 44 ns ≫ t c ≃ 2.1 ns.
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(a) Equivalent circuit of the flux detector based on the Josephson transmission line (JTL) and (b) diagram of scattering of the fluxon injected into the JTL with momentum k by the potential U x that is controlled by the measured qubit. The fluxons are periodically injected into the JTL by the generator and their scattering characteristics (transmission and reflection coefficients t k , r k ) are registered by the receiver.... Schematics of the QND fluxon measurement of a qubit which suppresses the effect of back-action dephasing on the qubit oscillations. The fluxon injection frequency f is matched to the qubit oscillation frequency Δ : f ≃ Δ / π , so that the individual acts of measurement are done when the qubit density matrix is nearly diagonal in the σ z basis, and the measurement back-action does not introduce dephasing in the oscillation dynamics.
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(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.... 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
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Energy-level diagram of a qubit and transitions created by a bichromatic field at double resonance ( ω 0 = ω , ω 1 = ω r f ).
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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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We fit the escape rates in Fig.  F031806DGtot (and additional data for other powers not shown) to a decaying sinusoid with an offset. The extracted frequencies are shown with circles in Fig.  F031806Dfosc. To compare to theory, Ω R , 0 1 , calculated using the rotating wave solution for a system with five levels, is shown with a solid line. The implied assumption that the oscillation frequencies of Γ and ρ 11 are equal, even at high power in a multilevel system, will be addressed in Sec.  SSummary. In plotting the data, we have introduced a single fitting parameter 117   n A / m W that converts the power P S  at the microwave source to the current amplitude I r f  at the qubit. Good agreement is found over the full range of power.... We next consider the time dependence of the escape rate for the data plotted in Fig.  F032106FN10. Here, a 6.2 GHz microwave pulse nominally 30 ns long was applied on resonance with the 0 → 1  transition of the qubit junction. The measured escape rate shows Rabi oscillations followed by a decay back to the ground state once the microwave drive has turned off. This decay appears to be governed by three time constants. Nontrivial decays have previously been reported in phase qubits and we have found them in several of our devices.... F031806Dfosc Rabi oscillation frequency Ω R , 0 1  at fixed bias as a function of microwave current I r f . Extracted values from data (including the plots in Fig.  F031806DGtot) are shown as circles, while the rotating wave solution is shown for two- (dashed line) and five- (solid) level simulations, calculated using I 01 = 17.930   μ A and C 1 = 4.50   p F with ω r f / 2 π = 6.2   G H z .... As I r f  increases in Fig.  F031806Dfosc, the oscillation frequency is smaller than the expected linear relationship for a two-level system (dashed line). This effect is a hallmark of a multilevel system and has been previously observed in a similar phase qubit. There are two distinct phenomena that affect 0 → 1  Rabi oscillations in such a device. To describe... FDeviceThe dc SQUID phase qubit. (a) The qubit junction J 1 (with critical current I 01 and capacitance C 1 ) is isolated from the current bias leads by an auxiliary junction J 2 (with I 02 and C 2 ) and geometrical inductances L 1 and L 2 . The device is controlled with a current bias I b and a flux current I f which generates flux Φ a through mutual inductance M . Transitions can be induced by a microwave current I r f , which is coupled to J 1 via C r f . (b) When biased appropriately, the dynamics of the phase difference γ 1 across the qubit junction are analogous to those of a ball in a one-dimensional tilted washboard potential U . The metastable state n  differs in energy from m  by ℏ ω n m and tunnels to the voltage state with a rate Γ n . (c) The photograph shows a Nb/AlO x /Nb device. Not seen is an identical SQUID coupled to this device intended for two-qubit experiments; the second SQUID was kept unbiased throughout the course of this work.... F032206MNstats (Color online) The (a) on-resonance Rabi oscillation frequencies Ω R , 0 1 m i n  and Ω R , 0 2 m i n  and (b) resonance frequency shifts Δ ω 0 1 = ω r f - ω 0 1 and Δ ω 0 2 = 2 ω r f - ω 0 2 are plotted as a function of the microwave current, for data taken at 110   m K with a microwave drive of frequency ω r f / 2 π = 6.5   G H z and powers P S = - 23 , - 20 , - 17 , - 15 , - 10   d B m . Values extracted from data for the 0 → 1  ( 0 → 2 ) transition are plotted as open circles (filled squares), while five-level rotating wave solutions for a junction with I 01 = 17.736   μ A and C 1 = 4.49   p F are shown as solid (dashed) lines. In (a), the dotted line is from a simulation of a two-level system.... F031806DGtot Rabi oscillations in the escape rate Γ were induced at I b = 17.746   μ A by switching on a microwave current at t = 0 with a frequency of 6.2 GHz (resonant with the 0 → 1  transition) and source powers P S  between -12 and -32 dBm, as labeled. The measurements were taken at 20 mK. The solid lines are from a five-level density-matrix simulation with I 01 = 17.930   μ A , C 1 = 4.50   p F , T 1 = 17   n s , and T φ = 16   n s .... F010206H1 (Color online) Multiphoton, multilevel Rabi oscillations plotted in the time and frequency domains. (a) The escape rate Γ (measured at 110 mK) is plotted as a function of the time after which a 6.5 GHz, -11 dBm microwave drive was turned on and the current bias I b of the qubit; Γ ranges from 0 (white) to 3 × 10 8   1 / s (black). (b) The normalized power spectral density of the time-domain data from t = 1 to 45 ns is shown with a grayscale plot. The dashed line segments indicate the Rabi frequencies obtained from the rotating wave model for transitions involving (from top to bottom) 1, 2, 3, and 4 photons, evaluated with junction parameters I 01 = 17.828   μ A and C 1 = 4.52   p F , and microwave current I r f = 24.4   n A . Corresponding grayscale plots calculated with a seven-level density-matrix simulation are shown in (c) and (d).... Figure  F010206H1(b) shows that the minimum oscillation frequency Ω R , 0 1 m i n / 2 π = 540   M H z of the first (experimental) band occurs at I b = 17.624   μ A , for which ω 0 1 / 2 π = 6.4   G H z . This again indicates an ac Stark shift of this transition, which we denote by Δ ω 0 1 ≡ ω r f - ω 0 1 ≈ 2 π × 100   M H z . In addition, the higher levels have suppressed the oscillation frequency below the bare Rabi frequency of Ω 0 1 / 2 π = 620   M H z [calculated with Eq. ( eqf)].... For this data set, the level spacing ω 0 1 / 2 π is equal to the microwave frequency ω r f / 2 π = 6.5   G H z at I b = 17.614   μ A . The band with the highest current in Fig.  F010206H1(b) is centered about I b = 17.624   μ A , suggesting that 0 → 1  Rabi oscillations are the dominant process near this bias. For slightly higher or lower I b , the oscillation frequency increases as Ω R , 0 1 ≈ Ω 01 ′ 2 + ω r f - ω 0 1 2 , in agreement with simple two-level Rabi theory, leading to the curved band in the grayscale plot.
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