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56057 results
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:... 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
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(Color online) (a) Nonlinear response A of the detector coupled to the qubit prepared in its ground state | ↓ (orange solid line) and in its excited state | ↑ (black dashed line) for the same parameters as in Fig.  fig2. The quadratic qubit-detector coupling induces a global frequency shift of the response by δ ω e x = 2 g . (b) Discrimination power D ω e x of the detector coupled to the qubit for the same parameters as in a). fig3... (Color online) (a) Asymptotic population difference P ∞ of the qubit states, and (b) the corresponding detector response A as a function of the external frequency ω e x for the same parameters as in Fig.  fig2. fig4... (Color online) (a) Relaxation rate Γ of the nonlinear quantum detector, (b) the measurement time T m e a s , and (c) the measurement efficiency Γ m e a s / Γ as a function of the external frequency ω e x . The parameters are the same as in Fig.  fig2. fig5... For a fixed value of g , the shift between the two cases of the opposite qubit states is given by the frequency gap δ ω e x ≃ 2 g . Figure fig3 (a) shows the nonlinear response of the detector for the two cases when the qubit is prepared in one of its eigenstates: | ↑ (orange solid line) and | ↓ (black dashed line).... (Color online) (a) Amplitude A of the nonlinear response of the decoupled quantum Duffing detector ( g = 0 ) as a function of the external driving frequency ω e x . (b) The corresponding quasienergy spectrum ε α . The labels N denote the corresponding N -photon (anti-)resonance. The parameters are α = 0.01 Ω , f = 0.006 Ω , T = 0.006 Ω , and γ = 1.6 × 10 -4 Ω . fig1... Before turning to the quantum detection scheme, we discuss the dynamical properties of the isolated detector, which is the quantum Duffing oscillator. A key property is its nonlinearity which generates multiphoton transitions at frequencies ω e x close to the fundamental frequency Ω . In order to see this, one can consider first the undriven nonlinear oscillator with f = 0 and identify degenerate states, such as | n and | N - n (for N > n ), when δ Ω = α N + 1 / 2 . For finite driving f > 0 , the degeneracy is lifted and avoided quasienergy level crossings form, which is a signature of discrete multiphoton transitions in the detector. As a consequence, the amplitude A of the nonlinear response signal exhibits peaks and dips, which depend on whether a large or a small oscillation state is predominantly populated. The formation of peaks and dips goes along with jumps in the phase of the oscillation, leading to oscillations in or out of phase with the driving. A typical example of the nonlinear response of the quantum Duffing oscillator in the deep quantum regime containing few-photon (anti-)resonances is shown in Fig.  fig1(a) (decoupled from the qubit), together with the corresponding quasienergy spectrum [Fig.  fig1(b)]. We show the multiphoton resonances up to a photon number N = 5 . The resonances get sharper for increasing photon number, since their widths are determined by the Rabi frequency, which is given by the minimal splitting at the corresponding avoided quasienergy level crossing. Performing a perturbative treatment with respect to the driving strength f , one can get the minimal energy splitting at the avoided quasienergy level crossing 0 N as... (Color online) Nonlinear response A of the detector as a function of the external driving frequency ω e x in the presence of a finite coupling g = 0.0012 Ω to the qubit (black solid line). The blue dashed line indicates the response of the isolated detector. The parameters are the same as in Fig.  fig1 and ϵ = 2.2 Ω and Δ = 0.05 Ω , in correspondence to realistic experimental parameters . fig2... Notice that g and α depend on the external flux ϕ e x , i.e., they are tunable in a limited regime with respect to the desired oscillator frequency Ω , where the coupling term is considered as a perturbation to the SQUID ( g oscillator to dominate. The dependence of the dimensionless ratios α / Ω and g / Ω is shown in Fig.  fig0.
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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 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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(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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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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(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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