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  • In the preceeding analysis we neglected the effect of the local environment by setting Y i n t ω = 0 . As a result, the low-frequency value of T 1 is substantially larger than obtained in experiment . By modeling the local environment with R 0 = 5000  ohms and L 0 = 0 we obtain the T 1 versus ω 01 plot shown in Fig.  fig:three. Notice that this value of R 0 brings T 1 to values close to 20 ns at T = 0 . The message to extract from Figs.  fig:two and  fig:three is that increasing R 0 as much as possible and increasing the qubit frequency ω 01 from 0.1 Ω to 2 Ω at fixed low temperature can produce a large increase in T 1 .... Schematic drawing of the phase qubit with an RLC isolation circuit.... The circuit used to describe intrinsic decoherence and self-induced Rabi oscillations in phase qubits is shown in Fig.  fig:one, which correponds to an asymmetric dc SQUID . The circuit elements inside the dashed box form an isolation network which serves two purposes: a) it prevents current noise from reaching the qubit junction; b) it is used as a measurement tool.... In the limit of T = 0 , we can solve for c 1 t exactly and obtain the closed form c 1 t = L -1 s + Γ - i ω 01 2 + Ω 2 - Γ 2 s s + Γ - i ω 01 2 + Ω 2 - Γ 2 - κ Ω 4 π i / Γ where L -1 F s is the inverse Laplace transform of F s , and κ = α / M ω 01 × Φ 0 / 2 π 2 ≈ 1 / ω 01 T 1 , 0 . The element ρ 11 = | c 1 t | 2 of the density matrix is plotted in Fig.  fig:four for three different values of resistance, assuming that the qubit is in its excited state such that ρ 11 0 = 1 . We consider the experimentally relevant limit of Γ ≪ ω 01 ≈ Ω , which corresponds to the weak dissipation limit. Since Γ = 1 / 2 C R the width of the resonance in the spectral density shown in Eq. ( eqn:sd-poles) is smaller for larger values of R . Thus, for large R , the RLC environment transfers energy resonantly back and forth to the qubit and induces Rabi-oscillations with an effective time dependent decay rate γ t = - 2 ℜ c ̇ 1 t / c 1 t .... fig:three T 1 (in nanoseconds) as a function of qubit frequency ω 01 . The solid (red) curves describes an RLC isolation network with parameters R = 50  ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and qubit parameters C 0 = 4.44 pF, R 0 = 5000  ohms and L 0 = 0 . The dashed curves correspond to an RL isolation network with the same parameters, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 GHz.... fig:four Population of the excited state of the qubit as a function of time ρ 11 t , with ρ 11 t = 0 = 1 for R = 50  ohms (solid curve), 350  ohms (dotted curve), and R = 550  ohms (dashed curve), and L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 .... fig:two T 1 (in seconds) as a function of qubit frequency ω 01 . The solid (red) curves describes an RLC isolation network with parameters R = 50  ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and qubit parameters C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 . The dashed curves correspond to an RL isolation network with the same parameters, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 GHz.... In Fig.  fig:two, T 1 is plotted versus qubit frequency ω 01 for spectral densities describing an RLC (Eq.  eqn:spectral-density-isolation) or Drude (Eq.  eqn:sd-drude) isolation network at fixed temperatures T = 0 (main figure) and T = 50 mK (inset), for J i n t ω = 0 corresponding to R 0 ∞ . In the limit of low temperatures k B T / ℏ ω 01 ≪ 1 , the relaxation time becomes T 1 ω 01 = M ω 01 / J ω 01 . From Fig.  fig:two (main plot) several important points can be extracted. First, in the low frequency regime ( ω 01 ≪ Ω ) the RL (Drude) and RLC environments produce essentially the same relaxation time T 1 , R L C 0 = T 1 , R L 0 = T 1 , 0 ≈ L 1 / L 2 R C 0 , because both systems are ohmic. Second, near resonance ( ω 01 ≈ Ω ), T 1 , R L C is substantially reduced because the qubit is resonantly coupled to its environment producing a distinct non-ohmic behavior. Third, for ( ω 01 > Ω ), T 1 grows very rapidly in the RLC case. Notice that for ω 01 > 2 Ω , the RLC relaxation time T 1 , R L C is always larger than T 1 , R L . Furthermore, in the limit of ω 01 ≫ m a x Ω , 2 Γ , T 1 , R L C grows with the fourth power of ω 01 behaving as T 1 , R L C ≈ T 1 , 0 ω 01 4 / Ω 4 , while for ω 01 ≫ Ω 2 / 2 Γ , T 1 , R L grows only with second power of ω 01 behaving as T 1 , R L ≈ 4 T 1 , 0 Γ 2 ω 01 2 / Ω 4 . Thus, T 1 , R L C is always much larger than T 1 , R L for sufficiently large ω 01 . Notice, however, that for parameters in the experimental range such as those used in Fig  fig:two, T 1 , R L C is two orders of magnitude larger than T 1 , R L , indicating a clear advantage of the RLC environment shown in Fig  fig:one over the standard ohmic RL environment. Thermal effects are illustrated in the inset of Fig.  fig:two where T = 50 mK is a characteristic temperature where experiments are performed . The typical values of T 1 at low frequencies vary from 10 -5 s at T = 0 to 10 -6 s at T = 50 mK, while the high frequency values remain essentially unchanged as the thermal effects are not important for ℏ ω 01 ≫ k B T .... These environmentally-induced Rabi oscillations are a clear signature of the non-Markovian behavior produced by the RLC environment, and are completely absent in the RL environment because the energy from the qubits is quickly dissipated without being temporarily stored. These environmentally-induced Rabi oscillations are generic features of circuits with resonances in the real part of the admittance. The frequency of the Rabi oscillations Ω R a = π κ Ω 3 / 2 Γ is independent of the resistance since Ω R a ≈ Ω π L 2 C / L 1 2 C 0 , and has the value of Ω R a = 2 π f R a ≈ 360 × 10 6  rad/sec for Fig.  fig:four.
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  • where we have defined the total spin operators J ̂ α = ∑ σ ̂ α / 2 . In the limit ℏ ω 0 / Δ → 0 , all the results concerning the low-energy spectrum of the resonator remain unchanged; one could say that the reduction of the coupling strength by the factor N is compensated by the strengthening of the spin raising and lowering operators by the same factor because of the collective behaviour of the qubits. In particular, the transition occurs at the critical coupling strength given by Eq. ( Eq:CriticalCouplingStrength). Because the qubits now have a larger total spin (when compared to the single-qubit case), spin states that are separated by small angles can be drastically different (i.e. have a small overlap). In particular, the overlap for N qubits is given by cos 2 N θ / 2 . By expanding this function to second order around θ = 0 , one can see that for small values of θ the relevant overlap is lower than unity by an amount that is proportional to N . This dependence translates into the dependence of the qubit-oscillator entanglement on the coupling strength just above the critical point. The entanglement therefore rises more sharply in the multi-qubit case (with the increase being by a factor N ), as demonstrated in Fig.  Fig:EntropyLogLog.... (Color online) The logarithm of the von Neumann entropy S as a function of the logarithm of the quantity λ / λ c - 1 , which measures the distance of the coupling strength from the critical value. The red solid line corresponds to the single-qubit case, whereas the other lines correspond to the multi-qubit case: N = 2 (green dashed line), 3 (blue short-dashed line), 5 (purple dotted line) and 10 (dash-dotted cyan line). All the lines correspond to ℏ ω 0 / Δ = 10 -7 . The slope of all lines is approximately 0.92 when λ / λ c - 1 = 10 -4 . The ratio of the entropy in the multi-qubit case to that in the single-qubit case approaches N for all the lines as we approach the critical point.... The energy level structure in the single-qubit case is simple in principle. In the limit ℏ ω 0 / Δ → 0 , one can say that the energy levels form two sets, one corresponding to each qubit state. Each one of these sets has a structure that is similar to that of a harmonic oscillator with some modifications that are not central in the present context. In particular the density of states has a weak dependence on energy, a situation that cannot support a thermal phase transition. If the temperature is increased while all other system parameters are kept fixed, qubit-oscillator correlations (which are finite only above the critical point) gradually decrease and vanish asymptotically in the high-temperature limit. No singular point is encountered along the way. This result implies that the transition point is independent of temperature. In other words, it remains at the value given by Eq. ( Eq:CriticalCouplingStrength) for all temperatures. If, for example, one is investigating the dependence of the correlation function C on the coupling strength (as plotted in Fig.  Fig:SpinFieldSignCorrelationFunction), the only change that occurs as we increase the temperature is that the qubit-oscillator correlations change more slowly when the coupling strength is varied.... where p ̂ is the oscillator’s momentum operator, which is proportional to i â † - â in our definition of the operators. The squeezing parameter mirrors the behaviour of the low-lying energy levels. In particular we can see from Fig.  Fig:SqueezingParameter that only when ℏ ω 0 / Δ reaches the value 10 -5 does the squeezing become almost singular at the critical point.... (Color online) The von Neumann entropy S as a function of the oscillator frequency ℏ ω 0 and the coupling strength λ , both measured in comparison to the qubit frequency Δ . One can see clearly that moving in the vertical direction the rise in entropy is sharp in the regime ℏ ω 0 / Δ ≪ 1 , whereas it is smooth when ℏ ω 0 / Δ is comparable to or larger than 0.1.... The tendency towards singular behaviour (in the dependence of various physical quantities on λ ) in the limit ℏ ω 0 / Δ → 0 is illustrated in Figs.  Fig:ColorPlot- Fig:SqueezingParameter. In these figures, the entanglement, spin-field correlation function, low-lying energy levels (measured from the ground state) and the oscillator’s squeezing parameter are plotted as functions of the coupling strength. It is clear from Figs.  Fig:EntropyLinear and Fig:SpinFieldSignCorrelationFunction that when ℏ ω 0 / Δ ≤ 10 -3 both the entanglement (which is quantified through the von Neumann entropy S = T r ρ q log 2 ρ q with ρ q being the qubit’s reduced density matrix) and the correlation function C = σ z s i g n a + a † rise sharply upon crossing the critical point . The low-lying energy levels, shown in Fig.  Fig:EnergyLevels, approach each other to form a large group of almost degenerate energy levels at the critical point before they separate again into pairs of asymptotically degenerate energy levels. This approach is not complete, however, even when ℏ ω 0 / Δ = 10 -3 ; for this value the energy level spacing in the closest-approach region is roughly ten times smaller than the energy level spacing at λ = 0 . The squeezing parameter is defined by the width of the momentum distribution relative to that in the case of an isolated oscillator. For consistency with Ref. , we define it as
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  • We performed a spectroscopy measurement of the qubit with long (50 ns) single-frequency microwave pulses. We observed multi-photon resonant peaks ( Φ q b 1.5 Φ 0 ) in the dependence of P s w on f M W 1 at a fixed magnetic flux Φ q b . We obtained the qubit energy diagram by plotting their positions as a function of Φ q b / Φ 0 (Fig.  Fig2(a)). We took the data around the degeneracy point Φ q b ≈ 1.5 Φ 0 by applying an additional dc pulse to the microwave line to shift Φ q b away from 1.5 Φ 0 just before the readout, because the dc-SQUID could not distinguish the qubit states around the degeneracy point. The top solid curve in Fig.  Fig2(a) represents a numerical fit to the resonant frequencies of one-photon absorption. From this fit, we obtain the qubit parameters E J / h = 213 GHz, Δ / 2 π = 1.73 GHz, and α = 0.8. The other curves in Fig.  Fig2(a) are drawn by using these parameters for n 1 = 2, 3, and 4.... Next, we used short single-frequency microwave pulses with a frequency of 10.25 GHz to observe the coherent quantum dynamics of the qubit. Figures  Fig2(b) and (c) show one- and four-photon Rabi oscillations observed at the operating points indicated by arrows in Fig.  Fig2(a) with various microwave amplitudes V M W 1 . These data can be fitted by damped oscillations ∝ exp - t p / T d cos Ω R a b i t p , except for the upper two curves in Fig.  Fig2(b). Here, t p and T d are the microwave pulse length and qubit decay time, respectively. To obtain Ω R a b i , we performed a fast Fourier transform (FFT) on the curves that we could not fit by damped oscillations. Although we controlled the qubit environment, there were some unexpected resonators coupled to the qubit, which could be excited by the strong microwave driving or by the Rabi oscillations of the qubit. We consider that these resonators degraded the Rabi oscillations in the higher V M W 1 range of Fig.  Fig2(b). Figure  Fig2(d) shows the V M W 1 dependences of Ω R a b i / 2 π up to four-photon Rabi oscillations, which are well reproduced by Eq. ( eq2). Here, we used only one scaling parameter a (10.25 GHz) = 0.013 defined as a f M W 1 ≡ 4 g 1 α 1 / ω M W 1 V M W 1 , because it is hard to measure the real amplitude of the microwave applied to the qubit at the sample position. The scaling parameter a f M W 1 reflects the way in which the applied microwave is attenuated during its transmission to the qubit and the efficiency of the coupling between the qubit and the on-chip microwave line. In this way, we can estimate the real microwave amplitude and the interaction energy between the qubit and the microwave 2 ℏ g 1 α 1 by fitting the dependence of Ω R a b i / 2 π on V M W 1 . These results show that we can reach a driving regime that is so strong that the interaction energy 2 ℏ g 1 α 1 is larger than the qubit transition energy ℏ ω q b .... Experimental results with single-frequency microwave pulses. (a) Spectroscopic data of the qubit. Each set of the dots represents the resonant frequencies f r e s caused by the one to four-photon absorption processes. The solid curves are numerical fits. The dashed line shows a microwave frequency f M W 1 of 10.25 GHz. (b) One-photon Rabi oscillations of P s w with exponentially damped oscillation fits. Both the qubit Larmor frequency f q b and the microwave frequency f M W 1 are 10.25 GHz. The external flux is Φ q b / Φ 0 = 1.4944. (c) Four-photon Rabi oscillations when f q b = 41.0 GHz, f M W 1 = 10.25 GHz, and Φ q b / Φ 0 = 1.4769. (d) The microwave amplitude dependence of the Rabi frequencies Ω R a b i / 2 π up to four-photon Rabi oscillations. The solid curves represent theoretical fits. Fig2... The measurements were carried out in a dilution refrigerator. The sample was mounted in a gold plated copper box that was thermalized to the base temperature of 20 mK ( k B T frequency microwave pulses, we added two microwaves MW1 and MW2 with frequencies of f M W 1 and f M W 2 , respectively by using a splitter SP (Fig.  Fig1(b)). Then we shaped them into microwave pulses through two mixers. We measured the amplitude of MW k V M W k at the point between the attenuator and the mixer with an oscilloscope. We confirmed that unwanted higher-order frequency components in the pulses, for example | f M W 1 ± f M W 2 | , 2 f M W 1 , and 2 f M W 2 are negligibly small under our experimental conditions. First, we choose the operating point by setting Φ q b around 1.5 Φ 0 , which fixes the qubit Larmor frequency f q b . The qubit is thermally initialized to be in | g by waiting for 300 μ s, which is much longer than the qubit energy relaxation time (for example 3.8 μ s at f q b = 11.1 GHz). Then a qubit operation is performed by applying a microwave pulse to the qubit. The pulse, with an appropriate length t p , amplitudes V M W k , and frequencies f M W k , prepares a qubit in the superposition state of | g and | e . After the operation, we immediately apply a dc readout pulse to the dc-SQUID. This dc pulse consists of a short (70 ns) initial pulse followed by a long (1.5 μ s) trailing plateau that has 0.6 times the amplitude of the initial part. For Φ q b qubit is detected as being in | e , the SQUID switches to a voltage state and an output voltage pulse should be observed; otherwise there should be no output voltage pulse. By repeating the measurement 8000 times, we obtain the SQUID switching probability P s w , which is directly related to P e t p for the dc readout pulse with a proper amplitude. For Φ q b > 1.5 Φ 0 , P s w is directly related to 1 - P e t p .... We next investigated the coherent oscillations of the qubit through the parametric processes by using short two-frequency microwave pulses. Figure  Fig3(a) [(b)] shows the Rabi oscillations of P s w when the qubit Larmor frequency f q b = 26.45 [7.4] GHz corresponds to the sum of the two microwave frequencies f M W 1 = 16.2 GHz, f M W 2 = 10.25 GHz [the difference between f M W 1 = 11.1 GHz and f M W 2 = 18.5 GHz] and the microwave amplitude of MW2 V M W 2 was fixed at 33.0 [50.1] mV. They are well fitted by exponentially damped oscillations ∝ exp - t p / T d cos Ω R a b i t p . The Rabi frequencies obtained from the data in Fig. 3(a) [(b)] are well reproduced by Eq. ( eq3) without any fitting parameters (Fig.  Fig3(c) [(d)]). Here, we used Δ , which was obtained from the spectroscopy measurement (Fig.  Fig2(a)) and used a (10.25 GHz) = 0.013 and a (16.2 GHz) = 0.0074 [ a (11.1 GHz) = 0.013 and a (18.5 GHz) = 0.0082], which had been obtained from Rabi oscillations by using single-frequency microwave pulses with each frequency. Those results provide strong evidence that we can achieve parametric control of the qubit with two-frequency microwave pulses.... (a) Scanning electron micrograph of a flux qubit (inner loop) and a dc-SQUID (outer loop). The loop sizes of the qubit and SQUID are 10.2 × 10.4  μ m 2 and 12.6 × 13.5  μ m 2 , respectively. They are magnetically coupled by the mutual inductance M ≈ 13 pH. (b) A circuit diagram of the flux qubit measurement system. On-chip components are shown in the dashed box. L ≈ 140 pH, C ≈  9.7 pF, R I 1 = 0.9 k Ω , R V 1 =  5 k Ω . Surface mount resistors R I 2 = 1 k Ω and R V 2 = 3 k Ω are set in the sample holder. We put adequate copper powder filters CP and LC filters F and attenuators A for each line. Fig1... Experimental results with two-frequency microwave pulses. (a) [(b)] Two-photon Rabi oscillations due to a parametric process when f q b = f M W 2 + -   f M W 1 . The solid curves are fits by exponentially damped oscillations. (c) [(d)] Rabi frequencies as a function of V M W 1 , which are obtained from the data in Fig.  Fig3(a) [(b)]. The dots represent experimental data when V M W 2 = 16.9, 23.5, 33.0, and 52.0 [50.1, 62.9, 79.1, and 124.7] mV from the bottom set of dots to the top one. The solid curves represent Eq. ( eq3). The inset is a schematic of the parametric process that causes two-photon Rabi oscillation when f q b = f M W 2 + -   f M W 1 . Fig3
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  • (Color online) Numerical (solid lines) and analytical (dashed lines) dependence of zero-frequency detector noise S a a 0 and cross-noise S a b 0 on the phase shift ϕ between the bias voltage combs for several values of the pulse width δ t a , b , and also for the harmonic biasing. Almost complete noise anticorrelation at ϕ = ± π indicate persistent Rabi oscillations.... Analyzed system: a double-quantum-dot qubit measured by two QPC detectors, which are biased by combs of short voltage pulses with frequency Ω coinciding with the Rabi frequency Ω R .
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  • The evolutions of reduced density matrix elements ρ12 (below) and ρ11 (up) in SB and SIB models in low-frequency bath. The parameters are the same as in Fig. 1. ... The spectral density functions Johm(ω) (b) and Jeff(ω) (a) versus the frequency ω of the bath modes, where Δ=5×109Hz,λκ=1,ξ=0.01,Ω0=10Δ,T=0.01K,Γ=2.6×1011Hz. ... The evolutions of reduced density matrix elements of ρ12 (below) and ρ11 (up) in SIB model in medium-frequency bath in different values of Ω0, the other parameters are the same as in Fig. 1. ... The response functions of the Ohmic bath in (a) low and (c) medium frequencies and effective bath in (b) low and (d) medium frequencies. The parameters are the same as in Fig. 1. The cut-off frequencies for the two cases are taken according to Fig. 2. ... The sketch map on the low-, medium-, and high-frequency baths.
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  • Sketch of the flux qubit (blue) coupled to a dc-SQUID. The interaction is characterised by the linear coupling g 1 , which depends linearly on the SQUID bias current I b , and the quadratic coupling g 2 . The SQUID with Josephson inductance L J is shunted by a capacitance C . The frequency shift of the resulting harmonic oscillator (green) can be probed by external resonant ac-excitation A cos Ω a c t via the transmission line (black), in which the quantum fluctuations ξ i n q m t are also present.... eq:in-out-total1. In an experiment, this can be achieved by lock-in techniques which we mimic in the following way : First, we focus on the associated spectrum ξ o u t ω depicted in figure  fig:qubit-osc-phase-spectrum(b). It reflects the qubit dynamics in terms of two sidebands around the central peak related to the oscillator frequency, here chosen as Ω = 10 ω q b . The dissipative influence of the environment, modelled by a transmission line (see figure fig:setup), is reflected in a broadening of this peak. The corresponding oscillator bandwidth is given as 2 α Ω , where α denotes the dimensionless damping strength; see app:QME. Here, we recall that the oscillator is driven resonantly by the external driving signal A cos Ω a c t , that is, Ω = Ω a c . In the time domain, the sidebands correspond to the phase-shifted signal ξ o u t t = A cos Ω t - ϕ e x p t with slowly time-dependent phase ϕ e x p t . In order to obtain this phase ϕ e x p t , we select the spectral data from a frequency window of size 2 Δ Ω centred at the oscillator frequency Ω , which means that ξ o u t ω is multiplied with a Gaussian window function exp - ω - Ω 2 / Δ Ω 2 . We choose for the window size the resonator bandwidth, Δ Ω = α Ω , which turns out to suppress disturbing contributions from the low-frequency qubit dynamics. Finally, we centre the clipped spectrum at zero frequency and perform an inverse Fourier transform to the time domain. If the phase shift φ e x t was constant, one could use a much smaller measurement bandwidth. Then the outcome of the measurement procedure would correspond to homodyne detection of a quadrature defined by the phase shift and yield a value ∝ cos φ e x p .... Time-resolved measurement of coherent qubit oscillations at the degeneracy point ϵ = 0 . The full qubit-oscillator state was simulated with the quantum master equation  eq:blochredfield with N = 10 oscillator states and the parameters Ω = Ω a c = 10 ω q b , g 1 = 0.1 ω q b , g 2 = 0.01 ω q b , A = 1.0 ω q b . The dimensionless oscillator dissipation strength is α = 0.12 . The resonator bandwidth is given by 2 α Ω = 2.4 ω q b . (a) Lock-in amplified phase ϕ e x p t (dashed green lines), compared to the estimated phase ϕ t (solid red line) of the outgoing signal ξ o u t t . Here, ϕ t ∝ σ x t [cf.  equation  eq:dr-osc-phase], which is corroborated by the inset showing that σ x t performs oscillations with (angular) frequency ω q b . (b) Power spectrum ξ o u t ω for the resonantly driven oscillator (blue solid line). The sidebands stemming from the qubit dynamics are visible at frequencies Ω ± ω q b . In order to extract the phase information, we apply a Gaussian window function with respect to the frequency window of half-width Δ Ω = 1.2 ω q b , which turns out to be the optimal value for the measurement bandwidth.... We consider a superconducting flux qubit coupled to a SQUID  as sketched in figure  fig:setup. The SQUID is modelled as a harmonic oscillator, which gives rise to the Hamiltonian... In figure  fig:qubit-osc-fid(a) we depict the fidelity defect δ F = 1 - F between ϕ e x p t and ϕ t as a function of the oscillator frequency Ω = Ω a c for different quadratic coupling coefficients g 2 . As expected, the overall fidelity is rather insufficient for small oscillator frequency Ω oscillator bandwidth is too small to resolve the qubit dynamics, i.e., if ω q b oscillator frequencies, we again observe an increase of the fidelity defect, which occurs the sooner the smaller g 2 . This latter effect, which is only visible for the smallest value of g 2 in figure  fig:qubit-osc-fid(a), is directly explained by a reduced maximum angular visibility of the phase ϕ t ∝ g 2 / Ω . Thus, figure  fig:qubit-osc-fid(a) provides a pertinent indication for the validity frame of our central relation ... If the qubit is only weakly coupled to the oscillator, and if the latter is driven only weakly, the qubit’s time evolution is rather coherent (see section  sec:sn on qubit decoherence). For this scenario, figure  fig:qubit-osc-phase-spectrum(a) depicts the time-dependent phase ϕ t computed with the measurement relation... (a) Fidelity defect δ F = 1 - F for the phases ϕ t and ϕ e x p t and (b) time-averaged trace distance D ̄ between the density operators of a qubit with finite coupling to the oscillator and a reference qubit without oscillator. Both quantities are depicted for various coupling strengths g 2 in dependence of the oscillator frequency Ω . All other parameters are as in figure  fig:qubit-osc-phase-spectrum.
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  • In our sample geometry, the qubit is spatially separated from the rest of the circuitry. The qubit is enclosed by a superconducting quantum interference device (SQUID) that is inductively coupled to the qubit. The SQUID functions as a detector for the qubit state: the switching current of the SQUID is sensitive to the flux produced by the current in the qubit. The qubit is also enclosed by a larger loop containing on–chip capacitors that provide a well–defined electromagnetic environment for the SQUID and filtering of the measurement leads. The lead inductance L and capacitance C in the outer loop constitute an LC oscillator [see Fig.  fig1(b)] with resonance frequency ω r = 2 π ν r = 1 / L C . The LC oscillator is described by a simple harmonic oscillator Hamiltonian: H o s c = ℏ ω r a † a + 1 / 2 , where a † ( a ) is the plasmon creation (annihilation) operator. The qubit is coupled to the LC oscillator via the mutual inductance M , giving an interaction Hamiltonian H I = h λ σ z a † + a , where the coupling constant is h λ = M I p ℏ ω r / 2 L . The total system is thus described by a Jaynes-Cummings type of Hamiltonian H = h / 2 ϵ σ z + Δ σ x + ℏ ω r a † a + 1 / 2 + h λ σ z a † + a . We denote the state of the system by | Q , i , with the qubit either in the ground ( Q = g ) or excited ( Q = e ) state, and the oscillator in the Fock state ( i = 0 , 1 , 2 , ). The parameters of the system can readily be engineered during fabrication; the qubit gap is determined by α and the junction resistance, the oscillator plasma frequency is fully determined by L and C , and the coupling between the qubit and the oscillator can be tuned by M and L .... fig1 (a) SEM micrograph of the sample. The qubit and the detector SQUID enclosing it are the small square loops in the lower center picture. The square plates at the top of the picture are the top plates of the on-chip capacitors separated by an insulator from the large bottom plate. (b) A close up of the qubit and the SQUID. The qubit dimension is 10.2 × 10.4   μ m 2 . (c) Equivalent circuit of the sample. The Josephson junctions are indicated by crosses: three in the inner qubit loop and two in the SQUID. The LC mode is indicated by the dashed line. The inductance and capacitances are calculated from the geometry to be L = 140  pH and C = 10  pF, and the qubit LC oscillator mutual inductance to be M = 5.7  pH. The current and voltage lines are filtered through a series combination of copper powder filters and lossy coaxial cables at mixing chamber temperature and on–chip resistors ( R V = 3 k Ω and R I = 1 k Ω ).... fig4 (a) Rabi oscillations when a 2 ns long pulse with frequency ν e x = 4.35 GHz and amplitude A M W ∝ 10 P o s c / 20 is inserted between the π pulse and the shift pulse. (b) Measured Rabi oscillations at different drive powers (symbols), and a fit (solid curve) to ∑ n = 0 3 a n cos n + 1   Ω R t exp - Γ t with a 0 , , a 3 and Ω R as the only fitting parameters ( Γ is fixed from a fit to the -100 dBm curve). (c) The weights of the different frequency components a 0 , , a 3 obtained from the fit as a function of the drive amplitude (The red arrows show the position of the curves in (b). The error-bars indicate the errors obtained from the fitting procedure.... fig2 (a) Spectroscopic characterization of the qubitoscillator system showing the LC oscillator at ν r = 4.35  GHz and the qubit dispersion around the gap of Δ = 2.1  GHz. (b) A close up of the region around 4.35 GHz (indicated by the red square in (a) showing an avoided crossing. The lines are guides for the eye. (c) Schematic of the pulse sequence used to obtain the spectroscopy in (b): the qubit operating point is fixed at 10.5 GHz and via the MW line a shift pulse of variable height moves the operating point to the vicinity of 4.35 GHz. Two 50 ns MW pulses separated by 2 ns are added to the shift pulse. Here the second MW pulse is phase shifted by 180 ∘ . The phase shifted second pulse damps the oscillations in the LC circuit , and is crucial in terms of resolving the relatively weak qubit signal in this region. After the MW pulses the qubit state is measured by applying a measurement pulse to the SQUID (green curve). The spectroscopy in Fig. 2(a) was obtained with the same scheme, but without the second phase shifted pulse.... Next we investigate the dynamics of the coupled system in the time domain. We performed a measurement cycle where we first excited the qubit and then brought the qubit and the oscillator into resonance where the exchange of a single energy quantum between the qubit and oscillator manifests itself as the vacuum Rabi oscillation | e , 0 ↔ | g , 1 (see Fig. 3). Figure. 3(c) is a schematic of the pulse sequence: We started by fixing the qubit operating point far from the resonance point [point 3 in Fig.  fig3(c)] and prepared the qubit in the excited state by employing a π pulse. The π pulse was followed by a shift pulse, which brought the qubit into resonance with the oscillator for the duration of the shift pulse. After the shift pulse the qubit and the oscillator were brought back into off–resonance and the measurement pulse was applied to detect the state of the qubit. It is important to note that the rise time of the shift pulse, τ r i s e = 0.8  ns, is adiabatic with respect to both the qubit and the oscillator, τ r i s e > 1 / E 2 π / ω r , but non-adiabatic with respect to the coupling of the two systems, τ r i s e oscillations is thus different from that of normal Rabi oscillations where the system is driven by an external classical field and oscillates between two energy eigenstates. Also, in the normal Rabi oscillations the Rabi frequency is determined by the drive amplitude whereas the vacuum Rabi oscillation frequency is determined only by the system‘s intrinsic parameters. The observed Rabi oscillations are in excellent agreement with those calculated numerically [solid line in Fig. 3(a)]. The numerical calculation uses the total Hamiltonian and incorporates the effects of the decoherence of the qubit and the oscillator. The calculation was performed with the known qubit and LC oscillator parameters (obtained from spectroscopy and qubit experiments: qubit dephasing rate Γ φ = 0.1  GHz, qubit relaxation rate Γ e = 0.2  MHz, Δ = 2.1  GHz, ω r / 2 π = 4.35  GHz) and by treating the coupling constant and oscillator dephasing and relaxation rates as fitting parameters. From the fit we extracted the coupling constant λ = 0.2  GHz, oscillator dephasing rate Γ φ = 0.3  GHz, and relaxation rate Γ e = 0.02  GHz. The coupling constant extracted from the fit agrees well with that calculated with the mutual inductance λ = 0.216  GHz.... fig3 (a) Vacuum Rabi oscillations (symbols) and a numerical fit (solid line). (b) The few lowest unperturbed and dressed energy levels when the system is in resonance. (c) The qubit energy level diagram and pulse sequence for the vacuum Rabi measurements. The π pulse (4.6 ns long at -16 dBm) on the qubit brings the system from state 1 to 2 and the shift pulse changes the flux in the qubit by Φ s h i f t , which, in turn, changes the operating point from 2 to 3 where the system undergoes free evolution between | e , 0 and | g , 1 at the vacuum Rabi frequency Ω R until the shift pulse ends and the system returns to the initial operating point where the state is measured to be either in 2 or 4.
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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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  • 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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  • 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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