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  • Schematic of the Josephson-junction qubit structure that enables measurements of the two non-commuting observables of the qubit, σ z and σ y , as required in the QND Hamiltonian ( 2). For discussion see text.... Spin representation of the QND measurement of the quantum coherent oscillations of a qubit. The oscillations are represented as a spin rotation in the z - y plane with frequency Δ . QND measurement is realized if the measurement frame (dashed lines) rotates with frequency Ω ≃ Δ .
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  • A contour plot indicating location of two-dimensional potential energy minima forming a symmetric double well potential when the cantilever equilibrium angle θ0=cos−1⁡[Φo/2BxA], ωi=2π×12000 rad/s, Bx=5×10−2 T. The contour interval in units of frequency (E/h) is ∼4×1011 Hz. ... A superconducting-loop-oscillator with its axis of rotation along the z-axis consists of a closed superconducting loop without a Josephson Junction. The superconducting loop can be of any arbitrary shape. ... A contour plot indicating location of a two-dimensional global potential energy minimum at (nΦ0=0, θn+=π/2) and the local minima when the cantilever equilibrium angle θ0=π/2, ωi=2π×12000 rad/s, Bx=5.0×10−2 T. The contour interval in units of frequency (E/h) is ∼3.9×1011 Hz. ... The potential energy profile of the superconducting-loop-oscillator when the intrinsic frequency is 10 kHz. (a) For external magnetic field Bx=0, a single well harmonic potential near the minimum is formed. (b) Bx=0.035 T. (c) For Bx=0.045 T, a double well potential is formed. ... A schematic of the flux-qubit-cantilever. A part of the flux-qubit (larger loop) is projected from the substrate to form a cantilever. The external magnetic field Bx controls the coupling between the flux-qubit and the cantilever. An additional magnetic flux threading through a dc-SQUID (smaller loop) which consists of two Josephson junctions adjusts the tunneling amplitude. The dc-SQUID can be shielded from the effect of Bx.
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  • Rabi oscillation of the double SQUID manipulated as a phase qubit by applying microwave pulses at 19 GHz. The oscillation frequency changes from 540 MHz to 1.2 GHz by increasing the power of the microwave signal by 10 dB.... Probability of measuring the state | L as a function of the pulse duration. The coherent oscillation shown here has a frequency of 14 GHz and a coherence time of approximately 1.2 ns.... Measurement of the relaxation time T 1 for the double SQUID operated as a phase qubit.... Measured oscillation frequencies versus amplitude of the short flux pulse (full dots). The solid curve is a numerical simulation using the measured parameters of the circuit.... Measured Rabi oscillation frequency versus the normalized amplitude of the microwave signal (solid dots). The dashed line is a linear fit taking into account slightly off-resonance microwave field, while the fit represented by the solid line considers a population of higher excited states.
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  • The QND character of the qubit measurement is studied by repeating the measurement. A perfect QND setup guarantees identical outcomes for the two repeated measurement with certainty. In order to fully characterize the properties of the measurement, we can initialize the qubit in the state | 0 , then rotate the qubit by applying a pulse of duration τ 1 before the first measurement and a second pulse of duration τ 2 between the first and the second measurement. The conditional probability to detect the qubit in the states s and s ' is expected to be independent of the first pulse, and to show sinusoidal oscillation with amplitude 1 in τ 2 . Deviations from this expectation witness a deviation from a perfect QND measurement. The sequence of qubit pulses and oscillator driving is depicted in Fig.  Fig1b). The conditional probability P 0 | 0 to detect the qubit in the state "0" twice in sequence is plotted versus τ 1 and τ 2 in Fig.  Fig1c) for Δ = 0 , and in Fig.  Fig1d) for Δ / ϵ = 0.1 . We anticipate here that a dependence on τ 1 is visible when the qubit undergoes a flip in the first rotation. Such a dependence is due to the imperfections of the mapping between the qubit state and the oscillator state, and is present also in the case Δ = 0 . The effect of the non-QND term Δ σ X results in an overall reduction of P 0 | 0 .... (Color online) Conditional probability to obtain a) s ' = s = 1 , b) s ' = - s = 1 , c) s ' = - s = - 1 , and d) s ' = s = - 1 for the case Δ t = Δ / ϵ = 0.1 and T 1 = 10 ~ n s , when rotating the qubit around the y axis before the first measurement for a time τ 1 and between the first and the second measurement for a time τ 2 , starting with the qubit in the state | 0 0 | . Correction in Δ t are up to second order. The harmonic oscillator is driven at resonance with the bare harmonic frequency and a strong driving together with a strong damping of the oscillator are assumed, with f / 2 π = 20 ~ G H z and κ / 2 π = 1.5 ~ G H z . Fig6... In Fig.  Fig5 we plot the second order correction to the probability to obtain "1" having prepared the qubit in the initial state ρ 0 = | 0 0 | , corresponding to F 2 t , for Δ t = Δ / ϵ = 0.1 . We choose to plot only the deviation from the unperturbed probability because we want to highlight the contribution to spin-flip purely due to tunneling in the qubit Hamiltonian. In fact most of the contribution to spin-flip arises from the unperturbed probability, as it is clear from Fig.  Fig3. Around the two qubit-shifted frequencies, the probability has a two-peak structure whose characteristics come entirely from the behavior of the phase ψ around the resonances Δ ω ≈ ± g . We note that the tunneling term can be responsible for a probability correction up to ∼ 4 % around the qubit-shifted frequency.... We now investigate whether it is possible to identify the contribution of different mechanisms that generate deviations from a perfect QND measurement. In Fig.  Fig7 we study separately the effect of qubit relaxation and qubit tunneling on the conditional probability P 0 | 0 . In Fig.  Fig7 a) we set Δ = 0 and T 1 = ∞ . The main feature appearing is a sudden change of the conditional probability P → 1 - P when the qubit is flipped in the first rotation. This is due to imperfection in the mapping between the qubit state and the state of the harmonic oscillator, already at the level of a single measurement. The inclusion of a phenomenological qubit relaxation time T 1 = 2 ~ n s , intentionally chosen very short, yields a strong damping of the oscillation along τ 2 and washes out the response change when the qubit is flipped during the first rotation. This is shown in Fig.  Fig7 b). The manifestation of the non-QND term comes as a global reduction of the visibility of the oscillations, as clearly shown in Fig.  Fig7 c).... (Color online) Comparison of the deviations from QND behavior originating from different mechanisms. Conditional probability P 0 | 0 versus qubit driving time τ 1 and τ 2 starting with the qubit in the state | 0 0 | , for a) Δ = 0 and T 1 = ∞ , b) Δ = 0 and T 1 = 2 ~ n s , and c) Δ = 0.1 ~ ϵ and T 1 = ∞ . The oscillator driving amplitude is f / 2 π = 20 ~ G H z and a damping rate κ / 2 π = 1.5 ~ G H z is assumed. Fig7... For driving at resonance with the bare harmonic oscillator frequency ω h o , the state of the qubit is encoded in the phase of the signal, with φ 1 = - φ 0 , and the amplitude of the signal is actually reduced, as also shown in Fig.  Fig3 for Δ ω = 0 . When matching one of the two frequencies ω i the qubit state is encoded in the amplitude of the signal, as also clearly shown in Fig.  Fig3 for Δ ω = ± g . Driving away from resonance can give rise to significant deviation from 0 and 1 to the outcome probability, therefore resulting in an imprecise mapping between qubit state and measurement outcomes and a weak qubit measurement.... (Color online) Schematic description of the single measurement procedure. In the bottom panel the coherent states | α 0 and | α 1 , associated with the qubit states | 0 and | 1 , are represented for illustrative purposes by a contour line in the phase space at HWHM of their Wigner distributions, defined as W α α * = 2 / π 2 exp 2 | α | 2 ∫ d β - β | ρ | β exp β α * - β * α . The corresponding Gaussian probability distributions of width σ centered about the qubit-dependent "position" x s are shown in the top panel. Fig2... The combined effect of the quantum fluctuations of the oscillator together with the tunneling between the qubit states is therefore responsible for deviation from a perfect QND behavior, although a major role is played, as expected, by the non-QND tunneling term. Such a conclusion pertains to a model in which the qubit QND measurement is studied in the regime of strong projective qubit measurement and qubit relaxation is taken into account only phenomenologically. We compared the conditional probabilities plotted in Fig.  Fig6 and Fig.  Fig7 directly to Fig. 4 in Ref. [... (Color online) a) Schematics of the 4-Josephson junction superconducting flux qubit surrounded by a SQUID. b) Measurement scheme: b1) two short pulses at frequency ϵ 2 + Δ 2 , before and between two measurements prepare the qubit in a generic state. Here, ϵ and Δ represent the energy difference and the tunneling amplitude between the two qubit states. b2) Two pulses of amplitude f and duration τ 1 = τ 2 = 0.1 ~ n s drive the harmonic oscillator to a qubit-dependent state. c) Perfect QND: conditional probability P 0 | 0 for Δ = 0 to detect the qubit in the state "0" vs driving time τ 1 and τ 2 , at Rabi frequency of 1 ~ G H z . The oscillator driving amplitude is chosen to be f / 2 π = 50 ~ G H z and the damping rate κ / 2 π = 1 ~ G H z . d) Conditional probability P 0 | 0 for Δ / ϵ = 0.1 , f / 2 π = 20 ~ G H z , κ / 2 π = 1.5 ~ G H z . A phenomenological qubit relaxation time T 1 = 10 ~ n s is assumed. Fig1
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  • fig:plots5 Analysis of echo data for extraction of the decoherence time T 2 . (a–c) Transconductance G L as a function of the base level of detuning ε and δ t (defined in the main text) for total free evolution times of t = 390  ps, t = 690  ps, and t = 990  ps, respectively. (d–f) Fourier transforms of the charge occupation P 1 2 as a function of detuning ε and oscillation frequency f for the data in (a–c), respectively. We obtain P 1 2 (not shown here) by integrating the transconductance data in (a–c) and normalizing by noting that the total charge transferred across the polarization line is one electron. Fast Fourier transforming the time-domain data of P 1 2 allows us to quantify the amplitude of the oscillations visible near δ t = 0 . The oscillations of interest appear as weight in the FFT that moves to higher frequency at more negative detuning (farther from the anti-crossing). For an individual detuning energy, the FFT has nonzero weight for a nonzero bandwidth. (g) Echo amplitude as a function of free evolution time t . The data points (dark circles) are obtained at ε = - 120   μ eV by integrating a horizontal line cut of the FFT data over a bandwidth range of 46 - 72  GHz, then normalizing by the echo oscillation amplitude of the first data point, as described in the supplemental text. The echo oscillation amplitudes, plotted for multiple free evolution times, decay with characteristic time T 2 as the free evolution time t is made longer. By fitting the decay to a Gaussian, we obtain T 2 = 760 ± 190  ps. (h–j) Fourier transforms of the transconductance G L as a function of ε and oscillation frequency f for (a–c), respectively. As t is increased, the magnitude for oscillations at a given frequency decays with characteristic time T 2 . We take the magnitude of the FFT at the point where the central feature (black line) intersects 65  GHz. (k) Measured FFT magnitudes at 65 GHz for multiple free evolution times (dark circles) with a Gaussian fit (red line), which yields T 2 = 620 ± 140  ps, in reasonable agreement with the result shown in (g).
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  • (Color online) Upper panel: Adiabatic energies during a LZ sweep of a qubit coupled to two oscillators. Parameters: γ = 0.25 ℏ v and Ω 2 = 100 ℏ v , both as in Fig.  fig:energylandscape; ℏ Ω 1 = 80 ℏ v . Lower panel: Probability P ↑ → ↑ t that the system stays in the initial state | ↑ 0 0 (solid), and corresponding exact survival final survival probability P ↑ → ↑ ∞ of Eq. ( centralresulttwoosc) (dotted).... cond1 this implies that an integral is non-vanishing only if the non-zero component of λ 2 ℓ - 1 is + 1 while the same component of λ 2 ℓ equals -1 . In other words, we obtain the selection rule that to the occupation probability at t = ∞ only those processes contribute in which the oscillator jumps (repeatedly) from the state | 0 to any state with a single photon (i.e. to b j | 0 ) and back; see Fig.  fig:perturbation. It follows that the oscillators not only start but also end in their ground state | 0 if the final qubit state is | ↑ . We call this dynamical selection rule the “no-go-up theorem” (see also ).... sec:largedetuning If the resonance energies of the cavities differ by much more than the qubit-oscillator coupling, then the dynamics can very well be approximated by two independent standard Landau-Zener transitions, see Figure fig:largedetuning.... (Color online) Upper panel: Adiabatic energies during a LZ sweep of a qubit coupled to two oscillators. Parameters: γ = 0.25 ℏ v , ℏ Ω 1 = 90 ℏ v and Ω 2 = 100 ℏ v . Viewed on this scale of oscillator energies, the differences between exact and avoided level crossings are invisible. Lower panel: for the same parameters, probability P ↑ → ↑ t that the system stays in the initial state | ↑ 0 0 (solid), and corresponding exact survival final survival probability P ↑ → ↑ ∞ of Eq. ( centralresulttwoosc) (dotted).... (Color online) Upper panel: Adiabatic energies during a LZ sweep of a qubit coupled to two oscillators with large energies, and with detunings of the order of the qubit-oscillator coupling γ . Parameters: γ = 0.25 ℏ v and ℏ Ω 2 = 100 ℏ v , as before; ℏ Ω 1 = 96 ℏ v . Lower panel: Probability P ↑ → ↑ t that the system stays in the initial state | ↑ 0 0 (solid), and corresponding exact survival final survival probability P ↑ → ↑ ∞ of Eq. ( centralresulttwoosc) (dotted).... In the following we are interested in the properties of the final qubit-two-oscillator state | ψ ∞ rather than merely the transition probability P ↑ ↓ ∞ of the qubit. In general not much can be said about this final state, but let us now make the realistic assumption ℏ Ω 1 , 2 ≫ γ : both oscillator energies ℏ Ω 1 , 2 are much larger than the qubit-oscillator couplings γ 1 = γ 2 = γ . Still, the frequency detuning δ ω = Ω 2 - Ω 1 may be larger or smaller than γ / ℏ . The adiabatic energies in this case are sketched in Fig.  fig:energylandscape.... (Color online) LZ dynamics of a qubit coupled to one oscillator, far outside the RWA regime: γ = ℏ Ω = 0.25 ℏ v . The red solid curve is the survival probability P ↑ ↑ t when starting in the initial state | ↑ 0 . The dotted black line is the exact survival probability P ↑ → ↑ ∞ based on Eq. ( centralresult2). The dashed purple curve depicts the average photon number in the oscillator if the qubit would be measured in state | ↓ ; the dash-dotted blue curve at the bottom shows the analogous average photon number in case the qubit would be measured | ↑ . fig:photon_averages... While P ↑ ↓ ∞ is determined by the ratio γ 2 / ℏ v , the coefficients c 2 n + 1 depend also on the oscillator frequency. In Fig.  fig:photon_averages we depict how for a small frequency (very small: equal to the coupling strength!) the average photon numbers in the oscillator depend on the state of the qubit.... (Color online) Upper panel: Adiabatic energies during a LZ sweep of a qubit coupled to two oscillators with degenerate energies. Parameters: γ = 0.25 ℏ v and ℏ Ω 2 = 100 ℏ v , as before; this time ℏ Ω 1 = ℏ Ω 2 . Lower panel: Probability P ↑ → ↑ t that the system stays in the initial state | ↑ 0 0 (solid), and corresponding exact survival final survival probability P ↑ → ↑ ∞ of Eq. ( centralresulttwoosc) (dotted).... (Color online) Sketch of adiabatic eigenstates during LZ sweep of a qubit that is coupled to one oscillator. Starting in the ground state | ↑ 0 and by choosing a slow LZ sweep, a single photon can be created in the oscillator. Due to cavity decay, the 1-photon state will decay to a zero-photon state. Then the reverse LZ sweep creates another single photon that eventually decays to the initial state | ↑ 0 . This is a cycle to create single photons that can be repeated.
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  • tau0 The tangle as a function of α in the symmetric case W = 0 for different values of the qubit tunnelling amplitude D . One can appreciate that the result of Eq. ( tangl) is indeed reached asymptotically.... Concerning the asymmetric case, our results for the ground state entanglement appear similar to those found by Costi and McKenzie in Ref. , where the interaction of a qubit with an ohmic environment was numerically analyzed. It turns out that, for a bath with finite band-width, the entanglement displays a behavior analogous to that reported in Figs. ( tau10)-( tau01), when plotted with respect to the value of the impedance of the bath. Here, instead, we concentrated on the dependence of the tangle on the coupling strength between the qubit and the environmental oscillator. Unfortunately, the coupling strength is not easily related to the coefficient of the spectral density used in Ref. , and therefore one cannot make a precise comparison between th... pot The lower adiabatic potential for D = 10 and α = 2 . The dashed line refers to the symmetric, W = 0 , case (dashed line), while the solid line refers to W = 1 . The case of frozen qubit ( W = D = 0 ) would have given a pair of independent parabolas instead of the adiabatic potentials U l , u of Eq. ( udq).... As we have shown, the procedure is easily extended to the asymmetric case and this is important since the entanglement changes dramatically for any finite (however small) value of the asymmetry in the qubit Hamiltonian. As mentioned in section sect2 above, this is due to the fact the this term modifies the symmetry properties of the Hamiltonian, so that the form of the ground state changes radically and the same occurs to the reduced qubit state. For example, for a large enough interaction strength, the qubit state is a complete mixture if W = 0 , while it becomes the lower eigenstate of σ z if W 0 . As a result, for large α , there is much entanglement if W = 0 , while the state of the system is factorized and thus τ = 0 if W 0 . This is seen explicitly in Fig. ( tau10). Furthermore, from the comparison of Figs. ( tau10), ( tau01), and ( tau0), one can see that, with increasing α , the tangle increases monotonically in the symmetric case, while it reaches a maximum before going down to zero if W 0 . This is due to the fact that, in the first case, the ground state of the system becomes a Schrödinger cat-like entangled superposition, approximately given by — 12 { — + —- - — - —+ }   ,   for 1   , schroca where | φ ± are the two coherent states for the oscillator defined in Eq. ( due1), centered in Q = ± Q 0 , respectively, and almost orthogonal if α ≫ 1 . In the presence of asymmetry, on the other hand, the oscillator localizes in one of the wells of its effective potential and this implies that, for large α , the ground state is given by just one of the two components superposed in Eq. ( schroca). This is, clearly, a factorized state and therefore one gets τ = 0 . Since τ is zero for uncoupled sub-systems (i.e., for very small values of α ), weather W = 0 or not, and since, for W 0 , it has to decay to zero for large α , it follows that a maximum is present in between. In fact, for intermediate values of the coupling, there is a competition between the α -dependences of the two non zero components of the Bloch vector. In particular, the length | b → | is approximately equal to one for both small and large α ’s, see Figs. ( asx)-( asz), but the vector points in the x direction for α ≪ 1 and in the z direction for α ≫ 1 . The maximum of the tangle in the asymmetric case occurs near the point in which b x ≈ b z . For the symmetric case, we were also able to derive analytically the sharp increase of the entanglement at α = 1 . This behavior appears to be reminiscent of the super-radiant transition in the many qubit Dicke model, which, in the adiabatic limit, shows exactly the same features described here, and which can be described along similar lines. Finally, we would like to comment on the relationship of this work with those of Refs. and . The approach proposed by Levine and Muthukumar, Ref. , employs an instanton description for the effective action. This has been applied to obtain the entropy of entanglement in the symmetric case, in the same critical limit described above. It turns out that this description is equivalent to a fourth order expansion of the lower adiabatic potential U l . This approximation, although retaining all the distinctive qualitative features discussed above, gives slight quantitative changes in the results. Concerning the asymmetric case, our results for the ground state entanglement appear similar to those found by Costi and McKenzie in Ref. , where the interaction of a qubit with an ohmic environment was numerically analyzed. It turns out that, for a bath with finite band-width, the entanglement displays a behavior analogous to that reported in Figs. ( tau10)-( tau01), when plotted with respect to the value of the impedance of the bath. Here, instead, we concentrated on the dependence of the tangle on the coupling strength between the qubit and the environmental oscillator. Unfortunately, the coupling strength is not easily related to the coefficient of the spectral density used in Ref. , and therefore one cannot make a precise comparison between the two results. At least qualitatively, however, we can say that the ground state quantum correlations induced by the coupling with an ohmic environment are already present when the qubit is coupled to a single oscillator mode. 99 weiss U. Weiss, Quantum Dissipative Systems, 2 nd ed., World Scientific 1999. yuma see, e.g., Yu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). levine G. Levine and V. N. Muthukumar, Phys. Rev. B 69, 113203 (2004). martinis R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and J. M. Martinis, Phys. Rev. Lett. 93 077003 (2005); P. R. Johnson, W. T. Parsons, F. W. Strauch, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, Phys. Rev. Lett. 94, 187004 (2005). pino E. Paladino, L. Faoro, G. Falci, and R. Fazio, Phys. Rev. Lett. 88, 228304 (2002); G. Falci, A. D’Arrigo, A. Mastellone, and E. Paladino, Phys. Rev. Lett. 94, 167002 (2005) hines A.P. Hines, C.M. Dawson, R.H. McKenzie and G.J. Milburn, Phys. Rev. A 70, 022303 (2004). blais A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004); A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M.H. Devoret, S. M. Girvin and R. J. Schoelkopf, Phys. Rev. Lett. 95, 060501 (2005). prb03 F. Plastina and G. Falci, Phys. Rev. B 67, 224514 (2003). costi T.A. Costi and R.H. McKenzie, Phys. Rev. A 68, 034301 (2003). ent1 A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002); T. J. Osborne, and M. A. Nielsen Phys. Rev. A 66, 032110 (2002). ent2 G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003); L. A. Wu, M. S. Sarandy, and D. A. Lidar, Phys. Rev. Lett. 93, 250404 (2004). ent3 T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and V. Tognetti, Phys. Rev. Lett. 94, 147208 (2005). ent4 N. Lambert, C. Emary, and T. Brandes, Phys. Rev. Lett. 92, 073602 (2004). crisp M.D. Crisp, Phys. Rev. A 46, 4138 (1992). Irish E.K. Irish, J. Gea-Banacloche, I. Martin, and K. C. Schwab, Phys. Rev. B 72, 195410 (2005). Rungta V. Coffman, J. Kundu, and W.K. Wootters, Phys. Rev. A 61, 052306 (2000); T. J. Osborne, Phys. Rev. A 72, 022309 (2005), see also quant-ph/0203087. Wallraff A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M.H. Devoret, S. M. Girvin and R. J. Schoelkopf, Phys. Rev. Lett. 95, 060501 (2005). Nakamura Y. Nakamura, Yu.A. Pashkin and J.S. Tsai, Phys. Rev. Lett. 87, 246601 (2001). armour A.D. Armour, M.P. Blencowe and K.C. Schwab, Phys. Rev. Lett. 88, 148301 (2002). Grajcar M. Grajcar, A. Izmalkov and E. Ilxichev, Phys. Rev. B 71, 144501 (2005). Chiorescu I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C.J.P.M. Harmans and J.E. Mooij, Nature 431, 159 (2004).... wf Normalized ground state wave function for the oscillator in the lower adiabatic potential, for D = 10 and α = 2 and with W = 0 (dashed line) and W = 0.1 (solid line).
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  • In practice, the cavity frequency Ω and the qubit-oscillator coupling γ are determined by the design of the setup, while the Josephson energy can be switched at a controllable velocity v — ideally from E J = - ∞ to E J = ∞ . In reality, however, E J is bounded by E J , m a x which is determined by the critical current. The condition E J , m a x > ℏ Ω is required so that the qubit comes into resonance with the oscillator sometime during the sweep. Moreover, inverting the flux through the superconducting loop requires a finite time 2 T m i n , so that v cannot exceed v m a x = E J , m a x / 2 T m i n . In order to study under which conditions the finite initial and final times can be replaced by ± ∞ , we have numerically integrated the Schrödinger equation in a finite time interval - T T . Results are presented in Fig.  fig:P_single.... Hint. The time evolution of the probability that the qubit is in state | ↓ is depicted in Fig.  fig:one-osc. It demonstrates that at intermediate times, the dynamics depends strongly on the oscillator frequency Ω , despite the fact that this is not the case for long times. For a large oscillator frequency, P ↑ ↓ t resembles the standard LZ transition with a time shift ℏ Ω / v .... Population dynamics of individual qubit-oscillator states for a coupling strength γ = 0.6 ℏ v and oscillator frequency Ω = 0.5 v / ℏ .... Hint correlates every creation or annihilation of a photon with a qubit flip, the resulting dynamics is restricted to the states | ↑ , 2 n and | ↓ , 2 n + 1 . Figure  fig:updown reveals that the latter states survive for long times, while of the former states only | ↑ , 0 stays occupied, as it follows from the relation that A n ∝ δ n , 0 , derived above. Thus, the final state exhibits a peculiar type of entanglement between the qubit and the oscillator, and can be written as... Landau-Zener dynamics for the coupling strength γ = 0.6 ℏ v for various cavity frequencies Ω . The dashed line marks the Ω -independent, final probability centralresult to which all curves converge.... centralresult. Thus we find that finite-time effects do not play a role as long as γ ≪ ℏ Ω . Our predicted transition probabilities based on analytical results for infinite propagation time are therefore useful to describe the finite-time LZ sweeps. Figure  fig:P_single also illustrates that the probability for single-photon production is highest in the adiabatic regime ℏ v / γ 2 ≪ 1 . Here the typical duration of a LZ transition is 2 γ / v . So in the regime of interest, the sought condition for a “practically infinite time interval” is v T = E J , m a x > ℏ Ω + 2 γ . For the unrealistically large qubit-oscillator coupling γ / ℏ Ω = 0.5 , reliable single-photon generation is less probable. This is so because (i) the LZ transition is incomplete within - T T ; (ii) more than two oscillator levels take part in the dynamics and more than one photon can be generated, as depicted in Fig.  fig:updown; and (iii) the approximation of the instantaneous ground state at t = - T by | ↑ , 0 is less accurate.... Probability of single-photon generation P | ↓ , 1 as a function of ℏ v / γ 2 , for LZ sweeps within the finite time interval - T T with T > T m i n chosen such that v T = 3 ℏ Ω / 2 . The initial state is | ↑ , 0 . Shown probabilities are averaged within the time interval 29 20 ℏ Ω / v and 3 2 ℏ Ω / v , whereby the small and fast oscillations that are typical for the tail of a LZ transition are averaged out.
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  • In conclusion, we have observed resonant tunneling in a macroscopic superconducting system, containing an Al flux qubit and a Nb tank circuit. The latter played dual control and readout roles. The impedance readout technique allows direct characterization of some of the qubit’s quantum properties, without using spectroscopy. In a range 50 ∼ 200 mK, the effective qubit temperature has been verified [Fig.  fig:Temp_dep(b)] to be the same as the mixing chamber’s (after Δ has been determined at low  T ), which is often difficult to confirm independently.... Our technique is similar to rf-SQUID readout. The qubit loop is inductively coupled to a parallel resonant tank circuit [Fig.  fig:schem(b)]. The tank is fed a monochromatic rf signal at its resonant frequency  ω T . Then both amplitude v and phase shift χ (with respect to the bias current  I b ) of the tank voltage will strongly depend on (A) the shift in resonant frequency due to the change of the effective qubit inductance by the tank flux, and (B) losses caused by field-induced transitions between the two qubit states. Thus, the tank both applies the probing field to the qubit, and detects its response.... (a) Tank phase shift vs flux bias near degeneracy and for V d r = 0.5 ~ μ V. From the lower to the upper curve (at f x = 0 ) the temperature is 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 350, 400 mK. (b) Normalized amplitude of tan χ (circles) and tanh Δ / k B T (line), for the Δ following from Fig.  fig:Bias_dep; the overall scale κ is a fitting parameter. The data indicate a saturation of the effective qubit temperature at 30 mK. (c) Full dip width at half the maximum amplitude vs temperature. The horizontal line fits the low- T ( < 200  mK) part to a constant; the sloped line represents the T 3 behavior observed empirically for higher  T .... (a) Quantum energy levels of the qubit vs external flux. The dashed lines represent the classical potential minima. (b) Phase qubit coupled to a tank circuit.... -dependence of ϵ t is adiabatic. However, it does remain valid if the full (Liouville) evolution operator of the qubit would contain standard Bloch-type relaxation and dephasing terms (which indeed are not probed) in addition to the Hamiltonian dynamics ( eq01), since the fluctuation–dissipation theorem guarantees that such terms do not affect equilibrium properties. Normalized dip amplitudes are shown vs T in Fig.  fig:Temp_dep(b) together with tanh Δ / k B T , for Δ / h = 650  MHz independently obtained above from the low- T width. The good agreement strongly supports our interpretation, and is consistent with Δ being T... Δ is the tunneling amplitude. At bias ϵ = 0 the two lowest energy levels of the qubit anticross [Fig.  fig:schem(a)], with a gap of  2 Δ . Increasing ϵ slowly enough, the qubit can adiabatically transform from Ψ l to Ψ r , staying in the ground state  E - . Since d E - / d Φ x is the persistent loop current, the curvature d 2 E - / d Φ x 2 is related to the qubit’s susceptibility. Hence, near degeneracy the latter will have a peak, with a width given by | ϵ | Δ . We present data demonstrating such behavior in an Al 3JJ qubit.
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  • (Color online) Upper panel: adiabatic energies during a LZ sweep of a qubit coupled to two oscillators. Parameters: γ=0.25ℏv and Ω2=100ℏv, both as in Fig. 4; ℏΩ1=80ℏv. Lower panel: probability P↑→↑(t) that the system stays in the initial state |↑00〉 (solid), and corresponding exact survival final survival probability P↑→↑(∞) of Eq. (20) (dotted). ... (Color online) LZ dynamics of a qubit coupled to one oscillator, far outside the RWA regime: γ=ℏΩ=0.25ℏv. The red solid curve is the survival probability P↑→↑(t) when starting in the initial state |↑0〉. The dotted black line is the exact survival probability P↑→↑(∞) based on Eq. (16). The dashed purple curve depicts the average photon number in the oscillator if the qubit would be measured in state |↓〉; the dash-dotted blue curve at the bottom shows the analogous average photon number in case the qubit would be measured |↑〉. ... (Color online) Upper panel: adiabatic energies during a LZ sweep of a qubit coupled to two oscillators with large energies, and with detunings of the order of the qubitoscillator coupling γ. Parameters: γ=0.25ℏv and ℏΩ2=100ℏv, as before; ℏΩ1=96ℏv. Lower panel: probability P↑→↑(t) that the system stays in the initial state |↑00〉 (solid), and corresponding exact survival final survival probability P↑→↑(∞) of Eq. (20) (dotted). ... (Color online) Upper panel: adiabatic energies during a LZ sweep of a qubit coupled to two oscillators. Parameters: γ=0.25ℏv, ℏΩ1=90ℏv, and Ω2=100ℏv. Viewed on this scale of oscillator energies, the differences between exact and avoided level crossings are invisible. Lower panel: for the same parameters, probability P↑→↑(t) that the system stays in the initial state |↑00〉 (solid), and corresponding exact survival final survival probability P↑→↑(∞) of Eq. (20) (dotted). ... (Color online) Sketch of adiabatic eigenstates during LZ sweep of a qubit that is coupled to one oscillator. Starting in the ground state |↑0〉 and by choosing a slow LZ sweep, a single photon can be created in the oscillator. Due to cavity decay, the one-photon state will decay to a zero-photon state. Then the reverse LZ sweep creates another single photon that eventually decays to the initial state |↑0〉. This is a cycle to create single photons that can be repeated.
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