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- The ac Stark shift Δ ω 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. The inset shows the
**oscillation****frequency**Ω ̄ R , 01 as a function of the level spacing ω 01 for I a c = 5.87 nA and the fit using ( rabif) to obtain Ω R , 01 and Δ ω 01 .... Experimental microwave spectroscopy of a Josephson phase**qubit**, scanned in**frequency**(vertical) and bias current (horizontal). Dark points indicate experimental microwave enhancement of the tunneling escape rate, while white dashed lines are quantum mechanical calculations of (from right to left) ω 01 , ω 02 / 2 , ω 12 , ω 13 / 2 , and ω 23 .... Rabi**frequency**Ω R , 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed lines are the lowest-order results ( rabi1) (top) and second-order ( rabi2) (bottom) perturbative results. The inset shows Rabi**oscillations**of the escape rate for I a c = 16.5 nA.... Rabi**frequency**Ω R , 02 of the two-photon 0 2 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. Inset shows Rabi**oscillations**of the escape rate for I a c = 16.5 nA.Data Types:- Image

- (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.Data Types:- Image

- (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.Data Types:- Image

- (color) Numerically optimized free-evolution period T for some of clock protocols considered here, when the
**oscillator**noise has an Allan deviation of 1 Hz.... (color) Probability (P) of measuring each basis state as a function of the atom-**oscillator**phase difference ( φ ). Shown are the various protocols for two and five atoms. Each differently colored curve corresponds to a basis state that ψ 1 is projected onto after free evolution. Vertical text near the curves’ peaks indicates the optimized phase estimate ( φ E s t ). In the simulations, the**frequency**corrections are φ E s t / 2 π T . Shaded in the background is the Gaussian distribution whose variance φ 2 represents the atom-**oscillator**phase differences that occur in the simulation. Also listed is the optimized probe period T , squeezing parameter κ where applicable, and long-term**frequency**variance of the clock extrapolated to 1 second. For long-term averages of n seconds, the variance is f 2 / n .... Numerical simulations of the clock protocols considered here are summarized in Figure figPerf. Ramsey’s protocol defines the standard quantum limit (SQL), and it is evident that entangled states of two or more**qubits**can reduce clock instability, although the GHZ states yield no gain for the noise model considered here, as has been noted previously . The spin-squeezed states suggested by André et al. yield the best performance for 3 to 15**qubits**, and improve upon the SQL variance by a factor of N -1 / 3 . For more**qubits**, the protocol of Bu... (color online) Long-term statistical variance of entangled clocks that contain different numbers of**qubits**, compared to the standard quantum limit (SQL). The most stable clocks found by the large-scale search are shown as black points. Each point is based on several hours of runtime on NISTxs computing cluster, where typically 2000 processor cores were utilized in parallel. Also shown is the simulated performance of analytically optimized clock protocols. Approximately 15**qubits**are required to improve upon the SQL by a factor of two.Data Types:- Image

- 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).Data Types:- Image

- (Color) Simulated energy for (a) the first
**qubit**, (b) the CPW cavity, (c) the second**qubit**. (Red): ϕ 1 = 0.8949 ϕ c 1 and ϕ 2 = 0.893 ϕ c 2 . The first**qubit**decays exponentially up to t ≈ 123 ~ n s . At this time the**frequency**of the**oscillation**in right well matches the CPW cavity resonant**frequency**and the**qubit**transfers part of its energy to the CPW cavity. The second**qubit**is resonating at a different**frequency**and it is minimally exited by the incoming microwave voltage. This corresponds to the red x of Fig. 2(b) (Black): ϕ 1 = 0.82 ϕ c 1 and ϕ 2 = 0.836 ϕ c 2 . In this case the first**qubit**transfers part of its energy to the CPW cavity at t ≈ 103 ~ n s because it starts at a lower energy in the deep well. At this flux the second**qubit**is in resonance with the cavity and it is excited up to the sixth quantized level. This corresponds to the white x of Fig. 2(b) fig:singlelinecut... For our experiment, we initially determine the optimal ’simultaneous’ timing between the two MPs that takes into account the different cabling and instrumental delays from the room-temperature equipment to the cold devices. Then, as a function of the flux applied to the two**qubits**, we measure the tunneling probability for the second (first)**qubit**after we purposely induce a tunneling event in the first (second)**qubit**. The results are shown in Fig. fig:experiment(a,c). The probability of finding the second (first)**qubit**in the excited state as a result of measurement crosstalk is significant only in a region around ϕ 2 / ϕ c 2 = Φ 2 / Φ c 2 ≈ Φ ¯ 2 / Φ c 2 ∼ 0.842 ( ϕ 1 / ϕ c 1 = Φ 1 / Φ c 1 ≈ Φ ¯ 1 / Φ c 1 ∼ 0.82 ) where the resonant**frequency**of the second (first)**qubit**is close to the CPW cavity**frequency**.... AltomareX2009, the resonant**frequency**of both**qubits**exhibits an avoided crossing at the CPW cavity**frequency**( ≈ 8.9 GHz). For the first**qubit**this happens at a flux Φ ¯ 1 = 0.82 Φ c 1 , and for the second at a flux Φ ¯ 2 = 0.842 Φ c 2 . For each**qubit**, Φ c i is the critical flux at which the left well of Fig. fig:QBpotential(b) disappears.... (Color) Measurement crosstalk: (a) Experimental tunneling probability for**qubit**2, after**qubit**1 has already tunneled as function of the (dimensionless) flux applied to the**qubits**. The left ordinate displays the resonant**frequency**as measured from the**qubit**spectroscopy. The right ordinate displays the ratio between the applied flux and the critical flux for**qubit**2. (b) Simulation: ratio between the maximum energy acquired by the second**qubit**and the resonant**frequency**in the left well ( N l ) as a function of the flux applied to the**qubits**. The left ordinate displays the**oscillation****frequency**as determined from the Fast Fourier Transform of the energy of**qubit**2. The right ordinate displays the ratio between the applied flux and the critical flux for**qubit**1. Temporal traces corresponding to the two x’s are displayed in Fig. fig:singlelinecut. (c-d) Same as (a-b) after reversing the roles of the two**qubits**. fig:experiment... From these initial conditions the phase of the first**qubit**(classically) undergoes damped**oscillations**in the anharmonic right well. Because of the anharmonicity of the potential, when the amplitude of the**oscillation**is large, the**frequency**of the**oscillations**is lower than the unmeasured**qubit****frequency**. As the system loses energy due to the damping, the**oscillation****frequency**increases as seen by the CPW cavity. When the crosstalk voltage has a**frequency**close to the CPW cavity**frequency**, it can transfer energy to the CPW cavity. If the second**qubit**’s**frequency**matches that of the CPW cavity then the cavity’s excitation can be transferred to the second**qubit**. In Fig. fig:experiment(b) we plot, for the second**qubit**, the ratio ( N l ) between the maximum energy acquired and ℏ ω p , where ω p is the plasma**frequency**of the**qubit**in the left well, as a function of the fluxes in the two**qubits**. The crosstalk, measured as the maximum energy transferred to the second**qubit**, is maximum at a flux ϕ 2 / ϕ c 2 ∼ 0.837 , where the second**qubit**’s**frequency**is ≈ 8.97 ~ G H z , determined by taking the Fast Fourier Transform of the**oscillations**in energy over time (see Fig. fig:singlelinecut (a-c)). Reversing the roles of the two**qubits**, we find that for the first**qubit**the crosstalk is maximum at a flux ∼ 0.825 ϕ c 1 , corresponding to an excitation**frequency**of ≈ 8.84 ~ G H z (Fig. fig:experiment(d)). These values were determined for**qubit**2 (**qubit**1) by performing a Gaussian fit of N l versus flux (or**frequency**) after averaging over the span of flux (or**frequency**) values for**qubit**1 (**qubit**2). Notice that the crosstalk transferred to**qubit**2 (**qubit**1) is flux independent of**qubit**1 (**qubit**2) and substantial only when the cavity**frequency**matches the**frequency**of**qubit**2 (**qubit**1). The results of the simulations are in good agreement with the experimental data. To gain additional insight into the dynamics of the system, we plot the time evolution of the energy for the**qubits**and the CPW cavity (Fig. fig:singlelinecut (a-c)) for two different sets of fluxes in the two**qubits**. At ϕ 1 = 0.895 ϕ c 1 and ϕ 2 = 0.893 ϕ c 2 (red x in Fig. fig:experiment(b)) the first**qubit**decays exponentially for a time t 123 ~ n s (Fig. fig:singlelinecut (a-c)-Red). At t = 123 ~ n s there is a downward jump in the energy of the first**qubit**while the energy of the CPW cavity exhibits an upward jump. At this time, the**frequency**of**oscillation**in the right well matches the CPW cavity resonant**frequency**, so part of the**qubit**energy is transferred to the CPW cavity. However, since the second**qubit**is not on resonance with the CPW cavity, it does not get significantly excited by the microwave current passing through the capacitor C x .... (a) Equivalent electrical circuit for two flux-biased phase**qubits**coupled to a CPW cavity (modelled as a lumped element harmonic**oscillator**). C i is the total i -**qubit**(or CPW cavity) capacitance, L i the geometrical inductance, L j , i the Josephson inductance of the JJ, R i models the dissipation in the system. (b) U ϕ ϕ e is the potential energy of the phase**qubit**as function of superconducting phase difference ϕ across the JJ and the dimensionless external flux bias ϕ e = Φ 2 π / Φ 0 . Δ U ϕ e is the difference between the local potential maximum and the local potential minimum in the left well at the flux bias ϕ e . (c) During the MP, the potential barrier Δ U ϕ e between the two wells is lowered for a few nanoseconds allowing the | 1 state to tunnel into the right well where it will (classically)**oscillate**and lose energy due to the dissipation. fig:QBpotential... At ϕ 1 = 0.82 ϕ c 1 and ϕ 2 = 0.836 ϕ c 2 (white x in Fig. fig:experiment(b)), the dynamics of the first**qubit**and the CPW cavity are essentially unchanged, except that the CPW cavity**frequency**is matched at a different time ( t = 103 ~ n s ) because the first**qubit**starts at a lower energy in the deep well (Fig. fig:singlelinecut (a-c)-Black). However, in this case, the second**qubit**is on resonance with the CPW cavity and is therefore excited to an energy N l ∼ 6 .Data Types:- Image

- Pulse sequence producing (trivial) diagonal gate: during time T 1 ,
**qubit**1 swaps its state onto the**oscillator**, then the**oscillator**interacts with**qubit**2 before swapping its state back onto**qubit**1; free evolution during time T 3 is added to annihilate two-photon state in the cavity.... Protocol for creating a Bell-pair: the cavity**frequency**is sequentially swept through resonances with both**qubits**; at the first resonance the**oscillator**is entangled with**qubit**1, at the next resonance the**oscillator**swaps its state onto**qubit**2 and ends up in the ground state. A Bell measurement is performed by applying Rabi pulses to non-interacting**qubits**, and projecting on the**qubit**eigenbasis, | g | e , by measuring quantum capacitance.... Equivalent circuit for the device in Fig. Sketch: chain of L C -**oscillators**represents the stripline cavity, φ 1 and φ N are superconducting phase values at the ends of the cavity, φ j and φ l are local phase values where the**qubits**are attached; attached dc-SQUID has effective flux-dependent Josephson energy, E J s f , and capacitance C s , control line for tuning the SQUID is shown at the right; SCB**qubits**are coupled to the cavity via small capacitances, C c 1 and C c 2 .... Sketch of the device: charge**qubits**(single Cooper pair boxes, SCB) coupled capacitively ( C c ) to a stripline cavity integrated with a dc-SQUID formed by two large Josephson junctions (JJ); cavity eigenfrequency is controlled by magnetic flux Φ through the SQUID.... Jonn,NewJP: the duration of the gate operation in the latter case is h / 8 in the units of inverse coupling energy, while it is 2.7 h for the protocol presented in Fig. fig_prot_SK. This illustrates the advantage of longitudinal, z z coupling (in the**qubit**eigenbasis), which is achieved for the charge**qubits**biased at the charge degeneracy point by current-current coupling. More common for charge**qubits**is the capacitive coupling, however there the situation is different: this coupling has x x symmetry at the charge degeneracy point, and because of inevitable difference in the**qubit****frequencies**, the gate operation takes much longer time, prolonged by the ratio between the**qubits****frequency**asymmetry and the coupling**frequency**. Recent suggestions to employ dynamic control methods to effectively bring the**qubits**into resonance can speed up the gate operation. For these protocols, the gate duration is ∼ h in units of direct coupling energy, which is longer than in the case of z z coupling, but somewhat shorter than in our case. However, the protocol considered in this paper might be made faster by using pulse shaping.... For a given eigenmode, the integrated stripline + SQUID system behaves as a lumped**oscillator**with variable**frequency**. Our goal in this section will be to derive an effective classical Lagrangian for this**oscillator**. To this end we consider in Fig. 2qubit_circuit an equivalent circuit for the device depicted in Fig. Sketch. A discrete chain of identical L C -**oscillators**, with phases φ i across the chain capacitors (i=1,…,N), represents the stripline cavity; the dc SQUID is directly attached at the right end of the chain, while the superconducting Cooper pair boxes (SCB) are attached via small coupling capacitors, C c 1 and C c 2 to the chain nodes with local phases, φ j and φ l (for simplicity we consider only two attached SCBs). The classical Lagrangian for this circuit,... Gate circuit for constructing a CNOT gate using the control-phase gate: a z-axis rotation is applied to**qubit**1, and Hadamard gates H are applied to the second**qubit**.... In this section we modify the Bell state construction to implementing a control-phase (CPHASE) two-**qubit**gate. This gate has the diagonal form: | α β 0 → exp i φ α β | α β 0 ( φ 00 = φ 01 = φ 10 = 0 , φ 11 = π ), and it is equivalent to the CNOT gate (up to local rotations). To generate such a diagonal gate, we adopt the following strategy: first tune the**oscillator**through resonance with both**qubits**performing π -pulse swaps in every step, and then reverse the sequence, as shown in figure fig_prot_naive. With an even number of swaps at every level, clearly the resulting gate will be diagonal.... The experimental setup with the**qubit**coupling to a distributed**oscillator**- stripline cavity possesses potential for scalability - several**qubits**can be coupled to the cavity. In this paper we investigate the possibility to use this setup for implementation of tunable**qubit**-**qubit**coupling and simple gate operations. Tunable**qubit**-cavity coupling is achieved by varying the cavity**frequency**by controlling magnetic flux through a dc-SQUID attached to the cavity (see Fig. Sketch). An advantage of this method is the possibility to keep the**qubits**at the optimal points with respect to decoherence during the whole two-**qubit**operation. The**qubits**coupled to the cavity must have different**frequencies**, and the cavity in the idle regime must be tuned away from resonance with all of the**qubits**. Selective addressing of a particular**qubit**is achieved by relatively slow passage through the resonance of a selected**qubit**, while other resonances are rapidly passed. The speed of the active resonant passage should be comparable to the**qubit**-cavity coupling**frequency**while the rapid passages should be fast on this scale, but slow on the scale of the cavity eigenfrequency in order to avoid cavity excitation. This strategy requires narrow width of the**qubit**-cavity resonances compared to the differences in the**qubit****frequencies**, determined by the available interval of the cavity**frequency**divided by the number of attached**qubits**. This consideration simultaneously imposes a limit on the maximum number of employed**qubits**. Denoting the difference in the**qubit**energies, Δ E J , the coupling energy, κ , the maximum variation of the cavity**frequency**, Δ ω k , and the number of**qubits**, N , we summarize the above arguments with relations, κ ≪ Δ E J , N ∼ ℏ Δ ω k / Δ E J . In the off-resonance state, the**qubit**-**qubit**coupling strength is smaller than the on-resonance coupling by the ratio, κ / ℏ ω k - E J ≪ 1 .Data Types:- Image

- (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**qubit**–**oscillator**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.Data Types:- Image

- 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. Fig1Data Types:- Image
- Tabular Data

- fig3 Experimental device: a transmon
**qubit**coupled to a nonlinear superconducting cavity. (a) Circuit diagram of the device; the anharmonic resonator is formed from a meander inductor embedded with a Josephson junction and an interdigitated capacitor. The resonator is isolated from the 50 Ω environment by coupling capacitors C c and coupled to a transmon**qubit**characterized by Josephson and charging energy scales E J and E c respectively. The coupling rate is g . (b) Scanning electron micrograph of the the device with the resonator and**qubit**junctions (lower and upper insets).... fig2 Energy levels and the effective nonlinearity λ of the strongly coupled system. (a) The measured coefficient of nonlinear response for a strongly coupled system versus**qubit**-cavity detuning for the**qubit**prepared in the ground state in the low excitation regime. Theory curves for the model Eq. ( eq:1) with N l = 2 (blue), N l = 3 (gray), and for a model of coupled nonlinear classical**oscillators**(red) are shown. The arrows indicate the locations of avoided crossings of the level pairs | 1 , 3 ↔ | 0 , 4 and | 1 , 4 ↔ | 0 , 5 . (Inset) The transmission of the resonator when driven with tone at ω d that occupied the resonator with = 0.4 (black) and = 10 (gray) off-resonant photons. (b) Energy levels of the**qubit**-**oscillator**model with N l = 7 show the avoided crossings in the 4 and 5 excitation manifold. (c) Quantum trajectory simulation of the system exhibits a general trends of increasing effective nonlinearity λ with diminishing**qubit**-cavity detuning ( Δ ) with abrupt reductions associated with avoided crossings in the 4 and 5 excitation manifold. For these simulations, the**qubit**energy levels were modeled as a Duffing nonlinearity.... fig2 Measured autoresonance and threshold sensitivity on the level structure and initial**qubit**state. (a) Color plot shows S | 1 versus**qubit**detuning. The dashed line indicates the AR threshold, V | 1 . AR measurements were not taken for small values of the detuning as indicated by the hatched region. (b) Color plot of S | 0 with V | 0 indicated as a solid black line. The AR threshold, V | 1 , is also plotted for comparison as a dashed black line. The two arrows indicate the location of avoided crossings in the 4 and 5 excitation manifold.... fig3duffing (a) The transmitted magnitude for chirp sequences with drive voltages that were above (black), near (dashed), and below (gray) the AR threshold. (Inset) Pulse sequence: the**qubit**manipulation pulse was applied immediately before the start of the chirp sequence. (b) The average transmitted magnitude near 400 ns versus drive voltage shows S | 0 (black) and S | 1 (red) for Δ / 2 π = 0.59 GHz.Data Types:- Image

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