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  • (a) Equivalent circuit of the flux detector based on the Josephson transmission line (JTL) and (b) diagram of scattering of the fluxon injected into the JTL with momentum k by the potential U x that is controlled by the measured qubit. The fluxons are periodically injected into the JTL by the generator and their scattering characteristics (transmission and reflection coefficients t k , r k ) are registered by the receiver.... Schematics of the QND fluxon measurement of a qubit which suppresses the effect of back-action dephasing on the qubit oscillations. The fluxon injection frequency f is matched to the qubit oscillation frequency Δ : f ≃ Δ / π , so that the individual acts of measurement are done when the qubit density matrix is nearly diagonal in the σ z basis, and the measurement back-action does not introduce dephasing in the oscillation dynamics.
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  • The 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.
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  • The average energy H of the system as a function of the driving frequency ω . The main peak ( ω 0 ≈ 0.6 ) corresponds to the resonance. The left peak at ω 0 / 2 is the nonlinear effect of the excitation by a subharmonic, similar to a multiphoton process in the quantum case. The right peak at 2 ω 0 is the first overtone and it has no quantum counterpart. Here ϕ e d = π ; ϕ e a = 0.05 ; γ = 10 -3 .... The dependence of the pseudo-Rabi frequency on the driving amplitude ϕ e a for ω = 0.6 , γ = 10 -3 . The solid line, Ω = 0.35 ϕ e a 2 + ω - 0.63 2 1 / 2 , is the best fit to the calculated data.... (Color online) The potential profile of Eq. ( eq_potential) with α = 0.8 , ϕ e d = π . The arrows indicate quantum (solid) and classical (dotted) oscillations.... A quantitative difference between this effect and true Rabi oscillations is in the different scale of the resonance frequency. To induce Rabi oscillations between the lowest quantum levels in the potential ( eq_potential), one must apply a signal in resonance with their tunneling splitting, which is exponentially smaller than ω 0 . Still, this is not a very reliable signature of the effect, since the classical effect can also be excited by subharmonics, ∼ ω 0 / n , as we can see in Fig.  fig4.... In the presence of the external field ( eq_external) the system will undergo forced oscillations around one of the equlibria. For α = 0.8  , which is close to the parameters of the actual devices , the values of the dimensionless frequencies become ω θ ≈ 0.612 , and ω χ ≈ 0.791 . Solving the equations of motion ( eq_motion) numerically, we see the appearance of slow oscillations of the amplitude and energy superimposed on the fast forced oscillations (Fig.  fig2), similar to the classical oscillations in a phase qubit (Fig. 2 in  ). The dependence of the frequency of these oscillations on the driving amplitude shows an almost linear behaviour (Fig.  fig3), which justifies the “Pseudo-Rabi” moniker.... The key observable difference between the classical and quantum cases, which would allow to reliably distinguish between them, is that the classical effect can also be produced by driving the system at the overtones of the resonance signal, ∼ n ω 0 (Fig.  fig4). This effect can be detected using a standard technique for RF SQUIDs . The current circulating in the qubit circuit produces a magnetic moment, which is measured by the inductively coupled high-quality tank circuit. For the tank voltage V T we have... where τ T = R T C T is the RC-constant of the tank, ω T = L T C T -1 / 2 its resonant frequency, M the mutual inductance between the tank and the qubit, and I q t the current circulating in the qubit. The persistent current in the 3JJ loop can be determined directly from ( eq_I). Its behaviour in the presence of an external RF field is shown in Fig.  fig2c. Note that the sign of the current does not change, which is due to the fact that the oscillations take place inside one potential well (solid arrow in Fig.  fig1), and not between two separate nearby potential minima like in the quantum case. (Alternatively, this would also allow to distinguish between the classical and quantum effects by measuring the magnetization with a DC SQUID.)... (a) Driven oscillations around a minimum of the potential profile of Fig. fig1 as a function of time. The driving amplitude is ϕ e a = 0.01 , driving frequency ω = 0.612 , and the decay rate γ = 10 -3 . Low-frequency classical beat oscillations are clearly seen. (b) Low-frequency oscillations of the persistent current in the 3JJ loop. (c) Same for the energy of the system.
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  • We fit the escape rates in Fig.  F031806DGtot (and additional data for other powers not shown) to a decaying sinusoid with an offset. The extracted frequencies are shown with circles in Fig.  F031806Dfosc. To compare to theory, Ω R , 0 1 , calculated using the rotating wave solution for a system with five levels, is shown with a solid line. The implied assumption that the oscillation frequencies of Γ and ρ 11 are equal, even at high power in a multilevel system, will be addressed in Sec.  SSummary. In plotting the data, we have introduced a single fitting parameter 117   n A / m W that converts the power P S  at the microwave source to the current amplitude I r f  at the qubit. Good agreement is found over the full range of power.... We next consider the time dependence of the escape rate for the data plotted in Fig.  F032106FN10. Here, a 6.2 GHz microwave pulse nominally 30 ns long was applied on resonance with the 0 → 1  transition of the qubit junction. The measured escape rate shows Rabi oscillations followed by a decay back to the ground state once the microwave drive has turned off. This decay appears to be governed by three time constants. Nontrivial decays have previously been reported in phase qubits and we have found them in several of our devices.... F031806Dfosc Rabi oscillation frequency Ω R , 0 1  at fixed bias as a function of microwave current I r f . Extracted values from data (including the plots in Fig.  F031806DGtot) are shown as circles, while the rotating wave solution is shown for two- (dashed line) and five- (solid) level simulations, calculated using I 01 = 17.930   μ A and C 1 = 4.50   p F with ω r f / 2 π = 6.2   G H z .... As I r f  increases in Fig.  F031806Dfosc, the oscillation frequency is smaller than the expected linear relationship for a two-level system (dashed line). This effect is a hallmark of a multilevel system and has been previously observed in a similar phase qubit. There are two distinct phenomena that affect 0 → 1  Rabi oscillations in such a device. To describe... FDeviceThe dc SQUID phase qubit. (a) The qubit junction J 1 (with critical current I 01 and capacitance C 1 ) is isolated from the current bias leads by an auxiliary junction J 2 (with I 02 and C 2 ) and geometrical inductances L 1 and L 2 . The device is controlled with a current bias I b and a flux current I f which generates flux Φ a through mutual inductance M . Transitions can be induced by a microwave current I r f , which is coupled to J 1 via C r f . (b) When biased appropriately, the dynamics of the phase difference γ 1 across the qubit junction are analogous to those of a ball in a one-dimensional tilted washboard potential U . The metastable state n  differs in energy from m  by ℏ ω n m and tunnels to the voltage state with a rate Γ n . (c) The photograph shows a Nb/AlO x /Nb device. Not seen is an identical SQUID coupled to this device intended for two-qubit experiments; the second SQUID was kept unbiased throughout the course of this work.... F032206MNstats (Color online) The (a) on-resonance Rabi oscillation frequencies Ω R , 0 1 m i n  and Ω R , 0 2 m i n  and (b) resonance frequency shifts Δ ω 0 1 = ω r f - ω 0 1 and Δ ω 0 2 = 2 ω r f - ω 0 2 are plotted as a function of the microwave current, for data taken at 110   m K with a microwave drive of frequency ω r f / 2 π = 6.5   G H z and powers P S = - 23 , - 20 , - 17 , - 15 , - 10   d B m . Values extracted from data for the 0 → 1  ( 0 → 2 ) transition are plotted as open circles (filled squares), while five-level rotating wave solutions for a junction with I 01 = 17.736   μ A and C 1 = 4.49   p F are shown as solid (dashed) lines. In (a), the dotted line is from a simulation of a two-level system.... F031806DGtot Rabi oscillations in the escape rate Γ were induced at I b = 17.746   μ A by switching on a microwave current at t = 0 with a frequency of 6.2 GHz (resonant with the 0 → 1  transition) and source powers P S  between -12 and -32 dBm, as labeled. The measurements were taken at 20 mK. The solid lines are from a five-level density-matrix simulation with I 01 = 17.930   μ A , C 1 = 4.50   p F , T 1 = 17   n s , and T φ = 16   n s .... F010206H1 (Color online) Multiphoton, multilevel Rabi oscillations plotted in the time and frequency domains. (a) The escape rate Γ (measured at 110 mK) is plotted as a function of the time after which a 6.5 GHz, -11 dBm microwave drive was turned on and the current bias I b of the qubit; Γ ranges from 0 (white) to 3 × 10 8   1 / s (black). (b) The normalized power spectral density of the time-domain data from t = 1 to 45 ns is shown with a grayscale plot. The dashed line segments indicate the Rabi frequencies obtained from the rotating wave model for transitions involving (from top to bottom) 1, 2, 3, and 4 photons, evaluated with junction parameters I 01 = 17.828   μ A and C 1 = 4.52   p F , and microwave current I r f = 24.4   n A . Corresponding grayscale plots calculated with a seven-level density-matrix simulation are shown in (c) and (d).... Figure  F010206H1(b) shows that the minimum oscillation frequency Ω R , 0 1 m i n / 2 π = 540   M H z of the first (experimental) band occurs at I b = 17.624   μ A , for which ω 0 1 / 2 π = 6.4   G H z . This again indicates an ac Stark shift of this transition, which we denote by Δ ω 0 1 ≡ ω r f - ω 0 1 ≈ 2 π × 100   M H z . In addition, the higher levels have suppressed the oscillation frequency below the bare Rabi frequency of Ω 0 1 / 2 π = 620   M H z [calculated with Eq. ( eqf)].... For this data set, the level spacing ω 0 1 / 2 π is equal to the microwave frequency ω r f / 2 π = 6.5   G H z at I b = 17.614   μ A . The band with the highest current in Fig.  F010206H1(b) is centered about I b = 17.624   μ A , suggesting that 0 → 1  Rabi oscillations are the dominant process near this bias. For slightly higher or lower I b , the oscillation frequency increases as Ω R , 0 1 ≈ Ω 01 ′ 2 + ω r f - ω 0 1 2 , in agreement with simple two-level Rabi theory, leading to the curved band in the grayscale plot.
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  • 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 .
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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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  • (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 .
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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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  • (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.
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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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