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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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  • (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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  • fig3 (Color online) Emergence of frequency locking for two-mode JT system shown in spectrum of the lowest five eigenvalues depending on the frequency difference Δ . (a) At Δ = 0 Rabi splitting of first energy levels occurs for k = 0.1 / 2 . Interaction between priviledged and disadvantaged mode can be tuned up to Δ = 0.1 in single effective mode. (b) Range of single mode regime extends up to Δ = 0.5 in ultrastrong regime k = 1.0 / 2... fig3 (Color online) Emergence of localization and synchronization transitions in weak, strong and ultrastrong regime. (a) shows snynchronous structure between damped oscillating population imbalance and correlation of priviledged mode in weak coupling k = 0.01 / 2 . (b) priviledged mode becomes synchronous with qubit and delocalization-localization transition occurs in population imbalance in strong coupling regime k = 0.1 / 2 . (c) presents the photon blockade in priviledged mode and fully trapped regime in population imbalance with k = 1.0 / 2 and γ φ = 0.1
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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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  • (Color online) FC driving of a transmon with an external flux. The transmon is modelled using the first four levels of the Hamiltonian given by Eq. ( eqn:duffing), using parameters E J / 2 π = 25 GHz and E C / 2 π = 250 MHz. We also have g g e / 2 π = 100 MHz and ω r / 2 π = 7.8 GHz, which translates to Δ g e / 2 π ≃ 2.1 GHz. a) Frequency of the transition to the first excited state obtained by numerical diagonalization of Eq. ( eqn:duffing). As obtained from Eqs. ( eqn:hamonic:1) to ( eqn:hamonic:4), the major component in the spectrum of ω g e t when shaking the flux away from the flux sweet spot at frequency ω F C also has frequency ω F C . However, when shaking around the sweet spot, the dominant harmonic has frequency 2 ω F C . Furthermore, the mean value of ω g e is shifted by G . b) Rabi frequency of the red sideband transition | 1 ; 0 ↔ | 0 ; 1 . The system is initially in | 1 ; 0 and evolves under the Hamiltonian given by Eq. ( eqn:H:MLS) and a flux drive described by Eq. ( eqn:flux:drive). Full red line: analytical results from Eq. ( eqn:rabi:freq) with m = 1 and φ i = 0.25 . Dotted blue line: m = 2 and φ i = 0 . Black dots and triangles: exact numerical results. c) Geometric shift for φ i = 0.25 (full red line) and 0 (dotted blue line). d) Increase in the Rabi frequency for higher coupling strengths with φ i = 0.25 and Δ φ = 0.075 . e) Behavior of the resonance frequency for the flux drive. As long as the dispersive approximation holds ( g g c r i t / 2 π = 1061 MHz), it remains well approximated by Eq. ( eqn:resonance), as shown by the full red line. The same conclusion holds for the Rabi frequency. fig:transmon... (Color online) Average error with respect to the perfect red sideband process | 1 ; 0 ↔ | 0 ; 1 . A gaussian FC pulse is sent on the first qubit at the red sideband frequency assuming the second qubit is in its ground state. Full red line: average error of the red sideband as given by Eq. ( eqn:FUV:simple) when the second qubit is excited. Blue dashed line: population transfer error 1 - P t , with P t given by Eq. ( eqn:pop:transfer). Black dots: numerical results for the average error. We find the evolution operator after time t p for each eigenstate of the second qubit. The fidelity is extracted by injecting these unitaries in Eq. ( eqn:trace). The qubits are taken to be transmons, which are modelled as 4-level Duffing oscillators (see Section  sec:Duffing) with E J 1 = 25 GHz, E J 2 = 35 GHz, E C 1 = 250 MHz, E C 2 = 300 MHz, yielding ω 01 1 = 5.670 GHz and ω 01 2 = 7.379 GHz, and g 01 1 = 100 MHz. The resonator is modeled as a 5-level truncated harmonic oscillator with frequency ω r = 7.8 GHz. As explained in Section  sec:transmon, the splitting between the first two levels of a transmon is modulated using a time-varying external flux φ . Here, we use gaussian pulses in that flux, as described by Eq. ( eqn:gaussian) with τ = 2 σ , σ = 6.6873 ns, and flux drive amplitude Δ φ = 0.075 φ 0 . The length of the pulse is chosen to maximize the population transfer. fig:FUV... This method is first applied to simulate a R 01 1 pulse by evolving the two-transmon-one-resonator system under the Hamiltonian of Eq. ( eqn:H:MLS), along with the FC drive Hamiltonian for the pulse. The simulation parameters are indicated in Table  tab:sequence. To generate the sideband pulse R 01 1 , the target qubit splitting is modulated at a frequency that lies exactly between the red sideband resonance for the spectator qubit in states | 0 or | 1 , such that the fidelity will be the same for both these spectator qubit states. We calculate the population transfer probability for | 1 ; 0 ↔ | 0 ; 1 after the pulse and find a success rate of 99.2% for both initial states | 1 ; 0 and | 0 ; 1 . This is similar to the prediction from Eq. ( eqn:pop:transfer), which yields 98.7%. The agreement between the full numerics and the simple analytical results is remarkable, especially given that with | δ ± / ϵ n | = 0.23 the small δ ± ≪ ϵ n assumption is not satisfied. Thus, population transfers between the transmon and the resonator are achievable with a good fidelity even in the presence of Stark shift errors coming from the spectator qubit (see Section  sec:SB).... In Fig.  fig:transmonb), the Rabi frequencies predicted by the above formula are compared to numerical simulations using the full Hamiltonian Eq. ( eqn:H:MLS), along with a cosine flux drive. The geometric shifts described by Eq. ( eq:G) are also plotted in Fig.  fig:transmonc), along with numerical results. In both cases, the scaling with respect to Δ φ follows very well the numerical predictions, allowing us to conclude that our simple analytical model accurately synthesizes the physics occurring in the full Hamiltonian. It should be noted that, contrary to intuition, the geometric shift is roughly the same at and away from the sweet spot. This is simply due to the fact that the band curvature does not change much between the two operation points. However, as expected from Eqs. ( eqn:hamonic:1) to ( eqn:hamonic:4), the Rabi frequencies are much larger for the same drive amplitude when the transmon is on average away from its flux sweet spot. In that regime, large Rabi frequencies ∼ 30 -40 MHz can be attained, which is well above dephasing rates in actual circuit QED systems, especially in the 3D cavity . However, the available power that can be sent to the flux line might be limited in the lab, putting an upper bound on achievable rates. Furthermore, at those rates, fast rotating terms such as the ones dropped between Eq. ( eq:eps:n) and ( eq:V) start to play a role, adding spurious oscillations in the Rabi oscillations that reduce the fidelity. These additional oscillations have been seen to be especially large for big relevant ε m ω / Δ ~ j , j + 1 n ratios, i.e. when the qubit spends a significant amount of time close to resonance with the resonator and the dispersive approximation breaks down.... We have also defined ω ' p = 8 E C E J Σ cos φ i , the plasma frequency associated to the operating point φ i . This frequency is illustrated by the black dots for two operating points on Fig.  fig:transmona). In addition, there is a frequency shift G , standing for geometric, that depends on the shape of the transmon energy bands. As is also illustrated on Fig.  fig:transmona), this frequency shift comes from the fact that the relation between ω j , j + 1 and φ is nonlinear, such that the mean value of the transmon frequency during flux modulation is not its value for the mean flux φ i . To fourth order in Δ φ , it is... In words, the infidelity 1 - F U V is minimized when the Rabi frequency that corresponds to the FC drive is large compared to the Stark shift associated to the spectator qubit. The average fidelity corresponding to the gate fidelity Eq. ( eqn:FUV:simple) is illustrated in Fig.  fig:FUV as as a function of S 2 (red line) assuming the second qubit to be in its excited state. We also represent as black dots a numerical estimate of the error coming from the spectator qubit’s Stark shift. The latter is calculated with Eqs. ( eqn:trace) and ( eqn:avg:fid). Numerically solving the system’s Schrödinger equation allows us to extract the unitary evolution operator that corresponds to the applied gate. Taking U to be that evolution operator for the spectator qubit in state | 0 and V the operator in state | 1 , we obtain the error caused by the Stark shift shown in Fig.  fig:FUV. The numerical results closely follow the analytical predictions, even for relatively large dispersive shifts S 2 .... Schemes for two-qubit operations in circuit QED. ϵ is the strength of the drive used in the scheme, if any. ∗ There are no crossings in that gate provided that the qubits have frequencies separated enough that they do not overlap during FC modulations. tab:gates... Amplitude of the gaussian pulse over time. Δ φ ' is such that the areas A + and 2 A - are equal. Then, driving the sideband at its resonance frequency for the geometric shift that corresponds to the flux drive amplitude Δ φ ' allows population inversion. fig:gaussian... (Color online) Sideband transitions for a three-level system coupled to a resonator. Applying an FC drive at frequency Δ i , i + 1 generates a red sideband transitions between states | i + 1 ; n and | i ; n + 1 , where the numbers represent respectively the MLS and resonator states. Similarly, driving at frequency Σ i , i + 1 leads to a blue sideband transition, i.e. | i ; n ↔ | i + 1 ; n + 1 . Transitions between states higher in the Fock space are not shown for reasons of readability. This picture is easily generalized to an arbitrary number of levels. fig:MLS:sidebands... Table  tab:gates summarizes theoretical predictions and experimental results for recent proposals for two-qubit gates in circuit QED. These can be divided in two broad classes. The first includes approaches that rely on anticrossings in the qubit-resonator or qubit-qubit spectrum. They are typically very fast, since their rate is equal to the coupling strength involved in the anticrossing. Couplings can be achieved either through direct capacitive coupling of the qubits with strength J C  , or through the 11-02 anticrossing in the two-transmon spectrum which is mediated by the cavity . The latter technique has been successfully used with large coupling rates J 11 - 02 and Bell-state fidelities of ∼ 94 % . However, since these gates are activated by tuning the qubits in and out of resonance, they have a finite on/off ratio determined by the distance between the relevant spectral lines. Thus, the fact that the gate is never completely turned off will make it very complicated to scale up to large numbers of qubits. Furthermore, adding qubits in the resonator leads to more spectral lines that also reduce scalability. In that situation, turning the gates on and off by tuning qubit transition frequencies in and out of resonance without crossing these additional lines becomes increasingly difficult as qubits are added in the resonator, an effect known as spectral crowding.
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  • Population difference for zero static bias. Further parameters are Δ / Ω = 0.5 , ℏ β Ω = 10 and g / Ω = 1.0 . The adiabatic approximation and VVP are compared to numerical results. The first one only covers the longscale dynamics, while VVP also returns the fast oscillations. With increasing time small differences between numerical results and VVP become more pronounced. Fig::P_e=0_D=0.5_g=1.0... As a first case, we consider in Fig. Fig::PF_e=Sqrt0.5_D=Sqrt0.5_g=1.0 a weakly biased qubit ( ε / Ω = 0.5 ) being at resonance with the oscillator ( Δ b = Ω ). For a coupling strength of g / Ω = 1.0 , we find a good agreement between the numerics and VVP. The adiabatic approximation, however, conveys a slightly different picture: Looking at the time evolution it reveals collapse and rebirth of oscillations after a certain interval. This feature does not survive for the exact dynamics. Like in the unbiased case, the adiabatic approximation gives only the first group of frequencies between the quasidegenerate subspaces, and thus yields a wrong picture of the dynamics. In order to cover the higher frequency groups, we need again to go to higher-order corrections by using VVP. For the derivation of our results we assumed that ε is a multiple of the oscillator frequency Ω , ε = l Ω . In this case we found that the levels E ↓ , j 0 and E ↑ , j + l 0 form a degenerate doublet, which dominates the long-scale dynamics through the dressed oscillations frequency Ω j l . For l being not an integer those doublets cannot be identified unambiguously anymore. For instance, we examine the case ε / Ω = 1.5 in Fig. Fig::PF_e=1.5_D=0.5_g=1.0. Here, it is not clear which levels should be gathered into one subspace: j and j + 1 or j and j + 2 . Both the dressed oscillation frequencies Ω j 1 and Ω j 2 influence the longtime dynamics.... Fourier transform of the population difference in Fig. Fig::P_e=0_D=0.5_g=1.0. The left-hand graph shows the whole frequency range. The lowest frequency peaks originate from transitions between levels of a degenerate subspace and are determined through the dressed oscillation frequency Ω j 0 . Numerical calculations and VVP predict group of peaks located around ν / Ω = 0 , 1.0 , 2.0 , 3.0 . The first group at ν / Ω = 0 is shown in the middle graph. One can identify frequencies Ω 0 0 and Ω 2 0 , which fall together, and Ω 1 0 . The small peak comes from the frequency Ω 3 0 . This first gr... Population difference and Fourier spectrum for a biased qubit ( ε / Ω = 0.5 ) at resonance with the oscillator Δ b = Ω in the ultrastrong coupling regime ( g / Ω = 1.0 ). Concerning the time evolution VVP agrees well with numerical results. Only for long time weak dephasing occurs. The inset in the left-hand figure shows the adiabatic approximation only. It exhibits death and revival of oscillations which are not confirmed by the numerics. For the Fourier spectrum, VVP covers the various frequency peaks, which are gathered into groups like for the unbiased case. The adiabatic approximation only returns the first group. Fig::PF_e=Sqrt0.5_D=Sqrt0.5_g=1.0... respectively. Concerning the population difference, we see a relatively good agreement between the numerical calculation and VVP for short timescales. In particular, VVP also correctly returns the small overlaid oscillations. For longer timescales, the two curves get out of phase. The adiabatic approximation only can reproduce the coarse-grained dynamics. The fast oscillations are completely missed. To understand this better, we turn our attention to the Fourier transform in Fig. Fig::F_e=0_D=0.5_g=1.0. There, we find several groups of frequencies located around ν / Ω = 0 , ν / Ω = 1.0 , ν / Ω = 2.0 and ν / Ω = 3.0 . This can be explained by considering the transition frequencies in more detail. We have from Eq. ( VVEnergies)... with ζ k , j l = 1 8 ε ↓ , k 2 - ε ↓ , j 2 + ε ↑ , j + l 2 - ε ↑ , k + l 2 being the second-order corrections. For zero bias, ε = 0 , the index l vanishes. The term k - j Ω determines to which group of peaks a frequency belongs and Ω j 0 its relative position within this group. The latter has Δ as an upper bound, so that the range over which the peaks are spread within a group increases with Δ . The dynamics is dominated by the peaks belonging to transitions between the same subspace k - j = 0 , while the next group with k - j = 1 yields already faster oscillations. To each group belong theoretically infinite many peaks. However, under the low temperature assumption only those with a small oscillator number play a role. For the used parameter regime, the adiabatic approximation does not take into account the connections between different manifolds. It therefore covers only the first group of peaks with k - j = 0 , providing the long-scale dynamics. For ε = 0 , the dominating frequencies in this first group are given by Ω 0 0 = | Δ e - α / 2 | , Ω 1 0 = | Δ 1 - α e - α / 2 | and Ω 2 0 = | Δ L 2 0 α e - α / 2 | , where Ω 0 0 and Ω 2 0 coincide. A small peak at Ω 3 0 = | Δ L 3 0 α e - α / 2 | can also be seen. Notice that for certain coupling strengths some peaks vanish; like, for example, choosing a coupling strength of g / Ω = 0.5 makes the peak at Ω 1 0 vanish completely, independently of Δ , and the Ω 0 0 and Ω 2 0 peaks split. The JCM yields two oscillation peaks determined by the Rabi splitting and fails completely to give the correct dynamics, see the left-hand graph in Fig. Fig::F_e=0_D=0.5_g=1.0. Now, we proceed to an even stronger coupling, g / Ω = 2.0 , where we also expect the adiabatic approximation to work better. From Fig. Fig::EnergyVSg_e=0_W=1_D=0.5 we noticed that at such a coupling strength the lowest energy levels are degenerate within a subspace. Only for oscillator numbers like j = 3 , we see that a small splitting arises. This splitting becomes larger for higher levels. Thus, only this and higher manifolds can give significant contributions to the long time dynamics; that is, they can yield low frequency peaks. Also the adiabatic approximation is expected to work better for such strong couplings . And indeed by looking at Figs. Fig::P_e=0_D=0.5_g=2.0 and Fig::F_e=0_D=0.5_g=2.0, we notice that both the adiabatic approximation and VVP agree quite well with the numerics. Especially the first group of Fourier peaks in Fig. Fig::F_e=0_D=0.5_g=2.0 is also covered almost correctly by the adiabatic approximation. The first manifolds we can identify with those peaks are the ones with j = 3 and j = 4 . This is a clear indication that even at low temperatures higher oscillator quanta are involved due to the large coupling strength. Also frequencies coming from transitions between the energy levels from neighboring manifolds are shown enlarged in Fig. Fig::F_e=0_D=0.5_g=2.0. The adiabatic approximation and VVP can cover the main structure of the peaks involved there, while the former shows stronger deviations. If we go to higher values Δ / Ω 1 , the peaks in the individual groups become more spread out in frequency space, and for the population difference dephasing already occurs at a shorter timescale. For Δ / Ω = 1 , at least VVP yields still acceptable results in Fourier space but gets fast out of phase for the population difference.... Fourier spectrum of the population difference in Fig. Fig::P_e=0_D=0.5_g=2.0. In the left-hand graph a large frequency range is covered. Peaks are located around ν / Ω = 0 , 1.0 , 2.0 , 3.0 etc. Even the adiabatic approximation exhibits the higher frequencies. The upper right-hand graph shows the first group close to ν / Ω = 0 . The two main peaks come from Ω 3 0 and Ω 4 0 and higher degenerate manifolds. Frequencies from lower manifolds contribute to the peak at zero. The adiabatic approximation and VVP agree well with the numerics. The lower right-hand graph shows the second group of peaks around ν / Ω = 1.0 . This group is also predicted by the adiabatic approximation and VVP, but they do not fully return the detailed structure of the numerics. Interestingly, there is no peak exactly at ν / Ω = 1.0 indicating no nearest-neighbor transition between the low degenerate levels. Fig::F_e=0_D=0.5_g=2.0... Population difference and Fourier spectrum for ε / Ω = 1.5 , Δ / Ω = 0.5 and g / Ω = 1.0 . Van Vleck perturbation theory is confirmed by numerical calculations, while results obtained from the adiabatic approximation deviate strongly. In Fourier space, we find pairs of frequency peaks coming from the two dressed oscillation frequencies Ω j 1 and Ω j 2 . The spacings in between those pairs is about 0.5 Ω . The adiabatic approximation only returns one of those dressed frequencies in the first pair. Fig::PF_e=1.5_D=0.5_g=1.0... Fourier transform of the population difference in Fig. Fig::P_e=0_D=0.5_g=1.0. The left-hand graph shows the whole frequency range. The lowest frequency peaks originate from transitions between levels of a degenerate subspace and are determined through the dressed oscillation frequency Ω j 0 . Numerical calculations and VVP predict group of peaks located around ν / Ω = 0 , 1.0 , 2.0 , 3.0 . The first group at ν / Ω = 0 is shown in the middle graph. One can identify frequencies Ω 0 0 and Ω 2 0 , which fall together, and Ω 1 0 . The small peak comes from the frequency Ω 3 0 . This first group of peaks is also covered by the adiabatic approximation. The other groups come from transitions between different manifolds. The adiabatic approximation does not take them into account, while VVP does. A blow-up of the peaks coming from transitions between neighboring manifolds is given in the right-hand graph. In the left-hand graph additionally the Jaynes-Cummings peaks are shown, which, however, fail completely. Fig::F_e=0_D=0.5_g=1.0
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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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  • Long Josephson junction, fluxon, Josephson vortex, flux qubit, qubit readout... We would like to employ fluxons for developing a fast and sensitive magnetic field detector for measurements of superconducting qubits. In this Letter, we report direct measurements of electromagnetic radiation from a fluxon moving in an annular Josephson junction (AJJ). The radiation is detected by using a microstrip antenna capacitively coupled to the AJJ. Furthermore, we place a flux qubit close to the long junction and couple them magnetically with a superconducting loop (see Fig.  AJJ+Qubit). This coupling scheme makes the fluxon interact with a current dipole formed by the electrodes of the loop coupled to the qubit. The time delay of the fluxon can be detected as a frequency shift of the electromagnetic radiation emitted from the junction. This shift provides information about the state of the flux qubit.... Optical photograph of the chip with the annular Josephson junction on the right part and experimental set-up schematics. Left part shows the zoom into the area with the flux qubit with a coupling loop (yellow loop) and control line (green loop). Red crosses indicate the positions of three Josephson junctions in the flux qubit loop.... Using the possibility to directly detect radiation of the fluxon resonant oscillations, we have performed systematic measurements of the dependence of the fluxon velocity versus bias current - the current-voltage characteristics - measured in the frequency domain (see Fig.  ZFS). This approach provides an easy access to study the fine structure of the current-voltage curve as the precision of frequency measurements is by several orders of magnitude greater than the resolution of direct dc voltage measurement.... Persistent current for a ground state of the flux qubit versus magnetic frustration (black line). Red line shows the corresponding fluxon shift calculated using the perturbation theory for.... An annular Josephson junction with a trapped fluxon coupled to a flux qubit.... By numerically solving ( FKE1)-( FKE2) with the additional condition u - l / 2 = u l / 2 one can calculate an equilibrium trajectory in phase space for fluxon oscillations in the AJJ with the current dipole and estimate a deviation of fluxon oscillation frequency from the unperturbed case δ ν = ν μ - ν 0 , where ν 0 is the oscillation frequency for μ = 0 . Black line in Fig.  FD shows the dependence of relative deviation δ ν / ν 0 versus bias current γ calculated from the perturbation theory for the following set of system parameters: l = 20 , α = 0.02 , μ = 0.05 , d = 2 . The deviation δ ν is large and negative for small bias currents γ ≪ 0.1 , what means that the fluxon is being slowed down by the current dipole and eventually can be pinned at the dipole if the bias current is too small. Surprisingly, for larger currents γ > 0.05 the sign of δ ν becomes positive meaning that the current dipole accelerates the fluxon. To understand this phenomenon, we need to look at the Eq. ( FKE1) and notice that the effective damping term α e = α u 1 - u 2 has a non-monotonic behavior. When increasing the fluxon velocity u , the effective damping is increasing for u ≤ 1 / 3 and then starts decreasing. This means that deceleration (acceleration) is favorable for low (high) bias currents.... Modulation of the fluxon’s oscillation frequency due to the coupling to the flux qubit. Every point consists of 100 averages. Bias current was set at γ = 0.521 , w ≃ 9.1 .... To couple a flux qubit to the fluxon inside an annular Josephson junction, it is necessary to engineer an interaction between two orthogonal magnetic dipoles. To facilitate this interaction, we have added a superconducting coupling loop embracing a flux qubit, as shown in Fig.  AJJ+Qubit. The current induced in the coupling loop attached to the AJJ is proportional to the persistent current in the flux qubit. Thus, the persistent current in the qubit manifests itself in the AJJ as a current dipole with an amplitude μ on top of the homogeneous background of bias current. When fluxon scatters on a positive current dipole - it first gets accelerated and then decelerated by the dipole poles. In the ideal case of absence of damping and bias current, the sign of frequency change δ ν is determined only by polarity of the dipole. In the presence of finite damping and homogeneous bias current, situation completely changes - as the total propagation time becomes dependent on the complex interplay between bias current, current dipole strength and damping.... Relative frequency deviation from equilibrium δ ν / ν 0 of the fluxon oscillation frequency versus bias current. Black line shows the result of perturbation approach, while the red line depicts results of direct numerical simulations of the PSGE equation ( PSGEm) with a d = 1 . The blue curve corresponds to the case with a d = 0.2 .... The experimental curve showing the reaction of the fluxon to the magnetic flux through the flux qubit are presented in Fig.  FM. The periodic modulation of the fluxon frequency versus magnetic flux through the qubit corresponds to the changing of the persistent currents in the qubit as Fig.  FQ_Icc suggests. We did not observe clear narrow peaks at the half flux quantum point, most probably due to excess fluctuations. Emerging dip-like peculiarities can be noted at presumed half flux quantum points which suggest that the dips may be there, covered by noise and insufficient resolution. Further improvements of experimental setup are required to resolve these peaks. The presented measurement curve has a convex profile which tells that indeed the deviation of frequency δ ν is positive, consistently with predictions made above by the perturbation approach and numerical simulations.
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  • (a) Circuit diagram for the Josephson junction qubit.  Junction current bias I is set by I φ and microwave source I μ w .  Parameters are I 0 ≃ 11.659 μ A , C ≃ 1.2 p F , L ≃ 168 p H , and L / M ≃ 81 .  (b) Potential energy diagram of qubit, showing qubit states and in cubic well at left.  Measurement of state performed by driving the 1 → 3 transition, tunneling to right well, then relaxation of state to bottom of right well.  Post-measurement classical states 0  and 1  differ in flux by Φ 0 , which is readily measured by readout SQUID.  (c) Schematic description of tunnel-barrier states A and B in a symmetric well.  Tunneling between states produces ground and excited states separated in energy by ℏ ω r .  (d) Energy-level diagram for coupled qubit and resonant states for ω 10 ≃ ω r .  Coupling strength between states and is given by H ˜ i n t .... (a) Measured probability of state 1  versus microwave excitation frequency ω / 2 π and bias current I for a fixed microwave power.  Data indicate ω 10 transition frequency.   Dotted vertical lines are centered at spurious resonances.  (b) Measured occupation probability of versus Rabi-pulse time t r and bias current I .  In panel (b), a color change from dark blue to red corresponds to a probability change of 0.4.  Color modulation in time t r (vertical direction) indicates Rabi oscillations.  ... superconductors, qubits, Josephson junction, decoherence... (a)-(c) Measured occupation probability of versus time duration of Rabi pulse t r for three values of microwave power, taken at bias I = 11.609 μ A in Fig. 2.  The applied microwave power for (a), (b), and (c) correspond to 0.1 , 0.33 , and 1.1 m W , respectively.  (d) Plot of Rabi oscillation frequency versus microwave amplitude.  A linear dependence is observed, as expected from theory.  
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  • (Color online) (a) Polarization P z t = 1 2 ρ 10 t + ρ 01 t and (b) Decoherence rate Γ d e c . t = - ρ ̇ 10 t + ρ ̇ 01 t ρ 10 t + ρ 01 t of the qubit with frequency lying inside ( δ / β < 0 ) and outside ( δ / β = 2 ) the PBG region .... (Color online) Dynamics of (a) the qubit’s excited-state probability P t and (b) relaxation rate Γ r e l a x . t of the qubit with different detuning frequencies δ / β = ω 10 - ω c / β from the band edge frequency ω c of the PhC reservoir.... (Color online) (a) A qubit with excited state and ground state . The transition frequency ω 10 is nearly resonant with the frequency range of the PhC reservoir. (b) Directional dependent dispersion relation near band edge expressed by the effective-mass approximation with the edge frequency ω c . (c) Photon DOS ρ ω of the anisotropic PhC reservoir exhibiting cut-off photon mode below the edge frequency ω c .
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