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  • fig3 Experimental device: a transmon qubit coupled to a nonlinear superconducting cavity. (a) Circuit diagram of the device; the anharmonic resonator is formed from a meander inductor embedded with a Josephson junction and an interdigitated capacitor. The resonator is isolated from the 50 Ω environment by coupling capacitors C c and coupled to a transmon qubit characterized by Josephson and charging energy scales E J and E c respectively. The coupling rate is g . (b) Scanning electron micrograph of the the device with the resonator and qubit junctions (lower and upper insets).... fig2 Energy levels and the effective nonlinearity λ of the strongly coupled system. (a) The measured coefficient of nonlinear response for a strongly coupled system versus qubit-cavity detuning for the qubit prepared in the ground state in the low excitation regime. Theory curves for the model Eq. ( eq:1) with N l = 2 (blue), N l = 3 (gray), and for a model of coupled nonlinear classical oscillators (red) are shown. The arrows indicate the locations of avoided crossings of the level pairs | 1 , 3 ↔ | 0 , 4 and | 1 , 4 ↔ | 0 , 5 . (Inset) The transmission of the resonator when driven with tone at ω d that occupied the resonator with = 0.4 (black) and = 10 (gray) off-resonant photons. (b) Energy levels of the qubit-oscillator model with N l = 7 show the avoided crossings in the 4 and 5 excitation manifold. (c) Quantum trajectory simulation of the system exhibits a general trends of increasing effective nonlinearity λ with diminishing qubit-cavity detuning ( Δ ) with abrupt reductions associated with avoided crossings in the 4 and 5 excitation manifold. For these simulations, the qubit energy levels were modeled as a Duffing nonlinearity.... fig2 Measured autoresonance and threshold sensitivity on the level structure and initial qubit state. (a) Color plot shows S | 1 versus qubit detuning. The dashed line indicates the AR threshold, V | 1 . AR measurements were not taken for small values of the detuning as indicated by the hatched region. (b) Color plot of S | 0 with V | 0 indicated as a solid black line. The AR threshold, V | 1 , is also plotted for comparison as a dashed black line. The two arrows indicate the location of avoided crossings in the 4 and 5 excitation manifold.... fig3duffing (a) The transmitted magnitude for chirp sequences with drive voltages that were above (black), near (dashed), and below (gray) the AR threshold. (Inset) Pulse sequence: the qubit manipulation pulse was applied immediately before the start of the chirp sequence. (b) The average transmitted magnitude near 400 ns versus drive voltage shows S | 0 (black) and S | 1 (red) for Δ / 2 π = 0.59 GHz.
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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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  • The measurement process that we used to observe coherent oscillations consists of several steps shown in Fig.  fig:3(a). Each step is realized by applying a combination of magnetic fluxes Φ x and Φ c as indicated by numbers in Fig.  fig:2(b). The first step in our measurement is the initialization of the system in a defined flux state (1). Starting from a double well at Φ x ≅ Φ 0 / 2 with high barrier, the potential is tilted by changing Φ x until it has only a single minimum (left or right, depending on the amplitude and polarity of the applied flux pulse). This potential shape is maintained long enough to ensure the relaxation to the ground state. Afterwards the potential is tuned back to the initial double-well state (2). The high barrier prevents any tunneling and the qubit is thus initialized in the chosen potential well. Next, the barrier height is lowered to an intermediate level (3) that preserves the initial state and allows to use just a small-amplitude Φ c flux pulse for the subsequent manipulation. The following Φ c -pulse transforms the potential into a single well (4). The Φ c -pulse duration Δ t is in the nanosecond range. The relative phase of the ground and the first excited states evolves depending on the energy difference between them. Once Φ c -pulse is over, the double well is restored and the system is measured in the basis | L | R (5). The readout of the qubit flux state is done by applying a bias current ramp to the dc SQUID and recording its switching current to the voltage state.... (a) The measured double SQUID flux Φ in dependence of Φ x , plotted for two different values of Φ c and initial preparation in either potential well. (b) Position of the switching points (dots) in the Φ c - Φ x parameter space. Numbered tags indicate the working points for qubit manipulation at which the qubit potential has a shape as indicated in the insets.... Calculated energy spacing of the first (solid line), second (dashed line) and third (dotted line) energy levels with respect to the ground state in the single well potential, plotted vs. the control flux amplitude Φ c 3 . Circles are the experimentally observed oscillation frequencies for the corresponding pulse amplitudes.... The oscillation frequency ω 0 depends on the amplitude of the manipulation pulse Δ Φ c since it determines the shape of the single well potential and the energy level spacing E 1 - E 0 . A pulse of larger amplitude Δ Φ c generates a deeper well having a larger level spacing, which leads to a larger oscillation frequency as shown in Fig.  fig:4(a). In Fig.  fig:5, we plot the energy spacing between the ground state and the three excited states (indicated as E k - E 0 / h with k=1,2,3) versus the flux Φ c 3 = Φ c 2 + Δ Φ c obtained from a numerical simulation of our system using the experimental parameters. In the same figure, we plot the measured oscillation frequencies for different values of Φ c (open circles). Excellent agreement between simulation (solid line) and data strongly supports our interpretation. The fact that a small asymmetry in the potential does not change the oscillation frequency, as shown in Fig.  fig:4(b), is consistent with the interpretation as the energy spacing E 1 - E 0 is only weakly affected by small variations of Φ x . This provides protection against noise in the controlling flux Φ x .... The flux pattern is repeated for 10 2 - 10 4 times in order to evaluate the probability P L = L | Ψ f i n a l 2 of occupation of the left state at the end of the manipulation. By changing the duration Δ t of the manipulation pulse Φ c , we observed coherent oscillations between the occupations of the states | L and | R shown in Fig.  fig:4(a). The oscillation frequency could be tuned between 6 and 21 GHz by changing the pulse amplitude Δ Φ c . These oscillations persist when the potential is made slightly asymmetric by varying the value Φ x 1 . As it is shown in Fig.  fig:4(b), detuning from the symmetric potential by up to ± 2.9 m Φ 0 only slightly changes the amplitude and symmetry of the oscillations. When the qubit was initially prepared in | R state instead of | L state we observed similar oscillations.... Probability to measure the state in dependence of the pulse duration Δ t for the qubit initially prepared in the state, and for (a) different pulse amplitudes Δ Φ c , resulting in the indicated oscillation frequency, and (b) for different potential symmetry by detuning Φ x from Φ 0 / 2 by the indicated amount.... Assuming identical junctions and negligible inductance of the smaller loop ( l ≪ L ), the system dynamics is equivalent to the motion of a particle with the Hamiltonian H = p 2 2 M + Φ b 2 L 1 2 ϕ - ϕ x 2 - β ϕ c cos ϕ , where ϕ = Φ / Φ b is the spatial coordinate of the equivalent particle, p is the relative conjugate momentum, M = C Φ b 2 is the effective mass, ϕ x = Φ x / Φ b and ϕ c = π Φ c / Φ 0 are the normalized flux controls, and β ϕ c = 2 I 0 L / Φ b cos ϕ c , with Φ 0 = h / 2 e and Φ b = Φ 0 / 2 π . For β qubit initialization and readout. The single well, or more exactly the two lowest energy states | 0 and | 1 in this well, is used for the coherent evolution of the qubit.... (a) Schematic of the flux qubit circuit. (b) The control flux Φ c changes the potential barrier between the two flux states | L and | R , here Φ x = 0.5 Φ 0 . (c) Effect of the control flux Φ x on the potential symmetry.... The investigated circuit, shown in Fig.  fig:1(a), is a double SQUID consisting of a superconducting loop of inductance L = 85 pH, interrupted by a small dc SQUID of loop inductance l = 6 pH. This dc SQUID is operated as a single Josephson junction (JJ) whose critical current is tunable by an external magnetic field. Each of the two JJs embedded in the dc SQUID has a critical current I 0 = 8 μ A and capacitance C = 0.4 pF. The qubit is manipulated by changing two magnetic fluxes Φ x and Φ c , applied to the large and small loops by means of two coils of mutual inductance M x = 2.6 pH and M c = 6.3 pH, respectively. The readout of the qubit flux is performed by measuring the switching current of an unshunted dc SQUID, which is inductively coupled to the qubit . The circuit was manufactured by Hypres  using standard Nb/AlO x /Nb technology in a 100 A/cm 2 critical current density process. The dielectric material used for junction isolation is SiO 2 . The whole circuit is designed gradiometrically in order to reduce magnetic flux pick-up and spurious flux couplings between the loops. The JJs have dimensions of 3 × 3 μ m 2 and the entire device occupied a space of 230 × 430 μ m 2 . All the measurements have been performed at a sample temperature of 15 mK. The currents generating the two fluxes Φ x and Φ c were supplied via coaxial cables including 10 dB attenuators at the 1K-pot stage of a dilution refrigerator. To generate the flux Φ c , a bias-tee at room temperature was used to combine the outputs of a current source and a pulse generator. For biasing and sensing the readout dc SQUID, we used superconducting wires and metal powder filters at the base temperature, as well as attenuators and low-pass filters with a cut-off frequency of 10 kHz at the 1K-pot stage. The chip holder with the powder filters was surrounded by one superconducting and two cryoperm shields.
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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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  • A list of atomic constants of several of the ions considered for quantum information processing. Here I is the nuclear spin, γ is the natural linewidth of the P 1 / 2 level , ω 0 is the frequency separation between the two qubit states set by the hyperfine splitting of the S 1 / 2 level , ω f is the fine-structure splitting , λ 1 / 2 and λ 3 / 2 are the wavelengths of the transitions between the S 1 / 2 and the P 1 / 2 and P 3 / 2 levels , respectively. The branching ratio of decay from the P levels to the D and the S levels is f .... Table Table3 lists ϵ R ∞ for different ion species for ω t r a p / 2 π =5 MHz and a single-circle ( K =1) gate. With the exception of 9 Be + , the error due to photon recoil in a two ion-qubit gate is below 10 -4 . For this error heavier ions benefit due to their smaller recoil.... Relevant energy levels (not to scale) in an ion-qubit, with nuclear spin I . The P 1 / 2 and P 3 / 2 excited levels are separated by an angular frequency ω f . The S 1 / 2 electronic ground state consists of two hyperfine levels F = I - 1 / 2 and F = I + 1 / 2 . The relative energies of these two levels depends on the sign of the hyperfine constant A h f and can vary between ion species (in this figure, A h f is negative). The qubit is encoded in the pair of m F = 0 states of the two F manifolds separated by an angular frequency ω 0 . Coherent manipulations of the qubit levels are performed with a pair of laser beams that are detuned by Δ from the transition to the P 1 / 2 level, represented by the two straight arrows. The angular frequency difference between the two beams equals the angular frequency separation between the qubit levels ω b - ω r = ω 0 . Some ion species have D levels with energies below the P manifold. Wavy arrows illustrate examples of Raman scattering events.... Gates are assumed to be driven by pairs of Raman beams detuned by Δ from the transition between the S 1 / 2 and the P 1 / 2 levels (See Fig. Levels). We further assume that the Raman beams are linearly polarized and Raman transitions are driven by both σ + photon pairs and σ - photon pairs. The two beams in a Raman pair are designated as red Raman ( r ) and blue Raman ( b ) by their respective frequencies. In the following we also assume that Δ is much larger than the hyperfine and Zeeman splitting between levels in the ground and excited states.... Here P t o t a l - g a t e is the probability that one of the ions scattered a photon during the two-qubit gate, and P t o t a l is the one-qubit gate scattering probability given in Eq. ( photon per pulse). Since both the Raman and the total scattering probabilities increase by the same factor as compared to the one-qubit gate, the ratio of the two errors ϵ D / ϵ S will remain the same as given by Eq. ( StoD error_ratio). Table Table2 lists ϵ D / ϵ S for the different ions when ϵ S = 10 -4 . Notice that for ϵ S = 10 -4 some ions require | Δ 0 | 2 ω f / 3 f . For those ions ϵ D is no longer negligible compared to ϵ S .... Most ion species considered for QIP studies have a single valence electron, with a 2 S 1 / 2 electronic ground state, and 2 P 1 / 2 and 2 P 3 / 2 electronic excited states. Some of the ions also have D levels with lower energy than those of the excited state P levels. Ions with a non zero nuclear spin also have hyperfine structure in all of these levels. A small magnetic field is typically applied to remove the degeneracy between different Zeeman levels. Here we consider qubits that are encoded into a pair of hyperfine levels of the 2 S 1 / 2 manifold. Figure Levels illustrates a typical energy level structure.... Schematic of Raman laser beam geometry assumed for the two-qubit phase gate. The gate is driven by two Raman fields, each generated by a Raman beam pair. Each pair consists of two perpendicular beams of different frequencies that intersect at the position of the ions such that the difference in their wave vector lies parallel to the trap axis. One beam of each pair is parallel to the magnetic field which sets the quantization direction. The beams’ polarizations in each pair are assumed to be linear, perpendicular to each other and to the magnetic field. Wavy arrows illustrate examples of photon scattering directions.... NIST Boulder, Time and Frequency division, Boulder, Colorado 80305 ... A list of different errors in a two-qubit phase gate due to spontaneous photon scattering. The error due to Raman scattering back into the S 1 / 2 manifold, ϵ S , is calculated assuming Gaussian beams with w 0 =20 μ m, a gate time τ g a t e =10 μ s, ω t r a p / 2 π = 5 MHz, a single circle in phase space ( K =1), and 10 mW in each of the four Raman beams. P 0 is the power in milliwatts needed in each of the beams, and Δ 0 / 2 π is the detuning in gigahertz for ϵ S = 10 -4 . The ratio between errors due to Raman scattering to the D and S manifolds, ϵ D / ϵ S , is given when ϵ S = 10 -4 . The asymptotic value of ϵ D in the | Δ | ≫ ω f limit is ϵ D ∞ . The Lamb-Dicke parameter η for the above trap frequency is also listed for different ions.... A list of errors in a single-qubit gate ( π rotation) due to spontaneous photon scattering. The error due to Raman scattering back into the S 1 / 2 manifold ϵ S is calculated with the same parameters as Fig. Power_vs_error: Gaussian beams with w 0 = 20 μ m, a single ion Rabi frequency Ω R / 2 π = 0.25 MHz ( τ π = 1 μ sec), and 10 mW in each of the Raman beams. P 0 is the power (in milliwatts) needed in each of the beams, and Δ 0 / 2 π is the detuning (in gigahertz) for ϵ S = 10 -4 . The ratio between errors due to Raman scattering to the D and S manifolds, ϵ D / ϵ S , is given when ϵ S = 10 -4 . The asymptotic value of ϵ D in the | Δ | ≫ ω f limit is ϵ D ∞ .... Assume that the ratio of the beam waist to the transition wavelength is constant for different ion species. In this case, the power needed to obtain a given Rabi frequency and to keep the error below a given value would scale linearly with the optical transition frequency. A more realistic assumption might be that the Raman beam waist is not diffraction limited and is determined by other experimental considerations, such as the inter-ion distance in the trap or beam pointing fluctuations. In this case, assuming that w 0 is constant, the required power would scale as the optical transition frequency cubed. Either way, ion species with optical transitions of longer wavelength are better suited in the sense that less power is required for the same gate speed and error requirements. In addition, high laser power is typically more readily available at longer wavelengths. Finally, we note that the error is independent of the fine-structure splitting as long as we have sufficient power to drive the transition. The transition wavelengths of different ions are listed in Table Table0.
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  • qubits
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  • Phase qubit coupled to a tank circuit.... In our method a resonant tank circuit with known inductance L T , capacitance C T and quality factor Q T is coupled with a target Josephson circuit through the mutual inductance M (Fig.  fig1). The method was successfully applied to a three-junction qubit in classical regime, when the hysteretic dependence of ground-state energy on the external magnetic flux was reconstructed in accordance to the predictions of Ref. 
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  • fig:plots5 Analysis of echo data for extraction of the decoherence time T 2 . (a–c) Transconductance G L as a function of the base level of detuning ε and δ t (defined in the main text) for total free evolution times of t = 390  ps, t = 690  ps, and t = 990  ps, respectively. (d–f) Fourier transforms of the charge occupation P 1 2 as a function of detuning ε and oscillation frequency f for the data in (a–c), respectively. We obtain P 1 2 (not shown here) by integrating the transconductance data in (a–c) and normalizing by noting that the total charge transferred across the polarization line is one electron. Fast Fourier transforming the time-domain data of P 1 2 allows us to quantify the amplitude of the oscillations visible near δ t = 0 . The oscillations of interest appear as weight in the FFT that moves to higher frequency at more negative detuning (farther from the anti-crossing). For an individual detuning energy, the FFT has nonzero weight for a nonzero bandwidth. (g) Echo amplitude as a function of free evolution time t . The data points (dark circles) are obtained at ε = - 120   μ eV by integrating a horizontal line cut of the FFT data over a bandwidth range of 46 - 72  GHz, then normalizing by the echo oscillation amplitude of the first data point, as described in the supplemental text. The echo oscillation amplitudes, plotted for multiple free evolution times, decay with characteristic time T 2 as the free evolution time t is made longer. By fitting the decay to a Gaussian, we obtain T 2 = 760 ± 190  ps. (h–j) Fourier transforms of the transconductance G L as a function of ε and oscillation frequency f for (a–c), respectively. As t is increased, the magnitude for oscillations at a given frequency decays with characteristic time T 2 . We take the magnitude of the FFT at the point where the central feature (black line) intersects 65  GHz. (k) Measured FFT magnitudes at 65 GHz for multiple free evolution times (dark circles) with a Gaussian fit (red line), which yields T 2 = 620 ± 140  ps, in reasonable agreement with the result shown in (g).
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  • For maximum generality, we first define a minimal model needed to describe the splitting of Fig.  fig:Splitting. To this end, we restrict ourselves to the lowest two states of the phase qubit circuit (the qubit subspace) and disregard the longitudinal coupling ∝ τ z . Within the rotating wave approximation (RWA) the Hamiltonian reads... (color online) (a) Analytically obtained transition spectrum of the Hamiltonian ( eq:H4Levels) in the minimal model for Ω q / h = 40  MHz and v ⊥ / h = 25  MHz. Dashed-dotted lines show the transition frequencies while the gray-scale intensity of the thicker lines indicates the weight of the respective Fourier-components in the probability P . The system shows a symmetric response as a function of the detuning δ ω . Two of the four lines are double degenerate. (b) The same as (a) but including the second order Raman process with Ω f v = v ⊥ Ω q / Δ . The two degenerate transitions in (a) split and the symmetry of the response is broken. Inset: Schematic representation of the structure of the Hamiltonian ( eq:H4Levels). We denote the ground and excited states of the qubit as and and those of the TLF as and . Arrows indicate the couplings between qubit and fluctuator v ⊥ and to the microwave field Ω q and Ω f v .... The sample investigated in this study is a phase qubit , consisting of a capacitively shunted Josephson junction embedded in a superconducting loop. Its potential energy has the form of a double well for suitable combinations of the junction’s critical current (here, I c = 1.1 μ A) and loop inductance (here, L = 720  pH). For the qubit states, one uses the two Josephson phase eigenstates of lowest energy which are localized in the shallower of the two potential wells, whose depth is controlled by the external magnetic flux through the qubit loop. The qubit state is controlled by an externally applied microwave pulse, which in our sample is coupled capacitively to the Josephson junction. A schematic of the complete qubit circuit is depicted in Fig.  fig:Splitting(a). Details of the experimental setup can be found in Ref. ... (color online) (a) Schematic of the phase qubit circuit. (b) Probability to measure the excited qubit state (color-coded) vs. bias flux and microwave frequency, revealing the coupling to a two-level defect state having a resonance frequency of 7.805 GHz (indicated by a dashed line).... superconducting qubits, Josephson junctions, two-level fluctuators, microwave spectroscopy, Rabi oscillations ... (color online) (a) Experimentally observed time evolution of the probability to measure the qubit in the excited state, P t , vs. driving frequency; (b) Fourier-transform of the experimentally observed P t . The resonance frequency of the TLF is indicated by vertical lines. (c) Time evolution of P and (d) its Fourier-transform obtained by the numerical solution of Eq. ( eq:master_eq) as described in the text, taking into account also the next higher level in the qubit. (As the anharmonicity Δ / h ∼ 100  MHz in our circuit is relatively small, this required going beyond the second order perturbation theory and analyze the 6-level dynamics explicitly). The qubit’s Rabi frequency Ω q / h is set to 48 MHz.... Spectroscopic data taken in the whole accessible frequency range between 5.8 GHz and 8.1 GHz showed only 4 avoided level crossings due to TLFs having a coupling strength larger than 10 MHz, which is about the spectroscopic resolution given by the linewidth of the qubit transition. In this work, we studied the qubit interacting with a fluctuator whose energy splitting was ϵ f / h = 7.805  GHz. From its spectroscopic signature shown in Fig.  fig:Splitting(b), we extract a coupling strength v ⊥ / h ≈ 25  MHz. The coherence times of this TLF were measured by directly driving it at its resonance frequency while the qubit was kept detuned. A π pulse was applied to measure the energy relaxation time T 1 , f ≈ 850  ns, while two delayed π / 2 pulses were used to measure the dephasing time T 2 , f * ≈ 110  ns in a Ramsey experiment. To read out the resulting TLF state, the qubit was tuned into resonance with the TLF to realize an iSWAP gate, followed by a measurement of the qubit’s excited state.... where δ ω = ϵ q - ϵ f . The level structure and the spectrum of possible transitions in the Hamiltonian ( eq:H4Levels) is illustrated in Fig.  fig:Transitionsa. The transition frequencies in the rotating frame correspond to the frequencies of the Rabi oscillations observed experimentally.... Figure  fig:DataRabi(a) shows a set of time traces of P taken for different microwave drive frequencies. Each trace was recorded after adjusting the qubit bias to result in an energy splitting resonant to the chosen microwave frequency. The Fourier transform of this data, shown in Fig.  fig:DataRabi(b), allows us to distinguish several frequency components. Frequency and visibility of each component depend on the detuning between qubit and TLF. We note a striking asymmetry between the Fourier components appearing for positive and negative detuning of the qubit relative to the TLF’s resonance frequency, which is indicated in Figs.  fig:DataRabi(a,b) by the vertical lines at 7.805 GHz. We argue below that this asymmetry is due to virtual Raman-transitions involving higher levels in the qubit.
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  • In conclusion, we have observed resonant tunneling in a macroscopic superconducting system, containing an Al flux qubit and a Nb tank circuit. The latter played dual control and readout roles. The impedance readout technique allows direct characterization of some of the qubit’s quantum properties, without using spectroscopy. In a range 50 ∼ 200 mK, the effective qubit temperature has been verified [Fig.  fig:Temp_dep(b)] to be the same as the mixing chamber’s (after Δ has been determined at low  T ), which is often difficult to confirm independently.... Our technique is similar to rf-SQUID readout. The qubit loop is inductively coupled to a parallel resonant tank circuit [Fig.  fig:schem(b)]. The tank is fed a monochromatic rf signal at its resonant frequency  ω T . Then both amplitude v and phase shift χ (with respect to the bias current  I b ) of the tank voltage will strongly depend on (A) the shift in resonant frequency due to the change of the effective qubit inductance by the tank flux, and (B) losses caused by field-induced transitions between the two qubit states. Thus, the tank both applies the probing field to the qubit, and detects its response.... (a) Tank phase shift vs flux bias near degeneracy and for V d r = 0.5 ~ μ V. From the lower to the upper curve (at f x = 0 ) the temperature is 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 350, 400 mK. (b) Normalized amplitude of tan χ (circles) and tanh Δ / k B T (line), for the Δ following from Fig.  fig:Bias_dep; the overall scale κ is a fitting parameter. The data indicate a saturation of the effective qubit temperature at 30 mK. (c) Full dip width at half the maximum amplitude vs temperature. The horizontal line fits the low- T ( < 200  mK) part to a constant; the sloped line represents the T 3 behavior observed empirically for higher  T .... (a) Quantum energy levels of the qubit vs external flux. The dashed lines represent the classical potential minima. (b) Phase qubit coupled to a tank circuit.... -dependence of ϵ t is adiabatic. However, it does remain valid if the full (Liouville) evolution operator of the qubit would contain standard Bloch-type relaxation and dephasing terms (which indeed are not probed) in addition to the Hamiltonian dynamics ( eq01), since the fluctuation–dissipation theorem guarantees that such terms do not affect equilibrium properties. Normalized dip amplitudes are shown vs T in Fig.  fig:Temp_dep(b) together with tanh Δ / k B T , for Δ / h = 650  MHz independently obtained above from the low- T width. The good agreement strongly supports our interpretation, and is consistent with Δ being T... Δ is the tunneling amplitude. At bias ϵ = 0 the two lowest energy levels of the qubit anticross [Fig.  fig:schem(a)], with a gap of  2 Δ . Increasing ϵ slowly enough, the qubit can adiabatically transform from Ψ l to Ψ r , staying in the ground state  E - . Since d E - / d Φ x is the persistent loop current, the curvature d 2 E - / d Φ x 2 is related to the qubit’s susceptibility. Hence, near degeneracy the latter will have a peak, with a width given by | ϵ | Δ . We present data demonstrating such behavior in an Al 3JJ qubit.
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