Filter Results

58523 results

- Generation and control of entangled states. a, Spectroscopic characterization of the energy levels (see Fig. 1b inset) after a π (upper scan) and a 2 π (lower scan) Rabi pulse on the
**qubit**transition. In the upper scan, the system is first excited to | 10 from which it decays towards the | 01 excited state (red sideband at 3.58 GHz) or towards the | 00 ground state ( F L = 6.48 GHz). In the lower scan, the system is rotated back to the initial state | 00 wherefrom it is excited into the | 10 or | 11 states (see, in dashed, the blue sideband peak at 9.48 GHz for 13 dB more power). b, Coupled Rabi**oscillations**: the blue sideband is excited and the switching probability is recorded as a function of the pulse length for different microwave powers (plots are shifted vertically for clarity). For large microwave powers, the resonance peak of the blue sideband is shifted to 9.15 GHz. When detuning the microwave excitation away from resonance, the Rabi**oscillations**become faster (bottom four curves). These**oscillations**are suppressed by preparing the system in the | 10 state with a π pulse and revived after a 2 π pulse (top two curves in Fig. 3b) c, Coupled Rabi**oscillations**: after a π pulse on the**qubit**resonance ( | 00 → | 10 ) we excite the red sideband at 3.58 GHz. The switching probability shows coherent**oscillations**between the states | 10 and | 01 , at various microwave powers (the curves are shifted vertically for clarity). The decay time of the coherent**oscillations**in a, b is ∼ 3 ns....**Oscillator**relaxation time. a, Rabi**oscillations**between the | 01 and | 10 states (during pulse 2 in the inset) obtained after applying a first pulse (1) in resonance with the**oscillator**transition. Here, the interval between the two pulses is 1 ns. The continuous line represents a fit using an exponentially decaying sinusoidal**oscillation**plus an exponential decay of the background (due to the relaxation into the ground state). The**oscillation**’s decay time is τ c o h = 2.9 ns, whereas the background decay time is ∼ 4 ns. b, The amplitude of Rabi**oscillations**as a function of the interval between the two pulses (the vertical bars represent standard error bars estimated from the fitting procedure, see a). Owing to the**oscillator**relaxation, the amplitude decays in τ r e l ≈ 6 ns (the continuous line represents an exponential fit)....**Qubit**- SQUID device and spectroscopy a, Atomic force micrograph of the SQUID (large loop) merged with the flux**qubit**(the smallest loop closed by three junctions); the**qubit**to SQUID area ratio is 0.37. Scale bar, 1 μ m . The SQUID (**qubit**) junctions have a critical current of 4.2 (0.45) μ A. The device is made of aluminium by two symmetrically angled evaporations with an oxidation step in between. The surrounding circuit shows aluminium shunt capacitors and lines (in black) and gold quasiparticle traps 3 and resistive leads (in grey). The microwave field is provided by the shortcut of a coplanar waveguide (MW line) and couples inductively to the**qubit**. The current line ( I ) delivers the readout pulses, and the switching event is detected on the voltage line ( V ). b, Resonant**frequencies**indicated by peaks in the SQUID switching probability when a long microwave pulse excites the system before the readout pulse. Data are represented as a function of the external flux through the**qubit**area away from the**qubit**symmetry point. Inset, energy levels of the**qubit**-**oscillator**system for some given bias point. The blue and red sidebands are shown by down- and up-triangles, respectively; continuous lines are obtained by adding 2.96 GHz and -2.90 GHz, respectively, to the central continuous line (numerical fit). These values are close to the**oscillator**resonance ν p at 2.91 GHz (solid circles) and we attribute the small differences to the slight dependence of ν p on**qubit**state. c, The plasma resonance (circles) and the distances between the**qubit**peak (here F L = 6.4 GHz) and the red/blue (up/down triangles) sidebands as a function of an offset current I b o f f through the SQUID. The data are close to each other and agree well with the theoretical prediction for ν p versus offset current (dashed line).... Rabi**oscillations**at the**qubit**symmetry point Δ = 5.9 GHz. a, Switching probability as a function of the microwave pulse length for three microwave nominal powers; decay times are of the order of 25 ns. For A = 8 dBm, bi-modal beatings are visible (the corresponding**frequencies**are shown by the filled squares in b). b, Rabi**frequency**, obtained by fast Fourier transformation of the corresponding**oscillations**, versus microwave amplitude. In the weak driving regime, the linear dependence is in agreement with estimations based on sample design. A first splitting appears when the Rabi**frequency**is ∼ ν p . In the strong driving regime, the power independent Larmor precession at**frequency**Δ gives rise to a second splitting. c, This last aspect is obtained in numerical simulations where the microwave driving is represented by a term 1 / 2 h F 1 cos Δ t and a small deviation from the symmetry point (100 MHz) is introduced in the strong driving regime (the thick line indicates the main Fourier peaks). Radiative shifts 20 at high microwave power could account for such a shift in the experiment.Data Types:- Image

- (color online). Rabi
**oscillations**for a squbit-fluctuator system. The probability p 1 t to find the squbit in state | 1 is obtained from numerical integration of Eq. ( eq:mastereq) (solid line) and the analytical solution Eq. ( eq:pdyn-3) (dashed line) which is valid for b x / J ≫ 1 . For weak decoherence of the fluctuator, γ / J 1 , damped single-**frequency****oscillations**are restored. The fluctuator leads to a reduction of the first maximum in p 1 t to ∼ 0.8 [(a) and (b)] and ∼ 0.9 [(c) and (d)], respectively. The parameters are (a) b x / J = 1.5 , γ / J = 0.5 ; (b) b x / J = 1.5 , γ / J = 1.5 ; (c) b x / J = 3 , γ / J = 0.5 ; (d) b x / J = 3 , γ / J = 1.5 .... exhibits single-**frequency****oscillations**with reduced visibility [Fig. Fig2(b)]. Part of the visibility reduction can be traced back to leakage into state | 2 . More subtly, off-resonant transitions from | 1 to | 2 induced by the driving field lead to an energy shift of | 1 , such that the transition from | 0 to | 1 is no longer resonant with the driving field, which also reduces the visibility. For b x / 2 ℏ Δ ω = 1 / 3 , corresponding to b x / h = 150 ~ M H z in Ref. ... Energy shifts induced by AC driving field. – Exploring the visibility reduction at a time-scale of 10 n s requires Rabi**frequencies**| b x | / h 100 M H z . We show next that, in this regime, transitions to the second excited squbit state lead to an oscillatory behavior in p 1 t with a visibility smaller than 0.7 . For characteristic parameters of a phase-squbit, the second excited state | 2 is energetically separated from | 1 by ω 21 = 0.97 ω 10 . Similarly to | 0 and | 1 , the state | 2 is localized around the local energy minimum in Fig. Fig2(a). For adiabatic switching of the AC current, transitions to | 2 can be neglected as long as | b x | ≪ ℏ Δ ω = ℏ ω 10 - ω 21 ≃ 0.03 ℏ ω 10 . However, for b x comparable to ℏ Δ ω , the applied AC current strongly couples | 1 and | 2 because 2 | φ ̂ | 1 ≠ 0 , where φ ̂ is the phase operator. For typical parameters, b x / ℏ Δ ω ranges from 0.05 to 1 , depending on the irradiated power . Taking into account the second excited state of the phase-squbit, the squbit Hamiltonian in the rotating frame isData Types:- Image

- These environmentally-induced Rabi
**oscillations**are a clear signature of the non-Markovian behavior produced by the RLC environment, and are completely absent in the RL environment because the energy from the**qubits**is quickly dissipated without being temporarily stored. In the RL environment the decay in time of ρ 11 t has the characteristic non-oscillatory Markovian behavior. These environmentally-induced Rabi**oscillations**are generic features of circuits with resonances in the real part of the admittance. The**frequency**of the Rabi**oscillations**Ω R a = π κ Ω 3 / 2 Γ is independent of the resistance since Ω R a ≈ Ω π L 2 C / L 1 2 C 0 , and has the value of Ω R a = 2 π f R a ≈ 360 × 10 6 rad/sec in Fig. fig:seven. This effect is similar to the so-called circuit quantum electrodynamics which has been of great experimental interest recently ... In Fig. fig:four, T 1 is plotted for the phase**qubit**as a function of the**qubit****frequency**ω 01 in the case of spectral densities describing an RLC [Eq. ( eqn:spectral-density-isolation)] or Drude [Eq. ( eqn:sd-drude)] isolation network at fixed temperatures T = 0 (main figure) and T = 50 mK (inset), with J i n t ω = 0 corresponding to R 0 ∞ . In the limit of low temperatures k B T / ℏ ω 01 ≪ 1 , the relaxation time becomes... fig:four (Color-online) T 1 (in seconds) as a function of**qubit****frequency**ω 01 . The solid (red) curves describe the phase**qubit**with RLC isolation network (Fig. fig:three) with parameters R = 50 ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and**qubit**parameters C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 . The dashed curves correspond to an RL isolation network with the same parameters, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 × 10 9 rad/sec.... As an illustration of the qualitative results discussed in this section, we show in Fig. fig:six the**frequency**shift (renormalization) of the phase**qubit**with RLC isolation network described in Fig. fig:three. We make the identification E = ω 01 and δ E = δ ω 01 . Near resonance ω 01 ≈ Ω , we find a**frequency**renormalization of about 2 % which is due to the term δ E r e s .... (Color-online) Schematic drawing of a phase**qubit**with an RLC isolation circuit. The phase**qubit**is shown inside the solid (red) box, the RLC isolation circuit is shown inside the dashed box to the left, and the internal admittance circuit is shown inside the dashed box to the right.... fig:six Renormalization of energy splitting for the phase**qubit**with RLC isolation network (Fig. fig:three) for the parameters R = 50 ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and**qubit**parameters C 0 = 4.44 pF, R 0 = 5000 ohms and L 0 = 0 , T = 0 , and Ω = 141 × 10 9 rad/sec.... (Color-online) Flux**qubit**measured by a dc-SQUID gray (blue) line. The**qubit**corresponds to the inner SQUID loop with critical current I c and capacitance C J for both Josephson junctions denoted by the large × symbol. The inner SQUID is shunted by a capacitance C s , and environmental resistance R and is biased by a ramping current I b . The dc-SQUID loop has junction capatitance C 0 and critical current I c 0 .... fig:five (Color-online) T 1 (in nanoseconds) as a function of**qubit****frequency**ω 01 . The solid (red) curves describe a phase**qubit**with RLC isolation network (Fig. fig:three) with same parameters of Fig. fig:two except that R 0 = 5000 ohms. The dashed curves correspond to an RL isolation network with the same parameters of the RLC network, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 × 10 9 rad/sec.Data Types:- Image

- model We consider a composed system built of a
**qubit**, -the system of interest-, coupled to a nonlinear quantum**oscillator**(NLO), see Fig. linearbath. To read-out the**qubit**state we couple the**qubit**linearly to the**oscillator**with the coupling constant g ¯ , such that via the intermediate NLO dissipation also enters the**qubit**dynamics.... Jeff The effective spectral density follows from Eqs. ( gl20) and ( chilarger). It reads: J s i m p l J e f f ω e x = g ¯ 2 γ ω e x n 1 0 4 2 Ω 1 | ω e x | + Ω 1 M γ 2 Ω 1 2 2 n t h Ω 1 + 1 2 n 1 0 4 + 4 M Ω 2 | ω e x | - Ω 1 2 . As in case of the effective spectral density J e f f H O , Eq. ( linearspecdens), we observe Ohmic behaviour at low**frequency**. In contrast to the linear case, the effective spectral density is peaked at the shifted**frequency**Ω 1 . Its shape approaches the Lorentzian one of the linear effective spectral density, but with peak at the shifted**frequency**, as shown in Fig. CompLorentz.... Schematic representation of the complementary approaches available to evaluate the**qubit**dynamics: In the first approach one determines the eigenvalues and eigenfunctions of the composite**qubit**plus**oscillator**system (yellow (light grey) box) and accounts afterwards for the harmonic bath characterized by the Ohmic spectral density J ω . In the effective bath description one considers an environment built of the harmonic bath and the nonlinear**oscillator**(red (dark grey) box). In the harmonic approximation the effective bath is fully characterized by its effective spectral density J e f f ω . approachschaubild... mapping The main aim is to evaluate theevolution described by q t . This can be achieved within an effective description using a mapping procedure. Thereby the**qubit**’s**oscillator**and the Ohmic bath are put together, as depicted in Figure approachschaubild, to form an effective bath. The effective Hamiltonian... The transition**frequencies**in Eqs. ( rc1) and ( rc2) coincide, and in Figs. Plowg and Flowg there is no deviation observed when comparing the three different approaches.... where the trace over the degrees of freedom of the bath and of the**oscillator**is taken. In Fig. approachschaubild two different approaches to determine the**qubit**dynamics are depicted. In the first approach, which is elaborated in Ref. [... Corresponding Fourier transform of P t shown in Fig. CompNLLP. The effect of the nonlinearity is to increase the resonance**frequencies**with respect to the linear case. As a consequence the relative peak heights change. CompNLLF... Schematic representation of the composed system built of a**qubit**, an intermediate nonlinear**oscillator**and an Ohmic bath. linearbathData Types:- Image

- 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.Data Types:- Image
- Tabular Data

- Figure fig:Ttrans shows the numerically obtained coherence times and whether the decay is predominantly Gaussian or Markovian. For the large
**oscillator**damping γ = ϵ , the conditions for the validity of the (Markovian) Bloch-Redfield equation stated at the end of Sec. sec:analytics-g2 hold. Then we observe a good agreement of the numerically obtained T ⊥ * and Eq. ... (Color online) Dephasing time for purely quadratic**qubit**-**oscillator**coupling ( g 1 = 0 ), resonant driving at large**frequency**, Ω = ω 0 = 5 ϵ , and various values of the**oscillator**damping γ . The driving amplitude is A = 3.5 γ , such that always n ̄ = 6.125 . Filled symbols mark Markovian decay, while stroked symbols refer to Gaussian shape. The solid line depicts the value obtained for γ = ϵ in the Markov limit. The corresponding numerical values are connected by a dashed line which serves as guide to the eye.... (Color online) Typical time evolution of the**qubit**operator σ x (solid line) and the corresponding purity (dashed) for Ω = ω 0 = 0.8 ϵ , g 1 = 0.02 ϵ , γ = 0.02 ϵ , and driving amplitude A = 0.06 ϵ such that the stationary photon number is n ̄ = 4.5 . Inset: Purity decay shown in the main panel (dashed) compared to the decay given by Eq. P(t) together with Eq. Lambda(t) (solid line).... Figure fig:timeevolution depicts the time evolution of the**qubit**expectation value σ x which exhibits decaying**oscillations**with**frequency**ϵ . The parameters correspond to an intermediate regime between the Gaussian and the Markovian dynamics, as is visible in the inset.... (Color online) Dephasing time for purely linear**qubit**-**oscillator**coupling ( g 2 = 0 ), resonant driving, Ω = ω 0 , and**oscillator**damping γ = 0.02 ϵ . The amplitude A = 0.07 ϵ corresponds to the mean photon number n ̄ = 6.125 . Filled symbols and dashed lines refer to predominantly Markovian decay, while for Gaussian decay, stroked symbols and solid lines are used.Data Types:- Image

- The two possible flux values at the readout spot leads to two possible
**frequencies**for the tunable cavity coupled to the**qubit**loop. Similar microwave readout schemes have been used with other rf-SQUID phase**qubits**. For our circuit design, the size of this**frequency**difference is proportional to the slope d f c / d φ c of the cavity**frequency**versus flux curve at a particular cavity flux φ c = Φ c / Φ o . The transmission of the cavity can be measured with a network analyzer to resolve the**qubit**flux (or circulating current) states. The periodicity of the rf SQUID phase**qubit**can be observed by monitoring the cavity’s resonance**frequency**while sweeping the**qubit**flux. This allows us to observe the single-valued and double-valued regions of the hysteretic rf SQUID. In Fig. Fig4(a), we show the cavity response to such a flux sweep for design A . Two data sets have been overlaid, for two different**qubit**resets ( φ q = ± 2 ) and sweep directions (to the left or to the right), allowing the double-valued or hysteretic regions to overlap. There is an overall drift in the cavity**frequency**due to flux crosstalk between the**qubit**bias line and the cavity’s rf SQUID loop that was not compensated for here. This helps to show how the**frequency**difference in the overlap regions increases as the slope d f c / d φ c increases.... (Color online) (a)**Qubit**spectroscopy (design A ) overlaid with cavity spectroscopy at two**frequencies**, f c = 6.58 GHz and 6.78 GHz. (b) Zoom-in of the split cavity spectrum in (a) when f c = 6.78 GHz with corresponding fit lines. (c) Zoom-in of the split cavity spectrum in (a) when f c = 6.58 GHz with corresponding fit lines. (d) Cavity spectroscopy (design B ) while sweeping the**qubit**flux with f c = 7.07 GHz showing a large normal-mode splitting when the**qubit**is resonant with the cavity. All solid lines represent the uncoupled**qubit**and cavity**frequencies**and the dashed lines show the new coupled normal-mode**frequencies**. Notice in (d) the additional weak splitting from a slot-mode just below the cavity, and in (c) and (d),**qubit**tunneling events are visible as abrupt changes in the cavity spectrum.... (Color online) (a) Cavity spectroscopy (design A ) while sweeping the cavity flux bias with the**qubit**far detuned, biased at its maximum**frequency**. The solid line is a fit to the model including the junction capacitance. (b) Zoom-in near the maximum cavity**frequency**showing a slot-mode. (c) Line-cut on resonance along the dashed line in (b) with a fit to a skewed Lorentzian (solid line).... In general, rf SQUID phase**qubits**have lower T 2 * (and T 2 ) values than transmons, specifically at lower**frequencies**, where d f 01 / d φ q is large and therefore the**qubit**is quite sensitive to bias fluctuations and 1/f flux noise . For example, 600 MHz higher in**qubit****frequency**, at f 01 = 7.98 GHz, Ramsey**oscillations**gave T 2 * = 223 ns. At this location, the decay of on-resonance Rabi**oscillations**gave T ' = 727 ns, a separate measurement of**qubit**energy decay after a π -pulse gave T 1 = 658 ns, and so, T 2 ≈ 812 ns, or T 2 ≈ 3.6 × T 2 * , a small, but noticeable improvement over the lower**frequency**results displayed Fig. Fig6. The current device designs suffer from their planar geometry, due to a very large area enclosed by the non-gradiometric rf SQUID loop (see Fig. Fig1). Future devices will require some form of protection against flux noise , possibly gradiometric loops or replacing the large geometric inductors with a much smaller series array of Josephson junctions .... (Color online) Coupling rate 2 g / 2 π (design A ) as a function of cavity**frequency**ω c / 2 π . The solid red (blue) line is the prediction from Eq. ( eq:g) (including L x and C J ’s). The (dotted) dashed line is the prediction for capacitive coupling with C = 15 fF ( C = 5 fF). The solid circles were measured spectroscopically (see text). At lowest cavity**frequency**, the solid ⋆ results from a fit to the Purcell data, discussed later in section TCQEDC. The gray region highlights where the phase**qubit**(design A ) remains stable enough for operation (see text).... Next, we carefully explore the size of the dispersive shifts for various cavity and**qubit****frequencies**. In order to capture the maximum dispersive**frequency**shift experienced by the cavity, we applied a π -pulse to the**qubit**. A fit to the phase response curve allows us to extract the cavity’s amplitude response time 2 / κ , the**qubit**T 1 , and the full dispersive shift 2 χ . Changing the cavity**frequency**modifies the coupling g and the detuning Δ 01 , while changes to the**qubit****frequency**change both Δ 01 and theanharmonicity α . In Fig. Fig9(a), we show the phase**qubit**’sanharmonicity as a function of its transition**qubit**’s**frequency**ω 01 / 2 π extracted from the spectroscopic data shown in Fig. Fig5 from section QBB for design A . The solid red line is a polynomial fit to the experimental data, used to calculate the three-level model curves in Fig. Fig9(b–d), while the blue line is a theoretical prediction of the relative anharmonicity (including L x , but neglecting C J ) using perturbation theory and the characteristic**qubit**parameters extracted section QBB. In Fig. Fig9(b–d), we find that the observed dispersive shifts strongly depend on all of these factors and agree well with the three-level model predictions . For comparison, in Fig. Fig9(b), we show the results for the two-level system model (bold dashed line) when f c = 6.58 GHz, which has a significantly larger amplitude for all detunings (outside the “straddling regime”). Notice that it is possible to increase the size of the dispersive shifts for a given | Δ 01 | / ω 01 by decreasing the cavity**frequency**f c , which increases the coupling rate 2 g / 2 π (as seen in Fig. Fig2 in section TCQED). Also, notice that decreasing the ratio of | Δ 01 | / ω 01 also significantly increases the size of the dispersive shifts, even when the phaserelative anharmonicity α r decreases as ω 01 increases. Essentially, the ability to reduce | Δ 01 | helps to counteract any reductions in α r . These results clearly demonstrate the ability to tune the size of the dispersive shift through selecting the relative**qubit**’s**frequency**of the**qubit**and the cavity. This tunability offers a new flexibility for optimizing dispersive readout of**qubits**in cavity QED architectures and provides a way for rf SQUID phase**qubits**to avoid the destructive effects of tunneling-based measurements.... (Color online) (a) Time domain measurements (design A ). Rabi**oscillations**for**frequencies**near f 01 = 7.38 GHz. (b) Line-cut on-resonance along the dashed line in (a). The fit (solid line) yields a Rabi**oscillation**decay time of T ' = 409 ns. (c) Ramsey**oscillations**versus**qubit**flux detuning near f 01 = 7.38 GHz. (d) Line-cut along the dashed line in (c). The fit (solid line) yields a Ramsey decay time of T 2 * = 106 ns. With T 1 = 600 ns, this implies a phase coherence time T 2 = 310 ns.... (Color online) (a) Pulse sequence. (b) Rabi**oscillations**(design A ) for various pulse durations obtained using dispersive measurement at f 01 = 7.18 GHz, with Δ 01 = + 10 g . (c) A single, averaged time trace along the vertical dashed line in (b). (d) Rabi**oscillations**extracted from the final population at the end of the drive pulse, along the dashed diagonal line in (b). (e) Zoom-in of dashed box in (b) showing Rabi**oscillations**observed during continuous driving.... We can explore the coupled**qubit**-cavity behavior described by Eq. ( eq:H) by performing spectroscopic measurements on either the**qubit**or the cavity near the resonance condition, ω 01 = ω c . Fig. Fig7(a) shows**qubit**spectroscopy for design A overlaid with cavity spectroscopy for two cavity**frequencies**, f c = 6.58 GHz and 6.78 GHz. Fig. Fig7(d) shows cavity spectroscopy for design B with the cavity at its maximum**frequency**of f c m a x = 7.07 GHz while sweeping the**qubit**flux bias φ q . In both cases, when the**qubit****frequency**f 01 is swept past the cavity resonance, the inductive coupling generates the expected spectroscopic normal-mode splitting.... The weak additional splitting just below the cavity in Fig. Fig7(d) is from a resonant slot-mode. We can determine the coupling rate 2 g / 2 π between the**qubit**and the cavity by extracting the splitting size as a function of cavity**frequency**f c from the measured spectra. Three examples of fits are shown in Fig. Fig7(b–d) with solid lines representing the bare**qubit**and cavity**frequencies**, whereas the dashed lines show the new coupled normal-mode**frequencies**. For design A ( B ), at the maximum cavity**frequency**of 6.78 GHz (7.07 GHz), we found a minimum coupling rate of 2 g m i n / 2 π = 78 MHz (104 MHz). Notice that the splitting size is clearer bigger in Fig. Fig7(c) than for Fig. Fig7(b) by about 25 MHz. The results for the coupling rate 2 g / 2 π as a function of ω c / 2 π for design A were shown in Fig. Fig2 in section TCQED. Also visible in Fig. Fig7(c–d) are periodic, discontinuous jumps in the cavity spectrum. These are indicative of**qubit**tunneling events between adjacent metastable energy potential minima, typical behavior for hysteretic rf SQUID phase**qubits**. Moving away from the maximum cavity**frequency**increases the flux sensitivity, with the**qubit**tunneling events becoming more visible as steps. This behavior is clearly visible in Fig. Fig7(c) and was already shown in Fig. Fig4 in Sec. QBA and, as discussed there, provides a convenient way to perform rapid microwave readout of traditional tunneling measurements . Next, we describe dispersive measurements of the phase**qubit**for design A . These results agree with the tunneling measurements across the entire**qubit**spectrum.Data Types:- Image
- Tabular Data

- fig:1fFig2 (a) Solid dots show temperature dependence of α 1 / 2 with a fixed bias current (the bias voltage is adjusted to keep the current constant). Open dots show α 1 / 2 derived from the measurement of
**qubit**dephasing during coherent**oscillations**. The coherent**oscillations**(solid line) as well as the envelope exp - t 2 / 2 T 2 * 2 with T 2 * = 180 ps (dashed line) are in the inset. (b) Solid dots show temperature dependence of α 1 / 2 for the SET on GaAs substrate.... Note that at a fixed bias voltage the average current through the SET increases with temperature (see Fig. fig:1fFig1(a)). However, it has almost no effect on the noise as we confirmed from the measurement of the current noise dependence. Nevertheless, to avoid possible contribution from the current dependent noise we adjust the bias voltage in the next measurements so that the average current is kept nearly constant at the measurement points for different temperatures. Fig. fig:1fFig2(a) shows the temperature dependences of α 1 / 2 for a different sample with a similar geometry taken in the**frequency**range from 0.1 Hz to 10 Hz with a bias current adjusted to about I = 12 ± 2 pA. The straight line in the plot is α 1 / 2 = η 1 / 2 T , which corresponds to T 2 -dependence of α with η ≈ ( 1.3 × 10 -2 e / K ) 2 .... Solid dots in Fig. fig:1fFig1(c) represent α 1 / 2 as a function of temperature. α 1 / 2 saturates at temperatures below 200 mK at the level of 2 × 10 -3 e and exhibits nearly linear rise at temperatures above 200 mK with α 1 / 2 ≈ η 1 / 2 T , where η ≈ 1.0 × 10 -2 e / K 2 (the solid line in Fig. fig:1fFig1(c)). T 2 dependence of α is observed in many samples, though sometimes the noise is not exactly 1 / f , having a bump from the Lorentzian spectrum of a strongly coupled low**frequency**fluctuator. In such cases, switches from the single two-level fluctuator are seen in time traces of the current .... The typical current**oscillation**as a function of t away from the degeneracy point ( θ ≠ π / 2 ) is exemplified in the inset of Fig. fig:1fFig2(a). If dephasing is induced by the Gaussian noise, the**oscillations**decay as exp - t 2 / 2 T 2 * 2 with... We use the**qubit**as an SET and measure the low**frequency**charge noise, which causes the SET peak position fluctuations. Temperature dependence of the noise is measured from the base temperature of 50 mK up to 900 - 1000 mK. The SET is normally biased to V b = 4 Δ / e ( ∼ 1 mV), where Coulomb**oscillations**of the quasiparticle current are observed. Figure fig:1fFig1(a) exemplifies the position of the SET Coulomb peak as a function of the gate voltage at temperatures from 50 mK up to 900 mK with an increment of 50 mK. The current noise spectral density is measured at the gate voltage corresponding to the slope of the SET peak (shown by the arrow), at the maximum (on the top of the peak) and at the minimum (in the Coulomb blockade). Normally, the noise spectra in the two latter cases are**frequency**independent in the measured**frequency**range (and usually do not exceed the noise of the measurement setup). However, the noise spectra taken on the slope of the peak show nearly 1 / f**frequency**dependence (see examples of the current noise S I at different temperatures in Fig. fig:1fFig1(b)) saturating at a higher**frequencies**(usually above 10 - 100 Hz depending on the device properties) at the level of the noise of the measurement circuit. The fact that the measured 1 / f noise on the slope is substantially higher than the noises on the top of the peak and in the blockade regime indicates that the noise comes from fluctuations of the peak position, which can be translated into charge fluctuations in the SET.... The solid line in the inset of Fig. fig:1fFig2(a) shows decay of coherent**oscillations**measured at T = 50 mK and the dashed envelope exemplifies a Gaussian with T 2 * = 180 ps. We derive α 1 / 2 from Eq. ( eq:Eq3) and plot it in Fig. fig:1fFig2(a) by open dots as a function of temperature. The low**frequency**integration limit and the high**frequency**cutoff are taken to be ω 0 ≈ 2 π × 25 Hz and ω 1 ≈ 2 π × 5 GHz for our measurement time constant τ = 0.02 s and typical dephasing time T 2 * ≈ 100 ps .Data Types:- Image

- Fast Fourier transform of at different amplitudes G / h of low-
**frequency**field.... As an example we show below the time evolution of the quantity σ Z t = Z t , obtained from the numerical solution of the equations ( sigmaZ), ( sigmaY), and ( sigmaX) where we take a low**frequency**excitations as G t = G c o s ω L t . The calculations have been performed with initial conditions σ Z 0 = 1 , σ X 0 = σ Y 0 = 0 for the following set of the parameters: F / h = 36 MHz, Δ / h = 1 GHz, Γ / 2 π = 4 MHz, Γ z / 2 π = 1 MHz, ϵ / Δ = 1 , Z 0 = - 1 , δ / 2 π = 6.366 MHz, ω L / Ω R = 1 . As is seen from Fig. fig1 in the absence of low**frequency**signal ( G = 0 ) the**oscillations**are damped out, while if G ≠ 0 the**oscillations**persist.... The Fourier spectra of these signals are shown on Fig. fig2 for different amplitudes of low**frequency**excitation. For G = 0 the Rabi**frequency**is positioned at approximately 26.2 MHz, which is close to Ω R = 26.24 MHz. With the increase of G the peak becomes higher. It is worth noting the appearance of the peak at the second harmonic of Rabi**frequency**. This peak is due to the contribution of the terms on the order of G 2 which we omitted in our theoretical analysis.... Time evolution of . (thick) G=0, (thin) G / h = 1 MHz. The insert shows the undamped**oscillations**of at G / h = 1 MHz.... The comparison of analytical and numerical resonance curves calculated for low**frequency**amplitude, G / h = 1 MHz and different dephasing rates, Γ are shown on Fig. fig3. The curves at the figure are the peak-to-peak amplitudes of**oscillations**of Z t calculated from Eq. ( ZOmega) with g ˜ ω = g δ ω + ω L + δ ω - ω L / 2 , where δ ω is Dirac delta function. The point symbols are found from numerical solution of Eqs. ( sigmaZ),( sigmaY),( sigmaX). The widths of the curves depend on Γ (see the insert) and the positions of the resonances coincide with the Rabi**frequency**. A good agreement between numerics and Eq. ZOmega, as shown at Fig. fig3, is observed only for relative small low**frequency**amplitude G / h , for which our linear response theory is valid.Data Types:- Image

- where the basis comprises the Fock, or number, states of the field. We note that is an eigenstate of the annihilation operator labelled by its eigenvalue α . In this scenario, although the
**qubit**initially exhibits Rabi**oscillations**analogous to those in the classical case, these apparently “decay”, and then subsequently revive . An example is shown in figure rabirev. Such collapse and revival of Rabi**oscillations**of a**qubit**is widely recognised as a characteristic of a**qubit**coupled to a quantum field mode. It is understood both theoretically and experimentally that the apparent decay of**qubit**coherence is due to entanglement with the field mode, generated by the coherent evolution of the coupled quantum systems. This is illustrated in figure rabirev through a plot of the**qubit**entropy, S t = - T r ( ρ t l n ρ t ) where ρ t is the reduced density matrix of the**qubit**resulting from a trace over the field (or vice versa, given the initial system state is pure). Clearly there is a sharp rise in entropy sympathetic with the initial collapse. The**qubit**then disentangles from the field at the “attractor time” , half way to the revival. The revival arises through oscillatory re-entanglement of the**qubit**and field, as seen through the subsequent entropy**oscillations**that coincide with the revival. Obviously there is no entanglement between the**qubit**and the field in the classical limit, because the field is classical, so the**qubit**entropy is zero for all times.... (Color online)**Qubit**inversion, σ z , as a function of dimensionless time ω t for the resonant cases of Rabi**oscillations**(dotted; light grey/magenta) in a classical field and collapse and revival (dark grey/red heavy line) in a quantum field. For the classical field case ν / ω = 8 is used, corresponding to the same dominant Rabi**frequency**(see ) as in the quantum field case, which uses a coherent state ( coh) with α = 15 and λ / ω = 1 . Also shown (in green/light grey solid line) is the**qubit**entropy (in Nats), which indicates the degree of entanglement with the field.... (Color online) Results (dark grey/red) showing**qubit**inversion, σ z , for different values of dissipation (and therefore drive) applied to the field. These illustrate collapse and revival, suppression of collapse and an approach to complete Rabi**oscillations**as dissipation and drive are increased. For each individual QSD run shown, the**qubit**entropy in Nats (light grey/green) is superimposed.... Further insight into the classical limit can be obtained from the phase space behaviour of the field. It is well known that in the pure quantum limit the**qubit**-field entanglement correlates distinct and localised (coherent-like) states of the field with different**qubit**amplitudes. Thus, when there is no entanglement at the “attractor time”, the interaction of the atom with the field mode generates a macroscopically distinct superposition of states in the**oscillator**—a Schrödinger cat state. As one would expect, and in order to render the field behaviour classical, the introduction of decoherence suppresses this phenomenon. We illustrate this in by providing two animations of the dynamics of the Wigner function and atomic inversion for the parameters of Fig. rabirev, one undamped and one with dissipation of γ / ω = 0.01 .... The detuning is defined as Δ = ω 0 - ω and the Rabi**oscillation****frequency**as Ω R = Δ 2 + ν 2 , so for the case of the field on resonance with the**qubit**(zero detuning, Δ = 0 ) the Rabi**frequency**is simply Ω R = ν . It is set by the amplitude of the field, not its**frequency**. In this resonant case, a**qubit**initially in state**oscillates**fully to at**frequency**Ω R . In the language of atomic and optical physics, the atomic inversion—in**qubit**language σ z —satisfies σ z = cos Ω R t . In the absence of any decoherence acting on the**qubit**, these Rabi**oscillations**persist and are a well known characteristic of a**qubit**resonantly coupled to an external classical field. An example is shown in figure rabirev.... It is well known that there are three timescales in the collapse and revival situation . For fields with large n = n ̄ , the Rabi time (or period) is given by t R = 2 π Ω R -1 = π / λ n ̄ , the collapse time that sets the Gaussian decay envelope of the**oscillations**by t c = 2 / λ and the (first) revival time that determines when the**oscillations**reappear, such as in the example of figure rabirev, by t r = 2 π n ̄ / λ . For the coherent state ( coh) the average photon, or excitation, number is n ̄ = | α | 2 . Note that the different dependencies of the times on n ̄ (which corresponds to the—e.g. electric—field strength of the coherent field state) allow a sort of “classical limit” to be taken. As n ̄ is increased, there are more Rabi periods packed in before the collapse—so this appears more like persistent Rabi**oscillations**—and the revival is pushed out further in time. However, the collapse (and revival) still occur eventually, and in any case the reason there are more Rabi**oscillations**before the collapse is due to the inverse scaling of t R with n ̄ , so the actual Rabi period is shortened as n ̄ is increased. The classical limit we consider in this work is quite different. We shall consider a fixed n ̄ , so the Rabi period of the**qubit**does not change in our various examples. We’ll show how the transition from quantum (collapse and revival) to classical (continuous Rabi**oscillations**) can be effected by introducing decoherence to the quantum field. Our work complements the dissipative, small- n , short-time study of Kim et al. , who show that such fields are sufficiently classical to provide Ramsey pulses to Rydberg atoms.... (Color online) Results (dark grey/red) showing**qubit**inversion, σ z , for different values of dissipation (and therefore drive) applied to the field. These plots focus on the parameter range where the revival of Rabi**oscillations**begins to emerge. For each individual QSD run shown, the**qubit**entropy in Nats (light grey/green) is superimposed.... In our approach, there are two ways in which the**qubit**state could become mixed. Firstly, it could entangle with the field , as happens in the pure quantum limit to generate the collapse. Such entanglement can be inferred from the**qubit**entropy in a single run of QSD (for which the full**qubit**-field state is pure). Secondly, the**qubit**could remain pure in an individual QSD run, but, when averaged over an ensemble, show mixture. For the classical limit of the top left plot of figure fig2, we have calculated that both of these effects are small for times in excess of the collapse time. The**qubit**-field entanglement remains very close to zero for all times in a single QSD run, as shown in the entropy plot presented. Independent QSD runs have been made and these show that the**qubit**mixture is still very small at the collapse time. Therefore the persistence of good Rabi**oscillations**well beyond the collapse time and all the way out to the revival time, as illustrated in the top left plot of figure fig2, provides a clear signature of the classical limit of the field. In this limit, the quantum field state is a localized lump in phase space (like a coherent state), following the expected classical trajectory and suffering negligible back-reaction from the**qubit**. However, the field coherence time is so short as to prevent entanglement with the**qubit**developing, unlike in the quantum limit . The resultant**qubit**Rabi**oscillations**are thus like those due to a classical field, and not like those that arise from entanglement with a single Fock state (which is a delocalized ring in phase space).Data Types:- Image

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