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  • 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.
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  • (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 is 
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  • 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.
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  • 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 the qubit’s evolution described by q t . This can be achieved within an effective description using a mapping procedure. Thereby the 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. linearbath
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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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  • 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.
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  • 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 the qubit’s anharmonicity α . In Fig.  Fig9(a), we show the phase qubit’s anharmonicity as a function of its transition 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 phase qubit’s relative 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 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.
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  • 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 .
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  • 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.
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  • 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).
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