### 54077 results for qubit oscillator frequency

Contributors: Omelyanchouk, A. N., Shevchenko, S. N., Zagoskin, A. M., Il'ichev, E., Nori, Franco

Date: 2007-05-12

The average energy H of the system as a function of the driving **frequency** ω . The main peak ( ω 0 ≈ 0.6 ) corresponds to the resonance. The left peak at ω 0 / 2 is the nonlinear effect of the excitation by a subharmonic, similar to a multiphoton process in the quantum case. The right peak at 2 ω 0 is the first overtone and it has no quantum counterpart. Here ϕ e d = π ; ϕ e a = 0.05 ; γ = 10 -3 ....The dependence of the pseudo-Rabi **frequency** on the driving amplitude ϕ e a for ω = 0.6 , γ = 10 -3 . The solid line, Ω = 0.35 ϕ e a 2 + ω - 0.63 2 1 / 2 , is the best fit to the calculated data....(Color online) The potential profile of Eq. ( eq_potential) with α = 0.8 , ϕ e d = π . The arrows indicate quantum (solid) and classical (dotted) **oscillations**....A quantitative difference between this effect and true Rabi **oscillations** is in the different scale of the resonance **frequency**. To induce Rabi **oscillations** between the lowest quantum levels in the potential ( eq_potential), one must apply a signal in resonance with their tunneling splitting, which is exponentially smaller than ω 0 . Still, this is not a very reliable signature of the effect, since the classical effect can also be excited by subharmonics, ∼ ω 0 / n , as we can see in Fig. fig4....Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux **qubit** in the _classical_ regime can produce low-**frequency** **oscillations** in the presence of an external field in resonance with the (high-**frequency**) harmonic mode of the system, $\omega$. Like in the case of_quantum_ Rabi **oscillations**, the **frequency** of these pseudo-Rabi **oscillations** is much smaller than $\omega$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external perturbation not only at the resonance **frequency** $\omega$ and its subharmonics ($\omega/n$), but also at its overtones, $n\omega$....In the presence of the external field ( eq_external) the system will undergo forced **oscillations** around one of the equlibria. For α = 0.8 , which is close to the parameters of the actual devices , the values of the dimensionless **frequencies** become ω θ ≈ 0.612 , and ω χ ≈ 0.791 . Solving the equations of motion ( eq_motion) numerically, we see the appearance of slow **oscillations** of the amplitude and energy superimposed on the fast forced **oscillations** (Fig. fig2), similar to the classical **oscillations** in a phase **qubit** (Fig. 2 in ). The dependence of the **frequency** of these **oscillations** on the driving amplitude shows an almost linear behaviour (Fig. fig3), which justifies the “Pseudo-Rabi” moniker....The key observable difference between the classical and quantum cases, which would allow to reliably distinguish between them, is that the classical effect can also be produced by driving the system at the overtones of the resonance signal, ∼ n ω 0 (Fig. fig4). This effect can be detected using a standard technique for RF SQUIDs . The current circulating in the **qubit** circuit produces a magnetic moment, which is measured by the inductively coupled high-quality tank circuit. For the tank voltage V T we have...where τ T = R T C T is the RC-constant of the tank, ω T = L T C T -1 / 2 its resonant **frequency**, M the mutual inductance between the tank and the **qubit**, and I q t the current circulating in the **qubit**. The persistent current in the 3JJ loop can be determined directly from ( eq_I). Its behaviour in the presence of an external RF field is shown in Fig. fig2c. Note that the sign of the current does not change, which is due to the fact that the **oscillations** take place inside one potential well (solid arrow in Fig. fig1), and not between two separate nearby potential minima like in the quantum case. (Alternatively, this would also allow to distinguish between the classical and quantum effects by measuring the magnetization with a DC SQUID.)...(a) Driven **oscillations** around a minimum of the potential profile of Fig. fig1 as a function of time. The driving amplitude is ϕ e a = 0.01 , driving **frequency** ω = 0.612 , and the decay rate γ = 10 -3 . Low-**frequency** classical beat **oscillations** are clearly seen. (b) Low-**frequency** **oscillations** of the persistent current in the 3JJ loop. (c) Same for the energy of the system. ... Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux **qubit** in the _classical_ regime can produce low-**frequency** **oscillations** in the presence of an external field in resonance with the (high-**frequency**) harmonic mode of the system, $\omega$. Like in the case of_quantum_ Rabi **oscillations**, the **frequency** of these pseudo-Rabi **oscillations** is much smaller than $\omega$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external perturbation not only at the resonance **frequency** $\omega$ and its subharmonics ($\omega/n$), but also at its overtones, $n\omega$.

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Contributors: Strauch, F. W., Dutta, S. K., Paik, Hanhee, Palomaki, T. A., Mitra, K., Cooper, B. K., Lewis, R. M., Anderson, J. R., Dragt, A. J., Lobb, C. J.

Date: 2007-03-02

The ac Stark shift Δ ω 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. The inset shows the **oscillation** **frequency** Ω ̄ R , 01 as a function of the level spacing ω 01 for I a c = 5.87 nA and the fit using ( rabif) to obtain Ω R , 01 and Δ ω 01 ....Experimental microwave spectroscopy of a Josephson phase **qubit**, scanned in **frequency** (vertical) and bias current (horizontal). Dark points indicate experimental microwave enhancement of the tunneling escape rate, while white dashed lines are quantum mechanical calculations of (from right to left) ω 01 , ω 02 / 2 , ω 12 , ω 13 / 2 , and ω 23 ....Rabi **frequency** Ω R , 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed lines are the lowest-order results ( rabi1) (top) and second-order ( rabi2) (bottom) perturbative results. The inset shows Rabi **oscillations** of the escape rate for I a c = 16.5 nA....Rabi **frequency** Ω R , 02 of the two-photon 0 2 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. Inset shows Rabi **oscillations** of the escape rate for I a c = 16.5 nA....Rabi **oscillations** have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (**qubits**) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase **qubit** (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi **frequency**, and two-photon Rabi **frequency** are compared to measurements made on a dc SQUID phase **qubit** with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment. ... Rabi **oscillations** have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (**qubits**) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase **qubit** (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi **frequency**, and two-photon Rabi **frequency** are compared to measurements made on a dc SQUID phase **qubit** with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment.

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Contributors: Chirolli, Luca, Burkard, Guido

Date: 2009-06-04

The QND character of the **qubit** measurement is studied by repeating the measurement. A perfect QND setup guarantees identical outcomes for the two repeated measurement with certainty. In order to fully characterize the properties of the measurement, we can initialize the **qubit** in the state | 0 , then rotate the **qubit** by applying a pulse of duration τ 1 before the first measurement and a second pulse of duration τ 2 between the first and the second measurement. The conditional probability to detect the **qubit** in the states s and s ' is expected to be independent of the first pulse, and to show sinusoidal **oscillation** with amplitude 1 in τ 2 . Deviations from this expectation witness a deviation from a perfect QND measurement. The sequence of **qubit** pulses and **oscillator** driving is depicted in Fig. Fig1b). The conditional probability P 0 | 0 to detect the **qubit** in the state "0" twice in sequence is plotted versus τ 1 and τ 2 in Fig. Fig1c) for Δ = 0 , and in Fig. Fig1d) for Δ / ϵ = 0.1 . We anticipate here that a dependence on τ 1 is visible when the **qubit** undergoes a flip in the first rotation. Such a dependence is due to the imperfections of the mapping between the **qubit** state and the **oscillator** state, and is present also in the case Δ = 0 . The effect of the non-QND term Δ σ X results in an overall reduction of P 0 | 0 ....We theoretically describe the weak measurement of a two-level system (**qubit**) and quantify the degree to which such a **qubit** measurement has a quantum non-demolition (QND) character. The **qubit** is coupled to a harmonic **oscillator** which undergoes a projective measurement. Information on the **qubit** state is extracted from the **oscillator** measurement outcomes, and the QND character of the measurement is inferred by the result of subsequent measurements of the **oscillator**. We use the positive operator valued measure (POVM) formalism to describe the **qubit** measurement. Two mechanisms lead to deviations from a perfect QND measurement: (i) the quantum fluctuations of the **oscillator**, and (ii) quantum tunneling between the **qubit** states $|0>$ and $|1>$ during measurements. Our theory can be applied to QND measurements performed on superconducting **qubits** coupled to a circuit **oscillator**....(Color online) Conditional probability to obtain a) s ' = s = 1 , b) s ' = - s = 1 , c) s ' = - s = - 1 , and d) s ' = s = - 1 for the case Δ t = Δ / ϵ = 0.1 and T 1 = 10 ~ n s , when rotating the **qubit** around the y axis before the first measurement for a time τ 1 and between the first and the second measurement for a time τ 2 , starting with the **qubit** in the state | 0 0 | . Correction in Δ t are up to second order. The harmonic **oscillator** is driven at resonance with the bare harmonic **frequency** and a strong driving together with a strong damping of the **oscillator** are assumed, with f / 2 π = 20 ~ G H z and κ / 2 π = 1.5 ~ G H z . Fig6...In Fig. Fig5 we plot the second order correction to the probability to obtain "1" having prepared the **qubit** in the initial state ρ 0 = | 0 0 | , corresponding to F 2 t , for Δ t = Δ / ϵ = 0.1 . We choose to plot only the deviation from the unperturbed probability because we want to highlight the contribution to spin-flip purely due to tunneling in the **qubit** Hamiltonian. In fact most of the contribution to spin-flip arises from the unperturbed probability, as it is clear from Fig. Fig3. Around the two **qubit**-shifted **frequencies**, the probability has a two-peak structure whose characteristics come entirely from the behavior of the phase ψ around the resonances Δ ω ≈ ± g . We note that the tunneling term can be responsible for a probability correction up to ∼ 4 % around the **qubit**-shifted **frequency**....We now investigate whether it is possible to identify the contribution of different mechanisms that generate deviations from a perfect QND measurement. In Fig. Fig7 we study separately the effect of **qubit** relaxation and **qubit** tunneling on the conditional probability P 0 | 0 . In Fig. Fig7 a) we set Δ = 0 and T 1 = ∞ . The main feature appearing is a sudden change of the conditional probability P → 1 - P when the **qubit** is flipped in the first rotation. This is due to imperfection in the mapping between the **qubit** state and the state of the harmonic **oscillator**, already at the level of a single measurement. The inclusion of a phenomenological **qubit** relaxation time T 1 = 2 ~ n s , intentionally chosen very short, yields a strong damping of the **oscillation** along τ 2 and washes out the response change when the **qubit** is flipped during the first rotation. This is shown in Fig. Fig7 b). The manifestation of the non-QND term comes as a global reduction of the visibility of the **oscillations**, as clearly shown in Fig. Fig7 c)....(Color online) Comparison of the deviations from QND behavior originating from different mechanisms. Conditional probability P 0 | 0 versus **qubit** driving time τ 1 and τ 2 starting with the **qubit** in the state | 0 0 | , for a) Δ = 0 and T 1 = ∞ , b) Δ = 0 and T 1 = 2 ~ n s , and c) Δ = 0.1 ~ ϵ and T 1 = ∞ . The **oscillator** driving amplitude is f / 2 π = 20 ~ G H z and a damping rate κ / 2 π = 1.5 ~ G H z is assumed. Fig7...For driving at resonance with the bare harmonic **oscillator** **frequency** ω h o , the state of the **qubit** is encoded in the phase of the signal, with φ 1 = - φ 0 , and the amplitude of the signal is actually reduced, as also shown in Fig. Fig3 for Δ ω = 0 . When matching one of the two **frequencies** ω i the **qubit** state is encoded in the amplitude of the signal, as also clearly shown in Fig. Fig3 for Δ ω = ± g . Driving away from resonance can give rise to significant deviation from 0 and 1 to the outcome probability, therefore resulting in an imprecise mapping between **qubit** state and measurement outcomes and a weak **qubit** measurement....(Color online) Schematic description of the single measurement procedure. In the bottom panel the coherent states | α 0 and | α 1 , associated with the **qubit** states | 0 and | 1 , are represented for illustrative purposes by a contour line in the phase space at HWHM of their Wigner distributions, defined as W α α * = 2 / π 2 exp 2 | α | 2 ∫ d β - β | ρ | β exp β α * - β * α . The corresponding Gaussian probability distributions of width σ centered about the **qubit**-dependent "position" x s are shown in the top panel. Fig2...The combined effect of the quantum fluctuations of the **oscillator** together with the tunneling between the **qubit** states is therefore responsible for deviation from a perfect QND behavior, although a major role is played, as expected, by the non-QND tunneling term. Such a conclusion pertains to a model in which the **qubit** QND measurement is studied in the regime of strong projective **qubit** measurement and **qubit** relaxation is taken into account only phenomenologically. We compared the conditional probabilities plotted in Fig. Fig6 and Fig. Fig7 directly to Fig. 4 in Ref. [...(Color online) a) Schematics of the 4-Josephson junction superconducting flux **qubit** surrounded by a SQUID. b) Measurement scheme: b1) two short pulses at **frequency** ϵ 2 + Δ 2 , before and between two measurements prepare the **qubit** in a generic state. Here, ϵ and Δ represent the energy difference and the tunneling amplitude between the two **qubit** states. b2) Two pulses of amplitude f and duration τ 1 = τ 2 = 0.1 ~ n s drive the harmonic **oscillator** to a **qubit**-dependent state. c) Perfect QND: conditional probability P 0 | 0 for Δ = 0 to detect the **qubit** in the state "0" vs driving time τ 1 and τ 2 , at Rabi **frequency** of 1 ~ G H z . The **oscillator** driving amplitude is chosen to be f / 2 π = 50 ~ G H z and the damping rate κ / 2 π = 1 ~ G H z . d) Conditional probability P 0 | 0 for Δ / ϵ = 0.1 , f / 2 π = 20 ~ G H z , κ / 2 π = 1.5 ~ G H z . A phenomenological **qubit** relaxation time T 1 = 10 ~ n s is assumed. Fig1 ... We theoretically describe the weak measurement of a two-level system (**qubit**) and quantify the degree to which such a **qubit** measurement has a quantum non-demolition (QND) character. The **qubit** is coupled to a harmonic **oscillator** which undergoes a projective measurement. Information on the **qubit** state is extracted from the **oscillator** measurement outcomes, and the QND character of the measurement is inferred by the result of subsequent measurements of the **oscillator**. We use the positive operator valued measure (POVM) formalism to describe the **qubit** measurement. Two mechanisms lead to deviations from a perfect QND measurement: (i) the quantum fluctuations of the **oscillator**, and (ii) quantum tunneling between the **qubit** states $|0>$ and $|1>$ during measurements. Our theory can be applied to QND measurements performed on superconducting **qubits** coupled to a circuit **oscillator**.

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Contributors: Poletto, S, Chiarello, F, Castellano, M G, Lisenfeld, J, Lukashenko, A, Carelli, P, Ustinov, A V

Date: 2009-10-23

Rabi **oscillation** of the double SQUID manipulated as a phase **qubit** by applying microwave pulses at 19 GHz. The **oscillation** **frequency** changes from 540 MHz to 1.2 GHz by increasing the power of the microwave signal by 10 dB....Probability of measuring the state | L as a function of the pulse duration. The coherent **oscillation** shown here has a **frequency** of 14 GHz and a coherence time of approximately 1.2 ns....Measurement of the relaxation time T 1 for the double SQUID operated as a phase **qubit**....Measured **oscillation** **frequencies** versus amplitude of the short flux pulse (full dots). The solid curve is a numerical simulation using the measured parameters of the circuit....We report on two different manipulation procedures of a tunable rf SQUID. First, we operate this system as a flux **qubit**, where the coherent evolution between the two flux states is induced by a rapid change of the energy potential, turning it from a double well into a single well. The measured coherent Larmor-like **oscillation** of the retrapping probability in one of the wells has a **frequency** ranging from 6 to 20 GHz, with a theoretically expected upper limit of 40 GHz. Furthermore, here we also report a manipulation of the same device as a phase **qubit**. In the phase regime, the manipulation of the energy states is realized by applying a resonant microwave drive. In spite of the conceptual difference between these two manipulation procedures, the measured decay times of Larmor **oscillation** and microwave-driven Rabi **oscillation** are rather similar. Due to the higher **frequency** of the Larmor **oscillations**, the microwave-free **qubit** manipulation allows for much faster coherent operations....Measured Rabi **oscillation** **frequency** versus the normalized amplitude of the microwave signal (solid dots). The dashed line is a linear fit taking into account slightly off-resonance microwave field, while the fit represented by the solid line considers a population of higher excited states. ... We report on two different manipulation procedures of a tunable rf SQUID. First, we operate this system as a flux **qubit**, where the coherent evolution between the two flux states is induced by a rapid change of the energy potential, turning it from a double well into a single well. The measured coherent Larmor-like **oscillation** of the retrapping probability in one of the wells has a **frequency** ranging from 6 to 20 GHz, with a theoretically expected upper limit of 40 GHz. Furthermore, here we also report a manipulation of the same device as a phase **qubit**. In the phase regime, the manipulation of the energy states is realized by applying a resonant microwave drive. In spite of the conceptual difference between these two manipulation procedures, the measured decay times of Larmor **oscillation** and microwave-driven Rabi **oscillation** are rather similar. Due to the higher **frequency** of the Larmor **oscillations**, the microwave-free **qubit** manipulation allows for much faster coherent operations.

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Contributors: Lisenfeld, Juergen, Mueller, Clemens, Cole, Jared H., Bushev, Pavel, Lukashenko, Alexander, Shnirman, Alexander, Ustinov, Alexey V.

Date: 2009-09-18

For maximum generality, we first define a minimal model needed to describe the splitting of Fig. fig:Splitting. To this end, we restrict ourselves to the lowest two states of the phase **qubit** circuit (the **qubit** subspace) and disregard the longitudinal coupling ∝ τ z . Within the rotating wave approximation (RWA) the Hamiltonian reads...(color online) (a) Analytically obtained transition spectrum of the Hamiltonian ( eq:H4Levels) in the minimal model for Ω q / h = 40 MHz and v ⊥ / h = 25 MHz. Dashed-dotted lines show the transition **frequencies** while the gray-scale intensity of the thicker lines indicates the weight of the respective Fourier-components in the probability P . The system shows a symmetric response as a function of the detuning δ ω . Two of the four lines are double degenerate. (b) The same as (a) but including the second order Raman process with Ω f v = v ⊥ Ω q / Δ . The two degenerate transitions in (a) split and the symmetry of the response is broken. Inset: Schematic representation of the structure of the Hamiltonian ( eq:H4Levels). We denote the ground and excited states of the **qubit** as and and those of the TLF as and . Arrows indicate the couplings between **qubit** and fluctuator v ⊥ and to the microwave field Ω q and Ω f v ....The sample investigated in this study is a phase **qubit** , consisting of a capacitively shunted Josephson junction embedded in a superconducting loop. Its potential energy has the form of a double well for suitable combinations of the junction’s critical current (here, I c = 1.1 μ A) and loop inductance (here, L = 720 pH). For the **qubit** states, one uses the two Josephson phase eigenstates of lowest energy which are localized in the shallower of the two potential wells, whose depth is controlled by the external magnetic flux through the **qubit** loop. The **qubit** state is controlled by an externally applied microwave pulse, which in our sample is coupled capacitively to the Josephson junction. A schematic of the complete **qubit** circuit is depicted in Fig. fig:Splitting(a). Details of the experimental setup can be found in Ref. ...(color online) (a) Schematic of the phase **qubit** circuit. (b) Probability to measure the excited **qubit** state (color-coded) vs. bias flux and microwave **frequency**, revealing the coupling to a two-level defect state having a resonance **frequency** of 7.805 GHz (indicated by a dashed line)....superconducting **qubits**, Josephson junctions, two-level
fluctuators, microwave spectroscopy, Rabi **oscillations**
...(color online) (a) Experimentally observed time evolution of the probability to measure the **qubit** in the excited state, P t , vs. driving **frequency**; (b) Fourier-transform of the experimentally observed P t . The resonance **frequency** of the TLF is indicated by vertical lines. (c) Time evolution of P and (d) its Fourier-transform obtained by the numerical solution of Eq. ( eq:master_eq) as described in the text, taking into account also the next higher level in the **qubit**. (As the anharmonicity Δ / h ∼ 100 MHz in our circuit is relatively small, this required going beyond the second order perturbation theory and analyze the 6-level dynamics explicitly). The ** qubit’s** Rabi

**frequency**Ω q / h is set to 48 MHz....Superconducting

**qubits**often show signatures of coherent coupling to microscopic two-level fluctuators (TLFs), which manifest themselves as avoided level crossings in spectroscopic data. In this work we study a phase

**qubit**, in which we induce Rabi

**oscillations**by resonant microwave driving. When the

**qubit**is tuned close to the resonance with an individual TLF and the Rabi driving is strong enough (Rabi

**frequency**of order of the

**qubit**-TLF coupling), interesting 4-level dynamics are observed. The experimental data shows a clear asymmetry between biasing the

**qubit**above or below the fluctuator's level-splitting. Theoretical analysis indicates that this asymmetry is due to an effective coupling of the TLF to the external microwave field induced by the higher

**qubit**levels....Spectroscopic data taken in the whole accessible

**frequency**range between 5.8 GHz and 8.1 GHz showed only 4 avoided level crossings due to TLFs having a coupling strength larger than 10 MHz, which is about the spectroscopic resolution given by the linewidth of the

**qubit**transition. In this work, we studied the

**qubit**interacting with a fluctuator whose energy splitting was ϵ f / h = 7.805 GHz. From its spectroscopic signature shown in Fig. fig:Splitting(b), we extract a coupling strength v ⊥ / h ≈ 25 MHz. The coherence times of this TLF were measured by directly driving it at its resonance

**frequency**while the

**qubit**was kept detuned. A π pulse was applied to measure the energy relaxation time T 1 , f ≈ 850 ns, while two delayed π / 2 pulses were used to measure the dephasing time T 2 , f * ≈ 110 ns in a Ramsey experiment. To read out the resulting TLF state, the

**qubit**was tuned into resonance with the TLF to realize an iSWAP gate, followed by a measurement of the

**excited state....where δ ω = ϵ q - ϵ f . The level structure and the spectrum of possible transitions in the Hamiltonian ( eq:H4Levels) is illustrated in Fig. fig:Transitionsa. The transition**

**qubit**’s**frequencies**in the rotating frame correspond to the

**frequencies**of the Rabi

**oscillations**observed experimentally....Figure fig:DataRabi(a) shows a set of time traces of P taken for different microwave drive

**frequencies**. Each trace was recorded after adjusting the

**qubit**bias to result in an energy splitting resonant to the chosen microwave

**frequency**. The Fourier transform of this data, shown in Fig. fig:DataRabi(b), allows us to distinguish several

**frequency**components.

**Frequency**and visibility of each component depend on the detuning between

**qubit**and TLF. We note a striking asymmetry between the Fourier components appearing for positive and negative detuning of the

**qubit**relative to the TLF’s resonance

**frequency**, which is indicated in Figs. fig:DataRabi(a,b) by the vertical lines at 7.805 GHz. We argue below that this asymmetry is due to virtual Raman-transitions involving higher levels in the

**qubit**. ... Superconducting

**qubits**often show signatures of coherent coupling to microscopic two-level fluctuators (TLFs), which manifest themselves as avoided level crossings in spectroscopic data. In this work we study a phase

**qubit**, in which we induce Rabi

**oscillations**by resonant microwave driving. When the

**qubit**is tuned close to the resonance with an individual TLF and the Rabi driving is strong enough (Rabi

**frequency**of order of the

**qubit**-TLF coupling), interesting 4-level dynamics are observed. The experimental data shows a clear asymmetry between biasing the

**qubit**above or below the fluctuator's level-splitting. Theoretical analysis indicates that this asymmetry is due to an effective coupling of the TLF to the external microwave field induced by the higher

**qubit**levels.

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Contributors: Greenberg, Ya. S.

Date: 2003-03-04

Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Here we present the results of detailed computer simulations of the interaction of a classical object (resonant tank circuit) with a quantum object (phase **qubit**). We explicitly account for the back action of a tank circuit and for the unpredictable nature of outcome of a single measurement. According to the results of our simulations the Rabi **oscillations** in MHz range can be detected using conventional NMR pulse Fourier technique....It is clearly seen that B **oscillates** with gap **frequency**, while the **frequency** of A is almost ten times smaller: (**oscillation** period of B: T B ≈ 2 × 10 -9 s, while the same quantity for A is T A ≈ 2 × 10 -8 s. The small distortions on A curve are due to a strong deviation of excitation signal from transverse rotating wave form, while B curve is clearly modulated with Rabi **frequency** Ω R (Fig. fig2)....As is seen from the Fig. fig3, A decays to 0.5 **oscillating** with Rabi **frequency**, while B (C) decays to zero. (Note: to be rigorous, the stable state solution for A is...Phase **qubit** coupled to a tank circuit....As the coupling is increased further the **qubit** wave function is completely destroyed. The quantity A is quenched to approximately 0.85 (Fig. fig10a). That is | C - | ≈ 0.92 . It might seem that we have here so called Zeno effect- as if **qubit** state is frozen in its ground state. However, in case of a strong coupling it is not correct to say about wave function of the **qubit** alone. This is shown in Fig. figcoh where for λ > 10 -2 the phase coherence is seen to be completely lost ....At every graph of the figures the results for one realization of random number generator ξ t are compared with the case when we replaced F ξ t in ( Q) with deterministic term 2 A - 1 - 2 C / 2 , which means that the tank measures the average current ( avcurr) in a **qubit** loop. As is seen from the Fig. fig4a, A **oscillates** with Rabi **frequency**. The voltage across tank circuit **oscillates** also with Rabi **frequency** which is equal to 50 MHz in our case (Fig. fig4b) which is modulated with the lower **frequency** the value of which is about 5 MHz....Phase loss-free **qubit** coupled to a loss-free tank circuit. **Oscillations** of A. Deterministic case (a) together with one realization (b) are shown. Small scale time **oscillations** correspond to Rabi **frequency**....Phase loss-free **qubit** coupled to a dissipative tank circuit. The evolution of A exhibits modulation of Rabi **oscillations** with lower **frequency**. Deterministic case (a) together with one realization (b) are shown....In conclusion we want to show the effect of **qubit** evolution as the coupling between the **qubit** and the tank is increased. We numerically solved the system consisting of the loss-free **qubit** coupled to the dissipative tank circuit. The system is described by Eqs. ( A2, B2, C2, flux_tank) and Eq. ( Q1). For the simulations we take the coupling parameter λ = 2.5 × 10 -2 . The results of simulations are shown on Figs. fig10a, fig10d for deterministic case. As is seen from the Figs. fig10a during Rabi period the quantity A became partially frozen at some level. At the endpoints of this period the system tries to escape to another level of A. Between the endpoints of Rabi period A **oscillates** with a high **frequency** which is about 10 GHz in our case. As expected, the evolution of B is suppressed approximately by a factor of ten below its free evolution amplitude which is equal to 0.5. As we show below, the strong coupling completely destroys the phase coherence between **qubit** states, nevertheless the voltage across the tank **oscillates** with Rabi **frequency**. Its amplitude is considerably increased and it does not reveal any peculiarities associated with the frozen behavior of A (Fig. fig10d)....Phase loss-free **qubit** coupled to a dissipative tank circuit. The voltage across the tank exhibits modulation of Rabi **frequency**. Deterministic case (a) together with one realization (b) are shown....Time evolution of A and B for **qubit** without dissipation. ... Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Here we present the results of detailed computer simulations of the interaction of a classical object (resonant tank circuit) with a quantum object (phase **qubit**). We explicitly account for the back action of a tank circuit and for the unpredictable nature of outcome of a single measurement. According to the results of our simulations the Rabi **oscillations** in MHz range can be detected using conventional NMR pulse Fourier technique.

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Contributors: Yoshihara, Fumiki, Nakamura, Yasunobu, Yan, Fei, Gustavsson, Simon, Bylander, Jonas, Oliver, William D., Tsai, Jaw-Shen

Date: 2014-02-06

Parameters in calculations and measurements in units of GHz. In the first column, cal: δ ω Ω R 0 stands for the calculation to study the shift of the resonant **frequency**, and cal: Γ R s t δ ω m w stands for the calculation to study the decay of Rabi **oscillations** due to quasistatic flux noise. “Optimal" in the last column means that at each ε m w , ω m w is chosen to minimize dephasing due to quasistatic flux noise....We infer the high-**frequency** flux noise spectrum in a superconducting flux **qubit** by studying the decay of Rabi **oscillations** under strong driving conditions. The large anharmonicity of the **qubit** and its strong inductive coupling to a microwave line enabled high-amplitude driving without causing significant additional decoherence. Rabi **frequencies** up to 1.7 GHz were achieved, approaching the **qubit**'s level splitting of 4.8 GHz, a regime where the rotating-wave approximation breaks down as a model for the driven dynamics. The spectral density of flux noise observed in the wide **frequency** range decreases with increasing **frequency** up to 300 MHz, where the spectral density is not very far from the extrapolation of the 1/f spectrum obtained from the free-induction-decay measurements. We discuss a possible origin of the flux noise due to surface electron spins....(Color online) Rabi **oscillation** curves with different Rabi **frequencies** Ω R measured at different static flux bias ε . At each Ω R , δ ω m w is chosen to minimize dephasing due to quasistatic flux noise. The red lines are the fitting curves. In the measurements shown in the middle and bottom panels, only parts of the **oscillations** are monitored so that we can save measurement time while the envelopes of Rabi **oscillations** are captured. The inset is a magnification of the data in the bottom panel together with the fitting curve....In the Rabi **oscillation** measurements, a microwave pulse is applied to the **qubit** followed by a readout pulse, and P s w as a function of the microwave pulse length is measured. First, we measure the Rabi **oscillation** decay at ε = 0 , where the quasistatic noise contribution is negligible. Figure GRfR1p5(d) shows the measured 1 / e decay rate of the Rabi **oscillations** Γ R 1 / e as a function of Ω R 0 . For Ω R 0 / 2 π up to 400 MHz, Γ R 1 / e is approximately 3 Γ 1 / 4 , limited by the energy relaxation, and S Δ Ω R 0 is negligible. For Ω R 0 / 2 π from 600 MHz to 2.2 GHz, Γ R 1 / e > 3 Γ 1 / 4 . A possible origin of this additional decoherence is fluctuations of ε m w , δ ε m w : Ω R 0 is first order sensitive to δ ε m w , which is reported to be proportional to ε m w itself. Next, the decay for the case ε ≈ Δ is studied. To observe the contribution from quasistatic flux noise, the Rabi **oscillation** decay as a function of ω m w is measured, where the contribution from the other sources is expected to be almost constant. Figure GRfR1p5(b) shows Γ R 1 / e at ε / 2 π = 4.16 GHz as a function of δ ω m w while keeping Ω R / 2 π between 1.5 and 1.6 GHz. Besides the offset and scatter, the trend of Γ R 1 / e agrees with that of the simulated Γ R s t . This result indicates that numerical calculation properly evaluates δ ω m w minimizing Γ R s t . Finally, the decay for the case ε ≈ Δ as a function of ε m w , covering a wide range of Ω R , is measured (Fig. Rabis)....(Color online) Power spectrum density of flux fluctuations S n φ ω extracted from the Rabi **oscillation** measurements in the first ( ε / 2 π = 4.16 GHz) and second cooldowns. The PSDs obtained from the spin-echo and energy relaxation measurements in the second cooldown are also plotted. The black solid line is the 1/ f spectrum extrapolated from the FID measurements in the second cooldown. The purple dashed line is the estimated Johnson noise from a 50 Ω microwave line coupled to the **qubit** by a mutual inductance of 1.2 pH and nominally cooled to 35 mK. The pink dotted line is a Lorentzian, S n φ m o d e l ω = S h ω w 2 / ω 2 + ω w 2 , and the orange solid line is the sum of the Lorentzian and the Johnson noise. Here the parameters are S h = 3.6 × 10 -19 r a d -1 s and ω w / 2 π = 2.7 × 10 7 H z ....Josephson devices, decoherence, Rabi **oscillation**, $1/f$ noise...(Color online) (a) Numerically calculated shift of the resonant **frequency** δ ω (black open circles) and the Bloch–Siegert shift δ ω B S (blue line). (b) Numerically calculated decay rate Γ R s t (black open circles) and Rabi **frequency** Ω R (red solid triangles) as functions of the detuning δ ω m w from ω 01 . The purple solid line is a fit based on Eq. ( fRfull). The measured 1/ e decay rates Γ R 1 / e at ε / 2 π = 4.16 GHz for the range of Rabi **frequencies** Ω R / 2 π between 1.5 and 1.6 GHz (blue solid circles) are also plotted. (c) Calculated Rabi **frequency** Ω R , based on Eq. ( fRfull), as a function of ε for the cases (i) ω m w = ω 01 + δ ω (black solid line) and (ii) ω m w / 2 π = 6.1 GHz (red dashed line). The upper axis indicates ω 01 , corresponding to ε in the bottom axis. (d) The measured 1 / e decay rate of the Rabi **oscillations**, Γ R 1 / e , at ε = 0 and as a function of Ω R 0 . The red solid line indicates 3 4 Γ 1 obtained independently....The condition, ∂ Ω R / ∂ ε = 0 , is satisfied when ε = 0 or δ ω m w = δ ω - Ω R 0 2 / ω 01 . For Ω R 0 / 2 π = 1.52 GHz and ω 01 / 2 π = 6.400 GHz, the latter condition is calculated to be δ ω m w / 2 π = - 295 MHz, slightly different from the minimum of Γ R s t seen in Fig. G R f R 1 p 5 (b). The difference is due to the deviation from the linear approximation in Eq. ( fRfull), Ω R 0 ∝ ε m w / ω 01 . Figure GRfR1p5(c) shows the calculation of Ω R as a function of ε , based on Eq. ( fRfull). The Rabi **frequency** Ω R 0 at the shifted resonance decreases as ε increases, while Ω R , for a fixed microwave **frequency** of ω m w / 2 π = 6.1 GHz, has a minimum of approximately ω 01 / 2 π = 6.4 GHz. Here in the first order, Ω R is insensitive to the fluctuation of ε ....In Fig. GRfR1p5(a), δ ω as a function of Ω R 0 is plotted together with the well-known Bloch–Siegert shift, δ ω B S = 1 4 Ω R 0 2 ω 01 , obtained from the second-order perturbation theory. Fixed parameters for the calculation are Δ / 2 π = 4.869 and ε / 2 π = 4.154 GHz ( ω 01 / 2 π = 6.400 GHz). We find that δ ω B S overestimates δ ω when Ω R 0 / 2 π 800 MHz. The deviation from the Bloch–Siegert shift is due to the component of the ac flux drive that is parallel to the ** qubit’s** energy eigenbasis; this component is not averaged out when Ω R is comparable to ω m w . ... We infer the high-

**frequency**flux noise spectrum in a superconducting flux

**qubit**by studying the decay of Rabi

**oscillations**under strong driving conditions. The large anharmonicity of the

**qubit**and its strong inductive coupling to a microwave line enabled high-amplitude driving without causing significant additional decoherence. Rabi

**frequencies**up to 1.7 GHz were achieved, approaching the

**qubit**'s level splitting of 4.8 GHz, a regime where the rotating-wave approximation breaks down as a model for the driven dynamics. The spectral density of flux noise observed in the wide

**frequency**range decreases with increasing

**frequency**up to 300 MHz, where the spectral density is not very far from the extrapolation of the 1/f spectrum obtained from the free-induction-decay measurements. We discuss a possible origin of the flux noise due to surface electron spins.

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Contributors: Higgins, Kieran D. B., Lovett, Brendon W., Gauger, Erik M.

Date: 2012-03-27

Our methodology can be used to predict dynamics of nanomechanical resonators connected to either quantum dots or superconducting **qubits**. The criterion for the single term approximation to be valid is readily met by current experiments such as those presented in Refs. and their parameters yield near perfect agreement between numerical and analytic results. Most experiments operate in a regime where the **qubit** dynamics are not greatly perturbed by the presence of the **oscillator**, which has a much lower **frequency** ( ϵ ≈ Δ ≈ 10 GHz, ω = 1 GHz). In Figure fig:1, we chose ϵ ≈ Δ ≈ 100 MHz, because this better demonstrates the effect of the **oscillator** on the **qubit**. These parameters can be achieved experimentally using the same **qubit** design but with an **oscillating** voltage applied to the CPB bias gate . However, we stress the accuracy of our method is not restricted to this regime....fig:2 Main panel: comparison of dynamics calculated from truncating ( Sin5) at N M A X = ± 10 (red) and a numerically exact approach (blue). Lower left: Fourier transform of the dynamics. Lower right: the numerical weight of the n t h term in the series expansion of ( Sin5), showing there are still only two dominant **frequencies** at n = 0 and n = - 1 . Parameters: ω = 0.5 , g = 0.1 , ϵ = 0 , Δ = 0.5 , T = 1 ~ m K , ℏ = 1 and k b = 1 ....Figure fig:3 demonstrates this idea, showing that by measuring Ω and fitting it to our expression ( eqn:rho3), we can obtain submilli-Kelvin precision in the experimentally relevant regime of 20-55 mK. At low temperatures the single term **frequency** plateaus, causing the accuracy to break down. In the higher temperature limit, we also see a deviation from the diagonal, this is to be expected as we leave the regime of validity described by ( eqn:crit). Naturally accuracy in this region could be improved by retaining higher order terms in ( Sin5), but this would become a more numeric than analytic approach. The upper inset shows the dependence of the accuracy of the prediction on the number of points (at a separation of 1ns) sampled from the dynamics. The accuracy increases initially as more points improve the fitted value of Ω , however after a certain length the accuracy is diminished by long term envelope effects in the dynamics not captured by the single term approximation. We note that the corresponding analysis in the **frequency** domain would not be equally affected by the long time envelope, however a large number of points in the FFT is then required in order to obtain the desired accuracy. The lower inset of Figure fig:3 shows the direct dependence of Ω on the temperature. The temperature range with steepest gradient and hence greatest **frequency** dependence on temperature varies with the coupling strength; thus the device could be specifically designed to have a maximal sensitivity in the temperature range of the most interest....Figure fig:1 shows a comparison of the dynamics predicted using these expressions and a numerically exact approach. The latter are obtained by imposing a truncation of the **oscillator** Hilbert space at a point where the dynamics have converged and any higher modes have an extremely low occupation probability. Our zeroth order approximation proves to be unexpectedly powerful, giving accurate dynamics well into the strong coupling regime ( g / ω = 0.25 ) and even beyond this it still captures the dominant oscillatory behaviour, see Figure fig:1. Stronger coupling increases the numerical weight of higher **frequency** terms in the series, causing a modulation of the dynamics. The approximation starts to break down at ( g / ω = 0.5 ). The equations ( eqn:rho0) and ( eqn:rho1) are obviously unable to capture the higher **frequency** modulations to the dynamics or any potential long time phenomena like collapse and revival, but these are unlikely to be resolvable in experiments in any case. Nonetheless, it is worth pointing out that even in this strong coupling case the base **frequency** of the **qubit** dynamics is still adequately captured by our single term approximation....fig:1 Comparison of the single term approximation (red) and a numerically exact approach (blue) for different coupling strengths. Uncoupled Rabi **oscillations** are also shown as a reference (green). Left: the population ρ 00 t in the time-domain. Right: the same data in the **frequency** domain. The full numerical solution was Fourier transformed using Matlab’s FFT algorithm. Other parameters are ω = 1 GHz, g = 0.1 GHz, ϵ = Δ = 100 MHz and T = 10 mK....fig:3 Demonstration of **qubit** thermometry: T i n is the temperature supplied to the numerical simulation of the system and T o u t is the temperature that would be predicted by fitting **oscillations** with **frequency** ( eqn:rho3) to it. The blue line is the data and red line shows the effect of a 10kHz error in the **frequency** measurement; the grey dashed line serves as a guide to the eye. The lower inset shows the variation of the **qubit** **frequency** Ω with temperature. The upper inset shows the dependence of the absolute error in the prediction against the signal length (see text). Other parameters are: ω = 1 GHz, g = 0.01 GHz, ϵ = 0 , Δ = 100 MHz...A quantum two level system coupled to a harmonic **oscillator** represents a ubiquitous physical system. New experiments in circuit QED and nano-electromechanical systems (NEMS) achieve unprecedented coupling strength at large detuning between **qubit** and **oscillator**, thus requiring a theoretical treatment beyond the Jaynes Cummings model. Here we present a new method for describing the **qubit** dynamics in this regime, based on an **oscillator** correlation function expansion of a non-Markovian master equation in the polaron frame. Our technique yields a new numerical method as well as a succinct approximate expression for the **qubit** dynamics. We obtain a new expression for the ac Stark shift and show that this enables practical and precise **qubit** thermometry of an **oscillator**....Including extra terms in the series expansion ( Sin5) makes the time dependence of the **qubit** dynamics analytically unwieldy, because the rational function form of the series leads to a complex interdepence of the positions of the poles in ( eqn:rsol1). However, if the values of the parameters are known the series can truncated at ( ± N M A X ) to give an efficient numerical method to obtain more accurate dynamics, extending the applicability of our approach beyond the regime described by ( eqn:crit). This is demonstrated in Fig. fig:2, where the dynamics are clearly dominated by two **frequencies** – an effect that could obviously never be captured by a single term approximation. There is a qualitative agreement between the many terms expansion and full numerical solution, particularly at short times. We would not expect a perfect agreement in this case because the simulations are of the dynamics in the large tunnelling regime ( Δ = 0.5 ), and the polaron transform makes the master equation perturbative in this parameter. Nonetheless, the rapid convergence of the series is shown in Fig. fig:2; N M A X = 5 - 10 is sufficient to calculate ρ 00 t and ρ 10 t with an accuracy only limited by the underlying Born Approximation. The asymmetry of the amplitudes of the terms in the series expansion of ( Sin5) is due to the exponential functions in the series. ... A quantum two level system coupled to a harmonic **oscillator** represents a ubiquitous physical system. New experiments in circuit QED and nano-electromechanical systems (NEMS) achieve unprecedented coupling strength at large detuning between **qubit** and **oscillator**, thus requiring a theoretical treatment beyond the Jaynes Cummings model. Here we present a new method for describing the **qubit** dynamics in this regime, based on an **oscillator** correlation function expansion of a non-Markovian master equation in the polaron frame. Our technique yields a new numerical method as well as a succinct approximate expression for the **qubit** dynamics. We obtain a new expression for the ac Stark shift and show that this enables practical and precise **qubit** thermometry of an **oscillator**.

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Contributors: Grajcar, M., Izmalkov, A., Il'ichev, E., Wagner, Th., Oukhanski, N., Huebner, U., May, T., Zhilyaev, I., Hoenig, H. E., Greenberg, Ya. S.

Date: 2003-03-31

In conclusion, we have observed resonant tunneling in a macroscopic superconducting system, containing an Al flux **qubit** and a Nb tank circuit. The latter played dual control and readout roles. The impedance readout technique allows direct characterization of some of the ** qubit’s** quantum properties, without using spectroscopy. In a range 50 ∼ 200 mK, the effective

**qubit**temperature has been verified [Fig. fig:Temp_dep(b)] to be the same as the mixing chamber’s (after Δ has been determined at low T ), which is often difficult to confirm independently....Our technique is similar to rf-SQUID readout. The

**qubit**loop is inductively coupled to a parallel resonant tank circuit [Fig. fig:schem(b)]. The tank is fed a monochromatic rf signal at its resonant

**frequency**ω T . Then both amplitude v and phase shift χ (with respect to the bias current I b ) of the tank voltage will strongly depend on (A) the shift in resonant

**frequency**due to the change of the effective

**qubit**inductance by the tank flux, and (B) losses caused by field-induced transitions between the two

**qubit**states. Thus, the tank both applies the probing field to the

**qubit**, and detects its response....We have observed signatures of resonant tunneling in an Al three-junction

**qubit**, inductively coupled to a Nb LC tank circuit. The resonant properties of the tank

**oscillator**are sensitive to the effective susceptibility (or inductance) of the

**qubit**, which changes drastically as its flux states pass through degeneracy. The tunneling amplitude is estimated from the data. We find good agreement with the theoretical predictions in the regime of their validity....(a) Tank phase shift vs flux bias near degeneracy and for V d r = 0.5 ~ μ V. From the lower to the upper curve (at f x = 0 ) the temperature is 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 350, 400 mK. (b) Normalized amplitude of tan χ (circles) and tanh Δ / k B T (line), for the Δ following from Fig. fig:Bias_dep; the overall scale κ is a fitting parameter. The data indicate a saturation of the effective

**qubit**temperature at 30 mK. (c) Full dip width at half the maximum amplitude vs temperature. The horizontal line fits the low- T ( < 200 mK) part to a constant; the sloped line represents the T 3 behavior observed empirically for higher T ....(a) Quantum energy levels of the

**qubit**vs external flux. The dashed lines represent the classical potential minima. (b) Phase

**qubit**coupled to a tank circuit....-dependence of ϵ t is adiabatic. However, it does remain valid if the full (Liouville) evolution operator of the

**qubit**would contain standard Bloch-type relaxation and dephasing terms (which indeed are not probed) in addition to the Hamiltonian dynamics ( eq01), since the fluctuation–dissipation theorem guarantees that such terms do not affect equilibrium properties. Normalized dip amplitudes are shown vs T in Fig. fig:Temp_dep(b) together with tanh Δ / k B T , for Δ / h = 650 MHz independently obtained above from the low- T width. The good agreement strongly supports our interpretation, and is consistent with Δ being T...Δ is the tunneling amplitude. At bias ϵ = 0 the two lowest energy levels of the

**qubit**anticross [Fig. fig:schem(a)], with a gap of 2 Δ . Increasing ϵ slowly enough, the

**qubit**can adiabatically transform from Ψ l to Ψ r , staying in the ground state E - . Since d E - / d Φ x is the persistent loop current, the curvature d 2 E - / d Φ x 2 is related to the

**susceptibility. Hence, near degeneracy the latter will have a peak, with a width given by | ϵ | Δ . We present data demonstrating such behavior in an Al 3JJ**

**qubit**’s**qubit**. ... We have observed signatures of resonant tunneling in an Al three-junction

**qubit**, inductively coupled to a Nb LC tank circuit. The resonant properties of the tank

**oscillator**are sensitive to the effective susceptibility (or inductance) of the

**qubit**, which changes drastically as its flux states pass through degeneracy. The tunneling amplitude is estimated from the data. We find good agreement with the theoretical predictions in the regime of their validity.

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Contributors: Shi, Zhan, Simmons, C. B., Ward, Daniel. R., Prance, J. R., Mohr, R. T., Koh, Teck Seng, Gamble, John King, Wu, Xian., Savage, D. E., Lagally, M. G.

Date: 2012-08-02

fig:plots5 Analysis of echo data for extraction of the decoherence time T 2 . (a–c) Transconductance G L as a function of the base level of detuning ε and δ t (defined in the main text) for total free evolution times of t = 390 ps, t = 690 ps, and t = 990 ps, respectively. (d–f) Fourier transforms of the charge occupation P 1 2 as a function of detuning ε and **oscillation** **frequency** f for the data in (a–c), respectively. We obtain P 1 2 (not shown here) by integrating the transconductance data in (a–c) and normalizing by noting that the total charge transferred across the polarization line is one electron. Fast Fourier transforming the time-domain data of P 1 2 allows us to quantify the amplitude of the **oscillations** visible near δ t = 0 . The **oscillations** of interest appear as weight in the FFT that moves to higher **frequency** at more negative detuning (farther from the anti-crossing). For an individual detuning energy, the FFT has nonzero weight for a nonzero bandwidth. (g) Echo amplitude as a function of free evolution time t . The data points (dark circles) are obtained at ε = - 120 μ eV by integrating a horizontal line cut of the FFT data over a bandwidth range of 46 - 72 GHz, then normalizing by the echo **oscillation** amplitude of the first data point, as described in the supplemental text. The echo **oscillation** amplitudes, plotted for multiple free evolution times, decay with characteristic time T 2 as the free evolution time t is made longer. By fitting the decay to a Gaussian, we obtain T 2 = 760 ± 190 ps. (h–j) Fourier transforms of the transconductance G L as a function of ε and **oscillation** **frequency** f for (a–c), respectively. As t is increased, the magnitude for **oscillations** at a given **frequency** decays with characteristic time T 2 . We take the magnitude of the FFT at the point where the central feature (black line) intersects 65 GHz. (k) Measured FFT magnitudes at 65 GHz for multiple free evolution times (dark circles) with a Gaussian fit (red line), which yields T 2 = 620 ± 140 ps, in reasonable agreement with the result shown in (g)....Fast quantum **oscillations** of a charge **qubit** in a double quantum dot fabricated in a Si/SiGe heterostructure are demonstrated and characterized experimentally. The measured inhomogeneous dephasing time T2* ranges from 127ps to ~2.1ns; it depends substantially on how the energy difference of the two **qubit** states varies with external voltages, consistent with a decoherence process that is dominated by detuning noise(charge noise that changes the asymmetry of the **qubit**'s double-well potential). In the regime with the shortest T2*, applying a charge-echo pulse sequence increases the measured inhomogeneous decoherence time from 127ps to 760ps, demonstrating that low-**frequency** noise processes are an important dephasing mechanism. ... Fast quantum **oscillations** of a charge **qubit** in a double quantum dot fabricated in a Si/SiGe heterostructure are demonstrated and characterized experimentally. The measured inhomogeneous dephasing time T2* ranges from 127ps to ~2.1ns; it depends substantially on how the energy difference of the two **qubit** states varies with external voltages, consistent with a decoherence process that is dominated by detuning noise(charge noise that changes the asymmetry of the **qubit**'s double-well potential). In the regime with the shortest T2*, applying a charge-echo pulse sequence increases the measured inhomogeneous decoherence time from 127ps to 760ps, demonstrating that low-**frequency** noise processes are an important dephasing mechanism.

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