### 56121 results for qubit oscillator frequency

Contributors: Bertet, P., Chiorescu, I., Harmans, C. J. P. M, Mooij, J. E.

Date: 2005-07-13

Contrary to the shift produced by the linear coupling term, the sign of this **frequency** shift now depends on ϵ . Since g 2 is negative (see figure fig:couplings), δ ν 0 2 actually has the same sign as ϵ . We also note that the quadratic term has no effect on the **qubit** when ϵ = 0 , since at that point the average flux generated by both **qubit** states | 0 and | 1 averages out to zero so that the SQUID Josephson inductance is unchanged....We will now discuss quantitatively the behaviour of g 1 and g 2 for actual sample parameters : I C = 3.4 μ A , M = 6.5 p H , I p = 240 n A , Δ = 5.5 G H z , ν p = 3.1 G H z , L J = 300 p H , f ' / 2 = 1.45 π . We will restrict ourselves to a range of bias conditions relevant for our conditions, supposing that I b varies between ± 300 n A and that f ' / 2 varies by d f ' = ± 4 ⋅ 10 -3 π around 1.45 π . We chose such an interval for f ' because it corresponds to changing the **qubit** bias point ϵ by ± 2 G H z around 0 . The constants g 1 and g 2 are plotted in figure fig:couplings as a function of I b for two different values of f ' ( g 1 is shown as a full line, g 2 as a dashed line, and the two different values of f ' are symbolized by gray for d f ' = - 2 π 4 ⋅ 10 -3 and black for d f ' = 0 ). It can be seen that the coupling constants only weakly depend on the value of the flux in this range, so that we will neglect this dependence in the following and consider that g 1 and g 2 only depend on the bias current I b . Moreover we see from figure fig:couplings that the approximations made in equation eq:g1g2approx are justified in this range of parameters since g 1 is closely linear in I b and g 2 nearly constant. We also note that g 1 = 0 for I b = 0 . This fact can be generalized to the case where the SQUID-**qubit** coupling is not symmetric and the junctions critical current are dissimilar : in certain conditions these asymmetries can be compensated for by applying a bias current I b * for which g 1 I b * = 0 . At the current I b * , the **qubit** is effectively decoupled from the measuring circuit fluctuations to first order....(a) **qubit** biased by Φ x and SQUID biased by current I b . (b) Simplified electrical scheme : the SQUID-**qubit** system is seen as an inductance L J connected to the shunct capacitor C s h through inductance L s h . Φ a is the flux across the two inductances L J and L s h in series....**Qubit** **frequency** ν q as a function of the bias ϵ for Δ = 5.5 G H z (minimum **frequency** in the figure). The dashed line indicates the phase-noise insensitive bias point ϵ = 0 where d ν q / d ϵ = 0...**Frequency** shift per photon δ ν 0 as a function of I b and ϵ . The white regions correspond to -15 M H z and the black to + 35 M H z . The solid line ϵ m I b indicates the bias conditions for which δ ν 0 = 0 . The dashed line indicates the phase noise insensitive point ϵ = 0 ; the dotted line indicates the decoupling current I b = I b * ....Decoherence in superconducting **qubits** is known to arise because of a variety of environmental degrees of freedom. In this article, we focus on the influence of thermal fluctuations in a weakly damped circuit resonance coupled to the **qubit**. Because of the coupling, the **qubit** **frequency** is shifted by an amount $n \delta \nu_0$ if the resonator contains $n$ energy quanta. Thermal fluctuations induce temporal variations $n(t)$ and thus dephasing. We give an approximate formula for the **qubit** dephasing time as a function of $\delta \nu_0$. We discuss the specific case of a flux-**qubit** coupled to the plasma mode of its DC-SQUID detector. We first derive a plasma mode-**qubit** interaction hamiltonian which, in addition to the usual Jaynes-Cummings term, has a coupling term quadratic in the **oscillator** variables coming from the flux-dependence of the SQUID Josephson inductance. Our model predicts that $\delta \nu_0$ cancels in certain non-trivial bias conditions for which dephasing due to thermal fluctuations should be suppressed....The hamiltonian eq:qubit_hamiltonian yields a **qubit** transition **frequency** ν q = Δ 2 + ϵ 2 . The corresponding dependence is plotted in figure fig:nuq for realistic parameters. An interesting property is that when the **qubit** is biased at ϵ = 0 (dashed line in figure fig:nuq), it is insensitive to first order to noise in the bias variable ϵ ....We stress that these biasing conditions are non-trivial in the sense that they do not satisfy an obvious symmetry in the circuit. This point is emphasized in figure fig:deltanu0 where we plotted as a dashed line the bias conditions ϵ = 0 for which the **qubit** is insensitive to phase noise (due to flux or bias current noise) ; and as a dotted line the decoupling current conditions I b = I b * for which the **qubit** is effectively decoupled from its measuring circuit. The ϵ m I b line shares only one point with these two curves : the point I b * ϵ which is optimal with respect to flux, bias current, and photon noise. For the rest, the three lines are obviously distinct. This makes it possible to experimentally discriminate between the various noise sources limiting the **qubit** coherence by studying the dependence of τ φ on bias parameters....The flux-**qubit** is a superconducting loop containing three Josephson junctions threaded by an external flux Φ x ≡ f Φ 0 / 2 π . It is coupled to a DC-SQUID detector shunted by an external capacitor C s h whose role is to limit phase fluctuations across the SQUID as well as to filter high-**frequency** noise from the dissipative impedance. The SQUID is threaded by a flux Φ S q ≡ f ' Φ 0 / 2 π . The circuit diagram is shown in figure fig1a. There, the flux-**qubit** is the loop in red containing the three junctions of phases φ i and capacitances C i ( i = 1 , 2 , 3 ). It also includes an inductance L 1 which models the branch inductance and eventually the inductance of a fourth larger junction . The two inductances K 1 and K 2 model the kinetic inductance of the line shared by the SQUID and the **qubit**. The SQUID is the larger loop in blue. The junction phases are called φ 4 and φ 5 and their capacitances C 4 and C 5 . The critical current of the circuit junctions is written I C i ( i = 1 to 5 ). The SQUID loop also contains two inductances K 3 and L 2 which model its self-inductance. The SQUID is connected to the capacitor C s h through superconducting lines of parasitic inductance L s . The phase across the stray inductance and the SQUID is denoted φ A . The whole circuit is biased by a current source I b in parallel with a dissipative admittance Y ω . Since our goal is primarily to determine the **qubit**-plasma mode coupling hamiltonian, we will neglect the admittance Y ω . ... Decoherence in superconducting **qubits** is known to arise because of a variety of environmental degrees of freedom. In this article, we focus on the influence of thermal fluctuations in a weakly damped circuit resonance coupled to the **qubit**. Because of the coupling, the **qubit** **frequency** is shifted by an amount $n \delta \nu_0$ if the resonator contains $n$ energy quanta. Thermal fluctuations induce temporal variations $n(t)$ and thus dephasing. We give an approximate formula for the **qubit** dephasing time as a function of $\delta \nu_0$. We discuss the specific case of a flux-**qubit** coupled to the plasma mode of its DC-SQUID detector. We first derive a plasma mode-**qubit** interaction hamiltonian which, in addition to the usual Jaynes-Cummings term, has a coupling term quadratic in the **oscillator** variables coming from the flux-dependence of the SQUID Josephson inductance. Our model predicts that $\delta \nu_0$ cancels in certain non-trivial bias conditions for which dephasing due to thermal fluctuations should be suppressed.

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Contributors: Saiko, A. P., Fedaruk, R.

Date: 2010-12-10

Energy-level diagram of a **qubit** and transitions created by a bichromatic field at double resonance ( ω 0 = ω , ω 1 = ω r f )....Multiplication of spin **qubits** arises at double resonance in a bichromatic field when the **frequency** of the radio-**frequency** (rf) field is close to that of the Rabi **oscillation** in the microwave field, provided its **frequency** equals the Larmor **frequency** of the initial **qubit**. We show that the operational multiphoton transitions of dressed **qubits** can be selected by the choice of both the rotating frame and the rf phase. In order to enhance the precision of dressed **qubit** operations in the strong-field regime, the counter-rotating component of the rf field is taken into account. ... Multiplication of spin **qubits** arises at double resonance in a bichromatic field when the **frequency** of the radio-**frequency** (rf) field is close to that of the Rabi **oscillation** in the microwave field, provided its **frequency** equals the Larmor **frequency** of the initial **qubit**. We show that the operational multiphoton transitions of dressed **qubits** can be selected by the choice of both the rotating frame and the rf phase. In order to enhance the precision of dressed **qubit** operations in the strong-field regime, the counter-rotating component of the rf field is taken into account.

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Contributors: Dial, O. E., Shulman, M. D., Harvey, S. P., Bluhm, H., Umansky, V., Yacoby, A.

Date: 2012-08-09

The **oscillations** in these FID experiments decay due to voltage noise from DC up to a **frequency** of approximately 1 / t . As the relaxation time, T 1 is in excess of 100 μ s in this regime, T 1 decay is not an important source of decoherence (Fig. S4). The shape of the decay envelope and the scaling of coherence time with d J / d ϵ (which effectively changes the magnitude of the noise) reveal information about the underlying noise spectrum. White (Markovian) noise, for example, results in an exponential decay of e - t / T 2 * where T 2 * ∝ d J / d ϵ -2 is the inhomogeneously broadened coherence time . However, we find that the decay is Gaussian (Fig. t2stard) and that T 2 * (black line in Fig. t2stare) is proportional to d J / d ϵ -1 (red solid line in Fig. t2stare) across two orders of magnitude of T 2 * . Both of these findings can be explained by quasistatic noise, which is low **frequency** compared to 1 / T 2 * . In such a case, one expects an amplitude decay of the form exp - t / T 2 * 2 , where T 2 * = 1 2 π d J / d ϵ ϵ R M S and ϵ R M S is the root-mean-squared fluctuation in ϵ (Eq. S3). From the ratio of T 2 * to d J / d ϵ -1 , we calculate ϵ R M S = 8 μ V in our device. At very negative ϵ , J becomes smaller than Δ B z , and nuclear noise limits T 2 * to approximately 90ns, which is consistent with previous work . We confirm that this effect explains deviations of T 2 * from d J / d ϵ -1 by using a model that includes the independently measured T 2 , n u c l e a r * and Δ B z (Eq. S1) and observe that it agrees well with measured T 2 * at large negative ϵ (dashed red line in Fig. t2stare)....Two level systems that can be reliably controlled and measured hold promise in both metrology and as **qubits** for quantum information science (QIS). When prepared in a superposition of two states and allowed to evolve freely, the state of the system precesses with a **frequency** proportional to the splitting between the states. In QIS,this precession forms the basis for universal control of the **qubit**,and in metrology the **frequency** of the precession provides a sensitive measurement of the splitting. However, on a timescale of the coherence time, $T_2$, the **qubit** loses its quantum information due to interactions with its noisy environment, causing **qubit** **oscillations** to decay and setting a limit on the fidelity of quantum control and the precision of **qubit**-based measurements. Understanding how the **qubit** couples to its environment and the dynamics of the noise in the environment are therefore key to effective QIS experiments and metrology. Here we show measurements of the level splitting and dephasing due to voltage noise of a GaAs singlet-triplet **qubit** during exchange **oscillations**. Using free evolution and Hahn echo experiments we probe the low **frequency** and high **frequency** environmental fluctuations, respectively. The measured fluctuations at high **frequencies** are small, allowing the **qubit** to be used as a charge sensor with a sensitivity of $2 \times 10^{-8} e/\sqrt{\mathrm{Hz}}$, two orders of magnitude better than the quantum limit for an RF single electron transistor (RF-SET). We find that the dephasing is due to non-Markovian voltage fluctuations in both regimes and exhibits an unexpected temperature dependence. Based on these measurements we provide recommendations for improving $T_2$ in future experiments, allowing for higher fidelity operations and improved charge sensitivity....Two level quantum systems (**qubits**) are emerging as promising candidates both for quantum information processing and for sensitive metrology . When prepared in a superposition of two states and allowed to evolve, the state of the system precesses with a **frequency** proportional to the splitting between the states. However, on a timescale of the coherence time, T 2 , the **qubit** loses its quantum information due to interactions with its noisy environment. This causes **qubit** **oscillations** to decay and limits the fidelity of quantum control and the precision of **qubit**-based measurements. In this work we study singlet-triplet ( S - T 0 ) **qubits**, a particular realization of spin **qubits** , which store quantum information in the joint spin state of two electrons. We form the **qubit** in two gate-defined lateral quantum dots (QD) in a GaAs/AlGaAs heterostructure (Fig. pulsesa). The QDs are depleted until there is exactly one electron left in each, so that the system occupies the so-called 1 1 charge configuration. Here n L n R describes a double QD with n L electrons in the left dot and n R electrons in the right dot. This two-electron system has four possible spin states: S , T + , T 0 , and T - . The S , T 0 subspace is used as the logical subspace for this **qubit** because it is insensitive to homogeneous magnetic field fluctuations and is manipulable using only pulsed DC electric fields . The relevant low-lying energy levels of this **qubit** are shown in Fig. pulsesc. Two distinct rotations are possible in these devices: rotations around the x -axis of the Bloch sphere driven by difference in magnetic field between the QDs, Δ B z (provided in this experiment by feedback-stabilized hyperfine interactions), and rotations around the z -axis driven by the exchange interaction, J (Fig. pulsesb) . A S can be prepared quickly with high fidelity by exchanging an electron with the QD leads, and the projection of the state of the **qubit** along the z -axis can be measured using RF reflectometery with an adjacent sensing QD (green arrow in Fig. pulsesa)....The device used in these measurements is a gate-defined S - T 0 **qubit** with an integrated RF sensing dot. a The detuning ϵ is the voltage applied to the dedicated high-**frequency** control leads pictured. b, The Bloch sphere that describes the logical subspace of this device features two rotation axes ( J and Δ B Z ) both controlled with DC voltage pulses. c, An energy diagram of the relevant low-lying states as a function of ϵ . States outside of the logical subspace of the **qubit** are grayed out. d, J ϵ and d J / d ϵ in three regions; the 1 1 region where J and d J / d ϵ are both small and S - T 0 **qubits** are typically operated, the transitional region where J and d J / d ϵ are both large where the **qubit** is loaded and measured, and the 0 2 region where J is large but d J / d ϵ is small and large quality **oscillations** are possible. pulses...Ramsey oscilllations reveal low **frequency** enivronmental dynamics. a, The pulse sequence used to measure exchange **oscillations** uses a stabilized nuclear gradient to prepare and readout the **qubit** and gives good contrast over a wide range of J . b, Exchange **oscillations** measured over a variety of detunings ϵ and timescales consistently show larger T 2 * as d J / d ϵ shrinks until dephasing due to nuclear fluctuations sets in at very negative ϵ . c, Extracted values of J and d J / d ϵ as a function of ϵ . d, The decay curve of FID exchange **oscillations** shows Gaussian decay. e, Extracted values of T 2 * and d J / d ϵ as a function of ϵ . T 2 * is proportional to d J / d ϵ -1 , indicating that voltage noise is the cause of dephasing of charge **oscillations**. f, Charge **oscillations** measured in 0 2 . This figure portrays the three basic regions we can operate our device in: a region of low **frequency** **oscillations** and small d J / d ϵ , a region of large **frequency** **oscillations** and large d J / d ϵ , and a region where **oscillations** are fast but d J / d ϵ is comparatively small. t2star...Since we observe J to be approximately an exponential function of ϵ , ( d J / d ϵ ∼ J ), we expect and observe the quality (number of coherent **oscillations**) of these FID **oscillations**, Q ≡ J T 2 * / 2 π ∼ J d J / d ϵ -1 , to be approximately constant regardless of ϵ . However, when ϵ is made very positive and J is large, an avoided crossing occurs between the 1 1 T 0 and the 0 2 T 0 state, making the 0 2 S and 0 2 T 0 states electrostatically virtually identical. Here, as ϵ is increased, J increases but d J / d ϵ decreases(Fig. pulsesd), allowing us to probe high quality exchange rotations and test our charge noise model in a regime that has never before been explored....Spin-echo measurements reveal high **frequency** bath dynamics. a, The pulse sequence used to measure exchange echo rotations. b, A typical echo signal. The overall shape of the envelope reflects T 2 * , while the amplitude of the envelope as a function of τ (not pictured) reflects T 2 e c h o . c, T 2 e c h o and Q ≡ J T 2 e c h o / 2 π as a function of J . A comparison of the two noise models: power law and a mixture of white and 1 / f noise. Noise with a power law spectrum fits over a wide range of **frequencies** (constant β ), but the relative contributions of white and 1 / f noise change as a function of ϵ . d, A typical echo decay is non-exponential but is well fit by exp - τ / T 2 e c h o β + 1 . e, T 2 e c h o varies with d J / d ϵ in a fashion consistent with dephasing due to power law voltage fluctuations. echo...Using a modified pulse sequence that changes the clock **frequency** of our waveform generators to achieve picosecond timing resolution (Fig. S1)), we measure exchange **oscillations** in 0 2 as a function of ϵ and time (Fig. t2stare) and we extract both J (Fig. t2starc) and T 2 * (Fig. t2stard) as a function of ϵ . Indeed, the predicted behavior is observed: for moderate ϵ we see fast **oscillations** that decay after a few ns, and for the largest ϵ we see even faster **oscillations** that decay slowly. Here, too, we observe that T 2 * ∝ d J d ϵ -1 (Fig. t2stard), which indicates that FID **oscillations** in 0 2 are also primarily dephased by low **frequency** voltage noise. We note, however, that we extract a different constant of proportionality between T 2 * and d J / d ϵ -1 for 1 1 and 0 2 . This is expected given that the charge distributions associated with the **qubit** states are very different in these two regimes and thus have different sensitivities to applied electric fields. We note that in the regions of largest d J / d ϵ (near ϵ = 0 ), T 2 * is shorter than the rise time of our signal generator and we systematically underestimate J and overestimate T 2 * (Fig. S1)....The use of Hahn echo dramatically improves coherence times, with T 2 e c h o (the τ at which the observed echo amplitude has decayed by 1 / e ) as large as 9 μ s , corresponding to qualities ( Q ≡ T 2 e c h o J / 2 π ) larger than 600 (Fig. echoc). If at high **frequencies** (50kHz-1MHz) the voltage noise were white (Markovian), we would observe exponential decay of the echo amplitude with τ . However, we find that the decay of the echo signal is non-exponential (Fig. echod), indicating that even in this relatively high-**frequency** band being probed by this measurement, the noise bath is not white. ... Two level systems that can be reliably controlled and measured hold promise in both metrology and as **qubits** for quantum information science (QIS). When prepared in a superposition of two states and allowed to evolve freely, the state of the system precesses with a **frequency** proportional to the splitting between the states. In QIS,this precession forms the basis for universal control of the **qubit**,and in metrology the **frequency** of the precession provides a sensitive measurement of the splitting. However, on a timescale of the coherence time, $T_2$, the **qubit** loses its quantum information due to interactions with its noisy environment, causing **qubit** **oscillations** to decay and setting a limit on the fidelity of quantum control and the precision of **qubit**-based measurements. Understanding how the **qubit** couples to its environment and the dynamics of the noise in the environment are therefore key to effective QIS experiments and metrology. Here we show measurements of the level splitting and dephasing due to voltage noise of a GaAs singlet-triplet **qubit** during exchange **oscillations**. Using free evolution and Hahn echo experiments we probe the low **frequency** and high **frequency** environmental fluctuations, respectively. The measured fluctuations at high **frequencies** are small, allowing the **qubit** to be used as a charge sensor with a sensitivity of $2 \times 10^{-8} e/\sqrt{\mathrm{Hz}}$, two orders of magnitude better than the quantum limit for an RF single electron transistor (RF-SET). We find that the dephasing is due to non-Markovian voltage fluctuations in both regimes and exhibits an unexpected temperature dependence. Based on these measurements we provide recommendations for improving $T_2$ in future experiments, allowing for higher fidelity operations and improved charge sensitivity.

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Contributors: Wallquist, M., Shumeiko, V. S., Wendin, G.

Date: 2006-08-09

Pulse sequence producing (trivial) diagonal gate: during time T 1 , **qubit** 1 swaps its state onto the **oscillator**, then the **oscillator** interacts with **qubit** 2 before swapping its state back onto **qubit** 1; free evolution during time T 3 is added to annihilate two-photon state in the cavity....Protocol for creating a Bell-pair: the cavity **frequency** is sequentially swept through resonances with both **qubits**; at the first resonance the **oscillator** is entangled with **qubit** 1, at the next resonance the **oscillator** swaps its state onto **qubit** 2 and ends up in the ground state. A Bell measurement is performed by applying Rabi pulses to non-interacting **qubits**, and projecting on the **qubit** eigenbasis, | g | e , by measuring quantum capacitance....Equivalent circuit for the device in Fig. Sketch: chain of L C -**oscillators** represents the stripline cavity, φ 1 and φ N are superconducting phase values at the ends of the cavity, φ j and φ l are local phase values where the **qubits** are attached; attached dc-SQUID has effective flux-dependent Josephson energy, E J s f , and capacitance C s , control line for tuning the SQUID is shown at the right; SCB **qubits** are coupled to the cavity via small capacitances, C c 1 and C c 2 ....Sketch of the device: charge **qubits** (single Cooper pair boxes, SCB) coupled capacitively ( C c ) to a stripline cavity integrated with a dc-SQUID formed by two large Josephson junctions (JJ); cavity eigenfrequency is controlled by magnetic flux Φ through the SQUID....Jonn,NewJP: the duration of the gate operation in the latter case is h / 8 in the units of inverse coupling energy, while it is 2.7 h for the protocol presented in Fig. fig_prot_SK. This illustrates the advantage of longitudinal, z z coupling (in the **qubit** eigenbasis), which is achieved for the charge **qubits** biased at the charge degeneracy point by current-current coupling. More common for charge **qubits** is the capacitive coupling, however there the situation is different: this coupling has x x symmetry at the charge degeneracy point, and because of inevitable difference in the **qubit** **frequencies**, the gate operation takes much longer time, prolonged by the ratio between the **qubits** **frequency** asymmetry and the coupling **frequency**. Recent suggestions to employ dynamic control methods to effectively bring the **qubits** into resonance can speed up the gate operation. For these protocols, the gate duration is ∼ h in units of direct coupling energy, which is longer than in the case of z z coupling, but somewhat shorter than in our case. However, the protocol considered in this paper might be made faster by using pulse shaping....For a given eigenmode, the integrated stripline + SQUID system behaves as a lumped **oscillator** with variable **frequency**. Our goal in this section will be to derive an effective classical Lagrangian for this **oscillator**. To this end we consider in Fig. 2qubit_circuit an equivalent circuit for the device depicted in Fig. Sketch. A discrete chain of identical L C -**oscillators**, with phases φ i across the chain capacitors (i=1,…,N), represents the stripline cavity; the dc SQUID is directly attached at the right end of the chain, while the superconducting Cooper pair boxes (SCB) are attached via small coupling capacitors, C c 1 and C c 2 to the chain nodes with local phases, φ j and φ l (for simplicity we consider only two attached SCBs). The classical Lagrangian for this circuit,...Gate circuit for constructing a CNOT gate using the control-phase gate: a z-axis rotation is applied to **qubit** 1, and Hadamard gates H are applied to the second **qubit**....In this section we modify the Bell state construction to implementing a control-phase (CPHASE) two-**qubit** gate. This gate has the diagonal form: | α β 0 → exp i φ α β | α β 0 ( φ 00 = φ 01 = φ 10 = 0 , φ 11 = π ), and it is equivalent to the CNOT gate (up to local rotations). To generate such a diagonal gate, we adopt the following strategy: first tune the **oscillator** through resonance with both **qubits** performing π -pulse swaps in every step, and then reverse the sequence, as shown in figure fig_prot_naive. With an even number of swaps at every level, clearly the resulting gate will be diagonal....The experimental setup with the **qubit** coupling to a distributed **oscillator** - stripline cavity possesses potential for scalability - several **qubits** can be coupled to the cavity. In this paper we investigate the possibility to use this setup for implementation of tunable **qubit**-**qubit** coupling and simple gate operations. Tunable **qubit**-cavity coupling is achieved by varying the cavity **frequency** by controlling magnetic flux through a dc-SQUID attached to the cavity (see Fig. Sketch). An advantage of this method is the possibility to keep the **qubits** at the optimal points with respect to decoherence during the whole two-**qubit** operation. The **qubits** coupled to the cavity must have different **frequencies**, and the cavity in the idle regime must be tuned away from resonance with all of the **qubits**. Selective addressing of a particular **qubit** is achieved by relatively slow passage through the resonance of a selected **qubit**, while other resonances are rapidly passed. The speed of the active resonant passage should be comparable to the **qubit**-cavity coupling **frequency** while the rapid passages should be fast on this scale, but slow on the scale of the cavity eigenfrequency in order to avoid cavity excitation. This strategy requires narrow width of the **qubit**-cavity resonances compared to the differences in the **qubit** **frequencies**, determined by the available interval of the cavity **frequency** divided by the number of attached **qubits**. This consideration simultaneously imposes a limit on the maximum number of employed **qubits**. Denoting the difference in the **qubit** energies, Δ E J , the coupling energy, κ , the maximum variation of the cavity **frequency**, Δ ω k , and the number of **qubits**, N , we summarize the above arguments with relations, κ ≪ Δ E J , N ∼ ℏ Δ ω k / Δ E J . In the off-resonance state, the **qubit**-**qubit** coupling strength is smaller than the on-resonance coupling by the ratio, κ / ℏ ω k - E J ≪ 1 ....We theoretically investigate selective coupling of superconducting charge **qubits** mediated by a superconducting stripline cavity with a tunable resonance **frequency**. The **frequency** control is provided by a flux biased dc-SQUID attached to the cavity. Selective entanglement of the **qubit** states is achieved by sweeping the cavity **frequency** through the **qubit**-cavity resonances. The circuit is scalable, and allows to keep the **qubits** at their optimal points with respect to decoherence during the whole operation. We derive an effective quantum Hamiltonian for the basic, two-**qubit**-cavity system, and analyze appropriate circuit parameters. We present a protocol for performing Bell inequality measurements, and discuss a composite pulse sequence generating a universal control-phase gate. ... We theoretically investigate selective coupling of superconducting charge **qubits** mediated by a superconducting stripline cavity with a tunable resonance **frequency**. The **frequency** control is provided by a flux biased dc-SQUID attached to the cavity. Selective entanglement of the **qubit** states is achieved by sweeping the cavity **frequency** through the **qubit**-cavity resonances. The circuit is scalable, and allows to keep the **qubits** at their optimal points with respect to decoherence during the whole operation. We derive an effective quantum Hamiltonian for the basic, two-**qubit**-cavity system, and analyze appropriate circuit parameters. We present a protocol for performing Bell inequality measurements, and discuss a composite pulse sequence generating a universal control-phase gate.

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Contributors: Dutta, S. K., Strauch, Frederick W., Lewis, R. M., Mitra, Kaushik, Paik, Hanhee, Palomaki, T. A., Tiesinga, Eite, Anderson, J. R., Dragt, Alex J., Lobb, C. J.

Date: 2008-06-28

We fit the escape rates in Fig. F031806DGtot (and additional data for other powers not shown) to a decaying sinusoid with an offset. The extracted **frequencies** are shown with circles in Fig. F031806Dfosc. To compare to theory, Ω R , 0 1 , calculated using the rotating wave solution for a system with five levels, is shown with a solid line. The implied assumption that the **oscillation** **frequencies** of Γ and ρ 11 are equal, even at high power in a multilevel system, will be addressed in Sec. SSummary. In plotting the data, we have introduced a single fitting parameter 117 n A / m W that converts the power P S at the microwave source to the current amplitude I r f at the **qubit**. Good agreement is found over the full range of power....We next consider the time dependence of the escape rate for the data plotted in Fig. F032106FN10. Here, a 6.2 GHz microwave pulse nominally 30 ns long was applied on resonance with the 0 → 1 transition of the **qubit** junction. The measured escape rate shows Rabi **oscillations** followed by a decay back to the ground state once the microwave drive has turned off. This decay appears to be governed by three time constants. Nontrivial decays have previously been reported in phase **qubits** and we have found them in several of our devices....F031806Dfosc Rabi **oscillation** **frequency** Ω R , 0 1 at fixed bias as a function of microwave current I r f . Extracted values from data (including the plots in Fig. F031806DGtot) are shown as circles, while the rotating wave solution is shown for two- (dashed line) and five- (solid) level simulations, calculated using I 01 = 17.930 μ A and C 1 = 4.50 p F with ω r f / 2 π = 6.2 G H z ....As I r f increases in Fig. F031806Dfosc, the **oscillation** **frequency** is smaller than the expected linear relationship for a two-level system (dashed line). This effect is a hallmark of a multilevel system and has been previously observed in a similar phase **qubit**. There are two distinct phenomena that affect 0 → 1 Rabi **oscillations** in such a device. To describe...FDeviceThe dc SQUID phase **qubit**. (a) The **qubit** junction J 1 (with critical current I 01 and capacitance C 1 ) is isolated from the current bias leads by an auxiliary junction J 2 (with I 02 and C 2 ) and geometrical inductances L 1 and L 2 . The device is controlled with a current bias I b and a flux current I f which generates flux Φ a through mutual inductance M . Transitions can be induced by a microwave current I r f , which is coupled to J 1 via C r f . (b) When biased appropriately, the dynamics of the phase difference γ 1 across the **qubit** junction are analogous to those of a ball in a one-dimensional tilted washboard potential U . The metastable state n differs in energy from m by ℏ ω n m and tunnels to the voltage state with a rate Γ n . (c) The photograph shows a Nb/AlO x /Nb device. Not seen is an identical SQUID coupled to this device intended for two-**qubit** experiments; the second SQUID was kept unbiased throughout the course of this work....F032206MNstats (Color online) The (a) on-resonance Rabi **oscillation** **frequencies** Ω R , 0 1 m i n and Ω R , 0 2 m i n and (b) resonance **frequency** shifts Δ ω 0 1 = ω r f - ω 0 1 and Δ ω 0 2 = 2 ω r f - ω 0 2 are plotted as a function of the microwave current, for data taken at 110 m K with a microwave drive of **frequency** ω r f / 2 π = 6.5 G H z and powers P S = - 23 , - 20 , - 17 , - 15 , - 10 d B m . Values extracted from data for the 0 → 1 ( 0 → 2 ) transition are plotted as open circles (filled squares), while five-level rotating wave solutions for a junction with I 01 = 17.736 μ A and C 1 = 4.49 p F are shown as solid (dashed) lines. In (a), the dotted line is from a simulation of a two-level system....We present Rabi **oscillation** measurements of a Nb/AlOx/Nb dc superconducting quantum interference device (SQUID) phase **qubit** with a 100 um^2 area junction acquired over a range of microwave drive power and **frequency** detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the **qubit** dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi **oscillations** were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed **oscillation** **frequencies** with a simplified model constructed from the full phase **qubit** Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi **frequency**, and leakage to the higher excited states....F031806DGtot Rabi **oscillations** in the escape rate Γ were induced at I b = 17.746 μ A by switching on a microwave current at t = 0 with a **frequency** of 6.2 GHz (resonant with the 0 → 1 transition) and source powers P S between -12 and -32 dBm, as labeled. The measurements were taken at 20 mK. The solid lines are from a five-level density-matrix simulation with I 01 = 17.930 μ A , C 1 = 4.50 p F , T 1 = 17 n s , and T φ = 16 n s ....F010206H1 (Color online) Multiphoton, multilevel Rabi **oscillations** plotted in the time and **frequency** domains. (a) The escape rate Γ (measured at 110 mK) is plotted as a function of the time after which a 6.5 GHz, -11 dBm microwave drive was turned on and the current bias I b of the **qubit**; Γ ranges from 0 (white) to 3 × 10 8 1 / s (black). (b) The normalized power spectral density of the time-domain data from t = 1 to 45 ns is shown with a grayscale plot. The dashed line segments indicate the Rabi **frequencies** obtained from the rotating wave model for transitions involving (from top to bottom) 1, 2, 3, and 4 photons, evaluated with junction parameters I 01 = 17.828 μ A and C 1 = 4.52 p F , and microwave current I r f = 24.4 n A . Corresponding grayscale plots calculated with a seven-level density-matrix simulation are shown in (c) and (d)....Figure F010206H1(b) shows that the minimum **oscillation** **frequency** Ω R , 0 1 m i n / 2 π = 540 M H z of the first (experimental) band occurs at I b = 17.624 μ A , for which ω 0 1 / 2 π = 6.4 G H z . This again indicates an ac Stark shift of this transition, which we denote by Δ ω 0 1 ≡ ω r f - ω 0 1 ≈ 2 π × 100 M H z . In addition, the higher levels have suppressed the **oscillation** **frequency** below the bare Rabi **frequency** of Ω 0 1 / 2 π = 620 M H z [calculated with Eq. ( eqf)]....For this data set, the level spacing ω 0 1 / 2 π is equal to the microwave **frequency** ω r f / 2 π = 6.5 G H z at I b = 17.614 μ A . The band with the highest current in Fig. F010206H1(b) is centered about I b = 17.624 μ A , suggesting that 0 → 1 Rabi **oscillations** are the dominant process near this bias. For slightly higher or lower I b , the **oscillation** **frequency** increases as Ω R , 0 1 ≈ Ω 01 ′ 2 + ω r f - ω 0 1 2 , in agreement with simple two-level Rabi theory, leading to the curved band in the grayscale plot. ... We present Rabi **oscillation** measurements of a Nb/AlOx/Nb dc superconducting quantum interference device (SQUID) phase **qubit** with a 100 um^2 area junction acquired over a range of microwave drive power and **frequency** detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the **qubit** dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi **oscillations** were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed **oscillation** **frequencies** with a simplified model constructed from the full phase **qubit** Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi **frequency**, and leakage to the higher excited states.

Data types:

Contributors: Averin, D. V.

Date: 2002-02-05

The concept of quantum nondemolition (QND) measurement is extended to coherent **oscillations** in an individual two-state system. Such a measurement enables direct observation of intrinsic spectrum of these **oscillations** avoiding the detector-induced dephasing that affects the standard (non-QND) measurements. The suggested scheme can be realized in Josephson-junction **qubits** which combine flux and charge dynamics....Schematic of the Josephson-junction **qubit** structure that enables measurements of the two non-commuting observables of the **qubit**, σ z and σ y , as required in the QND Hamiltonian ( 2). For discussion see text....Spin representation of the QND measurement of the quantum coherent **oscillations** of a **qubit**. The **oscillations** are represented as a spin rotation in the z - y plane with **frequency** Δ . QND measurement is realized if the measurement frame (dashed lines) rotates with **frequency** Ω ≃ Δ . ... The concept of quantum nondemolition (QND) measurement is extended to coherent **oscillations** in an individual two-state system. Such a measurement enables direct observation of intrinsic spectrum of these **oscillations** avoiding the detector-induced dephasing that affects the standard (non-QND) measurements. The suggested scheme can be realized in Josephson-junction **qubits** which combine flux and charge dynamics.

Data types:

Contributors: Mandip Singh

Date: 2015-07-14

A contour plot indicating location of two-dimensional potential energy minima forming a symmetric double well potential when the cantilever equilibrium angle θ0=cos−1[Φo/2BxA], ωi=2π×12000 rad/s, Bx=5×10−2 T. The contour interval in units of **frequency** (E/h) is ∼4×1011 Hz.
...In this paper a macroscopic quantum **oscillator** is proposed, which consists of a flux-**qubit** in the form of a cantilever. The net magnetic flux threading through the flux-**qubit** and the mechanical degrees of freedom of the cantilever are naturally coupled. The coupling between the cantilever and the magnetic flux is controlled through an external magnetic field. The ground state of the flux-**qubit**-cantilever turns out to be an entangled quantum state, where the cantilever deflection and the magnetic flux are the entangled degrees of freedom. A variant, which is a special case of the flux-**qubit**-cantilever without a Josephson junction, is also discussed....A superconducting-loop-**oscillator** with its axis of rotation along the z-axis consists of a closed superconducting loop without a Josephson Junction. The superconducting loop can be of any arbitrary shape.
...A contour plot indicating location of a two-dimensional global potential energy minimum at (nΦ0=0, θn+=π/2) and the local minima when the cantilever equilibrium angle θ0=π/2, ωi=2π×12000 rad/s, Bx=5.0×10−2 T. The contour interval in units of **frequency** (E/h) is ∼3.9×1011 Hz.
...The potential energy profile of the superconducting-loop-**oscillator** when the intrinsic **frequency** is 10 kHz. (a) For external magnetic field Bx=0, a single well harmonic potential near the minimum is formed. (b) Bx=0.035 T. (c) For Bx=0.045 T, a double well potential is formed.
...A schematic of the flux-**qubit**-cantilever. A part of the flux-**qubit** (larger loop) is projected from the substrate to form a cantilever. The external magnetic field Bx controls the coupling between the flux-**qubit** and the cantilever. An additional magnetic flux threading through a dc-SQUID (smaller loop) which consists of two Josephson junctions adjusts the tunneling amplitude. The dc-SQUID can be shielded from the effect of Bx.
... In this paper a macroscopic quantum **oscillator** is proposed, which consists of a flux-**qubit** in the form of a cantilever. The net magnetic flux threading through the flux-**qubit** and the mechanical degrees of freedom of the cantilever are naturally coupled. The coupling between the cantilever and the magnetic flux is controlled through an external magnetic field. The ground state of the flux-**qubit**-cantilever turns out to be an entangled quantum state, where the cantilever deflection and the magnetic flux are the entangled degrees of freedom. A variant, which is a special case of the flux-**qubit**-cantilever without a Josephson junction, is also discussed.

Data types:

Contributors: Il'ichev, E., Oukhanski, N., Izmalkov, A., Wagner, Th., Grajcar, M., Meyer, H. -G., Smirnov, A. Yu., Brink, Alec Maassen van den, Amin, M. H. S., Zagoskin, A. M.

Date: 2003-03-20

We use a small-inductance superconducting loop interrupted by three Josephson junctions (a 3JJ **qubit**) , inductively coupled to a high-quality superconducting tank circuit (Fig. fig1). This approach is similar to the one in entanglement experiments with Rydberg atoms and microwave photons in a cavity . The tank serves as a sensitive detector of Rabi transitions in the **qubit**, and simultaneously as a filter protecting it from noise in the external circuit. Since ω T ≪ Ω / ℏ , the **qubit** is effectively decoupled from the tank unless it **oscillates** with **frequency** ω T . That is, while wide-band (i.e., fast on the **qubit** time scale) detectors up to now have received most theoretical attention (e.g., ), we use narrow-band detection to have sufficient sensitivity at a single **frequency** even with a small coupling coefficient; cf. above Eq. ( S). The tank voltage is amplified and sent to a spectrum analyzer. This is a development of the Silver–Zimmerman setup in the first RF-SQUID magnetometers , and is effective for probing flux **qubits** . As such, it was used to determine the potential profile of a 3JJ **qubit** in the classical regime ....We plotted S V , t ω for different HF powers P in Fig. fig3. As P is increased, ω R grows and passes ω T , leading to a non-monotonic dependence of the maximum signal on P in agreement with the above picture. This, and the sharp dependence on the tuning of ω H F to the **qubit** **frequency**, confirm that the effect is due to Rabi **oscillations**. The inset shows that the shape is given by the second line of Eq. ( S) for all curves....Measurement setup. The flux **qubit** is inductively coupled to a tank circuit. The DC source applies a constant flux Φ e ≈ 1 2 Φ 0 . The HF generator drives the **qubit** through a separate coil at a **frequency** close to the level separation Δ / h = 868 MHz. The output voltage at the resonant **frequency** of the tank is measured as a function of HF power....Under resonant irradiation, a quantum system can undergo coherent (Rabi) **oscillations** in time. We report evidence for such **oscillations** in a _continuously_ observed three-Josephson-junction flux **qubit**, coupled to a high-quality tank circuit tuned to the Rabi **frequency**. In addition to simplicity, this method of_Rabi spectroscopy_ enabled a long coherence time of about 2.5 microseconds, corresponding to an effective **qubit** quality factor \~7000....The Al **qubit** inside the Nb pancake coil....(a) Comparing the data to the theoretical Lorentzian. The fitting parameter is g ≈ 0.02 . Letters in the picture correspond to those in Fig. fig3. (b) The Rabi **frequency** extracted from (a) vs the applied HF amplitude. The straight line is the predicted dependence ω R / ω T = P / P 0 . The good agreement provides strong evidence for Rabi **oscillations**....The spectral amplitude of the tank voltage for HF powers P a **qubit** modifying the tank’s inductance and hence its central **frequency**, and in principle similarly for dissipation in the **qubit** increasing the tank’s linewidth ; these are inconsequential for our analysis....Without an HF signal, the ** qubit’s** influence at ω T is negligible. Thus, the “dark” trace in Fig. fig3 is a quantitative measure of S b . ... Under resonant irradiation, a quantum system can undergo coherent (Rabi)

**oscillations**in time. We report evidence for such

**oscillations**in a _continuously_ observed three-Josephson-junction flux

**qubit**, coupled to a high-quality tank circuit tuned to the Rabi

**frequency**. In addition to simplicity, this method of_Rabi spectroscopy_ enabled a long coherence time of about 2.5 microseconds, corresponding to an effective

**qubit**quality factor \~7000.

Data types:

Contributors: Altomare, Fabio, Cicak, Katarina, Sillanpää, Mika A., Allman, Michael S., Sirois, Adam J., Li, Dale, Park, Jae I., Strong, Joshua A., Teufel, John D., Whittaker, Jed D.

Date: 2010-02-02

(Color) Simulated energy for (a) the first **qubit**, (b) the CPW cavity, (c) the second **qubit**. (Red): ϕ 1 = 0.8949 ϕ c 1 and ϕ 2 = 0.893 ϕ c 2 . The first **qubit** decays exponentially up to t ≈ 123 ~ n s . At this time the **frequency** of the **oscillation** in right well matches the CPW cavity resonant **frequency** and the **qubit** transfers part of its energy to the CPW cavity. The second **qubit** is resonating at a different **frequency** and it is minimally exited by the incoming microwave voltage. This corresponds to the red x of Fig. 2(b) (Black): ϕ 1 = 0.82 ϕ c 1 and ϕ 2 = 0.836 ϕ c 2 . In this case the first **qubit** transfers part of its energy to the CPW cavity at t ≈ 103 ~ n s because it starts at a lower energy in the deep well. At this flux the second **qubit** is in resonance with the cavity and it is excited up to the sixth quantized level. This corresponds to the white x of Fig. 2(b) fig:singlelinecut...For our experiment, we initially determine the optimal ’simultaneous’ timing between the two MPs that takes into account the different cabling and instrumental delays from the room-temperature equipment to the cold devices. Then, as a function of the flux applied to the two **qubits**, we measure the tunneling probability for the second (first) **qubit** after we purposely induce a tunneling event in the first (second) **qubit**. The results are shown in Fig. fig:experiment(a,c). The probability of finding the second (first) **qubit** in the excited state as a result of measurement crosstalk is significant only in a region around ϕ 2 / ϕ c 2 = Φ 2 / Φ c 2 ≈ Φ ¯ 2 / Φ c 2 ∼ 0.842 ( ϕ 1 / ϕ c 1 = Φ 1 / Φ c 1 ≈ Φ ¯ 1 / Φ c 1 ∼ 0.82 ) where the resonant **frequency** of the second (first) **qubit** is close to the CPW cavity **frequency**....AltomareX2009, the resonant **frequency** of both **qubits** exhibits an avoided crossing at the CPW cavity **frequency** ( ≈ 8.9 GHz). For the first **qubit** this happens at a flux Φ ¯ 1 = 0.82 Φ c 1 , and for the second at a flux Φ ¯ 2 = 0.842 Φ c 2 . For each **qubit**, Φ c i is the critical flux at which the left well of Fig. fig:QBpotential(b) disappears....(Color) Measurement crosstalk: (a) Experimental tunneling probability for **qubit** 2, after **qubit** 1 has already tunneled as function of the (dimensionless) flux applied to the **qubits**. The left ordinate displays the resonant **frequency** as measured from the **qubit** spectroscopy. The right ordinate displays the ratio between the applied flux and the critical flux for **qubit** 2. (b) Simulation: ratio between the maximum energy acquired by the second **qubit** and the resonant **frequency** in the left well ( N l ) as a function of the flux applied to the **qubits**. The left ordinate displays the **oscillation** **frequency** as determined from the Fast Fourier Transform of the energy of **qubit** 2. The right ordinate displays the ratio between the applied flux and the critical flux for **qubit** 1. Temporal traces corresponding to the two x’s are displayed in Fig. fig:singlelinecut. (c-d) Same as (a-b) after reversing the roles of the two **qubits**. fig:experiment...From these initial conditions the phase of the first **qubit**(classically) undergoes damped **oscillations** in the anharmonic right well. Because of the anharmonicity of the potential, when the amplitude of the **oscillation** is large, the **frequency** of the **oscillations** is lower than the unmeasured **qubit** **frequency**. As the system loses energy due to the damping, the **oscillation** **frequency** increases as seen by the CPW cavity. When the crosstalk voltage has a **frequency** close to the CPW cavity **frequency**, it can transfer energy to the CPW cavity. If the second **qubit**’s**frequency** matches that of the CPW cavity then the cavity’s excitation can be transferred to the second **qubit**. In Fig. fig:experiment(b) we plot, for the second **qubit**, the ratio ( N l ) between the maximum energy acquired and ℏ ω p , where ω p is the plasma **frequency** of the **qubit** in the left well, as a function of the fluxes in the two **qubits**. The crosstalk, measured as the maximum energy transferred to the second **qubit**, is maximum at a flux ϕ 2 / ϕ c 2 ∼ 0.837 , where the second **qubit**’s**frequency** is ≈ 8.97 ~ G H z , determined by taking the Fast Fourier Transform of the **oscillations** in energy over time (see Fig. fig:singlelinecut (a-c)). Reversing the roles of the two **qubits**, we find that for the first **qubit** the crosstalk is maximum at a flux ∼ 0.825 ϕ c 1 , corresponding to an excitation **frequency** of ≈ 8.84 ~ G H z (Fig. fig:experiment(d)). These values were determined for **qubit** 2 (**qubit** 1) by performing a Gaussian fit of N l versus flux (or **frequency**) after averaging over the span of flux (or **frequency**) values for **qubit** 1 (**qubit** 2). Notice that the crosstalk transferred to **qubit** 2 (**qubit** 1) is flux independent of **qubit** 1 (**qubit** 2) and substantial only when the cavity **frequency** matches the **frequency** of **qubit** 2 (**qubit** 1). The results of the simulations are in good agreement with the experimental data. To gain additional insight into the dynamics of the system, we plot the time evolution of the energy for the **qubits** and the CPW cavity (Fig. fig:singlelinecut (a-c)) for two different sets of fluxes in the two **qubits**. At ϕ 1 = 0.895 ϕ c 1 and ϕ 2 = 0.893 ϕ c 2 (red x in Fig. fig:experiment(b)) the first **qubit** decays exponentially for a time t 123 ~ n s (Fig. fig:singlelinecut (a-c)-Red). At t = 123 ~ n s there is a downward jump in the energy of the first **qubit** while the energy of the CPW cavity exhibits an upward jump. At this time, the **frequency** of **oscillation** in the right well matches the CPW cavity resonant **frequency**, so part of the **qubit** energy is transferred to the CPW cavity. However, since the second **qubit** is not on resonance with the CPW cavity, it does not get significantly excited by the microwave current passing through the capacitor C x ....(a) Equivalent electrical circuit for two flux-biased phase **qubits** coupled to a CPW cavity (modelled as a lumped element harmonic **oscillator**). C i is the total i - **qubit** (or CPW cavity) capacitance, L i the geometrical inductance, L j , i the Josephson inductance of the JJ, R i models the dissipation in the system. (b) U ϕ ϕ e is the potential energy of the phase **qubit** as function of superconducting phase difference ϕ across the JJ and the dimensionless external flux bias ϕ e = Φ 2 π / Φ 0 . Δ U ϕ e is the difference between the local potential maximum and the local potential minimum in the left well at the flux bias ϕ e . (c) During the MP, the potential barrier Δ U ϕ e between the two wells is lowered for a few nanoseconds allowing the | 1 state to tunnel into the right well where it will (classically) **oscillate** and lose energy due to the dissipation. fig:QBpotential...At ϕ 1 = 0.82 ϕ c 1 and ϕ 2 = 0.836 ϕ c 2 (white x in Fig. fig:experiment(b)), the dynamics of the first **qubit** and the CPW cavity are essentially unchanged, except that the CPW cavity **frequency** is matched at a different time ( t = 103 ~ n s ) because the first **qubit** starts at a lower energy in the deep well (Fig. fig:singlelinecut (a-c)-Black). However, in this case, the second **qubit** is on resonance with the CPW cavity and is therefore excited to an energy N l ∼ 6 ....We analyze the measurement crosstalk between two flux-biased phase **qubits** coupled by a resonant coplanar waveguide cavity. After the first **qubit** is measured, the superconducting phase can undergo damped **oscillations** resulting in an a.c. voltage that produces a **frequency** chirped noise signal whose **frequency** crosses that of the cavity. We show experimentally that the coplanar waveguide cavity acts as a bandpass filter that can significantly reduce the crosstalk signal seen by the second **qubit** when its **frequency** is far from the cavity's resonant **frequency**. We present a simple classical description of the **qubit** behavior that agrees well with the experimental data. These results suggest that measurement crosstalk between superconducting phase **qubits** can be reduced by use of linear or possibly nonlinear resonant cavities as coupling elements. ... We analyze the measurement crosstalk between two flux-biased phase **qubits** coupled by a resonant coplanar waveguide cavity. After the first **qubit** is measured, the superconducting phase can undergo damped **oscillations** resulting in an a.c. voltage that produces a **frequency** chirped noise signal whose **frequency** crosses that of the cavity. We show experimentally that the coplanar waveguide cavity acts as a bandpass filter that can significantly reduce the crosstalk signal seen by the second **qubit** when its **frequency** is far from the cavity's resonant **frequency**. We present a simple classical description of the **qubit** behavior that agrees well with the experimental data. These results suggest that measurement crosstalk between superconducting phase **qubits** can be reduced by use of linear or possibly nonlinear resonant cavities as coupling elements.

Data types:

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.

Data types: