### 56121 results for qubit oscillator frequency

Contributors: Wei, L. F., Liu, Yu-xi, Nori, Franco

Date: 2004-02-27

tab1 Typical settings of the controllable experimental parameters ( V k and Φ k ) and the corresponding time evolutions Û j t of the **qubit**-bus system. Here, C g k and 2 ε J k are the gate capacitance and the maximal Josephson energy of the k th SQUID-based charge **qubit**. ζ k is the maximum strength of the coupling between the k th **qubit** with energy ε k and the bus of **frequency** ω b . The detuning between the **qubit** and the bus energies is ℏ Δ k = ε k - ℏ ω b . n = 0 , 1 is occupation number for the number state | n of the bus. The various time-evolution operators are: Û 0 t = exp - i t H ̂ b / ℏ , Û 1 k t = exp - i t δ E C k σ ̂ x k / 2 ℏ ⊗ Û 0 t , Û 2 k = Â t cos λ ̂ n | 0 k 0 k | - sin λ ̂ n â / n ̂ + 1 | 0 k 1 k | + â † sin ξ ̂ n / n ̂ | 1 k 0 k | + cos ξ ̂ n | 0 k 0 k | , and Û 3 k t = Â t exp - i t ζ k 2 | 1 k 1 k | n ̂ + 1 - | 0 k 0 k | n ̂ / ℏ Δ k , with Â t = exp - i t 2 H ̂ b + E J k σ ̂ z k / 2 ℏ , λ ̂ n = 2 ζ k t n ̂ + 1 / ℏ , and ξ ̂ n = 2 ζ k t n ̂ / ℏ ....Josephson **qubits** without direct interaction can be effectively coupled by sequentially connecting them to an information bus: a current-biased large Josephson junction treated as an **oscillator** with adjustable **frequency**. The coupling between any **qubit** and the bus can be controlled by modulating the magnetic flux applied to that **qubit**. This tunable and selective coupling provides two-**qubit** entangled states for implementing elementary quantum logic operations, and for experimentally testing Bell's inequality....A pair of SQUID-based charge **qubits**, located on the left of the dashed line, coupled to a large CBJJ on the right, which acts as an information bus. The circuit is divided into two parts, the **qubits** and the bus. The dashed line only indicates a separation between these. The controllable gate voltage V k k = 1 2 and external flux Φ k are used to manipulate the **qubits** and their interactions with the bus. The bus current remains fixed during the operations. ... Josephson **qubits** without direct interaction can be effectively coupled by sequentially connecting them to an information bus: a current-biased large Josephson junction treated as an **oscillator** with adjustable **frequency**. The coupling between any **qubit** and the bus can be controlled by modulating the magnetic flux applied to that **qubit**. This tunable and selective coupling provides two-**qubit** entangled states for implementing elementary quantum logic operations, and for experimentally testing Bell's inequality.

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Contributors: Schmidt, Thomas L., Nunnenkamp, Andreas, Bruder, Christoph

Date: 2012-11-09

(Color online) Upper panel: Rabi **frequency** | Ω R | in units of c = n p h g c 2 / L for μ = - 100 ϵ L and κ = 5 ϵ L , where ϵ L = 2 m L 2 -1 . A large photon linewidth κ has been chosen to highlight the essential features. The crosses denote the Rabi **frequency** and damping determined numerically from Eq. ( eq:Dgamma2). Solid green and red lines correspond to the solutions for the limits Ω | μ | , see Eq. ( eq:GammaOmega_metallic), respectively. Lower panel: The ratio between Rabi **frequency** and damping, Ω R / Γ R , determines the fidelity of **qubit** rotations....These functions are plotted in Fig. fig:plots. The Rabi **frequency** is, as expected, exponentially suppressed in the length of the CR. However, as the photon **frequency** Ω approaches the critical value | μ | , the prefactor 1 - Ω / | μ | **qubit** state for a time t * = π / 4 Ω R . In the presence of damping, the fidelity of such an operation can be estimated as...(Color online) Upper panel: A semiconductor nanowire (along the x axis) hosting Majorana fermions is embedded in a microwave stripline cavity (along the y axis). The red lines show the amplitude of the electric field E → r → . Dark blue (light yellow) sections of the wire indicate topologically nontrivial (trivial) regions. MBSs (stars) exist at the edges of nontrivial (topological superconductor, TS) regions. The MBSs γ 1 and γ 2 can be braided using a T -junction . Lower panel: Band structure of the individual sections of the wire. The four MBSs γ 1 , 2 , 3 , 4 encode one logical **qubit**. The central MBSs γ 2 and γ 3 are tunnel-coupled ( t c ) to a topologically trivial, gapped central region (CR, light yellow) with length L . All energies are small compared to the induced gap Δ ....Majorana bound states have been proposed as building blocks for **qubits** on which certain operations can be performed in a topologically protected way using braiding. However, the set of these protected operations is not sufficient to realize universal quantum computing. We show that the electric field in a microwave cavity can induce Rabi **oscillations** between adjacent Majorana bound states. These **oscillations** can be used to implement an additional single-**qubit** gate. Supplemented with one braiding operation, this gate allows to perform arbitrary single-**qubit** operations. ... Majorana bound states have been proposed as building blocks for **qubits** on which certain operations can be performed in a topologically protected way using braiding. However, the set of these protected operations is not sufficient to realize universal quantum computing. We show that the electric field in a microwave cavity can induce Rabi **oscillations** between adjacent Majorana bound states. These **oscillations** can be used to implement an additional single-**qubit** gate. Supplemented with one braiding operation, this gate allows to perform arbitrary single-**qubit** operations.

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Contributors: Cao, Xiufeng, You, J. Q., Zheng, H., Nori, Franco

Date: 2010-01-26

(Color online) Time evolution of the coherence σ x t versus the time t multiplied by the **qubit** energy spacing Δ . (a) The case of weak interaction between the bath and the **qubit**, where the parameters of the low-**frequency** Lorentzian-type spectrum are α / Δ 2 = 0.01 , λ = 0.09 Δ (red solid curve); while for the high-**frequency** Ohmic bath with Drude cutoff the parameters are α o h = 0.01 , ω c = 10 Δ (green dashed-dotted curve). (b)The case of strong interaction between the bath and the **qubit**, where the parameters of the low-**frequency** bath are α / Δ 2 = 0.1 , λ = 0.3 (red solid line), and for the high-**frequency** Ohmic bath are α o h = 0.1 , ω c = 10 Δ (green dashed-dotted line). These results show that the decay rate for the low-**frequency** bath is shorter than for the high-**frequency** Ohmic bath. This means that the coherence time of the **qubit** in the low-**frequency** bath is longer than in the high-**frequency** noise case, demonstrating the powerful temporal memory of the low-**frequency** bath. Also, our results reflect the structure of the solution with branch cuts . The **oscillation** **frequency** for the low-**frequency** noise is ω 0 > Δ , in spite of the strength of the interaction. This can be referred to as a blue shift. However, in an Ohmic bath, the **oscillation** **frequency** is ω 0 **frequency** noises....(Color online) The spectral density J ω of the low- and high-**frequency** baths. (a) The case of weak interaction between the bath and the **qubit**, where the parameters of the low-**frequency** Lorentzian-like spectrum are α / Δ 2 = 0.01 and λ = 0.09 Δ (red solid curve), while for the high-**frequency** Ohmic bath with Drude cutoff the parameters are α o h = 0.01 and ω c = 10 Δ (green dashed-dotted curve). (b) The case of strong interaction between the bath and the **qubit**, where the parameters of the low-**frequency** bath are α / Δ 2 = 0.1 and λ = 0.3 Δ (red solid curve) and the parameters of the high-**frequency** Ohmic bath are α o h = 0.1 and ω c = 10 Δ (green dashed-dotted curve). The characteristic energy of the isolated **qubit** is indicated by a vertical blue dotted line. Here, and in the following figures, the energies are shown in units of Δ ....(Color online) The effective decay γ τ / γ 0 , versus the time interval τ between successive measurements, for a strong coupling between the **qubit** and the bath. The time-interval τ is multiplied by the **qubit** energy difference Δ . The curves in (a) correspond to the case of a low-**frequency** bath with parameters α / Δ 2 = 0.1 and λ = 0.3 Δ (red solid curve). (b) corresponds to the case of an Ohmic bath with parameters α o h = 0.1 and ω c = 10 Δ (red solid curve). The green dashed-dotted curves are the results under RWA when the same parameters are used. Note how different the RWA result is in (b), especially for any short measurement interval τ ....We use a non-Markovian approach to study the decoherence dynamics of a **qubit** in either a low- or high-**frequency** bath modeling the **qubit** environment. This approach is based on a unitary transformation and does not require the rotating-wave approximation. We show that for low-**frequency** noise, the bath shifts the **qubit** energy towards higher energies (blue shift), while the ordinary high-**frequency** cutoff Ohmic bath shifts the **qubit** energy towards lower energies (red shift). In order to preserve the coherence of the **qubit**, we also investigate the quantum Zeno effect in two cases: low- and high-**frequency** baths. For very frequent projective measurements, the low-**frequency** bath gives rise to the quantum anti-Zeno effect on the **qubit**. The quantum Zeno effect only occurs in the high-**frequency** cutoff Ohmic bath, after considering counter-rotating terms. For a high-**frequency** environment, the decay rate should be faster (without frequent measurements) or slower (with frequent measurements, in the Zeno regime), compared to the low-**frequency** bath case. The experimental implementation of our results here could distinguish the type of bath (either a low- or high-**frequency** one) and protect the coherence of the **qubit** by modulating the dominant **frequency** of its environment....(Color online) The effective decay γ τ / γ 0 , versus the time interval τ between consecutive measurements, for a weak coupling between the **qubit** and the bath. In the horizontal axis, the time-interval τ is multiplied by the **qubit** energy difference Δ . The curves in (a) correspond to the case of a low-**frequency** bath with parameters α / Δ 2 = 0.01 and λ = 0.09 Δ (red solid curve). (b) corresponds to the case of an Ohmic bath with parameters α o h = 0.01 and ω c = 10 Δ (red solid curve). The green dashed-dotted curves are the results under the RWA when the same parameters are used. Note how different the RWA result is in (b), especially for any short measurement interval τ . ... We use a non-Markovian approach to study the decoherence dynamics of a **qubit** in either a low- or high-**frequency** bath modeling the **qubit** environment. This approach is based on a unitary transformation and does not require the rotating-wave approximation. We show that for low-**frequency** noise, the bath shifts the **qubit** energy towards higher energies (blue shift), while the ordinary high-**frequency** cutoff Ohmic bath shifts the **qubit** energy towards lower energies (red shift). In order to preserve the coherence of the **qubit**, we also investigate the quantum Zeno effect in two cases: low- and high-**frequency** baths. For very frequent projective measurements, the low-**frequency** bath gives rise to the quantum anti-Zeno effect on the **qubit**. The quantum Zeno effect only occurs in the high-**frequency** cutoff Ohmic bath, after considering counter-rotating terms. For a high-**frequency** environment, the decay rate should be faster (without frequent measurements) or slower (with frequent measurements, in the Zeno regime), compared to the low-**frequency** bath case. The experimental implementation of our results here could distinguish the type of bath (either a low- or high-**frequency** one) and protect the coherence of the **qubit** by modulating the dominant **frequency** of its environment.

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Contributors: Plourde, B. L. T., Robertson, T. L., Reichardt, P. A., Hime, T., Linzen, S., Wu, C. -E., Clarke, John

Date: 2005-01-27

(a) SQUID switching probability vs. amplitude of bias current pulse near **qubit** 2 transition. The two curves represent the states corresponding to Φ Q 2 = 0.48 Φ 0 (red) and Φ Q 2 = 0.52 Φ 0 (blue); Φ S is held constant. Each curve contains 100 points averaged 8 , 000 times. (b) I s 50 % vs. Φ S . Each period of **oscillation** contains ∼ 5 , 000 flux values, and each switching current is averaged 8 , 000 times. (c) Dependence of I s 50 % on Φ Q 1 for constant Φ S . (d) **Qubit** flux map. fig:flux-map...We report measurements on two superconducting flux **qubits** coupled to a readout Superconducting QUantum Interference Device (SQUID). Two on-chip flux bias lines allow independent flux control of any two of the three elements, as illustrated by a two-dimensional **qubit** flux map. The application of microwaves yields a **frequency**-flux dispersion curve for 1- and 2-photon driving of the single-**qubit** excited state, and coherent manipulation of the single-**qubit** state results in Rabi **oscillations** and Ramsey fringes. This architecture should be scalable to many **qubits** and SQUIDs on a single chip....(a) Chip layout. Dark gray represents Al traces, light gray AuCu traces. Pads near upper edge of chip provide two independent flux lines; wirebonded Al jumpers couple left and right halves. Pads near lower edge of chip supply current pulses to the readout SQUID and sense any resulting voltage. (b) Photograph of center region of completed device. Segments of flux lines are visible to left and right of SQUID, which surrounds the two **qubits**. fig:layout...Spectroscopy of **qubit** 2. Enhancement and suppression of I s 50 % is shown as a function of Φ Q 2 and f m relative to measurements in the absence of microwaves. Dashed lines indicate fit to hyperbolic dispersion for 1- and 2-photon **qubit** excitations. The 2-photon fit is one-half the **frequency** of the 1-photon fit. Inset containing ∼ 23 , 000 points is at higher resolution. fig:spectroscopy...Coherent manipulation of **qubit** state. (a) Rabi **oscillations**, scaled to measured SQUID fidelity, as a function of width of 10.0 GHz microwave pulses. (b) Rabi **frequency** vs. 10.0 GHz pulse amplitude; line is least squares fit to the data. (c) Ramsey fringes for **qubit** splitting of 9.95 GHz, microwave **frequency** of 10.095 GHz. (d) Ramsey fringe **frequency** vs. microwave **frequency**. Lines with slopes ± 1 are fits to data. fig:rabi ... We report measurements on two superconducting flux **qubits** coupled to a readout Superconducting QUantum Interference Device (SQUID). Two on-chip flux bias lines allow independent flux control of any two of the three elements, as illustrated by a two-dimensional **qubit** flux map. The application of microwaves yields a **frequency**-flux dispersion curve for 1- and 2-photon driving of the single-**qubit** excited state, and coherent manipulation of the single-**qubit** state results in Rabi **oscillations** and Ramsey fringes. This architecture should be scalable to many **qubits** and SQUIDs on a single chip.

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Contributors: van Heck, B., Hyart, T., Beenakker, C. W. J.

Date: 2014-07-10

An alternative layout that has only Coulomb couplings needs three rather than two islands, forming a tri-junction as in Fig. fig_layoutb. A tri-junction pins a Majorana zero-mode , which can be Coulomb-coupled to each of the other three Majoranas . The tri-junction also binds higher-lying fermionic modes, separated from the zero mode by an excitation energy E M . This is the minimal design for a fully flux-controlled Majorana **qubit**. In Fig. fig_qubit we have worked it out in some more detail for the quantum spin-Hall insulator....The conservation of fermion parity on a single superconducting island implies a minimum of two islands for a Majorana **qubit**, each containing a pair of Majorana zero-modes. The minimal circuit that can operate on a Majorana **qubit** would then have the linear layout of Fig. fig_layouta. While the couplings between Majoranas on the same island are flux-controlled Coulomb couplings, the inter-island coupling is via a tunnel barrier, which would require microscopic control by a gate voltage....Topological **qubit** formed out of four Majorana zero-modes, on either two or three superconducting islands. Dashed lines indicate flux-controlled Coulomb couplings, as in the Cooper pair box of Fig. fig_transmon. In the linear layout (panel a) the coupling between Majoranas on different islands is via a tunnel barrier (thick horizontal line), requiring gate voltage control. By using a tri-junction (panel b) all three couplings can be flux-controlled Coulomb couplings....Energy spectrum of the top-transmon circuit of Fig. fig_qubit, obtained from numerical diagonalization of the Hamiltonian Htilde for E J = 300 GHz, E C = 5 GHz, Φ max = h / 4 e . The junction asymmetry was d = 0.1 , so that E J Φ m a x ≃ 30 GHz. In panel (a), the lowest eight energy levels for E M = 5 GHz are shown as a function of the induced charge q i n d 1 . They correspond to the eight eigenstates | σ , τ | f , where σ = ± 1 labels the excited/ground state of the charge **qubit**, τ = ± 1 labels the even/odd parity state of the topological **qubit**, and f = 0 , 1 the occupation number of the fermionic state in the constriction. As indicated by the colored arrows, the ground and excited state of the charge **qubit** are separated by an energy Ω 0 ± 2 Δ + ≃ 27.5 ± 1.7 GHz, depending on the state of the topological **qubit**. The inset shows the weak charge dispersion of the ground state doublet ( Δ m a x ≃ 120 MHz). In panel (b), the same energy levels are shown as a function of the tunnel coupling E M for a fixed value of q ind 1 = 0 . For a proper operation of the circuit it is required that the states f = 1 with an excited fermionic mode are well separated from both ground and excited states of the charge **qubit**. We have highlighted between grey panels a large energy window 3 G H z E M 8 G H z where this requirement is met....The basic building block of the transmon, shown in Fig. fig_transmon, is a Cooper pair box (a superconducting island with charging energy E C ≪ Josephson energy E J ) coupled to a microwave transmission line (coupling energy ℏ g ). The plasma **frequency** ℏ Ω 0 ≃ 8 E J E C is modulated by an amount Δ + cos π q i n d / e upon variation of the charge q i n d induced on the island by a gate voltage V . Additionally, there is a q i n d -dependent contribution Δ - cos π q i n d / e to the ground state energy. The charge sensitivity Δ ± ∝ exp - 8 E J / E C can be adjusted by varying the flux Φ enclosed by the Josephson junction, which modulates the Josephson energy E J ∝ cos 2 π e Φ / h . In a typical device , a variation of Φ between Φ m i n ≈ 0 and Φ m a x h / 4 e changes Δ ± by several orders of magnitude, so the charge sensitivity can effectively be switched on and off by increasing the flux by half a flux quantum....We construct a minimal circuit, based on the top-transmon design, to rotate a **qubit** formed out of four Majorana zero-modes at the edge of a two-dimensional topological insulator. Unlike braiding operations, generic rotations have no topological protection, but they do allow for a full characterization of the coherence times of the Majorana **qubit**. The rotation is controlled by variation of the flux through a pair of split Josephson junctions in a Cooper pair box, without any need to adjust gate voltages. The Rabi **oscillations** of the Majorana **qubit** can be monitored via **oscillations** in the resonance **frequency** of the microwave cavity that encloses the Cooper pair box....Schematic of a Cooper pair box in a transmission line resonator (transmon) containing a pair of Majorana zero-modes at the edge of a quantum spin-Hall insulator. This hybrid device (top-transmon) can couple charge **qubit** and topological **qubit** by variation of the flux Φ through a Josephson junction....Hya13 the initialization of the ancillas also requires that k B T ≪ Δ m a x , so the Coulomb coupling Δ m a x cannot be much smaller than 10 G H z . There is no such requirement for the simpler circuit of Fig. fig_qubit, because no ancillas are needed for the nontopological rotation of a Majorana **qubit**. This is one reason, in addition to the smaller number of Majoranas, that we propose this circuit for the first generation of experiments on Majorana **qubits**....Top-transmon circuit to rotate the **qubit** formed out of four Majorana zero-modes at the edge of a quantum spin-Hall insulator. One of the Majoranas ( γ B ) is shared by three superconductors at a constriction. The topological **qubit** is rotated by coupling it to a Cooper pair box in a transmission line resonator (transmon). The coupling strength is controlled by the magnetic flux Φ through a pair of split Josephson junctions. The diagrams at the top indicate how the Coulomb couplings of pairs of Majoranas are switched on and off: they are off (solid line) when Φ = 0 and on (dashed line) when Φ = Φ m a x h / 4 e . This single-**qubit** rotation does not have topological protection, it serves to characterize the coherence times of the Majorana **qubit**....Implementation of the braiding circuit of Ref. Hya13 in a quantum spin-Hall insulator. The two T-junctions are formed by a pair of constrictions. The flux-controlled braiding protocol requires four independently adjustable magnetic fluxes. The Majorana **qubit** formed out of zero-modes γ A , γ B , γ C , γ D is flipped at the end of the operation, as can be measured via a shift of the resonant microwave **frequency**. This braiding operation has topological protection. ... We construct a minimal circuit, based on the top-transmon design, to rotate a **qubit** formed out of four Majorana zero-modes at the edge of a two-dimensional topological insulator. Unlike braiding operations, generic rotations have no topological protection, but they do allow for a full characterization of the coherence times of the Majorana **qubit**. The rotation is controlled by variation of the flux through a pair of split Josephson junctions in a Cooper pair box, without any need to adjust gate voltages. The Rabi **oscillations** of the Majorana **qubit** can be monitored via **oscillations** in the resonance **frequency** of the microwave cavity that encloses the Cooper pair box.

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Contributors: Lupascu, A., Bertet, P., Driessen, E. F. C., Harmans, C. J. P. M., Mooij, J. E.

Date: 2008-10-03

fig1 (a) **Frequency** of the spectroscopy peaks p 1 (black squares), p 2 (black circles), and p 3 (triangles) versus Φ . The black lines are a fit for the peaks p 1 and p 2 with the expressions for E 01 q b + T L S and E 02 q b + T L S , yielding the following parameters: I p = 331 nA, Δ = 4.512 GHz, ν T L S = 4.706 GHz, and g = 0.104 GHz. The gray line is a plot of E 01 q b + T L S + E 02 q b + T L S / 2 with the above parameters. (b) Spectroscopy for different values of the microwave power P m w at Φ = 3 Φ 0 / 2 . The curves are vertically shifted for clarity....where σ x , y , z T L S are TLS operators and g is the coupling strength. This Hamiltonian is easily diagonalized, yielding the eigenenergies E n q b + T L S ( n = 0 to 3 ) and the transition energies E m n q b + T L S = E n q b + T L S - E m q b + T L S . The continuous lines in In Fig. fig1b are a combined fit of the Φ -dependent transition energies E 01 q b + T L S and E 02 q b + T L S with the **frequency** of the peaks p 1 and p 2 . This fit yields the parameters I p , Δ , ν T L S , and g . The agreement of the model with the data is very good. We note that the good agreement does not justify the specific model for the interaction in Eq. eq_Hamiltonian_interaction, as discussed in more detail below, but it justifies the model of resonant interaction and moreover it provides the value of the coupling g ....We observed the dynamics of a superconducting flux **qubit** coupled to an extrinsic quantum system (EQS). The presence of the EQS is revealed by an anticrossing in the spectroscopy of the **qubit**. The excitation of a two-photon transition to the third excited state of the **qubit**-EQS system allows us to extract detailed information about the energy level structure and the coupling of the EQS. We deduce that the EQS is a two-level system, with a transverse coupling to the **qubit**. The transition **frequency** and the coupling of the EQS changed during experiments, which supports the idea that the EQS is a two-level system of microscopic origin....fig3 Energy level structure for the **qubit** (qb) coupled to (a) a two level system (TLS) or (b) an harmonic **oscillator** (HO). The dotted lines indicates energy levels for the uncoupled system. Black/gray arrows indicate one-/two-photon transitions starting in the ground state and are labeled by p n , with n the final state of the coupled system....The observation of the two-photon transition brings important additional information about the coupled EQS. We observe (see Fig. fig1a) that the **frequency** of the peak p 3 is the average of the **frequencies** of peaks p 1 and p 2 . This is clearly shown by the gray line in Fig. fig1a, which is a plot of the average of the transition energies E 01 q b + T L S and E 02 q b + T L S , where E 01 q b + T L S and E 02 q b + T L S are given by the best fit to the **frequencies** of p 1 and p 2 . This particular position of the 2-photon peak is consistent with the hypothesis that the EQS is a TLS , but rules out that the EQS is a HO. This can be understood by considering the structure of levels for the coupled **qubit**-TLS and **qubit**-HO cases, as shown in Fig. fig3 a and b respectively. For the latter case the two-photon transition to the third excited state would have a **frequency** significantly lower than the average value of the one-photon transition **frequencies** to the first and second excited states. This conclusion holds for any type coupling between the **qubit** and the HO which is linear in the **oscillator** creation and annihilation operators. Making the distinction between coupled TLSs and HOs is important since HO modes coupled to the **qubit** can appear due to spurious resonances in the electromagnetic circuit used to control and read out the **qubit**....During the experiments we observed important changes of the energy-level structure of the combined **qubit**-TLS system. Our **qubit** sample was used in two experiments, A and B. Between these two experiments our cryostat was warmed up to room temperature. In experiment A a few different configurations of the energy-level structure were observed. In Fig. fig4a we present the spectroscopy data at small power (only lines p 1 and p 2 ) for two such configurations. The measured spectroscopy is in both cases well described by the **qubit**-TLS model (given by Eqs. eq_Hamiltonian_qubit, eq_Hamiltonian_TLS, and eq_Hamiltonian_interaction), but with different **frequency** and coupling of the TLS: ν T L S = 4.706 GHz and g = 0.104 GHz for the first configuration and ν T L S = 4.493 GHz and g = 0.099 GHz for the second configuration. During the first experiment we observed a few changes between such configurations. The change between two configurations was fast on the time scale of a few tens of minutes, which is the time necessary to acquire the data in order to characterize the spectroscopic structure. Each configuration was in turn stable over times of the order of days. We observed for each of these configurations the two-photon transition and Rabi **oscillations** on all the three transitions. In experiment B we observed a similar spectrum (see Fig. fig4b), with a configuration given by I p = 350 nA, Δ = 4.565 GHz, ν T L S = 5.039 GHz, and g = 0.036 GHz. The **frequency** and the coupling of the TLS are significantly different. In contrast to experiment A, the spectrum was stable over all the duration of the experiment (two months). These observations are consistent with other experiments and support the idea that the coupled TLS is of microscopic origin....fig2 Measurements of Rabi **oscillations** at Φ = 3 Φ 0 / 2 , for a **qubit**-TLS configuration given by I p = 331 nA, Δ = 4.47 GHz, ν T L S = 4.39 GHz, and g = 0.099 GHz, for transitions p 1 (squares), p 2 (circles), and p 3 (triangles). (a) Rabi **oscillations** for microwave power P m w = 7 dBm. (b) Rabi **frequency** F Rabi vs P m w . The lines are power law fits for the one- (black) and two-photon (gray) transitions. Only values of F Rabi smaller than 40 MHz are considered for the fit....We now discuss the origin of the peak p 3 in the spectroscopy signal shown in Fig. fig1. Further understanding on this peak is provided by the analysis of Rabi **oscillations**, observed at strong microwave driving. These are shown in Fig. fig2a for three different **frequencies**, corresponding respectively to the peaks p 1 , p 2 , and p 3 . It is interesting to note that the measurement of the Rabi **oscillations** shows that the EQS has coherence times comparable to those of the **qubit** . The microwave amplitude dependence of the Rabi **frequency** is shown in Fig. fig2b for the three different transitions. For low microwave power, we observe a power law behavior with exponent 1.0 for peaks p 1 and p 2 , and 1.8 for peak p 3 . This confirms that p 1 and p 2 are one-photon transitions. We attribute p 3 to a two-photon transition to the third excited state of the coupled system. The value of the exponent of the amplitude dependence, 1.8 , is smaller than the ideal value of 2 . This is consistent with numerical simulations of the driven dynamics. We attribute this difference to the partial excitation of the first two excited states of the coupled system....Spectroscopy is performed by repeating, typically 10 6 times, the following steps : the **qubit** is first prepared in the ground state by energy relaxation. Transitions to excited states are then induced with microwaves at power P m w and **frequency** f m w , applied for a time T m w ; for spectroscopy measurements we take T m w > > T 1 , T 2 . As a final step the driving of the resonant circuit used for readout is switched on and the amplitude V a c is measured. The information on the **qubit** state is provided by the average value of V a c , . In Fig. fig1a the position of the observed spectroscopy peaks for low power is shown as black squares and circles. Away from the symmetry point of the **qubit** ( Φ = 3 Φ 0 / 2 ) the spectrum is similar to the usual flux **qubit** spectrum: we observe a single peak at **frequency** f m w ≈ 2 I p / h Φ - 3 Φ 0 / 2 2 + Δ 2 , corresponding to the transition between the ground and excited states of the **qubit**. However, around Φ = 3 Φ 0 / 2 , we observe two peaks (labeled p 1 and p 2 ) with a Φ dependence characteristic of an anticrossing. This reveals the presence of an EQS with a **frequency** close to the **qubit** parameter Δ . At larger microwave power a third peak is observed (labeled p 3 ) between the peaks p 1 and p 2 . In Fig. fig1b we plot the average as a function of the microwave **frequency** at Φ = 3 Φ 0 / 2 , for increasing microwave power. ... We observed the dynamics of a superconducting flux **qubit** coupled to an extrinsic quantum system (EQS). The presence of the EQS is revealed by an anticrossing in the spectroscopy of the **qubit**. The excitation of a two-photon transition to the third excited state of the **qubit**-EQS system allows us to extract detailed information about the energy level structure and the coupling of the EQS. We deduce that the EQS is a two-level system, with a transverse coupling to the **qubit**. The transition **frequency** and the coupling of the EQS changed during experiments, which supports the idea that the EQS is a two-level system of microscopic origin.

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Contributors: Berg, J. W. G. van den, Nadj-Perge, S., Pribiag, V. S., Plissard, S. R., Bakkers, E. P. A. M., Frolov, S. M., Kouwenhoven, L. P.

Date: 2012-10-26

Utilizing the large difference in g -factors between the two dots we have achieved coherent control of both **qubits**. Here, we probe the **qubits** using similar microwave **frequencies**, but different magnetic fields. The insets in figure fig4 show the Rabi **oscillations** obtained for each of the **qubits**. The **frequency** of the Rabi **oscillations** (see insets of figure fig4) for the **qubit** corresponding to g -factor 48, was 96 MHz. For the other dot, with a corresponding g -factor of 36, a lower Rabi **frequency** of 47 MHz was achieved. This slower Rabi **oscillation** is consistent with a weaker coupling of the microwave electric field to this dot....Due to the strong spin-orbit interaction in indium antimonide, orbital motion and spin are no longer separated. This enables fast manipulation of **qubit** states by means of microwave electric fields. We report Rabi **oscillation** **frequencies** exceeding 100 MHz for spin-orbit **qubits** in InSb nanowires. Individual **qubits** can be selectively addressed due to intrinsic dierences in their g-factors. Based on Ramsey fringe measurements, we extract a coherence time T_2* = 8 +/- 1 ns at a driving **frequency** of 18.65 GHz. Applying a Hahn echo sequence extends this coherence time to 35 ns....fig4(color online) Main panel: Two well separated EDSR peaks for the spin-orbit **qubit** in each of the two dots. The microwave driving **frequency** is 20.9 GHz. Insets: Rabi **oscillations** for the corresponding EDSR peaks. Linear slopes (attributed to PAT) of 0.6 and 0.9 fA/ns respectively are subtracted to flatten the average. The Rabi data on the left was obtained at 31.2 mT B -field and 20.9 GHz driving **frequency**. From the fit (as in Fig. fig2) a Rabi **frequency** of 96 ± 2 MHz is obtained. On the right the field was 41.2 mT and driving **frequency** 21 GHz. The Rabi **frequency** obtained from the fit is 47 ± 3 MHz....fig1(color online) (a) Electron microscope image of the device consisting of an InSb nanowire contacted by source and drain electrodes, lying across a set of fine gates (numbered, 60 nm pitch) as well as a larger bottom gate (labeled BG). (b) Current through the device when applying microwaves with the double dot in spin blockade configuration (a vertical linecut near 0 mT has been subtracted to suppress resonances at constant **frequency**). When the microwave **frequency** matches the Larmor **frequency** (resonance highlighted by dashed blue box), blockade is lifted and current increases. (c) Schematic illustration of Pauli spin blockade on which read-out depends. Only anti-parallel states (right) can occupy the same dot, allowing current through the device. A parallel configuration (left) leads to a suppression of the current....To demonstrate coherent control over the **qubit** we apply microwave bursts of variable length. First, the **qubit** is initialized into a spin blocked charge configuration. This is accomplished by idling inside the bias triangle (Fig. fig2(a)). In order to prevent the electron from tunneling out of the dot during its subsequent manipulation, the double dot is maintained in Coulomb blockade in the same charge configuration. While in the Coulomb blockade regime, a microwave burst is applied. The double dot is then again quickly brought back to the spin blockade configuration by pulsing the plunger gates. By applying such microwave bursts (schematically depicted in figure fig2(b)), we perform a Rabi measurement. If the manipulation has flipped the electron spin-orbit state, the blockade is lifted and an electron can move from the first to the second dot and exit again through the outgoing lead. By continuously repeating the pulse sequence and measuring the (DC) current through the double dot, we measure the Rabi **oscillations** associated with the rotation of the spin-orbit state (Fig. fig2(c))....fig3(color online) (a) Ramsey experiment; an initial π / 2 -pulse rotates the spin to the xy-plane. After some delay a 3 π / 2 pulse is applied, restoring spin blockade or (partially) lifting it, depending on the phase of the pulse. (b) Decay of the Ramsey fringe contrast with increasing delay time τ for different driving **frequencies**. Solid line is a fit to exp - τ / T 2 * 2 at a driving **frequency** of 18.65 GHz, giving T 2 * = 8 ± 1 ns. (c) A Hahn echo sequence (top), extends the decay of the fringe contrast, to 30 ns in this case. (d) Decay of the fringe contrast in the Hahn echo sequence for different microwave **frequencies**. Solid line is a fit to exp - τ / T e c h o 3 for driving **frequency** 18.65 GHz, yielding T e c h o = 34 ± 2 ns....fig2(color online) (a) Bias triangle in which spin blockade was observed for a negative bias of -5 mV. This is the ( 2 m + 1 , 2 n + 1 ) → ( 2 m , 2 n + 2 ) transition (transition A, see ). (b) Sequence used for measuring Rabi **oscillations**. Pulses are applied to gates 2 and 4 to move the double dot along the detuning axis between Coulomb blockade (CB) and spin blockade (SB) configurations. In CB a microwave burst is applied via gate BG to rotate the spin. (c) Rabi **oscillation** obtained at a driving **frequency** of 18.65 GHz and source power of 11 (bottom) to 17 (top) dBm. Dashed lines are fits to a cos f R τ b u r s t + φ τ b u r s t - d + b , giving Rabi **frequencies** f R of 54 ± 1 ; 67 ± 1 ; 84 ± 1 and 104 ± 1 MHz. Linear slopes, attributed to photon assisted tunneling, of 0.5, 0.6, 0.4, and 0.6 fA/ns (top to bottom) were subtracted. d = 0.5 for the bottom trace and 0.4 for the others. Curves are offset by 0.5 pA for clarity. Inset: Rabi **frequencies** set out against driving amplitudes, including a linear fit through 0....When the delay time between the first and final pulse in the Ramsey sequence is increased, the **qubit** starts to dephase. The loss of phase coherence leads to decay of the Ramsey fringe contrast, as shown in figure fig3(b). By fitting the experimental data to exp - τ / T 2 * 2 we extract a dephasing time of T 2 * = 8 ± 1 ns, obtained at a driving **frequency** of 18.65 GHz. Other driving **frequencies** of 7.9 GHz and 31.91 GHz resulted in similar T 2 * values of 6 ± 1 and 9 ± 1 respectively. To extend the coherence of the **qubit**, we employ a Hahn echo technique : halfway between two π / 2 pulses an extra pulse is applied to flip the state over an angle π . Doing so partially refocuses the dephasing caused by the nuclear magnetic field, which varies slowly compared to the electron spin dynamics . From figure fig3(c), where the total delay has been extended to τ = 30 ns, it is clear that this technique can maintain contrast of the Ramsey fringes for considerably longer times. An increase in the coherence time to T e c h o = 35 ± 1 ns is obtained from the decay of the contrast (figure fig3(d)) for a driving **frequency** of 18.65 GHz. Similar values of 34 ± 2 and 32 ± 1 ns were obtained at driving **frequencies** 7.9 and 31.91 GHz respectively....The Rabi experiment demonstrates rotation of the **qubit** around a single axis. However, in order to be able to prepare the **qubit** in any arbitrary superposition, it is necessary to achieve rotations around two independent axes. We demonstrate such universal control by means of a Ramsey experiment, where the axis of **qubit** rotation is determined by varying the phase of the applied microwave bursts, as illustrated at the top of figure fig3(a). As in the Rabi experiment, the microwave bursts are applied while the dots are kept in Coulomb blockade, to maintain a well defined charge state and prevent the electrons from tunneling out during manipulation. In the Ramsey sequence an initial microwave burst rotates the state by π / 2 to the xy-plane of the Bloch sphere. We take this rotation axis to be the x-axis. A second burst is then applied after some delay τ , making a 3 π / 2 rotation. By varying the phase of this pulse with respect to the initial π / 2 pulse, we can control the axis of the second rotation (see figure fig3(a)). For example, if the two bursts are applied with the same phase, in total a 2 π rotation will have been made. This restores a spin blockade configuration, thus leading to a suppression of the current. A second burst with a phase π , however, would rotate the **qubit** in the opposite direction, ending up along the -direction on the Bloch sphere. For this case spin blockade is thus fully lifted and current increases to a maximum. ... Due to the strong spin-orbit interaction in indium antimonide, orbital motion and spin are no longer separated. This enables fast manipulation of **qubit** states by means of microwave electric fields. We report Rabi **oscillation** **frequencies** exceeding 100 MHz for spin-orbit **qubits** in InSb nanowires. Individual **qubits** can be selectively addressed due to intrinsic dierences in their g-factors. Based on Ramsey fringe measurements, we extract a coherence time T_2* = 8 +/- 1 ns at a driving **frequency** of 18.65 GHz. Applying a Hahn echo sequence extends this coherence time to 35 ns.

Data types:

Contributors: Allman, M. S., Altomare, F., Whittaker, J. D., Cicak, K., Li, D., Sirois, A., Strong, J., Teufel, J. D., Simmonds, R. W.

Date: 2010-01-06

We demonstrate coherent tunable coupling between a superconducting phase **qubit** and a lumped element resonator. The coupling strength is mediated by a flux-biased RF SQUID operated in the non-hysteretic regime. By tuning the applied flux bias to the RF SQUID we change the effective mutual inductance, and thus the coupling energy, between the phase **qubit** and resonator . We verify the modulation of coupling strength from 0 to $100 MHz$ by observing modulation in the size of the splitting in the phase **qubit**'s spectroscopy, as well as coherently by observing modulation in the vacuum Rabi **oscillation** **frequency** when on resonance. The measured spectroscopic splittings and vacuum Rabi **oscillations** agree well with theoretical predictions....The next step in the experiment is to demonstrate the effect of the coupler on the quantum mechanical interactions between the **qubit** and cavity. We first look for a cavity interaction using well-established spectroscopic techniques . By use of figure IGFCombined(a) the coupler is set to the desired coupling strength and then **qubit** spectroscopic measurements are performed. When the **qubit** transition **frequency** nears the resonant **frequency** of the resonator, an avoided crossing occurs, splitting the resonant peak into two peaks. When the **qubit** **frequency** exactly matches the resonator’s **frequency** ( Δ = 0 ) the size of the spectroscopic splitting is minimized to g Φ x / π . This whole cycle is repeated for different flux biases applied to the coupler. We observe the size of the zero-detuning splitting modulate from a maximum of g m a x / π ≈ 100 M H z down to no splitting (Figure CombinedWFall (a)). The spectroscopic measurements are a good indicator that the coupler is working, but we do not consider them to be proof of coherent coupling between the **qubit** and resonator, because the length of the microwave pulse is longer ( ≃ 500 n s ) than the lifetime of the **qubit**....The coupler is first calibrated by sweeping its external flux bias, Φ x , and measuring the effect on the tunneling probability of the | g state of the **qubit**. By tracking the required applied **qubit** flux Φ q , to maintain a constant total **qubit** flux φ q = Φ q + M q c I c / Φ 0 such that the | g state tunneling probability is approximately 10 % , we can determine the circulating current in the coupler as a function of Φ x . Figure IGFCombined(a) shows the measured coupler circulating current as a function of applied coupler bias flux....For our design parameters g r e s i d u a l ∼ 10 k H z , much too weak to account for the residual effect seen in the data. We believe the residual coupling effect is due to weakly coupled, spurious two-level system fluctuators (TLSs) interacting with the **qubit** at this **frequency** . We have used a scan of vacuum Rabi data that confirms these types of weak **oscillations** throughout the entire spectroscopic range, even at **frequencies** far detuned from the resonator. This indicates interactions with weakly coupled TLSs not seen in traditional spectroscopy measurements. Figure T1 compares the vacuum Rabi data taken at Φ x / Φ 0 = - 0.421 and the exponential and non-exponential T 1 data taken at **qubit** **frequencies** far detuned from the resonator and where no TLS splittings were visible in the spectroscopic data....Experimentally, we excite the | e 0 state by applying a short τ p ≃ 5 - 10 n s pulse with the **qubit** on resonance with the resonator. The pulse is fast enough that the resonator remains in its ground state during state preparation. We then measure the state of the **qubit** as a function of time. Figures IGFCombined(b,c) and CombinedWFall summarize the spectroscopic and time domain measurements. For g Φ x / π > 10 M H z , the vacuum Rabi data are used to determine the coupling strength by applying a Fast Fourier Transform (FFT) to the measured probability data. For g Φ x / π **oscillation** in the data (Figure T1)....where ω r 0 = 1 / L r C r . The measured resonator **frequency** is shown in Figure IGFCombined(b)....(a) Circuit diagram for the phase **qubit**, coupler and resonator. The **qubit** parameters are I q 0 ≃ 0.6 μ A , C q s ≃ 0.6 p F , L q ≃ 1000 p H , β q ≃ 1.8 , and M q c ≃ 60 p H . The coupler parameters are I c 0 ≃ 0.9 μ A , L c ≃ 200 p H , C j c ≃ 0.3 p F and β c ≃ 0.5 . The resonator parameters are L r ≃ 1000 p H , C r ≃ 0.4 p F , and M c r ≃ 60 p H . (b) Optical micrograph of the circuit....Spectroscopic and time-domain data over the range Φ x / Φ 0 = - 0.462 to Φ x / Φ 0 = - 0.366 bounded by the vertical dashed lines in Figure IGFCombined. (a) Waterfall plot of the spectroscopic measurements of the | ± states showing the splitting transition from g c -0.462 / π ≃ 50 M H z through g c -0.421 / π = 0 to g c -0.366 / π ≃ 40 M H z . The inset to the left is a 3D plot of the **qubit** spectroscopy showing the avoided crossing transition through zero for applied coupler flux values close to Φ x = - 0.421 . (b) The corresponding vacuum Rabi measurements demonstrating coherent modulation in the coupling strength g c Φ x ....Measurements of the dependence of I c , ω r , and g c on applied coupler flux, Φ x / Φ 0 . The vertical dashed lines bracket the applied flux ranges for the waterfall data shown in Figure CombinedWFall. (a) The measured circulating coupler current as a function of applied coupler flux along with the theoretical fit giving β c = 0.51 . (b) Measured resonator **frequency** as a function of applied coupler flux, along with theoretical fit using β c extracted from (a). The fit yields ω r 0 / 2 π = 7.710 G H z . (c) Measured coupling strength as a function of applied coupler flux along with the theoretical fit using parameters extracted from the theory fits in (a) and (b)....A higher resolution trace of the occupation probability of the | e 0 state when Φ x / Φ 0 = - 0.421 along with exponential T 1 and non-exponential T 1 measurements taken at a **qubit** **frequencies** largely detuned from the resonator. The non-exponential T 1 trace showed no evidence of a TLS interaction in the corresponding spectroscopy. ... We demonstrate coherent tunable coupling between a superconducting phase **qubit** and a lumped element resonator. The coupling strength is mediated by a flux-biased RF SQUID operated in the non-hysteretic regime. By tuning the applied flux bias to the RF SQUID we change the effective mutual inductance, and thus the coupling energy, between the phase **qubit** and resonator . We verify the modulation of coupling strength from 0 to $100 MHz$ by observing modulation in the size of the splitting in the phase **qubit**'s spectroscopy, as well as coherently by observing modulation in the vacuum Rabi **oscillation** **frequency** when on resonance. The measured spectroscopic splittings and vacuum Rabi **oscillations** agree well with theoretical predictions.

Data types:

Contributors: Wirth, T., Lisenfeld, J., Lukashenko, A., Ustinov, A. V.

Date: 2010-10-05

The the position of a dip in the amplitude of the reflected pulse is plotted in Fig. fig:2 as a function of microwave **frequency** and applied SQUID flux bias Φ S . Data points indicate the dependence of the tank circuit resonance **frequency** on the applied bias flux. The larger (red) circles correspond to the flux swept from negative to positive values, while the smaller (blue) dots stand for the flux swept in opposite direction. During the flux sweep, due to the crosstalk between Φ S and Φ Q flux lines approximately one flux quantum Φ 0 enters or leaves the **qubit** loop, which gives rise to abrupt shift of the dip **frequency** at specific flux bias values. The resonance **frequency** shift at a bias flux of -0.35 Φ 0 is about 22 MHz, which is larger than the tank circuit’s resonance line width of about 4 MHz. This scheme is thus capable of single-shot detection of the **qubit** flux state....(Color online) Scheme of the measurement setup. The SQUID with shunt capacitor C 1 coupled to the **qubit**. The pulsed microwave signal is applied via a cryogenic circulator, and the reflected signal is amplified by a cryogenic amplifier....(Color online) Shift of the resonance **frequency** of the SQUID resonator by 30 MHz due to the **qubit** changing its magnetic flux by approximately Φ 0 . (a) In the linear regime. (b) SQUID driven in the non-linear regime. Note the larger signal amplitude compared to the linear regime....We couple the **qubit** to a capacitively shunted dc-SQUID which forms a tank circuit having a resonance **frequency** around 2 GHz. It is connected to a microwave line by a coupling capacitor C 0 shown in Fig. fig:1. Our sample was fabricated in a standard niobium-aluminium trilayer process. Measurement of the amplitude and phase of a reflected microwave pulse allows one to determine the shift of the resonance **frequency** of the SQUID-resonator and by doing this deduce the magnetic flux of the **qubit** state....The SQUID resonator **frequency** shift induced by the **qubit** is shown in detail in Fig. fig:3(a). It displays two traces of the normalized reflected signal amplitude versus the applied microwave **frequency** in the vicinity of the **qubit**-state switching. Here, the resonance was located at around 1.9 GHz where the SQUID has higher sensitivity to the flux. The amplitude of the reflected signal drops at the resonance **frequency**. For this measurement, very low microwave power of -120 dBm was applied to SQUID to stay in the linear regime, giving rise to the Lorentzian shape of the resonance dips. Taking into account the line width of 4 MHz and the dependence of the resonance **frequency** on the flux, we achieve a flux resolution of 2-3 m Φ 0 of the detector at operating **frequency** of 1.9 GHz. As the two **qubit** states differ by magnetic flux of the order of Φ 0 , this allows for a very weak inductive coupling between SQUID and **qubit** for future experiments. Fig. fig:3 (b) shows the same **frequency** range as above, but now the power of the input signal is larger, -115 dBm, driving the SQUID into the nonlinear regime. This is revealed by the shape of the dips. The advantage of the non-linear regime is the sharper edge on the low **frequency** side which allows for an even better flux resolution of about 0.5-0.7 m Φ 0 ....superconducting **qubits**, phase **qubit**, dispersive readout, SQUID...We present experimental results on a dispersive scheme for reading out a Josephson phase **qubit**. A capacitively shunted dc-SQUID is used as a nonlinear resonator which is inductively coupled to the **qubit**. We detect the flux state of the **qubit** by measuring the amplitude and phase of a microwave pulse reflected from the SQUID resonator. By this low-dissipative method, we reduce the **qubit** state measurement time down to 25 microseconds, which is much faster than using the conventional readout performed by switching the SQUID to its non-zero dc voltage state. The demonstrated readout scheme allows for reading out multiple **qubits** using a single microwave line by employing **frequency**-division multiplexing....Figure fig:4 shows Rabi **oscillations** of the **qubit** measured for different driving powers of the **qubit** microwave driving. As it is expected, the **frequency** of Rabi **oscillations** increases approximately linearly with the driving field amplitude. The measured energy relaxation time of the tested **qubit** is rather short and is of order of T 1 = 5 ns. This time is it not limited by the chosen type of readout but rather determined by the intrinsic coherence of the **qubit** itself. We verified this fact by measuring the same **qubit** with the conventional SQUID switching current method, which yielded very similar T 1 . The observed short coherence time is likely to be caused by the dielectric loss in the silicon oxide forming the insulating dielectric layer around the **qubit** Josephson junction ....(Color online) Coherent **oscillations** of the **qubit** for different driving powers, from bottom to top: -18 dBm, -15 dBm, -12 dBm, -9 dBm and -6 dBm. Curves are offset by 0.1 for better visibility....(Color online) Microwave **frequency** applied to the SQUID vs. externally applied flux. The measurement points show the position of a dip in the reflected signal amplitude for two different directions of the flux sweep....On chip, there are two magnetic flux lines, one for flux Φ S biasing the SQUID, see Fig. fig:1, and another for flux Φ Q biasing the **qubit**. The **qubit** is controlled by microwave pulses which are applied via a separate line (not shown) attenuated at several low temperature stages. The SQUID flux bias line is equipped with a current divider and filter at the 1 K stage, and a powder filter at the sample holder. By taking the crosstalk of the two flux coils into account, we can independently change the flux that is seen by the **qubit** and the flux that is seen by the SQUID. This sample was designed with a large mutual inductance between **qubit** loop and dc-SQUID which allowed us to independently characterize the sample by the conventional switching-current technique. The dispersive readout results presented below were obtained without applying any dc-bias to the readout SQUID. ... We present experimental results on a dispersive scheme for reading out a Josephson phase **qubit**. A capacitively shunted dc-SQUID is used as a nonlinear resonator which is inductively coupled to the **qubit**. We detect the flux state of the **qubit** by measuring the amplitude and phase of a microwave pulse reflected from the SQUID resonator. By this low-dissipative method, we reduce the **qubit** state measurement time down to 25 microseconds, which is much faster than using the conventional readout performed by switching the SQUID to its non-zero dc voltage state. The demonstrated readout scheme allows for reading out multiple **qubits** using a single microwave line by employing **frequency**-division multiplexing.

Data types:

Contributors: Baur, M., Filipp, S., Bianchetti, R., Fink, J. M., Göppl, M., Steffen, L., Leek, P. J., Blais, A., Wallraff, A.

Date: 2008-12-23

depending linearly on the drive amplitude ε . Therefore, one would expect that the strong drive at the **qubit** transition **frequency** ω d ≈ ω g e should lead to a square-root dependence of the Autler-Townes and Mollow spectral lines on the drive power P d ∝ ε 2 . However, the Autler-Townes spectral lines show a clear power dependent shift, see Fig. fig:fig3, and the splitting of both pairs of lines scales weaker than linearly with ε ....We measure the Autler-Townes and the Mollow spectral lines according to the scheme shown in Fig. fig:fig1(b). First, we tune the **qubit** to the **frequency** ω g e / 2 π ≈ 4.811 G H z , where it is strongly detuned from the resonator by Δ / 2 π = 1.63 G H z . We then strongly drive the transition | g → | e with a first microwave tone of amplitude ε applied to the **qubit** at the fixed **frequency** ω d = 4.812 G H z . The drive field is described by the Hamiltonian H d = ℏ ε a † e - i ω d t + a e i ω d t where the drive amplitude ε is given in units of a **frequency**. The **qubit** spectrum is then probed by sweeping a weak second microwave signal over a wide range of **frequencies** ω p including ω g e and ω e f . Simultaneously, amplitude T and phase φ of a microwave signal applied to the resonator are measured . We have adjusted the measurement **frequency** to the **qubit** state-dependent resonance of the resonator under **qubit** driving for every value of ε . Figures fig:fig2(a) and (b) show the measurement response T and φ for selected values of ε . For drive amplitudes ε / 2 π > 65 M H z , two peaks emerge in amplitude from the single Lorentzian line at **frequency** ω e f corresponding to the Autler-Townes doublet, see Fig. fig:fig2(a). The signal corresponding to the sidebands of the Mollow triplet is visible at high drive amplitudes ε / 2 π > 730 M H z in phase, see Fig. fig:fig2(b). Black lines in Fig. fig:fig2 are fits of the data to Lorentzians from which the dressed **qubit** resonance **frequencies** are extracted....(a) Extracted splitting **frequencies** of the Mollow triplet sidebands (red dots) and the Autler-Townes doublet (blue dots) as a function of the drive field amplitude. Dashed lines: Rabi **frequencies** obtained with Eq. ( eq:1). Black solid lines: Rabi **frequencies** calculated by numerically diagonalizing the Hamiltonian Eq. ( eq:2) taking into account 5 transmon levels. (b) Zoom in of the region in the orange rectangle in (a). Orange dots: Rabi **frequency** Ω g e vs. drive amplitude ε extracted from time resolved Rabi **oscillation** experiments, lines as in (a). (c) Rabi **oscillation** measurements between states | g and | e with Ω R / 2 π = 50 M H z and 85 M H z ....fig:fig3 Measured Autler-Townes doublet (blue dots) and Mollow triplet sideband **frequencies** (red dots) vs. drive power P d at a fixed drive **frequency** ω d / 2 π = 4.812 G H z . Black solid lines are transition **frequencies** calculated by numerically diagonalizing the Hamiltonian ( eq:2) taking into account the lowest 5 transmon levels....The **frequencies** of the Autler-Townes doublet (blue data points) and of the Mollow triplet sidebands (red data points) extracted from the Lorentzian fits in Fig. fig:fig2(a) and (b) are plotted in Fig. fig:fig3. The splitting of the spectral lines in pairs separated by Ω R and 2 Ω R , respectively, is observed for Rabi **frequencies** up to Ω R / 2 π ≈ 300 M H z corresponding to about 6% of the **qubit** transition **frequency** ω g e ....(a) Simplified circuit diagram of the measurement setup analogous to the one used in Ref. . In the center at the 20 mK stage, the **qubit** is coupled capacitively through C g to the resonator, represented by a parallel LC **oscillator**, and the resonator is coupled to the input and output transmission lines over capacitances C i n and C o u t . Three microwave signal generators are used to apply the measurement ν r f and drive and probe tones ν d r i v e / p r o b e to the input port of the resonator. The transmitted measurement signal is then amplified by an ultra-low noise amplifier at 1.5 K, down-converted with an IQ-mixer and a local **oscillator** (LO) to an intermediate **frequency** at 300K and digitized with an analog-to-digital converter (ADC). (b) Energy-level diagram of a bare three-level system with states | g , | e , | f ordered with increasing energy. Drive and probe transitions are indicated by black and red/blue arrows, respectively. (c) Energy-level diagram of the dipole coupled dressed states with the coherent drive tone. Possible transitions induced by the probe tone between the dressed states and the third **qubit** level ( ν - , f , ν + , f ) and between the dressed states ( ν - , + , ν + , - ) are indicated with blue and red arrows....In the experiments presented here, we use a version of the Cooper pair box , called transmon **qubit** , as our multilevel quantum system. States of increasing energies are labelled | l with l = g , e , f , h , i , … The transition **frequency** ω g e between the ground | g and first excited state | e is approximated by ℏ ω g e ≈ 8 E C E J m a x | cos 2 π Φ / Φ 0 | - E C , where E C / h = 233 M H z is the charging energy and E J m a x / h = 32.8 G H z is the maximum Josephson energy. The transition **frequency** ω g e can be controlled by an external magnetic flux Φ applied to the SQUID loop formed by the two Josephson junctions of the **qubit**. The transition **frequency** from the first | e to the second excited state | f is given by ω e f = ω g e - α , where α ≈ 2 π E C / h is the **qubit** anharmonicity . The **qubit** is strongly coupled to a coplanar waveguide resonator with resonance **frequency** ω r / 2 π = 6.439 G H z and photon decay rate κ / 2 π ≈ 1.6 MHz. A schematic circuit diagram of the setup is shown in Fig fig:fig1(a)....We present spectroscopic measurements of the Autler-Townes doublet and the sidebands of the Mollow triplet in a driven superconducting **qubit**. The ground to first excited state transition of the **qubit** is strongly pumped while the resulting dressed **qubit** spectrum is probed with a weak tone. The corresponding transitions are detected using dispersive read-out of the **qubit** coupled off-resonantly to a microwave transmission line resonator. The observed **frequencies** of the Autler-Townes and Mollow spectral lines are in good agreement with a dispersive Jaynes-Cummings model taking into account higher excited **qubit** states and dispersive level shifts due to off-resonant drives....To confirm the direct relationship between the measured dressed state splitting **frequency** and the Rabi **oscillation** **frequency** of the excited state population we have also performed time resolved measurements of the Rabi **frequency** up to 100 M H z , see Fig. fig:fig4(c). The extracted Rabi **frequencies** (orange data points) are in good agreement with the spectroscopically measured Rabi **frequencies** (blue squares) over the range of accessible ε , as shown in Fig. fig:fig4(b) ... We present spectroscopic measurements of the Autler-Townes doublet and the sidebands of the Mollow triplet in a driven superconducting **qubit**. The ground to first excited state transition of the **qubit** is strongly pumped while the resulting dressed **qubit** spectrum is probed with a weak tone. The corresponding transitions are detected using dispersive read-out of the **qubit** coupled off-resonantly to a microwave transmission line resonator. The observed **frequencies** of the Autler-Townes and Mollow spectral lines are in good agreement with a dispersive Jaynes-Cummings model taking into account higher excited **qubit** states and dispersive level shifts due to off-resonant drives.

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