### 54067 results for qubit oscillator frequency

Contributors: Kim, Dohun, Ward, D. R., Simmons, C. B., Gamble, John King, Blume-Kohout, Robin, Nielsen, Erik, Savage, D. E., Lagally, M. G., Friesen, Mark, Coppersmith, S. N.

Date: 2014-07-28

Pulse sequence used for lock-in measurement of the **qubit** state. The orange dashed line shows the corresponding lock-in reference signal, which also serves as the probability calibration pulse....The ultimate test of experimental **qubit** control is the demonstration of repeatable quantum logic gates. Although π / 2 rotations that generate the Clifford group are commonly demonstrated, ac control allows direct implementation of any unitary. We therefore demonstrated (and validated with quantum tomography) two distinct gatesets: (1) high-fidelity approximations to X π / 2 Z π / 2 ; and (2) a set of three arbitrarily chosen near-unitary operations G 1 G 2 G 3 . We used standard quantum process tomography (QPT) to characterize the first gateset. Fig. fig4(a) shows the resulting process matrices ( χ ) expressed in the Pauli basis: solid bars represent the ideal “target" quantum processes, while open circles show the results of QPT. The process fidelities between the QPT estimates and the targets are F = 86 % and F = 91 % for the X π / 2 and Z π / 2 operations, respectively. QPT analysis is not entirely reliable, because QPT relies on prior knowledge of input states and final measurements that are implemented using the same logic gates that we seek to characterize (see Supplementary Information S3). Therefore, we also applied a technique called gate set tomography (GST) that avoids these assumptions. GST characterizes logic gates and state preparation/measurement simultaneously and self-consistently, by representing all of them as unknown process matrices. Because this frees us from any obligation to use carefully calibrated operations, we applied GST to a set of three repeatable but uncalibrated logic gates that we denote G 1 G 2 G 3 ....A most intuitive realization of a **qubit** is a single electron charge sitting at two well-defined positions, such as the left and right sides of a double quantum dot. This **qubit** is not just simple but also has the potential for high-speed operation, because of the strong coupling of electric fields to the electron. However, charge noise also couples strongly to this **qubit**, resulting in rapid dephasing at nearly all operating points, with the exception of one special 'sweet spot'. Fast dc voltage pulses have been used to manipulate semiconductor charge **qubits**, but these previous experiments did not achieve high-fidelity control, because dc gating requires excursions away from the sweet spot. Here, by using resonant ac microwave driving, we achieve coherent manipulation of a semiconductor charge **qubit**, demonstrating a Rabi **frequency** of up to 2GHz, a value approaching the intrinsic **qubit** **frequency** of 4.5GHz. Z-axis rotations of the **qubit** are well-protected at the sweet spot, and by using ac gating, we demonstrate the same protection for rotations about arbitrary axes in the X-Y plane of the **qubit** Bloch sphere. We characterize operations on the **qubit** using two independent tomographic approaches: standard process tomography and a newly developed method known as gate set tomography. Both approaches show that this **qubit** can be operated with process fidelities greater than 86% with respect to a universal set of unitary single-**qubit** operations....Data for GST are obtained from many repetitions of several specific experiments, each described by a specific sequence of operations: (i) initialize the **qubit** in state ρ ; (ii) perform a sequence of L ∈ 0 … 32 operations chosen from G 1 G 2 G 3 ; (ii) perform measurement M . Statistical analysis (a variant of maximum likelihood estimation) is used to find the estimates G ̂ 1 G ̂ 2 G ̂ 3 ρ M that are most consistent with the measurements (see the Supplementary Information S3 for more details). Since we did not set out to implement any particular rotations, we compute ex post facto the closest unitary rotations to these estimates, and define these as the “target" gates. The results are shown in Figs. fig4b-c. Fig. fig4b shows the elements of the process ( χ ) matrix for the GST estimates (triangles) and the closest-unitary “targets" (solid bars). Fig. fig4c portrays those same unitaries as rotations of the Bloch sphere....Quantum process tomography and gate set tomography of the ac-gated charge **qubit**. a, Real and imaginary parts of the elements of the process matrix χ in the Pauli basis { I , X , Y , Z } for X π / 2 and Z π / 2 processes : ideal “targets" (solid bars), and standard QPT estimates (open circles). b, χ for the uncalibrated operations G 1 , G 2 , and G 3 , obtained by gate set tomography (GST, triangles) and standard quantum process tomography (QPT, open circles), compared to target gates T 1 , T 2 , and T 3 (solid bars). Since these gates are not precalibrated, the target gates are defined to be the unitary processes closest to the GST estimates of G 1 , G 2 , and G 3 in Frobenius norm. GST self-consistently determines the state preparation, gate operations, and measurement processes . c, Rotation axes on the Bloch sphere, rotation angle θ , process fidelities obtained by GST ( F p , G S T ), and QPT ( F p , Q P T ) for three processes G 1 , G 2 , and G 3 . Here, the rotation axis and angle correspond to the closest unitary operations to the GST estimate ( T 1 , T 2 , and T 3 ) ; the process fidelities are also taken between the estimates and these target processes. The error on F p , Q P T was estimated by repeating QPT using 10 distinct sets of input and output states; standard deviations are reported. GST and QPT yield consistent results, with process fidelities ≥ 86 % for all gates....For the measurement of changes in the probabilities of charge occupation resulting from fast microwave bursts, we use the general approach described in , where we measure the difference between the QPC conductance with and without the manipulation pulse train. Fig. fig:S1 shows similar scheme adopted in this work. We alternate an appropriate number of manipulation and measurement sequences (measurement time ∼ 60 ns) with microwave bursts with adiabatic in and out sequences without microwave to form a low **frequency** signal with **frequency** on the order of ∼ 1 kHz. The adiabatic ramp sequence without microwave manipulation does not induce state population change, but greatly reduces background signal due to capacitive crosstalk. The manipulation sequence including the microwave bursts is generated using Tektronix AWG70002A arbitrary waveform generator with maximum sample rate of 25 Gs/s and analog bandwidth of about 13 GHz. The data are acquired using a lock-in amplifier with a reference signal corresponding to the presence and absence of the pulses, as shown schematically by the orange dashed line in Fig. fig:S1. We compare the measured signal level with the corresponding 2 1 - 1 2 charge transition signal level, calibrated by sweeping gate GL and applying the orange square pulse shown in Fig. fig:S1 to gate GL. Similarly to previous work , charge relaxation during the measurement phase is taken into account using the measured charge relaxation time T 1 ≈ 23.5 ns at the read-out detuning of ε r ≈ - 160 μ eV....We characterized decoherence times by implementing a Hahn echo of the ac-gated charge **qubit** by applying the pulse sequence shown in Fig. fig3a. Inserting an X π pulse between state initialization and measurement, which is performed with X π / 2 gates, corrects for noise that is static on the time scale of the pulse sequence. In Fig. fig3b and c, while keeping the total free evolution time τ fixed, we sweep the position of the decoupling X π pulse to reveal an echo envelope . The maximum amplitude of the observed envelope reveals the extent to which the state has dephased during the free evolution time τ , characterized by T 2 , whereas the amplitude decays as a function of δ t with inhomogeneous decay time T 2 * . The **oscillations** of P 1 in Fig. fig3b and c are observed as a function of δ t at twice the Ramsey **frequency** ( 2 f R a m s e y ≈ 9 GHz) and are well-fit by a Gaussian decay (red solid curve). Fig. fig3d shows the echo amplitude decay as a function of τ , where for each τ the echo amplitude is determined by fitting the echo envelop to Gaussian decay similar to Fig. fig3c. By fitting the echo amplitude decays to a Gaussian, we obtain the dephasing time T 2 ≈ 2.2 ns....Si/SiGe quantum dot device, **qubit** spectroscopy, and coherent Rabi **oscillation** measurements. a, SEM image of a device lithographically identical to the one used in the experiment, with the locations of the double dot indicated by white dashed circles . The current through the quantum point contact (QPC) I Q P C is used for charge sensing via a measurement of its change when microwave component is added to a voltage sequence applied to gate GR. b-c, **Qubit** energy levels and microwave spectroscopy. b, Probability P 1 of the state to be | 1 at the end of the driving sequence as a function of detuning ε and excitation **frequency** f e x of the microwave applied to gate GR. For this experiment, the repetition rate of the driving sequence is ≈ 15 MHz and the microwave pulses are of duration 10 ns. The base value of the detuning ε r used for both readout and initialization is ε = - 160 ~ μ e V . P 1 is large when the microwave pulse is resonant, so that it excites the system from the ground state to the excited state. Dashed green curve shows fit to calculated energy difference between ground state and lowest-energy excited state of the three-level model of Ref. (see also Supplementary Information S2). The third level of higher energy affects the dispersion of energy levels shown, but is otherwise unimportant because its occupation is negligible for all experiments shown. c, Diagram of the calculated energy levels E versus detuning ε , including the ground states of the 2 1 and 1 2 charge configuration, | L and | R respectively, and logical states | 0 = | L + | R / 2 and | 1 = | L - | R / 2 . Black solid line inset: pulse sequence used for Rabi **oscillation** and spectroscopy measurements. d-e, Rabi **oscillation**. d, P 1 as a function of the voltage V G L and microwave pulse duration t b with f e x = 4.54 GHz and excitation amplitude V a c = 70 mV. e, Line-cut of P 1 near V G L = -390 mV, showing ≈ 1 GHz coherent Rabi **oscillations**. Red solid curve shows a fit to an exponentially damped sine wave with best fit parameter T 2 * = 1.5 ns. f-g, Dependence of the Rabi **oscillation** **frequency** on the microwave amplitude. f, P 1 as a function of V a c and t b with f e x = 4.54 GHz, which demonstrates that the Rabi **oscillation** **frequency** f R a b i increases as the amplitude of the microwave driving is increased. g, Rabi **oscillation** **frequency** f R a b i as a function of V a c with fixed f e x = 4.54 GHz. The good agreement of a linear fit (red dashed line) to the data is strong evidence that the measured **oscillations** are indeed Rabi **oscillations**, with the Rabi **frequency** proportional to the driving amplitude....Hahn echo measurement. a, Schematic pulse sequence for the measurement of Hahn echo that corrects for noise that is static on the time scale of the pulse sequence . b-c, Typical echo measurement with fixed total evolution time τ = 1.4 ns. b, P 1 as a function of V G L and delay time δ t of the X π pulse with decoupling X π gate. The effects of static inhomogeneities are minimized at δ t = 0 , and **oscillations** of P 1 as function of δ t at twice the Ramsey **frequency** decay with δ t at the inhomogeneous decay rate 1 / T 2 * . The magnitude of the signal at δ t = 0 as the wait time τ is varied decays at the homogeneous decay rate 1 / T 2 . c, Line cut of P 1 near V G L = -390 mV showing oscillatory signal at twice the Ramsey **frequency**, ≈ 9 GHz. Solid red curve is a fit to gaussian envelope with fixed T 2 * = 1.3 ns, consistent with the Ramsey fringe measurement. d, Echo amplitude as a function of τ . The solid red curve is a gaussian fit with T 2 , e c h o = 2.1 ns. Applying the Hahn echo sequence increases the dephasing time, indicating that a significant component of the dephasing arises from low-**frequency** noise processes....Ramsey fringes and demonstration of three axes control of ac-gated charge **qubit**. a, Schematic of the pulse sequences used to perform universal control of the **qubit**. Both the delay t e and the phase φ of the second microwave pulse are varied in the experiment. b-c, Experimental measurement of Z-axis rotation. In a Ramsey fringe (Z-axis rotation) measurement, the first X π / 2 gate rotates the Bloch vector onto the X - Y plane, and the second X π / 2 gate ( φ = 0) is delayed with respect to the first gate by t e , during which time the state evolves freely around the Z-axis of the Bloch sphere. b, P 1 as a function of V G L and t e for states initialized near | Y . c, Line-cut of P 1 near V G L = -390 mV, showing ≈ 4.5 GHz Ramsey fringe. Red solid curve shows a fit to exponentially damped sine wave with best fit parameter T 2 * = 1.3 ns. d-e, Two axis control of the **qubit** d, P 1 as a function of φ and t e . e, Line-cut of d at φ = 0 (X-axis, black), 90 (Y-axis, red), and 180 (-X-axis, blue). The coherent z-axis rotation along with the rotation axis control with φ demonstrates full control of the **qubit** states around three orthogonal axes on the Bloch sphere. ... A most intuitive realization of a **qubit** is a single electron charge sitting at two well-defined positions, such as the left and right sides of a double quantum dot. This **qubit** is not just simple but also has the potential for high-speed operation, because of the strong coupling of electric fields to the electron. However, charge noise also couples strongly to this **qubit**, resulting in rapid dephasing at nearly all operating points, with the exception of one special 'sweet spot'. Fast dc voltage pulses have been used to manipulate semiconductor charge **qubits**, but these previous experiments did not achieve high-fidelity control, because dc gating requires excursions away from the sweet spot. Here, by using resonant ac microwave driving, we achieve coherent manipulation of a semiconductor charge **qubit**, demonstrating a Rabi **frequency** of up to 2GHz, a value approaching the intrinsic **qubit** **frequency** of 4.5GHz. Z-axis rotations of the **qubit** are well-protected at the sweet spot, and by using ac gating, we demonstrate the same protection for rotations about arbitrary axes in the X-Y plane of the **qubit** Bloch sphere. We characterize operations on the **qubit** using two independent tomographic approaches: standard process tomography and a newly developed method known as gate set tomography. Both approaches show that this **qubit** can be operated with process fidelities greater than 86% with respect to a universal set of unitary single-**qubit** operations.

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

Date: 2003-05-20

(A) Calculated energy diagram for the three-junction **qubit**, for E J / E C = 35 , E C = 7.4 GHz and α = 0.8 (11). Δ γ q indicates the phase shift induced by the SQUID bias current. (B) Ground-state transition step: The sinusoidal background modulation of the SQUID ( I b g ) is subtracted from the I b pulse amplitude corresponding to 50 % switching probability ( I s w ) and then normalized to I s t e p , the middle value (at the dashed line). A sharp peak and dip are induced by a long (1 μ s) MW radiation burst at 16 GHz, allowing the symmetry point to be found (midpoint of the peak/dip positions, dotted line). Data show I s w versus Δ Φ e x t , the deviation in external flux from this point. The transition step is displaced from this point by Δ γ q / 2 π . (C) **Frequency** of the resonant peaks/dips (dots) versus Δ Φ e x t ; the continuous line is a numerical fit with the same parameters as in (A) leading to a value of Δ = 3.4 GHz, whereas the dashed line depicts the case Δ = 0 ....(A) Rabi **oscillations** for a resonant **frequency** F = E 10 = 6.6 GHz and three different microwave powers A = 0 , -6 and -12 dBm, where A is the nominal microwave power applied at room temperature. The data are well fitted by exponentially damped sinusoidal **oscillations**. The resulting decay time is ∼ 150 ns for all powers. (B) Linear dependence of the Rabi **frequency** on the microwave amplitude, expressed as 10 A / 20 . The slope is in agreement with estimations based on sample design....(A) Scanning electron micrograph of a flux-**qubit** (small loop with three Josephson junctions of critical current ∼ 0.5 mA) and the attached SQUID (large loop with two big Josephson junctions of critical current ∼ 2.2 mA). Evaporating Al from two different angles with an oxidation process between them gives the small overlapping regions (the Josephson junctions). The middle junction of the **qubit** is 0.8 times the area of the other two, and the ratio of **qubit**/SQUID areas is about 1:3. Arrows indicate the two directions of the persistent current in the **qubit**. The mutual **qubit**/SQUID inductance is M ≈ 9 pH. (B) Schematic of the on-chip circuit; crosses represent the Josephson junctions. The SQUID is shunted by two capacitors ( ∼ 5 pF each) to reduce the SQUID plasma **frequency** and biased through a resistor ( ∼ 150 ohms) to avoid parasitic resonances in the leads. Symmetry of the circuit is introduced to suppress excitation of the SQUID from the **qubit**-control pulses. The MW line provides microwave current bursts inducing **oscillating** magnetic fields in the **qubit** loop. The current line provides the measuring pulse I b and the voltage line allows the readout of the switching pulse V o u t . The V o u t signal is amplified, and a threshold discriminator (dashed line) detects the switching event at room temperature....(A) Ramsey interference: The measured switching probability (dots) is plotted against the time between the two π / 2 pulses. The continuous line is a fit by exponentially damped **oscillations** with a decay time of 20 ns. The Ramsey interference period of 4.5 ns agrees with the inverse of the detuning from resonance, 220 MHz. The resonant **frequency** is 5.71 GHz and microwave power A = 0 dBm. (B) Spin-echo experiment: switching probability versus position of the π pulse between two π / 2 pulses. The period of ∼ 2.3 ns corresponds well to half the inverse of the detuning. The width and timing of microwave pulses in the MW line are shown in each graph. The readout pulse in the bias line immediately follows the last π / 2 pulse (see Fig. 1B)....We have observed coherent time evolution between two quantum states of a superconducting flux **qubit** comprising three Josephson junctions in a loop. The superposition of the two states carrying opposite macroscopic persistent currents is manipulated by resonant microwave pulses. Readout by means of switching-event measurement with an attached superconducting quantum interference device revealed quantum-state **oscillations** with high fidelity. Under strong microwave driving it was possible to induce hundreds of coherent **oscillations**. Pulsed operations on this first sample yielded a relaxation time of 900 nanoseconds and a free-induction dephasing time of 20 nanoseconds. These results are promising for future solid-state quantum computing. ... We have observed coherent time evolution between two quantum states of a superconducting flux **qubit** comprising three Josephson junctions in a loop. The superposition of the two states carrying opposite macroscopic persistent currents is manipulated by resonant microwave pulses. Readout by means of switching-event measurement with an attached superconducting quantum interference device revealed quantum-state **oscillations** with high fidelity. Under strong microwave driving it was possible to induce hundreds of coherent **oscillations**. Pulsed operations on this first sample yielded a relaxation time of 900 nanoseconds and a free-induction dephasing time of 20 nanoseconds. These results are promising for future solid-state quantum computing.

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Contributors: Forn-Díaz, P., Lisenfeld, J., Marcos, D., García-Ripoll, J. J., Solano, E., Harmans, C. J. P. M., Mooij, J. E.

Date: 2010-05-10

fig4(Color online) Bloch-Siegert shift. (a) Spectrum in proximity to the resonator **frequency** obtained using lower driving power than in Fig. fig3 and flux pulses . The solid black line is the fit of Eq. eq1 and the dashed green line is a plot of the JC model (Eq. eq1 without counter-rotating terms). The dotted line indicates the bare resonator **frequency** ω r . A clear deviation between the dashed line and the data can be observed around the symmetry point of the **qubit**. A transition associated with thermal population of the **qubit** excited state can be observed around 8 GHz. (b) Difference between measurement (blue dots) and the prediction of the JC model (dashed green line). The solid black curve is the same as the solid black curve in (a) and the dashed red curve represents λ 1 , g - λ 0 , g . All the curves are subtracted from the JC model. The blue dots are peak values extracted from Lorentzian fits to **frequency** scans at fixed flux, with the error bars representing the full width at half maximum of each Lorentzian....The spectral line of the resonator can be resolved when it is detuned several GHz away from the **qubit** [Fig. fig3]. This could be caused by the external driving when it is resonant with the **oscillator**. By loading photons in it, the **oscillator** can drive the **qubit** off-resonantly by their large coupling. Another possibility is an adiabatic shift during state readout through the anticrossing of the **qubit** and resonator energies. The **qubit** readout pulse produces a negative shift of -2 m Φ 0 in magnetic flux, making the spectral amplitude asymmetric with respect to the **qubit** symmetry point [Fig. fig3]. For our parameters, this shift is coincidental with the avoided level crossing with the **oscillator**. Then, a state containing one photon in the resonator (e. g., Φ / Φ 0 - 0.5 = 4 m Φ 0 in Fig. fig3) can be converted into an excited state of the **qubit** with very high probability, as the Landau-Zener tunneling rate is very low. Both effects, off-resonant driving and adiabatic shifting, would explain that the sign of the spectral line of the resonator coincides with the one of the **qubit** on both sides of the symmetry point. Irrespective of the mechanism, the spectral features of Fig. fig3 allow us to give a low bound for the quality factor of the resonator Q > 10 3 ....fig2(Color online) Measurement scheme and energy-level diagram. (a) Schematic of the measurement protocol to perform **qubit** spectroscopy. (b) JC ladder depicting the energy-level structure of the system of a flux **qubit** coupled to an L C resonator. The levels are drawn for the case δ = ω q - ω r **qubit** and resonator states. δ q and δ r are the dispersive shifts that the **qubit** and the resonator induce to each other....Our system consists of a four-Josephson-junction flux **qubit** , in which one junction is made smaller than the other three by a factor of approximately 0.5. The **qubit** is galvanically connected to a lumped-element L C resonator [Fig. fig1]. In previous work the employed L C resonators were strongly coupled to the flux **qubit** , but since they were loaded by the impedance of the external circuit their quality factor was low. Flux **qubits** have also been successfully coupled to high-quality transmission line resonators . In our experiment we use an interdigitated finger capacitor in series with a long superconducting wire, following the ideas from lumped-element kinetic inductance detectors . In order to read out the **qubit** state a dc-switching SQUID magnetometer was placed on top of the **qubit**. The detection procedure can be found in ....We prepare the **qubit** in the ground state by cooling it to 20 mK in a dilution refrigerator. Using the protocol shown in Fig. fig2 (a), we measure the spectrum of the **qubit**-resonator system [Fig. fig3]. To obtain a higher resolution in the relevant region around 8.15 GHz, we repeated the spectroscopy using lower driving power in combination with the application of flux pulses in order to equalize the **qubit** signal by reading out far from its degeneracy point [Fig. fig4] . We can identify the energy-level transitions on the basis of the JC ladder shown in Fig. fig2 (b). A large avoided crossing between states | g , 1 and | e , 0 is observed around a **frequency** of ∼ 8 GHz. This is very close to the estimated resonance **frequency** of the **oscillator**. The energy splitting 2 g / 2 π Δ / ω r [Fig. fig3 (inset)] is approximately 0.9 GHz. A combined least-squares fit of the full Hamiltonian [Eq. eq1] of the data from Figs. fig3, fig4 leads to Δ / h = 4.20 ± 0.02 GHz, I p = 500 ± 10 nA, ω r / 2 π = 8.13 ± 0.01 GHz and g / 2 π = 0.82 ± 0.03 GHz....fig3(Color online) Spectrum of the flux **qubit** coupled to the L C resonator. An avoided-level crossing is observed at a **frequency** of 8.13 GHz. The weak transition near 8 GHz is associated with excited photons due to thermal population of the **qubit** excited state ( T e f f ∼ 100 ~ mK at ∼ 4 - 5 ~ GHz energy splitting). (Inset) Zoom in around the resonance between **qubit** and **oscillator**. The splitting on resonance is 2 g sin θ / 2 π ≃ 0.9 GHz....fig1(Color online) Circuit layout and images of the device. (a) Schematic of the measurement setup. The interdigitated capacitor of the L C resonator can be seen in the center of the optical image, with the circuitry of the two SQUIDs next to it (top left and bottom right); C r / 2 ≃ 0.25 pF and L r ≃ 1.5 nH. (b) Scanning electron micrograph (SEM) picture of the SQUID circuit. The readout line is made to overlap with a big volume of AuPd and Au to thermalize the quasiparticles when the SQUID switches. (c) SEM picture of the **qubit** with the SQUID on top. On the right of the picture the coupling wire to the resonator of length l can be seen....We measure the dispersive energy-level shift of an $LC$ resonator magnetically coupled to a superconducting **qubit**, which clearly shows that our system operates in the ultrastrong coupling regime. The large mutual kinetic inductance provides a coupling energy of $\approx0.82$~GHz, requiring the addition of counter-rotating-wave terms in the description of the Jaynes-Cummings model. We find a 50~MHz Bloch-Siegert shift when the **qubit** is in its symmetry point, fully consistent with our analytical model....The **qubit** is galvanically attached to the resonator [Fig. fig1 (c)] with a coupling wire of length l = 5 ~ μ m, width w = 100 nm and thickness t = 50 nm. To achieve our coupling energy we use the kinetic inductance L K of the wire that can easily be made larger than the geometric contribution. The kinetic inductance for our narrow dirty wire is found from its normal state resistance L K = 0.14 ℏ R n / k B T c ≃ 25 ± 2 pH. The strength of the coupling can be approximated by ℏ g = I p I r m s L K . Since our ∼ 500 μ m L C resonator is much smaller than the wavelength at the resonance **frequency** ( λ r ≈ 20 mm), the current is uniform along the superconducting wires connecting the capacitor plates. Therefore the position of the **qubit** along the inductor will not affect the coupling strength. ... We measure the dispersive energy-level shift of an $LC$ resonator magnetically coupled to a superconducting **qubit**, which clearly shows that our system operates in the ultrastrong coupling regime. The large mutual kinetic inductance provides a coupling energy of $\approx0.82$~GHz, requiring the addition of counter-rotating-wave terms in the description of the Jaynes-Cummings model. We find a 50~MHz Bloch-Siegert shift when the **qubit** is in its symmetry point, fully consistent with our analytical model.

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Contributors: Buisson, O., Hekking, F. W. J.

Date: 2000-08-18

Using the above paramaters, we have plotted P 1 t , Eq. ( P1), as a function of time in Fig. rabi. We clearly see the Rabi **oscillations** with perdiodicity T R a b i ≈ 8 ns....The capacitance C c plays a crucial role in our proposed circuit since it couples the charge **qubit** and the resonator to each other. These two circuits are no longer independent and the system must be considered in its totality. Thus the proposed quantum circuit of Fig. device realizes the simple situation in which a two level system is coupled to a harmonic **oscillator**. In spite of its simplicity, such a system describes a great variety of interesting situations ....We study the dynamics of a quantum superconducting circuit which is the analogue of an atom in a high-Q cavity. The circuit consists of a Josephson charge **qubit** coupled to a superconducting resonator. The charge **qubit** can be treated as a two level quantum system whose energy separation is split by the Josephson energy $E_j$. The superconducting resonator in our proposal is the analogue of a photon box and is described by a quantum harmonic **oscillator** with characteristic **frequency** $\omega_r$. The coupling between the charge **qubit** and the resonator is realized by a coupling capacitance $C_c$. We have studied the eigenstates as well as the dynamics of the quantum circuit. Interesting phenomena occur when the Josephson energy equals the **oscillator** **frequency**, $E_j=\hbar\omega_r$. Then the quantum circuit is described by entangled states. We have deduced the time evolution of these states in the limit of weak coupling between the charge **qubit** and the resonator. We found Rabi **oscillations** of the excited charge **qubit** eigenstate. This effect is explained by the spontaneous emission and re-absorption of a single photon in the superconducting resonator. ... We study the dynamics of a quantum superconducting circuit which is the analogue of an atom in a high-Q cavity. The circuit consists of a Josephson charge **qubit** coupled to a superconducting resonator. The charge **qubit** can be treated as a two level quantum system whose energy separation is split by the Josephson energy $E_j$. The superconducting resonator in our proposal is the analogue of a photon box and is described by a quantum harmonic **oscillator** with characteristic **frequency** $\omega_r$. The coupling between the charge **qubit** and the resonator is realized by a coupling capacitance $C_c$. We have studied the eigenstates as well as the dynamics of the quantum circuit. Interesting phenomena occur when the Josephson energy equals the **oscillator** **frequency**, $E_j=\hbar\omega_r$. Then the quantum circuit is described by entangled states. We have deduced the time evolution of these states in the limit of weak coupling between the charge **qubit** and the resonator. We found Rabi **oscillations** of the excited charge **qubit** eigenstate. This effect is explained by the spontaneous emission and re-absorption of a single photon in the superconducting resonator.

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Contributors: Gustavsson, Simon, Yan, Fei, Bylander, Jonas, Yoshihara, Fumiki, Nakamura, Yasunobu, Orlando, Terry P., Oliver, William D.

Date: 2012-04-28

(a) Decay times of the coherent **oscillations**, measured with ( N =1) and without ( N =0) a refocusing pulse. At = 0 , the decay is limited by energy relaxation, but the coherence times decrease away from = 0 due to increased sensitivity to flux noise. For large | | , the refocusing sequence improves the decay time by more than a factor of four. The inset show examples of decay envelope h t , measured at = - 60 . For the N =1 case, we use τ 1 = τ 2 = t / 2 . (b) Decay times of multi-pulse refocusing sequences, showing the improvement coherence as the number of pulses increases. (c-d) Time evolution of the **qubit**-TLS system, measured with and without refocusing pulses. Note the echo signals appearing after each refocusing pulse, giving a strong enhancement of the coherence time. The data is taken at = - 84 ....We now extend the refocusing technique to implement dynamical decoupling protocols with multi-pulse sequences. For 1 / f -type noise, it has been shown that the Carr-Purcell sequence , consisting of equally spaced π -rotations, improves coherence times by filtering the noise at low **frequencies** . Figures fig:vsN(c-d) show the coherent evolution of the system when repeatedly applying refocusing pulses. Echo signals form between each pair of π pulses, giving considerable longer coherence times compared to the N = 0 case. The increase in decay time with the number of refocusing pulses N is plotted in...At = Φ * = ± 4.15 , the **qubit** becomes resonant with a TLS . The microscopic nature of the TLS is unknown, but studies of two-level systems in similar **qubit** designs show that the most likely origin is an electric dipole in one of the tunnel junctions . Figure fig:Setup(a) shows a magnification of the region around -4.15 , revealing a clear anticrossing with splitting S = 76 . We describe the system using the four states | 0 g | 1 g | 0 e | 1 e , where 0 1 are the **qubit** energy eigenstates and g e refer to the ground and excited state of the TLS. On resonance, | 1 g and | 0 e are degenerate and coupled by the coupling energy h S . To characterize the coupling, we use the pulse scheme depicted in...We implement dynamical decoupling techniques to mitigate noise and enhance the lifetime of an entangled state that is formed in a superconducting flux **qubit** coupled to a microscopic two-level system. By rapidly changing the **qubit**'s transition **frequency** relative to the two-level system, we realize a refocusing pulse that reduces dephasing due to fluctuations in the transition **frequencies**, thereby improving the coherence time of the entangled state. The coupling coherence is further enhanced when applying multiple refocusing pulses, in agreement with our $1/f$ noise model. The results are applicable to any two-**qubit** system with transverse coupling, and they highlight the potential of decoupling techniques for improving two-**qubit** gate fidelities, an essential prerequisite for implementing fault-tolerant quantum computing....Figure fig:Setup(c) shows the **qubit** state after the pulse sequence, measured versus interaction time τ 1 and flux detuning . At = 0 and for τ 1 = 1 / 2 S = 6 , the pulse sequence implements an iSWAP gate between **qubit** and TLS, taking | 0 e → i | 1 g and | 1 g → i | 0 e . The characteristic decay time of the **oscillations** is shown in...(a) Spectroscopy of the **qubit**-TLS system. The **qubit** and TLS are resonant at f = 7.08 , where the spectrum has an anticrossing with splitting S = 76 . The inset shows the **qubit** spectrum over a larger range, with the red circle indicating the region of interest. (b) Pulse sequence for probing the **qubit**-TLS interactions. The π -pulse generates a **qubit** excitation, which is coherently exchanged back and forth between **qubit** and TLS during the interaction time τ 1 . (c) Coherent **oscillations** between **qubit** and TLS, measured using the pulse sequence shown in (b). High switching probability P S W corresponds to the ** qubit’s** ground state | 0 , low P S W to the

**excited state | 1 . (d-e) Characteristic decay time t d e c a y and**

**qubit**’s**oscillation**

**frequency**, extracted from the data in (c). The

**oscillations**decay faster for ≠ 0 , a consequence of the increased sensitivity Φ to flux noise....(a) Pulse sequence and (b) Bloch sphere representation of the refocusing protocol. The blue arrows are state vectors, while the red arrows represent the Hamiltonian in eq:Hsub. After the

**qubit**π pulse, the system enters the | 1 g | 0 e subspace (step I). The coupling S rotates the state vector around the x -axis, but due to noise in the effective coupling the state vector fans out (II). The rapid flux pulse Φ r e f o c u s generates a large

**frequency**detuning , the system will start rotating around the z -axis (III) and eventually complete a π -rotation (IV). The inhomogeneous broadening now refocuses the state vector, giving an echo at V. (c-d) Evolution of the

**qubit**-TLS system, measured with and without a refocusing pulse. A clear echo appears after the refocusing pulse, with a maximum close to τ 2 = τ 1 . The traces were taken at = - 72 ....fig:Bloch(d) requires careful calibration of the refocusing pulse. Figure fig:Calib shows an example of a calibration experiment, where we fix τ 1 = 97 and = 550 and measure refocused

**oscillations**versus the refocusing time τ r e f o c u s . The data shows strong

**oscillations**whenever the refocusing pulse rotates the state vector by an odd integer of π , i.e. when τ r e f o c u s = 2 n + 1 × 0.5 / , in agreement with the schematics discussed in...fig:Bloch(a-b). The system is brought into the | 1 g | 0 e subspace by applying a π -pulse to the

**qubit**[step I in Figs. fig:Bloch(a-b)], followed by a non-adiabatic shift in to bring the

**qubit**and TLS close to resonance. | 1 g is not an eigenstate of the coupled system, so the interaction S will cause the system to rotate around the x -axis,

**oscillating**between | 1 g and | 0 e . Low-

**frequency**fluctuations in the effective coupling strength will cause the Bloch state vector to fan out (over many realizations of the experiment), and the system loses its phase coherence (step II). ... We implement dynamical decoupling techniques to mitigate noise and enhance the lifetime of an entangled state that is formed in a superconducting flux

**qubit**coupled to a microscopic two-level system. By rapidly changing the

**qubit**'s transition

**frequency**relative to the two-level system, we realize a refocusing pulse that reduces dephasing due to fluctuations in the transition

**frequencies**, thereby improving the coherence time of the entangled state. The coupling coherence is further enhanced when applying multiple refocusing pulses, in agreement with our $1/f$ noise model. The results are applicable to any two-

**qubit**system with transverse coupling, and they highlight the potential of decoupling techniques for improving two-

**qubit**gate fidelities, an essential prerequisite for implementing fault-tolerant quantum computing.

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Contributors: Bera, Soumya, Florens, Serge, Baranger, Harold, Roch, Nicolas, Nazir, Ahsan, Chin, Alex

Date: 2013-01-30

The coupling of a **qubit** to a macroscopic reservoir plays a fundamental role in understanding the complex transition from the quantum to the classical world. Considering a harmonic environment, we use both intuitive arguments and numerical many-body quantum tomography to study the structure of the complete wavefunction arising in the strong-coupling regime, reached for intense **qubit**-environment interaction. The resulting strongly-correlated many-body ground state is built from quantum superpositions of adiabatic (polaron-like) and non-adiabatic (antipolaron-like) contributions from the bath of quantum **oscillators**. The emerging Schr\"odinger cat environmental wavefunctions can be described quantitatively via simple variational coherent states. In contrast to **qubit**-environment entanglement, we show that non-classicality and entanglement among the modes in the reservoir are crucial for the stabilization of **qubit** superpositions in regimes where standard theories predict an effectively classical spin....Displacements and spin average in the many-mode case. A. Displacements determined variationally from the two-polaron ansatz Eq. ( trial), showing the emergence of an antipolaron component for low energies, with equal and opposite displacement to the polaron state. The antipolaron state merges smoothly onto the polaron state at high energy as the adiabaticity of the **oscillators** with respect to tunneling of the spin is recovered (the NRG logarithmic discretisation of the bath spectrum is used here, namely **frequency** points are evenly spaced on a logarithmic scale; Note that a point at higher energy is associated to a larger energy window of the continuum spectrum, leading to the saturation of f k pol. for high **frequencies**, instead of the fall off obtained for a linear energy mesh). [Parameters: α = 0.5 and Δ = 0.01 .] B. Ground state averaged spin amplitude - as a function of dissipation strength α computed with the NRG (circles) for Δ / ω c = 0.01 , and compared to the one-polaron (red line) and two-polaron (blue line) predictions. A clear breakdown of the one-polaron Silbey-Harris ansatz occurs at strong dissipation, while the two-polaron trial state accounts for the correct behavior up to the quantum critical point ( α c = 1 ), due to preserved tunneling amplitude via the antipolaron component of the wavefunction....Joint spin-mode entropies and **oscillator** wavefunctions for the two-mode spin-boson model. A. Joint spin-mode entropy S S p i n + 1 for a two-mode environment. For all these data ω 2 = 1.05 ω 1 , g 1 = g 2 = 2.5 ω 1 and Δ = 0.01 , sweeping the **frequencies** ω 1 of the first mode. Plot shows results for (top to bottom) exact diagonalisation (grey stars), three-polaron ansatz (green diamonds), two-polaron ansatz (yellow squares) and one-polaron Silbey-Harris theory (red dots). The horizontal line indicates the maximum entropy of a fully-mixed spin state ( ln 2 ). The main panel is a close-up on the entropy peak occuring near the resonance **frequency**, as discussed in the text, while the inset shows the whole entropy and **frequency** range. B. Spin up-projected two-mode **oscillator** wave functions along the diagonal coordinate x 1 = x 2 for the same parameters as the left panel, computed for ω 1 = 0.0023 (namely at the peak position of the exact diagonalisation entropy curve in A). Results are shown for ground states obtained by exact diagonalisation (dashed black line), one-polaron Silbey-Harris ansatz (solid red line), two-polaron ansatz (solid yellow line) and three-polaron ansatz (solid green line)....where E R = ∑ k g k 2 / 4 ω k is the reorganisation energy of the bath. For Δ = 0 , the ground state of H ˜ is doubly degenerate, and is given by the product of the bosonic vacuum and the spin states, | Ψ ˜ ↑ , 0 = | ↑ ⊗ | 0 and | Ψ ˜ ↓ , 0 = | ↓ ⊗ | 0 , in the transformed basis (denoted by tildes). It thus corresponds to polaronic wavefunctions in the original frame, where the positive/negative sign of the displacement is fully correlated to the spin projection (adiabatic response): | Ψ ↑ , g k / 2 ω k = | ↑ ⊗ | + g k / 2 ω k and | Ψ ↓ , - g k / 2 ω k = | ↓ ⊗ | - g k / 2 ω k . The two-fold degenerate ground state thus takes the form of a product of semiclassical coherent states (displaced **oscillators**) | ± f k ≡ e ± ∑ k f k a k † - a k † | 0 , with displacements f k = ± g k / 2 ω k which shift each **oscillator** to the minimum of its static spin-dependent potential. This potential is evident in Eq. ( ham) for Δ = 0 and is shown explicitly in Fig. antipolaronA. In the presence of spin tunneling ( Δ ≠ 0 ), one needs to understand the effect of the operators K ± ≡ Δ σ ± e ∓ ∑ k g k / ω k a k † - a k † in Eq. ( hamrot) which correlate spin flip processes with simultaneous displacements of all **oscillator** states. As we now show, these correlations ultimately control the ground state **qubit** superposition....Braak for an exact solution); note that a similar ansatz for the single-mode Rabi model (without reference to polaron theory) has been previously explored numerically . In Figure antipolaronD we compare the spatial wavefunctions of the **oscillator** correlated with each spin state with those obtained from the ansatz Eq. ( trial) following a numerical optimization of p , f 1 p o l . , and f 1 a n t i . to minimise the ground state energy. Choosing **oscillator** parameters where we expect non-adiabatic response, namely ω 1 < Δ , we find that both wavefunctions clearly show a superposition of polaron and antipolaron contributions, with much larger displacements compared to the prediction of the SH theory (single polaron case p = 0 ). The agreement of the diagonalised and the two-polaron ansatz ground state wavefunctions is extremely good, as well as the energies and spin observables, even for a coupling strength as large as g = 3 ω 1 (see also Supplementary Information). As motivated above, the emergence of an antipolaron component in the environment enhances the overlap of the tunneling states. The single polaron SH state fails in this regard (see Figures antipolaronB and 1D, and Supplementary Information) as it finds itself frustrated between minimizing the elastic energy and maintaining good overlap between the opposite spin states: the resulting displacements are thus totally wrong....Origins of polaron and antipolaron displacements in environment wavefunctions. In plots A-C, black dashed lines are the spin-dependent potential energies of a single harmonic **oscillator** in the absence of spin tunneling [see Eq. ( ham)], while blue (red) curves are the gaussian wavefunctions (in real space x ) of the **oscillator** on the σ x = 1 ~ -1 potential surfaces. A. Polarons. For a high **frequency** mode ( ω ≫ Δ ), transitions to other **oscillator** states on the potential surfaces are suppressed by the steep curvature of the potentials; **oscillator** displacement adiabatically tunnels with the spin between minima of the potentials, suppressing the tunneling amplitude by the reduced overlap of the displaced **oscillator** wave functions to a value Δ R . B. Non-adiabatic response in Silbey-Harris variational polaron theory. Low **frequency** modes ( ω ≪ Δ ) have shallow potentials, leading to well-separated minima. Poor wave function overlap prevents tunneling of the spin between minima, destroying spin superposition. Variationally-determined displacements adjust to smaller values, sacrificing their displacement energy to maintain the spin-tunneling energy through better overlap. C. Antipolaron response of non-adiabatic **oscillators**. For modes with ω ∼ Δ , spin flips that do not change the position of the **oscillators** (and thus have unsuppressed amplitude Δ ) may become low enough in energy to compete with the overlap-suppressed inter-minima tunneling. The **oscillator** wavefunctions correlated with spin are now superpositions of displaced coherent states with opposite signs. D. Ground state wavefunction components of a single **oscillator**. Spin up (blue) and spin down (red) components are shown for the exact ground state (circles), our variational polaron-antipolaron state (solid lines), and the Silbey-Harris ansatz (dashed lines) [ Δ / ω 1 = 4 , g / ω 1 = 3 ]. The exact result shows distinct antipolaron features which are well captured by the variational polaron-antipolaron state. The Silbey-Harris ansatz shows reduced displacements and thus poor agreement with the exact result....Joint entanglement entropy in the many-mode case. The joint entanglement entropy of the **qubit** and a given ω k mode, defined by Eq. ( Sspink), is calculated with NRG for Δ / ω c = 0.01 . The entropy of the **qubit** alone is subtracted. Negative correlations for α **qubit**....Two mode wavefunctions. A-B. Contour plots in real space of the spin-up projected joint **oscillator** wavefunctions of two modes obtained from exact diagonalisation. A. Polarons. For high **frequency** modes ( ω 2 = 2 ω 1 = 0.04 > Δ = 0.01 ), the wavefunction is a single, displaced gaussian, in qualitative agreement with Silbey-Harris theory. B. Entangled antipolarons. Low **frequency** modes ( ω 2 = 2 ω 1 = 0.004 < Δ = 0.01 ) show the development of an antipolaron component, visible in the ( X 1 < 0 , X 2 < 0 ) quadrant, in addition to the Silbey-Harris state. C. Product state. Hypothetical wavefunction obtained from a product state of polaron-antipolaron superpositions for each mode, showing symmetric off-diagonal peaks. These features are absent in B, indicating that the exact joint wavefunction is not a product state but, in contrast, is entangled as described in the text....Fig. twomodes shows the spin-up component of the two-mode wavefunction as a function of the two independent spatial coordinates of the modes ( x 1 and x 2 ) for two modes taken at different **frequencies** ω 2 = 2 ω 1 . The ground state wavefunctions were determined by exact numerical diagonalisation. We see the clear development of an antipolaron component to the wavefunction (Fig. twomodesB) for low-energy non-adiabatic modes, in contrast to the situation of high-energy adiabatic modes (Fig. twomodesA). However, we see that only two peaks appear in the wavefunction – those along the diagonal line x 1 = x 2 – indicating unambiguously that this two-mode wavefunction takes the inter-mode entangled form | f 1 p o l . ⊗ | f 2 p o l . + p | f 1 a n t i . ⊗ | f 2 a n t i . . This can be contrasted with a hypothetical polaron-antipolaron product state | f 1 p o l . + p | f 1 a n t i . ⊗ | f 2 p o l . + p | f 2 a n t i . which would rather display four peaks, as shown in Fig. twomodesC. The implications of this inter-mode entanglement for the entropy of the reservoir modes is given in Supplementary Information. Again, one can check that the variational energy of the two-mode ground state is remarkably close to the exact energy....Shown in Fig. osc2 are thus φ ↓ 0 x (red) and φ ↑ 0 x (blue) as a function of position, calculated from our polaron-antipolaron ground state (solid curves), from the Silbey-Harris ground state (dashed-dotted curves), and from a numerical diagonalisation of the full Hamiltonian (points). We take Δ / ω 1 = 4 here as a representative example. For g / ω 1 ∼ 1 , the displacement from x = 0 is found to be fairly small, | f | **oscillator** states starts to become extremely important. For example, at g / ω 1 = 3 we observe that the Silbey-Harris state is completely unable to reproduce the correct **oscillator** wavefunctions, due to the restriction to a single displacement associated with each spin state. In fact, for these parameters, the displacements obtained by the Silbey-Harris approach are much too small, and reproduce none of displacements seen in our polaron-antipolaron ansatz, which itself matches the numerical solution very well. Finally, as the coupling strength is increased further, the Silbey-Harris displacements eventually “jump" to those of the full polaron transformation, which then captures the dominant displacements in the exact states quite well (see the g / ω 1 = 4 plot ), but of course completely misses the smaller displacements in the opposite direction, which are still captured extremely well by our ansatz. Hence, the obtained ground state energy is still lower in our ansatz (and in the numerical diagonalisation) than from the Silbey-Harris state. The theories will eventually converge at larger coupling, however, when associating a single (polaron) displacement with each spin state finally becomes a good description....In Fig. eground1 we plot the dimensionless ground state energy E / ω 1 determined from our polaron-antipolaron variational ansatz as a function of the dimensionless spin-**oscillator** coupling strength g / ω 1 , and compare with the results from an exact numerical diagonalisation of the model, and from Silbey-Harris and polaron theories (Eqs. ( eSHgs) and ( epolgs) in the single mode case, respectively). In this figure, Δ / ω 1 ≥ 1 for all plots, and so we would expect standard polaron theory to break down in this regime, since the full **oscillator** displacement is no longer appropriate. From the dashed curves, this can indeed be seen to be the case, and polaron theory may even predict the incorrect trend as g / ω 1 increases. Silbey-Harris theory fixes this problem to a certain extent (at least at small g / ω 1 , see dashed-dotted curves), though again runs into problems as the coupling strength increases, deviating from the numerically-exact results, and even more worryingly predicting discontinuous behaviour in the ground-state energy at certain values of g / ω 1 . Our polaron-antipolaron variational ansatz, however, predicts ground-state energies in almost perfect agreement with the numerical results for all possible coupling strengths. Furthermore, the discontinuous behaviour seen in Silbey-Harris theory is removed in this more flexible variational state. ... The coupling of a **qubit** to a macroscopic reservoir plays a fundamental role in understanding the complex transition from the quantum to the classical world. Considering a harmonic environment, we use both intuitive arguments and numerical many-body quantum tomography to study the structure of the complete wavefunction arising in the strong-coupling regime, reached for intense **qubit**-environment interaction. The resulting strongly-correlated many-body ground state is built from quantum superpositions of adiabatic (polaron-like) and non-adiabatic (antipolaron-like) contributions from the bath of quantum **oscillators**. The emerging Schr\"odinger cat environmental wavefunctions can be described quantitatively via simple variational coherent states. In contrast to **qubit**-environment entanglement, we show that non-classicality and entanglement among the modes in the reservoir are crucial for the stabilization of **qubit** superpositions in regimes where standard theories predict an effectively classical spin.

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Contributors: Steffen, Matthias, Kumar, Shwetank, DiVincenzo, David, Keefe, George, Ketchen, Mark, Rothwell, Mary Beth, Rozen, Jim

Date: 2010-01-09

We present a novel readout scheme for phase **qubits** which eliminates the read-out SQUID so that the entire **qubit** and measurement circuitry only requires a single Josephson junction. Our scheme capacitively couples the phase **qubit** directly to a transmission line and detects its state after the measurement pulse by determining a **frequency** shift observable in the forward scattering parameter of the readout microwaves. This readout is extendable to multiple phase **qubits** coupled to a common readout line and can in principle be used for other flux biased **qubits** having two quasi-stable readout configurations....The **qubit** is calibrated by executing a pulse sequence similar to the one shown in Fig. fig:fig2. A reset pulse is applied to the flux line and added to the flux bias line via a DC-block at room temperature. However, neither the microwave pulse "rotate" nor the subsequent measurement pulse "measure" is applied to the **qubit**. The microwave readout signal μ R O is mixed with itself rather than a separate microwave source tuned to the same **frequency** to eliminate phase drift between the microwave generators. The resultant I and Q signals at DC are filtered and digitized using an Acqiris data acquisition board and averaged on board using sufficient averages for acceptable signal-to-noise ratios. We then plot the amplitude response | I + i Q | versus flux in Fig. fig:fig3. To better visualize the results we plot the negative amplitude for a negative reset pulse (black) and the positive amplitude for a positive reset pulse (white). Whenever the **qubit** is hysteretic (between 0 and ≈ 0.45 Φ 0 ) the **frequency** response is vastly different depending on which reset pulse was applied, showing that the L and R configurations have very different resonant **frequencies**....Experimental pulse sequence. The flux bias Φ and **qubit** microwave pulse μ q follow standard phase **qubit** protocols. The microwave read-out pulse μ R O is applied after the measurement pulse is executed for a duration of 400 μ s ....**Frequency** response of the phase **qubit** for negative (black) and positive (white) reset pulses. When the **qubit** is hysteretic two resonance **frequencies** are observed, consistent with the L and R configurations. The lower **frequency** response ω L corresponds to the configuration in which the **qubit** is operated, and the other, ω R , corresponds to the configuration in which the phase particle tunneled during the measurement pulse. The read-out is performed by observing the presence or absence of resonance response at ω R ....Rabi **oscillations** and an energy relaxation curve of the **qubit**. Note that no absolute scale is shown as we are presently not performing a single shot measurement....Outline (a) and micrograph (b) of the **qubit**. The phase **qubit** consists of the standard circuit elements in parallel - a Josephson junction, an inductor and a capacitor all capacitively coupled to a feedline via a coupling capacitor. An optical micrograph of the **qubit** depicts the actual layout of circuit. c) Shows the readout schematic, The **qubit** flux bias and microwave lines (S1,S2) are sufficiently filtered and attenuated to ensure sufficiently low electron temperatures. A HEMT at the 4K stage amplifies the outgoing microwave signal....The fabrication was similar to ref . An optical micrograph of the fabricated **qubit** is shown in Fig. fig:fig1b. The sample was mounted inside an RF-tight box and cooled down in a dilution refrigerator. The bias lines were configured as outlined in Fig. fig:fig1c. The flux bias line is filtered with a low pass bronze powder filter, matched to Z 0 = 50 Ω impedance, with a 1 GHz cut-off **frequency** ....We have successfully implemented the most basic operations of this microwave based read out technique. Here we present our experimental results and speculate on further improvements. Because the operation and "measurement" of the phase **qubit** does not change dramatically in our implementation we are able to base our design on published literature . The basic layout of the **qubit** circuit is shown in Fig. fig:fig1a. We chose a capacitively shunted phase **qubit** to minimize the number of junction two-level systems in the **qubit** spectroscopy and maximize measurement fidelity . In order to ensure long coherence times of the **qubit** we fabricated the shunting capacitor using an interdigitated comb with the ends shorted together to eliminate parasitic cavity modes . The target value for the shunting capacitor is C s = 2 pF. The **qubit** loop is designed gradiometrically with L = 625 pH to eliminate sensitivities to stray flux variations and reducing flux coupling to the microwave feedline. The coupling capacitor is chosen to be C c = 23 fF which should limit T 1 to about T 1 , f e e d = 2 C / Z 0 ω C C 2 ≈ 180 ns for a **qubit** **frequencies** near 4.6 GHz ....Following standard phase **qubit** pulse sequences we are able to characterize the **qubit**. We show in Fig. fig:fig4 the results for Rabi **oscillations** and a T 1 experiment for a **qubit** **frequency** of ω L / 2 π = 4.46 GHz ( ω R / 2 π = 5.6 GHz). The energy relaxation time T 1 is found to be approximately 83 ns. We believe this value is limited by T 1 , f e e d because a separate sample using a SQUID based read out gave T 1 , i n s t r i n s i c = 260 ns and because 1 / T 1 , i n t r i n s i c + 1 / T 1 , f e e d -1 = 106 ns is close to the observed coherence time. An improvement in T 1 should therefore be possible by simply reducing the size of the coupling capacitor. ... We present a novel readout scheme for phase **qubits** which eliminates the read-out SQUID so that the entire **qubit** and measurement circuitry only requires a single Josephson junction. Our scheme capacitively couples the phase **qubit** directly to a transmission line and detects its state after the measurement pulse by determining a **frequency** shift observable in the forward scattering parameter of the readout microwaves. This readout is extendable to multiple phase **qubits** coupled to a common readout line and can in principle be used for other flux biased **qubits** having two quasi-stable readout configurations.

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Contributors: Martin, Ivar, Shnirman, Alexander, Tian, Lin, Zoller, Peter

Date: 2003-10-09

In the opposite, strong driving case, Δ n > Γ r / 2 , there are coherent **oscillations** between the two levels. The appropriate description is then to say that doublets of new eigenstates, are formed which are split in energy by 2 Δ n , see Fig. Fig:Strong_Driving_Cooling. The doublets are defined for n ≥ 1 as ψ n ± ≡ ( and for n = 0 we have a single state ψ n = 0 ≡ ....Another driving-induced heating process, not characteristic for quantum optics, is due to the fact that in solid state systems there is strong noise at low **frequencies** ( 1 / f noise). Thus, processes like the one shown in Fig. Fig:Spin_Heating become relevant. This process excites the **qubit** with the rate...**Qubit** heating induced by the applied drive....We propose an application of a single Cooper pair box (Josephson **qubit**) for active cooling of nanomechanical resonators. Latest experiments with Josephson **qubits** demonstrated that long coherence time of the order of microsecond can be achieved in special symmetry points. Here we show that this level of coherence is sufficient to perform an analog of the well known in quantum optics ``laser'' cooling of a nanomechanical resonator capacitively coupled to the **qubit**. By applying an AC driving to the **qubit** or the resonator, resonators with **frequency** of order 100 MHz and quality factors higher than $10^3$ can be efficiently cooled down to their ground state, while lower **frequency** resonators can be cooled down to micro-Kelvin temperatures. We also consider an alternative setup where DC-voltage-induced Josephson **oscillations** play the role of the AC driving and show that cooling is possible in this case as well....Another way to achieve AC cooling is by applying radio **frequency** voltage bias to the gates. In Fig. Fig:SQUIDsys, apply a driving voltage V x = V 0 cos ω d t on the resonator and another driving voltage V g = - C x / C g V x on the CPB. The ac voltage V x generates resonant coupling between the mechanical resonator and the CPB when ω a c = E J - ω 0 , which corresponds to the first red sideband coupling in quantum optics. The voltage V x also generates an **oscillating** charge bias on the CPB with δ N g x = C x V x / 2 e ; however, it is balanced by the bias V g , which prevents harmful ac pumping of the CPB....While the Hamiltonians ( Eq:Spin_Hamiltonian_pumped) and ( Eq:Spin_Hamiltonian_V_rotated) look similar, there are two important differences. One, already discussed, is the fact that the pumping **frequency** ω J in ( Eq:Spin_Hamiltonian_V_rotated) is fundamentally noisy, while ω d in ( Eq:Spin_Hamiltonian_pumped) can be made coherent. The second (very important) difference is that in ( Eq:Spin_Hamiltonian_pumped) the pumping is applied to σ z only, while in ( Eq:Spin_Hamiltonian_V_rotated) it couples to σ z and σ y . Both these facts hinder the cooling. Indeed, the coupling to σ y gives a direct matrix element E J , R / 4 between the states and . This interaction repels the levels and we must choose E J , R ≪ 4 ω 0 so that the resonant detuning as in Fig. Fig:Stair_Cooling is possible. In addition, the noise of the transport voltage translates into the line width for the transition equal to Γ ϕ = 2 π α t r k B T / ℏ , where α t r ≡ R / R Q . The fluctuations of the transport voltage are not screened by the ratio of capacitances as it happens for the gate charge. Therefore α t r ≈ 10 -2 . Because of these additional constraints the applicability of the DC cooling scheme is limited to higher **frequency**/quality factor resonators. For an estimate, consider an **oscillator** with ω 0 = 2 π × 1 G H z ≈ 5 μ e V ≈ 50 mK at temperature T = 50 mK. We then obtain Γ ϕ ≈ 0.3 μ e V , which significantly exceeds Γ r . Hence, we have to substitute Γ r by Γ ϕ in all formulas. For the Josephson coupling in the right junction we take E J , R = 2 μ eV. Then, instead of Eq. ( Eq:Delta), we find Δ ≈ E J , R λ / 2 E J , L ≈ 2 ⋅ 10 -3 μ eV (we assume E J , L ≈ 50 μ eV). The cooling rate can again be represented as A - n , where A - ≈ 2 Δ 2 / Γ ϕ ≈ 2 ⋅ 10 -5 μ eV. Thus, cooling becomes possible only if Q > ω 0 / A - ≈ 2.5 ⋅ 10 5 ....The direct coupling between the bath and the **oscillator** gives the dissipative rates between the **oscillator** states : Γ n → n - 1 ≈ g 2 X ω = ω 0 2 n / ℏ 2 and Γ n → n + 1 ≈ g 2 X ω = - ω 0 2 n + 1 / ℏ 2 . In addition, the **oscillator** can relax via the virtual excitations of the **qubit**. The corresponding processes are shown in Fig. Fig:Add_Diss....Dissipative processes due to the presence of the **qubit**: a) n → n - 1 ; b) n → n + 1 . The spectra of the **oscillator** and the **qubit** are superimposed. ... We propose an application of a single Cooper pair box (Josephson **qubit**) for active cooling of nanomechanical resonators. Latest experiments with Josephson **qubits** demonstrated that long coherence time of the order of microsecond can be achieved in special symmetry points. Here we show that this level of coherence is sufficient to perform an analog of the well known in quantum optics ``laser'' cooling of a nanomechanical resonator capacitively coupled to the **qubit**. By applying an AC driving to the **qubit** or the resonator, resonators with **frequency** of order 100 MHz and quality factors higher than $10^3$ can be efficiently cooled down to their ground state, while lower **frequency** resonators can be cooled down to micro-Kelvin temperatures. We also consider an alternative setup where DC-voltage-induced Josephson **oscillations** play the role of the AC driving and show that cooling is possible in this case as well.

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Contributors: Ku, Jaseung, Yoscovits, Zack, Levchenko, Alex, Eckstein, James, Bezryadin, Alexey

Date: 2015-04-13

[fig:TransmVsB] fig:TransmVsB. This plot represents the heterodyne voltage, produced by mixing a microwave signal passing through the cavity containing the **qubit** and a reference signal. During this measurement the **qubit** remains in its ground state. Yet the cavity input power is chosen such that the transmission of the cavity is the most sensitive to the **qubit** transition from the ground to the excited state, i.e., the maximum-contrast power was used [...tab:table1 Comparison of the experimental and theoretical periods for five samples. Δ B is the measured period of **oscillation**. Δ B Y Z , Δ B Y Z + c X Z , and Δ B k Y Z + c X Z are the theoretical periods calculated using three different effective areas denoted by the subscripts....fig:highB(Color online) The **qubit** transition **frequencies** ( f 01 ) and three times scales ( T 1 , T 2 * , and T 2 ) were measured at the sweet spots over the wide range of magnetic field for the N1 (a) and N7 (b)....We present a new type of transmon split-junction **qubit** which can be tuned by Meissner screening currents in the adjacent superconducting film electrodes. The best detected relaxation time ($T_1$) was of the order of 50 $\mu$s and the dephasing time ($T_2$) about 40 $\mu$s. The achieved period of **oscillation** with magnetic field was much smaller than in usual SQUID-based transmon **qubits**, thus a strong effective field amplification has been realized. This Meissner **qubit** allows an efficient coupling to superconducting vortices. We present a quantitative analysis of the radiation-free energy relaxation in **qubits** coupled to Abrikosov vortices. The observation of coherent quantum **oscillations** provides strong evidence that vortices can exist in coherent quantum superpositions of different position states. According to our suggested model, the wave function collapse is defined by Caldeira-Leggett dissipation associated with viscous motion of the vortex cores....(a) Optical image of the Meissner transmon **qubit** fabricated on a sapphire chip, which is mounted in the copper cavity. (b) A zoomed-in optical image of the **qubit**. Two rectangular pads marked A1 and A2 act as an RF antenna and shunt capacitor. (c) Scanning electron microscope (SEM) image of the electrodes marked E1 and E2, and a pair of JJs. (d) Schematics of the Meissner **qubit**. The X, Y and Z denote the width, the distance between the electrodes, and the distance between two JJs, which are indicated by × symbols. The red dot and circular arrow around it in the bottom electrode represent a vortex and vortex current flowing clockwise, respectively. Θ v is a polar angle defined by two dashed lines connecting the vortex and two JJs. The orange rectangular loop on the boundary of the bottom electrode indicates the Meissner current circulating counterclockwise....[fig:EffArea] fig:EffArea the theoretical effective areas A t h e f f versus the “experimental” effective area A e x e f f = Φ 0 / Δ B for five samples. Here Δ B is the low-field, vortex-free period of the HV-**oscillation** for each sample. The black squares and the red triangles represent, respectively, the theoretical effective area calculated as A t h e f f = Y Z (geometric SQUID loop area approach) and as A t h e f f = Y Z + c X Z (Meissner current phase gradients approach). The dashed curve represents the ideal case, A t h e f f = A e x e f f . The red triangles appear much closer to the ideal dashed line. Therefore the **qubit** energy is controlled mostly by the Meissner currents, which produce a strong phase bias of the SQUID loop....fig:Spec(Color online) (a) Spectroscopy of Meissner transmon (N1) as a function of applied magnetic field. This is raw data. (b) The parabola-like dashed line shows a phenomenological fit to the **qubit** transition **frequency** f 01 versus magnetic field B ....[fig:vortexcount] fig:vortexcount_c]. The steps are made more noticeable by placing the horizontal dashed lines. The spacing between the lines is constant and they serve as guide to the eye. The step size turns out almost constant. We speculate that each step corresponds to the entrance of a single vortex which is effectively coupled to the **qubit**. This scenario assumes that not all vortices present in the loop are sufficiently well coupled to the supercurrent generated by the **qubit** but only those which enter the area near the SQUID loop. The physical reason for this is the fact that the current tends to be concentrated near the edges due to the Meissner effect. At the same time, many other vortices get pushed in the middle of the electrode, thus making their impact on the **qubit** very minimal. It is naturally expected that vortices entering the electrodes near the loop would make a relatively large impact on the change of the period, B n + 1 - B n and therefore can cause an sharp increase in the estimated N n number. Thus the steps apparent in Fig. ...fig:lowB(Color online) (a) Magnetic field dependence of the **qubit** **frequency** ( f 01 ), three measured time scales ( T 1 , T 2 * , and T 2 ) and two calculated time scales ( T P and T 1 c a l ) at low magnetic field much smaller than the SQUID **oscillation** period for the sample N1. T P (Purcell time) was calculated by T P = 1 / Γ P —inverse of Purcell rate, and T 1 c a l by 1 / T 1 c a l = 1 / T N P + 1 / T P (see texts) (b) The **qubit** **frequency** and three measured time scales for N7....[fig:lowB] fig:lowB_N7 show, at zero field the relaxation time T 1 was substantially larger for the sample N7 than for N1. Furthermore, when a small magnetic field was applied, the energy relaxation time for N1 increased, while hardly any change was observed for N7. Both of these effects can be understood as consequences of the Purcell effect in which the rate of spontaneous emission is increased when the cavity mode to which the **qubit** couples lies close by in **frequency**. The excitation **frequency** of N7 (4.97 GHz) was further from the cavity **frequency** than was the excitation **frequency** of N1 (6.583 GHz) and this almost completely determines the difference in T 1 . To see this we first compare the measured ratio Γ N 1 / Γ N 7 and the calculated ratio of Purcell rates for the two devices. (Since the next higher cavity resonance is more than 11 GHz above the fundamental, we ignore its contribution to the Purcell relaxation rate.) For the Purcell relaxation rate we use Γ P = κ 1 g / Δ 2 , where κ 1 = ω c / Q L is the cavity power decay rate, g is the **qubit**-cavity coupling rate, and Δ is the **qubit**-cavity detuning Δ = | ω 01 - ω c | = 2 π | f 01 - f c | . The ratio of the Purcell rates depends only on the **qubit**-cavity **frequency** differences which are easy to measure. We find that the ratio of the **qubits** measured lifetimes for samples N1 and N7 is T 1 , N 1 / T 1 , N 7 = 13 μ s / 44 μ s = 0.3 , while the ratio of the calculated Purcell times, T P = 1 / Γ P is Δ N 1 / Δ N 7 2 = 0.29 . Since the measured and the estimated ratios are very close to each other, one can conclude that the relaxation is Purcell limited....[fig:vortexcount] fig:vortexcount_a], B n (black open circle) increase linearly with n , as is expected for the situation in which the sweet spots occur periodically with magnetic field. Such exact periodicity is observed only in the low field regime, when there are no vortices in the electrodes, i.e., B n **oscillation** is perfectly periodic in this vortex-free regime, the positions of the sweet spots of the **qubit** can be approximated as B n = Δ B ⋅ n + B 0 , where Δ B is the unperturbed period of the HV-**oscillation** and B 0 is the position of the zero’s sweet spot. The linear fit [blue line in Fig. ... We present a new type of transmon split-junction **qubit** which can be tuned by Meissner screening currents in the adjacent superconducting film electrodes. The best detected relaxation time ($T_1$) was of the order of 50 $\mu$s and the dephasing time ($T_2$) about 40 $\mu$s. The achieved period of **oscillation** with magnetic field was much smaller than in usual SQUID-based transmon **qubits**, thus a strong effective field amplification has been realized. This Meissner **qubit** allows an efficient coupling to superconducting vortices. We present a quantitative analysis of the radiation-free energy relaxation in **qubits** coupled to Abrikosov vortices. The observation of coherent quantum **oscillations** provides strong evidence that vortices can exist in coherent quantum superpositions of different position states. According to our suggested model, the wave function collapse is defined by Caldeira-Leggett dissipation associated with viscous motion of the vortex cores.

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Contributors: Tian, Guihua, Zhong, Shuquan

Date: 2013-09-24

The dynamics of two **qubits** ultra-strongly coupled with a quantum **oscillator** is investigated by the adiabatic approximation method. The evolution formula of the initial four Bell states are studied under the control mechanism of the coherent state of the quantum **oscillator**. The influential parameters for the preservation of the entanglement are the four parameters: the average number of the coherent state, the ultra-strong coupling strength, the ratio of two **frequencies** of **qubit** and **oscillator**, and the inter-interaction coupling of the two **qubits**. The novel results show that the appropriate choice of these parameters can enable this mechanism to be utilized to preserve the entanglement of the two **qubits**, which is initially in the state |I_0> of the four Bell states. We give two different schemes to choose the respective parameters to maintain the entangled state |I_0> almost unchanged. The results will be helpful for the quantum information process....To initial entangled state | I 0 of the two **qubits** with control field mode in its coherent state, its evolution involves on the various parameters in Eq.( t0-1). It depends on the number N in an extremely nonlinear and intricate way. The kind perplexity makes it hard to study the Rabi model, never the less, it also provide the opportunity to preserve the entanglement of the two **qubits** by careful choice of the appropriate parameters. Because the coherent state has the probability of Poisson distribution, which will be approximated by a Gauss distribution if the average number | α | 2 is large enough. The intricacy of the Rabi model could be utilized to make the quantity Y ~ N , + 2 L ~ N , + 4 = B N in Eq.( t0-1) extremely small when N is in the neighborhood of | α | 2 by some selection of the appropriate parameters, which will guaranty the initial state of the **qubits** unchanging. This is shown in Fig.( fig8). It can be easy to see that whenever we select the parameters appropriate, for example, the parameters as | α | 2 = 55 , a = - 0.6 , β = 0.5599 , ω 0 ω = 0.24 , the Bell state | I 0 = | 1 , 0 = 1 2 | ↑ ↑ - | ↓ ↓ have the probability about 1 - 0.005 = 99.5 100 to remain unchanged....Figs.( fig11-3) show the general trait for the **qubit** remaining in its initial states | 1 , ± for different parameter a = 0.2 , 0 , - 0.2 . Obviously, P N 1 t is influenced by four parameters β , a , N , ω 0 ω . In Ref., it is shown that the coupling strength β ranges from 0.01 to 1 for the application of adiabatic approximation (weak coupling will not be discussed here). From Eqs.( e0)-( t0),( p1- omega00), we see that the parameter a will come to action apparently whenever β ≈ 0.01 - 0.6 . As stated before, the **qubits** is equivalent to a two **qubits** system and the non-equal-energy-level parameter a represents the coupling strength between the two **qubits**. This shows the coupling of the two **qubits** changes their dynamics considerately in the range of β ≈ 0.1 - 0.6 for the adiabatic approximation method to be applied, and this is our limit on the coupling parameter β...Schematic diagram of P N 1 t with the four parameters as N = 2 , ω 0 ω = 0.25 , β = 0.2 , a = 0.2 , 0 , - 0.2 from the top to bottom respectively. The apparent difference in these three figures strongly implies that the parameter a influences the **qubits** dynamically....The parameter a is connected with the inter-**qubit** coupling strength κ as a = ± κ in symmetric and asymmetric transition cases respectively. Study also shows that the parameter a negative is favorable for | T α t approaching zero, as Fig.( fig8) exhibits. So the inter-**qubit** coupling is in favor of preservation of the initial entanglement, especial with asymmetrical transition case ( a < 0 ). ... The dynamics of two **qubits** ultra-strongly coupled with a quantum **oscillator** is investigated by the adiabatic approximation method. The evolution formula of the initial four Bell states are studied under the control mechanism of the coherent state of the quantum **oscillator**. The influential parameters for the preservation of the entanglement are the four parameters: the average number of the coherent state, the ultra-strong coupling strength, the ratio of two **frequencies** of **qubit** and **oscillator**, and the inter-interaction coupling of the two **qubits**. The novel results show that the appropriate choice of these parameters can enable this mechanism to be utilized to preserve the entanglement of the two **qubits**, which is initially in the state |I_0> of the four Bell states. We give two different schemes to choose the respective parameters to maintain the entangled state |I_0> almost unchanged. The results will be helpful for the quantum information process.

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