### 54087 results for qubit oscillator frequency

Contributors: Agarwal, S., Rafsanjani, S. M. Hashemi, Eberly, J. H.

Date: 2012-01-13

Numerical and analytical evaluation of the entanglement dynamics between the two **qubits** for ω 0 = 0.15 ω , β = 0.16 and α = 3 . Entanglement between the **qubits** exhibits collapse and revival. The analytic expression agrees well with the envelope of the numerically evaluated entanglement evolution....Here, within the adiabatic approximation, we extend the examination to the two-**qubit** case. Qualitative differences between the single-**qubit** and the multi-**qubit** cases are highlighted. In particular, we study the collapse and revival of joint properties of both the **qubits**. Entanglement properties of the system are investigated and it is shown that the entanglement between the **qubits** also exhibits collapse and revival. We derive what we believe are the first analytic expressions for the individual revival signals beyond the RWA, as well as analytic expression for the collapse and revival dynamics of entanglement. In the quasi-degenerate regime, the invalidity of the RWA in predicting the dynamical evolution will clearly be demonstrated in Sec. s.collapse_rev (see Figs. f.collapse_revival_double and f.collapse_revival_single)....The three potential wells corresponding to the states | 1 , 1 | N 1 (left), | 1 , 0 | N 0 (middle) and | 1 , - 1 | N -1 (right). The factor Δ X z p is the zero point fluctuation of a harmonic **oscillator**. For an **oscillator** of mass M and **frequency** ω the zero point fluctuation is given by Δ X z p = ℏ / 2 M ω ....Collapse and revival dynamics for ω 0 = 0.1 ω , β = 0.16 and α = 3 . The first two panels show analytic evaluations of (a.) one-**qubit** and (b.) two-**qubit** probability dynamics, and (c.) shows that the two-**qubit** analytic formula matches well to the corresponding numerical evolution. In each case the initial state is a product of a coherent **oscillator** state with the lowest of the S x states. Note the breakup in the main revival peak of the two-**qubit** numerical evaluation, which comes from the ω - 2 ω beat note, not included in the analytic calculation, and not present for a single **qubit**....When squared, the probability shows two **frequencies** of **oscillation**, 2 Ω N ω and 2 2 Ω N ω . Since three new basis states are involved, we could expect three **frequencies**, but two are equal: | E N + - E N 0 | = | E N - - E N 0 | . This is in contrast to the single-**qubit** case where only one Rabi **frequency** determines the evolution . We show below in Fig. f.col_rev the way differences between one and a pair of **qubits** can be seen....If the average excitation of the **oscillator**, n ̄ = α 2 , is large one can evaluate the above sum approximately (see Appendix) and obtain analytic expressions and graphs of the evolution, as shown in Fig. f.col_rev. As expected, because of the double **frequency** in ( eqnP(t)), the revival time for S t 2 ω 0 is half the revival time for S t ω 0 . Thus there are two different revival sequences in the time series. Appropriate analytic formulas, e.g., ( a.Phi), agree well with the numerically evaluated evolution even for the relatively weak coherent excitation, α = 3 ....Entanglement dynamics between the **qubits**. (a) ω 0 = 0.1 ω , β = 0.16 , (b) ω 0 = 0.15 ω , β = 0.16 , (c) ω 0 = 0.1 ω , β = 0.2 and (d) ω 0 = 0.15 ω , β = 0.2 . For all the figures, α = 3 ....Collapse and revival dynamics for P 1 2 , - 1 2 α t , given ω 0 = 0.15 ω , β = 0.16 and α = 3 . Note the single revival sequence. Also, note that there are no breakups in the revival peaks in contrast to the two-**qubit** case (Fig. f.collapse_revival_double). The RWA fails to describe the dynamical evolution even for the single **qubit** case....With recent advances in the area of circuit QED, it is now possible to engineer systems for which the **qubits** are coupled to the **oscillator** so strongly, or are so far detuned from the **oscillator**, that the RWA cannot be used to describe the system’s evolution correctly . The parameter regime for which the coupling strength is strong enough to invalidate the RWA is called the ultra-strong coupling regime . Niemczyk, et al. and Forn-Díaz, et al. have been able to experimentally achieve ultra-strong coupling strengths and have demonstrated the breakdown of the RWA. Motivated by these experimental developments and the importance of understanding collective quantum behavior, we investigate a two-**qubit** TC model beyond the validity regime of RWA. The regime of parameters we will be concerned with is the regime where the **qubits** are quasi-degenerate, i.e., with **frequencies** much smaller than the **oscillator** **frequency**, ω 0 ≪ ω , while the coupling between the **qubits** and the **oscillator** is allowed to be an appreciable fraction of the **oscillator** **frequency**. In this parameter regime, the dynamics of the system can neither be correctly described under the RWA, nor can the effects of the counter rotating terms be taken as a perturbative correction to the dynamics predicted within the RWA by including higher powers of β . For illustration, systems are shown in Fig. f.model for which the RWA is valid, or breaks down, because the condition ω 0 ≈ ω is valid, or is violated. The regime that we will be interested in, for which ω 0 ≪ ω , is shown on the right....There is only one revival sequence for the single **qubit** system as a consequence of having only one Rabi **frequency** in the single **qubit** case. The analytic and numerically exact evolution of P 1 2 , - 1 2 α t is plotted in Fig. f.collapse_revival_single. The single revival sequence is evident from the figure. A discussion on the multiple revival sequences for the K -**qubit** TC model, within the parameter regime where the RWA is valid, can be found in ....The Tavis-Cummings model for more than one **qubit** interacting with a common **oscillator** mode is extended beyond the rotating wave approximation (RWA). We explore the parameter regime in which the **frequencies** of the **qubits** are much smaller than the **oscillator** **frequency** and the coupling strength is allowed to be ultra-strong. The application of the adiabatic approximation, introduced by Irish, et al. (Phys. Rev. B \textbf{72}, 195410 (2005)), for a single **qubit** system is extended to the multi-**qubit** case. For a two-**qubit** system, we identify three-state manifolds of close-lying dressed energy levels and obtain results for the dynamics of intra-manifold transitions that are incompatible with results from the familiar regime of the RWA. We exhibit features of two-**qubit** dynamics that are different from the single **qubit** case, including calculations of **qubit**-**qubit** entanglement. Both number state and coherent state preparations are considered, and we derive analytical formulas that simplify the interpretation of numerical calculations. Expressions for individual collapse and revival signals of both population and entanglement are derived. ... The Tavis-Cummings model for more than one **qubit** interacting with a common **oscillator** mode is extended beyond the rotating wave approximation (RWA). We explore the parameter regime in which the **frequencies** of the **qubits** are much smaller than the **oscillator** **frequency** and the coupling strength is allowed to be ultra-strong. The application of the adiabatic approximation, introduced by Irish, et al. (Phys. Rev. B \textbf{72}, 195410 (2005)), for a single **qubit** system is extended to the multi-**qubit** case. For a two-**qubit** system, we identify three-state manifolds of close-lying dressed energy levels and obtain results for the dynamics of intra-manifold transitions that are incompatible with results from the familiar regime of the RWA. We exhibit features of two-**qubit** dynamics that are different from the single **qubit** case, including calculations of **qubit**-**qubit** entanglement. Both number state and coherent state preparations are considered, and we derive analytical formulas that simplify the interpretation of numerical calculations. Expressions for individual collapse and revival signals of both population and entanglement are derived.

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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: 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.

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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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Contributors: Chudzicki, Christopher, Strauch, Frederick W.

Date: 2010-08-10

Entanglement distribution rate R (in units of 1 / T ) as function of the number of nodes N in a quantum network. Three distribution schemes are shown: the massively parallel (MP) and **qubit**-compatible schemes on the hypercube of dimension d (each with N = 2 d ), and the complete graph of size N . Each network was chosen to to have a coupling of Ω 0 / 2 π = 20 MHz with a bandwidth ω max - ω min / 2 π = 2 GHz ....Parallel state transfer on programmable quantum networks. Each node is an **oscillator** with a tunable **frequency**. Each line (solid or dashed) indicates a coupling between **oscillators**. Solid lines indicate couplings between **oscillators** with the same **frequency**; dashed lines indicate couplings between **oscillators** with different **frequencies**. High fidelity state transfer occurs for large detuning. (a) Hypercube network with d = 3 , programmed into two subcubes (red and blue squares). Each node is labeled by a bit-string of length d = 3 , here with the first m = 1 bits indicating the subcube. In the **qubit**-compatible scheme (QC), one entangled pair is sent on each subcube, as indicated by the arrow for the inner (red) square. (b) In the massively parallel scheme (MP) scheme, multiple entangled pairs are sent between every node of each subcube, as indicated by the arrows for the inner (red) square. (b) Completely connected network with N = 8 , programmed into N / 2 = 4 two-site networks....We study the routing of quantum information in parallel on multi-dimensional networks of tunable **qubits** and **oscillators**. These theoretical models are inspired by recent experiments in superconducting circuits using Josephson junctions and resonators. We show that perfect parallel state transfer is possible for certain networks of harmonic **oscillator** modes. We further extend this to the distribution of entanglement between every pair of nodes in the network, finding that the routing efficiency of hypercube networks is both optimal and robust in the presence of dissipation and finite bandwidth....The fidelity of entanglement transfer on the hypercube as a function of the detuning parameter η = 2 Ω 0 / Δ ω . The **qubit** curves are, from top to bottom (for small η ), numerical simulations for dimension d = 2 6 . Each network is split into M = 2 subcube channels with one sender and receiver per channel, with entanglement being sent in the same direction on each channel. Also shown also is the lower bound of Eq. ( ES:fid) for the **oscillator** network with m = 1 ....ES:fid was derived for entanglement transfer on **oscillator** networks. However, as long as only one sender-receiver pair uses each channel at a time, numerical calculations, shown in Fig. fig2, indicate that **qubit** networks behave similarly. For this reason we call the parallel state transfer protocol discussed so far the “**qubit**-compatible” (QC) protocol. There are some notable differences between **qubits** and **oscillators**, namely **qubits** do better on average, but do not exhibit perfect state transfer....**Qubit**, entanglement, quantum computing, superconductivity, Josephson junction....These three distribution rates are plotted as a function of the number of nodes in Fig. fig:ESplot, where we have fixed the bandwidth appropriate to recent superconducting **qubit** experiments . For the hypercube schemes, the massively parallel protocol is more than quadratically better than the **qubit**-compatible scheme. Entanglement transfer on the complete graph quickly fails due to significant cross-talk for N ≈ 20 . One might expect that this is due to the large number of couplings, but it is actually due to the finite bandwidth of the network. It is clear that studying extended coupling schemes such as the cavity grid is an important task....Parallel state transfer on the hypercube. Christandl et al. showed that one can perform perfect state transfer from corner-to-corner of a d -dimensional hypercube in constant time T = π / 2 Ω 0 . Here we analyze how this result can be extended to the transfer of quantum states in parallel, by splitting the cube into subcubes. Specifically, by tuning the **frequencies** of each node, the d -dimensional hypercube can be broken up into 2 m subcubes each of dimension d - m , as shown in Fig. fig1(a) for d = 3 and m = 1 . These subcubes can be made to act as good channels between their antipodal nodes by separating the **oscillator** **frequencies** for each channel from adjacent channels by an amount Δ ω . For fixed couplings, there is still the potential for cross-talk between channels, which we now analyze. ... We study the routing of quantum information in parallel on multi-dimensional networks of tunable **qubits** and **oscillators**. These theoretical models are inspired by recent experiments in superconducting circuits using Josephson junctions and resonators. We show that perfect parallel state transfer is possible for certain networks of harmonic **oscillator** modes. We further extend this to the distribution of entanglement between every pair of nodes in the network, finding that the routing efficiency of hypercube networks is both optimal and robust in the presence of dissipation and finite bandwidth.

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Contributors: Yu.A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D.V. Averin, T. Tilma, F. Nori, J.S. Tsai

Date: 2005-01-01

Probe current **oscillations** in the first (a) and the second (b) **qubit** when the system is driven non-adiabatically to the double-degeneracy point X for the case EJ1=9.1GHz and EJ2=13.4GHz. Right panels show the corresponding spectra obtained by Fourier transformation. Arrows and dotted lines indicate theoretically expected position of the peaks.
...EJ1 dependence of the spectrum components of Fig. 6. Solid lines: dependence of Ω+ε and Ω−ε obtained from Eq. (6) using EJ2=9.1GHz and Em=14.5GHz and varying EJ1 from zero to its maximum value of 13.4GHz. Dashed lines: dependence of the **oscillation** **frequencies** of both **qubits** in the case of zero coupling (Em=0).
...Schematic diagram of the two-coupled-**qubit** circuit. Black bars denote Cooper pair boxes.
...Probe current **oscillations** in the first (a) and the second (b) **qubit** when the system is driven non-adiabatically to the points R and L, respectively. Right panels show the corresponding spectra obtained by the Fourier transform. Peak position in the spectrum gives the value of the Josephson energy of each **qubit**, indicated by arrow. In both cases, the experimental data (open triangles and open dots) can be fitted to a cosine dependence (solid lines) with an exponential decay with 2.5ns time constant.
...We have analyzed and measured the quantum coherent dynamics of a circuit containing two-coupled superconducting charge **qubits**. Each **qubit** is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Two **qubits** are coupled electrostatically by a small island overlapping both Cooper pair boxes. Quantum state manipulation of the **qubit** circuit is done by applying non-adiabatic voltage pulses to the common gate. We read out each **qubit** by means of probe electrodes connected to Cooper pair boxes through high-Ohmic tunnel junctions. With such a setup, the measured pulse-induced probe currents are proportional to the probability for each **qubit** to have an extra Cooper pair after the manipulation. As expected from theory and observed experimentally, the measured pulse-induced current in each probe has two **frequency** components whose position on the **frequency** axis can be externally controlled. This is a result of the inter-**qubit** coupling which is also responsible for the avoided level crossing that we observed in the **qubits**’ spectra. Our simulations show that in the absence of decoherence and with a rectangular pulse shape, the system remains entangled most of the time reaching maximally entangled states at certain instances....Solid-state **qubits** ... We have analyzed and measured the quantum coherent dynamics of a circuit containing two-coupled superconducting charge **qubits**. Each **qubit** is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Two **qubits** are coupled electrostatically by a small island overlapping both Cooper pair boxes. Quantum state manipulation of the **qubit** circuit is done by applying non-adiabatic voltage pulses to the common gate. We read out each **qubit** by means of probe electrodes connected to Cooper pair boxes through high-Ohmic tunnel junctions. With such a setup, the measured pulse-induced probe currents are proportional to the probability for each **qubit** to have an extra Cooper pair after the manipulation. As expected from theory and observed experimentally, the measured pulse-induced current in each probe has two **frequency** components whose position on the **frequency** axis can be externally controlled. This is a result of the inter-**qubit** coupling which is also responsible for the avoided level crossing that we observed in the **qubits**’ spectra. Our simulations show that in the absence of decoherence and with a rectangular pulse shape, the system remains entangled most of the time reaching maximally entangled states at certain instances.

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Contributors: Lecocq, Florent, Pop, Ioan M., Matei, Iulian, Dumur, Etienne, Feofanov, A. K., Naud, Cécile, GUICHARD, Wiebke, Buisson, Olivier

Date: 2012-01-19

Energy spectrum. Escape probability P e s c versus **frequency** as a function of current bias (a) and flux bias (b) measured at Φ b = 0.48 Φ 0 and I b = 0 respectively. P e s c is enhanced when the **frequency** matches a resonant transition of the circuit. The microwave amplitude was tuned to keep the resonance peak amplitude at 10%. Dark and bright blue scale correspond to high and small P e s c . The red dashed lines are the transition **frequencies** deduced from the spectrum of the full hamiltonian with C = 510 fF (see text). The green diamond is the initial working point for the measurement of coherent free **oscillations** between the two modes, presented in Fig. **Oscillations**, and the green dotted square is the area where these **oscillations** take place. spectro...Description of the device. (a) A micrograph of the aluminium circuit. The two small squares are the two Josephson junctions (enlarged in the top right inset, 10 μ m 2 area, I c = 713 nA and C = 510 fF) decoupled by a large inductive loop ( L = 629 pH). The width of the two SQUID arms were adjusted to reduce the inductance asymmetry to about 10%. Very narrow current bias lines, with a large 15 nH-inductance, isolate the quantum circuit from the dissipative environment at high **frequencies** . The symmetric and antisymmetric **oscillation** modes are illustrated by blue and red arrows, respectively. (b) and (c) Potentials of the s and a-mode respectively, for the bias working point ( I b = 0 , Φ b = 0.37 Φ 0 ). (d) Schematic energy level diagram indexed by the quantum excitation number of the two modes n s , n a at the same working point. Climbing each vertical ladder one increases the excitation number of the s-mode, keeping the excitation number of the a-mode constant. The two first levels, 0 s , 0 a and 1 s , 0 a , realize a camelback phase **qubit**. fig1...By adding a large inductance in a dc-SQUID phase **qubit** loop, one decouples the junctions' dynamics and creates a superconducting artificial atom with two internal degrees of freedom. In addition to the usual symmetric plasma mode ({\it s}-mode) which gives rise to the phase **qubit**, an anti-symmetric mode ({\it a}-mode) appears. These two modes can be described by two anharmonic **oscillators** with eigenstates $\ket{n_{s}}$ and $\ket{n_{a}}$ for the {\it s} and {\it a}-mode, respectively. We show that a strong nonlinear coupling between the modes leads to a large energy splitting between states $\ket{0_{s},1_{a}}$ and $\ket{2_{s},0_{a}}$. Finally, coherent **frequency** conversion is observed via free **oscillations** between the states $\ket{0_{s},1_{a}}$ and $\ket{2_{s},0_{a}}$....We now discuss the coherent properties and measurement contrast in our device. The unexpectedly short coherence time of the a-mode can be explained by the coupling to spurious two-level systems (TLS) . With a junction area of 10 μ m 2 our device suffers from a large TLS density of about 12 TLS/GHz (barely visible in Fig. spectro). Therefore it is very difficult to operate the a-mode in a **frequency** window free of TLS since ν 01 a is only slightly flux dependent. However this is not a real issue as it can be solved easily by reducing the junction area . The minimum linewidth of both a-mode and s-mode, and therefore their coherence times, are limited in our experiment by low **frequency** flux noise. Operating the system at Φ b = 0 will lift this limitation since it is an optimal point with respect to flux noise. The small **oscillation** amplitude in Fig. **Oscillations** has two additionnals origins. First the duration t π of the π -pulse applied for preparation of the state 0 s , 1 a has to fulfill the condition t π -1 **frequency** conversion ....One of the opportunities given by the rich spectrum of this two DoF artificial atom is the observation of a coherent **frequency** conversion process using the x ̂ s 2 x ̂ a coupling of Eq. eq:2. The pulse sequence, similar to other states swapping experiment , is presented in (see Fig. Oscillationsa). At t = 0 , the system is prepared in the state 0 s , 1 a , at the initial working point Φ b = 0.37 Φ 0 (green diamond in Fig. spectrob). Immediately after, a non-adiabatic flux pulse brings the system to the working point defined by Φ i n t , close to the degeneracy point ( ν 02 s ≈ ν 01 a ). After the free evolution of the quantum state during the time Δ t i n t , we measure the escape probility P e s c . Fig. Oscillationsb presents P e s c as a function of Δ t i n t for Φ i n t = 0.515 Φ 0 . The observed **oscillations** have a 815 M H z -characteristic **frequency** (inset of Fig. Oscillationsb) that matches precisely the theoritical **frequency** splitting at this flux bias (red arrow). In Fig. Oscillationsc, we present these **oscillations** as function of Φ i n t . Their **frequency** varies with Φ i n t , showing a typical “chevron” pattern. In the inset of Fig. Oscillationsc, the **oscillation** **frequency** versus Φ i n t is compared to theoretical predictions. The good agreement between theory and experiment is a striking confirmation of the observation of swapping between the quantum states 0 s , 1 a and 2 s , 0 a . Instead of the well known linear coupling x ̂ s x ̂ a between two **oscillators**, which corresponds to a coherent exchange of single excitations between the two systems, here the coupling x ̂ s 2 x ̂ a is non-linear and produces a coherent exchange of a single excitation of the a-mode with a double excitation of the s-mode, i.e. a coherent **frequency** conversion. Starting from the state 0 s , 1 a , an excitation pair 2 s , 0 a is then deterministically produced in about a single nanosecond at the degeneracy point Φ i n t = 0.537 Φ 0 ....Free coherent **oscillations** between states 0 s , 1 a and 2 s , 0 a produced by a nonlinear coupling. (a) Schematic pulse sequence. The energy diagram, without coupling in blue/red and with coupling in black, is represented for both the preparation and interaction steps.(b) Escape probability P e s c versus interaction time Δ t i n t . The inset presents the Fourier transform of these **oscillations** with a clear peak at 815 MHz. The red arrow indicates the theoretically expected **frequency**. (c) P e s c versus interaction time Δ t i n t for different interaction flux Φ i n t close to the resonance condition between ν 02 s and ν 01 a . For clarity the data is numerically processed using 200 MHz high-pass filter(dashed line in inset of (b)). Inset : **oscillation** **frequency** as function of flux. The dashed red line shows the theoretical predictions....The readout of the circuit is performed using switching current techniques. For spectroscopy measurements we apply a microwave pulse field, through the current bias line (see Fig fig1a), followed by the readout nanosecond flux pulse that produces a selective escape depending on the quantum state of the circuit . The energy spectrum versus current bias at Φ b = 0.48 Φ 0 and versus flux bias at I b = 0 are plotted in Fig. spectroa and Fig. spectrob respectively. In the following we will denote ν n m α as the transition **frequency** between the states n α and m α , with the other mode in the ground state. The first transition **frequency** in Fig. spectro is the one of the camelback phase **qubit**, ν 01 s . With a maximum **frequency** at zero-current bias, the system is at an optimal working point with respect to current fluctuations . At higher **frequency**, the second transition of the s-mode is observed with ν 02 s ≈ 2 ν 01 s . In the flux biased spectrum the third transition ν 03 s is also visible. An additional transition is observed at about 14.6 GHz, with a very weak current dependence (Fig. spectroa) but a finite flux dependence (Fig. spectrob). It corresponds to the first transition of the a-mode, ν 01 a . The s-mode transition **frequencies** drop when Φ b / Φ 0 approaches 0.7 which is consistent with the critical flux Φ c / Φ 0 = 1 / 2 + L / 2 π L J = 0.717 for which ω p s → 0 and x a m i n → π / 2 . On the contrary ω p a remains finite when Φ b → Φ c with ω p a → 4 E J / m L J / L . One also observes a large level anti-crossing of about 700 MHz between the two transitions ν 02 s and ν 01 a . Additionally no level anti-crossing is measurable between ν 03 s and ν 01 a ....The circuit is a camelback phase **qubit** with a large loop inductance, i.e a dc-SQUID build by a superconducting loop of large inductance L interrupted by two identical Josephson junctions with critical current I c and capacitance C , operated at zero current bias (see Fig. fig1). As we will see in the following, the presence of a large loop inductance modifies dractically the quantum dynamics of this system. The two phase differences φ 1 and φ 2 across the two junctions correspond to the two degrees of freedom of this circuit, which lead to two **oscillating** modes: the symmetric ( s-) and the anti-symmetric (a-) plasma modes . The s-mode corresponds to the well-known in-phase plasma **oscillation** of the two junctions with the average phase x s = φ 1 + φ 2 / 2 , **oscillating** at a characteristic **frequency** given by the plasma **frequency** of the dc-SQUID, ω p s . The a-mode is an opposite-phase plasma mode related to **oscillations** of the phase difference x a = φ 1 - φ 2 / 2 , producing circulating current **oscillations** at **frequency** ω p a . In previous experiments , the loop inductance L was small compared to the Josephson inductance L J = Φ 0 / 2 π I c . Therefore the two junctions were strongly coupled and the dynamics of the phase difference x a was neglected and fixed by the applied flux. The quantum behavior of the circuit was described by the s-mode only, showing a one-dimensional motion of the average phase x s . Hereafter we will consider a circuit with a large inductance ( L ≥ L J ) that decouples the phase dynamics of the two junctions. This large inductance lowers the **frequency** of the a-mode and the dynamics of the system becomes fully two-dimensional. The a-mode was previously introduced to discuss the thermal and quantum escape of a current-biased dc-SQUID but its dynamics was never observed. We present measurements of the full spectrum of this artificial atom, independent coherent control of both modes and finally we exploit the strong nonlinear coupling between the two DoF to observe a time resolved up and down **frequency** conversion of the system excitations. ... By adding a large inductance in a dc-SQUID phase **qubit** loop, one decouples the junctions' dynamics and creates a superconducting artificial atom with two internal degrees of freedom. In addition to the usual symmetric plasma mode ({\it s}-mode) which gives rise to the phase **qubit**, an anti-symmetric mode ({\it a}-mode) appears. These two modes can be described by two anharmonic **oscillators** with eigenstates $\ket{n_{s}}$ and $\ket{n_{a}}$ for the {\it s} and {\it a}-mode, respectively. We show that a strong nonlinear coupling between the modes leads to a large energy splitting between states $\ket{0_{s},1_{a}}$ and $\ket{2_{s},0_{a}}$. Finally, coherent **frequency** conversion is observed via free **oscillations** between the states $\ket{0_{s},1_{a}}$ and $\ket{2_{s},0_{a}}$.

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Contributors: Serban, I., Solano, E., Wilhelm, F. K.

Date: 2007-02-28

Discrimination time as function of the coupling strength between **qubit** and **oscillator**. Here ℏ Ω / k B T = 2 , κ / Ω = 0.025 and ℏ ν / k B T = 1.95 ....In this section we present a different measurement protocol. It is based on the short time dynamics illustrated as follows: for the **qubit** initially in the state 1 / 2 | ↑ + | ↓ the probability distribution of momentum is plotted in Fig. probability (a) and (b)....As one can see for the parameters of Fig. comp, in the WQOC regime the measurement time is longer than the dephasig time. Their difference decreases as we increase Δ due to the onset of the strong coupling plateau in the dephasing rate, approaching the quantum limit where the measurement time becomes comparable to the dephasing time. Note that, for superstrong coupling either between **qubit** and **oscillator** or between **oscillator** and bath, corrections of the order κ / Ω ↓ 2 of the dephasing rate gain importance. These corrections are not treated in our Born approximation. Therefore the regions where the dephasing rate becomes lower than the measurement rate, in violation with the quantum limitation of Ref. , should be regarded as a limitation of our approximation....Dephasing rate dependence on driving: dependence on Δ for different driving strengths F 0 ( κ / Ω = 10 -4 and ν = 2 Ω ). Top inset: dependence of the decoherence rate on F 0 for different values of κ ( Δ / Ω = 5 ⋅ 10 -2 and ν = 2 Ω ). Bottom inset: dependence of the decoherence rate on driving **frequency** ν for different vales of κ ( Δ / Ω = 0.5 ). Here ℏ Ω / k B T = 2 and F ¯ 0 is the dimensionless force F 0 ℏ / k B T m k B T ....Probability density of momentum P p 0 t (a), snapshots of it at different times (b) and expectation value of momentum for the two different **qubit** states (c). Here ℏ Ω / k B T = 2 , Δ / Ω = 0.45 , κ / Ω = 0.025 and ℏ ν / k B T = 1.9 and p ¯ 0 is the dimensionless momentum p 0 / k B T m ....We consider a simplified version of the experiment described in Ref. . The circuit consists of a flux **qubit** drawn in the single junction version, the surrounding SQUID loop, an ac source, and a shunt resistor, as depicted in Fig. circuit. We note here that we later approximate the **qubit** as a two-level system. The **qubit** used in the actual experiment contains three junctions. An analogous but less transparent derivation would, after performing the two-state approximation, lead to the same model, parameterized by the two-state Hamiltonian, the circulating current, and the mutual inductance, in an identical way ....Motivated by recent experiments, we study the dynamics of a **qubit** quadratically coupled to its detector, a damped harmonic **oscillator**. We use a complex-environment approach, explicitly describing the dynamics of the **qubit** and the **oscillator** by means of their full Floquet state master equations in phase-space. We investigate the backaction of the environment on the measured **qubit** and explore several measurement protocols, which include a long-term full read-out cycle as well as schemes based on short time transfer of information between **qubit** and **oscillator**. We also show that the pointer becomes measurable before all information in the **qubit** has been lost....The dependence on the driving **frequency** has also been analyzed in Fig. antrieb. Here we observe two peaks at Ω ↑ and Ω ↓ . At ν = Ω the classical driven and undamped trajectory ξ t diverges. In terms of the calculation this means that the Floquet modes are not well-defined when the driving **frequency** is at resonance with the harmonic **oscillator** — we have a continuum instead. Physically this means that at t = 0 our **oscillator** has the **frequency** Ω because it has not yet "seen" the **qubit**, and we are driving it at resonance, and by amplifying the **oscillations** of x ̂ which is subject to noise we amplify the noise seen by the **qubit**. The dephasing rate is also expected to diverge. The peaks at Ω ↑ and Ω ↓ show the same effect after the **qubit** and the **oscillator** become entangled. The dephasing rate drops again for large driving **frequencies** to the value obtained in the case without driving....Simplified circuit consisting of a **qubit** with one Josephson junction (phase γ , capacitance C q and inductance L q ) inductively coupled to a SQUID with two identical junctions (phases γ 1 , 2 , capacitance C S ) and inductance L S . The SQUID is driven by an ac bias I B t and the voltage drop is measured by a voltmeter with internal resistance R . The total flux through the **qubit** loop is Φ q and through the SQUID is Φ S ....In Fig. probability one can see that the two peaks corresponding to the two states of the **qubit** split already during the transient motion of p ̂ t , much faster than the transient decay time. If the peaks are well enough separated, a single measurement of momentum gives the needed information about the **qubit** state, and has the advantage of avoiding decoeherence effects resulting from a long time coupling to the environment. Nevertheless the parameters we need to reduce the discrimination time also enhance the decoherence rate. ... Motivated by recent experiments, we study the dynamics of a **qubit** quadratically coupled to its detector, a damped harmonic **oscillator**. We use a complex-environment approach, explicitly describing the dynamics of the **qubit** and the **oscillator** by means of their full Floquet state master equations in phase-space. We investigate the backaction of the environment on the measured **qubit** and explore several measurement protocols, which include a long-term full read-out cycle as well as schemes based on short time transfer of information between **qubit** and **oscillator**. We also show that the pointer becomes measurable before all information in the **qubit** has been lost.

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Contributors: Saito, Keiji, Wubs, Martijn, Kohler, Sigmund, Kayanuma, Yosuke, Hanggi, Peter

Date: 2007-03-22

(Color online) Sketch of the diabatic energy levels as a function of time for a **qubit** coupled to a single harmonic **oscillator**. Energies of the states in the “up cluster” increase. These states correspond to the **qubit** state | ↑ . Energies decrease in the “down cluster”, where the **qubit** state is | ↓ . According to the “no-go-up theorem” app:nogoup, the initial state | ↑ , 0 + evolves to a superposition in which | ↑ , 0 + is the only “up” state. Energies within a band are separated by the **oscillator** energy ℏ Ω . For a **qubit** coupled to an **oscillator** bath, the corresponding crossing clusters would be continuous bands of states....(Color online) Landau-Zener dynamics for a **qubit** with Δ = 0 , in all three cases shown off-diagonally coupled via σ x to three **oscillators**. The various **oscillator** **frequencies** Ω j are given in units of v / ℏ . All coupling strengths have the same value γ j = ℏ v / 3 . The dotted line marks the analytical final transition probability corresponding to Eq. transverse result....(Color online) Sketch of spectral density for an atom near a local defect in a photonic crystal with a band gap. The quadratic free-space spectral density is modified by the crystal that creates a spectral gap around ω 0 . A narrow defect mode inside a broader band gap allows a controlled atom-defect interaction via LZ sweeps of the atomic transition **frequency** ω A t ....By making a controlled point defect or line defect defect in the vicinity of the atom that breaks the periodicity of the photonic crystal, a narrow defect mode may be created within the spectral gap, as sketched in Fig. fig:Jphotoniccrystal. Ideally, this would allow cavity QED experiments to be performed within a photonic crystal, and progress is made in this direction. We propose to do LZ sweeps of the atomic **frequency** ω A t around the defect **frequency** but within the band gap. This will allow the creation of atom-defect entanglement and of single photons in the defect mode, in quite the same way as in circuit QED....The diabatic energies cross, but the adiabatic energies ± 1 2 v 2 t 2 + Δ 2 for Δ ≠ 0 form an avoided crossing, as sketched in Fig. fig:energies. The adiabatic theorem states that the splitting Δ prevents transfer of population between the adiabatic eigenstates in the adiabatic limit ℏ v ≪ Δ 2 , in other words if the sweep occurs slowly enough. A **qubit** prepared at t = - ∞ in the initial ground state | ↑ will then end up in the final ground state | ↓ . Beyond the adiabatic regime, the dynamics can be rather complex. Nevertheless, the population of the diabatic states at t = ∞ can be calculated exactly and is determined by the Landau-Zener transition probability...Interestingly enough, the Landau-Zener tunneling probability is then fully determined by the integrated spectral density S . In particular, there is no dependence on the **oscillator** **frequencies** Ω j . This result is nicely illustrated in the simple example of Figure fig:three_osc, showing Landau-Zener dynamics of a **qubit** that is coupled to only three **oscillators**. The **oscillator** **frequencies** are varied, while the **qubit**-**oscillator** couplings are kept constant. The dynamics at intermediate times depends on the **oscillator** **frequencies**, but the final transition probability does not....(Color online) Time evolution of spin-flip probability for a **qubit** with Δ = 0.5 ℏ v diagonally coupled to two harmonic **oscillators**, two nonlinear **oscillators**, and seven spins, respectively. The harmonic **oscillators** are specified by Ω 1 = 0.1 v / ℏ , Ω 2 = 0.5 v / ℏ , γ 1 = 2 ℏ v , and γ 2 = 6 ℏ v , while the nonlinear **oscillators**, in addition, have β 1 = β 2 = 3 ℏ v . The values of the B j ν and the γ j z are randomly chosen from the range - ℏ v / 10 ℏ v / 10 . In all three cases, the transition probability converges to the universal value PLZuniversal....We study Landau-Zener transitions in a **qubit** coupled to a bath at zero temperature. A general formula is derived that is applicable to models with a non-degenerate ground state. We calculate exact transition probabilities for a **qubit** coupled to either a bosonic or a spin bath. The nature of the baths and the **qubit**-bath coupling is reflected in the transition probabilities. For diagonal coupling, when the bath causes energy fluctuations of the diabatic **qubit** states but no transitions between them, the transition probability coincides with the standard LZ probability of an isolated **qubit**. This result is universal as it does not depend on the specific type of bath. For pure off-diagonal coupling, by contrast, the tunneling probability is sensitive to the coupling strength. We discuss the relevance of our results for experiments on molecular nanomagnets, in circuit QED, and for the fast-pulse readout of superconducting phase **qubits**....General coupling. When the **oscillators** neither couple purely off-diagonally ( ϑ = π / 2 ) nor purely diagonally ( ϑ = 0 ), the Landau-Zener probability generally exhibits a non-monotonic dependence on the tunnel coupling Δ . This is shown in Figure fig2 for various angles ϕ and ϑ . Most interesting is the comparison to the non-dissipative case, which as we saw coincides with the result for diagonal coupling ( ϑ = 0 ): Any dissipative Landau-Zener probability lower than the curve for ϑ = 0 is counterintuitive. Such situations occur, however: for several values of ϑ and for a sufficiently large tunnel splitting Δ , the bath coupling reduces P ↑ → ↓ ∞ , i.e. dissipation enhances the population of the final excited **qubit** state....LZ sweeps pas resonances of nonlinear **oscillators** are of practical interest, since nonlinear **oscillators** are currently used for the readout of flux **qubits**. For a numerical test of the predicted final transition probability, we take the situation in which the **qubit** is diagonally coupled to two of these nonlinear **oscillators**. Figure fig3 shows the corresponding time-evolution of the probability P ↑ → ↓ t for the **qubit** to be in the “down” state. It also shows the dynamics in case the **qubit** couples to two linear **oscillators** with the same parameters, except that now β 1 , 2 = 0 . Furthermore, the effect of a diagonally coupled spin bath, a special case of Eq. ... We study Landau-Zener transitions in a **qubit** coupled to a bath at zero temperature. A general formula is derived that is applicable to models with a non-degenerate ground state. We calculate exact transition probabilities for a **qubit** coupled to either a bosonic or a spin bath. The nature of the baths and the **qubit**-bath coupling is reflected in the transition probabilities. For diagonal coupling, when the bath causes energy fluctuations of the diabatic **qubit** states but no transitions between them, the transition probability coincides with the standard LZ probability of an isolated **qubit**. This result is universal as it does not depend on the specific type of bath. For pure off-diagonal coupling, by contrast, the tunneling probability is sensitive to the coupling strength. We discuss the relevance of our results for experiments on molecular nanomagnets, in circuit QED, and for the fast-pulse readout of superconducting phase **qubits**.

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Contributors: Zorin, A. B.

Date: 2003-12-09

The persistent current is not, however, the only phase-dependent quantity characterizing the quantum state of the charge-phase **qubit**. Another useful quantity is the Josephson inductance of the double junction, which can be probed by small rf **oscillations** induced in the **qubit**. Recently, we proposed a transistor configuration of the Cooper pair box (see Fig. Scheme), in which the macroscopic superconducting loop closing the transistor terminals was inductively coupled to a radio-**frequency** tank circuit . Similar to the rf-SQUID-based method of measurement of the Josephson junction impedance , this setup makes it possible to measure the rf impedance (more specifically, the Josephson inductance) of the system of two small tunnel junctions connected in series and in doing so, to probe the macroscopic states of the **qubit**....(a) The electric circuit diagram of the charge-flux **qubit** inductively coupled to a tank circuit by mutual inductance M . The macroscopic superconducting loop of inductance L is interrupted by two small Josephson tunnel junctions positioned close to each other and forming a single charge transistor; the capacitively coupled gate polarizes the island of this transistor. The **qubit** is controlled by charge Q 0 generated by the gate and flux Φ m induced by coil L m . The tank circuit which is either of a parallel (b) or a serial (c) type is driven by a harmonic signal ( I r f or V r f , respectively) of **frequency** ω r f ≈ ω 0 , the resonant **frequency** of the uncoupled tank circuit....Evaluated **qubit** parameters derived on the assumption E J 0 = 2 E c = 80 μ eV (i.e., I c 0 ≈ 40 nA and 5 E c = Δ A l ≈ 200 μ eV, the energy gap of Al) and j 1 - j 2 = κ 1 - κ 2 = 0.1 . The tank circuit quality factor Q = 100 , **frequency** ω 0 = 2 π × 100 MHz, L T / C T 1 / 2 = 100 Ω , k 2 Q β L = 20 and temperature T * = 1 K ≫ T ∼ 20 mK. As long as the dephasing rate in the magic points is nominally zero, a 0.1% inaccuracy of the adjustment of the values φ = π and 0 was assumed....The charge-phase Josephson **qubit** based on a superconducting single charge transistor inserted in a low-inductance superconducting loop is considered. The loop is inductively coupled to a radio-**frequency** driven tank circuit enabling the readout of the **qubit** states by measuring the effective Josephson inductance of the transistor. The effect of **qubit** dephasing and relaxation due to electric and magnetic control lines as well as the measuring system is evaluated. Recommendations for operation of the **qubit** in magic points producing minimum decoherence are given....The principle of narrow-band radio-**frequency** readout of the **qubit**. (a) The resonance curves of the uncoupled (dotted line) and coupled to the **qubit** tank circuit biased at operation point A in the excited state (dashed line) and in the ground state (solid line). (b) Driving pulse applied to the tank circuit (top curve) and the response signal of the tank in resonance (the ground **qubit** state, bottom curve) and off-resonance (excited state, middle curve). A smooth envelope of the driving pulse is used to suppress transient **oscillations** and has a small effect on the rise time of the response signal. For clarity the curves are shifted vertically....The small tunnel junctions of the charge-flux **qubit** are characterized by self-capacitances C 1 and C 2 and the Josephson coupling strengths E J 1 and E J 2 . These junctions with a small central island in-between and a capacitively coupled gate therefore form a single charge transistor connected in our network as the Cooper pair box (see Fig. Scheme). Critical currents of the junctions are equal to I c 1 , c 2 = 2 π Φ 0 E J 1 , J 2 , where Φ 0 = h / 2 e is the flux quantum, and their mean value I c 0 = 1 2 I c 1 + I c 2 . The design enables magnetic control of the Josephson coupling in the box in a dc SQUID manner. The system therefore has two parameters, the total Josephson phase across the two junctions φ = ϕ 1 + ϕ 2 = 2 π Φ / Φ 0 controlled by flux Φ threading the loop and the gate charge Q 0 set by the gate voltage V g . The geometrical inductance of loop L is assumed to be much smaller than the Josephson inductance of the junctions L J 0 = Φ 0 / 2 π I c 0 ,...Terms composed of diagonal (a) and off-diagonal (b) matrix elements of operators cos χ and sin χ , respectively, calculated for different values of flux Φ e for the given **qubit** parameters (see caption of Fig. 2)....The terms composed of diagonal (a) and off-diagonal (b) matrix elements of operator V ̂ and entering Eqs. ( Vmod) and ( Vtan) are presented for different values of flux Φ e for the given **qubit** parameters (see caption of Fig. 2)....is nonzero for any Q 0 and Φ (see the plots of the two items in Fig. e2ME), only the condition E c ≤ Δ s c / 5 can ensure suppression of these transitions in arbitrary operation point of our **qubit**. Possibly insufficiently small value of E c was the reason of very short relaxation time (tens of ns) in the recent experiment with a charge **qubit** by Duty et al. . Their Al Cooper pair box had E c ≈ 0.8 Δ s c and E J ≈ 0.4 E c , so the energy gain in the chosen operation point ( Q 0 = 0.4 e ) was too large, i.e., about 2.2 E c ≈ 1.8 Δ s c > Δ s c (although in the ground state this sample nicely showed the pure Cooper pair characteristic)....For reading out the final state, the **qubit** dephasing is of minor importance, while the requirement of a sufficiently long relaxation time is decisive. Moreover, the relaxation rate may somewhat increase due to **oscillations** in the tank induced by a drive pulse (see Fig. Pulse), which leads to the development of **oscillations** around a magic point along φ axis, Eq. ( phase_osc). If the **frequency** of these **oscillations** is sufficiently low, ω r f ≪ Ω , they result only in a slow modulation of transition **frequency** Ω . Increase in amplitude of steady **oscillations** up to φ a ≈ π / 2 (determined by the amplitude of the drive pulse and detuning) yields a large output signal and still ensures the required resolution in the measurement provided the product k 2 Q β L > 1 is sufficiently large. (At larger amplitudes φ a , the circuit operates in a non-linear regime probing an averaged reverse inductance of the **qubit** whose value, as well as the produced **frequency** shift δ ω 0 , is smaller .) As points A and B lie on the axis Q 0 = 0 and are both characterized by a sufficiently long relaxation time, reading-out of the **qubit** state with the rf **oscillation** span ± π / 2 is preferable in either point. In the case of operation point C , the limited amplitude of the **oscillations** does not reduce much the relaxation time either. Significant reduction of the relaxation time occurs in the vicinity of point D . Due to this property which is due to the dependence of the transversal coupling strength on φ , Eqs. ( I-2)-( tan-n), the measurement of the Quantronium state using a switching current technique was possible in the middle of segment C D (see Fig. 2), where the maximum values of the circulating current in the excited and ground states were of different sign ....The two lowest energy levels E n q φ , i.e., n = 0 and 1 (see their dependencies on q and φ in Fig. Eigenenergies), form the basis | 0 | 1 suitable for **qubit** operation. In this basis, the Hamiltonian ( H0) is diagonal, ... The charge-phase Josephson **qubit** based on a superconducting single charge transistor inserted in a low-inductance superconducting loop is considered. The loop is inductively coupled to a radio-**frequency** driven tank circuit enabling the readout of the **qubit** states by measuring the effective Josephson inductance of the transistor. The effect of **qubit** dephasing and relaxation due to electric and magnetic control lines as well as the measuring system is evaluated. Recommendations for operation of the **qubit** in magic points producing minimum decoherence are given.

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