### 56121 results for qubit oscillator frequency

Contributors: Hua, Ming, Deng, Fu-Guo

Date: 2013-09-30

(color online) (a) The probability distribution of the two quantum Rabi **oscillations** ROT 0 (the blue solid line) and ROT 1 (the green solid line) of two charge **qubits** coupled to a resonator. (b) The probability distribution of the four quantum Rabi **oscillations** in our cc-phase gate on a three-charge-**qubit** system. Here, the blue-solid, green-solid, red-dashed, and Cambridge-blue-dot-dashed lines represent the quantum Rabi **oscillations** ROT 00 ( | 0 1 | 0 2 | 1 3 | 0 a ↔ | 0 1 | 0 2 | 0 3 | 1 a ), ROT 01 ( | 0 1 | 1 2 | 1 3 | 0 a ↔ | 0 1 | 1 2 | 0 3 | 1 a ), ROT 10 ( | 1 1 | 0 2 | 1 3 | 0 a ↔ | 1 1 | 0 2 | 0 3 | 1 a ), and ROT 11 ( | 1 1 | 1 2 | 1 3 | 0 a ↔ | 1 1 | 1 2 | 0 3 | 1 a ), respectively....The quantum entangling operation based on the SR can also help us to complete a single-step controlled-controlled phase (cc-phase) quantum gate on the three charge **qubits** q 1 , q 2 , and q 3 by using the system shown in Fig. fig2 except for the resonator R b . Here, q 1 and q 2 act as the control **qubits**, and q 3 is the target **qubit**. The initial state of this system is prepared as | Φ 0 = 1 2 2 | 0 1 | 0 2 | 0 3 + | 0 1 | 0 2 | 1 3 + | 0 1 | 1 2 | 0 3 + | 0 1 | 1 2 | 1 3 + | 1 1 | 0 2 | 0 3 + | 1 1 | 0 2 | 1 3 + | 1 1 | 1 2 | 0 3 + | 1 1 | 1 2 | 1 3 | 0 a . In this system, both q 1 and q 2 are in the quasi-dispersive regime with R a , and the transition **frequency** of q 3 is adjusted to be equivalent to that of R a when q 1 and q 2 are in their ground states. The QSD transition **frequency** on R a becomes...(color online) Sketch of a coplanar geometry for the circuit QED with three superconducting **qubits**. **Qubits** are placed around the maxima of the electrical field amplitude of R a and R b (not drawn in this figure), and the distance between them is large enough so that there is no direct interaction between them. The fundamental **frequencies** of resonators are ω r j / 2 π ( j = a , b ), the **frequencies** of the **qubits** are ω q i / 2 π ( i = 1 , 2 , 3 ), and they are capacitively coupled to the resonators. The coupling strengths between them are g i j / 2 π . We can use the control line (not drawn here) to afford the flux to tune the transition **frequencies** of the **qubits**....(color online) (a) The **qubit**-state-dependent resonator transition, which means the **frequency** shift of the resonator transition δ r arises from the state ( | 0 q or | 1 q ) of the **qubit**. (b) The number-state-dependent **qubit** transition, which means the **frequency** shift δ q takes place on the **qubit** due to the photon number n = 1 or 0 in the resonator in the dispersive regime....in which we neglect the direct interaction between the two **qubits** (i.e., q 1 and q 2 ), shown in Fig. fig2. Here σ i + = | 1 i 0 | is the creation operator of q i ( i = 1 , 2 ). g i is the coupling strength between q i and R a . The parameters are chosen to make q 1 interact with R a in the quasi-dispersive regime. That is, the transition **frequency** of R a is determined by the state of q 1 . By taking a proper transition **frequency** of q 2 (which equals to the transition **frequency** of R a when q 1 is in the state | 0 1 ), one can realize the quantum Rabi **oscillation** (ROT) ROT 0 : | 0 1 | 1 2 | 0 a ↔ | 0 1 | 0 2 | 1 a , while ROT 1 : | 1 1 | 1 2 | 0 a ↔ | 1 1 | 0 2 | 1 a occurs with a small probability as q 2 detunes with R a when q 1 is in the state | 1 1 . Here the Fock state | n a represents the photon number n in R a ( n = 0 , 1 ). | 0 i and | 1 i are the ground and the first excited states of q i , respectively....Eq.( stark) means the NSD **qubit** transition and Eq.( kerr) means the QSD resonator transition, shown in Fig. fig1(a) and (b), respectively....(color online) Simulated outcomes for the maximum amplitude value of the expectation about the quantum Rabi **oscillation** varying with the coupling strength g 2 and the **frequency** of the second **qubit** ω 2 . (a) The outcomes for ROT 0 : | 0 1 | 1 2 | 0 a ↔ | 0 1 | 0 2 | 1 a . (b) The outcomes for ROT 1 : | 1 1 | 1 2 | 0 a ↔ | 1 1 | 0 2 | 1 a . Here the parameters of the resonator and the first **qubit** q 1 are taken as ω a / 2 π = 6.0 GHz, ω q 1 / 2 π = 7.0 GHz, and g 1 / 2 π = 0.2 GHz....and it is shown in Fig. fig4 (a). In the simulation of our SR, we choose the reasonable parameters by considering the energy level structure of a charge **qubit**, according to Ref. . Here ω r a / 2 π = 6.0 GHz. The transition **frequency** of two **qubits** between and are chosen as ω 0 , 1 ; 1 / 2 π = E 1 ; 1 - E 0 ; 1 = 5.0 GHz, ω 1 , 2 ; 1 / 2 π = E 2 ; 1 - E 1 ; 1 = 6.2 GHz, ω 0 , 1 ; 2 / 2 π = E 1 ; 2 - E 0 ; 2 = 6.035 GHz, and ω 1 , 2 ; 2 / 2 π = E 2 ; 2 - E 1 ; 2 = 7.335 GHz. Here E i ; q is the energy for the level i of the **qubit** q , and σ i , i ' ; q + ≡ i q i ' . g i , j ; q is the coupling strength between the resonator R a and the **qubit** q in the transition between the energy levels | i q and | j q ( i = 0 , 1 , j = 1 , 2 , and q = 1 , 2 ). For convenience, we take the coupling strengths as g 0 , 1 ; 1 / 2 π = g 1 , 2 ; 1 / 2 π = 0.2 GHz and g 0 , 1 ; 2 / 2 π = g 1 , 2 ; 2 / 2 π = 0.0488 GHz....We numerically simulate the maximal expectation values (MAEVs) of ROT 0 and ROT 1 based on the Hamiltonian H 2 q , shown in Fig. fig3(a) and (b), respectively. Here, the expectation value is defined as | ψ | e - i H 2 q t / ℏ | ψ 0 | 2 . | ψ 0 and | ψ are the initial and the final states of a quantum Rabi **oscillation**, respectively. The MAEV s vary with the transition **frequency** ω 2 and the coupling strength g 2 . It is obvious that the amplified QSD resonator transition can generate a selective resonance (SR) when the coupling strength g 2 is small enough....We present a fast quantum entangling operation on superconducting **qubits** assisted by a resonator in the quasi-dispersive regime with a new effect --- the selective resonance coming from the amplified **qubit**-state-dependent resonator transition **frequency** and the tunable period relation between a wanted quantum Rabi **oscillation** and an unwanted one. This operation does not require any kind of drive fields and the interaction between **qubits**. More interesting, the non-computational third excitation states of the charge **qubits** can play an important role in shortening largely the operation time of the entangling gates. All those features provide an effective way to realize much faster quantum entangling gates on superconducting **qubits** than previous proposals. ... We present a fast quantum entangling operation on superconducting **qubits** assisted by a resonator in the quasi-dispersive regime with a new effect --- the selective resonance coming from the amplified **qubit**-state-dependent resonator transition **frequency** and the tunable period relation between a wanted quantum Rabi **oscillation** and an unwanted one. This operation does not require any kind of drive fields and the interaction between **qubits**. More interesting, the non-computational third excitation states of the charge **qubits** can play an important role in shortening largely the operation time of the entangling gates. All those features provide an effective way to realize much faster quantum entangling gates on superconducting **qubits** than previous proposals.

Data types:

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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### Temperature square dependence of the low **frequency** 1/f charge noise in the Josephson junction **qubits**

Contributors: Astafiev, O., Pashkin, Yu. A., Nakamura, Y., Yamamoto, T., Tsai, J. S.

Date: 2006-04-04

fig:1fFig2 (a) Solid dots show temperature dependence of α 1 / 2 with a fixed bias current (the bias voltage is adjusted to keep the current constant). Open dots show α 1 / 2 derived from the measurement of **qubit** dephasing during coherent **oscillations**. The coherent **oscillations** (solid line) as well as the envelope exp - t 2 / 2 T 2 * 2 with T 2 * = 180 ps (dashed line) are in the inset. (b) Solid dots show temperature dependence of α 1 / 2 for the SET on GaAs substrate....Note that at a fixed bias voltage the average current through the SET increases with temperature (see Fig. fig:1fFig1(a)). However, it has almost no effect on the noise as we confirmed from the measurement of the current noise dependence. Nevertheless, to avoid possible contribution from the current dependent noise we adjust the bias voltage in the next measurements so that the average current is kept nearly constant at the measurement points for different temperatures. Fig. fig:1fFig2(a) shows the temperature dependences of α 1 / 2 for a different sample with a similar geometry taken in the **frequency** range from 0.1 Hz to 10 Hz with a bias current adjusted to about I = 12 ± 2 pA. The straight line in the plot is α 1 / 2 = η 1 / 2 T , which corresponds to T 2 -dependence of α with η ≈ ( 1.3 × 10 -2 e / K ) 2 ....Solid dots in Fig. fig:1fFig1(c) represent α 1 / 2 as a function of temperature. α 1 / 2 saturates at temperatures below 200 mK at the level of 2 × 10 -3 e and exhibits nearly linear rise at temperatures above 200 mK with α 1 / 2 ≈ η 1 / 2 T , where η ≈ 1.0 × 10 -2 e / K 2 (the solid line in Fig. fig:1fFig1(c)). T 2 dependence of α is observed in many samples, though sometimes the noise is not exactly 1 / f , having a bump from the Lorentzian spectrum of a strongly coupled low **frequency** fluctuator. In such cases, switches from the single two-level fluctuator are seen in time traces of the current ....To verify the hypothesis about the common origin of the low **frequency** 1/f noise and the quantum f noise recently measured in the Josephson charge **qubits**, we study temperature dependence of the 1/f noise and decay of coherent **oscillations**. T^2 dependence of the 1/f noise is experimentally demonstrated, which supports the hypothesis. We also show that dephasing in the Josephson charge **qubits** off the electrostatic energy degeneracy point is consistently explained by the same low **frequency** 1/f noise that is observed in the transport measurements....The typical current **oscillation** as a function of t away from the degeneracy point ( θ ≠ π / 2 ) is exemplified in the inset of Fig. fig:1fFig2(a). If dephasing is induced by the Gaussian noise, the **oscillations** decay as exp - t 2 / 2 T 2 * 2 with...We use the **qubit** as an SET and measure the low **frequency** charge noise, which causes the SET peak position fluctuations. Temperature dependence of the noise is measured from the base temperature of 50 mK up to 900 - 1000 mK. The SET is normally biased to V b = 4 Δ / e ( ∼ 1 mV), where Coulomb **oscillations** of the quasiparticle current are observed. Figure fig:1fFig1(a) exemplifies the position of the SET Coulomb peak as a function of the gate voltage at temperatures from 50 mK up to 900 mK with an increment of 50 mK. The current noise spectral density is measured at the gate voltage corresponding to the slope of the SET peak (shown by the arrow), at the maximum (on the top of the peak) and at the minimum (in the Coulomb blockade). Normally, the noise spectra in the two latter cases are **frequency** independent in the measured **frequency** range (and usually do not exceed the noise of the measurement setup). However, the noise spectra taken on the slope of the peak show nearly 1 / f **frequency** dependence (see examples of the current noise S I at different temperatures in Fig. fig:1fFig1(b)) saturating at a higher **frequencies** (usually above 10 - 100 Hz depending on the device properties) at the level of the noise of the measurement circuit. The fact that the measured 1 / f noise on the slope is substantially higher than the noises on the top of the peak and in the blockade regime indicates that the noise comes from fluctuations of the peak position, which can be translated into charge fluctuations in the SET....The solid line in the inset of Fig. fig:1fFig2(a) shows decay of coherent **oscillations** measured at T = 50 mK and the dashed envelope exemplifies a Gaussian with T 2 * = 180 ps. We derive α 1 / 2 from Eq. ( eq:Eq3) and plot it in Fig. fig:1fFig2(a) by open dots as a function of temperature. The low **frequency** integration limit and the high **frequency** cutoff are taken to be ω 0 ≈ 2 π × 25 Hz and ω 1 ≈ 2 π × 5 GHz for our measurement time constant τ = 0.02 s and typical dephasing time T 2 * ≈ 100 ps . ... To verify the hypothesis about the common origin of the low **frequency** 1/f noise and the quantum f noise recently measured in the Josephson charge **qubits**, we study temperature dependence of the 1/f noise and decay of coherent **oscillations**. T^2 dependence of the 1/f noise is experimentally demonstrated, which supports the hypothesis. We also show that dephasing in the Josephson charge **qubits** off the electrostatic energy degeneracy point is consistently explained by the same low **frequency** 1/f noise that is observed in the transport measurements.

Data types:

Contributors: Eugene Grichuk, Margarita Kuzmina, Eduard Manykin

Date: 2010-09-26

A network of coupled stochastic **oscillators** is
proposed for modeling of a cluster of entangled **qubits** that is
exploited as a computation resource in one-way quantum
computation schemes. A **qubit** model has been designed as a
stochastic **oscillator** formed by a pair of coupled limit cycle
**oscillators** with chaotically modulated limit cycle radii and
**frequencies**. The **qubit** simulates the behavior of electric field of
polarized light beam and adequately imitates the states of two-level
quantum system. A cluster of entangled **qubits** can be associated
with a beam of polarized light, light polarization degree being
directly related to cluster entanglement degree. Oscillatory network,
imitating **qubit** cluster, is designed, and system of equations for
network dynamics has been written. The constructions of one-**qubit**
gates are suggested. Changing of cluster entanglement degree caused
by measurements can be exactly calculated....network of stochastic **oscillators** ... A network of coupled stochastic **oscillators** is
proposed for modeling of a cluster of entangled **qubits** that is
exploited as a computation resource in one-way quantum
computation schemes. A **qubit** model has been designed as a
stochastic **oscillator** formed by a pair of coupled limit cycle
**oscillators** with chaotically modulated limit cycle radii and
**frequencies**. The **qubit** simulates the behavior of electric field of
polarized light beam and adequately imitates the states of two-level
quantum system. A cluster of entangled **qubits** can be associated
with a beam of polarized light, light polarization degree being
directly related to cluster entanglement degree. Oscillatory network,
imitating **qubit** cluster, is designed, and system of equations for
network dynamics has been written. The constructions of one-**qubit**
gates are suggested. Changing of cluster entanglement degree caused
by measurements can be exactly calculated.

Data types:

Contributors: Shevchenko, S. N., Omelyanchouk, A. N., Zagoskin, A. M., Savel'ev, S., Nori, F.

Date: 2007-12-12

which determines the thermally activated escape probability from the local minimum of the potential in Eq. ( eq_U). The classical Rabi-like **oscillations** are displayed in Fig. Figure1. In Fig. Figure1(a) the modulated transient **oscillations** of the phase difference φ are plotted. These **oscillations** result in the **oscillating** behaviour of the energy of the system as shown in Fig. Figure1(b). Averaging over fast **oscillations**, we plot in Fig. Figure1(c) with green solid curve the damped **oscillations** of the energy, analogous to the quantum Rabi **oscillations** . These curves are plotted for the following set of the parameters: η = 0.95 , α = 10 -3 , ϵ = 10 -3 , and γ = γ 0 . For comparison we also plotted the energy averaged over the fast **oscillations** for different parameters, changing one of these parameters and leaving the others the same. The dashed black curve in Fig. Figure1(c) is for the smaller damping, α = 10 -4 ; the solid (violet) line and the dash-dotted (black) line in Fig. Figure1(d) demonstrate the change in the **frequency** and the amplitude of the **oscillations** respectively for ϵ = 2 ⋅ 10 -3 and η = 0.9 . We notice that the effect analogous to the classical Rabi **oscillations** exist in a wide range of parameters....(Color online). The time-averaged probability P ¯ of the upper level to be occupied versus the driving **frequency**. The parameters used here are: η = 0.95 , E J / ℏ ω p = 300 , Γ r e l a x / ℏ ω p = Γ φ / ℏ ω p = 3 ⋅ 10 -4 . Numbers next to the curves stand for ϵ multiplied by 10 3 . Upper inset: the time dependence of the probability P τ . Right inset: the shift of the principal resonance (at ℏ ω ≈ Δ E ), where Δ ω = ω - Δ E / ℏ ....Rabi **oscillations** are coherent transitions in a quantum two-level system under the influence of a resonant perturbation, with a much lower **frequency** dependent on the perturbation amplitude. These serve as one of the signatures of quantum coherent evolution in mesoscopic systems. It was shown recently [N. Gronbech-Jensen and M. Cirillo, Phys. Rev. Lett. 95, 067001 (2005)] that in phase **qubits** (current-biased Josephson junctions) this effect can be mimicked by classical **oscillations** arising due to the anharmonicity of the effective potential. Nevertheless, we find qualitative differences between the classical and quantum effect. First, while the quantum Rabi **oscillations** can be produced by the subharmonics of the resonant **frequency** (multiphoton processes), the classical effect also exists when the system is excited at the overtones. Second, the shape of the resonance is, in the classical case, characteristically asymmetric; while quantum resonances are described by symmetric Lorentzians. Third, the anharmonicity of the potential results in the negative shift of the resonant **frequency** in the classical case, in contrast to the positive Bloch-Siegert shift in the quantum case. We show that in the relevant range of parameters these features allow to confidently distinguish the bona fide Rabi **oscillations** from their classical Doppelganger....Superconducting phase **qubits** provide a clear demonstration of quantum coherent behaviour in macroscopic systems. They also have a very simple design: a phase **qubit** is a current-biased Josephson junction (see Fig. scheme(a)), and its working states | 0 , | 1 are the two lowest metastable energy levels E 0 , 1 in a local minimum of the washboard potential. The transitions between these levels are produced by applying an RF signal at a resonant **frequency** ω 10 = E 1 - E 0 / ℏ ≡ Δ E / ℏ . The readout utilizes the fact that the decay of a metastable state of the system produces an observable reaction: a voltage spike in the junction or a flux change in a coupled dc SQUID. In the three-level readout scheme (Fig. scheme(b)) both | 0 and | 1 have negligible decay rates. A pulse at a **frequency** ω 21 = E 2 - E 1 / ℏ transfers the probability amplitude from the state | 1 to the fast-decaying state | 2 . Its decay corresponds to a single-shot measurement of the **qubit** in state | 1 . Alternatively, instead of an RF readout pulse one can apply a dc pulse, which increases the decay rate of | 1 ....(Color online). Rabi-type **oscillations** in current-biased Josephson junctions: (a) and (b) show the time-dependence of the phase difference φ and of the energy H , (c) presents the time-dependence of the energy, averaged over the fast **oscillations** with period 2 π / ω . All energies are shifted by their stationary value: H 0 = 1 - 1 - η 2 - η arcsin η . The parameters for the blue, red, and green curves are: η = 0.95 , α = 10 -3 , ϵ = 10 -3 , and γ = γ 0 ; for the other curves in (c) and (d) only one parameter was different from the above, for comparison. Namely: (c) dashed black line α = 10 -4 , (d) solid violet line ϵ = 2 ⋅ 10 -3 , dash-dotted black line η = 0.9 ....(Color online). The time-averaged energy H ¯ - H 0 versus reduced **frequency** for relatively weak (a) and strong (b) driving. Different values of the driving amplitude ϵ (multiplied by 10 3 ) are shown by the numbers next to the curves. The parameters are: η = 0.95 and α = 10 -4 . In (b) the region between the vertical black lines corresponds to the escape from the phase-locked state....When the system is driven close to resonance, ω ≈ Δ E , the upper level occupation probability P τ exhibits Rabi **oscillations**. The damped Rabi **oscillations** are demonstrated in the upper inset in Fig. Figure3, which is analogous to the classical **oscillations** presented in Fig. Figure1. After averaging the time dependent probability, we plot it versus **frequency** in Fig. Figure3 for two values of the amplitude, demonstrating the multiphoton resonances. Figure Figure3 demonstrates the following features of the multiphoton resonances in the quantum case: (a) in contrast to the classical case, the resonances appear only at the subharmonics, at ℏ ω ≈ Δ E / n ; (b) the resonances have Lorentzian shapes (as opposed to the classical asymmetric resonances); (c) with increasing the driving amplitude the resonances shift to the higher **frequencies** – the Bloch-Siegert shift, which has the opposite sign from its classical counterpart. The Bloch-Siegert shift (the shift of the principal resonance at ℏ ω ≈ Δ E ) is plotted numerically in the right inset in Fig. Figure3. Analogous shifts of the positions of the resonances were recently observed experimentally ....In Fig. Figure2 the effect of the driving current on the time-averaged energy of the system is shown for different driving amplitudes: Fig. Figure2(a) for weaker amplitudes, close to the main resonance, to show the asymmetry and negative shift of the resonance; and Fig. Figure2(b) for stronger amplitudes, to show the resonances at γ 0 / 2 and 2 γ 0 (which are also shown closer in the insets). We note that the parametric-type resonance at 2 γ 0 originates from the third-order terms when the solution of the equation for ψ is sought by iterations ; when there are two or more terms responsible for this resonance, the respective resonance may become splitted, which is visible in Fig. Figure2(b) for the lowest curve. An analogous tiny splitting of the resonance was obtained for the driven flux **qubit** in Fig. 4 of Ref. [...Phase **qubit** (a) and its Josephson energy (b). The metastable states and can be used as **qubit** states. ... Rabi **oscillations** are coherent transitions in a quantum two-level system under the influence of a resonant perturbation, with a much lower **frequency** dependent on the perturbation amplitude. These serve as one of the signatures of quantum coherent evolution in mesoscopic systems. It was shown recently [N. Gronbech-Jensen and M. Cirillo, Phys. Rev. Lett. 95, 067001 (2005)] that in phase **qubits** (current-biased Josephson junctions) this effect can be mimicked by classical **oscillations** arising due to the anharmonicity of the effective potential. Nevertheless, we find qualitative differences between the classical and quantum effect. First, while the quantum Rabi **oscillations** can be produced by the subharmonics of the resonant **frequency** (multiphoton processes), the classical effect also exists when the system is excited at the overtones. Second, the shape of the resonance is, in the classical case, characteristically asymmetric; while quantum resonances are described by symmetric Lorentzians. Third, the anharmonicity of the potential results in the negative shift of the resonant **frequency** in the classical case, in contrast to the positive Bloch-Siegert shift in the quantum case. We show that in the relevant range of parameters these features allow to confidently distinguish the bona fide Rabi **oscillations** from their classical Doppelganger.

Data types:

Contributors: Huang, Ren-Shou, Dobrovitski, Viatcheslav, Harmon, Bruce

Date: 2005-04-18

Recent experiments on Josephson junction **qubits** have suggested the existence in the tunnel barrier of bistable two level fluctuators that are responsible for decoherence and 1/f critical current noise. In this article we treat these two-level systems as fictitious spins and investigate their influence quantum mechanically with both analytical and numerical means. We find that the Rabi **oscillations** of the **qubit** exhibit multiple stages of decay. New approaches are established to characterize different decoherence times and to allow for easier feature extraction from experimental data. The Rabi **oscillation** of a **qubit** coupled to a spurious resonator is also studied, where we proposed an idea to explain the serious deterioration of the Rabi osillation amplitude....Fig. ( fig:spurious) shows the comparison of the numerical result of a **qubit** dephased by 14 spins with and without a spurious resonator, which is also dephased by the same group of spins. We can see that the Rabi **oscillation** becomes somewhat irregular with reduced amplitude while the long-time slow decay still persists without the sign of fading away. This explains the experimental observation that decoherence time is not reduced by the coupling to the spurious resonator is because the time has already passed T φ and the Rabi **oscillations** have entered the featureless slow decay regime. Notice that the irregularity starts to appear only after t ∼ 2 π / g , which suggests the cause is due to the smearing of the beats....Numerical simulation for the Rabi **oscillation** of one **qubit** coupled with 14 spins at T = 200 mK and T = 10 mK. The left column are the real time Rabi **oscillations**, and the right column are their Fourier transforms. The dotted lines in the right column are the exact Fourier transform obtained in Eq. ( eq:FTsz) for the limit of infinite number of spins. T φ , calculated using Eq. ( eq:t_phi), are 84ns and 165ns respectively. Notice that the small bumps in the **oscillation** envelope in the T = 10 mK are due to the effects of finite spins and some spins are frozen....Time evolution of the **qubit** decohered by many spins undergoing Rabi **oscillation** by a coherent microwave source, σ z t . The Rabi **frequency** 2 α / 2 π = 200 MHz, and δ Ω is generated from a special case where all spin energy splittings are chosen for simplification to be ω k / 2 π = 1 GHz and ∑ k A k 2 / 2 π = 50 MHz at the temperature T = 150 mK. For Ω / 2 π = 10 GHz, these parameters correspond to δ Ω / Ω 0.005 . Notice that the amplitude of the **oscillation** envelope is already down to 50 % at t ∼ 20 ns, but it reaches 25 % only after t = 80 ns, while an exponential decay should reach 25 % around t ∼ 40 ns....Now the width of the peak ω ' = 0 , i.e. ω = 2 α , is controlled by the Gaussian function with width 2 δ Ω , as shown in Fig. ( fig:sz_w). When δ Ω → 0 , the spectrum becomes a delta function at the Rabi **frequency** ω = 2 α . At high **frequency** the Fourier spectrum is dominated by the Gaussian term, which means that σ z t has a Gaussian decay in the short time limit. The singularity at the Rabi **frequency** ω = 2 α implies a very slow decay in the long time limit. The result agrees with the that obtained for Eq. ( eq:sz) in the limit when α ≫ δ Ω , the **oscillation** envelope of σ z t is given by σ z 0 1 + 2 t δ Ω 2 / α 2 - 1 4 , where the evolution begins with a Gaussian (quadratic) damping then changes to a slow power law decay of ∼ α / δ Ω 2 t ....Simulation of the spectral probing of resonance on the **qubit**-spurious resonator system. In both graphs, the vertical axes are the value of σ z at the steady state, and the horizontal axes are the driving microwave **frequencies**. The spurious resonator here has an energy splitting at 10 GHz. The left graph shows when the **qubit** energy is detuned from the resonator at 9.95 GHz, the most visible peak is the **qubit** and the resonator peak is barely visible. When the **qubit** energy is tuned close to the resonator **frequency**, as shown in the right graph, level repulsion takes place. Notice that the spectral peaks here are clearly seperated even though δ Ω and δ Ω r e s are both greater than g ....Rabi **oscillations** of a **qubit** dephased and relaxed by a many-spin system. The parameters are the same as those used in the T = 10 mK graph in Fig. ( fig:tdep). The only additional parameter, the relaxation time T 1 , calculated from Fermi’s golden rule is 1 μ s. The solid lines are the numerical results, and the dashed lines are the approximations in Eq. ( eq:relax_fit) and Eq. ( eq:relax_ft_fit)....which is the ideal Rabi **oscillation**, because every spin is frozen to its own ground state. But as soon as we turn up the temperature, when the value B is allowed to fluctuate, the **oscillation** now has a decay pattern(see in Fig. ( fig:sz)). Notice that the **oscillation** lasts much longer than an ordinary exponential decay but has a rather drastic decrease of amplitude in the beginning....The simulation of both dephasing and relaxation present can be done easily in our program. The newly added spins remain non-interactive among themselves. Fig. ( fig:relax) shows a simulation result with the relaxation time T 1 ≫ T φ . In the real time evolution graph we can clearly see the three stages of the decay process, which starts with a fast Gaussian decay followed by slow decay, and then later the exponential decay finally takes over. Similar behavior has also been observed in the result produced by a **qubit** under the direct influence of 1/f noise. In the graph of the Fourier transform of the same data, the original sharp peak in the Rabi **frequency** is now smeared. Since this problem cannot be solved analytically, we made the approximation of multiplying the **oscillating** part of Eq. ( eq:sz) with an exponential factor e - t / 2 T 1 , so that it becomes...Effect on Rabi **oscillations** caused by a dephased spurious resonator. The solid line represents the case where the **qubit** is coupled to a spurious resonator, and the dotted line is not. The parameters for both are 2 α / 2 π = 100 MHz and δ Ω / 2 π = 145 MHz. Those for the coupled resonator are g / 2 π = 15 MHz and δ Ω r e s / 2 π = 30 MHz. Both **qubit** and spurious resonator couple to the same 14 spins. Notice that because δ Ω > h , the initial reduction of amplitude ends even before one period of Rabi **oscillation**....Fourier spectrum σ ~ z ω near the Rabi **frequency** ω = 2 α . We can use the width of the peak to estimate the decoherence time of the Rabi **oscillation**. δ Ω > and δ Ω **frequency** ω ≫ 2 α + 2 δ Ω the tail of the peak is dominated by the Gaussian term, therefore we can expect a Gaussian decay of σ z t when t ≪ ℏ / δ Ω . While the **frequency** ω is very near 2 α , the peak goes like ω - 2 α -1 / 2 , which implies a very slow decay at the long time limit. ... Recent experiments on Josephson junction **qubits** have suggested the existence in the tunnel barrier of bistable two level fluctuators that are responsible for decoherence and 1/f critical current noise. In this article we treat these two-level systems as fictitious spins and investigate their influence quantum mechanically with both analytical and numerical means. We find that the Rabi **oscillations** of the **qubit** exhibit multiple stages of decay. New approaches are established to characterize different decoherence times and to allow for easier feature extraction from experimental data. The Rabi **oscillation** of a **qubit** coupled to a spurious resonator is also studied, where we proposed an idea to explain the serious deterioration of the Rabi osillation amplitude.

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Contributors: Simmonds, R. W., Lang, K. M., Hite, D. A., Pappas, D. P., Martinis, John M.

Date: 2004-02-18

(a) Circuit diagram for the Josephson junction **qubit**. Junction current bias I is set by I φ and microwave source I μ w . Parameters are I 0 ≃ 11.659 μ A , C ≃ 1.2 p F , L ≃ 168 p H , and L / M ≃ 81 . (b) Potential energy diagram of **qubit**, showing **qubit** states and in cubic well at left. Measurement of state performed by driving the 1 → 3 transition, tunneling to right well, then relaxation of state to bottom of right well. Post-measurement classical states 0 and 1 differ in flux by Φ 0 , which is readily measured by readout SQUID. (c) Schematic description of tunnel-barrier states A and B in a symmetric well. Tunneling between states produces ground and excited states separated in energy by ℏ ω r . (d) Energy-level diagram for coupled **qubit** and resonant states for ω 10 ≃ ω r . Coupling strength between states and is given by H ˜ i n t ....(a) Measured probability of state 1 versus microwave excitation **frequency** ω / 2 π and bias current I for a fixed microwave power. Data indicate ω 10 transition **frequency**. Dotted vertical lines are centered at spurious resonances. (b) Measured occupation probability of versus Rabi-pulse time t r and bias current I . In panel (b), a color change from dark blue to red corresponds to a probability change of 0.4. Color modulation in time t r (vertical direction) indicates Rabi **oscillations**. ...superconductors, **qubits**, Josephson junction, decoherence...(a)-(c) Measured occupation probability of versus time duration of Rabi pulse t r for three values of microwave power, taken at bias I = 11.609 μ A in Fig. 2. The applied microwave power for (a), (b), and (c) correspond to 0.1 , 0.33 , and 1.1 m W , respectively. (d) Plot of Rabi **oscillation** **frequency** versus microwave amplitude. A linear dependence is observed, as expected from theory. ...Although Josephson junction **qubits** show great promise for quantum computing, the origin of dominant decoherence mechanisms remains unknown. We report Rabi **oscillations** for an improved phase **qubit**, and show that their coherence amplitude is significantly degraded by spurious microwave resonators. These resonators arise from changes in the junction critical current produced by two-level states in the tunnel barrier. The discovery of these high **frequency** resonators impacts the future of all Josephson **qubits** as well as existing Josephson technologies. We predict that removing or reducing these resonators through materials research will improve the coherence of all Josephson **qubits**. ... Although Josephson junction **qubits** show great promise for quantum computing, the origin of dominant decoherence mechanisms remains unknown. We report Rabi **oscillations** for an improved phase **qubit**, and show that their coherence amplitude is significantly degraded by spurious microwave resonators. These resonators arise from changes in the junction critical current produced by two-level states in the tunnel barrier. The discovery of these high **frequency** resonators impacts the future of all Josephson **qubits** as well as existing Josephson technologies. We predict that removing or reducing these resonators through materials research will improve the coherence of all Josephson **qubits**.

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

Date: 2004-02-27

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

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

Date: 2012-11-09

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

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Contributors: Zueco, David, Reuther, Georg M., Kohler, Sigmund, Hänggi, Peter

Date: 2009-07-20

**Oscillator** **frequency** shift as function or the **qubit** splitting ϵ = ω + Δ for the spin state | ↓ obtained (a) within RWA, Eq. shiftRWA, and (b) beyond RWA, Eq. wrnonRWA. The lines mark the analytical results, while the symbols refer to the numerically obtained splitting between the ground state and the first excited state in the subspace of the **qubit** state | ↓ ....We generalize the dispersive theory of the Jaynes-Cummings model beyond the frequently employed rotating-wave approximation (RWA) in the coupling between the two-level system and the resonator. For a detuning sufficiently larger than the **qubit**-**oscillator** coupling, we diagonalize the non-RWA Hamiltonian and discuss the differences to the known RWA results. Our results extend the regime in which dispersive **qubit** readout is possible. If several **qubits** are coupled to one resonator, an effective **qubit**-**qubit** interaction of Ising type emerges, whereas RWA leads to isotropic interaction. This impacts on the entanglement characteristics of the **qubits**....HRabi in the subspace of the **qubit** state | ↓ , where σ z | ↓ = - | ↓ . The results are depicted in Fig. fig:wr. ... We generalize the dispersive theory of the Jaynes-Cummings model beyond the frequently employed rotating-wave approximation (RWA) in the coupling between the two-level system and the resonator. For a detuning sufficiently larger than the **qubit**-**oscillator** coupling, we diagonalize the non-RWA Hamiltonian and discuss the differences to the known RWA results. Our results extend the regime in which dispersive **qubit** readout is possible. If several **qubits** are coupled to one resonator, an effective **qubit**-**qubit** interaction of Ising type emerges, whereas RWA leads to isotropic interaction. This impacts on the entanglement characteristics of the **qubits**.

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