### 20481 results for qubit oscillator frequency

Contributors: Griffith, E. J., Ralph, J. F., Greentree, Andrew D., Clark, T. D.

Date: 2005-10-04

In the example figure (Fig. fig:qubosc1d2d), the control bias is varied from left to right for a low **frequency** **oscillator** circuit (1.36GHz). For each bias point the simulation is reinitialised, the stochastic time evolution of the system density matrix is simulated over 1500 **oscillator** cycles. Then the **oscillator** and **qubit** charge expectation values are extracted to obtain the power spectrum for each component, with a **frequency** resolution of 4.01MHz. The power spectra for each time series are collated as an image such that the power axis is now represented as a colour, and the individual power spectra are vertical ‘slices’ through the image. The dominant **frequency** peaks become line traces, therefore illustrating the various avoided crossings, mergeings and intersections. The example figure shows the PSD ‘slice’ at Bias = 0.5187 , the broadband noise is readily apparent and is due to the discontinuous quantum jumps in the **qubit**. The bias **oscillator** peak (1.36GHz) is most prominent in the **oscillator** PSD, as would be expected, but it is also present in the **qubit** PSD. It should also be noted that most features are present in both the **qubit** and **oscillator**, including the noise which is generated by the quantum jumps and the quantum state diffusion processes. Interestingly, the **qubit** PSD is significantly stronger than the **oscillator** PSD, however, a larger voltage is generated by the smaller charge due to the extremely small island capacitance, V q = q / C q ....fig:mwRamp (Color online) **Oscillator** PSD as a function of the applied microwave drive **frequency** f m w , for microwave amplitudes A m w = 0.0050 (A) and A m w = 0.0100 (B). It is important to notice that there are now two **frequency** axes per plot, a drive (H) and a response (V). Of particular interest is the magnified section which shows clearly the distinct secondary splitting in the sub-GHz regime. This occurs due to a high **frequency** interaction seen in the upper plots, where the lower Rabi sideband of the microwave drive passes through the high **frequency** **oscillator** signal. The maximum splitting occurs when the Rabi amplitude is a maximum, hence this is observed for a very particular combination of bias and drive, which is beneficial for charactering the **qubit**. Most importantly, this would not be observed with a conventional low **frequency** **oscillator** configuration as the f m w - f o s c separation would be too large for the Rabi **frequency**. ( κ = 5 × 10 -5 )....Fig. fig:mwRamp is presented in a similar manner as Fig. fig:BiasRamp. However there are now two **frequency** axes: the horizontal axis represents the **frequency** of the applied microwave drive field, and the vertical axis is the **frequency** response. It should be remembered that the microwave **frequency** axis is focused near the **qubit** transition **frequency** ( f q u b i t ≈ 3.49GHz) and the diagonally increasing line is now the microwave **frequency**....Autler Townes effect, charge **qubit**, characterisation, **frequency** spectrum...fig:QubitOscEnergy A two level **qubit** is coupled to a many level harmonic **oscillator**, investigated for two different **oscillator** energies. Firstly, the **oscillator** resonant **frequency** is set to 1.36GHz, this more resembles the conventional configuration such that the fundamental component of the **oscillator** does not drive the **qubit**. However, we also investigate the use of a high **frequency** **oscillator** of 3.06GHz which can excite this **qubit**. In addition, **qubit** is constantly driven by a microwave field at 3.49GHz to generate Rabi **oscillations** and in this paper we examine the relation between these three fields....A theoretical spectroscopic analysis of a microwave driven superconducting charge **qubit** (Cooper-pair box coupled) to an RLC **oscillator** model is performed. By treating the **oscillator** as a probe through the backreaction effect of the **qubit** on the **oscillator** circuit, we extract **frequency** splitting features analogous to the Autler-Townes effect from quantum optics, thereby extending the analogies between superconducting and quantum optical phenomenology. These features are found in a **frequency** band that avoids the need for high **frequency** measurement systems and therefore may be of use in **qubit** characterization and coupling schemes. In addition we find this **frequency** band can be adjusted to suit an experimental **frequency** regime by changing the **oscillator** **frequency**....fig:qubosc1d2d (Color online) **Oscillator** and **Qubit** power spectra slices for Bias = 0.5187, using the low **frequency** **oscillator** circuit f o s c = 1.36 GHz. The solid lines overlay the energy level separations found in Fig. fig:EnergyLevel. ( κ = 5 × 10 -5 ). As one would expect, the bias **oscillator** peak at 1.36GHz is clearly observed in the **oscillator** PSD, but only weakly in the **qubit** PSD. Likewise the **qubit** Rabi **frequency** is found to be stronger in the **qubit** PSD. However it is important to note that the **qubit** dynamics such as the Rabi **oscillations** are indeed coupled to the bias **oscillator** circuit and so can be extracted. In addition, it is recommended to compare the layout of the most prominent features with Fig. fig:BiasRamp....fig:BiasRamp (Color online) **Oscillator** PSD as a function of bias, for microwave amplitudes A m w = 0.0025 (A) and A m w = 0.0050 (B). The red lines track the positions (in **frequency**) of significant power spectrum peaks (+10dB to +15dB above background), the overlaid black and blue lines are the **qubit** energy and microwave transition (Fig. fig:EnergyLevel). Unlike Fig. fig:qubosc1d2d, in these figures the 3.06GHz **oscillator** circuit can now drive the **qubit** (Fig. fig:EnergyLevel) and so creates excitations which mix with the microwave driven excitations creating a secondary splitting centred on f m w - f o s c (430MHz). This feature contains the Rabi **frequency** information in the sidebands of the splitting, but now in a different and controllable **frequency** regime. In addition, the intersection of the two differently driven excitations (illustrated in the magnified sections), opens the possibility of calibrating the biased **qubit** against a fixed engineered **oscillator** circuit, using a single point feature. ( κ = 5 × 10 -5 )....In a previous paper , a method was proposed by which the energy level structure of a charge **qubit** can be obtained from measurements of the peak noise in the bias/control **oscillator**, without the need of extra readout devices. This was based on a technique originally proposed for superconducting flux **qubits** but there are many similarities between the two technologies. The **oscillator** noise peak is the result of broadband noise caused by quantum jumps in the **qubit** being coupled back to the **oscillator** circuit. This increase in the jump rate becomes a maximum when the Rabi **oscillations** are at peak amplitude, this should only occur when the **qubit** is correctly biased and the microwave drive is driving at the transition **frequency**. Therefore by monitoring this peak as a function of bias, we can associate a bias position with a microwave **frequency** equal to that of the energy gap, hence constructing the energy diagram (Fig. fig:EnergyLevel)....fig:Jumps (Color online) (A) **Oscillator** power spectra when the coupled **qubit** is driven at f m w = 5.00 GHz. An increase in bias noise power ( f o s c = 1.36 GHz) can be observed when Rabi **oscillations** occur, the more frequent quantum jump noise couples back to the **oscillator**. (B) Bias noise power peak position changes as a function of f m w , the microwave drive **frequency**. Therefore, it is possible to probe the **qubit** energy level structure by using the power increase in the **oscillator** which is already in place, eliminating the need for additional measurement devices. However, it should be noted that the surrounding **oscillator** harmonics may mask the microwave driven peak. ( κ = 1 × 10 -3 ). ... A theoretical spectroscopic analysis of a microwave driven superconducting charge **qubit** (Cooper-pair box coupled) to an RLC **oscillator** model is performed. By treating the **oscillator** as a probe through the backreaction effect of the **qubit** on the **oscillator** circuit, we extract **frequency** splitting features analogous to the Autler-Townes effect from quantum optics, thereby extending the analogies between superconducting and quantum optical phenomenology. These features are found in a **frequency** band that avoids the need for high **frequency** measurement systems and therefore may be of use in **qubit** characterization and coupling schemes. In addition we find this **frequency** band can be adjusted to suit an experimental **frequency** regime by changing the **oscillator** **frequency**.

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Contributors: Kofman, A. G., Zhang, Q., Martinis, J. M., Korotkov, A. N.

Date: 2006-06-02

The first-**qubit** **oscillation** **frequency** f d as a function of time t (normalized by the energy relaxation time T 1 ) for C x = 0 (solid line) and C x = 6 fF (dashed line), assuming N l 1 = 1.355 and parameters of Eq. ( 2.16). Dash-dotted horizontal line, ω r 1 / 2 π = 15.3 GHz, shows the long-time limit of f d t . Two dotted horizontal lines show the plasma **frequency** for the second **qubit**: ω l 2 / 2 π = 10.2 GHz for N l 2 = 10 and ω l 2 / 2 π = 8.91 GHz for N l 2 = 5 . The arrow shows the moment t c of exact resonance in the case N l 2 = 5 ....The circuit schematic of a flux-biased phase **qubit** and the corresponding potential profile (as a function of the phase difference δ across the Josephson junction). During the measurement the state | 1 escapes from the “left” well through the barrier, which is followed by **oscillations** in the “right” well. This dissipative evolution leads to the two-**qubit** crosstalk....The **oscillating** term in Eq. ( 3.11a) describes the beating between the **oscillator** and driving force **frequencies**, with the difference **frequency** increasing in time, d t ~ 2 / d t = α t - t c , and amplitude of beating decreasing as 1 / t ~ (see dashed line in Fig. f4a). Notice that F 0 = 1 / 4 , F ∞ = 1 , and the maximum value is F 1.53 = 1.370 , so that E 0 is the long-time limit of the **oscillator** energy E 2 , while the maximum energy is 1.37 times larger:...The second **qubit** energy E 2 (in units of ℏ ω l 2 ) in the **oscillator** model as a function of time t (in ns) for (a) C x = 5 fF and T 1 = 25 ns and (b) C x = 2.5 fF and 5 fF and T 1 = 500 ns, while N l 2 = 5 . Dashed line in (a) shows approximation using Eq. ( 3.10). The arrows show the moment t c when the driving **frequency** f d (see Fig. f3) is in resonance with ω l 2 / 2 π = 8.91 GHz....mcd05, a short flux pulse applied to the measured **qubit** decreases the barrier between the two wells (see Fig. f0), so that the upper **qubit** level becomes close to the barrier top. In the case when level | 1 is populated, there is a fast population transfer (tunneling) from the left well to the right well. Due to dissipation, the energy in the right well gradually decreases, until it reaches the bottom of the right well. In contrast, if the **qubit** is in state | 0 the tunneling essentially does not occur. The **qubit** state in one of the two potential minima (separated by almost Φ 0 ) is subsequently distinguished by a nearby SQUID, which completes the measurement process....Now let us consider the effect of dissipation in the second **qubit**. ...We analyze the crosstalk error mechanism in measurement of two capacitively coupled superconducting flux-biased phase **qubits**. The damped **oscillations** of the superconducting phase after the measurement of the first **qubit** may significantly excite the second **qubit**, leading to its measurement error. The first **qubit**, which is highly excited after the measurement, is described classically. The second **qubit** is treated both classically and quantum-mechanically. The results of the analysis are used to find the upper limit for the coupling capacitance (thus limiting the **frequency** of two-**qubit** operations) for a given tolerable value of the measurement error probability....Dots: Rabi **frequencies** R k , k - 1 / 2 π for the left-well transitions at t = t c , for N l = 10 , C x = 6 fF, and T 1 = 25 ns. Dashed line shows analytical dependence 1.1 k GHz....2.16 Figure f2 shows the **qubit** potential U δ for N l = 10 (corresponding to φ = 4.842 ), N l = 5 ( φ = 5.089 ), and N l = 1.355 ( φ = 5.308 ); the last value corresponds to the bias during the measurement pulse (see below). The **qubit** levels | 0 and | 1 are, respectively, the ground and the first excited levels in the left well....Solid lines: log-log contour plots for the values of the error (switching) probability P s = 0.01 , 0.1, and 0.3 on the plane of relaxation time T 1 (in ns) and coupling capacitance C x (in fF) in the quantum model for (a) N l 2 = 5 and (b) N l 2 = 10 . The corresponding results for C x , T T 1 in the classical models are shown by the dashed lines (actual potential model) and the dotted lines [**oscillator** model, Eq. ( bound1)]. The numerical data are represented by the points, connected by lines as guides for the eye. The scale at the right corresponds to the operation **frequency** of the two-**qubit** imaginary-swap quantum gate....3.17 in the absence of dissipation in the second **qubit** ( T 1 ' = ∞ ) for N l 2 = 5 and 10, while T 1 = 25 ns. (In this subsection we take into account the mass renormalization m → m ' ' explicitly, even though this does not lead to a noticeable change of results.) A comparison of Figs. f4(a) and f7 shows that in both models the **qubit** energy remains small before a sharp increase in energy. However, there are significant differences due to account of anharmonicity: (a) The sharp energy increase occurs earlier than in the **oscillator** model (the position of short-time energy maximum is shifted approximately from 3 ns to 2 ns); (b) The excitation of the **qubit** may be to a much lower energy than for the **oscillator**; (c) After the sharp increase, the energy occasionally undergoes noticeable upward (as well as downward) jumps, which may overshoot the initial energy maximum; (d) The model now explicitly describes the **qubit** escape (switching) to the right well [Figs. f7(b) and f7(c)]; in contrast to the **oscillator** model, the escape may happen much later than initial energy increase; for example, in Fig. f7(b) the escape happens at t ≃ 44 ns ≫ t c ≃ 2.1 ns. ... We analyze the crosstalk error mechanism in measurement of two capacitively coupled superconducting flux-biased phase **qubits**. The damped **oscillations** of the superconducting phase after the measurement of the first **qubit** may significantly excite the second **qubit**, leading to its measurement error. The first **qubit**, which is highly excited after the measurement, is described classically. The second **qubit** is treated both classically and quantum-mechanically. The results of the analysis are used to find the upper limit for the coupling capacitance (thus limiting the **frequency** of two-**qubit** operations) for a given tolerable value of the measurement error probability.

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Contributors: Chirolli, Luca, Burkard, Guido

Date: 2009-06-04

The QND character of the **qubit** measurement is studied by repeating the measurement. A perfect QND setup guarantees identical outcomes for the two repeated measurement with certainty. In order to fully characterize the properties of the measurement, we can initialize the **qubit** in the state | 0 , then rotate the **qubit** by applying a pulse of duration τ 1 before the first measurement and a second pulse of duration τ 2 between the first and the second measurement. The conditional probability to detect the **qubit** in the states s and s ' is expected to be independent of the first pulse, and to show sinusoidal **oscillation** with amplitude 1 in τ 2 . Deviations from this expectation witness a deviation from a perfect QND measurement. The sequence of **qubit** pulses and **oscillator** driving is depicted in Fig. Fig1b). The conditional probability P 0 | 0 to detect the **qubit** in the state "0" twice in sequence is plotted versus τ 1 and τ 2 in Fig. Fig1c) for Δ = 0 , and in Fig. Fig1d) for Δ / ϵ = 0.1 . We anticipate here that a dependence on τ 1 is visible when the **qubit** undergoes a flip in the first rotation. Such a dependence is due to the imperfections of the mapping between the **qubit** state and the **oscillator** state, and is present also in the case Δ = 0 . The effect of the non-QND term Δ σ X results in an overall reduction of P 0 | 0 ....We theoretically describe the weak measurement of a two-level system (**qubit**) and quantify the degree to which such a **qubit** measurement has a quantum non-demolition (QND) character. The **qubit** is coupled to a harmonic **oscillator** which undergoes a projective measurement. Information on the **qubit** state is extracted from the **oscillator** measurement outcomes, and the QND character of the measurement is inferred by the result of subsequent measurements of the **oscillator**. We use the positive operator valued measure (POVM) formalism to describe the **qubit** measurement. Two mechanisms lead to deviations from a perfect QND measurement: (i) the quantum fluctuations of the **oscillator**, and (ii) quantum tunneling between the **qubit** states $|0>$ and $|1>$ during measurements. Our theory can be applied to QND measurements performed on superconducting **qubits** coupled to a circuit **oscillator**....(Color online) Conditional probability to obtain a) s ' = s = 1 , b) s ' = - s = 1 , c) s ' = - s = - 1 , and d) s ' = s = - 1 for the case Δ t = Δ / ϵ = 0.1 and T 1 = 10 ~ n s , when rotating the **qubit** around the y axis before the first measurement for a time τ 1 and between the first and the second measurement for a time τ 2 , starting with the **qubit** in the state | 0 0 | . Correction in Δ t are up to second order. The harmonic **oscillator** is driven at resonance with the bare harmonic **frequency** and a strong driving together with a strong damping of the **oscillator** are assumed, with f / 2 π = 20 ~ G H z and κ / 2 π = 1.5 ~ G H z . Fig6...In Fig. Fig5 we plot the second order correction to the probability to obtain "1" having prepared the **qubit** in the initial state ρ 0 = | 0 0 | , corresponding to F 2 t , for Δ t = Δ / ϵ = 0.1 . We choose to plot only the deviation from the unperturbed probability because we want to highlight the contribution to spin-flip purely due to tunneling in the **qubit** Hamiltonian. In fact most of the contribution to spin-flip arises from the unperturbed probability, as it is clear from Fig. Fig3. Around the two **qubit**-shifted **frequencies**, the probability has a two-peak structure whose characteristics come entirely from the behavior of the phase ψ around the resonances Δ ω ≈ ± g . We note that the tunneling term can be responsible for a probability correction up to ∼ 4 % around the **qubit**-shifted **frequency**....We now investigate whether it is possible to identify the contribution of different mechanisms that generate deviations from a perfect QND measurement. In Fig. Fig7 we study separately the effect of **qubit** relaxation and **qubit** tunneling on the conditional probability P 0 | 0 . In Fig. Fig7 a) we set Δ = 0 and T 1 = ∞ . The main feature appearing is a sudden change of the conditional probability P → 1 - P when the **qubit** is flipped in the first rotation. This is due to imperfection in the mapping between the **qubit** state and the state of the harmonic **oscillator**, already at the level of a single measurement. The inclusion of a phenomenological **qubit** relaxation time T 1 = 2 ~ n s , intentionally chosen very short, yields a strong damping of the **oscillation** along τ 2 and washes out the response change when the **qubit** is flipped during the first rotation. This is shown in Fig. Fig7 b). The manifestation of the non-QND term comes as a global reduction of the visibility of the **oscillations**, as clearly shown in Fig. Fig7 c)....(Color online) Comparison of the deviations from QND behavior originating from different mechanisms. Conditional probability P 0 | 0 versus **qubit** driving time τ 1 and τ 2 starting with the **qubit** in the state | 0 0 | , for a) Δ = 0 and T 1 = ∞ , b) Δ = 0 and T 1 = 2 ~ n s , and c) Δ = 0.1 ~ ϵ and T 1 = ∞ . The **oscillator** driving amplitude is f / 2 π = 20 ~ G H z and a damping rate κ / 2 π = 1.5 ~ G H z is assumed. Fig7...For driving at resonance with the bare harmonic **oscillator** **frequency** ω h o , the state of the **qubit** is encoded in the phase of the signal, with φ 1 = - φ 0 , and the amplitude of the signal is actually reduced, as also shown in Fig. Fig3 for Δ ω = 0 . When matching one of the two **frequencies** ω i the **qubit** state is encoded in the amplitude of the signal, as also clearly shown in Fig. Fig3 for Δ ω = ± g . Driving away from resonance can give rise to significant deviation from 0 and 1 to the outcome probability, therefore resulting in an imprecise mapping between **qubit** state and measurement outcomes and a weak **qubit** measurement....(Color online) Schematic description of the single measurement procedure. In the bottom panel the coherent states | α 0 and | α 1 , associated with the **qubit** states | 0 and | 1 , are represented for illustrative purposes by a contour line in the phase space at HWHM of their Wigner distributions, defined as W α α * = 2 / π 2 exp 2 | α | 2 ∫ d β - β | ρ | β exp β α * - β * α . The corresponding Gaussian probability distributions of width σ centered about the **qubit**-dependent "position" x s are shown in the top panel. Fig2...The combined effect of the quantum fluctuations of the **oscillator** together with the tunneling between the **qubit** states is therefore responsible for deviation from a perfect QND behavior, although a major role is played, as expected, by the non-QND tunneling term. Such a conclusion pertains to a model in which the **qubit** QND measurement is studied in the regime of strong projective **qubit** measurement and **qubit** relaxation is taken into account only phenomenologically. We compared the conditional probabilities plotted in Fig. Fig6 and Fig. Fig7 directly to Fig. 4 in Ref. [...(Color online) a) Schematics of the 4-Josephson junction superconducting flux **qubit** surrounded by a SQUID. b) Measurement scheme: b1) two short pulses at **frequency** ϵ 2 + Δ 2 , before and between two measurements prepare the **qubit** in a generic state. Here, ϵ and Δ represent the energy difference and the tunneling amplitude between the two **qubit** states. b2) Two pulses of amplitude f and duration τ 1 = τ 2 = 0.1 ~ n s drive the harmonic **oscillator** to a **qubit**-dependent state. c) Perfect QND: conditional probability P 0 | 0 for Δ = 0 to detect the **qubit** in the state "0" vs driving time τ 1 and τ 2 , at Rabi **frequency** of 1 ~ G H z . The **oscillator** driving amplitude is chosen to be f / 2 π = 50 ~ G H z and the damping rate κ / 2 π = 1 ~ G H z . d) Conditional probability P 0 | 0 for Δ / ϵ = 0.1 , f / 2 π = 20 ~ G H z , κ / 2 π = 1.5 ~ G H z . A phenomenological **qubit** relaxation time T 1 = 10 ~ n s is assumed. Fig1 ... We theoretically describe the weak measurement of a two-level system (**qubit**) and quantify the degree to which such a **qubit** measurement has a quantum non-demolition (QND) character. The **qubit** is coupled to a harmonic **oscillator** which undergoes a projective measurement. Information on the **qubit** state is extracted from the **oscillator** measurement outcomes, and the QND character of the measurement is inferred by the result of subsequent measurements of the **oscillator**. We use the positive operator valued measure (POVM) formalism to describe the **qubit** measurement. Two mechanisms lead to deviations from a perfect QND measurement: (i) the quantum fluctuations of the **oscillator**, and (ii) quantum tunneling between the **qubit** states $|0>$ and $|1>$ during measurements. Our theory can be applied to QND measurements performed on superconducting **qubits** coupled to a circuit **oscillator**.

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Contributors: Yoshihara, Fumiki, Nakamura, Yasunobu, Yan, Fei, Gustavsson, Simon, Bylander, Jonas, Oliver, William D., Tsai, Jaw-Shen

Date: 2014-02-06

Parameters in calculations and measurements in units of GHz. In the first column, cal: δ ω Ω R 0 stands for the calculation to study the shift of the resonant **frequency**, and cal: Γ R s t δ ω m w stands for the calculation to study the decay of Rabi **oscillations** due to quasistatic flux noise. “Optimal" in the last column means that at each ε m w , ω m w is chosen to minimize dephasing due to quasistatic flux noise....We infer the high-**frequency** flux noise spectrum in a superconducting flux **qubit** by studying the decay of Rabi **oscillations** under strong driving conditions. The large anharmonicity of the **qubit** and its strong inductive coupling to a microwave line enabled high-amplitude driving without causing significant additional decoherence. Rabi **frequencies** up to 1.7 GHz were achieved, approaching the **qubit**'s level splitting of 4.8 GHz, a regime where the rotating-wave approximation breaks down as a model for the driven dynamics. The spectral density of flux noise observed in the wide **frequency** range decreases with increasing **frequency** up to 300 MHz, where the spectral density is not very far from the extrapolation of the 1/f spectrum obtained from the free-induction-decay measurements. We discuss a possible origin of the flux noise due to surface electron spins....(Color online) Rabi **oscillation** curves with different Rabi **frequencies** Ω R measured at different static flux bias ε . At each Ω R , δ ω m w is chosen to minimize dephasing due to quasistatic flux noise. The red lines are the fitting curves. In the measurements shown in the middle and bottom panels, only parts of the **oscillations** are monitored so that we can save measurement time while the envelopes of Rabi **oscillations** are captured. The inset is a magnification of the data in the bottom panel together with the fitting curve....In the Rabi **oscillation** measurements, a microwave pulse is applied to the **qubit** followed by a readout pulse, and P s w as a function of the microwave pulse length is measured. First, we measure the Rabi **oscillation** decay at ε = 0 , where the quasistatic noise contribution is negligible. Figure GRfR1p5(d) shows the measured 1 / e decay rate of the Rabi **oscillations** Γ R 1 / e as a function of Ω R 0 . For Ω R 0 / 2 π up to 400 MHz, Γ R 1 / e is approximately 3 Γ 1 / 4 , limited by the energy relaxation, and S Δ Ω R 0 is negligible. For Ω R 0 / 2 π from 600 MHz to 2.2 GHz, Γ R 1 / e > 3 Γ 1 / 4 . A possible origin of this additional decoherence is fluctuations of ε m w , δ ε m w : Ω R 0 is first order sensitive to δ ε m w , which is reported to be proportional to ε m w itself. Next, the decay for the case ε ≈ Δ is studied. To observe the contribution from quasistatic flux noise, the Rabi **oscillation** decay as a function of ω m w is measured, where the contribution from the other sources is expected to be almost constant. Figure GRfR1p5(b) shows Γ R 1 / e at ε / 2 π = 4.16 GHz as a function of δ ω m w while keeping Ω R / 2 π between 1.5 and 1.6 GHz. Besides the offset and scatter, the trend of Γ R 1 / e agrees with that of the simulated Γ R s t . This result indicates that numerical calculation properly evaluates δ ω m w minimizing Γ R s t . Finally, the decay for the case ε ≈ Δ as a function of ε m w , covering a wide range of Ω R , is measured (Fig. Rabis)....(Color online) Power spectrum density of flux fluctuations S n φ ω extracted from the Rabi **oscillation** measurements in the first ( ε / 2 π = 4.16 GHz) and second cooldowns. The PSDs obtained from the spin-echo and energy relaxation measurements in the second cooldown are also plotted. The black solid line is the 1/ f spectrum extrapolated from the FID measurements in the second cooldown. The purple dashed line is the estimated Johnson noise from a 50 Ω microwave line coupled to the **qubit** by a mutual inductance of 1.2 pH and nominally cooled to 35 mK. The pink dotted line is a Lorentzian, S n φ m o d e l ω = S h ω w 2 / ω 2 + ω w 2 , and the orange solid line is the sum of the Lorentzian and the Johnson noise. Here the parameters are S h = 3.6 × 10 -19 r a d -1 s and ω w / 2 π = 2.7 × 10 7 H z ....Josephson devices, decoherence, Rabi **oscillation**, $1/f$ noise...(Color online) (a) Numerically calculated shift of the resonant **frequency** δ ω (black open circles) and the Bloch–Siegert shift δ ω B S (blue line). (b) Numerically calculated decay rate Γ R s t (black open circles) and Rabi **frequency** Ω R (red solid triangles) as functions of the detuning δ ω m w from ω 01 . The purple solid line is a fit based on Eq. ( fRfull). The measured 1/ e decay rates Γ R 1 / e at ε / 2 π = 4.16 GHz for the range of Rabi **frequencies** Ω R / 2 π between 1.5 and 1.6 GHz (blue solid circles) are also plotted. (c) Calculated Rabi **frequency** Ω R , based on Eq. ( fRfull), as a function of ε for the cases (i) ω m w = ω 01 + δ ω (black solid line) and (ii) ω m w / 2 π = 6.1 GHz (red dashed line). The upper axis indicates ω 01 , corresponding to ε in the bottom axis. (d) The measured 1 / e decay rate of the Rabi **oscillations**, Γ R 1 / e , at ε = 0 and as a function of Ω R 0 . The red solid line indicates 3 4 Γ 1 obtained independently....The condition, ∂ Ω R / ∂ ε = 0 , is satisfied when ε = 0 or δ ω m w = δ ω - Ω R 0 2 / ω 01 . For Ω R 0 / 2 π = 1.52 GHz and ω 01 / 2 π = 6.400 GHz, the latter condition is calculated to be δ ω m w / 2 π = - 295 MHz, slightly different from the minimum of Γ R s t seen in Fig. G R f R 1 p 5 (b). The difference is due to the deviation from the linear approximation in Eq. ( fRfull), Ω R 0 ∝ ε m w / ω 01 . Figure GRfR1p5(c) shows the calculation of Ω R as a function of ε , based on Eq. ( fRfull). The Rabi **frequency** Ω R 0 at the shifted resonance decreases as ε increases, while Ω R , for a fixed microwave **frequency** of ω m w / 2 π = 6.1 GHz, has a minimum of approximately ω 01 / 2 π = 6.4 GHz. Here in the first order, Ω R is insensitive to the fluctuation of ε ....In Fig. GRfR1p5(a), δ ω as a function of Ω R 0 is plotted together with the well-known Bloch–Siegert shift, δ ω B S = 1 4 Ω R 0 2 ω 01 , obtained from the second-order perturbation theory. Fixed parameters for the calculation are Δ / 2 π = 4.869 and ε / 2 π = 4.154 GHz ( ω 01 / 2 π = 6.400 GHz). We find that δ ω B S overestimates δ ω when Ω R 0 / 2 π 800 MHz. The deviation from the Bloch–Siegert shift is due to the component of the ac flux drive that is parallel to the ** qubit’s** energy eigenbasis; this component is not averaged out when Ω R is comparable to ω m w . ... We infer the high-

**frequency**flux noise spectrum in a superconducting flux

**qubit**by studying the decay of Rabi

**oscillations**under strong driving conditions. The large anharmonicity of the

**qubit**and its strong inductive coupling to a microwave line enabled high-amplitude driving without causing significant additional decoherence. Rabi

**frequencies**up to 1.7 GHz were achieved, approaching the

**qubit**'s level splitting of 4.8 GHz, a regime where the rotating-wave approximation breaks down as a model for the driven dynamics. The spectral density of flux noise observed in the wide

**frequency**range decreases with increasing

**frequency**up to 300 MHz, where the spectral density is not very far from the extrapolation of the 1/f spectrum obtained from the free-induction-decay measurements. We discuss a possible origin of the flux noise due to surface electron spins.

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Contributors: Beaudoin, Félix, da Silva, Marcus P., Dutton, Zachary, Blais, Alexandre

Date: 2012-08-09

(Color online) FC driving of a transmon with an external flux. The transmon is modelled using the first four levels of the Hamiltonian given by Eq. ( eqn:duffing), using parameters E J / 2 π = 25 GHz and E C / 2 π = 250 MHz. We also have g g e / 2 π = 100 MHz and ω r / 2 π = 7.8 GHz, which translates to Δ g e / 2 π ≃ 2.1 GHz. a) **Frequency** of the transition to the first excited state obtained by numerical diagonalization of Eq. ( eqn:duffing). As obtained from Eqs. ( eqn:hamonic:1) to ( eqn:hamonic:4), the major component in the spectrum of ω g e t when shaking the flux away from the flux sweet spot at **frequency** ω F C also has **frequency** ω F C . However, when shaking around the sweet spot, the dominant harmonic has **frequency** 2 ω F C . Furthermore, the mean value of ω g e is shifted by G . b) Rabi **frequency** of the red sideband transition | 1 ; 0 ↔ | 0 ; 1 . The system is initially in | 1 ; 0 and evolves under the Hamiltonian given by Eq. ( eqn:H:MLS) and a flux drive described by Eq. ( eqn:flux:drive). Full red line: analytical results from Eq. ( eqn:rabi:freq) with m = 1 and φ i = 0.25 . Dotted blue line: m = 2 and φ i = 0 . Black dots and triangles: exact numerical results. c) Geometric shift for φ i = 0.25 (full red line) and 0 (dotted blue line). d) Increase in the Rabi **frequency** for higher coupling strengths with φ i = 0.25 and Δ φ = 0.075 . e) Behavior of the resonance **frequency** for the flux drive. As long as the dispersive approximation holds ( g g c r i t / 2 π = 1061 MHz), it remains well approximated by Eq. ( eqn:resonance), as shown by the full red line. The same conclusion holds for the Rabi **frequency**. fig:transmon...(Color online) Average error with respect to the perfect red sideband process | 1 ; 0 ↔ | 0 ; 1 . A gaussian FC pulse is sent on the first **qubit** at the red sideband **frequency** assuming the second **qubit** is in its ground state. Full red line: average error of the red sideband as given by Eq. ( eqn:FUV:simple) when the second **qubit** is excited. Blue dashed line: population transfer error 1 - P t , with P t given by Eq. ( eqn:pop:transfer). Black dots: numerical results for the average error. We find the evolution operator after time t p for each eigenstate of the second **qubit**. The fidelity is extracted by injecting these unitaries in Eq. ( eqn:trace). The **qubits** are taken to be transmons, which are modelled as 4-level Duffing **oscillators** (see Section sec:Duffing) with E J 1 = 25 GHz, E J 2 = 35 GHz, E C 1 = 250 MHz, E C 2 = 300 MHz, yielding ω 01 1 = 5.670 GHz and ω 01 2 = 7.379 GHz, and g 01 1 = 100 MHz. The resonator is modeled as a 5-level truncated harmonic **oscillator** with **frequency** ω r = 7.8 GHz. As explained in Section sec:transmon, the splitting between the first two levels of a transmon is modulated using a time-varying external flux φ . Here, we use gaussian pulses in that flux, as described by Eq. ( eqn:gaussian) with τ = 2 σ , σ = 6.6873 ns, and flux drive amplitude Δ φ = 0.075 φ 0 . The length of the pulse is chosen to maximize the population transfer. fig:FUV...This method is first applied to simulate a R 01 1 pulse by evolving the two-transmon-one-resonator system under the Hamiltonian of Eq. ( eqn:H:MLS), along with the FC drive Hamiltonian for the pulse. The simulation parameters are indicated in Table tab:sequence. To generate the sideband pulse R 01 1 , the target **qubit** splitting is modulated at a **frequency** that lies exactly between the red sideband resonance for the spectator **qubit** in states | 0 or | 1 , such that the fidelity will be the same for both these spectator **qubit** states. We calculate the population transfer probability for | 1 ; 0 ↔ | 0 ; 1 after the pulse and find a success rate of 99.2% for both initial states | 1 ; 0 and | 0 ; 1 . This is similar to the prediction from Eq. ( eqn:pop:transfer), which yields 98.7%. The agreement between the full numerics and the simple analytical results is remarkable, especially given that with | δ ± / ϵ n | = 0.23 the small δ ± ≪ ϵ n assumption is not satisfied. Thus, population transfers between the transmon and the resonator are achievable with a good fidelity even in the presence of Stark shift errors coming from the spectator **qubit** (see Section sec:SB)....In Fig. fig:transmonb), the Rabi **frequencies** predicted by the above formula are compared to numerical simulations using the full Hamiltonian Eq. ( eqn:H:MLS), along with a cosine flux drive. The geometric shifts described by Eq. ( eq:G) are also plotted in Fig. fig:transmonc), along with numerical results. In both cases, the scaling with respect to Δ φ follows very well the numerical predictions, allowing us to conclude that our simple analytical model accurately synthesizes the physics occurring in the full Hamiltonian. It should be noted that, contrary to intuition, the geometric shift is roughly the same at and away from the sweet spot. This is simply due to the fact that the band curvature does not change much between the two operation points. However, as expected from Eqs. ( eqn:hamonic:1) to ( eqn:hamonic:4), the Rabi **frequencies** are much larger for the same drive amplitude when the transmon is on average away from its flux sweet spot. In that regime, large Rabi **frequencies** ∼ 30 -40 MHz can be attained, which is well above dephasing rates in actual circuit QED systems, especially in the 3D cavity . However, the available power that can be sent to the flux line might be limited in the lab, putting an upper bound on achievable rates. Furthermore, at those rates, fast rotating terms such as the ones dropped between Eq. ( eq:eps:n) and ( eq:V) start to play a role, adding spurious **oscillations** in the Rabi **oscillations** that reduce the fidelity. These additional **oscillations** have been seen to be especially large for big relevant ε m ω / Δ ~ j , j + 1 n ratios, i.e. when the **qubit** spends a significant amount of time close to resonance with the resonator and the dispersive approximation breaks down....We have also defined ω ' p = 8 E C E J Σ cos φ i , the plasma **frequency** associated to the operating point φ i . This **frequency** is illustrated by the black dots for two operating points on Fig. fig:transmona). In addition, there is a **frequency** shift G , standing for geometric, that depends on the shape of the transmon energy bands. As is also illustrated on Fig. fig:transmona), this **frequency** shift comes from the fact that the relation between ω j , j + 1 and φ is nonlinear, such that the mean value of the transmon **frequency** during flux modulation is not its value for the mean flux φ i . To fourth order in Δ φ , it is...In words, the infidelity 1 - F U V is minimized when the Rabi **frequency** that corresponds to the FC drive is large compared to the Stark shift associated to the spectator **qubit**. The average fidelity corresponding to the gate fidelity Eq. ( eqn:FUV:simple) is illustrated in Fig. fig:FUV as as a function of S 2 (red line) assuming the second **qubit** to be in its excited state. We also represent as black dots a numerical estimate of the error coming from the spectator ** qubit’s** Stark shift. The latter is calculated with Eqs. ( eqn:trace) and ( eqn:avg:fid). Numerically solving the system’s Schrödinger equation allows us to extract the unitary evolution operator that corresponds to the applied gate. Taking U to be that evolution operator for the spectator

**qubit**in state | 0 and V the operator in state | 1 , we obtain the error caused by the Stark shift shown in Fig. fig:FUV. The numerical results closely follow the analytical predictions, even for relatively large dispersive shifts S 2 ....Schemes for two-

**qubit**operations in circuit QED. ϵ is the strength of the drive used in the scheme, if any. ∗ There are no crossings in that gate provided that the

**qubits**have

**frequencies**separated enough that they do not overlap during FC modulations. tab:gates...Amplitude of the gaussian pulse over time. Δ φ ' is such that the areas A + and 2 A - are equal. Then, driving the sideband at its resonance

**frequency**for the geometric shift that corresponds to the flux drive amplitude Δ φ ' allows population inversion. fig:gaussian...Sideband transitions have been shown to generate controllable interaction between superconducting

**qubits**and microwave resonators. Up to now, these transitions have been implemented with voltage drives on the

**qubit**or the resonator, with the significant disadvantage that such implementations only lead to second-order sideband transitions. Here we propose an approach to achieve first-order sideband transitions by relying on controlled

**oscillations**of the

**qubit**

**frequency**using a flux-bias line. Not only can first-order transitions be significantly faster, but the same technique can be employed to implement other tunable

**qubit**-resonator and

**qubit**-

**qubit**interactions. We discuss in detail how such first-order sideband transitions can be used to implement a high fidelity controlled-NOT operation between two transmons coupled to the same resonator....(Color online) Sideband transitions for a three-level system coupled to a resonator. Applying an FC drive at

**frequency**Δ i , i + 1 generates a red sideband transitions between states | i + 1 ; n and | i ; n + 1 , where the numbers represent respectively the MLS and resonator states. Similarly, driving at

**frequency**Σ i , i + 1 leads to a blue sideband transition, i.e. | i ; n ↔ | i + 1 ; n + 1 . Transitions between states higher in the Fock space are not shown for reasons of readability. This picture is easily generalized to an arbitrary number of levels. fig:MLS:sidebands...Table tab:gates summarizes theoretical predictions and experimental results for recent proposals for two-

**qubit**gates in circuit QED. These can be divided in two broad classes. The first includes approaches that rely on anticrossings in the

**qubit**-resonator or

**qubit**-

**qubit**spectrum. They are typically very fast, since their rate is equal to the coupling strength involved in the anticrossing. Couplings can be achieved either through direct capacitive coupling of the

**qubits**with strength J C , or through the 11-02 anticrossing in the two-transmon spectrum which is mediated by the cavity . The latter technique has been successfully used with large coupling rates J 11 - 02 and Bell-state fidelities of ∼ 94 % . However, since these gates are activated by tuning the

**qubits**in and out of resonance, they have a finite on/off ratio determined by the distance between the relevant spectral lines. Thus, the fact that the gate is never completely turned off will make it very complicated to scale up to large numbers of

**qubits**. Furthermore, adding

**qubits**in the resonator leads to more spectral lines that also reduce scalability. In that situation, turning the gates on and off by tuning

**qubit**transition

**frequencies**in and out of resonance without crossing these additional lines becomes increasingly difficult as

**qubits**are added in the resonator, an effect known as spectral crowding. ... Sideband transitions have been shown to generate controllable interaction between superconducting

**qubits**and microwave resonators. Up to now, these transitions have been implemented with voltage drives on the

**qubit**or the resonator, with the significant disadvantage that such implementations only lead to second-order sideband transitions. Here we propose an approach to achieve first-order sideband transitions by relying on controlled

**oscillations**of the

**qubit**

**frequency**using a flux-bias line. Not only can first-order transitions be significantly faster, but the same technique can be employed to implement other tunable

**qubit**-resonator and

**qubit**-

**qubit**interactions. We discuss in detail how such first-order sideband transitions can be used to implement a high fidelity controlled-NOT operation between two transmons coupled to the same resonator.

Files:

Contributors: Ginossar, Eran, Bishop, Lev S., Girvin, S. M.

Date: 2012-07-19

In this book chapter we analyze the high excitation nonlinear response of the Jaynes-Cummings model in quantum optics when the **qubit** and cavity are strongly coupled. We focus on the parameter ranges appropriate for transmon **qubits** in the circuit quantum electrodynamics architecture, where the system behaves essentially as a nonlinear quantum **oscillator** and we analyze the quantum and semi-classical dynamics. One of the central motivations is that under strong excitation tones, the nonlinear response can lead to **qubit** quantum state discrimination and we present initial results for the cases when the **qubit** and cavity are on resonance or far off-resonance (dispersive)....quantum degenerate regime We now move to exploring what happens when the detuning between the **qubit** and the cavity is reduced such that the anharmonicity of quantum ladder of states becomes much larger than the corresponding linewidth κ (see Fig. gino:fig:context). In order to describe the response of the system to external drive in this regime it is important to take into account the quantum dynamics on the lower anharmonic part of the ladder. When the system is initialized in the ground state, there is a range of drive strengths for which the system will remain blockaded from excitations out of the ground state. However, since the anharmonicity of the JC ladder decreases with excitation number, the transition **frequency** for excitations between adjacent levels ultimately approaches the bare cavity **frequency**. Qualitatively, when the excitation level n is such that the anharmonicity becomes smaller than the linewidth κ , we expect the state dynamics to be semiclassical, similar to a driven-damped harmonic **oscillator**...gino:chirp_figure (Color) Readout control pulse (a) Time trace of the drive amplitude: a fast initial chirp **frequency** chirp( 10 n s ) can selectively steer the initial state, while the **qubit** is detuned from the cavity ( ω q - ω c / 2 π ≈ 2 g ). It is followed by a slow displacement to increase contrast and lifetime of the latching state, while the **qubit** is resonant with the cavity ( κ / 2 π = 2.5 MHz ). The drive amplitude ramp is limited so that the photon blockade photon blockadeis not broken, but the contrast is enhanced by additional driving at the highest drive amplitude. (b) A diagram of transition **frequencies** shows how the drive **frequency** chirps through the JC ladder **frequencies** of the (+) manifold, and how the manifold changes due to the time dependent **qubit** **frequency**. (c) Wave packet snapshots at selected times (indicated by bullet points on panel (b)) of the chirping drive **frequency** of panel (b) conditioned on the initial state of the **qubit**. (d) The temporal evolution of the reduced density matrix | ρ m n | (the x , y axes denote the quantum numbers m , n of the cavity levels) of the cavity with the control pulse (a) when the **qubit** initial state is superposition 1 2 | 0 + | 1 . The resonator enters a mesoscopic state of superposition around t = t c due to the entanglement with the **qubit** and the quantum state sensitivity of the protocol. At later times the off-diagonal parts of this superposition dephase quickly due to the interaction with the environment and the state of the system is being completely projected around t = 3 t c ....strong!driving( t **qubit** being detuned. Due to the interaction with the **qubit**, the cavity behaves as nonlinear **oscillator** with its set of transition **frequencies** depending on the state of the **qubit** (see the two distinct sets of lines in Fig. gino:chirp_figure(b)). The cavity responds with a ringing behavior which is different for the two cases (see Fig. gino:chirp_figure(c)). The ringing due to the pulse effectively maps the | ↓ and | ↑ to the dim and bright state basins, respectively (see Fig. 3(c)). Since κ t c ≪ 1 , an initial superposition α | ↑ + β | ↓ maps into a coherent superposition of the dim and bright states. Next, (2) a much weaker long pulse transfers the initially created bright state (for initial | ↑ ) to even brighter and longer lived states ( t c t c effects a projection of the pointer state. In designing such a pulse sequence we have the following physical considerations: (a) the initial fast selective chirp...(Color) Symmetry breaking. State-dependent transition **frequency** ω n , q = 2 π E n + 1 , q - E n , q versus photon number n , where E n , q denotes energy of the system eigenstate with n photons and **qubit** state q : (a) for the JC model, parameters as in Figs. gino:fig:latch000 and gino:fig:densclass; (b) for the model extended to 2 **qubits**, δ 1 / 2 π = - 1.0 G H z , δ 2 / 2 π = - 2.0 G H z , g 1 / 2 π = g 2 / 2 π = 0.25 G H z . Here, χ 2 denotes the 0-photon dispersive shift dispersive regimeof the second **qubit**; (c) for the model extended to one transmon **qubit** koch charge-insensitive 2007, tuned below the cavity, / 2 π = 7 G H z E C / 2 π = 0.2 G H z , E J / 2 π = 30 G H z , g / 2 π = 0.29 G H z . (For the given parameters, δ 01 / 2 π = - 0.5 G H z , δ 12 / 2 π = - 0.7 G H z , defining δ i j = E j - E i - , with E i the energy of the i th transmon level.) In all panels, the transition **frequency** asymptotically returns to the bare cavity **frequency**. In (a) the **frequencies** within the σ z = ± 1 manifolds are (nearly) symmetric with respect to the bare cavity **frequency**. For (b), if the state of one (‘spectator’) **qubit** is held constant, then the **frequencies** are asymmetric with respect to flipping the other (‘active’) **qubit**. In (c), the symmetry is also broken due the existence of higher levels in the weakly anharmonic transmon....The solution of eqn gino:eq:classic is plotted in Fig. gino:fig:densclass for the same parameters as in Fig. gino:fig:latch000b. For weak driving the system response approaches the linear response of the dispersively shifted cavity. Above the lower critical amplitude ξ C 1 the **frequency** response bifurcates, and the JC **oscillator** enters a region of bistability...**Qubit** state measurement in circuit QED Circuit QEDcan operate in different parameter regimes and relies on different dynamical phenomena of the strongly coupled transmon-resonator system strong!coupling. The dispersive readout is the least disruptive to the **qubit** state and it is realized where the cavity and **qubit** are strongly detuned. The high power readout operates in a regime where the system response can be described using a semi-classical model and yields an relatively high fidelity fidelitywith simple measurement protocol. When the cavity and **qubit** are on resonance (the quantum degenerate regime quantum degenerate regime) it is theoretically predicted that the photon blockade photon blockade can also be used to realize a high fidelity readout. gino:fig:context...matt-pc in Fig. gino:fig:latch000, where we show the average heterodyne amplitude a as a function of drive **frequency** and amplitude. Despite the presence of 4 **qubits** in the device, the fact that extensions beyond a two-level model would seem necessary since higher levels of the transmons...Solution to the semiclassical equation gino:eq:classic, using the same parameters as Fig. gino:fig:latch000b. (a) Amplitude response as a function of drive **frequency** and amplitude. The region of bifurcation bifurcationis indicated by the shaded area, and has corners at the critical points C 1 , C 2 . The dashed lines indicate the boundaries of the bistable region for a Kerr **oscillator** (Duffing **oscillator**) Duffing **oscillator**, constructed by making the power-series expansion of the Hamiltonian to second order in N / . The Kerr bistability bistability Kerr region matches the JC region in the vicinity of C 1 but does not exhibit a second critical point. (b) Cut through (a) for a drive of 6.3 ξ 1 , showing the **frequency** dependence of the classical solutions (solid line). For comparison, the response from the full quantum simulation of Fig. gino:fig:latch000b is also plotted (dashed line) for the same parameters. (c) Cut through (a) for driving at the bare cavity **frequency**, showing the large gain available close to C 2 (the ‘step’). Faint lines indicate linear response. (d) Same as (c), showing intracavity amplitude on a linear scale. gino:fig:densclass...Transmitted heterodyne amplitude a as a function of drive detuning (normalized by the dispersive shift dispersive regime χ = g 2 / δ ) and drive amplitude (normalized by the amplitude to put n = 1 photon in the cavity in linear response, ξ 1 = κ / 2 ). Dark colors indicate larger amplitudes. (a) Experimental data matt-pc, for a device with cavity at 9.07 G H z and 4 transmon **qubits** transmonat 7.0 , 7.5 , 8.0 , 12.3 G H z . All **qubits** are initialized in their ground state, and the signal is integrated for the first 400 n s ≃ 4 / κ after switching on the drive. (b) Numerical results for the JC model of eqn gino:eq:master, with **qubit** fixed to the ground state and effective parameters δ / 2 π = - 1.0 G H z , g / 2 π = 0.2 G H z , κ / 2 π = 0.001 G H z . These are only intended as representative numbers for circuit QED Circuit QEDand were not optimized against the data of panel (a). Hilbert space is truncated at 10,000 excitations (truncation artifacts are visible for the strongest drive), and results are shown for time t = 2.5 / κ ....(Color) Symmetry breaking. State-dependent transition **frequency** ω n , q = 2 π E n + 1 , q - E n , q versus photon number n , where E n , q denotes energy of the system eigenstate with n photons and **qubit** state q : (a) for the JC model, parameters as in Figs. gino:fig:latch000 and gino:fig:densclass; (b) for the model extended to 2 **qubits**, δ 1 / 2 π = - 1.0 G H z , δ 2 / 2 π = - 2.0 G H z , g 1 / 2 π = g 2 / 2 π = 0.25 G H z . Here, χ 2 denotes the 0-photon dispersive shift ... In this book chapter we analyze the high excitation nonlinear response of the Jaynes-Cummings model in quantum optics when the **qubit** and cavity are strongly coupled. We focus on the parameter ranges appropriate for transmon **qubits** in the circuit quantum electrodynamics architecture, where the system behaves essentially as a nonlinear quantum **oscillator** and we analyze the quantum and semi-classical dynamics. One of the central motivations is that under strong excitation tones, the nonlinear response can lead to **qubit** quantum state discrimination and we present initial results for the cases when the **qubit** and cavity are on resonance or far off-resonance (dispersive).

Files:

Contributors: Whittaker, J. D., da Silva, F. C. S., Allman, M. S., Lecocq, F., Cicak, K., Sirois, A. J., Teufel, J. D., Aumentado, J., Simmonds, R. W.

Date: 2014-08-08

We describe a tunable-cavity QED architecture with an rf SQUID phase **qubit** inductively coupled to a single-mode, resonant cavity with a tunable **frequency** that allows for both microwave readout of tunneling and dispersive measurements of the **qubit**. Dispersive measurement is well characterized by a three-level model, strongly dependent on **qubit** anharmonicity, **qubit**-cavity coupling and detuning. A tunable cavity **frequency** provides a way to strongly vary both the **qubit**-cavity detuning and coupling strength, which can reduce Purcell losses, cavity-induced dephasing of the **qubit**, and residual bus coupling for a system with multiple **qubits**. With our **qubit**-cavity system, we show that dynamic control over the cavity **frequency** enables one to avoid Purcell losses during coherent **qubit** evolutions and optimize state readout during **qubit** measurements. The maximum **qubit** decay time $T_1$ = 1.5 $\mu$s is found to be limited by surface dielectric losses from a design geometry similar to planar transmon **qubits**....The two possible flux values at the readout spot leads to two possible **frequencies** for the tunable cavity coupled to the **qubit** loop. Similar microwave readout schemes have been used with other rf-SQUID phase **qubits** . For our circuit design, the size of this **frequency** difference is proportional to the slope d f c / d φ c of the cavity **frequency** versus flux curve at a particular cavity flux φ c = Φ c / Φ o . The transmission of the cavity can be measured with a network analyzer to resolve the **qubit** flux (or circulating current) states. The periodicity of the rf SQUID phase **qubit** can be observed by monitoring the cavity’s resonance **frequency** while sweeping the **qubit** flux. This allows us to observe the single-valued and double-valued regions of the hysteretic rf SQUID. In Fig. Fig4(a), we show the cavity response to such a flux sweep for design A . Two data sets have been overlaid, for two different **qubit** resets ( φ q = ± 2 ) and sweep directions (to the left or to the right), allowing the double-valued or hysteretic regions to overlap. There is an overall drift in the cavity **frequency** due to flux crosstalk between the **qubit** bias line and the cavity’s rf SQUID loop that was not compensated for here. This helps to show how the **frequency** difference in the overlap regions increases as the slope d f c / d φ c increases....(Color online) (a) **Qubit** spectroscopy (design A ) overlaid with cavity spectroscopy at two **frequencies**, f c = 6.58 GHz and 6.78 GHz. (b) Zoom-in of the split cavity spectrum in (a) when f c = 6.78 GHz with corresponding fit lines. (c) Zoom-in of the split cavity spectrum in (a) when f c = 6.58 GHz with corresponding fit lines. (d) Cavity spectroscopy (design B ) while sweeping the **qubit** flux with f c = 7.07 GHz showing a large normal-mode splitting when the **qubit** is resonant with the cavity. All solid lines represent the uncoupled **qubit** and cavity **frequencies** and the dashed lines show the new coupled normal-mode **frequencies**. Notice in (d) the additional weak splitting from a slot-mode just below the cavity, and in (c) and (d), **qubit** tunneling events are visible as abrupt changes in the cavity spectrum....(Color online) (a) Cavity spectroscopy (design A ) while sweeping the cavity flux bias with the **qubit** far detuned, biased at its maximum **frequency**. The solid line is a fit to the model including the junction capacitance. (b) Zoom-in near the maximum cavity **frequency** showing a slot-mode. (c) Line-cut on resonance along the dashed line in (b) with a fit to a skewed Lorentzian (solid line)....In general, rf SQUID phase **qubits** have lower T 2 * (and T 2 ) values than transmons, specifically at lower **frequencies**, where d f 01 / d φ q is large and therefore the **qubit** is quite sensitive to bias fluctuations and 1/f flux noise . For example, 600 MHz higher in **qubit** **frequency**, at f 01 = 7.98 GHz, Ramsey **oscillations** gave T 2 * = 223 ns. At this location, the decay of on-resonance Rabi **oscillations** gave T ' = 727 ns, a separate measurement of **qubit** energy decay after a π -pulse gave T 1 = 658 ns, and so, T 2 ≈ 812 ns, or T 2 ≈ 3.6 × T 2 * , a small, but noticeable improvement over the lower **frequency** results displayed Fig. Fig6. The current device designs suffer from their planar geometry, due to a very large area enclosed by the non-gradiometric rf SQUID loop (see Fig. Fig1). Future devices will require some form of protection against flux noise , possibly gradiometric loops or replacing the large geometric inductors with a much smaller series array of Josephson junctions ....(Color online) Coupling rate 2 g / 2 π (design A ) as a function of cavity **frequency** ω c / 2 π . The solid red (blue) line is the prediction from Eq. ( eq:g) (including L x and C J ’s). The (dotted) dashed line is the prediction for capacitive coupling with C = 15 fF ( C = 5 fF). The solid circles were measured spectroscopically (see text). At lowest cavity **frequency**, the solid ⋆ results from a fit to the Purcell data, discussed later in section TCQEDC. The gray region highlights where the phase **qubit** (design A ) remains stable enough for operation (see text)....Next, we carefully explore the size of the dispersive shifts for various cavity and **qubit** **frequencies**. In order to capture the maximum dispersive **frequency** shift experienced by the cavity, we applied a π -pulse to the **qubit**. A fit to the phase response curve allows us to extract the cavity’s amplitude response time 2 / κ , the **qubit** T 1 , and the full dispersive shift 2 χ . Changing the cavity **frequency** modifies the coupling g and the detuning Δ 01 , while changes to the **qubit** **frequency** change both Δ 01 and the ** qubit’s** anharmonicity α . In Fig. Fig9(a), we show the phase

**anharmonicity as a function of its transition**

**qubit**’s**frequency**ω 01 / 2 π extracted from the spectroscopic data shown in Fig. Fig5 from section QBB for design A . The solid red line is a polynomial fit to the experimental data, used to calculate the three-level model curves in Fig. Fig9(b–d), while the blue line is a theoretical prediction of the relative anharmonicity (including L x , but neglecting C J ) using perturbation theory and the characteristic

**qubit**parameters extracted section QBB. In Fig. Fig9(b–d), we find that the observed dispersive shifts strongly depend on all of these factors and agree well with the three-level model predictions . For comparison, in Fig. Fig9(b), we show the results for the two-level system model (bold dashed line) when f c = 6.58 GHz, which has a significantly larger amplitude for all detunings (outside the “straddling regime”). Notice that it is possible to increase the size of the dispersive shifts for a given | Δ 01 | / ω 01 by decreasing the cavity

**frequency**f c , which increases the coupling rate 2 g / 2 π (as seen in Fig. Fig2 in section TCQED). Also, notice that decreasing the ratio of | Δ 01 | / ω 01 also significantly increases the size of the dispersive shifts, even when the phase

**relative anharmonicity α r decreases as ω 01 increases. Essentially, the ability to reduce | Δ 01 | helps to counteract any reductions in α r . These results clearly demonstrate the ability to tune the size of the dispersive shift through selecting the relative**

**qubit**’s**frequency**of the

**qubit**and the cavity. This tunability offers a new flexibility for optimizing dispersive readout of

**qubits**in cavity QED architectures and provides a way for rf SQUID phase

**qubits**to avoid the destructive effects of tunneling-based measurements....(Color online) (a) Time domain measurements (design A ). Rabi

**oscillations**for

**frequencies**near f 01 = 7.38 GHz. (b) Line-cut on-resonance along the dashed line in (a). The fit (solid line) yields a Rabi

**oscillation**decay time of T ' = 409 ns. (c) Ramsey

**oscillations**versus

**qubit**flux detuning near f 01 = 7.38 GHz. (d) Line-cut along the dashed line in (c). The fit (solid line) yields a Ramsey decay time of T 2 * = 106 ns. With T 1 = 600 ns, this implies a phase coherence time T 2 = 310 ns....(Color online) (a) Pulse sequence. (b) Rabi

**oscillations**(design A ) for various pulse durations obtained using dispersive measurement at f 01 = 7.18 GHz, with Δ 01 = + 10 g . (c) A single, averaged time trace along the vertical dashed line in (b). (d) Rabi

**oscillations**extracted from the final population at the end of the drive pulse, along the dashed diagonal line in (b). (e) Zoom-in of dashed box in (b) showing Rabi

**oscillations**observed during continuous driving....We can explore the coupled

**qubit**-cavity behavior described by Eq. ( eq:H) by performing spectroscopic measurements on either the

**qubit**or the cavity near the resonance condition, ω 01 = ω c . Fig. Fig7(a) shows

**qubit**spectroscopy for design A overlaid with cavity spectroscopy for two cavity

**frequencies**, f c = 6.58 GHz and 6.78 GHz. Fig. Fig7(d) shows cavity spectroscopy for design B with the cavity at its maximum

**frequency**of f c m a x = 7.07 GHz while sweeping the

**qubit**flux bias φ q . In both cases, when the

**qubit**

**frequency**f 01 is swept past the cavity resonance, the inductive coupling generates the expected spectroscopic normal-mode splitting....The weak additional splitting just below the cavity in Fig. Fig7(d) is from a resonant slot-mode. We can determine the coupling rate 2 g / 2 π between the

**qubit**and the cavity by extracting the splitting size as a function of cavity

**frequency**f c from the measured spectra. Three examples of fits are shown in Fig. Fig7(b–d) with solid lines representing the bare

**qubit**and cavity

**frequencies**, whereas the dashed lines show the new coupled normal-mode

**frequencies**. For design A ( B ), at the maximum cavity

**frequency**of 6.78 GHz (7.07 GHz), we found a minimum coupling rate of 2 g m i n / 2 π = 78 MHz (104 MHz). Notice that the splitting size is clearer bigger in Fig. Fig7(c) than for Fig. Fig7(b) by about 25 MHz. The results for the coupling rate 2 g / 2 π as a function of ω c / 2 π for design A were shown in Fig. Fig2 in section TCQED. Also visible in Fig. Fig7(c–d) are periodic, discontinuous jumps in the cavity spectrum. These are indicative of

**qubit**tunneling events between adjacent metastable energy potential minima, typical behavior for hysteretic rf SQUID phase

**qubits**. Moving away from the maximum cavity

**frequency**increases the flux sensitivity, with the

**qubit**tunneling events becoming more visible as steps. This behavior is clearly visible in Fig. Fig7(c) and was already shown in Fig. Fig4 in Sec. QBA and, as discussed there, provides a convenient way to perform rapid microwave readout of traditional tunneling measurements . Next, we describe dispersive measurements of the phase

**qubit**for design A . These results agree with the tunneling measurements across the entire

**qubit**spectrum. ... We describe a tunable-cavity QED architecture with an rf SQUID phase

**qubit**inductively coupled to a single-mode, resonant cavity with a tunable

**frequency**that allows for both microwave readout of tunneling and dispersive measurements of the

**qubit**. Dispersive measurement is well characterized by a three-level model, strongly dependent on

**qubit**anharmonicity,

**qubit**-cavity coupling and detuning. A tunable cavity

**frequency**provides a way to strongly vary both the

**qubit**-cavity detuning and coupling strength, which can reduce Purcell losses, cavity-induced dephasing of the

**qubit**, and residual bus coupling for a system with multiple

**qubits**. With our

**qubit**-cavity system, we show that dynamic control over the cavity

**frequency**enables one to avoid Purcell losses during coherent

**qubit**evolutions and optimize state readout during

**qubit**measurements. The maximum

**qubit**decay time $T_1$ = 1.5 $\mu$s is found to be limited by surface dielectric losses from a design geometry similar to planar transmon

**qubits**.

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Contributors: Kim, Mun Dae

Date: 2008-09-02

(Color online) Rabi-type **oscillations** of occupation probabilities of | ρ ρ ' states for strongly coupled **qubits** with the initial state ψ 0 = | 00 + | 10 / 2 . Here the parameters are J / h = 5 GHz, ω 0 / 2 π =4GHz, and Ω 0 / 2 π = 600 MHz at the degeneracy point where E = E in Fig. weak(a)....We study the coupled-**qubit** **oscillation** driven by an **oscillating** field. When the period of the non-resonant mode is commensurate with that of the resonant mode of the Rabi **oscillation**, we show that the controlled-NOT (CNOT) gate operation can be demonstrated. For a weak coupling the CNOT gate operation is achievable by the commensurate **oscillations**, while for a sufficiently strong coupling it can be done for arbitrary parameter values. By finely tuning the amplitude of **oscillating** field it is shown that the high fidelity of the CNOT gate can be obtained for any fixed coupling strength and **qubit** energy gap in experiments....In Fig. PC we show the Rabi-type **oscillation** for strongly coupled **qubits**. While the P 00 ( P 01 ) is reversed from 0.5 (0) to 0 (0.5) at Ω t = (odd) π , we can observe that the probabilities P 10 and P 11 remain their initial values 0.5 and 0, respectively. In this case the parameters need not satisfy the commensurate condition of Eq. ( condition) for the CNOT gate operation....The scheme for CNOT gate operation in this study uses the non-Rabi **oscillations** for | 10 and | 11 states which are commensurate with the Rabi **oscillation** for | 00 and | 01 states. In Fig. TwoRabi we display the numerical results obtained from the Hamiltonian in Eqs. ( tilH0) and ( tilH1), which show such commensurate mode **oscillations**. The initial state, | ψ 0 = | 00 + | 10 / 2 , is driven by an **oscillating** field with the resonant **frequency** ω = ω 0 < ω 1 ....The commensurate **oscillations** of resonant and non-resonant modes enable the high fidelity CNOT gate operation by finely tuning the **oscillating** field amplitude for any given values of **qubit** energy gap and coupling strength between **qubits**. While for a sufficiently strong coupling the CNOT gate can be achieved for any given parameter values, for a weak coupling a relation between the parameters should be satisfied for the fidelity maxima. For a sufficiently weak coupling compared to the **qubit** energy gap, J / ℏ ω 0 ≪ 1 , we have α 1 ≈ 1 and β 1 ≈ 0 , resulting in the expression for g in Eq. ( ga). For J / ℏ ω 0 ≪ 1 , Eq. ( ga) immediately gives rise to the relation g / J ≪ 1 and thus g / ℏ ω 0 ≪ 1 after some manipulation. This means that for a weak coupling J / ℏ ω 0 ≪ 1 the numerical results are well fit with the RWA as shown in Table table, because the RWA is good for g / ℏ ω 0 ≪ 1 . As a result, the high performance CNOT gate operation can be achieved as shown in Fig. dF....Let us consider a concrete example for comprehensive understanding. For superconducting flux **qubits**, g = m B is the coupling between the amplitude B of the magnetic microwave field and the magnetic moment m , induced by the circulating current, of the **qubit** loop. In order to adjust the value of g , actually we need to vary the microwave amplitude B , because the **qubit** magnetic moment is fixed at a specified degeneracy point. The Rabi-type **oscillation** occurs between the transformed states | 0 = | + | / 2 and | 1 = | - | / 2 . The states of **qubits** can be detected by shifting the magnetic pulse adiabatically . Since these **qubit** states are the superposition of the clockwise and counterclockwise current states, | and | , the averaged current of **qubit** states vanishes at the degeneracy point in Fig. weak(a). Thus, one can apply a finite dc magnetic pulse to shift the **qubits** slightly away from the degeneracy point to detect the **qubit** current states....(Color online) (a) Energy levels E ρ ρ ' of coupled **qubits**, where ρ , ρ ' ∈ 0 1 . E s s ' with s , s ' ∈ are shown as thin dotted lines. The distance between two degeneracy points corresponds to the coupling strength between two **qubits**. (b) Occupation probabilities of | ρ ρ ' states during Rabi-type **oscillations** at the lower degeneracy point where E = E . Here we use the parameter values such that coupling strength J / h = 0.6GHz, **qubit** energy gap ω 0 / 2 π =4GHz, and Rabi **frequency** Ω 0 / 2 π = 600 MHz. The initial state is chosen as ψ 0 = | 00 + | 10 / 2 and the CNOT gate is expected to be achieved at Ω t = (odd) π ....The values of g / h for the main fidelity maxima ( n = 1 ) obtained from numerical calculation and from the RWA of Eq. ( g) for various coupling J and **qubit** energy gap ω 0 . For small ω 0 and large J the **oscillations** are far from the Rabi **oscillation**. Here, the unit of all numbers is GHz....(Color online) (a) Commensurate **oscillations** of occupation probability of coupled-**qubit** states with the initial state, | ψ 0 = | 00 + | 10 / 2 for g / h = 0.265 GHz. The non-resonant **oscillation** modes ( P 10 and P 11 ) are commensurate with the resonant modes ( P 00 and P 01 ). At Ω t = (odd) π , P 10 and P 11 recover their initial values, thus the CNOT gate operation is achieved. Here Ω 0 = g / ℏ , J / h = 0.5 GHz, and ω 0 / 2 π = 4.0GHz. (b) Higher order commensurate modes for smaller g / h = 0.122 GHz with the same J and ω 0 ....Figure weak(a) shows the energy levels E s s ' as a function of κ b , where we choose κ a such that | E s s ' - E - s s ' | ≫ t q a and thus t q a can be negligible. In the figure there are two degeneracy points; lower degeneracy point where E = E and upper degeneracy point where E = E . By adjusting the variable κ b , the coupled-**qubit** states can be brought to one of these degeneracy points. Here the distance between these degeneracy points is related to the coupling strength between two **qubits**. ... We study the coupled-**qubit** **oscillation** driven by an **oscillating** field. When the period of the non-resonant mode is commensurate with that of the resonant mode of the Rabi **oscillation**, we show that the controlled-NOT (CNOT) gate operation can be demonstrated. For a weak coupling the CNOT gate operation is achievable by the commensurate **oscillations**, while for a sufficiently strong coupling it can be done for arbitrary parameter values. By finely tuning the amplitude of **oscillating** field it is shown that the high fidelity of the CNOT gate can be obtained for any fixed coupling strength and **qubit** energy gap in experiments.

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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: 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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