### 56121 results for qubit oscillator frequency

Contributors: Strauch, F. W., Dutta, S. K., Paik, Hanhee, Palomaki, T. A., Mitra, K., Cooper, B. K., Lewis, R. M., Anderson, J. R., Dragt, A. J., Lobb, C. J.

Date: 2007-03-02

The ac Stark shift Δ ω 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. The inset shows the **oscillation** **frequency** Ω ̄ R , 01 as a function of the level spacing ω 01 for I a c = 5.87 nA and the fit using ( rabif) to obtain Ω R , 01 and Δ ω 01 ....Experimental microwave spectroscopy of a Josephson phase **qubit**, scanned in **frequency** (vertical) and bias current (horizontal). Dark points indicate experimental microwave enhancement of the tunneling escape rate, while white dashed lines are quantum mechanical calculations of (from right to left) ω 01 , ω 02 / 2 , ω 12 , ω 13 / 2 , and ω 23 ....Rabi **frequency** Ω R , 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed lines are the lowest-order results ( rabi1) (top) and second-order ( rabi2) (bottom) perturbative results. The inset shows Rabi **oscillations** of the escape rate for I a c = 16.5 nA....Rabi **frequency** Ω R , 02 of the two-photon 0 2 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. Inset shows Rabi **oscillations** of the escape rate for I a c = 16.5 nA....Rabi **oscillations** have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (**qubits**) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase **qubit** (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi **frequency**, and two-photon Rabi **frequency** are compared to measurements made on a dc SQUID phase **qubit** with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment. ... Rabi **oscillations** have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (**qubits**) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase **qubit** (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi **frequency**, and two-photon Rabi **frequency** are compared to measurements made on a dc SQUID phase **qubit** with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment.

Data types:

Contributors: Omelyanchouk, A. N., Shevchenko, S. N., Zagoskin, A. M., Il'ichev, E., Nori, Franco

Date: 2007-05-12

The average energy H of the system as a function of the driving **frequency** ω . The main peak ( ω 0 ≈ 0.6 ) corresponds to the resonance. The left peak at ω 0 / 2 is the nonlinear effect of the excitation by a subharmonic, similar to a multiphoton process in the quantum case. The right peak at 2 ω 0 is the first overtone and it has no quantum counterpart. Here ϕ e d = π ; ϕ e a = 0.05 ; γ = 10 -3 ....The dependence of the pseudo-Rabi **frequency** on the driving amplitude ϕ e a for ω = 0.6 , γ = 10 -3 . The solid line, Ω = 0.35 ϕ e a 2 + ω - 0.63 2 1 / 2 , is the best fit to the calculated data....(Color online) The potential profile of Eq. ( eq_potential) with α = 0.8 , ϕ e d = π . The arrows indicate quantum (solid) and classical (dotted) **oscillations**....A quantitative difference between this effect and true Rabi **oscillations** is in the different scale of the resonance **frequency**. To induce Rabi **oscillations** between the lowest quantum levels in the potential ( eq_potential), one must apply a signal in resonance with their tunneling splitting, which is exponentially smaller than ω 0 . Still, this is not a very reliable signature of the effect, since the classical effect can also be excited by subharmonics, ∼ ω 0 / n , as we can see in Fig. fig4....Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux **qubit** in the _classical_ regime can produce low-**frequency** **oscillations** in the presence of an external field in resonance with the (high-**frequency**) harmonic mode of the system, $\omega$. Like in the case of_quantum_ Rabi **oscillations**, the **frequency** of these pseudo-Rabi **oscillations** is much smaller than $\omega$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external perturbation not only at the resonance **frequency** $\omega$ and its subharmonics ($\omega/n$), but also at its overtones, $n\omega$....In the presence of the external field ( eq_external) the system will undergo forced **oscillations** around one of the equlibria. For α = 0.8 , which is close to the parameters of the actual devices , the values of the dimensionless **frequencies** become ω θ ≈ 0.612 , and ω χ ≈ 0.791 . Solving the equations of motion ( eq_motion) numerically, we see the appearance of slow **oscillations** of the amplitude and energy superimposed on the fast forced **oscillations** (Fig. fig2), similar to the classical **oscillations** in a phase **qubit** (Fig. 2 in ). The dependence of the **frequency** of these **oscillations** on the driving amplitude shows an almost linear behaviour (Fig. fig3), which justifies the “Pseudo-Rabi” moniker....The key observable difference between the classical and quantum cases, which would allow to reliably distinguish between them, is that the classical effect can also be produced by driving the system at the overtones of the resonance signal, ∼ n ω 0 (Fig. fig4). This effect can be detected using a standard technique for RF SQUIDs . The current circulating in the **qubit** circuit produces a magnetic moment, which is measured by the inductively coupled high-quality tank circuit. For the tank voltage V T we have...where τ T = R T C T is the RC-constant of the tank, ω T = L T C T -1 / 2 its resonant **frequency**, M the mutual inductance between the tank and the **qubit**, and I q t the current circulating in the **qubit**. The persistent current in the 3JJ loop can be determined directly from ( eq_I). Its behaviour in the presence of an external RF field is shown in Fig. fig2c. Note that the sign of the current does not change, which is due to the fact that the **oscillations** take place inside one potential well (solid arrow in Fig. fig1), and not between two separate nearby potential minima like in the quantum case. (Alternatively, this would also allow to distinguish between the classical and quantum effects by measuring the magnetization with a DC SQUID.)...(a) Driven **oscillations** around a minimum of the potential profile of Fig. fig1 as a function of time. The driving amplitude is ϕ e a = 0.01 , driving **frequency** ω = 0.612 , and the decay rate γ = 10 -3 . Low-**frequency** classical beat **oscillations** are clearly seen. (b) Low-**frequency** **oscillations** of the persistent current in the 3JJ loop. (c) Same for the energy of the system. ... Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux **qubit** in the _classical_ regime can produce low-**frequency** **oscillations** in the presence of an external field in resonance with the (high-**frequency**) harmonic mode of the system, $\omega$. Like in the case of_quantum_ Rabi **oscillations**, the **frequency** of these pseudo-Rabi **oscillations** is much smaller than $\omega$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external perturbation not only at the resonance **frequency** $\omega$ and its subharmonics ($\omega/n$), but also at its overtones, $n\omega$.

Data types:

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**.

Data types:

Contributors: Poletto, S, Chiarello, F, Castellano, M G, Lisenfeld, J, Lukashenko, A, Carelli, P, Ustinov, A V

Date: 2009-10-23

Rabi **oscillation** of the double SQUID manipulated as a phase **qubit** by applying microwave pulses at 19 GHz. The **oscillation** **frequency** changes from 540 MHz to 1.2 GHz by increasing the power of the microwave signal by 10 dB....Probability of measuring the state | L as a function of the pulse duration. The coherent **oscillation** shown here has a **frequency** of 14 GHz and a coherence time of approximately 1.2 ns....Measurement of the relaxation time T 1 for the double SQUID operated as a phase **qubit**....Measured **oscillation** **frequencies** versus amplitude of the short flux pulse (full dots). The solid curve is a numerical simulation using the measured parameters of the circuit....We report on two different manipulation procedures of a tunable rf SQUID. First, we operate this system as a flux **qubit**, where the coherent evolution between the two flux states is induced by a rapid change of the energy potential, turning it from a double well into a single well. The measured coherent Larmor-like **oscillation** of the retrapping probability in one of the wells has a **frequency** ranging from 6 to 20 GHz, with a theoretically expected upper limit of 40 GHz. Furthermore, here we also report a manipulation of the same device as a phase **qubit**. In the phase regime, the manipulation of the energy states is realized by applying a resonant microwave drive. In spite of the conceptual difference between these two manipulation procedures, the measured decay times of Larmor **oscillation** and microwave-driven Rabi **oscillation** are rather similar. Due to the higher **frequency** of the Larmor **oscillations**, the microwave-free **qubit** manipulation allows for much faster coherent operations....Measured Rabi **oscillation** **frequency** versus the normalized amplitude of the microwave signal (solid dots). The dashed line is a linear fit taking into account slightly off-resonance microwave field, while the fit represented by the solid line considers a population of higher excited states. ... We report on two different manipulation procedures of a tunable rf SQUID. First, we operate this system as a flux **qubit**, where the coherent evolution between the two flux states is induced by a rapid change of the energy potential, turning it from a double well into a single well. The measured coherent Larmor-like **oscillation** of the retrapping probability in one of the wells has a **frequency** ranging from 6 to 20 GHz, with a theoretically expected upper limit of 40 GHz. Furthermore, here we also report a manipulation of the same device as a phase **qubit**. In the phase regime, the manipulation of the energy states is realized by applying a resonant microwave drive. In spite of the conceptual difference between these two manipulation procedures, the measured decay times of Larmor **oscillation** and microwave-driven Rabi **oscillation** are rather similar. Due to the higher **frequency** of the Larmor **oscillations**, the microwave-free **qubit** manipulation allows for much faster coherent operations.

Data types:

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.

Data types:

Contributors: Wubs, Martijn, Kohler, Sigmund, Hanggi, Peter

Date: 2007-03-15

(Color online) Upper panel: Adiabatic energies during a LZ sweep of a **qubit** coupled to two **oscillators**. Parameters: γ = 0.25 ℏ v and Ω 2 = 100 ℏ v , both as in Fig. fig:energylandscape; ℏ Ω 1 = 80 ℏ v . Lower panel: Probability P ↑ → ↑ t that the system stays in the initial state | ↑ 0 0 (solid), and corresponding exact survival final survival probability P ↑ → ↑ ∞ of Eq. ( centralresulttwoosc) (dotted)....cond1 this implies that an integral is non-vanishing only if the non-zero component of λ 2 ℓ - 1 is + 1 while the same component of λ 2 ℓ equals -1 . In other words, we obtain the selection rule that to the occupation probability at t = ∞ only those processes contribute in which the **oscillator** jumps (repeatedly) from the state | 0 to any state with a single photon (i.e. to b j | 0 ) and back; see Fig. fig:perturbation. It follows that the **oscillators** not only start but also end in their ground state | 0 if the final **qubit** state is | ↑ . We call this dynamical selection rule the “no-go-up theorem” (see also )....sec:largedetuning If the resonance energies of the cavities differ by much more than the **qubit**-**oscillator** coupling, then the dynamics can very well be approximated by two independent standard Landau-Zener transitions, see Figure fig:largedetuning....(Color online) Upper panel: Adiabatic energies during a LZ sweep of a **qubit** coupled to two **oscillators**. Parameters: γ = 0.25 ℏ v , ℏ Ω 1 = 90 ℏ v and Ω 2 = 100 ℏ v . Viewed on this scale of **oscillator** energies, the differences between exact and avoided level crossings are invisible. Lower panel: for the same parameters, probability P ↑ → ↑ t that the system stays in the initial state | ↑ 0 0 (solid), and corresponding exact survival final survival probability P ↑ → ↑ ∞ of Eq. ( centralresulttwoosc) (dotted)....(Color online) Upper panel: Adiabatic energies during a LZ sweep of a **qubit** coupled to two **oscillators** with large energies, and with detunings of the order of the **qubit**-**oscillator** coupling γ . Parameters: γ = 0.25 ℏ v and ℏ Ω 2 = 100 ℏ v , as before; ℏ Ω 1 = 96 ℏ v . Lower panel: Probability P ↑ → ↑ t that the system stays in the initial state | ↑ 0 0 (solid), and corresponding exact survival final survival probability P ↑ → ↑ ∞ of Eq. ( centralresulttwoosc) (dotted)....In the following we are interested in the properties of the final **qubit**-two-**oscillator** state | ψ ∞ rather than merely the transition probability P ↑ ↓ ∞ of the **qubit**. In general not much can be said about this final state, but let us now make the realistic assumption ℏ Ω 1 , 2 ≫ γ : both **oscillator** energies ℏ Ω 1 , 2 are much larger than the **qubit**-**oscillator** couplings γ 1 = γ 2 = γ . Still, the **frequency** detuning δ ω = Ω 2 - Ω 1 may be larger or smaller than γ / ℏ . The adiabatic energies in this case are sketched in Fig. fig:energylandscape....(Color online) LZ dynamics of a **qubit** coupled to one **oscillator**, far outside the RWA regime: γ = ℏ Ω = 0.25 ℏ v . The red solid curve is the survival probability P ↑ ↑ t when starting in the initial state | ↑ 0 . The dotted black line is the exact survival probability P ↑ → ↑ ∞ based on Eq. ( centralresult2). The dashed purple curve depicts the average photon number in the **oscillator** if the **qubit** would be measured in state | ↓ ; the dash-dotted blue curve at the bottom shows the analogous average photon number in case the **qubit** would be measured | ↑ . fig:photon_averages...A **qubit** may undergo Landau-Zener transitions due to its coupling to one or several quantum harmonic **oscillators**. We show that for a **qubit** coupled to one **oscillator**, Landau-Zener transitions can be used for single-photon generation and for the controllable creation of **qubit**-**oscillator** entanglement, with state-of-the-art circuit QED as a promising realization. Moreover, for a **qubit** coupled to two cavities, we show that Landau-Zener sweeps of the **qubit** are well suited for the robust creation of entangled cavity states, in particular symmetric Bell states, with the **qubit** acting as the entanglement mediator. At the heart of our proposals lies the calculation of the exact Landau-Zener transition probability for the **qubit**, by summing all orders of the corresponding series in time-dependent perturbation theory. This transition probability emerges to be independent of the **oscillator** **frequencies**, both inside and outside the regime where a rotating-wave approximation is valid....While P ↑ ↓ ∞ is determined by the ratio γ 2 / ℏ v , the coefficients c 2 n + 1 depend also on the **oscillator** **frequency**. In Fig. fig:photon_averages we depict how for a small **frequency** (very small: equal to the coupling strength!) the average photon numbers in the **oscillator** depend on the state of the **qubit**....(Color online) Upper panel: Adiabatic energies during a LZ sweep of a **qubit** coupled to two **oscillators** with degenerate energies. Parameters: γ = 0.25 ℏ v and ℏ Ω 2 = 100 ℏ v , as before; this time ℏ Ω 1 = ℏ Ω 2 . Lower panel: Probability P ↑ → ↑ t that the system stays in the initial state | ↑ 0 0 (solid), and corresponding exact survival final survival probability P ↑ → ↑ ∞ of Eq. ( centralresulttwoosc) (dotted)....(Color online) Sketch of adiabatic eigenstates during LZ sweep of a **qubit** that is coupled to one **oscillator**. Starting in the ground state | ↑ 0 and by choosing a slow LZ sweep, a single photon can be created in the **oscillator**. Due to cavity decay, the 1-photon state will decay to a zero-photon state. Then the reverse LZ sweep creates another single photon that eventually decays to the initial state | ↑ 0 . This is a cycle to create single photons that can be repeated. ... A **qubit** may undergo Landau-Zener transitions due to its coupling to one or several quantum harmonic **oscillators**. We show that for a **qubit** coupled to one **oscillator**, Landau-Zener transitions can be used for single-photon generation and for the controllable creation of **qubit**-**oscillator** entanglement, with state-of-the-art circuit QED as a promising realization. Moreover, for a **qubit** coupled to two cavities, we show that Landau-Zener sweeps of the **qubit** are well suited for the robust creation of entangled cavity states, in particular symmetric Bell states, with the **qubit** acting as the entanglement mediator. At the heart of our proposals lies the calculation of the exact Landau-Zener transition probability for the **qubit**, by summing all orders of the corresponding series in time-dependent perturbation theory. This transition probability emerges to be independent of the **oscillator** **frequencies**, both inside and outside the regime where a rotating-wave approximation is valid.

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Contributors: Greenberg, Ya. S.

Date: 2003-03-04

Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Here we present the results of detailed computer simulations of the interaction of a classical object (resonant tank circuit) with a quantum object (phase **qubit**). We explicitly account for the back action of a tank circuit and for the unpredictable nature of outcome of a single measurement. According to the results of our simulations the Rabi **oscillations** in MHz range can be detected using conventional NMR pulse Fourier technique....It is clearly seen that B **oscillates** with gap **frequency**, while the **frequency** of A is almost ten times smaller: (**oscillation** period of B: T B ≈ 2 × 10 -9 s, while the same quantity for A is T A ≈ 2 × 10 -8 s. The small distortions on A curve are due to a strong deviation of excitation signal from transverse rotating wave form, while B curve is clearly modulated with Rabi **frequency** Ω R (Fig. fig2)....As is seen from the Fig. fig3, A decays to 0.5 **oscillating** with Rabi **frequency**, while B (C) decays to zero. (Note: to be rigorous, the stable state solution for A is...Phase **qubit** coupled to a tank circuit....As the coupling is increased further the **qubit** wave function is completely destroyed. The quantity A is quenched to approximately 0.85 (Fig. fig10a). That is | C - | ≈ 0.92 . It might seem that we have here so called Zeno effect- as if **qubit** state is frozen in its ground state. However, in case of a strong coupling it is not correct to say about wave function of the **qubit** alone. This is shown in Fig. figcoh where for λ > 10 -2 the phase coherence is seen to be completely lost ....At every graph of the figures the results for one realization of random number generator ξ t are compared with the case when we replaced F ξ t in ( Q) with deterministic term 2 A - 1 - 2 C / 2 , which means that the tank measures the average current ( avcurr) in a **qubit** loop. As is seen from the Fig. fig4a, A **oscillates** with Rabi **frequency**. The voltage across tank circuit **oscillates** also with Rabi **frequency** which is equal to 50 MHz in our case (Fig. fig4b) which is modulated with the lower **frequency** the value of which is about 5 MHz....Phase loss-free **qubit** coupled to a loss-free tank circuit. **Oscillations** of A. Deterministic case (a) together with one realization (b) are shown. Small scale time **oscillations** correspond to Rabi **frequency**....Phase loss-free **qubit** coupled to a dissipative tank circuit. The evolution of A exhibits modulation of Rabi **oscillations** with lower **frequency**. Deterministic case (a) together with one realization (b) are shown....In conclusion we want to show the effect of **qubit** evolution as the coupling between the **qubit** and the tank is increased. We numerically solved the system consisting of the loss-free **qubit** coupled to the dissipative tank circuit. The system is described by Eqs. ( A2, B2, C2, flux_tank) and Eq. ( Q1). For the simulations we take the coupling parameter λ = 2.5 × 10 -2 . The results of simulations are shown on Figs. fig10a, fig10d for deterministic case. As is seen from the Figs. fig10a during Rabi period the quantity A became partially frozen at some level. At the endpoints of this period the system tries to escape to another level of A. Between the endpoints of Rabi period A **oscillates** with a high **frequency** which is about 10 GHz in our case. As expected, the evolution of B is suppressed approximately by a factor of ten below its free evolution amplitude which is equal to 0.5. As we show below, the strong coupling completely destroys the phase coherence between **qubit** states, nevertheless the voltage across the tank **oscillates** with Rabi **frequency**. Its amplitude is considerably increased and it does not reveal any peculiarities associated with the frozen behavior of A (Fig. fig10d)....Phase loss-free **qubit** coupled to a dissipative tank circuit. The voltage across the tank exhibits modulation of Rabi **frequency**. Deterministic case (a) together with one realization (b) are shown....Time evolution of A and B for **qubit** without dissipation. ... Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Here we present the results of detailed computer simulations of the interaction of a classical object (resonant tank circuit) with a quantum object (phase **qubit**). We explicitly account for the back action of a tank circuit and for the unpredictable nature of outcome of a single measurement. According to the results of our simulations the Rabi **oscillations** in MHz range can be detected using conventional NMR pulse Fourier technique.

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

Date: 2006-03-07

In practice, the cavity **frequency** Ω and the **qubit**-**oscillator** coupling γ are determined by the design of the setup, while the Josephson energy can be switched at a controllable velocity v — ideally from E J = - ∞ to E J = ∞ . In reality, however, E J is bounded by E J , m a x which is determined by the critical current. The condition E J , m a x > ℏ Ω is required so that the **qubit** comes into resonance with the **oscillator** sometime during the sweep. Moreover, inverting the flux through the superconducting loop requires a finite time 2 T m i n , so that v cannot exceed v m a x = E J , m a x / 2 T m i n . In order to study under which conditions the finite initial and final times can be replaced by ± ∞ , we have numerically integrated the Schrödinger equation in a finite time interval - T T . Results are presented in Fig. fig:P_single....Hint. The time evolution of the probability that the **qubit** is in state | ↓ is depicted in Fig. fig:one-osc. It demonstrates that at intermediate times, the dynamics depends strongly on the **oscillator** **frequency** Ω , despite the fact that this is not the case for long times. For a large **oscillator** **frequency**, P ↑ ↓ t resembles the standard LZ transition with a time shift ℏ Ω / v ....Population dynamics of individual **qubit**-**oscillator** states for a coupling strength γ = 0.6 ℏ v and **oscillator** **frequency** Ω = 0.5 v / ℏ ....Hint correlates every creation or annihilation of a photon with a **qubit** flip, the resulting dynamics is restricted to the states | ↑ , 2 n and | ↓ , 2 n + 1 . Figure fig:updown reveals that the latter states survive for long times, while of the former states only | ↑ , 0 stays occupied, as it follows from the relation that A n ∝ δ n , 0 , derived above. Thus, the final state exhibits a peculiar type of entanglement between the **qubit** and the **oscillator**, and can be written as...We study a **qubit** undergoing Landau-Zener transitions enabled by the coupling to a circuit-QED mode. Summing an infinite-order perturbation series, we determine the exact nonadiabatic transition probability for the **qubit**, being independent of the **frequency** of the QED mode. Possible applications are single-photon generation and the controllable creation of **qubit**-**oscillator** entanglement....Landau-Zener dynamics for the coupling strength γ = 0.6 ℏ v for various cavity **frequencies** Ω . The dashed line marks the Ω -independent, final probability centralresult to which all curves converge....centralresult. Thus we find that finite-time effects do not play a role as long as γ ≪ ℏ Ω . Our predicted transition probabilities based on analytical results for infinite propagation time are therefore useful to describe the finite-time LZ sweeps. Figure fig:P_single also illustrates that the probability for single-photon production is highest in the adiabatic regime ℏ v / γ 2 ≪ 1 . Here the typical duration of a LZ transition is 2 γ / v . So in the regime of interest, the sought condition for a “practically infinite time interval” is v T = E J , m a x > ℏ Ω + 2 γ . For the unrealistically large **qubit**-**oscillator** coupling γ / ℏ Ω = 0.5 , reliable single-photon generation is less probable. This is so because (i) the LZ transition is incomplete within - T T ; (ii) more than two **oscillator** levels take part in the dynamics and more than one photon can be generated, as depicted in Fig. fig:updown; and (iii) the approximation of the instantaneous ground state at t = - T by | ↑ , 0 is less accurate....Probability of single-photon generation P | ↓ , 1 as a function of ℏ v / γ 2 , for LZ sweeps within the finite time interval - T T with T > T m i n chosen such that v T = 3 ℏ Ω / 2 . The initial state is | ↑ , 0 . Shown probabilities are averaged within the time interval 29 20 ℏ Ω / v and 3 2 ℏ Ω / v , whereby the small and fast **oscillations** that are typical for the tail of a LZ transition are averaged out. ... We study a **qubit** undergoing Landau-Zener transitions enabled by the coupling to a circuit-QED mode. Summing an infinite-order perturbation series, we determine the exact nonadiabatic transition probability for the **qubit**, being independent of the **frequency** of the QED mode. Possible applications are single-photon generation and the controllable creation of **qubit**-**oscillator** entanglement.

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Contributors: Higgins, Kieran D. B., Lovett, Brendon W., Gauger, Erik M.

Date: 2012-03-27

Our methodology can be used to predict dynamics of nanomechanical resonators connected to either quantum dots or superconducting **qubits**. The criterion for the single term approximation to be valid is readily met by current experiments such as those presented in Refs. and their parameters yield near perfect agreement between numerical and analytic results. Most experiments operate in a regime where the **qubit** dynamics are not greatly perturbed by the presence of the **oscillator**, which has a much lower **frequency** ( ϵ ≈ Δ ≈ 10 GHz, ω = 1 GHz). In Figure fig:1, we chose ϵ ≈ Δ ≈ 100 MHz, because this better demonstrates the effect of the **oscillator** on the **qubit**. These parameters can be achieved experimentally using the same **qubit** design but with an **oscillating** voltage applied to the CPB bias gate . However, we stress the accuracy of our method is not restricted to this regime....fig:2 Main panel: comparison of dynamics calculated from truncating ( Sin5) at N M A X = ± 10 (red) and a numerically exact approach (blue). Lower left: Fourier transform of the dynamics. Lower right: the numerical weight of the n t h term in the series expansion of ( Sin5), showing there are still only two dominant **frequencies** at n = 0 and n = - 1 . Parameters: ω = 0.5 , g = 0.1 , ϵ = 0 , Δ = 0.5 , T = 1 ~ m K , ℏ = 1 and k b = 1 ....Figure fig:3 demonstrates this idea, showing that by measuring Ω and fitting it to our expression ( eqn:rho3), we can obtain submilli-Kelvin precision in the experimentally relevant regime of 20-55 mK. At low temperatures the single term **frequency** plateaus, causing the accuracy to break down. In the higher temperature limit, we also see a deviation from the diagonal, this is to be expected as we leave the regime of validity described by ( eqn:crit). Naturally accuracy in this region could be improved by retaining higher order terms in ( Sin5), but this would become a more numeric than analytic approach. The upper inset shows the dependence of the accuracy of the prediction on the number of points (at a separation of 1ns) sampled from the dynamics. The accuracy increases initially as more points improve the fitted value of Ω , however after a certain length the accuracy is diminished by long term envelope effects in the dynamics not captured by the single term approximation. We note that the corresponding analysis in the **frequency** domain would not be equally affected by the long time envelope, however a large number of points in the FFT is then required in order to obtain the desired accuracy. The lower inset of Figure fig:3 shows the direct dependence of Ω on the temperature. The temperature range with steepest gradient and hence greatest **frequency** dependence on temperature varies with the coupling strength; thus the device could be specifically designed to have a maximal sensitivity in the temperature range of the most interest....Figure fig:1 shows a comparison of the dynamics predicted using these expressions and a numerically exact approach. The latter are obtained by imposing a truncation of the **oscillator** Hilbert space at a point where the dynamics have converged and any higher modes have an extremely low occupation probability. Our zeroth order approximation proves to be unexpectedly powerful, giving accurate dynamics well into the strong coupling regime ( g / ω = 0.25 ) and even beyond this it still captures the dominant oscillatory behaviour, see Figure fig:1. Stronger coupling increases the numerical weight of higher **frequency** terms in the series, causing a modulation of the dynamics. The approximation starts to break down at ( g / ω = 0.5 ). The equations ( eqn:rho0) and ( eqn:rho1) are obviously unable to capture the higher **frequency** modulations to the dynamics or any potential long time phenomena like collapse and revival, but these are unlikely to be resolvable in experiments in any case. Nonetheless, it is worth pointing out that even in this strong coupling case the base **frequency** of the **qubit** dynamics is still adequately captured by our single term approximation....fig:1 Comparison of the single term approximation (red) and a numerically exact approach (blue) for different coupling strengths. Uncoupled Rabi **oscillations** are also shown as a reference (green). Left: the population ρ 00 t in the time-domain. Right: the same data in the **frequency** domain. The full numerical solution was Fourier transformed using Matlab’s FFT algorithm. Other parameters are ω = 1 GHz, g = 0.1 GHz, ϵ = Δ = 100 MHz and T = 10 mK....fig:3 Demonstration of **qubit** thermometry: T i n is the temperature supplied to the numerical simulation of the system and T o u t is the temperature that would be predicted by fitting **oscillations** with **frequency** ( eqn:rho3) to it. The blue line is the data and red line shows the effect of a 10kHz error in the **frequency** measurement; the grey dashed line serves as a guide to the eye. The lower inset shows the variation of the **qubit** **frequency** Ω with temperature. The upper inset shows the dependence of the absolute error in the prediction against the signal length (see text). Other parameters are: ω = 1 GHz, g = 0.01 GHz, ϵ = 0 , Δ = 100 MHz...A quantum two level system coupled to a harmonic **oscillator** represents a ubiquitous physical system. New experiments in circuit QED and nano-electromechanical systems (NEMS) achieve unprecedented coupling strength at large detuning between **qubit** and **oscillator**, thus requiring a theoretical treatment beyond the Jaynes Cummings model. Here we present a new method for describing the **qubit** dynamics in this regime, based on an **oscillator** correlation function expansion of a non-Markovian master equation in the polaron frame. Our technique yields a new numerical method as well as a succinct approximate expression for the **qubit** dynamics. We obtain a new expression for the ac Stark shift and show that this enables practical and precise **qubit** thermometry of an **oscillator**....Including extra terms in the series expansion ( Sin5) makes the time dependence of the **qubit** dynamics analytically unwieldy, because the rational function form of the series leads to a complex interdepence of the positions of the poles in ( eqn:rsol1). However, if the values of the parameters are known the series can truncated at ( ± N M A X ) to give an efficient numerical method to obtain more accurate dynamics, extending the applicability of our approach beyond the regime described by ( eqn:crit). This is demonstrated in Fig. fig:2, where the dynamics are clearly dominated by two **frequencies** – an effect that could obviously never be captured by a single term approximation. There is a qualitative agreement between the many terms expansion and full numerical solution, particularly at short times. We would not expect a perfect agreement in this case because the simulations are of the dynamics in the large tunnelling regime ( Δ = 0.5 ), and the polaron transform makes the master equation perturbative in this parameter. Nonetheless, the rapid convergence of the series is shown in Fig. fig:2; N M A X = 5 - 10 is sufficient to calculate ρ 00 t and ρ 10 t with an accuracy only limited by the underlying Born Approximation. The asymmetry of the amplitudes of the terms in the series expansion of ( Sin5) is due to the exponential functions in the series. ... A quantum two level system coupled to a harmonic **oscillator** represents a ubiquitous physical system. New experiments in circuit QED and nano-electromechanical systems (NEMS) achieve unprecedented coupling strength at large detuning between **qubit** and **oscillator**, thus requiring a theoretical treatment beyond the Jaynes Cummings model. Here we present a new method for describing the **qubit** dynamics in this regime, based on an **oscillator** correlation function expansion of a non-Markovian master equation in the polaron frame. Our technique yields a new numerical method as well as a succinct approximate expression for the **qubit** dynamics. We obtain a new expression for the ac Stark shift and show that this enables practical and precise **qubit** thermometry of an **oscillator**.

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

Date: 2012-05-10

Figure fig:Ttrans shows the numerically obtained coherence times and whether the decay is predominantly Gaussian or Markovian. For the large **oscillator** damping γ = ϵ , the conditions for the validity of the (Markovian) Bloch-Redfield equation stated at the end of Sec. sec:analytics-g2 hold. Then we observe a good agreement of the numerically obtained T ⊥ * and Eq. ...(Color online) Dephasing time for purely quadratic **qubit**-**oscillator** coupling ( g 1 = 0 ), resonant driving at large **frequency**, Ω = ω 0 = 5 ϵ , and various values of the **oscillator** damping γ . The driving amplitude is A = 3.5 γ , such that always n ̄ = 6.125 . Filled symbols mark Markovian decay, while stroked symbols refer to Gaussian shape. The solid line depicts the value obtained for γ = ϵ in the Markov limit. The corresponding numerical values are connected by a dashed line which serves as guide to the eye....(Color online) Typical time evolution of the **qubit** operator σ x (solid line) and the corresponding purity (dashed) for Ω = ω 0 = 0.8 ϵ , g 1 = 0.02 ϵ , γ = 0.02 ϵ , and driving amplitude A = 0.06 ϵ such that the stationary photon number is n ̄ = 4.5 . Inset: Purity decay shown in the main panel (dashed) compared to the decay given by Eq. P(t) together with Eq. Lambda(t) (solid line)....Figure fig:timeevolution depicts the time evolution of the **qubit** expectation value σ x which exhibits decaying **oscillations** with **frequency** ϵ . The parameters correspond to an intermediate regime between the Gaussian and the Markovian dynamics, as is visible in the inset....(Color online) Dephasing time for purely linear **qubit**-**oscillator** coupling ( g 2 = 0 ), resonant driving, Ω = ω 0 , and **oscillator** damping γ = 0.02 ϵ . The amplitude A = 0.07 ϵ corresponds to the mean photon number n ̄ = 6.125 . Filled symbols and dashed lines refer to predominantly Markovian decay, while for Gaussian decay, stroked symbols and solid lines are used....We study **qubit** decoherence under generalized dispersive readout, i.e., we investigate a **qubit** coupled to a resonantly driven dissipative harmonic **oscillator**. We provide a complete picture by allowing for arbitrarily large **qubit**-**oscillator** detuning and by considering also a coupling to the square of the **oscillator** coordinate, which is relevant for flux **qubits**. Analytical results for the decoherence time are obtained by a transformation of the **qubit**-**oscillator** Hamiltonian to the dispersive frame and a subsequent master equation treatment beyond the Markov limit. We predict a crossover from Markovian decay to a decay with Gaussian shape. Our results are corroborated by the numerical solution of the full **qubit**-**oscillator** master equation in the original frame. ... We study **qubit** decoherence under generalized dispersive readout, i.e., we investigate a **qubit** coupled to a resonantly driven dissipative harmonic **oscillator**. We provide a complete picture by allowing for arbitrarily large **qubit**-**oscillator** detuning and by considering also a coupling to the square of the **oscillator** coordinate, which is relevant for flux **qubits**. Analytical results for the decoherence time are obtained by a transformation of the **qubit**-**oscillator** Hamiltonian to the dispersive frame and a subsequent master equation treatment beyond the Markov limit. We predict a crossover from Markovian decay to a decay with Gaussian shape. Our results are corroborated by the numerical solution of the full **qubit**-**oscillator** master equation in the original frame.

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