### 54077 results for qubit oscillator frequency

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: Liberti, G., Zaffino, R. L., Piperno, F., Plastina, F.

Date: 2005-11-21

tau0 The tangle as a function of α in the symmetric case W = 0 for different values of the **qubit** tunnelling amplitude D . One can appreciate that the result of Eq. ( tangl) is indeed reached asymptotically....Concerning the asymmetric case, our results for the ground state entanglement appear similar to those found by Costi and McKenzie in Ref. , where the interaction of a **qubit** with an ohmic environment was numerically analyzed. It turns out that, for a bath with finite band-width, the entanglement displays a behavior analogous to that reported in Figs. ( tau10)-( tau01), when plotted with respect to the value of the impedance of the bath. Here, instead, we concentrated on the dependence of the tangle on the coupling strength between the **qubit** and the environmental **oscillator**. Unfortunately, the coupling strength is not easily related to the coefficient of the spectral density used in Ref. , and therefore one cannot make a precise comparison between th...pot The lower adiabatic potential for D = 10 and α = 2 . The dashed line refers to the symmetric, W = 0 , case (dashed line), while the solid line refers to W = 1 . The case of frozen **qubit** ( W = D = 0 ) would have given a pair of independent parabolas instead of the adiabatic potentials U l , u of Eq. ( udq)....As we have shown, the procedure is easily extended to the asymmetric case and this is important since the entanglement changes dramatically for any finite (however small) value of the asymmetry in the **qubit** Hamiltonian. As mentioned in section sect2 above, this is due to the fact the this term modifies the symmetry properties of the Hamiltonian, so that the form of the ground state changes radically and the same occurs to the reduced **qubit** state. For example, for a large enough interaction strength, the **qubit** state is a complete mixture if W = 0 , while it becomes the lower eigenstate of σ z if W 0 . As a result, for large α , there is much entanglement if W = 0 , while the state of the system is factorized and thus τ = 0 if W 0 . This is seen explicitly in Fig. ( tau10). Furthermore, from the comparison of Figs. ( tau10), ( tau01), and ( tau0), one can see that, with increasing α , the tangle increases monotonically in the symmetric case, while it reaches a maximum before going down to zero if W 0 . This is due to the fact that, in the first case, the ground state of the system becomes a Schrödinger cat-like entangled superposition, approximately given by — 12 { — + —- - — - —+ } , for 1 , schroca where | φ ± are the two coherent states for the **oscillator** defined in Eq. ( due1), centered in Q = ± Q 0 , respectively, and almost orthogonal if α ≫ 1 . In the presence of asymmetry, on the other hand, the **oscillator** localizes in one of the wells of its effective potential and this implies that, for large α , the ground state is given by just one of the two components superposed in Eq. ( schroca). This is, clearly, a factorized state and therefore one gets τ = 0 . Since τ is zero for uncoupled sub-systems (i.e., for very small values of α ), weather W = 0 or not, and since, for W 0 , it has to decay to zero for large α , it follows that a maximum is present in between. In fact, for intermediate values of the coupling, there is a competition between the α -dependences of the two non zero components of the Bloch vector. In particular, the length | b → | is approximately equal to one for both small and large α ’s, see Figs. ( asx)-( asz), but the vector points in the x direction for α ≪ 1 and in the z direction for α ≫ 1 . The maximum of the tangle in the asymmetric case occurs near the point in which b x ≈ b z . For the symmetric case, we were also able to derive analytically the sharp increase of the entanglement at α = 1 . This behavior appears to be reminiscent of the super-radiant transition in the many **qubit** Dicke model, which, in the adiabatic limit, shows exactly the same features described here, and which can be described along similar lines. Finally, we would like to comment on the relationship of this work with those of Refs. and . The approach proposed by Levine and Muthukumar, Ref. , employs an instanton description for the effective action. This has been applied to obtain the entropy of entanglement in the symmetric case, in the same critical limit described above. It turns out that this description is equivalent to a fourth order expansion of the lower adiabatic potential U l . This approximation, although retaining all the distinctive qualitative features discussed above, gives slight quantitative changes in the results. Concerning the asymmetric case, our results for the ground state entanglement appear similar to those found by Costi and McKenzie in Ref. , where the interaction of a **qubit** with an ohmic environment was numerically analyzed. It turns out that, for a bath with finite band-width, the entanglement displays a behavior analogous to that reported in Figs. ( tau10)-( tau01), when plotted with respect to the value of the impedance of the bath. Here, instead, we concentrated on the dependence of the tangle on the coupling strength between the **qubit** and the environmental **oscillator**. Unfortunately, the coupling strength is not easily related to the coefficient of the spectral density used in Ref. , and therefore one cannot make a precise comparison between the two results. At least qualitatively, however, we can say that the ground state quantum correlations induced by the coupling with an ohmic environment are already present when the **qubit** is coupled to a single **oscillator** mode. 99 weiss U. Weiss, Quantum Dissipative Systems, 2 nd ed., World Scientific 1999. yuma see, e.g., Yu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). levine G. Levine and V. N. Muthukumar, Phys. Rev. B 69, 113203 (2004). martinis R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and J. M. Martinis, Phys. Rev. Lett. 93 077003 (2005); P. R. Johnson, W. T. Parsons, F. W. Strauch, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, Phys. Rev. Lett. 94, 187004 (2005). pino E. Paladino, L. Faoro, G. Falci, and R. Fazio, Phys. Rev. Lett. 88, 228304 (2002); G. Falci, A. D’Arrigo, A. Mastellone, and E. Paladino, Phys. Rev. Lett. 94, 167002 (2005) hines A.P. Hines, C.M. Dawson, R.H. McKenzie and G.J. Milburn, Phys. Rev. A 70, 022303 (2004). blais A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004); A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M.H. Devoret, S. M. Girvin and R. J. Schoelkopf, Phys. Rev. Lett. 95, 060501 (2005). prb03 F. Plastina and G. Falci, Phys. Rev. B 67, 224514 (2003). costi T.A. Costi and R.H. McKenzie, Phys. Rev. A 68, 034301 (2003). ent1 A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002); T. J. Osborne, and M. A. Nielsen Phys. Rev. A 66, 032110 (2002). ent2 G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003); L. A. Wu, M. S. Sarandy, and D. A. Lidar, Phys. Rev. Lett. 93, 250404 (2004). ent3 T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and V. Tognetti, Phys. Rev. Lett. 94, 147208 (2005). ent4 N. Lambert, C. Emary, and T. Brandes, Phys. Rev. Lett. 92, 073602 (2004). crisp M.D. Crisp, Phys. Rev. A 46, 4138 (1992). Irish E.K. Irish, J. Gea-Banacloche, I. Martin, and K. C. Schwab, Phys. Rev. B 72, 195410 (2005). Rungta V. Coffman, J. Kundu, and W.K. Wootters, Phys. Rev. A 61, 052306 (2000); T. J. Osborne, Phys. Rev. A 72, 022309 (2005), see also quant-ph/0203087. Wallraff A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M.H. Devoret, S. M. Girvin and R. J. Schoelkopf, Phys. Rev. Lett. 95, 060501 (2005). Nakamura Y. Nakamura, Yu.A. Pashkin and J.S. Tsai, Phys. Rev. Lett. 87, 246601 (2001). armour A.D. Armour, M.P. Blencowe and K.C. Schwab, Phys. Rev. Lett. 88, 148301 (2002). Grajcar M. Grajcar, A. Izmalkov and E. Ilxichev, Phys. Rev. B 71, 144501 (2005). Chiorescu I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C.J.P.M. Harmans and J.E. Mooij, Nature 431, 159 (2004)....wf Normalized ground state wave function for the **oscillator** in the lower adiabatic potential, for D = 10 and α = 2 and with W = 0 (dashed line) and W = 0.1 (solid line)....We discuss the ground state entanglement of a bi-partite system, composed by a **qubit** strongly interacting with an **oscillator** mode, as a function of the coupling strenght, the transition **frequency** and the level asymmetry of the **qubit**. This is done in the adiabatic regime in which the time evolution of the **qubit** is much faster than the **oscillator** one. Within the adiabatic approximation, we obtain a complete characterization of the ground state properties of the system and of its entanglement content. ... We discuss the ground state entanglement of a bi-partite system, composed by a **qubit** strongly interacting with an **oscillator** mode, as a function of the coupling strenght, the transition **frequency** and the level asymmetry of the **qubit**. This is done in the adiabatic regime in which the time evolution of the **qubit** is much faster than the **oscillator** one. Within the adiabatic approximation, we obtain a complete characterization of the ground state properties of the system and of its entanglement content.

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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: 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.

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Contributors: Fedorov, Kirill G., Shcherbakova, Anastasia V., Schäfer, Roland, Ustinov, Alexey V.

Date: 2013-01-22

Experiments towards realizing a readout of superconducting **qubits** by using ballistic Josephson vortices are reported. We measured the microwave radiation induced by a fluxon moving in an annular Josephson junction. By coupling a flux **qubit** as a current dipole to the annular junction, we detect periodic variations of the fluxon's **oscillation** **frequency** versus magnetic flux through the **qubit**. We found that the scattering of a fluxon on a current dipole can lead to the acceleration of a fluxon regardless of a dipole polarity. We use the perturbation theory and numerical simulations of the perturbed sine-Gordon equation to analyze our results....Long Josephson junction, fluxon, Josephson vortex, flux **qubit**, **qubit** readout...We would like to employ fluxons for developing a fast and sensitive magnetic field detector for measurements of superconducting **qubits**. In this Letter, we report direct measurements of electromagnetic radiation from a fluxon moving in an annular Josephson junction (AJJ). The radiation is detected by using a microstrip antenna capacitively coupled to the AJJ. Furthermore, we place a flux **qubit** close to the long junction and couple them magnetically with a superconducting loop (see Fig. AJJ+**Qubit**). This coupling scheme makes the fluxon interact with a current dipole formed by the electrodes of the loop coupled to the **qubit**. The time delay of the fluxon can be detected as a **frequency** shift of the electromagnetic radiation emitted from the junction. This shift provides information about the state of the flux **qubit**....Optical photograph of the chip with the annular Josephson junction on the right part and experimental set-up schematics. Left part shows the zoom into the area with the flux **qubit** with a coupling loop (yellow loop) and control line (green loop). Red crosses indicate the positions of three Josephson junctions in the flux **qubit** loop....Using the possibility to directly detect radiation of the fluxon resonant **oscillations**, we have performed systematic measurements of the dependence of the fluxon velocity versus bias current - the current-voltage characteristics - measured in the **frequency** domain (see Fig. ZFS). This approach provides an easy access to study the fine structure of the current-voltage curve as the precision of **frequency** measurements is by several orders of magnitude greater than the resolution of direct dc voltage measurement....Persistent current for a ground state of the flux **qubit** versus magnetic frustration (black line). Red line shows the corresponding fluxon shift calculated using the perturbation theory for....An annular Josephson junction with a trapped fluxon coupled to a flux **qubit**....By numerically solving ( FKE1)-( FKE2) with the additional condition u - l / 2 = u l / 2 one can calculate an equilibrium trajectory in phase space for fluxon **oscillations** in the AJJ with the current dipole and estimate a deviation of fluxon **oscillation** **frequency** from the unperturbed case δ ν = ν μ - ν 0 , where ν 0 is the **oscillation** **frequency** for μ = 0 . Black line in Fig. FD shows the dependence of relative deviation δ ν / ν 0 versus bias current γ calculated from the perturbation theory for the following set of system parameters: l = 20 , α = 0.02 , μ = 0.05 , d = 2 . The deviation δ ν is large and negative for small bias currents γ ≪ 0.1 , what means that the fluxon is being slowed down by the current dipole and eventually can be pinned at the dipole if the bias current is too small. Surprisingly, for larger currents γ > 0.05 the sign of δ ν becomes positive meaning that the current dipole accelerates the fluxon. To understand this phenomenon, we need to look at the Eq. ( FKE1) and notice that the effective damping term α e = α u 1 - u 2 has a non-monotonic behavior. When increasing the fluxon velocity u , the effective damping is increasing for u ≤ 1 / 3 and then starts decreasing. This means that deceleration (acceleration) is favorable for low (high) bias currents....Modulation of the fluxon’s **oscillation** **frequency** due to the coupling to the flux **qubit**. Every point consists of 100 averages. Bias current was set at γ = 0.521 , w ≃ 9.1 ....To couple a flux **qubit** to the fluxon inside an annular Josephson junction, it is necessary to engineer an interaction between two orthogonal magnetic dipoles. To facilitate this interaction, we have added a superconducting coupling loop embracing a flux **qubit**, as shown in Fig. AJJ+**Qubit**. The current induced in the coupling loop attached to the AJJ is proportional to the persistent current in the flux **qubit**. Thus, the persistent current in the **qubit** manifests itself in the AJJ as a current dipole with an amplitude μ on top of the homogeneous background of bias current. When fluxon scatters on a positive current dipole - it first gets accelerated and then decelerated by the dipole poles. In the ideal case of absence of damping and bias current, the sign of **frequency** change δ ν is determined only by polarity of the dipole. In the presence of finite damping and homogeneous bias current, situation completely changes - as the total propagation time becomes dependent on the complex interplay between bias current, current dipole strength and damping....Relative **frequency** deviation from equilibrium δ ν / ν 0 of the fluxon **oscillation** **frequency** versus bias current. Black line shows the result of perturbation approach, while the red line depicts results of direct numerical simulations of the PSGE equation ( PSGEm) with a d = 1 . The blue curve corresponds to the case with a d = 0.2 ....The experimental curve showing the reaction of the fluxon to the magnetic flux through the flux **qubit** are presented in Fig. FM. The periodic modulation of the fluxon **frequency** versus magnetic flux through the **qubit** corresponds to the changing of the persistent currents in the **qubit** as Fig. FQ_Icc suggests. We did not observe clear narrow peaks at the half flux quantum point, most probably due to excess fluctuations. Emerging dip-like peculiarities can be noted at presumed half flux quantum points which suggest that the dips may be there, covered by noise and insufficient resolution. Further improvements of experimental setup are required to resolve these peaks. The presented measurement curve has a convex profile which tells that indeed the deviation of **frequency** δ ν is positive, consistently with predictions made above by the perturbation approach and numerical simulations. ... Experiments towards realizing a readout of superconducting **qubits** by using ballistic Josephson vortices are reported. We measured the microwave radiation induced by a fluxon moving in an annular Josephson junction. By coupling a flux **qubit** as a current dipole to the annular junction, we detect periodic variations of the fluxon's **oscillation** **frequency** versus magnetic flux through the **qubit**. We found that the scattering of a fluxon on a current dipole can lead to the acceleration of a fluxon regardless of a dipole polarity. We use the perturbation theory and numerical simulations of the perturbed sine-Gordon equation to analyze our results.

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

Date: 2004-07-30

In the emerging field of quantum computation and quantum information, superconducting devices are promising candidates for the implementation of solid-state quantum bits or **qubits**. Single-**qubit** operations, direct coupling between two **qubits**, and the realization of a quantum gate have been reported. However, complex manipulation of entangled states - such as the coupling of a two-level system to a quantum harmonic **oscillator**, as demonstrated in ion/atom-trap experiments or cavity quantum electrodynamics - has yet to be achieved for superconducting devices. Here we demonstrate entanglement between a superconducting flux **qubit** (a two-level system) and a superconducting quantum interference device (SQUID). The latter provides the measurement system for detecting the quantum states; it is also an effective inductance that, in parallel with an external shunt capacitance, acts as a harmonic **oscillator**. We achieve generation and control of the entangled state by performing microwave spectroscopy and detecting the resultant Rabi **oscillations** of the coupled system....Generation and control of entangled states. a, Spectroscopic characterization of the energy levels (see Fig. 1b inset) after a π (upper scan) and a 2 π (lower scan) Rabi pulse on the **qubit** transition. In the upper scan, the system is first excited to | 10 from which it decays towards the | 01 excited state (red sideband at 3.58 GHz) or towards the | 00 ground state ( F L = 6.48 GHz). In the lower scan, the system is rotated back to the initial state | 00 wherefrom it is excited into the | 10 or | 11 states (see, in dashed, the blue sideband peak at 9.48 GHz for 13 dB more power). b, Coupled Rabi **oscillations**: the blue sideband is excited and the switching probability is recorded as a function of the pulse length for different microwave powers (plots are shifted vertically for clarity). For large microwave powers, the resonance peak of the blue sideband is shifted to 9.15 GHz. When detuning the microwave excitation away from resonance, the Rabi **oscillations** become faster (bottom four curves). These **oscillations** are suppressed by preparing the system in the | 10 state with a π pulse and revived after a 2 π pulse (top two curves in Fig. 3b) c, Coupled Rabi **oscillations**: after a π pulse on the **qubit** resonance ( | 00 → | 10 ) we excite the red sideband at 3.58 GHz. The switching probability shows coherent **oscillations** between the states | 10 and | 01 , at various microwave powers (the curves are shifted vertically for clarity). The decay time of the coherent **oscillations** in a, b is ∼ 3 ns....**Oscillator** relaxation time. a, Rabi **oscillations** between the | 01 and | 10 states (during pulse 2 in the inset) obtained after applying a first pulse (1) in resonance with the **oscillator** transition. Here, the interval between the two pulses is 1 ns. The continuous line represents a fit using an exponentially decaying sinusoidal **oscillation** plus an exponential decay of the background (due to the relaxation into the ground state). The **oscillation**’s decay time is τ c o h = 2.9 ns, whereas the background decay time is ∼ 4 ns. b, The amplitude of Rabi **oscillations** as a function of the interval between the two pulses (the vertical bars represent standard error bars estimated from the fitting procedure, see a). Owing to the **oscillator** relaxation, the amplitude decays in τ r e l ≈ 6 ns (the continuous line represents an exponential fit)....**Qubit** - SQUID device and spectroscopy a, Atomic force micrograph of the SQUID (large loop) merged with the flux **qubit** (the smallest loop closed by three junctions); the **qubit** to SQUID area ratio is 0.37. Scale bar, 1 μ m . The SQUID (**qubit**) junctions have a critical current of 4.2 (0.45) μ A. The device is made of aluminium by two symmetrically angled evaporations with an oxidation step in between. The surrounding circuit shows aluminium shunt capacitors and lines (in black) and gold quasiparticle traps 3 and resistive leads (in grey). The microwave field is provided by the shortcut of a coplanar waveguide (MW line) and couples inductively to the **qubit**. The current line ( I ) delivers the readout pulses, and the switching event is detected on the voltage line ( V ). b, Resonant **frequencies** indicated by peaks in the SQUID switching probability when a long microwave pulse excites the system before the readout pulse. Data are represented as a function of the external flux through the **qubit** area away from the **qubit** symmetry point. Inset, energy levels of the **qubit** - **oscillator** system for some given bias point. The blue and red sidebands are shown by down- and up-triangles, respectively; continuous lines are obtained by adding 2.96 GHz and -2.90 GHz, respectively, to the central continuous line (numerical fit). These values are close to the **oscillator** resonance ν p at 2.91 GHz (solid circles) and we attribute the small differences to the slight dependence of ν p on **qubit** state. c, The plasma resonance (circles) and the distances between the **qubit** peak (here F L = 6.4 GHz) and the red/blue (up/down triangles) sidebands as a function of an offset current I b o f f through the SQUID. The data are close to each other and agree well with the theoretical prediction for ν p versus offset current (dashed line)....Rabi **oscillations** at the **qubit** symmetry point Δ = 5.9 GHz. a, Switching probability as a function of the microwave pulse length for three microwave nominal powers; decay times are of the order of 25 ns. For A = 8 dBm, bi-modal beatings are visible (the corresponding **frequencies** are shown by the filled squares in b). b, Rabi **frequency**, obtained by fast Fourier transformation of the corresponding **oscillations**, versus microwave amplitude. In the weak driving regime, the linear dependence is in agreement with estimations based on sample design. A first splitting appears when the Rabi **frequency** is ∼ ν p . In the strong driving regime, the power independent Larmor precession at **frequency** Δ gives rise to a second splitting. c, This last aspect is obtained in numerical simulations where the microwave driving is represented by a term 1 / 2 h F 1 cos Δ t and a small deviation from the symmetry point (100 MHz) is introduced in the strong driving regime (the thick line indicates the main Fourier peaks). Radiative shifts 20 at high microwave power could account for such a shift in the experiment. ... In the emerging field of quantum computation and quantum information, superconducting devices are promising candidates for the implementation of solid-state quantum bits or **qubits**. Single-**qubit** operations, direct coupling between two **qubits**, and the realization of a quantum gate have been reported. However, complex manipulation of entangled states - such as the coupling of a two-level system to a quantum harmonic **oscillator**, as demonstrated in ion/atom-trap experiments or cavity quantum electrodynamics - has yet to be achieved for superconducting devices. Here we demonstrate entanglement between a superconducting flux **qubit** (a two-level system) and a superconducting quantum interference device (SQUID). The latter provides the measurement system for detecting the quantum states; it is also an effective inductance that, in parallel with an external shunt capacitance, acts as a harmonic **oscillator**. We achieve generation and control of the entangled state by performing microwave spectroscopy and detecting the resultant Rabi **oscillations** of the coupled system.

Files:

Contributors: Martijn Wubs, Sigmund Kohler, Peter Hänggi

Date: 2007-10-01

(Color online) Upper panel: adiabatic energies during a LZ sweep of a **qubit** coupled to two **oscillators**. Parameters: γ=0.25ℏv and Ω2=100ℏv, both as in Fig. 4; ℏΩ1=80ℏv. Lower panel: probability P↑→↑(t) that the system stays in the initial state |↑00〉 (solid), and corresponding exact survival final survival probability P↑→↑(∞) of Eq. (20) (dotted).
...(Color online) LZ dynamics of a **qubit** coupled to one **oscillator**, far outside the RWA regime: γ=ℏΩ=0.25ℏv. 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. (16). 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 |↑〉.
...(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=0.25ℏv and ℏΩ2=100ℏv, as before; ℏΩ1=96ℏv. Lower panel: probability P↑→↑(t) that the system stays in the initial state |↑00〉 (solid), and corresponding exact survival final survival probability P↑→↑(∞) of Eq. (20) (dotted).
...(Color online) Upper panel: adiabatic energies during a LZ sweep of a **qubit** coupled to two **oscillators**. Parameters: γ=0.25ℏv, ℏΩ1=90ℏv, and Ω2=100ℏv. 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 |↑00〉 (solid), and corresponding exact survival final survival probability P↑→↑(∞) of Eq. (20) (dotted).
...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....(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 one-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: 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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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: Oxtoby, Neil P., Gambetta, Jay, Wiseman, H. M.

Date: 2007-06-24

fig:dqdqpc Schematic of an isolated DQD **qubit** and capacitively coupled low-transparency QPC between source (S) and drain (D) leads....In simple dyne detection (see schematic in Fig. fig:rfcircuit), the output signal V o u t t is amplified, and mixed with a local **oscillator** (LO). The LO for homodyne detection of the amplitude quadrature is V L O t ∝ cos ω 0 t , where the LO **frequency** is the same as the signal of interest (or very slightly detuned). The resulting low-**frequency** beats due to mixing the signal with the LO are easily detected....Equivalent circuit for continuous monitoring of a charge **qubit** coupled to a classical L C **oscillator** with inductance L and capacitance C . We consider the charge-sensitive detector that loads the **oscillator** circuit to be a QPC (see Fig. fig:dqdqpc for details). Measurement is achieved using reflection with the input voltage, V i n t , and the output voltage, V o u t t , being separated by a directional coupler. The output voltage is then amplified and mixed with a local **oscillator**, L O , and then measured. fig:rfcircuit...The extension of quantum trajectory theory to incorporate realistic imperfections in the measurement of solid-state **qubits** is important for quantum computation, particularly for the purposes of state preparation and error-correction as well as for readout of computations. Previously this has been achieved for low-**frequency** (dc) weak measurements. In this paper we extend realistic quantum trajectory theory to include radio **frequency** (rf) weak measurements where a low-transparency quantum point contact (QPC), coupled to a charge **qubit**, is used to damp a classical **oscillator** circuit. The resulting realistic quantum trajectory equation must be solved numerically. We present an analytical result for the limit of large dissipation within the **oscillator** (relative to the QPC), where the **oscillator** slaves to the **qubit**. The rf+dc mode of operation is considered. Here the QPC is biased (dc) as well as subjected to a small-amplitude sinusoidal carrier signal (rf). The rf+dc QPC is shown to be a low-efficiency charge-**qubit** detector, that may nevertheless be higher than the dc-QPC (which is subject to 1/f noise)....The choice e > 0 corresponds to defining current in terms of the direction of electron flow. That is, in the opposite direction to conventional current. The DQDs are occupied by a single excess electron, the location of which determines the charge state of the **qubit**. The charge basis states are denoted | 0 and | 1 (see Fig. fig:dqdqpc). We assume that each quantum dot has only one single-electron energy level available for occupation by the **qubit** electron, denoted by E 1 and E 0 for the near and far dot, respectively....The two conjugate parameters we use to describe the **oscillator** state are the flux through the inductor, Φ t , and the charge on the capacitor, Q t . The dynamics of the **oscillator** are found by analyzing the equivalent circuit of Fig. fig:rfcircuit using the well-known Kirchhoff circuit laws. Doing this we find that the classical system obeys the following set of coupled differential equations...Consider the equivalent circuit of Fig. fig:rfcircuit. The **oscillator** circuit consisting of an inductance L and capacitance C terminates the transmission line of impedance Z T L = 50 Ω . The voltages (potential drops) across the **oscillator** components can be written as ... The extension of quantum trajectory theory to incorporate realistic imperfections in the measurement of solid-state **qubits** is important for quantum computation, particularly for the purposes of state preparation and error-correction as well as for readout of computations. Previously this has been achieved for low-**frequency** (dc) weak measurements. In this paper we extend realistic quantum trajectory theory to include radio **frequency** (rf) weak measurements where a low-transparency quantum point contact (QPC), coupled to a charge **qubit**, is used to damp a classical **oscillator** circuit. The resulting realistic quantum trajectory equation must be solved numerically. We present an analytical result for the limit of large dissipation within the **oscillator** (relative to the QPC), where the **oscillator** slaves to the **qubit**. The rf+dc mode of operation is considered. Here the QPC is biased (dc) as well as subjected to a small-amplitude sinusoidal carrier signal (rf). The rf+dc QPC is shown to be a low-efficiency charge-**qubit** detector, that may nevertheless be higher than the dc-QPC (which is subject to 1/f noise).

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