### 56121 results for qubit oscillator frequency

Contributors: Meier, Florian, Loss, Daniel

Date: 2004-08-26

(color online). Rabi **oscillations** for a squbit-fluctuator system. The probability p 1 t to find the squbit in state | 1 is obtained from numerical integration of Eq. ( eq:mastereq) (solid line) and the analytical solution Eq. ( eq:pdyn-3) (dashed line) which is valid for b x / J ≫ 1 . For weak decoherence of the fluctuator, γ / J 1 , damped single-**frequency** **oscillations** are restored. The fluctuator leads to a reduction of the first maximum in p 1 t to ∼ 0.8 [(a) and (b)] and ∼ 0.9 [(c) and (d)], respectively. The parameters are (a) b x / J = 1.5 , γ / J = 0.5 ; (b) b x / J = 1.5 , γ / J = 1.5 ; (c) b x / J = 3 , γ / J = 0.5 ; (d) b x / J = 3 , γ / J = 1.5 ....Coherent Rabi **oscillations** between quantum states of superconducting micro-circuits have been observed in a number of experiments, albeit with a visibility which is typically much smaller than unity. Here, we show that the coherent coupling to background charge fluctuators [R.W. Simmonds et al., Phys. Rev. Lett. 93, 077003 (2004)] leads to a significantly reduced visibility if the Rabi **frequency** is comparable to the coupling energy of micro-circuit and fluctuator. For larger Rabi **frequencies**, transitions to the second excited state of the superconducting micro-circuit become dominant in suppressing the Rabi **oscillation** visibility. We also calculate the probability for Bogoliubov quasi-particle excitations in typical Rabi **oscillation** experiments....exhibits single-**frequency** **oscillations** with reduced visibility [Fig. Fig2(b)]. Part of the visibility reduction can be traced back to leakage into state | 2 . More subtly, off-resonant transitions from | 1 to | 2 induced by the driving field lead to an energy shift of | 1 , such that the transition from | 0 to | 1 is no longer resonant with the driving field, which also reduces the visibility. For b x / 2 ℏ Δ ω = 1 / 3 , corresponding to b x / h = 150 ~ M H z in Ref. ...Energy shifts induced by AC driving field. – Exploring the visibility reduction at a time-scale of 10 n s requires Rabi **frequencies** | b x | / h 100 M H z . We show next that, in this regime, transitions to the second excited squbit state lead to an oscillatory behavior in p 1 t with a visibility smaller than 0.7 . For characteristic parameters of a phase-squbit, the second excited state | 2 is energetically separated from | 1 by ω 21 = 0.97 ω 10 . Similarly to | 0 and | 1 , the state | 2 is localized around the local energy minimum in Fig. Fig2(a). For adiabatic switching of the AC current, transitions to | 2 can be neglected as long as | b x | ≪ ℏ Δ ω = ℏ ω 10 - ω 21 ≃ 0.03 ℏ ω 10 . However, for b x comparable to ℏ Δ ω , the applied AC current strongly couples | 1 and | 2 because 2 | φ ̂ | 1 ≠ 0 , where φ ̂ is the phase operator. For typical parameters, b x / ℏ Δ ω ranges from 0.05 to 1 , depending on the irradiated power . Taking into account the second excited state of the phase-squbit, the squbit Hamiltonian in the rotating frame is ... Coherent Rabi **oscillations** between quantum states of superconducting micro-circuits have been observed in a number of experiments, albeit with a visibility which is typically much smaller than unity. Here, we show that the coherent coupling to background charge fluctuators [R.W. Simmonds et al., Phys. Rev. Lett. 93, 077003 (2004)] leads to a significantly reduced visibility if the Rabi **frequency** is comparable to the coupling energy of micro-circuit and fluctuator. For larger Rabi **frequencies**, transitions to the second excited state of the superconducting micro-circuit become dominant in suppressing the Rabi **oscillation** visibility. We also calculate the probability for Bogoliubov quasi-particle excitations in typical Rabi **oscillation** experiments.

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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: 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: Singh, Mandip

Date: 2014-07-01

To estimate the **oscillation** **frequencies**, consider a flux-**qubit**-cantilever made of niobium which is a type-II superconductor of transition temperature 9.26 ~ K . Consider niobium has a square cross-section with thickness t = 0.5 μ m, l = 6 μ m, w = 4 μ m ( A = l × w ). For these dimensions the mass of the cantilever is 3.64 × 10 -14 Kg and moment of inertia is I m ≃ 7.28 × 10 -25 Kgm 2 . The critical current of Josephson junction I c = 5 μ A, capacitance C = 0.1 pF and self inductance L = 100 pH are of the same order as described in Ref . The quantity β L = 2 π L I c / Φ o ≃ 1.52 . Consider intrinsic **frequency** of the cantilever is ω i = 2 π × 12000 rad/s. For an equilibrium angle θ 0 = θ n + = cos -1 n Φ o / B x A there exists a single global potential energy minimum. If we consider n = 0 and B x = 5 × 10 -2 T the global potential energy minimum is located at ( 0 , π / 2 ). For parameters described above ω φ ≃ 2 π × 7.99 × 10 10 rad/s, ω δ = 2 π × 25398.1 rad/s, κ = 0.012 A. The eigen **frequencies** of the flux-**qubit**-cantilever are ω X ≃ 2 π × 7.99 × 10 10 rad/s and ω Y = 2 π × 21122.5 rad/s. A contour plot of potential energy of the flux-**qubit**-cantilever, indicating a two dimensional global minimum located at ( 0 , π / 2 ) and two local minima, is shown in Fig. fig2. Even if we consider intrinsic **frequencies** to be zero the restoring force is still nonzero due to a finite coupling constant. For ω i = 0 , the angular **frequencies** are ω φ ≃ 2 π × 7.99 × 10 10 rad/s, ω δ = 2 π × 22384.5 rad/s, ω X ≃ 2 π × 7.99 × 10 10 rad/s and ω Y = 2 π × 17382.8 rad/s. The **frequencies** can be increased by increasing external magnetic field and by decreasing the dimensions of the cantilever that reduces mass and moment of inertia....In this paper a macroscopic quantum **oscillator** is introduced that consists of a flux **qubit** in the form of a cantilever. The magnetic flux linked to the flux **qubit** and the mechanical degrees of freedom of the cantilever are naturally coupled. The coupling is controlled through an external magnetic field. The ground state of the introduced flux-**qubit**-cantilever corresponds to a quantum entanglement between magnetic flux and the cantilever displacement....fig1 A schematic of a flux-**qubit**-cantilever. A part of the flux **qubit** (larger loop) is in the form of a cantilever. External magnetic field B x controls the coupling between the flux **qubit** and the cantilever. An additional magnetic flux threading a DC-SQUID (smaller loop) that consists of two Josephson junctions adjusts the tunneling amplitude. DC-SQUID can be shielded from the effect of B x ....The potential energy of the flux-**qubit**-cantilever corresponds to a symmetric double well potential (i.e. two global minima and m Φ o **qubit**-cantilever is biased at a half of a flux quantum, Φ o / 2 . A contour plot indicating a two dimensional symmetric double well potential is shown in Fig. fig3. Consider the nonseparable ground states of the left and the right well are | α L and | α R , respectively. The barrier height between the wells of the double well potential which is less than 2 E j reduces when ω i is increased. The barrier height controls the tunneling between potential wells and it can also be tuned through an external magnetic flux applied to a DC-SQUID of the flux-**qubit**-cantilever. When tunneling between wells is introduced the ground state of flux-**qubit**-cantilever is | Ψ E = | α L + | α R / 2 . The state | Ψ E is an entangled state of distinct magnetic flux and distinct cantilever deflection states. The state | Ψ E can be realised by cooling the flux-**qubit**-cantilever to its ground state....fig3 A contour plot indicating location of two dimensional potential energy minima forming a symmetric double well potential for cantilever equilibrium angle θ 0 = cos -1 Φ o / 2 B x A , ω i = 2 π × 12000 ~ r a d / s , B x = 5 × 10 -2 T. The contour interval in units of **frequency** ( E / h ) is ∼ 4 × 10 11 Hz....Consider a schematic of a flux-**qubit**-cantilever shown in Fig. fig1 where a part of a superconducting loop of a flux **qubit** forms a cantilever. The larger loop is interrupted by a smaller loop consisting of two Josephson junctions - a DC Superconducting Quantum Interference Device (DC-SQUID). The Josephson energy that is constant for a single Josephson junction can be varied by applying a magnetic flux to a DC-SQUID loop. However, for calculations a flux **qubit** with a single Josephson junction is considered throughout this paper. The external magnetic flux applied to the cantilever is Φ a = B x A cos θ , where B x is the magnitude of an uniform external magnetic field along x -axis and area vector A → substends an angle θ with the magnetic field direction ( x -axis). Consider the cantilever **oscillates** about an equilibrium angle θ 0 with an intrinsic **frequency** of **oscillation** ω i i.e. the **frequency** in absence of magnetic field. The external magnetic flux applied to the flux-**qubit**-cantilever depends on the cantilever deflection therefore, the flux **qubit** whose potential energy depends on an external flux is coupled to the cantilever degrees of freedom. The potential energy of the flux-**qubit**-cantilever corresponds to a two dimensional potential V Φ θ and the Hamiltonian of the flux-**qubit**-cantilever interrupted by a single Josephson junction is...fig2 A contour plot indicating location of a two dimensional global potential energy minimum at ( 0 , π / 2 ) and local minima for cantilever equilibrium angle θ 0 = π / 2 , ω i = 2 π × 12000 rad/s, B x = 5 × 10 -2 T. The contour interval in units of **frequency** ( E / h ) is ∼ 3.9 × 10 11 Hz. ... In this paper a macroscopic quantum **oscillator** is introduced that consists of a flux **qubit** in the form of a cantilever. The magnetic flux linked to the flux **qubit** and the mechanical degrees of freedom of the cantilever are naturally coupled. The coupling is controlled through an external magnetic field. The ground state of the introduced flux-**qubit**-cantilever corresponds to a quantum entanglement between magnetic flux and the cantilever displacement.

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Contributors: Shi, Zhan, Simmons, C. B., Ward, Daniel. R., Prance, J. R., Mohr, R. T., Koh, Teck Seng, Gamble, John King, Wu, Xian., Savage, D. E., Lagally, M. G.

Date: 2012-08-02

fig:plots5 Analysis of echo data for extraction of the decoherence time T 2 . (a–c) Transconductance G L as a function of the base level of detuning ε and δ t (defined in the main text) for total free evolution times of t = 390 ps, t = 690 ps, and t = 990 ps, respectively. (d–f) Fourier transforms of the charge occupation P 1 2 as a function of detuning ε and **oscillation** **frequency** f for the data in (a–c), respectively. We obtain P 1 2 (not shown here) by integrating the transconductance data in (a–c) and normalizing by noting that the total charge transferred across the polarization line is one electron. Fast Fourier transforming the time-domain data of P 1 2 allows us to quantify the amplitude of the **oscillations** visible near δ t = 0 . The **oscillations** of interest appear as weight in the FFT that moves to higher **frequency** at more negative detuning (farther from the anti-crossing). For an individual detuning energy, the FFT has nonzero weight for a nonzero bandwidth. (g) Echo amplitude as a function of free evolution time t . The data points (dark circles) are obtained at ε = - 120 μ eV by integrating a horizontal line cut of the FFT data over a bandwidth range of 46 - 72 GHz, then normalizing by the echo **oscillation** amplitude of the first data point, as described in the supplemental text. The echo **oscillation** amplitudes, plotted for multiple free evolution times, decay with characteristic time T 2 as the free evolution time t is made longer. By fitting the decay to a Gaussian, we obtain T 2 = 760 ± 190 ps. (h–j) Fourier transforms of the transconductance G L as a function of ε and **oscillation** **frequency** f for (a–c), respectively. As t is increased, the magnitude for **oscillations** at a given **frequency** decays with characteristic time T 2 . We take the magnitude of the FFT at the point where the central feature (black line) intersects 65 GHz. (k) Measured FFT magnitudes at 65 GHz for multiple free evolution times (dark circles) with a Gaussian fit (red line), which yields T 2 = 620 ± 140 ps, in reasonable agreement with the result shown in (g)....Fast quantum **oscillations** of a charge **qubit** in a double quantum dot fabricated in a Si/SiGe heterostructure are demonstrated and characterized experimentally. The measured inhomogeneous dephasing time T2* ranges from 127ps to ~2.1ns; it depends substantially on how the energy difference of the two **qubit** states varies with external voltages, consistent with a decoherence process that is dominated by detuning noise(charge noise that changes the asymmetry of the **qubit**'s double-well potential). In the regime with the shortest T2*, applying a charge-echo pulse sequence increases the measured inhomogeneous decoherence time from 127ps to 760ps, demonstrating that low-**frequency** noise processes are an important dephasing mechanism. ... Fast quantum **oscillations** of a charge **qubit** in a double quantum dot fabricated in a Si/SiGe heterostructure are demonstrated and characterized experimentally. The measured inhomogeneous dephasing time T2* ranges from 127ps to ~2.1ns; it depends substantially on how the energy difference of the two **qubit** states varies with external voltages, consistent with a decoherence process that is dominated by detuning noise(charge noise that changes the asymmetry of the **qubit**'s double-well potential). In the regime with the shortest T2*, applying a charge-echo pulse sequence increases the measured inhomogeneous decoherence time from 127ps to 760ps, demonstrating that low-**frequency** noise processes are an important dephasing mechanism.

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Contributors: Grajcar, M., Izmalkov, A., Il'ichev, E., Wagner, Th., Oukhanski, N., Huebner, U., May, T., Zhilyaev, I., Hoenig, H. E., Greenberg, Ya. S.

Date: 2003-03-31

In conclusion, we have observed resonant tunneling in a macroscopic superconducting system, containing an Al flux **qubit** and a Nb tank circuit. The latter played dual control and readout roles. The impedance readout technique allows direct characterization of some of the ** qubit’s** quantum properties, without using spectroscopy. In a range 50 ∼ 200 mK, the effective

**qubit**temperature has been verified [Fig. fig:Temp_dep(b)] to be the same as the mixing chamber’s (after Δ has been determined at low T ), which is often difficult to confirm independently....Our technique is similar to rf-SQUID readout. The

**qubit**loop is inductively coupled to a parallel resonant tank circuit [Fig. fig:schem(b)]. The tank is fed a monochromatic rf signal at its resonant

**frequency**ω T . Then both amplitude v and phase shift χ (with respect to the bias current I b ) of the tank voltage will strongly depend on (A) the shift in resonant

**frequency**due to the change of the effective

**qubit**inductance by the tank flux, and (B) losses caused by field-induced transitions between the two

**qubit**states. Thus, the tank both applies the probing field to the

**qubit**, and detects its response....We have observed signatures of resonant tunneling in an Al three-junction

**qubit**, inductively coupled to a Nb LC tank circuit. The resonant properties of the tank

**oscillator**are sensitive to the effective susceptibility (or inductance) of the

**qubit**, which changes drastically as its flux states pass through degeneracy. The tunneling amplitude is estimated from the data. We find good agreement with the theoretical predictions in the regime of their validity....(a) Tank phase shift vs flux bias near degeneracy and for V d r = 0.5 ~ μ V. From the lower to the upper curve (at f x = 0 ) the temperature is 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 350, 400 mK. (b) Normalized amplitude of tan χ (circles) and tanh Δ / k B T (line), for the Δ following from Fig. fig:Bias_dep; the overall scale κ is a fitting parameter. The data indicate a saturation of the effective

**qubit**temperature at 30 mK. (c) Full dip width at half the maximum amplitude vs temperature. The horizontal line fits the low- T ( < 200 mK) part to a constant; the sloped line represents the T 3 behavior observed empirically for higher T ....(a) Quantum energy levels of the

**qubit**vs external flux. The dashed lines represent the classical potential minima. (b) Phase

**qubit**coupled to a tank circuit....-dependence of ϵ t is adiabatic. However, it does remain valid if the full (Liouville) evolution operator of the

**qubit**would contain standard Bloch-type relaxation and dephasing terms (which indeed are not probed) in addition to the Hamiltonian dynamics ( eq01), since the fluctuation–dissipation theorem guarantees that such terms do not affect equilibrium properties. Normalized dip amplitudes are shown vs T in Fig. fig:Temp_dep(b) together with tanh Δ / k B T , for Δ / h = 650 MHz independently obtained above from the low- T width. The good agreement strongly supports our interpretation, and is consistent with Δ being T...Δ is the tunneling amplitude. At bias ϵ = 0 the two lowest energy levels of the

**qubit**anticross [Fig. fig:schem(a)], with a gap of 2 Δ . Increasing ϵ slowly enough, the

**qubit**can adiabatically transform from Ψ l to Ψ r , staying in the ground state E - . Since d E - / d Φ x is the persistent loop current, the curvature d 2 E - / d Φ x 2 is related to the

**susceptibility. Hence, near degeneracy the latter will have a peak, with a width given by | ϵ | Δ . We present data demonstrating such behavior in an Al 3JJ**

**qubit**’s**qubit**. ... We have observed signatures of resonant tunneling in an Al three-junction

**qubit**, inductively coupled to a Nb LC tank circuit. The resonant properties of the tank

**oscillator**are sensitive to the effective susceptibility (or inductance) of the

**qubit**, which changes drastically as its flux states pass through degeneracy. The tunneling amplitude is estimated from the data. We find good agreement with the theoretical predictions in the regime of their validity.

Data types:

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

Date: 2014-02-06

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

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

**qubit**by studying the decay of Rabi

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

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

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

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

**frequency**range decreases with increasing

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

Data types:

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.

Data types:

Contributors: Murch, K. W., Ginossar, E., Weber, S. J., Vijay, R., Girvin, S. M., Siddiqi, I.

Date: 2012-08-22

fig3 Experimental device: a transmon **qubit** coupled to a nonlinear superconducting cavity. (a) Circuit diagram of the device; the anharmonic resonator is formed from a meander inductor embedded with a Josephson junction and an interdigitated capacitor. The resonator is isolated from the 50 Ω environment by coupling capacitors C c and coupled to a transmon **qubit** characterized by Josephson and charging energy scales E J and E c respectively. The coupling rate is g . (b) Scanning electron micrograph of the the device with the resonator and **qubit** junctions (lower and upper insets)....When a **frequency** chirped excitation is applied to a classical high-Q nonlinear **oscillator**, its motion becomes dynamically synchronized to the drive and large **oscillation** amplitude is observed, provided the drive strength exceeds the critical threshold for autoresonance. We demonstrate that when such an **oscillator** is strongly coupled to a quantized superconducting **qubit**, both the effective nonlinearity and the threshold become a non-trivial function of the **qubit**-**oscillator** detuning. Moreover, the autoresonant threshold is sensitive to the quantum state of the **qubit** and may be used to realize a high fidelity, latching readout whose speed is not limited by the **oscillator** Q....fig2 Energy levels and the effective nonlinearity λ of the strongly coupled system. (a) The measured coefficient of nonlinear response for a strongly coupled system versus **qubit**-cavity detuning for the **qubit** prepared in the ground state in the low excitation regime. Theory curves for the model Eq. ( eq:1) with N l = 2 (blue), N l = 3 (gray), and for a model of coupled nonlinear classical **oscillators** (red) are shown. The arrows indicate the locations of avoided crossings of the level pairs | 1 , 3 ↔ | 0 , 4 and | 1 , 4 ↔ | 0 , 5 . (Inset) The transmission of the resonator when driven with tone at ω d that occupied the resonator with = 0.4 (black) and = 10 (gray) off-resonant photons. (b) Energy levels of the **qubit**-**oscillator** model with N l = 7 show the avoided crossings in the 4 and 5 excitation manifold. (c) Quantum trajectory simulation of the system exhibits a general trends of increasing effective nonlinearity λ with diminishing **qubit**-cavity detuning ( Δ ) with abrupt reductions associated with avoided crossings in the 4 and 5 excitation manifold. For these simulations, the **qubit** energy levels were modeled as a Duffing nonlinearity....fig2 Measured autoresonance and threshold sensitivity on the level structure and initial **qubit** state. (a) Color plot shows S | 1 versus **qubit** detuning. The dashed line indicates the AR threshold, V | 1 . AR measurements were not taken for small values of the detuning as indicated by the hatched region. (b) Color plot of S | 0 with V | 0 indicated as a solid black line. The AR threshold, V | 1 , is also plotted for comparison as a dashed black line. The two arrows indicate the location of avoided crossings in the 4 and 5 excitation manifold....fig3duffing (a) The transmitted magnitude for chirp sequences with drive voltages that were above (black), near (dashed), and below (gray) the AR threshold. (Inset) Pulse sequence: the **qubit** manipulation pulse was applied immediately before the start of the chirp sequence. (b) The average transmitted magnitude near 400 ns versus drive voltage shows S | 0 (black) and S | 1 (red) for Δ / 2 π = 0.59 GHz. ... When a **frequency** chirped excitation is applied to a classical high-Q nonlinear **oscillator**, its motion becomes dynamically synchronized to the drive and large **oscillation** amplitude is observed, provided the drive strength exceeds the critical threshold for autoresonance. We demonstrate that when such an **oscillator** is strongly coupled to a quantized superconducting **qubit**, both the effective nonlinearity and the threshold become a non-trivial function of the **qubit**-**oscillator** detuning. Moreover, the autoresonant threshold is sensitive to the quantum state of the **qubit** and may be used to realize a high fidelity, latching readout whose speed is not limited by the **oscillator** Q.

Data types:

Contributors: Xia, K., Macovei, M., Evers, J., Keitel, C. H.

Date: 2008-10-14

To study the fidelity of the population transfer between | a and | s , in Fig. subfig:a2s15P, we choose the anti-symmetric state as the initial condition. Applying a continuous-wave TDMF with Rabi **frequency** Ω 0 = 15 γ 0 and detuning δ = 0 , the symmetric state | s reaches its maximum population of 0.90 at time 0.1 γ 0 -1 . After this maximum, the population continues to **oscillate** between | a and | s due to the applied field. This **oscillation** is damped by an overall decay as we include damping with a rate γ 0 . The corresponding concurrence is shown in in Fig. subfig:a2s15C. The concurrence **oscillates** at twice the **frequency** of the population **oscillation**, since both | s and | a are maximally entangled. The local maximum values of the entanglement occur at times n π / 2 δ 2 + Ω 0 2 where either | s or | a is occupied....Coherent control and the creation of entangled states are discussed in a system of two superconducting flux **qubits** interacting with each other through their mutual inductance and identically coupling to a reservoir of harmonic **oscillators**. We present different schemes using continuous-wave control fields or Stark-chirped rapid adiabatic passages, both of which rely on a dynamic control of the **qubit** transition **frequencies** via the external bias flux in order to maximize the fidelity of the target states. For comparison, also special area pulse schemes are discussed. The **qubits** are operated around the optimum point, and decoherence is modelled via a bath of harmonic **oscillators**. As our main result, we achieve controlled robust creation of different Bell states consisting of the collective ground and excited state of the two-**qubit** system....In Fig. subfig:figg2sP, we show results for the population of | s for continuous driving. It can be seen that the population reaches a maximum value, but afterwards exhibits rapid **oscillations** at **frequency** 2 2 Ω 0 , while the amplitude of the subsequent maxima in the population decays as an exponential function exp - γ 0 + γ 12 t until the system approaches its stationary state. This result can be understood by reducing the system to a two-state system only involving in the states | s and | g . The numerical results can be well fitted by the solution of this two-state approximation,...(Color online) Time evolution of the population (solid red line) in the antisymmetric state | a . The idea is to prepare the anti-symmetric state while the two **qubits** are non-degenerate, and only afterwards render the two **qubits** degenerate. For this, the parameters are chosen such that γ 12 = 0.9986 γ 0 , λ = 50 γ 0 , δ = 50 γ 0 , Ω 0 = 50 γ 0 . The two **qubit** transition **frequencies** are adjusted via time-dependent bias fluxes, such that the **frequency** difference Δ t (dashed black line) changes from 18 γ 0 to zero as a cosine function during the time period 120 γ 0 -1 to 160 γ 0 -1 . The driving field (dash-dotted blue line) is turned off from its initial value Ω 0 in the period 165 γ 0 -1 to 175 γ 0 -1 ....Our model system consists of two flux **qubits** coupled to each other through their mutual inductance M and to a reservoir of harmonic **oscillators** modeled as an LC circuit, see Fig. fig:system....An example for this is shown in Fig. fig:figPadtn. Initially, the two **qubits** have a **frequency** difference Δ t = 0 = Δ 0 = 18 γ 0 . Applying a continuous TDMF Ω during 0 ≤ γ 0 t ≤ 165 allows to populate the antisymmetric state, as can be seen in Fig. fig:figPadtn. After a certain time ( γ 0 t = 120 in our example), the bias fluxes are continuously adjusted such that the two **qubits** become degenerate, Δ γ 0 t ≥ 160 = 0 . It can be seen from Fig. fig:figPadtn that a preparation fidelity for the antisymmetric state in the degenerate two-**qubit** system of about F = 0.94 is achieved. Finally, the TDMF is switched off as well in the time period 165 ≤ γ 0 t ≤ 175 , demonstrating that it is not required to preserve the population in the antisymmetric state. It should be noted that this scheme does not rely on a delicate choice and control of parameters, as it is the case, e.g., for state preparation via special-area pulses....fig:BellSCRAPdtn(Color online) Creation of superpositions of Bell states controlled by the static detuning δ 0 . The populations ρ ± in states | Φ + (dashed red line) and | Φ - (dash-dotted green line) exhibit periodic **oscillations** as a function of δ 0 . The maximum concurrence C is larger than 0.95 (solid blue line)....fig:BellSCRAP(Color online) Creation of Bell states | Φ ± using the SCRAP technique from the ground state. The solid red line denotes population ρ + in | Φ + , while the dashed green line shows population ρ - in | Φ - . The concurrence (dash-dotted blue line) has a maximum value of 0.94 . The black dash double dotted line indicates the applied SCRAP pulse Rabi **frequency**, while the thick blue line shows the time-dependent Stark detuning....fig:SCRAPg2s(Color online) Robust populating the symmetric state from the ground state via the SCRAP technique for γ 12 = 0.9986 γ 0 , and λ = 50 γ 0 . The solid black line shows the population of the desired symmetric state, while the thick solid blue line and the dash-dotted red line are the time-dependent Rabi **frequency** and detuning required for SCRAP....fig:systemTwo superconducting flux **qubits** interacting with each other through their mutual inductance M and damped to a common reservoir modeled as an LC circuit. The individual bias fluxes are varied dynamically in order to control the **qubit** transition **frequencies** around the optimum point. In the figure, crosses indicate Josephson junctions, whereas the bottom circuit loop visualizes the bath....where Φ 0 = h / 2 e is the flux quantum, the system in Fig. fig:system can be described by the total Hamiltonian H = H Q + H B . The two-**qubit** Hamiltonian H Q in two-level approximation and rotating wave approximation is given by ... Coherent control and the creation of entangled states are discussed in a system of two superconducting flux **qubits** interacting with each other through their mutual inductance and identically coupling to a reservoir of harmonic **oscillators**. We present different schemes using continuous-wave control fields or Stark-chirped rapid adiabatic passages, both of which rely on a dynamic control of the **qubit** transition **frequencies** via the external bias flux in order to maximize the fidelity of the target states. For comparison, also special area pulse schemes are discussed. The **qubits** are operated around the optimum point, and decoherence is modelled via a bath of harmonic **oscillators**. As our main result, we achieve controlled robust creation of different Bell states consisting of the collective ground and excited state of the two-**qubit** system.

Data types: