### 63448 results for qubit oscillator frequency

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

Date: 2008-10-14

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 ....Robust creation of entangled states of two coupled flux **qubits** via dynamic control of the transition **frequencies**...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....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: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 ...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....(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 ....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,...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 ...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....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....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....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: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: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: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. ... 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.

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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. ...Non-Markovian **qubit** decoherence during dispersive readout...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....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. ... 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: Liberti, G., Zaffino, R. L., Piperno, F., Plastina, F.

Date: 2005-11-21

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-Bana...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)....Entanglement of a **qubit** coupled to a resonator in the adiabatic regime...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....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. ... 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 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 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) 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) 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.
...(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.
...(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.

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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 ....(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 ....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. ...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....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 ...Reduced Visibility of Rabi Oscillations in Superconducting **Qubits** ... 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: 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....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...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....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....Macroscopic quantum **oscillator** based on a flux **qubit**...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** **flux**-**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....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....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: 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....**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)....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....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....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)....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)....Oscill...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....Coherent dynamics of a flux **qubit** coupled to a harmonic **oscillator**...Qub ... 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.

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Contributors: Oxtoby, Neil P., Gambetta, Jay, Wiseman, H. M.

Date: 2007-06-24

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...Model for monitoring of a charge **qubit** using a radio-**frequency** quantum point contact including experimental imperfections...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)....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....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...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 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....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...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...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).

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Contributors: Greenberg, Ya. S., Izmalkov, A., Grajcar, M., Il'ichev, E., Krech, W., Meyer, H. -G.

Date: 2002-08-07

In our method a resonant tank circuit with known inductance L T , capacitance C T and quality factor Q T is coupled with a target Josephson circuit through the mutual inductance M (Fig. fig1). The method was successfully applied to a three-**junction** **qubit** in classical regime, when the hysteretic dependence of ground-state energy on the external magnetic flux was reconstructed in accordance to the predictions of Ref. ...Phase **qubit** coupled to a tank circuit....Method for direct observation of coherent quantum oscillations in a superconducting phase **qubit**...Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems were so far restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a new method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Detailed calculation for zero and non-zero temperature are made for the case of persistent current **qubit**. According to the estimates for dephasing and relaxation times, the effect can be detected using conventional rf circuitry, with Rabi **frequency** in MHz range....Time-domain observations of coherent oscillations between quantum states in mesoscopic superconducting systems were so far restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a new method for direct observation of Rabi oscillations in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi oscillations of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Detailed calculation for zero and non-zero temperature are made for the case of persistent current **qubit**. According to the estimates for dephasing and relaxation times, the effect can be detected using conventional rf circuitry, with Rabi **frequency** in MHz range....In our method a resonant tank circuit with known inductance L T , capacitance C T and quality factor Q T is coupled with a target Josephson circuit through the mutual inductance M (Fig. fig1). The method was successfully applied to a three-junction **qubit** in classical regime, when the hysteretic dependence of ground-state energy on the external magnetic flux was reconstructed in accordance to the predictions of Ref. ... Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems were so far restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a new method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Detailed calculation for zero and non-zero temperature are made for the case of persistent current **qubit**. According to the estimates for dephasing and relaxation times, the effect can be detected using conventional rf circuitry, with Rabi **frequency** in MHz range.

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

Date: 2008-09-08

We experimentally demonstrate the coherent **oscillations** of a tunable superconducting flux **qubit** by manipulating its energy potential with a nanosecond-long pulse of magnetic flux. The occupation probabilities of two persistent current states **oscillate** at a **frequency** ranging from 6 GHz to 21 GHz, tunable via the amplitude of the flux pulse. The demonstrated operation mode allows to realize quantum gates which take less than 100 ps time and are thus much faster compared to other superconducting **qubits**. An other advantage of this type of **qubit** is its insensitivity to both thermal and magnetic field fluctuations....The measurement process that we used to observe coherent **oscillations** consists of several steps shown in Fig. fig:3(a). Each step is realized by applying a combination of magnetic fluxes Φ x and Φ c as indicated by numbers in Fig. fig:2(b). The first step in our measurement is the initialization of the system in a defined flux state (1). Starting from a double well at Φ x ≅ Φ 0 / 2 with high barrier, the potential is tilted by changing Φ x until it has only a single minimum (left or right, depending on the amplitude and polarity of the applied flux pulse). This potential shape is maintained long enough to ensure the relaxation to the ground state. Afterwards the potential is tuned back to the initial double-well state (2). The high barrier prevents any tunneling and the **qubit** is thus initialized in the chosen potential well. Next, the barrier height is lowered to an intermediate level (3) that preserves the initial state and allows to use just a small-amplitude Φ c flux pulse for the subsequent manipulation. The following Φ c -pulse transforms the potential into a single well (4). The Φ c -pulse duration Δ t is in the nanosecond range. The relative phase of the ground and the first excited states evolves depending on the energy difference between them. Once Φ c -pulse is over, the double well is restored and the system is measured in the basis | L | R (5). The readout of the **qubit** flux state is done by applying a bias current ramp to the dc SQUID and recording its switching current to the voltage state....(a) The measured double SQUID flux Φ in dependence of Φ x , plotted for two different values of Φ c and initial preparation in either potential well. (b) Position of the switching points (dots) in the Φ c - Φ x parameter space. Numbered tags indicate the working points for **qubit** manipulation at which the **qubit** potential has a shape as indicated in the insets....Calculated energy spacing of the first (solid line), second (dashed line) and third (dotted line) energy levels with respect to the ground state in the single well potential, plotted vs. the control flux amplitude Φ c 3 . Circles are the experimentally observed **oscillation** **frequencies** for the corresponding pulse amplitudes....We experimentally demonstrate the coherent oscillations of a tunable superconducting flux **qubit** by manipulating its energy potential with a nanosecond-long pulse of magnetic flux. The occupation probabilities of two persistent current states oscillate at a **frequency** ranging from 6 GHz to 21 GHz, tunable via the amplitude of the flux pulse. The demonstrated operation mode allows to realize quantum gates which take less than 100 ps time and are thus much faster compared to other superconducting **qubits**. An other advantage of this type of **qubit** is its insensitivity to both thermal and magnetic field fluctuations....The flux pattern is repeated for 10 2 - 10 4 times in order to evaluate the probability P L = L | Ψ f i n a l 2 of occupation of the left state at the end of the manipulation. By changing the duration Δ t of the manipulation pulse Φ c , we observed coherent **oscillations** between the occupations of the states | L and | R shown in Fig. fig:4(a). The **oscillation** **frequency** could be tuned between 6 and 21 GHz by changing the pulse amplitude Δ Φ c . These **oscillations** persist when the potential is made slightly asymmetric by varying the value Φ x 1 . As it is shown in Fig. fig:4(b), detuning from the symmetric potential by up to ± 2.9 m Φ 0 only slightly changes the amplitude and symmetry of the **oscillations**. When the **qubit** was initially prepared in | R state instead of | L state we observed similar **oscillations**....Calculated energy spacing of the first (solid line), second (dashed line) and third (dotted line) energy levels with respect to the ground state in the single well potential, plotted vs. the control **flux **amplitude Φ c 3 . Circles are the experimentally observed oscillation frequencies for the corresponding pulse amplitudes....Assuming identical junctions and negligible inductance of **the** smaller loop ( l ≪ L ), **the** system dynamics is equivalent to **the** motion of a particle with **the** Hamiltonian H = p 2 2 M + Φ b 2 L 1 2 ϕ - ϕ x 2 - β ϕ c cos ϕ , where ϕ = Φ / Φ b is **the** spatial coordinate of **the** equivalent particle, p is **the** relative conjugate momentum, M = C Φ b 2 is **the** effective mass, ϕ x = Φ x / Φ b and ϕ c = π Φ c / Φ 0 are **the** normalized flux controls, and β ϕ c = 2 I 0 L / Φ b cos ϕ c , with Φ 0 = h / 2 e and Φ b = Φ 0 / 2 π . For β **l **shape is used for **qubit** initialization and readout. **The** single well, or more exactly **the** two lowest energy states | 0 and | 1 in this well, is used for **the** coherent evolution of **the** **qubit**....Assuming identical junctions and negligible inductance of the smaller loop ( l ≪ L ), the system dynamics is equivalent to the motion of a particle with the Hamiltonian H = p 2 2 M + Φ b 2 L 1 2 ϕ - ϕ x 2 - β ϕ c cos ϕ , where ϕ = Φ / Φ b is the spatial coordinate of the equivalent particle, p is the relative conjugate momentum, M = C Φ b 2 is the effective mass, ϕ x = Φ x / Φ b and ϕ c = π Φ c / Φ 0 are the normalized flux controls, and β ϕ c = 2 I 0 L / Φ b cos ϕ c , with Φ 0 = h / 2 e and Φ b = Φ 0 / 2 π . For β **qubit** initialization and readout. The single well, or more exactly the two lowest energy states | 0 and | 1 in this well, is used for the coherent evolution of the **qubit**....(a) Schematic of the flux **qubit** circuit. (b) The control flux Φ c changes the potential barrier between the two flux states | L and | R , here Φ x = 0.5 Φ 0 . (c) Effect of the control flux Φ x on the potential symmetry....Coherent oscillations in a superconducting tunable flux **qubit** manipulated without microwaves...**The** investigated circuit, shown in Fig. fig:1(a), is a double SQUID consisting of a superconducting loop of inductance L = 85 pH, interrupted by a small dc SQUID of loop inductance l = 6 pH. This dc SQUID is operated as a single Josephson junction (JJ) whose critical current is tunable by an external magnetic field. Each of **the** two JJs embedded in **the** dc SQUID has a critical current I 0 = 8 μ A and capacitance C = 0.4 pF. **The** **qubit** is manipulated by changing two magnetic fluxes Φ x and Φ c , applied to **the** large and small loops by means of two coils of mutual inductance M x = 2.6 pH and M c = 6.3 pH, respectively. **The** readout of **the** **qubit** flux is performed by measuring **the** switching current of an unshunted dc SQUID, which is inductively coupled to **the** **qubit** . **The** circuit was manufactured by Hypres using standard Nb/AlO x /Nb technology in a 100 A/cm 2 critical current density process. **The** dielectric material used for junction isolation is SiO 2 . **The** whole circuit is designed gradiometrically in order to reduce magnetic flux pick-up and spurious flux couplings between **the** loops. **The** JJs have dimensions of 3 × 3 μ m 2 and **the** entire device occupied a space of 230 × 430 μ m 2 . All **the** measurements have been performed at a sample temperature of 15 mK. **The** currents generating **the** two fluxes Φ x and Φ c were supplied via coaxial cables including 10 dB attenuators at **the** 1K-pot stage of a dilution refrigerator. To generate **the** flux Φ c , a bias-tee at room temperature was used to combine **the** outputs of a current source and a pulse generator. For biasing and sensing **the** readout dc SQUID, we used superconducting wires and metal powder filters at **the** base temperature, as well as attenuators and low-pass filters with a cut-**off** **frequency** of 10 kHz at **the** 1K-pot stage. **The** chip holder with **the** powder filters was surrounded by one superconducting and two cryoperm shields....The **oscillation** **frequency** ω 0 depends on the amplitude of the manipulation pulse Δ Φ c since it determines the shape of the single well potential and the energy level spacing E 1 - E 0 . A pulse of larger amplitude Δ Φ c generates a deeper well having a larger level spacing, which leads to a larger **oscillation** **frequency** as shown in Fig. fig:4(a). In Fig. fig:5, we plot the energy spacing between the ground state and the three excited states (indicated as E k - E 0 / h with k=1,2,3) versus the flux Φ c 3 = Φ c 2 + Δ Φ c obtained from a numerical simulation of our system using the experimental parameters. In the same figure, we plot the measured **oscillation** **frequencies** for different values of Φ c (open circles). Excellent agreement between simulation (solid line) and data strongly supports our interpretation. The fact that a small asymmetry in the potential does not change the **oscillation** **frequency**, as shown in Fig. fig:4(b), is consistent with the interpretation as the energy spacing E 1 - E 0 is only weakly affected by small variations of Φ x . This provides protection against noise in the controlling flux Φ x ....Probability to measure the state in dependence of the pulse duration Δ t for the **qubit** initially prepared in the state, and for (a) different pulse amplitudes Δ Φ c , resulting in the indicated oscillation **frequency**, and (b) for different potential symmetry by detuning Φ x from Φ 0 / 2 by the indicated amount....**The** oscillation **frequency** ω 0 depends on **the** amplitude of **the** manipulation pulse Δ Φ c since it determines **the** shape of **the** single well potential and **the** energy level spacing E 1 - E 0 . A pulse of larger amplitude Δ Φ c generates a deeper well having a larger level spacing, which leads to a larger oscillation **frequency** as shown in Fig. fig:4(a). In Fig. fig:5, we plot **the** energy spacing between **the** ground state and **the** three excited states (indicated as E k - E 0 / h with k=1,2,3) versus **the** flux Φ c 3 = Φ c 2 + Δ Φ c obtained from a numerical simulation of our system using **the** experimental parameters. In **the** same figure, we plot **the** measured oscillation frequencies for different values of Φ c (open circles). Excellent agreement between simulation (solid line) and data strongly supports our interpretation. **The** fact that a small asymmetry in **the** potential does not change **the** oscillation **frequency**, as shown in Fig. fig:4(b), is consistent with **the** interpretation as **the** energy spacing E 1 - E 0 is only weakly affected by small variations of Φ x . This provides protection against noise in **the** controlling flux Φ x ....Probability to measure the state in dependence of the pulse duration Δ t for the **qubit** initially prepared in the state, and for (a) different pulse amplitudes Δ Φ c , resulting in the indicated **oscillation** **frequency**, and (b) for different potential symmetry by detuning Φ x from Φ 0 / 2 by the indicated amount....**The** flux pattern is repeated for 10 2 - 10 4 times in order to evaluate **the** probability P L = L | Ψ f i n a **l **2 of occupation of **the** left state at **the** end of **the** manipulation. By changing **the** duration Δ t of **the** manipulation pulse Φ c , we observed coherent oscillations between **the** occupations of **the** states | L and | R shown in Fig. fig:4(a). **The** oscillation **frequency** could be tuned between 6 and 21 GHz by changing **the** pulse amplitude Δ Φ c . These oscillations persist when **the** potential is made slightly asymmetric by varying **the** value Φ x 1 . As it is shown in Fig. fig:4(b), detuning from **the** symmetric potential by up to ± 2.9 m Φ 0 only slightly changes **the** amplitude and symmetry of **the** oscillations. When **the** **qubit** was initially prepared in | R state instead of | L state we observed similar oscillations....(a) The measured double SQUID **flux **Φ in dependence of Φ x , plotted for two different values of Φ c and initial preparation in either potential well. (b) Position of the switching points (dots) in the Φ c - Φ x parameter space. Numbered tags indicate the working points for **qubit** manipulation at which the **qubit** potential has a shape as indicated in the insets....(a) Schematic of the flux **qubit** circuit. (b) The control **flux **Φ c changes the potential barrier between the two **flux **states | L and | R , here Φ x = 0.5 Φ 0 . (c) Effect of the control **flux **Φ x on the potential symmetry....The investigated circuit, shown in Fig. fig:1(a), is a double SQUID consisting of a superconducting loop of inductance L = 85 pH, interrupted by a small dc SQUID of loop inductance l = 6 pH. This dc SQUID is operated as a single Josephson junction (JJ) whose critical current is tunable by an external magnetic field. Each of the two JJs embedded in the dc SQUID has a critical current I 0 = 8 μ A and capacitance C = 0.4 pF. The **qubit** is manipulated by changing two magnetic fluxes Φ x and Φ c , applied to the large and small loops by means of two coils of mutual inductance M x = 2.6 pH and M c = 6.3 pH, respectively. The readout of the **qubit** flux is performed by measuring the switching current of an unshunted dc SQUID, which is inductively coupled to the **qubit** . The circuit was manufactured by Hypres using standard Nb/AlO x /Nb technology in a 100 A/cm 2 critical current density process. The dielectric material used for junction isolation is SiO 2 . The whole circuit is designed gradiometrically in order to reduce magnetic flux pick-up and spurious flux couplings between the loops. The JJs have dimensions of 3 × 3 μ m 2 and the entire device occupied a space of 230 × 430 μ m 2 . All the measurements have been performed at a sample temperature of 15 mK. The currents generating the two fluxes Φ x and Φ c were supplied via coaxial cables including 10 dB attenuators at the 1K-pot stage of a dilution refrigerator. To generate the flux Φ c , a bias-tee at room temperature was used to combine the outputs of a current source and a pulse generator. For biasing and sensing the readout dc SQUID, we used superconducting wires and metal powder filters at the base temperature, as well as attenuators and low-pass filters with a cut-off **frequency** of 10 kHz at the 1K-pot stage. The chip holder with the powder filters was surrounded by one superconducting and two cryoperm shields. ... We experimentally demonstrate the coherent **oscillations** of a tunable superconducting flux **qubit** by manipulating its energy potential with a nanosecond-long pulse of magnetic flux. The occupation probabilities of two persistent current states **oscillate** at a **frequency** ranging from 6 GHz to 21 GHz, tunable via the amplitude of the flux pulse. The demonstrated operation mode allows to realize quantum gates which take less than 100 ps time and are thus much faster compared to other superconducting **qubits**. An other advantage of this type of **qubit** is its insensitivity to both thermal and magnetic field fluctuations.

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