### 63603 results for qubit oscillator frequency

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

Date: 2012-08-02

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

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

Date: 2014-02-06

The condition, ∂ Ω R / ∂ ε = 0 , is satisfied when ε = 0 or δ ω m w = δ ω - Ω R 0 2 / ω 01 . For Ω R 0 / 2 π = 1.52 GHz and ω 01 / 2 π = 6.400 GHz, the latter condition is calculated to be δ ω m w / 2 π = - 295 MHz, slightly different from the minimum of Γ R s t seen in Fig. G R f R 1 p 5 (b). The difference is due to the deviation from the linear approximation in Eq. ( **fRfull**), Ω R 0 ∝ ε m w / ω 01 . Figure GRfR1p5(c) shows the calculation of Ω R as **a** function of ε , based on Eq. ( **fRfull**). The Rabi **frequency** Ω R 0 at the shifted resonance decreases as ε increases, while Ω R , for **a** fixed microwave **frequency** of ω m w / 2 π = 6.1 GHz, has **a** minimum of approximately ω 01 / 2 π = 6.4 GHz. Here in the first order, Ω R is insensitive to the fluctuation of ε ....(Color online) Power spectrum density of flux fluctuations S n φ ω extracted from the Rabi **oscillation** measurements in the first ( ε / 2 π = 4.16 GHz) and second cooldowns. The PSDs obtained from the spin-echo and energy relaxation measurements in the second cooldown are also plotted. The black solid line is the 1/ f spectrum extrapolated from the FID measurements in the second cooldown. The purple dashed line is the estimated Johnson noise from a 50 Ω microwave line coupled to the **qubit** by a mutual inductance of 1.2 pH and nominally cooled to 35 mK. The pink dotted line is a Lorentzian, S n φ m o d e l ω = S h ω w 2 / ω 2 + ω w 2 , and the orange solid line is the sum of the Lorentzian and the Johnson noise. Here the parameters are S h = 3.6 × 10 -19 r a d -1 s and ω w / 2 π = 2.7 × 10 7 H z ....In Fig. GRfR1p5(**a**), δ ω as **a** function of Ω R 0 is plotted together with the well-known Bloch–Siegert shift, δ ω B S = 1 4 Ω R 0 2 ω 01 , obtained from the second-order perturbation theory. Fixed parameters for the calculation are Δ / 2 π = 4.869 and ε / 2 π = 4.154 GHz ( ω 01 / 2 π = 6.400 GHz). We find that δ ω B S overestimates δ ω when Ω R 0 / 2 π 800 MHz. The deviation from the Bloch–Siegert shift is due to the component of the ac flux drive that is parallel to the qubit’s energy eigenbasis; this component is not averaged out when Ω R is comparable to ω m w ....(Color online) Rabi oscillation curves with different Rabi frequencies Ω R measured at different static flux bias ε . At each Ω R , δ ω m w is chosen to minimize dephasing due to quasistatic flux noise. The red lines are the fitting curves. In the measurements shown in the middle and bottom panels, only parts of the oscillations are monitored so that we can save measurement time while the envelopes of Rabi oscillations are captured. The inset is a magnification of the data in the bottom panel together with the fitting curve....(Color online) (a) Numerically calculated shift of the resonant **frequency** δ ω (black open circles) and the Bloch–Siegert shift δ ω B S (blue line). (b) Numerically calculated decay rate Γ R s t (black open circles) and Rabi **frequency** Ω R (red solid triangles) as functions of the detuning δ ω m w from ω 01 . The purple solid line is a fit based on Eq. ( fRfull). The measured 1/ e decay rates Γ R 1 / e at ε / 2 π = 4.16 GHz for the range of Rabi **frequencies** Ω R / 2 π between 1.5 and 1.6 GHz (blue solid circles) are also plotted. (c) Calculated Rabi **frequency** Ω R , based on Eq. ( fRfull), as a function of ε for the cases (i) ω m w = ω 01 + δ ω (black solid line) and (ii) ω m w / 2 π = 6.1 GHz (red dashed line). The upper axis indicates ω 01 , corresponding to ε in the bottom axis. (d) The measured 1 / e decay rate of the Rabi **oscillations**, Γ R 1 / e , at ε = 0 and as a function of Ω R 0 . The red solid line indicates 3 4 Γ 1 obtained independently....Flux **qubit** noise spectroscopy using Rabi oscillations under strong driving conditions...In Fig. GRfR1p5(a), δ ω as a function of Ω R 0 is plotted together with the well-known Bloch–Siegert shift, δ ω B S = 1 4 Ω R 0 2 ω 01 , obtained from the second-order perturbation theory. Fixed parameters for the calculation are Δ / 2 π = 4.869 and ε / 2 π = 4.154 GHz ( ω 01 / 2 π = 6.400 GHz). We find that δ ω B S overestimates δ ω when Ω R 0 / 2 π 800 MHz. The deviation from the Bloch–Siegert shift is due to the component of the ac flux drive that is parallel to the ** qubit’s** energy eigenbasis; this component is not averaged out when Ω R is comparable to ω m w ....Parameters in calculations and measurements in units of GHz. In the first column, cal: δ ω Ω R 0 stands for the calculation to study the shift of the resonant

**frequency**, and cal: Γ R s t δ ω m w stands for the calculation to study the decay of Rabi

**oscillations**due to quasistatic flux noise. “Optimal" in the last column means that at each ε m w , ω m w is chosen to minimize dephasing due to quasistatic flux noise....We infer the high-

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

**qubit**by studying the decay of Rabi

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

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

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

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

**frequency**range decreases with increasing

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

**frequency**δ ω (black open circles) and the Bloch–Siegert shift δ ω B S (blue line). (b) Numerically calculated decay rate Γ R s t (black open circles) and Rabi

**frequency**Ω R (red solid triangles) as functions of the detuning δ ω m w from ω 01 . The purple solid line is a fit based on Eq. ( fRfull). The measured 1/ e decay rates Γ R 1 / e at ε / 2 π = 4.16 GHz for the range of Rabi frequencies Ω R / 2 π between 1.5 and 1.6 GHz (blue solid circles) are also plotted. (c) Calculated Rabi

**frequency**Ω R , based on Eq. ( fRfull), as a function of ε for the cases (i) ω m w = ω 01 + δ ω (black solid line) and (ii) ω m w / 2 π = 6.1 GHz (red dashed line). The upper axis indicates ω 01 , corresponding to ε in the bottom axis. (d) The measured 1 / e decay rate of the Rabi oscillations, Γ R 1 / e , at ε = 0 and as a function of Ω R 0 . The red solid line indicates 3 4 Γ 1 obtained independently....(Color online) Rabi

**oscillation**curves with different Rabi

**frequencies**Ω R measured at different static flux bias ε . At each Ω R , δ ω m w is chosen to minimize dephasing due to quasistatic flux noise. The red lines are the fitting curves. In the measurements shown in the middle and bottom panels, only parts of the

**oscillations**are monitored so that we can save measurement time while the envelopes of Rabi

**oscillations**are captured. The inset is a magnification of the data in the bottom panel together with the fitting curve....(Color online) Power spectrum density of flux fluctuations S n φ ω extracted from the Rabi oscillation measurements in the first ( ε / 2 π = 4.16 GHz) and second cooldowns. The PSDs obtained from the spin-echo and energy relaxation measurements in the second cooldown are also plotted. The black solid line is the 1/ f spectrum extrapolated from the FID measurements in the second cooldown. The purple dashed line is the estimated Johnson noise from a 50 Ω microwave line coupled to the qubit by a mutual inductance of 1.2 pH and nominally cooled to 35 mK. The pink dotted line is a Lorentzian, S n φ m o d e l ω = S h ω w 2 / ω 2 + ω w 2 , and the orange solid line is the sum of the Lorentzian and the Johnson noise. Here the parameters are S h = 3.6 × 10 -19 r a d -1 s and ω w / 2 π = 2.7 × 10 7 H z ....In the Rabi

**oscillation**measurements, a microwave pulse is applied to the

**qubit**followed by a readout pulse, and P s w as a function of the microwave pulse length is measured. First, we measure the Rabi

**oscillation**decay at ε = 0 , where the quasistatic noise contribution is negligible. Figure GRfR1p5(d) shows the measured 1 / e decay rate of the Rabi

**oscillations**Γ R 1 / e as a function of Ω R 0 . For Ω R 0 / 2 π up to 400 MHz, Γ R 1 / e is approximately 3 Γ 1 / 4 , limited by the energy relaxation, and S Δ Ω R 0 is negligible. For Ω R 0 / 2 π from 600 MHz to 2.2 GHz, Γ R 1 / e > 3 Γ 1 / 4 . A possible origin of this additional decoherence is fluctuations of ε m w , δ ε m w : Ω R 0 is first order sensitive to δ ε m w , which is reported to be proportional to ε m w itself. Next, the decay for the case ε ≈ Δ is studied. To observe the contribution from quasistatic flux noise, the Rabi

**oscillation**decay as a function of ω m w is measured, where the contribution from the other sources is expected to be almost constant. Figure GRfR1p5(b) shows Γ R 1 / e at ε / 2 π = 4.16 GHz as a function of δ ω m w while keeping Ω R / 2 π between 1.5 and 1.6 GHz. Besides the offset and scatter, the trend of Γ R 1 / e agrees with that of the simulated Γ R s t . This result indicates that numerical calculation properly evaluates δ ω m w minimizing Γ R s t . Finally, the decay for the case ε ≈ Δ as a function of ε m w , covering a wide range of Ω R , is measured (Fig. Rabis)....We infer the high-

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

**qubit**by studying the decay of Rabi oscillations under strong driving conditions. The large anharmonicity of the

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

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

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

**frequency**range decreases with increasing

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

**oscillation**, $1/f$ noise...Parameters in calculations and measurements in units of GHz. In the first column, cal: δ ω Ω R 0 stands for the calculation to study the shift of the resonant

**frequency**, and cal: Γ R s t δ ω m w stands for the calculation to study the decay of Rabi oscillations due to quasistatic flux noise. “Optimal" in the last column means that at each ε m w , ω m w is chosen to minimize dephasing due to quasistatic flux noise....The condition, ∂ Ω R / ∂ ε = 0 , is satisfied when ε = 0 or δ ω m w = δ ω - Ω R 0 2 / ω 01 . For Ω R 0 / 2 π = 1.52 GHz and ω 01 / 2 π = 6.400 GHz, the latter condition is calculated to be δ ω m w / 2 π = - 295 MHz, slightly different from the minimum of Γ R s t seen in Fig. G R f R 1 p 5 (b). The difference is due to the deviation from the linear approximation in Eq. ( fRfull), Ω R 0 ∝ ε m w / ω 01 . Figure GRfR1p5(c) shows the calculation of Ω R as a function of ε , based on Eq. ( fRfull). The Rabi

**frequency**Ω R 0 at the shifted resonance decreases as ε increases, while Ω R , for a fixed microwave

**frequency**of ω m w / 2 π = 6.1 GHz, has a minimum of approximately ω 01 / 2 π = 6.4 GHz. Here in the first order, Ω R is insensitive to the fluctuation of ε ....In the Rabi oscillation measurements,

**a**microwave pulse is applied to the qubit followed by

**a**readout pulse, and P s w as

**a**function of the microwave pulse length is measured. First, we measure the Rabi oscillation decay at ε = 0 , where the quasistatic noise contribution is negligible. Figure GRfR1p5(d) shows the measured 1 / e decay rate of the Rabi oscillations Γ R 1 / e as

**a**function of Ω R 0 . For Ω R 0 / 2 π up to 400 MHz, Γ R 1 / e is approximately 3 Γ 1 / 4 , limited by the energy relaxation, and S Δ Ω R 0 is negligible. For Ω R 0 / 2 π from 600 MHz to 2.2 GHz, Γ R 1 / e > 3 Γ 1 / 4 . A possible origin of this additional decoherence is fluctuations of ε m w , δ ε m w : Ω R 0 is first order sensitive to δ ε m w , which is reported to be proportional to ε m w itself. Next, the decay for the case ε ≈ Δ is studied. To observe the contribution from quasistatic flux noise, the Rabi oscillation decay as

**a**function of ω m w is measured, where the contribution from the other sources is expected to be almost constant. Figure GRfR1p5(b) shows Γ R 1 / e at ε / 2 π = 4.16 GHz as

**a**function of δ ω m w while keeping Ω R / 2 π between 1.5 and 1.6 GHz. Besides the offset and scatter, the trend of Γ R 1 / e agrees with that of the simulated Γ R s t . This result indicates that numerical calculation properly evaluates δ ω m w minimizing Γ R s t . Finally, the decay for the case ε ≈ Δ as

**a**function of ε m w , covering

**a**wide range of Ω R , is measured (Fig. Rabis). ... We infer the high-

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

**qubit**by studying the decay of Rabi

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

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

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

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

**frequency**range decreases with increasing

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

Files:

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

Date: 2013-01-22

Experiments towards realizing a readout of superconducting **qubits** by using ballistic Josephson vortices are reported. We measured the microwave radiation induced by a fluxon moving in an annular Josephson junction. By coupling a flux **qubit** as a current dipole to the annular junction, we detect periodic variations of the fluxon's **oscillation** **frequency** versus magnetic flux through the **qubit**. We found that the scattering of a fluxon on a current dipole can lead to the acceleration of a fluxon regardless of a dipole polarity. We use the perturbation theory and numerical simulations of the perturbed sine-Gordon equation to analyze our results....Long Josephson junction, fluxon, Josephson vortex, flux **qubit**, **qubit** readout...Optical photograph of the chip with the annular Josephson junction on the right part and experimental set-up schematics. Left part shows the zoom into the area with the flux **qubit** with a coupling loop (yellow loop) and control line (green loop). Red crosses indicate the positions of three Josephson junctions in the flux **qubit** loop....Relative **frequency** deviation from equilibrium δ ν / ν 0 of the fluxon oscillation **frequency** versus bias current. Black line shows the result of perturbation approach, while the red line depicts results of direct numerical simulations of the PSGE equation ( PSGEm) with a d = 1 . The blue curve corresponds to the case with a d = 0.2 ....We would like to employ fluxons for developing a fast and sensitive magnetic field detector for measurements of superconducting qubits. In this Letter**, **we report direct measurements of electromagnetic radiation from a fluxon moving in an annular Josephson junction (AJJ). The radiation is detected by using a microstrip antenna capacitively coupled to** the **AJJ. Furthermore**, **we place a **flux **qubit close to** the **long junction and couple them magnetically with a superconducting loop (see Fig. AJJ+Qubit). This coupling scheme makes** the **fluxon interact with a current dipole formed by** the **electrodes of** the **loop coupled to** the **qubit. The time delay of** the **fluxon can be detected as a frequency shift of** the **electromagnetic radiation emitted from** the **junction. This shift provides information about** the **state of the **flux **qubit....Using the possibility to directly detect radiation of the fluxon resonant **oscillations**, we have performed systematic measurements of the dependence of the fluxon velocity versus bias current - the current-voltage characteristics - measured in the **frequency** domain (see Fig. ZFS). This approach provides an easy access to study the fine structure of the current-voltage curve as the precision of **frequency** measurements is by several orders of magnitude greater than the resolution of direct dc voltage measurement....An annular Josephson junction with a trapped fluxon coupled to a flux **qubit**....Modulation of the fluxon’s oscillation **frequency** due to the coupling to the** flux** qubit. Every point consists of 100 averages. Bias current was set at γ = 0.521 , w ≃ 9.1 ....Josephson vortex coupled to a flux **qubit**...To couple a **flux **qubit to** the **fluxon inside an annular Josephson junction**, **it is necessary to engineer an interaction between two orthogonal magnetic dipoles. To facilitate this interaction**, **we have added a superconducting coupling loop embracing a **flux **qubit**, **as shown in Fig. AJJ+Qubit. The current induced in** the **coupling loop attached to** the **AJJ is proportional to** the **persistent current in the **flux **qubit. Thus**, **the persistent current in** the **qubit manifests itself in** the **AJJ as a current dipole with an amplitude μ on top of** the **homogeneous background of bias current. When fluxon scatters on a positive current dipole - it first gets accelerated and then decelerated by** the **dipole poles. In** the **ideal case of absence of damping and bias current**, **the sign of frequency change δ ν is determined only by polarity of** the **dipole. In** the **presence of finite damping and homogeneous bias current**, **situation completely changes - as** the **total propagation time becomes dependent on** the **complex interplay between bias current**, **current dipole strength and damping....By numerically solving ( FKE1)-( FKE2) with the additional condition u - l / 2 = u l / 2 one can calculate an equilibrium trajectory in phase space for fluxon **oscillations** in the AJJ with the current dipole and estimate a deviation of fluxon **oscillation** **frequency** from the unperturbed case δ ν = ν μ - ν 0 , where ν 0 is the **oscillation** **frequency** for μ = 0 . Black line in Fig. FD shows the dependence of relative deviation δ ν / ν 0 versus bias current γ calculated from the perturbation theory for the following set of system parameters: l = 20 , α = 0.02 , μ = 0.05 , d = 2 . The deviation δ ν is large and negative for small bias currents γ ≪ 0.1 , what means that the fluxon is being slowed down by the current dipole and eventually can be pinned at the dipole if the bias current is too small. Surprisingly, for larger currents γ > 0.05 the sign of δ ν becomes positive meaning that the current dipole accelerates the fluxon. To understand this phenomenon, we need to look at the Eq. ( FKE1) and notice that the effective damping term α e = α u 1 - u 2 has a non-monotonic behavior. When increasing the fluxon velocity u , the effective damping is increasing for u ≤ 1 / 3 and then starts decreasing. This means that deceleration (acceleration) is favorable for low (high) bias currents....An annular Josephson junction with a trapped fluxon coupled to a** flux** qubit....To couple a flux **qubit** to the fluxon inside an annular Josephson junction, it is necessary to engineer an interaction between two orthogonal magnetic dipoles. To facilitate this interaction, we have added a superconducting coupling loop embracing a flux **qubit**, as shown in Fig. AJJ+**Qubit**. The current induced in the coupling loop attached to the AJJ is proportional to the persistent current in the flux **qubit**. Thus, the persistent current in the **qubit** manifests itself in the AJJ as a current dipole with an amplitude μ on top of the homogeneous background of bias current. When fluxon scatters on a positive current dipole - it first gets accelerated and then decelerated by the dipole poles. In the ideal case of absence of damping and bias current, the sign of **frequency** change δ ν is determined only by polarity of the dipole. In the presence of finite damping and homogeneous bias current, situation completely changes - as the total propagation time becomes dependent on the complex interplay between bias current, current dipole strength and damping....The experimental curve showing** the **reaction of** the **fluxon to** the **magnetic **flux **through the **flux **qubit are presented in Fig. FM. The periodic modulation of** the **fluxon frequency versus magnetic **flux **through** the **qubit corresponds to** the **changing of** the **persistent currents in** the **qubit as Fig. FQ_Icc suggests. We did not observe clear narrow peaks at** the **half **flux **quantum point**, **most probably due to excess fluctuations. Emerging dip-like peculiarities can be noted at presumed half **flux **quantum points which suggest that** the **dips may be there**, **covered by noise and insufficient resolution. Further improvements of experimental setup are required to resolve these peaks. The presented measurement curve has a convex profile which tells that indeed** the **deviation of frequency δ ν is positive**, **consistently with predictions made above by** the **perturbation approach and numerical simulations....We would like to employ fluxons for developing a fast and sensitive magnetic field detector for measurements of superconducting **qubits**. In this Letter, we report direct measurements of electromagnetic radiation from a fluxon moving in an annular Josephson junction (AJJ). The radiation is detected by using a microstrip antenna capacitively coupled to the AJJ. Furthermore, we place a flux **qubit** close to the long junction and couple them magnetically with a superconducting loop (see Fig. AJJ+**Qubit**). This coupling scheme makes the fluxon interact with a current dipole formed by the electrodes of the loop coupled to the **qubit**. The time delay of the fluxon can be detected as a **frequency** shift of the electromagnetic radiation emitted from the junction. This shift provides information about the state of the flux **qubit**....Using** the **possibility to directly detect radiation of** the **fluxon resonant oscillations**, **we have performed systematic measurements of** the **dependence of** the **fluxon velocity versus bias current - the current-voltage characteristics - measured in** the **frequency domain (see Fig. ZFS). This approach provides an easy access to study** the **fine structure of** the **current-voltage curve as** the **precision of frequency measurements is by several orders of magnitude greater than** the **resolution of direct dc voltage measurement....The zero-field step measured in the **frequency** domain for the ambient temperature T = 4.2 K and injection current I C I = 3.973 mA. The inset shows the sample spectrum for the fixed bias current I b = 5.3 mA with the respective Lorentz fit....Persistent current for a ground state of the flux **qubit** versus magnetic frustration (black line). Red line shows the corresponding fluxon shift calculated using the perturbation theory for....Optical photograph of the chip with the annular Josephson junction on the right part and experimental set-up schematics. Left part shows the zoom into the area with the** flux** qubit with a coupling loop (yellow loop) and control line (green loop). Red crosses indicate the positions of three Josephson junctions in the** flux** qubit loop....Modulation of the fluxon’s **oscillation** **frequency** due to the coupling to the flux **qubit**. Every point consists of 100 averages. Bias current was set at γ = 0.521 , w ≃ 9.1 ....Relative **frequency** deviation from equilibrium δ ν / ν 0 of the fluxon **oscillation** **frequency** versus bias current. Black line shows the result of perturbation approach, while the red line depicts results of direct numerical simulations of the PSGE equation ( PSGEm) with a d = 1 . The blue curve corresponds to the case with a d = 0.2 ....Experiments towards realizing a readout of superconducting **qubits** by using ballistic Josephson vortices are reported. We measured the microwave radiation induced by a fluxon moving in an annular Josephson junction. By coupling a flux **qubit** as a current dipole to the annular junction, we detect periodic variations of the fluxon's oscillation **frequency** versus magnetic flux through the **qubit**. We found that the scattering of a fluxon on a current dipole can lead to the acceleration of a fluxon regardless of a dipole polarity. We use the perturbation theory and numerical simulations of the perturbed sine-Gordon equation to analyze our results....The experimental curve showing the reaction of the fluxon to the magnetic flux through the flux **qubit** are presented in Fig. FM. The periodic modulation of the fluxon **frequency** versus magnetic flux through the **qubit** corresponds to the changing of the persistent currents in the **qubit** as Fig. FQ_Icc suggests. We did not observe clear narrow peaks at the half flux quantum point, most probably due to excess fluctuations. Emerging dip-like peculiarities can be noted at presumed half flux quantum points which suggest that the dips may be there, covered by noise and insufficient resolution. Further improvements of experimental setup are required to resolve these peaks. The presented measurement curve has a convex profile which tells that indeed the deviation of **frequency** δ ν is positive, consistently with predictions made above by the perturbation approach and numerical simulations. ... Experiments towards realizing a readout of superconducting **qubits** by using ballistic Josephson vortices are reported. We measured the microwave radiation induced by a fluxon moving in an annular Josephson junction. By coupling a flux **qubit** as a current dipole to the annular junction, we detect periodic variations of the fluxon's **oscillation** **frequency** versus magnetic flux through the **qubit**. We found that the scattering of a fluxon on a current dipole can lead to the acceleration of a fluxon regardless of a dipole polarity. We use the perturbation theory and numerical simulations of the perturbed sine-Gordon equation to analyze our results.

Files:

Contributors: Martijn Wubs, Sigmund Kohler, Peter Hänggi

Date: 2007-10-01

(Color online) Upper panel: adiabatic energies during a LZ sweep **of a **qubit coupled to two oscillators 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.

Files:

Contributors: Greenberg, Ya. S.

Date: 2003-03-04

Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Here we present the results of detailed computer simulations of the interaction of a classical object (resonant tank circuit) with a quantum object (phase **qubit**). We explicitly account for the back action of a tank circuit and for the unpredictable nature of outcome of a single measurement. According to the results of our simulations the Rabi **oscillations** in MHz range can be detected using conventional NMR pulse Fourier technique....As the coupling is increased further the **qubit** wave function is completely destroyed. The quantity A is quenched to approximately 0.85 (Fig. fig10a). That is | C - | ≈ 0.92 . It might seem that we have here so called Zeno effect- as if **qubit** state is frozen in its ground state. However, in case of a strong coupling it is not correct to say about wave function of the **qubit** alone. This is shown in Fig. figcoh where for λ > 10 -2 the phase coherence is seen to be completely lost ....Loss**-free** **qubit** coupled to a dissipative tank circuit Q T = 100 , λ = 2.5 × 10 -2 . The evolution of the voltage across the tank for the deterministic case....At every graph of the figures the results for one realization of random number generator ξ t are compared with the case when we replaced F ξ t in ( Q) with deterministic term 2 A - 1 - 2 C / 2 , which means that the tank measures the average current ( avcurr) in a **qubit** loop. As is seen from the Fig. fig4a, A **oscillates** with Rabi **frequency**. The voltage across tank circuit **oscillates** also with Rabi **frequency** which is equal to 50 MHz in our case (Fig. fig4b) which is modulated with the lower **frequency** the value of which is about 5 MHz....At every graph of the figures the results for one realization of random number generator ξ t are compared with the case when we replaced F ξ t in ( Q) with deterministic term 2 A - 1 - 2 C / 2 , which means that the tank measures the average current ( avcurr) in a qubit loop. As is seen from the Fig. fig4a, A oscillates with Rabi frequency. The voltage across tank circuit oscillates also with Rabi frequency which is equal to 50 MHz in our case (Fig. fig4b) which is modulated with the lower frequency the value of which is about 5 MHz....Phase loss-free **qubit** coupled to a dissipative tank circuit. The evolution of A exhibits modulation of Rabi **oscillations** with lower **frequency**. Deterministic case (a) together with one realization (b) are shown....In conclusion we want to show the effect of **qubit** evolution as the coupling between the **qubit** and the tank is increased. We numerically solved the system consisting of the loss-free **qubit** coupled to the dissipative tank circuit. The system is described by Eqs. ( A2, B2, C2, flux_tank) and Eq. ( Q1). For the simulations we take the coupling parameter λ = 2.5 × 10 -2 . The results of simulations are shown on Figs. fig10a, fig10d for deterministic case. As is seen from the Figs. fig10a during Rabi period the quantity A became partially frozen at some level. At the endpoints of this period the system tries to escape to another level of A. Between the endpoints of Rabi period A **oscillates** with a high **frequency** which is about 10 GHz in our case. As expected, the evolution of B is suppressed approximately by a factor of ten below its free evolution amplitude which is equal to 0.5. As we show below, the strong coupling completely destroys the phase coherence between **qubit** states, nevertheless the voltage across the tank **oscillates** with Rabi **frequency**. Its amplitude is considerably increased and it does not reveal any peculiarities associated with the frozen behavior of A (Fig. fig10d)....L...Phase loss-free **qubit** coupled to a dissipative tank circuit. The voltage across the tank exhibits modulation of Rabi **frequency**. Deterministic case (a) together with one realization (b) are shown....Phase loss**-free** **qubit** coupled to a dissipative tank circuit. The voltage across the tank exhibits modulation of Rabi **frequency**. Deterministic case (a) together with one realization (b) are shown....Time-domain observations of coherent oscillations between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi oscillations in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi oscillations of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Here we present the results of detailed computer simulations of the interaction of a classical object (resonant tank circuit) with a quantum object (phase **qubit**). We explicitly account for the back action of a tank circuit and for the unpredictable nature of outcome of a single measurement. According to the results of our simulations the Rabi oscillations in MHz range can be detected using conventional NMR pulse Fourier technique....P...It is clearly seen that B **oscillates** with gap **frequency**, while the **frequency** of A is almost ten times smaller: (**oscillation** period of B: T B ≈ 2 × 10 -9 s, while the same quantity for A is T A ≈ 2 × 10 -8 s. The small distortions on A curve are due to a strong deviation of excitation signal from transverse rotating wave form, while B curve is clearly modulated with Rabi **frequency** Ω R (Fig. fig2)....In conclusion we want to show the effect of qubit evolution as the coupling between the qubit and the tank is increased. We numerically solved the system consisting of the loss-free qubit coupled to the dissipative tank circuit. The system is described by Eqs. ( A2, B2, C2, flux_tank) and Eq. ( Q1). For the simulations we take the coupling parameter λ = 2.5 × 10 -2 . The results of simulations are shown on Figs. fig10a, fig10d for deterministic case. As is seen from the Figs. fig10a during Rabi period the quantity A became partially frozen at some level. At the endpoints of this period the system tries to escape to another level of A. Between the endpoints of Rabi period A oscillates with a high frequency which is about 10 GHz in our case. As expected, the evolution of B is suppressed approximately by a factor of ten below its free evolution amplitude which is equal to 0.5. As we show below, the strong coupling completely destroys the phase coherence between qubit states, nevertheless the voltage across the tank oscillates with Rabi frequency. Its amplitude is considerably increased and it does not reveal any peculiarities associated with the frozen behavior of A (Fig. fig10d)....As is seen from the Fig. fig3, A decays to 0.5 **oscillating** with Rabi **frequency**, while B (C) decays to zero. (Note: to be rigorous, the stable state solution for A is...Phase **qubit** coupled to a tank circuit....Phase loss**-free** **qubit** coupled to a loss**-free** tank circuit. Oscillations of A. Deterministic case (a) together with one realization (b) are shown. Small scale time oscillations correspond to Rabi **frequency**....In conclusion we want to show the effect of qubit evolution as the coupling between the qubit and the tank is increased. We numerically solved the system consisting of the loss-free qubit coupled to the dissipative tank circuit. The system is described by Eqs. ( A2, B2, C2, flux_tank) and Eq. ( Q1). For the simulations we take the coupling parameter λ = 2.5 × 10 -2 . The results of simulations are shown on Figs. fig10a, fig10d for deterministic case. As is seen from the Figs. fig10a during Rabi period the quantity A became partially frozen at some level. At the endpoints of this period the system tries to escape to another level of A. Between the endpoints of Rabi period A oscillates with a high frequency which is about 10 GHz in our case. As expected, the evolution of B is suppressed approximately by a factor of ten below its free evolution amplitude which is equal to 0.5. As we show below, the strong coupling completely destroys the phase coherence between qubit states, neverthe...Phase loss-free **qubit** coupled to a loss-free tank circuit. **Oscillations** of A. Deterministic case (a) together with one realization (b) are shown. Small scale time **oscillations** correspond to Rabi **frequency**....Method for direct observation of coherent quantum oscillations in a superconducting phase **qubit**. Computer simulations...If we switch on the interaction with a tank we may not, strictly speaking, consider qubit as having definite wave function. However, if the interaction is rather weak the qubit wave function could be well defined. We showed before that for relatively weak coupling the dissipation resulted in quenching A to the 0.5 level (see Figs. fig5, fig7a, fig8a). That means | C + | = | C - | → 1 2 . However, as is seen from Fig. figcoh, the condition of phase coherence is still valid up to λ ≈ 10 -3 ....Time evolution of A and B for **qubit** without dissipation....Phase loss**-free** **qubit** coupled to a dissipative tank circuit. The evolution of A exhibits modulation of Rabi oscillations with lower **frequency**. Deterministic case (a) together with one realization (b) are shown. ... Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Here we present the results of detailed computer simulations of the interaction of a classical object (resonant tank circuit) with a quantum object (phase **qubit**). We explicitly account for the back action of a tank circuit and for the unpredictable nature of outcome of a single measurement. According to the results of our simulations the Rabi **oscillations** in MHz range can be detected using conventional NMR pulse Fourier technique.

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

Date: 2006-03-07

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

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

Date: 2007-03-15

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

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

Date: 2012-03-27

fig:2 Main panel: comparison of dynamics calculated from truncating ( Sin5) at N M A X = ± 10 (red) and a numerically exact approach (blue). Lower left: Fourier transform of the dynamics. Lower right: the numerical weight of the n t h term in the series expansion of ( Sin5), showing there are still only two dominant **frequencies** at n = 0 and n = - 1 . Parameters: ω = 0.5 , g = 0.1 , ϵ = 0 , Δ = 0.5 , T = 1 ~ m K , ℏ = 1 and k b = 1 ....fig:1 Comparison of the single term approximation (red) and a numerically exact approach (blue) for different coupling strengths. Uncoupled Rabi oscillations are also shown as a reference (green). Left: the population ρ 00 t in the time-domain. Right: the same data in the frequency domain. The full numerical solution was Fourier transformed using Matlab’s FFT algorithm. Other parameters are ω = 1 GHz, g = 0.1 GHz, ϵ = Δ = 100 MHz and T = 10 mK....fig:1 Comparison of the single term approximation (red) and a numerically exact approach (blue) for different coupling strengths. Uncoupled Rabi **oscillations** are also shown as a reference (green). Left: the population ρ 00 t in the time-domain. Right: the same data in the **frequency** domain. The full numerical solution was Fourier transformed using Matlab’s FFT algorithm. Other parameters are ω = 1 GHz, g = 0.1 GHz, ϵ = Δ = 100 MHz and T = 10 mK....fig:3 Demonstration of qubit thermometry: T i n is the temperature supplied to the numerical simulation of the system and T o u t is the temperature that would be predicted by fitting oscillations with frequency ( eqn:rho3) to it. The blue line is the data and red line shows the effect of a 10kHz error in the frequency measurement; the grey dashed line serves as a guide to the eye. The lower inset shows the variation of the qubit frequency Ω with temperature. The upper inset shows the dependence of the absolute error in the prediction against the signal length (see text). Other parameters are: ω = 1 GHz, g = 0.01 GHz, ϵ = 0 , Δ = 100 MHz...Including extra terms in the series expansion ( Sin5) makes the time dependence of the **qubit** dynamics analytically unwieldy, because the rational function form of the series leads to a complex interdepence of the positions of the poles in ( eqn:rsol1). However, if the values of the parameters are known the series can truncated at ( ± N M A X ) to give an efficient numerical method to obtain more accurate dynamics, extending the applicability of our approach beyond the regime described by ( eqn:crit). This is demonstrated in Fig. fig:2, where the dynamics are clearly dominated by two **frequencies** – an effect that could obviously never be captured by a single term approximation. There is a qualitative agreement between the many terms expansion and full numerical solution, particularly at short times. We would not expect a perfect agreement in this case because the simulations are of the dynamics in the large tunnelling regime ( Δ = 0.5 ), and the polaron transform makes the master equation perturbative in this parameter. Nonetheless, the rapid convergence of the series is shown in Fig. fig:2; N M A X = 5 - 10 is sufficient to calculate ρ 00 t and ρ 10 t with an accuracy only limited by the underlying Born Approximation. The asymmetry of the amplitudes of the terms in the series expansion of ( Sin5) is due to the exponential functions in the series....Our methodology can be used to predict dynamics of nanomechanical resonators connected to either quantum dots or superconducting **qubits**. The criterion for the single term approximation to be valid is readily met by current experiments such as those presented in Refs. and their parameters yield near perfect agreement between numerical and analytic results. Most experiments operate in a regime where the **qubit** dynamics are not greatly perturbed by the presence of the **oscillator**, which has a much lower **frequency** ( ϵ ≈ Δ ≈ 10 GHz, ω = 1 GHz). In Figure fig:1, we chose ϵ ≈ Δ ≈ 100 MHz, because this better demonstrates the effect of the **oscillator** on the **qubit**. These parameters can be achieved experimentally using the same **qubit** design but with an **oscillating** voltage applied to the CPB bias gate . However, we stress the accuracy of our method is not restricted to this regime....Our methodology can be used to predict dynamics of nanomechanical resonators connected to either quantum dots or superconducting qubits. The criterion for the single term approximation to be valid is readily met by current experiments such as those presented in Refs. and their parameters yield near perfect agreement between numerical and analytic results. Most experiments operate in a regime where the** qubit** dynamics are not greatly perturbed by the presence of the

**oscillator**, which has a much lower

**frequency**( ϵ ≈ Δ ≈ 10 GHz, ω = 1 GHz). In Figure fig:1, we chose ϵ ≈ Δ ≈ 100 MHz, because this better demonstrates the effect of the

**oscillator**on the

**. These parameters can be achieved experimentally using the same**

**qubit****design but with an oscillating voltage applied to the CPB bias gate . However, we stress the accuracy of our method is not restricted to this regime....Figure fig:3 demonstrates this idea, showing that by measuring Ω and fitting it to our expression ( eqn:rho3), we can obtain submilli-Kelvin precision in the experimentally relevant regime of 20-55 mK. At low temperatures the single term**

**qubit****frequency**plateaus, causing the accuracy to break down. In the higher temperature limit, we also see a deviation from the diagonal, this is to be expected as we leave the regime of validity described by ( eqn:crit). Naturally accuracy in this region could be improved by retaining higher order terms in ( Sin5), but this would become a more numeric than analytic approach. The upper inset shows the dependence of the accuracy of the prediction on the number of points (at a separation of 1ns) sampled from the dynamics. The accuracy increases initially as more points improve the fitted value of Ω , however after a certain length the accuracy is diminished by long term envelope effects in the dynamics not captured by the single term approximation. We note that the corresponding analysis in the

**frequency**domain would not be equally affected by the long time envelope, however a large number of points in the FFT is then required in order to obtain the desired accuracy. The lower inset of Figure fig:3 shows the direct dependence of Ω on the temperature. The temperature range with steepest gradient and hence greatest

**frequency**dependence on temperature varies with the coupling strength; thus the device could be specifically designed to have a maximal sensitivity in the temperature range of the most interest....Figure fig:1 shows a comparison of the dynamics predicted using these expressions and a numerically exact approach. The latter are obtained by imposing a truncation of the

**oscillator**Hilbert space at a point where the dynamics have converged and any higher modes have an extremely low occupation probability. Our zeroth order approximation proves to be unexpectedly powerful, giving accurate dynamics well into the strong coupling regime ( g / ω = 0.25 ) and even beyond this it still captures the dominant oscillatory behaviour, see Figure fig:1. Stronger coupling increases the numerical weight of higher

**frequency**terms in the series, causing a modulation of the dynamics. The approximation starts to break down at ( g / ω = 0.5 ). The equations ( eqn:rho0) and ( eqn:rho1) are obviously unable to capture the higher

**frequency**modulations to the dynamics or any potential long time phenomena like collapse and revival, but these are unlikely to be resolvable in experiments in any case. Nonetheless, it is worth pointing out that even in this strong coupling case the base

**frequency**of the

**qubit**dynamics is still adequately captured by our single term approximation....fig:3 Demonstration of

**qubit**thermometry: T i n is the temperature supplied to the numerical simulation of the system and T o u t is the temperature that would be predicted by fitting

**oscillations**with

**frequency**( eqn:rho3) to it. The blue line is the data and red line shows the effect of a 10kHz error in the

**frequency**measurement; the grey dashed line serves as a guide to the eye. The lower inset shows the variation of the

**qubit**

**frequency**Ω with temperature. The upper inset shows the dependence of the absolute error in the prediction against the signal length (see text). Other parameters are: ω = 1 GHz, g = 0.01 GHz, ϵ = 0 , Δ = 100 MHz...A quantum two level system coupled to a harmonic

**oscillator**represents a ubiquitous physical system. New experiments in circuit QED and nano-electromechanical systems (NEMS) achieve unprecedented coupling strength at large detuning between

**qubit**and

**oscillator**, thus requiring a theoretical treatment beyond the Jaynes Cummings model. Here we present a new method for describing the

**qubit**dynamics in this regime, based on an

**oscillator**correlation function expansion of a non-Markovian master equation in the polaron frame. Our technique yields a new numerical method as well as a succinct approximate expression for the

**qubit**dynamics. We obtain a new expression for the ac Stark shift and show that this enables practical and precise

**qubit**thermometry of an

**oscillator**....Including extra terms in the series expansion ( Sin5) makes the time dependence of the

**dynamics analytically unwieldy, because the rational function form of the series leads to a complex interdepence of the positions of the poles in ( eqn:rsol1). However, if the values of the parameters are known the series can truncated at ( ± N M A X ) to give an efficient numerical method to obtain more accurate dynamics, extending the applicability of our approach beyond the regime described by ( eqn:crit). This is demonstrated in Fig. fig:2, where the dynamics are clearly dominated by two frequencies – an effect that could obviously never be captured by a single term approximation. There is a qualitative agreement between the many terms expansion and full numerical solution, particularly at short times. We would not expect a perfect agreement in this case because the simulations are of the dynamics in the large tunnelling regime ( Δ = 0.5 ), and the polaron transform makes the master equation perturbative in this parameter. Nonetheless, the rapid convergence of the series is shown in Fig. fig:2; N M A X = 5 - 10 is sufficient to calculate ρ 00 t and ρ 10 t with an accuracy only limited by the underlying Born Approximation. The asymmetry of the amplitudes of the terms in the series expansion of ( Sin5) is due to the exponential functions in the series....Figure fig:1 shows a comparison of the dynamics predicted using these expressions and a numerically exact approach. The latter are obtained by imposing a truncation of the**

**qubit****oscillator**Hilbert space at a point where the dynamics have converged and any higher modes have an extremely low occupation probability. Our zeroth order approximation proves to be unexpectedly powerful, giving accurate dynamics well into the strong coupling regime ( g / ω = 0.25 ) and even beyond this it still captures the dominant oscillatory behaviour, see Figure fig:1. Stronger coupling increases the numerical weight of higher

**frequency**terms in the series, causing a modulation of the dynamics. The approximation starts to break down at ( g / ω = 0.5 ). The equations ( eqn:rho0) and ( eqn:rho1) are obviously unable to capture the higher

**frequency**modulations to the dynamics or any potential long time phenomena like collapse and revival, but these are unlikely to be resolvable in experiments in any case. Nonetheless, it is worth pointing out that even in this strong coupling case the base

**frequency**of the

**dynamics is still adequately captured by our single term approximation. ... A quantum two level system coupled to a harmonic**

**qubit****oscillator**represents a ubiquitous physical system. New experiments in circuit QED and nano-electromechanical systems (NEMS) achieve unprecedented coupling strength at large detuning between

**qubit**and

**oscillator**, thus requiring a theoretical treatment beyond the Jaynes Cummings model. Here we present a new method for describing the

**qubit**dynamics in this regime, based on an

**oscillator**correlation function expansion of a non-Markovian master equation in the polaron frame. Our technique yields a new numerical method as well as a succinct approximate expression for the

**qubit**dynamics. We obtain a new expression for the ac Stark shift and show that this enables practical and precise

**qubit**thermometry of an

**oscillator**.

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

Date: 2003-03-31

Our technique is similar to rf-SQUID readout. The **qubit** loop is inductively coupled to a parallel resonant tank circuit [Fig. fig:schem(b)]. The tank is fed a monochromatic rf signal at its resonant **frequency** ω T . Then both amplitude v and phase shift χ (with respect to the bias current I b ) of the tank voltage will strongly depend on (A) the shift in resonant **frequency** due to the change of the effective **qubit** inductance by the tank flux, and (B) losses caused by field-induced transitions between the two **qubit** states. Thus, the tank both applies the probing field to the **qubit**, and detects its response....In conclusion, we have observed resonant tunneling in a macroscopic superconducting system, containing an Al flux **qubit** and a Nb tank circuit. The latter played dual control and readout roles. The impedance readout technique allows direct characterization of some of **the** **qubit**’s quantum properties, without using spectroscopy. In a range 50 ∼ 200 mK, **the** effective **qubit** temperature has been verified [Fig. fig:Temp_dep(b)] to be **the** same as **the** mixing chamber’s (after Δ has been determined at low T ), which is often difficult to confirm independently....(a) Quantum energy levels of **the** **qubit** vs external flux. The dashed lines represent **the** classical potential minima. (**b**) Phase **qubit** coupled to a tank circuit....Low-**frequency** measurement of the tunneling amplitude in a flux **qubit**...(a) Quantum energy levels of the **qubit** vs external flux. The dashed lines represent the classical potential minima. (b) Phase **qubit** coupled to a tank circuit....-dependence of ϵ t is adiabatic. However, it does remain valid if the full (Liouville) evolution operator of the **qubit** would contain standard Bloch-type relaxation and dephasing terms (which indeed are not probed) in addition to the Hamiltonian dynamics ( eq01), since the fluctuation–dissipation theorem guarantees that such terms do not affect equilibrium properties. Normalized dip amplitudes are shown vs T in Fig. fig:Temp_dep(b) together with tanh Δ / k B T , for Δ / h = 650 MHz independently obtained above from the low- T width. The good agreement strongly supports our interpretation, and is consistent with Δ being T...-dependence of ϵ t is adiabatic. However, it does remain valid if **the** full (Liouville) evolution operator of **the** **qubit** would contain standard Bloch-type relaxation and dephasing terms (which indeed are not probed) in addition to **the** Hamiltonian dynamics ( eq01), since **the** fluctuation–dissipation theorem guarantees that such terms do not affect equilibrium properties. Normalized dip amplitudes are shown vs T in Fig. fig:Temp_dep(b) together with tanh Δ / k B T , for Δ / h = 650 MHz independently obtained above from **the** low- T width. The good agreement strongly supports our interpretation, and is consistent with Δ being T...In conclusion, we have observed resonant tunneling in a macroscopic superconducting system, containing an Al flux **qubit** and a Nb tank circuit. The latter played dual control and readout roles. The impedance readout technique allows direct characterization of some of the ** qubit’s** quantum properties, without using spectroscopy. In a range 50 ∼ 200 mK, the effective

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

**qubit**loop is inductively coupled to a parallel resonant tank circuit [Fig. fig:schem(b)]. The tank is fed a monochromatic rf signal at its resonant frequency ω T . Then both amplitude v and phase shift χ (with respect to

**the**bias current I b ) of

**the**tank voltage will strongly depend

**on**(A)

**the**shift in resonant frequency due to

**the**change of

**the**effective

**qubit**inductance by

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

**the**two

**qubit**states. Thus,

**the**tank both applies

**the**probing field to

**the**

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

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

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

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

**the**tunneling amplitude. At bias ϵ = 0

**the**

**two**lowest energy levels of

**the**

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

**the**

**qubit**can adiabatically transform from Ψ l to Ψ r , staying in

**the**ground state E - . Since d E - / d Φ x is

**the**persistent loop current,

**the**curvature d 2 E - / d Φ x 2 is related to

**the**

**qubit**’s susceptibility. Hence, near degeneracy

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

**3JJ**

**qubit**....(a) Tank phase shift vs flux bias near degeneracy and for V d r = 0.5 ~ μ V. From

**the**lower to

**the**upper curve (at f x = 0 )

**the**temperature is 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 350, 400 mK. (b) Normalized amplitude of tan χ (circles) and tanh Δ / k B T (line), for

**the**Δ following from Fig. fig:Bias_dep;

**the**overall scale κ is a fitting parameter. The data indicate a saturation of

**the**effective

**qubit**temperature at 30 mK. (c) Full dip width at half

**the**maximum amplitude vs temperature. The horizontal line fits

**the**low- T ( < 200 mK) part to a constant;

**the**sloped line represents

**the**T 3 behavior observed empirically for higher T ....(a) Tank phase shift vs flux bias near degeneracy and for V d r = 0.5 ~ μ V. From the lower to the upper curve (at f x = 0 ) the temperature is 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 350, 400 mK. (b) Normalized amplitude of tan χ (circles) and tanh Δ / k B T (line), for the Δ following from Fig. fig:Bias_dep; the overall scale κ is a fitting parameter. The data indicate a saturation of the effective

**qubit**temperature at 30 mK. (c) Full dip width at half the maximum amplitude vs temperature. The horizontal line fits the low- T ( < 200 mK) part to a constant; the sloped line represents the T 3 behavior observed empirically for higher T ....(a) Tank phase shift vs flux bias near degeneracy and for V d r = 0.5 ~ μ V. From

**the**lower to

**the**upper curve (at f x = 0 )

**the**temperature is 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 350, 400 mK. (

**b**) Normalized amplitude of tan χ (circles) and tanh Δ / k B T (line), for

**the**Δ following from Fig. fig:Bias_dep;

**the**overall scale κ is a fitting parameter. The data indicate a saturation of

**the**effective

**qubit**temperature at 30 mK. (c) Full dip width at half

**the**maximum amplitude vs temperature. The horizontal line fits

**the**low- T ( < 200 mK) part to a constant;

**the**sloped line represents

**the**T 3 behavior observed empirically for higher T ....Δ is the tunneling amplitude. At bias ϵ = 0 the two lowest energy levels of the

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

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

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

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

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

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

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

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