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Design of a Lambda system for population transfer in superconducting nanocircuits

G. Falci, A. La Cognata, M. Berritta, A. D’Arrigo, E. Paladino, and B. Spagnolo
Phys. Rev. B 87, 214515 – Published 20 June 2013
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  • INTRODUCTION
  • COHERENT POPULATION TRANSFER IN…
  • STIRAP IN THE COOPER PAIR BOX
  • EFFECT OF BROADBAND COLORED NOISE ON…
  • OPTIMAL DESIGN OF THE DEVICE
  • COMPARISON OF DIFFERENT MECHANISMS OF…
  • CONCLUSIONS
  • ACKNOWLEDGEMENTS
  • APPENDICES
    • References

    Abstract

    The implementation of a Lambda scheme in superconducting artificial atoms could allow detection of stimulated Raman adiabatic passage (STIRAP) and other quantum manipulations in the microwave regime. However, symmetries which on one hand protect the system against decoherence yield selection rules which may cancel coupling to the pump external drive. The tradeoff between efficient coupling and decoherence due to broad-band colored noise (BBCN), which is often the main source of decoherence, is addressed in the class of nanodevices based on the Cooper pair box (CPB) design. We study transfer efficiency by STIRAP, showing that substantial efficiency is achieved for off-symmetric bias only in the charge-phase regime. We find a number of results uniquely due to non-Markovianity of BBCN, namely (a) the efficiency for STIRAP depends essentially on noise channels in the trapped subspace; (b) low-frequency fluctuations can be analyzed and represented as fictitious correlated fluctuations of the detunings of the external drives; and (c) a simple figure of merit for design and operating prescriptions allowing the observation of STIRAP is proposed. The emerging physical picture also applies to other classes of coherent nanodevices subject to BBCN.

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    • Received 25 February 2013

    DOI:https://doi.org/10.1103/PhysRevB.87.214515

    ©2013 American Physical Society

    Authors & Affiliations

    G. Falci1,2,3,*, A. La Cognata3, M. Berritta1,†, A. D’Arrigo2, E. Paladino1,2,3, and B. Spagnolo4

    • 1Dipartimento di Fisica e Astronomia, Università di Catania, Via Santa Sofia 64, 95123 Catania, Italy
    • 2CNR-IMM UOS Università (MATIS), Consiglio Nazionale delle Ricerche, Via Santa Sofia 64, 95123 Catania, Italy
    • 3Centro Siciliano di Fisica Nucleare e Struttura della Materia, Via Santa Sofia 64, 95123 Catania, Italy
    • 4Dipartimento di Fisica e Chimica, Università di Palermo, Group of Interdisciplinary Physics and CNISM, Unità di Palermo, Viale delle Scienze, Ed.18, I-90128 Palermo, Italy

    • *gfalci@dmfci.unict.it
    • †Also at Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom.

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    Issue

    Vol. 87, Iss. 21 — 1 June 2013

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    Images

    • Figure 1
      Figure 1
      (a) Three-level system driven with ac fields in Λ configuration. (b) The counterintuitive sequence: the Stokes field is switched on before the pump field (here Ω0T=20,τ=0.6T). (c) Instantaneous eigenvalues {ε0(t),ε±(t)}, for δ=0, δp=−0.2Ω0, and κ=1. (d) Population histories ρii(t)=|〈i|ψ(t)〉|2 for ideal STIRAP (δ=0): the system prepared in |0〉 follows the Hamiltonian along the ε0 adiabatic path yielding complete population transfer to |1〉.Reuse & Permissions
    • Figure 2
      Figure 2
      Sensitivity to detunings of the efficiency of STIRAP. Here Ω0T=20, τ=T/2, and κ=1. In the white zone the efficiency is larger than 90%. Efficiency is very sensitive to a nonzero two-photon detuning δ, and much less sensitive to δp≠0 (notice the different scale of the axis). The two lines on the plot represent correlated stray detunings induced in the CPB by charge noise for two different values of qg=0.47,0.49 (see Secs. 3d and 4a).Reuse & Permissions
    • Figure 3
      Figure 3
      Nonideal STIRAP (Ω0T=20,τ=0.6T) with δ≠0, shows different classes of patterns of instantaneous eigenstates. (a) Top: instantaneous eigenstates for δ=0.25Ω0 and δp=−0.4Ω0. These LZ patterns with a single avoided crossing during the pump-induced EIT phase result from the effect of low-frequency charge noise in CPBs (or flux noise in flux qubits). Bottom: population histories of the diabatic states. (b) Top: generic LZ pattern (δ=0.5Ω0, δp=0.5Ω0); bottom: population histories.Reuse & Permissions
    • Figure 4
      Figure 4
      In the CPB design the state of the superconducting island is a superpositions of states with a well defined number n of extra Cooper pairs. The device is biased by the gate voltage Vg determining the operating point of qg=Vg/2eCg; control is operated by an ac component of Vg. Charge fluctuations are equivalent to voltage fluctuations δVx. The effective Josephson energy can be tuned via the flux Φg of the magnetic field threading the loop, EJ=EJ(Φg).Reuse & Permissions
    • Figure 5
      Figure 5
      (a) Energy spectrum Ei of a charge-phase CPB for J=1.32 [corresponds to the quantronium (Ref. 23)], relative to the ground state E0=0, vs the bias qg. (b) Matrix elements of n̂ involved in the Λ scheme vs qg for J=1.32: the element n02 vanishes at the symmetry point qg=1/2; large n23 may be a potential source of leakage from the three-level subspace. (c) Matrix elements vs J=EJ/EC for qg=0.48; notice that n02 is much smaller than other elements (it vanishes at qg=1/2) and it has nonmonotonous behavior for increasing J.Reuse & Permissions
    • Figure 6
      Figure 6
      Population histories in the quantronium at qg=0.475, averaged over fluctuations with σx=0.004. Charge fluctuations determine anticorrelated stray detunings, δp=−23.5δ. Drives are symmetrized, κ=1. The total time for the quantronium corresponds to ∼250ns. The resulting efficiency is P1(tf)=0.77.Reuse & Permissions
    • Figure 7
      Figure 7
      Population histories ρii(t) in the presence of high-frequency noise (P1 is in absence), at resonance (δ=δp=0, left panel) and for finite anticorrelated detunings (δ=0.05, δp=−25δ, right panel), for κ=1. Solid lines are obtained by inserting only γ01=1/T1 in Eq. (8), describing relaxation 1→0 only, and the associated secular dephasing. We have chosen a rather large γ01/Ω0=0.01 to emphasize the effect. Dashed lines take into account all the other low-temperature emission and drive-induced absorption channels (the chosen rates overestimate these processes), which are seen to have a limited impact on the efficiency. Physical scales for Ω0=3.46×108rad/s (the value we use for the Quantronium at qg=0.48) are T≈43ns, for the overall protocol TT≈290ns, and the chosen T1≈TT.Reuse & Permissions
    • Figure 8
      Figure 8
      Efficiency of STIRAP P1(tf) as a function of the bias qg in the presence of low-frequency and BBCN for the quantronium (EJ/EC=1.32). Here Ω0T=15, σx=0.004, ν01=600 MHz. Upper curves show effects of low-frequency noise, whereas the lower curve (black dashed) includes also high-frequency noise. Low-frequency noise is analyzed by adding different components, namely linear and quadratic correlated fluctuations of detunings (red solid curve and red squares), linear and quadratic fluctuations of n02 (blue solid curve and blue dots). For off-symmetry bias (qg<0.9), only linear detuning noise is important (see Appendix for the behavior near qg<1/2). In the inset the population P2(0) at intermediate times is shown.Reuse & Permissions
    • Figure 9
      Figure 9
      The figure of merit Ωp/σδ is plotted in the (qg,EJ/EC) plane. We have chosen σx=0.004 and Ω0 produced by an external field, which would determine Rabi oscillations with νR=600MHz in the first doublet. The analysis is valid far enough from the charge-parity symmetry point, which is not an interesting regime since the efficiency is suppressed. Dashed lines correspond to the values of EJ/EC checked in this paper (Figs. 8 and 10).Reuse & Permissions
    • Figure 10
      Figure 10
      Efficiency P1(tf) vs bias qg for J=0.7,1,2 (see Fig. 9). Parameters are the same as in Fig. 8, where T1=1000ns (black short-dashed curves). In the two upper panels efficiency for smaller T1=500ns (gray short-dashed curve) is also shown.Reuse & Permissions
    • Figure 11
      Figure 11
      Efficiency of STIRAP (final populations) as a function of the drive amplitudes Ω0. We compare the case of Markovian (ρii) pure dephasing[49] with the non-Markovian (Pi) model studied here. In both cases we let T=T2=57ns, which for non-Markovian noise is obtained by taking σx=0.004 in a device with J=1.32 at qg=0.48. It is seen that the effects of non-Markovian dephasing can be attenuated and suppressed by using larger Ω0, whereas for Markovian noise STIRAP, when effective, does not depend on Ω0.Reuse & Permissions
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