Coherent manipulation of noise-protected superconducting artificial atoms in the Lambda scheme

We propose a protocol for the manipulation of a three-level artificial atom in Lambda ($\mathrm{\Lambda }$) configuration. It allows faithful, selective and robust population transfer analogous to stimulated Raman adiabatic passage ($\mathrm{\Lambda }$-STIRAP), in last-generation superconducting artificial atoms, where protection from noise implies the absence of a direct pump coupling. It combines the use of a two-photon pump pulse with suitable advanced control, operated by a slow modulation of the phase of the external fields, leveraging on the stability of semiclassical microwave drives. This protocol is a building block for manipulation of microwave photons in complex quantum architectures. Its demonstration would be a benchmark for the implementation of a class of multilevel advanced control procedures for quantum computation and microwave quantum photonics in systems based on artificial atoms.

Figure 1

(a) A three-level system with splittings ${\omega }_{ij}:=E_j-E_i$ driven by two quasiresonant ac pump (${\mathrm{\Omega }}_p$) and Stokes (${\mathrm{\Omega }}_s$) fields, in the usual $\mathrm{\Lambda }$ scheme (${\omega }_p:={\omega }_{02}-{\delta }_p, {\omega }_s={\omega }_{12}-{\delta }_s$). In 2+1 STIRAP the pump is operated by two pulses ${\mathrm{\Omega }}_{pk}$ at frequencies ${\omega }_{p1}:={\omega }_{01}-{\delta }_2, {\omega }_{p2}={\omega }_{12}-{\delta }_p+{\delta }_2$ such that ${\omega }_{p1}+{\omega }_{p2}={\omega }_{02}-{\delta }_p$. (b) Pulses in conventional STIRAP (dashed lines) in the counterintuitive sequence, i.e., the Stokes pulse is shined before the pump pulse. Real part of the pulses in 2+1-STIRAP (solid lines): here ${\mathrm{\Omega }}_s{\text{e}}^{i{\varphi }_s\left(t\right)}$ shows the phase modulated control of Eq. (6).

Figure 2

Population histories ${\rho }_{00}\left(t\right)$ (blue), ${\rho }_{11}\left(t\right)$ (red), and ${\rho }_{22}\left(t\right)$ (green). (a) For a flux AA biased at the symmetry point $f=1/2$ (spectrum in the inset), with the phase modulation Eq. (6). We used ${\mathrm{\Omega }}_r/2\pi =200\text{MHz}$ and ${\delta }_2=-5{\mathrm{\Omega }}_r$, for the two-photon pump, yielding ${\mathrm{\Omega }}_0=|{\mathrm{\Omega }}_r/2{\delta }_2|=20\text{MHz}$. Good adiabaticity, ${\mathrm{\Omega }}_{0}T=15$, is obtained with $T=0.12\mu \mathrm{s}$ and $\tau =0.6T$. Results refer to the device in Ref. [18] and account for leakage and effects of noise. In the absence of phase modulation, population histories ${\sigma }_{ii}$ in the absence of decoherence (dashed lines) show no population transfer. (b) Same results for a transmon (spectrum in the inset) with phase modulation Eq. (13). Here ${\mathrm{\Omega }}_0=3.9\text{MHz}$ from Eq. (11), $T=0.6\mu \mathrm{s}$, and $\tau =0.6T$. For both designs the approximate effective dynamics (gray thin lines above the exact population histories), obtained respectively from Eqs. (5) and (10), reproduces remarkably well the coarse-grained time evolution.

Figure 3

Parametric robustness of the protocol. (a) Effective duration of the protocol $T=15/{\mathrm{\Omega }}_0$ ($\mu \mathrm{s}$ units) vs ${\mathrm{\Omega }}_r/2\pi$ ($\mathrm{GHz}$ units) for the transmon of Ref. [19] [Eq. (11), upper curve] and for the flux qudit of Ref. [18] (limit $\alpha \gg {\delta }_2$, lower curve), at fixed $|{\delta }_2|/{\mathrm{\Omega }}_r=5$. (b) Efficiency versus stray detunings $\tilde{\delta },{\tilde{\delta }}_p$ for the transmon design, showing the robustness of the protocol Eq. (13). The solid black inner curve encloses the region of efficiency $>95\%$; results of the approximation $H_{\text{eff}}$, Eq. (10), are also reported (dashed curve) which show again the remarkable accuracy of the effective theory. The white straight line ${\tilde{\delta }}_p=2\tilde{\delta }$ represents the correlated quasitatic fluctuations of the stray detunings. The red outer curve encloses the $>95\%$ efficiency area for $r=2$: it is seen that robustness along the line further increases using $r>1$. (c) Efficiency vs $T_2$ for the transmon, $r=1$ (lower line) and $r=2$ (upper line).