Tunable $\mathrm{\Lambda }$-type system made of a superconducting qubit pair

Two transversely coupled and resonant qubits form symmetric and antisymmetric states as their eigenstates. In this paper, we show that parametric modulation of an individual qubit enables direct Rabi swapping between the two states. Its application to set up a $\mathrm{\Lambda }$-type system with a pair of strongly coupled superconducting transmon qubits is discussed. The excited state is made of the symmetric state, and the metastable state is the antisymmetric state. The coherence of the metastable state is only limited by the pure dephasing mechanism. Based on this scheme, $\mathrm{\Lambda }$-type electromagnetically induced transparency, Autler-Townes splitting, and stimulated Raman adiabatic passage are numerically demonstrated. We highlight the large frequency tunability in such superconducting $\mathrm{\Lambda }$-type systems.

Figure 1

Level diagram of the tunable $\mathrm{\Lambda }$-type system. (a) Representation of the control protocol in the basis of two resonant qubits $Q_a$ and $Q_b$. The transition frequency ${\omega }_a$ is modulated sinusoidally (red wavy, see main text), while ${\omega }_b$ stays static. $J$ represents the interqubit coupling. (b) Level diagram described in the system eigenstates, with corresponding transition driven by the parametric modulation in (a) (red circular). It is also referred to as the coupling beam in Sec. 4 or the Stokes tone in Sec. 5. The blue arrow from $|G\rangle$ to $|B\rangle$ indicates the dipole-allowed transition, named as the probe beam in Sec. 4 or the pump tone in Sec. 5.

Figure 2

Tunable $\mathrm{\Lambda }$-type system constructed with two transmon qubits. (a) Two capacitively coupled and resonant split-junction transmon qubits, with time-dependent flux ${\mathrm{\Phi }}_a\left(t\right)$ threading the loop of $Q_a$. Level spacing of the on-resonance system is set by the static fluxes ${\overline{\mathrm{\Phi }}}_a={\mathrm{\Phi }}_b=\overline{\mathrm{\Phi }}$ threading the two qubits. The parametric drive ${\mathrm{\Omega }}_{\mathrm{\Phi }}\left(t\right)$ is through the local flux line nearby $Q_a$. The electromagnetic excitation ${\mathrm{\Omega }}_p$ is through the waveguide that is capacitively coupled to $Q_a$ and $Q_b$. (b) Frequency tunability of the system. The transition frequency of the bright state ${\omega }_B$ (thick solid), the dark state ${\omega }_D$ (dotted), and the bare qubit frequency ${\omega }_0$ (dashed green) as a function of dc flux bias $\overline{\mathrm{\Phi }}$.

Figure 3

Numerical demonstration of parametrically induced transition between $|B\rangle$ and $|D\rangle$, started with $\rho (t=0)=|B\rangle \langle B|$. The external flux ${\mathrm{\Phi }}_a\left(t\right)=\overline{\mathrm{\Phi }}+\delta \mathrm{\Phi }cos\left({\omega }_{\mathrm{\Phi }}t\right)$ is applied to $Q_a$. (a), (b) $\overline{\mathrm{\Phi }}=0.25{\mathrm{\Phi }}_0$. $\delta \mathrm{\Phi }={10}^{-3}{\mathrm{\Phi }}_0$ corresponds to ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0/2\pi =$ 2.32 MHz. The parametric driving frequency ${\omega }_{\mathrm{\Phi }}$ is set around ${\omega }_B-{\omega }_D\cong 2J$. (a) Rabi oscillation chevron. ${\rho }_{DD}$ (colored) is shown as a function of time, against detuning ${\mathrm{\Delta }}_{\mathrm{\Phi }}$. (b) Density matrix elements ${\rho }_{DD}$ (violet sinusoidal, starting at 0), ${\rho }_{BB}$ (yellow sinusoidal, starting at 1), and ${\rho }_{GG}$ (gray, stepwise) are shown as a function of time at ${\mathrm{\Delta }}_{\mathrm{\Phi }}=0$. The decay envelope (dashed) refers to the overall excitation decay rate ${\mathrm{\Gamma }}_B^1/2$ to the ground state. (c) Extracted Rabi frequency ${\mathrm{\Omega }}_{\mathrm{\Phi }}^{\mathrm{fit}}$ (hollow symbols) against flux modulation amplitude $\delta \mathrm{\Phi }$ at different flux bias points $\overline{\mathrm{\Phi }}$ with ${\mathrm{\Delta }}_{\mathrm{\Phi }}=0$. They agree well with the evaluation ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0=\frac{1}{2}\frac{\partial {\omega }_a}{\partial \overline{\mathrm{\Phi }}}\delta \mathrm{\Phi }$ (solid lines). Refer to Appendix pp5 for $\rho \left(t\right)$ under different $J/{\mathrm{\Omega }}_{\mathrm{\Phi }}^0$ ratios. The detailed mapping of ${\mathrm{\Omega }}_{\mathrm{\Phi }}^{\mathrm{fit}}$ and the validity of applying RWA against $\overline{\mathrm{\Phi }}$ are given in Appendix pp6. Calculation results that include only the lowest three levels of a TCQ are presented in (b) (black lines) and (c) (solid symbols) as well.

Figure 4

Demonstration of $\mathrm{\Lambda }$-type EIT and ATS in a degenerate TCQ. Refer to Fig. 1 for the setup and Sec. 3 for the simulation parameters. Steady-state atomic absorption response $\mathrm{Im}({\rho }_{GB}^{\mathrm{SS}}$) is shown as a function of probe detuning ${\delta }_p$ for (a) ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0/2\pi =13.1$ MHz and (c) ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0/2\pi =69.0$ MHz, respectively. (b, d) Corresponding dispersion responses Re(${\rho }_{GB}^{\mathrm{SS}}$) of (a) and (c), respectively. The symbols indicate the results of numerical simulation from Eq. (8). The results that include the six lowest levels of the system are also shown as crosses here for comparison. The fitting curve adapted from Eq. (10) is presented as black solid lines. (e) Extracted fitting parameters ${\gamma }_{BB}/2\pi$ (black star), ${\gamma }_{DD}/2\pi$ (black cross), and ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0/2\pi$(red square) of Eq. (10) to Im(${\rho }_{GB}^{\mathrm{SS}}$) are shown as a function of ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0$. The lines indicate the parameter values used in the simulation. (f) ${\overline{\omega }}_{\mathrm{EIT}}$ and ${\overline{\omega }}_{\mathrm{ATS}}$ against ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0$. ${\overline{\omega }}_{\mathrm{EIT}}>0.5$ indicates where the EIT spectrum fits better. The shaded region indicates the corresponding EIT regime ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0<|{\gamma }_{BB}-{\gamma }_{DD}|$.

Figure 5

$\mathrm{\Lambda }$-type STIRAP to swap population from $|G\rangle$ to $|D\rangle$ state, starting with $\rho \left(t=0\right)=|G\rangle \langle G|$ and stopping at time $t_{\mathrm{f}}=0.8\mu \mathrm{s}=2.57\times 2\pi /{\gamma }_{BB}$. Refer to Fig. 1 for the setup. (a) Control sequence, with the parametric Stokes pulse ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0\left(t\right)$ (red) followed by the pump pulse ${\mathrm{\Omega }}_p^0\left(t\right)$ (blue). The pulse separation $2\tau =182$ ns and the pulse duration $2T=200$ ns. The black dotted line indicates ${\mathrm{\Omega }}_{\mathrm{rms}}^0\left(t\right)$ with peak Rabi frequency 67.2 MHz. (b) Corresponding atomic response as a function of time. ${\rho }_{DD}\left(t_{\mathrm{f}}\right)=0.969$ (dotted) and ${\rho }_{GG}\left(t_{\mathrm{f}}\right)=0.031$ (gray solid). The intermediated state ${\rho }_{BB}$ (thick solid black) reaches 0.014 during the process. Simulation results that include the six lowest levels of a TCQ are presented as colored shaded lines for comparison. (c) Transfer efficiency as a function of peak pulse amplitude ${\mathrm{\Omega }}^{\mathrm{pk}}$. Colors represent different pulse delays $\tau$. Solid lines represent the analytical modeling accounting for nonadiabaticity [46, 47].

Figure 6

Energy levels and selection rules of a degenerate TCQ. (a) Six lowest eigenstates $\left\{\right|j\rangle \}$ of a degenerate TCQ. They are presented in the basis of {$n_a,n_b$}, with truncated net cooper pair number $n_i\in \{-7,7\}$. The color bar represents the probability amplitude $\psi (n_a,n_b)$. The opposite sign (red and blue) only indicates the relative phase. (b) The selection rules between state $|j\rangle$ and $|k\rangle$, determined by the transition moment $\langle j\left|\widehat{N}\right|k\rangle$, with $\widehat{N}$ the number operator of Cooper pairs. The allowed transition (black arrows) and parametrically available driving channel (red circular arrow) are indicated.

Figure 7

Simulation of parametric drive with an overestimated flux crosstalk $C_{\mathrm{f}}=0.1$ between $Q_a$ and $Q_b$. (a) Effective Rabi oscillation of ${\rho }_{BB}\left(t\right)$. The black line indicates $C_{\mathrm{f}}=0$. The crosstalk values with ${\mathrm{\Omega }}_{\mathrm{\Phi }}^b=0.1{\mathrm{\Omega }}_{\mathrm{\Phi }}^a$(red dashed) and ${\mathrm{\Omega }}_{\mathrm{\Phi }}^b=-0.1{\mathrm{\Omega }}_{\mathrm{\Phi }}^a$(blue dotted) are presented as well. (b) Fitted Rabi frequency ${\mathrm{\Omega }}_{\mathrm{\Phi }}^{\mathrm{fit}}$ as a function of parametric driving amplitude $\delta {\mathrm{\Phi }}_a$ on $Q_a$. The symbols represent the simulation results and the lines give the deduction from Eq. (C4). The setting parameters here are identical to those of Fig. 3 (a), (b).

Figure 8

Examination of the rotating wave approximation. The initial condition $\rho (t=0)=|B\rangle \langle B|$, bias point $\overline{\mathrm{\Phi }}=0.25{\mathrm{\Phi }}_0$, and driving frequency at zero detuning ${\mathrm{\Delta }}_{\mathrm{\Phi }}=0$ are set in the simulation. (a–c) Snapshots of ${\rho }_{BB}\left(t\right)$ (black solid lines) at different $2J/{\mathrm{\Omega }}_{\mathrm{\Phi }}^0$. The dashed red lines are the sinusoidal fitting. The parametric driving amplitude is fixed at ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0/2\pi =23.3$ MHz and the interqubit coupling $J$ is varied. (a) $2J/{\mathrm{\Omega }}_{\mathrm{\Phi }}^0=15.8$, resulting in ${\epsilon }_s=2.6\times {10}^{-4}$. (b) $2J/{\mathrm{\Omega }}_{\mathrm{\Phi }}^0=5.8, {\epsilon }_s=2.1\times {10}^{-3}$. (c) $2J/{\mathrm{\Omega }}_{\mathrm{\Phi }}^0=2.1, {\epsilon }_s=4.6\times {10}^{-2}$. (d) The sinusoidal likelihood ${\epsilon }_s$ (colored) as a function of ${\mathrm{\Omega }}_{\mathrm{\Phi }}^0$ and interqubit coupling $J$.

Figure 9

Search over the tunability of the parametric drive with $J/2\pi =700$ MHz. (a) Flux tunability $\partial {\omega }_0/\partial \mathrm{\Phi }$ as a function of bias point $\overline{\mathrm{\Phi }}$. (b) ${\mathrm{\Omega }}_{\mathrm{\Phi }}^{\mathrm{fit}}$ (colored). (c) RWA deviation factor ${\epsilon }_s$ (colored) as a function of $\overline{\mathrm{\Phi }}$ and modulation amplitude $\delta \mathrm{\Phi }$.