Optical Selection Rules and Phase-Dependent Adiabatic State Control in a Superconducting Quantum Circuit

We analyze the optical selection rules of the microwave-assisted transitions in a flux qubit superconducting quantum circuit (SQC). We show that the parities of the states relevant to the superconducting phase in the SQC are well defined when the external magnetic flux ${\Phi }_{\mathrm{e}}={\Phi }_0/2$; then the selection rules are the same as the ones for the electric-dipole transitions in usual atoms. When ${\Phi }_{\mathrm{e}}\ne {\Phi }_0/2$, the symmetry of the potential of the artificial “atom” is broken, a so-called $\Delta$-type “cyclic” three-level atom is formed, where one- and two-photon processes can coexist. We study how the population of these three states can be selectively transferred by adiabatically controlling the electromagnetic field pulses. Different from $\Lambda$-type atoms, the adiabatic population transfer in our three-level $\Delta$ atom can be controlled not only by the amplitudes but also by the phases of the pluses.

Figure 1

(a) Energy levels, in units of $E_{\mathrm{J}}$, of the flux qubit vs reduced flux $f$ for states $|0⟩$ to $|5⟩$. (b) $f$-dependent ratios of transition frequencies: $D_{ij}=({\epsilon }_3-{\epsilon }_2)/({\epsilon }_i-{\epsilon }_j)$ with $j<i<3$. (c) Moduli $|t_{ij}|$ of the transition matrix elements between states $|i⟩$ and $|j⟩$ vs $f$, for the lowest three levels. Transition diagram of three energy levels vs $f$: (d) corresponds to $f=0.5$ and (e) to $f\ne 0.5$, where the green solid (red dashed) line means allowed (forbidden) transitions. (e) A triangle-shaped or $\Delta$-type energy diagram for $f\ne 0.5$, where all photon-assisted transitions are possible and the electric-dipole selection rules do not hold.

Figure 2

(a) The rescaled time $t/\tau$ dependence of the eigenvalues $E_k$ of $H_{\mathrm{int}}$ with the detuning ${\omega }^{\prime }=0$ for different pulse central times: ${\tau }_1=2{\tau }_2/3={\tau }_3/2=2\tau$ (dash-dotted blue curves); ${\tau }_1={\tau }_3/2=2\tau$ and ${\tau }_2=3.7\tau$ (solid red curves), and ${\tau }_1={\tau }_2/2={\tau }_3/2=2\tau$ (dotted black curves). (b) The phase-dependent eigenvalues $E_k$ at $t=3\tau$ (solid red curves) and $t=3.5\tau$ (dotted blue curves), with time delays ${\tau }_1={\tau }_2/2={\tau }_3/2=2\tau$. Here eigenvalues are given in order $E_1$, $E_3$, and $E_2$ from the top curve to the bottom curve for the same color.

Figure 3

(a) The rescaled time $t/\tau$ and (phase) $\varphi$-dependent probabilities $P_{3m}$ of the diabatic components in the third adiabatic eigenstate $|E_3⟩$. (b) The representative top curves of $F_{kl}(t)$ are plotted for two phases $\varphi =\pi$ (dashed blue curve) and $\varphi =\pi /2$ (solid black curve) with the pulses given in Fig. 2 for central times ${\tau }_1=2{\tau }_2/3={\tau }_3/2=2\tau$; here, the detunings are taken as ${\Delta }_{mn}=0$. The dash-dotted red line denotes $F_{kl}(t)=1$.