Resonator-assisted quantum bath engineering of a flux qubit

We demonstrate quantum bath engineering for preparation of any orbital state with the controllable phase factor of a superconducting flux qubit assisted by a microwave coplanar waveguide resonator. We investigate the polarization efficiency of the arbitrary direction rotating on the Bloch sphere, and obtain an effective Rabi frequency by using the convergence condition of the Markovian master equation. The processes of polarization can be implemented effectively in a dissipative environment created by resonator photon loss when the spectrum of the microwave resonator matches with the specially tailored Rabi and resonant frequencies of the drive. Our calculations indicate that state-preparation fidelities in excess of 99% and the required time on the order of magnitude of a microsecond are in principle possible for experimentally reasonable sample parameters. Furthermore, our proposal could be applied to other systems with spin-based qubits.

Figure 1

(a) A superconducting flux qubit is coupled to a CPW resonator via the induced magnetic field. The blue sinusoidal curves describe the microwave drive. (b) The superconducting flux qubit is labeled with the new eigenstates $\left|0\right.〉$ and $\left|1\right.〉$ with the energy splitting ${\omega }_{\mathrm{sc}}$. (c) Bloch sphere diagrams indicate that polarization direction z (pink arrow) defined by an around-$z$-axis rotation with angle $\varphi -R_z\left(\varphi \right)$ followed by an around-$y$-axis rotation with angle $\theta -R_y\left(\theta \right)$, is determined by the rate of the detuning of the drive (blue arrow), the real part (red arrow), and the imaginary part (green arrow) of the Rabi frequency [as illustrated by Eq. (20)].

Figure 2

(a) Normalized expectation value $-〈{\sigma }_{\mathbf{z}}〉$ of the flux qubit as a function of the dimensionless parameter ${\Gamma }_{\mathbf{z}}t$ for various equilibrium temperatures of the resonator ranging from $\overline{n}=0$ to $\overline{n}=0.5$. (b) Effective dissipation rate in the units of $g^2/\kappa =1$ versus the dimensionless parameters $\Delta /\kappa$ and $\theta$. (c) The infidelity of the generated state as a function of the dimensionless parameters ${\Delta }_R=\delta \mathrm{Re}\left(\Omega \right)/\mathrm{Re}\left(\Omega \right)$ and ${\Delta }_I=\delta \mathrm{Im}\left(\Omega \right)/\mathrm{Im}\left(\Omega \right)$ at equilibrium, i.e., $I{F}_{\mathbf{z}}$ vs ${\Delta }_R$ and ${\Delta }_I$, for parameters $[\mathrm{Re}\left(\Omega \right),\mathrm{Im}\left(\Omega \right),\delta \varpi /2]/2\pi =[100,100,100]/\sqrt{3}$ MHz and $[\theta ,\varphi ]=[arccos(1/\sqrt{3}),3\pi /4]$. (d) The evolution of the fidelity of the ground state of ${\sigma }_{\mathbf{z}}=({\sigma }_x+{\sigma }_y+{\sigma }_z)/\sqrt{3}$ for different deviations of parameters ${\Delta }_R$ and ${\Delta }_I$, where the deviation situations of $[{\Delta }_R,{\Delta }_I]=[0.0,0.0],[-0.2,+0.2],[+0.2,-0.2]$, which respond to red, green, and blue curves, respectively, almost overlap. The dimensionless parameter $\tau =2\overline{\Omega }t$ is defined, while other parameters are the same as (c). Here $k$ means ${10}^3$.

Figure 3

(a) The fidelity of the prepared state of the flux qubit, at equilibrium, versus the dimensionless parameters $\gamma =\Gamma /2\overline{\Omega }$ and $\theta$, i.e., $F_{\mathbf{z}}$ vs $\gamma$ and $\theta$, for $\eta =0.52,\zeta =0.30$, and $\varphi =\pi$; (b) $F_{\mathbf{z}}$ vs $\theta$ and $\varphi$, for $\eta =0.25,\zeta =0.17$, and $\gamma =0.02$.

Figure 4

The evolution of the fidelity of the ground state of ${\sigma }_z$ and ${\sigma }_{x,y}$ (inset) for various dissipative decay rates of the flux qubit ranging from $\gamma =0$ to $\gamma =0.05$, corresponding to the enclosed optimized parameters $\eta$ and $\zeta$. Here $k$ means ${10}^3$.