Robust coherent preparation of entangled states of two coupled flux qubits via dynamic control of the transition frequencies

Coherent control and the creation of entangled states are discussed in a system of two superconducting flux qubits interacting with each other through their mutual inductance and identically coupling to a reservoir of harmonic oscillators. We present different schemes using continuous-wave control fields or Stark-chirped rapid adiabatic passages, both of which rely on a dynamic control of the qubit transition frequencies via the external bias flux in order to maximize the fidelity of the target states. For comparison, also special area pulse schemes are discussed. The qubits are operated around the optimum point, and decoherence is modeled via a bath of harmonic oscillators. As our main result, a coherent control scheme is presented which is robust against imperfections in the driving fields, and that enables one to prepare different Bell states consisting of the collective ground and excited states of the two-qubit system.

Figure 1

Two superconducting flux qubits interacting with each other through their mutual inductance $M$ and damped to a common reservoir modeled as an LC circuit. The individual bias fluxes are varied dynamically in order to control the qubit transition frequencies around the optimum point. In the figure, crosses indicate Josephson junctions, whereas the bottom circuit loop visualizes the bath.

Figure 2

(Color online) Time evolution of the population (solid red line) in the antisymmetric state $|a⟩$. The idea is to prepare the antisymmetric state while the two qubits are nondegenerate, and only afterward render the two qubits degenerate. For this, the parameters are chosen such that ${\gamma }_{12}=0.9986{\gamma }_0$, $\lambda =50{\gamma }_0$, $\delta =50{\gamma }_0$, and ${\Omega }_0=50{\gamma }_0$. The two qubit transition frequencies are adjusted via time-dependent bias fluxes, such that the frequency difference $\Delta \left(t\right)$ (dashed black line) changes from $18{\gamma }_0$ to zero as a cosine function during the time period $120{\gamma }_0^{-1}–160{\gamma }_0^{-1}$. The driving field (dash-dotted blue line) is turned off from its initial value ${\Omega }_0$ in the period $165{\gamma }_0^{-1}–175{\gamma }_0^{-1}$.

Figure 3

(Color online) (a) Time-dependent population of the symmetric state $|s⟩$ directly from the ground state. The dynamics is induced by a continuous TDMF with parameters ${\Omega }_0=15{\gamma }_0$ and $\delta =-50{\gamma }_0$, and the other parameters are ${\gamma }_{12}=0.9986{\gamma }_0$, $\lambda =50{\gamma }_0$, and $\Delta =0$. The numerical result (solid red line) is well fitted by Eq. (19) shown as the black dash-dotted line. A small part of the population is transferred to the collective excited state (dashed green line). (b) Corresponding results for the concurrence.

Figure 4

(Color online) Robust populating the symmetric state from the ground state via the SCRAP technique for ${\gamma }_{12}=0.9986{\gamma }_0$, and $\lambda =50{\gamma }_0$. The thin solid black line shows the population of the desired symmetric state, while the thick solid blue line and the dash-dotted red line are the time-dependent Rabi frequency and detuning required for SCRAP.

Figure 5

(Color online) Population of the symmetric state $|s⟩$ from the antisymmetric state $|a⟩$. The two panels show the time-evolution of (a) population and (b) concurrence for ${\gamma }_{12}=0.9986{\gamma }_0$, $\lambda =50{\gamma }_0$, $\Delta =0$, $\delta =0$, and ${\Omega }_0=15{\gamma }_0$. The populations in the state $|g⟩$ (black dashed line) and $|s⟩$ (solid red line) are well fitted by ${\rho }_{gg}\left(t\right)$ and ${\rho }_{ss}\left(t\right)$ (fits shown by green dotted lines), respectively.

Figure 6

(Color online) Direct population transfer from the collective ground to the collective excited state (solid red thin line) using the SCRAP technique. A Gaussian pump pulse (dashed black line) is followed by a delayed Gaussian time-dependent detuning (blue thick line).

Figure 7

(Color online) Creation of Bell states $|{\Phi }_{\pm }⟩$ using the SCRAP technique from the ground state. The solid thin red line denotes population ${\rho }_{+}$ in $|{\Phi }_{+}⟩$, while the dashed green line shows population ${\rho }_{-}$ in $|{\Phi }_{-}⟩$. The concurrence (dash-dotted blue line) has a maximum value of 0.94. The black dash double dotted line indicates the applied SCRAP pulse Rabi frequency, while the thick blue line shows the time-dependent Stark detuning.

Figure 8

(Color online) Creation of superpositions of Bell states controlled by the static detuning ${\delta }_0$. The populations ${\rho }_{\pm }$ in states $|{\Phi }_{+}⟩$ (dashed red line) and $|{\Phi }_{-}⟩$ (dash-dotted green line) exhibit periodic oscillations as a function of ${\delta }_0$. The maximum concurrence $C$ is larger than 0.95 (solid blue line).