Stationary entanglement in strongly coupled qubits

The dynamics of two superconducting flux qubits coupled to each other and to a common bath is discussed. We focus on the case in which the qubit-qubit coupling strength dominates over the respective qubit transition frequencies. We derive the master equation including collective effect by modeling the bath as one-dimensional open space in this ultrastrong coupling regime, and find that the coupling greatly modifies both the coherent and the incoherent dynamics of the system, giving rise to qualitatively different properties. By analyzing the steady-state and the dynamics governed by the master equation, we show that ground-state entanglement and maximum coherence between the two qubits can be induced by the environment alone. By employing in addition a single external driving field, both the entangled antisymmetric and symmetric collective states can be populated and preserved with high fidelity. Similarly, entangled states can be prepared using adiabatic passage techniques using two external fields. Our results could find applications in entangling quantum gates and quantum memories free from the decoherence.

Figure 1

The considered model system. Two superconducting flux qubits are coupled through their mutual effective inductance $M$ and coupled to a common reservoir modeled as an $LC$ circuit via the mutual inductance $M_B$. We consider the case of strong coupling between the qubits, in which the always-on-coupling strength between the two qubits exceeds their respective transition frequencies.

Figure 2

The transition dipole matrix element as a function of bias $f_b={\Phi }_e/{\Phi }_0$ for $\xi =0.8\text{and}{E}_J/E_C=35$.

Figure 3

The relevant relaxation channels in the collective dressed state basis. The transition $|1\rangle \to |4\rangle$ is a two-photon channel.

Figure 4

The system in the collective Dicke state basis. The green downward arrows denote dissipation, whereas the red upward arrows indicate pumping, which is introduced in the collective state basis due to relaxation in the dressed-state basis shown in Fig. 3.

Figure 5

The collective decay rates ${\gamma }_{1\to 2}$ [(i) thin blue line], ${\gamma }_{2\to 4}$ [(ii) red line], ${\gamma }_{1\to 3}$ [(iii) green line], and ${\gamma }_{3\to 4}$ [(iv) brown line] as a function of the AOC coupling $J$. The distance is chosen as $r_{12}={\lambda }_0/50$. In (a), the two transition dipole moments are parallel ($p=1$). The result for antiparallel dipole moments ($p=-1$) is obtained by mirroring the figure at the $J=0$ axis.

Figure 6

The collective decay rates ${\gamma }_{1\to 2}$ [(i) thin blue line], ${\gamma }_{2\to 4}$ [(ii) red line], ${\gamma }_{1\to 3}$ [(iii) green line], and ${\gamma }_{3\to 4}$ [(iv) brown line] as a function of the coupling $J$ for $r={\lambda }_0/1000$ and $p=1$.

Figure 7

The dipole-dipole induced shifts ${\Omega }_d^{+}$ [(i) blue line], ${\Omega }_d^{-}$ [(ii) dashed red line], and ${\Omega }_d$ [(iii) dash-dotted green line] as functions of the qubit separation $r_{21}/{\lambda }_0$ at a fixed frequency ${\omega }_0$.

Figure 8

(a) Pictorial representation of a photon emission from qubit Q1 in an open two-dimensional (2D) space. (b) The corresponding situation in 1D space. The results are obtained by numerically solving Maxwell’s equations for an oscillating source with the finite-difference time-domain method (FDTD).[73]

Figure 9

(a) Fidelities of the Bell states $|{\Phi }_{+}\rangle$ [thin blue line (i)], $|{\Phi }_{-}\rangle$ [green lines (iii)], and entanglement [red lines (ii)] as a function of the AOC $J$. (b) Fidelities of the product states $|e_1,e_2\rangle$ [blue lines (i)], $|g_1,g_2\rangle$ [green lines (ii)], and coherence [thick cyan lines (iii)] as a function of the AOC $J$. In both panels, the parameters are $p=1$ and $r={\lambda }_0/1000$. Note that the coherence was shifted upward by $0.5$. The solid lines indicate the case without pure dephasing, while the dashed lines include a pure dephasing of ${\Gamma }_{\phi }=0.01{\gamma }_0$.

Figure 10

Time-evolution of entanglement for time-dependent couplings between the qubits. The time-dependent coupling strength $J$ is shown as thin blue line (i). The amount of entanglement is shown as red line (ii). The fidelity of the Bell state $|{\Phi }_{-}\rangle$ is indicated as the dashed green line (iii) and the total excitation from the ground state is the dot-dashed cyan line (iv). Panel (a) shows the case of slow switching of the AOC $J$ with switching time $t_{\text{sw}}={\gamma }_0^{-1}$. (b) shows the corresponding results with fast switching $t_{\text{sw}}=0.01{\gamma }_0^{-1}$ (b). The results are obtained for $r_{21}={\lambda }_0/1000$ and include a pure dephasing of ${\Gamma }_{\phi }=0.01{\gamma }_0$.

Figure 11

(a) Amount of entanglement as a function of the switching time $t_{\text{sw}}$ and the coupling $J$. The white dashed lines indicate pulses with area $3\pi /2,5\pi /2,7\pi /2,9\pi /2,11\pi /2,13\pi /2,\text{and}15\pi /2$, respectively. (b) shows the time evolution of the entanglement generation for a pulsed AOC $J$. The time evolution of the coupling $J$ is shown as a blue line (i). The resulting entanglement is plotted as a red line (ii). The fidelity of the state $|{\Phi }_{-}\rangle$ is shown as a green line (iii) and that of the state $|{\Phi }_{+}\rangle$ as a cyan line (iv). Finally, the state population of $|G\rangle$ is shown as a green dashed line (v), and the population of $|E\rangle$ as a cyan dashed line (vi). Note that the lines (i), (v), and (vi) are shifted down by $-1$ for better clarity. In (b), the switching time is $t_{\text{sw}}=0.0027{\gamma }_0^{-1}$. The other parameters are ${\Gamma }_{\phi }=0.01{\gamma }_0$ and $r_{21}={\lambda }_0/1000$.

Figure 12

Level diagram and relevant coherent and incoherent processes for the generation of stationary state entanglement with a single continuous wave driving field. As an example, we use the parameters that correspond to those indicated by the point P in Fig. 14a.

Figure 13

Optimizing the entanglement generation with a single cw field. The figure shows the fidelity of the Bell state $|{\Psi }_{-}\rangle$ as a function of the detuning $\delta$. The solid lines are drawn for a distance $r_{21}={\lambda }_0/50$, the dashed lines show a smaller separation $r_{21}={\lambda }_0/1000$. The two upper blue lines correspond to the ideal system without pure dephasing, whereas the two lower red lines show the results taking into account a pure dephasing ${\Gamma }_{\phi }=0.01{\gamma }_0$.

Figure 14

Steady-state entanglement with a single continuous-wave driving field as a function of the driving field Rabi frequency $\Omega$ and the coupling $J$. The upper panel (a) shows the results for parallel dipole moments, $p=1$, of the two qubits. The lower panel (b) shows the corresponding results for antiparallel dipole moments, $p=-1$. In both cases, the frequency of the driving field is chosen as half the transition transition frequency of the $|1\rangle \to |4\rangle$ transition. Thus, for large coupling $J$, all transitions are far off-resonant with the field. The phase is chosen as ${\phi }_{12}=0$.

Figure 15

Interpretation of the steady-state entanglement shown in Fig. 14 in terms of the population of the magic-state basis states. The four panels show the fidelity of the states (a) $|{\Phi }_{-}\rangle$, (b) $|{\Psi }_{-}\rangle$, (c) $|{\Phi }_{+}\rangle$, and (d) $|{\Psi }_{+}\rangle$. The figure is drawn for parallel dipole moments ($p=1$).

Figure 16

Steady-state entanglement and interpretation in terms of Bell-state populations against the driving field Rabi frequency $\Omega$ and the coupling $J$. The figure is similar to Fig. 14, but includes pure dephasing ${\Gamma }_{\phi }=0.01{\gamma }_0$. (a) shows the amount of entanglement, (b) the fidelity of the Bell state $|{\Phi }_{-}\rangle$, and (c) that of the Bell state $|{\Psi }_{-}\rangle$. The results are shown for antiparallel dipole moments ($p=-1$).

Figure 17

Steady-state entanglement and interpretation in terms of Bell-state populations against the driving field Rabi frequency $\Omega$ and the coupling $J$. The figure is similar to Fig. 16, but generated for larger qubit separation $r_{21}={\lambda }_0/50$. (a) shows the amount of entanglement and (b) the fidelity of the state $|{\Psi }_{-}\rangle$.

Figure 18

Time evolution of steady-state entanglement generation with a single cw field. The red curve shows (ii) the amount of entanglement, the blue thin line (i) shows the fidelity of state $|{\Psi }_{-}\rangle$, the green dashed line (iii) that of $|{\Phi }_{-}\rangle$, the cyan line (iv) that of $|{\Psi }_{+}\rangle$, and the orange dot-dashed line (v) that of $|{\Phi }_{+}\rangle$. The parameters are $r_{21}={\lambda }_0/50$, $p=1$, $\delta =2.5{\gamma }_0$, ${\Gamma }_{\phi }=0.01{\gamma }_0$, and $\Omega =80{\gamma }_0$.

Figure 19

Level diagram and relevant coherent and incoherent processes for entanglement generation using dark-state techniques.

Figure 20

Entanglement generation with the dark-state technique. In (a), the results are shown against the detuning ${\delta }_p$. The red curve (ii) shows the amount of entanglement, the blue thin curve (i) the fidelity of $|{\Psi }_{-}\rangle$, the green dashed line (iii) that of $|4\rangle$, the cyan line (iv) that if $|2\rangle$, and the orange dot-dashed line (v) that of $|1\rangle$. The parameters are ${\delta }_c=0$ and ${\Omega }_p={\gamma }_0$. Corresponding results shown against the control field Rabi frequency ${\Omega }_c$. The parameters are ${\Omega }_p=4{\gamma }_0$ and ${\delta }_p={\delta }_c=0$. In both panels, the other parameters are $r_{21}={\lambda }_0/50$, ${\Gamma }_{\phi }=0.01{\gamma }_0$, ${\tilde{\gamma }}_0=0.02{\gamma }_0$, $J=5{\omega }_0$, and $p=1$.

Figure 21

Coherent population transfer from the ground state to state $|3\rangle$ via the STIRAP technique. The Stokes pulse ${\Omega }_c$ (blue thin dashed line) precedes the pumping pulse ${\Omega }_p$ (red thin dashed line) by $15{\gamma }_0^{-1}$. In the figure, both pulses are shown normalized. The other curves show the amount of entanglement [red line (ii)], and the fidelity of states $|{\Psi }_{-}\rangle$ [blue thin line (i)], $|4\rangle$ [green dashed line (iii)], $|2\rangle$ [cyan line (iv)], and $|1\rangle$ [orange dot-dashed line (v)]. The durations of both pulses are chosen as $\tau =5{\gamma }_0^{-1}$. The detunings are ${\delta }_p={\delta }_c=\delta$. The other parameters are $J=5{\omega }_0$, $r_{21}={\lambda }_0/50$, ${\Gamma }_{\phi }=0.01{\gamma }_0$, ${\tilde{\gamma }}_0=0.02{\gamma }_0$, and $p=1$.