Measurement-induced two-qubit entanglement in a bad cavity: Fundamental and practical considerations

An entanglement-generating protocol is described for two qubits coupled to a cavity field in the bad-cavity limit. By measuring the amplitude of a field transmitted through the cavity, an entangled spin-singlet state can be established probabilistically. Both fundamental limitations and practical measurement schemes are discussed, and the influence of dissipative processes and inhomogeneities in the qubits are analyzed. The measurement-based protocol provides criteria for selecting states with an infidelity scaling linearly with the qubit-decoherence rate.

Figure 1

The physical system under consideration. Two qubits are coupled to a cavity field, ${\widehat{a}}_{\mathrm{c}}$, which is driven externally by a constant coherent field, $\beta$ (in the frame rotating at the driving frequency, ${\omega }_{\mathrm{L}}$). The field-decay rates through the cavity mirrors are denoted by ${\kappa }_1$ and ${\kappa }_2$. The output field, ${\widehat{a}}_{\mathrm{T}}$, is subjected to a balanced homodyne measurement using a local oscillator, ${\alpha }_{\mathrm{LO}}=\left|{\alpha }_{\mathrm{LO}}\right|e^{i\theta }$. The differential and summed photocurrents are denoted by $i_{-}\left(t\right)$ and $i_{+}\left(t\right)$, respectively.

Figure 2

(a) The photocurrent (top part) and the singlet-state overlap (bottom part) as a function of time for two individual instances of the simulations. One (blue crosses) eventually collapses into $|-\rangle$, whereas the other (black circles) collapses into the triplet space. Panel (b) shows for each integration time, $T$, the distribution of the singlet-state overlap on a relative scale set by the shaded bar.

Figure 3

All panels correspond to ${\Delta }_{\mathrm{q}}=10{\gamma }_{\mathrm{p}}$, $\chi =1.65{\Delta }_{\mathrm{q}}$, $\theta =-\pi /2$, and $\eta =1$. (a) Two instances of the integrated measurement signal, ${\zeta }_{\langle I\rangle }$, as a function of the integration time, $T$. These examples are based on the photocurrents shown in Fig. 2a, which led to the singlet state (blue crosses) or the triplet state (black circles). The red dotted lines mark a chosen threshold condition, $|{\zeta }_{\langle I\rangle }|\le {\zeta }_{\langle I\rangle }^{\mathrm{thr}}=1.96$, separating accepted and rejected states after the integration time, ${\gamma }_{\mathrm{p}}T=10$. Panel (b) shows from 10 000 simulations the distribution of the measurement signal, ${\zeta }_{\langle I\rangle }$, in case of $\langle -\left|\widehat{\rho }\right|-\rangle \ge 0.8$ (crosses) and $\langle -\left|\widehat{\rho }\right|-\rangle \le 0.2$ (circles). The solid lines show Gaussian distributions with unit variance and with mean values given by the approximate estimate of Eq. (12). The acceptance window from panel (a) is marked with red dotted lines. (c) The obtained fidelity, $F$ (dashed line), and success probability, $p_{\mathrm{suc}}$ (solid line), as functions of the threshold value, ${\zeta }_{\langle I\rangle }^{\mathrm{thr}}$, of the acceptance window. The red dotted lines correspond to the choice of panels (a) and (b) leading to $p_{\mathrm{suc}}=50\%$ and $F=94\%$. (d) Solid line, the fidelity, $F$, as a function of the success probability, $p_{\mathrm{suc}}$; red dotted line, $p_{\mathrm{suc}}=50\%$; blue dashed line, Theoretical limit for completely separate singlet- and triplet-space-measurement signals and perfect state overlap.

Figure 4

Panels (a) and (b) show as a function of the resonant Rabi frequency, $\chi$, the obtained fidelity, $F$, with a success probability, $p_{\mathrm{suc}}=50\%$. Panel (a) dc analysis of $I\left(t\right)$ with $\theta =-\pi /2$, and ${\Delta }_{\mathrm{q}}/{\gamma }_{\mathrm{p}}$ given by 10 (blue crosses), 1 (green circles), and $0.3$ (red squares). Panel (b): ac analysis of $I\left(t\right)$ with $\theta =0$, and ${\Delta }_{\mathrm{q}}/{\gamma }_{\mathrm{p}}$ equal to 0 (blue crosses), 1 (green circles), 3 (red squared), and 10 (black diamonds). The simulations corresponding to the blue crosses have been performed twice in order to depict the statistical uncertainty of $F$. (c) The magnitude of the triplet-space steady-state value of ${\widehat{S}}_x$ as given by Eq. (13) for ${\Delta }_{\mathrm{q}}/{\gamma }_{\mathrm{p}}=10$ (blue solid line), 1 (green dash-dotted line), and $0.3$ (red dashed line). (d) The height of the spectral peak in $S_I\left(\Delta \right)$ when ${\Delta }_{\mathrm{q}}/{\gamma }_{\mathrm{p}}=0$ (blue solid line), 1 (green dash-dotted line), 3 (red dashed line), and 10 (black dotted line).

Figure 5

All graphs are based on 10 000 simulations with $\chi =10{\gamma }_{\mathrm{p}}$, ${\Delta }_{\mathrm{q}}=0$, ${\gamma }_{\mathrm{p}}T=10$. (a) The simulated spectrum (dots) using the periodogram, $P_I\left(\Delta \right)$ [Eq. (16)], on simulation instances with $\langle -\left|\widehat{\rho }\right|-\rangle \le 0.2$, compared to the analytical spectrum, $S_I\left(\Delta \right)$ (solid line), of Eq. (14). The dashed line at ${\Delta }_0=9.89{\gamma }_{\mathrm{p}}$ marks the maximum of $S_I\left(\Delta \right)$, and $\Gamma$ denotes the FWHM of $S_I\left(\Delta \right)$. (b) The distribution of ${\zeta }_{\mathrm{lockin}}$ distinguished by a high ($\ge$$0.8$, crosses) and a low ($\le$$0.2$, circles) singlet-state overlap. The lockin parameters, ${\Delta }_l={\Delta }_0$ and ${\tau }_l={\Gamma }^{-1}$, follow the characteristics of $S_I\left(\Delta \right)$ from panel (a), and the red dotted line marks an acceptance criterion, ${\zeta }_{\mathrm{lockin}}\le {\zeta }_{\mathrm{lockin}}^{\mathrm{thr}}$, with $p_{\mathrm{suc}}=50\%$. Panels (c) and (d) show the obtained fidelity (with $p_{\mathrm{suc}}=50\%$) for varying lockin parameters, ${\Delta }_l$ and ${\tau }_l$, respectively. The dashed lines mark the parameters used in panel (b). Panels (e) and (f) show the variations in fidelity and success probability for varying acceptance thresholds. The red dotted lines mark the obtained fidelity $F=93\%$ at $p_{\mathrm{suc}}=50\%$.

Figure 6

The obtained fidelity versus integration time when $p_{\mathrm{suc}}=50\%$ (a) or $p_{\mathrm{suc}}=10\%$ (b). Fidelities obtained by optimal analysis based on $\langle -|\widehat{\rho }|-\rangle \ge F_{\min }$ are shown for $\theta =0$ (open circles) and $\theta =-\pi /2$ (solid circles). The lockin analysis, ${\zeta }_{\mathrm{lockin}}\le {\zeta }_{\mathrm{lockin}}^{\mathrm{thr}}$ from Sec. 4a2, gives rise to the open triangles, while the integrated-photocurrent analysis, $|{\zeta }_{\langle I\rangle }|\le {\zeta }_{\langle I\rangle }^{\mathrm{thr}}$ from Sec. 4a 1, leads to the solid triangles. The solid squares represent the displaced, time-weighted measurement signal described in the text.

Figure 7

All panels show simulation results for ${\Delta }_{\mathrm{q}}=10{\gamma }_{\mathrm{p}}$, $\chi =16.5{\gamma }_{\mathrm{p}}$, and $\theta =-\pi /2$. (a) The obtained fidelity when $p_{\mathrm{suc}}=50\%$ using $|{\zeta }_{\langle I\rangle }|\le {\zeta }_{\langle I\rangle }^{\mathrm{thr}}$ while varying ${\gamma }_{\parallel }/{\gamma }_{\mathrm{p}}=1\times {10}^{-3}$ (red squares), $2\times {10}^{-3}$ (green circles), $5\times {10}^{-3}$ (blue crosses), $1\times {10}^{-2}$ (black diamonds), $2\times {10}^{-2}$ (cyan tip-up triangles), $5\times {10}^{-2}$ (magenta tip-down triangles). (b) The optimum fidelity when $p_{\mathrm{suc}}=50\%$ for various characteristic decoherence times ${\tau }_{\mathrm{c}}$. Red squares, population decay with ${\tau }_{\mathrm{c}}={\gamma }_{\parallel }^{-1}$, green circles, phase decoherence with ${\tau }_{\mathrm{c}}=\tau$, blue triangles: Inhomogeneous coupling strength with ${\tau }_{\mathrm{c}}=\delta \chi$, black diamonds, inhomogeneous qubit frequency with ${\tau }_{\mathrm{c}}=\delta {\omega }_{\mathrm{q}}$. Open and closed symbols are obtained using $|{\zeta }_{\langle I\rangle }|\le {\zeta }_{\langle I\rangle }^{\mathrm{thr}}$ and $\langle -|\widehat{\rho }|-\rangle \ge F_{\min }$, respectively (this holds also for panels c and d). (c) Fidelity versus ${\gamma }_{\parallel }^{-1}$ for $p_{\mathrm{suc}}=50\%$ (squares), $p_{\mathrm{suc}}=30\%$ (circles), and $p_{\mathrm{suc}}=10\%$ (diamonds). Solid and dashed lines denote $F_{\mathrm{opt}}=1-\sqrt{{\gamma }_{\parallel }/{\gamma }_{\mathrm{p}}}$ and $F_{\mathrm{opt}}=1-{\gamma }_{\parallel }/{\gamma }_{\mathrm{p}}$, respectively. (d) Fidelity versus $\delta {\chi }^{-1}$ for $p_{\mathrm{suc}}=50\%$ (triangles) and $p_{\mathrm{suc}}=10\%$ (squares). Solid and dashed lines correspond to slopes of $-1$ and $-2$, respectively. In panels (e) and (f) $|\psi \rangle =|-\rangle$ at $t=0$ and ${\gamma }_{\parallel }/{\gamma }_{\mathrm{p}}=0.01$. (e) Singlet-state overlap at ${\gamma }_{\mathrm{p}}T=1$ (blue crosses), 2 (green circles), 5 (red squares), and 20 (black diamonds). (f) Average value of $\langle -\left|\widehat{\rho }\right|-\rangle$ for all incidences (blue crosses) and for those with $\langle -\left|\widehat{\rho }\right|-\rangle \ge 0.5$ (green circles). Red squares, most probable value of $\langle -\left|\widehat{\rho }\right|-\rangle$; black diamonds, fraction of states with $\langle -\left|\widehat{\rho }\right|-\rangle \ge 0.5$; solid line, $\exp (-{\gamma }_{\parallel }T)$.