Quantum master equations for entangled qubit environments

We study the Markovian dynamics of a collection of $n$ quantum systems coupled to an irreversible environmental channel consisting of a stream of entangled qubits. Within the framework of repeated quantum interactions, we derive the master equation for the joint-state dynamics of the $n$ quantum systems. We investigate the evolution of the joint state for two-qubit environments where the presence of antidiagonal coherences in the state of the bath qubits (in the local energy basis) is essential for preserving and generating entanglement between two remote quantum systems. However, maximally entangled bath qubits, such as Bell states, exhibit exceptional behavior, where the master equation does not have a unique steady state and can destroy entanglement between the systems. For the general case of $n$-qubit environments we show that antidiagonal coherences that arise from multibody entanglement in the bath qubits do not affect the composite system evolution in the weak-coupling regime.

Figure 1

Conceptual diagram of the physical model where a stream of entangled qubits sequentially interacts with separate quantum systems. Shown here is the case of $n=2$ qubits.

Figure 2

Dissipative preparation of a two-mode squeezed vacuum state across two remote cavities. The joint cavity state, initially prepared in two-mode vacuum, evolves under the ME in Eq. (10) with jump operators given by Eq. (18). The associated squeezing parameter $r$ is related to the bath qubit coefficients; see Eq. (21). For the plotted values of $r$, the ground-state coefficient magnitudes are $|b_{\mathrm{g}\mathrm{g}}|\approx \{0.908,0.796,0.720,0.709,0.707\}$ and the squeezing angle is set to $\vartheta =0$. Smaller $|b_{\mathrm{g}\mathrm{g}}|$ corresponds to larger squeezing and infinite squeezing to $|b_{\mathrm{g}\mathrm{g}}|=1/\sqrt{2}$. (a) Fidelity, Eq. (B13), of the state with the two-mode squeezed vacuum state Eq. (24) for various values of $r$. (b) Time at which the fidelity $\mathcal{F}$ surpasses 0.98 as a function of the squeezing parameter $r$.

Figure 3

Entanglement and purity of the atomic state when the two-qubit bath is prepared in the $|{\mathrm{\Phi }}_E^{+}\rangle$ Bell state. The initial two-atom state ${\widehat{\rho }}_0\left(\theta \right)$ is parametrized by $\theta$, Eq. (32). We also include plots for nonmaximally entangled atomic states, $|{\mathrm{\Phi }}^{\pm }\left(\epsilon \right)\rangle$, that deviate from Bell states by $\epsilon =0.001$, Eq. (37). (a) Entanglement as quantified by the logarithmic negativity, Eq. (30), for which $LN\left(\widehat{\rho }\right)\approx 1$. (b) State purity, $\mathrm{Tr}\left[{\widehat{\rho }}^2\right]$. Note that, for $\theta <{\theta }_c=\pi /4$, the gray curve and dashed blue curve exactly coincide. That is, initial entanglement is preserved despite the fact that the state becomes mixed. Beyond ${\theta }_c$ the steady state is no longer entangled and the minimum purity, 0.25, occurs at $\theta =\pi /3$.

Figure 4

Steady-state entanglement of two remote atoms interacting with a two-qubit bath, quantified by the logarithmic negativity $LN\left({\widehat{\rho }}_{\mathrm{ss}}\right)$. The qubit bath is parametrized by Eq. (8a) with $b_{\mathrm{e}\mathrm{e}}$ and $b_{\mathrm{g}\mathrm{g}}$ taken to be real. At the exceptional points, where the bath is prepared in a Bell state, $b_{\mathrm{e}\mathrm{e}}=\pm 1/\sqrt{2}$ indicated by red and green dots, the atomic steady state is not unique. In this case, the steady-state entanglement is a function of the initial atomic state (see Fig. 3).

Figure 5

Color-coded representation of the state matrix for three- and two-qubit environments. Matrix elements in the local energy basis contribute to different terms in the multiqubit ME, Eqs. (39) and (40). Matrix elements labeled in gray () are zero because of the form of the environment's state, Eq. (8). Those labeled in dark blue () contribute to local dissipators, $\mathcal{D}\left[\widehat{o}\right]\widehat{\rho }$, and those labeled in magenta () contribute to the two-body superoperators, $\mathcal{S}[{\widehat{o}}_{\ell },{\widehat{o}}_m]\widehat{\rho }$. Matrix elements labeled in black () do not contribute to the ME dynamics. Note the similarities to Fig. 3 in Ref. [29].