Entangled-state cycles from conditional quantum evolution

A system of cascaded qubits interacting via the one-way exchange of photons is studied. While for general operating conditions the system evolves to a superposition of Bell states (a dark state) in the long-time limit, under a particular resonance condition no steady state is reached within a finite time. We analyze the conditional quantum evolution (quantum trajectories) to characterize the asymptotic behavior under this resonance condition. A distinct bimodality is observed: for perfect qubit coupling, the system either evolves to a maximally entangled Bell state without emitting photons (the dark state) or executes a sustained entangled-state cycle—random switching between a pair of Bell states while emitting a continuous photon stream; for imperfect coupling, two entangled-state cycles coexist, between which a random selection is made from one quantum trajectory to another.

Figure 1

(a) A pair of cascaded cavities, cavity 1 and cavity 2, each contain a single trapped atom; a unidirectional coupling between the cavities is realized by Faraday isolators $F$. (b) The atomic excitation scheme couples two stable ground states, $\mid 0⟩$ and $\mid 1⟩$, to three excited states, $\mid r⟩$, $\mid s⟩$, and $\mid t⟩$.

Figure 2

The relaxation time $\tau =\mid \mathrm{Re}\left({\lambda }_2\right){\mid }^{-1}$ plotted as a function of $\mid {\beta }_s∕{\beta }_r\mid$. Note the singularity at resonance, $\mid {\beta }_s∕{\beta }_r\mid =1$.

Figure 3

Conceptual photon detectors, detectors 1 and 2, used for unraveling the master equation. The Faraday isolators are omitted for clarity.

Figure 4

Evolution of the ensemble average (i) compared with that of a single quantum trajectory (ii), for $ϵ=1$. The off-resonance case $({\beta }_s\ne {\beta }_r)$ is shown to the left and compared with the resonant case $({\beta }_s={\beta }_r)$ to the right. Time is measured in units of $({\beta }_r∕\sqrt{\kappa }{)}^{-1}$.

Figure 5

(Color online) Examples of phase space trajectories for $r=0.5$ with initial state $\mid {\varphi }^{+}\left(0\right)⟩=\mid 00⟩$ and $ϵ=1$. See text for description.

Figure 6

(Color online) Examples of phase space trajectories for $r=0.5$ with initial state $\mid {\varphi }^{-}\left(0\right)⟩=\mid 10⟩$ and $ϵ=1$. See text for description.

Figure 7

(Color online) Examples of phase space trajectories for $r=1$ with initial state $\mid {\varphi }^{+}\left(0\right)⟩=\mid 00⟩$ and $ϵ=1$. See text for description.

Figure 8

(Color online) Examples of phase space trajectories for $r=1$ with initial state $\mid {\varphi }^{-}\left(0\right)⟩=\mid 10⟩$ and $ϵ=1$. See text for description.

Figure 9

(Color online) (a) Example of a phase space trajectory for $r=1$ with initial state $\mid {\varphi }^{+}\left(0\right)⟩=\mid 00⟩$ and $ϵ=0.5$; the system eventually settles into the $\mid {\Phi }^{-}⟩\leftrightarrow \mid {\Psi }^{-}⟩$ entangled-state cycle. (b) and (c) Photon counts for detectors 1 and 2, respectively.

Figure 10

(Color online) (a) Example of a phase space trajectory for $r=1$ with initial state $\mid {\varphi }^{+}\left(0\right)⟩=\mid 00⟩$ and $ϵ=0.5$; the system eventually settles into the $\mid {\Phi }^{+}⟩\leftrightarrow \mid {\Psi }^{+}⟩$ entangled-state cycle. (b) and (c) Photon counts for detectors 1 and 2, respectively.