Spontaneous generation of phononic entanglement in quantum dark-soliton qubits

We show that entanglement between two solitary qubits in quasi-one-dimensional Bose-Einstein condensates can be spontaneously generated due to quantum fluctuations. Recently we have shown that dark solitons are an appealing platform for qubits thanks to their appreciable long lifetime. We investigate the spontaneous generation of entanglement between dark-soliton qubits in the dissipative process of spontaneous emission. By driving the qubits with the help of oscillating magnetic field gradients, we observe the formation of long distance steady-state concurrence. Our results suggest that dark-soliton qubits are good candidates for quantum information protocols based purely on matter-wave phononics.

Figure 1

(a) Schematic representation of two dark-soliton qubits placed at distance $d$ in a cigar shaped quasi-one-dimensional BEC, surrounded by a dilute gas of impurities. (b) Collective states of two dark-soliton qubits. Due to the coherent coupling, the two intermediate states $|s\rangle$ and $|a\rangle$ are maximally entangled. (c) Qubit amplitudes in the ground $\left(|{\phi }_0{\left(x\right)|}^2\right)$ and excited $\left(|{\phi }_1{\left(x\right)|}^2\right)$ states.

Figure 2

Collective damping $\mathrm{\Gamma }$ and qubit-qubit interaction parameter $\eta$ (inset) as a function of the soliton separation $d$. We have chosen $\nu =0.75$, for which dark-soliton qubits are well defined.

Figure 3

Time evolution of the concurrence $C\left(t\right)$ in the absence of driving. (a) $C\left(t\right)$ for the superposition of maximally entangled Dicke states. $d\simeq \xi$ (dashed curve) and $d\simeq 5\xi /2$ (solid curve). (b) The population of symmetric state $\left|s\right.〉$ (dashed curve), antisymmetric state $\left|a\right.〉$ (dotted-dashed curve), and time evolution of concurrence $C\left(t\right)$ (solid curve) at distance $d\simeq 5\xi /2$.

Figure 4

Time evolution of the concurrence $C\left(t\right)$ for symmetric pumping $\left({\mathrm{\Omega }}_1={\mathrm{\Omega }}_2\right)$ at the distance $d=5\xi /2$. We have chosen $\mathrm{\Omega }=0.25\gamma$ (dashed curve) and $\mathrm{\Omega }=0.35\gamma$ (solid curve) for illustration.

Figure 5

(a) The steady-state concurrence $C\left(\infty \right)$ as a function of distance $d$ between DS qubits, with $\mathrm{\Omega }=0.25\gamma$ (dashed curve) and $\mathrm{\Omega }=0.35\gamma$ (solid curve). (b) The variation of $C\left(\infty \right)$ with the Rabi frequency $\mathrm{\Omega }$.

Figure 6

Qubits in a possible experimental situation: Numerical profiles of the dark soliton (black lines) and the impurity eigenstates (blue lines). From left to right, we depict the ground $\left[{\phi }_0\left(x\right)\right]$ and the excited $\left[{\phi }_1\left(x\right)\right]$ states, respectively, of a fermionic ${}^{134}\mathrm{Cs}$ impurity trapped in a ${}^{85}\mathrm{Rb}$ BEC dark soliton. The solid lines are the numerical solutions, while the dashed lines are the analytical expression described in the text. We have used the following parameters: $m_{\phi }=1.56m_{\psi },\chi =0.88g$ (corresponding to $\nu \simeq 0.75$, as considered in the paper). We fix the number of depleted condensate atoms by the dark soliton to be $n_0\xi \simeq 50$, although our numerical simulations (not shown) indicate that the solutions are not very sensitive to its variation.

Figure 7

On-site $\left(i=j\right)$ Interband $g_{01,k}=g_{10,k}^{*}\equiv g_k$ (solid line) and intraband $g_{00,k}$ and $g_{11,k}$ (dashed and dot-dashed lines, respectively) coupling amplitudes. Near resonance $\left(k\sim 0.9{\xi }^{-1}\right)$, the interband terms clearly dominate over the intraband transitions, allowing us to neglect the latter within the rotating wave approximation.

Figure 8

A box potential of size $L=100\mu \mathrm{m}$ containing 24 solitons. Noticeable displacement is only found for the outer pair of solitons, while the inner 20 solitons stay almost during the lapsed simulation time, $\tau =100$ ms, larger than the concurrence build-up time of $\sim 80$ ms described in the paper.