Degradation of electron-hole entanglement by spin-orbit coupling

Electron-hole pairs produced by tunneling in a degenerate electron gas lose their spin entanglement by spin-orbit coupling, which transforms the fully entangled Bell state into a partially entangled mixed density matrix of the electron and hole spins. We calculate the dependence of the entanglement (quantified by the concurrence) on the spin-orbit coupling time tau_so and on the diffusion time (or dwell time) tau_dwell of electron and hole in the conductors (with conductances >> e^2/h) at the two sides of the tunnel barrier (with conductance << e^2/h). The entanglement disappears when the ratio tau_dwell/tau_so exceeds a critical value of order unity. The results depend on the type of conductor (disordered wire or chaotic quantum dot), but they are independent of other microscopic parameters (number of channels, level spacing). Our analytical treatment relies on an "isotropy approximation" (no preferential basis in spin space), which allows us to express the concurrence entirely in terms of spin correlators. We test this approximation for the case of chaotic dynamics with a computer simulation (using the spin kicked rotator) and find good agreement.