Dynamics of entanglement via propagating microwave photons

We propose a simple circuit quantum electrodynamics (QED) experiment to test the generation of entanglement between two superconducting qubits. Instead of the usual cavity QED picture, we study qubits which are coupled to an open transmission line and get entangled by the exchange of propagating photons. We compute their dynamics using a full quantum field theory beyond the rotating-wave approximation and explore a variety of regimes which go from a weak coupling to the recently introduced ultrastrong-coupling regime. Due to the existence of single photons traveling along the line with finite speed, our theory shows a light cone dividing the space time in two different regions. In one region, entanglement may only arise due to correlated vacuum fluctuations while in the other, the contribution from exchanged photons shows up.

Figure 1

(Color online) (a) Qubits that interact via traveling photons with finite velocity $v$ can be spacelike (I, white) or timelike (II, shaded) separated, depending on the value of $\xi =vt/r$. While only in II, they are causally connected, entanglement may appear already in region I. (b) A possible implementation of these ideas consists of flux qubits ultrastrongly coupled to a common transmission line. (c) With a slight modification, the coupling of the qubits to the line can be dynamically tuned via fast magnetic fluxes, $\Phi$.

Figure 2

(Color online) (a) Concurrence vs dimensionless separation $\xi$ for $r=\pi v/4\Omega \sim \lambda /8$ and couplings $K=K_0$, $10K_0$, $100K_0$, and $1000K_0$ (bottom to top). (b) Zoom around $\xi =1$ for the strongest coupling $K=1000K_0$.

Figure 3

(Color online) Concurrence (dash) and probability of excitation of atom $B$ (solid) vs dimensionless time, $\Omega t$. Qubits are separated by $r=\lambda /12$ (circles) and $\lambda /8$ (crosses) and have a coupling strength $K=1000K_0$. Note that, following microcausality, the excitation probabilities do not depend on the separation $r$ outside the light cone.