Gibbs entropy from entanglement in electric quenches

In quantum electrodynamics with charged fermions, a background electric field is the source of the chiral anomaly which creates a chirally imbalanced state of fermions. This chiral state is realized through the production of entangled pairs of right-moving fermions and left-moving antifermions (or vice versa, depending on the orientation of the electric field). Here we show that the statistical Gibbs entropy associated with these pairs is equal to the entropy of entanglement between the right-moving particles and left-moving antiparticles. We then derive an asymptotic expansion for the entanglement entropy in terms of the cumulants of the multiplicity distribution of produced particles and explain how to re-sum this asymptotic expansion. Finally, we study the time dependence of the entanglement entropy in a specific time-dependent pulsed background electric field, the so-called “Sauter pulse”, and illustrate how our resummation method works in this specific case. We also find that short pulses (such as the ones created by high energy collisions) result in an approximately thermal distribution for the produced particles.

Figure 1

Entanglement entropy as a function of time, shown for different values of the dimensionless ratio $\gamma =\frac{\mathcal{E}\tau }{m}$, where $\mathcal{E}$ is the electric field strength and $\tau$ is the duration of the pulse. We observe that as the pulse is getting shorter, the entropy possesses less distinct features and the equilibration becomes more efficient, see text for discussion.

Figure 2

The momentum spectrum of the produced particles. On the lhs, we show the asymptotic spectrum in the regime of small $\gamma$ and small $m\tau$, while on the rhs we keep $m\tau$ finite. For illustration purposes, in the latter case, we also show the approximated spectrum without the polynomial “gray factor”.

Figure 3

Entanglement entropy for particles produced by Sauter pulse and the thermodynamical entropy. Both quantities are normalized by the number of particles: for the case of the Sauter pulse, these are the produced particles, and for the thermodynamical entropy the number of particles is computed from the Boltzmann distribution at temperature $T=\frac{1}{2\pi \tau }$. We fix $m=1$ and $\mathcal{E}=1$ and plot as a function of $\gamma =m\tau /4$.

Figure 4

Entanglement entropy as a function of time obtained from the moments of the distribution of created particles. On the l.h.s., we show the entropy obtained from the asymptotic expansion (63) truncated after $L$ terms. On the rhs, we show the results obtained with the resummed expression (89).