Decoherence and Thermalization of a Pure Quantum State in Quantum Field Theory

We study the real-time evolution of a self-interacting $O(N)$ scalar field initially prepared in a pure, coherent quantum state. We present a complete solution of the nonequilibrium quantum dynamics from a $1/N$ expansion of the two-particle-irreducible effective action at next-to-leading order, which includes scattering and memory effects. We demonstrate that, restricting one’s attention (or ability to measure) to a subset of the infinite hierarchy of correlation functions, one observes an effective loss of purity or coherence and, on longer time scales, thermalization. We point out that the physics of decoherence is well described by classical statistical field theory.

Figure 1

The dots indicate that each diagram of the series is obtained from the previous one by adding another bubble.

Figure 2

The function $\ln [1+1/n_p(t)]$ as a function of ${ϵ}_p(t)$ for times $M_{0}t=2.5\times 2^n$ for $n=0,\dots ,14$. A straight line at late times corresponds to a Bose-Einstein distribution. The inset shows the rapid decrease of $\mathrm{tr}[D_p^2(t)]$ as a function of time for modes (from bottom to top) $p/a=1,5,10,15,20,24$, where $a/M_0=0.15$. Higher momentum modes, $p/a=25,27,30$, are also shown for comparison.

Figure 3

Exponential decay of the coherence parameter ${\gamma }_p(t)$ as a function of rescaled time $\tau =({\lambda }_{\mathrm{eff}}/6{)}^2(M_{0}t)$ (see text), for the mode $p/M_0=0.15$, for four couples of parameters (${\gamma }_0,\lambda$): (0.99, 3.05) (solid line) and (0.95, 6.62) (long-dashed line), for which ${\lambda }_{\mathrm{eff}}=21.7$; (0.95, 2.2) (dashed line), for which ${\lambda }_{\mathrm{eff}}=7.215$; and (0.95, 0.915) (dotted line), for which ${\lambda }_{\mathrm{eff}}=3.0$. The inset shows the corresponding decay rates, obtained from exponential fits, for modes $p\le p_c$. The agreement of the first two runs illustrates the ${\lambda }_{\mathrm{eff}}$ scaling of early-time decoherence. The overall agreement shows the approximate ${\lambda }_{\mathrm{eff}}^2$ dependence of the decoherence rate.