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arXiv:0910.2570 · Physical Review Letters

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

A. Giraud, Julien Serreau

Published Jun 9, 2010• 58 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.104.230405

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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.
We consider the time evolution of nonequilibrium quantum scalar fields in the O(N) model, using the next-to-leading order 1/N expansion of the two-particle irreducible effective action. A comparison with exact numerical simulations in 1+1 dimensions in the classical limit shows that the 1/N expansion gives quantitatively precise results already for moderate values of N. For sufficiently high initial occupation numbers the time evolution of quantum fields is shown to be accurately described by classical physics. Eventually the correspondence breaks down due to the difference between classical and quantum thermal equilibrium.
Nonperturbative dynamics of quantum fields out of equilibrium is often described by the time evolution of a hierarchy of correlation functions, using approximation methods such as Hartree, large Nf,{N}_{f},Nf​, and nPIn\mathrm{PI}nPI-effective action techniques. These truncation schemes can be implemented equally well in a classical statistical system, where results can be tested by comparison with the complete nonlinear evolution obtained by numerical methods. For a (1+1)(1+1)(1+1)-dimensional scalar field we find that the early-time behavior is reproduced qualitatively by the Hartree dynamics. The inclusion of direct scattering improves this to the quantitative level. We show that the emergence of nonthermal temperature profiles at intermediate times can be understood in terms of the fixed points of the evolution equations in the Hartree approximation. The form of the profile depends explicitly on the initial ensemble. While the truncated evolution equations do not seem to be able to get away from the fixed point, the full nonlinear evolution shows thermalization with a (surprisingly) slow relaxation.
We consider the nonequilibrium evolution of an O(N)-symmetric scalar quantum field theory using a systematic two-particle irreducible 1/N-expansion to next-to-leading order, which includes scattering and memory effects. The corresponding ``full Kadanoff-Baym equations'' are solved numerically without further approximations. This allows one to obtain a controlled nonperturbative description of far-from-equilibrium dynamics and the late-time approach to quantum thermal equilibrium. Employing in addition a first-order gradient expansion for the Wigner transformed correlators we derive kinetic equations. In contrast to standard descriptions based on loop expansions, our equations remain valid for nonperturbatively large fluctuations. As an application, we discuss the fluctuation dominated regime following parametric resonance in quantum field theory.
I review the use of the 2PI effective action in nonequilibrium quantum field theory. The approach enables one to find approximation schemes which circumvent long-standing problems of non-thermal or secular (unbounded) late-time evolutions encountered in standard loop or 1/N expansions of the 1PI effective action. It is shown that late-time thermalization can be described from a numerical solution of the three-loop 2PI effective action for a scalar ϕ4ϕ^4ϕ4--theory in 1+1 dimensions (with Jurgen Cox, hep-ph/0006160). Quantitative results far from equilibrium beyond the weak coupling expansion can be obtained from the 1/N expansion of the 2PI effective action at next-to-leading order (NLO), calculated for a scalar O(N) symmetric quantum field theory (hep-ph/0105311). Extending recent calculations in classical field theory by Aarts et al. (hep-ph/0007357) and by Blagoev et al. (hep-ph/0106195) to N>1N>1N>1 we show that the NLO approximation converges to exact (MC) results already for moderate values of NNN (with Gert Aarts, hep-ph/0107129). I comment on characteristic time scales in scalar quantum field theory and the applicability of classical field theory for sufficiently high initial occupation numbers.
We derive the nonequilibrium real-time evolution of an O(N)O(N)O(N)-invariant scalar quantum field theory in the presence of a nonvanishing expectation value of the quantum field. Using a systematic 1/N1/N1/N expansion of the 2PI effective action to next-to-leading order, we obtain nonperturbative evolution equations which include scattering and memory effects. The equivalence of the direct method, which requires the resummation of an infinite number of skeleton diagrams, with the auxiliary-field formalism, which involves only one diagram at next-to-leading order, is shown.
The 1/N1/N1/N expansion of the two-particle irreducible effective action offers a powerful approach to study quantum field dynamics far from equilibrium. We investigate the effective convergence of the 1/N1/N1/N expansion in the O(N)O(N)O(N) model by comparing results obtained numerically in 1+11+11+1 dimensions at leading, next-to-leading and next-to-next-to-leading order in 1/N1/N1/N as well as in the weak coupling limit. A comparison in classical statistical field theory, where exact numerical results are available, is made as well. We focus on early-time dynamics and quasiparticle properties far from equilibrium and observe rapid effective convergence already for moderate values of 1/N1/N1/N or the coupling.