Abstract
We present a method to probe the Out-of-Time-Order Correlators (OTOCs) of a general system by coupling it to a harmonic oscillator probe. When the system’s degrees of freedom are traced out, the OTOCs imprint themselves on the generalized influence functional of the oscillator. This generalized influence functional leads to a local effective action for the probe whose couplings encode OTOCs of the system. We study the structural features of this effective action and the constraints on the couplings from microscopic unitarity. We comment on how the OTOCs of the system appear in the OTOCs of the probe.
Highlights
Such Lyapunov exponents for thermal states in quantum systems have been shown to be constrained by an upper bound [2].1 for large N gauge theories which have holographic duals, the Lyapunov exponent saturates this bound [1, 2]
We present a method to probe the Out-of-Time-Order Correlators (OTOCs) of a general system by coupling it to a harmonic oscillator probe
This generalized influence functional leads to a local effective action for the probe whose couplings encode OTOCs of the system
Summary
Suppose we are interested in the OTOCs of this operator. These OTOCs can be extracted from the OTOCs of a probe coupled to the system. We will assume that the system and the probe are initially unentangled and the interaction between them is switched on at a time t0. The density matrix of the system and the probe at time t0 is given by ρ(t0) = ρS(t0) ⊗ ρprobe(t0) ,. Where ρS(t0) and ρprobe(t0) are the initial density matrices of the sytem and the probe respectively. In the corresponding path integrals we will integrate out the system’s degrees of freedom to obtain a generalized influence phase for the probe which would allow computation of its OTOCs
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