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Register a new userQuantum Mechanical Out-Of-Time-Ordered-Correlators for the Anharmonic (Quartic) Oscillator
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Authors
Paul Romatschke
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Out-of-time-ordered correlators (OTOCs) have been suggested as a means to study quantum chaotic behavior in various systems. In this work, I calculate OTOCs for the quantum mechanical anharmonic oscillator with quartic potential, which is classically integrable and has a Poisson-like energy-level distribution. For low temperature, OTOCs are periodic in time, similar to results for the harmonic oscillator and the particle in a box. For high temperature, OTOCs exhibit a rapid (but power-like) rise at early times, followed by saturation consistent with $2langle x^2rangle_T langle p^2rangle_T$ at late times. At high temperature, the spectral form factor decreases at early times, bounces back and then reaches a plateau with strong fluctuations.
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We study the dynamics of a quantum Brownian particle weakly coupled to a thermal bath. Working in the Schwinger-Keldysh formalism, we develop an effective action of the particle up to quartic terms. We demonstrate that this quartic effective theory is dual to a stochastic dynamics governed by a non-linear Langevin equation. The Schwinger-Keldysh effective theory, or the equivalent non-linear Langevin dynamics, is insufficient to determine the out of time order correlators (OTOCs) of the particle. To overcome this limitation, we construct an extended effective action in a generalised Schwinger-Keldysh framework. We determine the additional quartic couplings in this OTO effective action and show their dependence on the baths 4-point OTOCs. We analyse the constraints imposed on the OTO effective theory by microscopic reversibility and thermality of the bath. We show that these constraints lead to a generalised fluctuation-dissipation relation between the non-Gaussianity in the distribution of the thermal noise experienced by the particle and the thermal jitter in its damping coefficient. The quartic effective theory developed in this work provides extension of several results previously obtained for the cubic OTO dynamics of a Brownian particle.
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 systems 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.
Out-of-time-ordered correlators (OTOC) have been proposed to characterize quantum chaos in generic systems. However, they can also show interesting behavior in integrable models, resembling the OTOC in chaotic systems in some aspects. Here we study the OTOC for different operators in the exactly-solvable one-dimensional quantum Ising spin chain. The OTOC for spin operators that are local in terms of the Jordan-Wigner fermions has a shell-like structure: after the wavefront passes, the OTOC approaches its original value in the long-time limit, showing no signature of scrambling; the approach is described by a $t^{-1}$ power law at long time $t$. On the other hand, the OTOC for spin operators that are nonlocal in the Jordan-Wigner fermions has a ball-like structure, with its value reaching zero in the long-time limit, looking like a signature of scrambling; the approach to zero, however, is described by a slow power law $t^{-1/4}$ for the Ising model at the critical coupling. These long-time power-law behaviors in the lattice model are not captured by conformal field theory calculations. The mixed OTOC with both local and nonlocal operators in the Jordan-Wigner fermions also has a ball-like structure, but the limiting values and the decay behavior appear to be nonuniversal. In all cases, we are not able to define a parametrically large window around the wavefront to extract the Lyapunov exponent.
We consider a quantum Brownian particle interacting with two harmonic baths, which is then perturbed by a cubic coupling linking the particle and the baths. This cubic coupling induces non-linear dissipation and noise terms in the influence functional/master equation of the particle. Its effect on the Out-of-Time-Ordered Correlators (OTOCs) of the particle cannot be captured by the conventional Feynman-Vernon formalism.We derive the generalised influence functional which correctly encodes the physics of OTO fluctuations, response, dissipation and decoherence. We examine an example where Markovian approximation is valid for the OTO dynamics. If the original cubic coupling has a definite time-reversal parity, the leading order OTO influence functional is completely determined by the couplings in the usual master equation via OTO generalisation of Onsager-Casimir relations. New OTO fluctuation-dissipation relations connect the non-Gaussianity of the thermal noise to the thermal jitter in the damping constant of the Brownian particle.
The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time.
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