We work with reference to a well-known semiclassical model, in which quantum degrees of freedom interact with classical ones. We show that, in the classical limit, it is possible to represent classical results (e.g., classical chaos) by means a pure-state density matrix.
In this paper, a reference to the semiclassical model, in which quantum degrees of freedom interact with classical ones, is considered. The classical limit of a maximum-entropy density matrix that describes the temporal evolution of such a system is analyzed. Here, it is analytically shown that, in the classical limit, it is possible to reproduce classical results. An example is classical chaos. This is done by means a pure-state density matrix, a rather unexpected result. It is shown that this is possible only if the quantum part of the system is in a special class of states.
We examine the classical limit of a fairly general nonlinear semiclassical hybrid system within a MaxEnt framework. The consistency of the hybrid dynamics requires algebraic constraints on quantum operators and smoothness conditions for the classical variables. Analytically, we demonstrate that the classical limit is characterized by a pure density matrix representing a single state, which reproduces the dynamics of its classical analogue. To illustrate the methodology, we revisit and synthesize two previously studied examples.
Understanding the emergence of chaos in many-body quantum systems away from semi-classical limits, particularly in spatially local interacting spin Hamiltonians, has been a long-standing problem. In these intrinsically quantum regimes, quantum chaos has been primarily understood through the correspondence between the eigensystem statistics of midspectrum eigenstates and the universal statistics described by random matrix theory (RMT). However, this correspondence no longer holds for finite-temperature eigenstates. Here we show that the statistical properties of finite-temperature eigenstates of quantum chaotic Hamiltonians can be accurately described by pure random states constrained by a local charge, with the average charge density of the constrained random state ensemble playing the same role as the average energy density of the eigenstates. By properly normalizing the energy density using a single Hamiltonian-dependent parameter that quantifies the typical energy per degree of freedom, we find excellent agreement between the entanglement entropy statistics of eigenstates and that of constrained random states. Interestingly, in small pockets of Hamiltonian parameter phase space which we previously identified as `maximally chaotic' [PRX 14, 031014 (2024)], we find excellent agreement not only at the level of the first moment, including O(1) corrections, but also at the level of statistical fluctuations. These results show that notions of maximal chaos -- in terms of how much randomness eigenstates contain -- can still be defined at finite temperature in physical Hamiltonian models away from semi-classical and large-N limits.
This paper clarifies the limits of representing spontaneous emission with a classical electromagnetic field sourced only by an atom’s mean dipole. Extending a pure-state hydrogen result to an arbitrary two-level density matrix, it separates total radiated power from the portion attributable to optical coherence. Under the electric-dipole, rotating-wave, Markov, and free-space approximations, positivity of the density matrix places a strict upper bound on the coherent fraction of the emitted intensity. The remainder is necessarily associated with quantum dipole fluctuations rather than the classical mean field. A hydrogen benchmark and ensemble structure-factor analysis illustrate how excited-state population, coherence, and radiation can be measured and tested independently. The result provides a diagnostic boundary for semiclassical Maxwell–Scretching Chain (MSC) and Scretching Quantum Chain (SQC) source models while remaining consistent with established spontaneous-emission theory rather than proposing a new emission law.