Dephasing representation: Employing the shadowing theorem to calculate quantum correlation functions

Abstract

Due to the Heisenberg uncertainty principle, various classical systems differing only on the scale smaller than Planck's cell correspond to the same quantum system. This fact is used to find a unique semiclassical representation without the Van Vleck determinant, applicable to a large class of correlation functions expressible as quantum fidelity. As in the Feynman path integral formulation of quantum mechanics, all contributing trajectories have the same amplitude: that is why it is denoted the ``dephasing representation.'' By relating the present approach to the problem of existence of true trajectories near numerically-computed chaotic trajectories, the approximation is made rigorous for any system in which the shadowing theorem holds. Numerical implementation only requires computing actions along the unperturbed trajectories and not finding the shadowing trajectories. While semiclassical linear-response theory was used before in quasi-integrable and chaotic systems, here its validity is justified in the most generic, mixed systems. Dephasing representation appears to be a rare practical method to calculate quantum correlation functions in nonuniversal regimes in many-dimensional systems where exact quantum calculations are impossible.

5 pages, 1 figure, to appear in Phys. Rev. E (R)

References in corpus (6)

• Decoherence and the Loschmidt echo
• A random matrix formulation of fidelity decay
• Semiclassical evaluation of quantum fidelity
• Anomalous power law of quantum reversibility for classically regular dynamics
• On the stability of classical chaotic motion under system's perturbations
• The survival probability and the local density of states for one-dimensional Hamiltonian systems

Cited by in corpus (40)

• Dynamics of Loschmidt echoes and fidelity decay
• Loschmidt Echo
• Decoherence, Entanglement and Irreversibility in Quantum Dynamical Systems with Few Degrees of Freedom
• Dephasing representation of quantum fidelity for general pure and mixed states
• Decoherence of two qubit systems: A random matrix description
• Relation of exact Gaussian basis methods to the dephasing representation: Theory and application to time-resolved electronic spectra
• On-the-fly ab initio semiclassical evaluation of third-order response functions for two-dimensional electronic spectroscopy
• Semiclassical propagation of Wigner functions
• Evaluation of the nondiabaticity of quantum molecular dynamics with the dephasing representation of quantum fidelity
• Uniform Semiclassical Approach to Fidelity Decay: from Weak to Strong Perturbation
• Measuring nonadiabaticity of molecular quantum dynamics with quantum fidelity and with its efficient semiclassical approximation
• Quantum to classical transition in the work distribution for chaotic systems
• Loschmidt Echo and the Local Density of States
• Loschmidt echo for local perturbations: non-monotonous cross-over from the Fermi-golden-rule to the escape-rate regime
• Displacement Echoes: Classical Decay and Quantum Freeze
• Beating the efficiency of both quantum and classical simulations with semiclassics
• An initial value representation for the Loschmidt echo
• Loschmidt echo in quantum maps: the elusive nature of the Lyapunov regime
• Stability of quantum motion in regular systems: a uniform semiclassical approach
• Universal Response of Quantum Systems with Chaotic Dynamics
• Evaluation of the importance of spin-orbit couplings in the nonadiabatic quantum dynamics with quantum fidelity and with its efficient "on-the-fly" ab initio semiclassical approximation
• Improving the accuracy and efficiency of time-resolved electronic spectra calculations: Cellular dephasing representation with a prefactor
• Efficient evaluation of accuracy of molecular quantum dynamics using dephasing representation
• Decay of the Loschmidt echo in a time-dependent environment
• Path integral approach to the quantum fidelity amplitude
• Semiclassical approach to the work distribution
• Semiclassical evolution of correlations between observables
• Loschmidt echoes in two-body random matrix ensembles
• Quantum dephasing and decay of classical correlation functions in chaotic systems
• Semiclassical approach to fidelity amplitude
• The quantum fidelity for the time-dependent singular quantum oscillator
• Non-monotonous short-time decay of the Loschmidt echo in quasi-one-dimensional systems
• Quantum non-Markovian behavior at the chaos border
• Stability of Fock states in a two-component Bose-Einstein condensate with a regular classical counterpart
• Lyapunov decay in quantum irreversibility
• Sensitivity of Quantum Motion to Perturbation in Triangle Map
• Loschmidt echo and scrambling of systematic errors in tomography -- a quantum signature of chaos
• Breaking the exponential wall in classical simulations of fidelity
• Perturbations and chaos in quantum maps
• Refined approach to cellularization: going from Heller's thawed Gaussian approximation to the Herman--Kluk propagator