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Symplectic evolution of Wigner functions in Markovian open systems
O Brodier1, A M Ozorio de Almeida
1Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil. brodier@cbpf.br
Summary
The Wigner function evolves classically with quadratic Hamiltonians. Environmental interactions cause broadening, making the Wigner function positive at a set time, independent of the initial state.
Area of Science:
- Quantum mechanics
- Statistical mechanics
- Quantum optics
Background:
- The Wigner function describes quantum states in phase space.
- Quadratic Hamiltonians lead to classical Wigner function evolution.
- Environmental interactions introduce dissipation and decoherence.
Purpose of the Study:
- Analyze Wigner function evolution under environmental interactions.
- Investigate the conditions for Wigner function positivity.
- Derive formulas for linear entropy and its long-time behavior.
Main Methods:
- Analysis of Wigner function dynamics with Lindblad operators.
- Case studies for elliptic, hyperbolic, and parabolic Hamiltonians.
- Derivation of exact formulas for linear entropy.
Main Results:
- Wigner function evolution is a convolution of classical propagation and a broadening Gaussian.
- Wigner function becomes positive at a time independent of the initial state.
- Derived exact and asymptotic formulas for linear entropy growth.
Conclusions:
- Environmental interactions drive Wigner functions towards positivity.
- Classical dynamics and dissipation influence the positivity threshold.
- Discussed the possibility of recovering initial quantum states.