Abstract
The nonlinear generalization of the von Neumann equation preserving convexity of the state space is studied in the nontrivial case of the qutrit. This equation can be cast into the integrable classical Riccati system of nonlinear ordinary differential equations. The solutions of such system are investigated in both the linear case corresponding to the standard von Neumann equation and the nonlinear one referring to the generalization of this equation. The analyzed dynamics of the qutrit is rich and includes quasiperiodic motion, multiple equilibria and limit cycles.
Highlights
In a recent paper [1], the evolution of the density matrix was studied of the form ρ (t ) =et(G−i H )ρ0et(G+i H ) Tr[et(G−i H)ρ0et(G+i H )] (1.1)where ρ0 is the N × N matrix and H and G are Hermitian
We study the nonlinear generalization of the von Neumann equation
An advantage of such nonlinear generalization in comparison with alternative approaches is that it preserves the convexity of the space of quantum states
Summary
In a recent paper [1], the evolution of the density matrix was studied of the form ρ (t ). The local form of (1.1) is the nonlinear generalization of the von Neumann equation given by ρ(t) = −i[H , ρ(t)] + {G − Tr[ρ(t)G], ρ(t)}, ρ(0) = ρ0,. The quantum dynamics described by the nonlinear equation (1.2) was illustrated in [1] by the example of the qubit. An interesting property of the nonlinear dynamics of the system (1.12) was found—the global asymptotic stability of stationary (equilibrium) solutions corresponding to evolution of the qubit from mixed states to pure ones. The vectors a and b from (1.10) are identified with the external magnetic and electric field, respectively Another interesting physical example is the quantum dynamics with the su(1, 1) Hamiltonian given by (1.10) with time-dependent a and b introduced in Ref. All the necessary identities corresponding to the su(3) algebra are collected in “Appendix”
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