Controlled quantum evolutions and transitions

We study the nonstationary solutions of Fokker-Planck equations associated to either stationary or non stationary quantum states. In particular, we discuss the stationary states of quantum systems with singular velocity fields. We introduce a technique that allows arbitrary evolutions ruled by these equations to account for controlled quantum transitions. As a first signficant application we present a detailed treatment of the transition probabilities and of the controlling time-dependent potentials associated to the transitions between the stationary, the coherent, and the squeezed states of the harmonic oscillator.

[1]  P. Blanchard,et al.  Mathematical and Physical Aspects of Stochastic Mechanics , 1987 .

[2]  Theory of controlled quantum dynamics , 1997, quant-ph/9702040.

[3]  V. Hutson.  Integral Equations , 1967, Nature.

[4]  Lagrangian variational principle in stochastic mechanics: Gauge structure and stability , 1989 .

[5]  R. Ho,et al.  A remark on the connection between stochastic mechanics and the heat equation , 1974 .

[6]  Edward Nelson.  Dynamical Theories of Brownian Motion , 1967 .

[7]  Quantum mechanical states as attractors for Nelson processes , 1995 .

[8]  Stochastic mechanics and quantum interference , 1989 .

[9]  Nicola Cufaro Petroni.  Asymptotic Behaviour of Densities for Nelson Processes , 1995 .

[10]  J. P. Vigier,et al.  Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations , 1954 .

[11]  J. Vigier,et al.  Single-particle trajectories and interferences in quantum mechanics , 1992 .

[12]  H. Risken.  Fokker-Planck Equation , 1984 .

[13]  N. Petroni.  Conditioning in quantum mechanics , 1991 .

[14]  D. Bohm.  A SUGGESTED INTERPRETATION OF THE QUANTUM THEORY IN TERMS OF "HIDDEN" VARIABLES. II , 1952 .

[15]  Stochastic variational approach to minimum uncertainty states , 1995, hep-th/9501034.

[16]  Edward Nelson.  Derivation of the Schrodinger equation from Newtonian mechanics , 1966 .

[17]  F. Guerra,et al.  Origin of the quantum observable operator algebra in the frame of stochastic mechanics , 1983 .

[18]  Francesco Guerra,et al.  Structural aspects of stochastic mechanics and stochastic field theory , 1981 .

[19]  C. Ross.  Found , 1869, The Dental register.

[20]  Holland,et al.  Realistic physical origin of the quantum observable operator algebra in the frame of the causal stochastic interpretation of quantum mechanics: The relativistic spin-zero case. , 1985, Physical review. D, Particles and fields.

[21]  Salvatore De Martino,et al.  Exact solutions of Fokker-Planck equations associated to quantum wave functions , 1998 .

[22]  Francesco Guerra,et al.  Quantization of Dynamical Systems and Stochastic Control Theory , 1983 .