The damping of the harmonic oscillator is studied in the framework of the Lindblad theory for open quantum systems. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger, Heisenberg and Weyl-WignerMoyal representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that not all of these equations are satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions and a comparative study is made for the Glauber P representation, the antinormal ordering Q representation and the Wigner W representation. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided. The solution of the master equation in the Weyl-Wigner-Moyal representation is of Gaussian type if the initial form of the Wigner function is taken to be a Gaussian corresponding (for example) to a coherent wavefunction. The damped harmonic oscillator is applied for the description of the charge equilibration mode observed in deep inelastic reactions. For a system consisting of two harmonic oscillators the time dependence of expectation values, Wigner function and Weyl operator are obtained and discussed. In addition models for the damping of the angular momentum are studied. Using this theory to the quantum tunneling through the nuclear barrier, besides Gamow's transitions with energy conservation, additional transitions with energy loss, are found. The tunneling spectrum is obtained as a function of the barrier characteristics. When this theory is used to the resonant atom-field interaction, new optical equations describing the coupling through the environment of the atomic observables are obtained. With these equations, some characteristics of the laser radiation absorption spectrum and optical bistability are described.

o r due to a characteristic of the mechanism independent of this s e t of values.After extrapolating ( d~/ d i 2 )~" at l a r g e r angles, a s indicated by the dashed line in Fig. 3, the integrated c r o s s section for the production of quasi-Kr i s about 500 mb. It is a large part of the reaction c r o s s section (calculated value 870 mb), and this s e e m s to confirm the idea that the quasifission reactions occur instead of the complete-fusion reactions. One should remember, however, that the method used for the angular distribution is based on a hypothesis which i s reasonable but not entirely proved for angles f a r away from 553.More work-with more intense beams if possible-has to be done in o r d e r to get more detailed information on the features of the quasifission reactions induced by K r ions. Nevertheless we hope the indications we have obtained will stimulate theoretical work which i s now underway for understanding the reaction mechanism. In particular, the angular distribution should help in deciding among the various pict u r e s for the potential energy between the two nuclei which can be considered in order to explain the quasifission reactions."We thank N. Rowley for discussions and for reading the manuscript.Note added.*e have recently made angulardistribution measurements a t large angles. The maximum a t 65" has been confirmed but the descent above 80" is not as steep as the dashed curve on Fig. 3 Nuclear Shock Waves in Heavy-Ion CollisionsWerner Scheid, Hans Müller, and Walter Greiner k s t i t u t für Theoretische Physik der Universität Frankfurt, Frankfurt arn Main, Germany (Received 19 November 1973) It is shown that nuclear matter is compressed during the encounter of heavy ions. If the relative velocity of the nuclei is larger than the velocity of first sound in nuclear matter (compression sound for isospin T =O), nuclear shock waves occur. They lead to densities which are 3-5 times higher than the nuclear equilibrium density P,,, depending on the energy of the nuclei. The implications of this phenomenon are discussed.The possibility of compression of nuclear matt e r in nucleus-nucleus collisions in one of the most interesting aspects of heavy-ion physics. It has been discussed e a r l i e r in connection with the sudden nucleus-nucleus potentialsl and their energy d e p e n d e n~e ,~ which s e e m s to confirm the experimentally deduced heavy-ion potentials of the Yale g r o~p .~ These e a r l i e r considerations a r e valid a s long a s the relative heavy-ion velocity U , does not exceed the velocity of f i r s t sound in nuclear matter C,; i.e., for V , < C,. Here the f i r s t sound is an isospin T = 0 compression wave while the second sound describes an isospin T = 1 wave where a proton-neutron separation travels in constant nuclear matter density p, = p, + P , .~ In o r d e r to study local compression effects for

A new procedure is developed for calculating the charge, mass, and kinetic energy distributions of quasifission products. The quasifission is treated within a transport model which describes a master equation for the evolution of the dinuclear system in charge and mass asymmetries and its decay along the internuclear distance. The calculated yields of quasifission products and their distributions in kinetic energy are in agreement with recent experimental data of hot fusion reactions leading to superheavy nuclei. The importance of shell and deformation effects in quasifission is noted. The preneutron and postneutron emissions as well as the fission of a heavy nucleus in the dinuclear system are considered.The DNS model ͓12-14͔ assumes that the compound nucleus is reached by a series of transfers of nucleons or PHYSICAL REVIEW C 68, 034601 ͑2003͒

With the quantum diffusion approach the unexpected behavior of fusion cross section, angular momentum, and astrophysical S-factor at sub-barrier energies has been revealed. Out of the region of short-range nuclear interaction and action of friction at turning point the decrease rate of the cross section under the barrier becomes smaller. The calculated results for the reactions with spherical nuclei are in a good agreement with the existing experimental data. PACS numbers: 25.70.Ji, 24.10.Eq, 03.65.-w The capture cross section is a sum of partial capture cross sections σ c (E c.m. ) = J σ c (E c.m. , J) = πλ 2 J

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