Open quantum systems

Wikipedia: An open quantum system is a quantum system which is found to be in interaction with an external quantum system, the environment. The open quantum system can be viewed as a distinguished part of a larger closed quantum system, the other part being the environment. INTERDISCIPLINARY Small quantum systems, whose properties are profoundly affected by environment, i.e., continuum of scattering and decay channels, are intensely studied in various fields of physics: nuclear physics, atomic and molecular physics, nanoscience, quantum optics, etc. http://www.phy.ornl.gov/theory/MBOQS/Manifesto_09.html OQS environment Example: EM radiation

Quasistationary States (alpha decay) For the description of a decay, we demand that far from the force center there be only the outgoing wave. The macroscopic equation of decay is N is a number of radioactive nuclei, i.e., number of particles inside of sphere r=R:

relation between decay width and decay probability We should thus seek a solution of the form J.J. Thompson, 1884 G. Gamow, 1928 relation between decay width and decay probability The time dependent equation can be reduced by the above substitution to the stationary equation The boundary condition …takes care of the discrete complex values of E

This relation has been checked in numerous precision experiments. In principle, resonances and decaying particles are different entities. Usually, resonance refers to the energy distribution of the outgoing particles in a scattering process, and it is characterized by its energy and width. A decaying state is described in a time dependent setting by its energy and lifetime. Both concepts are related by: This relation has been checked in numerous precision experiments. See more discussion in R. de la Madrid, Nucl. Phys. A812, 13 (2008) U. Volz et al., Phys. Rev. Lett 76, 2862(1996)

A comment on the time scale… Time Dependent Schrödinger Equation Can one calculate G with sufficient accuracy using TDSE? 238U: T1/2=1016 years 256Fm: T1/2=3 hours For very narrow resonances, explicit time propagation impossible!

Radial Schrödinger equation How do resonances appear? A square well example square-well potential spherical symmetry l=0 (s-wave) Radial Schrödinger equation Region I: Region II: Region III:

defines “virtual” levels in region I: In almost all cases |cIII| is much larger than |cI|. We are now interested in those situations where |cIII| is as small as possible. The condition defines “virtual” levels in region I: particle is well localized; very small penetrability through the barrier

When c+=0, the penetrability becomes proportional to This is the semi-classical WKB result:

Width of a narrow resonance open closed time-dependent Hamiltonian …expansion of  in the basis of Ho As initial conditions, let us assume that at t=0 the system is in the state 0 If the perturbation is weak, in the first order, we obtain:

Furthermore, if the time variation of V is slow compared with exp(iwkot), we may treat the matrix element of V as a constant. In this approximation: The probability for finding the system in state k at time t if it started from state 0 at time t=0 is: The total probability to decay to a group of states within some interval labeled by f equals:

Fermi’s golden rule The transition probability per unit time is Since the function sin(x)/x oscillates very quickly except for x~0, only small region around E0 can contribute to this integral. In this small energy region we may regard the matrix element and the state density to be constant. This finally gives: Fermi’s golden rule Although named after Fermi, most of the work leading to the Golden Rule was done by Dirac, who formulated an almost identical equation 1927. It is given its name because Fermi called it "Golden Rule No. 2." in 1950.

mean lifetime half-life transition probability Fermi’s golden rule normalized amplitude uncertainty principle

When can we talk about “existence” of an unbound nuclear system? A typical time associated with the s.p. nucleonic motion Compute the half-life of: 141Ho proton emitter (G=2･10-20 MeV) 3- state in 10Be at E=10.16 MeV (G=296 keV) First 2+ state in 6He at E=1.797MeV (G=113 keV) Hoyle state in 12C at E=7.654 MeV (G=8.5 eV) 8Be ground state (G=5.57 eV) Baryon N(1440)1/2+ (G=350 MeV) Discuss the result.

Rep. Prog. Phys. 67 (2004) 1187–1232

Prog. Part. Nucl. Phys. 59, 432 (2007)

11Li Ikeda Diagram Excitation energy Mass number (7.27) (14.44) (19.17) (28.48) (7.16) (11.89) (21.21) (14.05) Ikeda Diagram (4.73) Excitation energy (13.93) (9.32) 11Li Mass number http://www.nndc.bnl.gov/qcalc/ sequence of reaction channels 1

Beyond the Neutron Drip-Line http://www.tandfonline.com/doi/pdf/10.1080/10619127.2014.882735 http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.152501

Tetraneutron??? http://www.cnrs.fr/cw/en/pres/compress/noyau.htm http://www.gamefaqs.com/pc/944906-mass-effect-2/answers/157357-where-is-the-best-planet-to-find-element-zero-resources

Can Modern Nuclear Hamiltonians Tolerate a Bound Tetraneutron? http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.90.252501

Viewpoint: Can Four Neutrons Tango? BUT..... Kisamori et al. http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.052501 Viewpoint: Can Four Neutrons Tango? http://physics.aps.org/articles/v9/14

Baryon and meson resonances Lots of unbound states! N* D*