Spin Coherent State in Real Parameterization SU(4)

Spin Coherent State in Real Parameterization SU(4)Coherent domains in SU(4) of spin systems and calculate the cull form for qudit with spin 3/2 particle in SU(4) in quantum mechanicsYadollah Farahmand*, ZABIALAH HEIDARNEZHAD**, Fatemeh Heidarnezhad***,Fatemeh Heydari*** and Kh . Kh Muminov*Abstract In this paper, we develop the formulation of the spin coherent state in real argumentization SU(4).we obtain Berry phase from Schrdinger equation. For vector states, basic kets atomic number 18 coherent states in real parameterization. Wecalculate Berry phase for qudit with spin S=3/2 in SU(3) separate and Berry phase.Key words quantum mechanics, Schrdinger equation ,coherent state ,SU(n)group , Quadrupole moment , Berry phase.IntroductionIn 1984 Berry published a paper 1 which has until now deeply influenced the physical community. In mechanics (including classical mechanics as well as quantum mechanics), theGeometric phase, or the Pancharatnam-Berry phase (named after S. Pancharat nam and Sir Michael Berry), also known as the Pancharatnam phase or, more commonly, Berry phase2, Therein he meets cyclic evolutions of systems under special conditions, namely adiabatic ones. He finds that an additional phase factor occursin contrast to the well-known dynamical phase factor. is a phase acquired over the course of a cycle, when the system is subjected to cyclic adiabatic processes, resulting from the geometrical properties of the parameter space of the Hamiltonian. Apart from quantum mechanics, it arises in a variety of former(a) wave systems, such as classical optics 3.As a rule of thumb, it occurs when ever there are at least two parameters affecting a wave, in the vicinity of some sort of singularity or some sort of hole in the topology. In nonrelativistic quantum mechanics, the state of a system is described by the vector of the Hilbert space (the wave function) H which depends on time and some set of other variable stars depending on the considered problem. The evolution of a quantum system in time t is described by the Schrodinger equationWe consider a quantum system described by a Hamiltonian H that depends ona multidimensional real parameter R which parameterizes the environment of the system. The time evolution is described by the timedependent Schrodinger equationWe can choose at any instant a basis of eigen statesfor the Hamiltonian labelled by the quantum number n such that the eigen value equation is fulfilledWe assume that the energy spectrum of H is discrete, that the eigen values are not degenerated and that no level crossing occurs during the evolution. Suppose the environment and therefore R(t) is adiabatically varied, that means the changes happen slowly in time compared to the typical time scale of the system. The system starts in the n-the nergy eigen statethen according to the adiabatic theorem the system stays over the whole evolution in the n-the igen state of the instant Hamiltonian. But it is possible that the sta te gains some phase factor which does not affect the physical state. Therefore the state of the system can be written asOne would expect that this phase factor is identical with the dynamical phase factorwhich is the integral over the energy eigenvaluesbut it is not veto by the adiabatic theorem and the Schrodinger equation to add another term which is called the Berry phase 4-8We can determine this additional term by inserting the an sat z (4) together with equation (6) into the Schrodinger equation (1). This yields with the simplifying notation R R(t)After taking the inner product (which should be normalized) with we getand after the integrationwhere we introduced the notation accordingly the total change in the phase of the wave function is equal to theintegraThe respective local form of the curvature has only two nonzero componentsThe expression for the Berry phase (14) can be rewritten as a surface integral of the components of the local curvature form. Using Stokes formulae, we obtain the following expressionwhere S is a surface in and are components of the local curvature form .9Berrys phase for coherent state in SU(4) group for a spin particle (qudit)We consider reference state as for a spin-3/2 particle (qudit) in SU(4) in nonrelativistic quantum mechanics. Coherent state in real parameter in this group is in the following form 10-12where 0i is reference state andis Wigner function. Quadrupole moment isOctupole moment isIf we insert all above calculation in coherent state, obtain newsGeometric phases are important in quantum physical science and are now central to fault tolerant quantum computation. We have presented a detailed analysis of geometrical phase that can arise within general representations of coherent states in real parameterization in SU(4). Berry phase also change in like method. We can continues this method to obtain Berry phase in SU(N) group, where N 5 . we can also obtain Berry phase from complex variable base ket, we conclusio n that result in two different base ket is similar. Berry phase application in optic, magnetic resonance, molecular and atomic physics 13,14 .References1 M. V. Berry, Quantal phase factors accompanying adiabatic changes,Proc.R. Soc. Lond. A 392 (1984) 4557.2 S. Pancharatnam, Proc. Ind. Acad. Sci. A44, 247-262 (1956).3 M.V. Berry, J. Mod. Optics 34, 1401-1407 (1987).4 M. V. Berry. Quantal phase factors accompanying adiabatic changes. Proc.Roy. Soc. London, A329(1802)45-57, (1984).5 J. J. Sakurai. Modern quantum mechanics, (1999).6 Yadollah Farahmand, Zabialah Heidarnezhad, Fatemeh Heidarnezhad, Kh Kh Muminov, Fatemeh Heydari, A Study of Quantum Information and Quantum Computers. repoint J Chem., Vol. 30 (2), Pg. 601-606 ( 2014)7 Yadollah Farahmand, Zabialah Heidarnezhad, Fatemeh Heidarnezhad, Fatemeh Heydari, Kh Kh Muminov, Presentation Quantum Computation Based on Many Level Quantum System and SU(n) Cohered States and Qubit, Qutrit and Qubit Using nuclear Magnetic Resonance Techniq ue and Nuclear Quadrupole Resonance. Chem Sci Trans.,vol 3(4), 1432-1440(2014)8 Yadollah Farahmand, Zabialah Heidarnezhad, Fatemeh Heidarnezhad, Fatemeh Heydari, Kh Kh Muminov,Seyedeh Zeinab Hoosseinirad, Presentation Entanglemen States and its Application in Quantum Computation. Orient J Chem., Vol. 30 (2), Pg. 821-826 ( 2014)9 M. O. Katanaev, arXiv0909.0370v2 math-ph 18 Nov (2009).10 V.S. Ostrovskii, Sov. Phys. JETP 64(5), 999, (1986).11 Kh. O. Abdulloev, Kh. Kh. Muminov. Coherent states of SU(4) groupin real parameterization and Hamiltonian equations of motion. Reports ofTajikistan Academy of science V.36, N6, I993 (in Russian).12 Kh. O. Abdulloev, Kh. Kh. Muminov. Accounting of quadrupole kineticsof magnets with spin . Proceedings of Tajikistan Academy of Sciences, N.1,1994, P.P. 28-30 (in Russian).13 T. Bitter and D. Dubbers. Manifestation of berry,s topology phase in neutronspin rotation. Phys. Rev. Lett, 59251-254, (1987).14 D. Suter, Gerard. C, Chingas, Robert. A, Harris an d A. Pines, MolecularPhys, 1987, V. 61, NO. 6, 1327-1340.