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Resonances and background scattering in gedanken experiment with varying projectile flux Petra Zdanska, IOCB June 2004 – Feb 2006 Personal acknowledgement • Milan Sindelka and Nimrod Moiseyev • Vlada Sychrovsky and people attending my unfinished Summer course of resonances 2004 • Nimrod’s group and conferences IOCB February 20, 2006 2 Resonance and direct scattering as two mechanisms • Direct – density of states changes evenly smooth spectrum IOCB February 20, 2006 • Resonance – metastable states – density of states includes peaks 3 Simultaneous occurrence of direct and resonance scattering mechanisms? IOCB February 20, 2006 4 Question: • Are direct and resonance scattering mechanisms separable at near resonance energy ? • Mathematical answer: yes by complex scaling transformation. • Physical answer: IOCB February 20, 2006 ? 5 Complex scaling method (CS) • useful non-hermitian states – “resonance poles” – purely outgoing condition is a cause to exponential divergence and complex energy eigenvalue • complex scaling transformation of Hamiltonian – non-unitary similarity transformation for taming diverging states IOCB February 20, 2006 6 Ougoing condition for resonances and CS • Problem: exp ipx exp ix Re p x Im p • Solution: exp ipxe i exp ix cos Re p sin Im p exp x sin Re p cos Im p Im p c arctan arctan Re p Re p IOCB February 20, 2006 7 Outgoing condition for resonances and CS IOCB February 20, 2006 8 Separation of direct and resonance scattering by CS transformation Re E bound states resonance rotated continuum Im E IOCB February 20, 2006 9 States obtained by CS as scattering states for varying projectile flux IOCB February 20, 2006 10 • Connection between gamma and theta: IOCB February 20, 2006 11 Proofs by semiclassical and quantum simulations • Why semiclassical and not just quantum mechanics – only way to prove a correspondence between the classical notion of flux of particles and quantum wavefunctions • Cases I and II: – I. analytical proof for free-particle scattering – II. numerical evidence for direct scattering problem • Case III: – a quantum simulation of resonance scattering for varying projectile flux displaying the new effects IOCB February 20, 2006 12 Case I: Free-particle Hamiltonian • non-hermitian solutions of CS Hamiltonian: Re E 2 2 2 ˆ ˆ p p Hˆ Hˆ e 2i 2 2 Hˆ Im E IOCB February 20, 2006 E e2i exp ipx exp ipxe i 13 Wavefunctions of rotated continuum • exponentially modulated plane waves: decays in time grows in x IOCB February 20, 2006 14 • time-dependence: i i exp p xe i i ˆ 2 i t exp Ht exp E e t i 2 i i exp p xe E te IOCB February 20, 2006 15 Semiclassical solution to the expected physical process behind these non-hermitian states: • step I: construction of a corresponding density probability in classical phase space – 1st order emission in an asymptotic distance xe with the rate : IOCB February 20, 2006 16 – density of particles in a close neighborhood of the emitter: – analytical integration of the classical Liouville equation with the above boundary condition: IOCB February 20, 2006 17 Classical density for free particles: IOCB February 20, 2006 18 Step II: transformation of classical phase space density to a quantum wavefunction – non-approximate, in the case of freeHamiltonian IOCB February 20, 2006 19 IOCB February 20, 2006 20 Exact comparison with nonhermitian wavefunction as a proof • the non-hermitian and scattering wavefunctions have the same form and are equivalent supposed that, – which was to be proven. IOCB February 20, 2006 21 Case II: Rotated complex continuum of Morse oscillator • potential: D 1a.u., 1a.u.1 , 10a.u. • semiclassical simulation of scattering experiment with parameters: – particles arrive with classical energy: – decay rate of the emitter: IOCB February 20, 2006 22 Construction of classical phase space density • classical orbit [x(t),p(t)] is evaluated • phase space density: IOCB February 20, 2006 23 Construction of semiclassical wavefunction • dividing to incoming and outgoing parts: • transformation of density to wf: IOCB February 20, 2006 24 IOCB February 20, 2006 25 IOCB February 20, 2006 26 The expected quantum counterpart • Non-hermitian solution of CS Hamiltonian with the energy: IOCB February 20, 2006 27 Solution of CS Hamiltonian in finite box: • box: • N=200 basis functions • solution of CS Hamiltonian: • back scaled solution: IOCB February 20, 2006 28 Comparison of scattering wavefunction and rotated continuum state: IOCB February 20, 2006 29 Case III: near resonance scattering • Potential: • Examined scattering energies: – resonance hit – very slightly off-resonance IOCB February 20, 2006 30 in complex energy plane: V(x) 0.7126 0.716 Re E -0.002 -0.0034 -0.004 Im E x IOCB February 20, 2006 31 Quantum dynamical simulations of scattering experiments • “particles” added as Gaussian wavepackets in an asymptotic distance, 40 a.u. • beginning of simulation: scattering experiment does not start abruptly but the intensity I(t) is modulated as follows: IOCB February 20, 2006 32 slow change of gamma 0.7126 0.716 Re E -0.002 -0.0034 -0.004 Im E IOCB February 20, 2006 33 Resonance hit: IOCB February 20, 2006 34 Off-resonance: IOCB February 20, 2006 35 Off-resonance IOCB February 20, 2006 36 What is going on: • We reach stationary-like scattering states, which are characterized by a constant scattering matrix and by a constant (and complex) expectation energy value. • Are these states the non-hermitian solutions to Hamiltonian obtained by CS method? IOCB February 20, 2006 37 Calculations of scattering matrix: • comparison of dynamical simulations with stationary solutions of complex scaled Hamiltonian • gamma<Gamma_res : – rotated continuum • gamma>Gamma_res : – resonance hit resonance pole – slightly off-resonance rotated continuum IOCB February 20, 2006 38 Scattering matrix from simulations: IOCB February 20, 2006 39 Inverted control over dynamics for gamma>Gamma_res • incoming flux decays faster than the wavefunction trapped in resonance • natural control: incoming flux disappears faster than outgoing flux – this occurs for discrete resonance energies • inverted control: outgoing flux decays according to gamma and not Gamma_res. Reason: destructive quantum interference removes the trapped particle. IOCB February 20, 2006 40 • empirical rule in CS: rotated continuum for θ> θc (γ>Γres) is not responsible for resonance cross-sections. IOCB February 20, 2006 41 Conclusions: • resonance phenomenon studied in a new context of scattering dynamics • new light shed into complex scaling method, interference effect behind the long accepted empirical rule • first physical realization of complex scaling eventually interesting for experiment IOCB February 20, 2006 42