Ultracold Quantum Gases beyond Equilibrium Natal, Brasil, September 27 October 1, 2010 Dynamic properties of interacting bosons and magnons Peter Kopietz, Universität Frankfurt collaboration: A. Kreisel, T. Kloss, L. Bartosch, F. Sauli, J. Hick (Frankfurt) A. Sinner (Augsburg) N. Hasselmann (Natal) A. Serga, B. Hillebrands (Kaiserslautern, experiment) references: Eur. Phys. J. B 71, 59 (2009) Phys. Rev. B 81,104308 (2010) arxiv:1007.3200 and 1008.4521 Phys. Rev. Lett. 102, 120601 (2009) outline: 1. Towards a non equilibrium many body theory for YIG 2. Spectral function of interacting bosons in 2D 1
1.Towards a non-equilibrium many-body theory for YIG Motivation: collaboration with experimental group of B. Hillebrands (Kaiserslautern) non-equilibrium dynamics of interacting magnons in YIG Experiment: microwave-pumping of magnons in YIG measurement of magnon distriubution via Brillouin light scattering 2
Parallel pumping of magnons in YIG by microwaves what is parallel pumping of magnons? 3
Parallel pumping of magnons in YIG by microwaves what is parallel pumping of magnons? two pumps in parallel: 4
Parallel pumping of magnons in YIG by microwaves what is parallel pumping of magnons? two pumps in parallel: in context of YIG: oscillating magnetic field is parallel to magnetization: Hamiltonian in Bogoliubov basis: time-dependent off-diagonal terms: 5
Parametric resonance what is parametric resonance? classical harmonic oscillator with harmonic frequency modulation: resonance condition: oscillator absorbs energy at a rate proportional to the energy it already has! history: discovered: Melde experiment, 1859 excite oscillations of string by periodically varying its tension at twice its resonance frequency theoretically explained: Rayleigh 1883 6
History: parametric resonance in YIG H. Suhl, 1957, E. Schlömann et al, 1960s, V. E. Zakharov, V. S. L vov, S. S. Starobionets, 1970s minimal model: S-theory : time-dependent self-consistent Hartree- Fock approximation for magnon distributions functions weak points: no microscopic description of dissipation and damping possibility of BEC not included! goals: consistent quantum kinetic theory for magnons in YIG beyond Hartree-Fock include time-evolution of Bose-condendsate develop functional renormalization group for non-equilibrium 7
Toy model for parametric resonance T. Kloss, A. Kreisel, PK, PRB 2010 anharmonic oscillator with off-diagonal pumping: rotating reference frame: instability of non-interacting system for large pumping: for oscillator has negative mass non-interacting Hamiltonian not positive definite interactions stabilize ground state for large pumping 8
Time-dependent Hartree-Fock approximation ( S-theory ) order parameter Gross-Pitaevskii equation: connected correlation functions: kinetic equations: order parameter: Hamiltonian dynamics in effective potential (Hartree-Fock) 9
Functional integral formulation of the Keldysh technique T. Kloss, PK, in preparation 6 types of non-equilibrium Green functions: retarded: advanced: Keldysh: Keldysh component at equal times gives distribution function functional integral formulation (Kamenev 2004) 10
Non-perturbative method: functional renormalization group exact equation for change of generating functional of irreducible vertices as IR cutoff is reduced (Wetterich 1993) exact RG flow equations for all vertices flow of self-energy: 11
FRG for bosons out of equilibrium introduce RG flow parameter which cuts off long-time behavior eventually exact RG flow equation for non-equilibrium self-energy: see also Gasenzer, Pawlowski 2008 self-energy defines collision integral in quantum kinetic equation: 12
Out-scattering rate cutoff scheme problem: perturbation theory fails for long time dynamics (secular terms: U*t) solution: non-equilibrium dynamics from the functional RG! RG flow parameter: out scattering rate: make infinitesimal imaginary part finite parameter for the FRG and use it as flow 13
Comparison of exact solution of toy model with FRG toy model can be solved numerically exactly by solving time-dependent Schrödinger equation comparison with various approximations: 14
2. Interacting bosons: Bogoliubov theory (1947) Bogoliubov-shift: Bogoliubov mean-field Hamiltonian: condensate density: excitation energy: long wavelength excitations: sound! 15
Beyond mean-field: infrared divergencies Bogoliubov mean-field Hamiltonian: normal self-energy: anomalous self-energy: mean-field fails due to infrared divergent fluctuation corrections: for Ginzburg scale 16
Exact result: Nepomnyashi-identity (1975) anomalous self-energy vanishes at zero momentum/frequency: Bogoliubov approximation wrong: need non-perturbative methods! 17
Origin of infrared divergencies reason for IR divergence: coupling between transverse and longitudinal fluctuations and resulting divergence of longitudinal susceptibility (Patashinskii+Pokrovskii, JETP 1973) consequence: critical continuum in longitudinal part of spectral function (for bosons not directly measurable!) experimentally accessible: longitudinal spin structure factor in quantum antiferromagnets: BEC of magnons! (Kreisel, Hasselmann, PK, PRL 2007) 18
FRG flow equations for order parameter and self-energies order parameter: normal self-energy: anomalous self-energy: 19
Low-density truncation: retain only two-body interactions bare action: ansatz for generating functional of irreducible vertices: depends on condensate density momentum-dependent functions and two frequency- and and analytic non-analytic 20
Parametrization of threeand four-point vertices: normal and anomalous self-energy: Hugenholtz-Pines relation Nepomnyashchy identity three-legged vertices: effective interaction: 21
Results: spectral function and quasi-particle damping in 2D spectral function: quasi-particle damping: 22
What about experiments? Cold atom experiments can now measure renormalized excitation spectrum of strongly interacting bosons: 23
Summary+Outlook magnon gas in yttrium-iron garnet: model system for non-equilibrium physics of interacting bosons functional RG out of equilibrium: non-perturbative calculation of time-evolution functional RG in equilibrium: non-perturbative calculation of spectral function of interacting bosons 24