Exotic topological states of ultra-cold atomic matter Lecture 1: - PowerPoint PPT Presentation

Exotic topological states of ultra-cold atomic matter Lecture 1: Topolgical and non- topological solitons Joachim Brand Canberra International Physics Summer School: 2018 – Topological Matter

To be covered: Solitons in quantum gases Lecture 1: Solitons and topological solitons • – solitons in water: the KdV equa;on, iintegrability – solitons of the nonlinear Schrodinger equa;on – solitons of the sine Gordon equa;on - topological solitons – Bose Josephson vor;ces in linearly coupled BECs Lecture 2: Semitopological solitons in mul;ple dimension • – Solitons as quasipar;cles: effec;ve mass – solitons in the strongly-interac;ng Fermi gas – snaking instability – vortex rings – solitonic vor;ces Lecture 3: Quantum solitons and Majorana solitons • – solitons in strongly-correlated 1D quantum gas – solitons with Majorana quasipar;cles in fermionic superfluids

Solitons Water Optics Coupled Pedula Tikhonenko et al. (1996) Sengstock group (2008) 5 BEC credit: Alex Kasman

Topologial solitons So if a soliton is a localised wave, then what is a topological soliton?

Hagfish makes a knot Credit: Stefan Siebert, Sophia Tintory, Casey Dunn hRps://vimeo.com/7825337

Topologial solitons So if a soliton is a localised wave, then what is a topological soliton? Wikipedia: “A topological soliton or a topological defect is a solu;on of a system of par;al differen;al equa;ons or of a quantum field theory homotopically dis;nct from the vacuum solu;on.” Homotopy: a con;nuous deforma;on

� � � � � � � ������ Solitons appear spontaneously e.g. when cooling through the Bose-Einstein condensa;on phase transi;on • Nature Physics 2013: ARTICLES PUBLISHED ONLINE: 8 SEPTEMBER 2013 | DOI: 10.1038/NPHYS2734 Spontaneous creation of Kibble–Zurek solitons in a Bose–Einstein condensate Giacomo Lamporesi, Simone Donadello, Simone Serafini, Franco Dalfovo and Gabriele Ferrari * Also: proposal to observe Josephson vor;ces (topological solitons) by rapidly cooling a double-ring Bose- J Einstein condensate. �� � � � � � � SW Su, SC Gou, AS Bradley, O Fialko, JB , Phys. Rev. LeR. 110 , 215302 (2013)

From linear to nonlinear waves: shallow water Linear wave equa;on φ ( x, t ) = A sin( x − ct ) ∂ t φ + c ∂ x φ = 0 Nonlinear waves: wave speed depends on amplitude: Inviscid Burgers equa;on ∂ t φ + φ ∂ x φ = 0 Add dispersive (higher order deriva;ve term): Source: Leon van Dommelen, FSU Korteweg – de Vries equa;on (1895) ∂ t φ + ∂ 3 x φ + 6 φ ∂ x φ = 0 Soliton solu;on  √ c � φ ( x, t ) = 1 2 c sech 2 2 ( x − ct − a ) Source: Wikipedia

KdV: an integrable soliton equa;on 1965: Zabusky and Kruskal discover robust collision in numerics, invent the term “soliton” 1967: Inverse scaRering transform (Gardner, Greene, Kruskal, Miura) is based on the existence of a Lax pair L = − ∂ 2 x + φ A = 4 ∂ 3 x − 3[2 φ∂ x + ( ∂ x φ )] ∂ t L = [ L, A ] scaRering data Ini;al condi;on L φ ( x, 0) S ( t = 0) A inverse scaRering φ ( x, t ) S ( t ) Inverse scaRering transform method

The scaRering problem The linear Schrödinger equa;on L = − ∂ 2 with L ψ ( x ) = λψ ( x ) x + φ has bound state solu;ons “solitons” λ i < 0 and scaRering states “radia;on” λ ≥ 0 The nature of the scaRering problem does not change as ;me evolves, thus solitons are eternal. Moreover, there is an infinite number of constants of the mo;on – the problem is integrable . Long term fate of a localised ini;al state (finite support) φ ( x, 0) For with φ ( x, t ) t → ∞ • Solitons will persist, separate • Radia;on will decay to zero amplitude

Examples of integrable soliton equa;ons Korteweg – de Vries equa=on : • ∂ t φ + ∂ 3 x φ + 6 φ ∂ x φ = 0 real wave func;on, bright solitons only Nonlinear Schrödinger equa=on: • i ∂ t u = − ∂ 2 x u ± | u | 2 u complex wave func;on, bright and dark solitons Sine Gordon equa=on: • ∂ 2 t φ − ∂ 2 x φ + sin( φ ) = 0 rela;vis;c covariant wave equa;on (Lorentz transforma;on); real wave func;on, topological solitons

Theory: Bose-Einstein Condensate (BEC) • Bose gas in an external poten;al For BECs we may use the classical Interaction becomes a or mean field (Hartree) approximation: tunable parameter Gross-Pitaevskii equation s-wave scattering length The GP equation is a nonlinear Schrödinger equation Is GP valid for soliton phenomena? Criterium of validity: healing length particle distance length scale for solitons

Solitons as sta;onary solu;ons of the nonlinear Schrödinger equa;on cos(x) sin(x) g=0 g=0 cn(x|k) sn(x|k) g<0 g>0 bright soliton dark soliton tanh(x) sech(x) For a tutorial-style introduc;on see Reinhardt 1988

Solitons in the nonlinear Schrödinger equation (NLS) Dispersion Nonlinearity bright dark solitons soliton u ( x, t ) = u 0 { Ai + B tanh[ u 0 B ( x − Au 0 t )] } e iu 2 0 t A 2 + B 2 = 1 Phase step ✓ A ◆ ∆ φ = 2 tan − 1 B From: Kivshar (1998)

Solitons in quantum gases • Bose-Einstein condensate in quasi-1D trap: Gross-Pitaevskii equa;on -> NLS – Dark solitons with repulsive interac;ons – Bright solitons with aRrac;ve interac;ons • Superfluid Fermi gas in BEC – BCS crossover – BEC regime -> dark solitons as above (NLS) in quasi 1D – BCS regime -> Bogoliubov-de Gennes equa;on with dark soliton solu;ons in 1D – Unitary regime, 3D, strictly 1D -> to be discussed • Linearly coupled 1D BECs -> coupled 1D GPEs – Not integrable but features both NLS and sine Gordon soli;ons

Josephson vor;ces in superconductor Long Josephson juc;on

Solitons of the sine Gordon equa;on The sine Gordon equa;on ) 2 − cos( φ ) Field poten;al ∂ 2 t φ − ∂ 2 x φ + sin( φ ) = 0 0 2π corresponds to the energy density φ w = 1 2( ∂ x φ ) 2 − cos( φ ) vaccum 1 vaccum 2 The sine Gordon kink is a topological soliton. It connects two vacuua.

Classifica;on of solitons • Non-topological soliton: relies on the balance of nonlinearity and dispersion • Topological soliton: owes its existence to a mul;plicity of ground states that allow topologically non-trivial field configura;ons Topological charge for sine-Gordon : Q = 1 2 π [ φ ( x = ∞ ) − φ ( x = −∞ )] Associated conserved current: Z ∞ ∂φ j = 1 Q = j dx 2 π ∂ x −∞

� � � � � ������ � � Two coupled Bose fields i ~ ∂ t ψ 1 = − ~ 2 2 m ∂ xx ψ 1 − µ ψ 1 + g | ψ 1 | 2 ψ 1 − J ψ 2 i ~ ∂ t ψ 2 = − ~ 2 2 m ∂ xx ψ 2 − µ ψ 2 + g | ψ 2 | 2 ψ 2 − J ψ 1 J is tunnel coupling µ is the chemical potential g>0 interaction between atoms J �� � � � � � � ν = J Important parameter: µ Could be realised in double ring trap or two linear traps with narrow barrier (Schmiedmayer experiments).

Field potential for coupled BECs Field potential for coupled BEC fields • Relative phase and amplitude yield sine-Gordon equation – a relativistic field theory! • Total phase and density yield nonlinear Schrödinger equation – with dark solitons and phonons. B Opanchuk, R Polkinghorne, O Fialko, JB, P Drummond, Ann Phys. (Berlin) (2013)

Josephson vortex and dark soliton i ~ ∂ t ψ 1 = − ~ 2 2 m ∂ xx ψ 1 − µ ψ 1 + g | ψ 1 | 2 ψ 1 − J ψ 2 i ~ ∂ t ψ 2 = − ~ 2 2 m ∂ xx ψ 2 − µ ψ 2 + g | ψ 2 | 2 ψ 2 − J ψ 1 Josephson vortex Dark soliton arg ψ 1 − arg ψ 2 arg ψ 1 arg ψ 2 The stationary solutions were found by Kaurov and Kuklov PRA (2005) Related: JB,T Haigh, U Zuelicke PRA 2009 L Wen, H Xiong, B Wu PRA 2010

Josephson vortex vs dark soliton Dark soliton (unstable) Energy Josephson vortex (stable) ν = J coupling parameter µ

Josephson vortex dispersion Josephson vortices can move dark soliton v = dE dP c 1 = 2 dE dP 2 m eff c Josephson vortex They are quasiparticles with tunable effective mass Sophie Shamailov and JB, arXiv:1709.00403

Breathers and oscillons • Breathers in the sine Gordon equation are not topological, but live forever Small-amplitude breather Stationary large-amplitude breather • In the coupled BECs, instead we find oscillons : breather-like excitations that live a long time 1.5 4 600 1.4 3 1.3 energy 400 t 2 1.2 1.1 200 1 1 0 0 500 1000 − 100 0 100 z t S-W Su, S-C Gou, I-K Liu, AS Bradley, O Fialko, JB, PRA (2015)

Examples of integrable soliton equa;ons Korteweg – de Vries equa=on : water waves • ∂ t φ + ∂ 3 x φ + 6 φ ∂ x φ = 0 Focusing nonlinear Schrödinger equa=on: • i ∂ t u = − ∂ 2 x u − | u | 2 u ARrac;ve Bose-Einstein condensates in quasi-1D waveguide Experiments by Hulet, Salomon, Cornish, Kasevich Defocusing nonlinear Schrödinger equa=on: • i ∂ t u = − ∂ 2 x u + | u | 2 u Repulsively interac;ng Bose-Einstein condensates Experiments by Sengstock, Phillips, Oberthaler Sine Gordon equa=on: • ∂ 2 t φ − ∂ 2 x φ + sin( φ ) = 0 Realised by linearly coupled Bose-Einstein condensates (Schmiedmayer experiments?)