Multi-Body Interacting Bosons Dmitry Petrov Laboratoire Physique Théorique et Modèles Statistiques (Orsay)
Outline Effective multi-body interactions Multi-body interacting systems. Why interesting? How to make (engineer) them? 3-body interacting dipolar molecules in bi-layers with tunneling 3- and 4-body interacting 39K in optical lattices
Effective multi-body interactions Parasites which appear when we want to simplify our life: Simple 2-body pot. +... Simple 2-body pot. + 3-body + Hammer et al., Rev. Mod. Phys. (2013)
Examples Lee, Huang, Yang'57, Wu'59... Hard sphere gas Zero-range interacting gas + three-body interaction This workshop & program Bosons with van der Waals two-body forces Zero-range interacting gas + three-body parameter Beyond LHY effects. Three-body force important at large a Braaten et al.'99, Bulgac'02,...
Examples Quasi-1D bosons 1D Lieb-Liniger model l0 2-body Olshanii'98 + 3-body Muryshev et al.'02 N 2 1 2a 4 H = x x 12 log i j 2 i=1 xi2 l 0 i j 3 a2 xi x j x j x k 2 l 0 i j k Perturbative, second order, weak, attractive, but BREAKS INTEGRABILITY!
Examples All quantum Hall numerics Quantum Hall layer Single Landau level model + higher order interactions Landau levels E E Fermi level Nayak et al.'09,'13, Sodemann&MacDonald'13 x Weak, higher order in g/, but important due to high Landau level degeneracy!
Examples Jaksch et al.'98... Bosons in an optical lattice Bose-Hubbard model + higher order interactions Niu et al.'06, Tiesinga et al.'09,11, Hazzard&Mueller'10 t t Weak, high order in g /, but measurable Campbell et al.'06, Will et al.'10
g~a 3/ 2 g 2 / ~ ~ N N 1 N N 1 N 2 N N 1 N 2 N 3 U 3 U 4... 2! 3! 4! ~ E N =U 2 g 3 / 2
Multi-body interacting systems N-body interacting g3 g N 1 = g2 = = = U 2=U 3 =...=U N 1 =0 AND U N 0 gn E n2 n3 n4 =g2 g3 g 4... Volume 2! 3! 4!
Why interesting? Bosons with g2 0 is a bad combination, but if g2 is small and if we add g3 0 Cubic-quintic GP (Schroedinger) equation in a trap, gas-liquid transition Gammal et al.'00- [ 2 r /2 g2 n r g3 n2 r /2 V ext r ] r =0 Free space self-trapped droplet state Bulgac'02: Neglecting surface tension, flat density profile n=3 g2 /2 g3 Including surface tension surface modes Staying dilute requires large g3 and small g2 close to zero crossing!
Why interesting? Above critical g2 bosons pair topological transition, not crossover! Nozieres&Saint James'82 Radzihovsky et al., Romans et al., Lee&Lee'04 repel because of g3 g 2 critical g 2 Pairing on a lattice with three-body constraint: Daley et al.'09-, Ng&Yang'11, Bonnes&Wessel'12,... g3 is necessary! = Pauli pressure in the BCS-BEC crossover!
Why interesting? Bosons with dipolar interactions L. Santos, M. Lewenstein, P. Zoller, G. Shlyapnikov C. Eberlein, S. Giovanazzi, O Dell K. Góral, M. Brewekyk, K. Rzatenski S. Yi and L. You, S. Giovanazzi, A. Gorliz, T. Pfau g3 > 0 stabilizes weakly interacting supersolid phase, Lu et al., to be published
Why interesting? Local repulsive gk+1 is the ``parent'' Hamiltonian for the k-th state of the ReadRezayi series of quantum Hall states, Nayak et al., Rev. Mod. Phys. (2008) k=1 (2-body int.) Laughlin state (abelian anyons) k=2 (3-body int.) Moore-Read state (non-abelian anyons, some topologically protected operations) k=3 (4-body int.) Read-Rezayi state (non-abelian anyons, universal quantum computing) Ground state degeneracy protected by gap ~ gk+1 Important to maximize!
Some previous work
3-body interacting case: perturb. prosp. E N =U 2 N N 1 N N 1 N 2 N N 1 N 2 N 3 U 3 U 4... 2! 3! 4! Perturbative approach Johnson et al.'09 V 0 2 E 2 =U 2= 0 V 0... 0 Higher order terms 2 V 0 E 3 =3 E 2 O V 2... 0 Double counting compensation Higher order terms U 3 ~V 2 / 0 1
3-body interacting case: perturb. prosp. E N =U 2 N N 1 N N 1 N 2 N N 1 N 2 N 3 U 3 U 4... 2! 3! 4! Perturbative approach Johnson et al.'09 V 0 2 E 2 =U 2= 0 V 0... 0 Higher order terms 2 V 0 E 3 =3 E 2 O V 2... 0 Double counting compensation Additional non-additive higher order terms U 3 ~V 2 / 0 1 This talk U 2=0 AND U 3 0 AND STRONG!
Where to look? 0 V 0 0 vanishing on-shell scattering... OK V 0 0 large off-shell contribution... which should repel the third particle...??? R 1 Re R V R 3 4 Just two-body zero crossing is not enough! g 3D ~R a~r 3 e e
Dipoles on layers Repulsive intralayer potential V =r / 3 No bound state =1 Interlayer potential averages to zero V =r 2 2 / 2 1 5 /2 At least one bound state
O ne -b od y pr ob le m Bilayer with tunneling t =0 =2t 2 2 coupling Tw obo d y pr ob le m V V 2t V r / 3 4t V r / 3 4t V V 2t
Vertex function and bound state Vertex function for 2D scattering with weakly-bound state + dipolar tails: Baranov et al.'11 4 E, k, k ' 2 r k k ' ln 4t / E 4 / g 2 i 0 =4t exp 4 / g2 Critical t Exponentially weakly bound state for small negative g2
S-wave scattering at finite collision energy E, k, k ' 4 2 r k k ' ln 4t / E 4 / g2 i 4 tan s q g 2 8 r q s-wave and on shell Two-body zero crossing + dipolar tail rotonization, density wave, etc
3-body coupling constant g 3 = free 3 V true 3 free2 V true 2 adiabatic hyperspherical method on the line g2=0 Always repulsive!
Large r* dipolar frustration (cf. Volosniev et al.'12) V V 2t coupling V r / 3 4t 4t V r / 3 V V 2t
Small r* large off shell contribution coupling 1/ 4t 4t 1/ 4t Zero-range model 24 g3= tc 2 [ 3 ln 4 / 3 1 4... ln E / E ln E / E 3 ]
Three-body repulsion and trimers B 0 3 =16.522 688 1 B 2 r =1 1.1 1.2 1.3 1.4 1.5 1.6 t Bruch&Tjon'79,Nielsen et al.'99, Hammer&Son'04
Frustration is good! Lattice bosons with on-site Hamiltonian g ' H = b b b b b b b b, ' b b b b 2 2 2 ' ' Bilayer dipoles =0, =2t, g = g 0, and g 0 Simple example = =g =0, g = g 0
39 K: F=1,mF=0> and F=1,mF=-1> [MHz ] Couple them with RF (~50MHz) =Rabi frequency (~khz) =Detuning (~khz) 2 2=spin splitting T B [G] g ' =? Spin is virtually excited during collisions! D'Errico et al.'07, Lysebo&Veseth'10 Many thanks to A. Simoni and M. Lysebo for the data!
Optimization problem Find E(2) and E(3) by diagonalizing g H = b b b b b b b b, ' ' b b ' b b ' 2 2 2 U 2 and U 3 Find parameters for which The result is: reached for U 2=0 and U 3 max { g g g U 3, max = min g, g, max g, g g, g, g, g =0 && =g sign g g U 3, max 0 requires min g, g 0 && max g, g g 0
Nice window of B a a Bohr B [G] Data from A. Simoni
V0 lattice constant=532 nm V 0 =15 E R on-site osc. freq.=2 35 khz l x =l y =l z =86 nm tunneling amp.=2 30 Hz a =9.4 nm g =2 3.05 khz a =1.7 nm g =2 0.55 khz a = 2.8 nm g = 2 0.91 khz g =0 =2 0.5 khz =2 3.33 khz g g
U2 U3 =0 =2 0.5 khz =2 3.33 khz
U2 U3 =0 =2 0.5 khz =2 3.33 khz
4-body interacting case =2 1.7 khz
Perspectives Bosonic dipoles... Have to wait a bit... 39 K on a lattice - collapse and revivals - solitonic-like self-trapping - Mott lobes Chen et al.'08 Gap=U 2 Gap=U 2 U 3 Frustration is local, large-size off-shell effects are cooler! requires small t (bilayers) or small 2 2 (RF coupling)
Thank you! Merci beaucoup!