Breakdown and restoration of integrability in the Lieb-Liniger model

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1 Breakdown and restoration of integrability in the Lieb-Liniger model Giuseppe Menegoz March 16, 2012 Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 1 / 16

2 Outline 1 1-d bosons 2 Breaking integrability 3 Restoring integrability Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 2 / 16

3 Integrable systems Integrals of motion degrees of freedom Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 3 / 16

4 Integrable systems Integrals of motion degrees of freedom System always ƒremembers initial state Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 3 / 16

5 Integrable systems Integrals of motion degrees of freedom System always ƒremembers initial state NO THERMALIZATION Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 3 / 16

6 Integrable systems Integrals of motion degrees of freedom System always ƒremembers initial state NO THERMALIZATION Kinoshita et al., Nature 440 (2006), 900 Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 3 / 16

7 1D Bosons H = 2 2m N i=1 2 i + 2c i<j δ(x i x j ) Hardcore bosons on a line This system is integrable via Bethe Ansatz c n 1 Bogoliubov approximation c n 1 Fermion-like behavior Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 4 / 16

8 1D Bosons How do we realize this model? Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 5 / 16

9 1D Bosons Strongly elongated harmonic trap ω r ω z µ ω r and k B T ω r Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 5 / 16

10 1D Bosons Strongly elongated harmonic trap ω r ω z µ ω r and k B T ω r Integrability thermalization is prevented System is not really 1D thermalization is possible Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 5 / 16

11 3D system Ĥ 3D = + 2π 2 α s m [ d 3 r ˆψ (r) ( 2 2m 2 z 2 + Ĥr ] ˆψ (r) ˆψ (r) ˆψ(r) ˆψ(r) ) ˆψ(r)+ Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 6 / 16

12 3D system Ĥ 3D = + 2π 2 α s m [ d 3 r ˆψ (r) ( 2 2m 2 z 2 + Ĥr ] ˆψ (r) ˆψ (r) ˆψ(r) ˆψ(r) ) ˆψ(r)+ ( ) Ĥ r = 2 2 2m x y 2 + mω2 r 2 (x 2 + y 2 ) Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 6 / 16

13 3D system Ĥ 3D = + 2π 2 α s m [ d 3 r ˆψ (r) ( 2 2m 2 z 2 + Ĥr ] ˆψ (r) ˆψ (r) ˆψ(r) ˆψ(r) ) ˆψ(r)+ ˆψ(r) = 1 â nlk φ nl (x, y) e ıkz L nlk Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 6 / 16

14 3D system Ĥ 3D = + 2π 2 α s m [ d 3 r ˆψ (r) ( 2 2m 2 z 2 + Ĥr ] ˆψ (r) ˆψ (r) ˆψ(r) ˆψ(r) ) ˆψ(r)+ ˆψ(r) = 1 â nlk φ nl (x, y) e ıkz L nlk Ĥ r φ nl = (n + 1) ω r φ nl (x, y) and L z φ nl = lφ nl Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 6 / 16

15 Two-body collisions (n, l) Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 7 / 16

16 Two-body collisions (n, l) n = 0, 1, 2,... Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 7 / 16

17 Two-body collisions n = 0, 1, 2,... (n, l) l = mod (n, 2), mod (n, 2) + 2,..., n 2, n Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 7 / 16

18 Two-body collisions (n, l) (0, 0); (0, 0) (0, 0); (0, 0) Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 7 / 16

19 Two-body collisions (n, l) (0, 0); (0, 0) (0, 0); (0, 0) NO THERMALIZATION (because of energy and momentum conservation) Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 7 / 16

20 Two-body collisions (n, l) (0, 0); (0, 0) (0, 0); (0, 0) NO THERMALIZATION (0, 0); (0, 0) (0, l); (0, l) Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 7 / 16

21 Two-body collisions Fermi Golden rule T i f = 2π f Ĥ i 2 ρ Γ 2b n 1d l 2 r αs 2 e 2 ωr k B T ml r Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 8 / 16

22 Two-body collisions Fermi Golden rule T i f = 2π f Ĥ i 2 ρ Γ 2b n 1d l 2 r αs 2 e 2 ωr k B T ml r 3d density Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 8 / 16

23 Two-body collisions Fermi Golden rule T i f = 2π f Ĥ i 2 ρ cross section Γ 2b n 1d l 2 r 3d density αs 2 e 2 ωr k B T ml r Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 8 / 16

24 Two-body collisions Fermi Golden rule T i f = 2π f Ĥ i 2 ρ cross section Γ 2b n 1d l 2 r αs 2 e 2 ωr k B T ml r 3d density collision speed Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 8 / 16

25 Two-body collisions Fermi Golden rule T i f = 2π f Ĥ i 2 ρ cross section Boltzmann factor Γ 2b n 1d αs 2 e 2 ωr k B T ml r l 2 r 3d density collision speed Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 8 / 16

26 Two-body collisions Fermi Golden rule T i f = 2π f Ĥ i 2 ρ Γ 2b n 1d l 2 r αs 2 e 2 ωr k B T ml r Γ 2b 0.01 s 1 Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 8 / 16

27 Three-body collisions (0, 0); (0, 0) (0, 0); (2p, 0) Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 9 / 16

28 Three-body collisions (0, 0); (0, 0) (n 1, l); (n 2, l) Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 9 / 16

29 substitute ansatz for ˆψ into Ĥ3D Integrate out x and y dependence (φ n0 are Laguerre polynomials) eliminate excited states Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 10 / 16

30 Effective Hamiltonian Ĥ 1D = k k 2 2m â kâk + ω r α s L ξ ω r α 2 s 2L 2 k,k,q â k 1 â k 2 â k 3 â k1 â k2 â k3 {k} â k+qâ k qâk â k+ Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 11 / 16

31 three-body collision rates k 1, k 2, k 3 = â k 1 â k 2 â k 3 vac k 1, k 2, k 3 Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 12 / 16

32 three-body collision rates k 1, k 2, k 3 = â k 1 â k 2 â k 3 vac k 1, k 2, k 3 dk 1 dk 2 dk 3 Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 12 / 16

33 three-body collision rates k 1, k 2, k 3 = â k 1 â k 2 â k 3 vac k 1, k 2, k 3 dk 1 dk 2 dk 3 Γ k1,k 2,k 3 = ζ ω r α 4 s L 2 l 2 r Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 12 / 16

34 three-body collision rates k 1, k 2, k 3 = â k 1 â k 2 â k 3 vac k 1, k 2, k 3 dk 1 dk 2 dk 3 Γ 3b = 1 3! Γ k1,k 2,k 3 = ζ ω r α 4 s L 2 l 2 r dkγ k1,k 2,k 3 (Nfk ) = ζ 3! ω ( n1d l 2 r ) 2 α 4 s No T dependence Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 12 / 16

35 Comparison Γ 3b Γ 2b ( n1d α 2 S l r ) 2 e 2 ωr K B T Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 13 / 16

36 Comparison Γ 3b can dominate in the appropriate experimental conditions In the weak interaction regime this ratio correspond also to the ratio between thermalization rates Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 13 / 16

37 Restored integrability Strong interaction regime fermion-like behavior ψ ψ ψ ψψψ c g 3 = 1 c 6 n 3 1D Low probability of a 3-body collision Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 14 / 16

38 Conclusions Radially confined 1D gases are never perfectly 1D, so radial motion can be virtually excited. These processes lead to thermalization. Integrability of confined 1D systems is not caused by the freeze out of 2-body collisions but is ensured by quantum correlations in the strongly interacting regime. Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 15 / 16

39 References I.E. Mazets, T. Schumm, J. Schmiedmayer, Breakdown of integrability in a quasi-one-dimensional ultracold bosonic gas, Phys. Rev. Lett. 100, (2008) I.E. Mazets, J. Schmiedmayer, Restoring integrability in one-dimensional quantum gases by two-particle correlations, Phys. Rev. A 79, (R) (2009) Giuseppe Menegoz () Breakdown and restoration of integrability in the Lieb-Liniger model 16 / 16