Lieb-Liniger , Tonks -Girardeau and super Tonks -Girardeau gases

1. Lieb-Liniger, Tonks-Girardeau and super Tonks-Girardeau gases Departament de Fìsica i Enginyeria Nuclear, Campus Nord B4-B5, Universitat Politècnica de Catalunya, Barcelona, Spain G.E. Astrakharchik Summer school, Trier, August 18 (2012)

2. CONTENTS •Introduction •Why low-dimensional systems are special? • Density scaling: potential vs kinetic energy •Lieb-Liniger gas •Energy expansion in Bogoliubov theory •Tonks-Girardreu gas •Unitary gas •Super Tonks-Girardeau gas •Dynamic form factor •Conclusions

3. ONE-DIMENSIONAL QUANTUM WIRES • One-dimensional quantum systems have been experimentally realized and studied already in 80's in GaAs/AlGaAs semiconductor structures. • • Systems with free electrons confined to one dimension • (quantum wires) has been realized. • •Conductance properties, elementary excitations, etc have been measured. Figure with dispersion of excitations in a quantum wire taken from 1991 experiment by A. R. Goñi et al. in AT&T Bell Laboratories, New Jersey

4. 1D BOSE GAS IN A MICROCHIP • Atoms can be confined to one-dimensional geometry • heavier and larger • easier to detect • can be bosons or fermions • microchip experiments: cold atoms trapped in a combination of two magnetic fields: external magnetic field and field generated by an electric current in wire. taken from “BEC on a microchip”, J. Reichel et al. MPI, München

5. COLD GASES IN OPTICAL LATTICES • Advanced and unique features of dilute ultracold gas experiments with optical lattices: • • Possibility to fine-tune the interaction strength by using Feshbach resonances • - contrary interactions between electrons in a quantum wire can be hardly changed • • highly controllable geometry of the confinement parameters • - for example spacing and height of an optical lattice • - can be changed dynamically • • extremely pure systems • - no defects as in quantum wires • - no condensate fragmentation as might happen in micro chip traps

6. CONDITION FOR ONE-DIMENSIONALITY The gas behaves dynamically as one-dimensional when the excitations of the levels of the transverse confinement are not possible: • Condition for the energy • Condition for the temperature Figure is taken from T.Esslinger et al./Zurich Comparison of the frequencies of the dipole and breathing modes confirms the achievement of the quasi-one-dimensional regime.

7. CONTENTS •Introduction •Why low-dimensional systems are special? • Density scaling: potential vs kinetic energy •Lieb-Liniger gas •Energy expansion in Bogoliubov theory •Tonks-Girardreu gas •Unitary gas •Super Tonks-Girardeau gas •Dynamic form factor •Conclusions

8. WHY LOW-DIMENSIONAL SYSTEMS ARE SPECIAL? • From theoretical point of view low-D systems are interesting as: • • The role of quantum effects is increased as the dimensionality is lowered. • • In particular, in one dimension, quantum fluctuations destroy • - crystalline order (no solid in 1D) • - Bose-Einstein condensation (no BEC in 1D) • • Quantum statitstics are topologically interconnected • - Bose-Fermi mapping from Tonks-Girardeau gas to an ideal Fermi gas. • • There is a number of exactly-solvablemany-body Hamiltonians • •Integrability in Hamiltonian might lead to absence of thermalization, as shown in recent experiments

9. EXACTLY SOLVABLE 1D MODELS

10. CONTENTS •Introduction •Why low-dimensional systems are special? • Density scaling: potential vs kinetic energy •Lieb-Liniger gas •Energy expansion in Bogoliubov theory •Tonks-Girardreu gas •Unitary gas •Super Tonks-Girardeau gas •Dynamic form factor •Conclusions

11. POTENTIAL ENERGY: TYPICAL SCALING WITH DENSITY

12. KINETIC ENERGY: TYPICAL SCALING WITH DENSITY

13. POTENTIAL vs KINETIC ENERGY

14. THREE-DIMENSIONAL GEOMETRY

15. TWO-DIMENSIONAL GEOMETRY

16. ONE-DIMENSIONAL GEOMETRY

17. MEAN-FIELD vs QUANTUM REGIMES

18. CONTENTS •Introduction •Why low-dimensional systems are special? • Density scaling: potential vs kinetic energy •Lieb-Liniger gas •Energy expansion in Bogoliubov theory •Tonks-Girardreu gas •Unitary gas •Super Tonks-Girardeau gas •Dynamic form factor •Conclusions

19. LIEB-LINIGER HAMILTONIAN Nbosonicparticles of mass minteracting with contact δ-function pseudopotentialin a one dimensional system are described by the Lieb-Liniger Hamiltonian:The effective quasi-one-dimensional coupling constant is inversely proportional to the one-dimensional s-wave scattering length • The model solved exactly by • E. H. Lieb and W. Liniger in (1963) • Phys. Rev. 130, 1605 (1963) Elliott H. Lieb

20. BETHE-ANSATZ SOLUTION

21. ENERGY OF LIEB-LINIGER GAS

22. CORRELATION FUNCTIONS

23. ANALYTIC RESUTLS • LIEB-LINIGER GAS(arbitrary value of n|a1D|). Many properties of the system can be obtained from the Bethe ansatz. 1) The ground-state energy  / N = e(n|a1D|)2n2 / 2m • [Lieb, LinigerPhys. Rev. 130, 1605 (1963)] • 2) The expansion of the OBDM g1(z) at zerog1(z) = 1 - 1/2 (e + (n|a1D|) e’(n|a1D|)) |nz|2 + e’(n|a1D|)/6 |nz|3 • [Olshanii, Dunjko PRL. 91, 090401 (2003)] • 3) The value of the pair distribution function g2(z) at zerog2(0) = - (n|a1D|) e’(n|a1D|) /2 • [Gangardt, ShlyapnikovPRL 90, 010401 (2003)] • 4) Relation between short-distance(2-body) physics and equation of state for δ-interaction: analog of Tan’s contact M. Barth, W. Zwerger (2011) • 5) Large-range asymptotic g1(z) are predicted from hydrodynamic theory [Haldane PRL 47, 1840 (1981)] g1(z) = C / |n z|α with the coefficient α = mc/ 2πn = c / 2cFrelated to the speed of sound mc2 =μ/n

24. PAIR DISTRIBUTION FUNCTION Pair distribution function function for different values of the gas parameter. Arrows indicate the value of g2(0) as obtained from the equation of state. At n|a1D|=10-3 the g2(z) function is similar to the one of the a Tonks-Girardeau gas.

25. LOCAL PAIR CORRELATION FUNCTION

26. ONE-BODY DENSITY MATRIX One-body density matrix g1(z)(solid lines), power-law fits (dashed lines). The long-range asymptotic value of OBDM gives the condensate fraction. • i.e. condensate is absent in all cases.

27. TRAPPED TG GAS: EXPERIMENTS

28. ENERGY EXPANSION In the mean-field description the chemical potential is linear with the density: Beyond mean-field terms can be found perturbatively within Bogliubov theory. • for 3D Bose gas the correction was found by K.W. Huang and Nobel prize winners C. N. Yang and T. D. Lee. • for 1D Bose gas the correction can be found from Bogoliubov theory and it coincides with expansion of the Bethe ansatz result: This coincidence is not trivial as condensate fraction is zero!

29. THREE-BODY CORRELATION FUNCTION DEFINITIONThe three-body correlation function (its value at zero) is defined as This function is related to the probability of three-body collisions LIMITING EXPRESSIONS1)In the Tonks-Girardeau regime (n|a1D| 1) the three-body collisions are highly suppressed 2)In the limit of weakly interacting gas (Bogoliubov theory),(n|a1D| 1) 3)Approximation (arbitrary density n|a1D|)

30. THREE-BODY CORRELATION FUNCTION Value at zero of the three-body correlation function g3(0) (black squares) on the log-log scale, TG limit (blue dashed line), Bogoliubov limit (red dashed line), (g2(0))3 (solid black line), experimental result of Tolra et al.’04 (green diamond).

31. THREE-BODY CORRELATION FUNCTION Value at zero of the three-body correlation function g3(0)on the log-log scale, as a function of γ). Solid line V.V. Cheianov et al, JSTAT 8, P08015. (2006). Experimental data: H.-C. Nägerl group, PRL 107, 230404 (2011).

32. STATIC STRUCTURE FACTOR: LIEB LINIGER Static structure factor S(k) for different values of the gas parameter (solid lines). The dashed lines are the corresponding long-wavelength asymptotics.At n|a1D|=10-3 the static structure factor is similar to the one of the a Tonks-Girardeau gas.

33. CONTENTS •Introduction •Why low-dimensional systems are special? • Density scaling: potential vs kinetic energy •Lieb-Liniger gas •Energy expansion in Bogolubov theory •Tonks-Girardreu gas •Unitary gas •Super Tonks-Girardeau gas •Dynamic form factor •Conclusions

34. TONKS-GIRARDEAU GAS

35. BOSE-EINSTEIN STATISTICS

36. FERMI-DIRAC STATISTICS

37. ONE-DIMENSIONAL FERMI GAS

38. MAPPING OF THE WAVE FUNCTION

39. TONKS-GIRARDEAU GAS: WF

40. TG GAS: PDF and STATIC STRUCTURE FACTOR

41. TG GAS: MOMENTUM DISTRIBUTION

42. CONTENTS •Introduction •Why low-dimensional systems are special? • Density scaling: potential vs kinetic energy •Lieb-Liniger gas •Energy expansion in Bogolubov theory •Tonks-Girardreu gas •Unitary gas •Super Tonks-Girardeau gas •Dynamic form factor •Conclusions

43. UNITARY AND TG GASES: 3D

44. UNITARY AND TG GASES: BOSONS IN 3D

45. UNITARY AND TG GASES: BOSONS IN 1D

46. GOING BEYOND TONKS-GIRARDEAU LIMIT

47. TWO-BODY SCATTERING SOLUTION 1.0 Y ( ) x mean-field Gross-Pitaevskii 0.8 strong repulsion 0.6 Tonks-Girardeau > 0 g ® + ¥ 1D g 0.4 1D 0.2 superTonks-Girardeau < 0 g 1D 0.0 -0.2 hard rods > 0 a < 0 a 1D 1D -0.4 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 , [a.u.] x

48. CONFINEMENT INDUCED RESONANCE

49. CONTENTS •Introduction •Why low-dimensional systems are special? • Density scaling: potential vs kinetic energy •Lieb-Liniger gas •Energy expansion in Bogolubov theory •Tonks-Girardreu gas •Unitary gas •Super Tonks-Girardeau gas •Dynamic form factor •Conclusions

50. SUPER TONKS-GIRARDEAU SYSTEM