Exactly solvable models and ultracold atoms

Transcription

1 Exactly solvable models and ultracold atoms p. 1/51 Exactly solvable models and ultracold atoms Angela Foerster Universidade Federal do Rio Grande do Sul Instituto de Física talk presented at "Recent Advances in Quantum Integrable Systems" September, Angers, 2012

2 Exactly solvable models and ultracold atoms p. 2/51 OUTLINE 1- INTRODUCTION review some experimental results 2- INTEGRABLE GENERALISED BEC MODELS main emphasis: present the mathematical construction 3- ULTRACOLD ATOMIC FERMI GASES main emphasis: discuss the physical properties 4- CONCLUSIONS outlook of the area

3 1-INTRODUCTION Exactly solvable models and ultracold atoms p. 3/51

4 Exactly solvable models and ultracold atoms p. 4/51 Prediction and experiments: S. N. Bose (1924) Theoretical prediction: A. Einstein ( ) Experimental realization (70 years later) E. A. Cornell et al. (1995 ) 87 Rb W. Ketterle et al. (1995) 23 Na C. C. Bradley et al. (1995) 7 Li

5 Exactly solvable models and ultracold atoms p. 5/51 Direct observation of tunneling and self-trapping: Albiez, M. et al., Phys. Rev. Lett. 95 (2005) In Section 2: Integrable model that describes qualitatively tunneling X self-trapping

6 Donley, E. A., Nature 417 (2002) 529 Exactly solvable models and ultracold atoms p. 6/51 Atom-molecule BEC Since the experimental realization of a BEC using atoms, a significant effort was made to produce a stable BEC in a gas of molecules Zoller, P., Nature 417 (2002) 493

7 Exactly solvable models and ultracold atoms p. 7/51 Fermionic pair condensation D. Jin et al, Nature 424 (2003) The first ultracold Fermi gas of 40 K atoms was created in 1999 by the group of D. Jin at JILA. A breakthrough in the area was the creation of a molecular condensate in an ultracold degenerate Fermi gas by the groups of D. Jin, W. Ketterle and R. Grimm in After that the condensation of fermionic pairs was detected and proved to be a superfluid.

8 Fermions with Polarization Superfluidity can persist with polarization? P = N N N +N 0 P 1 A. Cho, Science 319 (2008) Superfluidity may still occur in a mismatched system and some theories with unusual pairing, exotic phases have been proposed: Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, etc. Experimental studies in ultracold Fermi gases with unequal spin population have been conducted, searching for these new phases (groups at MIT and RICE - 3D) Exactly solvable models and ultracold atoms p. 8/51

9 Section 3: Integrable model that predicts these 3 phases - qualitative and quantitative agreement Exactly solvable models and ultracold atoms p. 9/51 Two-component Fermi gases: 1D experiments Crossed beam optical trap 1D gas with spin imbalance Spin-imbalance in a one-dimensional Fermi gas, R. Hulet et al, Nature 467 (2010) 568 Basic result:observation of a partially polarized core (FFLO-like phase) surrounded by either a completely paired BCS superfluid or a fully polarized Fermi gas, depending on the polarization

10 Section 3: Integrable model that describes trions, pairs Exactly solvable models and ultracold atoms p. 10/51 Three-component Fermi gases Experimental techniques allow to explore 3-component Fermi gases: Jochim et al PRL(2010) and Science(2010); R. Grimm et al Nature(2009); Ueda et al PRL(2009) Theoretically, 3 atoms with different internal quantum states can, depending on the strength of the interactions between them, form: a) bound units of trions or b) some form of Cooper pairs: Wilzeck, Nature(2007) Wilzeck, Nature Physics 2007

11 2- INTEGRABLE MODELS OF BEC Exactly solvable models and ultracold atoms p. 11/51

12 Exactly solvable models and ultracold atoms p. 12/51 Two-site Bose Hubbard Hamiltonian: H = K 8 (N 1 N 2 ) 2 µ 2 (N 1 N 2 ) E J 2 (a 1 a 2+a 2 a 1) N i = a i a i: number of atoms in the well (i = 1, 2) K: atom-atom interaction term µ: external potential E J : tunneling strength G. Milburn et al, Phys. Rev. A 55 (1997) 4318; A. Leggett, Rev. Mod. Phys. 73 (2001) 307 A. Foerster, J. Links and H.Q. Zhou, Class. and Quant. Nonlinear Integ. Systems (2003) The quantum dynamics of the model exhibits tunneling X self-trapping - experiment of Albiez et al The model describes recent experiments of Oberthaler et al on bifurcations and entanglement /12

13 Exactly solvable models and ultracold atoms p. 13/51 Integrability and exact solution: R-matrix: R(u) = b(u) c(u) 0 0 c(u) b(u) 0, b(u) = u u + η Yang-Baxter algebra: c(u) = η u + η R 12 (x y)r 13 (x)r 23 (y) = R 23 (y)r 13 (x)r 12 (x y)

14 Exactly solvable models and ultracold atoms p. 14/51 Monodromy-matrix: T(u) = ( ) A(u) B(u) C(u) D(u) Yang-Baxter algebra: R 12 (u v)t 1 (u)t 2 (v) = T 2 (v)t 1 (u)r 12 (u v) Realization of the monodromy matrix: L(u) = π(t(u)) = L a 1(u + w)l a 2(u w) L a i (u) = ( ) u + ηni a i a i η 1 i = 1, 2

15 Exactly solvable models and ultracold atoms p. 15/51 Transfer matrix: τ(u) = π(tr(t(u))) = π(a(u) + D(u)) Integrability: [τ(u),τ(v)] = 0 [H,τ(v)] = 0

16 Exactly solvable models and ultracold atoms p. 16/51 Hamiltonian and transfer matrix: H = κ (τ(u) 14 ) (τ (0)) 2 uτ (0) η 2 + w 2 u 2 with the identification: K 4 = κη2 2, µ 2 = κηw, E J 2 = κ H = K 8 (N 1 N 2 ) 2 µ 2 (N 1 N 2 ) E J 2 (a 1 a 2+a 2 a 1)

17 Exactly solvable models and ultracold atoms p. 17/51 Applying the algebraic Bethe ansatz method: Energy: E = κ(η 2 N i=1 ( 1 + η ) η2 N 2 v i w 4 wηn η 2 ) Bethe Ansatz Equations: η 2 (v 2 i w 2 ) = N j i v i v j η v i v j + η the solutions of the BAE can be used to study the Quantum Phase Transitions of the model D. Rubeni, A. Foerster, E. Mattei and I. Roditi, NPB 2012

18 INTEGRABLE GENERALISED MODELS: Basic idea: We can construct integrable generalised models in the BEC context exploring different representations of some algebra, such as the gl(n) algebra and gl(m/n) superalgebra. Models for homo and hetero atom-molecular BEC Three-coupled BEC model... A. Foerster, J. Links and H.Q. Zhou, Class. and Quant. Nonlinear Integ. Systems (2003); A. Foerster and E. Ragoucy, Nuclear Phys. B777 (2007) 373; A. Tonel, G. Santos, A. Foerster, I. Roditi, Z. Santos, Physical Review A79 (2009) ; Exactly solvable models and ultracold atoms p. 18/51

19 3 - ULTRACOLD ATOMIC FERMI GASES Exactly solvable models and ultracold atoms p. 19/51

20 Exactly solvable models and ultracold atoms p. 20/51 1D 2-component attractive Fermi gas with polarization: Hamiltonian H = 2 2m N i=1 2 x 2 i + g 1D 1 i<j N δ(x i x j ) H 2 (N N ) N spin 1/2 fermions of mass m constrained by PBC to a line of length L H: external field g 1D = 2 c: 1D interaction strength: m attractive for g 1D < 0 and repulsive for g 1D > 0 HERE: ATTRACTIVE REGIME γ c (n = N ) : dimensionless interaction; n L

21 Exactly solvable models and ultracold atoms p. 21/51 Bethe ansatz method C.N. Yang, PRL 19(1967)1312; M. Gaudin, Phys. Lett. 24 (1967) 55 Energy: BAE: N l=1 E = 2 2m exp(ik j L) = N kj, 2 j=1 M l=1 k j Λ l + ic/2 k j Λ l ic/2 M Λ α k l + ic/2 Λ α k l ic/2 = Λ α Λ β + ic Λ α Λ β ic β=1 {k j, j = 1,..., N} are the quasimomenta for the fermions; {Λ α, α = 1,..., M} are the rapidities for the internal spin degrees of freedom The solutions to the BAE give the GS properties and provide a clear pairing signature

22 Exactly solvable models and ultracold atoms p. 22/51 BA-root configuration for the GS BA configuration of quasimomenta k in the complex plane for the GS: for a given polarization, the system is described by bound states ("Cooper pairs") and unpaired fermions Weak Regime E L 2 n 3 2m γ «2 (1 P2 ) + π π2 4 P2 Strong Regime E L 2 n 3 2m j γ2 (1 P) 4 + P3 π «4(1 P) + π2 (1 P) 3 γ P γ + 4P «ff γ

23 Exactly solvable models and ultracold atoms p. 23/51 Thermodynamical Bethe Ansatz - TBA elegant method to study thermodynamical properties convenient formalism to analyse QPT at T = C. N. Yang and C. P. Yang, "Yang-Yang approach" 1972 M. Takahashi, string hypothesis thermodynamic limit: L, N with N/L finite: consider a distribution function for the BA-roots; the equilibrium state is determined by the condition of minimizing the Gibbs free energy: G = E HM z µn TS

24 Exactly solvable models and ultracold atoms p. 24/51 TBA - equations: set of coupled nonlinear integral equation: ǫ b (k) = 2(k 2 µ 1 4 c2 ) + Ta 2 ln(1 + e ǫb (k)/t ) + Ta 1 ln(1 + e ǫu (k)/t ) ǫ u (k) = k 2 µ 1 2 H + Ta 1 ln(1 + e ǫb (k)/t ) T X n=1 a n ln(1 + η 1 n (k)) ln η n (λ) = nh T + a n ln(1 + e ǫu (λ)/t ) + X n=1 T nm ln(1 + η 1 m (λ)) The dressed energies: ǫ b (k) := T ln(σ h (k)/σ(k)) and ǫ u (k) := T ln(ρ h (k)/ρ(k)) for paired and unpaired fermions; the function η n (λ) := ξ h (λ)/ξ(λ) is the ratio of string densities. The Gibbs free energy per unit length (Takahashi s book): G = T π Z dk ln(1 + e ǫb (k)/t ) T 2π Z dk ln(1 + e ǫu (k)/t )

25 Exactly solvable models and ultracold atoms p. 25/51 Limit T 0: dressed energy equations ǫ b (Λ) = ǫ u (k) = Z 2(Λ 2 µ c2 B 4 ) a 2 (Λ Λ )ǫ b (Λ )dλ B Z Q Q (k 2 µ H 2 ) Z B B a 1 (Λ k)ǫ u (k)dk a 1 (k Λ)ǫ b (Λ)dΛ a m (x) = 1 2π m c (m c/2) 2 + x 2, ǫb (±B) = ǫ u (±Q) = 0 The Gibbs free energy per unit length at zero temperature is given by G(µ, H) = 1 π Z B B ǫ b (Λ)dΛ + 1 2π Z Q Q ǫ u (k)dk From the Gibbs free energy per unit length we have the relations G(µ, H)/ µ = n, G(µ, H)/ H = m z = np/2

26 Exactly solvable models and ultracold atoms p. 26/51 Strong attraction POLARIZATION: H 2 2 2m c µu µ b µ u 2 n 2 π 2 2m µ b 2 n 2 π 2 j P 2 + (1 P)(49P2 2P + 1) 12 γ + (1 P)2 (93P 2 + 2P + 1) (1 P) 8γ 2 ˆ1441π γ 3 P P 4 324π 2 P P P π 2 P 2 120P 4π 2 P 30 + π 2 2m j (1 P) (3P + 1)(6P2 3P + 1) 12 γ + (1 P)(5 + 17P P P 3 ) 64γ γ 3 ˆ15(1 + 2P 2 ) P π 2 P 2 180π 2 P π 2 P 4 420π 2 P P 4 π P P 5 CRITICAL FIELDS: H c1 2 n 2 2m H c2 2 n 2 2m γ 2 2 π2 8 γ π γ 2 1 ««γ 3 ««1 4 3 γ + 16π2 15 γ 3

27 Exactly solvable models and ultracold atoms p. 27/51 Phase diagram and schematic representation: X-W. Guan et al, PRB2007; J. He, A. Foerster, X-W. Guan, M. Batchelor, NJP2009

28 Exactly solvable models and ultracold atoms p. 28/51 Magnetization M z 2(H H c1) nπ γ + 11 «2γ 2 M z n 2 1 H c2 H 4n 2 π 2 (1 + 4 γ + 12 «γ 2 ) The analytical results (dashed lines) coincide well with the numerical solution (solid lines)

29 Exactly solvable models and ultracold atoms p. 29/51 For T=0: The TBA predicts 3 phases in the strong coupling regime: a superfluid phase,a fully polarized phase and a partially polarized phase, in agreement with experiments For low T: Theoretical predictions from low-t TBA + LDA in the strong coupling regime are in quantitative agreement with experimental measurements of density profiles of a 2-spin mixture of ultracold 6 Li atoms in 1D tubes Liao et al. Nature 2010; the black (blue) circles are the density of fermions in the state 1 > ( 2 >) and the red squares the difference between these two states; the solid lines are the predictions from TBA+LDA; the polarizations are 0.015, 0.055, 0.10 and 0.33.

30 II-Three-component attractive Fermi gas: Hamiltonian H = 2 2m N i=1 2 x 2 i + g 1D 1 i<j N δ(x i x j ) Energy and BAE N l=1 E = 2 2m N kj 2, exp(ik jl) = j=1 M1 Λ α k l + ic/2 Λ α k l ic/2 = M 1 β=1 β=1 Λ α Λ β + ic Λ α Λ β ic M 1 l=1 M 2 β=1 M2 λ µ Λ β + ic/2 λ µ Λ β ic/2 = k j Λ l + ic/2 k j Λ l ic/2, Λ α λ β ic/2 Λ α λ β + ic/2, β=1 λ µ λ β + ic λ µ λ β ic B. Sutherland PRL 20(68)98; M. Takahashi, Prog. Theor. Phys. 44(70)899 Exactly solvable models and ultracold atoms p. 30/51

31 Phase diagram in the strong regime: ( c = 10) Exhibits a rich scenario with new phases: TRIONS: three-body bound states Unpaired phase A, pairing phase B, trion phase C and 4 different mixtures of these phases; Above: unpaired fermion density n 1, pair density n 2 and GS energy vs the fields H 1,H 2 X-W. Guan et al,prl100(2008); M.T.Batchelor, A. Foerster,X-W. Guan and Kuhn, JSTAT(2010)P12014 Exactly solvable models and ultracold atoms p. 31/51

32 Exactly solvable models and ultracold atoms p. 32/51 III-Few-fermion system in a 1D harmonic trap: H = 2 2m N i=1 2 x 2 i + g 1D 1 i<j N δ(x i x j ) N i=1 mω 2 x i 2 Recent experiments: Jochim et al, Science2011;PRL2012 They prepared and controlled with high precision a system of 2 atoms of 6 Li in a 1D harmonic trap: the GS-energy was measured as a function of the interaction strength and compared with good agreement with the analytical result: -Th. Busch et al, Found.Phys. 1998; N=2 Alternative approach: combining the Bethe ansatz with the variational principle, we calculate the GS-energy of the system with good agreement with the analytical result. the central part of the trial function is the BA-wavefunction for the integrable model

33 Exactly solvable models and ultracold atoms p. 33/51 Comparison: BA + variational X analytical solution Energies for the GS of the relative motion for 2 distinct fermions: the analytical results (black dotted line) are compared to results obtained by the combination of the BA with the variational principle (red straight line); D. Rubeni, A. Foerster and I. Roditi, preprint 2012 The BA+variational approach has the potentiality to be extended to the 3 fermions case.

34 Exactly solvable models and ultracold atoms p. 34/51 Other exactly solvable model of interest: Spin-1 bosons with repulsive density-density and antiferromagnetic spin-exchange interactions: H = 2 2m N i=1 2 x 2 i + i<j [c 0 + c 2 S i S j ]δ(x i x j ) T. L. Ho PRL1998 S i is the spin-1 operator with z-component (m s = 1, 0, 1) N bosons with mass m in 3 different hyperfine states Integrable for c = c 0 = c 2 Important contributions: J.Cao et al, EPL 2007, F.Essler et al, JSTAT2009

35 Exactly solvable models and ultracold atoms p. 35/51 Theory: equation of state, phase diagram, density profiles in a 1D harmonic trap (TBA + LDA),... C.Kuhn, X.W Guan, A.Foerster and M.Batchelor PRA(2012); arxiv: Experiment: N. Robbins et al, Atom laser Group, ANU

36 Exactly solvable models finding their way into the lab: (everything old is new again...) In the past few years striking experimental achievements in trapping and cooling atoms in 1D have provided remarkable realisations of exactly solved models in the lab. Some examples: the Lieb-Liniger Bose gas T. Kinoshita et al Science 2004, PRL 2005, Nature 2006; A. van Amerongen et al PRL2008; T. Kitagawa et al PRL 2010; J. Armijo et al PRL 2010 the super Tonks-Girardeau gas E. Haller et al Science 2009 the degenerate spin-1/2 Fermi gas Y. Liao et al Nature 2010; S. Jochim et al, Science 2011,PRL 2012: Deterministic preparation of few-fermion system; 2 fermions in a 1D harmonic trap the two-component spinor Bose gas J. van Druten et al arxiv: Exactly solvable models and ultracold atoms p. 36/51

37 Exactly solvable models and ultracold atoms p. 37/51 Concluding remarks: Exactly solvable models provide a precise description of the quantum phases and thermodynamics which are applicable to experiments with ultracold atoms confined to 1D tubes. It is clear that the Bethe Ansatz will continue to prosper as a valuable tool in the description of ultracold atoms.

38 Exactly solvable models and ultracold atoms p. 38/51 Collaborators Prof. Murray T. Batchelor, ANU-Australia Prof. Xiwen Guan, ANU-Australia Prof. Jon Links, UQ-QLD-Australia Prof. Eric Ragoucy, LAPTH-France Prof. Itzhak Roditi, CBPF-Brazil Dr. Gilberto S. Filho, CBPF-Brazil Dr. Eduardo Mattei, UQ-QLD-Australia Carlos Kuhn, ANU-Australia and UFRGS-Brazil Diefferson Lima, UFRGS-Brazil Jardel Cestari, UFRGS-Brazil David Carvalho, UFRGS-Brazil

39 THANK YOU! Exactly solvable models and ultracold atoms p. 39/51

40 Schematic representation of the normalized probability density, Ψ(x) 2 in the relative coordinates system. The variational parameter L may be used to delineate three regions according to the boundaries with respect to the harmonic potential. Exactly solvable models and ultracold atoms p. 40/51 Variational principle: E GS ψ H rel ψ, ψ ψ Ansatz for the trial wavefunction: ψ I = e α(x+l)2 ψ II ( L) < x < L ψ II = (e ikl e ikx + e ikx )Θ(x) ψ = +(e ikl e ikx + e ikx )Θ( x) L < x < +L ψ III = e α(x L)2 ψ II (+L) +L < x < + (1) ψ(x) 2 -L 0 +L x Region I Region II Region III

41 bosons F, and a mixed phase of the pairs and unpaired bosons. V stands for the vacuum. Exactly solvable models and ultracold atoms p. 41/51 Phase diagram: M spin +1 spin 0 spin -1 µ/ε b 0 S V F H/ε b Phase diagram of the spin-1 Bose gas at T = 0 in the plane H µ in the strong regime. The model exhibits three quantum phases: spin-singlet paired bosons S, ferromagnetic spin-aligned

42 Exactly solvable models and ultracold atoms p. 42/51 Equation of State: p 1 = T 3 2 ( 4π 2 2m p 2 = T 3 2 ( 2π 2 where f i s = Li s 2m e A i T )1 2 )1 2 f f ( 1 p 1 2m c p ) 2 2m, c 3 2 ( p 1 2m 125 c p ) 2 2m, c 3 2 «, with i = 1, 2 and A 1 = µ + H + 2p 1 c 16p 2 5 c A 2 = 2 2m + Te H T e K 4 I 0 K 4 c µ 32p 1 5 c «+ p 2 3 c T «3 c π f « π f2 5, 2 2m T c m 3968 « π f « π f2 5. 2

43 Exactly solvable models and ultracold atoms p. 43/51 Density Profiles The equation of state can be reformulated within the local density approximation (LDA) by a replacement µ(x) = µ(0) 1 2 mω2 xx 2 in which x is the position and ω x is the trapping frequency, the total particle number and the polarization are given by: N a 2 xc 2 = P = ñ( x)d x, ñ 1 ( x)d x/(n/(a 2 xc 2 )).

44 n (x)/(n 1/2 a x ) Exactly solvable models and ultracold atoms p. 44/ (a) x/(n 1/2 a x ) (b) x/(n 1/2 a x ) n 1 n 2 n 1 +n 2 The density distribution profiles for trapped bosons at zero temperature with N/a 2 xc for (a) P 12.22% (or h = 0.49) and (b) P 22.33% (or h = 0.51); a x = mw x is the harmonic characteristic radius

45 Exactly solvable models and ultracold atoms p. 45/51 Appendix:Bethe ansatz A solvable or integrable quantum many-body system is one in which N-particle wave function may be explicitly constructed. In general, N! plane waves are N-fold products of individual exponential phase factors e ik ix j, where the N distinct wave numbers, k i, are permuted among the N distinct coordinates, x j. Each of the N! plane waves have an amplitude coefficient in each of regions. For example, in the domain 0 < x Q1 < x Q2 <... < x QN < L, the wave function is written as ψ = X P A σ1...σ N (P 1,..., P N Q 1,..., Q N )exp i(k P1 x Q k P N x QN ) Continuity: ψxqi =x Qj = ψ xqi =x + Qj Schrödinger equation: Hψ = Eψ two-body scattering relation:a σ1...σ N (P i P j Q i Q j ) = [Y ij ] σ 1...σ N σ 1...σ N A σ 1...σ N (P jp i Q i Q j ) boundary conditions: ψ(x1,..., x i,..., x N ) = ψ(x 1,..., x i + L,..., x N )

46 BA-root configuration for the GS in the complex plane N = 29 with N 1 = 6, N 2 = 4 and N 3 = 5 Exactly solvable models and ultracold atoms p. 46/51

47 Exactly solvable models and ultracold atoms p. 47/51 Experimental Quantities N1 120, is the number of atoms per 1D tube in state 1 >; t k B 17nK here t is tunnelling rate; ε F kb 1.2µK, where ε F = N 1 ω z is 1D Fermi energy; ω /ω z = 1000 where ω z and ω are the axial and transverse confinement frequencies of an individual tube; T 175nK. It is necessary that ε F > t and T > t to ensure that Fermi surface is one dimensional and avoid inter-tube coupling respectively.

48 Exactly solvable models and ultracold atoms p. 48/51 Three-coupled BEC model H = Ω 2 (a 2 a 1 + a 1 a 2 + a 2 a 3 + a 3 a 2) +Ω (a 1 a 3 + a 3 a 1) + µn 1 + µn 3 + µ 2 n 2, (1): left well (2): middle well (3): right well Ω: tunneling between the left and the right wells Ω 2 : left-middle and middle-right tunneling µ 2, µ: external potentials. A. Foerster and E. Ragoucy, Nuclear Phys. B777 (2007) 373;

49 Exactly solvable models and ultracold atoms p. 49/51 Appendix: Quantum dynamics: Temporal operator U: determines the time evolution of any physical quantity U = NX n=0 e iλ nt ψ n ψ n {λ n } ; { ψ n }: eigenvalues and eigenvectors of H Temporal evolution of any state: ψ(t) = U φ = P N n=0 a ne iλ nt ψ n, a n = ψ n φ and φ : initial state Expectation value of any operator A A = ψ(t) A ψ(t) Imbalance population A = (N 1 N 2 )/N Plot the time evolution of the expectation value of the imbalance population for different ratios of the coupling K/E J

50 Exactly solvable models and ultracold atoms p. 50/51 Appendix: Dynamical regimes: N = 100 N = <N 1 -N 2 >/N t K E J = 1 1 N, 2 N, 1, N, N2

51 Exactly solvable models and ultracold atoms p. 51/ <N 1 -N 2 >/N t t K E J = 1 N, 2 N, 3 N, 4 N, 5 N, 10 N, 50 N, 1 λ t = 2 K E J = 4 N, ; µ = 0 Tunneling X Self-trapping