Bose-Einstein Condensate

Bose-Einstein Condensate Prof. Robert Seiringer, McGill University Notes by Dana Mendelson and Alexandre Tomberg Summer 2010 Contents 1 Quantum Many-Body Systems 1 1.1 Grand-canonical Ensemble . 2 1.2 Bose-Einstein Condensate . 4 2 Interacting Systems 6 2.1 The Criterion of BEC . 6 2.2 Hard-core Lattice Gas . 7 3 Dilute Bose Gas 9 3.1 Model . 10 3.2 Two Particle Case . 10 3.3 Dyson, Lieb and Yngvason Theorem . 11 4 Dilute Trapped Gases 13 1 Quantum Many-Body Systems Classically, we have N particles, and a Hamiltonian H(p1; : : : ; pN ; x1 : : : ; xN ) defined on our phase space. Typically we will have N X p2 H = i + V (x ; : : : x ) 2m 1 N i=1 i P P where V could be i W (xi) (for an external force), i<j W (xi; xj) (for pairwise interaction), or some other more complicated interaction. We will 1 usually assume that X X V = W (xi) + v(jxi − xjj): i i<j In the quantum setting, pi = −irxi and we obtain N X −1 H = ∆ + V (x ; : : : x ) 2m xi 1 N i=1 i which, with the appropriate boundary conditions, is a self-adjoint operator 2 3N on the Hilbert space L (R ). Quantities of Interest: • Ground state energy, E0 = inf spec(H). • Free energy at inverse temperature β, 1 F = − ln Tr e−βH β 2 3N Remark 1.1. All of L (R ) may not be relevant. The property of particles being indistinguishable imposes a certain permutation symmetry on the 2 3N wave function π 2 L (R ). If x1 and x2 are coordinates of two particles of the same kind, (x1; x2;:::) = (x2; x1;:::) [Bosons] (x1; x2;:::) = − (x2; x1;:::) [Fermions]: Therefore, the actual underlying Hilbert space is the subspace Hphys of 2 3N L (R ) corresponding to functions satisfying the above symmetry relations. 1.1 Grand-canonical Ensemble Here, we also average the above quantities over the number of particles in our system. Our Hilbert space is now the Fock space 1 M N 0 F = Hphys; Hphys := C N=0 2 L1 N with Hamiltonian H = N=0 H . We now have that −β(H−µN) X N −βHN X −β(H−µN) TrF e = Z TrHN e = e N N where Z = eβµ is called the fugacity, where µ is the chemical potential, and N is the number operator. Hence 1 PV (µ, β) = Tr e−β(H−µN): β F Exercise 1 (Ideal quantum gas). All particles of the same kind, mass m, N PN (i) H = i=1 H0 , for example with 1 1 H(i) = p2 = − ∆ 0 2m i 2m xi 3 on the cube [0; 1] with Dirichlet boundary conditions. H0 has a discrete spectrum, f"0 ≤ "1 ≤ "2 :::g: N 2 3N 2 3N We wish to calculate PV (µ, β), and here Hphys = Lsym(R ) or Lasym(R ), the space of totally symmetric or totally antisymmetric wave functions. • The difference between bosonic and fermionic gas lies in the Fock space. • The traces are taken over difference eigenvalues (depending on the eigenfunction being symmetric or antisymmetric). N PN Hence the spectrum of H is E = i=1 "ji for ji 2 N[f0g. The occupation number, nj is the number of particles having energy "j, hence we have 1 1 X X E = nj"j; nj = N: j=0 j=0 Summing over all possible occupation numbers is the same as summing over all eigenstates, hence we have N P −βH X −β j nj "j TrHN e = e : P fnj g: nj =N For Fermions, nj 2 f0; 1g, and for Bosons, nj 2 N [ f0g. Now notice that −β P n " Y e j j j = e−βj "j : j 3 P The problem is the \ j nj = N " constraint, but passing to the Grand- Canonical ensemble and absorbing µ into each "j we can drop it. X −βHN −βH TrHN e = TrF e N X Y = e−βnj "j fnj g j Y X = e−β"j n j n ( Q 1 for bosons = j 1−e−β"j Q −β"j j(1 + e ) for fermions Hence ( P −β"j −βH − j ln(1 − e ) for bosons ln TrF e = P −β"j j ln(1 − e ) for fermions 1.2 Bose-Einstein Condensate Consider and ideal gas in a box of sides L with periodic boundary conditions. The spectrum of n2 + n2 + n2 −∆ : (2π)2 x y z L2 3 i~p·~x ~n with ~n = (nx; ny; nz) 2 Z . We have eigenstates e , where ~p = L (2π). The pressure P is given by 1 P = ln Tr e−β(H−µN): L3β F 1 2π 2 2 Let m = 2 , then "j = L ~n − µ hence 2 −1 X −β 2π ~n2+βµ P = ln(1 − e ( L ) ) L3β ~n −1 X 2 = ln(1 − e−β~p +βµ): L3β 2π 3 ~p2( L Z ) In the thermodynamic limit, 1 Z lim P = ln(1 − Zeβp)dp: L!1 3 3 (2π) β R 4 P1 −β"n −β" −1 Recall that n=1 e = (1 − e ) only if " > 0 () µ < 0 () Z < 1. Average particle number: Average of any observable in the Gibbs state ρ = e−βH is Tr(Ae−β(H−µN)) hAi = : Tr(e−β(H−µN)) Notice that d Tr(Ne−β(H−µN)) (ln(Tr(e−β(H−µN)))) = β = βhNi: dµ Tr(e−β(H−µN)) For the ideal Bose gas, X ln(Tr(e−β(H−µN))) = − ln(1 − e−β"j µ) j so 1 @ X e−β("j −µ) X 1 hNi = ln Tr e−β(H−µN) = = : β @µ 1 − e−β(j −µ) eβ(j −µ) − 1 j j | {z } hnj i Hence ( 1 µ " 0 hNi −! : 0 µ # −∞ In the thermodynamic limit, the density hNi Z 1 ρ = −! dp = ρ L 3 β(p2−µ) L R3 e − 1 and Z 1 lim ρ = 2 dp := ρc(β) < 1! µ"0 3 βp R e − 1 What is happening here? If ρ ≤ ρc(β), then µL ! µ < 0 in the thermodynamic limit. But when ρ > ρc(β), µL ! 0! In this case, the limits L ! 1 and µ ! 0 must be taken simultaneously, and the calculation above fails. We then consider the quantity −βµ −1 hn0i = (e − 1) : Whereas β("j −µ) −1 2 3 hnpi = (e − 1) L L ; 5 Figure 1: Particle density at infinite volume depending on µ. 3 3 −1 we want hn0i ∼ L (ρ − ρc). Hence µ ∼ −(β(ρ − ρc)L ) . If ρ > ρc(β), then 3 hn0i lim = ρ − ρc(β); L!1 L which is what we refer to as Bose-Einstein Condensate. The number of particles in the ground state increases as L3 2 Interacting Systems 2.1 The Criterion of BEC Definition 2.1. The one particle reduced density matrix is defined by y hgjγjfi = ha (f)a(g)iβ,µ 2 3 where γ is a positive trace class operator on L (R ). Let us briefly recall what the annihilation and creation operators are. L 2 3N L Given a Fock space F = Lsym(R ) =: HN . Then y a (f): HN −! HN+1 N+1 y 1 X (a (f) )(x1; : : : ; xN+1) = p f(xi) (x1;:::; xî; : : : ; xN+1): N + 1 j=1 6 and p Z (a(f) )(x2; : : : ; xN+1) = N + 1 f(x1) (x1; : : : ; xN+1)dx1: Hence y hfjγjfi = ha (f)a(f)iβ,µ = \ number particles in state f": Then, by definition, BEC is the state when 1 y lim 3 sup ha (f)a(f)i > 0: L!1 L kfk=1 | {z } largest eigenvalue of γ For translation invariant systems of the form N X 2 H = pi + V (x1; : : : ; xN ) i=1 we have that Z hfjγjgi = f(x)γ(x − y)g(y)dxdy with γ(x − y) ≥ 0 8x; y. Then X 1 γ(x − y) = eip(x−y)γ L3 p 2π 3 p2( L Z ) ip(x−y) where e are plain wave eigenfunctions with eigenvalues γp, and γ0 the largest eigenvalue. Hence the largest eigenvalue is associated to a constant eigenfunction! 2.2 Hard-core Lattice Gas For interacting Bose gases, the only known proof of the existence of BEC 3 3 concerns the hard-core lattice gas, i.e. replacing R by Z with particles 2 3 2 3 hopping on the lattice. Our Hilbert space L (R ) then becomes ` (Z ). By discretizing the laplacian, we get the kinetic energy X (∆ )(x) = ( (x + e) − (x)): e where e points to the nearests neighbours on the lattice. Hardcore interac- tions imply very strong but very short range repulsions, like billiard balls, 7 no two can occupy the same spot, but they don't interact when they do not touch. We can rewrite the Hamiltonian as X y X X y H = − axay + u nx(nx − 1) − µ axax hx;yi x | {z } N=number operator where nx are the number of particles at x and hx; yi represents summing over nearest neighbour pairs. We have u > 0 for a repulsive force and in the 2 3 3 L3 hardcore case, u = 1. We have now that ` (Z \ [0; l) ) = C = H1, and 1 M F = Hn: n=0 We write F as O F = Fx; Fx = span(nx): x2[0;L)3 By imposing the hard-core interaction, from nx 2 f0; 1; 2;:::g we obtain nx 2 f0; 1g because there cannot be two particles at the same site.

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