Superfluidity versus Bose-Einstein Condensation

Transcription

1 Superfluidity versus Bose-Einstein Condensation Jakob Yngvason University of Vienna IHES, Bures-sur-Yvette, March 19, 2015 Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 1 / 42

2 Concepts Bose Einstein Condensation: Macroscopic occupation of a single 1-particle state Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 2 / 42

3 Concepts Bose Einstein Condensation: Macroscopic occupation of a single 1-particle state Superfluidity can mean two things: Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 2 / 42

4 Concepts Bose Einstein Condensation: Macroscopic occupation of a single 1-particle state Superfluidity can mean two things: Flow without friction (non-equilibrium phenomenon) Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 2 / 42

5 Concepts Bose Einstein Condensation: Macroscopic occupation of a single 1-particle state Superfluidity can mean two things: Flow without friction (non-equilibrium phenomenon) Non-classical response to an infinitesimal boost or rotation (equilibrium phenomenon) Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 2 / 42

6 Some history 1924 Einstein: Theoretical prediction of BEC in an ideal gas Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 3 / 42

7 Some history 1924 Einstein: Theoretical prediction of BEC in an ideal gas 1938 Kapitza; Allen & Misner: Superfluidity in liquid He 4 Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 3 / 42

8 Some history 1924 Einstein: Theoretical prediction of BEC in an ideal gas 1938 Kapitza; Allen & Misner: Superfluidity in liquid He London: Suggests connection to BEC Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 3 / 42

9 Some history 1924 Einstein: Theoretical prediction of BEC in an ideal gas 1938 Kapitza; Allen & Misner: Superfluidity in liquid He London: Suggests connection to BEC 1941 Landau: Two fluid model; elementary excitations. No mention of BEC Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 3 / 42

10 Some history 1924 Einstein: Theoretical prediction of BEC in an ideal gas 1938 Kapitza; Allen & Misner: Superfluidity in liquid He London: Suggests connection to BEC 1941 Landau: Two fluid model; elementary excitations. No mention of BEC 1946 Bogoliubov: Seminal work on microscopic theory of Bose gases and superfluidity; assumes BEC Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 3 / 42

11 Some history 1924 Einstein: Theoretical prediction of BEC in an ideal gas 1938 Kapitza; Allen & Misner: Superfluidity in liquid He London: Suggests connection to BEC 1941 Landau: Two fluid model; elementary excitations. No mention of BEC 1946 Bogoliubov: Seminal work on microscopic theory of Bose gases and superfluidity; assumes BEC 1946 Andronikashvili: Reduction of moment of inertia Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 3 / 42

12 Some history 1924 Einstein: Theoretical prediction of BEC in an ideal gas 1938 Kapitza; Allen & Misner: Superfluidity in liquid He London: Suggests connection to BEC 1941 Landau: Two fluid model; elementary excitations. No mention of BEC 1946 Bogoliubov: Seminal work on microscopic theory of Bose gases and superfluidity; assumes BEC 1946 Andronikashvili: Reduction of moment of inertia 1950 s 1960 s Lee, Huang, Yang,...; Dyson; Lieb; Gross; Pitaevskii: Theoretical work on interacting Bose gas; Lieb-Liniger model; Gross-Pitaevskii equation Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 3 / 42

13 Some history 1924 Einstein: Theoretical prediction of BEC in an ideal gas 1938 Kapitza; Allen & Misner: Superfluidity in liquid He London: Suggests connection to BEC 1941 Landau: Two fluid model; elementary excitations. No mention of BEC 1946 Bogoliubov: Seminal work on microscopic theory of Bose gases and superfluidity; assumes BEC 1946 Andronikashvili: Reduction of moment of inertia 1950 s 1960 s Lee, Huang, Yang,...; Dyson; Lieb; Gross; Pitaevskii: Theoretical work on interacting Bose gas; Lieb-Liniger model; Gross-Pitaevskii equation 1995 Ketterle; Cornell, Wieman: Experimental realization of BEC in dilute, trapped gases Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 3 / 42

14 Definitions 1-body density matrix of a (pure or mixed) many-body state : ρ (1) (x, x ) = a (x)a(x ) = N i ψ i (x)ψ i (x ), i=0 N 0 N 1, N i = N. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 4 / 42

15 Definitions 1-body density matrix of a (pure or mixed) many-body state : ρ (1) (x, x ) = a (x)a(x ) = N i ψ i (x)ψ i (x ), i=0 N 0 N 1, N i = N. BEC means by definition that the condensate fraction, N 0 /N, is O(1). Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 4 / 42

16 Definitions 1-body density matrix of a (pure or mixed) many-body state : ρ (1) (x, x ) = a (x)a(x ) = N i ψ i (x)ψ i (x ), i=0 N 0 N 1, N i = N. BEC means by definition that the condensate fraction, N 0 /N, is O(1). The superfluid mass density ρ s (at rest) is defined through the response of the free energy to a small boost v: F (v) = F (0) mn (ρ s/ρ)v 2 + o(v 2 ) Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 4 / 42

17 The boost is mathematically implemented by the substitution p i p i mv in the Hamiltonian, assuming periodic boundary conditions in the direction e of v = ve with period Λ, say. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 5 / 42

18 The boost is mathematically implemented by the substitution p i p i mv in the Hamiltonian, assuming periodic boundary conditions in the direction e of v = ve with period Λ, say. Equivalently one can consider the original Hamiltonian but with twisted boundary conditions: Ψ(..., x i 1, Λe, x i,... ) = e iϕ Ψ(..., x i 1, 0, x i,... ) with ϕ = mvλ/. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 5 / 42

19 The boost is mathematically implemented by the substitution p i p i mv in the Hamiltonian, assuming periodic boundary conditions in the direction e of v = ve with period Λ, say. Equivalently one can consider the original Hamiltonian but with twisted boundary conditions: Ψ(..., x i 1, Λe, x i,... ) = e iϕ Ψ(..., x i 1, 0, x i,... ) with ϕ = mvλ/. The free energy as a function of v is periodic with period ϕ. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 5 / 42

20 Convenient experimental realization: Container in the form of a thin annulus, radius R, thickness d R. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 6 / 42

21 Convenient experimental realization: Container in the form of a thin annulus, radius R, thickness d R. A boost corresponds to a rotation of the walls of the container: v RΩ with Ω the angular velocity. Since Λ = 2πR the phase is ϕ = 2πmR 2 Ω/ Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 6 / 42

22 Reduction of moment of inertia Hess-Fairbank experiment (1967): Rotate slowly a thin annulus of liquid He 4 at angular velocity Ω < Ω c /mr 2. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 7 / 42

23 Reduction of moment of inertia Hess-Fairbank experiment (1967): Rotate slowly a thin annulus of liquid He 4 at angular velocity Ω < Ω c /mr 2. 1) T > T λ : Liquid rotates classically with angular momentum L = I classical Ω, I classical = NmR 2 Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 7 / 42

24 Reduction of moment of inertia Hess-Fairbank experiment (1967): Rotate slowly a thin annulus of liquid He 4 at angular velocity Ω < Ω c /mr 2. 1) T > T λ : Liquid rotates classically with angular momentum L = I classical Ω, I classical = NmR 2 2) T < T λ : Liquid rotates with reduced moment of inertia I(T ) = (ρ n /ρ)i classical < I classical, I(T = 0) = 0. Only the normal fluid rotates while the superfluid remains stationary in the laboratory frame. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 7 / 42

25 Metastable superfluid flow 1) Rotate rapidly, Ω > Ω c 1/mR 2, at T > T λ : Liquid rotates classically with angular momentum L = I classical Ω. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 8 / 42

26 Metastable superfluid flow 1) Rotate rapidly, Ω > Ω c 1/mR 2, at T > T λ : Liquid rotates classically with angular momentum L = I classical Ω. 2) Continue rotating but cool to T < T λ : Liquid still rotates classically. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 8 / 42

27 Metastable superfluid flow 1) Rotate rapidly, Ω > Ω c 1/mR 2, at T > T λ : Liquid rotates classically with angular momentum L = I classical Ω. 2) Continue rotating but cool to T < T λ : Liquid still rotates classically. 3) Stop rotating the annulus. Liquid keeps rotating but (after a short decay time) with reduced moment of inertia I s = (ρ s /ρ)i classical Only the superfluid rotates in the lab frame, the normal component is stationary. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 8 / 42

28 Metastable superfluid flow 1) Rotate rapidly, Ω > Ω c 1/mR 2, at T > T λ : Liquid rotates classically with angular momentum L = I classical Ω. 2) Continue rotating but cool to T < T λ : Liquid still rotates classically. 3) Stop rotating the annulus. Liquid keeps rotating but (after a short decay time) with reduced moment of inertia I s = (ρ s /ρ)i classical Only the superfluid rotates in the lab frame, the normal component is stationary. The superfluid flow is metastable but with a huge lifetime due to energy barriers in a macroscopic system with a repulsive interaction. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 8 / 42

29 Is there a connection between BEC and SF? It is certainly not true that the condensate fraction and superfluid fraction are the same: Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 9 / 42

30 Is there a connection between BEC and SF? It is certainly not true that the condensate fraction and superfluid fraction are the same: Close to T = 0 liquid He 4 is almost completely superfluid, but the condensate fraction is only about 10%. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 9 / 42

31 Is there a connection between BEC and SF? It is certainly not true that the condensate fraction and superfluid fraction are the same: Close to T = 0 liquid He 4 is almost completely superfluid, but the condensate fraction is only about 10%. A 1D hard core Bose gas at T = 0 has no BEC but is superfluid. Same holds for a 2D Kosterlitz-Thouless superfluid. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 9 / 42

32 Is there a connection between BEC and SF? It is certainly not true that the condensate fraction and superfluid fraction are the same: Close to T = 0 liquid He 4 is almost completely superfluid, but the condensate fraction is only about 10%. A 1D hard core Bose gas at T = 0 has no BEC but is superfluid. Same holds for a 2D Kosterlitz-Thouless superfluid. What about the converse, i.e., does BEC imply superfluidity? Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 9 / 42

33 A rigorous study of this question is hampered by the fact that it is notoriously difficult to prove BEC for systems with interactions. The only known examples so far are: Hard core Bose lattice gas at exactly half filling (Dyson, Lieb Simon, 1978; Lieb Kennedy and Shastry 1988) Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 10 / 42

34 A rigorous study of this question is hampered by the fact that it is notoriously difficult to prove BEC for systems with interactions. The only known examples so far are: Hard core Bose lattice gas at exactly half filling (Dyson, Lieb Simon, 1978; Lieb Kennedy and Shastry 1988) Ground state of a dilute Bose gas with repulsive interaction in the Gross-Pitaevskii limit (Lieb, Seiringer 2002) Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 10 / 42

35 A rigorous study of this question is hampered by the fact that it is notoriously difficult to prove BEC for systems with interactions. The only known examples so far are: Hard core Bose lattice gas at exactly half filling (Dyson, Lieb Simon, 1978; Lieb Kennedy and Shastry 1988) Ground state of a dilute Bose gas with repulsive interaction in the Gross-Pitaevskii limit (Lieb, Seiringer 2002) Complete superfluidity in the ground state has also been proved in the GP limit in the translationally (rotationally) invariant situation (Lieb, Seiringer, JY, 2003). Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 10 / 42

36 On the other hand its has long been claimed (and supported by numerical and heuristic arguments) that an external random potential may strongly reduce superfluidity. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 11 / 42

37 On the other hand its has long been claimed (and supported by numerical and heuristic arguments) that an external random potential may strongly reduce superfluidity. A state the has BEC but no superfluidity has been called a Bose Glass (S. Yukalov). Until recently there have been no mathematical results on the existence of such states. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 11 / 42

38 On the other hand its has long been claimed (and supported by numerical and heuristic arguments) that an external random potential may strongly reduce superfluidity. A state the has BEC but no superfluidity has been called a Bose Glass (S. Yukalov). Until recently there have been no mathematical results on the existence of such states. The effect of a random potential on BEC and superfluidity can be investigated mathematically in a simple model: R.Seiringer, J. Yngvason, V. Zagrebnov, Disordered Bose Einstein Condensates with Interaction in One Dimension, J. Stat. Mech. P11007, 2012 M. Könenberg, T. Moser, Robert Seiringer, and J.Yngvason, Superfluid Behavior of a Bose Einstein Condensate in a Random Potential, New J. Phys (2015). Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 11 / 42

39 The Model The model is a gas of 1D bosons with contact interaction (Lieb-Linger model) on the unit interval but with an additional external random potential V ω. The Hamiltonian on the Hilbert space L 2 ([0, 1], dz) sn is H = N ( 2 zi + V ω (z i ) ) + γ δ(z i z j ) N i=1 i<j with γ 0 and periodic boundary conditions. This can be regarded as a model of a gas in a thin annulus. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 12 / 42

40 The Model The model is a gas of 1D bosons with contact interaction (Lieb-Linger model) on the unit interval but with an additional external random potential V ω. The Hamiltonian on the Hilbert space L 2 ([0, 1], dz) sn is H = N ( 2 zi + V ω (z i ) ) + γ δ(z i z j ) N i=1 i<j with γ 0 and periodic boundary conditions. This can be regarded as a model of a gas in a thin annulus. The random potential will be taken to be V ω (z) = σ δ(z z ω j ) with σ 0 independent of the random sample ω while the obstacles } are Poisson distributed with density ν 1. {zj ω Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 12 / 42

41 Parameters Besides N, the model has three parameters: Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 13 / 42

42 Parameters Besides N, the model has three parameters: γ: Strength of the interaction Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 13 / 42

43 Parameters Besides N, the model has three parameters: γ: Strength of the interaction ν: Density of obstacles Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 13 / 42

44 Parameters Besides N, the model has three parameters: γ: Strength of the interaction ν: Density of obstacles σ: Strength of the random potential Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 13 / 42

45 Parameters Besides N, the model has three parameters: γ: Strength of the interaction ν: Density of obstacles σ: Strength of the random potential In the N limit the ground state energy and the wave function of the condensate is described by a Gross Pitaevskii (GP) energy functional. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 13 / 42

46 Parameters Besides N, the model has three parameters: γ: Strength of the interaction ν: Density of obstacles σ: Strength of the random potential In the N limit the ground state energy and the wave function of the condensate is described by a Gross Pitaevskii (GP) energy functional. Moreover, the energy becomes deterministic if the parameters satisfy ν 1, γ ν (ln ν) 2, σ ν 1 + ln (1 + ν 2 /γ). We shall refer to these conditions as the standard conditions and assume them throughout. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 13 / 42

47 Main Results In the whole parameter range there is complete BEC. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 14 / 42

48 Main Results In the whole parameter range there is complete BEC. If γ ν 2 the superfluid fraction is arbitrarily small, i.e., it goes to zero under the standard conditions. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 14 / 42

49 Main Results In the whole parameter range there is complete BEC. If γ ν 2 the superfluid fraction is arbitrarily small, i.e., it goes to zero under the standard conditions. The same holds for ν 2 γ ν 4 provided σ (γ/ν 2 ) 2 γ 1/2. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 14 / 42

50 Main Results In the whole parameter range there is complete BEC. If γ ν 2 the superfluid fraction is arbitrarily small, i.e., it goes to zero under the standard conditions. The same holds for ν 2 γ ν 4 provided σ (γ/ν 2 ) 2 γ 1/2. If γ (σν) 2 there is complete superfluidity, i.e., the superfluid fraction tends to 1. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 14 / 42

51 14 12 log Σ log Ν log Γ log Ν Figure : Red: Absence of superfluidity. Green: Complete superfluidity. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 15 / 42

52 The general heuristic picture is that Strong repulsive interaction between the particles (large γ) tends to make the density uniform and favours superfluidity. Strong randomness (large ν and σ) leads to fractionation of the density that is unfavorable for superfluidity. It is remarkable, however, that BEC survives the fractionation of the density, i.e. the long range corelations prevail although superfluidity may be strongly suppressed. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 16 / 42

53 BEC Bose-Einstein condensation in the ground state holds in the GP limit, where N and γ is fixed, or does not grow too fast with N. This holds for an arbitrary positive potential V. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 17 / 42

54 BEC Bose-Einstein condensation in the ground state holds in the GP limit, where N and γ is fixed, or does not grow too fast with N. This holds for an arbitrary positive potential V. The wave function of the condensate is the minimizer ψ 0 of the Gross Pitaevskii (GP) energy functional E GP [ψ] = 1 with the normalization 1 0 ψ 2 = 1. 0 ( ψ (z) 2 + V (z) ψ(z) 2 + γ 2 ψ(z) 4) dz Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 17 / 42

55 BEC Bose-Einstein condensation in the ground state holds in the GP limit, where N and γ is fixed, or does not grow too fast with N. This holds for an arbitrary positive potential V. The wave function of the condensate is the minimizer ψ 0 of the Gross Pitaevskii (GP) energy functional E GP [ψ] = 1 with the normalization 1 0 ψ 2 = 1. 0 ( ψ (z) 2 + V (z) ψ(z) 2 + γ 2 ψ(z) 4) dz The minimizer ψ 0 is also the ground state of the mean field Hamiltonan with eigenvalue e 0 = E GP [ψ 0 ]. h = 2 z + V (z) + γ ψ 0 2 γ ψ 0 4 Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 17 / 42

56 The average occupation of the one particle state ψ 0 in the many-body ground state Ψ 0 of H is N 0 = Ψ 0, a (ψ 0 )a(ψ 0 )Ψ 0. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 18 / 42

57 The average occupation of the one particle state ψ 0 in the many-body ground state Ψ 0 of H is N 0 = Ψ 0, a (ψ 0 )a(ψ 0 )Ψ 0. Bose-Einstein condensation in the GP limit follows from the estimate of the depletion of the condensate, ( 1 N ) 0 N e 0 (const.) N 1/3 min{γ 1/2, γ} e 1 e 0 where e 1 is the second lowest eigenvalue of the mean field Hamiltonian h. Moreover, the ground state energy per particle of H converges to the GP energy e 0. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 18 / 42

58 Superfluidity With an imposed velocity field v (moving walls) the Hamiltonian becomes: H v = N { (i zj + v) 2 + V (z j ) } + γ δ(z i z j ) N j=1 on L 2 ([0, 1], dz) N symm with periodic boundary conditions. i<j Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 19 / 42

59 Superfluidity With an imposed velocity field v (moving walls) the Hamiltonian becomes: H v = N { (i zj + v) 2 + V (z j ) } + γ δ(z i z j ) N j=1 on L 2 ([0, 1], dz) N symm with periodic boundary conditions. Denote its ground state energy by E QM 0 (v), and by e 0 (v) corresponding ground state energy of the modified GP functional E GP v [ψ] = 1 0 i<j ( iψ (z) + vψ(z) 2 + V (z) ψ(z) 2 + γ 2 ψ(z) 4) dz. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 19 / 42

60 For small enough v, Ev GP has a unique minimizer, denoted by ψ v, and e 0 (v) is equal to the ground state energy of the mean field Hamiltonian h v = (i z + v) 2 + V (z) + γ ψ v (z) 2 γ ψ v 4. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 20 / 42

61 For small enough v, Ev GP has a unique minimizer, denoted by ψ v, and e 0 (v) is equal to the ground state energy of the mean field Hamiltonian h v = (i z + v) 2 + V (z) + γ ψ v (z) 2 γ ψ v 4. Using the diamagnetic inequality one shows in an analogous way to the v = 0 case: E QM 0 (v)/n e 0 (v)(1 (const. )N 1/3 min{γ 1/2, γ}). Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 20 / 42

62 We conclude that in the GP limit the superfluid fraction 1 ρ s /ρ = lim v 0 v 2 lim N 1 N (EQM 0 (v) E QM 0 (0)) is the same as the corresponding quantity derived from the GP energy, i.e., 1 ρ s /ρ = lim v 0 v 2 (e 0(v) e 0 (0)). Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 21 / 42

63 A closed formula for the superfluid fraction We claim that ( 1 ρ s /ρ = 0 ) 1 ψ 0 (z) 2 dz This provides and explicit connection between the wave function of the BE condensate and the superfluid fraction. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 22 / 42

64 A closed formula for the superfluid fraction We claim that ( 1 ρ s /ρ = 0 ) 1 ψ 0 (z) 2 dz This provides and explicit connection between the wave function of the BE condensate and the superfluid fraction. Proof. Start with the variational equation for ψ v : (i z + v) 2 ψ v (z) + V (z)ψ v (z) + γ ψ v (z) 2 ψ v (z) = µψ v (z). We multiply this by ψ v and take the imaginary part, to obtain z ( v ψv (z) 2 I[ ψ v (z)dψ v (z)/dz] ) = 0. Hence there exists a constant C R such that I[ ψ v (z)dψ v (z)/dz] = v ψ v (z) 2 C. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 22 / 42

65 Since de 0 (v)/dv = 2v I[ ψ v (z)dψ v (z)/dz]dz we actually see that C = 1 2 de 0(v)/dv. Now ψ v has no zeroes for small v so we can divide by ψ v (z) 2 and obtain S (z) := I[ ψ v (z)dψ v (z)/dz] C ψ v (z) 2 = v ψ v (z) 2. Since S is, in fact, the derivative of the phase of ψ v we have, due to the periodic boundary conditions, 1 0 S (z)dz = 2πn with n Z, and in fact n = 0 for small enough v. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 23 / 42

66 Therefore This gives and thus v = C 1 0 ψ v (z) 2 dz. ( 1 ) 1 e 0(v) = 2C = 2v ψ v (z) 2 dz 0 e 0 ρ s /ρ = lim (v) ( 1 1 = ψ 0 (z) dz) 2. v 0 2v 0 Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 24 / 42

67 Complete superfluidity for γ (σν) 2 Using that ψ 0 is a GP minimizer and the GP energy is V ω + γ 2 one can show that Hence we see: ψ /2 1 + ψ0 2 1 γ The superfluid fraction tends to 1 in probability if γ (σν) 2. 0 V ω (σν)/γ 1/2. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 25 / 42

68 Absence of superfluidity If I is any (measurable) subset of [0, 1] with length I we have and hence ( I 2 = I ) 2 ψ 0 ψ 0 1 I ρ s ψ 0(z) 2 dz I 2. I ψ 0 (z) 2 ψ 0 (z) 2 I Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 26 / 42

69 Absence of superfluidity If I is any (measurable) subset of [0, 1] with length I we have and hence ( I 2 = I ) 2 ψ 0 ψ 0 1 I ρ s ψ 0(z) 2 dz I 2. I ψ 0 (z) 2 ψ 0 (z) 2 I To prove that superfluidity is small we have therefore to identify subsets such that I ψ 0(z) 2 dz is small, while I is not too small. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 26 / 42

70 Choice of I The random points zj ω split the intervall [0, 1] into subintervals I j = [zj ω, zω j+1 ] of various lengths l j = zi+1 ω zω j that are i.i.d. random variables with probability distribution dp ν (l) = νe νl dl. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 27 / 42

71 Choice of I The random points zj ω split the intervall [0, 1] into subintervals I j = [zj ω, zω j+1 ] of various lengths l j = zi+1 ω zω j that are i.i.d. random variables with probability distribution dp ν (l) = νe νl dl. We shall take I = j:l j l I j with a suitably chosen l. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 27 / 42

72 Choice of I The random points zj ω split the intervall [0, 1] into subintervals I j = [zj ω, zω j+1 ] of various lengths l j = zi+1 ω zω j that are i.i.d. random variables with probability distribution dp ν (l) = νe νl dl. We shall take I = j:l j l I j with a suitably chosen l. The average length of I is L = ν l 0 ν ldp ν (l) = 1 (1 + (ν l))e l). In particular it tends to 1 if and only if l ν 1. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 27 / 42

73 With the notation we define N small,ω = n GP j = ψ 0 (z) 2 dz I j I ψ 0 (z) 2 dz = lj l n GP j. Note that ψ 0 and n GP j also depend on ω but we have suppressed this in the notation for simplicity. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 28 / 42

74 With the notation we define N small,ω = n GP j = ψ 0 (z) 2 dz I j I ψ 0 (z) 2 dz = lj l n GP j. Note that ψ 0 and n GP j also depend on ω but we have suppressed this in the notation for simplicity. Our estimate on N small,ω is based on estimates on the GP energy. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 28 / 42

75 Auxiliary energy Define E κ,α [φ] = 1 0 dx ( φ (x) 2 + κ 2 φ(x) 4) + α ( φ(0) 2 + φ(1) 2) 2 with κ 0 and α 0. Let e(κ, α) denote the auxiliary GP energy e(κ, α) = inf E κ,α[φ]. φ 2 =1 Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 29 / 42

76 Auxiliary energy Define E κ,α [φ] = 1 0 dx ( φ (x) 2 + κ 2 φ(x) 4) + α ( φ(0) 2 + φ(1) 2) 2 with κ 0 and α 0. Let e(κ, α) denote the auxiliary GP energy e(κ, α) = inf E κ,α[φ]. φ 2 =1 The corresponding energy for an interval of length l with mass Interval φ 2 = n, coupling constant γ and strength σ of the obstacle potential is then, by scaling, n e(nlγ, lσ). l2 Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 29 / 42

77 We use the following bounds on e(κ, α) : e(κ, ) e(κ, α) e(κ, ) (1 Kα 1/2) Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 30 / 42

78 We use the following bounds on e(κ, α) : e(κ, ) e(κ, α) e(κ, ) (1 Kα 1/2) and e(κ, α) e(0, α) Cα 1 + α. with constants K and C independent of κ and α. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 30 / 42

79 The interval density functional With n(l) 0 a mass distribution on intervals of various lengths l we define an interval density functional as E IDF [n( )] = ν with corresponding energy { e IDF (ν, γ) = inf E IDF [n( )] : ν 0 dp ν (l) n(l) e(n(l)lγ, ) l2 0 } dp ν (l)n(l) = 1. This energy is the deterministic limit (in probability) of the GP energy under our standard conditions on the parameters. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 31 / 42

80 The Lagrange multiplier µ for the normalization condition fulfills µ γ f(ν 2 /γ), where f : R + R + denotes the function { 1 for x 1 f(x) = x for x 1. (1+ln x) 2 Also e IDF (γ, ν) γ f(ν 2 /γ). Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 32 / 42

81 The Lagrange multiplier µ for the normalization condition fulfills µ γ f(ν 2 /γ), where f : R + R + denotes the function { 1 for x 1 f(x) = x for x 1. (1+ln x) 2 Also e IDF (γ, ν) γ f(ν 2 /γ). A further result is that the minimizing n(l) of the interval density functional is nonzero if and only if µl 2 > π 2. We can therefore expect that the GP mass is small in intervals I j such that l j (const. )/ µ. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 32 / 42

82 Absence of superfluidity for γ ν 2 Split the GP energy e GP ω (γ, ν, σ) into contributions from the large and the small intervals: e GP ω (γ, ν, σ) l j l n GP j l 2 j e(n GP j l j γ, l j σ) + l j < l n GP j l 2 e(n GP j l j γ, l j σ) j where l = s/ µ with a suitable s and (because γ/ν 2 1) µ ν 2 (1 + ln(ν 2 /γ)) 2. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 33 / 42

83 Absence of superfluidity for γ ν 2 Split the GP energy e GP ω (γ, ν, σ) into contributions from the large and the small intervals: e GP ω (γ, ν, σ) l j l n GP j l 2 j e(n GP j l j γ, l j σ) + l j < l n GP j l 2 e(n GP j l j γ, l j σ) j where l = s/ µ with a suitable s and (because γ/ν 2 1) µ ν 2 (1 + ln(ν 2 /γ)) 2. Note that, since σ ν/(1 + ln(1 + ν 2 /γ)) we have for γ ν 2 lσ 1. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 33 / 42

84 Energy estimate for small intervals The sum over the small intervals can be estimated as l j < l n GP j l 2 j e(n GP j l j γ, l j σ) l j < l N small,ω n GP j l 2 e(0, l j σ) j lj< l n GP j l 2 j C l j σ 1 + l j σ C σ l(1 + lσ) = N small,ω µ C s 2 σ l 1 + σ l. Here we have used the estimates for the energy in intervals between obstacles. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 34 / 42

85 Energy Estimate for large intervals For the large intervals we have l j l n GP j l 2 e(n GP n j j l j γ, l j σ) inf j ni =1 N small,ω l 2 e(n j l j γ, l j σ) j l j l n j inf e(n j l j γ, )(1 K( lσ) 1/2 ). ni =1 N small,ω j l 2 j Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 35 / 42

86 Energy Estimate for large intervals For the large intervals we have l j l n GP j l 2 e(n GP n j j l j γ, l j σ) inf j ni =1 N small,ω l 2 e(n j l j γ, l j σ) j l j l n j inf e(n j l j γ, )(1 K( lσ) 1/2 ). ni =1 N small,ω Apart from the factor (1 K( lσ) 1/2 ) the right side is the GP energy for σ = with normalization ψ 2 = 1 N small,ω instead of ψ 2 = 1. By simple scaling this is 1 N small,ω times the the GP energy with normalization 1 and γ replaced by (1 N small,ω )γ, which in turn is not smaller than (1 N small,ω ) 2 times e GP ω (γ, ν, ). j l 2 j Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 35 / 42

87 We can further estimate (1 N small,ω ) 2 e GP ω (γ, ν, ) (1 2N small,ω )e GP ω (γ, ν, σ) and putting everything together we obtain e GP ω (γ, ν, σ) N small,ω C µ s 2 σ l 1 + σ l + (1 2N small,ω ) egp ω (γ, ν, σ) (1 K( lσ) 1/2 ). µ Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 36 / 42

88 We can further estimate (1 N small,ω ) 2 e GP ω (γ, ν, ) (1 2N small,ω )e GP ω (γ, ν, σ) and putting everything together we obtain e GP ω (γ, ν, σ) N small,ω C µ s 2 Important point: σ l 1 + σ l + (1 2N small,ω ) egp ω (γ, ν, σ) (1 K( lσ) 1/2 ). µ If ν, γ and σ tend to infinity under the standard conditions, the ratio (γ, ν, σ)/µ stays bounded (in probability). e GP ω Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 36 / 42

89 Mass in small intervals Moreover, for γ ν 2 we have σ l = sσ/ µ 1. For C/s 2 > 2e GP ω (γ, ν, σ)/µ we thus arrive at an estimate for the mass in the small intervals:. ( N small,ω (const.) egp ω (γ, ν, σ) µ µ 1/2 σ ) 1/2, Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 37 / 42

90 Mass in small intervals Moreover, for γ ν 2 we have σ l = sσ/ µ 1. For C/s 2 > 2e GP ω (γ, ν, σ)/µ we thus arrive at an estimate for the mass in the small intervals:. ( N small,ω (const.) egp ω (γ, ν, σ) µ Since l = s/µ 1/2 and lσ 1, we have µ 1/2 σ 1, µ 1/2 σ ) 1/2, and we we have shown that N small,ω 0 in probability if γ ν 2 and the standard conditions hold. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 37 / 42

91 The superfluid fraction is bounded from above by N small,ω /L 2 ω where L ω = I is the total length of intervals of length l. The latter converges in probability to the expectation value L = ν l provided the fluctuations remain small. 0 ν ldp ν (l) = 1 (1 + (ν l))e l), Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 38 / 42

92 The superfluid fraction is bounded from above by N small,ω /L 2 ω where L ω = I is the total length of intervals of length l. The latter converges in probability to the expectation value L = ν l provided the fluctuations remain small. 0 ν ldp ν (l) = 1 (1 + (ν l))e l), For γ ν 2 we have lν 1 and the length L converges to 1 as ν, while for γ ν 2 the length stays bounded away from 0 because lν is O(1). The fluctuations are O(ν 1/2 ). Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 38 / 42

93 The superfluid fraction is bounded from above by N small,ω /L 2 ω where L ω = I is the total length of intervals of length l. The latter converges in probability to the expectation value L = ν l provided the fluctuations remain small. 0 ν ldp ν (l) = 1 (1 + (ν l))e l), For γ ν 2 we have lν 1 and the length L converges to 1 as ν, while for γ ν 2 the length stays bounded away from 0 because lν is O(1). The fluctuations are O(ν 1/2 ). The conclusion is: The superfluid fraction tends to 0 in probability if γ ν 2. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 38 / 42

94 The case γ ν 2 Here µ γ and we take l µ 1/2 γ 1/2 ν 1. We need in any case σ l 1, i.e., σ γ 1/2, which is compatible with the standard conditions. In the same way as above we obtain the estimate this time with µ γ. N small,ω (const.)(µ 1/2 /σ) 1/2, Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 39 / 42

95 The case γ ν 2 Here µ γ and we take l µ 1/2 γ 1/2 ν 1. We need in any case σ l 1, i.e., σ γ 1/2, which is compatible with the standard conditions. In the same way as above we obtain the estimate this time with µ γ. N small,ω (const.)(µ 1/2 /σ) 1/2, Since ν l ν/γ 1/2 1, however, the average length of the small intervals is now L (ν/γ 1/2 ) 2 1 rather than O(1) as for γ ν 2. To exclude superfluidity we need N small,ω /L 2 (γ 1/4 /σ 1/2 )(γ/ν 2 ) 1 which holds for σ (γ/ν 2 ) 4 γ 1/2. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 39 / 42

96 This condition is still not sufficient, however, because the estimate L ω (ν/γ 1/2 ) 2 can only be claimed to be true in probability as long as the fluctuations of the random variable L ω = l j l l j are small compared to its average value, L. A sufficient condition for this is that ν l 0 l2 dp ν (l) L 2, which holds for γ ν 4. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 40 / 42

97 This condition is still not sufficient, however, because the estimate L ω (ν/γ 1/2 ) 2 can only be claimed to be true in probability as long as the fluctuations of the random variable L ω = l j l l j are small compared to its average value, L. A sufficient condition for this is that ν l 0 l2 dp ν (l) L 2, which holds for γ ν 4. Altogether we conclude: The superfluid fraction tends to 0 in probability, if and ν 2 γ ν 4. σ (γ/ν 2 ) 4 γ 1/2 Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 40 / 42

98 Concluding remarks We have studied superfluidity in the ground state of a one-dimensional model of bosons with a repulsive contact interaction and in a random potential generated by Poisson distributed point obstacles. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 41 / 42

99 Concluding remarks We have studied superfluidity in the ground state of a one-dimensional model of bosons with a repulsive contact interaction and in a random potential generated by Poisson distributed point obstacles. In the Gross Pitaevskii (GP) limit this model always shows complete BEC, but depending on the parameters, superfluidity may or may not occur. In the course of the analysis we derived a closed formula for the superfluid fraction, expressed in terms of the GP wave function. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 41 / 42

100 The advantage of this model is that it is amenable to a rigorous mathematical analysis leading to unambiguous statements. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 42 / 42

101 The advantage of this model is that it is amenable to a rigorous mathematical analysis leading to unambiguous statements. The model has its limitations: Nothing is claimed about positive temperatures and the proof of BEC requires that the ratio between the coupling constant for the interaction and the density tends to zero as N. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 42 / 42

102 The advantage of this model is that it is amenable to a rigorous mathematical analysis leading to unambiguous statements. The model has its limitations: Nothing is claimed about positive temperatures and the proof of BEC requires that the ratio between the coupling constant for the interaction and the density tends to zero as N. Nevertheless, to our knowledge this is the only model where a Bose glass phase in the sense of complete BEC but absence of superfluidity, has been rigorously established so far. Jakob Yngvason (Uni Vienna) Superfluidity vs BEC 42 / 42