trento, june 4, 2009

Trento, June 4, 2009

Gentaro Watanabe, Franco Dalfovo, Giuliano Orso, Francesco Piazza, Lev P. Pitaevskii, and Sandro Stringari

Summary

Equation of state and effective mass of the unitary Fermi gas in a 1D periodic potential [Phys. Rev. A 78, 063619 (2008)]

Critical velocity of superfluid flow [work in progress]

We were stimulated by:

6Li Feshbach res. @ B=834GN=106

weak lattice

Superfluidity is robust at unitarity

vc is maximum around unitarity

What we have done first:

Calculation of

energy density

chemical potential

inverse compressibility

effective mass

sound velocity

Key quantities for the characterization of the collective properties of the superfluid

The theory that we have used:

Bogoliubov – de Gennes equation

Order parameter

A. J. Leggett, in Modern Trends in the Theory of Condensed Matter, edited by A. Pekalski and R. Przystawa (Springer-Verlag, Berlin, 1980);M. Randeria, in Bose Einstein Condensation, edited by A. Griffin, D. Snoke, and S. Stringari (Cambridge University Press, Cambridge, England, 1995).

Refs:

We numerically solve the BdG equations. We use a Bloch wave decomposition.

From the solutions we get the energy density:

Caveat: a regularization procedure must be used to cure ultraviolet divergence (pseudo-potential, cut-off energy) as discussed by Randeria and Leggett. See also: G. Bruun, Y. Castin, R. Dum, and K. Burnett, Eur. Phys. J D 7, 433 (1999)A. Bulgac and Y. Yu, Phys. Rev. Lett. 88, 042504 (2002).

Results: compressibility and effective mass

Results: compressibility and effective mass

Two-bodyresults byOrso et al.,PRL 95, 060402(2005)

when EF << ER :

The lattice favors the formation of molecules (bosons).

The interparticle distance becomes larger than the molecular size.

In this limit, the BdG equations describe a BEC of molecules.

The chemical potential becomes linear in density.

s=5

s=0

Results: compressibility and effective mass

Two-bodyresults byOrso et al.,PRL 95, 060402(2005)

when EF << ER :

The lattice favors the formation of molecules (bosons).

The interparticle distance becomes larger than the molecular size.

In this limit, the BdG equations describe a BEC of molecules.

The chemical potential becomes linear in density.

The system is highly compressible.

The effective mass approaches the solution of the two-body problem.

The effects of the lattice are larger than for bosons!

Results: compressibility and effective mass

when EF >> ER :

Both quantities approach their values for a uniform gas.

Analytic expansions in the small parameter (sER/EF)

Results: compressibility and effective mass

when EF ~ ER :

Both quantities have a maximum, caused by the band structure of the quasiparticle spectrum.

Sound velocity

Significant reduction of sound velocity the by lattice !

aspect ratio = 1ħω/ER = 0.01N=5105

s=5

Density profile of a trapped gas

From the results for μ(n) and using a local density approximation, we find the density profile of the gas in the harmonic trap + 1D lattice

Thomas-Fermi for fermions

Bose-likeTF profile !

Summary of the first part

We have studied the behavior of a superfluid Fermi gas at unitarity in a 1D optical lattice by solving the BdG equations.

The tendency of the lattice to favor the formation of molecules results in a significant increase of both the effective mass and the compressibility at low density, with a consequent large reduction of the sound velocity.

For trapped gases, the lattice significantly changes:

the density profile the frequency of the collective oscillations

[See Phys. Rev. A 78, 063619 (2008)]

Equation of state and effective mass of the unitary Fermi gas in a 1D periodic potential [Phys. Rev. A 78, 063619 (2008)]

Critical velocity of superfluid flow [work in progress]

The concept of critical current plays a fundamental role in the physics of superfluids.

Examples:

- Landau critical velocity for the breaking of superfluidity and the onset of dissipative effects. This is fixed by the nature of the excitation spectrum: • phonons, rotons, vortices in BEC superfluids• single particle gap in BCS superfluids

-Critical current for dynamical instability(vortex nucleation in rotating BECs, disruptionof superfluidity in optical lattices)

- Critical current in Josephson junctions. This is fixed by quantum tunneling

In ultracold atomic gases:

Motion of macroscopic impurities has revealed the onset of heating effect (MIT 2000)

Quantum gases in rotating traps have revealed the occurrence of both energetic and dynamic instabilities (ENS 2001)

Double well potentials are well suited to explore Josephson oscillations (Heidelberg 2004)

Moving periodic potentials allow for the investigation of Landau critical velocity as well as for dynamic instability effects (Florence 2004, MIT 2007)

Questions:

Can we obtain a unifying view of critical velocity phenomena driven by different external potentials and for different quantum statistics (Bose vs. Fermi)?

Can we theoretically account for the observed values of the critical velocity?

NOTE: The mean-field calculations by Spuntarelli et al. [PRL 99, 040401 (2007)] for fermions through a single barrier show a similar dependence of the critical velocity on the barrier height.

Our goal:

establishing an appropriate framework in which general results can be found in order to compare different situations (bosons vs. fermions and single barrier vs. optical lattice) and extract useful indications for available and/or feasible experiments.

L

Vmax

d

Vmax

Assumption: the system behaves locally as a uniform gas of density n, with energy density e(n) and local chemical potential, μ(n). The density profile of the gas at rest in the presence of an external potential is given by the Thomas-Fermi relation

If the gas is flowing with a constant current density j=n(x)v(x), the Bernoulli equation for the stationary velocity field v(x) is

The simplest approach: Hydrodynamics in Local Density Approximation (LDA)

This equation gives the density profile, n(x), for any given current j, once the equation of state μ(n) of the uniform gas of density n is known.

The simplest approach: Hydrodynamics in Local Density Approximation (LDA)

The system becomes energetically unstable when the local velocity, v(x), at some point x becomes equal to the local sound velocity, cs[n(x)].

For a given current j, this condition is first reached at the point of minimum density, where v(x) is maximum and cs(x) is minimum.

here the density has a minimum and the local velocity has a maximum !

The simplest approach: Hydrodynamics in Local Density Approximation (LDA)

The same happens in a periodic potential

here the density has a minimum and the local velocity has a maximum !

The simplest approach: Hydrodynamics in Local Density Approximation (LDA)

To make calculations, one needs the equation of state μ(n) of the uniform gas!

We use a polytropic equation of state:

The simplest approach: Hydrodynamics in Local Density Approximation (LDA)

To make calculations, one needs the equation of state μ(n) of the uniform gas!

We use a polytropic equation of state:

Bosons (BEC) Unitary Fermions

α= (1+β)(3π2)2/3ħ2/2mα= g = 4πħ2as/m

The simplest approach: Hydrodynamics in Local Density Approximation (LDA)

To make calculations, one needs the equation of state μ(n) of the uniform gas!

We use a polytropic equation of state:

Bosons (BEC) Unitary Fermions

Local sound velocity:

α= (1+β)(3π2)2/3ħ2/2mα= g = 4πħ2as/m

The simplest approach: Hydrodynamics in Local Density Approximation (LDA)

Inserting the critical condition

into the Bernoulli equation

one gets an implicit relation for the critical current: Universal !!Bosons and Fermions in any 1D potential

Note: for bosons through a single barrier see also Hakim, and Pavloff et al.

The simplest approach: Hydrodynamics in Local Density Approximation (LDA)

LDA

Bosons in a lattice

Bosons through a barrier

Fermions in a lattice

Fermions through a barrier

bosons fermions

LDA

The limit Vmax << μ corresponds to the usual Landau criterion for a uniform superfluid flow in the presence of a small external perturbation, i.e., a critical velocity equal to the sound velocity of the gas.

LDA

The limit Vmax << μ corresponds to the usual Landau criterion for a uniform superfluid flow in the presence of a small external perturbation, i.e., a critical velocity equal to the sound velocity of the gas.

the critical velocity decreases because the density has a local depletion and the velocity has a corresponding local maximum

LDA

The limit Vmax << μ corresponds to the usual Landau criterion for a uniform superfluid flow in the presence of a small external perturbation, i.e., a critical velocity equal to the sound velocity of the gas.

the critical velocity decreases because the density has a local depletion and the velocity has a corresponding local maximum

When Vmax = μ thedensity vanishes and the critical velocity too.

LDA

Question: when is LDA reliable?

LDA

Question: when is LDA reliable?

Answer: the external potential must vary on a spatial scalemuch larger than the healing length of the superfluid.

L >> ξ

LDA

Question: when is LDA reliable?

Answer: the external potential must vary on a spatial scalemuch larger than the healing length of the superfluid.

For a single square barrier, L is just its width.

For an optical lattice, L is of the order of the lattice spacing (we choose L=d/2).

For bosons with density n0, the healing length is ξ=ħ/(2mgn0)1/2.

For fermions at unitarity, one has ξ ≈ 1/kF, where kF = (3π2n0)1/3

L >> ξ

LDA

Question: when is LDA reliable?

Answer: the external potential must vary on a spatial scalemuch larger than the healing length of the superfluid.

Quantum effects beyond LDA become important when - ξ is of the same order or larger than L; they cause a smoothing of both density and velocity distributions, as well as the emergence of solitonic excitations (and vortices in 3D).- Vmax > µ ; in this case LDA predicts a vanishing current, while quantum tunneling effects yield Josephson current.

Quantitative estimates of the deviations from the predictions of LDA can be obtained by using quantum many-body theories, like Gross-Pitaevskii theory for dilute bosons and Bogoliubov-de Gennes equations for fermions.

LDA

Bosons in a lattice

Bosons through a barrier

Fermions in a lattice

Fermions through a barrier

LDA vs. GP/BdG

Bosons in a lattice

Bosons through a barrier

Fermions in a lattice

Fermions through a barrier

LkF=4 [Spuntarelli et al.]

fermionsbosons

L/ξ=1

105

LkF=0.5

0.89

1.11

1.571.92

2.5

L/ξ=0.5

1

35

10

1.57

Bosons (left) and Fermions (right) through single barrier

LkF=4

bosons fermions

L/ξ=1

105

Bosons (left) and Fermions (right) through single barrier

LkF=4

bosons fermions

L/ξ=1

105

L >> ξ : Hydrodynamic flow in LDA

Bosons (left) and Fermions (right) through single barrier

LkF=4

bosons fermions

L/ξ=1

105

L >> ξ : Hydrodynamic flow in LDA

L < ξ : Macroscopic flow with quantum effects beyond LDA

Bosons (left) and Fermions (right) through single barrier

LkF=4

bosons fermions

L/ξ=1

105

L >> ξ : Hydrodynamic flow in LDA

Vmax> μ : quantum tunneling between weakly coupled superfluids (Josephson regime)

Bosons (left) and Fermions (right) in a periodic potential

The periodic potential gives results similar to the case of single barrier

bosons fermions

LkF=0.5

0.89

1.11

1.571.92

2.5

L/ξ=0.5

1

35

10

1.57

Bosons (left) and Fermions (right) in a periodic potential

bosons fermions

LkF=0.5

0.89

1.11

1.571.92

2.5

L/ξ=0.5

1

35

10

1.57

L >> ξ : Hydrodynamic flow in LDA

L < ξ : Macroscopic flow with quantum effects beyond LDA

Vmax>> μ : quantum tunneling between weakly coupled supefluids (Josephson current regime)

Bosons in a lattice

Bosons through a barrier

L/ξ=1

105

L/ξ=0.5

1

35

10

1.57

Differences between single barrier and periodic potential (bosons)

Single barrier (bosons)

For a single barrier, μ and ξ are fixed by the asymptotic density n0 only. They are unaffected by the barrier. All quantities behave smoothly when plotted as a function of L/ ξ or Vmax/ μ.

For L/ ξ >> 1: vc/cs 1 – const (Vmax )1/2

For L/ ξ << 1: vc/cs 1 – const (LVmax )2/3

L/ξ=1

105

L/ξ=0.5

1

35

10

1.57

Periodic potential (bosons)

In a periodic potential the barriers are separated by distance d. The energy density, e, and the chemical potential, µ, are not fixed by the average density n0 only, but they depend also on Vmax. They exhibit a Bloch band structure.

Periodic potential (bosons)

Bloch band structure.p = quasi-momentumpB= Bragg quasi-momentumER= p2

B/2m = recoil energyVmax=sER = lattice strength

Energy density vs. quasi-momentum

Lowest Bloch band for the same gn0=0.4ER and different s

Periodic potential (bosons)

Bloch band structure.p = quasi-momentumpB= Bragg quasi-momentumER= p2

B/2m = recoil energyVmax=sER = lattice strength

Energy density vs. quasi-momentum

Periodic potential (bosons)

Bloch band structure.p = quasi-momentumpB= Bragg quasi-momentumER= p2

B/2m = recoil energyVmax=sER = lattice strength

Energy density vs. quasi-momentum

The curvature at p=0 gives the effective mass:

e = n0 p2/2m*

Periodic potential (bosons)

Bloch band structure.p = quasi-momentumpB= Bragg quasi-momentumER= p2

B/2m = recoil energyVmax=sER = lattice strength

Energy density vs. quasi-momentum

Tight-binding limit (Vmax >> µ):

e = δJ [1-cos(πp/pB)]

(2ERn0/π2)(m/m*) = δJ = tunnelling energy(Josephson current)

Critical p: pc=0.5pB

Critical velocity: vc = (2/π)(m/m*) ER/pB

m/m* proportional to e-2√s

L/ξ=0.5

1

35

10

1.57

Differences between single barrier and periodic potential (fermions)

Fermions in a lattice

Fermions through a barrier

LkF=0.5

0.89

1.11

1.571.92

2.5

Same qualitative behavior as for bosons

LkF=4

What about experiments?

BOSONS: Experiments at LENS-Florence

L/ξ=0.5

1

35

10

Weak lattice (energetic vs. dynamic instability)

L. De Sarlo, L. Fallani, J. E. Lye, M. Modugno, R. Saers, C. Fort, M. Inguscio, Unstable regimes for a Bose-Einstein condensate in an optical lattice Phys. Rev. A 72, 013603 (2005)

BOSONS: Experiments at LENS-Florence

L/ξ=0.5

1

35

10

Weak lattice (energetic vs. dynamic instability)

L. De Sarlo, L. Fallani, J. E. Lye, M. Modugno, R. Saers, C. Fort, M. Inguscio, Unstable regimes for a Bose-Einstein condensate in an optical lattice Phys. Rev. A 72, 013603 (2005)

Strong lattice (Josephson current regime):

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio Josephson Junction arrays with Bose-Einstein Condensates Science 293, 843 (2001)

L/ξ ≈ 0.7 and Vmax/µ ≈ 10 - 25

What about experiments?

FERMIONS: Experiments at MIT

D. E. Miller, J. K. Chin, C. A. Stan, Y. Liu, W. Setiawan, C. Sanner, W. Ketterle Critical velocity for superfluid flow across the BEC-BCS crossover PRL 99, 070402 (2007)]

Problem: Which density n0? Which Vmax?

What about experiments?

LkF=0.5

0.89

1.11

1.571.92

2.5

?

FERMIONS: Experiments at MIT

D. E. Miller, J. K. Chin, C. A. Stan, Y. Liu, W. Setiawan, C. Sanner, W. Ketterle Critical velocity for superfluid flow across the BEC-BCS crossover PRL 99, 070402 (2007)]

Problem: Which density n0? Which Vmax?

n0

Vext

Depending on how one chooses n0 (and hence EF) one gets

0.5 < EF/ER < 1 or 1 < LkF < 1.5

What about experiments?

LkF=0.5

0.89

1.11

1.571.92

2.5

?

FERMIONS: Experiments at MIT

D. E. Miller, J. K. Chin, C. A. Stan, Y. Liu, W. Setiawan, C. Sanner, W. Ketterle Critical velocity for superfluid flow across the BEC-BCS crossover PRL 99, 070402 (2007)]

If EF is by the density at e-2 beam waist: EF/ER ≈ 0.5 (LkF ≈ 1)

BdG theoryExpt

LDA

Significant discrepancies between theory and MIT data. Additional dissipation mechanisms? Nonuniform nature of the gas/lattice ?

What about experiments?

Unifying theoretical picture of critical velocity phenomena in 1D geometries (including single barrier and periodic potentials as well as Bose and Fermi statistics).

Different scenarios considered, including LDA hydrodynamics and Josephson regime

Comparison with experiments reveals that the onset of energetic instability is affected by nonuniform nature of the gas (also in recent MIT experiment).

Need for more suitable geometrical configurations. For example toroidal geometry with rotating barrier would provide cleaner and more systematic insight on criticality of superfluid phenomena (including role of quantum vorticity)

Summary of the second part