Comparison and Contrast:Cold Atoms and Dilute Neutron Matter

J. Carlson - LANL

C. Regal et al. PRL 2004

Fermi Condensates

Introduction

Homogeneous Matter : Equation of State (T=0)

Contact Gap

Linear Response: RF Response Spin Response Density Response

Inhomogeneous (trapped) Matter Local Density approximation and Density functional Theory

Future Outlook

MIT Optical Trap

roadrunner

Interactions:

Diagram from Innsbruck

Fermions: 6Li, 40KDensity ~ 1/ μm3

Temperature ~ 200 nK ~ 0.1 Ef

NN phase shifts (1S0)Density ~ 1 / (10 fm)3

T ~ 0

BCS-BEC Transition (T=0):

Interaction Strength

Pola

rizat

ion

Free (1 component) Fermi Gas

Free

(2

com

pone

nt)

Ferm

i Gas

Bose

-Fer

mi M

ixtu

reBEC BCS

Uni

tary

Reg

ime

Neutron Matter (T=0)

Density

Pola

rizat

ion

Free

(2

com

pone

nt)

Ferm

i Gas

Bose

-Fer

mi M

ixtu

re

BEC BCS

Uni

tary

Reg

ime

a ~ -18 fm |a|/re ~ 10

(nearly) Free Fermions (nearly) Free Bosons ‘Universality’ and the BCS-BEC transition Polarons Efimov States Superfluid Fermions (s-, p-, d-wave,... pairing) Exotic Polarized Superfluids (FFLO, breached pair,...) PseudoGap States Itinerant Ferromagnetism `Perfect’ Fluids Reduced Dimensionality More than pairing (3-,4-body condensates, ...) Bose, Fermi Hubbard Models, .....

Cold Fermi Atoms: Physics Interests

Unitarity

Interaction

Pola

rizat

ion

Unitarity = limit of 0 pair binding a = ∞

All quantities multiples of Fermi Gas at same ρAt zero polarization, expect strong pairing

∆ = δ�2k2

F

2m

E = ξ EFG = ξ35

�2k2F

2m

Tc = t�2k2

F

2m

Values of ξ, δ, t are independent of ρ

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 20120.0

0.2

0.4

0.6

0.8

1.0

time �yrs�Ξ

simulation

experiment

analytical

FIG. 13: Historical results for the Bertsch parameter determined experimentally, by analytic cal-

culation, and by numerical simulation. Numerical values and citations are tabulated in Table VII;

our value, based on simulations at L = 14 is indicated as the latest simulation data point.

considered. References for the historical results are provided in Table VII.

V. CONCLUSION

We have studied up to 66 unpolarized unitary fermions in a periodic box by applying a

lattice Monte Carlo method developed for studying large numbers of strongly interacting

nonrelativistic spin-1/2 fermions [49]. Our method differs from methods used in the past

in that it does not make use of importance sampling, nor is it variational. Our approach

also allows us to study the energy levels of systems with unequal numbers of spin up and

spin down fermions. One of the obstacles in the calculation of ground state energies of large

numbers of fermions is a severe distribution overlap problem which results in inefficient

estimation of the correlation function. A key part of our approach is to use the cumulant

expansion for the logarithm of the correlators [58], allowing us to avoid this difficulty and

determine the energies in a reliable manner.

Our main findings are as follows:

1. The exact diagonalization of the transfer matrix for two, three, and four fermions en-

30

Endres, Kaplan, Lee and Nicholson, arXiv:1203.3169 (2012)

ξ: Experiments, Analytic, and Computational

Zero Temperature Simulations

Quantum Monte Carlo: Ψ = exp[−Hτ ] Ψ0

Ψ0 = ΨBCS =�

k

[vk/uk] a†↑(k)a†↓(−k) |0�

Branching Random Walk: DMC coordinate space AFMC orbitals

AFMC exact for unpolarized systems (finite lattices) good for finite temperature DMC in continuum, but fixed-node approximation good for polarized systems

Computational requirements from workstation (Total Energy) to largest supercomputers

(exotic superfluids)

Experiments at Unitarity: # = #Cloud Size and Sound Velocity

Cloud Sizevs E (B)

c0

vf=

ξ1/4

�(5)

Joseph, et al., PRL 2007Sound Propagation

scaling verified as ρ varied by 30

ξ = 0.39(02)

Energy vs. Entropy

Luo and Thomas, JLTP, 2009

Improved Lattice (AFMC) Methods for Unitary Gas BCS importance function no sign problem control of lattice size, N, effective range

At finite (small) effective range:

Can measure neutron matter EOS (including effective range corrections) in cold atoms

E / EFG = ξ + S kF re

and are universal parametersξ S

TIext

0 0.1 0.2 0.3 0.41/3 = N1/3/L

0.32

0.34

0.36

0.38

0.40

0.42

!!

N = 38N = 48N = 66

k(4)

k(h)

k(2)

Unitary Fermi Gas (lattice)

K.E. Schmidt, S. Zhang, S. Gandolfi, JC, PRA 2011

E/EFG

up to 273 lattice, 66 particles

ξ = 0.372(0.005)

from experiment (MIT) ξ = 0.376(0.005)arXiv:1110.3309 (Hu, et al)

kF re ~0.3A. Gezerlis, J. C., 2008,2010

Equation of State: Cold Atoms vs. Neutron Matter

Neutron Matter EOS

Neutron Matter EOS strongly constrained at low-moderate densities

Contact from the EOS

Tan, Annals of Phys. 2008

Gandolfi, et al, PRA 2011

Probability of

at same point

Pairing Gap at Unitarity from the polarized EOS

Spin up, down densities in a trap

Zweirleinρ

ρ Polarization

radius1

0

Quasiparticle Dispersion in cold AtomsAdd one to fully-paired systemEnergy cost for an unpaired particle: μ + Δ

∆ = δ�2k2

F

2m

0 0.2 0.4 0.6 0.8 1 1.2( k / kF)2

0

0.2

0.4

0.6

0.8

1

Δ /

EF

δ = 0.50 (03)

(kmin/kf )2 = 0.80(10)

JC and Reddy, PRL 2005

Pairing Gap at Unitarity - Experiment

4

0

0.2

0.4

0.6

0.8

1

!=0.5, tc=0.25

!=0.38, tc=0.25

!=0.43, tc=0.07

(MIT Expt.)

0 0.2 0.4 0.6 0.8 1r/R

0

0.2

0.4

0.6

0.8

1

!=0.46, tc=0.1

!(r)

Pola

rization

T’=0.03

T’=0.05

!(r)

FIG. 3: Polarization versus radius, theory and experiment, fordi!erent values of ! and tc at T ! = 0.03 and T ! = 0.05. Thedashed curves show the local finite temperature gap. Theresults indicate that the data provide both an upper and alower bound on the gap: 0.5 ! ! ! 0.4.

order transition somewhere in between. In contrast, thecomparison between theory and data at T ! = 0.05 sug-gests that the superfluid extends further out. Polariza-tion in the superfluid state (dotted curve) extrapolatedto p ! 0.4 provides a better description of the data thanthe normal state. A clear signature of a first-order tran-sition is also absent. In both cases there appears to beevidence for a mild decrease in the gap with increasingT/EF and polarization.

For a fixed central density and R", our analysis pre-dicts that the phase-boundary Rc moves outward in thetrap with increasing temperature. This behavior is sen-sitive to the thermal properties of both phases at lowtemperature. At small temperature and polarization, thethermal response of the superfluid phase in the vicinity ofthe transition is stronger than that of the normal phase– driven entirely by the fact that spin-up quasiparticlesare easy to excite and have a large density of states.

The comparison in Fig. 3 provides compelling lowerand upper bounds for the superfluid gap. Even if thetemperature was extracted incorrectly from the exper-iment, the extracted gap cannot be too small. A gapsmaller than " 0.4EF would produce a shell of polarizedsuperfluid before the transition even at zero temperature.Furthermore, the radial dependence of this polarizationwould be quite di!erent than observed experimentally,rising abruptly from the point where " = !µ and beingconcave rather than convex. A gap larger than " 0.5EF

would be unable to produce the observed polarization inthe superfluid phase. We have also examined the depen-dence of our results on the universal parameters " and#. Both of these are expected to be uncertain by 0.02.These uncertainties, as well as the uncertainties in the

superfluid quasiparticle dispersion relation do not signif-icantly alter the extracted bounds on the superfluid gap.

In summary, we have extracted the pairing gap frommeasurements of spin up and spin down densities in po-larized Fermi gases in the unitary regime. These systemshave an extremely large gap of almost one-half the Fermienergy – the value extracted in this work is clearly thelargest gap measured in any Fermi system. Future moreprecise experiments extending over the BCS-BEC transi-tion region would allow an experimental determination ofthe evolution of the pairing gap from the weak-couplingregime of traditional superfluids and superconductors tothe strongly-interacting regime. This could resolve long-standing issues regarding, for example, the pairing gapin neutron matter and the cooling of neutron stars.

We would like to thank M. Alford, A. Gezerlis and Y.Shin for useful comments on the manuscript. The work ofS.R. and J.C. is supported by the Nuclear Physics O#ceof the U.S. Department of Energy and by the LDRDprogram at Los Alamos National Laboratory.

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Polarization in a trap

δ = 0.45(05)∆ = δ

�2k2F

2m

Largest Δ/Ef !

JC and Reddy, PRL 2007analyzing MIT data

0 0.2 0.4 0.6 0.8 1 1.2( k / kF)2

0

0.2

0.4

0.6

0.8

1

Δ /

EF

Beyond the Equation of State:Structure and Dynamics

Linear Response:

RF response (q=0): Flip atom to a new HF state

Spin response (large q): Flip atom between states and give ‘kick’ of momentum q

Density response (large q): Give atom `kick’ of momentum q

Credit: Greg Kuebler, JILA

Response sensitive to pairing gap, `contact’, and more

S(k,ω) = �0|O†(k)|f��f |O(k)|0�δ(ω − (Ef − E0))

Density Response: Bragg Spectroscopyat large momentum transfer

S(k,ω) = �0|O†(k)|f��f |O(k)|0�δ(ω − (Ef − E0))

O =�

i

exp[ik · r]

Moving Pairs Breaking pairsBEC

BCS

Spin Responsefrom Bragg Spectroscopy

S. Hoinka, M. Lingham, M. Delehaye, and C. J. Vale, arXiv 1203.4657

Large momentum transfer : sensitive to contact, gap,...

Spin response in neutron matter critical for neutrinoslow q important, depends upon L.S, tensor interactions

Pairing Gap at Unitarity - ExperimentRF response

Credit: Greg Kuebler, JILA

Shin, Ketterle, ... 2008

0 0.2 0.4 0.6 0.8 1 1.2( k / kF)2

0

0.2

0.4

0.6

0.8

1

Δ /

EF

Shift of response of paired vs. unpaired atoms

Neutron Matter and Cold Atom Pairing Gap

Transition from weak pairing to near unitarity

0 0.1 0.2 0.3 0.4 0.5 0.6kF [fm-1]

0

0.5

1

1.5

2

2.5

Δ (M

eV)

Reuter-Schwenk, in prep.BCSChen 93Wambach 93Schulze 96Schwenk 03Cao 06Fabrocini 05dQMC 09AFDMC 08QMC AV4

0 2 4 6 8 10 12- kF a

Pairing Gap: Cold Atoms and Neutron Matter

At small |kF a| consistent with Gorkov polarization suppression of BCSAt large densities consistent withcold atom results

0 2 4 6 8

-60

-40

-20

0

20

40

Inhomogeneous Matter

Atoms: nearly 1D Huletexternal potentials

Neutron Star Matter

(Chamel)

Inhomogeneous Neutron Matter

N = 6 to 50 NeutronsHarmonic Oscillator and

Wood-Saxon external wells

Explore very large isospin limit of the density functional.Examine gradient, spin-orbit, and pairing terms

UNEDF SCIDAC

Gandolfi, et al, PRL 2011

Repulsive gradient terms required to fit neutron dropsalso smaller spin-orbit, pairing interactions

Improved Density FunctionalsNeutron Drops, Masses, Fission,...

0 20 40 60 80N

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

E / E

TF2

KaplanBlumeGandolfi

ETF = ω(3N)4/3/4

Results for Trapped Fermions at Unitarity

0 20 40 60N

0.5

0.6

0.7

0.8

0.9

1

1.1

E / (

ω N

4/3 )

ω = 5 MeV ω = 10 MeV

Neutrons

Unitary Fermi Gas

Trapped Neutrons and Unitary Fermi Gas

neutrons: Gandolfi, Pieper, JC, PRL 2011atoms: Gandolfi, Gezerlis, Forbes, preliminary

Effective range impacts: EOS, shell structure, pairing gaps

Future

Transition from 3D to 2D in cold atom systems

Pairing in inhomogeneous systems in strong interaction regime

Spin/Density response at small/moderate q

Spin response in neutron matter

Additional response: viscosity,...

Low-energy excitations