R = 10 R = 1k k R = 0k [E-E (N)]/2E [E-E (N)]/2E [E-E (N)]/2E

THREE-DIMENSIONAL IMPURITY PROBLEMS WITH COLD ATOMS

Yvan Castin, Christian Trefzger

LKB, Ecole normale sup´erieure (Paris, France) Mauro Antezza

Lab. Charles Coulomb, Universit´e de Montpellier (France) David Hutchinson

University of Otaga, Dunedin, New Zealand

A RICH PROBLEM

An impurity atom A (mass m_A) interacting with another species (or spin state) B (mass m_B) [no interaction among B atoms]:

1. monomeron/dimeron pb: a Fermi sea of B atoms on a (narrow) Feshbach resonance [with C. Trefzger]

2. strong localisation of A: the B atoms are randomly pinned at the nodes of an optical lattice [with M. An- tezza, D. Hutchinson, P. Massignan, U. Gavish]

3. photonic band gaps: A is a photon; one B atom per node of an optical lattice [with M. Antezza]

1 Monomeron-dimeron problem

1.1 Physical motivation

• Monomerons and dimerons are quasi-particles belonging to the general class of Fermi polarons [neutral objects dressed by the Fermi sea, rather than electrons dressed by phonons in solids]

• The ground state of strongly polarized spin 1/2 Fermi gas at unitarity [N_A ≪ N_B, A =↓, B =↑] is a Fermi gas of monomerons [Chevy; Lobo, Recati, Giorgini, Stringari;

Combescot, Giraud, Leyronas; Mora]

• monomeron = A dressed by particle-hole excitations of B Fermi sea. It is a quasi-particle of dispersion relation

∆E(P) =

P→0 ∆E(0) + P ²

2m_∗ + O(P⁴)

• At unitarity, measured equation of state of monomerons

≃ an ideal Fermi gas [Navon, Nascimb`ene, Chevy, Sa- lomon]

1.2 Monomeron vs dimeron

• The ground state of A, as the function of the AB scat- tering length a, has two branches with a cusp [Prokof ’ev, Svistunov]

• Dimeron = AB dimer dressed by particle-hole excita- tions of B Fermi sea

• transition from monomeron to dimeron at a_c

• a_c > 0 (broad Feshbach resonance) “intuitive”: dimer exists in free space for a > 0 only.

0.5 1 1.5 2 -20

-10 0

[E-E FS(N)]/2E F

0.6 0.7 0.8 0.9 1

-1.2 -1 -0.8 -0.6

-0.2 0 0.2 0.4 0.6

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1

1 1.5 2 2.5 3

-7 -6 -5 -4 -3 -2

[E-E FS(N)]/2E F

0 0.2 0.4 0.6 0.8 1 -0.8

-0.6 -0.4 -0.2

-1 -0.8 -0.6 -0.4 -0.2 0 -0.3

-0.2 -0.1

0 0.5 1 1.5 2

1/(k_Fa) -1

-0.8 -0.6 -0.4

[E-E FS(N)]/2E F

-4.5 -4.25 -4 -3.75 -3.5 1/(k_Fa)

-0.1 -0.05

-9 -8.8 -8.6 -8.4 -8.2 -8 1/(k_Fa)

-0.05 -0.04 -0.03 -0.02 -0.01

(a1)

(a2)

(a3)

(b1)

(b2)

(b3)

(c1)

(c2)

(c3)

M/m = 0.1505 M/m = 1 M/m = 6.6439

k FR * = 0k FR * = 1k FR * = 10

monomeron

dimeron 1/(k_Fa)_c = 1.75

1/(kFa)

c = 2.06

1/(kFa)

c = 1.07 1/(k

Fa)

c = -4.13 1/(k

Fa)

c = -8.38 1/(kFa)

c = -0.58 1/(k_Fa)_c = 0.25 1/(k_Fa)_c = 0.84

1/(kFa)

c = 0.36

1.3 Two-channel model

H = X

k

~²k²

2m_Aa^†_ka_k+ ~²k²

2m_Bb^†_kb_k+(E_mol+ ~²k²

2(m_A + m_B))γ_k^†γ_k + Λ

L^3/2

X

k_A,k_B

χ(k_rel)[γ_k^†

A+k_Ba_kAb_kB + h.c.]

Take cut-off to ∞ for a and Λ fixed (E_mol → +∞). Fesh- bach length R_∗ = π~⁴/(µ²Λ²). Free-space dimer iff a > 0.

1.4 How to solve

For Λ = 0:

one A, zero molecule zero A, one molecule E₀ = E_FS(N_B) E₀ = E_FS(N_B − 1) + E_mol

∆E_pol = 0 ∆E_dim = E_mol − E_F

For Λ > 0: Expand in the number of particle-hole pairs [Chevy ; Combescot, Giraud], here up to one pair.

ψ_mono〉= +

ψ_dim〉=

+ +

+ + +

...

...

B:

A:

B:

A:

Mol:

Mol:

k_A=0

vacuum

vacuum

vacuum k_mol=q

kA=q-k o

x

q oq

H H k H

H H H

kmol=0 vacuum

kA=q-k

kmol=q-k

vacuum k vacuum

A=-k

xk

oq

+

o H

xk x

k x k’

k_A=q-(k+k’)

vacuum q Generalized Chevy Ansätze

1.5 Numerical result

a_c < 0 at large k_FR_∗ ! Paradoxical. Stable dimeronic branch extends to a regime where no free space dimer.

0 1 2 3 4 5 6 7 8 9 10

k

R

* -8

-6 -4 -2 0 2

1/(k

a)

m_A/m_B=0.1505

m_A/m_B=1

m_A/m_B=6.6439

(a)

1.6 Analytics

• Two-body scattering amplitude [Petrov]:

f_krel = − 1

a1 + ik_rel + k_rel² R_∗

• Usual weakly attractive limit: a → 0⁻, R_∗ fixed

• Appropriate weakly attractive limit: a → 0⁻, aR_∗ fixed

• Define α = _m ^mA

A+m_B and

s ≡ αk_F (−aR_∗)^1/2

Note that s → 0 in the weakly attractive limit a → 0⁻.

lim

∆ E/E

1/2

s

monomeron dimeron

N_mol=0 N_mol=1

N_mol=0 N_mol=1

m_A

m_A+m_B 1

m_B m_A

m_A m_A+m_B

m_A m_A+m_B

Critical scattering length on a narrow Feshbach resonance:

1

k_F a_c =

k_FR_∗→∞ −αk_F R_∗ +2

π

"

1 − α⁻² + 1 2

α^−5/2 − α^1/2

ln 1 + α^1/2 1 − α^1/2

#

+ O( 1

k_F R_∗)

1.7 Physical interpretation of 1/(kFa)c < 0

• Stabilization of molecule by Fermi sea

• Molecule energy renormalized by Lambshift:

E˜_mol = E_mol +

Z d³k (2π)³

χ²(k)Λ²

0 − ~²k²/2µ = − Λ²µ 2π~²a

• Free space: molecule stable iff E˜_mol < 0 that is a > 0

• Fermi sea: molecule stable iff E˜_mol < E_F that is 1

k_F a > −αk_F R_∗

• For a < 0 this imposes

s > α^1/2 =

m_A

m_A + m_B

_1/2

2 Strong (Anderson) localisation of A matterwave

2.1 Physical motivation and configuration

• B randomly filling the nodes of an optical lattice (p_occ ≪ 1) with no tunneling

• A does not see the lattice potential, it sees a disordered ensemble of scatterers

• ~²k_A² /2m_A ≪ ~ω_B so elastic AB scattering

• A solvable alternative to laser speckle (Aspect). Opti- misation of localisation by tuning the A − B scattering length

• One expects (in 3D) an Anderson transition [mobility edge] between extended states (continuous spectrum) and localized states (point-like spectrum).

2.2 Model

• Each trapped B atom replaced by Wigner-Bethe-Peierls contact conditions at lattice node on A wavefunction, with effective scattering length a_eff :

φ(r_A) =

r_AB→0 D(r_B) ×

1

r_AB − 1 a_eff

| {z }

zero energy scattering state for r_AB≫a_ho

+ O(r_AB)

• Scatterers occupy a sphere of finite but large diameter

2.3 How to solve

• single particle Green’s function in presence of a number N_B of B scatterers at energy E = ~²k_A² /2m_A exactly given by N_B × N_B matrix inversion

G(r, r₀) = g(r−r₀)+2π~² m_A

N_B

X

i,j=1

g(r−r_i)[M⁻¹]_ijg(r_j−r₀)

M_ij =











−2π~²

m g(r_i − r_j) = e^ikArij/r_ij if i 6= j,

ik_A + a⁻¹_eff if i = j.

where free-space Green’s function g(r − r₀) is transla- tionally invariant.

• Gives access to localisation length ξ: decay length of field radiated by a source of A in the B medium:

G(r, r₀) ≃

|r−r₀| large Ae^{−|r−r0|/ξ}

|r − r₀|^α

• Gives access to density ρ_E of states (E < 0) [real poles of G] or resonances [complex poles of G] (E=Re z₀ > 0,

~Γ/2 = − Im z₀ > 0)

2.4 Matrix M is not an Anderson problem

• off-diagonal disorder only

• at E > 0, M_ij decays slowly for r_ij → ∞: more slowly than 1/r_ij³ . So no localized states ? (Boris Altshuler, informal discussion)

2.5 Numerical results

• p_occ = 1/10, diameter= 140d; hN_Bi ≃ 1.4 × 10⁵

• for a_eff < 0, for a_eff → +∞, no evidence of localisation [van Tiggelen, Lagendijk]

• The best for localisation is to take a_eff ≃ half mean inter-B distance

• Rich situation because we consider also E < 0. E.g.

for a_eff = d we shall find three mobility edges, one at positive energy and two at negative energy. Is certainly sensitive to the presence of the lattice.

LOCALISATION LENGTH

-2 -1.6 -1.2 -0.8 -0.4 0

0 2 4 6 8 10 12

ξ / d

0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 12

0 0.1 0.2 0.3 0.4 0.5 0

0.3 0.6 0.9 1.2 1.5 1.8

-2 -1.6 -1.2 -0.8 -0.4 0

0 0.3 0.6 0.9 1.2 1.5 1.8

κ d

(a)

(b)

E / E₀

a_eff/d =0.1 (black), 0.2 (red), 0.7 (green), 1 (blue), 1.3 (violet)

DOS WITHOUT/WITH SPATIAL FILTER

Oblique line = mean-field bottom E = ρ_Bg_eff

-2 -1.6 -1.2 -0.8 -0.4 0 0

2 4 6 8 10 12

ξ / d

0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 12

-0.2 0.6

mobility edge mobility edge mobility edge

spectral gap ρ_E=0

localized ρ_E^loc>0

ρ E>0

extended

states

states ρ_E>0 ρ_E^loc=0

ρ E

loc >0

localized states ρ_E>0

a_eff=d

extended states

ρ_E>0 ρ_E^loc=0 -1

E / E

0

-1.3

3 Photonic band gaps

3.1 Physical motivation

• find matter lattices leading to omnidirectional band gaps (OBG) for light

• first studies with extended objects (dielectric spheres) [Ho, Cha Soukoulis, 1990]

• for atoms: Lagendijk et al., 1996 predict OBG for fcc lattice

• Knoester et al. , 2006: Lagendijk’s sum is divergent;

addition by hand of a regularizing term; no OBG for fcc. But this no longer solves the original Hamiltonian problem.

3.2 Case of classical physics

• for a stationary state of the field at frequency ω, atom in R carries a mean electric dipole D~ (R):

D~ (R) = ǫ₀α(ω) X

R^′6=R

g(R − R^′)D~ (R^′)

• α(ω) ∝ 1/(ω − ω₀ + iΓ/2) is atomic polarisability

• g_ij(r) is component along i of electric field radiated in r by a unit dipole oscillating along j and located at the origin of coordinates:

g_ij(r) ∝ [(ω/c)²δ_ij + ∂_ri∂_rj]

| {z }

ensures transversality

e^i(ω/c)r

| {zr }

scalar field

• periodic case, Bloch theorem: ~ω = ǫ_q with D~ (R) = D~ e^iq·R

3.3 How to calculate the sum ?

• “formule sommatoire de Poisson” : X

R∈DL

f(R) = 1 V

X

K∈RL

f˜(K)

• but g(0) = ∞ and P

K K_iK_j/K² not absolutely conver- gent

• Physical regularising effect: atomic positions delocalized over a_ho, provides a Gaussian cut-off in Fourier space

g(R − R^′) → hg(R + u − R^′ − u^′)i_u,u′ = ¯g(R − R^′)

3.4 Looking for band gaps

• none for Bravais lattices (Knoester was right)

• the historical work of Soukoulis predicted OBG for a di- amond lattice of dielectric spheres (fcc+translated copy by (^d₄, ^d₄, ^d₄)). May be it also works with atoms ?

• we indeed predict a gap for an atomic diamond lattice

• important to check for the absence of free wave (field vanishes on all lattice sites) using

ω_free

c ≥ inf

K6=0

K 2

Photonic density of states for diamond, k₀d = 2 (k₀ = ω₀/c)

-6 -4 -2 0 2 4 6

(ω−ω₀)/Γ 0

1 2

3 4 5

ρ(ω) [1/ΓV L]

diamant extrapole a portee nulle

k0=1.2599 (k0a=2) & a_ho=0.091287 & nq=50

1 2 3 4 5 k

d

0 5 10 15

∆/Γ

diamant

aho=0.08 ; extrapole a portee nulle

4.7 4.8 4.9 5 5.1 5.2 k₀d

0 0.2 0.4 0.6 0.8

∆/Γ