Massless Dirac Fermions with cold atoms

Benoît Grémaud1 Christian Miniatura2

Berthold-Georg Englert3 Kean Loon Lee3 Rui Han3

1Laboratoire Kastler Brossel (Paris)

2Institut Non-Linéaire de Nice

3National University of Singapore

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Two level atom in laser field

ωω

0L

∆

|0>

|1>I ω0 atomic frequency (λ0 ≈ 671nm for

Lithium)I Γ spontaneous emisssion rate≈ 2π × 6MHz

I ωL laser frequency, detuning ∆ = ωL − ω0

For ∆ Γ (typically 103 − 104Γ), spontaneous emission is negligible⇒ effective potential V ∝ Γ

∆ I(R)

Residual spontaneous rate ≈ 1~

Γ∆V (can be less than 1 per second)

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Optical lattices - Graphene

x

y

θ

I

I

I

1

2

3-4

-4-2

-2

0 0y

x2 2

4 4

V (R) ∝∑

ij

√Ii Ij exp (i [ki − kj ] · R)

⇒ the reciprocal lattice is generated by ki − kj

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Band structure - I

Natural units:I length: λL

2π ≈ 0.1µmI momentum: ~kL

I energy: Er =~2k2

L2M (recoil energy) ≈ 3µK for 6Li

⇒ scaled hamiltonian:

H = −∆

2+ V0

∑ij

αiαj exp(

i [ni − nj ] · R)

experimentally: V0 values up to 30Er

Reciprocal lattice basis:

a∗1 = n1 − n2

a∗2 = n1 − n3

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Band structure - II

V0 = 25ErE0 ≈ 10.35Er

V0 Er ⇒ tight-binding approximation

H ≈ −tX

<i,j>

“A†

i Bj + h.c”

t ≈„

V0

Er

«exp

"−

„V0

Er

«1/2#

The Fermi velocity ∝ t at the crossing can be easily tuned (≈ few cm/s)

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Band structure - III

Unbalanced laser intensities (or misalignment)I3 = I2 = 1.001I1 ⇒ gap ∆ ≈ 3.10−6Er (green and blue curves)

0.45 0.5 0.55kx

I3 = I2 = I1 (black and red curves)

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

effective DC field

φi(t , R) = ni · R− ωLt + φ0i

V (R) ∝∑

ij

αiαj exp(

i [ni − nj ] · R + φ0i − φ0

j

)⇒ translation δR of the lattice, with:

δR = −φ0

1 − φ02

2πa1 −

φ01 − φ0

3

2πa2

Chirping the laser frequencies: ωi(t) = ωL + δωi(t)⇒ δφ0

i (t) = −δωi(t)t

I δωi(t) =Cste ⇒ uniform translationI δωi(t) ∝ t ⇒ uniformly accelerated translation⇒ in the accelerated frame, an effective constant force

experimentally: δω = αt , with α ≈ few MHz/s ⇒ acceleration of theorder of few ms−2

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

effective DC field

φi(t , R) = ni · R− ωLt + φ0i

V (R) ∝∑

ij

αiαj exp(

i [ni − nj ] · R + φ0i − φ0

j

)⇒ translation δR of the lattice, with:

δR = −φ0

1 − φ02

2πa1 −

φ01 − φ0

3

2πa2

Chirping the laser frequencies: ωi(t) = ωL + δωi(t)⇒ δφ0

i (t) = −δωi(t)t

I δωi(t) =Cste ⇒ uniform translationI δωi(t) ∝ t ⇒ uniformly accelerated translation⇒ in the accelerated frame, an effective constant force

experimentally: δω = αt , with α ≈ few MHz/s ⇒ acceleration of theorder of few ms−2

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Cold fermions - Preparation

I Sympathetic Cooling between Bosonand Fermion (ex: 7Li and 6Li)⇒ N ≈ 104 spin-polarized fermions|F = 3

2 , mF = 32 〉 in (asymmetric)

harmonic potentialω0 ≈ 2π.102 − 103Hz(i.e.Tho ≈ few nK)

I Final temperature T ≈ 0.2µK≈ 0.2TF

⇒ Can be transfered in|F = 1

2 , mF = ± 12 〉, with controlled

population number.

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Cold fermions - Lattices

−100 −50 0 50 1000

0.5

1

1.5

2

Filling factor: ρ = NFd3

ζ3

with ζ =√

2t/Mω20 , the “size” of the single particle wavefuntion

d is the size of a site, t the tunneling rate, ω0 the harmonic trapfrequency

ρ can be tuned with NF , t and ω0

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Cold fermions - Measurement

Time of flight proceduren(R, t) ∝ n(k = RM/~t)

adiabatic release of the trap/lattice⇒ maps the band-structure to thefree particle spectrum⇒ direct measurement of thequasi-momentum distribution

M.Köhl, et al. PRL, 94, 80403 (2005)

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Controlling the interactionsVan der Waals interaction with size ≈ 1.5nm ⇒ almost interaction free

Interaction potential between two fermions:VI(R12) = VS(R12)Psinglet + VT (R12, B)Ptriplet

Hyperfine interaction couples the singlet and triplet statesFeshbach resonance occurs when δ = 0 ⇒ stronger interaction

The scattering length between the two fermions can be tunned(from few nm up to µm)

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Quantum Hall effect

Lattice in rotation:

S. Tung, V. Schweikhard and E.A. Cornell, PRL 92 240402 (2006)In the rotating frame:

H =(P−Mω0ez × R)2

2M+ (ω0 − Ω)Lz + V (R)

ω0 is the harmonic trapping frequency (102 − 103Hz)Quantum Hall effect ω0 = 2ωc

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07

Outlook

I Experimental aspects (true geometry, temperature, interaction...)for realistic predictions.

I Tuning the tunnelling rates perpendicular to the lattice ⇒multilayer graphene

I Total spin I can be larger than 12

I Klein paradox

Massless Dirac Fermions with cold atoms Physique du graphène - 22 et 23 mai 07