Ultracold dipolar atoms in two dimensions: From Wigner crystal to - - PowerPoint PPT Presentation

Ultracold dipolar atoms in two dimensions: From Wigner crystal to pair superfluidity and ferromagnetism

• S. Giorgini (BEC Trento)

CNR – Istituto Nazionale di Ottica Research and Development Center on Bose-Einstein Condensation

Dipartimento di Fisica – Università di Trento

Frontiers in Two-Dimensional Quantum Systems Trieste ICTP, November 13 – 17 2017

Outline

Ø Introduction to ultracold dipolar gases Ø Single layer of dipolar fermions

• QPT from Fermi liquid to Wigner crystal

Ø Dipolar Fermi polaron in bilayers

• Interlayer coupling between impurity and FL or WC

Ø Bilayer of dipolar fermions and bosons

• Fermions: Novel type of BCS-BEC crossover
• Bosons: Single-particle to pair superfluidity

Ø Single layer of two-component dipolar fermions

• Ferromagnetic instability

Cold gases: interactions are s-wave and short range

typical range of interaction typical interparticle distance

s-wave scattering is sufficient to describe interactions

• With dipoles interactions are anisotropic and long range

Leads to new interesting many-body effects

V(r) = d 2 r3 1-3cos2θ

( )

Dipoles aligned along z

è electric dipole d è magnetic dipole d=µ R0 ≈ 10 nm Strength of interaction typical length r0=md2/ħ2 1/kF≈ 100 nm

Science, 345 (2014)

• Phys. Rev. Lett., 116 (2016)

Science, 352 (2016) arXiv:1705.06914 (2017)

• Atomic species with large magnetic moment
• Chromium: Stuttgart – µ=6µB è d=0.06D - (r0= 2.4 nm)
• Dysprosium: Stanford, Stuttgart – µ=10µB è d=0.09D - (r0= 21 nm)
• Erbium: Innsbruck – µ=7µB (r0=10 nm)
• Heteronuclear molecules with large electric moment
• 40K-87Rb: JILA è d=0.57D - (r0= 611 nm)
• 23Na-40K: MIT, Hannover è d=2.7D - (r0=6800 nm)
• 6Li-133Cs: Heidelberg è d=5.5D - (r0= 62 µm)
• ….

kFr0= 0.02 - 0.2 kFr0= 6 – 600

In 2D enhanced stability

V(r) = d 2 r3 1-3sin2θ0 cos2ϕ

( )

(from Yamaguchi et al. 2010)

if θ0=0 interaction purely repulsive i. avoids bad chemistry ii. avoids clusterization due to head to tail attraction KRb + KRb è K2 + Rb2 + energy

Single-layer systems

• perpendicular dipoles

– fluid to solid transition – hexatic phase (Lechner et al.)

• tilted dipoles

– CDW (stripe) phase

(Bruun and Taylor, Parish and Marchetti)

– p-wave Fermi superfluidity

(Sieberer and Baranov) For bosons: Astrakharchik et al., Buechler et al.

Hamiltonian

(r0>>az transverse confinement)

One dimensionless parameter: kFr0 Use FN-DMC: projection method Nodal surface of ψT kept fixed during time evolution è E0 upper bound of ground-state energy

H = − 2 2m ∇i

2 +

d 2 r

ij 3 i<j

∑

i=1 N

∑

ψ0e−τE0 = limτ→∞ e−τHψT = limn→∞ e−δτH... e−δτH

n times

       ψT

kF = 4πn r0 = md 2 2

Fermi-liquid phase Crystal phase

2 4 6 8 10 12 kF x 2 4 6 8 10 12 kF y 0.5 1 1.5 2 2.5

ψT (r

1,...,rN ) =

f (r

ij) det e−(ri−Rm )2/α2

( )

i<j

∏

Rm are the lattice points of the WC

crystal liquid

ψT (r

1,...,rN ) =

f (r

ij) det e ikα⋅ri

( )

i<j

∏

Equation of state

0.5 0.6 0.7 0.8 0.9 1 0.1 1 10 E/EHF kFr0

• 0.002

0.002 0.004 0.006 20 30 40 50 60 70 !E/EHF kFr0

• FL to WC transition at kFr0=25±3 (in bosons kFr0 ≈ 60)

WC classical energy + z.p. motion of phonons

(Mora et al. 2007)

Bilayer system (no interlayer tunneling)

• bound state of two particles (analogy with electron-hole exciton)
• Fermions: interlayer superfluidity and BCS-BEC crossover as a function
• f separation λ (Pikovski et al.)

(analogy with electron-hole bilayer and two bilayer graphene – quest for high-Tc superconductivity)

Polaron problem in bilayer system

H = − 2 2m ∇i

2 +

d 2 r

ij 3 i<j

∑

i=1 N

∑

+ V(r

ip) i=1 N

∑

V(r

ip) = d 2(r ip 2 − 2λ 2)

(r

ip 2 + λ 2)5/2

where

• Bound state always exists for 2 particles
• Many-body problem depends on:

a. kFr0 (interaction in lower layer) b. kFλ (interlayer coupling)

Polaron energy

µP = EN+pol − EN

a) In units of Fermi energy varies by orders of magnitude as a function of kFl b) At strong interlayer coupling (small kFl) è 2-body binding energy

Polaron effective mass

a) very different behavior at large interlayer coupling in FL and WC phase b) polaron “localization” in WC phase

Bilayer system with balanced populations (Na=Nb)

Mean-field result

• 𝜈 = 𝜁$ +

&' (

• Δ =

2𝜁$|𝐹-|

• H = − 2

2m ∇i

2 i=1 Na

∑

+ ∇α

2 j=1 Nb

∑

       + d 2 r

i ′ i 3 i< ′ i

∑

+ d 2 rj ′

j 3 j< ′ j

∑

+ V(r

ij) i, j

∑

V(r

ij) = d 2(r ij 2 − 2λ 2)

(r

ij 2 + λ 2)5/2

where

Fermions: Effective 2D system (always dimer bound state)

Equation of state

Ø weak intra-layer repulsion kFr0=0.5 Ø dimer binding energy Eb is the largest scale in the BEC regime Single layer of fermions Single layer of composite bosons

Pairing gap

unbalanced populations: P = Na − Nb

Na + Nb

In the BEC regime Eb provides dominant contribution to gap

Schematic phase diagram

• BCS to BEC separation when µsl~|Eb|/2
• At small kFl critical density of WC transition reduced by factor 8 with

respect to Bose single layer (kFr0~60)

Bosons (DMC method provides exact ground state)

T=0 equation of state: in-plane interaction nr0

2=1

Energy per particle as a function of interlayer distance h

At small interlayer distance: stable gas of pairs

Single layer of atoms Single layer of pairs mass=2m dipole moment=2d

Quantum phase transition from single-particle to pair superfluidity

• Superfluid response of single atoms

from winding number (super-counterfluid density)

• Atomic condensate from OBDM

ψu(d)

+

( r)ψu(d)(′ r ) → n0

• Intrinsic molecular condensate

from TBDM

ψu

+(

r)ψd

+(

r)ψd(′ r )ψu(′ r ) − n0

2 → nM

Pairing gap in single-particle excitations

0. 2 0. 3 0. 4 0. 5 0. 6 10 20 30 40

0. 00 0. 05 0. 10 0. 15 50 100 150 200

!

|

"b|

/ 2

h/ r = 0. 4

!,

[

!

2/

m r

2 0]

h/ r = 0. 2

( E( P)

• E(

0) ) / N, [

!

2/

m r

2 0]

h / r

P

3 6 9 12 15

Δgap≠0 in the pair superfluid

P = Na − Nb Na + Nb

T=0 schematic phase diagram

S-P SF Pair SF

Freezing of single atomic layer Freezing of single pair layer

Single-layer two-component Fermi gas (Na=Nb) Itinerant ferromagnetism Spin symmetric Hamiltonian

H = − 2 2m ∇i

2 +

d 2 r

ij 3 i<j

∑

i=1 N

∑

Paramagnetic state Ferromagnetic state

• No competition with pairing instability
• Ferromagnetism driven by exchange effects

Analogy with Coulomb gas

Preliminary results using VMC Compare FM with PM ground state

2 4 6 8 10 12 14 16

kFM

F r0 0.65 0.70 0.75 0.80 0.85 0.90

EVMC [EFM

HF ]

(N↑, N↓) = (61, 61) (N↑, N↓) = (121, 0)

• Use DMC with fixed node approximation
• Add backflow to improve PM wave function

Thank you for your attention! Collaborators Trento BEC group UPC Barcelona

Natalia Matveeva Grigori Astrakharchik Jordi Boronat Ferran Mazzanti Adrian Macia Tommaso Comparin

LPMMC Grenoble

Markus Holzmann