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) Frontiers in Two-Dimensional Quantum Systems Trieste ICTP, November 13 – 17 2017 CNR – Istituto Nazionale di Ottica Research and Development Center on Bose-Einstein Condensation Dipartimento di Fisica – Università di Trento

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 R 0 ≈ 10 nm typical range of interaction 1/k F ≈ 100 nm typical interparticle distance s-wave scattering is sufficient to describe interactions • With dipoles interactions are anisotropic and long range Dipoles aligned along z V ( r ) = d 2 ( ) r 3 1-3cos 2 θ è electric dipole d è magnetic dipole d=µ Strength of interaction typical length r 0 =md 2 / ħ 2 Leads to new interesting many-body effects

Science, 345 (2014) Phys. Rev. Lett., 116 (2016) Science, 352 (2016) arXiv:1705.06914 (2017)

o Atomic species with large magnetic moment • Chromium: Stuttgart – µ=6µ B è d=0.06D - (r 0 = 2.4 nm) • Dysprosium: Stanford, Stuttgart – µ=10µ B è d=0.09D - (r 0 = 21 nm) • Erbium: Innsbruck – µ=7µ B (r 0 =10 nm) k F r 0 = 0.02 - 0.2 o Heteronuclear molecules with large electric moment o 40 K- 87 Rb: JILA è d=0.57D - (r 0 = 611 nm) o 23 Na- 40 K: MIT, Hannover è d=2.7D - (r 0 =6800 nm) o 6 Li- 133 Cs: Heidelberg è d=5.5D - (r 0 = 62 µm) o …. k F r 0 = 6 – 600

In 2D enhanced stability V ( r ) = d 2 ( ) r 3 1-3sin 2 θ 0 cos 2 ϕ if θ 0 =0 interaction purely repulsive (from Yamaguchi et al. 2010) i. avoids bad chemistry KRb + KRb è K 2 + Rb 2 + energy ii. avoids clusterization due to head to tail attraction

Single-layer systems • perpendicular dipoles – fluid to solid transition For bosons: Astrakharchik et al., Buechler et al. – hexatic phase (Lechner et al.) • tilted dipoles – CDW (stripe) phase (Bruun and Taylor, Parish and Marchetti) – p-wave Fermi superfluidity (Sieberer and Baranov)

Hamiltonian (r 0 >>a z transverse confinement ) N H = −  2 d 2 2 + ∑ ∑ ∇ i 3 2 m r i = 1 i < j ij 4 π n r 0 = md 2 One dimensionless parameter: k F r 0 k F =  2 Use FN-DMC: projection method ψ 0 e − τ E 0 = lim τ →∞ e − τ H ψ T = lim n →∞ e − δτ H ... e − δτ H  ψ T       n times Nodal surface of ψ T kept fixed during time evolution è E 0 upper bound of ground-state energy

( ) ∏ i k α ⋅ r i Fermi-liquid phase ψ T ( r 1 ,..., r N ) = f ( r ij ) det e i < j Crystal phase ( ) ij ) det e − ( r i − R m ) 2 / α 2 ∏ ψ T ( r 1 ,..., r N ) = f ( r i < j R m are the lattice points of the WC liquid crystal 2.5 12 2 10 8 1.5 k F y 6 1 4 0.5 2 0 0 0 2 4 6 8 10 12 k F x

Equation of state 1 0.9 0.8 E/E HF 0.006 0.7 ! E/E HF 0.004 0.002 0.6 0 -0.002 WC classical energy 0.5 20 30 40 50 60 70 k F r 0 + z.p. motion of phonons 0.1 1 10 (Mora et al. 2007) k F r 0 • FL to WC transition at k F r 0 =25±3 (in bosons k F r 0 ≈ 60)

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 of separation λ (Pikovski et al.) (analogy with electron-hole bilayer and two bilayer graphene – quest for high-Tc superconductivity)

Polaron problem in bilayer system N N H = −  2 d 2 2 + ∑ ∑ ∑ V ( r ip ) ∇ i + 3 2 m r ij i = 1 i < j i = 1 where 2 − 2 λ 2 ) ip ) = d 2 ( r ip V ( r 2 + λ 2 ) 5/2 ( r ip • Bound state always exists for 2 particles • Many-body problem depends on: a. k F r 0 (interaction in lower layer) b. k F λ (interlayer coupling)

Polaron energy µ P = E N + pol − E N a) In units of Fermi energy varies by orders of magnitude as a function of k F l b) At strong interlayer coupling (small k F l ) 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 (N a =N b )   N a N b H = −  2 d 2 d 2 ∑ 2 ∑ 2 ∑ ∑ ∑   V ( r ij ) ∇ i + ∇ α  + + +  3 3 2 m r r j ′   i ′ i j i = 1 j = 1 i < ′ i j < ′ j i , j where 2 − 2 λ 2 ) ij ) = d 2 ( r ij V ( r 2 + λ 2 ) 5/2 ( r ij Fermions: Effective 2D system (always dimer bound state) Mean-field result & ' • 𝜈 = 𝜁 $ + ( • Δ = 2𝜁 $ |𝐹 - |

Equation of state Ø weak intra-layer repulsion k F r 0 =0.5 Ø dimer binding energy E b is the largest scale in the BEC regime Single layer of fermions Single layer of composite bosons

Pairing gap unbalanced populations: P = N a − N b N a + N b In the BEC regime E b provides dominant contribution to gap

Schematic phase diagram BCS to BEC separation when µ sl ~|E b |/2 • At small k F l critical density of WC transition reduced by factor 8 with • respect to Bose single layer (k F r 0 ~60)

Bosons (DMC method provides exact ground state) 2 =1 T=0 equation of state: in-plane interaction nr 0 Energy per particle as a function of interlayer distance h At small interlayer distance: stable gas of pairs Single layer of pairs mass=2m dipole moment=2d Single layer of atoms

Quantum phase transition from single-particle to pair superfluidity (  r ) ψ u ( d ) (  ′ + r ) → n 0 ψ u ( d ) • Atomic condensate from OBDM + (  + (  r ) ψ d (  ′ r ) ψ u (  ′ 2 → n M • Intrinsic molecular condensate r ) ψ d r ) − n 0 ψ u from TBDM • Superfluid response of single atoms from winding number (super-counterfluid density)

Pairing gap in single-particle excitations Δ gap ≠ 0 in the pair superfluid 40 0 ] 0 ] 2 m r 2 15 m r P = N a − N b 200 2 / 2 / h/ r 0 = 0. 2 ! 12 ! [ [ N a + N b ! , 150 N, 30 9 / ) 100 0) 6 E( - 50 3 P) h/ r 0 = 0. 4 20 E( 0 0 ( 0. 00 0. 05 0. 10 0. 15 P 10 ! | " b | / 2 0 0. 2 0. 3 0. 4 0. 5 0. 6 h / r 0

T=0 schematic phase diagram Freezing of single atomic layer Freezing of single pair layer S-P SF Pair SF

Single-layer two-component Fermi gas (N a =N b ) Itinerant ferromagnetism Spin symmetric Hamiltonian Paramagnetic state H = −  2 N d 2 2 + ∑ ∑ ∇ i 3 2 m r ij i = 1 i < j • No competition with pairing instability Ferromagnetic state • Ferromagnetism driven by exchange effects

Analogy with Coulomb gas

Preliminary results using VMC Compare FM with PM ground state ( N ↑ , N ↓ ) = (61 , 61) 0 . 90 ( N ↑ , N ↓ ) = (121 , 0) 0 . 85 HF ] [ E FM 0 . 80 E VMC 0 . 75 0 . 70 0 . 65 2 4 6 8 10 12 14 16 k FM F r 0 • Use DMC with fixed node approximation • Add backflow to improve PM wave function

Thank you for your attention! Collaborators Trento BEC group LPMMC Grenoble Tommaso Comparin Natalia Matveeva Markus Holzmann UPC Barcelona Grigori Astrakharchik Jordi Boronat Adrian Macia Ferran Mazzanti