Dipolar quantum gases Barcelona, May 2010

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1 Barcelona, May D DQG Quasi-D DQG Institut für Theoretische Physik, Johannes Kepler Universität, Linz, Austria May,

2 Outline D DQG Quasi-D DQG : D polarization polarization : Slabs : weakly/unpolarized dipoles

3 to dipolar QGs experiments: permanent magnetic dipole moments of atoms (Cr): Pfau group (Lahaye et al, Nature 8, 67 (7)) permanent electric dipole moments of heteronuclear dimers (RbK, etc): transfer atom pairs to weakly bound state by Feshbach resonance transfer to rovibrational g.s. by STIRAP laser pulses (Innsbruck; JILA, NIST: Ni et al., Science, (8),...) Diatomic molecules in optical lattices (Danzl et al., Nature Physics 6, 65 (): Cs ) D DQG Quasi-D DQG dipole-dipole interaction: polarized, D: v dd (r) = d r polarized, D: v dd (r) = d cos θ r ê ê (ê ˆr)(ê ˆr) unpolarized, D: v dd (r ) = d r units: length r = md ; energy ǫ = mr

4 our interest: strong interactions effects strong correlations between pairs effects of anisotropy of V dd D DQG Quasi-D DQG effects of rotational degrees of freedom of molecular BEC methodology: quantum many-body method: hypernetted chain Euler-Lagrange for ground state (HNC-EL) and excited state (TDHNC-EL) recent progress by Campbell and Krotscheck on the TDHNC-EL front QMC: path integral ground state MC (PIGSMC) for ground state and path integral MC (PIMC) for T > recent progress in group here (quasi-6th order,... ) and by REZ and Chin on high order propagators ( any-order )

5 (time-dependent) hyper-netted chain Euler-Lagrange ground state: HNC-EL Φ (R) = Y i ϕ(r i) Y i<j δ H δu (r) =, f (r i, r j) = e P i u (r i ) e P i<j u (r i,r j )... δ H δu (r, r ) =, u (r i) only (& effective δ-potential): Hartree (GP) u (r i, r j): minimal requirement for repulsive interaction u (r i, r j, r k ): even better... δ H δu (r, r, r ) = D DQG Quasi-D DQG excitations: TDHNC-EL Ψ(R; t) = e ie t e δu(r;t) with δu(r; t) = X i Z δ Φ(R) Ψ Ψ / δu (r i; t) + X i<j δu (r i, r j; t) +... dt Ψ(t) H(t) i Ψ(t) = t δu (r i; t) only: Bjil-Feynman approximation (Bogoliubov-deGennes/linearized GP) δu (r i, r j; t) & some approximations: CBF-BW δu (r i, r j, r k ; t) triplets & less approximation: Krotscheck & Campbell

6 Perpendicular dipoles in D recent work with Ferran, Gregory, and Jordi: excitation spectrum by combining DMC with CBF-BW (PRL, 5 (9)) dynamic structure function S(k, ω): increase density: 5 roton energy not going to in vicinity of solidification phonon-roton splits off from Bogoliubov mode hω hωnr nr= 9 nr= nr= nr=6 Bogoliubov-deGennes (linearzed GP) ~ k ms (k) hω hωnr Bijl-Feynman approximation CBF-BW approximation 6 8 k/n/ w/triplets δu.. 6 k/n/.. 8 D DQG Quasi-D DQG

7 Tilted dipoles in D with HNC/-EL anisotropy is not probed in D with polarization axis tilt polarization axis to form homogeneous anisotropic D quantum gas (i.e. nematic quantum gas) HNC/-EL ground state calculation (no elementaries, no triplets): test system to study well-defined instability (at angle α cr = 5.6?) coupling of excitations: rotons in strongly correlated direction, but not in weakly correlated direction anisotropic solidification? gas state: isotropic speed of sound solid state: anisotropic speed of sound....8 g(r).6... y g(r) x D DQG Quasi-D DQG S(k) ǫ F (k) 5 S(k) 5 ε(k) k y 6 k x k y k x all plots for ρ = 6, α = α cr

8 Tilted dipoles in D with HNC/-EL increase density towards solidification: ρ = 56, α =. HNC-EL/ not reliable anymore: ρ = 8 S(k) without elementaries & triplets DMC simulation with elementaries & triplets 5 5 kn -/ g(r) y g(r) x D DQG Quasi-D DQG S(k) ǫ F (k) S(k) 8 6 k y 6 k x ε(k) k y k x roton energy? QMC + CBF-BW (or better)!

9 quasi-d: Slabs of dipolar quantum gases relax infinitely strong confinement in z-direction system unstable via tunneling towards head-to-tail configurations stabilize with repulsive interaction H = X i σ = : h m i + mω i z i + X i<j hv dd (r ij) + σ i r ij σ : D DQG Quasi-D DQG z z x x studied in mean field approximation (GP + linearized GP) by Santos et al., PRL 9 5 (): rotonization, followed by instability completely different roton than roton in dense, strongly interacting systems like LHe!

10 Roton in dilute system like in mean field: rotonization system unstable towards σ, ρ+, ω HNC-EL does not reach point of instability where roton energy Eroton (presumably) vanishes in He: Feynman roton too high by factor of beyond Bijl-Feynman approximation dynamic structure function S(k, E ) with CBF-BW approximation of TDHNC-EL (no triplet correlations; convolution approximation) roton energy almost unchanged strong damping at Eroton D DQG S(k, E ) for ρ =, ω =, σ =.:. Quasi-D DQG.9 E/ω Bijl-Feynman approximation k losc.8. system (meta)stable up to Eroton =, but HNC-EL collapses to lower-energy phase before that happens...

11 Pair distribution function: dimerization? pair distribution function g(z, z, r )... probability to find particle at (x, y, z) and (another) particle at (x, y, z ) (r = p (x x ) + (y y ) ), divided by ρ(z i) g(z,-z,r ) σ =.: z = -z r g(z,-z,r ) σ =.: z = -z r D DQG Quasi-D DQG z = z correlation hole at (,, ) due to repulsion (σ/r) strong peak at r = expected for dimer ( bound dipoles) dimerization? check other side of phase transition with QMC E roton small: system stable or just metastable energetics with QMC (D. Hufnagl, E. Krotscheck, REZ, JLTP 58, 85 ())

12 Stability analysis: static response function tracking E roton not practical in HNC-EL... static response function χ(z, z, r )...density response to a (weak) perturbation P i Upert(ri) F.T. w/resp to r : χ(z, z, k) diagonalize w/resp to z, z : χ n(k) maximal response: χ max[χ n(k)] D DQG Quasi-D DQG max[g] ω=.6 ω=. ω=. ω=5.8 ω=. /χ σ (extrapolation by fitting a(σ σ ) b )

13 Outlook on unpolarized / partially polarized DBG molecular DBG has inner degree of freedom: molecule rotation, ê i v dd (r ) = d ê ê (ê ˆr)(ê ˆr) r D DQG Quasi-D DQG Weak interactions mean field estimate: GP equation for Ψ (r, Ω) coupling rotational degrees of freedom of molecules by dipole-dipole interaction: splitting of j = state by ǫ nd = O( cm ) for d = 5Debye and n = cm Strong interactions HNC-EL w/rotations PIGSMC w/rotations

14 Acknowledgements collaborators: D DBG Ferran Mazzanti, Gregory Astrakharchik, Jordi Boronat D DQG Quasi-D DQG quasi-d DBG Diana Hufnagl (HNC-EL), Rainer Kaltseis (mean field), Vesa Apaja (CBF-BW), Eckhard Krotscheck (HNC-EL) unpolarized molecular BG Brendan Abolins, K. Birgitta Whaley: PIGSMC eee: Austrian Science Foundation FWF #6 Acciones Integradas ES-9/9