Laboratoire Kastler Brossel Coll` ege de France, ENS, UPMC, CNRS - PowerPoint PPT Presentation

Laboratoire Kastler Brossel Coll` ege de France, ENS, UPMC, CNRS Anomalous momentum diffusion of strongly interacting bosons in optical lattices Fabrice Gerbier J´ erˆ ome Beugnon Rapha¨ el Bouganne Manel Bosch Aguilera Alexis Ghermaoui Humboldt Kolleg, Vilnius, August 1st 2018

Ytterbium team at LKB M. Bosch J. Beugnon R. Bouganne E. Soave FG Aguilera (now Innsbruck) Former members : Q. Beaufils A. Dareau D. Doering A. Ghermaoui M. Scholl E. Soave Some recent works : • Revealing the Topology of Quasicrystals with a Diffraction Experiment [Dareau et al. , Phys. Rev. Lett. 119 , 21530 (2017) . • Clock spectroscopy of interacting bosons in deep optical lattices [Bouganne et al. , New J. Phys. 19 , 113006 (2017) .

Superfluid-Mott insulator transition for bosons Optical lattices : interference pattern can be used to trap atoms in a periodic structure y Bosons in the Bose-Hubbard regime : 1 Quantum tunneling favor delocalization √ 2 J 2 Repulsive on-site interactions favor localization J U Quantum phase transition from a superfluid, Bose-condensed 3 U ground state to a Mott insulator x Greiner et al. , Nature 2002. Energy/Temperature scales : nanoKelvin Time scales ∼ 10 ms

Quantifying phase coherence in a bosonic gas Time-of-flight interferences : essentially a free flight revealing the momentum distribution � � K = M r n tof ( r , r ) ≈ n = G ( K ) S ( K ) � t • smooth “Wannier” enveloppe G ( K ) i,j e i K · ( r j −· r i ) � ˆ a † • structure factor S ( K ) = � i a j � Single-particle density matrix : ρ (1) ( r , r ′ ) = � ˆ Ψ † ( r )ˆ Ψ( r ′ ) � Contrast of interference fringes when two matter waves overlap. a † Lattice version : C ( i, j ) = � ˆ i ˆ a j � Momentum distribution n ( k ) : Fourier transform of ρ (1) Lattice version : S ( K )

Quantum-degenerate 174 Yb atoms in a 3D optical lattice How does spontaneous emission destroy the spatial coherences initially present in the superfluid ? Absorption/spontaneous emission cycles with rate γ sp

Momentum diffusion in resonant atom-light interaction Momentum diffusion : • Random momentum kicks after SE ω L , k L k 1 • Random walk in momentum space: √ k 2 ∆ k = 2 Dt D : Diffusion coefficient k 3 • Central role in laser cooling : limit D temperature T ∝ friction Equivalent point of view : destruction of spatial coherences ρ (1) ( r , r ′ ) • Spatial correlations beyond λ 0 / (2 π ) strongly suppressed [ Pfau et al. , PRL 1994 ] • Exponential decay in time for given r , r ′ • Interpretation : Modern version of Heisenberg’s microscope continuous, weak measurements of the atom position [ Marsteiner et al. , PRL 1996 ]

Anomalous momentum diffusion Basic analysis : monitor n k =0 ≡ n 0 (proxy for condensed fraction) • Exponential (linear ...) decay at short times t ≤ t cross • Algebraic decay at long times: ∆ k ∼ t α , n k =0 ∼ 1 /t 2 α with α < 1 / 2 • Normal momentum diffusion : exponential decay

A more precise analysis of the momentum distribution � C R e i k · R S 0 ( k ) = R ∈ ❩ 2 � � ≈ 1 + C nn cos( k x d ) + cos( k y d ) + · · · Nearest-neighbor correlation function : � a † C nn = � ˆ r i ˆ a r i + e x � r i

Continuous, weak measurement theory Dissipative Bose-Hubbard model : � ˆ d ρ = 1 n i − 1 ρ − 1 − γ ˆ ˆ � n 2 n 2 � � � � � dt ˆ H, ˆ ρ L ρ ˆ , L ρ ˆ = ˆ n i ˆ ρ ˆ 2 ˆ i ˆ 2 ˆ ρ ˆ i . i � i Poletti et al. , PRL 2012, PRL 2013 (Kollath/Georges group) See also Pichler et al. , PRA 2010 (Zoller group), Yanay and Mueller, PRA 2012 N bosons in two wells L, R . Fock basis : | n � = | n L = n, n R = N − n � Populations : ρ n,n = � n | ˆ ρ | n � Coherences : ρ m,n = � m | ˆ ρ | n � , m � = n • Fock states are pointer states: � n | ˆ L [ˆ ρ ] | n � = 0 2 ( n − m ) 2 ρ m,n • Coherences decay : � m | ˆ ρ ] | n � = − 1 L [ˆ • Role of tunneling : partial restoration of short-range spatial coherence = ⇒ allows relaxation of populations

Effective Pauli master equation in the decoherent regime Adiabatic elimination of fast variables (coherences) Effective master equation for the probability p n to find n atoms per site ( ∆ t ≫ γ − 1 ) : p n ≡ ∆ p n ˙ = W n +1 p n +1 + W n − 1 p n − 1 − 2 W n p n ∆ t Poletti et al. , PRL 2012, PRL 2013 1 . 0 0 . 8 0 . 6 p ( n ) 0 . 4 0 . 2 0 . 0 0 2 4 6 8 n Three successive stages in the relaxation : 1 t � γ − 1 : initial relaxation of coherences (not described by the master equation), 2 γ − 1 < γt ≪ t ∗ : algebraic regime with slow decay of populations, 3 t � t ∗ : final relaxation to the (infite temperature) steady state.

From regular to anomalous diffusion n − N Mapping to a Fokker-Planck equation for N ≫ 1 : n → x = 2 , p n → Np ( x ) N ∂p ( x, t ) = ∂ D ( x ) ∂ � � ∂x p ( x, t ) ∂t ∂x � � 1 x Scaling solution: p ( x ) = τ β f u = τ β 1 If D ∼ x − η , scaling exponent β = 2+ η � γ ≫ NU : D ≈ 2 J 2 J 2 γ 1 � γ ≪ NU : D ≈ ( NU ) 2 × � 2 γ x 2 • “Quantum Zeno effect” • “Interaction-induced impeding of decoherence” Patil, Chakram, Vengalattore, PRL 2015 • D uniform : • Power-law tail : regular diffusion Anomalous (sub-)diffusion • Scaling exponent β = 1 / 2 • Scaling exponent β = 1 / 4 Poletti et al. , PRL 2012, PRL 2013

Coherence decay exponents versus lattice depth V ⊥ Phase coherence, scaling regime : n = 2 U/J = 20 10 − 1 γ/U = 0 1 a † C nn C nn = � ˆ i ± 1 ˆ a i � ∝ t 2 β � γ → 0 10 − 2 c 0 10 0 10 1 10 2 C nn ≈ √ 2 zγt if n → ∞ γt Comparison with experimental decay exponents of C nn :

Direct observation of Fock space dynamics using three-body losses 1 . 0 0 . 8 0 . 6 p ( n ) 0 . 4 0 . 2 0 . 0 0 2 4 6 8 n V ⊥ = 8 . 8 E R 0 . 15 0 . 10 p 3 0 . 05 0 . 00 10 − 3 10 − 2 10 − 1 10 0 γ sp t diss

Conclusion Decoherence of a bosonic many-body system under spontaneous emission Main message : Interactions slow down decoherence √ • Observation of anomalous momentum diffusion : ∆ k ∼ t 1 / 4 instead of ∆ k ∼ t • Interpretation as a signature of an underlying anomalous diffusion in Fock space • Direct observation of Fock space dynamics using three-body losses • Why is the “Poletti at al. ” model working ? Many effects left out : dipole-dipole interactions, superradiance, interband transitions ... • Numerical study of XXZ chains [ Cai and Barthel, PRL 2013 ] under dephasing : similar power-law slowdown of behavior as we observed (but not the Ising chain) Universality classes also relevant for non-equilibrium phenomena/decoherence ?

Loss dynamics after freezing • Same experiment as before : illumination with near-resonant laser for t diss • then raise the horizontal lattice to “freeze” the density distribution • wait for t hold : three-body losses empty sites with n ≥ 3 • monitor losses to extract p ( n ≥ 3) SF 5Er – t diss = 2500 µs 30000 25000 20000 15000 0 1000 2000 3000 4000 5000 t [ms] Solid lines : fit with known loss time constants to extract N 3 = 3 p (3) , etc ...

Evolution of triply-occupied sites V ⊥ = 5 E R 15000 10 4 12500 10000 N 3 N 3 7500 5000 2500 10 3 0 10 − 1 10 0 0 1 2 3 4 5 6 t diss (Γ − 1 sp ) Γ sp t diss V ⊥ = 9 E R 15000 10 4 12500 10000 N 3 N 3 7500 5000 2500 10 3 0 10 − 1 10 0 0 1 2 3 4 5 6 t diss (Γ − 1 sp ) Γ sp t diss

Excited band populations From fits to momentum profiles :