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

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Laboratoire Kastler Brossel Coll` ege de France, ENS, UPMC, CNRS Anomalous momentum diffusion

f strongly interacting bosons in optical lattices

Fabrice Gerbier J´ erˆ

me Beugnon

Manel Bosch Aguilera Rapha¨ el Bouganne Alexis Ghermaoui Humboldt Kolleg, Vilnius, August 1st 2018

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Ytterbium team at LKB

M. Bosch

Aguilera

J. Beugnon
R. Bouganne
E. Soave

(now Innsbruck) FG

A. Ghermaoui

Former members :

Q. Beaufils
A. Dareau
D. Doering
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).

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Superfluid-Mott insulator transition for bosons

Optical lattices : interference pattern can be used to trap atoms in a periodic structure Bosons in the Bose-Hubbard regime :

1 Quantum tunneling favor delocalization 2 Repulsive on-site interactions favor localization

Quantum phase transition from a superfluid, Bose-condensed ground state to a Mott insulator

x y J √ 2J U 3U Greiner et al., Nature 2002.

Energy/Temperature scales : nanoKelvin Time scales ∼ 10 ms

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Quantifying phase coherence in a bosonic gas

Time-of-flight interferences : essentially a free flight revealing the momentum distribution ntof(r, r) ≈ n

K = Mr

t

= G (K) S (K)
smooth “Wannier” enveloppe G(K)
structure factor S(K) =

i,j eiK·(rj−·ri)ˆ

a†

i aj

Single-particle density matrix : ρ(1)(r, r′) = ˆ Ψ†(r)ˆ Ψ(r′) Contrast of interference fringes when two matter waves overlap. Lattice version : C(i, j) = ˆ a†

i ˆ

aj Momentum distribution n(k): Fourier transform of ρ(1) Lattice version : S(K)

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Quantum-degenerate 174Yb 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 γsp

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Momentum diffusion in resonant atom-light interaction

Momentum diffusion :

Random momentum kicks after SE
Random walk in momentum space:

∆k = √ 2Dt D : Diffusion coefficient

Central role in laser cooling : limit

temperature T ∝

D friction

ωL, kL

k1 k2 k3

Equivalent point of view : destruction of spatial coherences ρ(1)(r, r′)

Spatial correlations beyond λ0/(2π) 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]

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Anomalous momentum diffusion

Basic analysis : monitor nk=0 ≡ n0 (proxy for condensed fraction)

Exponential (linear ...) decay at short times t ≤ tcross
Algebraic decay at long times: ∆k ∼ tα, nk=0 ∼ 1/t2α with α < 1/2
Normal momentum diffusion : exponential decay

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A more precise analysis of the momentum distribution

S0(k) =

R∈❩2

CReik·R ≈ 1 + Cnn

cos(kxd) + cos(kyd)
+ · · ·

Nearest-neighbor correlation function : Cnn =

ri

ˆ a†

riˆ

ari+ex

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Continuous, weak measurement theory

Dissipative Bose-Hubbard model : d dt ˆ ρ = 1 i ˆ H, ˆ ρ

− γ ˆ

L

ˆ

ρ

,

ˆ L

ˆ

ρ

=
i

ˆ ni ˆ ρˆ ni − 1 2 ˆ n2

i ˆ

ρ − 1 2 ˆ ρˆ n2

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 = |nL = n, nR = N − n Populations : ρn,n = n|ˆ ρ|n Coherences : ρm,n = m|ˆ ρ|n, m = n

Fock states are pointer states: n| ˆ

L[ˆ ρ]|n = 0

Coherences decay : m| ˆ

L[ˆ ρ]|n = − 1

2 (n − m)2 ρm,n

Role of tunneling : partial restoration of short-range spatial coherence

= ⇒ allows relaxation of populations

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Effective Pauli master equation in the decoherent regime

Adiabatic elimination of fast variables (coherences) Effective master equation for the probability pn to find n atoms per site (∆t ≫ γ−1) : ˙ pn ≡ ∆pn ∆t = Wn+1pn+1 + Wn−1pn−1 − 2Wnpn

Poletti et al., PRL 2012, PRL 2013

2 4 6 8 n 0.0 0.2 0.4 0.6 0.8 1.0 p(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.

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From regular to anomalous diffusion

Mapping to a Fokker-Planck equation for N ≫ 1 : n → x =

n− N

2

N

, pn → Np(x) ∂p(x, t) ∂t = ∂ ∂x

D(x) ∂

∂x p(x, t)

Scaling solution: p(x) =

1 τβ f

u =

x τβ

If D ∼ x−η, scaling exponent β =

1 2+η

γ ≫ NU : D ≈ 2J2

2γ

“Quantum Zeno effect”

Patil, Chakram, Vengalattore, PRL 2015

D uniform :

regular diffusion

Scaling exponent β = 1/2

γ ≪ NU: D ≈

J2γ (NU)2 × 1 x2

“Interaction-induced impeding of

decoherence”

Power-law tail :

Anomalous (sub-)diffusion

Scaling exponent β = 1/4

Poletti et al., PRL 2012, PRL 2013

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Coherence decay exponents versus lattice depth V⊥

Phase coherence, scaling regime : Cnn = ˆ a†

i±1ˆ

ai ∝ 1 t2β Cnn ≈

c0 √2zγt if

γ → 0

n → ∞ 100 101 102

γt

10−2 10−1

Cnn

n= 2 U/J= 20 γ/U= 0 Comparison with experimental decay exponents of Cnn :

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Direct observation of Fock space dynamics using three-body losses

2 4 6 8 n 0.0 0.2 0.4 0.6 0.8 1.0 p(n)

10−3 10−2 10−1 100

γsptdiss

0.00 0.05 0.10 0.15

p3 V⊥ = 8.8 ER

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Conclusion

Decoherence of a bosonic many-body system under spontaneous emission Main message : Interactions slow down decoherence

Observation of anomalous momentum diffusion : ∆k ∼ t1/4 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 ?

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Loss dynamics after freezing

Same experiment as before : illumination with near-resonant laser for tdiss
then raise the horizontal lattice to “freeze” the density distribution
wait for thold : three-body losses empty sites with n ≥ 3
monitor losses to extract p(n ≥ 3)

1000 2000 3000 4000 5000

t [ms]

15000 20000 25000 30000

SF 5Er – tdiss = 2500 µs Solid lines : fit with known loss time constants to extract N3 = 3p(3), etc ...

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Evolution of triply-occupied sites

1 2 3 4 5 6

tdiss (Γ−1

sp )

2500 5000 7500 10000 12500 15000

N3

V⊥ = 5 ER

10−1 100

Γsptdiss

103 104

N3