Dynamics of impurities in a one-dimensional Bose gas Francesco Minardi Istituto Nazionale di Ottica-CNR European Laboratory for Nonlinear Spectroscopy New quantum states of matter in and out of equilibrium Galileo Galilei Institute – May 14, 2012 Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 1 / 30
Acknowledgements University of Geneva BEC 3 group at LENS, Firenze A. Kantian, T. Giamarchi J. Catani G. Lamporesi D. Naik FM, M. Inguscio Scuola Normale Superiore, Pisa M. Gring (U. Vienna) S. Peotta, D. Rossini, M. Polini R. Fazio Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 2 / 30
One-dimensional systems ⊲ large quantum fluctuations + exactly solvable models (Lieb-Liniger, . . . ) + powerful numerics; time-dependent dynamics, out-of-equilibrium calculations ⊲ real 1D systems do exist in our 3D world carbon nanotubes spin chains in cuprates Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 3 / 30
One-dimensional systems ⊲ Quantum gases - experiments on (quasi)1D BEC: MIT, Hamburg, NIST, Orsay/Palaiseau, Amsterdam, ETHZ, Vienna . . . - strongly interacting (Tonks-Girardeau) regime: T. Kinoshita et al., Science 305, 1125 (2004); B. Parades et al. , Nature 429, 277 (2004); E. Haller, Science 325, 1124 (2009) - relaxation dynamics: S. Trotzky et al. , Nature Physics (2012) transport of spin impurities through a Tonks gas impurity subject to constant force (gravity) + drag force due to host atoms S. Palzer et al. , PRL 103, 150601 (2009) Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 4 / 30
Outline ⊲ diffusion and oscillations of an initially localized impurity (K atoms) in a harmonically trapped 1D Bose gas (Rb atoms), ⊲ control of interaction of impurities (K) with host atoms (Rb), through Feshbach resonance Analogous to spin excitation in a ferro-magnetic chain Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 5 / 30
Outline ⊲ diffusion and oscillations of an initially localized impurity (K atoms) in a harmonically trapped 1D Bose gas (Rb atoms), horizontal ⊲ control of interaction of impurities (K) with host atoms (Rb), through Feshbach resonance Analogous to spin excitation in a ferro-magnetic chain Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 5 / 30
Spin chain, Yang-Gaudin model Lieb-Liniger model: N H = − � 2 ∂ 2 � � γ = mg / ( � 2 n ) + g δ ( x i − x j ) , ∂ x 2 2 m i i =1 i < j extended to (iso)spin = 1/2 → Yang-Gaudin model, SU(2) symmetric, only one coupling strength g C. N. Yang, PRL 19, 1312 (1967); M. Gaudin, Phys. Lett. A 24, 55 (1967); J. N. Fuchs et al. , PRL 95, 150402 (2005) Starting from ferromagnetic ground state: – density excitations (phonons) ǫ p = v s p m / m ∗ = 1 − 2 √ γ/ (3 π ) for weak coupling, γ ≪ 1 – spin excitations ǫ p = p 2 / (2 m ∗ ), m / m ∗ = 1 / N + 2 π 2 / (3 γ ) for strong coupling, γ ≫ 1 Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 6 / 30
Effective mass, slow diffusion Effective mass for spin excitations For γ ≫ 1 impurities move slowly, actually “subdiffuse” at short time, x rms ∼ log( t ) M. B. Zvonarev et al., PRL 99, 240404 (2009) Beyond Luttinger-liquid description J. N. Fuchs et al. , PRL (2005) About impurity motion in 1D also: G. E. Astrakharchik et al. , PRA 70, 013608 (2004); M. D. Girardeau et al. , PRA 79, 033610 (2009); D. M. Gangardt et al. , PRL 102, 070402 (2009); A. Yu. Cherny et al. , PRA 80, 043604 (2009); T. H. Johnson et al. PRA 84, 023617 (2011) Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 7 / 30
Scattering of two unequal particles in 1D Extension of Olshanii’s analysis on Confinement Induced Resonances: V. Peano et al., NJP 7, 192 (2005) No closed analytical expression for coupling strength of δ -potential: � |� 0 | e n �| 2 1 2 � � g 1 D = a µ = 2 µπ a 2 λ n + 1 / (4 π a ) µ ( ω 1 + ω 2 ) n µ where λ n , | e n � eigenvalues/vectors of regular part of the Green’s function Interspecies g 1 D Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 8 / 30
Experiment Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 9 / 30
Sample preparation, harmonic trap Evaporation, both species in lowest hf state | f = 1 , m f = 1 � featuring Feshbach resonances B field controls of interspecies (K-Rb) interactions, while intraspecies (K-K, Rb-Rb) fixed ω/ 2 π = (39 , 87 , 81)Hz for Rb ( × 1.47 for K) At this point: T ≃ 140nK N Rb ≃ 1 . 5 × 10 5 , N K ≃ 5 × 10 3 Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 10 / 30
Sample preparation, 2D lattice 2D lattice V =60(26) E r for Rb(K) 1st excited band gap = 29 kHz i.e. 1.4 µ K tunneling time � / J =57(0.27)s Non-homogenous 1D tubes, ω x / 2 π =57(80)Hz Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 11 / 30
Sample preparation, 2D lattice Max filling = 180 (2) atoms/tube for Rb(K) Rb n 1 D = 7 atoms/ µ m Lieb-Liniger parameter γ Rb = g 1 D , Rb m / ( � 2 n 1 D ) ≃ . 5 T=(350 ± 50) nK (from Rb time-of-flight images) Rb degeneracy temperature T d = � ω x N = 520nK → weakly interacting Rb condensates in central tubes Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 12 / 30
Sample preparation, 2D lattice + ”light-blade” “Light-blade” λ = 770nm, elliptic 75 × 15 µ m Species selective: V ≃ 0 on Rb, ≃ 6 µ K on K linear ramp in 50 ms Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 13 / 30
Sample preparation, 2D lattice + ”light-blade” “Light-blade” λ = 770nm, elliptic 75 × 15 µ m Species selective: V ≃ 0 on Rb, ≃ 6 µ K on K linear ramp in 50 ms Initial configuration, t = 0 after light-blade off abruptly K Rb initial K size < imaging resolution (8 µ m) Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 13 / 30
Impurity oscillations Longitudinal confinement along tubes → oscillations of K impurity rms size σ ( t ) g 1 D ( Rb ) = 2 . 36 · 10 − 37 Jm Interspecies interaction parameter: η ≡ g 1 D ( KRb ) / g 1 D ( Rb ) J. Catani et al., PRA 85, 023623 (2012) Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 14 / 30
Impurity oscillations Longitudinal confinement along tubes → oscillations of K impurity rms size σ ( t ) g 1 D ( Rb ) = 2 . 36 · 10 − 37 Jm Interspecies interaction parameter: η ≡ g 1 D ( KRb ) / g 1 D ( Rb ) J. Catani et al., PRA 85, 023623 (2012) Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 14 / 30
Impurity oscillations Longitudinal confinement along tubes → oscillations of K impurity rms size σ ( t ) g 1 D ( Rb ) = 2 . 36 · 10 − 37 Jm Interspecies interaction parameter: η ≡ g 1 D ( KRb ) / g 1 D ( Rb ) J. Catani et al., PRA 85, 023623 (2012) Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 14 / 30
Impurity oscillations Longitudinal confinement along tubes → oscillations of K impurity rms size σ ( t ) g 1 D ( Rb ) = 2 . 36 · 10 − 37 Jm Interspecies interaction parameter: η ≡ g 1 D ( KRb ) / g 1 D ( Rb ) J. Catani et al., PRA 85, 023623 (2012) Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 14 / 30
Impurity oscillations Longitudinal confinement along tubes → oscillations of K impurity rms size σ ( t ) g 1 D ( Rb ) = 2 . 36 · 10 − 37 Jm Interspecies interaction parameter: η ≡ g 1 D ( KRb ) / g 1 D ( Rb ) J. Catani et al., PRA 85, 023623 (2012) ⊲ larger interactions → smaller oscillation amplitude of σ ( t ) Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 14 / 30
Impurity oscillations Longitudinal confinement along tubes → oscillations of K impurity rms size σ ( t ) g 1 D ( Rb ) = 2 . 36 · 10 − 37 Jm Interspecies interaction parameter: η ≡ g 1 D ( KRb ) / g 1 D ( Rb ) J. Catani et al., PRA 85, 023623 (2012) ⊲ larger interactions → smaller oscillation amplitude of σ ( t ) ⊲ tilted oscillations Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 14 / 30
Oscillation frequency, damping and slope Fitting function: σ ( t ) = σ 1 + β t − A e − γω t cos( � 1 − γ 2 ω ( t − t 0 )) Fit results: Oscillation frequency constant within errorbars Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 15 / 30
Amplitude of first oscillation Focus on the peak value of 1st oscillation: σ p ≡ σ (t=3ms) vs g 1 D (exp. B field) - σ p sensitive to coupling with Rb bath - σ p least affected by Rb inhomogeneous density Francesco Minardi (INO-CNR and LENS) Dynamics of impurities in a 1D Bose gas GGI, 14/05/2012 16 / 30
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