SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMENTARY INFORMATION doi: /nature10748 Contents 1 Quenches to U /J = 5.0 and Quasiparticle model 3 3 Bose Hubbard parameters 6 4 Ramp of the lattice depth for the quench 7 1

2 RESEARCH SUPPLEMENTARY INFORMATION 1 Quenches to U /J = 5.0 and 7.0 We also recorded the time evolution of the two-point parity correlations (2) after quenches to U/J =5.0(2) and 7.0(3), and compared the experimental results to DMRG simulations of an infinite, homogeneous system at zero temperature (Supp. Fig. 1). The experimental sequence was identical to the one we used for the quench to U/J =9.0(3), apart from the different end point of the quench. The data presented here are those used in Fig. 3. U/J = 5.0 U/J = 7.0 Supplementary Figure 1 Time evolution of the two-point parity correlations. Left panel: quench to U/J =5.0(2). Right panel: quench to U/J =7.0(3). The circles indicate the correlations measured experimentally and the line is derived from the numerical simulations for an infinite, homogeneous system at zero temperature. The experimental and numerical values were obtained in the same way as described in the legend of Fig. 2 and in the Methods Summary section. 2

3 SUPPLEMENTARY INFORMATION RESEARCH 2 Quasiparticle model In the Bose Hubbard model, bosonic atoms in an optical lattice are confined to a single Bloch band and obey the Hamiltonian Ĥ = j { J ( ) â j âj+1 + h. c. + U } 2 ˆn j(ˆn j 1), (S1) where â j and â j represent the annihilation and creation operator of an atom at site j and ˆn j =â jâj counts the number of atoms at that site. The model is entirely parametrised by the effective interaction strength U/J. In order to analytically treat the time evolution of correlations after a sudden decrease of U/J, we developed a novel approach based on fermionized quasiparticles. The initial state being close to a Fock state with one atom per lattice site, an effective description of the evolution at sufficiently large final interaction strengths can be obtained within a local basis formed by empty states, j, singly occupied states, j, and doubly occupied states, j. Using generalised Jordan Wigner transformations [1], we then introduced fermionic creation operators for the excess particles, ˆd j j j, and the holes, ĥ j j j, as well as the corresponding annihilation operators. Within the truncated Hilbert space, the original Hamiltonian (S1) can be exactly written in terms of these operators: Ĥ = j ˆP { 2J ˆd ˆd j j+1 J ĥ j+1 ĥj J ( 2 ˆd j ĥ j+1 ĥj ˆd ) j+1 + h. c + U 2 (ˆn d,j +ˆn h,j ) } ˆP, (S2) with ˆn d,j = ˆd j ˆd j and ˆn h,j = ĥ jĥj. The complexity of the model is hidden in the projector ˆP = j (Î ˆn d,j ˆn h,j ) that eliminates the unphysical situation of having an excess particle and a hole at the same site (Î is the identity operator). As multiple occupancies of equal species are naturally avoided due to their statistics, one still 3

4 RESEARCH SUPPLEMENTARY INFORMATION obtains a good description of the system when setting ˆP Î, provided the density of excitations ˆn d,j (t)+ˆn h,j (t) remains low. This is in contrast to the usual bosonic representations 2,3. The Hamiltonian (S2) with ˆP Î is quadratic and can be diagonalised by a Bogolyubov transformation. The eigenmodes are doublons and holons with well defined momentum k: ˆγ d,k = u(k) ˆd k + v(k) ĥ k, ˆγ h, k = u(k) ĥ k v(k) ˆd k, (S3) (S4) with u(k) = cos[θ(k)/2], v(k) = i sin[θ(k)/2] and θ(k) =atan [ ] 32J sin(kalat ) U 6J cos(ka lat ). (S5) Their respective eigenenergies are given by ɛ d (k) = J cos(ka lat )+ 1 [U 6J cos(ka lat )] J 2 sin 2 (ka lat ), (S6) ɛ h ( k) =J cos(ka lat )+ 1 [U 6J cos(ka lat )] J 2 sin 2 (ka lat ). (S7) Within this model, the initial state ψ 0 evolves in time according to: ψ(t) = e iĥt/ ψ 0 = k { } ū(k) v(k) e i[ɛ d(k)+ɛ h ( k)]t/ ˆγ d,k ˆγ h, k vac, (S8) with ū(k) =u(k)u 0 (k) v(k)v 0 (k) and v(k) =v(k)u 0 (k) u(k)v 0 (k). Here the subscript 0 denotes quantities corresponding to the initial interaction strength, whereas no label is used for the quantities corresponding to the final interaction strength. Further, vac represents the quasiparticle vacuum at the final interaction 4

5 SUPPLEMENTARY INFORMATION RESEARCH strength (ˆγ d,k vac =ˆγ h,k vac =0). Equation (S8) describes wave packets of entangled quasiparticle pairs that travel in opposite directions with different velocities. One can extract a maximal velocity for the spreading of the correlations from the dispersion relation of a quasiparticle pair (black line in Fig. 3b): v max = 1 { d max ɛ d (k)+ɛ h (k) } k dk (S9) = (6Ja lat / ) [ 1 16J 2 /(9U 2 ) ] + O ( J 4 /U 3). (S10) Additionally, we derived from equation (S8) the time evolution of the two-point parity correlations. In agreement with DMRG simulations, it displays a clear positive signal, the position of which increases with time (Fig. 2). We extracted the instantaneous propagation velocity v(d 0 ) from a linear fit through the signal positions d 0 d d 0 +4(Supp. Fig. 2). The case d 0 =2corresponds to the data shown in Fig. 3. At short distances, we find velocities about 10 % smaller than v max, in good agreement with the velocities measured experimentally. At large distances, the velocity converges algebraically to v max. This latter behaviour can be understood from the expansion of the correlation functions to first order in J/U, which can be expressed in terms of the Airy function Ai(x): C d (t) d 1 [ 2 7/3 d 2/3 3 Ut Ai [ (2/d) 1/3 (d 6Jt/ ) ]] 2. (S11) We checked the validity of our model of freely propagating fermionic quasiparticles by comparing it with DMRG simulations. The propagation velocity of the two-point parity correlations is very accurate in the strongly interacting limit (e.g. U/J = 20), and remains in fairly good agreement down to U/J =9, where the quasiparticle density is about 0.1 per site (Supp. Fig. 2). At lower interaction strengths, we found that the approximation ˆP Î breaks down. Nevertheless, the experiment and the simulations show that the spreading of correlations is still characterised 5

6 RESEARCH SUPPLEMENTARY INFORMATION Supplementary Figure 2 Instantaneous propagation velocity. We compare the instantaneous velocity v(d 0 ) predicted by our analytical model (points) with the one derived from numerical simulations (circles). The agreement is excellent at U/J = 20 (left panel) and qualitatively good at U/J =9 (right panel). Error bars denote the 68 % confidence interval of the fit. The dashed line represents the asymptotic expression (S11). Arrows point to the maximum velocity v max at the given interaction strength. The signal position was obtained using the procedure described in the Methods Summary section for the numerical simulations. by a well defined velocity, for which v max remains a relevant upper bound. We verified numerically that the truncation of the local Hilbert space to three states is reasonable down to U/J 6. 3 Bose Hubbard parameters For a given lattice depth V, we calculated the tunnel coupling and the on-site interaction of the Bose Hubard model (S1) in the single-particle picture using their expressions as overlap integrals of the Wannier functions. In Supp. Tab. 1, we provide the values of V, J and U for the effective interaction strengths mentioned in the main text. 6

7 SUPPLEMENTARY INFORMATION RESEARCH U/J V (E r ) J/ (s 1 ) U/h (Hz) 40(2) 13.5(2) 113(5) 724(4) 9.0(3) 7.70(11) 422(11) 607(3) 7.0(3) 6.85(11) 522(14) 584(3) 5.0(2) 5.75(13) 691(23) 550(4) Supplementary Table 1 Lattice depth and Bose Hubbard parameters for the data reported in the main text. The lattice depth in the perpendicular axes is 20.0(5) E r. The uncertainties on J, U and U/J follow from the calibration uncertainty on the lattice depth (see Methods Summary). 4 Ramp of the lattice depth for the quench The ramp of the lattice depth for the quench follows the functional form: V (t) =V 0 +(V V 0 ) e t/τ 1 e T/τ 1. (S12) Here V 0 = 13.5 E r is the initial lattice depth, T is the duration of the ramp and τ determines the rate at which the lattice depth is decreased. These parameters have to be chosen such that the ramp is fast compared to the time scale of the dynamics following the quench, which is given by /J, but adiabatic with regard to transitions to higher Bloch bands. The latter condition requires the parameter Γ= ( / 2 ) d /dt to be much smaller than 1, where is the energy gap between the two Bloch bands considered. For each quench, we have chosen the ramp parameters such that T is the shortest ramp duration compatible with Γ < 0.3 (see Supp. Tab. 2). U/J T (µs) τ (µs) Supplementary Table 2 Lattice depth ramp parameters used for the quenches reported in the main text, as defined in equation (S12). 7

8 RESEARCH SUPPLEMENTARY INFORMATION References 1. Batista, C. D. & Ortiz, G. Generalized Jordan Wigner Transformations. Phys. Rev. Lett. 86, (2001). 2. Huber, S. D., Altman, E., Büchler, H. P. & Blatter, G. Dynamical properties of ultracold bosons in an optical lattice. Phys. Rev. B 75, (2007). 3. Altman, E. & Auerbach, A. Oscillating Superfluidity of Bosons in Optical Lattices. Phys. Rev. Lett. 89, (2002). 8