Propagation velocity

A natural way to introduce a velocity is to replace Jex/¯h by v/alat in the fit

robust with respect to temperature of the bath. It is extracted from the cor- responding density distribution by a fitting function (see Equation 4.1) to the simulated data and the velocity changes within the simulated temperature range by only 4 %. The experimental data support this observation because the results of both imaging techniques show consistent velocities even though their effective temperature differs.

a b c x y 2 µm −10 −50 0 5 10 0.3 0.6

Position (lattice sites)

Probability

0 ms

−10 −50 0 5 10

0.1 0.2

Position (lattice sites)

1 ms

−10 −50 0 5 10

0.1 0.2

Position (lattice sites) 2 ms

Figure 4.3: Spin impurity dynamics in the SF regime. Density distribution at (J/U =0.32) close to the critical point [(J/U)c ≈0.3] for three different

hold times. The upper panels display fluorescence images of the impurity spins after removing the other spin component (‘positive image’). The 1D chains containing more than one atom were excluded from the data analysis. The white vertical stripe highlights the initial position of the flipped spin im- purity. The lower panels show the position distribution averaged over about 300 chains (blue bars) together with a t-DMRG simulation at T = 0.11U/kB

(red line).

The mapping of the two-component Bose gas to the spin-1/2 model is only valid deep in the MI phase with unity filling. In that regime, only thermal fluctuations alter the coherent propagation. Towards the superfluid regime and especially in it, this mapping loses its validity. The |↓i bath has strong quantum fluctuation with low-energy excitations and the interference struc- ture gets washed out as seen in Figure 4.3.

In the superfluid phase, the quantum fluctuations lead to a Fröhlich-type Hamiltonian and it is natural to treat the|↑iimpurity together with the de- formation of the bath as a polaron. We study the crossover to this regime by lowering the lattice depth during the evolution, which increases the ra- tio of J/U. In the superfluid regime, only the positive imaging technique

MI SF 0 0.1 0.2 0.3 0.4 0.5 0 1 2 J/U Velocity v ( Ja lat / ħ )

Figure 4.4: Spin dynamics across the superfluid-to-Mott-insulator transi- tion. Measured velocities of the spin impurity for different values of J/U

extracted from negative (green circles) and positive images (blue circles); hor- izontal and vertical error bars indicate the 1σuncertainties of the lattice depth

and the combination of fit error and uncertainties of J, respectively. The dark gray line shows scaling with 4J2/U, whereas the brown line indicates the propagation velocity of a single free particle (J/U =∞_{). The gray shaded}

region shows results from a t-DMRG simulation atT =0 taking into account varying initial atom numbers. The area denotes the 1σfit error to the simu-

lated distributions.

can be used, as it is not possible in negative imaging to identify the spin po- sition out of many quantum fluctuations, which show up as holes as well. After preparing the initial state at high lattice depth, the lattice depth is re- duced relatively slowly in 50 ms to avoid any heating. The spin|↑iimpurity cannot move during this ramp as the addressing beam creates an attractive potential which pins the impurity. Equation 4.1 is the analytical solution for the Heisenberg spin chain. No full analytical solution exists which describes the propagation of the impurity in the Bose-Hubbard model. A fit of Pj(t)

with the velocity as the only free parameter still captures the velocity of the edges in the position distribution, which is given by the maximum propaga- tion velocity. We determine the velocity over the full range of J/U, which is accessible in the experiment spanning a full order of magnitude from 0.05 to 0.5. For small J/U the velocity divided by the tunneling rate J increases linearly with a slope of 4, which perfectly agrees with the theoretical expec- tation for the spin exchange coupling strength of 4J2/U. Close to the phase transition point of the superfluid-Mott-insulator transition in a homogeneous

system, the normalized velocity saturates. In the superfluid regime, it is fi- nally significantly smaller thanv/J =4J/U. Here, the velocity of the polaron is approximately half the velocity of a free particle moving in the same lattice, given byv/J =2alatJ/¯h. This points out that the strong interactions modify

the motion of the particle and increase the effective mass of the impurity. Such a behavior is expected and agrees with simulations of the two-species Bose-Hubbard model. Therefore the propagation velocity is extracted from density distributions, calculated with t-DMRG simulations [165–169], with the same method as used for the experimental data. The simulations take into account the harmonic trap as well as atom number fluctuations, which both can influence the propagation significantly. The obtained velocity as well as the numerically determined density distributions agree well with our experimental data.