Determination of Power-Law Exponents

After the presentation of the three theoretical methods in the previous sections, in this section, I describe how we extracted the power-law exponents for both experimental and theoretical data. The experimentally measured coherence length is plotted in Fig.6.17for several final interaction strengths (U/J)_f. In this plot, one can distinguish several distinct

6.2 Experimental and Numerical Methods

0.1 0.5

τ_ramp ξ (dlat)

1

0.5 2

Lieb-Robinson bound doublon-holon model

0.05

Figure 6.16: Emergence of coherence for the Lieb-Robinson bound provided by Eq. 6.19 and for the doublon-holon fermionic model for (U/J)f = 3 in 1D. The vertical dashed line indicates the ramp time at which the DHFM deviates from the DMRG calculation (Fig.6.15).

regimes. For very short ramp times, τ_ramp . 0.1, the evolution can be approximated by the sudden approximation. The measured coherence length is therefore given by the initial coherence length ξ_i at the beginning of the lattice ramp. This coherence length is considerably below one lattice spacing, ξi < dlat, and increases with decreasing (U/J)i

closer to the superfluid regime (cf. Figs.6.20and6.21). Thus, already during the Feshbach quench correlations build up between atoms, albeit only on a low level. For larger ramp times 0.1 .τ_ramp .1, the coherence length quickly increases up to several lattice sites.

In this project, we mainly focus on this regime, as it gives the major contribution to the emergence of coherence. Within almost one order of magnitude difference, the coherence length versus ramp time shows a power-law increase

ξ(τramp) =a τ_ramp^b , (6.20)

which I focus on in this section.

For larger ramp timesτramp&1, the coherence length starts to deviate from the power-law increase and, at some point, decreases. The reason for this deviation is the external trapping potential present in the experiment which requires global mass and entropy re-distributions when ramping from the Mott insulating into the superfluid regime. Details about the intricate effects of the external trapping potential are given in Section 6.3.3.

For very long ramp timesτramp &100, heating due to technical noise in the laser beams as well as by photon scattering decreases the coherence length even further: From hold time experiments in a shallow optical lattice potential, we know that phase coherence be-tween lattice sites decreases on a timescale of several hundred ms (cf., e.g., Section5.2.1), corresponding to ramp times τramp of several hundred.

To obtain a reliable value for the exponentb of the power-law growth of the coherence

0.1 1 10 1

5

0.5 2

τramp

ξ (dlat)

p_x (h/λ_lat)

n (a.u.)

0 2 -2 0 1

Figure 6.17: Emergence of coherence. Extracted coherence lengthξfor the 1D system and (U/J)i= 47, (U/J)_f = 2 versus ramp timeτrampin a double-logarithmic plot. For very shortτramp, the coherence length acquires a small, finite valueξi. The power-law increase for intermediate τrampis qualitatively highlighted by the straight line. For largeτramp, the coherence length decreases due to the external trapping potential and heating (see main text). The insets show sample time-of-flight profiles (black) with the corresponding fitted calculated interference pattern (red).

length, it is desirable to include as many data points as possible into the fitting procedure.

When fitting a pure power-law function ξ(τ_ramp) = a τ_ramp^b , however, the range of data points for the fit is limited by two effects: For small ramp times, the coherence length is given by the initial coherence length ξ_i, while for long ramp times τ_ramp & 1, the data deviates from a pure power-law increase due to the influence of the trap. To improve the stability of the fit, we include the initial coherence lengthξ_i in the fitting procedure via an empirical function

ξ(τ_ramp) = ξ_i^q+ (a τ_ramp^b )^q1/q

, (6.21)

which approaches the pure power-law increase for large τ_ramp. Here,a, b, andξ_i serve as free fit parameters; only for (U/J)f>(U/J)c,ξi is fixed to numerically calculated values (see below). With this fit function, we can include all data points for short and intermediate ramp times up to a maximum valueτ_ramp^max. As in some data sets, clear deviations from the pure power-law behavior due to the trap appear for τ_ramp >1, we chose τ_ramp^max = 1.0, as it guarantees that for all data sets in any dimension and at both repulsive and attractive interactions the influence of the trap on the fitted exponentbis negligible.

To obtain a reasonable value for the parameter q, we perform sample fits on the data sets for (U/J)_f= 1.0 and 1.9 in 1D for varyingq, for both experimental as well as exact diagonalization data (Section6.2.6). As a measure of how close the fits are to the data, we determine the sum of squared residuals (SSR) of the fit and find that q= 4 is a good compromise (Fig. 6.18). Figure 6.19 indicates that the fitted exponents are robust with respect to the choice ofq and Fig. 6.20shows that the choice q= 4 indeed captures the

6.2 Experimental and Numerical Methods

emergence of coherence well.

q

SSR of best fit

1 2 3 4 5 6 7 8 9 10

0 0.01 0.02 0.03 0.04

(U/J)_f = 1.0, experiment 1.9, experiment 1.0, theory 1.9, theory

Figure 6.18: Sum of squared residuals (SSR) of the general power-law fit (Eq.6.21) versus value ofqfor some 1D data. Light-blue and dark-blue are the results of fits to experimental data with (U/J)f= 1.0 and 1.9, respectively. Light-red and dark-red are the corresponding results for exact diagonalization data. A choice of q= 4 leads to close-to-minimum SSR values in all four cases.

q= 3 4 5

(U/J)_f

0 4 8

0.2 0.6 1.0 0.8

Exponent b 0.4

0 4 8

Figure 6.19: Fitted exponents in 1D forτ_ramp^max = 1 and variable (U/J)f for variousqvalues. Left, experi-mental data, right, DMRG data. The error bars are fit uncertainties, and the vertical dashed line indicates (U/J)c. For details of the fitting procedure, see Figs.6.20and6.21.

In the 1D case, we also recorded data sets for which the system does not cross the phase transition during the lattice ramp, but where the final interaction strength is still larger than the critical value, at (U/J)f > (U/J)c. In these cases, the power-law increase of the coherence length is slow, such that it is difficult to distinguish the power-law regime from the regime that is dominated by the initial coherence length ξi, and the fit does not capture the behavior reliably anymore. Exact diagonalization calculations (Section 6.2.6) provide theory values for the initial coherence lengthξi. In the case of the data sets with (U/J)_f < (U/J)_c (Fig. 6.20), these agree well with the fittedξ_i. For the data sets with (U/J)f >(U/J)c, we fix the initial coherence lengths to the calculated ones and thereby improve the stability of the fit. The resulting fits (Fig. 6.21) match the data well; only

0.1 1

τ_ramp ξ (dlat)

0.01 1 5

0.5

0.1

(U/J)_f = 1 2 3

0.1 1

0.01 0.01 0.1 1

Figure 6.20: Power-law fits (solid lines) for q = 4 for experimental data in 1D with (U/J)f < (U/J)c

and the initial coherence lengthξias a free fit parameter. The vertical dashed lines indicate the upper endτ_ramp^max = 1.0 of the fitting range. The horizontal dashed lines denoteξiat the beginning of the lattice ramp obtained from exact diagonalization calculations that show good agreement with the extrapolated fitted valuesξ(τramp→0).

for very large interaction strengths (U/J)f (U/J)c do systematic deviations from the simple power-law behavior become relevant.

0.1 1

τ_ramp ξ (dlat)

0.01

(U/J)_f = 4 5 7 10

1 5

0.5

0.10.01 0.1 1 0.01 0.1 1 0.01 0.1 1

Figure 6.21: Power-law fits (solid lines) forq= 4 for experimental data in 1D with (U/J)f>(U/J)c. The initial coherence lengthsξiin the fit are fixed to the value obtained by exact diagonalization calculations, indicated by the horizontal dashed lines. The vertical dashed lines indicate the upper endτ_ramp^max = 1.0 of the fitting range. The fitting model captures the behavior well for not too large (U/J)f. For very large (U/J)f, systematic deviations of the model from the data appear.

With this procedure, we were able to reduce the problem of defining an appropriate fitting range. The upper limit τ_ramp^max, however, is still arbitrary. In the 1D case, where the phase transition at the multicritical point is of Kosterlitz-Thouless type [187], we expect the power-law exponent to depend slightly on the ramp time and therefore also on the upper limit for the fit (Section6.3.1). As mentioned, we chose τ_ramp^max = 1.0, as it excludes data points that are considerably influenced by the trap, for all data sets in any dimension. To estimate the uncertainty of the resulting exponents b, we also performed fits for different values of τ_ramp^max = 0.9 and 0.7. In Fig.6.22, all resulting exponents in the 1D repulsive interactions case are plotted with the corresponding fit errors. As a measure for the uncertainty associated with the choice ofτ_ramp^max, we determined the total amplitude of the fitting errors for the three different choices ofτ_ramp^max.