insulator with variable atom number. The experimentally measured data is an average over many such tubes of different lengths, where the length of each tube is determined by the ellipsoid created by the external harmonic confinement. For longer ramp times τramp>1, the calculated two-point correlators obtain a significant imaginary contribution from dephasing due to the spatially varying chemical potential (Fig.6.32).
x (dlat)
0 20
-20
τramp = 0.52
y (dlat) 0 20
-20
1.97
1.09 3.07
Re Im Re Im
Re Im Re Im
1.1
-0.5 0
Figure 6.32: Real (Re) and imaginary (Im) part of the final two-point correlator for (U/J)f= 3 in a trapped system withN= 24 particles for various ramp times.
To extract the coherence length, the two-point correlators were fitted under the assump-tion of exponentially decaying correlaassump-tions (Eq. 3.47). The resulting coherence lengths demonstrate that the deviation of the coherence length from the power-law behavior is indeed caused by the trap and that this effect is dominated by the influence of the shorter tubes (Fig.6.33). A more precise modeling of the trapped system is not feasible because of uncertainties regarding the initial state: Since the loading of the lattice is not perfectly adiabatic and the system is not guaranteed to be in thermal equilibrium in the deep lat-tice, a complete dynamical simulation of the 3D loading procedure would be necessary to precisely predict the in situ distribution of the system in the deep lattice.
6.4 Emergence of Coherence in Higher Dimensions and
6.4 Emergence of Coherence in Higher Dimensions and for Attractive Interactions
0.1 τramp
ξ (dlat) 1
0.5 5
1 10
2
N = 44 36 28 24 20
Figure 6.33: Emergence of coherence in a trapped system for (U/J)f = 3 in 1D. The solid curves are DMRG calculations for various particle numbersNand the points are experimental data.
6.4.1 Emergence of Coherence in 2D and 3D
The experimental sequences for the 2D and 3D measurements are similar to the 1D case and are outlined in Section 6.2.2. The extraction of the coherence length from the individual images is performed in the same way as in the 1D case (Section 6.2.3). The resulting coherence lengths are shown in Fig. 6.34 as a function of ramp time. We found that, for short ramp times τramp . 1, the emergence of coherence is almost independent of dimensionality, showing similar curves for 1D, 2D and 3D up to the power-law regime.
This is rather surprising, as the Kibble-Zurek mechanism predicts exponents that strongly depend on dimensionality (Section 6.1.2). Apparently, in the regime where the coherence length is not larger than a few lattice sites, dimensionality only has a minor effect on the spreading of correlations. For longer ramp times, the higher-dimensional systems continue the emergence of coherence to largerξthan in lower dimensions. This difference might be explained by the different critical values (U/J)c ≈3.3 [102–104] in 1D, 16.7 [105] in 2D, and 29.3 [106] in 3D: A fixed (U/J)fin the 1D case is closer to or even deeper in the Mott insulating regime than for higher dimensions. Thereby, the maximum achievable coherence length, even for adiabatic ramps, is fundamentally limited by the final interaction (U/J)f, in addition to potential dephasing effects.
Similarly to 1D, we find that the extracted power-law exponents in both 2D and 3D depend on the interaction (U/J)f (Fig. 6.35), in contrast to the prediction of the Kibble-Zurek mechanism. The precise dependence, however, looks different in the various dimen-sionalities. It may be interesting to perform a detailed analysis of the measured coherence lengths and exponents, also with the help of theoretical models. Due to the difficulty of simulations in higher dimensions, however, such a study is very challenging. The presented measurements already reveal complex dynamics in this regime where, currently, there are
ξ (dlat) 1 10
τramp
(U/J)f = 2 3 4
0.1 1 0.1 1 0.1 1
3D2D 1D
Figure 6.34: Experimentally measured coherence length in 1D, 2D, and 3D versusτrampfor various (U/J)f. For short ramp timesτramp.1, the emergence of coherence is almost independent of dimen-sionality. For largerτramp, the maximum achievableξis limited by the ratio (U/J)f/(U/J)c, which is different for the three dimensionalities (see main text).
no theoretical methods available. This may encourage future theoretical efforts. Detailed investigations have to include the precise ramp schedules employed in the experiment, the dimension-dependent critical values (U/J)c, as well as the different nature of equilibrium correlations: Whereas quasi-long-range order is expected in 1D, true long-range order pre-vails in 2D in the case ofT = 0 and in 3D.
(U/J)f
0 4 8 0 4 8
Exponent b
0.2 0.6
1.0 2D 3D
12 KZ 2D tip of lobe
KZ 3D tip of lobe
Figure 6.35: Power-law exponentsbfor the 2D (left) and 3D (right) cases. The fitting procedure to the experimental data, including error bars, is identical to the 1D case (Section6.2.8). Also in higher dimensions, the exponent depends on the final interaction (U/J)f, in contrast to a typical Kibble-Zurek type prediction (dotted lines).
6.4.2 Emergence of Coherence for Attractive Interactions
Qualitatively, the rather fast timescale for the emergence of coherence has already been observed in previous experiments [34]. In these experiments, similarly to the experiments presented above, the phase order of the initial superfluid at quasimomentum~q= 0 before loading into the deep lattice is identical to that of the final superfluid after ramping down the lattice again, namely with an identical phase factoreiqx= 1 at each lattice site (cf. Fig.
3.3). Even in the deep optical lattice, if entropy is not too large, phase coherence is still present in the superfluid shell around the Mott insulating core, representing a remnant of the initial superfluid phase order. One might conjecture that the establishment of phase coherence in the final superfluid could be facilitated by this remnant: Instead of creating phase coherence out of completely scrambled local phases in a Mott insulating state, the surviving phase order in the superfluid shell could potentiallyseed phase order
6.4 Emergence of Coherence in Higher Dimensions and for Attractive Interactions
across the system. With the help of negative temperatures, however, we can show that the timescale for the emergence of coherence is indeed generic for this particular phase transition, independent of the preparation procedure. The general idea is to choose different phase orders for the initial and the final superfluid such that the above speculation can be falsified.
The experimental sequence for the emergence of coherence measurements for attractive interactions is described in Section 6.2.2. With our experimental setup we were able to perform such measurements in 1D and 2D. The resulting emergence of coherence in 2D (Fig.
6.36) is almost identical to the repulsive interactions case, consistent with the symmetry of the Bose-Hubbard Hamiltonian for repulsive and attractive interactions (cf. Section 4.3.5). Only for strong interactions are deviations visible. These can qualitatively be explained by multi-band effects that lead to changes in the effective local Wannier functions [235]: Attractive and repulsive interactions lead to an effectively deeper or shallower lattice potential, respectively, that modifies the effective lattice ramp [236]. The corresponding exponents of the power-law increase (Fig.6.37) agree within the error bars.
τramp ξ (dlat)
1 10
0.1 1 0.1 1 0.1 1
(U/J)f = ±2 ±3 ±5
T>0 T<0
Figure 6.36: Experimental data of coherence length versus ramp time for repulsive and attractive interac-tions in 2D. For not too strong interacinterac-tions (U/J)f, the emergence of coherence for the two cases is very similar. The deviations for larger interactions can be attributed to self-trapping effects (see main text).
Also in 1D, we obtain very good agreement between the emergence of coherence for attractive and repulsive interactions, at least for large interactions (Fig. 6.38). We can-not reliably measure the emergence of coherence for small attractive interactions, as this requires ramping the Feshbach field over the range where the scattering length vanishes.
This leads to a crossing from the Mott to the superfluid regime in the deep lattice. As the system cannot follow this quench due to the long timescales in the deep lattice, a lot of entropy is created and the system is heated.
Our theory collaborators performed exact diagonalization simulations of the 1D emer-gence of coherence for attractive interactions. These are identical to the calculations for repulsive interactions (Section 6.2.6). The simulated Feshbach ramp, in this case, crosses the Feshbach resonance such that the highest excited state in the lowest band is populated, consistent with a negative temperature state in the case of thermalization. The correlator contains an additional phase factor in comparison with the positive temperature state (Eq.
3.49). Just like in the repulsive interactions case, the resulting emergence of coherence signals show excellent agreement with the experimentally measured values (Fig.6.39).
The quasimomentum distributions of negative and positive temperature states, as
mea-|(U/J)f|
0 4 8
Exponent b
0.2 0.6 1.0
0.8
0.4
2 6 10
T>0 T<0
Figure 6.37: Power-law exponents for repulsive and attractive interactions in 2D, extracted from the data in Fig.6.36. The fitting procedure is the same as described in Section6.2.8. For not too strong interactions|(U/J)f|, the values for the two cases are consistent with each other.
ξ (dlat) 1 0.5
0.1 1 0.1 1 0.1 1
τramp 0.1 1
2 (U/J)f = ±3(1) ±4 ±5
T>0 T<0
±6(1)
Figure 6.38: Experimental data of coherence length versus ramp time for repulsive and attractive interac-tions in 1D. For the available (U/J)f values, the emergence of coherence in the two cases is very similar.
ξ (dlat) 1 0.5
τramp 5
0.1 1
(U/J)f = -3(1) -4 -5 -6
0.1 1 0.1 1 0.1 1
Figure 6.39: Emergence of coherence in 1D for attractive interactions for various (U/J)f. The points are experimental data and the solid curves exact diagonalization calculations.
sured in time-of-flight images, are fundamentally different, corresponding to different phase factors at the lattice sites (Fig. 3.3). Nonetheless, we experimentally observe the same timescale for the emergence of coherence in these different superfluid states. The super-fluid shell at the beginning of the lattice ramp, if any, still contains the phase order with identical phase factor at each lattice site: The switching of both interactions and external confinement cannot lead to the fast establishment, still in the deep lattice, of phase coher-ence with an alternating phase factor between lattice sites in the superfluid shell, as the