Fitting a Bose-Einstein Distribution

To obtain an estimate for the temperature of the negative and positive temperature states, we fitted the experimentally extracted quasimomentum distribution nexp(qx, qy) with a Bose-Einstein distribution function for the kinetic energy (neglecting interaction and po-tential energy),

n_fit(q_x, q_y) = 1

e^{(Ekin(qx,qy)−µ)/kBT} −1 +o, (5.4) where independent fitting parameters are the chemical potentialµ, the absolute tempera-tureT, and a constant offset o. We convolved this distribution with an elliptical Gaussian function to take into account the convolution with the in situ distribution after finite TOF and the expansion of the cloud along the vertical direction during time-of-flight: Since the vertical lattice axis, along which the cloud expands, and the imaging axis are not perfectly parallel, the vertical expansion of the cloud is projected into the imaging plane and appears in the TOF images as a convolution of the cloud with an elliptical Gaussian.

In the case of the negative temperature image, we fixed both the aspect ratio and the rotation angle of the elliptical Gaussian by fitting the central peak in a positive temperature image. Thus, only the width σG of the elliptical Gaussian remains as free parameter for the fit. As an additional fitting parameter, we also included the length l_BZ of the first Brillouin zone after TOF. Since the distribution extends beyond the first Brillouin zone due to the Gaussian convolution, we also included pixels on the outside of the first Brillouin zone in the fit. Therefore, there are five free parameters in the negative temperature fit, µ, T, o, σ_G, and l_BZ. To obtain reliable values for each free parameter in the fit, it is necessary that the effects of the different parameters on the fit function are distinguishable.

Both higher temperature T and larger widthσ_G increase the width of the four peaks in the negative temperature fit. However, as the profiles of the unconvolved peaks in the corners of the Brillouin zone are shaped as quadrants instead of being circular, the effect of a larger width σG, which broadens the peaks in the convolved image in all directions simultaneously, is different from the effect of increasing temperature T, which broadens the peaks only towards the center of the Brillouin zone. Therefore, fitting both parameters simultaneously yields reliable results. Monitoring the residual sum of squares shows that this fit is stable, as there is only a single global minimum.

For the positive temperature image, it is not possible to use both σ_G andT as free pa-rameters, because they are not independent when fitting only a single round peak. Instead, by assuming that the Gaussian convolution function is the same in the negative and pos-itive temperature case, we fixed σG to the value obtained from the negative temperature fit. As also l_BZ cannot be extracted from a single peak, we fixed its value by fitting the first order coherence peaks in a positive temperature image without band-mapping. Thus, in the positive temperature case, the free parameters are reduced toµ, T, and o.

As Bose enhancement increases with filling in the lattice, the fitted temperature ex-tracted from a measured quasimomentum distribution increases with assumed filling in the fit (cf. Fig.5.11). The filling enters the fitting process through the chemical potential, which determines the overall atom number, and through the discretization of the first Brillouin zone: The discretization defines the number of available quasimomentum states, which in turn equals the number of contributing lattice sites. While the atom number can be determined from TOF images, the precise number of contributing lattice sites can only be roughly estimated due to the inherent integration in absorption images and the inhomoge-neous, decreasing filling at the edge of the cloud. In an initial attempt to obtain an upper bound for the absolute value of temperature in both negative and positive temperature cases (see below), we normalized the experimental data to the number of quasimomentum states used in the fits, corresponding to unity filling in the optical lattice. The resulting fitted distributions are shown in Fig. 5.9 as the lower two insets. They include the con-volution with an elliptical Gaussian and reproduce well the measured experimental data above. From these fitted distributions, we extract the occupation per Bloch waveρfit(Ekin) analogously to the experimental data. The result is shown as the solid lines in Fig.5.9and is in excellent agreement with the experimental data. The temperatures that we extract from the fits are T = −2.2J/k_B and T = 2.7J/k_B, where the errors are dominated by systematic uncertainties about the filling and discussed in the following. The very good agreement between data and a thermal Bose-Einstein distribution function, together with the great stability, indicate that the final negative and positive temperature states are indeed thermalized.

The fitting procedure neglects interaction as well as potential energy. By using a man-ageable, homogeneous model system and by assuming unity filling for this system, we over-estimate the average filling of the real system: In the experiment, the atoms are trapped in a harmonic trap and we expect unity filling to be reached only in the center of the cloud, while the filling decreases at the edge of the cloud. The TOF images therefore average over many two-dimensional systems with different fillings each (Section3.4). As the den-sity distribution realized in the experiment is not precisely known, it is very challenging to perform the whole fitting procedure for this real distribution. Instead, we estimate a more realistic average filling of the in situ cloud. The final lattice ramp of 2.5 ms from the Mott insulating to the superfluid regime allows no more than 2 tunneling events (Section 6.2.1), limiting the overall mass transport possible during the ramp. Therefore it is an acceptable approximation to assume the same average filling ¯n at the end of the lattice ramp in the superfluid regime as at the beginning of the lattice ramp in the Mott insulating regime. The average filling in the Mott insulating regime depends on the average entropy per particle, S/N. In our experiment, we expect the entropy to lie somewhere between zero, which implies ¯n= 1, and an upper bound ofS/N ≈1.5k_B, which gives ¯n≈0.5. An intermediate value ofS/N ≈0.8kB corresponds to ¯n≈0.7.

In both negative and positive temperature cases we performed the fit to the same ex-perimental data several times, fixing the filling each time to different values resulting in

5.3 Extraction of Energy Distribution and of Absolute Temperature

Filling n_

Filling n_

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

μ (J)

8.0 8.1 8.2

-0.14 -0.12 -0.10 -0.08 -0.06 -0.04

μ (J)

A B

Figure 5.10: Chemical potential for various assumed fillings in the fit. The chemical potentials µ are adjusted in the Bose-Einstein fits and result in different fillings. The kinetic energies in the lowest band range from 0 to 8J.A, Fit results for the negative temperature case. B, Positive temperature case.

different chemical potentials (Fig.5.10). For these fillings, we extracted the temperatures and found an approximately linear scaling of temperature with filling in both cases (Fig.

5.11). From the linear fits, we extract that in the negative temperature case, a filling of

¯

n= 0.7 results in a fitted temperature whose absolute value is 17(2) % lower than the one for ¯n = 1, and a filling of ¯n = 0.5 gives a temperature which is 29(2) % lower. In the positive temperature case, the temperatures are 23(1) % and 38(1) %, respectively, lower.

Therefore, the fitted temperatures ofT =−2.2J/kB andT = 2.7J/kB that assume ¯n= 1 are systematically too large (considering the absolute values) and represent only upper bounds for the real temperatures realized in the experiment.

Filling n_

Filling n_

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

T (J)

-2.4 -2.2 -2.0 -1.8 -1.6 -1.4 -1.2

T (J)

1.0 1.5 2.0 2.5

A B 3.0

Figure 5.11: Fitted temperature for various assumed fillings in the fit. The temperatures T show an approximately linear dependence on the filling that is fixed for the fitting process via the chemical potential. A realistic average filling ¯n <1 leads to a reduction of the absolute value of the absolute temperature, compared to ¯n= 1. The values obtained from the fits assuming

¯

n= 1 are therefore upper bounds for the temperature. A, Negative temperature case. B, Positive temperature case.