Observing coherent evolution of spin correlations

The spin spiral technique introduced in previous section allows us to rotate the transverse correlation basis (see Eq. (5.20) and Fig.5.8b). Our measurement in spin correlations is no longer limited to the special caseq =0, as the imprinted spiral permits the measurement ofS(q)in the momentum

space. In this section, we present the measurement of the spin structure factor in the 2D Hubbard model resulting from the coherent evolution of spin correlations. Using the knowledge of the spin correlation in momentum space, we reconstruct individual spatial correlators and observe excellent agreement with the theoretical predictions. In addition, the access to the staggered structure factor S(±π/a,±π/a)allows us to perform local correlation thermometry and compare to thermometry based on fitting the global profile of the atomic distribution.

5.4.1 Decomposition of individual correlators

As shown in Eq. (5.21), spin correlations evolves periodically upon imprinting the spin spiral.

This corresponds to measuring the static spin structure factor S(q) along the 1^st Brillouin zone.

Fig.5.10ashows the experimental data of the uniform spin structure factorS(q=0)(recorded without manipulating the spins) and the staggered spin structure factorS(q_AFM)as a function of the chemical potentialµ, which is inferred in a local density approximation (LDA) from the precisely calibrated trapping potential [112]. Using Eq. (5.5), the measured local momentC_ii ≡ C₀₀ =(S_↑+S_↓)/4 is also plotted in gray, which reaches a maximum at half fillingµ=0.

The uniform spin structure factor is smaller than the local momentC₀₀owing to the presence of negative nearest-neighbour correlations. In contrast, the staggered spin structure factor exceeds the local moment significantly at lowering temperature. This asymmetry with respect to the local moment clearly indicates the presence of beyond nearest-neighbour AFM spin correlations. By direct comparison to numerical linked-cluster expansion (NLCE) calculations for the staggered structure factor at half-filling andU/t=8 [130], we deduce a temperature ofk_BT_s =0.57(3)tin the spin sector at half-filling. In addition, we extract the global density temperaturek_BT_d=0.63(3)t, obtained from fitting NLCE results to the singles density profiles.

In Figs.5.10b andd, we plotted the spin structure factor S(q) measured along the diagonal of the 1^st BZ for different fillings. At half filling µ= 0, the structure factor exhibits a minimum at q=0 and peaks atq_AFM, in nice agreement with the theoretical calculation in Fig.5.3. We observe a qualitatively similarq-dependence of the structure factor away from half filling, however, the build-up of spin correlations is suppressed. Note, that we can not invert the direction of the in-plane gradient. Hence, the data of the structure factor shown in the intervalq∈

−^π_a,0

corresponds to

|k_sp| ∈ _π

a,²_a^π

. The symmetry of the measured structure factor aroundq= 0 therefore indicates that we are not affected by incoherent processes on the time-scale of the spin manipulationt_sp. In addition, recording the spin structure factor as a function ofq further provides access to the individual spatial spin correlatorsC_{i j}. The value of the correlation length deduced at half-filling suggests that spin correlators with|i−j| ≥√

8 do not contribute significantly to the measured spin structure factor. RewritingS(q,q)along the diagonal of the BZ as a Fourier series we obtain

S(q,q) ≈ Õ3 n=0

f_ncos(nqa), (5.25)

with f₀ =C₀₀+2C₁₁, f₁ =4C₀₁+4C₁₂, f₂=2C₁₁+4C₀₂, and f₃=4C₁₂. Using the measurement of the local momentC₀from the density profile alone, we are able to extract all higher spin correlators C_ijup to a distance of|i− j|=√

5 from the first four Fourier components of the measured structure factor. The first three spin correlators are shown in Fig.5.10cas a function of µ.

a

d b

c

ij

Figure 5.10:Coherent evolution of spin correlations. aUniform (squares) and staggered (diamonds) spin structure factor recorded as a function of chemical potentialµ. The solid gray line shows the measured local momentC₀. The two dashed lines highlight the potential bins used for Figs. 3band 3c.b ,dSpin structure factor (circles) and local moment (solid gray line) recorded along the diagonal of the BZ at half fillingµ=U/2 and away from half filling µ=U/2−3.6t, respectively. We plot the fit using the Fourier series given in Eq.5.25(solid red line), and additionally, at half filling, the fit assuming an exponential decay of the magnitude of the spatial correlators according to Eq.5.6(black dashed line).cExtracted spin correlatorsC_{i j} (circles) as a function of chemical potential. The gray and yellow shadings show NLCE data for a temperature interval k_BT =[0.54,0.6]tof the local moment and nearest-neighbour correlator, respectively.

We compare the local momentC₀₀ and the nearest-neighbour correlatorC₀₁ to data from NLCE calculations for a temperature interval of k_BT_s = [0.54,0.6]t. The next-to-nearest neighbour correlator C₁₁ is observed to contribute significantly to the measured spin structure factor and possesses the correct positive sign. We estimate the length scale over which AFM correlations extend in our system and obtain a spin correlation length ofξ =0.43(3)aat half-filling at the lowest achieved temperature.

5.4.2 Local and global thermometry

As discussed in Sec.5.2.2, thermometry in strongly interacting fermionic lattice systems is especially difficult at low temperatures, when the density degree of freedom is essentially frozen. The uniform structure factor is not an ideal temperature probe since it reaches zero in theT →0 limit. In contrast, the staggered spin structure factor is a sensitive measure of the emerging spin order with respect to temperature. We compare the temperatureT_s extracted from the measured staggered structure factor at half-filling to the temperatureT_dobtained from fitting NLCE data [112,130] to the singles density profiles. We note thatT_s measures the local temperature of the gas around half-filling whereasT_dis a global measure of the temperature since it is extracted from the entire profile of the cloud. Fig.5.11

0 0.25 0.5 0.75 1 1.25 0.5

0.75 1

a b

time

tramp(s) tramp

lattice depth

tramp time

lattice depth

Figure 5.11:Thermalisation in spin and density sector. aTwo possible lattice ramps. The upper one shows a sine-squared shape ramp, which leads to a thermalisation profile as shown inb. The lower one shows the control setup at which the lattice beams are turned on and hold for the same amount of time.bThermalisation profile with sine-squared ramp. Shown are the density temperatureT_d(blue) extracted by fitting the global density profile with NLCE calculation, and the spin temperatureT_sextracted by comparing AFM correlation at half-filling with NLCE results. We observe thatT_d andT_sconverges for sufficient ramp time, indicating the local correlation builds up at a rate faster than global thermal equilibrium. The red line is a linear fit toT_sfor t_ramp>0.5 s, which we compare to an extrapolation using the half the heating rate in lattice holding as shown in lower part ina.

shows the measured temperaturesT_sandT_d as a function of the durationt_quench within which the in-plane lattice depth is increased from 0E_recto 6E_recusing a sine-squared ramp shape.

The two independent thermometers agree well for ramp times larger than 0.5 s, where both linearly increase with a slope of 0.18(5)t/s. We interpret this as genuine heating caused by the lattice beams.

To further support this, we compare this slope to the background heating rate 0.37(5)t/s, obtained by holding the equilibrated cloud at the final lattice depth. Under the assumption that residual heating is proportional to the integrated intensity of the lattice beams seen by the atoms, we expect the heating rate during holding to be twice the heating rate when quenching the lattice depth (sine-squared ramp shape) as shown in the upper half of Fig.5.11a. The linear increase in the spin temperatureT_s over the full range oft_rampsuggests that the spins are inlocal equilibriumfor all our measurement data.

For short ramp times, the density temperature deviates from the linear trend. SinceT_dis extracted from the entire density profile, we conclude that the cloud has not reached global equilibrium. This is due to the slight change in the harmonic confinement when quenching the lattice depth. With this equilibration study, we confirm that the spin correlation results presented above are obtained from a thermalised cloud with a reliable temperature estimate for arbitrary filling. More importantly, we observe that the distinct time-scales at which spin-ordering (formation of correlations locally) and density-ordering (density redistribution globally). This measurement thus offers valuable information for future quantum gas experiments in designing novel cooling schemes. A more detailed discussion about potential cooling schemes can be found in Chapter7.