Thermometry for Laughlin States of Ultracold Atoms

Cooling atomic gases into strongly correlated quantum phases requires estimates of the entropy to perform thermometry and establish viability. We construct an ansatz partition function for models of Laughlin states of atomic gases by combining high temperature series expansions with exact diagonalization. Using the ansatz we find that entropies required to observe Laughlin correlations with bosonic gases are within reach of current cooling capabilities.

Figure 1

Energy of $N=8$ fermions (bosons) at $\nu =1/3$ $(\nu =1/2)$ as a function of total wave vector for $n=1$ ($n=0$) in Eq. (1) in the lowest Landau level. The ground state is the $k=0$ Laughlin state set to zero energy. ${\mathrm{\Delta }}_T$ is the transport gap. The schematic depicts example excitations. The bosonic Laughlin state can be thought of as one filled level of CFs (bosons attached to one flux quantum). Low energy excitations are CF particle-hole pairs modeled as excitations with energy $E_1\sim {\mathrm{\Delta }}_T$. We consider additional excitations at higher energies, e.g., $E_2$, where bosons are not necessarily bound to flux quanta. The histograms show nearly Gaussian state counting.

Figure 2

Heat capacity versus temperature for $N=8$ bosons. The symbols are obtained from diagonalization of Eq. (1) and the solid line follows from the ansatz (2). The ansatz parameters were obtained by fitting the lowest six cumulants for $N=8$. The agreement at high $T$ follows by construction but the low $T$ agreement demonstrates the accuracy of the ansatz.

Figure 3

Top: entropy per particle versus temperature for bosons. The solid line is obtained from the ansatz (2), with parameters derived from the cumulants in Table 1. The circles (crosses) are obtained from finite $N$ extrapolations of the entropy at fixed low (high) $T$ [48]. The agreement between the symbols and the lines shows that the ansatz accurately captures the low and high $T$ limits. Top inset: low temperature enlargement. Bottom: the same but the dashed line plots the entropy per particle for the naive model [48]. The dot-dashed line plots the high $T$ limit from the first term of the cumulant expansion ($S={\kappa }_0$). Bottom inset: heat capacity from the ansatz in the thermodynamic limit. The vertical dotted line denotes the temperature $T_L$. The horizontal dotted line in the main panel plots the corresponding entropy $S_L$ determined from the ansatz.